Advances in Boundary Element & Meshless ...

The Conferences on Boundary Element and Meshless Techniques are devoted to fostering the continued involvement of the research community in identifying new problem areas, mathematical procedures, innovative applications, and novel solution techniques. Previous conferences devoted to Boundary Element and Meshless Techniques were held in London, UK (1999), New Jersey, USA (2001), Beijing, China (2002), Granada, Spain (2003), Lisbon, Portugal (2004), Montreal, Canada (2005), Paris, France (2006), Naples, Italy (2007), Seville, Spain (2008), Athens, Greece (2009), Berlin, Germany (2010), Brasilia, Brazil (2011), Prague, Czech Republic (2012) and Paris, France (2013)

ISBN 978‐0‐9576731‐1‐3

Advances in Boundary Element & Meshless Techniques XV

EC ltd


Edited by V Mallardo M H Aliabadi

Advances In Boundary Element and Meshless Techniques XV

Advances In Boundary Element and Meshless Techniques XV

Edited by V Mallardo M H Aliabadi

EC

ltd

Published by EC, Ltd, UK Copyright © 2014, Published by EC Ltd, P.O.Box 425, Eastleigh, SO53 4YQ, England Phone (+44) 2380 260334

All Rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, except under terms of the Copyright, Designs and Patents Act 1988. Requests to the Publishers should be addressed to the Permission Department, EC, Ltd Publications, P.O.Box 425, Eastleigh, Hampshire, SO53 4YQ, England.

ISBN: 978-0-9576731-1-3

The text of the papers in this book was set individually by the authors or under their supervision. No responsibility is assumed by the editors or the publishers for any injury and/or damage to person or property as a matter of product liability, negligence or otherwise, or from any used or operation of any method, instructions or ideas contained in the material herein.

International Conference on Boundary Element and Meshless Techniques XIV 15-17 July 2014, Florence, Italy Organising Committee: Prof. Vincenzo Mallardo

University of Ferrara Ferrara, Italy Prof. Ferri M H Aliabadi Department of Aeronautics Imperial College London London. UK International Scientific Advisory Committee Abe,K (Japan) Baker,G (USA) Benedetti,I (Italy) Beskos,D (Greece) Blasquez,A (Spain) Chen, Weiqiu (China) Chen, Wen (China) Cisilino,A (Argentina) Darrigrand, E (France) De Araujo, F C (Brazil) Denda,M (USA) Dong,C (China) Dumont,N (Brazil) Estorff, O.v (Germany) Gao,X.W. (China) Garcia-Sanchez,F (Spain) Hartmann,F (Germany) Hematiyan,M.R. (Iran)

Hirose, S (Japan) Kinnas,S (USA) Liu,G-R (Singapore) Mallardo,V (Italy) Mansur, W. J (Brazil) Mantic,V (Spain) Marin, L (Romania) Matsumoto, T (Japan) Mesquita,E (Brazil) Millazo, A (Italy) Minutolo,V (Italy) Ochiai,Y (Japan) Panzeca,T (Italy) Perez Gavilan, J J (Mexico) Pineda,E (Mexico) Prochazka,P (Czech Republic) Qin,Q (Australia) Saez,A (Spain) Sapountzakis E.J. (Greece) Sellier, A (France) Semblat, J-F (France) Seok Soon Lee (Korea) Shiah,Y (Taiwan) Sladek,J (Slovakia) Saldek, V (Slovakia) Sollero.P. (Brazil) Stephan, E.P (Germany) Taigbenu,A (South Africa) Tan,C.L (Canada) Telles,J.C.F. (Brazil) Wen,P.H. (UK)

PREFACE The Conferences on Boundary Element and Meshless Techniques are devoted to fostering the continued involvement of the research community in identifying new problem areas, mathematical procedures, innovative applications, and novel solution techniques in both boundary element methods (BEM) and boundary integral equation methods (BIEM). Previous successful conferences devoted to Boundary Element Techniques were held in London, UK (1999), New Jersey, USA (2001), Beijing, China (2002), Granada, Spain (2003), Lisbon, Portugal (2004), Montreal, Canada (2005), Paris, France (2006), Naples, Italy (2007), Seville, Spain (2008), Athens, Greece (2009), Berlin, Germany (2010), Brasilia, Brazil (2011), Prague, Czech Republic (2012) and Paris, France (2013) The present volume is a collection of edited papers that were accepted for presentation at the Boundary Element Techniques Conference held at the Grand Hotel Baglioni, Florence, Italy during th 15-17 July 2014. Research papers received from 18 countries formed the basis for the Technical Program. The themes considered for the technical program included solid mechanics, fluid mechanics, potential theory, composite materials, fracture mechanics, damage mechanics, contact and wear, optimization, heat transfer, dynamics and vibrations, acoustics and geomechanics. The conference organizers would also like to express their appreciation to the International Scientific Advisory Board for their assistance in supporting and promoting the objectives of the meeting and for their assistance in the form of reviews of the submitted papers. Editors

CONTENT Polycrystalline materials with pores: effective properties through a boundary element homogenization scheme. F. Trentacoste, I. Benedetti and M.H. Aliabadi A symmetric BEM approach to strain gradient elasticity for 2D static boundary-value problems Panzeca T, Terravecchia S. and Polizzotto C Comparison of Green element solutions for inverse heat conduction problems using the time-dependent and logarithmic fundamental solutions Akpofure E. Taigbenu Analysis of elastic problems by the fast multipole boundary element method A. F. Dias. Jr, E. L. Albuquerque Analysis of Damped Waves Using Energetic BEM-FEM Coupling A. Aimi, L. Desiderio, M. Diligenti, C. Guardasoni A topological optimization procedure applied to multiple region problems Carla Anflor, Éder L. Albuquerque and Luiz C. Wrobel Direct Interpolation Technique using Radial Basis Functions Applied to the Helmholtz Problem C. F. Loeffler, P. V. M. Pereira, H. M. Barcelos A two-scale three-dimensional boundary element framework for degradation and failure in polycrystalline materials I. Benedetti, M.H. Aliabadi A Boundary Element - Response Matrix method for 3D neutron diffusion and transport problems V.Giusti, B.Montagnini A new interface damage model with frictional contact. An SGBEM formulation and implementation Jozef Kšiňan, Vladislav Mantič , Roman Vodička Isotropic-BEM coupled with strong form Local radial point interpolation for the solution of 3D geometrically nonlinear elasticity problems Richard Kouitat Njiwa Multidomain BEM for crack analysis in stiffened anisotropic plates D. Flauto, I. Benedetti, A. Milazzo Biomagnetic fluid flow in a channel under the effect of a uniform localized magnetic field Ö. Türk, Canan Bozkaya and M. Tezer-Sezgin Ö. Türk1, Canan Bozkaya2 and M. Tezer-Sezgin1,2

1 7 15

23 28 34 40 46 52 60 68

75 81

Analysis of 3D anisotropic solids using fundamental solutions based on Fourier series and the Adaptive Cross Approximation method R. Q. Rodríguez, C. L. Tan, P. Sollero and E. L. Albuquerque Voxel-Based Analysis of Electrostatic Fields in Virtual-Human Model Duke using Indirect Boundary Element Method with Fast Multipole Method Shoji Hamada Development of the Boundary Element Method for 3D General Anisotropic Thermoelasticity Y.C. Shiah and C.L. Tan BEM Solution of MHD Pipe Flow Around a Conducting Cylindrical Solid and Inside an Insulating or Conducting Medium M. Tezer-Sezgin and S. Han Aydın MHD Flow in Rectangular Ducts of Partly Conducting Walls under an Inclined Magnetic Field Canan Bozkaya and M. Tezer-Sezgin An ACA accelerated isogeometric Boundary Element analysis for two-dimensional potential problems L. S. Campos, E.L. Albuquerque and L. C. Wrobel Using Eulerlets to give a Boundary Integral formulation in Euler flow. Edmund Chadwick and Apostolis Kapoulas The Regularized Method of Fundamental Solutions Applied to Interface Problems of Potential Equations Csaba Gáspár Half-space Fundamental Solution for harmonic wave propagation E.Puertas and R Gallego Time Dependent Fracture Non-Linear Problems with the Boundary Element Method E. Pineda León, A. Rodríguez-Castellanos, M.H. Aliabadi Advanced Beam Element for the Analysis of Engineering Structures E.J. Sapountzakis and I.C. Dikaros Torsional Vibration Analysis of Bars Including Secondary Torsional Shear Deformation Effect by BEM E.J. Sapountzakis, V.J. Tsipiras and A.K. Argyridi Generalized Warping Analysis of Composite Beams of Arbitrary Cross Section by BEM I.C. Dikaros and E.J. Sapountzakis An iterative coupling based on Green’s function to solve embedded crack problems E.F. Fontes Jr., J.A.F. Santiago and J.C.F. Telles Fracture with nonlocal elasticity: analytical approaches P.H. Wen, X.J. Huang and M.H. Aliabadi

87 93

101 109 115 121 129 138 146 152 158 167 173 181 187

A Dual Boundary Element Method for Measurement of Electromechanical Impedance Fangxin Zou, M. H. Aliabadi Study on the Water Coning Phenomenon in Oil Wells Using the Boundary Element Method G. S. V. Gontijo, E. L. Albuquerque, E. L. F. Fortaleza Investigations of dynamic interface crack problems in active bimaterials Felipe García-Sánchez, Michael Wünsche, Andrés Sáez and Chuanzeng Zhang A (constrained) microstretch approach in living tissue modelling: a numerical investigation by the local point interpolation – boundary element method. Jean-Philippe Jehl, Richard Kouitat Njiwa Boundary Element analysis of Mild Slope Equation problems with a monotonic bed profile A. Cerrato, J.A. González, L. Rodríguez-Tembleque Analysis of wear on fiber-reinforced composites using boundary elements L. Rodríguez-Tembleque and M.H. Aliabadi Analysis of crack onset and propagation at elastic interfaces by using Finite Fracture Mechanics M. Muñoz-Reja, L. Távara, V. Mantič, P. Cornetti Simplified Assessement and Evaluation Procedure of Finite-Part Hypersingular Integrals Ney Augusto Dumont Fracture Analysis of Viscoelastic Nonhomogeneous Media Using Boundary Element Method Hugo Luiz Oliveira and Edson Denner Leonel Solution of time-domain problems using Convolution Quadrature methods and BEM++ T.Betcke, N.Salles, W.Smigaj Sensitivity analysis by the fast multipole boundary element method V. Mallardo, M. H. Aliabadi Auxiliary Relations for the Corners in Coupled Stretching-Bending Boundary Elements Chyanbin Hwu and H.W. Chang Coupling of Boundary and Finite Elements for the Problems of Multiple Polygon-like Holes Chyanbin Hwu, C.C. Li and Shao-Tzu Huang

195 201 207

214

221 228 236 242 248 257 263 269 275

Elastostatic analysis by a BEM-NURBS approach V. Mallardo, V. Minutolo, E. Ruocco Boundary Element Analysis of Fibre-Reinforced Composites and Adhesion Joints with Bridged Cracks Mikhail Perelmuter Radon-Stroh formalism for 3D theory of anisotropic elasticity Federico C. Buroni and Mitsunori Denda Three-dimensional linear elastic boundary element method with direct evaluation of singular integrals Cristiano J. Brizzi Ubessi and Rogério José Marczak Enriched BEM for fracture in anisotropic materials G.Hattori, A. Sáez, J. Trevelyan and F. García-Sánchez The Method of Fundamental Solutions coupled with a Genetic Algorithm to Optimize Cathodic Protection Systems in Infinite Regions W. J. Santos, J. A. F. Santiago and J. C. F. Telles Stabilized FEM-BEM Solutions of MHD Flow in an Annular Pipe S. Han Aydın A new boundary approach for the 2D slow viscous MHD flow of a conducting liquid about a solid particle A. Sellier, M. Tezer-Sezgin and S. H. Aydin Direct Volume-to-Surface Integral Transformation for 2D BEM Analysis of Anisotropic Thermoelasticity H Y.C. Shiah, Chung-Lei Hsu, and Chyanbin Hwu Slow gravity-driven migration and interaction of a bubble and a solid particle near a free surface M. Guémas, A. Sellier and F. Pigeonneau A Fast 2D-3D BEM Approach to Dynamic Ride-Sharing A. Brancati and S.M. Siniscalchi Meshfree Modelling of Elastodynamic Response of Woven Fabric Composites. Y. H. Chen, M.H. Aliabadi, and P. H. Wen Boundary element method applied for folded thick plates D. I. G. Costa, E. L. Albuquerque, P. M. Baiz Vertical Vibration of Rigid and Flexible Foundations Presenting Inertia Properties and Interacting with Transversely Isotropic Layered Media Josue Labaki and Euclides Mesquita A 2D BEM-FEM model of thin structures for time harmonic fluid-soil-structure interaction analysis including poroelastic media J.D.R. Bordón, J.J. Aznárez and O. Maeso

281 287 295 301 309 315

323 329 335 342 348 354 361 367

375


1

Polycrystalline materials with pores: effective properties through a boundary element homogenization scheme. F. Trentacoste1, I. Benedetti1a and M.H. Aliabadi2b 1

Dipartimento di Ingegneria Civile, Ambientale, Aerospaziale e dei materiali, Università degli Studi di Palermo, Viale delle Scienze, Edificio 8, 90128, Palermo, Italy 2

Department of Aeronautics, Imperial College London, South Kensington Campus,SW7 2AZ, London, UK a

[email protected], [email protected]

Keywords: Polycrystalline materials, Micromechanics, Porosity, Boundary element method. Abstract. In this study, the influence of porosity on the elastic effective properties of polycrystalline materials is investigated using a formulation built on a boundary integral representation of the elastic problem for the grains, which are modeled as 3D linearly elastic orthotropic domains with arbitrary spatial orientation. The artificial polycrystalline morphology is represented using 3D Voronoi tessellations. The formulation is expressed in terms of intergranular fields, namely displacements and tractions that play an important role in polycrystalline micromechanics. The continuity of the aggregate is enforced through suitable intergranular conditions. The effective material properties are obtained through material homogenization, computing the volume averages of micro-strains and stresses and taking the ensemble average over a certain number of microstructural samples. In the proposed formulation, the volume fraction of pores, their size and distribution can be varied to better simulate the response of real porous materials. The obtained results show the capability of the model to assess the macroscopic effects of porosity. Introduction Polycrystalline materials, such as metals, ceramics and alloys, are widely used in many applications of engineering interest. The internal structure of a polycrystalline material is determined by the size and shape of the grains, by their crystallographic orientation and by different type of defects within them. For this reason the investigation of the link between their micro and macro-properties has attracted considerable attention [1]. Depending on the manufacturing conditions, several polycrystalline materials present, at the microscale, a regular arrangement of their grains: their microstructure is characterized by grains with uniform size and identical morphology. This holds in particular for several polycrystalline films. Examples of materials with regular microstructure are In-N and MgF2 films, Yttria and Alumina. In this paper, a computational model for the homogenization of polycrystalline materials with regular microstructure (grain size distribution and morphology) and pores is developed. Even if the study is referred to polycrystals with regular microstructure, the formulation is able to take into account polycrystals with a random size distribution and morphology of their grains. The results were compared with the trend predicted by the Spriggs equation, which is widely used to fit experimental measurements when porous material are considered. Generation of artificial regular microstructure The microstructure of polycrystalline materials can be reconstructed experimentally or it can be artifcially generated using algorithms able to retain the main statistical topological, morphological and crystallographic features of the polycrystalline aggregates. Although the experimental characterization may provide fundamental information, it generally requires expensive equipment and complicated and time consuming post-processing. The use of suitable computer models offers the opportunity of simulating large numbers of microstructures, complementing and reducing the experimental effort. In the case of polycrystalline materials, the Voronoi tessellation is widely recognized and used for the generation of the microstructural model [2]. The Voronoi cells are convex polyhedra bounded by at polygonal faces, which makes them particularly suitable for numerical treatment. For such reasons, threedimensional Voronoi tessellations are used in the present study to generate microstructures with regular grain

2

Eds V Mallardo & M H Aliabadi

distribution, size and orientation. The topology and morphology of the tessellation is controlled by the arrangement of the initial seeds. On the basis of this consideration, it is possible to generate different regular morphologies. In particular, common regular morphologies can be obtained by using regularly spaced seeds distributions: a cubic array of seeds provides cubes, collocating the seeds on the sites of a face-centered cubic structure provides runcated octahedra, while the body-centered structure provides dodecahedra. Considering the microstructural morphologies, it is possible to distinguish among: - Cubic structure; - Hexarhombic dodecahedron structure; - Rhombic dodecahedron structure; - Truncated octahedra structure. In Fig.(1) samples of the different types of regular microstructure are shown.

Fig.1: Samples of the different types of regular microstructures. Three-dimensional Grain boundary element formulation In the present work the numerical model for the single crystal is obtained by using the Boundary Element Method (BEM) for three-dimensional anisotropic elasticity. The polycrystalline aggregate is modeled as a multi-region boundary element problem, so that different elastic properties and crystallographic orientation can be assigned to each grain. Each grain is represented as a Voronoi cell ‫ܩ‬௞ bounded by the surface ‫ܤ‬௞ , which is formed by the union of at convex polygonal faces ‫ܨ‬௝௞ . On the surfaces of the domain boundary grains generated by the intersection with the external boundary ‫ ܤ‬of the analysis region some quantities, either displacements or tractions, are prescribed as external boundary conditions (BCs) that is [3]: ‫ݑ‬௜௞ ൌ ‫ݑ‬ത௜௞ ‫ݎ݋‬ ‫ݑ‬௜௞

‫ݐ‬௜௞ ൌ ‫ݐ‬௜ҧ ௞

݅ ൌ ͳǡ ǥ ǡ͵‫ܩ݊݋‬௞

‫ݐ‬௜௞

(1)

௞

where e are components of displacement and tractions on ‫ ܤ‬and the overbar denotes prescribed quantities. The remaining unknown quantities are determined by the solution of the numerical model. On the other hand, both displacements and tractions are unknown at the interfaces between two adjacent grains ‫ܩ‬௔ and ‫ܩ‬௕ . For this reason, displacement continuity and traction equilibrium equations must be enforced:

‫ݑ‬෤௜௔ െ ‫ݑ‬෤௜௕ ൌ Ͳ‫ݐ‬௜ǁ ௔ െ ‫ݐ‬ǁ௜௕ ൌ Ͳ

ሺʹሻ

where the tilde indicates quantities expressed in the local grain face reference system: opposite reference systems are associated to the two faces, of two different grains, meeting at a given interface. Considering a given grain, with its own crystallographic orientation, the displacement boundary integral equation can be written, in the face local reference system, as [4]: ே೑ೖ

ே೑ೖ

௞ ሺ࢟ሻ‫ ݑ‬௞ ሺ࢟ሻ ෩௜௝௞ ሺ࢞ǡ ࢟ሻ‫ݐݑ‬ ෦௝௞ ሺ࢞ሻ݀‫ܨ‬ሺ࢞ሻ ෤௝ ൅ σ௤ୀଵ ‫׬‬ிೖ ܶ෨௜௝௞ ሺ࢞ǡ ࢟ሻ‫ݑ‬෤௝௞ ሺ࢞ሻ݀‫ܨ‬ሺ࢞ሻ ൌ σ௤ୀଵ ‫׬‬ிೖ ܷ ܿǁ௜௝ ೜

೜

(3)

෩௜௝௞ and ܶ෨௜௝௞ are the displacement and the traction fundamental solution, see Benedetti and Aliabadi where the ܷ for further details [3]. To solve the polycrystalline problem eq (3) written for each grain ‫ܩ‬௞ ǡ ݇ͳǢ ǥ Ǣܰ௚ , is discretized following the classical BEM discretization procedure, which leads to the equation for each grain:


‫ܓ‬ ෩ ‫܋ܖ‬ ሾ۶

෩ ‫ ܓ܋‬ሿ ቈ ۶

෥ ‫܋ܖܓ‬ ‫ܝ‬ ‫ܓ‬ ෩‫܋ܖ‬ ቉ ൌ ሾ۵ ෥ ‫܋ܓ‬ ‫ܝ‬

‫ܜ‬ǁ ‫ܓ‬ ቉ ۵‫ ܓ܋‬ሿ ቈ ‫܋ܖ‬ ‫ܜ‬ǁ ‫܋ܓ‬

3

(4)

The system of equations for the entire polycrystalline aggregate is obtained by writing the previous set of equation for each grain and enforcing the boundary and interface conditions on the overall aggregate. The final system of equation is written: ࡭૚ ‫ۍ‬ ‫ێ‬૙ ‫ڭ ێ‬ ‫ ێ‬૙ ‫ێ‬ ‫ێ‬ ‫ۏ‬

࡮૚ ࡭૛

૙ ࡮૛

‫ڰ‬ ૙

ǥ

‫ڮ‬ ‫ڮ‬ ࡭

ࡺࢍ

۷۱‫ܝ‬ ۷۱‫ܜ‬

૚ ૙ ۱ ૚ ‫܋ܖ ܠ‬ ‫ܠ ې‬૚ ‫ ۍ ې‬૛ ૚ ‫ې‬ ૙ ‫܋ܖ ۍ ۑ‬ ૚ ‫ ێ‬۱ ‫ۑ ܋ܠ‬ ‫܋ܠ‬ ‫ۑ ڭ ێ ۑ ێۑ‬ ࡺࢍ ‫ ۑ ڭ ێ‬ൌ ࡮ ‫ۑ ࢍࡺ ࢍࡺ ێ ۑ ܏ۼ ێ ۑ‬ ‫ێ ۑ ܋ܖ ܠێ ۑ‬۱ ‫ۑ ܋ܖܡ‬ ‫ ێ‬૙ ‫ۑ‬ ‫܏ۼ ܠ ۑ‬ ‫ۏ ے ۏ‬ ‫܋ ے‬ ૙ ‫ے‬

(6)

‫ܓ‬ ‫ܓ‬ ෩‫܋ܖ‬ ෩ ‫܋ܖ‬ where the matrix blocks ࡭࢑ contains columns from the matrices ۶ and െ۵ corresponding to the ‫ܓ‬ ‫ܓ‬ ෩ ‫܋ܖ‬ ෥ ‫ ܋ܖܓ‬and ‫ܜ‬ǁ ‫ ܋ܖܓ‬that are collected in ‫ܠ‬෤ ‫܋ܖ‬ unknown components of ‫ܝ‬ , the blocks࡯࢑ collect columns from െ۶ and ‫ܓ‬ ‫ܓ‬ ‫ܓ‬ ‫ܓ‬ ෩ ǁ ෥ ‫ ܋ܖ‬and ‫ ܋ܖ ܜ‬, i.e. the BCs, that are collected in ‫ ܋ܖܡ‬, ࡮࢑ ൌ ۵‫ ܋ܖ‬corresponding to the known components of ‫ܝ‬ ‫ܓ‬ ‫ܓ‬ ෩‫܋ܖ‬ ෩ ‫܋ܖ‬ െ۵ ȁǡthe vectors ‫ ܓ܋ ܠ‬collect the unknown interface displacements and tractions of the k-th grain and ȁ۶ the matrices۷۱‫ ܝ‬and ۷۱‫ ܜ‬, whose lines contain only 1 and -1 and zeros, implement respectively the displacement and traction interface conditions in the system matrix. System (6) is highly sparse and the use of specialized sparse solvers is then desirable to speed up its solution. In this work PARDISO has been used as a solver. Details about the discretization process are given in Benedetti and Aliabadi [3].

Computational material homogenization An important goal of the mechanics and physics of heterogeneous materials is the derivation of their properties from the constitutive laws and spatial distribution of their micro-components. This estimation is referred to as material homogenization and it is the main task of micromechanics [5]. The material homogenization is usually performed by evaluating volume and ensemble averages of some relevant fields over one or more realizations of the microstructure subjected to suitable boundary conditions, and then linking these averaged fields by means of effective properties [6,7]. Given a polycrystalline realization ܴሺܰ௚ ሻǡ consisting of ܰ௚ grains subjected to consistent boundary conditions, since the material is supposed to not develop microcracks, stress and strain volume averages can be used to extract the apparent elastic moduli. Stress and strain volume averages are defined by: ଵ

ଵ

Ȟ௜௝ ൌ ‫׬‬௏ ߛ௜௝ ሺ‫ݔ‬ሻܸ݀ሺ‫ݔ‬ሻ ൌ ‫׬‬డ௏ሺ‫ݑ‬௜ ݊௝ ൅ ‫ݑ‬௜ ݊௝ ሻ݀ܵ ଶ௏ ௏ ଵ

ଵ

ȭ௜௝ ൌ ‫׬‬௏ ߪ௜௝ ሺ‫ݔ‬ሻܸ݀ሺ‫ݔ‬ሻ ൌ ‫׬‬డ௏ሺ‫ݔ‬௜ ‫ݐ‬௝ ሻ݀ܵ ௏ ௏

(7) (8)

where the integrals over the microstructure volume V are transformed into surface integrals over the volume boundary ߲ܸ , as consequence of the divergence theorem. Lower case letters in Eq.(7,8) refer to micro-scale quantities, i.e. displacements, tractions, strains and stresses in the polycrystalline microstructure, while capital Greek letters refer to homogenized quantities. Once the stress ઱ and strain ડ macro-fields are known, the apparent macro-properties are defined by: ෡ ડ ઱ ൌ ࡯

(9)

where the hat denotes apparent overall properties, to distinguish them from the microscopic ones. This relationship is the homogenized stress-strain equation and forms the basis for the homogenization procedure. The apparent properties, eq.(9), are computed for each considered microstructural realization and an ensemble average is eventually performed over the number of realizations. In the present work, the effective properties of polycrystalline yttria are first analyzed. Single crystals of Yttria present cubic morphology at the grain scale and cubic material symmetry [8]. It is characterized by three elastic constants‫ܥ‬ଵଵ , ‫ܥ‬ଵଶ and ‫ܥ‬ସସ being ‫ܥ‬ଵସ = ‫ܥ‬ଵହ = ‫ܥ‬ଵ଺ = 0, ‫ܥ‬ଶସ = ‫ܥ‬ଶହ = ‫ܥ‬ଶ଺ = 0 , ‫ܥ‬ଷସ = ‫ܥ‬ଷହ = ‫ܥ‬ଵଷ଺ ൌ Ͳ; the stiffness matrix in the material reference system can be written in Voigt notation as:

4


‫ܥ‬ ‫ ۍ‬ଵଵ ‫ܥ‬ ‫ ێ‬ଵଵ ‫ܥ‬ ۱ ൌ ‫ ێ‬ଵଵ ‫Ͳ ێ‬ ‫Ͳ ێ‬ ‫Ͳ ۏ‬

‫ܥ‬ଵଵ ‫ܥ‬ଵଵ ‫ܥ‬ଵଵ Ͳ Ͳ Ͳ

‫ܥ‬ଵଵ Ͳ ‫ܥ‬ଵଵ Ͳ ‫ܥ‬ଵଵ Ͳ Ͳ ‫ܥ‬ଵଵ Ͳ Ͳ Ͳ Ͳ

Ͳ Ͳ Ͳ Ͳ ‫ܥ‬ଵଵ Ͳ

Ͳ Ͳ ‫ې‬ ‫ۑ‬ Ͳ ‫ۑ‬ Ͳ ‫ۑ‬ Ͳ ‫ۑ‬ ‫ܥ‬ଵଵ ‫ے‬

(10)

To assess the capability and accuracy of the formulation, it is interesting to compute all the entries of the effective stiffness matrix, Eq.(10). For this purpose, it is necessary to perform six different computations, corresponding to the six linearly independent load cases for the unit cell. ܰ௧ = 10 aggregates with ܰ௚ = 125 grains and subjected to kinematic uniform boundary conditions have been simulated. The apparent obtained stiffness matrix is: ʹ͵͵Ǥͳ ͳͲ͵Ǥ͹ ͳͲ͵Ǥ͸ ͳǤͲ͵͸ െͲǤͲʹ െͲǤͳͲͶ ‫͵Ͳͳۍ‬Ǥ͹ ʹ͵͵Ǥͳ ͳͲ͵Ǥ͸ െͲǤͳ͵ͺ ͲǤͻʹ ͲǤͲͳͺ ‫ې‬ ‫ێ‬ ‫ۑ‬ ͳͲ͵Ǥ͸ ͳͲ͵Ǥ͸ ʹ͵͵Ǥʹ െͲǤͲʹ͹ െͲǤͲ͸ ͲǤͻ͵ͺ ‫ۑ‬ ෠ ൌ ‫ێ‬ (11) ͲǤͲʹ ͲǤ͸ʹͺ െͲǤʹ͵ ͸ͶǤͻʹ ͲǤ͹͵ ͲǤͲʹ͹ ‫ێ‬ ‫ۑ‬ ‫Ͳ ێ‬Ǥ͹Ͷ െͲǤͲ͸ െͲǤʹͷ ͲǤ͹ͳ ͸ͷ ͲǤͳ͵͵ ‫ۑ‬ ‫Ͳ ۏ‬Ǥ͸͸ െͲǤʹ ͲǤͲͳ ͲǤͷ͸ ͲǤͳʹ ͸ͷǤͲ͵ ‫ے‬ It is interesting to observe that the isotropy of the microstructural aggregate is confirmed by the structure of the effective stiffness matrix. Given an unit cell with ܰ௚ crystals, in general also for a isotropic material, the assumption of overall isotropic behavior of the unit cell may be restrictive and should be verified. However, if a suitable number of realizations/grains is considered the effective stiffness matrix assumes soon an isotropic structure. The values of the effective elastic moduli ‫ ܧ‬and ‫ ܩ‬for polycrystalline ܻଶ ܱଷ have been reported by various authors. In this study, the overall values, are ‫ ܧ‬ൌ ͳ͹ͷ GPa and ‫ ܩ‬ൌ ͸ͷ GPa, which are in very good agreement with the values ‫ ܧ‬ൌ ͳ͹Ͷ GPa and ‫ ܩ‬ൌ ͸͸Ǥͷ GPa, reported by Palko at al.[8]. The aggregate properties of alumina are also computationally investigated. Alumina ‫݈ܣ‬ଶ ܱଷ mono-crystals are characterized by the following stiffness matrix: ‫ ۍ‬ଵଵ ‫ ێ‬ଶଵ ൌ ‫ ێ‬ଷଵ ‫ێ‬ସଵ ‫Ͳ ێ‬ ‫Ͳ ۏ‬

ଵଶ ଶଶ ଷଶ ସଶ Ͳ Ͳ

ଵଷ ଵସ ଶଷ ଶସ ଷଷ Ͳ Ͳ ସସ Ͳ Ͳ Ͳ Ͳ

Ͳ Ͳ Ͳ Ͳ ହହ ଺ହ

Ͳ Ͳ ‫ې‬ ‫ۑ‬ Ͳ ‫ۑ‬ Ͳ ‫ۑ‬ ହ଺ ‫ۑ‬ ଺଺ ‫ے‬

(12)

being ‫ܥ‬ଵଵ = ‫ܥ‬ଶଶ ; ‫ܥ‬ଵଷ = ‫ܥ‬ଶଷ ; ‫ܥ‬ସସ = ‫ܥ‬ହହ ; ‫ܥ‬ଵସ = -‫ܥ‬ଶସ = ‫ܥ‬ହ଺ ; ‫ = ଺଺ܥ‬1/2(‫ܥ‬ଵଵ -‫ܥ‬ଶଶ ). The homogenization procedure leads, for the alumina polycrystalline to the following effective stiffness matrix: Ͷ͹͵Ǥͳ ͳͶ͵Ǥ͵ ͳͶ͵ǤͶ ͳǤͳ͹ͻ െͲǤͳͲͳ െͲǤ͸Ͷͳ ‫ͳۍ‬Ͷ͵Ǥ͵ Ͷ͹ʹǤͶ ͳͶ͵Ǥ͹ ͲǤʹͲͻ ͳǤ͵Ͷͺ ͲǤ͵Ͳʹ ‫ې‬ ‫ێ‬ ‫ۑ‬ ͳͶ͵ǤͶ ͳͶ͵Ǥ͹ Ͷ͹ʹǤͺ െͲǤͶͷͺ െͲǤ͸ʹͶ ͳǤʹͳͶ ‫ۑ‬ (13) ෨ ൌ ‫ێ‬ െͲǤͳ ͳǤͻͶʹ െͲǤͺ͸͵ ͳ͸ͷǤʹ ͳǤ͸ͺ ͲǤͶʹͶ ‫ێ‬ ‫ۑ‬ ‫ͳ ێ‬Ǥ͸͹ ͲǤͲͳ െͳǤͲͷ ͳǤ͸ͻʹ ͳ͸ͶǤ͸ ͲǤͲ͹ʹ ‫ۑ‬ ‫ͳ ۏ‬Ǥʹ െͲǤʹ͸ െͲǤͲͲʹ ͳǤͶͺ ͲǤͲ͵ ͳ͸ͶǤ͹ ‫ے‬ Again, macroscopic isotropy is observed. The values of the effective elastic moduli E and G for polycrystalline ‫݈ܣ‬ଶ ܱଷ have been reported by various authors [8]. In this study, the average values, calculated over ܰ௥ ൌ ͳͲ realizations of aggregates with ܰ௚ =189 grains, are ‫ ܧ‬ൌ ͶͲͷǡͷ GPa and ‫ ܩ‬ൌ ͳ͸ͷGPa, which are in very good agreement with the values ‫ ܧ‬ൌ ͶͲʹǡ͹ GPa and ‫ ܧ‬ൌ ͳ͸͵ǡͶGPa, reported by Pabst et al. who extrapolated the values to zero porosity. Polycrystalline materials with pores: the Spriggs law for the estimation of the effective properties Material porosity is related to the lack of solidification due to the fact that there are not enough atoms present to fill the available space. This is different from the case of void inclusion, related to the presence of gas atoms during the nucleation process. The effects of porosity on the elastic properties of polycrystalline materials have been studied by various investigators and a number of expressions have been proposed. H. Spriggs suggested an exponential equation


5

able to fit a wide set of data for aluminum oxide over a wide range of porosity [10]. According to his equation, the Young and shear moduli of a porous polycrystalline material can be expressed as: ‫ܧ‬ሺܲሻ ൌ ‫ܧ‬଴ ݁ ି௕ಶ ௉ ‫ܩ‬ሺܲሻ ൌ ‫ܩ‬଴ ݁ ି௕ಸ௉

(14)

where ‫ܧ‬ሺܲሻ and ‫ܩ‬ሺܲሻare the values of the elastic moduli at the current value of porosity ܲ, expressed as pores volume fraction, and ‫ܧ‬଴ and ‫ܩ‬଴ are the values for the zero porosity case. Eqs.(14) have been widely used to predict the elastic moduli of porous brittle solids. The empirical constants ܾா and ܾீ , associated with a given manufacturing technique, are related to the proportions of closed and open pores within the polycrystalline aggregates. In the aforementioned grain boundary formulation, to take into account the presence of pores, in the proposed formulation the following steps are followed: (i) The missing grains (pores) are localized and given a specific flag within the generated tessellation; (ii) The computation of the BE matrices H and G is skipped for the aged grains and they are not considered in the population of the final system, leading to a consequent reduction of its order; (iii) Free traction boundary conditions are enforced, instead of interface continuity and equilibrium equations, on the faces of the grains adjacent to the missing ones (pores). From the numerical point of view, when pores are considered, the number of rows and columns of the final system is reduced according to SO = ܰ଴ - 6ܰ௜ - 3ܰ௘

(15)

where ܰ଴ is the system order when no grains have been removed and ܰ௜ and ܰ௘ are the number of interface and external boundary nodes related to the missing grains. To simulate porosity for yttria, ܰ௥ ൌ ͷͲ microstructures of ܰ௚ ൌ512 cubic grains have been considered. In each of then a different distribution of pores has been chosen to better simulate the response of a real porous material. Then, the computed effective properties have been compared with the values predicted by the Spriggs law, over a given range of porosity (Fig.2), showing the capability of the model to assess the macroscopic effect of pores.

Fig.2: Young and shear moduli of ܻଶ ܱଷ as function of volume fraction of porosity. For alumina ܰ௥ ൌ ͷͲ microstructures with ܰ௚ ൌ189 truncated octahedral grains have been considered. As it possible to see in (Fig. 3), also in this case what is obtained is in very good agreement with the trend predicted by the Spriggs equation.

6


Fig.3: Young and shear moduli of ‫݈ܣ‬ଶ ܱଷ as function of volume fraction of porosity. Conclusions In the present work, a three dimensional grain boundary formulation has been proposed for studying the effect of porosity on the elastic properties of polycrystalline materials. The formulation is based on the boundary integral representation of the three-dimensional anisotropic elastic problem for the crystals of the aggregate. The method has been applied to the determination of the effective properties of some polycrystals with a regular microstructural morphologies and the results are in good agreement with literature data in the framework of numerical homogenization. The results obtained in case of porosity have been successfully compared with the Spriggs equation for the analyzed polycrystalline materials. Further studies may consider the microstructural damage evolution when the presence of pores is considered. References

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

B. Adams, T. Olson, Progress in Materials Science 43, 188 (1998). C. Rycroft, Chaos 19,(2009) 041111. I. Benedetti, M.H. Aliabadi, Computational Material Science 67, 249260 (2013). M. Aliabadi, The Boundary Element Method: Application in Solids and Structures Vol.2 (John Wiley and Sons Ltd., England, 2002). S. Nemat-Nasser, M. Hori, Micromechanics: Overall Properties of Heterogeneous Materials, Second Revised Edition. (North-Holland, Elsevier, Amsterdam, The Netherlands,1999). M. Ostoja-Starzewski, Probabilistic Engineering Mechanics 21, 112132 (2006). T. Kanit, S. Forest, I. Galliet, V. Mounoury, D. Jeulin, International Journal of Solids and Structures 40, 36473679, (2003). J.W. Palko, W.M. Kriven, S. V. Sinogeikin, J.D. Bass and A. Sayir Journal of appliedphysics 89, 7792-7796, (2001). W. Pabst, G. Tichà, E. Gregorovà, Ceramics 48, 41-48, (1961). R.M. Spriggs, Journal of the American Ceramic Society 44, 628-629, (1961).


7

A symmetric BEM approach to strain gradient elasticity for 2D static boundary-value problems Panzeca T1, Terravecchia S.2 and Polizzotto C3 1

DICAM, Viale delle Scienze, 90128 Palermo, [email protected]

2


3


Keywords: Strain gradient elasticity, Symmetric Galerkin BEM.

Abstract. The symmetric Galerkin Boundary Element Method is used to address a class of strain gradient elastic materials featured by a free energy function of the (classical) strain and of its (first) gradient. With respect to the classical elasticity, additional response variables intervene, such as the normal derivative of the displacements on the boundary, and the work-coniugate double tractions. The fundamental solutions featuring a fourth order partial differential equations (PDEs) system - exhibit singularities which in 2D may be of the order 1 / r 4 . New techniques are developed, which allow the elimination of most of the latter singularities. The present paper has to be intended as a research communication wherein a part of the results, being elaborated within a more general paper [1], are reported. Introduction After the pioneering work of Mindlin [2], theories of strain gradient elasticity have become very popular, particularly within the domain of nano-technologies, that is, for problems where the ratio surface/volume tends to become very large and there is a need to introduce at least one internal length. However the model introduced by Mindlin and then improved by Mindlin et al. [3] and Wu [4] leads to an excessive number of material coefficients, which at the best for isotropic materials reduce to the number of seven. In the early 1990’s, Aifantis [5] introduced a signified material model of strain gradient elasticity which requires only three material coefficients, that is, the Lame' constants and one length scale parameter. The latter model was then developed further following the so-called Form II format given by Mindlin et al. [3], that is a theory centered on the existence of a free energy function of the (classical) strain and of its first gradient, which leads to the generation of symmetric stress fields (see Askes et al. [6] for historical details about the latter formulations and its applications). Formulations in the boundary element method based on the strain gradient elasticity were pioneered by Polyzos et al. [7], Karlis et al. [8], who provide a collocation BEM formulation where the simplified constitutive equation by Aifantis [5], has been adopted. In latter papers only the fundamental solutions used in the collocation approach to BEM are provided. In the case of the symmetric formulation of the BEM, Somigliana Identities (SIs) for the tractions and for the double tractions are also needed. These new SIs are necessary in order to get, through the process of modeling and weighing, a solving equation system having symmetric operators. In Polizzotto et al. [1] all the set of fundamental solutions is derived starting from the displacement fundamental solution given in [7,8]. The symmetric formulation is motivated by the high efficiency achieved within classical elasticity by the method in [9] with regard to the techniques used to eliminate the singularities of the fundamental solutions, the evaluation of the coefficients of the solving system and the computational procedures characterized by great implementation simplicity. This has already led to the birth of the computer code Karnak sGbem [10] operating in the classical elasticity. The objective of this paper is to experiment new techniques and procedures that, applied in the context of strain gradient elastic materials, may permit one to obtain the related solving system. 1. Basic relations in 2D The class of strain gradient elastic materials herein considered is featured by the following strain elastic energy

8

W


2 1 ε : E : ε E :: ª¬ª((ε)T ε º¼ 2 2

(1)

that is a function of the 2nd order strains ε and of the strain gradient where

is the internal length and

ε (u)S .

(2)

Eq.(1) provides the "primitive" stresses σ(0)

E ε;

σ(1)

2

2σ(0)

(3a,b)

where E is the classic isotropic elasticity tensor. By the principle of the virtual power, the indefinite equilibrium equation and the following boundary conditions prove to be

σ b 0 in : ,

t

on * ,

t

r

r

on * ,

p p

on the corner P ,

(4a,b,c,d)

where the total stresses σ , the tractions t , the double tractions r and the force at the corner P are so defined

σ σ(0) 22σ(0) ,

t n σ C (n σ(1) ) ,

r nn : σ(1) ,

p

sn : σ (1) .

(5a,b,c,d)

In eq.(5b), the symbol C denotes the "reduced tangent gradient" defined as C ( s ) Kn with K ( s ) n being the boundary curvature at the considered point and ( s ) (I nn) is the operator of the tangent gradient on the boundary having normal n ; in eq.(5d) s denotes a unit vector tangent to the two boundary portions convergent in the corner P and the brackets x indicate that the enclosed quantity is the difference between the related values taken on two sides of the corner P. For a more exhaustive understanding of the jump x to see [2]. The constitutive equation relating the total stress σ to the strain is

σ E : (ε 22ε)

(6)

where ε is the classical strain. 2. The foundamental solutions. The introduction of eq.(6) in (4a) valid in :f allows to express the latter in terms of displacements only [7,8] and to determine the fundamental solution of the displacements that for 2D solids proves to be ª º ½ ª2 2 ° § r ·º ° « ®2 4 « 2 K 2 ¨ ¸ » ¾ I » r © ¹ ° «° » ¬ ¼¿ 1 ¯ G uu = « » 2 16 S P (1 Q ) « ° ½ ª º » ª º r 2 r ° § · § · « ®2(3 4Q ) « Log[r ] K 0 ¨ ¸ » 2 « r 2 K 2 ¨ ¸ » ¾r r » © ¹ ¼ °¿ © ¹¼ ¬ ¬ »¼ ¯ ¬« °

(7)

where K0 x and K 2 x are Bessel functions of the second kind and of order 0 and 2, respectively, and

r ξ x is the distance between effect ξ and cause x points. One can note that the singularity of the fundamental solution G uu does not depend on r . Indeed, by performing the limit r o 0 one obtains G uu (ξ

x)

1 ª 2(3 4Q ) >ln(2 ) J @ 1º¼ I 16 S P (1 Q ) ¬

(8)

where J is the Eulero constant. So the limit for ξ o x ( r o 0 ) the fundamental solution for isotropic gradient elasticity shows singularity of type ln( ) . In eq.(7) the limit of G uu for of type ln(r ) .

o 0 gives the classic isotropic elasticity solution (Kelvin) with singularity


9

By the fundamental solution G uu in (7), taking in account eqs.(5b,c,d) and using the well-known procedure given in [11], it is possible, by exploiting the known properties of symmetry of the fundamental solutions, to derive the entire tableau provided below.

Table 1. Fundamentals solutions for strain gradient elasticity.

The fundamental solutions G hk of Table I are characterized by two subscripts: the first indicates the effect at ξ , i.e. displacement for h u , traction for h t , displacement normal derivative for h g , double traction for h r , corner displacement for h v , corner force for h p , stress for h V ; whereas the second subscript indicates – through a work-conjugate rule – the cause applied at x , i.e. a unit concentrated force for k u , a unit surface relative displacement for k t , a unit concentrated double force for k g , a unit higher order surface distortion for k r , a unit corner force for k v , a unit corner displacement for k p , a unit imposed strain for k V . To consider the way in which the fundamental solutions included in the single

and double

brackets to see [2].

While the fundamental solutions represent the response in a point of the unlimited domain, the response to the distributed actions is provided by the SIs that in [1] are obtained by a generalization of the Betti theorem for the gradient elastic materials. 3. A case study. The fundamental solutions present in the Table I show various orders of singularity, up to 1/ r 4 in the column related to u , up to 1 / r 3 in that related to u E ; furthermore the singularities present in the column u E present an order lower than that of u , relatively to the same effect. In order to investigate the techniques useful to remove the singularities from the blocks of coefficients, a simple application in two-dimensional space is shown, but at this stage involving the study of fundamental solutions having the maximum order of singularity equal to 1/ r 2 ; it has to be remembered that in the symmetric BEM the coefficients are obtained by a double integration, the first (inner) regarding the modeling of the cause through appropriate shape functions and the second (external) regarding the effect weighed through shape functions, but dual in the energetic sense. For this purpose, the example shown concerns a plate completely constrained on the boundary, subjected to the simple distortions u , u E and distortions of higher order g wu / wn . The greater difficulties consist in removing the singularities when the cause is focused at the corner. But the simple observation that the vector u regards all the nodes of the boundary, including the vector u E at the corner, allows to eliminate the higher order singularities after the first integration and, successively summing the effects, those remaining through the second integration. The singularities present in the fundamental solutions used in the plate require shape functions of type C (0) . This example is also used to develop the techniques of rigid motion that could be used in other applications to compute the coefficient blocks in which the techniques available are not sufficient to suppress the residual singularities. The field response is derived by the SIs after the solution being obtained.

10


Let us analyze the two-dimensional plate of Fig.1 having the following physical and mechanical characteristics: E 1, Q 0.3, g 0.1, s 1 . The plate is subjected to a linear displacement distribution u , including the nodal displacement u E , and to the linear displacement normal derivatives g wu / wn , all imposed on the boundary (Fig.1c,d).

Fig.1 Plate: a) geometry and constraints; b) discretization into boundary elements and linear modelling of the boundary quantities; c) modelling of the displacements imposed; d) modelling of the displacement gradients imposed.

The displacement and displacement gradient fields imposed on the boundary are shown in Fig.1c,d. The displacements imposed on the boundary are obtained by displacement field which is not the solution of the problem, but is able to define the nodal values U x on the boundary of the plate ux

(1 x)(1 y) 4

(9a)

and as a consequence to obtain a normal derivative field gx =

wux wn

ª (1 y) º ª (1 x) º nx « » ny « 4 » . ¬ 4 ¼ ¬ ¼

(9b)

The two previous relations have been used to evaluate the nodal values of the displacements U x and of the displacement derivatives Gx , as in Fig.1c,d. For the plate of Fig.1a the boundary conditions are

u u on * u

g

g on * g with *u { * g

(10a,b)

Let us proceed to write the SIs on the boundary following the compact notation proposed by Panzeca et al. [9]:

1 u u[f , r ] u[uCPV ] u[u E ] u[g ] u 2

on * u

(11a)

1 g g[f , r] g[u] g[u E ] u[g CPV ] g 2

on * g

(11b)

In the SIs: x the superscripts are known quantities; x the apex CPV indicates that the corresponding integral is evaluated as Cauchy Principal Value to which the related free term is added; x u E is a vector containing only the displacement of the corner; x u[u E ] and g g[u E ] are, respectively, the displacement and normal derivative of displacement distribution on the boundary, computed as the difference between the response to the values of displacement imposed on the two portions of boundary afferent to the corner. Introducing the latter SIs given in eqs.(11a,b) into the boundary conditions eqs.(10a,b) one has:

1 u[f , r] u[uCPV ] u[u E ] u[g ] u 0 2

on * u

(12a)


1 g[f , r] g[u] g[u E ] u[g CPV ] g 2

11

on * g

0

(12b)

Let us now discretize the plate boundary into boundary elements (Fig.1b) and introduce the modeling of the quantities on the boundary (Fig.1c) as functions of the nodal variables through suitable matrices of shape functions N h , with h f , r , u, g

N f F;

f

r Nr R;

u Nu U;

NgG .

g

(13a,b,c,d)

Since the vector u E collects quantities to not be modeled, the following identity may be imposed uE

UE .

(14)

In eqs.(13a-d) the quantities F , R , U , G have the meaning of nodal quantities and precisely: force and double traction as unknowns, displacements and normal derivative of displacements as known quantities. To evaluate the vector G , because the discontinuity of the normal derivative at the corner P, a double node belonging respectively to the two portions of boundary afferent to P has to be considered. Introducing the modeling (13a-d) and the relation (14) into eqs.(12a,b) and performing the weighing second Galerkin, using the shape functions in dual form, the following system in compact form is obtained as in Panzeca et al.[9]

Aug º ª F º ª Aut Cut A gu »¼ «¬ R »¼ «¬ A gt

ª Auu «A ¬ gu

Aur º ª U º ª aupp « A gr C gr »¼ «¬ G »¼ « a ¬ gp

º ª0º » ( U E ) « » » ¬0¼ ¼

(15)

Since some components of the vector U and of the vector U E coincide, it is opportune to introduce the following relation UE

(16)

HE U

where the low rectangular matrix H E is a topological matrix made by I 2u2 and 02u2 blocks. The previous relation can be rewritten

Aug º ª F º ª«( Aut Cut ) aupp A gu »¼ «¬ R »¼ « A a * gt gp ¬«

ª Auu «A ¬ gu

*

º » (U) ª Aur º (G ) ª0º «A C » «0» » gr ¼ ¬ ¼ ¬ gr ¼»

(17)

where

aup

*

aup H E ;

a gp

*

a gp H E .

(18a,b)

Finally, the solving system is rewritten in compact way with obvious meaning of symbols K X Lu L g

0

(19)

L

with K symmetric flexibility matrix of the system, X vector of nodal unknowns, L u and L g load vectors due to U e G respectively, being L Lu L g the total load vector. In order to evaluate the coefficients of the equation system the numerical techniques for removing the singularities and the rigid motion strategy have to be employed. x

Observation about rigid motions

Let the plate of Fig.2 be subjected to a) a rigid motion of translation obtained as the sum of constant displacement fields u x and u y (Fig.2a), b) a rigid motion of rotation having wideness M around any point (Fig.2b), c) a rigid motion of roto-translation obtained as a combination of a) and b).

12


In all these cases the displacement and normal derivative of the displacement fields are entirely known both in the domain and on the boundary of the plate.

Fig.2 Rigid motions: a) translation, b) rotation.

Particularly - for the case of rigid motion of translation (Fig.2a) wu u x c1, gx x 0 wn wu y u y c2, gy 0 wn with c1 and c 2 being the wideness of the displacements;

(20a,b,c,d)

- for the case of rigid motion of rotation having width f (Fig.2b) M f ux uy

x

wu f y g x x f ny wn wu y f nx f x gy wn Observations about the load vector as a consequence of the rigid motion

(21a)

(22b,c,d,e)

1) If the solid is subject to a rigid motion of translation (Fig.2a) having assigned values of the displacements c1 and c 2 , one has U z 0 and G 0 but the total load vector has to be L Lu L g 0 with Lu

ª( A C ) a ut up p « ut * « «¬ A gt a gp

*

º » U » »¼

ª0 º «0 » ¬ ¼

ª Aur º ª0º «A C » G « » gr gr ¬0¼ ¬ ¼ and as a consequence the solution vector X will be null.

(23)

Lg

(24)

2) If the solid is subject to a rigid motion of rotation (Fig.2a) having assigned rotation vector φ

f , in

the generic node i one has Ui ri u φ z 0 Gi φ u ni

φ si z 0

(25a,b)

with ri vector distance between the instantaneous center of rotation and the node i , n i e s i being respectively the unit normal and tangent vectors to the boundary elements on which the node i lies, but the total load vector has to be L Lu L g 0 with Lu

ª( A C ) a ut up p « ut * « A a gt gp ¬«

*

º » (U) z ª0 º «0 » » ¬ ¼ ¼»

(26)


13

ª Aur º ª0º «A C » G z « » gr ¼ ¬0¼ ¬ gr with Lu L g and as a consequence the solution vector X will be null. Lg

(27)

3) If the solid is subjected to a combination of the rigid motions 1) and 2), the condition L Lu L g has to be always valid and as a consequence Lu

L g .

For the example considered the vectors U , U E e G are the following, as shown in Fig1c,d. UT

1 1 ª º « 0 0 2 0 1 0 2 0 0 0 0 0 0 0 0 0 » ¬ ¼

UTE

>

GT

1 1 1 1 1 1 1 1 ª º « 0 0 4 0 2 0 2 0 4 0 0 0 2 0 4 0 0 0 0 0 4 0 2 0 » ¬ ¼

0 0 1 0 0 0 0 0

@

The results are summarized in Table 2. In Fig.2 the displacement u x within the domain along a line placed at y 0.9 is shown.

Table 2 . Results wedged plate

Fig.2 Displacements ux at y=0.9

0

14


Conclusions The table of fundamental solutions for isotropic strain gradient elasticity is similar to an analogous table used within classical boundary element method. Computational techniques have been pursued in order to eliminate the singularities of order 1 / r , 1 / r 2 , in the blocks of the coefficients related to the corners of the solid and new techniques based on the rigid motion strategy have been introduced in order to test the coefficients of the blocks of the solving system. The displacement and internal deformation fields were obtained. Numerical techniques in order to remove the singularity of higher order like 1 / r 3 and 1 / r 4 are in advanced study. References. [1] Polizzotto, C., Panzeca, T., Terravecchia, S. (2014). "A symmetric Galerkin BEM formulation for a class of gradient elastic materials of Mindlin type. Part I: Theory". In preparation. [2] Mindlin, R.D.(1965). "Second gradient of strain and surface tension in linear elasticity". Int. J. Solids Struct., 1, 417-438. [3] Mindlin, R.D., Eshel, N.N., (1968). "On first strain-gradient theories in linear elasticity". Int. J. Solids Struct., 28, 845-858. [4] Wu, C.H. (1992). "Cohesive elasticity and surface phenomena". Quart. Appl. Math., L(1), 73-103. [5] Aifantis, E.C. (1992). " On the role of gradients in the localization of deformation and fracture". Int. J. Eng. Sci., 30, 1279-1299. [6] Askes, H., Aifantis, E.C. (2011). " Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results". Int. J. Solids Struct., 48, 1962-1990. [7] Polyzos, D., Tsepoura, K.G., Tsinopoulos, S.V., Beskos, D.E. (2003). "A boundary element method for solving 2-D and 3-D static gradient elastic problems. Part.I: integral formulation". Comput. Meth. Appl. Mech. Engng., 192, 2845-2873. [8] Karlis, G. F. , Charalambopoulos, A., Polyzos, D. (2010). "An advanced boundary element method for solving 2D and 3D static problems in Mindlin's strain gradient theory of elasticity". Int. J. Numer. Meth. Engng., 83, 1407-1427. [9] Panzeca, T., Cucco, F., Terravecchia S. (2002). "Symmetric Boundary Element Method versus Finite Element Method". Comp. Meth. Appl. Mech. Engrg., 191, 3347-3367. [10] Cucco F., Panzeca T., Terravecchia S. (2002). "The program Karnak.sGbem Release 2.1". Palermo University. [11] Polizzotto C. (1988) . "An energy approach to the boundary element method. Part.I: elastic solids". Comput. Meth. Appl. Mech. Engng., 69, 167-184.


15

Comparison of Green element solutions for inverse heat conduction problems using the time-dependent and logarithmic fundamental solutions Akpofure E. Taigbenu School of Civil and Environmental Engineering University of the Witwatersrand. P. Bag 3, Johannesburg, WITS 2050. South Africa. [email protected]

Keywords: Inverse heat conduction problems; Green element method; time-dependent fundamental solution; logarithmic fundamental solution; singular value decomposition; Tikhonov regularization

Abstract Inverse heat conduction problems (IHCPs) are addressed in this paper, and they relate to the recovery of the temperature, heat flux and heat source. They are solved in 2-D homogeneous media for transient cases by the Green element method (GEM). Comparisons are made when the time-dependent fundamental solution of the diffusion differential operator and the logarithmic fundamental solution of the Laplace operator are used in the numerical formulations. With both fundamental solutions, the over-determined coefficient matrices are decomposed by the singular value decomposition (SVD) method and solved by the least square method, while the ill-conditioned nature of the matrices are regularized by the Tikhonov regularization method. Four numerical examples of transient IHCPs are solved. Using the same spatial and temporal discretizations, the GEM with the logarithmic fundamental solution is generally more superior in accuracy and computational speed than the formulation with the time-dependent fundamental solution. Introduction The considerable interest that continues to attend the solution of inverse heat conduction problems (IHCPs) is primarily due to their many practical applications in the fields of science and engineering where heat, mass and energy transport processes occur and, in a broader sense, the intrigues and challenges posed by inverse problems in a wide variety of endeavours. Two classes of IHCPs addressed in this paper are the recovery of boundary temperature and heat flux [1,2,3], and the recovery temporal distribution of heat sources/sinks [4,5,6,7]. It is widely known that inverse problems are ill-posed so that the matrices resulting from solving them are usually ill-conditioned, in contrast to direct solution methods which produce well conditioned matrices. The performances of two formulations of the Green element method (GEM) are examined for these two classes of IHCPs. In the first, the differential equation is treated as a Poisson equation to which the logarithmic fundamental solution of the Laplace operator is used and the temporal part of the equation is effected by finite differencing in time [8,9], while the second approach uses the time-dependent fundamental solution of the heat equation. For both formulations, the internal normal fluxes are approximated by a second-order polynomial relationship in terms of the temperature, and the resulting over-determined matrices are decomposed by the singular value decomposition (SVD) method and solved by the least square method. The challenge posed by the ill-conditioned nature of the matrices is addressed by the Tikhonov regularization technique. Four numerical examples of which two address the first class of problems and the other two the second class of IHCPs are solved by the two GEM formulations. The GEM with the logarithmic fundamental solution generally achieves higher accuracy using about 3% of the computing speed of the formulation with the time-dependent fundamental solution.

16


Governing Equation The initial-boundary value problem that is addressed in this paper is governed by the differential equation wT (1) K 2T c Q(t ) wt where 2 is the 2-D Laplacian operator, t is the time dimension, T is the temperature, K and c are respectively the thermal conductivity and heat capacity of the medium which is considered homogeneous, and Q represents heat sources and sinks whose strengths are not known but have only temporal variation. The initial data of the temperature are specified everywhere in the domain Ω at time t0, (2a) T ( x, y, t0 ) T0 ( x, y) while Dirichlet, Neumann, and Cauchy-type conditions are specified on boundary segments Γ1, Γ2 and Γ3. That is: (2b) T ( x, y, t ) T1 on Γ1 KT n

q2 on Γ2

(2c)

J 1T J 2 KT n g3 on Γ3

(2d)

where n is the unit outward pointing normal on the boundary, γ1 and γ2 are known constants. The domain Ω with the boundary Γ=Γ1 Γ2 Γ3 Γ4 is shown in Fig. 1, and neither the temperature T nor heat flux q is specified on Γ4. For both types of IHCPs, temperature measurements are available at Ni internal points (xm,ym) in the domain and denoted as Tm=T(xm,ym,t). In practice these measurements are attended with errors which can be described by the relationship ~ (3) Tm Tm >1 V u RN (m)@ where σ is the error magnitude and RNϵ[-1,1] are random numbers. The formulation of GEM which uses the logarithmic fundamental solution, herein referred to as Formulation 1, had earlier been presented in Taigbenu [8] and as such it is not repeated here. It transforms the governing equation into a Poisson equation to which it applies Green’s identity to obtain the integral equation. The second formulation, referred to as Formulation 2, uses the fundamental solution of the transient diffusion differential operator T3(x,y,t) q 3(x,y,t)

T1 (x,y,t) *1

*3

T0 (x,y)

q2(x,y,t)

:

*2 T(xm,ym,t)

*4 Figure 1: Domain and problem statement representation Applying Green’s theorem to eq (1), and integrating in time between t0 and any time t yields the integral equation [10]


17

OTi D ³ ³ >T (r ,W )G * (r , ri , t ,W ) G (r , ri , t ,W )T * (r ,W )@dsdW t

t0 *

(4)

t

³³ G (r , ri , t ,0)T (r ,0) dA ³ Q(W ) ³³ G (r , ri , t ,W ) dAdW :

0

:

t0

Where G (r , ri , t ,W )

ª (r ri ) º H (t W ) exp « » D(t W ) ¬ 4 D(t W ) ¼

is the fundamental solution of D2G wG / wt

(5) G (r ri )G (t W ) , D=K/c, λ is twice the node angle at the

source or collocation node ri = (xi,yi), and G* and T* are respectively the normal derivatives of G and T. The boundary and domain integrals in eq (4) are implemented in the Green element sense over sub-domains or elements that are used to discretize the computational domain. On these elements, Lagrange-type interpolations are prescribed in space and in time for the unknown quantities. In this paper, linear interpolations are employed in time and space. Implementing eq (4) for each element Ωe yields

RijmTjm Lmijq mj U ijTj1 Fi mQm

0

(6)

where t2

D ³Q m (W ) ³ M j (r )G * (r , ri , t ,W ) dsdW G ijO ; Lmij

Rijm

*e

t1

m ³³ M j (r )G (r , ri , 't ,0) dA; Fi

U ij

:e

1 t2 m ³Q (W ) ³ M j (r )G (r , ri , t ,W ) dsdW ; c t1 *e

1 t2 m ³Q ³³ G (r , ri , t ,W ) dAdW c t1 :e

(7)

The index m represents the time levels of the previous and current times t1 and t2, q=–KT* is the normal heat flux, and ν and φ are respectively the interpolating functions in time and space. The presentation of eq (6) reflects one of two time-marching schemes in which the integrations are done at each time step; the other scheme requires that the time integration always restarts at the initial time t0 [10]. The discrete element equations represented by eq (6) are aggregated for all the elements that are employed in discretizing the computational domain, resulting in the matrix equation (8) Ap b Where p={T2,q2,Q2}tr is an N×1 vector of unknowns at t2 (T and/or q at external nodes, T at internal nodes and the heat sources, Q) and tr is the transpose. The right side b of eq (8) comprises the initial data and the specified boundary and interior data. The matrix A is an M×N matrix, where M is the number of nodes in the computational domain (which equals the number of discrete equations generated by the GEM formulation) and M ≥ N. Eq (8) is over-determined and its solution is amenable to the least square method, while the matrix A is usually ill-conditioned and it is regularized by the Tikhonov regularization method. The decomposition of A is facilitated by the singular value decomposition (SVD) method [11] A

N

tr ¦\ i ui vi

UDVt

(9)

i 1

where U and V are, respectively, M×M and N×N orthogonal matrices and D is an M×N diagonal matrix with N non-negative diagonal elements ψ1, ψ2, ..., ψN. The least square solution of eq (9) minimises the Euclidian norm ║Ap – b║2, resulting in the solution for the unknowns p N u tr b p B 1s ¦ i vi (10) i 1

\i

Where B=AtrA, s=Atrb, and ui and vi are the ith column of the matrices U and V, respectively. The small singular values ψi cause instability of the solution for p, and this is overcome by using the Tikhonov regularization method which minimizes ║Ap-b║2+α2║Ip║2 in calculating the solution for p by [12] p(D )

N

¦ i 1

\i u trbv D 2 \ i2 i i

(11)

18


where α is the regulation parameter whose choice is carefully made so that it is not too small to retain the instability of the numerical solution or too large to have smooth unrealistic results. Numerical Examples Four numerical examples of transient IHCPs are solved by the two GEM formulations. The first two only address the recovery of T and q, while the others address the recovery of Q. The first two examples had been solved by Lesnic et al. [3], while the other two had been addressed by Yan et al. [13] using the method of fundamental solutions (MFS). Example 1 This is a transient example in one spatial dimension. It is solved by the GEM formulations in a 2-D rectangular domain with insulated boundaries at the top and bottom. With the test function T(x,t)=2t+x2 that satisfies the governing eq (1) in xϵ[0,1], K=1, c=1 and Q=0, the temperature distribution T(x,0)=x2 is prescribed at the initial time t=0 and a Γ3 boundary is along x=1 where the temperature and flux are specified, T(1,t)+q(1,t)=2t+3. The boundary along x=0 is a Γ4 boundary where neither T nor q is specified, and three locations with available temperature measurements are examined: (i) xm=1, (ii) xm=0.5 and (iii) xm=0.25. The GEM simulations of this example used only four rectangular elements and a time step, ∆t=0.025, while the values of the regularization parameter for both formulations for the four examples are presented in Table 1. The numerical results from the two GEM formulations are presented Figs. 2a and 2b in terms of the relative error, calculated by eq (12) for T(x,t), and q(x=0,t) along the Γ4 boundary for the three cases. The superiority of Formulation 1 over Formulation 2 is evident by its accuracy from the results. M

H

1 M

cal exact 2 ¦ (Ti Ti ) i 1

M

¦ (Ti

exact 2

)

i 1

(a) (b) Figure 2: GEM simulation results of Example 1: (a) relative error of T(x,t) and (b) q(x=0,t)

(12)


19

Example 2 This is a transient IHCP with a more stiff test function that is prescribed in xϵ[0,1] and given by [3,14] u ( x, t ), t [0,0.5) °u ( x, t ) 2u ( x, t 0.5), t [0.5,1) ° (13) T ( x, t ) ® °u ( x, t ) 2u ( x, t 0.5) 2u ( x, t 1), t [1,1.5) ¯°u ( x, t ) 2u ( x, t 0.5) 2u ( x, t 1) 2u ( x, t 1.5), t [1.5,2] where u ( x, t )

f ( 1) n 2 2 3(1 x) 2 1 t 2¦ 2 2 cos[nS (1 x)]e n S t n 1 n S 6

(14)

Table 1: Values of the regularization parameter, α, used in the simulations of the four examples

α

Examples and cases 1 (i) 1 (ii) 1 (iii)

Formulation 1 2.2×10-4 3×10-3 7.1×10-4

Formulation 2 3×10-3 10-2 1.5×10-2

2 3: σ = 0% 3: σ = 1% 3: σ = 3% 3: σ = 5%

3.2×10-4 2.2×10-4 3.2×10-4 3.2×10-4 3.2×10-4

5×10-2 10-4 10-4 10-4 10-4

4: σ = 0% 4: σ = 1% 4: σ = 3% 4: σ = 5%

2.2×10-4 3.2×10-4 3.2×10-4 3.2×10-4

10-4 10-4 10-4 10-4

Along the boundary x=1, T and q are specified (q(1,t)=0) and measurements of T are available at xm=0.25, while the boundary along x=0 is a Γ4 boundary where neither T nor q is known. Using only 4 rectangular elements, the GEM simulations are carried in a 2-D domain with a uniform time step ∆t=0.025. The temporal variations of ε and the flux at x=0 for the two GEM formulations are respectively presented in Figs. 3 and 4, and the results indicate superior accuracy of Formulation 1.

Figure 3: Error plots of T(x,t) with time from the two GEM formulations for Example 2

20


Figure 4: GEM simulations of the variation of the flux at x=0 with time for Example 2 Example 3 In this example, the recovery of the strength of the heat source is sought. The exact solution to eq (1) in 1-D spatial domain xϵ[0,1] that is used as the test function is x4 (15) 3tx 2 sin( x)e t 4 It gives an expression for the heat source, Q(t) = –6t. Dirichlet boundary conditions are specified along x=0 and x=1, while temperature measurements are available at xm=0.5 for all times. The GEM simulations are implemented in a rectangular domain and discretized with 10 linear rectangular elements. A uniform time step of 0.025, and the temperature data at x=0, 0.5 and 1 are perturbed randomly with noise levels of σ=1, 3 and 5%. The values of the regularization parameters used in both formulations are found in Table 1. The numerical results from both GEM formulations for the heat source are presented in Fig. 5 for various noise levels. The results from both formulations when there is no noise in the data are left out of Fig. 5 because they correctly reproduced the exact solution. The plots of the relative error, ε, from both formulations are presented in Fig. 6. Formulation 2 produced more accurate and stable solutions than Formulation 1 for the various noise levels. T ( x, t )

Figure 5: GEM simulations for the heat source Q(t) of Example 3


21

Figure 6: Error plots of T(x,t) with time from the two GEM formulations for Example 3 Example 4 The test function that satisfies eq (1) in xϵ[0,1] used in this example is T ( x, t ) x 2 2t sin(2St )

(16)

The corresponding heat source expression is Q(t) = 2πsin(2πt). Along x=0 and x=1 the temperature is specified, and its measurements are available at xm=0.5 for all times. The GEM simulations use 10 linear rectangular elements, a uniform time step ∆t=0.025, and the temperature data at x=0, 0.5 and 1 are perturbed randomly with noise levels of σ=1, 3 and 5%. The GEM and exact solutions for the heat source are presented in Fig. 7 for various noise levels. Though not shown in Fig. 7, the numerical results from the two GEM formulations reproduced the exact solution when there is no noise in the data. The values of the regularization parameter used in the two formulations are found in Table 1. The plots of the relative error, ε, from the two formulations are presented in Fig. 8. The results from Formulation 2 are slightly more accurate than those of Formulation 1, but they are oscillatory for both formulations at noise levels of 3% and 5%.

Figure 7: GEM simulations for the heat source Q(t) of Example 4

22


Figure 8: Error plots of T(x,t) with time from the two GEM formulations for Example 4

Conclusion Two Green element formulations have been used to solve transient IHCPs in 2-D homogeneous and heterogeneous domains. Both formulations produced over-determined and ill-condition system of discrete equations which are solved by the least square method with Tikhonov regularization. Formulation 1, which uses the fundamental solution of the Laplace operator, produced more accurate results than Formulation 2, which uses the fundamental solution of transient diffusion equation, for the first two examples of temperature and heat flux recovery, and vice-versa for the other two examples of heat source recovery. The computing speed of Formulation 1 is faster by about 30 times than that of Formulation 2 for the four simulated examples.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

M.J. Ciałkowski and K. Grysa, J. Theo. and App. Mech., 48, 111-134 (2010). H.J. Reinhardt, D.N. Hao, J. Frohne and F.T. Suttmeier, J. Inverse & Ill-posed Prob., 15, 181-198 (2007). D. Lesnic, L. Elliot, D.B. Ingham, Application of the boundary element method to inverse heat conduction problems, Int. J. Heat Mass Trans., 39 (1996) 1503-1517. T. Wei and J.C.Wang, Engrg. Anal. Bound. Elem. 36, 1848–1855 (2012). C-H. Huang, J-X. Li and S. Kim, App. Math. Mod. 32, 417-431 (2008). L. Yan, C-L. Fu and F-L. Yang, Engrg. Anal. Bound. Elem. 32, 216-222 (2008). A. Farcas and D. Lesnic, J. Engrg. Math., 22, 1289-1305, 2006. A.E. Taigbenu, Engrg. Anal. Bound. Elem., 35, 125-136 (2012). A.E. Taigbenu, The Green Element Method, Kluwer, Boston, USA (1999). C.A. Brebbia, J.C.F. Telles and L.C. Wrobel, Boundary Element Techniques, Springer-Verlag, NY, 1984. G.H. Golub and V.F. Van Loan, Matrix Computations, John Hopkins Univ. Press, Baltimore, USA, 1996. P.C. Hansen, Numer. Algorithms, 6 1-35 (1994). L. Yan, C-L. Fu and F-L. Yang, Engrg. Anal. Bound. Elem., 32, 216-222 (2008). H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids, Oxford Univ. Press, UK, 1959.


23

Analysis of elastic problems by the fast multipole boundary element method A. F. Dias. Jr1 , E. L. Albuquerque1 1

University of Brasilia - Darcy Ribeiro Campus, North Wing - Brasilia - Brazil [email protected] // [email protected]

Keywords: Boundary Element Method, Plane Elasticity, Fast Multipole.

Abstract. This paper presents a Fast Multipole Boundary Element Method (FMMBEM) for large-scale analysis of plane elastic problems. Integral equations for plane elastic problems should be written in a complex form. The Taylor series will be used to expanded the fundamental solution. For far elements the integration are written considering only locations of element and expansion point. We will use the hierarchical tree structure to assemble the points that are near or far of the element. For points near the elements, the direct application Boundary Element Method (BEM) is applied. However, for points that are far of the elements, the fast multipole expansion is used. To study the accuracy and efficiency of the fast multipole BEM formulation, we will present an numerical example that will show the the potential of the fast multipole BEM for solving large-scale problems from plane elasticity. Introduction. Normally, the conventional BEM is not efficient to solve large-scale problems. To achieve higher run-time and memory storage efficiency, the Fast Multipole Method (FMM) proposed by [1] is applied to the BEM. The same discretization that is used in the BEM is used for the fast multipole. Furthermore, should be used a quad-tree (for two-dimensional problems) or an oct-tree (for three-dimensional problems) for computational efficiency and storage saving. The matrix vector product will be obtained by recursive operations on the tree structure without explicitly forming the coefficient matrices. The applications fo FMMBEM were used by many authors, including [3] for 2D elastostatics, [4] for 3D elastostatics, [8, 9] for 3D elastostatic crack problems, [2] for the modelling of carbon-nanotube composites, [5, 6] for the simulation of composite materials, [7] for the analysis of fatigue crack growth. In this paper, a fast multipole BEM for 2D elastic formulation will be presented to solve large-scale plane elastic problems. This result will demonstrate the potential of the fast multipole BEM for solving large-scale plane elastic problems. Boundary Element Equations. For elastostatic problems, we have the convencional boundary integral equation, CBIE: ci j (x)u j (x) =

u∗i j (x, y)t j (y) − ti∗j (x, y)u j (y)dΓ + q j (y)u∗i j (x, y)dΩ, Γ

Ω

(1)

where x = (x1 , x2 ) ∈ Γ and y = (y1 , y2 ) ∈ Γ are source point and field point, respectively, u∗i j and ti∗j are the fundamental solutions given by: 1 1 1 (2) (3 − 4ν )δi j log + r,i r, j − δi j , u∗i j (x, y) = 8πμ (1 − ν ) r 2

∂r −1 ti∗j (x, y) = [(1 − 2ν )δi j + 2r,i r, j ] − (1 − 2ν )(r,i n j − r, j ni ) . (3) 4π (1 − ν )r ∂ n and r = (x1 − y1 )2 + (x2 − y2 )2 . It is possible to write the two integrals in CBIE (1) in complex variables. To make this, we have to write the fundamental solution u∗i j (x, y) and ti∗j (x, y) in the complex notation. Considering that (x) is the coordinate of source point an (y) is the coordinate of field point. Then, for a smoth boundary, we have:

24


1 2

u1 0 0 u2

+

∗ t11 u1 ∗ u t21 1

Γ

∗ u t12 2 ∗ u t22 2

dΓ =

Γ

u∗11 t1 u∗12 t2 u∗21 t1 u∗22 t2

dΓ

(4)

writing u(z) = u1 + iu2 , we have this: 1 (u1 + iu2 ) + 2

Γ

∗ ∗ ∗ ∗ (t11 u1 + t12 u2 ) + i (t21 u1 + t22 u2 ) dΓ =

Γ

(u∗11 t1 + u∗12t2 ) + i(u∗21 t1 + u∗22t2 )dΓ

(5)

or 1 u(z) + Du = Dt 2

(6)

where: =

Dt

Du =

Γ Γ

∗ ∗ ∗ ∗ (t11 u1 + t12 u2 ) + i (t21 u1 + t22 u2 ) dΓ

(7)

(u∗11 t1 + u∗12t2 ) + i(u∗21 t1 + u∗22 t2 )dΓ

(8)

Now, if you use de fundamental solution (2) and (3) above, we will get this equations: Dt (z0 ) =

Du (z0 ) =

1 1 + κ)

1 2μ (1 − κ )

S

κ [G(z0 , z)t(z) + G(z0 , z)t(z)] − (z0 − z)t¯G (z¯0 , z) dS(z)

(9)

S

¯ κ (G (z0 , z))n(z)u(z) − (z0 − z)G (z0 , z)n(z)u(z) + G (z¯0 , z)[nz u¯z + n(z)u(z)] dS(z) (10)

where: G(z0 , z) =

−1 log(z0 − z) 2π

(11)

and that G (z0 , z) and G (z0 , z) is the first and second derivatives of G(z0 , z). Now, we will use the operation multipole to expand equations (Dt ) and (Du ). However, we will expand equation (Dt ) first. 0.1 Multipole Expansion for the Dt . Consider the point zc . This point is near to z, i.e., | z − zc || z0 − zc |. The multipole expansion will be carried out around zc , Figure (1). So, we have this following equation:

Dt (z0 ) =

∞ 1 κ ∑ Ok (z0 − zc )Mk (zc ) 4πμ (1 + κ ) k=0 ∞

∞

+z0 ∑ Ok+1 (z0 − zc )Mk (zc ) + ∑ Ok (z0 − zc )Nk (zc ) ; k=0

(12)

k=0

and Mk (zc ) = Nk (zc ) =

Sc

Ik (z − zc )t(z)dS(z), to k ≥ 0,

(13)

κ Ik (z − zc )t(z) − Ik−1 (z − zc )zt(z) dS(z) to k ≥ 1

(14)

Sc

where Ik (z) and Ok (z) are two auxiliary functions, define as:


25

S0 n z zc r zc zL zL

z0

Figure 1: Complex notation and the related points for fast multipole expansions.

Ik (z) =

zk , f or k ≥ 0; k!

(15)

(k − 1)! , f or k ≥ 1; (16) zk It is important to say that Mk and Nk are called the moments about zc , and Sc is a subset the of S. This subset is far away from the point z0 . Ok (z) =

0.2 Moment-to-Moment translation (M2M) for the Dt . Now, consider that the point zc moved to other position zc . We will get two new moments:

Mk (zc ) =

l=0

∑ Ik−l (z − zc )Ml (zc )

(17)

k

Nk (zc ) =

l=0

∑ Ik−l (z − zc )Nl (zc )

(18)

k

These new results will replace the two moments Mk and Nk in equation (12). 0.3 Moment-to-local translation (M2L) for the Dt In this new situation, considere the point zL . Now, this point is near to z0 , i.e., | z0 − zL || zc − zL |. The local expansion will be carried out around zL , Figure (1). So, we have this following equation:

Dt (z0 ) = ∞

∞ 1 κ ∑ Ll (zL )IL (z0 − zL ) 4πμ (1 + κ ) k=0 ∞

+z0 ∑ Lk+1 (zL )Il−1 (z0 − zL ) + ∑ Kl (zL )Il (z0 − zL ) ; k=0

(19)

k=0

where the coefficients Ll (zL ) and Kl (zL ) are given by the (M2L) translation: k=0

Ll (zL ) = (−1l ) ∑ Ol+k (zL − zc )Mk (zc ), to l ≥ 0

(20)

∞

k=0

Kl (zL ) = (−1l ) ∑ Ol+k (zL − zc )Nk (zc ), to l ≥ 0 ∞

(21)

26


0.4 Local-to-local expansion (L2L) for the Dt .Now, consider that the point zL moved to other position zL . We will get two new moments:

Ll (zL ) =

Kl (zL ) =

m=l

∑ Im−l (zL − zL )Lm (zL ), ∞

m=l

∑ Im−l (zL − zL )Km(zL ),

∞

to l ≥ 0

(22)

to l ≥ 0

(23)

These new results will replace the two moments Ll and Kl in equation (19). 0.5 Operation for the Du . Now, we will show all the operations above for the equation (Du ). So, the multipole expansion for the is: ∞ 1 κ ∑ Ok (zo − zc )M˜ k (zc ) Du (zo ) = 4πμ (1 + κ ) k=0

∞

∞

k=0

k=0

+zo ∑ Ok+1 (zo − zc )M˜ k (zc ) + ∑ Ok (zo − zc )N˜ k (zc ) ;

(24)

where the moments are: M˜ k (zc ) =

Sc

N˜ 1 = N˜ k (zc ) =

Sc

Ik−1 (z − zc )n(z)u(z)dS(z), f or k ≥ 1; Sc

n(z)u(z) + n(z)u(z) dS(z);

Ik−1 (z − zc ) n(z)u(z) + n(z)u(z) − Ik−2 (z − zc )zn(z)u(z) dS(z),

(25) (26) (27)

Now, to calculate (M2M), you have to use the equations (17) and (18) in the equation (24). This equations will replace the equations (25) and (27). To calculate the local expansion, you have to use the equation bellow:

∞

∞ 1 κ ∑ Ll (zL )IL (z0 − zL ) Du (z0 ) = 2πμ (1 + κ ) k=0 ∞

+z0 ∑ Lk+1 (zL )Il−1 (z0 − zL ) + ∑ Kl (zL )Il (z0 − zL ) ; k=0

(28)

k=0

So, for you calculate the (M2L), you have to use the equations (20) and (21) in the equation (28). Lastly, for you calculate the (L2L), you have to use the equations (22) and (23) in the equation (28) Numerical results. The aim of this section is to compare the use of memory to solve a problem in plane elasticity with FMMBEM and conventional BEM. The problem to be analysed is a square plate with 100 holes. An edge was fixed while in other is under tension (see Figure 2). The number of elements on each edge of the square plate is 20. For each hole, 16 constant elements were used. Is important to say that the area of holes is 12.47% percent of the area of the plate. The plate has an area of 1 m2 . The maximun number of elements per leaf was 15. For this problem, the number of degrees of freedom was 3360. The number of terms in Taylor series was 15. So, when we solver this problem using FMM, the computer used . The same problem was solved using BEM and, the computer used 838.457 Mbytes. With these parameters, solutions obtained by FMMBEM used 713.2734 Mbytes of memory while the conventional BEM used 838.457 Mbytes (17 % larger than the FMMBEM). This result shows that the FMMBEM is very efficient in saving memory, allowing the analysis of larger problems in a shorter time.


27

1.2

1

0.8

y

0.6

0.4

0.2

0

−0.2 −0.2

0

0.2

0.4

0.6

0.8

1

1.2

x

Figure 2: Plate with 100 holes. Conclusions. The main objective of this paper was show the gain of memory when you solve large scale problems. For this, we solved a problem that has 3360 degrees of freedom. With the results, it is possible notice that the FMMBEM consumes less memory than conventioanl BEM for large scale problems because the reduced number of interactions between source and field points. In future works, it will be possible to show the potential of the fast multipole BEM for solving large-scale problems. Acknowledgements . The authors would like to thank CAPES for the financial support of this work.

References [1] Rokhlin V. Greengard L. A fast algorithm for particle simulations. Journal of Computational Physics, 73:325–348, 1987. [2] Otani Y. Liu YJ, Nishimura N. Large-scale modeling of carbon-nanotube composites by a fast multipole boundary element method. Computational Materials Science, 34:173–187, 2005. [3] Napier JAL. Peirce AP. A spectral multipole method for efficient solutions of large scale boundary element models in elastostatics. International Journal for Numerical Methods in Engineering, 38:4009–4034, 1995. [4] Power H. Popov V. An o(n) taylor series multipole boundary element method for three-dimensional elasticity problems. Engineering Analysis with Boundary Elements, 25:718, 2001. [5] Yao ZH. Wang HT. A new fast multipole boundary element method for large scale analysis of mechanical properties in 3d particle-reinforced composites. Computer Modeling in Engineering and Sciences, 7:85–95, 2005. [6] Yao ZH. Wang HT. Application of a new fast multipole bem for simulation of 2d elastic solid with large number of inclusions. Acta Mechanica Sinica, 20:613–622, 2006. [7] Yao ZH. Wang PB. Fast multipole dbem analysis of fatigue crack growth. Computational Mechanics, 38:223–234, 2006. [8] Kobayashi S. Yoshida K, Nishimura N. Application of fast multipole galerkin boundary integral equation method to elastostatic crack problems in 3d. International Journal for Numerical Methods in Engineering, 50:525–547, 2001. [9] Kobayashi S. Yoshida K, Nishimura N. Application of new fast multipole boundary integral equation method to crack problems in 3d. Engineering Analysis with Boundary Elements, 25:239–247, 2001.

28


Analysis of Damped Waves Using Energetic BEM-FEM Coupling A. Aimi1 , L. Desiderio2 , M. Diligenti3 , C. Guardasoni4 1 Dept.

of Mathematics and Computer Science, University of Parma, 43124 Parma, Italy, [email protected]

2 POEMS 3 Dept. 4 Dept.

(UMR 7231,CNRS-ENSTA-INRIA) ENSTA, Paris, France, [email protected]



Keywords: damped wave propagation, multi-domain, energetic BEM-FEM coupling.

Abstract. Starting from a recently developed energetic space-time weak formulation of boundary integral equations related to wave propagation problems defined on single and multi domains, a coupling algorithm is presented, which allows a flexible use of finite and boundary element methods as local discretization techniques, in order to efficiently solve damped wave equation over unbounded multidomains. Partial differential equations associated to boundary integral equations will be weakly reformulated by the energetic approach and a particular emphasis will be given to the experimental analysis of the stability of the proposed method. In fact, recently proved theoretical properties of stability and convergence are crucial to ensure accurate numerical solutions for simulations even over long time intervals.

1 Introduction The subject of this paper is the one-dimensional, damped wave equation including both viscous damping, where resistance is proportional to velocity, and material damping, where resistance is proportional to displacement. The analysis of damping phenomena that occur, for example, in fluid dynamics, in kinetic theory and semiconductors, is of particular interest. The boundary element method (BEM) is a very effective numerical tool for analysis of this large class of problems both on bounded and unlimited domains. The main advantage of this method is that the discretization is done only on the boundary, yielding smaller meshes and systems of equations. Another advantage is that this method automatically satisfies Sommerfeld’s radiation conditions at infinity and there is no need to model the far field in problems with semi-infinite or infinite domains. Classical numerical methods (finite difference, finite element) cannot efficiently determine the solution, having to insert artificial boundaries, and consequently non-reflective conditions in order to try to significantly reduce spurious reflections of the wave propagating from the fictitious boundary towards the interior of the domain. When one deals with regions having different material properties (e.g. layered soils [5, 8]) or even different physics (e.g. in solid-fluid coupling [7] or wave-soil-structure interaction [9]) domain decomposition is needed. In this framework, BEM is nowadays understood to be complementary rather than concurrent to finite element method (FEM). It is clear that time domain BEM in comparison with frequency domain can be extended to the nonlinear behavior and it can be combined with FEM, too. Within engineering calculations, the BEM and the FEM are well established tools for the numerical approximation of problems for which analytical solutions are mostly unknown or available under unrealistic modeling only. In the last decades, contributions to BEM-FEM coupling, in the context of hyperbolic problems, started to appear [1, 3, 10], especially analyzing stability and convergence issues [4]. In this paper, taking advantage of a recently developed energetic space-time weak formulation of BIEs related to wave propagation problems defined on single and multi domains (see in particular [2] and references therein), a coupling algorithm is presented, which allows a flexible use of FEM and BEM as


29

local discretization techniques, in order to efficiently treat unbounded multilayered media. The approximation technique presented in this paper is based on a weak formulation directly expressed in the space-time domain, this avoiding the use of the Laplace transform of the kernel function, instead of the kernel function itself, and its inversion. Partial differential equations associated to BIEs will be weakly reformulated by the energetic approach and a particular emphasis will be given to experimental analysis of the stability of the proposed method. Some significant numerical results, which confirm theoretical results, are presented and discussed.

2 Model problem and energetic coupling Let Ω = (0, +∞) ⊂ R be an open unbounded one dimensional domain (modeling a rod with a dimension much greater than the remaining ones: the length along the x-direction) with boundary ΓN := {x = 0}. Let Ω1 ∪ Ω2 = Ω be a decomposition of Ω, with Ω1 = (0, L) bounded and Ω2 = (L, +∞) unbounded non-overlapping subdomains such that Ω1 ∩ Ω2 = {x = L} =: Γ. Note that the boundary of the unbounded subdomain Ω2 is just the interface point Γ. Having denoted with ui (x,t) the unknown function in the i-th subdomain, which ∂ui (x,t), which depends on a unitary outward vector nx w.r.t. depends on space and time, and with pi (x,t) := ∂nx the transversal section of the rod, therefore directed along the x-axis, i.e. nx = (nx , 0) , we want to solve the following wave propagation model problem: 1 2Di Pi ui,xx − 2 uï − 2 u˙i − 2 ui (x,t) = fi (x,t), x ∈ Ωi , t ∈ [0, T ], i = 1, 2 (2.1) ci ci ci x ∈ Ωi , i = 1, 2 (2.2) ui (x,t) = 0, u˙i (x, 0) = 0, p1 (0,t) = p(t),

x ∈ Ωi , i = 1, 2

(2.3)

t ∈ [0, T ],

(2.4)

where overhead dots indicate derivatives with respect to time, ci is the propagation velocity of a perturbation in the i-th subdomain, Di > 0 and Pi > 0 represent viscous and material damping coefficients, respectively, in Ωi , p(t) ¯ is a given function, and, at last, the assigned PDE right-hand sides f1 (x,t) and f2 (x,t) ≡ 0 are suitably connected at interface. Moreover, at the common endpoint x = L, the matching conditions read: u1 (L,t) = u2 (L,t),

c21 p1 (L,t) = −c22 p2 (L,t),

t ∈ [0, T ].

(2.5)

H 1 ([0, T ]; H 1 (Ωi )).

The unknown functions ui are understood in a weak sense, i.e. ui ∈ Since the goal of this work is to approximate u1 using a FEM approach and u2 using a BEM technique, we have to obtain a boundary integral reformulation of the problem (2.1) in Ω2 . Using classical arguments (see [6]) and the fundamental solution of the 1D damped wave operator √D2 − P c G(x, ξ;t − τ) := e−D(t−τ) I0 (2.6) c2 (t − τ)2 − (x − ξ)2 H[c(t − τ)− | x − ξ | ] , 2 c the problem (2.1) − (2.3) in the subdomain Ω2 can be rewritten as a system of two BIEs in the boundary unknowns the functions p2 (L,t) and u2 (L,t): u2 (L,t) = (V p2 )(L,t) (2.7) p2 (L,t) = (D u2 )(L,t) , where t

D22 − P2 (t − τ) p2 (L, τ)dτ , 0 D22 − P2 t e−D2 (t−τ) D2 1 I1 (D u2 )(L,t) = u2 (x,t) + u˙2 (x,t) − D22 − P2 (t − τ) u2 (L, τ)dτ . c2 c2 c2 t −τ 0

(V p2 )(L,t) = c2

e−D2 (t−τ) I0

(2.8)

30


In (2.6) I0 [·] and H[·] denote the modified Bessel function of order 0 and the Heaviside function, respectively; in (2.8) I1 (ξ) is the modified Bessel function of order 1. Of course, the problem (2.7) has to be coupled with the differential one specified for Ω1 , under the coupling conditions (2.5) at the interface. In particular, here we are interested in a direct space-time energetic weak formulation for the coupling of the integro-differential problem on Ω1 ∪ Ω2 , and this will be done extending what has been done in [3, 4]. At first, let us note that the solution of the damped wave equation in Ω2 satisfies the following energy identity: T 1 1 P2 2D2 T 2 EΩ2 (u2 , T ) := u˙2 (x,t)dt dx = u˙2 (L,t)p2 (L,t) dt , u22,x (x, T ) + 2 u˙22 (x, T ) + 2 u22 (x, T ) + 2 2 Ω2 c2 c2 c2 0 0 (2.9) which can be obtained multiplying equation (2.1) by u˙2 and integrating by parts over Ω2 × [0, T ]. Then, taking int account the nature of the two equations in the system (2.7), their energetic weak formulation consists in finding u2 (L,t) ∈ H 1 ([0, T ]) and p2 (L,t) ∈ L2 ([0, T ]) such that < u˙2 , q2 >=< (V ˙p2 ), q2 > (2.10) < p2 , v˙2 >=< D u2 , v˙2 >, where < ·, · >=< ·, · >L2 ([0,T ]) and q2 (L,t), v2 (L,t) are test functions, belonging to the same functional space of p2 (L,t) and u2 (L,t), respectively. In particular, the first equation in (2.7) has been differentiated with respect to time and projected with the L2 ([0, T ]) scalar product by means of functions belonging to L2 ([0, T ]), while the second equation in (2.7) has been projected with the L2 ([0, T ]) scalar product by means of functions belonging to H 1 ([0, T ]), derived with respect to time. For the energetic weak formulation in Ω1 , let us multiply the differential equation (2.1) for the time derivative of test function v1 (x,t) ∈ H 1 ([0, T ]; H 1 (Ω1 )) and integrate by parts in space obtaining: − A (u1 , v1 )+ < v˙1|Γ , p1|Γ >= F (v1 )− < v˙1|ΓN , p¯ >, where

A (u1 , v1 ) :=

T

Ω1

0

v˙1,x u1,x +

F (v1 ) :=

1 2D1 P1 v˙1 u¨1 + 2 v˙1 u˙1 + 2 v˙1 u1 (x,t) dx dt , 2 c1 c1 c1

T 0

Ω1

v˙1 (x,t) f1 (x,t) dx dt .

(2.11) (2.12) (2.13)

Now, remembering the interface condition (2.5) and using the further coupling condition at interface for test functions: v1 (L,t) = v2 (L,t), combining (2.11) with the second weak BIE in (2.10), multiplied by c22 , we finally obtain the following energetic weak formulation of the coupled problem: ⎧ ⎨ < (V ˙p ), q > − < u˙ , q >= 0 2

2

1|Γ

2

⎩ − < p2 , v˙1|Γ > − < D u2|Γ , v˙1|Γ > −2

c21 c2 c2 A (u1 , v1 ) = 2 12 F (v1 ) − 2 12 < v˙1|ΓN , p¯ > . c22 c2 c2

(2.14)

At every time instant, the unknowns are u1 in Ω1 and p2 at the interface point x = L. Let us conclude this Section with some energy considerations. At first, let us consider system (2.10) with q2 = p2 and v2 = u2 ; then, summing up the two equation and remembering (2.9) one obtains: 1 < u˙2 , p2 >= (< (V ˙p2 ), p2 > + < D u2 , u2 >) = EΩ2 (u2 , T ). 2 On the other side, considering v1 = u1 in (2.12), one gets:

A (u1 , u1 ) = EΩ1 (u1 , T ).

(2.15)

(2.16)

Following similar arguments, starting from (2.14) the following relation appears c21 EΩ1 (u1 , T ) + c22 EΩ2 (u2 , T ) = −c21 F (u1 ) + c21 < u˙1|ΓN , p¯ >,

(2.17)

from which one can deduce a-priori stability estimates for regular solutions u1 and u2 bounding from above the related energies by means of the problem data [4].


31

3 Space-time Galerkin discretization For time discretization we consider a uniform decomposition of the time interval [0, T ] with time step Δt = T /NΔt , NΔt ∈ N+ , generated by the NΔt + 1 time-knots: tk = kΔt, k = 0, . . . , NΔt , and we choose piecewise constant shape function for the time approximation of p1 and piecewise linear shape functions for the time approximation of u2 , although higher degree shape functions can be used. In particular, time shape functions, for k = 0, . . . , NΔt − 1, will be defined as: ¯ k (t) = H[t − tk ] − H[t − tk+1 ] , ψ

ˆ k (t) = R(t − tk ) − R(t − tk+1 ), ψ

(3.18)

t − tk H[t − tk ] is the ramp function. for the approximation of p2 and u1 , where: R(t − tk ) = Δt For the space discretization we consider the bounded subdomain Ω1 suitably decomposed by means of a mesh TΔx = {e1 , . . . , eMΔx } constituted by MΔx segments, with length(ei ) ≤ Δx, ei ∩ e j = 0/ if i = j and such that Δx ¯ ∪M i=1 e¯i = Ω1 . The functional background compels one to choose spatially shape functions belonging to C0 (Ω2 ) for the approximation of u1 . Hence, we will choose piece-wise linear continuous functions ϕˆ j (x), j = 1, . . . , M1 related to TΔx for the approximation of u1 in Ω1 . The approximate solutions of the problem at hand, for t ∈ [0, T ], will be expressed as p˜2 (L,t) =

NΔt −1

∑

¯ k (t), αk ψ

k=0

u˜1 (x,t) =

NΔt −1

∑

k=0

M1

ˆ k (t) ∑ αˆ k j ϕˆ j (x), ψ

x ∈ Ω1 .

(3.19)

j=1

The Galerkin BEM-FEM discretization coming from energetic weak formulation (2.14) produces the linear system: Aα = b, where matrix A has a NΔt blocks lower triangular Toeplitz structure, since its elements depend on the difference th − tk and in particular they vanish if th < tk . Each block has dimension M1 + 1. In [4], it has been proved that the numerical scheme coming from energetic BEM-FEM coupling is unconditionally stable and convergent, under suitable regularity assumptions for the problem data. In particular, it holds: max u˜1 (·,tk ) − u1 (·,tk )H 1 (Ω1 ) + p˜2 (L, ·) − p2 (L, ·)L2 ([0,T ]) = O(Δx) + O(Δt) . (3.20) k

4 Numerical Results 1. Homogeneous parameters in BEM and FEM sub-domains, smooth data. The first two configurations concern semi-infinite rod split in two portions, with non-zero source force f1 (x,t) = 2 3 6t 2 (1 − x) − c−2 1 (2 + 4D1t + P1t )(1 − x) over the domain Ω1 . At the end-point x = 0 of the bounded portion, the smooth Neumann boundary condition p(t) ¯ = 3t 2 is given and for the parameters in the two subdomains we have chosen different values but with the constraints c1 = c2 ; D1 = D2 ; P1 = P2 . The observation time interval is [0, 5]. For the discretization, we have used different temporal and spatial steps, Δt and Δx, respectively. In this case, problem (2.1) − (2.5) has solution given by u1 (x,t) = t 2 (1 − x)3 ∀ 0 ≤ x ≤ 1 . u(x,t) = u2 (x,t) = 0 ∀x≥1 In Table 1, the L∞ (Ω1 )-norm error evaluated w.r.t. the exact solution, obtained by varying the mesh of the domain Ω1 and of the time step are reported. The results show both the efficiency of the numerical scheme proposed and the stability of the numerical solution. For T = 5, in Table 2 we show the error Es defined in the left-hand side of (3.20) and the experimental convergence order, obtained by energetic BEM-FEM coupling approach, starting with a fixed mesh in time and space and then halving each time Δt and Δx (Δt = Δx = 0.1 × 2−s , s = 0, · · · , 6). Results are in accordance with the

32


Δx 0.1

c1 = c2 = 100 D1 = D2 = 0 P1 = P2 = 0 3.0 · 10−3 6.6 · 10−4 3.0 · 10−4 2.4 · 10−4 3.0 · 10−3 6.5 · 10−4 1.2 · 10−4 5.8 · 10−5 3.0 · 10−3 6.5 · 10−4 1.2 · 10−4 3.1 · 10−5 3.0 · 10−3 6.5 · 10−4 3.1 · 10−5 1.2 · 10−5

Δt 0.1 0.05 0.025 0.0125 0.1 0.05 0.025 0.0125 0.1 0.05 0.025 0.0125 0.1 0.05 0.025 0.0125

0.05

0.025

0.0125

c1 = c2 = 1 D1 = D2 = 100 P1 = P2 = 0 1.1 · 10−1 −− −− −− 2.9 · 10−2 2.8 · 10−2 −− −− 7.4 · 10−3 7.1 · 10−3 7.0 · 10−3 −− 2.1 · 10−3 1.8 · 10−3 1.7 · 10−3 −−

c1 = c2 = 1 D1 = D2 = 0 P1 = P2 = 100 1.1 · 10−1 −− −− −− 3.0 · 10−2 2.9 · 10−2 2.8 · 10−2 −− 8.7 · 10−3 7.4 · 10−3 7.1 · 10−3 7.0 · 10−3 4.3 · 10−3 2.8 · 10−3 1.9 · 10−3 1.8 · 10−3

Table 1: L∞ (Ω1 )-norm error w.r.t. the exact solution u(x,t) = t 2 (1 − x)3 . P1 = P2 = 1, D1 = D2 = 0 Es log2 Es−1 /Es 3.143 · 10−2 1.759 · 10−2 0.84 9.344 · 10−3 0.91 4.826 · 10−3 0.95 2.455 · 10−3 0.98 1.239 · 10−3 0.99 6.223 · 10−4 0.99

s 0 1 2 3 4 5 6

P1 = P2 = 0, D1 = D2 = 1 Es log2 Es−1 /Es 1.833 · 10−2 1.037 · 10−2 0.82 5.525 · 10−3 0.91 2.855 · 10−3 0.95 1.453 · 10−3 0.98 7.329 · 10−4 0.99 3.681 · 10−4 0.99

P1 = P2 = 1, D1 = D2 = 1 Es log2 Es−1 /Es 1.722 · 10−2 9.813 · 10−3 0.81 5.247 · 10−3 0.90 2.717 · 10−3 0.95 1.384 · 10−3 0.97 6.984 · 10−4 0.99 3.509 · 10−4 0.99

Table 2: Errors Es defined in (3.20) and experimental convergence order varying damping parameters. estimate (3.20). In Figure 1 the time history of recovered u(0,t) and u(L,t) are shown on the time interval [0, 5]. P =P =1 D =D =1 1

0.4

2

1

1

0.4

u(0,t) u(L,t)

0.3

P =P =10 D =D =1

P =P =10 D =D =10

2

2

1

2

0.3 0.2

0.2

0.1

0.1

0.1

0

0

0

1

2

3

4

5

−0.1 0

1

2

3

4

2

1

5

−0.1 0

Figure 1: Numerical solution with c1 = c2 = 1.

2

u(0,t) u(L,t)

0.3

0.2

−0.1 0

1

0.4

u(0,t) u(L,t)

1

2

3

4

5


33

2. Non-homogeneous parameters in BEM and FEM sub-domains. Keeping the configuration of previous examples (Δt = Δx = 0.05) with the source force f1 (x,t) = 0 and choosing the Neumann boundary condition p(t) ¯ = c−2 1 (H[t] − H[t − 0.25]) we start diversifying parameters in BEM-FEM subdomains. In Figs. 2 and 3, we plot the time history u(0,t) of the obtained numerical solutions until time T = 10. P1=P2=0 D1=D2=0

0.3

c =1;c =2 1 1

1

2

2

0 −0.1

0

−0.2 −0.1 −0.2 0

2

0.1

2

c =2;c =1 0.1

1

0.2

c1=c2=2

0.2

c =c =1 D =D =0

0.3

c1=c2=1

−0.3 2

4

6

8

10

−0.4 0

P1=P2=0 P1=P2=1 P =0 P =1 1

2

P1=1 P2=0 2

4

6

8

10

Figure 2: On the left, variation of c1 and c2 ; on the right, variation of P1 and P2 c =c =1 P =P =0 1

2

1

2

1

1

0.2

2

D1=D2=1 D1=0 D2=1

0.15

D1=D2=1 P1=P2=0 D1=0 P1=20 D2=1 P2=0

0.15

D =D =0

D1=1 D2=0 0.1

2

D1=D2=0 P1=P2=20

0.2

0.25

D1=1 P1=0 D2=0 P2=20

0.1 0.05 0

0.05 0 0

c =c =1

0.25

−0.05 2

4

6

8

10

−0.1 0

2

4

6

8

10

Figure 3: On the left, variation of D1 and D2 ; on the right, variation of all parameters

References [1] T. Abboud, P. Joly, J. Rodriguez, I. Terrasse, J. Comput. Physiscs, 230 (15), 5877–5907, (2011). [2] A. Aimi, S. Gazzola, C. Guardasoni, Mathematical Methods in the Applied Sciences, 35, 1140–1160, (2012). [3] A. Aimi, M. Diligenti, C. Guardasoni, S. Panizzi, Communications in Applied and Industrial Mathematics, 3, (2), 1-21, (2012). [4] A. Aimi, S. Panizzi, Numer. Methods Partial Differential Equations, in press, (2014). [5] V.S. Almedida, J.B. Paiva, Adv. Eng. Software, 38, 835-845, (2007). [6] M. Costabel, in: Encyclopedia of Computational Mechanics, Stein E., de Borst R., Hughes TJR (eds), Wiley, (2004). [7] O. Czygan, O. von Estorff, Engineering Analysis with Boundary Elements, 26, 773–779, (2002). [8] M. Sari, I. Demir, J. Appl. Sciences, 6 (8), 1703–1711, (2006). [9] J.L. Wegner, M.M. Yao, X. Zhang, Computer and Structures, 83, 2206–2214, (2005). [10] G.Y. Yu, Journal of Applied Mechanics, 70, 451–454, (2003).

34


A topological optimization procedure applied to multiple region problems Carla Anflor1, Éder L. Albuquerque2 and Luiz C. Wrobel 3 1 2

Universidade de Brasília, FGA, Campus Gama, Brazil, [email protected]

Universidade de Brasília, FT, Campus Universitário Darcy Ribeiro, Brazil, [email protected] 3

School of Engineering and Design, Brunel University, UK, [email protected]

Keywords: Topological optimization, BEM, Multiple materials, Inclusions, Anisotropy

Abstract. The main objective of this work is the application of the topological optimization procedure to heat transfer problems considering multiple materials. The topological derivative (DT) is employed for evaluating the domain sensitivity when perturbed by inserting a small inclusion. Electronic components such as printed circuit boards (PCBs) are an important area for the application of topological optimization. Generally, geometrical optimization involving heat transfer in PCBs considers only isotropic behaviour and/or a single material. Multiple domains with anisotropic characteristics take an important role on many industrial products, for instance when considering PCBs which are often connected to other components of different materials. In this sense, a methodology for solving topological optimization problems considering anisotropy and multiple regions is developed in this paper. A direct boundary element method (BEM) is employed for solving the proposed numerical problem. Introduction. Materials with anisotropic properties have been employed for a great number of applications since the earlier 1960s, Shiah et al. [1]. High performance is achieved when a composite is constructed by combining two or more materials. Despite its importance, very few works are found in the literature considering topology optimization for anisotropic materials applied for heat transfer, when compared to elasticity problems. Li et al. [2] developed a computational procedure based on FEM and ESO for the topology design of heat conduction in isotropic fields. Zhang and Liu [3] developed a new method based on topology optimization for solving heat conduction problems for isotropic media with distributed heat sources. Another concept recently employed for topology optimization using BEM instead of FEM is the topological derivative [4]. Some results employing DT were also performed using FEM, but it is worth noting that the BEM characteristics are attractive for optimization procedures once this method has no mesh dependence and low computational cost. The DT measures the sensitivity of a given shape functional when the domain is perturbed by an infinitesimal perturbation, such as the insertion of holes, inclusions and even cracks. This concept has been successfully used for a wide range of problems in addition to topology optimization, such as image processing and fracture mechanics. Many efforts for extending this concept for more complex problems have been done in the last years. Recently, Giusti et al. [5] obtained a closed form for the DT considering the total potential energy associated to an anisotropic and heterogeneous heat diffusion problem, when a small circular inclusion of the same nature of the bulk phase is introduced at an arbitrary point of the domain. This closed formula was derived in its general form and particularized for heterogeneous and isotropic media, showing agreement with the Amstutz [6] derivation. This paper aims at extending an optimization procedure for heat transfer problems considering heterogeneous materials using BEM and DT. The derivation of an integral equation valid over all the external boundaries of the different material regions, without due consideration for the interfaces between them, is not impossible, despite the complexity involved. Alternatively, a conforming mapping technique is implemented to reduce the steady-state anisotropic field to an equivalent isotropic domain, avoiding some new derivations. The influence of the anisotropic conductivity properties imposed to the inclusion and matrix on the final topology will be investigated.


35

BEM treatment of multi-domain mapping. Many research works have devoted substantial efforts on developing efficient and robust topology optimization procedures during the last decades. The present work is focused on anisotropic multi-regions for heat diffusion problems. When dealing with non-homogeneous composites, it is usually necessary to split the domain into several different materials held together, and treat each one in turn. The derivation of an integral equation valid over all the external boundaries of the different material regions without due consideration for the interfaces between them is not impossible, despite the complexity involved. Alternatively, a conforming mapping technique can be used to reduce the steady-state anisotropic field to an isotropic equivalent domain, avoiding some new derivations. Some works have successfully employed a linear coordinate transformation for solving anisotropic thermal field problems with FEM and/or BEM. One of the first works devoted to the mapping technique was presented by [7]. Recently, Shiah and Tan [8] presented a method for reducing a three-dimensional steady-state anisotropic field problem to an equivalent isotropic one, governed by the Laplace equation in a mapped domain. Despite the mapping technique formulation being available for 3D problems, only 2D problems have been considered. Furthermore, the DT formula for calculating the domain sensitivities due to a perturbation caused by the insertion of a small inclusion is only valid for 2D problems [5]. Problems governed by the Poison equation are well known in the BEM literature [9]. The boundary integral equation for this problem is as follows: c(K ) I (K )

_

_

³ q([ ) U (K, [ ) d *([ ) ³ I ([ ) T (K, [ ) d *([ ) ¦ *

*

n

b U (K, M n )

m 1 m

(1)

where I and q represent the temperature and its normal gradient, respectively. The variable n represents _

the unit outward normal vector along the boundary while * is used to denote the boundary on the mapped domain. The source and field points are designated by the variables η and ξ. The value of c(η) depends on the geometry at η. The variable bm is the mth internal heat-source point. The fundamental solutions for the potential and its gradient are represented by U(η,ξ) and T(η,ξ) and given by, U ([ )

1 §1· ln ¨ ¸ 2S ©r¹

T ([ )

and

1 ni r,i 2S r

for

2D

(2)

After the usual discretization of the boundary into * boundary elements, eq. (1) is applied at each

boundary nodal point K , generating the system of equations for a single domain. The original * is mapped into an isotropic equivalent domain by using a linear coordinate

geometry

transformation, > X1 X 2 @ ª¬ F ( Kij )º¼ > x1 x2 @ and > x1 x2 @ ª¬ F 1 ( Kij )º¼ > X1 X 2 @ , where Kij are the conductivity coefficients. The collocation process for solving the integral eq. (1) should be done on the distorted domain and this procedure also requires a mapping of the Neumann boundary conditions, as depicted in Fig. 1. It is important to note that the nodal potentials remain unchanged for corresponding points between the physical ( x1 , x2 ) and mapped coordinate systems ( X1 , X 2 ) , since the temperature is a scalar field. The Neumann boundary conditions are mapped according to the relation, T

T

dT dn

§ wT K xx wT K xy · § wT · ¨ ¸ n1 ¨ ¸ n2 © wx1 ' wx2 ' ¹ © wx2 ¹

(3)

Taking into account the boundary conditions imposed to the problem, a set of simultaneous equations for unknown and known temperature or its normal derivative at nodal points may now be solved by standard methods. Once the system of equations is solved, an inverse mapping must be further employed for recovering the potential gradients on the original domain, as follows, dT dn

T

Mi F

FT n T

T

F n

(4)

where n is the outward normal vector on the mapped domain. The variables M i represent the temperature gradients along the mapped boundary, and are defined as,

36


§ wT ¨ © wX 1

T

Mi

n

Γ

wT · ¸ wX 2 ¹

(5) n

Mapping

Γ Ω1

Ω2

Heat source Inclusion

qa

Ω1 x2

Ω2 X2

Anisotropic multi-region

qb

Thermal equilibrium qa = qb

x1

X1

Isotropic multi-region Figure 1. Direct domain mapping of a composite material with each temperature gradient of eq.(5) calculated by

dT

wT wx1

wT wx2

d n n1 ;

dT

(6)

d n n2

When dealing with non-homogeneous media, appropriate thermal compatibility and equilibrium conditions along the interfaces of conjoint materials must be supplied. As explained before, when the anisotropic domain is transformed to an equivalent isotropic one, it results in a deformed geometry. The inserting of an inclusion with different properties inside the matrix will result in an overlap or separation of the interfaces of the conjoint materials (see Figure 1). For a non-cracked interface between isotropic materials, the compatibility equation is imposed as T1=T2, where the superscripts denote materials 1 and 2, respectively. The above relation still holds for anisotropic materials, but special conditions are required for the temperature gradient due to the misalignment of the mapped interfaces. As a consequence of their distortion, the thermal equilibrium of the normal heat fluxes must be reformulated accordingly. The sum of the normal fluxes across the interfaces must vanish in order to satisfy the thermal equilibrium between adjacent materials, as follows (see Shiah et al. [1]), 2 (7) ¦ Kij( m )T, (jm ) nim 0 m 1

The unit outward normal vectors along the boundary surface of the coordinate systems are related by, n1

n

1

' K11 n2 K12 K11 :

n2

;

n2 :

;

:

n

1

' K11 n2 K12 K11

2

n2

(8)

Substituting eq.(8) into eq.(7) and carrying out some algebraic processes leads to a general thermal equilibrium form which takes into account the balance of the normal heat fluxes across the two adjacent anisotropic materials, 2

¦: m 1

'( m) dT ( m ) (m) (m) K11 dn

(m)

0

(9)

It is worth noting that only a 2D formulation was presented here, as the topology optimization is only performed for two-dimensional problems. Furthermore, to the authors’ knowledge, there is no DT available for evaluating the domain sensitivity for 3D problems perturbed by the insertion of inclusions. A closed formula proposed by [9] for DT is employed for determining the sensitivity of the conductivity tensor when an inclusion is introduced at an arbitrary point of the domain. The DT is a scalar field over Ω that depends only on the conductivity tensor of the matrix and the inclusion, and on the solution of the thermal equilibrium problem for the original unperturbed domain, k * ( y ) ^ k i J km y :m (10) where y denotes the centre of the circular perturbation, while J defines the ratio between the m i conductivity of the matrix (k ) and the conductivity of the perturbation (k ) . The sensitivity of the


37

macroscopic conductivity tensor to the topological change due to the insertion of a small inclusion is presented in closed form as, DT ( y )

2km

2 km ki u( y ) km ki

y :

(11)

It is important to note that DT is a scalar field over : that depends only the conductivity parameters and on the solution of the thermal equilibrium problem for the original domain : . Numerical Results Some tests are now presented in order to assess the proposed algorithm. All conduction fields to be designed consist of two materials, one with thermal conductivity {Kxx, Kyy, Kxy} and another insulating material with conductivity {kxx, kyy, kxy}. The proposed example represents a three heat sources conductor where a fully isotropic optimization is performed, and another case taking into account the conversion from anisotropic to isotropic behaviour. The main goal during the optimization process relies on achieve the best topology for the heat transfer process. This improvement is achieved by inserting a second material, with low conductive properties in those areas with less efficiency.The percentage amount of inclusions will be calculated as ((1-V/Vo)∙100), where V is the volume at a specific iteration and Vo is the initial volume. This provides a simplified criterion to compare the topologies generated for isotropic, orthotropic and anisotropic media, starting from the same design. Three heat sources conductor A region of 30x30 mm is taken into account here, with three heat sources and a heat flux prescribed on its boundaries, as illustrated in Fig. 2. The heat source temperature and the heat flux are set as u=100°C and q=30 W/mm2, respectively. The remaining external boundaries are all insulated. u u u = 100 °C u q =30 W/m2 insulated

kxx, kyy, kxy= thermal conductivity of inclusions Kxx, Kyy, Kxy= tThermal conductivity of matrix

Kxx, Kyy, Kxy kxx, kyy, kxy q Figure 2. Initial design for the three heat sources conductor Isotropic matrix and inclusions. For this case the thermal conductivity matrix was set as {kxx=100, kyy=100, kxy=0} while for the inclusions it was set as {kxx=1, kyy=1 and kxy=0}. The changes of the temperature field as the percentage of inclusions is increased is shown in the colour map presented in Fig.3, which also shows the evolution history at iterations 5, 13 and 17.

Iteration #5 - Vol = 89.83%



Figure 3. Evolution history for isotropic three heat sources conductor

38


Anisotropic matrix and inclusions. This example is employed to show the anisotropic behaviour produced by imposing the thermal properties as {Kxx=75, Kyy=75, Kxy=25, kxx=1, kyy=1, kxy=0.5}. Figure 4 depicts the evolution history until the percentage of inserting inclusions reach a target close to 53%. At iteration number 57, it is possible to see that a large amount of insulating material was inserted on the right side, resulting in a delay of the heat transfer in that region.


Iteration #32 - Vol = 62%


Figure 4. Evolution history for anisotropic three heat sources conductor During the optimization process, a temperature control point was set at the middle bottom of the plate. The evolution history at the temperature control point, as well as the amount of inclusion inserted per iteration, are presented in Figure 5(a) and Figure 5(b), respectively. As the material insertion was more concentrated at the rigth side a temperature reduction was achieved for the anisotropic case, see Figure 5(a). 50

Isotropic Anisotropic

Remaining volume ((1-V/Vo)*100)

Normalized temperature [°C]

1

0.995

0.99

0.985

0.98 0

10

20

30 40 Iteration number

50

60

40

30

20

Isotropic Anisotropic

10

0 0

10

20

30 40 Iteration number

50

60

(a) (b) Figure 5. Evolution history for three heat sources conductor: (a) Temperature and (b) Amount of inclusions inserted Another interesting issue relies on the amount of material inserted. For the isotropic case the influence of the insertion of the inclusions was not so strong when comparing to the resulting topology with anisotropic characteristics. This observation can be verified in Figure 5(b), where the isotropic case took only 17 iterations against 57 iterations performed for the anisotropic case. Anisotropic matrix and isotropic inclusions. The main goal of this example is to show the transformation of an initially anisotropic domain into an isotropic one. In this case, an anisotropic matrix will be filled with inclusions of isotropic thermal properties. The conductive thermal tensor was set as {Kxx=100, Kyy=100 and Kxy=50, kxx=1, kyy=1, kxy=0}. Figure 6 shows the optimization process history after the initial domain has been almost fully filled with 96.66% of isotropic inclusions.


39

Iteration # 4 - Inclusions = Iteration # 24 - Inclusions = Iteration # 83 - Inclusions = 3.85% 33.50% 96.66% Figure 6. Evolution history: Anisotropic to isotropic behaviour Conclusions A topology optimization procedure was extended to multiple materials for anisotropic problems. The derivation of an integral equation valid over all the external boundaries of the different material regions without due consideration for the interfaces between them is not impossible, despite the complexity involved. In this work, an alternative procedure involving a conformal mapping technique was implemented to reduce the steady-state anisotropic field to an equivalent isotropic domain. Furthermore, the BEM optimization procedure was successful from the point of view of computational cost, since the BEM presents no internal mesh dependency as one of its main characteristics. The proposed methodology should be useful for designing high performance heat transfer topologies involving the coupling of different materials. Finally, the application of designed anisotropic materials for the efficient energy management of heat conduction fields will be explored. References [1] [2]

[3] [4]

[5]

[6] [7] [8] [9]

Y.C. Shiah, P. Hwang, R. Yang, Heat conduction in multiply adjoined anisotropic media with embedded point heat sources. Journal of Heat Transfer 128, 207-214 (2006). Q. Li, G.P. Steven, Y.M. Xie, O.M. Querin, Evolutionary topology optimization for temperature reduction of heat conducting fields. International Journal of Heat and Mass Transfer 47, 50715083 (2004). Z. Zhang, S. Liu, Design of conducting paths based on topology optimization. Heat and Mass Transfer 44, 1217-1227, (2008). C.T.M. Anflor, R.J. Marczak, Topological optimization of anisotropic heat conducting devices using Bezier-smoothed boundary representation. Computer Modeling in Engineering & Sciences 78, 151-168, (2011). S.M. Giusti, A.A Novotny, E.A. de Souza Neto, R.A. Feijóo RA, Sensitivity of the macroscopic thermal conductivity tensor to topological microstructural changes. Computer Methods in Applied Mechanical and Engineering. 198, 727-739, (2009). S. Amstutz S, Sensitivity analysis with respect to a local perturbation of the material property. Asymptotic Analysis 49, 87–108, (2006). K.C. Poon, R.H.C. Tsou, Y.P. Chang, Solution of anisotropic problems of first class by coordinate-transformation. Journal of Heat Transfer 101, 340–345, (1979). Y.C. Shiah, C.L. Tan, BEM treatment of three-dimensional anisotropic field problems by direct domain mapping, Engineering Analysis with Boundary Elements 28, 43-52, (2004). L.C. Wrobel, The Boundary Element Method, Vol.1: Applications in Thermo-Fluids and Acoustics, Wiley (2002).

40


Direct Interpolation Technique using Radial Basis Functions Applied to the Helmholtz Problem C. F. Loeffler1, P. V. M. Pereira2, H. M. Barcelos3 Universidade Federal do Espírito Santo - UFES – Programa de Pós-Graduação em Engenharia Mecânica - CEP 29075 910 –Goiabeiras – Vitória – ES – Brazil 1 [email protected], 2 [email protected], 3 [email protected] Keywords: Boundary Element Method, Helmholtz Problems, Radial Basis Functions Interpolation

Abstract. In this work the goal is to solve the integral term which refers to the inertia term of the Helmholtz Equation, avoiding the standard techniques of domain integration like cells or the Dual Reciprocity technique (DRBEM). The procedure presented here is named as the DIBEM. It consists of an interpolation procedure with radial basis functions, similar to the DRBEM, but the DIBEM transforms the domain integral into a single boundary integral more directly. The proposed methodology is still more general and robust than the DRBEM allowing the use of any radial basis functions class without convergence problems or lack of monotonicity.

Introduction Nowadays, with the intense development of new techniques involving multivariate functions, new classes of radial basis functions have been tested and successfully applied in problems of interpolation, curve fitting and partial differential equations solution, particularly within the Finite Element method approach [1]. This same effort has been expended in improving of the Boundary Element Method (BEM), seeking a better solution for mathematically inhomogeneous problems, as the Poisson problems and also time dependent cases. In this context, the first major contribution came with the DRBEM [2], in which is still currently the most general alternative to overcame mathematical difficulties to solve with simplicity integral terms that have not auto-adjoint properties. However, despite the satisfactory results in some applications, the DRBEM presents certain numerical inaccuracies in cases in which many internal poles are required to represent domain properties as the inertia, for example. Thus, the aim of this paper is to present the DIBEM technique to solve the integral term that refers to the inertia in the Helmholtz equation. The procedure has been used successfully in Poisson problems [3]. This technique employs an approximation procedure with radial basis functions [4], relatively similar to the DRBEM, but it is more simple, general and robust, since that the domain action is interpolated directly.

Mathematical modeling The Helmholtz Equation in its inverse integral form [5] is given by:

c([)u ([) ³ u (X)q * ([; X)d* ³ q(X)u * ([; X)d* *

*

1 2 w n ³ u (X)u * ([; X)d: : c2

(1)

In eq.(1), u(X) is the scalar potential and q(X) is its normal derivative; u*(ξ,X) is the fundamental solution and q* (ξ,X) is its normal derivative; wn correspond to the frequencies associated. The coefficient c(ξ) depends on the positioning of the source point with respect to the domain Ω(X) [6].

The DIBEM interpolates the kernel of the domain integral directly using radial basis functions F i, according the following equation:


u(X)u * ([; X)

ξ

41

α i Fi (X i ; X)

(2)

For each source point ξ given by Eq. (2) is done a scan of all the base points Xi in relation to the domain points X, weighted by the coefficients ξαi. It must be pointing out that the number of basis points Xi must be

equal to the discrete nodal values. Thus, the coefficients ξαi may be calculated by solving a system of algebraic equations, after carrying out the following mathematical steps. First, consider the basic interpolation sentence, given by:

[F]α

(3)

u

Consider the matrix ξΛ as composed by values of the fundamental solution for each source point ξ. Multiplying both sides of the eq. (3) by ξΛ, it has:

[ ξ Λ]u [ ξ Λ][F]D

(4)

Eq. (4) also may be written in the following similar form:

[F][ [ α] [ [/]u

(5)

Thus, the last two equations can be equaled, resulting in:

[ ξ α] [F] 1 [ ξ Λ][F]α [F] 1 [ ξ Λ][u]

(6)

In the DIBEM, since the fundamental solutions composes de kernel to be interpolated, the source point ξ must have different positions than the basis points Xi to avoid singularities. Considering linear boundary elements, basis points are located initially centered between the nodal points. Later, the data referring these points are relocated to its original positions. Similarly to the DRBEM, the proposed method transforms the domain integral into a boundary integral using a primitive interpolation function ψj as follows:

³ u(X)u * ([; X)d: ³

:

:

[

³(

D i Fi (X i ; X)d:

:

[

D j \ ,jii (X))d:

[

D j ³ K j (X)d* *

(7)

The evaluation of integrals given by eq. (2) using standard BEM discretization procedures is very simple, resulting in:

H11u 1 ......H1n u n G 11q 1 ......G 1n q n H 21u 1 ......H 2 n u n G 21q1 ......G 2 n q n

1

D1 N1 1 D 2 N 2 .....1 D n N n 2

D1 N1 2 D 2 N 2 ..... 2 D n N n

n

D1 N1 n D 2 N 2 ..... n D n N n

........................ H n1 u 1 ......H nn u n G n1q 1 ......G nn q n

(8)

These last equations may be rewritten in matrix form, as follows:

42


· · § § ¨ H cc H ci ¸§ u · ¨ G cc 0 ci ¸§ q · c ¸ ¸¨ c ¸ ¨ ¨ ¸¨ ¸¨© u i ¸¹ ¨ ¸¨ q ¸ ¨ ¨ H ic I ii ¸ ¨ G ic 0 ii ¸© i ¹ © © ¹ ¹

§ 1 D1 ....1 D n ·§ N1 · ¸¨ ¨ ¸ ¨ .............. ¸¨ ... ¸ ¨ n 1 n n ¸¸¨ N ¸ © D .... D ¹© n ¹

§ A1 · ¨ ¸ ¨ ... ¸ Ä ¸ © n¹

(9)

For the Helmholtz problems, the DIBEM considers the nodal values of the potential u(X) in an implicit form, that is, in eq. (10) the nodal values of the potential are included in the vector Ai. Explicitation of the potential u(X) allows constructing an inertia matrix. For this purpose, the vector Ai must be rewritten as follows:

A[

( N1

§ [ D1 · ¸ ¨ N 2 .....N n )¨ ...... ¸ ¨ [ n ¸¸ © D ¹

(10)

Considering Eq. (11), the Eq. (7) must be written:

A[

( N1

· §1 1 ¨ F .....1 F n ¸§ [ /1 ....0 · §uc · ¸ § ) cc ...0 ·¨ 1 ¸ ¸¨ ¨ ¸¨ .... ¸ N 2 .....N n )¨ ................. ¸¨ ..............¸.¨¨ 0.....) ii ¸¹¨ i ¸ ¨ n 1 n n ¸¨ 0....[ /n ¸¸ © ¸ ©un ¹ ¨ F ¹ .... F ¸¹© ©

(11)

The matrix Φ is an interpolation matrix of nodal values, since that to avoid singularities, the points Xj, and the source points ξ are located in different positions. As mentioned, the points Xj are taken centered on the boundary elements, to allow the implementation of the DIBEM procedure. Thus, this matrix has simple composition for the m auxiliary boundary points, being an average of the nodal values at the extremes. As for the internal basis points, it is necessary a more elaborate strategy, in which radial interpolations were used to relate the internal source points with internal basis points, the latter serving exclusively as auxiliary interpolation points. Considering eq. (11) the vector ξA may be completely rewritten and the final system of equations is given by:

· · § § ¨ H cc H ci ¸§ u · ¨ G cc 0 ci ¸§ q · ¸¨ c ¸ ¨ ¸¨ c ¸ ¨ ¸¨© u i ¸¹ ¨ ¸¨ q ¸ ¨ ¨ H ic I ii ¸ ¨ G ic 0 ii ¸© i ¹ © © ¹ ¹

· § ¨ M cc M ci ¸§ u · ¸¨ c ¸ ¨ Z ¸¨ u ¸ ¨ ¨ M ic M ii ¸© i ¹ © ¹ 2

(12)

Numerical Simulations The first example consists in a one-dimensional model governed by the following equation:

c2

w 2 u(x) u(x) wx 2

w 2u

(13)

Fig. 1 shows its geometrical features and the boundary conditions prescribed. The values of c and w are taken as being unitary.


43

Figure 1 – Geometric features and boundary conditions of the first example In this case, the analytical solution is an exponential function, given by:

u(x)

senhx cosh 1

(14)

Two meshes with 32 and 80 linear boundary elements are used for testing. Different numbers of internal interpolating points are introduced and the results of the average value of the percentage average error are plotted in Fig. 2. For these simulations, the number of basis points and source points inside the domain are taken equal.

Figure 2 – Average percentage error curve for two boundary element meshes with different number of internal interpolation points. Confirming the expectation observed in simulations of the Poisson’s problems, the errors committed with DIBEM using finer boundary meshes without internal points are higher than poorer boundary meshes in the same condition. However, with the introduction of internal points, the error in the finer boundary meshes is quickly reduced. Unlike the Poisson’s problems, in which the internal poles are used exclusively for interpolation, now these points are also employed as source points. Thus, the error may increased if an excessive number of internal interpolation points are introduced, due solely to the integration problems, that is, many internal interpolation points being placed very close to the boundary without using an effective strategy for calculating the integrals. The second example still approach one-dimensional cases, but now the governing equation is given by: c2

w 2 u(x) u(x) wx 2

w 2 u(x)

(15)

Considering the same boundary conditions of the first example, the harmonic solution is given as follows:

44

u(x)


sen ( wx) w cos(w1)

(16)

For testing, the values of frequency w are increased gradually evaluating the accuracy of the DIBEM for generate the inertia matrix. Fig. (3) shows the behavior of the average percentage error as a function of the square of excitation frequencies. Two meshes, with 80 and 160 boundary elements are used, with 81 and 144 internal poles respectively.

Figure 3 – Average percentage error curve for two boundary element meshes with different values of the excitation frequency . It may be pointing out the good accuracy of the DIBEM results for the low frequencies. Although the errors grow to higher values of frequency, it is noteworthy that the inertia matrix generated by the DIBEM is multiplied by the square of these values, resulting in higher share of this matrix with respect to H and G matrices, that represent the stiffness. These results are much accurate than those obtained with the DRBEM for the same range of variation shown [7]. Conclusions Despite the simplicity of the examples here examined, the DIBEM results here obtained solving the Helmholtz Equation presented good accuracy. These preliminary results motivate the improvement of the formulation to solve more complex problems, particularly eingenvalue problems. The good performance of the DIBEM is mainly due to the fact that the Laplacian operator is mathematically transformed into inverse integral terms according to typical BEM procedures, which are admittedly efficient. Just the inertia term, free of spatial derivatives, is approximated by radial functions through a practically direct interpolation procedure. Among the applications of the radial basis functions for numerical approximation theory, the interpolation procedure is those who perform best. References [1] J. G. Wang, G. R. Liu, A point interpolation meshless method based on radial basis functions, Int. J. Numerical Methods in Engeneering, 54, 1623–1648 (2002). [2] P. W. Partridge, C. A. Brebbia, L. C. Wrobel, The Dual Reciprocity Boundary Element Method, first ed., Computational Mechanics Pub., (1992).


45

[3] C. F. Loeffler, A. L. Cruz, A. Bulcão, Direct Use of Radial Basis Interpolation Functions for Modeling Source Terms with the Boundary Element Method, submitted to Eng. Analysis with Boundary elements, (2014). [4] M. D. Buhmann, Radial Basis Functions: Theory and Implementations, 1ed. New York, Cambridge University Press. , (2003) [5] C.A. Brebbia, S. Walker, Boundary Element Techniques in Engineering, Newnes-Butterworths, London, (1980). [6] C. A.Brebbia, J. C.Telles, and L.C.Wrobel, Boundary Element Techniques, Springer-Verlag, (1984). [7] C. F. Loeffler, W. J. Mansur, Free Vibrations in Rods and Membranes using the Boundary Element Method (in portuguese), Revista Brasileira de Engenharia, 4, 2, 5-23 (1986).

46


A two-scale three-dimensional boundary element framework for degradation and failure in polycrystalline materials I. Benedetti1,a, M.H. Aliabadi2,b 1

Dipartimento di Ingegneria Civile, Ambientale, Aerospaziale e dei materiali, Università degli Studi di Palermo, Viale delle Scienze, Edificio 8, 90128, Palermo, Italy 2

Department of Aeronautics, Imperial College London, South Kensington Campus,SW7 2AZ, London, UK. a


Keywords: Polycrystalline materials; Multiscale modelling; Micromechanics; Non-linear boundary element method.

Abstract. A fully three-dimensional two-scale boundary element approach to degradation and failure in polycrystalline materials is proposed. The formulation involves the engineering component level (macroscale) and the material grain scale (micro-scale). The damage-induced local softening at the macroscale is modelled employing an initial stress approach. The microscopic degradation processes are explicitly modelled by associating Representative Volume Elements (RVEs) to relevant points of the macro continuum and employing a three-dimensional grain-boundary formulation to simulate intergranular degradation and failure in the microstructural Voronoi-type morphology through cohesive-frictional contact laws. The scales coupling is achieved downscaling macro-strains as periodic boundary conditions for the RVE, while overall macro-stresses are obtained via volume averages of the micro-stress field. The comparison between effective macro-stresses for the damaged and undamaged RVE allows to define a macroscopic measure of material degradation. Some attention is devoted to avoiding pathological damage localization at the macro-scale. The multiscale processing algorithm is described and some preliminary results are illustrated. Introduction Understanding materials degradation and failure is relevant for several modern structural applications. Damage and fracture can be considered at different length scales: it is widely recognized that the macroscopic material properties depend on the material microstructure[1]. Polycrystalline materials, either metals, alloys or ceramics, are commonly employed in Engineering. The microstructure is determined by the grains morphology, size distribution, anisotropy and crystallographic orientation, stiffness and toughness mismatch and by the physical and chemical properties of the intergranular interfaces. These aspects have a direct influence on the initiation and evolution of microstructural damage, which is also sensitive to the presence of imperfections, flaws or porosity. The microstructure of polycrystalline materials, and its failure mechanisms, can be investigated using experimental [2] and computational [3] techniques. However, a truly three-dimensional (3D) experimental characterization still poses relevant challenges; on the other hand, the present-day availability of cheaper and more powerful computational resources and facilities, namely High Performance Computing (HPC), is favoring the advancement of Computational Micromechanics [4]. The explicit simulation of the material microstructure and its evolution finds a remarkable application in the multiscale analysis of solids, in which a considered component is analyzed simultaneously at the component level, for which the load history is defined, and at the grain scale level, which provides the constitutive material evolution. The term multiscale can assume a variety of meanings [5]: however, here we focus on simulations involving two spatial scales, the continuum level and the grain scale level. The objective of these studies is the analysis of both the behavior of the macro-component and the processes happening at the microscale during the loading history. The multiscale analysis becomes particularly useful when, during the loading history, the microstructure undergoes transformations or damage, so that a simple constitutive model assumed at the macro-level could not be used to simulate the behavior of the structure.


47

In this work a fully three-dimensional two-scale boundary element approach to degradation and failure in polycrystalline materials is proposed. At the macroscale, the damage-induced local softening is modelled employing a classical initial stress approach, while the microscopic degradation processes are explicitly modelled employing a cohesive-frictional three-dimensional grain-boundary formulation to simulate intergranular degradation and failure in the microstructural Voronoi-type morphology [2,6]. The strategies for coupling the two scales as well as avoiding pathological damage localization at the macroscale are briefly described. Some preliminary results are eventually discussed. Two-scale formulation for polycrystalline degradation and failure Macroscale model. The component level is described using a classical non-linear incremental 3D BEM formulation, where the presence of regions experiencing material softening, due to microstructural degradation, is taken into account introducing an initial stress approach [7,8]. The total macro-stress tensor components at a macroscopic material points are given by

6 ij

6 ijel 6 ijD

Cijlk * ij 6 ijD

(1)

where 6 ijD are the components of the decremental macro-stress tensor contributing to the total macro-stress components 6 ij by reducing the value of the elastic macro-stress components 6 ijel that would correspond to the local macro-strain components * ij in absence of damage. The boundary integral equation used to model the macro-scale is

³ U x, y t y dS ³ < x, Y 6 Y dV

cij x u j x, y ³ Tij x, y u j y dS

ij

S

j

D jk

ijk

S

(2)

VD

where the last integral is performed over internal regions experiencing damage evolution. At a given macrostep, associated with a distribution of internal damage, Eq.(2) provides the values of boundary displacements and tractions that are subsequently used to compute the macro-strain components through the integral equation [7,8]

* ij X ³ TijkJ X, y uk y dS S

J ³ U X, y t y dS ijk

k

S

J J ³ < X, Y 6 Y dV f ¬ª6 X ¼º ijlk

D lk

ij

D lk

(3)

VD

The macro-strain components at an internal macro-point X are subsequently used as boundary conditions provided by for the corresponding associated micro-RVE. However, the direct use of the components * ij Eq.(3) may induce pathological localization of damage at the macro-scale. For this reason, a non-local

, denoted here with *ˆ , is used for providing the RVE boundary conditions, integral counterpart of * ij ij

D in Eqs.(2-3) are provided by suitable ensuring uniqueness and reproducibility of results. The terms 6 ij homogenization performed over the micro-scale RVEs, as will be shown. Microscale model. The micro-scale grain boundary formulation employed for following the microstructural material degradation is described in detail in [2,6]. Here, it is briefly recalled for the sake of completeness. The microstructure morphology is generated using Voronoi tessellations. Each grain is modeled as a 3D linear elastic orthotropic domain with arbitrary spatial orientation, using the BEM for 3D anisotropic elasticity [9]. The polycrystalline aggregate is seen as a multi-region problem [2]. Given a volume bounded by an external surface and containing N g grains, two kinds of grains can be distinguished: the boundary grains, intersecting the external boundary, and the internal grains, completely surrounded by other grains. Boundary conditions are prescribed on the surface of the boundary grains lying on the external boundary, while interface equations and equilibrium conditions are forced on interfaces between adjacent grains, to restore the integrity of the aggregate. The boundary integral equation for a generic grain G k is written

48


cijk x u kj x

³

BC BNC

Ti kj x, y u kj y dB k y

³

BC BNC

U ikj x, y tjk y dB k y

(4)

where u kj and tjk represent components of displacements and tractions of points belonging to the surface of the grain G , the tilde refers to quantities expressed in a local reference system set on the grain surface, U k k

ij

and Ti kj are the 3D displacement and traction fundamental solutions for the anisotropic elastic problem. The integrals in Eq.(4) are defined over the surface of the grain, that is generally given by the union of contact interfaces BC and external non-contact surfaces BNC . The model for the polycrystalline aggregate is obtained by discretizing Eq.(4) for each grain and complementing the system so obtained with a set of suitable boundary and interface equations. The interface between two grains can be either pristine, damaged or failed. When an interface is pristine continuity equations hold. Damage is introduced at the interface when the value of a suitable effective traction overcomes the interface cohesive strength Tmax [6]. When such condition is fulfilled, the following traction-separation laws are introduced at the interface

ª t1 º « » «t2 » ¬« t3 ¼»

ªD G utc 0 0 º ª G u1 º Tmax 1 d * « »« » c 0 D G u 0 t « » «G u2 » ; d* c» « 0 0 1 G un ¼ ¬«G u3 ¼» ¬

d

*

° max ®d Load Hist ° ¯

ª Gu n « c «¬ G un

2

1 2 2½ § G ut · º ° E ¨ c ¸ » ¾ © G ut ¹ ¼» ° ¿ 2

(5)

where d * [0,1] is an interface damage parameter, G un and G ut are the normal and tangential interface opening displacements and G unc and G utc represent their critical values in pure mode I and II respectively and d is the effective opening displacement. Upon interface failure, the traction-separation laws are replaced by the laws of the frictional contact mechanics. After discretization and classical BEM implementation of Eqs.(4) and the associated boundary and evolving interface conditions a sparse system is obtained and an incremental/iterative algorithm is employed to track the microstructural evolution. A load increment is applied and the system solution is iterated until no violation of the interface equations is detected and convergence is then reached. The interested reader is referred to [6] for further details about the microstructural model. Scales coupling: down-scaling and up-scaling. The macro- and micro-scale models must be suitably coupled to capture the damage evolution. The macro-scale may be representative of an engineering component or coupon, which is progressively loaded by external loads. Boundary conditions are defined at this level as a problem input in terms of a macro load factor / . The macro-component is assumed initially pristine and no macro-damage is present. When the initial elastic problem is solved, the internal points experience a macroscopic strain field * ij that can be computed through Eq.(3). Such strain field provides

the boundary conditions for the micro-RVEs associated with the relevant internal points. Different kind of boundary conditions can be applied to the RVEs: in this work the macro-strains are used to provide periodic micro boundary conditions expressed by uiS

uiM *ˆ ij x Sj x Mj ;

tiS

tiM

(6)

where the hat expresses non-local counterparts of the local macro-strains. Eqs.(6) express the downscaling of the macro-strains. Once periodic BCs coming from the macro-scale simulation are available to the microRVEs, the micro-scale simulations can start. From a threshold value of the macro load factor on, some RVEs start experiencing microstructural damage. The micro-structural damage is reflected at the macroscale by some local softening. To define the macro-damage, i.e. to up-scale damage, the following technique is employed. Given a generic RVE subjected to some macro-strain periodic BCs *ˆ , it is ij

possible to associate a macro-stress measure to *ˆ ij through computational homogenization:

6 ij

V ijP

1 VP

³ V

V

P

P ij

dV P

1 SP

³ x t i

VP

P j

dV P

(7)


49

where the integrals are defined over the RVE volume. If the RVE is pristine, the value of macro-stress should coincide with that obtained by the elastic material effective properties (overall properties), i.e. 6 ijel . On the other hand, if micro-damage is present, the macro-stress will be degraded. The degradation is expressed, in the present model, through the decremental component of stress 6 ijD

6 ijel 6 ij

§ 6 ij ¨¨1 el © 6ij

· el ¸¸ 6 ij ¹

Dij 6 ijel

(8)

where 0 d Dij d 1 is a macroscopic damage coefficient expressing the degradation of the material at the macro-point associated with the considered RVE. To determine the damage coefficient then the following practical stress are followed: a) a suitable measure of macro-strain is down-scaled; b) the micro-RVEs are simulated; c) the macro-stress components 6 ij are computed and compared with 6 ijel to estimate macrodamage Dij ; d) the macro-damage is used to define the decremental components of macro-stress.

Macro-micro algorithm. The solution of the two-scale problem involves an incremental-interative macromicro iterative solution strategy, which is briefly discussed here. The analysis starts with the determination of the macro load factor that initiates micro-structural damage. From then on, the analysis is fully non linear. The macro-strain field is downscaled and provides periodic BCs for the micro-RVEs. The microstructural RVEs are simulated and a macro-damage measure is defined for each active RVE though homogenization. The macro-damage is then used to compute the components of the decremental stress used in Eqs.(2-3). Convergence is checked by assessing the convergence of macro internal energy for each relevant macro-cell experiencing damage. When a micro-RVE is too damaged, the corresponding macrocell is removed and a macro-crack is initiated. The two-scale analysis strategy is illustrated in Fig.(1).

Fig. 1: Multiscale analysis scheme: the macro-scale analysis provides the boundary conditions for the micro-RVEs, whose evolution provides the constitutive behaviour for the macro-scale.

Some preliminary numerical results Some qualitative preliminary results are reported here. One of the main challenges, when dealing with fully 3D multiscale simulations, is the computational burden of the analysis. Several simulations are currently being carried out to test the capability of the formulation and complete results will be reported soon. However, some partial results are shown here, mainly to illustrate the aim of the method. The analyzed structural macro-scale component is depicted in Fig.(2), where also the specimen size, mesh features and macro-scale boundary conditions are given. The specimen is loaded in displacement control.

50


Micro-scale data. The considered material is polycrystalline alumina. In Voigt notation, for the alumina single crystals, the elatsic constant (to be used in the micro-model, are: C11 496.8 GPa , C12 163.6 GPa , C13 110.9 GPa , C14 23.5 GPa , C33 498.1 GPa , C44 147.4 GPa . The grain size is ASTM G 10 .

The cohesive-frictional inter-granular properties [6] are: K IC 4 MPa m1/ 2 , Tmax 500 MPa , D E 1 , P 0.2 . Micro-RVEs with 20 grains are considered. Macro-scale data. The considered macro-mesh has 15 u 8 u 4 volume macro-cells, with the corresponding associated micro-RVEs. The macro-scale material properties are E 407 GPa and Q 0.24 .

Fig. 2: Scheme of the performed numerical test.

Fig.(3) reports the distribution of macro-damage after 80 complete load increments, completed after approximately five days on 12 core machines. The relative magnitude of damage is shown. Few observation are worth. The color intensity is proportional to max Dij in Eq.(8) for the macro-points associated with i, j

the micro-RVEs. The absolute damage values are very low and associated mainly with shear loads acting on the micro-RVE. The different behavior of the RVEs to shear acting in opposite directions explains also the apparent lack of symmetry in damage distribution. It is expected that the lack of symmetry will be overcome at higher values of damage. Moreover, no removal of macro-cells occurs within the simulated range, as the damage has not reached critical values in any internal point.

Fig. 3: Damage activation after 80 macro-micro increments.


51

An important parameter in the simulations is the number of grains used in the RVEs: the lower limit is imposed by considerations of material representativity of the microstructure; the upper limit, at least in the present framework, is imposed by the need of maintaining acceptable computational requirements. The effect of this and other parameters is being currently investigated in on-going multiscale simulations. Summary

A two-scale three-dimensional framework for degradation and failure of polycrystalline material engineering components has been presented. The formulation is quasi-static and fully three-dimensional. The macroscale accounts for the presence of material damage by using a classical boundary element incremental initial stress approach, analogous to the method used in elasto-plastic analyses. The threedimensional micro-RVEs are analyzed employing a three-dimensional grain-boundary cohesive-frictional approach. The coupling between the two scales is achieved downscaling macro-strains as periodic RVE boundary conditions and up-scaling damage through volume stress averages. Some preliminary results are shown to illustrate the aim of the technique. Intensive simulations on several aspects and challenging issues are currently being carried out, to highlight the capability of the technique. Acknowledgements

This research was partially supported by a Marie Curie Intra-European Fellowship within the 7th European Community Framework Programme (Project No 274161). References

[1] S. Nemat-Nasser, M. Hori, Micromechanics: overall properties of heterogeneous materials, NorthHolland, Elsevier, The Netherlands, second revised edition edition, (1999). [2] I. Benedetti, M. H. Aliabadi, A three-dimensional grain boundary formulation for microstructural modelling of polycrystalline materials, Computational Materials Science, 67, 249–260, (2013). [3] T. I. Zohdi, P. Wriggers, An introduction to computational micromechanics, Lecture Notes in Applied and Computational Mechanics, vol. 20, Springer, Berlin, (2005). [4] I. Simonovski, L. Cizelj, Computational multiscale modeling of intergranular cracking, Journal of Nuclear Materials, 414, 243 – 250, (2011). [5] Ghoniem, N. M., Busso, E. P., Kioussis, N., Huang, H., Multiscale modelling of nanomechanics and micromechanics: an overview, Philosophical magazine, 83(31-34), 3475-3528, (2003). [6] Benedetti, I., Aliabadi, M.H., A three-dimensional cohesive-frictional grain-boundary micromechanical model for intergranular degradation and failure in polycrystalline materials. Computer Methods in Applied Mechanics and Engineering, 265, 36-62, (2013). [7] P.K. Banerjee, The boundary element methods in engineering, McGraw-Hill, 1994. [8] M.H. Aliabadi, The boundary element method: applications in solids and structures., Vol. 2, John Wiley & Sons Ltd, England, (2002). [9] R.B. Wilson, T.A. Cruse, Efficient implementation of anisotropic three-dimensional boundary-integral equation stress analysis, International Journal for Numerical Methods in Engineering, 12, 1383–1397, (1978).

52


A Boundary Element - Response Matrix method for 3D neutron diffusion and transport problems V.Giusti1 , B.Montagnini2 1

Department of Civil and Industrial Engineering, Pisa University, Largo Lucio Lazzarino 2, I-56126, Pisa, ITALY, [email protected] 2 Department of Civil and Industrial Engineering, Pisa University, Largo Lucio Lazzarino 2, I-56126, Pisa, ITALY, [email protected]

Keywords:

Response matrix, Neutron diffusion, Neutron transport, 3D criticality problems

Abstract. An application of a 3D Boundary Element Method (BEM) coupled with the Response Matrix (RM) technique to solve neutron diffusion and transport equations for multi-region domains is presented. The discussion is here limited to steady state problems, in which the neutrons have a wide energy spectrum, which leads to systems of several diffusion or transport equations. Moreover, the number of regions with different physical constants can be very large. The boundary integral equations concerning each region are solved via a polynomial moment expansion and the multi-fold integrals there involved are reduced to single or double integrals, taking advantage of suitable recurrence formulas. The usual unknowns (the boundary particle density and its normal derivative) are here replaced by the partial currents entering or leaving each cell. The intuitive physical meaning of such quantities facilitates the application of the response matrix technique. Only eigenvalue (criticality) problems will be here considered. As it regards the transport equation, the use of the so called Simplified Spherical Harmonics method allows, through suitable approximations, to cast the problem into a system of differential elliptic equations of the diffusion type, which can still be solved by BEM. Introduction. The computational methods based on the Boundary Element Method (BEM) here presented have been devised for the calculations of the nuclear reactor cores, a subject which is almost a taboo in Italy but not in other developed or emerging countries (e.g. France, Russia or China), also as a valid option to contrast the persisting (and even increasing) use of coal, oil and gas. An exhaustive collection of historical references on the application of BEM to neutronic problems is outside the aim of this paper. Thus we limit ourselves to list some relevant works. The first paper, with a rather complete account of the basic theory, goes back to Koskinen [1]. At the beginning of eighties, further developments [2, 3] gave a strong impulse to BEM, owing to the introduction of the Multiple Reciprocity Method (MRM) [4, 5, 6, 7, 8, 9, 10]. Despite of its elegance, the MRM method has not completely ruled out other classical methods for systems of second order partial differential equations (also pointed out in some of the aforementioned papers), i.e. the diagonalization method and the direct domain (volume) integration. A second step was represented by the decoupling of the global problem into two levels [11, 12, 13, 14]. Another approach will be sketched here below [15, 16, 17]. This paper aims to give a quick account of BEM as applied to practical reactor calculations where the neutron traffic, φ (r), (i.e. neutron density times neutron speed) commonly referred to as neutron flux, has to be determined in the reactor core under steady state conditions. The simplest model is thus based on a system of diffusion equations, one for each speed (or energy) group into which the neutron population can be divided:


53

Figure 1: Face numbering on the rectangular node.

∇ · Dg (r) ∇φg (r) − Σa,g φg (r) +

G

Σs,gg φg (r)

g =1

+

G χg νg Σf,g φg (r) (g = 1, . . . , G) k g =1

(1)

where Dg , Σa,g , Σs,gg and νg Σf,g are, respectively, the diffusion coefficient, the absorption cross section, the scattering cross section from group g to group g and the fission cross section times νg , the number of secondary neutrons emitted by this process whose energy falls inside the group g. Finally, χg is the fraction of secondary neutrons whose energy is within the group g, while k represents the multiplication constant, which plays the role of eigenvalue of the equation system, and it is equal to one under steady state (critical) conditions. The commonly used conditions on the reactor boundary are vanishing conditions for each φg . The main difficulties as regards the application of BEM to such problems are: (i) the problem domain is, in general, a 3D domain, (ii) the number of equations, G, is rather large, (iii) the reactor core is physically divided into homogeneous (i.e. uniform) regions (or made homogeneous through suitable averaging procedures), but such regions might be thousands. The lucky circumstance is that the regions or ’cells’ (as we will refer to them from now on) to be considered during the calculation have a rather simple shape: usually right prisms with a rectangular, hexagonal or triangular base, which are arranged accordingly to a regular lattice. As pointed out above the calculation procedure can be divided into two steps or levels: - the first level: one considers the neutrons entering each cell from the boundary, which is accomplished by giving φg or its normal derivative, ∂φg /∂n. Making use of BEM it is then possible to determine those neutrons which leave the cell and will therefore be considered as entering the adjacent ones; - the second level: on the basis of the ingoing and outgoing particles, an iterative procedure is established in order to determine the neutron distribution over the global reactor system as well as the multiplication constant. The first level: solution of the diffusion equation system for a single cell. Let us consider for simplicity a lattice cell, V , with the shape of a prism with rectangular base and sides a, b, and c and homogeneous physical properties (see Fig.1). Equation system (1) can be rewritten in compact form Δφ + Qφ = 0

(2)

(the eigenvalue k is embedded in the matrix Q, in which all the physical properties are contained, and is provisionally understood). Through a diagonalization of Q the system is reduced to the form Δψ + Λψ = 0

Ξ−1 QΞ = Λ

(3)

54


where the equations are now uncoupled. The eigenvalues λh are in general complex. However, in the simplest case, λh = −γh2 (with γh real and positive) and the fundamental solution of equation (3) is a familiar one, namely exp (−γh |r − r |) . ψ˜h r, r = 4π|r − r |

(4)

The usual procedure of the BEM direct method allows to obtain the following integral form c (r) ψh (r) +

˜ ∂ ψh

∂nS

S

r, rS ψ rS − ψ˜h r, rS

∂ψh r dS S

(5)

∂nS

(c (r) = characteristic function of the V cell). Applying once again the matrices Ξ and Ξ−1 we go back to the physical unknowns. c (r) φg (r) +

G ∂ φ˜gg

∂nS

g =1 S

r, rS φg rS − φ˜gg r, rS

∂φg r dS = 0 S

∂nS

(g = 1, . . . , G) ,

(6)

where r is any point of V , while rS denotes a boundary point. Taking r on the boundary, too, one obtains the set of the Boundary Integral Equations (BIE) that replaces the differential system (1) for a single cell: c (rS ) φg (rS ) +

G ∂ φ˜gg

∂nS

g =1 S

∂φg rS , rS φg rS − φ˜gg rS , rS r dS = 0 ∂nS S

(g = 1, . . . , G) .

(7)

A rearrangement of the unknowns, suggested by the physics of the diffusion process, is useful. It is related to the notion of outward and inward partial currents at a point of the boundary surface S, namely: ∂φg 1 1 Jg± (rS ) = φg (rS ) ∓ Dg (rS ) . 4 2 ∂nS

(8)

With these variables, system (7) is replaced by the following one G c (rS ) + Jg (rS ) + 2 g =1

+

S

G

g =1

S

c (rS ) − + + Jg (rS ) J˜gg rS , rS Jg rS dS = − 2 − − (g = 1, . . . , G) . J˜gg rS , rS Jg rS dS

(9)

Let the ingoing currents Jg− be known (so that the r.h.s is known). Then the response of the cell in + terms of the outgoing currents Jg+ will be determined by solving the BIE’s with the kernels Jgg (rS , rS ). The second or global level, in which all the cells are connected all together, can be then based on the following iteration steps G c (rS ) t+1 + Jg (rS ) + 2 g =1

+

G

S

g =1 S

+ J˜gg rS , rS

t+1 + c (rS ) t − Jg rS dS = − Jg (rS )

2

− t − Jg rS dS (g = 1, . . . , G) . J˜gg rS , rS

(10)


55

Let us return, however, to the single cell problem. To perform the numerical solution, the inward and outward partial currents are expanded into polynomials ws,mn (r) over the sM faces (sM = 6 in our case) of the parallelepiped V : Jg± (rS )

nM sM m M s=1 m=0 n=0

± Jg,smn ws,mn (rS )

(11)

± and similar expressions are adopted for the kernels J˜gg (rS , rS ). The BIE are then replaced by a linear system as follows s

m

n

G M M M 1 + + + Jg,smn + J˜gg ,ss mm nn Jg ,s m n = 4 g =1 s =1 m =0 n =0 s

m

n

G M M M 1 − − − + J˜gg − Jg,smn ,ss mm nn Jg ,s m n 4 g =1 s =1 m =0 n =0

(12)

The weight functions ws,mn are, in the present example, products of Legendre polynomials, e.g. Pm (x) Pn (z) for the face 1 in Fig.1. Note that this moment method clearly overcomes (although in a somewhat brute force way) the problems due to the edges and vertices of the body V . By regrouping the various quantities into vectors and matrices, equations (10) can be given a compact form, namely M+ J+ = M− J− , where to

± Mgg ,ss mm nn

(13) =

± J˜gg ,ss mm nn

±

1 4 δgg δss δmm δnn .

After inversion of

M+ ,

equations (13) leads

˜ −, J+ = RJ

(14)

˜ = M+ −1 M− . with R Equation (14) expresses what is usually meant as a response matrix approach. The evaluation of the elements of the matrices M± is not easy, even for a cell with the shape of a rectangular prism. Actually, considering the interaction, so to say, of the faces 1 and 4 in Fig.??, we are led e.g. to the evaluation of the following four-fold integrals √ a c b c 2 2 2 e−γh x +y +(z−z ) I14,mm nn = dx dz dy dz xm z n y m z n . (15) 2 0 0 0 0 2 2 4π x + y + (z − z ) However, elementary (although very tedious) procedures, based on several integrations by parts, allow to obtain some recurrence relations which end into easy-to-compute one-dimensional integrals. Only in a few cases one must recur to a double numerical integration. The symmetry of the cell problem (for instance, in the case of a prism with a square base) can also be exploited, in particular by means of the circulant properties of the basic matrices. The second level of the calculation. We now give some details about the second level of the calculation. If the reactor is divided into N homogeneous cells we can write J+ = Z (k) J− ,

(16) J+

J−

and are made where the dependence on the eigenvalue k is explicitly mentioned and the vectors of N blocks, which correspond to the outward and inward partial currents from each cell. The matrix Z (k) is therefore a block (N × N ) diagonal matrix where each block corresponds to a cell response ˜ Defining a suitable coupling matrix Π it is possible to write the partial currents entering matrix R. each cell in terms of the partial currents leaving the neighbouring ones: J− = ΠJ+ .

(17)

56


Thus, combining equations (17) and (16), we get J+ = Z (k) ΠJ+ = Θ (k) J+ .

(18)

This homogeneous system admits a non trivial solution only for specific values of the multiplication constant k. The following procedure can be devised in order to determine, in particular, the minimum value of k, which corresponds to the criticality constant. Let us consider the following auxiliary eigenvalue problem αJ+ = Θ (k) J+ .

(19)

We must look for the value of k that gives an α-eigenvalue equal to unity [15, 16]. This means that the J + ’s must be invariant under the transformation (19), as expected for a steady, critical state. Thus, an iterative procedure (actually the classical power method) can be exploited to determine the α-eigenvalue for a fixed value of k: (n+1) +

J = Θ (k) (n) J+

(20)

⎫1 ⎧ ⎨ (n+1) J+ , (n+1) J+ ⎬ 2 (n+1) α= , (n) J+ , (n) J+ ⎭ ⎩

(21)

where equation (20) is iterated until a suitable convergence on the estimated value of (n+1) α, given by equation (21), is achieved. As α (k) turns out to be a monotonic function of k [15], the new value of k can be estimated e.g, by means of the Newton’s chord method. The process will stop when a suitable convergence on the value of the multiplication constant k is obtained. To accelerate the convergence of the above procedure, a multi-step approach was chosen: in the first step only the first moment of the partial current Legendre expansion is considered; once the convergence is achieved the procedure is then repeated considering the first two moments of the Legendre expansions and so on up to the maximum number of moments used (e.g. five). It was found that such an approach was able to cut down the computational time by a factor 20 with respect to a direct use, from the beginning of the calculation, of the maximum number of the available Legendre moments. Finally, once the convergence over the multiplication constant k has been achieved, in order to obtain also an accurate eigenvector J+ it is necessary to perform some extra iterations according to ˆ = I − Θ (k). To this purpose, instead of the ˆ + = 0, where Θ Eq. (20), written now in the form ΘJ power method, which may result quite lengthy, an algorithm based on a Krylov subspace projection method like the Generalized Minimal RESidual algorithm (GMRES) [18] has been adopted. The neutron transport equation. The linear Boltzmann equation (or transport equation), here again written already in a group-energy discretized form, is as follows: Ω · gradr φg (r, Ω) + Σt,g (r) φg (r, Ω) −

+

G

Σs,gg r, Ω, Ω

g =1

χg νg Σf,g (r) φg (r, Ω) dΩ = 0 (g = 1, . . . , G) . k

(22)

The (capital) improvement obtained by solving this equation, as compare with the much less accurate diffusion approach, is due by the intervention of the supplementary angular variable Ω, representing the direction of the particle beams in which the neutron field can be subdivided, indeed an important deepening of the description of the physical process. The variable Ω can be discretized or, in a more elegant form, the fluxes φg (r, Ω) can be expanded in terms of spherical harmonics Ylm (Ω) (or Ylm (θ, ϕ), with the usual notation), as well as the kernels Σs,gg (r, Ω, Ω ).


57

Reector Reector + Control Rods Fuel + Control rods

20.0 cm

Fuel 1 Fuel 2

20.0 cm

Figure 2: Geometrical configuration and material composition of the IAEA 3D benchmark problem. For plane-parallel problems (r is then simply replaced by x) the spherical harmonics reduce to the Legendre polynomials and it is not difficult to show that the system so obtained has the form of a system of (ordinary) linear differential equations of the first order or also, after suitable manipulations, of the second order. In the general 3D case, even for a moderate value of the order L of the differential spherical harmonics equation system turns out to be overwhelming. A successful, although rude, idea in order to simplify the matter is to ignore the dependence on the azimuthal angle ϕ. Then it is again possible to arrive at a system of 3D diffusion equations (their number is now L × G) to which BEM can still be applied [19, 20? , 21, 22]. The example in the next section illustrate the improvement obtainable by such simplified spherical harmonics method with respect to the diffusion method. Numerical examples The first numerical example is a transport version prepared by Hébert [23] of the classical IAEA 3D benchmark problem of the Argonne Code Center [24] (see Fig.2). As stated by the author, the original cross section data have been converted in order to allow transport-like calculations consistent with the diffusion theory results. The IAEA 3D benchmark concerns a full 3D simplified version of a typical LWR core, where nine assemblies have fully inserted control rods while four assemblies have partially inserted control rods. The active part of the reactor core is made of 17 layers, 20 cm high. The four control rods partially inserted are dipped from the top of the active core by 80 cm. Finally, a reflector layer, 20 cm high, is present at the top and the bottom of the reactor core. A vacuum boundary condition is adopted on the external surface of the lateral and axial reflector. The values of the multiplication constant obtained with the diffusion and SP3 (Simplified spherical harmonics with order N =3) approximations through the Boundary Element Response Matrix (BERM) method are compared in Table 1 with the reference value obtained by a suitable two energy group MCNP Monte Carlo calculation [25]. The second example is derived from a benchmark problem [26] concerning the calculation of a 3D reactor core made of 16 fuel assemblies (quarter core simmetry), half of which contain mixed-oxide (MOX) fuel rods, and completely surrounded by a water reflector (see Fig.3). In the present case the number of mixed oxide mixtures has been reduced from three to one. Each fuel assembly is made of 17x17 square pin cells the side length of which is 1.26 cm. In this application, each pin cell has been spatially homogenized weighting the cross section over the volumes, with a preliminary estimate of the neutron flux within each cell. The sets of cross sections so obtained are reported in [21, Appendix B]. There are control rods inserted 2/3 of the way into the inner UO2 assembly and 1/3 of the way

58


Reflective B.C.

21.42

Vacuum B.C.

MOX

UO2

MOX

Reflector

Vacuum B.C.

UO2

14.28

MOX

14.28

UO2

21.42

14.28

21.42

Reective B.C.

Vacuum B.C.

Reflective B.C.

Reflector

Reector 21.42

Reective B.C. Vacuum B.C. UO2 fuel 4.3% MOX fuel

Fission chamber channel Control rod tube cell

Figure 3: Horizontal (left) and vertical (right) sections of the 3D MOX reactor core (dimensions are in cm). In the present configuration the control rods are crossing the upper reflector and inserted by 2/3 of the way into the inner UO2 fuel assembly and 1/3 of the way into the two MOX fuel assemblies (hatched region). into both MOX assemblies, as indicated by the hatched region in Fig. 3. Finally, a vacuum boundary condition is applied on the external surface of the reflector and, in order to reduce the computational burden, an axial symmetry is also assumed. In order to define the reference multiplication constant, again a suitable MCNP Monte Carlo calculation was run making use of the same sets of cross sections used by the deterministic code.

Table 1: The multiplication constant k for the transport version of the IAEA 3D benchmark. Code MCNP6a (ref.) BERM-diffusion BERM-SP3 a

k

Δk (pcm)

1.02955 1.02907 1.02956

− -48 1

with an estimated standard deviation of ±0.00002.

Table 2: The multiplication constant k for the 3D MOX reactor core. Code MCNP6a (ref.) BERM-SP3 BERM-SP7 a

k

Δk (pcm)

1.05932 1.05755 1.05838

− -177 -94

with an estimated standard deviation of ±0.00004.

Table 2 compares the reference value of the multiplication constant with those obtained by two BERM calculations in the SP3 and SP7 transport approximations (i.e. Simplified Spherical Harmonics with order N =3 and N =7, respectively). No results obtained with the diffusion approximation are shown for this example because, due to the too small size of the computational cells, they turned out to be rather inaccurate, as expected. Conclusions The present paper shows an application to nuclear reactor cores of the Boundary Element method associated with a Response Matrix approach. The calculation efficiency is improved by the use of suitable recurrence formulas for the evaluation of the boundary integrals and by other


59

acceleration techniques to determine the neutron flux distribution in the core and the criticality constant. Results show that the method is very accurate and represents a good alternative to the usual Finite Element methods. References [1] Koskinen, H. (1965) In UN, (ed.), Proceedings of the 3rd International Conference on the Peaceful Uses of Atomic Energy, Geneva, volume 4, : pp. 67–73. [2] Itagaki, M. (1985) Journal of Nuclear Science and Technology 22(7), 565–583. [3] Itagaki, M. and Brebbia, C. A. (1991) Nuclear Science Engineering 107(3), 246 – 264. [4] Novak A.J. and Neves A.C., (ed.) The multiple reciprocity boundary element method chapter 6 Computational Mechanics Publications, Southampton (1994). [5] Itagaki, M. (1995) Engineering Analysis with Boundary Elements 15(3), 289 – 293. [6] Itagaki, M. (2000) Engineering Analysis with Boundary Elements 24(2), 169 – 176. [7] Itagaki, M. (2002) Engineering Analysis with Boundary Elements 26(9), 807 – 812. [8] Ozgener, B. (1998) Annals of Nuclear Energy 25(6), 347–357. [9] Ozgener, B. and Ozgener, H. (2000) Engineering Analysis with Boundary Elements 24(3), 259– 269. [10] Cavdar, S. and Ozgener, H. A. (2004) Annals of Nuclear Energy 31(14), 1555 – 1582. [11] Purwadi, M. D., Tsuji, M., Narita, M., and Itagaki, M. (1997) Engineering Analysis with Boundary Elements 20(3), 197 – 204. [12] Purwadi, M. D., Tsuji, M., Narita, M., and Itagaki, M. (1998) Nuclear Science and Engineering 129(1), 88–96. [13] Chiba, G., Tsuji, M., and Shimazu, Y. (2001) Journal of Nuclear Science and Technology 38(8), 664–673. [14] Chiba, G., Tsuji, M., and Shimazu, Y. (2001) Annals of Nuclear Energy 28(9), 895–912. [15] Maiani, M. and Montagnini, B. (1999) Annals of Nuclear Energy 26(15), 1341 – 1369. [16] Maiani, M. and Montagnini, B. (2004) Annals of Nuclear Energy 31(13), 1447 – 1475. [17] Cossa, G., Giusti, V., and Montagnini, B. (2010) Annals of Nuclear Energy 37(7), 953–973. [18] Saad, Y. and Schultz, M. H. (1986) SIAM Journal of Scientific and Statistical Computing 7(3), 856–869. [19] Larsen, E. W., Morel, J. E., and McGhee, J. M. (1996) Nuclear Science Engineering 123, 328 – 342. [20] McClarren, R. G. (2011) Transport Theory and Statistical Physics 39(2-4), 73–109. [21] Giusti, V. and Montagnini, B. (2012) Annals of Nuclear Energy 42, 119–130. [22] Giusti, V., Montagnini, B., and Ravetto, P. (2013) Annals of Nuclear Energy 57, 350–367. [23] Hébert, A. (2010) Annals of Nuclear Energy 37(4), 498 – 511. [24] ANL-7416 (1977) Benchmark Problem Book, Supplement 2, Argonne National Laboratory, Argonne IL, USA. [25] Pelowitz D.B. (Ed.) MCNP6 User’s Manual, Ver.1 LA-CP-13-00634 Los Alamos National Laboratory (2013). [26] Smith, M. A., Lewis, E. E., and Na, B.-C. Benchmark on Deterministic Transport Calculations Without Spatial Homogenisation NEA/NSC/DOC(2005)16 OECD Nuclear Energy Agency (2005).

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A new interface damage model with frictional contact. An SGBEM formulation and implementation ˇ 1 , Vladislav Mantiˇc2 , Roman Vodiˇcka1 Jozef Kšinan 1

Technical University of Košice, Civil Engineering Faculty, Vysokoškolská 4, 042 00 Košice, Slovakia

[email protected], [email protected] 2

University of Seville, School of Engineering, Camino de los Descubrimientos s/n, 41092 Seville, Spain

[email protected] Keywords: interface damage, quasi-static delamination, cohesive interface, Coulomb friction, energy formulation, quadratic programming, BEM.

Abstract. A new numerical model able to predict interface damage considering Coulomb friction contact between debonded surfaces is developed and implemented. A comparison of two variants of this model is presented from the mathematical and engineering point of view, the linear elastic-brittle interface model and a cohesive interface model, both combined with frictional contact. The developed mathematical model exploits the concept of a quasi-static maximally dissipative local solution that describes the process of interface debonding. This solution is approximated by a time-stepping procedure and a boundary element approach. In particular, a Symmetric Galerkin Boundary Element Method (SGBEM) code has been employed, leading to an application of algorithms of quadratic programming to deal with the optimality character of the approximated solution. Different model responses, assuming either cohesive or linear elastic-brittle interface variants of the model, are studied in a 2D numerical example of a fibre reinforced composite under transverse compression. Introduction Analysis of many damage mechanisms in composites on micro-, meso- and macro-scale which include debonds and delaminations may require considering friction or frictionless contact at partially or fully damaged interfaces. A numerical analysis of such an interface damage problem with frictional contact is a quite challenging problem, for which several approaches are currently under development by researchers. With reference to the solution of contact problems by Boundary Element Method (BEM), different approaches have been developed in the past, see e.g. [1, 2, 6, 8, 9, 12] and references therein. The present study tries to further develop the energetic model of the debonding and delamination problems proposed in [10] in order to cover also the frictional contact between the debonded solids. Two models, referred to as the Linear Elastic-Brittle Interface Model (LEBIM) and Cohesive Interface Model (CIM), are introduced, taking into account the possibility of frictional contact, and implemented in a Symmetric Galerkin BEM (SGBEM) code. The former model is sometimes considered as a non-smooth limit of cohesive zone models. The frictional law is regularized to cope with the energetic character of the model, see [12]. The regularization is proposed so that convex quadratic energy functionals are obtained and algorithms of quadratic programming can successfully be applied. In the following sections the proposed model is briefly described, its numerical solution is outlined, and a numerical example documents the applicability of the proposed approach to study problems of interface damage onset and propagation in fibre reinforced composites (FRC). A particular micro-scale damage mechanism of FRC, fibre-matrix debonding under compression transverse to the fibres [3, 4], is analysed numerically. Frictional Contact Model For the sake of simplicity, only 2D contact problems between two solids Ω η (η=A, B) will be considered in the model description. Let t and [u] denote the traction and relative displacement vectors at the contact zone Γc , with σ and τ being the normal and shear stresses and [u]n and [u]s the normal and tangential relative displacements, e.g., [u]n =(uB −uA )·nA . The standard Signorini condition of unilateral contact σ [u]n =0, σ ≤0, [u]n ≥0 at Γc is


61

− replaced by the normal compliance penalization condition σ =kg [u]− n , where [u]n =([u]n − |[u]n |)/2 ≤ 0 denotes the negative part of the relative normal displacement. This penalization can also be explained by presence of a very thin layer of a very high normal stiffness kg > 0, which is compressed in contact and stress-free out of contact. The present model also includes the classical Coulomb friction law |τ|≤μ|σ | with a constant friction coefficient μ≥0. The solution of the contact problem is based on the evolution of energies during the loading process: the energies stored in the bulks and at interfaces and the energy dissipated due to friction. The elastic energy stored at 2 Γc corresponding to the the normal compliance penalization condition is given by the integral Γc 12 kg ([u]− n ) dΓ , − ˙ s dΓ , where the whereas the (rate of) energy dissipated due to friction is given by the integral Γc μkg |[u]n |·[u] ˙ s. rate of the tangential relative displacement [u]s is denoted as [u]

Linear Elastic-Brittle and Cohesive Interface Models (LEBIM and CIM) The Linear Elastic-Brittle Interface Model (LEBIM) [11] provides a simple and useful model of thin linear elastic adhesive layer between two surfaces which breaks in a brittle way. This model yields initially (for an undamaged layer) a linear response, given by the following relations between the normal and shear stresses, σ and τ, and normal and tangential displacements jumps across the layer, [u]n and [u]s : σ =ζ kn [u]n and τ=ζ ks [u]s , where ζ is a damage parameter, 0 ≤ ζ ≤ 1, initially ζ =1 representing an undamaged layer, whereas ζ =0 corresponds to a cracked layer, not able to transmit tensile stress. Considering, e.g., pure opening Mode I, the failure of a layer point occurs when the driving force (energy release rate) G reaches the activation threshold, usually referred to as fracture energy, Gd , and correspondingly both, √ the mechanical stress σ and relative normal displacement [u]n , achieve their critical values, respectively, σc = 2kn Gd and uc = 2Gd /kn . Consequently, at that instant, the damage parameter and mechanical stress jump down to zero abruptly, leading to a discontinuous response of the model, see Figure 1(a). The schematic illustrations present the model response in pure Mode I under a displacement-controlled 1D experiment. (a)

(b)

Figure 1: Model response for the driving force G, the damage parameter ζ and the mechanical stress σ : (a) LEBIM, (b) CIM. In engineering and computational mechanics practise another approach is usually preferred, with a nonlinear but continuous response of the interface model, including the so-called softening period. This kind of model is referred to as Cohesive Interface Model (CIM). An efficient and versatile approach to achieve the continuous non-linear interface response is based on the energy formulation, employing the stored energy functional E , as proposed in [10]. The expression of the stored interface energy is modified by an additional term including ζ 2 and some stiffnessparameters. The mechanical stress, e.g., in opening Mode I, decays with decreasing ζ following the law σ = kn1 ζ + kn2 ζ 2 [u]n . In the first, linear elastic, part of the stress-relative √ n2 displacement diagram, this stress is a linear function of [u]n up to achieving its critical value σc = √kkn1 +k 2Gd , n1 +2kn2 whereas in the second, softening, part of this diagram it evolves non-linearly until it vanishes for the critical relative (opening) displacement uc = 2Gd /kn1 , see Figure 1(b). The main feature of the proposed CIM is that the interface energy functional is separately quadratic both in the [u] and ζ variables. This fact enables to apply very efficient quadratic programming algorithms to the solution of the pertinent minimization problem, see [5, 12], also to this case.

62


Energetic Formulation of LEBIM and CIM Based on the above assumptions, a quasi-static evolution of LEBIM and CIM in presence of friction contact is governed by the following inclusions: ˙ ζ˙ ) + δu F (t,u) 0, ∂u E (t, u, ζ ) + Ru˙ (u, ζ ; u, ˙ ζ˙ ) ∂ E (t, u, ζ ) + R ˙ (u, ζ ; u, 0, ζ

ζ

(1)

where the symbol ∂ refers to partial subdifferential relying on the convexity of the energy functionals, see [10]. The stored energy functional [10] for the LEBIM is defined as

E (t,u, ζ ) =

ΩA

1 A A A ε :C :ε dΩ + 2

ΩB

1 B B B ε :C :ε dΩ + 2

Γc

1 2 ζ kn [u]2n + ζ ks [u]2s + kg ([u]− n ) dΓ , (2) 2

whereas for the CIM as E (t,u, ζ ) =

ΩA

1 A A A ε :C :ε dΩ + 2

ΩB

1 B B B ε :C :ε dΩ 2 1 2 + ζ (kn1 + ζ kn2 )[u]2n + ζ (ks1 + ζ ks2 )[u]2s + kg ([u]− n ) dΓ , (3) Γc 2

with the admissible displacements uη = wη (t) on Γuη , the small strain tensor ε η =ε(uη ), the potential energy of external forces (acting only along the boundary in the present work) F (t,u) = −

Γt A

fA ·uA dΓ −

Γt B

fB ·uB dΓ ,

(4)

and the dissipation potential ˙ ζ˙ ) = R(u, ζ ; u,

Γc

˙ s | + Gd |ζ˙ | dΓ , f (ζ )μkg |[u]− n |·|[u]

(5)

where 0 ≤ f (ζ ) ≤ 1 is a dimensionless function characterizing the switch between the interface shear stresses due to cohesive and friction forces. f (ζ ) is increasing with decreasing ζ (0 ≤ ζ ≤ 1) and f (0) = 1. This function should be determined by experiments. A simple but versatile expression of such a function, depending on two parameters, is f (ζ )=1 − qζ p where 0 < p < ∞ and 0 ≤ q ≤ 1 is considered in the present work. Numerical solution and computer implementation The numerical procedure devised to solve the above problem is based on the concept of Maximally-Dissipative Local Solution [14] and considers time and spatial discretizations separately, as usual [10]. The procedure is formulated in terms of the boundary data only using the Symmetric Galerkin BEM (SGBEM) for the spatial discretization. Spatial discretization and SGBEM The role of the SGBEM in the present computational procedure is to provide a complete boundary-value solution for given boundary data in each solid Ω η in order to calculate the elastic strain energy stored in these solids and at their interface Γc . For this, it is convenient to change the bulk integrals in (2) and (3) to boundary integrals Ωη

ε(uη ):Cη :ε(uη )dΩ =

Γη

tη (uη ) · uη dΓ .

(6)

In the present procedure, the SGBEM code calculates unknown tractions along Γc ∪ Γuη and unknown displacements along Γt η , assuming the displacements jump at Γc to be known from the used minimization procedure, in the same way as proposed and tested in [12, 13].


63

Time discretization The semi-implicit time-stepping scheme is defined by a fixed time step size t0 such that k k−1 ˙ u −u t k =kt0 for k=1, 2, . . . . The displacement rate is approximated by the finite difference u≈ , where uk t0 k denotes the solution at the discrete time t . Similarly the damage-parameter rate can be approximated by k k−1 . The differentiation with respect to the displacement and damage-parameter rates can be replaced ζ˙ ≈ ζ −ζ t0 by the differentiation with respect to u and ζ , respectively, as well, i.e. u − uk−1 ζ − ζ k−1 , ), t0 t0 u − uk−1 ζ − ζ k−1 ˙ ζ˙ ) ≈ t0 ∂ζ R(uk−1 , ζ k−1 ; , ). ∂ζ˙ R(uk−1 , ζ k−1 ; u, t0 t0 ˙ ζ˙ ) ≈ t0 ∂u R(uk−1 , ζ k−1 ; ∂u˙ R(uk−1 , ζ k−1 ; u,

(7) (8)

It means that the inclusions (1) are approximated at discrete times t k by the first order optimality condition for the total energy functional H k H k (u, ζ ) = E (kt0 , u, ζ ) + t0 R(uk−1 , ζ k−1 ;

u−uk−1 ζ −ζ k−1 , ) + F (kt0 ,u). t0 t0

(9)

assuming uη = wη (kt0 ) on Γuη and 0 ≤ ζ ≤ ζ k−1 on Γc .

(10)

The optimality solution, is denoted by (uk , ζ k ). Substituting the results of the previous time-step k − 1 into the dissipation potential due to friction, makes the functional H k (u, ζ ) separately convex with respect to the unknowns u and ζ . This leads to the following fractional-step-like strategy in the time step k: first H k (u, ζ k−1 ) is minimized with respect to u defining uk , and second H k (uk , ζ ) is minimized with respecto to ζ defining ζ k . The above formulation has been implemented in an SGBEM code [12, 13] in M ATLAB by using a conjugate gradient based method for constrained minimization [5]. Numerical example In this section, the above numerical procedure to analyse a quasi-static evolution of LEBIM and CIM in presence of friction contact is tested in a plane strain problem of crack onset and growth at the fibre-matrix interface under remote compression transverse to fibres. The objective of this study is to show the capabilities of the procedure, compare the solutions obtained by the LEBIM and CIM and capture the influence of friction in the analysed problem. Model description In particular, the response of a fibre reinforced composite given by a symmetric bundle of four fibres (represented by circular inclusions) embedded in a matrix, see Figure 2 (a), is studied. In the numerical solution, the square matrix cell of side length 2L with four fibres subjected to compressions along the top and bottom sides is replaced due to symmetry by a square matrix cell of side length L=30 μm including one fibre of radius r=7.5 μm in the center, with uniform displacements originating a compressive load prescribed at the top side of the matrix cell, see Figure 2 (b). The prescribed displacements u¯2 are increasing during the loading process, namely u¯k2 = − ut k for k=1, 2, . . . with u=0.1μm and t k =kt0 , t0 =1s. The elastic properties of epoxy matrix (m) and glass fibre (f) are Young’s moduli Em =2.79 GPa and E f =70.8 GPa, and Poisson’s ratios νm =0.33 and ν f =0.22. The interface stiffness parameters of the LEBIM are kn =2025MPa/μm, ks =675MPa/μm. In the CIM both normal and tangential stiffnesses are defined by relations kn =kn1 +kn2 , ks =ks1 +ks2 , kn1 =0.01 × kn , kn2 =0.99 × kn , ks1 =0.01 × ks , ks2 =0.99 × ks . The parameters that govern the crack growth are the fracture energy Gd =2 Jm−2 , and the critical tension of the interface σ¯ c =90 MPa and 63.8 MPa, respectively, in the LEBIM and CIM. In order to verify the effect of friction in the present problem, two values of the Coulomb friction coefficient μ=0 and 1 are considered. The dimensionless function f (ζ )=1 − qζ in (5) is used with q = 1 and 0, respectively, in the LEBIM and CIM.

64


Figure 2: Bundle of four fibres embedded in a matrix under transverse compression (a) Schematic of geometry and boundary conditions, (b) Unit cell obtained by applying symmetries. Numerical results Figures 3-6 show the distributions of σ , τ, [u]n , [u]s and ζ along the fibre-matrix interface for the LEBIM and CIM with friction coefficient μ=0 and 1, respectively. These figures confirm a quite relevant influence of friction on the solution of both interface models. The results of the LEBIM, Figures 3 and 4, show a discontinuous response of this interface model. When an interface part reaches the required amount of stored energy per unit length Gd the corresponding damage parameter ζ jumps from 1 to 0 abruptly representing a total breakage of this interface part. Four debonds are observed simultaneously for both μ = 0 and 1 in time step k = 10 at the following intervals of angle θ : (40◦ , 58◦ ), (126◦ , 147◦ ), (215◦ , 235◦ ) and (303◦ , 322◦ ). The debonds on the left hand side of fibre merge producing a quite large lateral debond in time step k=13 and k=15, respectively, for μ=0 and 1. ((a))

(b)

Figure 3: Response of the LEBIM without friction (μ=0) at time step k=12, (a) Distributions of σ , τ, and ζ , (b) Distributions of [u]n , [u]s and ζ . According to Figures 5 and 6, showing the results for the CIM, ζ changes from 1 to 0 continuously, which is characteristic for the softening stage in the CIM. Values of 0 < ζ < 1 correspond to the so-called softening zone. The first relevant interface-damage is observed for μ = 0 in time step k = 7 at angles θ =136◦ and 226◦ , simultaneously, whereas for μ=1 in time step k = 12 at angles θ =154◦ and 202◦ . These growing damaged zones merge originating a large lateral debond at time step k=13 and k=18, respectively, for μ=0 and 1. Thus, friction has a relevant influence on the value of the critical load initiating a debond or damage at the fibre-matrix interface, its location, and also on the subsequent debond or damage propagation along the interface. In the presence of friction the relevant damage in the CIM appears for larger loads and closer to the angle θ =180◦ than in the LEBIM. This, might be, at least partially, explained by different expressions of the function switching between cohesive and frictional shear stresses, f (ζ ) = 1 − ζ in the LEBIM and f (ζ ) = 1 in the CIM.


((a))

65

((b))

Figure 4: Response of the LEBIM with friction coefficient μ=1 at time step k=14, (a) Distributions of σ , τ, and ζ , (b) Distributions of [u]n , [u]s and ζ . ((a))

((b))

Figure 5: Response of the CIM without friction (μ=0) at time step k=11, (a) Distributions of σ , τ, and ζ , (b) Distributions of [u]n , [u]s and ζ . ((a))

(b)

Figure 6: Response of the CIM with friction coefficient μ=1 at load step k=15, (a) Distributions of σ , τ, and ζ , (b) Distributions of [u]n , [u]s and ζ .

66


Polar plots of distributions of the opening displacement [u]n along the interface are shown in Figures 7-10 for four different time steps in each case studied. Their common feature is that the debonding process begins with the evolution of two small debonded or damaged zones on the left hand side of the fibre. For increasing load these zones are growing until they merge in a quite large debonded or damaged zone, as described above. Simultaneously a couple of small debonded or damaged zones appear also on the right hand side of the fibre, but in the CIM requiring higher loads to be applied, due to a slight asymmetry in single-fibre problem configuration. In both models the propagation of debonds and damaged zones is slower in presence of friction requiring higher loads than in the cases without friction.

Figure 7: Evolution of [u]n × 20 for the LEBIM with μ = 0.

Figure 8: Evolution of [u]n × 20 for the LEBIM with μ = 1.

Figure 9: Evolution of [u]n × 20 for the CIM with μ = 0.

Figure 10: Evolution of [u]n × 20 for the CIM with μ = 1.


67

Conclusions Linear-Elastic Brittle and Cohesive Interface Models in presence of Coulomb friction contact have been suitably formulated in terms of functionals of stored and dissipated energies, and implemented in a multidomain SGBEM code [13] by using a quadratic programming approach to solve the required minimization problems in each time step. The code has been tested on a problem of a fibre reinforced composite under transverse compression, showing a behaviour expected according to some previous studies by other authors and different methods, and allowed us to study the influence of the interface model type and friction on the onset and propagation of interface damage. Acknowledgement J. K and R. V. acknowledge support from the grant VEGA 1/0201/11. V.M. acknowledges support from the Junta de Andalucía and European Social Fund (TEP-4051) and the Spanish Ministry of Economy and Competitiveness (MAT2012-37387). J. K. also acknowledges support from the Slovak Academic Information Agency through the National Scholarship Programme. References [1] A. Blázquez, F. París, V. Mantiˇc. BEM solution of two dimensional contact problems by weak application of contact conditions with non-conforming discretizations. Int. J. of Solids and Structures, 35:3259–3278, 1998. [2] A. Blázquez, R. Vodiˇcka, F. París, V. Mantiˇc. Comparing the conventional displacement BIE and the BIE formulations of the first and the second kind in frictionless contact problems. Engineering Analysis with Boundary Elements, 26:815–826, 2002. [3] E. Correa, V. Mantiˇc, F. París. Numerical characterisation of the fibre-matrix interface crack growth in composites under transverse compression. Engineering Fracture Mechanics, 75:4085–4103, 2008. [4] E. Correa, V. Mantiˇc, F. París. A micromechanical view of inter-fibre failure of composite materials under compression transverse to the fibres. Composites Science and Technology, 68:2010–2021, 2008. [5] Z. Dostál. Optimal Quadratic Programming Algorithms, volume 23 of Springer Optimization and Its Applications. Springer, Berlin, 2009. [6] C. Eck, O. Steinbach, W.L. Wendland. A symmetric boundary element method for contact problems with friction. Mathematics and Computers in Simulation, 50:43 – 61, 1999. [7] M. Koˇcvara, A. Mielke, T. Roubíˇcek. A rate-independent approach to the delamination problem. Math. Mech. Solids, 11:423–447, 2006. [8] C.G. Panagiotopoulos, V. Mantiˇc, I.G. García, E. Graciani. Quadratic programing for minimization of the total potential energy to solve contact problems using the collocation BEM In Advances in Boundary Element Techniques XIV, A. Sellier and M.H. Aliabadi (Eds.), pages 292–297, EC ltd, Eastleigh, 2013. [9] L. Rodríguez-Tembleque, F.C. Buroni, R. Abascal, A. Sáez. 3D frictional contact of anisotropic solids using BEM European Journal of Mechanics-A/Solids, 30:95–104, 2013. [10] T. Roubíˇcek, M. Kružík, J. Zeman. Delamination and adhesive contact models and their mathematical analysis and numerical treatment (Chapter 9). In V. Mantiˇc (Ed.), Mathematical Methods and Models in Composites, Imperial College Press, London, 2014. [11] L. Távara, V. Mantiˇc, E. Graciani, F. París. BEM analysis of crack onset and propagation along fibermatrix interface under transverse tension using a linear elastic-brittle interface model. Engineering Analysis with Boundary Elements, 35:207–222, 2011. [12] R. Vodiˇcka, V. Mantiˇc. An SGBEM implementation with quadratic programming for solving contact problems with Coulomb friction. In Advances in Boundary Element Techniques XIV, A. Sellier and M.H. Aliabadi (Eds.), pages 444–449, EC ltd, Eastleigh, 2013. [13] R. Vodiˇcka, V. Mantiˇc, F. París. Symmetric variational formulation of BIE for domain decomposition problems in elasticity – an SGBEM approach for nonconforming discretizations of curved interfaces. CMES – Comp. Model. Eng. Sci., 17:173–203, 2007. [14] R. Vodiˇcka, V. Mantiˇc, T. Roubíˇcek. Energetic versus maximally-dissipative local solutions of a quasistatic rate-independent mixed-mode delamination model. Meccanica (submitted).

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Abstract A boundary element based solution procedure for 3D geometrically nonlinear elastic problems is presented. The method called LPI-BEM (local point interpolation- boundary element method) has already proved efficient for the solution of anisotropic and multi-physics problems. It uses the partition of the displacement field into complementary and particular parts. The complementary field solves a Navier’s type equation and is obtained by conventional BEM. The local radial point interpolation is applied to a strong form differential equation in order to obtain the particular solution. No volume integral is needed in the procedure. The effectiveness and accuracy of the proposed approach are assessed by using some simple examples with three types of material models within the total Lagrangian framework. 1. Introduction In many and various engineering fields, it is necessary to consider geometrical and/or material nonlinearities in the analysis of structural members. These engineering fields cover the range from forming processes to that of the MEMS deformation via the motion and growth of biological tissues. In the presence of nonlinearities, it is difficult, if not impossible, to determine analytical solution of the field equations. Consequently, the analysis of the structural member is based on numerical results. The latter are usually obtained by the popular, versatile and powerful finite element method (FEM). It is a domain mesh-based method which may face some difficulties when dealing with three-dimensional problems involving large mesh distortion. There is a dense literature on the alternatives to the FEM. One can mention the more and more growing methods known as meshless methods which can be classified into two main categories: meshless methods based on weak-forms of the partial differential equations and those based on the strong form [1]. The first category has been successfully applied to structural problems at large strains with material nonlinearities (e.g. [2–6]). The point collocation method belongs to the second category. It has been successful for small strain problems (e.g.[7–9]). The method is accurate as long as the boundary conditions are of Dirichlet type. In the presence of Neuman type boundary condition which is practically inevitable in solid mechanics, the solution accuracy deteriorates. A way to overcome this difficulty is to use the weak form at boundary points with Neumann type boundary conditions. This method is known as the meshfree weak–strong form and has been proposed by Liu and Gu [8]. To our knowledge, this attractive point collocation method has not yet been applied to the solution of problems involving large strains. The boundary element method (BEM) is already well established as a powerful alternative to the FEM in the case of linear problems with wellestablished fundamental solutions (see e.g. [10–12]). In the presence of geometrical and/or material nonlinearities, the boundary integral formulation involves a domain integral term. The method is then known as the field boundary element method (e.g.[13–15]). The solution procedure uses volume cells in the same manner as in the FEM. Then, the main appeal of the BEM (reduction of the domain dimension by one) is tarnished to some extent. A number of strategies have been developed in order to convert volume integrals into surface integrals. One can mention the dual reciprocity method [16], the radial integration method [17], the generalized boundary element method [18]. Other boundary element based approaches have proved effective for a variety of problems: the local integral equation[19] and the analog equation method [20]. In this work, we propose a boundary element based solution procedure for 3D geometrically nonlinear elasticity problem without volume cells. The strategy combines the advantages of the strong form point interpolation method and the conventional (isotropic small strain) boundary element method. The method which is herein called the LPI-BEM (local point interpolation – boundary element method) has been successful for anisotropic small strain elastic and piezoelectric problems [21,22]. It uses the partition of the


69

displacement field into complementary and particular parts. The complementary field solves a Navier's type equation and is obtained by the well-known isotropic-BEM. The particular integral is obtained by solving the corresponding strong form differential equation using local radial point collocation. The final system of nonlinear equations to be solved is similar in form to the one obtained by methods such as the field boundary element method. The LPI-BEM for large strain elastic problems is outlined in section 2 below with emphasis on the determination of the particular solution by the meshfree approach. Since the main object of the paper is to prove the potential of the LPI-BEM in this context, in section 3, the effectiveness of this simple to implement approach is assessed by considering tension and/or bending of a Saint Venant-Kirchhoff, a neo-Hookean and a Mooney-Rivlin solid. 2.

Governing equations and solution method

We consider an elastic solid undergoing large elastic deformation. The total Lagrangian formulation is adopted and the equilibrium equations are expressed in the undeformed stress free initial configuration ȳ଴ with boundary Ȟ଴ . The position vector to a particle is denoted by ܺ. After deformation, the body occupies the domain : and the position vector to a particle is denoted by ‫ݔ‬. The displacement vector ‫ ݑ‬is given by ‫ݑ‬൫ܺ൯ ൌ ‫ ݔ‬െ ܺ. A measure of the deformation is described by the deformation gradient ‫ ܨ‬relative to ܺ given by ‫ ܨ‬ൌ

డఞ൫௑൯ డ௑

ൌ

డ௨൫௑൯ డ௑

൅ ͳ. The identity tensor is denoted ͳ. The right Cauchy-Green tensor (‫ )ܩ‬can be

found from ‫ܨ‬according to ‫ ܩ‬ൌ ‫ܨ ் ܨ‬. Adopting cartesian coordinates system and indicial notation with associated summation convention over repeated indices, when body forces are neglected, the equilibrium equations with respect to the reference configuration are: డ௉ೕೖ ൫௑൯ డ௑ೕ

ൌͲ

(1)

ܲ is the first Piola-Kirchhoff stress tensor which is non-symmetric. Eq. (1) must be supplemented by properly defined boundary conditions in terms of known displacement and traction. The stress tensor ܲ is expressed in terms of the symmetric second Piola-Kirchhoff stress tensor ܵ by ܲ௝௞ ൌ ‫ܨ‬௞௜ ܵ௝௜ . In the case of a డௐ

hyper-elastic law, ܵ is defined from an elastic potential ܹ by ܵ ൌ ʹ డீ . In this work, three material models are considered. Model 1 (Saint Venant-Kirchhoff solid) ܵ ൌ ߣ‫ ݎݐ‬ቀ‫ܧ‬ቁ ൅ ʹߤ‫ܧ‬

(2)

Model 2 (neo-Hookean solid) ܵ ൌ ߣ݈݊‫ି ܩܬ‬ଵ ൅ ߤ ቀͳ െ ‫ି ܩ‬ଵ ቁ

(3)

Model 3 (a modified Mooney-Rivlin material) ܵ ൌ ‫ܥ‬ଵ ͳ ൅ ‫ܥ‬ଶ ‫ ܩ‬൅ ‫ܥ‬ଷ ‫ି ܩ‬ଵ

(4)

ିଵȀଷ

ିଶȀଷ

൅ ʹ‫ܣ‬଴ଵ ‫ܫ‬ଷ ‫ܫ‬ଵ ‫ܥ‬ଵ ൌ ʹ‫ܣ‬ଵ଴ ‫ܫ‬ଷ ିଶȀଷ ‫ܥ‬ଶ ൌ െʹ‫ܣ‬଴ଵ ‫ܫ‬ଷ ʹ Ͷ ିଵȀଷ ିଶȀଷ ଵȀଶ ‫ܥ‬ଷ ൌ െ൬ ‫ܣ‬ଵ଴ ‫ܫ‬ଷ ‫ܫ‬ଵ ൅ ‫ܣ‬଴ଵ ‫ܫ‬ଷ ‫ܫ‬ଶ െ ‫ܭ‬ሺ‫ ܬ‬െ ͳሻ‫ܫ‬ଷ ൰ ͵ ͵ In Eq. (2) and (3) ‫ ܧ‬ൌ ቀ‫ ܩ‬െ ͳቁ Ȁʹ is the Green-Lagrange strain tensor, ‫ି ܩ‬ଵ is the inverse of ‫ ܩ‬and ‫ ܬ‬ൌ ݀݁‫ݐ‬ሺ‫ܨ‬ሻ O and P are the Lamé constants. In Eq. (4) ‫ܫ‬ଵ , ‫ܫ‬ଶ and ‫ܫ‬ଷ are respectively the first, second and third invariant of ‫ܩ‬. The material parameters ‫ܣ‬ଵ଴ and ‫ܣ‬଴ଵ are related to the shear modulus by ߤ ൌ ʹሺ‫ܣ‬଴ଵ ൅ ‫ܣ‬ଵ଴ ሻ. ‫ ܭ‬is the bulk modulus. Solution method by the LPI-BEM In this section we present the main lines of the method as applied for the solution of elastic large deformation problems. First, let us rewrite equation (1) as:

70


೛

೎ డቀఙೖೕ ାఙೖೕ ା்ೖೕ ቁ

డ௑ೕ ௖ where ߪ௞௝

ൌͲ

௖ ൌ ‫ܥ‬௞௝௠௡ ‫ݑ‬௠ǡ௡

(5) ௣ ߪ௞௝

௣ ൌ ‫ܥ‬௞௝௠௡ ‫ݑ‬௠ǡ௡

, and ܶ௞௝ ൌ ܲ௞௝ െ ‫ܥ‬௞௝௠௡ ‫ݑ‬௠ǡ௡ The displacement field is partitioned into a complementary part (‫ݑ‬௖ ) and a particular one (‫ݑ‬௣ ). That is ‫ ݑ‬ൌ ‫ݑ‬௖ ൅ ‫ݑ‬௣ . ‫ܥ‬௞௝௠௡ is the fourth order tensor of the elastic constants of an isotropic homogeneous material. The complementary solution is defined as the solution of the following differential equation: ೎ ൯ డ൫஼ೖೕ೘೙ ௨೘ǡ೙

డ௑ೕ

ൌͲ

(6)

Accordingly, the particular integral is obtained by solving: ೛

డ൫஼ೖೕ೘೙ ௨೘ǡ೙ ା்ೖೕ ൯ డ௑ೕ

ൌͲ

(7)

Equation (6) is similar to that of the classical small strain isotropic elastostatic. Following conventional steps, its solution by the boundary element method leads to a system of equations of the form: ሾ‫ܪ‬ሿሼ‫ݑ‬௖ ሽ ൌ ሾ‫ܩ‬ሿሼ‫ ݐ‬௖ ሽ

(8)

For the solution procedure of Eq. 7, we aimed to apply a local radial point collocation. Interpolation Let us start by a short remind on the local radial point interpolation. The domain : 0 and its boundary * 0 are represented by properly scattered collocation centres (nodes). A field v( x) is approximated as (see e.g. [7]): ெ ‫ݒ‬ሺ‫ݔ‬ሻ ൌ σே ௜ୀଵ ܴ௜ ሺ‫ݎ‬ሻܽ௜ ൅ σ௝ୀଵ ‫݌‬௝ ሺ‫ݔ‬ሻܾ௝ with the constraints:

(9)

σே ௜ୀଵ ‫݌‬௝ ሺ‫ݔ‬ሻܽ௜ ൌ Ͳ , ݆ ൌ ͳ െ ‫ ܯ‬and ݅ ൌ ͳ െ ܰ ܴ௜ ሺ‫ݎ‬ሻ is the selected radial basis functions, ܰ the number of nodes in the neighbourhood of point ‫ ݔ‬and ‫ ܯ‬is the number of monomials ‫݌‬௝ ሺ‫ݔ‬ሻ in the basis of the selected augmented polynomial degree. Recall that ‫ݎ‬ denotes the Euclidean distance between the point ‫ ݔ‬and the collocation centre ‫ ݔ‬௜ . Enforcing approximation (9) to be satisfied at all centres in the support domain, coefficients ai and b j are determined and the approximation written in the compact form: ‫ݒ‬ሺ‫ݔ‬ሻ ൌ ሾȰሺ‫ݔ‬ሻሿ൛‫ݒ‬Ȁ௅ ൟ where ൛‫ݒ‬Ȁ௅ ൟ denotes the vector of nodal values of ‫ݒ‬ሺ‫ݔ‬ሻ. Application ௉ ௉ ௉ ௉ ௉ ௉ ሻ, Introduce the following vectors: ሼߪ ௉ ሽ் ൌ ሺߪଵଵ ߪଷଷ ߪଶଶ ߪଵଶ ߪଵଷ ߪଶଷ ሼܶሽ் ൌ ሺܶଵଵ ܶଶଶ ܶଷଷ ܶଵଶ ܶଶଵ ܶଵଷ ܶଷଵ ܶଶଷ ܶଷଶ ሻ. Rewrite equation (7) as follows: ሾ‫ܤ‬ଵ ሺ‫׏‬ሻሿሼߪ ௉ ሽ ൅ ሾ‫ܤ‬ଶ ሺ‫׏‬ሻሿሼܶሽ ൌ Ͳ ሼ‫׏‬ሽ ൌ ሺ߲Ȁ߲ܺଵ ߲Ȁ߲ܺଶ ߲Ȁ߲ܺଷ ሻ் and matrices ‫ܤ‬ଵ ሺ‫ݖ‬ሻand ‫ܤ‬ଶ ሺ‫ݖ‬ሻ are defined from the vector ሼ‫ݖ‬ሽ ൌ ሺ‫ݖ‬ଵ ‫ݖ‬ଶ ‫ݖ‬ଷ ሻ் as ‫ݖ‬ଵ ሾ‫ܤ‬ଵ ሺ‫ݖ‬ሻሿ ൌ ቌ Ͳ Ͳ

Ͳ ‫ݖ‬ଶ Ͳ

Ͳ ‫ݖ‬ଶ ‫ݖ‬ଷ Ͳ Ͳ ‫ݖ‬ଵ Ͳ ‫ݖ‬ଷ ቍ ‫ݖ‬ଷ Ͳ ‫ݖ‬ଵ ‫ݖ‬ଶ

‫ݖ‬ଵ ሾ‫ܤ‬ଶ ሺ‫ݖ‬ሻሿ ቌ Ͳ Ͳ

Ͳ ‫ݖ‬ଶ Ͳ

(10)

(11)

Ͳ ‫ݖ‬ଶ Ͳ ‫ݖ‬ଷ Ͳ Ͳ Ͳ Ͳ Ͳ ‫ݖ‬ଵ Ͳ Ͳ ‫ݖ‬ଷ Ͳ ቍ ‫ݖ‬ଷ Ͳ Ͳ Ͳ ‫ݖ‬ଵ Ͳ ‫ݖ‬ଶ

Adopting interpolation (10) for each component of the vectors ሼߪ ௉ ሽ and ሼܶሽ eq. (11) becomes: ௉ ሾ‫ܤ‬ଵ ሺ‫׏‬ሻሿሾȰଵ ሿ൛ߪȀ௘௟ ൟ ൅ ሾ‫ܤ‬ଶ ሺ‫׏‬ሻሿሾȰଶ ሿ൛ܶȀ௘௟ ൟ ൌ Ͳ

(12)

In relation (12), >)1 @ and >) 2 @ are properly constructed matrix from interpolation functions defined by (10) and ൛‫ݖ‬Ȁ௘௟ ൟ ൌ ሺ‫ݖ‬ଵଵ ‫ݖ‬ଶଵ ǥ ‫ݖ‬௡ଵ ǥ ǥ ǥ Ǥ ‫ݖ‬ଵே ‫ݖ‬ଶே ǥ Ǥ Ǥ ‫ݖ‬௡ே ሻ் with ݊ = 6 or 9.


71

Now, using the definition of the particular stress tensor (cf. eq. (7)) and adopting interpolation (10) for each component of the particular displacement vector, one obtains: ௉ ሾ‫ܤ‬ଵ ሺ‫׏‬ሻሿሾȰଵ ሿሾ‫ܥ‬ሿሾ‫ܤ‬ଵ ሺ‫׏‬ሻሿ் ሾȰଷ ሿ൛‫ݑ‬Ȁ௘௟ (13) ൟ ൅ ሾ‫ܤ‬ଶ ሺ‫׏‬ሻሿሾȰଶ ሿ൛ܶȀ௘௟ ൟ ൌ Ͳ Eq. (13) can be rewritten in the compact form ௉ ሾܳ௖ ሿ൛‫ݑ‬Ȁ௘௟ (14) ൟ ൅ ሾߜܳሿ൛ܶȀ௘௟ ൟ ൌ Ͳ Collecting equations (14) for all internal collocation centres leads to: ‫ݑ‬௉ ሾ‫ܣ‬஻ ‫ܣ‬ூ ሿቊ ஻௉ ቋ ൅ ሾ‫ܤ‬ሿሼܶ ே௅ ሺ‫ݑ׏‬ሻሽ ൌ Ͳ (15) ‫ݑ‬ூ P

In relation (15), indices B and I stand respectively for boundary and internal centres. Since u is a particular solution, it can be chosen so that its value is zero at all boundary centres. It then follows that ሾ‫ܣ‬ூ ሿሼ‫ݑ‬ூ௉ ሽ ൅ ሾ‫ܤ‬ሿሼܶ ே௅ ሺ‫ݑ׏‬ሻሽ ൌ Ͳ Finally, the particular integral is given by: ሼ‫ݑ‬ூ௉ ሽ ൌ ሾܷܵሿሼܶ ே௅ ሺ‫ݑ׏‬ሻሽ

(16)

Denote by ܰ௝ the outward unit normal to the boundary in the reference configuration. The nominal traction is defined as ܲ௞ ൌ ܲ௞௝ ܰ௝ . With respect to the introduced displacement partition, one has ௉ ܰ௝ ൅ ܶ௞௝ ܰ௝ ൌ ‫ݐ‬௞௖ ൅ ‫ݐ‬௞௉ ൅ ܶ௞ . Adopting the radial point interpolation for the displacement ܲ௞ ൌ ‫ݐ‬௞௖ ൅ ߪ௞௝ field, after some algebraic manipulations, the following vectors of traction like densities can be constructed: ሼ‫ ݐ‬௉ ሽ ൌ ሾ‫ܭ‬ଵ ሿሼܶ ே௅ ሺ‫ݑ׏‬ሻሽ and ሼܶሽ ൌ ሾ‫ܭ‬ଶ ሿሼܶ ே௅ ሺ‫ݑ׏‬ሻሽ (17) 2.2.3 Final equations: Equation (9) is now rewritten as ሾ‫ܪ‬ሿሼ‫ ݑ‬െ ‫ݑ‬௉ ሽ ൌ ሾ‫ܩ‬ሿሼܲ െ ‫ ݐ‬௉ െ ܶሽ Using relations (16) and (17) a system of equations of the following form is obtained: ሾ‫ܪ‬ሿሼ‫ݑ‬ሽ െ ሾܴሿሼܶ ே௅ ሺ‫ݑ׏‬ሻሽ ൌ ሾ‫ܩ‬ሿሼܲሽ

(18)

Note that all the matrices in the nonlinear system of equations (18) are computed only once and stored. In the following, the case of follower loads is not considered. Usually the solution to a geometrically nonlinear mechanical problem is obtained by an incremental loading strategy. Let ο‫ ݑ‬and οܲ be the displacement and traction increments between two loading steps. The corresponding incremental form of eq. (18) reads: ሾ‫ܪ‬ሿሼο‫ݑ‬ሽ െ ሾܴሿ൛ܶ ே௅ ൫‫ݑ׏‬௡ ൅ ‫׏‬ሺ‫ݑ׏‬ሻ൯ െ ܶ ே௅ ሺ‫ݑ׏‬௡ ሻൟ ൌ ሾ‫ܩ‬ሿሼοܲሽ (19) In relation (19), ‫ݑ‬௡ stands for the displacement obtained at the preceding loading step. The nonlinear system of equations (19) should be solved by an appropriate iterative method such as the standard Newton-Raphson method or the Levenberg-Marquardt method. 3.

Numerical examples

The generalized multi-quadrics radial basis functions ܴ௜ ሺ‫ݎ‬ሻ ൌ ሺ‫ ݎ‬ଶ ൅ ܿ ଶ ሻ௤ are adopted. ‫ ݎ‬ൌ ԡ‫ ݔ‬െ ‫ݔ‬௜ ԡ is the Euclidian distance between the field point x and the collocation centre ‫ݔ‬௜ . The constants ܿ and ‫ ݍ‬are known as shape parameters. The reader is reminded that these parameters are known to affect the accuracy of the solution in methods such as the LRPIM [7,8], the analog equation method [20]. Accordingly, results from these methods are usually presented with the associated optimal values of the shape parameters. As already mentioned, three types of material models are considered in this work: a Saint-Venant solid, a neo-Hookean rubber and a Mooney-Rivlin solid. Only few numerical examples are presented below since the number of pages is limited. 3.1 Uni-axial loading. Let us first consider the case of uniaxial loading. This academic case is considered in order to assess the effectiveness of the proposed solution method and to analyse the impact of the multi-quadrics shape parameters on the solution accuracy. A unit cube is simply supported at its lower end and is uniformly loaded at its upper surface in the third direction. The other faces of the specimen are free of traction.

72


Assuming that either ܲଷଷ or ‫ܨ‬ଷଷ ൌ ߣଷ is known, the analytical solutions can be calculated from the following relations where ߤ and ߥ are respectively the shear modulus and the Poisson ratio (ߣଵ ൌ ‫ܨ‬ଵଵ ሻ: Model 1: ‫ܧ‬ ܲଷଷ ൌ ߣଷ ሺߣଶଷ െ ͳሻ ʹ ఔ ሺߣଵଶ െ ͳሻ ൌ െ ሺߣଶଷ െ ͳሻ ଶሺଵାఔሻఓ Model 2: ଶఔ ௌ ௌ ଷఔ ݈݊ ቀͳ െ ఓయయ ቁ ൅ ቀͳ െ ఓయయ ቁ ߣଶଷ ൅ ଵିଶఔ ݈݊ሺߣଶଷ ሻ ൌ ͳ ଵିଶఔ ߣଵଶ ൌ ൬ͳ െ

ܵଷଷ ଶ ൰ ߣଷ ߤ

Model 3: ߣଶଷ ܵଷଷ ൌ ‫ܥ‬ଵ ሺߣଶଷ െ ߣଵଶ ሻ ൅ ‫ܥ‬ଶ ሺߣସଷ െ ߣଵସ ሻ ሺ‫ܥ‬ଵ ൅ ‫ܥ‬ଶ ߣଵଶ ሻߣଶଵ ൅ ‫ܥ‬ଷ ൌ Ͳ The material parameters considered for the calculations are: ߤ = 5MPa, ߥ = 0.2. The material parameters used for model 3 are A10 = 0, A01 = 2.5 MPa and K = 20/3. For the results presented hereafter, the boundary of the unit cube is subdivided into 24 nine nodes elements, that is 4 elements per face. The boundary nodes are supplemented with 27 internal collocation centres. We consider two loading states which produce an axial elongation of 20% and 90 %. That is ‫ܨ‬ଷଷ ൌ ߣଷ ൌ ͳǤʹܽ݊݀ͳǤͻ. The corresponding nominal stresses applied at the top surface of the cube are collected in table 2 below. ߣଷ 1.2 1.9

Model 1 3.168 MPa 29.753 MPa

Model 2 2.1327 MPa 7.50693 MPa

Model 3 1.77078 MPa 3.748125 MPa

Table 1: Tension load applied at the top surface of the cube In this case of uniform loading, equation (21) has been solved efficiently by the Newton-Raphson method. The iterative process is stopped when the infinity norm of the difference between two consecutive iterates is less than ߝ= 10-7. The results obtained in one loading step with the multiquadrics shape parameters q = 1.03 and c = 0.005 are practically analytical solutions. These results are undisturbed when q is varied in the range (0.5, 1.5) and c in the range (0.0005, 0.05). Adopting incremental loading, analytical results are obtained after every converged loading increment. It is concluded that the proposed approach is effective and accurate in this case of unidirectional loading. The mean number of iterations for one loading step is 10. Consider a tubular specimen with height 2 mm, an outer radius of 2 mm and an inner radius of 1mm, similar accurate results are obtained. 3.2. Bending of a prismatic bar Now, the effectiveness of the approach with regards to the loading is checked. Then we consider the problem of bending of a prismatic bar. The bar with a square cross section (1 mm × 1mm) has 10 mm length. It is clamped at one end and submitted at the other end to a tangential loading as shown in figure 1. The boundary of the bar is subdivided into 48 quadrilateral elements. The results presented hereafter are obtained with 171 regularly spaced internal collocation centres. In this case with localized region of strong geometrical nonlinearities, the Newton –Raphson method sometimes fails at a certain load increment during the loading process. In all cases presented here, the Levenberg-Marquardt method has been successful. The deformation of the mean line of the specimen is represented in figures 2a and 2b for the Saint-Venant Kirchhoff solid and the Mooney Rivlin solid. The effectiveness of the approach is assessed by comparing the results obtained with two different loading steps. As can be observed, in this case also, the method provides stable results.


73

y p A

1 mm

z

10mm

Figure 1: Geometry of the bending specimen in the initial configuration

Saint Venant Kirchhoff solid

8

Load increment 'p = 0,24 10

-3

7

150 'p

7

6

A

vertcal displacement (mm)

vertical displacement (mm)

Mooney- Rivlin solid -3 load increment 'p = 0,24 10 MPa

8

150 'p

MPa

100 'p

5 4

50 'p

3 2 1

100 'p

B

6 5 4

50 'p 3 2 1

'p

'p

0

0

0

2

4

6

8

longitudinal displacement (mm)

10

12

0

2

4

6

8

10

longitudinal displacement (mm)

Figure 3: Bending load. Displacement of the center line of : A/ a Saint Venant Kirchhoff solid, B/ a Mooney-Rivlin material under bending

4. Conclusion Consider small strain isotropic elastic solid, the conventional BEM lead to highly accurate results. In the case of geometrically nonlinear elasticity, the field boundary element method is effective. In this case, the main appeal of the BEM (reduction of dimension by 1) is tarnished to some extent. In this work, the approach called LPI-BEM, which has already proved effective and accurate for anisotropic, nonlocal and piezoelectric elasticity is extended to the case of geometrically nonlinear elasticity. The method couples conventional isotropic BEM with local radial point interpolation applied to strong form differential equations. The solution procedure uses a nonlinear displacement boundary integral equation and its implementation requires only little modifications of existing BEM code. The effectiveness and accuracy of the method is demonstrated in the case of uniform loading and bending type loading with different constitutive equations. This simple to implement approach seems robust and promising. Further investigations will be carried out on different geometries and loading states. The next step will be the consideration of material nonlinearity as well as dynamic loading. References [1] Liu GR. Meshfree methods: moving beyond the finite element method. 2nd ed. Boca Raton: CRC Press; 2010. [2] Belytschko T, Lu YY, Gu L. Element-free Galerkin methods. Int J Numer Meth Engng 1994;37:229– 56. [3] Chen J-S, Pan C, Wu C-T, Liu WK. Reproducing Kernel Particle Methods for large deformation analysis of non-linear structures. Computer Methods in Applied Mechanics and Engineering 1996;139:195–227.

12

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[4] Han ZD, Rajendran AM, Atluri SN. Meshless Local Petrov-Galerkin (MLPG) approaches for solving nonlinear problems with large deformations and rotations. Computer Modeling in Engineering and Sciences 2005;10:1. [5] Zhang X, Yao Z, Zhang Z. Application of MLPG in Large Deformation Analysis. Acta Mech Mech Sinica 2006;22:331–40. [6] Liew KM, Ng TY, Wu YC. Meshfree method for large deformation analysis–a reproducing kernel particle approach. Engineering Structures 2002;24:543–51. [7] Liu GR, Gu YT. A local radial point interpolation method (LRPIM) for free vibration analyses of 2-D solids. J Sound Vib 2001;246:29–46. [8] Liu GR, Gu YT. A meshfree method: meshfree weak-strong (MWS) form method, for 2-D solids. Comput Mech 2003;33:2–14. [9] Lee S-H, Yoon Y-C. Meshfree point collocation method for elasticity and crack problems. International Journal for Numerical Methods in Engineering 2004;61:22–48. [10] Balas J, Sladek J, Sladek V. Stress Analysis by Boundary Element Method. Elsevier 1989. [11] Bonnet M. Boundary Integral Equation Methods for Solids and Fluids. New York: John Wiley & Sons; 1999. [12] Brebbia CA, Dominguez J. Boundary Elements: An introductory course. Computational Mechanics Publications. Southampton: WIT Press; 1992. [13] Foerster A, Kuhn G. A field boundary element formulation for material nonlinear problems at finite strains. International Journal of Solids and Structures 1994;31:1777–92. [14] Al-Gahtani HJ, Altiero NJ. Application of the boundary element method to rubber-like elasticity. Applied Mathematical Modelling 1996;20:654–61. [15] Prieto I, Ibán AL, Garrido JA. 2D analysis for geometrically non-linear elastic problems using the BEM. Engineering Analysis with Boundary Elements 1999;23:247–56. [16] Nardini D, Brebbia CA. A new approach to free vibration analysis using boundary elements. Appl Math Model 1983;7:157–62. [17] Gao X-W. The radial integration method for evaluation of domain integrals with boundary-only discretization. Eng Anal Bound Elem 2002;26:905–16. [18] Chen L, Kassab AJ, Nicholson DW, Chopra MB. Generalized boundary element method for solids exhibiting nonhomogeneities. Engineering Analysis with Boundary Elements 2001;25:407–22. [19] Zhu T, Zhang JD, Atluri SN. A local boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approach. Comput Mech 1998;21:223–35. [20] Katsikadelis J., Nerantzaki M. The boundary element method for nonlinear problems. Engineering Analysis with Boundary Elements 1999;23:365–73. [21] Kouitat Njiwa R. Isotropic-BEM coupled with a local point interpolation method for the solution of 3Danisotropic elasticity problems. Engineering Analysis with Boundary Elements 2011;35:611–5. [22] Thurieau N, Kouitat Njiwa R, Taghite M. A simple solution procedure to 3D-piezoelectric problems: Isotropic BEM coupled with a point collocation method. Engineering Analysis with Boundary Elements 2012;36:1513–21.


75

Multidomain BEM for crack analysis in stiffened anisotropic plates D. Flauto, I. Benedetti, A. Milazzo1,a 1

Dipartimento di Ingegneria Civile, Ambientale, Aerospaziale e dei Materiali - DICAM Università degli Studi di Palermo Viale delle Scienze,Edificio 8, 90128, Palermo, Italy a

[email protected]

Keywords: Bonded stiffener; fracture mechanics; multi-domain method; anisotropic panels

Abstract. The present paper is concerned with the application of a boundary element model for the analysis of cracks in stiffened composite panels. The panel stiffeners are reduced to equivalent strips and the multidomain technique is used to model panel zones presenting different properties (skin and stiffeners equivalent strip). Also the crack is modeled exploiting the multidomain formulation. Evaluation of stress intensity factors is performed for representative problems. Introduction Aircraft structures are typically built with thin sheets reinforced by stiffeners, i.e. stiffened panels. The stiffeners can be mechanically attached to the skin either by rivets or bolts, or adhesively bonded to it, or machined to form a single integral structure. This concept allows to conjugate weight and strength requirements. In cracked panels, stiffeners influence crack behavior: a crack in a stiffened panel propagates at the same speed of a crack contained in a non-stiffened panel, if both experience the same SIF. The stiffening elements thus influence the behavior of damaged panels by attenuating the stress field at the crack tip. Stiffeners and their attachments are then designed to optimize the damage tolerance performance of the aircraft structure. The presence of stiffening elements reduces the value of the stress intensity factor, since part of the load is transferred to the stiffeners. This circumstance is expressed by introducing a reduction factor in the relationship that describes the trend of the stress intensity factor as function of the crack length for a panel of finite size. The determination of the stress intensity factor is therefore fundamental because it is the essential parameter controlling crack behavior. Fatigue crack propagation is the most important crack growth mechanism, which is actually described by laws depending on the SIF [1,2]. Numerical analysis is a viable alternative to full-scale testing for assessing the effectiveness of damage tolerance design. Both the Finite Element Method (FEM) [3,4] and Boundary Element Method (BEM) can be used to perform analyses based on LEFM concepts. The BEM is a well-established powerful alternative to the FEM, particularly for problems involving cracks. Its most attractive features are: (i) reduction of the problem dimensionality; (ii) capability to model more efficiently, in comparison with the FEM, high stress gradients near the crack tip. For crack problems, different BEM solutions have been proposed by using both single- and multi-domain approaches [5]. In particular, for stiffened plates, Salgado [6] proposed a dual boundary formulation where the stiffener are modeled as beams. In the present paper, a model for the analysis of cracked stiffened panel is presented based on the multidomain boundary element technique. The stiffeners are modeled using an equivalent strip, so as to build a variable stiffness effective panel consisting of a set of homogeneous domains. Thus, for the analysis, the panel is divided into different regions, referring respectively to the areas concerned with the skin only and areas concerned the equivalent stiffener strip, whose properties are a suitable combination of those of the panel and of the stiffener. Multidomain Boundary Element Method. The Multi-domain Boundary Element Method is based on the displacement boundary integral equation [7] that, in absence of body forces, reads as

76


࡯௜௝ ሺ૆ሻ࢛௝ ሺ૆ሻ ൅ න ࢀ௜௝ ሺ૆ǡ ࢞ሻ࢛௝ ሺ࢞ሻ ݀ܵ ൌ න ࢁ௜௝ ሺ૆ǡ ࢞ሻ࢚௝ ሺ࢞ሻ ݀ܵ ௌ

(1)

ௌ

where ࢁ௜௝ and ࢀ௜௝ are the fundamental solution kernels, whereas ࡯௜௝ ሺ૆ሻ is the free term coefficient. For an anisotropic 2D problem, defined in the ‫ݔ‬ଵ ‫ݔ‬ଶ plane, the fundamental solutions kernels can be written as [8] ଶ

ͳ ஞ ୶ ࢁ௜௝ ሺ૆ǡ ࢞ሻ ൌ െ ࣬݁ ൥ ෍ ࡭௝ெ ࡽெ௜ ݈݊ ቀ‫ݖ‬ெ െ ‫ݖ‬ெ ቁ൩ ߨ ெୀଵ ଶ

(2)

ͳ ߤெ ݊ଵ െ ݊ଶ ࢀ௜௝ ሺ૆ǡ ࢞ሻ ൌ ࣬݁ ൥ ෍ ࡸ௝ெ ࡽெ௜ ൩ ஞ ߨ ‫ ୶ݖ‬െ ‫ݖ‬ ெ

ெୀଵ

ெ

where ࢞ and ૆ are the observation and source points, respectively, ݊௜ are the direction cosines of the boundary outer normal at the observation point and ‫ݖ‬ெఈ ൌ ߙͳ ൅ ߤெ ߙʹ. The coefficients ߤெand the columns of the matrices A and L, can be evaluated solving the following eigenvalue problem [9] ିଵ ் ࡾ ቂ െࢀ ࡾࢀିଵ ࡾ் െ ࡼ

ࢀିଵ ቃ ൜࡭௞ ൠ ൌ ߤ ൜࡭௞ ൠ ௞ ࡸ െࡾࢀିଵ ࡸ௞ ௞

(3)

where ‫ ܂‬ൌ ࡵଶ ் ࡱࡵଶ Ǣ

‫ ۾‬ൌ ࡵଵ ் ࡱࡵଵ Ǣ

‫ ܀‬ൌ ࡵଵ ் ࡱࡵଶ

(4)

Being E the elasticity matrix, whereas I1 and I2 are defined as ͳ ࡵଵ ൌ ൥Ͳ Ͳ

Ͳ Ͳ Ͳ Ͳ൩, ࡵଶ ൌ ൥Ͳ ͳ൩ ͳ ͳ Ͳ

(5)

Finally, the matrix Q is obtained from ഥ ሻିଵ ሿିଵ Ǣ ࡽ ൌ ࡭ିଵ ሾࡹିଵ ൅ ሺࡹ

ࡹ ൌ ݅ࡹࡸିଵ

(6)

For applying the multi-domain technique, the initial domain is subdivided into a finite number N of homogeneous sub-regions, for which the standard collocation technique for Eq.(1) provides the following algebraic system ே

ே

෍ ࡴ௜௜௝ ઢ௜௜௝ ൌ ෍ ࡳ௜௜௝ ࢀ௜௜௝

௝ୀଵ

ሺ݅ ൌ ͳǡʹǡ Ǥ Ǥ ǡ ܰሻ

(7)

௝ୀଵ

where ઢ௜௜௝ and ࢀ௜௜௝ are vectors containing the nodal displacements and tractions. In Eq. (7) the following notation is employed: superscript i denotes quantities referring to the i-th sub-region whereas subscripts ij refer to quantities pertaining the nodes belonging to the interface between the i-th and j-th sub-regions [9], with the convention that the external boundary of the i-th subdomain is denoted by the ii subscripts. The original domain solution is then obtained by forcing displacement continuity and traction equilibrium at the interface between two adjacent sub-regions ௝

௝

ઢ௜௜௝ ൌ ઢ௜௝ Ǣ ࢀ௜௜௝ ൌ െࢀ௜௝ Ǣ ݅ ൌ ͳǡʹǡ Ǥ Ǥ ǡ ܰ െ ͳǢ ݆ ൌ ݅ ൅ ͳǡ Ǥ Ǥ ǡ ܰ

(8)

Numerical implementation. In the present work, the multi-domain boundary element method has been implemented with the following fundamental features. Quadratic continuous boundary elements are employed for the subregions discretization. The influence coefficients are computed by using standard Gauss quadrature for regular integrals, whereas the transformation of variables technique is used for weakly singular integrations. Finally, the numerical technique proposed by Kutt [10] and Aliabadi [7], was used for strongly singular integrations.


77

Validation of the BE model. To validate the proposed modeling approach, it is firstly necessary to assess the assumptions made about the use of panel zones with equivalent properties between the panel and the stiffeners. To this aim, a FE model is used and its results are compared with those obtained by the present BEM. The panel configuration is shown in Fig.1, where a square aluminum plate with side length W=1m is reinforced through steel stiffeners.

Fig. 1: Finite Element Model used for validation purposes.

In the FEM solution, the stiffeners are modeled as one-dimensional beam elements and the panel is discretized by using two-dimensional shell elements. For the BEM solution, the multi-domain scheme shown in Fig.2 has been adopted, where domains labeled as 1, 3, 4 and 6 exhibit the aluminum skin geometric and material properties. On the other hand, for the domains labeled as 2 and 5, equivalent geometric and mechanical properties are attributed according to the scheme of Fig.3, where p stands for panel, s stands for stiffener and e stands for equivalent strip.

Fig. 2: Boundary Element Model domain subdivision.

Fig. 3: Geometrical and mechanical property for stiffened panel.

78


The boundary conditions used for the analysis are ‫ݑ‬ത௫ ሺ‫ ݕ‬ൌ Ͳሻ ൌ ͲǢ‫ݑ‬ത௫ ሺ‫ ݕ‬ൌ ܹሻ ൌ ͲǢ ‫ݑ‬ത௬ ሺ‫ ݕ‬ൌ Ͳሻ ൌ െͳǢ‫ݑ‬ത௬ ሺ‫ ݕ‬ൌ ܹሻ ൌ ͳ Fig.4 shows the comparison between normal stresses at the plate midline, between the finite element model and the multi-domain boundary elements model. The results obtained are in satisfactory agreement and this support the underlying idea of studying the behavior of stiffened panel with a two-dimensional model, thereby taking into account the influence of stiffeners on the fracture mechanics.

Fig. 4: Normal Stress at y/W=0.5.

Numerical results The fracture behavior of stiffened panels in a LEFM framework is evaluated by computing the reduction factor Cr with respect to the unstiffened panel stress intensity factor [11]. By using the multi-domain approach, cracks are modeled by introducing artificial interfaces, which connect the crack tips to the external boundary. By so doing the original cracked domain is divided in subregions whose boundaries contain the crack surfaces over which displacements are unknown. The multidomain boundary element solution thus directly supply the crack tip tractions and displacement, which can be used for the appraisal of the stress intensity factor. In the present work, for the determination of the SIF, the modified crack closure integral (MCCI) technique is used [12]. With reference to Fig.5, for quadratic elements, the displacement and traction around the crack tip are given by

v t

v j 1 0.5v j 2[ (0.5v j 2 v j 1 )[ 2

t j 1 0.5(t j 2 t j )[ [0.5(t j 2 t j ) t j 1 ][ 2

(9)

Accordingly, the crack closure work takes the form l

W

1 vtdx [v j 1 (c1t j c 2 t j 1 c3 t j 2 ) v j 2 (c 4 t j c5 t j 1 c 6 t j 2 )] / 60 2 ³0

(10)

where the values of the coefficients ܿ௜ can be found in Ref [12]. This work corresponds to the Energy Release Rate G, from which the SIF can be directly calculated.


79

Fig. 5 Cracked quadratic elements used for crack modelling.

In the following, some representative results for stiffened isotropic and anisotropic panels are presented. The isotropic panel case refer to an aluminium plate with steel stiffener, whereas the anisotropic panel consists of skin and stiffener made of a graphite-epoxy T300/914C laminate with the ሾേͶͷହ ȀͻͲସ ȀേͶͷȀͲଶ Ȁ േͶͷሿ௦ stacking sequence. Fig. 6 shows the trends of the SIF versus the crack length for both the cases of stiffened and unstiffened panel.

b. Anisotropic Case

a. Isotropic Case Fig. 6 SIF versus crack length.

For the evaluation of Cr several analyses were carried out for different values of the stiffener's stiffness. For the isotropic case the stiffener stiffness was varied by considering different values of its Young modulus (E=70, 93, 110 and 230 GPa). For the anisotropic analysis, different laminate stacking sequences have been considered for the stiffener with the extensional stiffness matrix coefficients (in GPa) as given in Table 1. The obtained results are shown in Fig. 7. A11

A12

A13

A21

A22

A23

A31

A32

A33

skin Stiffeners Ae1 Stiffeners Ae2

53,28 305,72 230,13

20,79 23,99 32,62

0 0 0

20,79 23,99 32,62

53,29 62,85 84,71

0 0 0

0 0 0

0 0 0

22,22 26,53 35,48

Stiffeners Ae3

192,33

37,1

0

37,1

96,64

0

0

0

39,96

Stiffeners Ae4

154,53

41,58

0

41,58

106,57

0

0

0

44,45

Table 1 Extensional stiffness matrix for skin and stiffeners

80


b. Anisotropic Case

a. Isotropic Case Fig.7: Cr versus crack length.

Conclusions A multi-domain boundary element solution for the analysis of anisotropic cracked stiffened panels has been presented. It is based on modelling of the stiffeners through a mechanically equivalent strip, which enables the use of 2D boundary element formulations. The obtained results confirm the ability of the approach in the assessment of the effect of the reinforcement on the crack behavior. References [1] P.C. Paris, F. Erdogan, (1963). A critical analysis of crack propagation laws, Journal of basic engineering, 85(4), 528-533. [2] F. Erdogan , G.C. Sih, (1963), On the crack extension in plates under plane loading and transverse shear, Journal of basic engineering, 85(4), 519-525. [3] H. Vlieger, (1973), The residual strength characteristics of stiffened panels containing fatigue cracks, Engineering Fracture Mechanics, 5(2), 447-477. [4] S.V. Shkarayev, E.T. Mover Jr, (1987), Edge cracks in stiffened plates, Engineering Fracture Mechanics, 27(2), 127-134. [5] Aliabadi, M.H., (1997), Boundary element formulations in fracture mechanics, Applied Mechanics Review, 50(2), 83-96. [6] N.K. Salgado, M.H. Aliabadi, (1998), Boundary Element Analysis of Fatigue Crack Propagation in Stiffened Panels, Journal of Aircraft, 35(1), 122-130. [7] Aliabadi M.H., (2002), The boundary element methods. Applications in solids and structures, Vol.2. Chichester, UK, John Wiley & Sons, Ltd. [8] F. Garcia-Sanchez, Andrés Sáez, J. Dominguez, (2005), Anisotropic and piezoelectric materials fracture analysis by BEM, Computers & Structures, 83(10), 804-820. [9] G. Davı,̀ A. Milazzo, (2001), Multidomain boundary integral formulation for piezoelectric materials fracture mechanics, International Journal of Solids and Structures, 38(40–41), 7065-7078. [10] H.R. Kutt, (1975), The numerical evaluation of principal value integrals by finite-part integration, Numerische Mathematik, 24(3), 205-210. [11] Broek, D., (1984), Elementary Engineering Fracture Mechanics, Noordhoff International Publishing [12] Rybicki E, Kanninen M., (1977), A finite element calculation of stress intensity factors by a modified crack closure integral, Engineering Fracture Mechanics, 9(4), 931–938.


81

Biomagnetic fluid flow in a channel under the effect of a uniform localized magnetic field Ö. Türk1 , Canan Bozkaya2 and M. Tezer-Sezgin1,2 1 Institute of Applied Mathematics, Middle East Technical University, 06800, Ankara, Turkey, e-mail: [email protected] 2 Department of Mathematics, Middle East Technical University, 06800, Ankara, Turkey,

Keywords: DRBEM, FEM, Biomagnetic fluid flow.

Abstract. In this study, the biomagnetic fluid flow between parallel plates under the influence of nodal or uniform externally applied magnetic fields is given numerically. The flow is steady, two-dimensional and laminar, and the biomagnetic fluid is assumed to be incompressible, viscous and electrically conducting. The plates are kept at different constant temperatures. The flow enters the channel with parabolic and linear profiles for the velocity and temperature, respectively. No-slip boundary conditions for the velocity are imposed on the plates. At the exit, all the unknowns satisfy zero normal derivative conditions. The mathematical modeling of the problem results in a coupled nonlinear system of partial differential equations in terms of the velocity, pressure and the temperature of the biofluid. The equations are solved in terms of stream function-vorticity and temperature using both dual reciprocity boundary element method (DRBEM) and the finite element method (FEM) for several values of characteristic flow parameters such as Reynolds and magnetic numbers. Treatment of nonlinear terms as inhomogeneity enables one to use only the fundamental solution of the Laplace equation in DRBEM. The discretization of only the boundary of the region is the main advantage of DRBEM giving small algebraic systems to be solved at a small expense. Finite element method on the other hand, is capable of giving more accurate results by discretizing the region affected by the magnetic field very finely, but it results in large sized algebraic systems requiring high computational cost. The results indicate that the flow is appreciably affected with the presence of a nodal or uniform magnetic field in terms of vortices. As the magnetic numbers increase, two circular vortices in opposite directions are formed at the area of the end points of the magnet for both the stream function and the vorticity. The temperature also increases at these areas.

Introduction Biomagnetic fluid dynamics (BFD) investigates the flow of biological fluids under the effect of magnetic fields. It has a wide range of applications in bioengineering and medical sciences, such as development of the magnetic devices for cell separation, use of magnetic fluids to produce blood flow stasis during surgeries, magnetic drug targeting, accelerating blood flow and provocation of occlusion of tumor feeding vessels tumors. Blood is considered to be a typical biomagnetic fluid due to the interaction of intercellular proteins, membrane and the hemoglobin. The blood flow plays an important role in regulating the temperature distribution between tissues and delivering oxygen and nutrients to the cells in arteries. The extended BFD model [1], takes both magnetization consistent with ferrohydrodynamics (FHD) principles and the Lorentz force, due to the induced electric current of magnetohydrodynamics (MHD) principles into account. Since the governing equations of the BFD model are highly nonlinear, analytical solutions are not available. Thus, it is important to devise suitable mathematical modeling and numerical simulations providing insights into the interaction between blood flow and magnetic field driven in a channel. There are several numerical techniques such as finite difference method (FDM) [1, 2], finite volume method (FVM) [3, 4], and FEM producing physical numerical results in arteries. Tezer-Sezgin et al. [5, 6] investigated the 2D biomagnetic fluid flow between parallel plates subject to a nodal magnetic source using FEM for both steady and unsteady flows in straight and multi-irregularly stenosed channels. DRBEM approach is also given for the solution of biomagnetic fluid flow in a straight channel [5]. The present study adapts the technique of Türk et.al. [5] to investigate numerically the effect of the intensity and the type (either uniform or nodal) of the magnetic field on the biomagnetic fluid flow through a straight channel.

82


Basic equations The steady, two-dimensional flow of a biomagnetic fluid is considered between parallel plates. The governing equations are given in stream function ψ , vorticity w and temperature T as ∇2 ψ = −w

(1)

∂w ∂ψ ∂w ∂ψ ∂H ∂T − ) − MnF ReH − MnM H 2 2 ∂x ∂y ∂y ∂x ∂x ∂y ∂y ∂T ∂ψ ∂T ∂ψ ∂H ∂ψ ∂ψ 2 = PrRe − MnF PrReEcH(ε + T ) − − MnM PrEcH 2 ∂x ∂y ∂y ∂x ∂x ∂y ∂y 2 2 ∂ 2ψ ∂ 2ψ ∂ 2ψ −PrEc − 2 +4 ∂ y2 ∂x ∂ x∂ y ∇2 w =

∇2 T

∂ 2ψ

Re(

(2)

(3)

where Re, Pr, Ec and ε are the nondimensional Reynolds, Prandtl, Eckert and temperature numbers, respectively. The magnetic numbers arising from ferrohydrodynamics and magnetohydrodynamics are denoted by MnF and MnM . The nondimensional intensity H of the nodal and uniform magnetic fields are defined respectively as, [7, 8], |b| H(x, y) = (nodal) , (x − a)2 + (y − b)2

1 H(x, y) = (tanh[a1 (x − x1 )] − tanh[a2 (x − x2 )]) (uniform) 2

(4)

where (a, b) is the point where the nodal magnetic source is located, whereas (x1 , 0), (x2 , 0) are the points between of which the uniform magnetic field is applied. The coefficients a1 and a2 determine the magnetic field gradient at the points x1 and x2 , respectively. Flow enters the channel with parabolic and linear profiles for the velocity and temperature, respectively. No-slip boundary conditions for the velocity are imposed on the plates, and the plates are kept at different constant temperatures. At the exit, all the unknowns obey exit condition. Hence, the boundary conditions are defined as Inflow Outflow Upper plate Lower plate

(x = 0, 0 ≤ y ≤ 1), (x = 18, 0 ≤ y ≤ 1), (y = 1, 0 ≤ x ≤ 18), (y = 0, 0 ≤ x ≤ 18),

ψ = 2y2 − (4/3)y3 , ∂ ψ /∂ x = 0, ψ = 2/3, ψ = 0,

T = y, ∂ T /∂ x = 0, T = 1, T = 0.

w = 8y − 4, ∂ w/∂ x = 0,

(5)

Numerical Methods DRBEM formulation of steady biomagnetic fluid flow The DRBEM aims to transform the governing equations of the problem into boundary integral equations by using the fundamental solution of the Laplace equation. The terms except Laplacian will be treated as inhomogeneity, [9]. Thus, the equations in system (1)-(3) are weighted with the two-dimensional fundamental solution of Laplace equation, u∗ = 1/2π ln(1/r). Then, the application of the Green’s second identity results in, [9], ci ψi +

Γ

(q∗ ψ − u∗

∂ψ )dΓ = − ∂n

Ω

(−w)u∗ dΩ

(6)

∂w ∂ψ ∂w ∂ψ ∂H ∂T ∂ 2ψ − ) − MnF ReH − MnM H 2 2 u∗ dΩ ∂x ∂y ∂y ∂x ∂x ∂y ∂y ∂ T ∂ T ∂ ψ ∂ T ∂ ψ ∂H ∂ψ ci Ti + (q∗ T − u∗ PrRe( )dΓ = − − ) − MnF PrReEcH(ε + T ) ∂n ∂x ∂y ∂y ∂x ∂x ∂y Γ Ω

ci wi +

Γ

(q∗ w − u∗

∂w )dΓ = − ∂n

Ω

Re(

−MnM PrEcH 2 (

∂ψ 2 ∂ 2ψ ∂ 2ψ ∂ 2ψ 2 ) − PrEc(( 2 − 2 )2 + 4( ) ) u∗ dΩ ∂y ∂y ∂x ∂ x∂ y

(7)

(8)

where q∗ = ∂ u∗ /∂ n, Γ is the boundary of the domain Ω and the subscript i denotes the source point. The constant ci = θi /2π with the internal angle θi at the source point.


83

The domain integrals on the right hand side of Equations (6), (7) and (8) are transformed into boundary N+L N+L integrals by the use of the approximations ∑N+L j=1 α j f j (x, y), ∑ j=1 β j f j (x, y) and ∑ j=1 γ j f j (x, y), respectively. The radial basis functions f j (x, y) are linked with the particular solutions uˆ j to the equation ∇2 uˆ j = f j , [9]. The coefficients α j , β j and γ j are undetermined constants and the numbers of the boundary and the internal nodes are denoted by N and L, respectively. Now, the right hand sides of Equations (6)-(8) also involve the multiplication of the Laplace operator with the fundamental solution u∗ , which can be treated in a similar manner by the use of DRBEM, [9], to obtain the boundary only integrals. The use of constant elements for the discretization of the boundary leads to the corresponding matrix-vector form of Equations (6)-(8) ∂ψ ˆ −1 {−w} , H¯ ψ − G = (H¯ Uˆ − GQ)F ∂n 2 ˆ −1 Re( ∂ w ∂ ψ − ∂ w ∂ ψ ) + MnF ReH ∂ H ∂ T − MnM H 2 ∂ ψ ¯ − G ∂ w ) = (H¯ Uˆ − GQ)F (Hw 2 ∂n ∂x ∂y ∂y ∂x ∂x ∂y ∂y

and ¯ −G (HT

(9) (10)

∂T ˆ −1 PrRe( ∂ T ∂ ψ − ∂ T ∂ ψ ) − MnF PrReEcH(ε + T ) ∂ H ∂ ψ ) = (H¯ Uˆ − GQ)F ∂n ∂x ∂y ∂y ∂x ∂x ∂y (11)

∂ψ 2 ∂ 2ψ ∂ 2ψ ∂ 2ψ 2 −MnM PrEcH ( ) − PrEc(( 2 − 2 )2 + 4( ) ) ∂y ∂y ∂x ∂ x∂ y 2

where the components of the matrices H¯ and G are calculated by integrating q∗ and u∗ , respectively, over each boundary element [9]. The (N + L) × (N + L) matrix F contains the coordinate functions f j ’s (which are taken as polynomial type in our simulations) as columns. H is the (N + L) × 1 vector obtained collocating H(x, y) at (N + L) points. In order to solve the resulting DRBEM equations, which are nonlinear and coupled, we need to introduce an iterative process similar to the work of Bozkaya and Tezer [10] with initial estimates of vorticity and temperature. This process will reduce Equations (9)-(11) to a set of linear algebraic equations in each iteration. On the other hand, in each iteration the required space derivatives of the unknowns ψ , w and T are obtained by using coordinate matrix F as [9] ∂ψ ∂ F −1 = F ψ, ∂x ∂x

∂ψ ∂ F −1 = F ψ, ∂y ∂y

∂ w ∂ F −1 = F w, ∂x ∂x

∂ w ∂ F −1 = F w, ∂y ∂y

∂T ∂ F −1 = F T, ∂x ∂x

∂T ∂ F −1 = F T. ∂y ∂y

This iterative procedure will stop when a preassigned tolerance is reached between two successive iterations.

FEM formulation of steady biomagnetic fluid flow The FEM formulation of Equations (1)-(3) is obtained first obtaining the weak form. The weak form is developed by multiplying Equations (1)-(3) with the weight functions ω1 , ω2 and ω3 which are assumed to be twice differentiable with respect to x and y, and taken as equal to the shape functions used for an element approximation in Galerkin approach [11]. Then, the application of the divergence theorem results in, −

Ω

∂ ω1 ∂ ψ ∂ ω1 ∂ ψ + ∂x ∂x ∂y ∂y

dΩ +

Ω

ω1 wdΩ +

∂Ω

ω1

∂ψ ds = 0 ∂n

∂ ω2 ∂ w ∂ ω2 ∂ w ∂ψ ∂w ∂w ∂ψ dΩ + Re ω2 dΩ + − ∂x ∂x ∂y ∂y ∂x ∂y ∂y ∂x Ω Ω ∂H ∂T ∂H ∂T ∂ ∂ψ ∂w −MnF Re ω2 H ω2 dΩ − MnM ω2 H2 dΩ − − ds = 0 ∂x ∂y ∂y ∂x ∂y ∂y ∂n Ω Ω ∂Ω

(12)

(13)

84


1 Pr

Ω

∂ ω 3 ∂ T ∂ ω3 ∂ T + ∂x ∂x ∂y ∂y

dΩ + Re

Ω

w3

∂T ∂ψ ∂T ∂ψ − ∂x ∂y ∂y ∂x

dΩ

∂H ∂ψ ∂H ∂ψ ∂ψ 2 dΩ − MnM Ec ω3 H 2 dΩ − ∂x ∂y ∂y ∂x ∂y Ω Ω

2 2 2 ∂ 2ψ ∂ 2ψ ∂ ψ ∂T 1 −Ec ω3 ω3 dΩ − − 2 +4 ds = 0 2 ∂y ∂x ∂ x∂ y Pr ∂Ω ∂n Ω −MnF ReEc

where H(x, y) and

ω3 (ε + T )H

(14)

∂H ∂H , are known values from the definition (4). The region is discretized by using 6∂x ∂y

nodal triangular elements and quadratic shape functions are used in the approximation of ψ , w and T over each element. Assembly procedure for all elements results in matrix-vector system of equations [K] {ψ } = [M] {w}

(15)

[K] {w} + Re [A] {w} = MnF Re {F3 } + MnM {F4 }

(16)

1 (17) [K] {T } + Re [A] {T } − MnF ReEc [A3 ] {T } = MnF ReEc ε {F1 } + MnM Ec {F5 } + Ec {F2 } Pr where the matrices [K] and [M] and are the stiffness and mass matrices, respectively. The matrix [A] contains the convective terms and [A3 ] involves magnetic field intensity H. The right hand side vectors {Fk }, k = 1, 2, 3, 4, are computed from previously obtained ψ , w and T values. In the computations of the element matrices and vectors, the integrations are carried out using an isoparametric interpolation and Gaussian quadrature. The equations are solved iteratively in stream function-vorticitytemperature form, and the missing vorticity boundary values on the plates are calculated using Taylor series expansion of the stream function with four inner values.

Results and discussion The steady biomagnetic fluid flow problem in a channel under the influence of a magnetic field is solved bye two effective numerical techniques, DRBEM and FEM. The flow is subject to uniform localized magnetic field and a magnetic field generated by a point source where in the definitions (4), (a, b) = (5.5, −0.05), x1 = 3, x2 = 8 and a1 = a2 = 5 are taken. The obtained numerical results are presented including a comparison of these two methods. A realistic blood flow case is considered, and the characteristic flow parameters are taken as Pr = 25, ε = 9, and Ec = 7.43 × 10−7 . Numerical tests are carried out for different combinations of Re, MnF and MnM . The FEM solutions are obtained by using 5000 quadratic triangular elements, whereas in DRBEM solutions 380 constant boundary elements are used. Stream function

Stream function

Vorticity

Vorticity

Temperature

Temperature

(a)

(b)

Figure 1: Streamlines, vorticity contours and isotherms for Re = 50, MnF = 11.39, MnM = 0.006: (a) FEM, (b) DRBEM. The FEM and DRBEM solutions to the system (1)-(3) are compared in Figure 1, (a) FEM and (b) DRBEM, under the effect of the uniform magnetic field. The stream function-vorticity-temperature contours are presented where Re = 50, MnF = 11.39 and MnM = 0.006. The presence of the localized uniform magnetic source leads


85

to the formation of two vortices in the opposite directions around x1 and x2 for stream function and temperature. Vorticity contours are also significantly distorted at the end points of the uniform magnetic source as expected. In Figure 2, (a) FEM and (b) DRBEM solutions are illustrated in terms of stream function-vorticitytemperature contours where the magnetic field is generated from the nodal source for Re = 100, MnF = 113.93 and MnM = 0.63. It is observed that, the nodal magnetic source causes a formation of single vortex in all variables. When Figure 1 and Figure 2 are compared, it is observed that the effect of the localized uniform magnetic field is more pronounced. In both cases, the effect of the magnetic field vanishes downstream the magnetic source area, and the flow regains its inlet fully developed profile. The obtained DRBEM and FEM results are in good agreement generally on the whole channel including the region around the magnetic source. However, the actual advantage of DRBEM lies in obtaining solutions with small number of elements compared to domain discretization methods. Stream function

Stream function

Vorticity

Vorticity

Temperature

Temperature

(a)

(b)

Figure 2: Streamlines, vorticity contours and isotherms for Re = 100, MnF = 113.93, MnM = 0.63: (a) FEM, (b) DRBEM.

The effect of the increase in the magnetic numbers is investigated for the localized uniform magnetic field case, and the results are illustrated in Figure 3. The solutions in terms of streamlines, vorticity contours and isotherms are obtained using FEM, when Re = 300 for the sets of magnetic numbers (a) MnF = 11.39, MnM = 0.006, and (b) MnF = 113.93, MnM = 0.63. It is observed that, the flow is shifted towards the upper plate due to the circulation of the fluid in the area where the magnetic field is applied even for the low magnetic numbers, MnF = 11.39, MnM = 0.006. The results are in good agreement with the ones reported in [7, 8]. The length of the oppositely directed vortices decreases as the magnetic numbers are increased, and the major effect of the magnetic field is observed at x1 and x2 where the magnetic field is applied. In all of the cases considered above, the flow regains its inlet profile downstream as the magnetic field effect vanishes. Stream function

Stream function

Vorticity

Vorticity

Temperature

Temperature

(a)

(b)

Figure 3: Streamlines, vorticity contours and isotherms for Re = 300 : (a) MnF = 11.39, MnM = 0.006, (b) MnF = 113.93, MnM = 0.63.

References [1] E. E. Tzirtzilakis. A mathematical model for blood flow in magnetic field. Physics of Fluids, 17:077103– 1–077103–15, 2005. [2] N. Rusli, Hongm AKB., EH. Kasiman, AYM. Yassin, and N. Amin. Numerical computation of a twodimensional biomagnetic channel flow. International Journal of Modern Physics: Conference Series, 9:178–192, 2012.

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[3] S. Kenjeres. Numerical analysis of blood flow in realistic arteries subjected to strong non-uniform magnetic fields. International Journal of Heat and Fluid Flow, 29:752–764, 2008. [4] A. Alshare, B. Tashtoush, and H.H. El-Khalil. Computational modeling of non-newtonian blood flow through stenosed arteries in the presence of magnetic field. Journal of Biomechanical Engineering, 135(11):114503–1–114503–6, 2013. [5] M. Tezer-Sezgin, C. Bozkaya, and Ö. Türk. BEM and FEM based numerical simulations for biomagnetic fluid flow. Engineering Analysis with Boundary Elements, 37:1127–1135, 2013. [6] Ö. Türk, C. Bozkaya, and M. Tezer-Sezgin. A fem approach to biomagnetic fluid flow in multiple stenosed channels. Computers & Fluids, 97:40–51, 2014. [7] E. E. Tzirtzilakis. A simple numerical methodology for BFD problems using stream function vorticity formulation. Communications in Numerical Methods in Engineering, 24:683–700, 2008. [8] EE. Tzirtzilakis and VC. Loukopoulos. Biofluid flow in a channel under the action of a uniform localized magnetic field. Computational Mechanics, 36:360–374, 2005. [9] C. A. Brebbia, P. W. Partridge, and L. C. Wrobel. The Dual Reciprocity Boundary Element Method. Computational Mechanics Publications, Southampton, Boston, 1992. [10] C. Bozkaya and M. Tezer-Sezgin. Solution to transient Navier-stokes equations by the coupling of differential quadrature time integration scheme with dual reciprocity boundary element method. International Journal for Numerical Methods in Fluids, 59:215–234, 2009. [11] J. N. Reddy. An introduction to the finite element method. The McGraw-Hill Companies, New York, 2006.


87

Analysis of 3D anisotropic solids using fundamental solutions based on Fourier series and the Adaptive Cross Approximation method R. Q. Rodr´ıguez1,2 , C. L. Tan2 , P. Sollero1 and E. L. Albuquerque3 1

2

Faculty of Mechanical Engineering, University of Campinas, Brazil, {reneqr87, sollero}@fem.unicamp.br

Department of Mechanical and Aerospace Engineering, Carleton University, Canada, [email protected] 3

Faculty of Technology, University of Brasilia, Brazil, [email protected]

Keywords: Boundary Element Method, Anisotropy, Fourier Series, Adaptive Cross Approximation.

Abstract. The efficient evaluation of the fundamental solution for 3D general anisotropic elasticity is a subject of great interest in the BEM community due to its mathematical complexity. Recently, C.L. Tan et al. [1] have represented the algebraically explicit form of it developed by Ting and Lee [2; 3] by a computational efficient double Fourier series. The Fourier coefficients are numerically evaluated only once for a specific material and are independent of the number of field points in the BEM analysis. This work deals with the application of hierarchical matrices and low rank approximations, applying the Adaptive Cross Approximation (ACA) to treat 3D general anisotropic solids in BEM using this Green’s function based on Fourier series. The use of hierarchical format is aimed at reducing the storage requirements of the system matrices and the computational effort in the BEM analysis of large systems. Numerical examples are presented to show the successful implementation of using ACA and the formulation based on Fourier series for for BEM analysis of 3D anisotropic solids. Introduction. The evaluation of the fundamental solution and its derivatives is a necessary step in the direct BEM formulation. The efficient and accurate computation of these quantities is a concern for the case of 3D general anisotropic solids because of their mathematical complexity. Also, for very large numerical problems, the fully populated and non-symmetric system matrices impose relatively high memory requirements and high solution times. This work deals with these two issues, the fundamental anisotropic solution and the speed-up of the BEM process. Fundamental solutions for 2D and 3D isotropic elastostatics can be represented in relatively simple explicit forms. That is not the case for general anisotropic solids, particularly in 3D. Ting and Lee [2; 3] have derived exact, explicit expressions for the fundamental solution and its derivatives. However, the presence of high-order tensors and highly complex mathematical expressions for the derivatives, although straightforward to implement, may be less than ideal for very efficient computations. These solutions were first implemented into a BEM code by Tan et al. [4]. More recently, these authors have expressed this Greens function and its derivatives by a double Fourier Series [1; 5], demonstrating the much superior efficiency for their computations, and the relative simplicity of their implementation. Many research studies have, in recent years, also focused in improving the solution process in BEM [6; 7; 8]. Bebendorf [8], for example, suggested the use of purely algebraic algorithms to generate the approximation of suitable blocks of the collocation matrix from only a few entries of the original blocks. This technique is referred to as the Adaptive Cross Approximation (ACA). The ACA uses matrix hierarchization to reduce the storage requirement and the computational effort in the BEM analysis. In this paper, the application of hierarchical matrices and ACA for treating 3D anisotropic solids using BEM with the fundamental solution based on double Fourier series is illustrated. In what follows, the anisotropic fundamental solution and its derivatives are first reviewed. The use of hierarchical matrices and ACA is then briefly discussed before numerical examples are presented.

88


3D fundamental solution for displacements and its derivatives in anisotropic elasticity The 3D fundamental solution for a generally anisotropic material can be expressed in terms of the Barnett-Lothe tensor H [x], see, e.g., [2]. The Barnett-Lotte tensor could also be expressed in spherical coordinates as follows, U (r, θ , φ ) =

1 H (θ , φ ) 4πr

(1)

where r represent the radial distance between the source and the field point. As this expression depends only on the spherical angles (θ , φ ), it can be expressed in terms of the Stroh’s eigenvalues as H (θ , φ ) =

1 4 ∑ qn Γˆ (n) |T| n=0

(2)

The explicit expressions for |T|, Γˆ (n) and qn can be found in [1; 2]. Due to its periodic nature, H (θ , φ ) can be expressed as double Fourier series around θ and φ , as follows, ∞

(m,n)

λuv √

=

∞

∑ ∑

Huv (θ , φ ) =

4π 2

(m,n) i(mθ +nφ )

λuv

m=−∞n=−∞ π π 1 −π −π

e

, (u, v = 1, 2, 3), (3)

Huv (θ , φ )e−i(mθ +nφ ) dθ dφ

,

(m,n) λuv

where i is −1. can be numerically integrated by Gaussian quadrature. Thus, the fundamental solution for displacements can also be written as Uuv (r, θ , φ ) =

α α 1 (m,n) ∑ ∑ λuv ei(mθ +nφ ) , 4πr m=−α n=−α

(4)

where α is an integer number, large enough to yield the desired accuracy. Lee [9]showed that very high order tensors in the derivatives of the Greens function can be avoided if the partial differentiations are first carried out in the spherical coordinate system and then employing the chain rule. This was described and explicitly obtained in [10]. The displacement derivatives can be written in spherical coordinates as Ui j,l =

∂Ui j ∂ r ∂Ui j ∂ θ ∂Ui j ∂ φ + + . ∂ r ∂ xl ∂ θ ∂ xl ∂ φ ∂ xl

(5)

Carrying out the indicated differentiating in equation (5), the partial derivatives of Ui j can be expressed in closed form in terms of the Stroh’s eigenvalues [1; 10]. With the partial differentiations carried out on the Fourier series, substituting all previous expressions into equation (5) yields

Ui j,l

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎨ = 4πr2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

α

α

∑ ∑

m=−α n=−α α

α

∑ ∑

(m,n) i(mθ +nφ )

λi j

(m,n) i(mθ +nφ )

λi j

e

∑ ∑

λi j

m=−α n=−α α α

(m,n)

m=−α n=−α

e

− cos θ (sin φ − in cos φ ) −imsin θ sin φ − sin θ (sin φ − in cos φ ) +imcos θ sin φ

(m,n) i(mθ +nφ )

e

[− (cos φ + in sin φ )]

for l = 1 for l = 2

(6)

for l = 3

where the coefficients λuv are evaluated in advance, only once, for a given material. Similarly, 2nd order derivatives can be obtained by applying the chain rule. All explicit forms are available in [1; 10].


89

Hierarchical Matrices and ACA. The objective of applying hierarchical matrices and ACA is to reduce the storage requirements as well as to speed up the time required to complete all matrix operations. In this scheme the matrix is represented as a collection of blocks, some of which admit a particular approximated representation that can be obtained by computing only few entries from the original blocks. These special blocks are called low rank blocks. The existence of low rank approximants is based on the asymptotic smoothness of the kernel functions, i.e., on the fact that kernels Ui∗j and Ti∗j are singular only when x = y [7; 8; 11]. This is a sufficient condition for the existence of low rank approximants. A hierarchical approximation of large dense matrices arising from some generating function having diagonal singularity consist of three basic steps [12]: (i) construction of clusters; (ii) finding of possible admissible blocks; and (iii) low rank approximation of admissible blocks. The construction of clusters was implemented based on the algorithm showed in [12]. First, the ”mass and its centre of each cluster are stored, and the covariance matrix C of the cluster is obtained. The eigenvector v1 corresponding to the largest eigenvalue of C shows the direction of the largest extension of the cluster. The separation plane goes through the centre X of the cluster and is orthogonal to v1 . This algorithm is applied recursively to the ”sons until they contain less than or equal to some prescribed number nmin of elements. Next, cluster pairs which are geometrically well separated are identified and regarded as admissible cluster pairs. An appropriate admissibility criterion is the following simple geometrical condition. A pair of clusters (Clx ,Cly ) with nx > nmin and my > nmin elements is admissible if min(diam(Clx ), diam(Cly )) ≤ ηdist(Clx ,Cly ),

(7)

where η is called the admissibility parameter. This parameter influences the number of admissible blocks and the convergence speed of the adaptive approximation of low rank blocks [13]. Once the clusters are defined and all admissible blocks are detected, the Adaptive Cross Approximation (ACA) is applied to approximate by low rank these blocks. Results obtained after the low rank approximation of the admissible blocks by ACA, can be further recompressed, taking advantage of the reduced singular value decomposition (SVD) [11], thereby decreasing the storage requirements. This also serves as a good pre-conditioner for iterative numerical solvers. Some works related to this topic may be found in [11; 14; 15]. Numerical results. Two numerical examples are presented to demonstrate the proper implementation of the 3D anisotropic formulation of the Greens function based on Fourier series and the application of the ACA to the BEM analysis. In the first example, a relatively beam with a square cross-section under pressure load is analyzed. Normalized displacements and normal stresses are compared with the analytical simple beam theory solution and with the isotropic formulation. The stiffness coefficients were set to match an isotropic material to allow the proper comparison, but the analysis was carried out through the algorithm based on the anisotropic formulation. In the second example, an internally pressurized cylinder with a generally anisotropic material is analyzed. The normalized displacement is compared with the results using the finite element method (FEM) as reported by Tan et al. [1]. Moreover, the application of the ACA is tested, and solution times are compared, with the conventional anisotropic BEM formulation. Example A The proper implementation of the algorithm for the anisotropic formulation based on Fourier series is first tested. A short beam of length 2L and square cross-section of side H, where L = 5H, is subjected to a uniformly distributed pressure load on its top surface, as shown in Fig. 1(b). Both ends of the beam are constrained in the three coordinate directions. Advantage is taken of symmetry and only half the beam was modeled in the BEM analysis, as shown in Fig. 1(a). A total of 88 quadratic quadrilateral elements and 266 nodes are used to model the problem. For the Fourier series representation, α = 16 and 64 Gauss integration points were used. Normalized transverse displacements along the x3 -direction are computed and compared with the isotropic BEM formulation and simple beam theory, as shown in Fig. 2(a). The normalized direct stress is also compared for cross-sections

90


(a)

(b)

Figure 1: (a) BEM mesh (Symmetry); (b) Equivalent beam model. corresponding to x3 = 2H, 3H, 4H, 5H, as can be seen in Fig. 2(b). Even with the coarse BEM mesh, good agreement of the results is observed, with a maximum normalized displacement error of 6.3% at x3 = L. 10

40 0

Beam theory, x3 = 4H BEM anisotropic BEM isotropic Beam theory, x3 = 5H BEM anisotropic BEM isotropic

30 −10

20 −20

σ33 /P

u2 E/P L

10 −30

−40

0

Beam theory, x3 = 2H BEM anisotropic BEM isotropic Beam theory, x3 = 3H BEM anisotropic BEM isotropic

−10 −50

Beam theory BEM anisotropic formulation BEM isotropic formulation

−60

−20 −30

−70

−40 −80

0

0.2

0.4

x3 /L

0.6

0.8

−1

−0.5

1

(a)

0

2x2 /H

0.5

1

(b)

Figure 2: (a) Normalized transverse displacements; (b) Normalized normal stresses at x1 = 2H, 3H, 4H, 5H. Example B The physical problem in this second example is an internally pressurized cylinder made of an alpha-quartz crystal, as treated in [1]. However, the main objective here is to verify the ACA scheme. The problem considered is a cylinder with internal pressure, P, with radius ratio R2 /R1 = 2 and total length 2H = 8R1 , as shown in Fig. 3(c). The external circumferential surface is constrained in the radial direction, while its two ends are fixed in the x3 direction. The principal material axes of the alpha-quartz crystal are successively rotated about the global Cartesian x1 , x2 and x3 axis by 30o , 45o and 60o clockwise, respectively. These successive rotations yield a fully populated stiffness matrix, as follows, ⎤ ⎡ 111.8 14.8 −5.2 −0.3 11.0 −14.0 ⎢ 14.8 101.8 −7.6 0.4 −0.6 18.9 ⎥ ⎥ ⎢ ⎢ −5.2 −7.6 129.7 4.4 1.6 0.6 ⎥ ⎥ GPa C=⎢ (8) ⎢ −0.3 0.4 4.4 31.3 2.5 3.6 ⎥ ⎥ ⎢ ⎦ ⎣ 11.0 −0.6 1.6 2.5 37.9 1.3 −14.0

18.9

0.6

3.6

1.3

55.2

Six different meshes (96, 216, 418, 680, 960 and 1232 quadratic quadrilateral elements) were analyzed. Figure 3(a) shows the coarsest mesh, while Fig. 3(b) shows the most refined one employed.


91

00

0

00 0

0

00

0 00

0

00

00000300000

0000000300300000000

(a)

(b)

(c)

Figure 3: (a) BEM most refined mesh; (b) BEM most coarse mesh; (c) geometry of example B. The ACA accuracy is set to εc = 10−4 . Moreover, a SVD recompression is accounted in order to create a pre-conditioner matrix for the iterative solver, the generalized minimum residual method (GMRES). The recompression accuracy is set to εc = 10−2 . The maximum number of elements per cluster was set to 60 and the admissibility parameter (η) to 0.8. More details of the choice of these parameters are available in [14; 16; 17]. For the most refined case (1232 elements) there were 55 clusters and 576 blocks created, from which 96 were admissible pairs. Results from the BEM anisotropic formulation using the ACA (for the most refined mesh) are compared with the FEM results obtained by the commercial software ANSYS, carried out in [1]. The normalized total displacement (uT C11 /PR1 ) at r = 1.5R1 on both ends is compared along the circumferential position θ . The results are shown in Fig. 4(a). Solution times are also compared and are shown in Fig. 4(b). 0.32

12000

0.3 10000 0.28 8000

time [s]

uT C11/R1 P

0.26

0.24

6000

0.22 4000 0.2

ANSYS x3 /H = +1 ACA BEM x3 /H = +1 ANSYS x3 /H = −1 ACA BEM x3 /H = −1

0.18

0.16

0

50

100

150

200

θ (Degrees)

(a)

250

300

350

BEM time ACA time

2000

0 0

200

400

600

800

Number of elements

1000

1200

(b)

Figure 4: (a) Normalized total displacement comparison; (b) BEM and ACA solution times. Conclusions. In this work, the use of hierarchical matrices and low-rank approximations applied to the anisotropic 3D formulation based on Fourier series has been presented. Low rank approximations were accomplished by the use of ACA. This method is suitable for memory and time savings, especially in the case of large-scale problems. The ACA works better beyond a certain number of elements in the mesh

92


(approximately 920 elements). After this point the solution time reported by the ACA will be less than the conventional BEM formulation. Acknowledgment. The authors would like to thank the Sao Paulo Research Foundation (FAPESP) for the financial support.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

C. L. Tan, Y. C. Shiah, C. Y. Wang, International Journal of Solids and Structures 50, 2701 (2013). T. Ting, V. Lee, J. Mech. Appl. Math. 50, 407 (1997). V. Lee, Mech. Res. Comm. 30, 241 (2003). C. Tan, Y. Shiah, C. Lin, CMES 41, 195 (2009). Y. C. Shiah, C. L. Tan, C. Y. Wang, Engineering Analysis with Boundary Elements 36, 1746 (2012). H.Rokhlin, Journal of Computational Physics 60, 187 (1985). M. Bebendorf, Numerishce Mathematik 86, 565 (2000). M. Bebendorf, S. Rjasanow, Computing 70, 1 (2003). V. Lee, International Journal for Solids and Structures 46, 3471 (2009). Y. C. Shiah, C. L. Tan, R. F. Lee, CMES 69, 167 (2010). L. Grasedyck, Computing 74, 205 (2005). O. R. S. Kurz, S. Rjasanow, Fast boundary element methods in computational Electromagnetism (Springer, 2007). S. Borm, L. Grasedick, W. Hackbusch, Engineering Analysis with Boundary Elements 27, 405 (2003). I. Benedetti, A. Milazzo, M. Aliabadi, International Journal for Numerical Methods in Engineering 80, 1356 (2009). W. Hackbusch, B. Khoromskij, R. Kriemann, Computing 73, 207 (2004). R. Rodr´ıguez, P. Sollero, E. Albuquerque, Advances in Boundary Element & Meshless Techniques (2012), pp. 303–310. R. Rodr´ıguez, A. F. Galvis, P. Sollero, E. Albuquerque, Advances in Boundary Element & Meshless Techniques (2013), pp. 225 – 230.


93

Voxel-Based Analysis of Electrostatic Fields in Virtual-Human Model Duke using Indirect Boundary Element Method with Fast Multipole Method Shoji Hamada 1 1

Department of Electrical Engineering, Kyoto University, Kyoto-Daigaku-katsura, Nishikyo-ku, Kyoto 615-8510, Japan, [email protected]

Keywords: Voxel-Based Analysis, Electrostatic Field, Eddy Current, Indirect Boundary Element Method, Virtual-Human Model, Fast Multipole Method

Abstract. A voxel-based indirect boundary element method (IBEM) with the fast multipole method (FMM) was applied to analyze electrostatic fields induced by 50-Hz homogeneous magnetic field in human anatomical models. The analyzed models were the voxel-version of adult male models provided by the IT'IS Foundation. These models have different voxel sizes but the same structural feature. By analyzing the models with voxel-side lengths of 5.0, 2.0, 1.0, and 0.5 mm, the O(N) and O(D2) dependencies of calculation times and amount of memory were confirmed, where N is the number of boundary elements and D is the reciprocal of the voxel-side length. In addition, a measure to improve the convergence performance of the linear equation solver for the FMM-IBEM was proposed and its effectiveness was demonstrated successfully. Introduction Numerical electromagnetic field analyses based on voxel models are widely conducted with the use of the finite difference method, the finite element method, etc. [1-3]. The chief merits of the voxel-based analysis include the easiness of making a realistic model from three-dimensional image data, and the simplicity in data structure suitable for easy storage, handling, and visualization. Such merits enable us to readily analyze relatively large-scale realistic models, a typical example of which is the human anatomical model made from magnetic resonance images. The voxel-based analysis also has some demerits, a major demerit being the numerical field error that is caused by the staircase approximation of the model shape. Despite such errors, the usefulness of voxel-based analyses, for example voxel-based FDTD analyses, is widely accepted. The voxel-based analyses with the boundary element methods (BEM), which analyzed relatively smallscale problems, have often been reported in the past. On the other hand, a voxel-based indirect BEM (IBEM) applied to relatively large-scale problems is reported in [4-6] that adopt the Laplace-kernel fast multipole method (FMM). This IBEM analyzed the electrostatic fields in anatomical cubic-voxel models, which describe the conductivities of biological tissues. In general, the IBEM can be applied to static field problems. The resolution of anatomical voxel models used in computational dosimetry is increasingly getting finer. For example, the IT’IS Foundation provides an adult male model called Duke [7]. Duke is originally a surface-based model, and it has derivational voxel-version models that are available for voxel-based analyses. The finest voxel-version Duke has a voxel-side length of 0.5 mm, 2.2 billion voxels, and 61 million square boundary elements. An important property of voxel-version Duke models is that they have different voxel sizes but the same structural feature. This property is useful to examine the O(N) and O(D2) dependencies of calculation times and amount of memory of the FMM-IBEM, where N and D are the number of boundary elements and the reciprocal of the voxel-side length, respectively. In this study, the O(N) and O(D2) dependencies of the voxel-based FMM-IBEM are confirmed by measuring calculation times and amount of memory required to analyze Duke models whose voxel-side lengths are 5.0, 2.0, 1.0, and 0.5 mm. In addition, a measure to improve the convergence performance of the linear equation solver for the FMM-IBEM is proposed and its effectiveness is demonstrated. The proposed measure considers both the non-uniqueness of the electric potential and the existence of isolated voxel clusters.

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Voxel-Based Indirect BEM with Fast Multipole Method Magnetically Induced Electrostatic Field Analysis in Biological Samples The basic equations of magnetically induced low-frequency faint currents in a biological sample were provided by, for example, Dawson [1], assuming that the displacement current is negligibly smaller than the conductive current, and the secondary magnetic field caused by the primarily induced eddy current is negligibly small. When an external magnetic flux density B0 and a vector potential A0 , which satisfy B0 u A0 , are applied, the magnetically induced electric field E and the electric current density J satisfy the following equations: (1), E jZA0 I , J VE , 2I 0 where j, Z , V , and I are the imaginary unit, angular frequency, conductivity, and scalar potential, respectively. The boundary normal components of the electrostatic fields E x n on both the sides of the boundary surface satisfy the following boundary equation: (2). V E x n V E x n The subscripts r indicate the plus or minus side with respect to a unit normal vector n on the boundary. An indirect BEM is used to analyze the Laplace equation in eq (1) by solving the simultaneous linear equations that are composed of eq (2) [4]. Voxel-Based Indirect Boundary Element Method The voxel-based IBEM comprises three stages. In the first stage, the voxel model is converted into an equivalent boundary element model by regarding a square-shaped boundary sandwiched by two voxels having different conductivities as a square boundary element. These N pieces of square boundary elements are assumed to have uniform surface charge densities xi (i = 1 to N), which numerically simulate I and I in eq (1). The surface integral of E r x n on the jth element is represented by the following equation:

³

Sj

E r x n dS

N ½° ° x xi dS ¾ x n dS r j S j ³ jZA0 x ndS ¦ ³ ®³ Sj Sj S i 4SH r r 2 H0 i 1 °¿ °¯ i 0 j

(3),

where ri is the position vector on the ith element. Each boundary equation is formulated as a surface integral of eq (2) on each square boundary element.

V ³ E x ndS V ³ E x ndS Sj

Sj

(4).

The simultaneous linear equations Cx b are composed of eq (4), in which eq (3) is substituted, where C is an N u N coefficient matrix, x is an N u 1 unknown vector, and b is an N u 1 constant vector. In the second stage, the surface charge densities x are determined by numerically solving Cx b . In the third stage, that is, in the post-processing stage, E and J are calculated at all voxel centers by an integral equation similar to eq (3) with the determined x . A Measure to Obtain Unique Solution Eq (4) corresponds to the Neumann boundary condition, thus the unknown charge densities x have non-uniqueness owing to the non-uniqueness of the electric potential. To obtain a unique solution, some measures are required. First, the ith row of C and b are divided by the diagonal component cii of C , thus new cii are scaled to unity. Second, Cx b are separated into two parts, that is, an arbitrary kth row and the other rows. It should be noted that the latter can regenerate the kth row by elementary raw operations because of the linear dependence owing to the non-uniqueness. Third, the kth row is replaced by the following equation that states the total sum of the charges is zero: (5). 1 1 x 0

N

This condition is imposed by the fact that the total amount of polarization charges is zero. The obtained simultaneous equations give a unique solution mathematically [4]. However, the numerically solved kth unknown, xk , tends to show a measurable deviation owing to the diagonally non-dominant nature of eq (5). Fourth, we transform the previous simultaneous equations into the following form by the elementary raw operations:


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(6), Cx DUx b where D is a scalar constant, and U is an N u N constant matrix, all entries of which are 1. The constant D is empirically set to the reciprocal of the size of U , that is, 1/N; thus, each row of the additional term represents the average of the unknown charges. Finally, by rescaling the diagonal component of the coefficient matrix to unity, the following simultaneous equations are obtained:

N N §1 · Cx ¨ U ¸x N 1 N 1© N ¹

N b N 1

(7).

Eq (6) or (7) gives a unique numerical solution without any deviation of xk , and is easily implemented in iterative solvers for simultaneous linear systems. However, a voxel model is sometimes composed of plural clusters of voxels that are isolated by the air region, the conductivity of which is zero. Even in such cases, eq (6) is easily modified by considering the total amount of charges in each isolated cluster as zero. For example, when we consider two clusters composed of n1 and n2 surface elements, respectively, eq (6) is modified to the following form:

0 · §U n ¸x b Cx ¨¨ 1 1 (8), U 2 n2 ¸¹ © 0 where the size of U matrices U 1 and U 2 are n1 u n1 and n2 u n2 , respectively. Finally, by rescaling the diagonal components to unity, the following simultaneous equations are obtained:

0 · § b n n 1 · §U n 1 § C11n1 n1 1 C12 n1 n1 1 · ¸¸ ¸¸ x ¨¨ 1 1 1 ¸¸ x ¨¨ 1 1 ¨¨ (9), C n n C n n U n 1 1 0 1 21 2 2 22 2 2 2 2 ¹ © b2 n2 n2 1 ¹ ¹ © © where the size of Cij and bi are ni u n j and ni u 1 , respectively. Note that if we analyze only the closed

region problems, the field in each cluster can alternatively be solved with eq (7) by considering just the voxels in the cluster. However, eq (9) is available without any such restriction. Complexity of Boundary Element Method with Fast Multipole Method The fast multipole method (FMM) [8] is one of the most famous algorithms that can be used to accelerate iterative solvers for simultaneous linear equations of BEM analysis. The FMM calculates interacting field among elements and the calculations are separated into two parts: (i) the direct interaction calculation of nearby elements, and (ii) the non-straightforward interaction calculation with multipole and local expansion coefficients. Here, we call the former and the latter as “near-part” and “far-part” calculations, respectively. The FMM is easily applied to the voxel-based BEM by regarding a cubic-shaped cluster of voxels, for example 6 × 6 × 6 = 63 cubic voxels, as a leaf cell defined in the FMM algorithm. The details of the application of the FMM to the voxel-based IBEM are found in references [4–6]. Both the calculation time and the amount of memory required for solving Cx b are expected to show O(N) dependency when using the Laplace kernel FMM. Besides, the number of surface elements N is expected to approximately exhibit O(D2) dependency, where D is the reciprocal of the voxel-side length, when the full size of an analyzed model is fixed. An example is explained in the next section. Virtual-Human Model Duke The IT’IS Foundation (www.itis.ethz.ch/vip) offers a series of anatomical virtual human models named the Virtual Population for research purposes. A subset of the Virtual Population with two adults and two children is named the Virtual Family. The adult male model of the Virtual Family is called Duke [7] (see Fig. 1). Duke is 34 years old, 1.77 m in height, and composed of 77 types of tissues. Although these anatomical models are originally defined as triangular surface models, their direct utilization as surface models in general program codes is contractually restricted. However, we can derive voxel-version Duke models by using support software provided by the IT’Is Foundation. We can set the voxel-side length between 5.0 mm to 0.5 mm, and we can utilize voxel-version models in general codes. In addition, there are four types of voxel-version models provided in the distribution DVD (version 1.2) with the voxel-side length of 5.0 mm, 2.0 mm, 1.0 mm, and 0.5 mm. In this study, these four Duke models are used as a series of models having the same structural feature to examine the O(N) and O(D2) dependencies of the calculation time and the amount of memory of the voxel-based FMM-IBEM.

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The conductivities of tissues are assumed to be isotropic, and the values are assumed by mainly referring to the values in [9]. Figs. 1 (a) and 1 (c) show the tissue numbers classified from 1 to 77 in a coronal slice of the model with 5.0 mm and 0.5 mm voxel-side lengths, respectively. Figs. 1 (b) and 1 (d) show the calculated | E | and Fig. 1 (d) shows finer details of the field distribution compared with Fig. 1 (b).

(a) 5 mm, tissue no. (b) 5 mm, |E|

(c) 0.5 mm, tissue no. (d) 0.5 mm, |E|

Fig. 1 Tissue number and calculated |E| in Duke models. The number of voxels, number of boundary elements, and number of isolated clusters are summarized in Table 1. The x, y, and z coordinates are defined as shown in Fig. 1. We can observe in Table 1 the following: (i) the maximum number of total voxels exceeds 2 billion, and the number of voxels shows O(D3) dependencies; (ii) the maximum number of boundary elements reaches 61 million, and the number of elements approximately shows O(D2) dependency, which is approximately O(D2.1); and (iii) the number of fragment clusters is non-predictable; thus, it has to be counted by a subprogram developed for this purpose. Voxel-side length 5.0 mm 2.0 mm 1.0 mm 0.5 mm Voxels in x u y u z 112 u 58 u 362 272 u 143 u 904 544 u 285 u 1806 1084 u 566 u 3610 Total voxels 2,351,552 35,161,984 280,002,240 2,214,893,840 Tissue voxels 548,164 8,567,668 68,549,358 548,386,439 Boundary elements, N 472,284 3,617,128 15,070,962 61,492,477 Clusters; main + fragments 1+3 1+0 1+1 1+9 Table 1 The numbers of voxels, boundary elements, and isolated clusters in Duke models. Computing Environment and Settings for Calculation Computing Environment A personal computer running 64-bit Microsoft Windows 7 was used for the calculations. It had an Intel Core i7-4960X CPU (6 CPU cores, 3.6 GHz) with 64 GiB of RAM. GPU acceleration [5, 6] was not applied. The software was compiled with Intel Visual FORTRAN Composer XE 2013, and the number of OpenMP threads was set up to 12. Settings for Calculation The applied homogeneous magnetic field B0 B0 k was 50-Hz AC, 1 PT in strength, and parallel to the vertical axis. The vector potential was defined as A0 i, j, and k are the unit vectors parallel to the x, y, and z axes, respectively.

0.5B0 yi xj , where


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In the program code of the FMM algorithm, the order of the multipole and local expansions was set to ten. The translation algorithm that converts multipole-expansion coefficients to local-expansion coefficients was a diagonal-form translation algorithm [8]. The leaf-cell size that is defined as the voxel-cluster size was 53, 63, or 73. The iterative solver used is the GBi-CGSTAB(s, L) [10], where s = 3 and L = 2. This solver requires L(s + 1) matrix-vector-product calculations before judging the convergence of the residual norm once. Here, we regard the number of matrix-vector products as the number of iteration steps. Thus, the convergence is judged every L(s + 1) iteration steps. The convergence was decided when the relative residual norm became less than 10-6 except in the case of the first subsection of the following section. Results

0

10

10

: Relative residual : |Ȋxi|/STD(x i)

10

0 200 400

(a)

Without eq (9)

Convergence property of linear equations solver

Convergence property of linear equations solver

Convergence Performance of the Linear Equation Solver The convergence performance of the linear equation solver was checked to confirm the validity of eq (9). The voxel-side length of the Duke model used was 5.0 mm and the leaf-cell size was 63 voxels. Fig. 2 (a) shows the result when using no measure to obtain a unique solution, and Fig. 2 (b) shows the result of using eq (9). The horizontal axis shows the iteration steps, and the residual norms, etc. are output once every L(s + 1) = 2(3 + 1) = 8 steps. The thin line in the figure shows the relative residual norm. In the first case, as shown in Fig. 2 (a), the calculation was forcibly stopped when the iteration steps exceeded four hundred. In the second case, as shown in Fig. 2 (b), the calculation was terminated at the two-hundredth step to avoid a singular calculation in the GBi-CGSTAB(s, L) solver owing to a residual that is too small. The thick line shows the absolute value of the sum of the solution xi over the standard deviation of xi , which is an index of the deviation of total charges from zero. Fig. 2 (a) shows that the convergence of the relative residual norm stagnated once around 10-5, and it jumped up in synchronization with the jump-up of the index of deviation. Empirically speaking, an increase in the deviation deteriorates the quality of the solution. Fig. 2 (b) shows that the relative residual norm and the index of deviation decreased almost smoothly, and the latter at least did not show upward trend. These results demonstrate that eq (9) is effective in achieving smooth convergence given the constraint of zero total charges. 0

10

: Relative residual : |Ȋxi|/STD(x i)

10

10

0 200 400

(b) With eq (9)

Fig. 2 Effectiveness of eq (9) in improving the convergence performance of the linear equation solver. Calculation Time and Required Amount of Memory in the Second Stage Calculation times and amount of memory required in the second stage are shown in Table 2 and Figs. 3, 4, and 5. The numbers of required iteration steps, listed in Table 2, are almost similar with the values ranging from 96 to 120. The solver took 6,587 s and 31.2 GiB in the finest model analysis with the setting of 73 voxels in a leaf cell. Figs. 3 (a) and 3 (b) show the dependencies of one-step calculation time on N and D, respectively. Fig. 3 (a) shows the least square line of the fastest data at each N. The slope shows that the dependency is approximately O(N0.957). In a similar way, Fig. 3 (b) shows that the dependency is approximately O(D2.023). Figs. 4 (a) and 4 (b) show the dependencies of near-part and far-part calculation times on N, respectively. The dependencies are O(N0.847) and O(N1.07), respectively. The difference in these dependencies makes it necessary to adjust the number of voxels in a leaf cell to obtain the best performance as shown in Fig. 3 (a).

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Fig. 5 shows the amount of memory required in the second stage. The dependency is approximately O(N0.915). Thus, it is confirmed that the voxel-based IBEM approximately exhibits the O(N1) and O(D2) dependencies of calculation times and amount of memory as expected. Total time Total Time for Time for Voxels Time for iteration for solver far-part near-part in a leaf one step /s steps /s /s /s cell 53 0.647 0.344 0.269 112 72.4 3 6 0.739 0.532 0.177 104 76.8 73 0.928 0.775 0.125 112 104 53 4.43 2.08 1.99 104 461 63 4.61 3.03 1.28 104 479 73 5.57 4.38 0.924 104 579 53 18.5 6.71 9.81 112 2,070 63 17.4 9.36 6.34 120 2,084 73 19.1 13.1 4.54 104 1,990 53 79.6 21.4 49.3 96 7,643 63 68.6 28.4 32.6 104 7,134 73 68.6 38.7 22.9 96 6,587 Table 2 Calculation times and memory usage in the second stage.

Voxelside length 5.0 mm

2.0 mm

1.0 mm

0.5 mm

10

2

10 : : : :

1

5*5*5 voxels in a leaf 6*6*6 voxels in a leaf 7*7*7 voxels in a leaf The fastest data Timeper step / s

Time per step / s

10

0.957

O(N 10

10

)

0

10

10

Memory for solver / GiB 0.461 0.386 0.358 2.63 2.15 1.86 10.1 8.25 7.21 41.0 34.8 31.2

2

: : : :

1

5*5*5 voxels in a leaf 6*6*6 voxels in a leaf 7*7*7 voxels in a leaf The fastest data

2.023

O(D 10

)

0

5

6

7

10 10 10 The number of surface elements (N)

10 10

8

0

10 10 10 The reciprocal of voxel side length (D)

1

(a) Time versus N. (b) Time versus D. Fig. 3 Dependencies of one-step calculation time on N and D in the second stage. 2

10

1

10

0

10 : : : :


Time for far part of FMM per step / s

Time for near part of FMM per step / s

10

0.847

O(N

10

)

10

5

6

7


8

2

10

1

10

0

: : : :


1.07

O(N

10

)

10

5

6

7


8

(a) Near-part calculation time versus N. (b) Far-part calculation time versus N. Fig. 4 Dependencies of near- and far-part calculation time per iteration step on N in the second stage.


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Memory Voxel Voxels Total time for post -side in a 10 for post / GiB length leaf : 5*5*5 voxels in a leaf cell /s : 6*6*6 voxels in a leaf 3 5.0 5 1.98 0.417 : 7*7*7 voxels in a leaf mm 63 2.78 0.382 : The least data 1 10 73 2.93 0.293 2.0 53 16.6 2.50 mm 63 22.5 2.02 0.915 3 O(N ) 7 23.8 1.76 0 10 1.0 53 84.2 10.8 mm 63 106 9.03 73 111 8.21 0.5 53 462 56.0 3 mm 6 574 48.3 10 5 6 7 8 10 10 10 10 73 539 46.8 The number of surface elements (N) Fig. 5 Dependency of memory in the second stage. Table 3 Time and memory usage in the third stage. Memory for solver / GiB

2

Calculation Time and Required Amount of Memory in the Third Stage In the third stage, the static electric fields at all gravity centers of tissue voxels are calculated by using the FMM. The field strengths at all voxels, that is, not only tissue voxels but also air voxels, are output into a file to visualize the field distribution as shown in Fig. 1. Calculation times and amount of memory required in the third stage are shown in Table 3 and Fig. 6. Figs. 6 (a) and 6 (b) show the dependencies of the post-processing calculation time and the required amount of memory on N, respectively. The dependencies are O(N1.118) and O(N1.038), respectively. If N increased further, the dependencies would approach O(D3) = O(N1.5), which should be proportional to the number of voxels. However, such O(D3) dependencies were not observed, indicating that the proportional constant of the O(D2) part related to the FMM is larger than that of the O(D3) part at the present scale of problems. In addition, the amount of memory required in the post-processing stage can be reduced by not storing the calculated value in arrays but directly writing them on a file, further reducing the proportional constant of the O(D3) part related to the amount of memory. 3

10

2

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1

: : : :

Memory for post processing / GiB

Time for post processing / s

10


1.118

O(N

)

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: : : :

5*5*5 voxels in a leaf 6*6*6 voxels in a leaf 7*7*7 voxels in a leaf The least data

1.038

O(N

10

)

10

5

6

7


8

Fig. 6 Dependencies of calculation times and amount of memory on N in the third stage.

Summary A voxel-based FMM-IBEM was applied to analyze electrostatic fields induced by a 50-Hz homogeneous magnetic field in human anatomical voxel models. The analyzed models were voxel-version Duke models provided by the IT'IS Foundation. The voxel-side lengths were 5.0, 2.0, 1.0, and 0.5 mm. The linear

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equation solver took 6,587 s and 31.2 GiB on a personal computer for the finest model with 61 million boundary elements and 2.2 billion voxels. An important property of the voxel-version Duke models is that these models have different voxel sizes but the same structural feature. Taking advantage of this property, the O(N) and O(D2) dependencies of calculation times and amount of memory required for the voxel-based FMM-IBEM were confirmed. In addition, a measure to improve the convergence performance of the linear equation solver for the FMM-IBEM was proposed and its effectiveness was successfully demonstrated. Acknowledgments This work was supported by JSPS KAKENHI Grant Number 25390153. References [1] Trevor W. Dawson, Kris Caputa, and Maria A. Stuchly, Influence of human model resolution on computed currents induced in organs by 60-Hz magnetic fields, Bioelectromagn., 18(7), 478-490 (1997). [2] Trevor W. Dawson, Kris Caputa, and Maria A. Stuchly, High-resolution organ dosimetry for human exposure to low-frequency electric fields, IEEE Trans. on Power Deliv., 13(2), 366-373, (1998). [3] Tomohisa Hatada, Masaki Sekino, and Shoogo Ueno, Finite element method-based calculation of the theoretical limit of sensitivity for detecting weak magnetic fields in the human brain using magneticresonance imaging, J. Appl. Phys., 97, 10E109 (2005). [4] Shoji Hamada and Tetsuo Kobayashi, IEEJ Trans. FM, 126, 355-362 (2006) (in Japanese) (translation: Analysis of electric field induced by ELF magnetic field utilizing fast-multipole surface-charge simulation method for voxel data, Electr. Eng. Japan, 165, 1-10 (2008)). [5] Shoji Hamada, GPU-accelerated indirect boundary element method for voxel model analyses with fast multipole method, Comput. Phys. Commun., 182, 1162-1168 (2011). [6] Shoji Hamada, Performance comparison of three types of GPU-accelerated indirect boundary element method for voxel model analysis, Int. J. Numer. Model., 26, 337-354 (2013). [7] Andreas Christ, Wolfgang Kainz, Eckhart G. Hahn, Katharina Honegger, Marcel Zefferer, Esra Neufeld, Wolfgang Rascher, Rolf Janka, Werner Bautz, Ji Chen, Berthold Kiefer, Peter Schmitt, Hans-Peter Hollenbach, Jianxiang Shen, Michael Oberle, Dominik Szczerba, Anthony Kam, Joshua W. Guag, and Niels Kuster, The virtual family – Development of surface-based anatomical models of two adults and two children for dosimetric simulations, Phys. Med. Biol., 55 (2), N23-N38 (2010). [8] Leslie Greengard and Vladimir Rokhlin, A new version of the fast multipole method for the Laplace equation in three dimensions, Acta Numerica, 6, 229-269 (1997). [9] Akimasa Hirata, Kenichi Yamazaki, Shoji Hamada, Yoshitsugu Kamimura, Hiroo Tarao, Kanako Wake, Yukihisa Suzuki, Noriyuki Hayashi, and Osamu Fujiwara, Intercomparison of induced fields in Japanese male model for ELF magnetic field exposures: effect of different computational methods and codes, Radiat. Protect. Dosim., 138(3), 237-244 (2010). [10] Masaaki Tanio and Masaaki Sugihara, GBi-CGSTAB(s,L): IDR(s) with higher-order stabilization polynomials, Comput. Appl. Math., 235, 765-784 (2010).


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Development of the Boundary Element Method for 3D General Anisotropic Thermoelasticity Y.C. Shiah1and C.L. Tan2* 1. 2

Department of Aeronautics and Astronautics National Cheng Kung University, Tainan 701, Taiwan, R.O.C. Department of Mechanical & Aerospace Engineering Carleton University, Ottawa, Canada K1S 5B6 (* Corresponding author; Email: [email protected])

Keywords: Anisotropic thermoelasticity, Green’s function, Fourier series. Abstract. In the boundary element method (BEM) for linear elasticity, it is well known that thermal effects manifest themselves as an additional domain integral in the integral equation. For isotropy, and 2D general anisotropy, this domain integral has been transformed analytically in an exact manner into surface integrals, thereby restoring the integral equation into a boundary one. This re-establishes the BEM as a boundary solution analysis tool without the need to invoke simplifying schemes, such as the dual reciprocity method. Because of the relative mathematical complexity of the Green’s function for 3D general anisotropic elasticity, similar exact analytical transformation of the domain integral has, hitherto, not been reported in the literature. This study deals with the successful development of this integral transformation. It makes use of the Fourier series representation of the Ting & Lee’s closed-form Green’s function for 3D general anisotropy that is recently proposed by the authors. The approach follows the same basic steps in the formulation previously derived for 2D anisotropy, also by the authors. Some numerical examples are presented to demonstrate the veracity of the exact volume-to-surface integral transformation due to the thermal effects. Introduction It is well known that in linear elastic stress analysis using the boundary element method (BEM), thermal effects manifest themselves as an additional domain integral in the integral equation in its primary form. This domain integral needs to be transformed into surface ones to restore the notion of the BEM as a boundary solution technique. The exact analytical transformation to this end, unlike some approximation techniques, represents a truly general scheme that can be applied to all geometric cases without further modification of the formulation, such as when re-entrant corners or cracks are present. This is well documented in the literature for isotropic, and 2D general anisotropic elasticity, see, e.g., [1] – [5]. Being not of closed form, the fundamental solution of Lifshitz and Rozenzweig [6], used in BEM for several decades, has been the main obstacle to this exercise for its extension to treat thermoelastic problems in 3D general anisotropy. More recently, however, the authors have implemented an explicit form of the Green’s function, derived by Ting and Lee [7], for BEM in 3D general anisotropic elasticity [8]. They have also very recently reported further progress in this development, showing very significant savings in the computational effort for the numerical evaluation of this Green’s function and its derivatives by representing this fundamental solution by a double Fourier series [9]. Although still mathematically quite involved, its explicit form allows the above-mentioned exact transformation method (ETM) to be performed; a preliminary successful attempt has been made for the special case of transverse isotropy [10]. This paper presents an extension to treat thermoelastic problems in 3D general anisotropy in BEM. It follows the same steps as seen previously for 2D general anisotropic elasticity by the authors [5]. A key part in this scheme is to re-define the governing equation for the thermal field problem in a mapped domain thereby reducing it into a canonical form [11]. The volume-to-surface integral transformation of the domain integral associated with the thermal effects is carried out in this mapped domain. More details of these will be described below, followed by a numerical example to demonstrate the veracity of the formulation. Boundary integral equation for thermoelasticity In BEM for thermoelasticity, the integral equation in its primary form is well-known and can be expressed in indicial notation as follows:

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C ij (P) u i (P) + ³s u i (Q) T ij (P, Q) dS(Q) = ³ t i (Q) U ij (P, Q) dS(Q) ³ J ik nk (Q) 4(Q)U ij (P, Q) dS(Q) ³ J ik 4 ,k (Q) U ij (P, q) d: s s :

(1)

It relates the displacements, ui, to the tractions, ti, on the surface S of the domain under a temperature field . In Eq. (1), Uij(P,Q) U(x) and Tij(P,Q) represent the fundamental solutions of displacements and tractions, respectively, in the xi-direction at the field point Q (or q) due to a unit load in the xj-direction at P in a homogeneous infinite body; Cij(P) depends on the geometry of the surface at P. Also, ik represent the thermal moduli of the anisotropic material. This equation is not truly a boundary integral equation (BIE) as the last term is a domain integral, unless it is transformed into surface ones. This clearly has implications for the numerical formulation of the BEM. The domain integral may be represented by Vj, i.e. Vj

³ J ik 4,k (q) U ij (P,q) d : :

(2)

Following the similar treatment as in [5], the volume integral in Eq.(2) can be redefined in a transformed coordinate system such that the temperature field is governed by the standard Laplace equation, i.e. 4,kk

0,

(3)

where the underline denotes the transformed coordinates. As has been shown in [11], the transformation of the corresponding governing equation for field problems in general anisotropy in three dimensions (namely, Euler’s equation) to Eq. (3) can be achieved by the following simple coordinate transformation:

xˆ T = F x T .

(4)

The transformation matrix F above is given by § ' / K11 ¨ F ¨ K12 / K11 ¨ D ©

0 1

E

0 · ¸ 0 ¸, F ¸¹

(5)

where Kij are the thermal conductivity coefficients; the other coefficients ', D, E, and F are defined by

' K11K22 K122 ,

D

( K12 K 23 K13 K 22 ) / Z ,

(6a) (6b)

E

( K12 K13 K 23 K11 ) / Z ,

(6c)

F

'/ Z,

(6d)

Z

K11 K33' K11 K 22 K132 2 K11 K12 K13 K 23 K 232 K112 .

(6e)

Equation (2) can be rewritten in this transformed coordinate system as follows

Vj where Zik are defined by

³ Zik 4 ,k (q) U ij (P,q) d : , :

(7)


Z ik

K11

§ J 11 J 12 J 13 · § ¨ Z ¨ J 21 J 22 J 23 ¸¸ ¨ '3 ¨¨ ¸¨ © J 31 J 32 J 33 ¹ ©

103

' / K11 K12 / K11 D · ¸ E¸. 0 1 F ¸¹ 0 0

(8)

Following the steps described in [5], Vj can be written in terms of the transformed boundary integrals as follows: (9) V j ³ Zik ª 4 Wijk ,t Wijk 4 ,t nt 4 U ij nk º d S , S ¬ ¼

where Wijk is a new kernel function which satisfies

Wijk ,tt

U ij ,k .

(10)

The exact analytical volume-to-surface integral transformation is not complete yet; the explicit expression of the new function Wijk needs to be first determined. The complex form of the exact analytical expression for U ij ,k , which has in fact been derived by the authors previously [12], presents a very serious challenge to carry out the exact transformation above, however. Very recently [9], the present authors have shown that they can be rewritten into a Fourier series as follows:

U uv ,l

º»

for l 1

º»

for l

2,

for l

3

D D ª cos Tˆ sin Iˆ i n cos Iˆ ˆ ˆ ° ¦ ¦ Oûv( m ,n ) ei mT nI « ° m D n D « i m sin Tˆ / sin Iˆ ¬ ° ° a a ª sin Tˆ sin Iˆ i n cos Iˆ ˆ ˆ 1 ° ( m , n ) i mT n I « ® ¦ ¦ Oûv e 4S rˆ 2 ° m a n a « i m cos Tˆ / sin Iˆ ¬ ° a a ˆ ˆ ( m , n ) i mT n I ª ° ˆ cos Iˆ i n sin Iˆ º Ouv e ° m¦ a n¦ ¼ ¬ a ° ¯

» ¼

» ¼

(11)

where ( rˆ , Tˆ , Iˆ ) represent the spherical coordinates for the mapped domain; Oûv( m ,n ) are the coefficients, computed by

1 4S 2

Oûv( m,n )

S

S

³S³S

H uv (T c, I c) i mTˆ nIˆ ˆ ˆ e dT dI . rc

(12)

In Eq. (12), H uv are defined by

U (r , T , I )=

H (T , I ) , 4S r

(13)

and ( rc , T c , I c ) are all intrinsic functions of ( Tˆ , Iˆ ), defined by

rc where

x1c x2c x3c 2

2

2

, Tc

§ xc · § xc · tan 1 ¨ 2 ¸ , I c cos 1 ¨ 3 ¸ , c x rc ¹ © © 1¹

(13)

104


x1c x2c x3c

' sin Iˆ cos Tˆ, K11 K12 sin Iˆ cos Tˆ sin Iˆ sin Tˆ, K11 D sin Iˆ cos Tˆ E sin Iˆ sin Tˆ F cos Iˆ.

(14)

The details of expressing the Green's function U in Eq. (13) may be found in [12]; thus, no further explication on its formulations needs to be provided here. Returning to Eq. (10) that is used to define the new third order tensor, Wijk , it may be expressed in the spherical coordinate system for the mapped domain as follows:

sin 2 Iˆ where N ijk (Tˆ, Iˆ)

w 2Wijk (Tˆ, Iˆ) sin 2Iˆ wWijk (Tˆ, Iˆ) w 2Wijk (Tˆ, Iˆ) 2 wIˆ 2 wIˆ wTˆ 2

N ijk (Tˆ, Iˆ) sin 2 Iˆ ,

(15)

U uv ,l (1, Tˆ, Iˆ) . By taking advantage of the periodical nature of the spherical angles,

Wijk (Tˆ, Iˆ) can be expressed as a Fourier series, ,

where

(16)

are unknown coefficients to be determined. Substituting Eq. (16) directly into Eq. (15) yields .

(17)

To determine the unknown coefficients, both sides of Eq. (17) are integrated as follows:

.

(18)

These integrations for all values of p, q ranging from -a to +a result in a system of equations, expressed in the following banded matrix form:


105

,

where Cijk(u ) is used to denote the u-th coefficient of each set of

(19)

(v) , numbered in the sequential order; bijk

໧

(v) are defined as represents the RHS value of Eq. (18) for the v-th equation. In Eq. (19), M u , M u , M u , bijk

follows:

Mu ໦ Mu ໧ Mu

2(2m2 n 2 ) (for m

p,n

q),

n(n 1)

(for m

p,n

q 2),

n(n 1)

(for m

p,n

(4m n 3) °(n 1) ° °(4m n 3) ( m,n ) ° ˆ i O ij °(n 1) bij( v1) ® 4 °(4m n 3) °(n 1) ° °(4m n 3) °(n 1) ¯ (4m n 3) °(n 1) ° °(4m n 3) ( m,n ) ° ˆ i O ij °(n 1) bij( v2) = ® 4 °(4m n 3) °(n 1) ° °(4m n 3) °(n 1) ¯

(3n 1) ˆ ( m ,n ) °(n 1) i O ij ° bij( v3) = ® 4 °(3n 1) °¯(n 1)

(20a)

q 2).

(for m

p 1, n

q 1)

(for m

p 1, n

q 3)

(for m (for m

p 1, n q 1) p 1, n q 3) ,

(for m (for m (for m

p 1, n q 1) p 1, n q 3)

(for m (for m (for m (for m (for m (for m (for m (for m (for m

p 1, n p 1, n p 1, n p 1, n p 1, n p 1, n p 1, n p 1, n p 1, n p 1, n

(for m (for m (for m (for m

q 1) q 3) q 1) q 3) q 1) q 3) , q 1) q 3) q 1) q 3)

p , n q 1) p , n q 3) . p , n q 1) p , n q 3)

(20b)

(20c)

(20d)

106


Equation (19) can be solved relatively quickly for the Fourier coefficients due to the banded form of the matrix. With all coefficients thus obtained, the explicit expression for Wijk (Tˆ, Iˆ) is now of Fourier series form, Eq. (16). By performing partial differentiations in the spherical coordinate system, its first-order derivatives are given by

.

(21)

This new tensor and its derivatives are well defined in the transformed coordinate system, and the transformed boundary integrals in Eq. (9) can now be computed in the usual manner in BEM analysis. Numerical verification Due to space limitation, only one numerical test example will be presented here to verify the exactness of Eq. (2) and the transformed surface integrals of Eq. (9) for the volume integral Vj. The following material property values of C (stiffness coefficients) are used; they correspond to those of Al2O3 with counterclockwise rotations of the material principal axes about the x1-, x2-, and x3-axes by 300, 450, and 750, respectively [13]:

ª « « « C « « « « ¬

488.3176 112.0742 115.1898 9.5042 20.5593 24.2035 112.0742 550.3789 90.8415 20.4306 15.6416 10.8253 115.1898 90.8415 534.0938 29.2473 13.1636 13.7425 9.5042 20.4306 29.2473 199.5245 2.5146 1.8322 20.5593 15.6416 13.1636 2.5146 195.0693 21.1779 24.2035 10.8253 13.7425 1.8322 21.1779 202.0117

º » » » , » GPa » » » ¼

(22a)

The values for the other thermal properties are taken to be:

K

§ 21.4210 7.5438 6.8134 · § 0.26220 0.02879 0.05556 · ¨ ¸ ¨ ¸ 0 6 0 . (22b) / 7.5438 21.9565 0.7030 W m C, ¨ ¸ ¨ 0.02879 0.21250 0.06302 ¸ u10 / C ¨ 6.8134 0.7030 25.6925 ¸ ¨ 0.05556 0.06302 0.23030 ¸ © ¹ © ¹

With these values, the thermal moduli of the original domain are determined from the basic theory of thermoelasticity to be:

§ 0.1809E 02 0.1924E 01 0.2831E 01 · ¨ ¸ 5 0 . ¨ 0.1924E 01 0.1687E 02 0.3391E 01 ¸ u 10 GPa / C ¨ 0.2831E 01 0.3391E 01 0.1768E 02 ¸ © ¹

(23)

§ 0.1991E 02 -0.5155E 01 0.3950E 01 · ¨ ¸ 5 0 . ¨ 0.2117E 01 0.1878E 02 -0.2454E 01¸ u10 GPa / C ¨ 0.3116E 01 0.2775E 01 0.1765E 02 ¸ © ¹

(24)

From Eq. (8), one obtains

Z


107

Consider a hollow cylinder (Figure 1), with height of 2 units, inner radius of unity, and outer radius of 2 units. Purely for the purpose to show the veracity of above formulations, the temperature distribution is arbitrarily assumed to be

4 (2.847593x1 1)(-1.760842x1 5x2 3)(-0.634276x1 0.159739x2 1.817269x3 3) ,

(25)

which satisfies the corresponding anisotropic heat conduction equation. The temperature distribution in the mapped domain can be easily verified to be

4 (3xˆ 1 1)(5xˆ 2 3)(2 xˆ 3 3) . The surface integrals over each of the boundary elements in the BEM mesh (Figure 1) were computed with 8-points Gauss quadrature. For the direct numerical evaluation of the domain integral, the 100-points Gauss quadrature was employed. Table 1 lists the computed values of Vj using Eq. (2) directly and using the transformed surface integral sin Eq. (9). It can be seen that there is very good agreement between both sets of results. The small discrepancies may be attributed to the errors introduced in the numerical modeling of the geometry by the assemblage of discrete boundary elements and in the numerical integrations over them. Although not presented here, these small deviations have been verified to decrease further when higher order Gauss quadrature schemes were used for integration over the elements.

(26)

x3

x2 x1

2

1 2

Conclusions Figure 1: A hollow cylinder for The domain integral present in the primary form of the integral the numerical verification equation for BEM in elastic stress analysis with thermal loads needs to be transformed into surface ones in order to re-establish the BEM as a boundary solution technique. Hitherto, this has been achieved in an exact manner for all but 3D general anisotropy in linear elasticity. This is because of the analytical complexity of the fundamental solution required in the BIE formulation. Following the same approach as has been done in 2D anisotropy [5] and employing a Fourier series representation of the fundamental solution very recently proposed by the authors as well, the domain integral associated with the thermal effects has been transformed into surface ones, albeit in a mapped domain. The veracity of this transformation has been demonstrated by an example. With this exact transformation method, the BIE is analytically exact and no further simplifying approximations are needed in the numerical formulation of the BEM.

Acknowledgement The authors gratefully acknowledge the financial support from the National Science and Engineering Research Council of Canada and the National Science Council of Taiwan (NSC 102-2221-E-006-290MY3).

108


Table 1: Comparison of the integrated values of Vj – hollow cylinder. Source pt. (0.0,-2.0,-1.0) (2.0,0.0,-1.0) (0.0,2.0,-1.0) (-2.0,0.0,-1.0) (0.0,-2.0,1.0) (2.0,0.0,1.0) (0.0,2.0,1.0) (-2.0,0.0,1.0)

j=1

Vj computed by Eq.(2) j=2

Vj computed by Eq.(9) j=2

j=3

j=1

0.1572E+5 (0.06%) 0.8423E+04 (0.01%) 0.6056E+03 (1.46%) 0.4093E+04 (0.15%) 0.1903E+05 (0.16%) 0.1689E+05 (1.11%) -0.1201E+04 (1.64%) 0.3785E+04 (0.13%)

0.1573E+05

0.1556E+04

0.1185E+05

0.8422E+04

-0.3565E+04

0.1261E+05

0.5969E+03

-0.2249E+04

0.4943E+00

0.4087E+04

0.6131E+04

0.6112E+04

0.1906E+05

0.9037E+01

0.1149E+05

0.1708E+05

-0.6516E+04

0.1589E+05

-0.1221E+04

-0.3057E+04

0.2332E+04

0.3780E+04

0.7977E+04

0.5494E+04

0.1565E+04 (0.58%) -0.3564E+04 (0.03%) -0.2239E+04 (0.45%) 0.6123E+04 (0.13%) 0.6824E+02 (655.12%) -0.6519E+04 (0.05%) -0.3018E+04 (1.28%) 0.7970E+04 (0.09%)

j=3

0.1185E+05 (0.00%) 0.1258E+05 (0.24%) 0.4926E+04 (0.35%) 0.6112E+04 (0.00%) 0.1141E+05 (0.70%) 0.1571E+05 (1.13%) 0.2374E+04 (1.80%) 0.5484E+04 (0.18%)

References [1] Rizzo, F. I., Shippy, D.J., Int. J. Numer. Methods Engng., 11, pp. 1753-1768 (1977). [2] Rashed, Y.F. (Ed.), Advances in Boundary Elements - Transformation of Domain Effects to the Boundary, (Ed. Y.F. Rashed), WIT Press (U.K.), ISBN 1-85312-896-1 (2003). [3] Deb, A., Henry, Jr., D.P., Wilson, Int. J. Numer. Methods Engng., 27, pp. 1721-1738 (1992). [4] Kogl, M., Gaul, L., Arch. Appl. Mech., 73, 377-398 (2003). [5] Shiah, Y.C., Tan, C.L., Computational Mech., 23, 87-96 (1999). [6] Lifshitz, I.M., Rozenzweig, L.N., Zh. Eksp. Teor. Fiz. 17, pp.783-791 (1947). [7] Ting, T.C.T., Lee, V.G., Q. J. Mech. Appl. Math. 50, 407-426 (1997). [8] Tan, C.L., Shiah, Y.C., Lin, C.W., CMES – Comp. Modeling Engng. & Sc., 41, 195-214 (2009). [9] Tan, C.L., Shiah, Y.C., Wang, C.Y., Int. J. Solids Struct., 50, 2701-2711 (2013). [10] Shiah, Y.C., Tan, C.L., Int. J. Solids Struct., 49, 2924-2933 (2012). [11] Shiah, Y.C., Tan, C.L., Engng. Analysis Boundary Elem., 28, 43-52 (2004). [12] Shiah, Y.C., Tan, C.L., CMES – Comp. Modeling Engng. & Sc., 78, 95-108 (2011). [13] J.F. Nye, Physical Properties of Crystals. Their Representations by Tensors and Matrices, Clarendon, Oxford (1960).


109

BEM Solution of MHD Pipe Flow Around a Conducting Cylindrical Solid and Inside an Insulating or Conducting Medium M. Tezer-Sezgin 1 and S. Han Aydın 2 1 2

Department of Mathematics, Middle East Technical University, Ankara, Turkey Department of Mathematics, Karadeniz Technical University, Trabzon, Turkey email: 1 [email protected], 2 [email protected]

Keywords: BEM, MHD pipe flow.

Abstract. Magnetohydrodynamic (MHD) flow is considered in an annular pipe around a conducting solid and in an insulating or conducting medium. An external magnetic field is applied perpendicular to the axis of the pipe. MHD equations in the pipe are coupled with the Laplace equations of the solid and external medium, through continuity conditions on the pipe walls. Coupled MHD equations for the fluid are transformed to decoupled modified Helmholtz equations and then are solved together with the two Laplace equations of the solid and outside medium by using boundary element method (BEM) as one discretized system of equations. Numerical results show the well known MHD characteristics as the boundary layer formation for both the fluid velocity and the induced magnetic field of the fluid, and the flattening tendency of the fluid velocity for increasing values of Reynolds and magnetic Reynolds numbers. The continuation of three induced magnetic fields on the pipe walls are maintained accordingly with the values of magnetic Reynolds numbers of the solid, fluid and external medium.

Introduction and Mathematical Problem MHD pipe flow finds some engineering and biomedical applications as MHD generators, pumps and instruments for measuring blood pressure. An annular pipe around a cylindrical conducting solid and in a conducting medium takes the attention of MHD researchers since it models the MHD turbines and nuclear fusion apparats. The MHD pipe flow problem without a solid in the pipe but in a conducting medium has been solved by using BEM [1] for square and circular pipes, by using DRBEM [2] and FEM [3] for a circular pipe with the assumption that B ex → 0 as x2 +y 2 → ∞ (B ex : induced magnetic field of outside conducting medium). Mathematical problem is derived from Navier-Stokes equations of conducting fluids and Maxwell’s equations of electromagnetic field through Ohm’s law. These are given in the annular cross-section Ωf of the pipe around a solid Ωs and in an insulating or conducting outside region Ωex as Fig. (1)

1

110


whereas applied magnetic field is in y -direction, [4]

∇2 B s = 0 ∇2 V f + Re · Rh · ∇2 B f + Rmf ∇2 B ex = 0

in Ωs

∂B f = −1 ∂y

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ∂V f ⎪ ⎭ · =0 ⎪ ∂y

Ωex

(1)

in Ωf

(if Ωex is conducting)

(2)

(3)

with the coupled boundary conditions ⎫

V f (x, y) = 0 ⎪ ⎪ ⎪ B f (x, y) = B s (x, y) s

f

1 ∂B 1 ∂B = Rms ∂n Rmf ∂n B f (x, y) = V f (x, y) = 0

⎫ B (x, y) = B (x, y) ⎪ ⎪ ⎪ ⎪ ⎬ f

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(x, y) ∈ Γ2

(x, y) ∈ Γ1

(if Ωex is insulated)

(4)

(5)

ex

1 ∂B f 1 ∂B ex = Rmf ∂n Rmex ∂n

⎪ ⎪ ⎪ ⎪ ⎭

(x, y) ∈ Γ2

(if Ωex is conducting)

(6)

V and B are the velocity and induced magnetic field, respectively. Superscripts f, s, ex refer to fluid, solid, external medium, and Re, Rh and Rm are the Reynolds number, magnetic pressure and magnetic Reynolds numbers, respectively.

Figure 1: Annular pipe around a solid

2


111

Boundary Element Application Equations (2) are decoupled by taking U1 = V f +

∇ 2 U1 + M

ReRh f ReRh f B , U2 = V f − B as M M ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

∂U1 = −1 ∂y

in Ωf

⎪ ⎪ ⎪ ∂U2 ⎪ = −1. ⎪ ∇ U2 − M ⎭ ∂y

(7)

2

where M =

ReRhRmf . These equations are further transformed to modified Helmholtz equations ⎫

∇2 u1 − k 2 u1 = 0 ⎪ ⎪ ⎬

in Ωf

⎪

⎪ ∇2 u2 − k 2 u2 = 0. ⎭

where u1 = (U1 +

(8)

1 1 y)eky , u2 = (U2 − y)e−ky and k = M/2. M M

1. Insulating External Medium Ωex In this case Equation (3) and boundary relations (6) on (Γ2 ) drop, and V f = B f = 0 on Γ2 implies u1 and u2 are known on Γ2 . Thus, discretization of the pipe walls Γ1 and Γ2 with constant elements gives the system of equations for one element

[H s ]{B s } = [Gs ]{ ⎡ ⎢ ⎢ ⎣ ⎡ ⎢ ⎢ ⎣

1 1 H11 H12 1 1 H21 H22 2 2 H11 H12 2 H21

2 H22

⎤⎧ Γ ⎪ ⎪ u 1 ⎥⎨ 1 ⎥ ⎦⎪ ⎪ ⎩ uΓ2

⎫ ⎪ ⎪ ⎬

⎤⎧ Γ ⎪ ⎪ u 1 ⎥⎨ 2 ⎥ ⎦⎪ ⎪ ⎩ uΓ2

⎫ ⎪ ⎪ ⎬

1

2

⎡

∂B s } ∂n

G1

⎢ 11 =⎢ ⎣ ⎪ ⎪ ⎭ G121

⎪ ⎪ ⎭

⎡ ⎢

=⎢ ⎣

G112 G122

G211 G212 G221 G222

(9) ⎤⎧ Γ ⎪ ⎪ q 1 ⎥⎨ 1 ⎥ ⎦⎪ ⎪ ⎩ q Γ2

⎫ ⎪ ⎪ ⎬

⎤⎧ Γ ⎪ ⎪ q 1 ⎥⎨ 2 ⎥ ⎦⎪ ⎪ ⎩ q Γ2

⎫ ⎪ ⎪ ⎬

1

2

⎪ ⎪ ⎭

⎪ ⎪ ⎭

(10)

(11)

and discretized boundary relations on Γ1

[ K ]{ uΓ1 1 } + [ K ]{ uΓ2 1 } = {0} 2ReRh 2 [ K ]{ uΓ1 1 } − [ K ]{ uΓ2 1 } − [ I ]{ B s } = {y} M M s 2M ∂B 2 ∂y [ I ]{ } } = { [ K ]{ q1Γ1 } − [ K ]{ q2Γ1 } + Rms ∂n M ∂n

(12) (13) (14)

where H, G, H s , Gs are BEM matrices constructed from the fundamental solutions of modified Helmholtz and Laplace equations, respectively. K, K are diagonal matrices with Kii = ekyi , K ii = e−kyi and qi denotes normal derivative of ui . The whole system (9)-(14) will be solved for 8 unknowns on Γ1 and Γ2 .

3

112


2. Conducting External Medium Ωex In this case Equation (3) and boundary relations (6) on (Γ2 ) stay but B f = 0 in Equation (5) drops. As an addition to system of equations (9)-(11) we have

[H ex ]{B ex } = [Gex ]{

∂B ex } ∂n

(15)

∂B s and three more boundary relations on Γ2 replacing B s and in Equations (12)-(14) with B ex ∂n ex ∂B and , and uΓi 1 and qiΓ1 with uΓi 2 and qiΓ2 , respectively are written. There are 12 unknowns ∂n now since uΓ1 2 and uΓ2 2 are also not known.

Numerical Results Numerical results are obtained from the assembled global system of equations for B s , V f , B f and B ex if the external medium is insulating or conducting. Fig. (2) represents fluid velocity and induced V

-3

0 X

-3

0 X

B

-3

-3

0 X

3

0 X

3

-3

0 X

3

0.104 0.0624 0.0208 -0.0208 -0.0624 -0.104

3

Y

0

-3

B

0.187 0.1122 0.0374 -0.0374 -0.1122 -0.187

3

Y

Y

0

-3

B

0.076 0.0456 0.0152 -0.0152 -0.0456 -0.076

3

0

-3

3

0.444 0.3554 0.2668 0.1782 0.0896 0.001

3

Y

0

-3

3

V

0.111 0.0898 0.0676 0.0454 0.0232 0.001

3

Y

Y

0

-3

V

0.434 0.3498 0.2626 0.1754 0.0882 0.001

3

0

-3

-3

0 X

3

Figure 2: Velocity of the fluid and induced currents for Re = 1, Rh = 10 and for Rmf = Rms = 1(left), Rmf = 50, Rms = 1 (center), and Rmf = 1, Rms = 50(right) magnetic fields of the solid and fluid for increasing values of magnetic Reynolds numbers for the case of Ωex is insulating. Boundary layers are formed for fluid velocity in the direction of the applied magnetic field (y -direction) as Rmf increases. Continuation of B s and B f is very well observed especially for Rmf = Rms . As Rmf increases B f suppresses B s behaving as induced current in an annular pipe. On the contrary, when Rms is higher than Rmf , induced current B s expands through the pipe and increases fluid velocity action.

4


V 0.082 0.0658 0.0496 0.0334 0.0172 0.001

V 0.22 0.177 0.133 0.089 0.045 0.001

-3

0 X

0

-3 -3

3

0

-3

0 X

3

0

-3 -3

0

0

3

X

3

B 0.0047 0.00282 0.00094 -0.00094 -0.00282 -0.0047

3

Y

3

Y

Y

-3

-3 -3

3

B 0.02 0.012 0.004 -0.004 -0.012 -0.02

B

0

0

X

0.128 0.0768 0.0256 -0.0256 -0.0768 -0.128

3

V 0.041 0.033 0.025 0.017 0.009 0.001

3

Y

0

-3

3

Y

Y

3

113

0

-3 -3

X

0

3

X

Figure 3: Velocity of the fluid and induced currents for Rmf = 10, Rms = 1, Rh = 10 and for Re = 1(left), Re = 10 (center), and Re = 50(right) An increase in the Re has the same effect on the fluid velocity as the formation of boundary layer in the direction of the applied magnetic field. Induced magnetic field of the fluid forms also boundary layers around the solid but in the y -direction from bottom to top of the pipe. Also, magnitudes of both fluid velocity and induced current drop as Re increases (Fig. 3). In Fig. (4) induced magnetic fields B s , B f and B ex are shown for the case of Ωex is conducting. Again continuation of induced currents on the pipe walls is observed for equal magnetic Reynolds numbers. As Rmf increases B f tries to close itself in the annular pipe but still connecting to solid and outside induced currents. If Rms is higher than Rmf induced magnetic field of the fluid forms boundary layer perpendicular to applied magnetic field squizing to the top and bottom of the outer pipe wall with the strong effect of solid induced current.

References [1] M. Tezer-Sezgin, S. Han Aydın Computing. 95(1) 751–770 (2013). [2] S. Han Aydın, M. Tezer-Sezgin J Comput Appl Math 259(B) 720–729 (2014). [3] M. Tezer-Sezgin, S. Han Aydın, S, FEM Solution of MHD Flow Equations Coupled on a Pipe Wall in a Conducting Medium, PAMIR 2014. [4] L. Drago¸s, Magnetofluid Dynamics, Abacus Pres (1975).

5

114


B

4

4

3

3

2

2

1

1

0

0

-1

-1

-2

-2

-3

-3

-4

-4

-5

-4

-2

0 X

2

0.175 0.105 0.035 -0.035 -0.105 -0.175

5

Y

Y

B

0.151 0.0906 0.0302 -0.0302 -0.0906 -0.151

5

-5

4

-4

-2

0 X

2

4

B

4

4

3

3

2

2

1

1

0

0

-1

-1

-2

-2

-3

-3

-4

-4

-5

-4

-2

0 X

2

0.169 0.1014 0.0338 -0.0338 -0.1014 -0.169

5

Y

Y

B

0.202 0.1212 0.0404 -0.0404 -0.1212 -0.202

5

-5

4

-4

-2

0 X

2

4

Figure 4: Induced currents for Rmex = 1, Re = 1, Rh = 10 and for Rmf = Rms = 1 (left-top), Rmf = 10, Rms = 1 (right-top), Rmf = 50, Rms = 10 (left-bottom) and Rmf = 10, Rms = 50 (right-bottom)

6


115

MHD Flow in Rectangular Ducts of Partly Conducting Walls under an Inclined Magnetic Field Canan Bozkaya1 and M. Tezer-Sezgin2 1

2

Department of Mathematics, Middle East Technical University, 06800, Ankara, Turkey, e-mail: [email protected] Department of Mathematics, Middle East Technical University, 06800, Ankara, Turkey, e-mail: [email protected]

Keywords: MHD flow, BEM, inclined magnetic field.

Abstract. The magnetohydrodynamic (MHD) flow which is laminar and steady of a viscous, incompressible conducting fluid flowing down a rectangular duct of thin wall under the effect of an external magnetic field is considered. The flow is driven by the current produced by electrodes placed symmetrically around y-axis in each of the walls (conducting parts) of the duct parallel to x-axis. The external magnetic field is applied at different angles to the insulated walls which are kept at the same magnetic field value in magnitude but opposite in sign. A boundary element method (BEM) solution has been obtained by using a fundamental solution which enables to threat the MHD equations directly in their coupled form with general wall conductivities. Numerical calculations are carried out for several values of Hartmann number M by applying the external magnetic field in different angles. The results are presented in terms of equivelocity and induced magnetic field contours. It is noted that as M increases, boundary layers are formed for both the velocity and the induced magnetic field, and the velocity becomes uniform at the center of the duct in the direction of the externally applied magnetic field.

Introduction The effect of the magnetic field on the flow of an incompressible, viscous, electrically conducting fluid is a significant problem and has been investigated by many scientist. It has many practical applications in MHD generators, pumps, accelerators, flowmeters and nuclear reactors. The external magnetic field determines the appearance within the fluid of an induced current which can be made to flow in an external circuit through the conducting walls. In this manner, some of the internal energy of the fluid is given up to the exterior as utilizable electrical energy. The MHD equations are coupled convection-diffusion type equation in velocity and induced magnetic field due to the coupling of the hydrodynamic flow equations and Maxwell equations of electrodynamics through Ohm’s law. The analytical solutions of these equations are available only for some simple geometries under simple boundary conditions [1, 2]. Thus, it is significant to devise numerical techniques to obtain approximate solutions of MHD flow in ducts with mixed type of wall conditions. Several numerical techniques such as FDM [3], FEM [4, 5] and BEM [6, 7] have been employed to produce numerical solutions for MHD flow in ducts for several configuration of interest. In all these numerical studies, the walls of the ducts are taken as either completely insulated or conducting. On the other hand, a meshless method has been used for solving MHD flow equations in channels of arbitrary cross-section and for arbitrary wall conductivities in the work [8]. Recently, Hsieh et.al. [9] has solved the MHD flow in ducts for high Hartmann number values by an upwinding difference scheme. The present study focuses on the direct BEM application to the MHD duct flow equations in their original coupled form when the walls of the duct are partly insulated and partly conducting. Thus, the BEM formulation of MHD flow in a semi-infinite duct by the work of Bozkaya et.al. [10] is adapted for MHD rectangular duct flow subject to an inclined magnetic field when the horizontal duct walls are partly insulated-partly conducting whereas the vertical duct walls are insulated. However, in previous BEM applications given in the literature the walls were taken insulated, which enables to decouple the equations, and the solutions were given only for small or moderate values of Hartmann number. Therefore, in this study the fundamental solution derived in [11] is employed to treat the MHD flow equations in their original coupled form through the BEM application.

116


Basic equations We consider the two-dimensional, steady, laminar flow of a viscous, incompressible conducting fluid in a rectangular duct subject to a constant and uniform inclined magnetic field. The flow is driven by the current produced by electrodes with length 2l placed symmetrically around y-axis in each of the walls of the duct parallel to x-axis. The governing equations are derived from Navier-Stokes equations of fluid dynamics interacting Maxwell’s equations of electromagnetics through Ohm’s law. Thus, the coupled equations of steady MHD flows are given in non-dimensional form by [1] ∂B ∂B + My ∂x ∂y ∂V ∂V 2 + My ∇ B + Mx ∂x ∂y ∇2V + Mx

=

−1

=

0

in Ω

(1)

where Ω is the rectangular duct, {(x, y) | − a ≤ x ≤ a, −b ≤ y ≤ b} shown in Fig. 1. The magnetic field is M = (Mx , My ) with the components Mx = M sin α and My = M cos α , where M is the Hartmann number and α is the angle between the applied magnetic field of intensity B0 and the positive y-axis. The dimensionless velocity V (x, y) and the induced magnetic field B(x, y) are in the direction of the duct axis (z-axis). The boundary conditions which are suitable in practice for the MHD flow in a rectangular duct can be expressed as B(±a, y) = ±k B(x, ±b) = −k B(x, ±b) = k ∂ B/∂ y(x, ±b) = 0

−b < y < b, −a < x < −l, l 6=> (8 # 8 " # !-*P3-*X3.*!,551

152


Time Dependent Fracture Non-Linear Problems with the Boundary Element Method E. Pineda León1, A. Rodríguez-Castellanos2, M.H. Aliabadi3 1

Escuela Superior de Ingeniería y Arquitectura, Instituto Politécnico Nacional, México D. F., email: [email protected], 2 Instituto Mexicano del Petróleo, México D F. Eje Central Lázaro Cárdenas 152, Gustavo A Madero, México D F, e-mail: [email protected] 3 Department of Aeronautical Engineering, Imperial College London,

Abstract. The Application of the Dual Boundary Element Method (DBEM) visco-plasticity an J-Integral is presented. The specimens analyzed are a square plate, a plate with a hole and a notched plate, all of them with different crack length. The boundary is discretized with quadratic continuous and semi-discontinuous elements, but the domain with nine nodes internal cells. In the case of plasticity and visco-plasticity only the part susceptible to yielding was discretized, but the creep case required the discretization in the whole domain. The Von Misses yield criterion is applied in the cases of plasticity and visco-plasticity so the material used for these sort of analysis are metals. In creep analysis the materials used are also metals. I. - Introduction For many years, problems of stress analysis in industry have been solved using Finite Difference Method and the Finite Element Method (FEM). FEM and the Boundary Element Method (BEM) have attained a level of development that has made them necessary tools for modern design engineers. The FEM is routinely used as a general analysis tool. The BEM.s applicability at present is not as wide ranging as FEM, however the method has become established as an e¤ective alternative to FEM in several important áreas of engineering which include acoustics and fracture mechanics. The attraction of BEM can be attributed to the reduction in dimensionality of the problem; for two-dimensional problems, only the line-boundary of the domain needs to be discretized into elements. This means that, compared to domain type analysis techniques, a boundary element analysis can result in substantial reduction in modelling e¤ort. In BEM, for certain nonlinear problems such as plasticity and creep part or the whole of the domain also needs to be discretized. However,only the boundary displacements and tractions are treated as unknown and hence the system matrix remains the same size as an equivalent elastic problem (see for example [1]). BEM has been applied to elastoplastic problems since the early seventies with the work of Swedlow and Cruse [24] and Richardella [21] who implemented the von Mises criterion for 2D problems using piecewise constant interpolation for the plastic strains. Later, Telles and Brebbia [25] and others had, by the beginning of the eighties, developed and implemented BEM formulations for 2D and 3D inelastic, viscoelastic and elastoplastic problems (see [1] for further details). In recent years, Aliabadi and co-workers [2] have introduced a new generation of boundary element method for solution of fracture mechanics problems. The method which was originally proposed for linear elastic problems [4][19][16] has since been extended to many other .elds including problems involving nonlinear material and geometric behaviour [7-9,12]. In this paper, the DBEM formulation for inelastic materials is presented. The dual equations of the method are the displacement and traction boundary integral equations. When the displacement equation is applied on one of the crack surfaces and the traction equation on the other, general mixed-mode problems can be solved. The nonlinear creep behaviour is modelled through the use of initial strain approach. The creep region is discretized with internal quadratic quadrilateral cells. The C-integral is implemented and evaluated as the fracture parameter. Several examples are presented to demonstrate the accuracy and effiency of the proposed method. This work is dedicated to the visco-plastic and creep analysis by applying the Dual Boundary Element Method (DBEM) in order to find a more efficient and accurate analysis technique in fracture mechanics problems. 2. - Discontinuous Boundary Elements and Internal Cells To keep the simplicity of the boundary element and for the sake of efficiency, discontinuous quadratic elements are used for the crack modelling. The general modelling strategy can be summarized as follows:


153

x

Discontinuous quadratic elements are used to model the crack boundaries.

x

Continuous quadratic elements are used to model the remaining boundary of the structure, except at the intersection between an edge and the crack mouth, where semi-discontinuous or discontinuous elements are used on the edge in order to avoid common nodes at the intersection.

x

The displacement boundary integral equation is applied for collocation on the upper crack surface, described as follows

cij u j ³ tijc u j d*

³ uc t d* ³

*

*

ij j

:

c H ajk d: V ijk

(1)

3. - DUAL BOUNDARY INTEGRAL EQUATIONS 3.1 Displacement and traction equations for crack surfaces In order to apply DBEM in crack analysis we consider two equations; the displacement equation located in the upper surface of the crack and the traction equation located in the lower surface of the crack. Consider a cracked plate where Γ^{±} are the boundaries of the crack and Γ is the rest of the boundary. Before the deformation takes place, the boundaries Γ^{±} are coincident. If the boundary integral displacement from equation (1) is applied to both crack boundaries the result is identical, which leads to a singular system. To avoid such system another set of independent equations needs to be employed. This extra set is provided by the boundary traction integral equations; these equations are found to be independent from the boundary displacement integral equations and can be applied to either of the crack boundaries. The displacement integral equation for the lower crack surface can be written as: ૚ ૚ ࢛ሶ ൫࢞Ʋ ൯ ൅ ૛ ࢛ሶ ࢏ ൫࢞Ʋି ൯ ൅ ‫׬‬ડ ࢚࢏࢐ ൫࢞Ʋା ǡ ࢞൯࢛ሶ ࢏ ሺ࢞ሻࢊડ ૛ ࢏ ା

ൌ ‫׬‬ડ ࢛ሶ ࢏࢐ ൫࢞Ʋା ǡ ࢞൯࢚ሶ࢏ ሺ࢞ሻࢊડ ൅ ‫׬‬ષ ࣌Ʋ࢏࢐࢑ ሺ࢞Ʋǡ ࢠሻࢿሶ ࢇ࢐࢑ ሺࢠሻࢊષ

(2)

From the definition of tractions followed by the application of Hooke's law and through the differentiation of the displacement boundary integral equation it is possible to define the time-dependent traction equation as: ࢚ሶ࢏ ൌ ࣌ሶ ࢏࢐ ࢔࢐ (3) Finally, from the definition of traction, multiplying the above equation by the outward unit normal nj and noticing that ࢔࢏ ሺ࢞ᇱ ି ሻ ൌ െ࢔࢏ ሺ࢞ᇱ ା ሻ the following equation is obtained

݊௜ ሺ‫ ି ݔ‬ሻ ‫ ܦ ୻׬‬ᇱ ௜௝௞ ሺ‫ିݔ‬ᇱ ǡ ‫ݔ‬ሻ‫ݐ‬ሶ௞ ሺ‫ݔ‬ሻ݀Ȟ െ

ଵ ଵ ‫݌‬ሶ ሺ‫ ݔ‬ᇱ ሻ െ ଶ ‫݌‬ሶ௝ ሺ‫ݔ‬ାᇱ ሻ ൌ ଶ ௝ ି ఈ ሺ‫ݔ‬ሻ݀ȳ ݊௜ ሺ‫ ି ݔ‬ሻ ‫ ܵ ୻׬‬ᇱ ௜௝௞ ሺ‫ିݔ‬ᇱ ǡ ‫ݔ‬ሻ‫ݑ‬ሶ ௞ ሺ‫ݔ‬ሻ݀Ȟ ൅ ቂ‫׬‬ஐ ȭᇱ ௜௝௞௟ ሺ‫ିݔ‬ᇱ ǡ ‫ݖ‬ሻߝሶ௞௟ ൅ ଵ ఈ ଶ ݂௜௝ ሺߝሶ௞௟ ሺ‫ݖ‬ሻሻቃ ݊௜ ሺ‫ݖ‬ሻ (4)

Where the integral ‫׬‬ષ stands for a Hadamard principal value Integral. Equation (4) represents the traction boundary equation on the lower crack surface for the source point xΖ′‫א‬Γ·. The displacement equation (38) is applied to all non-cracked surfaces. The displacement equation applied to all the required boundary and domain points leads to a system of linear algebraic equations. 4. - Examples In this section some visco-plastic problems are presented in order to demonstrate the validity of DBEM in fracture mechanic problems. A centre cracked tension specimen with a crack length a = 2 mm and geometry according to Figure 1 is considered in this example. The boundary of the problem is discretized with quadratic elements and the domain with 9-nodes internal quadratic cells. Visco-plastic behavior and plane stress analysis are assumed and a constant load of 250.09 MPa is applied. The boundary conditions are the same as in Figure 1a, the material properties and geometry are:

154


Geometry H = 35.6 mm W= 25.4 MM a = 6.492 mm

Material Properties Young’s Modulus: E=181,200 MPa Poisson’s ratio: v=0.3 Visco-plastic coefficient ɀ୮ ൌ ͲǤͲͶͷ Yield stress, ߪ௬ ൌ ͺͻͷǤͻ Applied stress, ߪ௔ ൌ ʹͷͲǤʹͻ

Figure 1. Centre cracked plate. a) Geometry and boundary conditions; b) BEM mesh, boundary elements and internal cells; c) FEM mesh. Benchmark problem for FEM/BEM Table 1 summarizes some aspects related to FEM and BEM models, including type, number of elements, size of matrices and time of analysis. It is clear from this table the system of equations for BEM is almost 20-times smaller than FEM. In the case of BEM for this particular problem the time of analysis is 30 seconds, while FEM takes up to 2 minutes by using the same computer. It is clear that BEM is faster to do the analysis than FEM, this difference is very important specially for the case when refined meshes are needed and the time will play a very important role. Another important difference is the fact that special elements around the carck tip (see the detail in Figure 1c), called “quarter point nodes”, have to be considered in FEM analisis, while with BEM special elements are not required in the cracked region. Table 1. Comparison between BEM and FEM mesh and size of matrix. Type of

Boundary

Boundary

Internal

Internal

Total of

Matrix size

Time of


Mesh BEM FEM

Elements 22

Nodes 44

cells 42

Nodes 111 3360

nodes 155 3360

155

310 x 310 6720 x 6720

analysis 30 sec. 2.0 min

The stress profiles in Figure 2 show the von Misses stresses against the distance from the crack tip. It is clear from Figure 2 that the maximum stresses are located close to the crack tip, where the plastic zone is developed. Two important parameters in order to reach convergence and accurate results are the tolerance and the initial time step. The tolerance applied in this case was 0.0001 sec. and the initial time step was Δt = 0.01 sec, which generated accurate results.

Figure 2. Von Misses stresses in centre cracked plate calculated at the distances x/a, for the visco-plastic analysis. 5 J-integral for visco-plasticity The material properties are similar to the above example, the boundary conditions, geometry and loads are shown in Figure 3 a. In this case σ = 12 N/mm2 which remains constant for the complete analysis, while the shear load τ is varying. The graphs of the Figure 3 b, c and d show results for the fracture parameters KI, KII and J-Integral in a visco-plastic analysis. They display the expected behaviour for these cases. Conclusions The contribution of this work is the development of an effective formulation to analyze fracture mechanics problems based on the dual boundary element to deal with stress-stain relations for visco-plastic behaviors. To accomplish this, the analysis of an existing non-linear time-independent boundary element formulation was extended to non-linear time-dependent problems. In this work, elasto-plastic two-dimensional boundary element formulation based on an initial strain approach was used to develop a visco-plastic formulation. The boundary integral equations for visco-plasticty have the same form as those for elasto-plasticity except that the plastic strains rates are replaced by visco-plastic strains rates. As was mentioned before, the BEM for elasto-plasticity is an incremental process (time-independent form), while the BEM for visco-palsticty is a rate dependent process (time-dependent). Several models with a crack, including a square plate, are analyzed. Special attention is taken when the discretization of the domain is done. In Fact, for the plasticity and visco-plasticity cases only the region susceptible to yield was discretized. The proposed formulation is presented as an alternative way to study these kind of non-linear problems. Results from the present formulation are compared to those of the well-established Finite Element Technique, and they are, generally, in good agreement.

156


Figure 3. a) Boundary conditions, geometry and loads; b) Results for KI; c) Results for KII and d) Results for J-Integral. References 1. Aliabadi, M.H., The Boundary Element Method. Applications in Solids and Structures. Vol. 2. JohnWiley & Sons, Ltd, West Sussex, England (2002). 2. Aliabadi,M.H. A new generation of boundary element methods in fracture mechanics, International Journal of Fracture, 86, 91-125, 1997. 3. Aliabadi,M.H. Boundary element formulations in fracture mechanics, Appl. Mech. Review, 50, 83-96, (1997). 4. Aliabadi,M.H. and Portela,A. Dual boundary element incremental analysis of crack growth in rotating disc. Boundary Element Technology VII, Computational Mechanics Publica-tions, Southampton, 607-616, (1992). 5. Bassani, J.L., and McClintock, F.A., Creep Relaxation of Stress Around a Crack tip, International Journal of Solids and Structures, 17, 479-492, (1981). 6. Becker, A.A., and Hyde, T.H., Fundamental Tests of Creep Behaviour, NAFEMS report R0027. (1993). 7. Chao Y.J., Zhu, X.K., Zhang, L., Higher-Order Asymptotic Crack-Tip fields in a power-law Creeping material, International Journal of Solids and Structures, 38, (2001). 8. Cisilino, A.P. and Aliabadi, M.H., Three-dimensional BEM Analysis for Fatigue Crack Growth in Welded Components, International Journal for Pressure Vessel and Piping, 70, 135-144, (1997). 9. Cisilino, A.P., Aliabadi, M.H. and Otegui, J.L., A Three-dimensional Element Formulation for the Elasto-Plastic analysis of Cracked Bodies, International Journal for Numerical Methods in Engineering, 42, 237-256, (1998). 10. Cisilino, A.P. and Aliabadi, M.H., Three-dimensional Boundary Element Analysis for Fa-tigue Crack in Linear and Non-Linear Fracture Problems, Engineering Fracture Mechanics, 63, 713-733, (1999).


157

11. Ehlers, R., and Riedel H., A Finite Element Analysis of Creep Deformation in a Specimen Containing a Microscopic Crack, in D. Francois(ed,), Advances in Fracture Research, Proc.Fifth. Int. Conf. on Fracture, Vol 2, Pergamon, New York, pp. 691-698, (1981). 12. Hutchinson, J.W., Singular behaviour at the end of a tensile crack in a hardening material, J. Mech. Phys. Solids, 16, 13-31, (1968). 13. Leitao, V., Aliabadi, M.H., Rooke, D.P., The Dual Boundary Element Method for Elasto-plastic Fracture Mechanics, Int. J. Num. Meth. Engng. (1994). 14. Li, F.Z., Needlemen, A., Shih, C.F., Characterization of Near Tip Stress and deformation fields in creeping solids. International Journal of Fracture, 36, 163-186, (1988). 15. Mendelson A., Boundary Integral Methods in Elasticity and Plasticity. Report No. NASA TN D-7418, NASA. (1973). 16. Mi,Y. and Aliabadi,M.H. Dual boundary element method for three-dimensional fracture mechanics analysis, Engng Anal. with Boundary Elem., 10, 161-171, (1992). 17. Oden, J.T., Finite Elements of Nonlinear Continua. McGraw-Hill, New York, (1972). 18. Ohji, K., Ogura, Kubo, S., Stress-Strain .eld and modi.ed integral J-Integral the vecinity of a crack tip under transient creep conditions. Japanese Society of Mechanical Engineer, 790-13 18-20 (1979). 19. Portela, A., Aliabadi, M.H., Rooke, D.P., The Dual Boundary Element Method: E¤ec-tive Implementation for Crack Problems, Int. Journ. Num. Meth. Engng., 33, 1269-1287, (1992). 20. [20] Providakis, C.P., and Kourtakis, S.G., Time-dependent Creep Fracture Using Singular Boundary Elements, Computational Mechanics, 29, 298-306, (2002). 21. Riccardella, P. An Implementation of the Boundary Integral Technique for plane problems of Elasticity and Elastoplasticity, PhD Thesis, Carnegie Mellon University, Pitsburg, PA (1973). 22. [22] Rice, J.R., Rosengren, G.F., Plane Strain Deformation Near a Crack Tip in a Power Law Hardening Material, J. Mech. Phys. Solids, 16, 1-12, (1968). 23. Riedel,H., and Rice,J.R. Tensil Cracks in Creeping Solids. Fracture Mechanics; 12th Congress, ASTM STP 700, American Society for Testing and Materials, pp112-130, (1980). 24. Swedlow, J. L. and Cruse, T. A. Formulation of the boundary integral equation for three-dimensional elastoplastic .ow, International Journal of Solids and Structures, 7, 1673-1681 (1971). 25. Telles, J. C. F., and Brebbia,C.A. Elastic/visco-plastic Problems using Boundary Elements, International Journal of Mechanical Sciences, 24, 605-618, (1982).

158


Advanced Beam Element for the Analysis of Engineering Structures E.J. Sapountzakis1 and I.C. Dikaros2 1 School of Civil Engineering, National Technical University, Zografou Campus, GR-157 80 Athens, Greece, [email protected] 2 School of Civil Engineering, National Technical University, Zografou Campus, GR-157 80 Athens, Greece, [email protected]

Keywords: Elastic stiffened plate, plate reinforced with beams, nonlinear analysis, bending, nonuniform torsion, warping, ribbed plate, boundary element method, slab-and-beam structure

Abstract. In the present study, a boundary element formulation is developed for the construction of an “advanced” 20x20 stiffness matrix and the corresponding nodal load vector of a member of arbitrary cross section taking into account shear lag effects due to both flexure and torsion, subjected to arbitrary external loading including warping moments. Introduction In engineering practice the analysis of spatial frames is frequently encountered. The involved beam members of such structures are usually analyzed employing Euler-Bernoulli or Timoshenko beam theories. Both theories maintain the assumption that cross sections remain plane after deformation. Thus, the formulation remains simple; however it fails to capture “shear lag” phenomenon which is associated with a significant modification of normal stress distribution due to nonuniform shear warping [1,2]. In up-to-date regulations, the significance of shear lag effect in flexure is recognized. However in order to simplify the analysis, the “effective breadth” concept is recommended. This simplifying approach may fail to capture satisfactorily the actual structural behavior of the member. Therefore, it is necessary to include nonuniform shear warping effects in the analysis. Similar considerations with the ones made for flexure could be also adopted for the problem of torsion which is also very often encountered in the analysis of spatial frames (e.g. in curved-in-plan bridges, buildings of complex geometry etc.). It is well-known, that a beam undergeneral twisting loading and boundary conditions, is leaded to nonuniform torsion. The major characteristic of this problem is the presence of normal stress due to primary torsional warping. In an analogy with Timoshenko beam theory when shear deformation is important, Secondary Torsional Shear Deformation Effect (STSDE) [3,4] has to be taken into account as well. The additional secondary torsional warping due to STSDE causes similar effects with shear lag in flexure, i.e. a modification of the initial normal stress distribution. In the present study, a boundary element formulation is developed for the construction of an advanced 20x20 stiffness matrix and the corresponding nodal load vector of a member of arbitrary cross section taking into account shear lag effects due to both flexure and torsion, subjected to arbitrary external loading including warping moments. Nonuniform warping distributions, which are responsible for shear lag effects are taken into account by employing four independent warping parameters, multiplying a shear warping function in each direction [1] and two torsional warping functions, which are obtained by solving corresponding boundary value problems. Ten boundary value problems with respect to kinematical components are formulated and solved using the Analog Equation Method [5], a BEM based technique. The warping functions and the geometric constants including the additional ones due to warping are evaluated employing a pure BEM approach. The aforementioned problems are formulated employing an improved stress field arising from the correction of shear stress components. Statement of the problem Stiffness matrix and nodal load vector Consider a prismatic element of length l with an arbitrarily shaped cross section occupying the region : of yz plane (Fig.1), the material of which is considered homogeneous, isotropic and linearly elastic with modulus of elasticity E and shear modulus G . In Fig.1 CXYZ is the principal bending coordinate system through centroid C , while yC , zC are its coordinates with respect to Sxyz reference coordinate system through shear center S .


159

The beam element is subjected to the combined action of arbitrarily distributed or concentrated axial loading px (X ) , transverse loading p y (x) and pz (x) , twisting moments mx (x) , bending moments mY (X ) , mZ (X ) , as well as to warping moments (bimoments) mM P (x) , mM P (x) , mM P (x) and mM S (x) (Fig.1a) [1]. x

Mxj, θxj Nj, uxj

MZj, θZj

Y

x

Z

Qzj, uzj

Z

Z

z pz

Qyj, uyj MYj, θΥj Υ

y Y

mx

mZ

t

yC

py

n a ω

y

S Qzk, uzk

mY

px l C C: Centroid S: Shear center

z

Qyk, uyk MYk, θΥk

S X

x

MZk, θZk Mxk, θxk

(a)

(b)

zC

Y

C Ω

E, G

Nk, uxk

Figure 1. Prismatic beam element (a) with a composite cross section of arbitrary shape occupying the two dimensional region : (b). Under the action of the aforementioned arbitrary external loading and of possible restraints, the beam member is leaded to nonuniform flexure and/or nonuniform torsion. Starting with the flexural behavior of the beam, the following remarks can be made. It is well-known that the bending moment at a beam cross section represents the distribution of normal stresses due to bending (primary normal stresses V xxP ). Due to the aforementioned bending moment variation along the beam length (nonuniform bending), shear stresses arise on horizontal sections of an infinitesimal beam element equilibrating the variation of normal stresses due to bending. Cauchy principle dictates that corresponding shear stresses arise on the plane of the cross section as well. If the assumption that plane sections remain plane after deformation (Euler-Bernoulli or Timoshenko beam theories) is maintained, the arising shear stresses obtain a uniform distribution over the section. However, this distribution violates local equilibrium since the requirement of vanishing tractions W xn on the lateral surface of the beam is not satisfied. Thus, the aforementioned shear stresses exhibit a nonuniform distribution over the cross section’s domain so that both local equilibrium and vanishing tractions W xn on the lateral surface of the beam are satisfied. These nonuniform shear stresses will be referred to as primary (or St.Venant) shear stresses ( W xyP , W xzP ) and lead the cross section to warp (Fig.2b). Furthermore, due to the nonuniform character of this warping along the beam length a secondary normal stress distribution V xxS is developed (Fig.2c). This normal stress distribution is responsible for the well-known shear lag phenomenon and it is taken into account by employing an independent warping parameter multiplying the warping function, which depends on the cross sectional configuration. The nonuniform distribution of secondary normal stresses V xxS along the length of the beam results in the development of secondary shear stresses W xyS , W xzS (Fig.2d), which equilibrate the variation of V xxS at an infinitesimal beam element. However, from Fig.3d it can be concluded in analogy to Timoshenko beam theory that the arising secondary shear stresses due to the use of the independent warping parameter fail to fulfill the boundary condition related to vanishing tractions W xn on the lateral surface of the beam. Thus, in the present study a modified stress field is applied with the aid of an additional warping function in order to “correct” secondary shear stresses W xyS ,

W xzS (Fig.3e). It is worth here noting that in order to remove the aforementioned inconsistency, instead of performing this “correction” the sequence of stress generation could be extended taking into account the contribution of higher order stresses (development of tertiary normal stresses V xxT due to varying secondary warping along the length of the beam, subsequent development of tertiary shear stresses W xyT , W xzT , subsequent quaternary normal stresses V xxQ etc.) until the error introduced in each stage due to the unfulfilled boundary conditions is negligible. However, this consideration leads to the formulation of highly complex differential equations with respect to an increased number of kinematic unknowns (DOFs). On the opposite side, the applied “correction” of stress distribution leads to sufficiently accurate evaluation of stresses [2], without the additional

160


increase in the number of the kinematic unknowns. The above remarks are also valid for the problem of nonuniform torsion taking into account secondary torsional shear deformation effect – STSDE [3,4]. In order to take into account torsional shear lag effects as well, the normal stress distribution due to secondary torsional warping M xS [4] is also taken into account (secondary warping normal stress V xxS ). This distribution is equilibrated by corresponding tertiary shear stresses W xzT , W xyT which, similarly with the case of shear lag analysis in flexure, require a correction. In the present study this is achieved by adding an additional torsional warping function. P V xx

=

+

P W xz

(b) (a) Primary normal stress due to bending ( V xxP ETY , x Z )

Primary shear stress distribution ( W xzP GJ ZP )YP, z )

S V xx S W xz

(d) (c) Secondary normal stress due to warping ( V xxS EKY , xMYP )

Secondary shear stress distribution according to model A ( W xzS GJ ZSMYP, z )

+

=

S W xz

(e)

Secondary shear stress distribution according to model B ( W xzS GJ ZS )YS , z ) Figure 2. Sequence of stress generation along the height of a rectangular cross section of a beam under flexure, according to models A, B. Within the above described context, in order to take into account nonuniform flexural and torsional warping (including shear lag effect due to both flexure and torsion), in the study of the aforementioned element in each node at the element ends, four additional degrees of freedom are added to the well-known six DOFs of the classical three-dimensional frame element. The additional DOFs include four independent parameters, namely K x , KY , K Z , [ x multiplying a shear warping function in each direction ( KY , K Z ) and two torsional warping functions ( K x , [ x ), respectively. These DOFs describe the “intensities” of the corresponding cross sectional warpings along the beam length, while these warpings are defined by the corresponding warping function ( MYP , M ZP , M xP , M xS ), depending only on the cross sectional configuration. Thus, the “actual” deformed configurations of the cross section due to primary (in each direction) shear and primary, secondary torsional warpings are given as KY x MYP y, z , KZ x MZP y, z , Kx x MxP y, z and [ x x M xS y, z at any position along the beam longitudinal axis, respectively. The corresponding stress resultants of the aforementioned additional DOFs are the warping moments M M P , Y

M M P , M M P , M M S (bimoments) along the beam length, arising from corresponding normal stress distributions. Z

x

x

These bimoments due to the aforementioned warpings constitute additional “higher order” stress resultants, which are developed in the nonuniform shear and torsion theories. By this modification the developed element is enhanced with the capability of taking into account both shear deformation and shear lag effects due to both flexure and torsion. Thus, the generalized nodal displacement vector in the local coordinate system, as shown in Fig.1a, can be written as


^D `

ª¬uxij

i

u ijy

uzij T xij TYij T Zij K xij KYij KZij

[ xij uxik

u iky

uzik

161

T xik TYik T Zik K xik KYik KZik [ xik º¼

T

(1)

and the respective nodal load vector as

^F ` i

ª N ij Qyij Qzij M xij M Yij M Zij M ijP M ijP M ijP M ijS N ik Qyik Qzik M xik M Yik M Zik M ikP M ikP M ikP M ikS º Mx MY MZ Mx Mx MY MZ Mx ¼ ¬

T

(2)

In eqn. (1) u x is the “average” axial displacement of the cross section and u y , u z describe the transverse displacements of shear center S . Moreover, TY , T Z are the angles of rotation due to bending along the centroidal Y , Z axes, respectively, while K x , [ x are the aforementioned independent warping parameters introduced to describe the nonuniform distribution of primary and secondary torsional warping and KY , K Z the nonuniform distribution of primary warping due to shear. Index i denotes the i-th beam element, while indices j , k refer to each element end. In eqn. (2) N , Qy , Qz , M x are the axial force, the shear forces and the twisting moment, respectively at the element ends given as N

EAux , x

QyP QyS

Qy

QzP QzS

Qz

M xP M xS M xT

Mx

(3a,b,c,d)

and M M P , M M P , M M P , M M S are the warping moments (bimoments) due to independent warping parameters K x , x

Y

x

Z

KY , K Z , [ x , respectively defined as

³V

MM P

:

x

M xP d :

MM S

xx

x

³V :

M xS d :

³V

MM P

xx

:

Y

MYP d :

³V

MM P

xx

:

Z

MZP d :

(4a,b,c,d)

xx

at the same sections and given as [1]

EI

E IM PM PK x , x IM PM PKY , x IM PM PKZ , x

MM P x

MM P Y

x

x

Y

K

MYPMYP Y , x

x

Z

x

IM PM PK x , x IM PM S [ x , x Y

x

Y

x

EI

E IM SM S [ x , x IM PM S KY , x IM PM S KZ , x

MM S x

MM P Z

x

x

MZPMZP

Y

x

Z

x

K Z , x IM M K x , x I M M [ x , x P P Z S

P S Z S

where ,i denotes differentiation with respect to i . In eqns. (3) Qi j ( i

y,z , j

(5a,b) (5c,d)

P, S ) are the primary and

j x

secondary parts of shear forces, while M ( j P, S ,T ) are the primary, secondary and tertiary parts of total twisting moment arising from the corresponding shear stress components presented in detail in [1]. These stress resultants are given as QyP

GAYPJ YP

QyS

QzP

GAZPJ ZP

QzS

M

P x

GI T

P x x, x

where J YP

M

G A J G I J

G AYS J YS D)S )S J xS D)S )T J xT

S x

u y , x T Z , J YS

Z

x

Z

x

S Z

S Z

D)S )S J xS D)S )T J xT

S x

S x

D)S )S J D)S )S J

Y

Z

x

x

Y

S Y

KY u y , x TZ , J ZP

Y

x

x

S Z

uz , x TY , J ZS

(6a,b) (6c,d) M

T x

G I J D)S )T J D)S )T J T x

T x

Z

x

S Y

Y

x

S Z

(6e,f,g)

KZ uz , x TY , J xP T x , x , J xS K x T x , x and

J xT

[ x K x T x , x are “average” shear strain quantities. Moreover, the geometric constants of the beam appearing in eqns. (5), (6) are given as Iij

³ i j d :,i, j :

M xP ,M xS ,MYP ,MZP

I xP

³

:

ª¬ y 2 z 2 zM xP, y yM xP, z º¼d :

(7a,b)

162

³

Dij

:


ª ¬ i j º ¼ d : , i, j

)Sx , )7x , )YP , )YS , ) ZP , ) SZ

(7c)

while it holds that AYP { D) P )P , AYS { D)S )S , AZP { D) P ) P , AZS { D)S )S , I xS { D)S )S , I xT { D)T )T . { , y i y , z i z Z

Z

Z

Z

Y

Y

Y

x

Y

x

x

x

is the gradient operator, i y , i z are the unit vectors along y , z axes, respectively. MYP , M ZP are the primary shear warping functions with respect to the centroid C [1], while M xP y, z , M xS are the primary and secondary torsional warping functions with respect to the center of twist S [4]. Finally, ) Sx , )Tx , )YP , )YS , ) ZP , ) SZ are warping functions given from

MxP MxS Z MYP

) Sx )

P Y

MxS MxT MYP MYS

)Tx )

S Y

(8a,b) )

Y M

P Z

)

P Z

S Z

M M P Z

S Z

(8c,d,e,f)

where MYS y, z , MZS y, z , M xT y, z are additional warping functions introduced in order to “correct” shear stress distribution arising directly from the kinematical assumptions [1]. Warping functions (8) can be determined by corresponding two-dimensional boundary value problems exploiting the longitudinal local equilibrium equation and the associated boundary condition, which will be given in next sections. The nodal displacement and load vectors given in eqns. (1), (2) are related with the 20x20 local stiffness matrix ª k i º the coefficients k ijk ( j, k 1,2,...,20 ) of which are established by expressions (3), (5), (6) obtained by ¬ ¼ solving the following system of differential equations [1] EAux, xx

px

(9a)

G A A

u G A A u EI T G A EI T G A P Y

S Y

P Z

S Z

ZZ

E I

x

MYPMxP

z , xx

TY , x GA K

pz

(9c)

A

S Y

S x

S Z

MZPMZP

Kx, xx

G I I P x

(9b)

Z

x

)YS ) Sx

Z

u u

D)S )T

MYPMxS S x

x , xx

S Z Y

Y

MZPM xS

Z , xx

GD

x , xx

S Y

x

x , xx

S Z

T x

T x

x, x

Y

x

Z

Z

x, x

x

x

T x , xx GD)S )T [ x, x

Z

)YS ) Sx

Z

y,x

Y

z,x

Y

x

x

Z

)YS )Tx

x

) SZ ) Sx

Z

)YS ) Sx

Y

) SZ ) Sx

x,x

) SZ )Tx

x

S x

T x

K

x

x

Y

x

) SZ )Tx

x

)YS )Tx

x

Z ,x

x

y , xx

(9e)

Y

[x

mM P (9f)

[x

mM P (9g)

) SZ )Tx

x,x

)YS )Tx

x,x

)YS )Sx

Z ,x

Z

Y

D)S )T Y

x

T x , x GI [ x G D)S )S D)S )T KZ u y , x TZ T x

G D)S )S D)S )T KY uz , x TY mM P Y

)YS )Tx

x, x

(9d)

Z

x

(9h)

E IM PM PKx , xx IM PM PKY , xx IM PM PKZ , xx G I I x

Y

Z

T Z GA KZ G D)S )S D)S )T Kx T x , x GD)S )T [ x

z,x

MYPMYP Y , xx

T x

y, x

S Y

KY , x uz , xx TY , x mx x

x

m A T GA K D K T GD [ m I K I [ GA K u T G D D K T GD I K I [ GA K u T G D D K T GD I T G I I K GI [ G D D K u T G D

E IM PM PK x , xx Z

py

S Z Y ,x

P Z

YY Y , xx

K

TZ , x GA KZ , x G D)S )S D)S )T Kx , x T x , xx GD)T )T [ x , x

P Y

Z , xx

GD

y , xx

S Y

Z

x

Z

x

(9i)

x

E IM PM S KY , xx IM PM S KZ , xx IM SM S [ x , xx GI xT [ x Kx T x , x GD)S )T KZ u y , x TZ GD)S )T KY uz , x TY mM S Y

x

Z

x

x

x

Z

x

Y

x

x

(9j) subjected to the corresponding boundary conditions which are given as a1ux D 2 N D3 E1u y E2Qy E3

J 1uz J 2Qz

E1TZ E2 M Z

J 1TY J 2 MY

E3

J3

J3

(10a) (10b,c) (10d,e)


E1KZ E2 MM

E3

P Z

J 1KY J 2 MM

J3

P Y

G1K x G 2 MM

G1T x G 2 M x G3

163

(10f,g)

G3

P x

G1[ x G 2 MM

G3

S x

(10h,i,j)

setting the external loading quantities equal to zero and applying appropriate values to functions Di ,Ei ,Ei ,Ei J i ,J i ,J i ,G i ,G i ,G i ( i 1,2,3 ) (e.g. for a unitary u y displacement at x 0 it is:

D1

E1 E1 E1 J 1 J 1 J 1 G1 G1 G1 1 ,

E3 1 , D 2 D3

E2

E2

E3

E3 J 2 J 3 J 2

E2

J3

G 2 G 3 G 2 G 3 G 2 G 3 0 ). In eqns. (9) the externally applied warping moments mi ( i M ,M ,M ,M ) are related to the component t x of the traction vector applied on the lateral surface of the beam as P x

mi x

³ t i d s,i

P Y

P Z

S x

MxP ,MYP ,MZP ,MxS . According to the nodal load vector, assuming that the span of the beam is

* x

subjected to arbitrary loading as described above, the evaluation of the elements of

^F `

establishing again the solution of the boundary value problems (9), (10) Di ,Ei ,Ei ,Ei J i ,J i ,J i ,G i ,G i ,G i ( i 1,2,3 ) and more specifically, for

E1 E1

J 1 G1 G1 G1 1 , D 2 D3

E2

E2

E3

E3

E2

E3 J 2 J 3 J 2 J 3 G 2

i

is accomplished

for appropriate values of

D1

E1

G3 G 2 G3 G 2

J1 J1 G3

0 at

S S , ) CZ , x 0, l . Finally, it is noted that model A can be derived by modifying eqns. (6), (7), (9) by replacing ) CY P P , MCZ , M SS , respectively [1]. )TS with MCY

Warping functions due to shear )YP , ) ZP , )YS , ) SZ The analysis described in the previous section presumes that the warping functions )YP , ) ZP , )YS , ) SZ with respect to CXYZ are already established. According to the procedure presented in [1], examining each primary warping function ) iP ( i Y ,Z ) separately, a boundary value problem in each direction is formulated as 2 )YP )

P Y ,n

2 ) ZP

Z 0on the boundary

where )YP

) ZP ,n

Y

(11a,b)

0on the boundary

(11c,d)

I ZZ ) ZP AYP , while ,n indicates the derivative with respect to the outward

IYY )YP AZP , ) ZP

normal vector to the boundary n . After establishing ) iP ( i Y ,Z ), primary shear areas can be evaluated as AYP

2 IYY D)P )P . Moreover, after evaluating ) iP ( i Y ,Z ), warping functions MiP ( i Y ,Z )

2 I ZZ D)P )P , AZP Z

Z

Y

Y

can be established through eqns. (8c,e). Subsequently, MiS ( i Y ,Z ) employed in order to correct shear stresses are indirectly computed through ) iS ( i Y ,Z ) which can be established by formulating the following boundary value problem in each direction 2 )iS

MiP ,

where )YS

( i Y ,Z )

)iS,n

0on the boundary

)YS IM PM P Z AZP , ) SZ Y

Y

(12a,b)

) SZ IM PM P Y AYP and )YS Z

IM PM P )YS AZS , ) SZ

Z

establishing ) , secondary shear areas can be evaluated as A S i

S Y

Y

Y

2

S Z

IM PM P D)S )S , A Z

Z

Z

Z

IM PM P ) SZ AYS . After Z

Z

2

IM PM P D)S )S . Y

Y

Y

Y

Warping functions due to torsion M xP , ) Sx , )Tx Similarly with the previous section, the torsional warping functions are established independently as follows. M xP with respect to Sxyz is given by the following boundary value problem (St. Venant problem of uniform torsion) as 2M xP

0

MxP,n

zny ynz on the boundary

(13a,b)

164


Following the strategy presented in the previous section, ) Sx can be established by formulating the following boundary value problem as 2 ) Sx

M xP

where ) Sx

) Sx,n

0on the boundary

IM PM P ) Sx I xS . I xS is evaluated as I xS x

x

(14a,b) IM2PM P D)S )S while, after evaluating ) Sx , M xS can be x

x

x

x

established through eqn. (8a). The final boundary value problem for )Tx is constructed as 2 )Tx

M xS

while it holds that )Tx

)TS ,n

0o on the boundary

)Tx IM SM S M xP I xS and )Tx x

x

(15a,b)

IM SM S )Tx I xT . Finally, I xT is evaluated as I tT x

x

IM2SM S D)T )T . S

S

S

S

Numerical solution According to the precedent analysis, the formulation of the advanced 20x20 stiffness matrix and corresponding nodal load vector reduces in establishing the components u x , u y , u z , T x , TY , T Z , K x , KY , K Z , [ x having continuous derivatives up to the second order with respect to x at the interval 0,l and up to the first order at x 0, l , satisfying the boundary value problem described by the coupled governing differential equations of equilibrium (eqns. (9)) along the beam and the boundary conditions (eqns. (10)) at the beam ends x 0, l . Equations (9), (10) are solved using the Analog Equation Method (AEM) Error! Reference source not found., a BEM based method. Application of the boundary element technique yields a system of linear coupled algebraic equations which can be solved without any difficulty. The geometric constants of the cross section [1,2] are evaluated employing a pure BEM approach, i.e. only boundary discretization of the cross section is used. Numerical example A two-span steel beam of a two-cell rectangular cross section (Fig.3), ( E

2.1u108 kN m2 ,

G 1.05 u108 kN m2 , A 0.1304m2 , 40 beam elements) subjected to a distributed pz 5.0kN m and a concentrated Pz 10.0kN loading (Fig.3a), is examined. In Table 1 the geometric constants of the cross section as computed employing BEM are presented. In order to assess the behavior of models A, B, comparisons were made employing FEM software FEMAP [6,7] using 8-noded hexahedral solid finite elements (63000 solid elements). In order to ensure the absence of distortional phenomena of the cross section on the FEM model, rigid diaphragms were placed in regular distances [7]. In Fig.4 deflection u z obtained employing 20x20 (models A, B) and classic 12x12 beam stiffness matrices and in Fig.5 the contour lines of longitudinal displacement over the upper flange defined as ux x, y,0.25 0.25TY x KY x MYP y,0.25 are presented. The accuracy of both models A, B of the present study as compared to FEM solid model is noteworthy, while the influence of transverse shear deformation and shear warping can be easily observed (classic 12x12 stiffness matrix exhibits significant discrepancies). In Fig.6 bending M Y and warping M M P moment diagrams of the examined structure Y

are shown, while in order to assess the efficiency of models A, B in evaluating stress components, in Fig.7 the distribution of V xx along the boundary of the cross section at x 0 are also presented and in Table 2 the extreme values of kinematical and stress components are given. From the latter figure, the significant influence of shear lag phenomenon which cannot be predicted by classical beam elements can be verified. From Table 2 it can be also observed that model B converges more satisfactorily to solid FEM results, while model A overestimates stress values.


z

pz = -5.0 kN/m

Pz = -10.0 kN

165

z

C1 t = 0.04

x

C≡S

h = 0.5 m

(a) l1 = 5.0 m

l2 = 2.5 m

t = 0.04 m

(b)

y

l2 = 2.5 m b = 0.5 m

b = 0.5 m

Figure 3. Beam structure (a) of hollow rectangular cross section. Table 1: Geometric constants of hollow rectangular cross section. Model A

IYY m4

IM PM P m4 Y

Model B

0.004982

0.004982

7.6274E–05

7.6274E–05

Model A A m P Z

AZS m2

Y

Table 2: Extreme values of u z m , TY

2

Model B

0.0481

0.0481

0.082313

0.004128

rad , ux x, yC1 , zC1 m , V xxC1 kPa , O

of beam structure.

Classic 12x12 stiffness matrix

20x20 stiffness matrix – model A

20x20 stiffness matrix – model B

Solid FEM model [6]

uz x max

1.0492E–05

1.2942E–05

1.2868E–05

1.2630E–05

TY x max

7.1111E–06

7.6465E–06

7.5816E–06

–

ux x, yC1 , zC1 max

1.7778E–06

1.9551E–06

1.9386E–06

1.9512E–06

5.3633E+02

1.0417E+03

7.0932E+02

7.2603E+02

V

C1 xx max

Figure 4. Deflection u z of beam structure of example 1. 12x12 stiffness matrix: 1.7778E – 06m

ux max 0

0

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

20x20 stiffness matrix – model B: ux max 1.9386E – 06m

u max Figure 5. Contour lines of ux x, y,0.25 of beam structure.

Solid FEM: 1.9512E – 06m

166


Figure 6. Bending M Y and warping M M P moment diagrams of beam structure employing model B. Values in Y

parentheses correspond to classic 12x12 stiffness matrix. Values in brackets correspond to model A.

12x12 matrix: V xx max

5.3633E 02kPa

20x20 matrix - model B: V xx max

7.0932E 02kPa

(error: 26.12%) (error: 2.3%) Figure 7. Distribution of V xx over the boundary of the cross section at x=0. Error value in parentheses has been computed employing solid FEM solution as reference value. Acknowledgements This research has been co-financed by the European Union (European Social Fund ESF) and Greek national funds through the Operational Program "Education and Lifelong Learning" of the National Strategic Reference Framework (NSRF) Research Funding Program: THALES: Reinforcement of the interdisciplinary and/or inter-institutional research and innovation. References [1] I.C. Dikaros, E.J. Sapountzakis, Generalized Warping Analysis of Composite Beams of Arbitrary Cross Section by BEM Part I: Theoretical Considerations and Numerical Implementation. Journal of Engineering Mechanics ASCE, in press, DOI: 10.1061/(ASCE)EM.1943-7889.0000775. [2] I.C. Dikaros, E.J. Sapountzakis, Generalized Warping Analysis of Composite Beams of Arbitrary Cross Section by BEM Part II: Numerical Applications. Journal of Engineering Mechanics ASCE, in press, DOI: 10.1061/(ASCE)EM.1943-7889.0000776. [3] V.G. Mokos, E.J. Sapountzakis, Secondary Torsional Moment Deformation Effect by BEM, International Journal of Mechanical Sciences, 53 (2011), 897-909. [4] V.J. Tsipiras, E.J. Sapountzakis, Secondary Torsional Moment Deformation Effect in Inelastic Nonuniform Torsion of Bars of Doubly Symmetric Cross Section by BEM, International Journal of Non-linear Mechanics, 47 (2012), 68-84. [5] J.T. Katsikadelis, The Analog Equation Method. A Boundary – only Integral Equation Method for Nonlinear Static and Dynamic Problems in General Bodies, Theoretical and Applied Mechanics, 27 (2002), 13-38. [6] FEMAP for Windows. Finite element modeling and post-processing software. Help System Index, Version 10, (2008). [7] Siemens PLM Software Inc., NX Nastran User’s Guide, (2008).


167

Torsional Vibration Analysis of Bars Including Secondary Torsional Shear Deformation Effect by BEM E.J. Sapountzakis1, V.J. Tsipiras2 and A.K. Argyridi3 1,2,3

School of Civil Engineering, National Technical University, Zografou Campus, GR-157 80 Athens, Greece [email protected], [email protected], [email protected]

Keywords: shear stresses, warping, bar, beam, nonuniform torsion, secondary torsional shear deformation effect, torsional vibrations, boundary element method.

Abstract. In this paper a BE method is developed for the torsional vibration problem of bars of arbitrary doubly symmetric constant cross section, taking into account the secondary torsional shear deformation effect (STSDE). The bar is subjected to arbitrarily distributed or concentrated dynamic torsional loading along its length, while its edges are subjected to the most general torsional and warping boundary conditions. Apart from the angle of twist, the primary angle of twist per unit length is considered as an additional 1-D degree of freedom in order to account for the STSDE in the equations of motion of the bar. The warping shear stress distribution and the pertinent secondary torsional rigidity are computed by satisfying local equilibrium considerations under dynamic conditions without adhering to assumptions of Thin Tube Theory (TTT). By employing a distributed mass model system accounting for rotatory and warping inertia, an initial boundary value and two boundary value problems with respect to the variable along the bar time-dependent 1-D kinematical components, to the primary and secondary warping functions, respectively, are formulated. The latter are solved employing a pure BE method, requiring exclusively boundary discretization of the bar’s cross section. The numerical solution of the initial boundary value problem is performed through a BE method leading to a system of differential equations with displacement only unknowns. Both open- and closed- shaped cross section bars are examined and the benefits from the inclusion of nonuniform torsional and STSD effects in dynamic analysis of bars are demonstrated, especially in the latter case. 1 Introduction When arbitrary torsional boundary conditions are applied either at the edges or at any other interior point of a bar, this bar under the action of general twisting loading is leaded to nonuniform torsion. In this case, apart from the well known primary (St. Venant) shear stress distribution, normal and secondary (warping) shear stresses arise formulating the warping and secondary torsional moments, respectively [1]. Besides, many loading conditions occurring in engineering practice require taking into account inertial effects in torsional analysis of bars. The inclusion of such effects, especially through distributed mass models in which warping inertia arises, requires a rigorous analysis. The STSDE [2-3], associated with the inclusion of warping shear stresses in global equilibrium, generally necessitates the introduction of an additional kinematical component (apart from the angle of twist) in the displacement field of the bar, increasing the difficulty of the problem at hand. Moreover, if suitable warping shear stress distribution is not employed, the longitudinal local equilibrium equation and its corresponding boundary condition are not satisfied under dynamic or even static conditions, affecting also the secondary torsional rigidity of global equilibrium (see [3-4]). To the authors' knowledge, the inclusion of all of the aforementioned features in torsional vibration analysis of bars, without adhering to the assumptions of TTT, has not been achieved in the research literature. In this paper a BE method is developed for the torsional vibration problem of bars of arbitrary doubly symmetric constant cross section, taking into account the STSDE. By employing a distributed mass model system accounting for rotatory and warping inertia, an initial boundary value and two boundary value problems with respect to the variable along the bar time-dependent 1-D kinematical components, to the primary and secondary warping functions, respectively, are formulated. The latter are solved employing a pure BE method, requiring exclusively boundary discretization of the bar’s cross section. The numerical solution of the initial boundary value problem is performed through a BE method leading to a system of differential equations with displacement only unknowns. The essential features and novel aspects of the present formulation are summarized as follows. i. The cross section is an arbitrarily shaped doubly symmetric thin or thick walled one. The formulation does not stand on the assumption of a thin walled structure and therefore the cross section’s primary, secondary torsional and warping rigidities are evaluated “exactly” in a numerical sense. ii. The warping shear stress distribution and the pertinent secondary torsional rigidity are computed by satisfying local equilibrium considerations under dynamic conditions.

168


iii. The developed procedure retains most of the advantages of a BEM solution over a domain approach, although it requires longitudinal domain discretization. It is based on kinematical unknowns exclusively, successfully alleviating shear locking effects through a linked interpolation scheme [5]. 2 Statement of the Problem Let us consider a prismatic bar of length l (Fig. 1) with an arbitrarily shaped doubly symmetric cross section. The homogeneous isotropic and linearly elastic material of the bar cross section, with modulus of elasticity E , shear modulus G and mass density U occupies the two dimensional multiply connected region : of the y,z plane and is bounded by the * j j 1,2,...,K boundary curves, which are piecewise smooth. The bar is

subjected to the combined action of the arbitrarily distributed or concentrated time dependent twisting mt mt x,t and warping mw mw x,t moments acting in the x direction (Fig. 1b).

ΓΚ Γ2 C{S{G

s y

Z E

n

mw x

C≡S≡G

(Ω)

P r Pq q t *

y,v

mt x

z,w

Γ1

l

z

x,u

S: center of twist { C: centroid { G: center of gravity

Kj 1 * j

(a)

(b)

Fig. 1. Prismatic bar of an arbitrarily shaped cross section (a) subjected to torsional loading (b). Assuming small twisting rotations and that the cross section maintains its shape at the transverse directions, under the aforementioned loading the displacement field of the bar taking into account STSDE is assumed to be given as

T c x,t I

u x, y,z,t

P x

P S

§

·

y,z ¨ T xP c x,t T xc x,t ¸ISS y,z

v x, y,z,t zT x x,t

©

¹ yT x x,t

w x, y,z,t

(1a) (1b,c)

where u , v , w are the axial and transverse bar displacement components with respect to the Syz system of axes;

c is the primary angle of twist per unit length [6] which is in general not equal

T x is the (total) angle of twist; T xP

to the total angle of twist per unit length; ISP , ISS are the primary and secondary warping functions with respect to the center of twist S (see [1]). Employing the 3-D linear strain-displacement relations and the Hooke's stress-strain law, the work contributing stress components can be expressed as

V xx

cc E T xP ISP

W xy

§ wI P · § · w) SS c GT xc ¨ S z ¸ G ¨ T xP T xc ¸ ¨ wy ¸ © ¹ wy © wy ¹

(2a,b)

secondary

primary

W xz

P S W xy W xy

§ wI P · § · w) SS c GT xc ¨ S y ¸ G ¨ T xP T xc ¸ ¨ wz ¸ wz © ¹ wz © wz ¹

P S W xz W xz

(2c)

secondary

primary

where ) SS ) SS y,z ISP ISS is an auxiliary secondary warping function. Exploiting the principle of virtual work, after some algebra, the following equations of motion of the bar are formulated

U I PT x G ItP ItS T xcc GItSK xc

mt

UCSK x ECSK xcc GItS K x T xc mw

(3a,b)


169

subjected to the initial conditions ( x 0,l )

T x xx,00 T x0 x

T x x,0 T x0 x

K x x,0 K x0 x

K x xx,00 K x0 x

(4a,b,c,d)

together with the boundary conditions at the bar ends x 0,l

D1M t D 2T x D3 where

E1M w E2Kx

denotes differentiation with respect to time, K x

E3

(5a,b)

T c is an independent warping parameter, a , E P x

i

i

( i 1,2,3 ) are time dependent functions specified at the boundary of the bar, noting that all types of the conventional boundary conditions can be derived from equations (5) by specifying appropriately the functions ai ,

Ei , while M w , M t are the warping and total torsional moments, respectively, at the bar ends and I P , I tP , CS are geometric constants [4]. Moreover, eqns(3-5) fully conform to the corresponding ones presented in [4] if geometrically nonlinear terms are neglected in the latter, with the exception of the expression of the secondary

³: «¬ w) S ª

w)

2º wz » d : . It is also noticed that in ¼ the case that nonuniform torsional effects (normal and secondary shear strains and stresses) are ignored in the presented equations, calling it for simplicity as “modified” uniform torsion theory (MUTT), the above established

torsion constant I tS , which is expressed here as ItS

S

wy

2

S S

initial boundary value problem is reduced to eqns(3a, 4a,b, 5a), in which it is set ItS 0 (see also [3]). Finally, it is pointed out that exploiting eqns(2, 3b), the longitudinal local equilibrium equation under dynamic conditions (see also [7]) and its corresponding boundary condition are satisfied (after some simplifications) by evaluating ISP from the relation ) SS

from the boundary value problem presented in [4] and ) SS

CS 2 ³: ª« w) SS ¬

w) 2

ItS CS ) SS

2º

wz » d : ) , where ) SS is an auxiliary secondary warping function given ¼ from the following boundary value problem

( ItS

wy

2) SS

S S

w) SS wn 0 on * j

ISP in :

(6a,b)

3 Integral Representations Numerical Solution The evaluation of ISP and ) SS (eqns(6)) is performed employing a pure BE method, requiring exclusively boundary discretization of the bar’s cross section. This method along with the evaluation of I tS and the rest geometric constants are outlined in [1, 3]. After the determination of the aforementioned quantities, the 1-D kinematical components are determined through the solution of the initial boundary value problem of eqns(3-5), which is accomplished employing the BEM, as this is developed in [8] for the solution of the corresponding inelastic static boundary value problem, after modifying it appropriately. According to this method, let u1 x,t T x x,t , u2 x,t K x x,t be the sought solution of the problem. The solution of the second order differential equations w 2u1 / wx2 are given in integral form as

³0 w

ui [ ,t

l

2

l

ui wx 2 u* dx ªu* wui wx wu* wx ui º ¬ ¼0

i

T xcc , w 2u2 / wx2 K xcc

1,2

(7)

where u* is the fundamental solution given in [8]. Following the methodology presented in [8] and exploiting eqns(3), eqns(7) can be written, after some algebraic operations, as

G ItP ItS u1 [ ,t

l

l

0

0

l

³ U I Pu1/2 dx ³ GIt u2 /1dx ³ mt /2 dx G It S

0

P

ItS > /1u1 @0 > M t /2 @0 l

l

(8a)

170


ECS u2 [ ,t

where the kernels / j

l

l

0

0

d ³ GIt ³ UCS u2 /2 dx

S

l

du1 · l § l ¨ u2 dx ¸ /2 dx ³ mw /2 dx > M w /2 @0 ECS > /1u2 @0 © ¹ 0

/ j r ( j 1,2 ), with r

(8b)

x [ , x , [ points of the bar are given in [8]. If STSD effects

are negligible, then u2 | uc1 . In such cases, locking effects could arise if proper care is not undertaken [5]. In the

present work, shear locking is alleviated by dividing the interval 0,l into L elements, on each of which u1 and

u2 are assumed to vary according to a linked interpolation scheme [5]. Employing the aforementioned procedure and a collocation technique, a set of 2L 2 semidiscretized equations of motion is obtained with respect to 2L 6 unknowns, namely the values of u1 i , u2 i

( i 2,3,...,L ) at the L 1 internal nodal points and the values of u1 j , u2 j , M t j , M w j ( j 1,L 1 ) at

the bar ends. Four additional semidiscretized equations of motion are obtained by applying the integral representations (8) at the bar ends [ 0,l . These 2L 2 equations along with the four boundary conditions (eqns(5)) yield a linear system of 2L 6 simultaneous Differential-Algebraic equations depending on both kinematical quantities and stress resultants at the bar ends. After some algebraic manipulations exploiting the boundary conditions and the applied integral representations (8) at the bar ends, the unknown stress resultants at the bar ends are expressed with respect to the unknown kinematical components at internal and boundary nodal points. After further elaboration, the equations of motion of the problem can be expressed as

> M @^d` > K @^d` ^ p`

in which > K @ and > M @ are 2 L 1 N f u 2 L 1 N f

^d `

and

^ p`

(9)

stiffness and mass matrices, respectively, while

are 2 L 1 N f u 1 column matrices of generalized unknown kinematical and known loading

quantities, respectively ( N f is the number of natural boundary conditions, with 0 d N f d 4 ). Eqns(9), together

^`

with the initial conditions ^d `0 , d 0 (which are trivially determined through eqns(4)), form an initial value problem of differential equations which can be solved using any efficient solver. After the solution of this problem, the unknown boundary stress resultants can be readily computed. Moreover, a post-processing step is required to obtain the derivatives uc1 , uc2 at any nodal point, which are useful for calculating stress resultants along the bar. To this end, eqns(8) are differentiated with respect to [ and the arising expressions yield uc1 , uc2 . 4 Numerical Example The forced vibrations of a double cell box shaped cross section (Fig. 2a) aluminium ( E 70GPa , G 26.9GPa , U 2.701kN sec 2 / m4 ) bar of length l 5m , clamped at both ends have been examined through the proposed method (namely STSDE) and MUTT. The bar is subjected to a concentrated torsional moment M t ,external ( t ) 10kNm (for t t 0 and vanishing initial conditions) at its midpoint. Two 2-D quadrilateral shell FEM models [9] have also been employed, namely one with rigid diaphragms at every set of nodes having the same longitudinal coordinate (FEM - diaph) and another with a single rigid diaphragm at the set of nodes situated at x l 2 (FEM - no diaph). The diaphragms are designed so as each set of nodes has the same translation and rotation in the cross section plane, corresponding to the assumption employed in the proposed method that the cross section maintains its shape at the transverse directions during deformation (no distortion). In Fig. 2b, the time history of the torsional rotation at the midpoint of the bar is presented as obtained from the aforementioned cases of analyses and FEM solutions. From the discrepancy between the shell FEM solutions, the significance of distortional deformations in predicting kinematical components is concluded in the specific example. Moreover, it is observed that the FEM-diaph and the STSDE responses practically coincide, demonstrating the accuracy of the proposed method. It is also indirectly concluded that shear locking is successfully alleviated through the proposed numerical technique. The small variation between STSDE and MUTT models indicate that nonuniform torsional effects influence slightly the response of kinematical components of thin walled closed shaped cross section bars.


171

θx(t) - middle of span 8.2E-05

7.2E-05

STSDE

6.2E-05

MUTT

θx(t) (rad)

FEM-no diaph 5.2E-05 FEM- diaph 4.2E-05 3.2E-05

I p ( m4 ) 1.940 u 10-2

2.2E-05

I tP ( m4 ) 1.133 u 10-2

1.2E-05

I tS ( m4 ) 1.336×10-3

2.0E-06

Cs ( m6 ) 6.288 u 10-5

(a)

0.000 -8.0E-06

0.002

0.004

0.006

0.008

0.010

(b)

t (s)

Fig. 2. Cross section (a) and time history of T x at the midpoint (b) of the bar of numerical example. Torsional Moments - left end

Mw(t) - left end - STSDE 14.00 MtpSTSDE 9.00

0.30 Mt(KNm)

Mw(t) (KNm)

0.40

0.20

MtsSTSDE MtpMUTT

4.00

Mtp+MtsSTSDE

0.10 0.00 0.000 -0.10

-1.000.000

0.002

0.004 t(s)

0.006

0.008

0.002

0.004

0.006

0.008

0.010

0.010 -6.00

t(s)

(a) Fig. 3. Cross section (a) and time history of T x at the midpoint (b) of the bar of numerical example.

(b)

In Figs. 3a,b the time histories of warping and torsional stress resultants at the left end of the bar are presented for the aforementioned cases of analysis, respectively, demonstrating the efficiency of the proposed formulation. Moreover, in order to directly compare stress results from the cases of analysis with the FEM solutions, the latter are exploited in order to resolve the position along the bar at which maximum Von Mises stress is developed. This position is found to be at the corner of the clamped edge of the bar for both FEM solutions (depicted with a circle in Fig. 2). In Fig. 4a, the time histories of Von Mises stress at the aforementioned position are presented as obtained from the aforementioned FEM solutions. Distortional deformations are proved to be significant in predicting the time varying response of stress components as well. Finally, the stress resultants of Figs 3a,b are exploited in order to determine the maximum Von Mises stress of the respective cases of analysis at the same position within the time interval 0 d t d 0.01sec . These results are presented in Fig. 4b along with the corresponding values of the FEM solutions as obtained from the time histories of Fig. 3, noting that these occur at different time instants. A very good agreement between the proposed method (STSDE) and the FEM-diaph solution is observed (3.05% discrepancy), demonstrating the accuracy of the proposed method in predicting stress components as well. Moreover, it is concluded that maximum Von Mises stress is significantly underestimated by the MUTT model (27.92% discrepancy with FEM-diaph solution), highlighting the benefits from the inclusion of nonuniform torsional effects in stress analysis of thin walled closed shaped cross section bars. This is in accordance with up-to-date codes regulating the design of aluminium structures (Eurocode 9 – part 1.3) and has also been reported in the literature for static analysis [2-3] of such bars.

172

1400


1323

1346

1313

VM Stess - diaph

992

1000

VM Stess - no diaph

1600 1400

800

1346 1323

1305

STSDE

1200

600

stress (kPa)

Von Mises Stress (kPa)

1200

400 200

1000 800

970

MUTT FEM - diaph FEM - no diaph

600 400

0 0

0.005 t (s)

0.01

200 0

(a) (b) Fig. 4. Time history of Von Mises stress at the corner of the left end of the bar of numerical example as obtained from FEM solutions [9] (a) and corresponding comparison of maximum Von Mises stress (b). 5 Concluding remarks The main conclusions that can be drawn from this investigation are a. The developed BE numerical technique is based on kinematical unknowns exclusively, successfully alleviating shear locking effects. Its accuracy in computing kinematical and stress components as compared to shell FEM solutions is noteworthy. b. Nonuniform torsional effects do not influence significantly kinematical components of thin walled closed shaped cross section bars undergoing torsional vibrations. However, stress analysis of such bars is greatly affected. Acknowledgements Financial support provided by the Project “THALIS”. The Project “THALIS” is implemented under the Operational Project “Education and Life Long Learning” and is co-funded by the European Union (European Social Fund) and National Resources (ESPA). References [1] E.J.Sapountzakis and V.G.Mokos, Warping shear stresses in nonuniform torsion by BEM, Computational Mechanics, 30, 131-142 (2003). [2] J.Murín and V. Kutis, An effective finite element for torsion of constant cross-sections including warping with secondary torsion moment deformation effect, Engineering Structures, 30, 2716-2723 (2008). [3] V.J. Tsipiras and E.J. Sapountzakis, Bars Under Nonuniform Torsion - Application to Steel Bars, Assessment of EC3 Guidelines, Engineering Structures, 60, 133-147 (2014). [4] E.J. Sapountzakis and V.J. Tsipiras, Shear Deformable Bars of Doubly Symmetrical Cross Section Under Nonlinear Nonuniform Torsional Vibrations – Application to Torsional Postbuckling Configurations and Primary Resonance Excitations, Nonlinear Dynamics, 62, 967-987 (2010). [5] O.C. Zienkiewicz and R.L. Taylor The Finite Element Method for Solid and Structural Mechanics, Elsevier (2005). [6] V.G. Mokos and E.J. Sapountzakis, Secondary torsional moment deformation effect by BEM. International Journal of Mechanical Sciences, 53, 897-909 (2011). [7] E.J. Sapountzakis and V.J. Tsipiras, Warping Shear Stresses in Nonlinear Nonuniform Torsional Vibrations of Bars by BEM, Engineering Structures, 32, 741-752 (2010). [8] V.J. Tsipiras and E.J. Sapountzakis (2012) “Secondary Torsional Moment Deformation Effect in Inelastic Nonuniform Torsion of Bars of Doubly Symmetric Cross Section by BEM”, International Journal of NonLinear Mechanics, 47, 68-84. [9] NX Nastran User’s Guide, Siemens PLM Software Inc. (2007).


173

Generalized Warping Analysis of Composite Beams of Arbitrary Cross Section by BEM I.C. Dikaros1 and E.J. Sapountzakis2 1 School of Civil Engineering, National Technical University, Zografou Campus, GR-157 80 Athens, Greece, [email protected] 2 School of Civil Engineering, National Technical University, Zografou Campus, GR-157 80 Athens, Greece, [email protected]

Keywords: Nonuniform warping; Shear lag; Flexure; Torsion; Shear deformation; Composite beams; Boundary Element Method.

Abstract. A boundary element method for the nonuniform warping analysis of composite beams of arbitrary cross section taking into account shear lag effects due to both flexure and torsion is presented. The beam can be subjected to arbitrary external loading including warping moments. Introduction In engineering practice the analysis of beam-like members under flexure is frequently encountered. In most cases, these members are analyzed employing Euler-Bernoulli or Timoshenko beam theories. However, both theories maintain the assumption that plane cross sections remain plane after deformation. Thus, the formulation remains simple; however it fails to capture “shear lag” phenomenon which is associated with a significant modification of normal stress distribution due to nonuniform shear warping [1,2]. In up-to-date regulations, the significance of shear lag effect in flexure is recognized. However in order to simplify the analysis, the “effective breadth” concept is recommended. This simplifying approach may fail to capture satisfactorily the actual structural behavior of the member. Therefore, it is necessary to include nonuniform shear warping effects in the analysis. Similar considerations with the ones made for flexure could be also adopted for the problem of torsion. It is well-known, that a beam under general twisting loading and boundary conditions is leaded to nonuniform torsion. The major characteristic of this problem is the presence of normal stress due to primary torsional warping. In an analogy with Timoshenko beam theory when shear deformation is important, Secondary Torsional Moment Deformation Effect (STMDE) [3,4] has to be taken into account as well. The additional secondary torsional warping due to STMDE causes similar effects with shear lag in flexure, i.e. a modification of the initial normal stress distribution. In the present study a general boundary element formulation for the nonuniform warping analysis of composite beams of arbitrary cross section taking into account shear lag effects due to both flexure and torsion is presented. The composite beam consists of materials in contact, each of which can surround a finite number of inclusions. The beam can be subjected to arbitrary external loading including warping moments. Nonuniform warping distributions are taken into account by employing four independent warping parameters multiplying a shear warping function in each direction [1] and two torsional warping functions, obtained by solving corresponding boundary value problems, formulated exploiting the longitudinal local equilibrium equation. A shear stress “correction” is also performed in order to improve the stress field arising from the employed kinematical considerations. Ten boundary value problems are formulated with respect to displacement and rotation components as well as to independent warping parameters and solved using AEM [5], a BEM based technique. Warping functions and geometric constants including the additional ones due to warping are evaluated employing a pure BEM approach. Statement of the problem Displacement, strain and stress components Consider a prismatic beam of length l , of arbitrarily shaped composite cross section consisting of materials in contact, each of which can surround a finite number of inclusions. The materials, occupying the simply or multiply connected regions : m ( m 1,2,..., M ) of the yz plane (Fig.1) are firmly bonded together and are

174


assumed homogeneous, isotropic and linearly elastic with modulus of elasticity Em , shear modulus Gm and Poisson ratio Q m . Let also the boundaries of the non-intersecting regions : m be denoted by * :m ( m 1,2,..., M ). These boundary curves are piecewise smooth, i.e. they may have a finite number of corners. In Fig.1 CXYZ is the principal bending coordinate system through the cross section’s centroid C , while yC , zC are its coordinates with respect to Sxyz system through the cross section’s center of twist S . The beam is subjected to the combined action of the arbitrarily distributed or concentrated axial loading px (X ) , transverse loading p y (x) and pz (x) , twisting moments mt (x) , bending moments mY (x) , mZ (x) and warping moments mM P (x) , mM S (x) , mM P (x) and S

S

CY

mM P (x) . CZ

*:m

pz

y Y

px Z

mt

z

mZ

mM P S

l

*

py

: mY

mM P CZ mM S C

E,G

y

1

mM P

CY

1

*1:1

Y

x X

En, Gn

M : m 1 m

* 0:1

SS

C: Centroid S:Center of twist

k i i 1 * :m M * m 1 :m

S z

C

C Ω

2 *: 1

*3:1

Ω

yC

1

E ,G m

m

Z

m

z

(a) (b) Figure.2. Prismatic beam under axial-flexural-torsional loading (a) of an arbitrary composite cross section occupying the two dimensional region : (b). Under the action of the aforementioned general loading and of possible restraints, the beam is leaded to flexure and/or torsion. Starting with the flexural behavior of the beam, the following remarks can be made. It is wellknown that in case of non-constant bending moment distributions, shear stresses arise on horizontal sections of an infinitesimal beam element equilibrating the variation of primary normal stresses due to bending ( V xxP ). Cauchy principle dictates that corresponding shear stresses arise on the plane of the cross section as well. Contrary to Timoshenko beam theory prediction, these shear stresses have a nonuniform distribution over the domain of the cross section to satisfy local equilibrium and the requirement of vanishing tractions over the lateral surface. These shear stresses constitute the primary or St.Venant shear stresses ( W xzP , W xyP ) and lead the cross section to warp. Furthermore, due to the nonuniform character of this warping along the beam length a secondary warping normal stress distribution V xxS is developed. This additional normal stress distribution is responsible for the well-known shear lag phenomenon and is taken into account by employing an independent warping parameter multiplying a corresponding warping function depending only on the cross sectional configuration [1]. The described nonuniform warping results in the development of secondary shear stresses W xzS , W xyS which equilibrate the variation of V xxS . However, in an analogy with Timoshenko beam theory, the secondary shear stress distribution arising from the use of the aforementioned independent warping parameter fails to fulfil zero-traction condition on the lateral surface of the beam. In the present study, in order to remove this inconsistency a shear stress correction is performed modifying the stress field by adding an additional warping function to “correct” W xzS , W xyS . The above remarks are also valid for the problem of nonuniform torsion with STMDE [3,4]. In the present study, in order to take into account torsional shear lag effects as well, normal stress distribution due to secondary torsional warping M SS [4] is also taken into account (secondary normal stress V xxS ). This distribution is equilibrated by corresponding tertiary shear stresses W xzT , W xyT which, similarly with the case of shear lag analysis in flexure, require a correction. In the present study this is achieved by adding an additional torsional warping function. Within the context of the above considerations, the displacement components of an arbitrary point of the beam are given as


175

P u x, y, z u x TY x Z T Z x Y K x x ISP y, z KY x ICY y, z KZ x ICZP y, z [ x x ISS y, z primary

(1a)

secondary

v x, y, z v x zT x x

w x, y, z w x yT x x

(1b,c)

where u , v , w are the axial and transverse beam displacement components with respect to the Sxyz system of axes. Moreover, v , w describe the deflection of the center of twist S , while u denotes the “average” axial displacement of the cross section. TY , T Z are the angles of rotation due to bending about Z , Y axes, respectively. K x , [ x are independent warping parameters introduced to describe the nonuniform distribution of primary and secondary torsional warping, while KY , K Z are independent warping parameters introduced to describe the nonuniform distribution of primary warping due to shear [1]. M SP , M SS are the primary and secondary P P , MCZ are the primary shear torsional warping functions with respect to the center of twist S [4], while MCY warping functions with respect to the centroid C . Finally, it holds that Z z zC , Y y yC . Employing eqns. (1), strain-displacement relations of three-dimensional elasticity for small displacements and the Hooke’s stress-strain law, the non-vanishing components of Cauchy stress tensor in : m ( m 1,2,..., M ) are obtained. In the present study, a correction of stress components is performed without increasing the number of S S , MCZ , M ST are introduced in global kinematical unknowns. To this end, three additional warping functions MCY stress expressions, which are written as P Em ªu, x TY , x Z T Z , xY K x , x MSP º Em ªKY , x MCY m KZ , x MCZP m [ x, x MSS m º¼ m¼ ¬ ¬

V xx m

primary

^

(2a)

secondary

`

P P P P ª P º Gm J ZP )CY , y m J Y ) CZ , y m J x ¬ z M S , y m ¼

W

xy m

primary

^

S S S S S Gm J ZS )CY , y J Y ) CZ , y J x ) S , y m

m

m

`G

m

secondary

^

ªJ xT )TS , y º m¼ ¬

`

(2b)

tertiary

P P P P ª P º Gm J ZP )CY , z m J Y )CZ , z m J x ¬ y M S , z m ¼

W xz m

primary S S S S S º ª T T º Gm ªJ ZS )CY , z m J Y ) CZ , z m J x ) S , z m ¼ Gm ¬J x ) S , z m ¼ ¬ secondary

(2c)

tertiary

where P Z MCY

P )CY

)

S S

M M P S

S S

S )CY

)

T S

P S MCY MCY S T MS MS

P Y MCZ

P )CZ

S )CZ

P S MCZ MCZ

(3a,b,c,d) (3e,f)

Warping functions (3) can be determined by formulating boundary value problems exploiting the longitudinal local equilibrium equation and the associated boundary condition, which will be given in next section. In what follows, a distinction will be made according to which stress description will be employed to derive global equations of equilibrium. The formulation employing stress expressions stemming right from the kinematical assumption will be referred to as model A, while the formulation based on corrected shear stresses (2) will be referred to as model B. Global equations of equilibrium In order to establish the differential equations of equilibrium based on model B, the principle of virtual work

³ V V

GH xx W xyGJ xy W xzGJ xz dV

xx

³ t G u t G v t G w d F Lat

x

y

z

(4)

176


is employed, where G denotes virtual quantities; t x , t y , t z are the components of the traction vector applied on the lateral surface of the beam including end cross sections denoted by F and V is the volume of the beam. The stress resultants of the beam as defined in [1], employing eqns. (1), (2) and the definitions of geometric constants [1] are expressed in terms of the kinematical components as N

Eref Au,x

(5a)

Eref IYYTY , x

MY

I

MM P

Eref IM PM PK x , x IM P M PKY , x IM P M PKZ , x

MM P

Eref

S

CY

QyP

S

S

P MCY MSP

CZ S

K x , x IM

P P CY MCY

KY , x IM

P S CY MS

[ x, x

Gref AYPJ YP

Eref I ZZTZ , x

MM S

Eref IM P M S KY , x IM P M S KZ , x IM SM S [ x , x

Gref A J

M tP

Gref I tPT x, x

P P Z Z

CY S

P MCZ

Gref I tS J xS D)S

S Y

S CZ ) S

S )CY ) SS

CZ S

P MCZ MSP

ref

S z

M tS

(5b,c)

M E I K I K I [ Q G A J D J D J Q G A J D J D J J D J M G I J D J D S

S y

Q

P z

CY S

MZ

ref

S S Y Y

ref

S Z

P P MCZ MCZ

x, x

S Z

S )CY ) SS

T t

S )CZ )TS

S x

S )CY )TS

T t

ref

T x

(5d,e)

S

P MCZ MSS

Z ,x

S x

S )CZ ) SS

S Z

S

(5f,g)

x, x

T x

(5h,i)

T x

(5j,k)

S Y

S )CZ )TS

S )CY )TS

J ZS (5l,m,n)

Eref , Gref are the modulus of elasticity and shear modulus, respectively of a reference material. Using the expressions of the strain components, the definitions of the stress resultants (eqns. (5)) and applying the principle of virtual work (eqn. (4)), the differential equations of equilibrium of the beam can be derived as

Eref Au,xx

px

Gref A A P Y

S Y

(6a)

v

, xx

TZ , x Gref A KZ , x Gref D)S S Y

S CZ ) S

Gref AZP AZS w, xx TY , x Gref AZSKY , x Gref D)S Eref I ZZT Z , xx Gref A A P Y

Gref D)S

T CZ ) S

[x

S Y

v

S CY ) S

D)S

K

D)S

K

T CZ ) S

T CY ) S

T Z Gref A KZ Gref D)S S Y

,x

S CZ ) S

T CY ) S

T x , xx Gref D)S

[ x, x

py

T x , xx Gref D)S

[x, x

pz

T CZ ) S

(6b) x, x

T CY ) S

D)S

T CZ ) S

K

(6c) x

Tx,x

mZ

(6d)

Eref IYYTY , xx Gref AZP AZS w, x TY Gref AZSKY Gref D)S

Gref D)S

x, x

[x

S CY ) S

D)S

T CY ) S

K

x

Tx,x

mY

(6e)

Eref IM P M PK x, xx IM P M P KZ , xx IM P M S [ x , xx Gref AYS KZ v, x TZ Gref D)S

CZ S

CZ CZ

Gref D)S

T CZ ) S

[x

CZ S

mM P

D)S

S CZ ) S

T CZ ) S

CY CY

Gref D)S

T CY ) S

Gref I I I P t

S t

T t

[x

T

Gref D)S

S CY ) S

CY S

S Z

mM P

Gref I I Kx , x G I [ x , x Gref D)S S t

D)S

T CY ) S

Tx,x

D)S

S CY ) S

T CY ) S

K

x

T x, x

(6g)

CY

x , xx

x

(6f)

CZ

Eref IM P M PK x, xx IM P M P KY , xx IM P M S [ x , xx Gref A KY w, x TY Gref D)S CY S

K

T t

K

Y ,x

T ref t

w, xx TY , x mt

S CZ ) S

D)S

T CZ ) S

K

Z ,x

v, xx TZ , x

(6h)

Eref IM PM PKx , xx IM P M PKY , xx IM P M PKZ , xx Gref I tS I tT K x T x , x Gref I tT [ x

S

S

CY S

Gref D)S

S CZ ) S

CZ S

D)S

T CZ ) S

K

Z

v, x T Z Gref D)S

S CY ) S

D)S

T CY ) S

K

Y

Eref IM P M S KY , xx IM P M S KZ , xx IM SM S [ x , xx Gref I tT [ x K x T x , x Gref D)S CY S

CZ S

S

S

w, x TY mM P

T CZ ) S

S

K

Z

v, x T Z

(6i)


Gref D)S

T CY ) S

K

Y

177

w, x TY mM S

(6j)

S

where the externally applied loads are related to the components t x , t y , t z as pi x

³ t d s,i x, y, z m x ³ t i d s,i M

mt x

* i

i

* x

P S

mY x

³ t y t zd s * z

y

mZ x ³ t xY d s

³ t Z d s * x

*

(7a,b,c,d)

,M ,M ,M S S

P CY

P CZ

(7e)

The above differential equations (eqns. (6)) are subjected to the corresponding boundary conditions of the problem at hand, which are given as a1u D 2 Nb E1v E2Vby

D3 E3

E1TZ E2 M bZ

J 1w J 2Vbz

E3

E1KZ E2 M bM

P CZ

G1T x G 2 M bt

J 1TY J 2 M bY

E3

(8a) (8b,c)

J3

J3

J3

G1K x G 2 M bM

G3

P CY

G3

(8d,e)

J 1KY J 2 M bM

P S

(8f,g)

G1[ x G 2 M bM

S S

G3

(8h,i,j)

at the beam ends x 0, l , where the reaction forces N b , Vby , Vbz , M bZ , M bY , M bM P , M bM P , M bt , M bM P , M bM S CZ

CY

S

S

are given by relations (5), when applied at x 0, l . Finally, D k ,Ek ,Ek ,Ek J k ,J k ,J k ,G k ,G k ,G k ( k 1,2,3 ) are functions specified at x 0, l . Boundary conditions (8) are the most general boundary conditions for the problem at hand, including also the elastic support. All types of conventional boundary conditions (clamped, simply supported, free or guided edge) can be derived from these equations by specifying appropriately these functions (e.g. for a clamped edge: D1 E1 E1 E1 J 1 J 1 J 1 G1 G1 G1 1 , D 2 D3 E2 E3 E2 E3 E 2

E3 J 2 J 3 J 2 J 3

G 2 G3

G 2 G 3 G 2 G 3 0 ). Finally, it is noted that model A can be derived

P P S S , ) CZ , )TS with MCY , MCZ , M SS , respectively and applying the same modifying eqns. (2) by replacing ) CY procedure.

P P S S Warping functions due to shear ) CY , ) CZ , ) CY , ) CZ P P S S , ) CZ , ) CY , ) CZ with The analysis described in the previous section presumes that warping functions ) CY respect to CXYZ are already established. According to the procedure presented in [1], examining each primary P ( i Y ,Z ) separately, a boundary value problem in each direction is formulated as warping function ) Ci

) G ) G ) 2

P CY

m

m

P CY , n m

m

P CY , n m

P where )CY

) G ) G ) 2

Z 0on the free surface of the beam P Gn )CY , n on the interfaces n

P CZ

m

m

P CZ , n m

m

P CZ , n m

Y

(9a,b) 0on the free surface of the beam

(9c,d)

P Gn )CZ , n on the interfaces

(9e,f)

n

P I ZZ )CZ AYP , while ,n indicates the derivative with respect to the outward

P P IYY )CY AZP , )CZ

P normal vector to the boundary n (directional derivative). After establishing ) Ci ( i Y ,Z ), primary shear areas

can be evaluated as AYP

2 I ZZ D)P

P CZ )CZ

, AZP

2 IYY D)P

P CY )CY

P . Moreover, after evaluating ) Ci ( i Y ,Z ), warping

functions MCiP ( i Y ,Z ) can be established through eqns. (3c,e). Subsequently, MCiS ( i Y ,Z ) employed in order S ( i Y ,Z ) which can be established by formulating to correct shear stresses are indirectly computed through ) Ci the following boundary value problem in each direction

178


) M , ( i Y ,Z ) G ) 0on the free surface of the beam 2

S Ci m

P Ci m

(10a) Gm )

S Ci , n m

m

S where )CY

S P S )CY C IM P M P Z AZ , )CZ CY CY

Gn )

S Ci , n m

S P S )CZ C IM P M P Y AY and )CY CZ CZ

S After establishing ) Ci , secondary shear areas can be evaluated as AYS

on the interfaces

S Ci , n n

S S IM P M P )CY AZS , )CZ CY CY

IM2P M P

CZ CZ

D)S

S CZ ) CZ

, AZS

(10b,c)

S IM P M P )CZ AYS . CZ CZ

IM2P M P

CY CY

D)S

S CY )CY

.

Warping functions due to torsion M SP , ) SS , )TS Similarly with the previous section, torsional warping functions are established independently as follows. M SP with respect to Sxyz is given by the following boundary value problem (St. Venant problem of uniform torsion) as

M G M G M 2

P S m

m

P S ,n m

m

P S ,n m

0

(11a) Gm zny ynz on the free surface of the beam

Gn M

G

P S ,n n

m

(11b)

Gn zny ynz on the interfaces

(11c)

Following the strategy presented in previous section, ) SS is established by formulating the following boundary value problem as

) M G ) 0on the free surface of the beam 2

m

S S m

P S m

(12a) Gm )

S S ,n m

where ) SS

IM PM P ) SS ItS . I tS is evaluated as I tS S

S

Gn )

S S ,n m

on the interfaces

S S ,n n

(12b,c)

IM2PM P D)S )S while, after evaluating ) SS , M SS can be S

S

S

S

established through eqn. (3e). The final boundary value problem for )TS is constructed as

) M G ) 0on the free surface of the beam 2

m

T S

m

S S m

T S ,n m

while it holds that )TS

)TS IM SM S MSP ItS and )TS S

S

(13a) Gm )

T S ,n m

Gn )

on the interfaces

T S ,n n

IM SM S )TS I tT . Finally, I tT is evaluated as I tT S

S

(13b,c) IM2SM S D)T )T . S

S

S

S

It is worth here noting that, since the problems (9), (10), (11), (12), (13) have Neumann type boundary conditions, each warping function contains an arbitrary integration constant indicating a parallel displacement of the cross section which is evaluated as in [1]. Numerical solution According to the precedent analysis, the nonuniform shear problem of composite beams reduces in establishing the components u x , v x , w x , T x x , TY x , T Z x , K x x , KY x , KZ x and [ x x

having continuous derivatives up to the second order with respect to x at the interval 0,l and up to the first order at x 0, l , satisfying the boundary value problem described by the coupled governing differential equations of equilibrium (eqns. (6)) along the beam and the boundary conditions (eqns. (8)) at the beam ends x 0, l . Equations (6), (8) are solved using the Analog Equation Method (AEM) Error! Reference source not found., a BEM based method. Application of the boundary element technique yields a system of linear coupled algebraic equations which can be solved without any difficulty. The geometric constants of the cross section [1,2] are evaluated employing a pure BEM approach, i.e. only boundary discretization of the cross section is used.


179

Numerical example In this example, in order to examine the flexural-torsional response of a monosymmetric beam, a homogeneous cantilever beam of a monosymmetric box-shaped cross section, as this is shown in Fig.2, ( E { Eref 4 u107 kN m2 , G { Gref 2 u107 kN m2 , l 10m ) under two load cases, is examined. More specifically, the beam is subjected to a concentrated force Py 1000kN (case I, Fig.2) or Pz 1000kN (case II, Fig.2) eccentrically applied at its tip cross section, while in Table 1 the geometric constants of the cross section are presented as computed employing model B. The aforementioned load cases have also been analyzed in [6] and in this study proper adjustments have been made so as to account for different coordinate systems. In Fig.3 the kinematical components w , TY , T x for load case II are presented as compared to the ones obtained from the FEM beam model presented in [6]. It is worth mentioning that the beam model of [6] is equivalent to model A of the present study (shear stresses are not corrected in each level), however it can gradually converge to the actual solution since multiple warping functions for shear and torsion (and corresponding independent warping parameters) can be incorporated into global equilibrium equations. From the above figure it can be easily observed that the present results and the corresponding ones of [6] are in excellent agreement. Furthermore, in order to verify the efficiency of model B in evaluating stress components, in Fig.4 the distribution of warping normal stresses (primary normal stress due to torsional warping + secondary normal stress due to shear and torsional warping) along the midline of the upper plate of the cross section for load cases I, II at x 0.05m and in Table 2 the corresponding stress values at the left upper joint of the box (Fig.2) are presented. From these figures and table, it can be observed that model A converges to the first solution of [6] (one parameter for shear in each direction and one parameter for torsion) exhibiting discrepancies from the actual solution due to inaccurate shear stress distribution. On the other hand, model B (involving the same global parameters with model A) converges satisfactorily to the results of [6] obtained by employing four parameters for shear in each direction and four parameters for torsion, as well as to shell model results [6]. This indicates that shear stress correction performed in this study leads to acceptable results without increasing the complexity of the formulation. Table 1: Geometric constants of box-shaped beam.

I ZZ

5.6570E 02m4

IYY

1.9962E 02m4

S

IM P M P

7.0222E 04m

IM PM P

9.2289E 04m6

S

S

4 4

AZS

2.3014E 03m2

IM P M P

2.8455E 04m5

ItP

2.0184E 02m4

IM P M S

1.5685E 04m5

ItS

3.7487 E 03m4

6.3665E 02m2

ItT

1.4397 E 04m4

A

3.2130E 02m

2

D)S

9.2153E 04m3

AYS

7.2076E 03m2

D)S

3.0309E 04m3

CY S

9.9893E 04m

CZ CZ

S

CY S

IM P M P

CY CY

7.7889E 05m6

IM S M S

A 1.1960E 01m2

AYP P Z

S CY ) S

T CY ) S

Table 2: Warping normal stress ( kN m2 ) at the left upper joint of the box-shaped cross section of the cantilever beam at x 0.05m for load cases I, II. Load case

Present study – model A

Present study – model B

Ferradi et al. (2013) [6] – one warping function

Ferradi et al. (2013) [6] – four warping functions

Ferradi et al. (2013) [6] – shell model

I

16571.9963

9369.0931

14069.0000

9360.0000

8774.0000

II

–35675.4695

–15848.8700

–32755.0000

–22026.0000

–22528.0000

Acknowledgements This research has been co-financed by the European Union (European Social Fund ESF) and Greek national funds through the Operational Program "Education and Lifelong Learning" of the National Strategic Reference Framework (NSRF) - Research Funding Program: THALES: Reinforcement of the interdisciplinary and/or inter-institutional research and innovation.

180


Load case II: Pz=1000 kN

z Z b =3.0 m f2

t=0.02 m

Load case I: P = –1000 kN

S C

y

h=1.0 m 0.7152 m

y Y 0.03963 m

bf1=1.0 10m

(a) (b) Figure 3. Monosymmetric box-shaped cross section of the cantilever beam (a) and kinematical components w x , TY x , T x x for load case II (b).

(a)

(b)

Figure 4. Distribution of warping normal stress along the midline of the upper plate of the cantilever beam at x 0.05m for load case I (a) and load case II (b).

References [1] I.C. Dikaros, E.J. Sapountzakis, Generalized Warping Analysis of Composite Beams of Arbitrary Cross Section by BEM Part I: Theoretical Considerations and Numerical Implementation. Journal of Engineering Mechanics ASCE, in press, DOI: 10.1061/(ASCE)EM.1943-7889.0000775. [2] I.C. Dikaros, E.J. Sapountzakis, Generalized Warping Analysis of Composite Beams of Arbitrary Cross Section by BEM Part II: Numerical Applications. Journal of Engineering Mechanics ASCE, in press, DOI: 10.1061/(ASCE)EM.1943-7889.0000776. [3] V.G. Mokos, E.J. Sapountzakis, Secondary Torsional Moment Deformation Effect by BEM, International Journal of Mechanical Sciences, 53 (2011), 897-909. [4] V.J. Tsipiras, E.J. Sapountzakis, Secondary Torsional Moment Deformation Effect in Inelastic Nonuniform Torsion of Bars of Doubly Symmetric Cross Section by BEM, International Journal of Non-linear Mechanics, 47 (2012), 68-84. [5] J.T. Katsikadelis, The Analog Equation Method. A Boundary – only Integral Equation Method for Nonlinear Static and Dynamic Problems in General Bodies, Theoretical and Applied Mechanics, 27 (2002), 13-38. [6] M.K. Ferradi, X. Cespedes, M. Arquier, A higher Order Beam Finite Element with Warping Eigenmodes, Engineering Structures, 46 (2013), 748-762.


181

An iterative coupling based on Green’s function to solve embedded crack problems E.F. Fontes Jr.1,a, J.A.F. Santiago1,b. J.C.F. Telles1,c 1

Civil Engineering Programme, COPPE/UFRJ, CaixaPostal 68506, CEP21941-972, Rio de Janeiro, RJ, Brazil, [email protected], [email protected], [email protected]

Keywords: MFS, Green’s function, iterative coupling, MLPG

Abstract An iterative coupling procedure using different meshless methods is presented to solve linear elastic fracture mechanic (LEFM) problems. The problem is decomposed into sub-problems, where each one is addressed using an appropriate meshless method. The method of fundamental solutions (MFS) using the numerical Green’s function (NGF) procedure as the fundamental solution has been chosen for modeling embedded cracks in an elastic medium and the meshless local Petrov-Galerkin (MLPG) method has been chosen for modeling the remaining sub-domain. Each meshless method run independently, coupled with an iterative update of interface variables to achieve the final convergence. The coupling procedure is easy to apply for any MFLE problems with one or more cracks and can save time in the construction of the problem representation by points. The iterative solution procedure presented yields good results as compared with the boundary element method and analytical solutions for stress intensity factor computations. 1 Introduction Modeling of computational mechanics problems by meshless methods have increasingly attracted the attention of researchers. In some cases, meshless methods can be an alternative to mesh type methods like the boundary element method (BEM) and the finite element method (FEM). Different types of meshless methods can be found in the works by Belytschko et al. [1], Golberg and Chen [2], Atluri [3], and Nguyen et al. [4]. Interesting coupling procedures between meshless method can be adopted, improving not only efficiency, but also solution accuracy for different coupled engineering problems as shown in the works [5-7]. In this work an efficient iterative coupling procedure to solve linear elastic fracture mechanics (LEFM) is developed. Here, the iterative coupling between the method of fundamental solutions (MFS) and the meshless local Petrov-Galerkin method (MLPG) is considered. The problem domain is divided in sub-domains. The MFS is adopted for modeling regions with embedded cracks [8] while for standard elastic regular sub-domains the MLPG is selected. In order to construct the solution, the MFS, firstly developed by Kupradze and Aleksidze [9], uses only a superposition of fundamental solutions associated to the problem. In the case of LEFM problems one can use the numerical Green's function (NGF) procedure developed by Telles et al. [10]. This strategy permits to solve the principal problem in a decoupled manner, without need to introduce a large number of near crack tip points, to capture accurate stress intensity factors (SIF), as opposed to the standard MLPG approach for fracture mechanics applications found in the works [5,11,12].

182


2 Iterative coupling between the MFS and the MLPG The goal of the iterative coupling is considering the global domain partitioned in sub-domains represented by different meshless methods and solving the coupling iteratively until a convergence criteria is satisfied. In this work the sub-domains are represented in the form of Fig. 1 and a sequential iterative algorithm proposed in [13] is employed. The sub-domains ષ۴‫ ܁‬comprehends a crack (or cracks) embedded and is solved by the MFS-NGF, whereas sub-domain ષ‫۾‬۵ is to be solved by the MLPG method.

Figure 1: Overview of the iterative coupling procedure between the MFS-NGF and the MLPG. 2.1 The method of fundamental solutions Consider the boundary value problem of an elastic solid of domain Ω enclosed by a boundary Γ governed by the Navier Equation, subjected to mixed boundary conditions in the absence of body forces. The method of fundamental solutions (MFS) [9] establish that the approximate solution can be constructed by a summation of similar problems solution given by the following superposition ே ‫כ‬ሺ ߦǡ ߯ሻ݀௜ ሺߦ௡ ሻ ‫ݑ‬෤௝ ሺ‫ݔ‬ሻ ൌ ෍ ‫ݑ‬௜௝

(1)

௡ୀଵ ே ‫כ‬ሺ ߦǡ ߯ሻ݀௜ ሺߦ௡ ሻ ‫݌‬෤௝ ሺ‫ݔ‬ሻ ൌ ෍ ‫݌‬௜௝

(2)

௡ୀଵ

where ‫ݑ‬෤௝ ሺ‫ݔ‬ሻ and ‫݌‬෤௝ ሺ‫ݔ‬ሻ are the approximations for displacements and the tractions at points ‫ א ݔ‬Ω ‫ ׫‬Γ, ‫כ‬ ‫כ‬ and ‫݌‬௜௝ are components of the numerical Green's function [10]. As usual, ߦ௡ ‫ ב‬Ω ‫ ׫‬Γ respectively and ‫ݑ‬௜௝ is the source point (virtual sources) and ݀௜ is an intensity factor associated with each direction i. The MFS used in this work is based on the numerical Green’s function developed in [10]. For crack problems the NGF can be written by ‫כ‬ሺ ௞ሺ ௖ ሺ ߦǡ ߯ሻ ൌ ‫ݑ‬௜௝ ߦǡ ߯ሻ ൅ ‫ݑ‬௜௝ ߦǡ ߯ሻ ‫ݑ‬௜௝

(3)

‫כ‬ሺ ߦǡ ߯ሻ ‫݌‬௜௝

(4)

ൌ

௞ሺ ‫݌‬௜௝ ߦǡ ߯ሻ

௖ሺ ൅ ‫݌‬௜௝ ߦǡ ߯ሻ

‫כ‬ሺ ‫כ‬ሺ ߦǡ ߯ሻ and ‫݌‬௜௝ ߦǡ ߯ሻ are the fundamental displacements and tractions in j direction at the field where ‫ݑ‬௜௝ ௞ሺ point ߯ due to unit point loads applied at the source point ߦ in i direction, and ‫ݑ‬௜௝ ߦǡ ߯ሻ and ௞ሺ ‫݌‬௜௝ ߦǡ ߯ሻrepresent the infinite elastic plane where the known Kelvin fundamental solution is calculated.


183

௖ሺ ௖ሺ ߦǡ ߯ሻ and ‫݌‬௜௝ ߦǡ ߯ሻ stand for complementary components of the fundamental problem defined as Also, ‫ݑ‬௜௝ an infinite space containing traction-free cracks of arbitrary geometry. Here, an indirect problem must be solved to compute the intensity factors ݀௜ . The boundary Γ is represented using M field points ‫ݔ‬௠ and N source points ߦ௡ are chosen to be distributed over an outside contour forming the fictitious boundary surrounding Γ. From the application of either Eqs. (1) or (2), depending on the boundary condition prescribed, for the N discrete field points ߦ௡ , a linear system of equations can be written:

‫ ܌ۯ‬ൌ ‫܊‬

(5)

where ࡭ is the coefficient matrix, ࢊ the unknown intensity factor vector and ࢈ the right-hand side vector. Once all the values of ݀௜ are determined, the displacement and the traction at any point on the boundary can be evaluated using Eq. (1) and Eq. (2), respectively. In addition, the displacement at any point inside the domain can be evaluated using Eq. (1). Note that matrix ࡭ is not necessarily a square matrix. Additionally, the stress intensity factors can be directly calculated by the superposition of the fundamental generalized openings of the crack ܿ௜௝ and the intensity factors ݀௜ [8]. 2.2 The meshless local Petrov-Galerkin The MLPG methods are a class of truly meshless methods that do not require any kind of element or mesh to generate approximations for the field variables or even cells to compute numerical integrals [14]. Among the several formulations of the MLPG methods, here we use the version known by MLPG-1, as described in [3] with the displacement field ࢛ being approximated by the moving least square (MLS) approximation [15]. For a point ࢞ in the global domain, the following approximation can be written: ࢛ሺ࢞ሻ ൌ ઴் ሺ࢞ሻ࢛௝௉ீ

(6)

where ઴் comes from the MLS approximation and needs to be carried out for each point ࢞within a local sub-domain using a compact support (equivalent to a circle of radius r centered at ࢞) to permit a local approximating function. In the MLPG-1, the same weight function as in the MLS approximation is taken to be the test function in all local sub-domains over which integrals are calculated. The generalized local weak form of the Navier equation and the boundary conditions over the local sub-domains ȳ௦ cover the whole global domain given by each point ࢞. Taking into account the stressstrain relations and the strain-displacement relationships at a local boundary Ȟ௦ ൌ ‫ܮ‬௦ ‫ ׫‬Ȟ௦௨ ‫ ׫‬Ȟ௦௧ , a system of linear equations can be written: (7) ۹࢛௉ீ ൌ ࢌ where, for ݅ǡ ݆ ൌ ͳǡʹǡ ‫ ڮ‬ǡ ݊ and ࢑ܑ‫ ܒ‬ൌ න ࣕ‫ ܟ‬୧ ۲۰୨ ݀Ω ൅ ߙ න ࢝୧ ሺ࢞ሻ઴୨ ݀Γ െ න ࢝୧ ሺ࢞ሻ‫ۼ‬۲۰୨ ݀Ȟ

(8)

ࢌܑ ൌ න ࢝୧ ሺ࢞ሻ‫݌‬ҧ ݀Γ ൅ ߙ න ࢝୧ ሺ࢞ሻ‫ݑ‬ത݀Γ െ න ࢝୧ ሺ࢞ሻ࢈݀Ω

(9)

Ωೞ

Γೞೠ

Γೞ೟

Γೞೠ

୻ೞೠ

Ωೞ

where ࢝୧ ሺ࢞ሻ is the test function associated with a point i,۲ is a matrix that depends on whether the problem is of plane stress or plane strain, ઴୨ is the shape function from the MLS approximation, ࣕ‫ ܟ‬୧ and ۰୨ represent partial derivatives of the test functions and shape functions, respectively, ‫ ۼ‬is a matrix composed of terms related to the outward normal direction to the boundary and ߙ is a penalty factor

184


employed to enforce satisfaction of the essential boundary conditions. The superscript PG represents the MLPG method variables. 2.3 The iterative coupling procedure Considering the compatibility and equilibrium conditions at the common interface ડ ۷‫ ܔ‬between the meshless methods: ௉ீ (10) ࢛ிௌ ூ೗ ൌ ࢛ ூ೗ ௉ீ ࢖ிௌ ூ೗ ൌ െ࢖ூ೗

(11) ୘

ிௌ ௉ீ where the subscript ୪ represents the common interface and the vectors ࢛ிௌ ൌ ൫࢛ிௌ ൌ ிௌ ࢛ூ೗ ൯ , ࢛ ୘

୘

୘

௉ீ ிௌ ௉ீ ிௌ ௉ீ ൫࢛௉ீ ൌ ൫࢖ிௌ ൌ ൫࢖௉ீ ௉ீ ࢛ூ೗ ൯ , ࢖ ிௌ ࢖ூ೗ ൯ and ࢖ ௉ீ ࢖ூ೗ ൯ represent the decomposed displacement and tractions for each meshless method. The iterative coupling algorithm comes from a successive update of the interface variables defined in Eqs. (10-11) as follow:

(i) The main problem is decomposed in two or more sub-problems and each one is modeled by either the MLPG method or the MFS if in the presence of cracks; (ii) Choose over the common interface ࢛ிௌ ூ೗ ൌ ૙ for the MFS; ிௌ (iii) Solve Eq. (5) and obtain the tractions ࢖ிௌ ூ೗ using Eq. (2) for sub-domain ષ ; ୘

ிௌ (iv) Assembly matrix and nodal force vector Eqs. (8-9) using ࢖௉ீ ൌ ൫࢖௉ீ ௉ீ െ࢖ூ೗ ൯ ;

(v) Solve Eq. (7) for displacements࢛௉ீ ூ೗ ; (vi) Check for convergence at interface values, i.e. ிௌ ቛ࢛ிௌ ூ೗ǡ೙శభ െ ࢛ூ೗ǡ೙ ቛ

ቛ࢛ிௌ ூ೗ǡ೙శభ ቛ

൑ ͳͲି଺

(12)

if yes then stop; ிௌ ௉ீ (vii) Otherwise set ࢛ிௌ ூ೗ǡ೙శభ ൌ ሺͳ െ ߚ ሻ࢛ூ೗ǡ೙ ൅ ߚ࢛ூ೗ǡ೙ , where ߚ is the same optimal relaxation parameter

used in [13]; (viii) Return to step (iii) until convergence is achieved at step (vi). 3 Numerical results The virtual sources were distributed in a circular way. The number of field points on the MFS has been always the same as the number of virtual sources. The NGF generation procedure described in [10] was strictly followed. The MLPG results have been obtained by using quartic splines as the weight function with circular compact supports and the numerical integral carried out on local circular sub-domains by the traditional Gaussian quadrature, using in all examples ͳʹ ൈ ͳʹintegration points for a sub-domain ȳ௦ and 12 integration points for the boundaries Ȟ௦௧ and Ȟ௦௨ .


185

Consider the Brazilian disk test developed by professor Carneiro as presented in [16], which is a useful method for determining the tensile strength of concrete materials. Here, a bi-material disk specimen, with a central crack of length 2a, is subjected to a compression load in plane stress. The dimensions are shown in Fig. 2. The problem is modeled by MFS with the NGF procedure in the interior circle using 52 boundary points and the outer circle is modeled by the MLPG method using 1000 points (no symmetry). The problem was solved for various values of crack length a with fixed radius r=40 and R=75. The SIF is normalized, calculated for several ratios of ߤଵ Ȁߤଶ, and ߥଵ ൌ ߥଶ ൌ ͲǤͳ.

Figure 2: Brazilian bi-material case. Fig. 2 shows the SIF versus the ratio of the shear modulus for several values of a/r. It is noted that as the crack distance ratio a/r increases the SIF increases. The result for ߤଵ Ȁߤଶ ൌ ͳ and some values of a/r exhibit less than 2% of relative error, as seen in Table 1, when compared to the closed form solutions presented in [17]. The number of iterations for the coupling procedure is also shown in Table 1. ‫ܭ‬ூ Ȁ‫ܭ‬଴ ‫ܭ‬ூ Ȁ‫ܭ‬଴ error #iter (current paper) (Yarema) 0.1 1.0227 1.0150 0.76 16 0.2 1.0645 1.0600 0.42 20 0.3 1.1257 1.1355 0.86 24 0.4 1.2565 1.2431 1.08 26 0.5 1.3852 1.3869 0.12 23 Table 1: Normalized SIF for the Brazilian test with ߤଵ ൌ ߤଶ a/r

4 Concluding remarks In this paper an iterative coupling between two meshless methods is presented to solve LEFM problems. The use of NGF in the MFS sub-domains allows for accurately solving crack problems saving computational cost. Good results for SIF computations were obtained in the example presented. Further improvements can be obtained if one takes advantage of parallel computing, a natural procedure for the coupling routine presented here. Acknowledgements: The National Council for Scientific and Technological Development (CNPq). References [1] Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P. Meshless methods: An overview and recent developments, Comput Methods Appl Mech Eng 1996; 139:3-47.

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[2] M.A. Golberg, C.S. Chen. The method of fundamental solutions for potential, Helmholtz and diffusion problems, in Boundary Integral Methods - Numerical and Mathematical Aspects, ed. M.A. Golberg, Computational Mechanics Publications, 1998, pp.103-176. [3] S.N. Atluri, S. Shen. The meshless local Petrov-Galerkin (MLPG) method. Tech. Science 2002. [4] V.P. Nguyen, T. Rabczuk, S. Bordas, M. Duflot. Meshless methods: A review and computer implementation aspects, Mathematics and Computers in Simulation 2008, 79(3):763-813. [5] Y.T. Gu, L.C. Zhang. Coupling of the meshfree and finite element methods for determination of the crack tip fields. Engineering fracture mechanics 2008; 75:986-1004. [6] L. Godinho, D. Soares Jr. Frequency Domain Analysis of Fluid-Solid Interaction Problems by Means of Iteratively Coupled Meshless Approaches. Computer Modeling in Engineering \& Sciences 2012; 87:327-354. [7] L. Godinho, D. Soares Jr. Frequency domain analysis of interacting acoustic-elastodynamic models taking into account optimized iterative coupling of different numerical methods. Eng Anal Boundary Elem 2013; 32:1074-1088. [8] E.F. Fontes Jr, J.A.F. Santiago, J.C.F. Telles. On a regularized method of fundamental solutions coupled with the numerical Green's function procedure to solve embedded crack problems, Eng Anal Boundary Elem 2013; 37(1):1-7. [9] V.D. Kupradze, M.A. Aleksidze. The method of functional equations for the approximate solution of certain boundary value problems. U.S.S.R. Computational Mathematics and Mathematical Phisics 1964;4(4):82-126. [10] J.C.F. Telles, G.S. Castor, S. Guimaraes. A numerical Green’s function approach for boundary elements applied to fracture mechanics. International Journal for Numerical Methods in Engineering 1995; 38:3259-74. [11] H-K. Ching, R.C. Batra. Determination of crack tip field in linear elastostatics by the meshless local Petrov-Galerkin (MLPG) method. CMES 2001; 2:273-289. [12] L.S. Miers, J.C.F. Telles. On NGF Applications to LBIE Potential and Displacement Discontinuity Analyses. Key Engineering Materials 2001; 454:127-135. [13] Lin CC, Lawton EC, Caliendo JA, Anderson LR. An iterative finite element-boundary element algorithm, Computers \& Structures 1996; 59(5):899--909. [14] S.N. Atluri, T. Zhu. A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics. Comput. Mech. 1998; 22:117-127. [15] P. Lancaster, K. Salkauskas. Surfaces generated by moving least squares methods. Mathematics of Computation 1981; 37:141-158. [16] F.L.L.B. Carneiro. A new method to determine the tensile strength of concrete. In: proceedings of the fifth meeting of the Brazilian association for technical rules 1943; pp: 126-129. [17] S. Yarema, G.S. Ivanitskaya, A.L. Maistrenko, A.I. Zboromirskii. Crack development in a sintered carbide in combined deformation of types I and II. Strength of Materials 1984; 16(8):1121-1128.


187

Fracture with nonlocal elasticity: analytical approaches P.H. Wen1,a, X.J. Huang1,b and M.H. Aliabadi2,c 1

School of Engineering and Material Sciences, Queen Mary University of London, UK 2

Department of Aeronautics, Imperial College, London, UK

a

[email protected], [email protected], [email protected]

Keywords: Nonlocal elasticity, fracture mechanics, Eringen’s model, analytical estimation Abstract. In this paper, the fracture analysis for two-dimensional nonlocal elasticity is presented by the analytical approach. Based on the Eringen’s model, the nonlocal stresses at the crack tip is regular. An approximate estimation stresses at the crack tip is proposed in a closed form by using the stress intensity factors of the classical theory. A rectangular cracked plate subjected to tensile load is observed analytically. Introduction It is well known that the classical continuum theories like the linear theory of elasticity are intrinsically size independent. Although the development of classical theories of linear elasticity has been quite successful for solving most engineering problems at intrinsically size independent, the need for an efficient and accurate analytical method is increasingly demanded for problems with nonlocal elasticity. According to the classical theories, the elastic strains and stresses are singular at the tips of crack (dislocation) and at the corner of the notch. The continuum damage mechanics has been established to fill the gap between the classical continuum mechanics and fracture mechanics [1-4]. A continuum model for micro-cracking in these materials leads inevitably to strain softening. To prevent such physical unreality, the mathematical models with localization limiters have been proposed in [5]. A nonlocal elastic model proposed by Erigen [6, 7] and reviewed by Altan [8] is based on the key idea that the long-range forces are adequately described by a constitutive relation. A comprehensive review on the nonlocal elasticity theory can be found in [9] by Pisano et al. A cracked plate for nonlocal elasticity theory is investigated analytically in this paper. The stresses at the crack tip are approved to be regular and estimated by using the mixed mode stress intensity factors of the classical theory. Analytical estimation of stress fields near crack tip A nonlocal elastic model proposed by using nonlocal elasticity [6, 7, 8] is based on the key idea that the long-range forces are adequately described by a constitutive relation of the form, for two-dimension isotropic medium, as

V ij , j (x) fi (x) 0 σ ( x)

[1σ (x) [ 2 ³ D (x, x' , l )Dε(x' )dV (x' )

(1)

V

σ {V 11, V 22 , V 12}T , ε {H 11, H 22 , H 12}T , σ

Dε, H ij

(ui , j u j ,i ) / 2

where [ 1 and [ 2 are portion factors and [1 [ 2 1 , V represents the volume of domain, f i body forces, D a nonlocal kernel defined as the influence coefficient, l the characteristic length or influence distance; x( x, y), x' ( x' , y' ) are collocation and domain integration points and u i displacements; σ , σ and ε are vectors of nonlocal stress, local stress (classical stress) and strain; D denotes the elastic modulus matrix. The nonlocal kernel D (x, x' , l ) D ( x x' / l ) D (r / l ) has to satisfy the normalization condition as

³ D x x' / l dV '

1

(2)

Vf

in which Vf indicates the infinite domain embedding V. For two dimensional problem, one option of nonlocal kernels is D A (x, x' , l )

K 0 ( x x' / l ) / 2Sl 2 , where K 0 is the modified Bessel function of the

first kind. For analytical estimation, one particular case [1

0 is considered. For an infinite plate with a

188


straight line crack, the local displacement and stress fields in the domain are proved to be the same as elasticity [7] and one has

σ ( x)

³D

A

(x, x' , l )σ (x' )dV (x' )

(3)

V

where σ (x) is classical local stress solution under the same boundary condition due to the infinite solid the displacement fields of the classical elastostatics are exactly the same as the nonlocal elastostatics [7]. Hence, we can borrow the displacement field from well-known classical solutions. Then the constitutive equations of the nonlocal theory give the stress field, i.e. (4) σ (x) σ c (x) where σ c (x) indicates the stress tensor of classical theory under the same boundary condition. Firstly we consider a uniform tensile load V 0 at infinite and the elasticity solution at point x' ( x' , y' ) is given, in complex variable [10], as

§

2z'

V 22 (x' ) V 11(x' ) V 0 Re¨¨

2 2 © z ' a

· 1¸¸ ¹

§ 2ia 2 y ' · 1¸ V 22 (x' ) V 11(x' ) 2iV 12 (x' ) V 0 ¨ ¨ ( z '2 a 2 ) 3 ¸ © ¹ where complex variable z ' x'iy ' and i 1 . Therefore, the solution of nonlocal elasticity becomes

· ¸K ( x x' / l )rdTdr ¸ 0 ¹ x' (a r cos T , r sin T ), z ' a reiT . Moreover, the nonlocal elasticity solution of stress V 22 along axis x is obtained

V 22 (x)

V 22 ( x,0)

z' ia 2 y ' V 0 f 2S §¨ Re 2 ³ ³ 2 2 2Sl 0 0 ¨© z ' a ( z '2 a 2 ) 3

V0 2Sl 2

f 2S

§

0 0

© z ' a

³ ³ Re¨¨

z' 2

2

R

· ¸K ( R / l )rdTdr 0 ( z ' a ) ¸¹

(5)

(6)

ia 2 y ' 2

2 3

(7)

(a r cos T x) r sin T . 2

2

2

Consider only the singular stress field, one approximate solution can be obtained for small ratio of r / a . In this case, the classical local elasticity gives

V 22 (r , T )

KI 2Sr

3T · T§ T cos ¨1 sin cos ¸ 2© 2 2 ¹

(8)

Then estimation of nonlocal stress distribution is obtained * ( x,0) V 22

V 0 a f 2S T § 3T · T ³ ³ cos 2 ¨©1 sin 2 cos 2 ¸¹K 0 ( R / l ) r dTdr 2Sl 2 2 0 0 R

(9)

( a r cos T x ) 2 r 2 sin 2 T .

* ( x,0) / V 0 along axis x( a [ ) is shown in The variation of normalized stresses V 22 ( x,0) / V 0 and V 22

Fig. 1(a), (b) and (c) for different ratios l / a . It is apparent that the stress distribution for nonlocal theory is regular along axis x including the crack tip. At the crack tip, from eq (7), we have the stress distribution exactly

V 22 (a,0)

V0 2Sl 2

f 2S

§ iT ¨ a re

0 0

iT © 2a re

³ ³ Re¨¨

e iT / 2

ia 2 sin T

2a re

iT 3

· ¸ e 3iT / 2 ¸K 0 (r / l ) r dTdr. ¸ ¹

(10)

Consider the singular stress field of the classical theory, approximated stress at the crack tip is obtained, from eq (9), as


189

10

8

Exact (7) Series1

σ22(x,0)/σ0

Approximate (9) Series2

6

Classical theory (8) Series3

4

2 Crack tip

(a)

0 0

4

8

[ /l

12

16

20

4.5

σ22(x,0)/σ0

4.0 3.5

Series1 Exact (7)

3.0


2.5

Series3 Classical theory (8)

2.0 1.5 1.0 Crack tip

0.5

(b)

0.0 0

4

8

[ /l

12

16

20

3.5

σ22(x,0)/σ0

3.0

Exact (7) Series1

2.5


2.0

Classical Series3 theory (8)

1.5 1.0 0.5

Crack tip

(c)

0.0 0

4

8

[ /l

12

16

20

Figure 1. Normalized stress V 22 ( x,0) / V 0 verse x (= a [ ): (a) l / a

l/a

0.05 ; (c) l / a

0.1 .

0.01 ; (b)

190


* V 22 (a,0)

where K I and N

f

KI 2S Sl

³K

0

(O ) O dO

0.5736V 0 / N

(11)

0

V 0 Sa denotes the stress intensity factor for a central crack under uniform tensile at infinite

l / a . It was given in [7] by Erigon as 0.5744V 0 / N by integral equation method, which is very

close to the solution in eq (11). Fig. 2 shows the variation of the normalized stress V 22 (a,0) /[V 0 / N ] at crack tip against the ratio l / a . The exact results from eq (10) can be presented approximately as

V 22 (a,0)

0.574V 0

N

.(1 0.4315N )

(12)

It is clear that the relative error of approximation solution in eq (11) is 0.4315N . For example, when N l / a 0.05 , the relative error of estimation is about 2%. Secondly, consider a uniform shear stress W 0 at infinite, the elasticity solution is given in [10]

§ · iz ' ¸ 2W 0 Re¨¨ 2 2 ¸ © z a ¹ a 2 z ' z ' (2 z ' 2 3a 2 ) V 22 (x' ) V 11 (x' ) 2iV 12 (x' ) iW 0 . ( z ' 2 a 2 ) 3 where z ' x'iy ' . Thus, the distribution of shear stress along x axis is obtained

V 22 (x' ) V 11 (x' )

V 12 ( x,0)

§ a 2 z ' z ' (2 z ' 2 3a 2 ) · ¸K ( R / l )rdTdr 2 2 3 ¸ 0 ( z ' a ) © ¹

f 2S

W0 4Sl 2

³ ³ Re¨¨ 0 0

R

(13)

(14)

(a r cos T x) r sin T 2

2

2

For small ratio of r / a , one has the classical elastic stress as

V 12 (r ,T )

T§ T K II 3T · cos ¨1 sin cos ¸ 2 ¹ 2 2 2Sr ©

(15)

and then the estimated nonlocal shear stress at the crack tip is

V 12* (a,0)

K II 2S Sl

f

³K

0

(O ) O dO

0.3824W 0 / N

(16)

0

where K II W 0 Sa denotes the shear mode stress intensity factor. In addition, the exact nonlocal shear stress at crack tip from eq (14) is shown in Fig. 2 and can be written approximately as

V 12 (a,0)

0.3812W 0

N

.(1 1.4743N )

(17)

Same as the tensile stress, the relative error of approximation solution for shear loading in eq (16) is 1.4743N . To determine the stress at the crack tip, we can extend this computation procedure for a finite cracked plate. For mixed-mode crack problems, in general cases, if the stress intensity factors of the classical theory are denoted as K I and K II respectively, the nonlocal stresses at the crack tip are regular and can be estimated from eq (11) and eq (16) for small ratio of l / a ( 0.05) , as * * (18) V 22 (a,0) 0.5736K I / Sl , V 12 (a,0) 0.3824K II / Sl . Obviously the degree of accuracy depends on the ratio N and the dimension of geometry of cracked plate, which can be seen from Fig. 2. In addition, maximum normal and shear stresses exist at [ / l 0.5 are given by Eringen [7] approximately, for small ratio of N , as V 22, max (a [ ,0) 1.12V 22 (a,0). (19)

Next, more nonlocal kernels (Modes) are observed. Select the following coefficient functions as

D B (r , l )

1 r / l D C (r , l ) e 2Sl 2 ,

1

Sl 2

e r

2

/ l2


3 ° 2 ®Sl ° ¯

D D (r , l )

§ r· ¨1 ¸ r d l © l¹ 0 r tl

D E (r , l ) ,

2 ° 2 ®Sl ° ¯

§ r2 ¨¨1 2 © l 0

191

· ¸¸ r d l ¹ rtl

(20)

x x' . From eq (3), eq (8) and eq (15), for Mode B, one has the estimation of stress at the crack

where r tip

f

6 2K I 5S Sl

* (a,0) V 22

³

t e t dt

0.4787

0

f

KI * ˈV 12 (a,0) Sl

4 2 K II 5S Sl

KI * ˈV 12 (a,0) Sl

2 2 K II 5S Sl

³

t e t dt

0.3192

0

K II Sl

(21)

and for Mode C, f

3 2K I 5S Sl

* (a,0) V 22

³

t e t dt 2

0.6620

0

f

³

t e t dt 2

0.4413

0

K II . Sl

(22)

0.65 0.60

σ22(a,0)√κ/σ0

0.55

Exact (10) Tensile load

Approximate (11)

0.50 Exact (14)

0.45

Shear stress

0.40 0.35

Approximate (16)

0.30 0.00

0.04

0.08

0.12

0.16

0.20

l/a Figure 2. Comparison of normalized stresses N V 22 (a,0) / V 0 and crack tip with exact and approximate solutions.

N V 12 (a,0) / W 0 at the

It is not difficult to obtain the normal and shear stresses at the crack tip for the rest of coefficient functions. * * Normalized stresses V 22 Sl / K I and V 12 Sl / K II are presented in Table 1 for different modes. Moreover, for Mode C, the shear stress at the crack tip by Erigen [7] is derived to be 0.4243W 0 / N . Compared with these two solutions (0.4243 and 0.4413), the relative error is less than 4%. However, the estimation in this paper should be more accurate for small ratio of l / a . Table 2 and Table 3 show the exact maximum nonlocal stresses and their locations when l / a 0.01 from (7) and (14) for different nonlocal kernels (Mode). Table 1. Nonlocal stresses at crack tip with different modes by eq (12).

Sl / K I

A 0.5736

B 0.4787

C 0.6620

D 0.8643

E 0.8231

* V 12 Sl / K II

0.3824

0.3192

0.4413

0.5762

0.5488

Mode

V

* 22

192


Table 2. Maximum nonlocal normal stress V 22,max (a [ ,0) / V 22 (a,0) when l / a Mode

[ /l V 22,max / V 22 (a,0)

A 0.45

B 0.85

C 0.61

D 0.40

E 0.47

1.1029

1.1378

1.1830

1.1838

1.1947

Table 3. Maximum nonlocal shear stress V 12,max (a [ ,0) / V 12 (a,0) when l / a Mode

[ /l V 12,max / V 12 (a,0)

0.01 .

0.01 .

A 0.55

B 1.30

C 0.88

D 0.54

E 0.65

1.0898

1.1203

1.1548

1.1553

1.1742

Rectangular plate with central crack

A rectangular plate of width 2b and height 2h containing a central crack of length 2a subjected to tensile load on the top is shown in Fig. 3(a). As symmetry of the plate and load condition, quarter of plate is analyzed. V 0

h

a

Crack tip

Δmin b (a)

(b)

Figure 3. Rectangular plate with central crack: (a) quarter of plate and boundary conditions; (b) collocation point for local integral equation method and grid for domain integrals.

Firstly, the effects of nonlocal kernel are considered. Normalized stress V 22 ( x,0) / V 0 for different nonlocal kernel D (r , l ) while l / a 0.05 , h b 2a and h b 3a are presented in Fig. 4 and Fig. 5. In this case, we can estimate the degree of accuracy for approximation of stress at the crack tip by eq (11). The estimation of stresses at crack tip for two modes (A and B) are shown in Table 4. Moreover, the effects of the characteristic length l is observed with nonlocal kernel D A (r , l ) . Normalized stress V 22 ( x,0) / V 0 for different characteristic length l / a 0.05 and 0.1 while h b 2a and h b 3a are presented in Fig. 6 and Fig. 7 respectively. It is clear that the maximum stress is located about [ / l 0.5 for each case, which agrees with the conclusion of the analytical solution.


193

Table 4. Estimations of stress at the crack tip and comparison with numerical solutions. Mode

A

B

b/a

KI

V 22 / V 0 by eq (11)

2.0

1.325

3.399

3.0

1.130

2.899

2.0

1.325

2.847

3.0

1.130

2.419

4.0

4.5 4.0

3.5

3.5

σ22(x,0)/σ0

3.0

1 K 0 (r / l ) 2Sl 2 1 D B (r , l ) exp( r / l ) 2Sl 2

2.5 2.0 1.5

1 K 0 (r / l ) 2Sl 2 1 exp( r / l ) 2Sl 2

D A (r , l )

3.0

σ22(x,0)/σ0

D A (r , l )

D B (r , l )

2.5 2.0 1.5 1.0

1.0 Crack tip

Crack tip

0.5

0.5 0.0

0.0 1.0

1.2

1.4

1.6

1.8

2.0

1.0

1.4

1.8

2.2

2.6

3.0

x/a

x/a Figure 4. Normalized stress V 22 ( x,0) / V 0 for different nonlocal kernel D (r , l ) while

Figure 5. Normalized stress V 22 ( x,0) / V 0 for different nonlocal kernel D (r , l ) while

l / a 0.05 and h b 2a .

l / a 0.05 and h b 3a .

4.5

4.0

l / a 0.05

4.0

Series1 3.0

l / a 0.1

Series2

σ22(x,0)/σ0

3.0

σ22(x,0)/σ0

l / a 0.05

3.5

Series1

3.5

2.5 2.0 1.5

Series2

2.5

l / a 0.1

2.0 1.5 1.0

1.0 Crack tip

Crack tip

0.5

0.5 0.0

0.0

1.0

1.2

1.4

1.6

1.8

2.0

1.0

1.5

2.0

2.5

3.0

x/a

x/a

Figure 6. Normalized stress V 22 ( x,0) / V 0 for different portion l / a while h b 2a .

Figure 7. Normalized stress V 22 ( x,0) / V 0 for different portion l / a while h b 3a .

Conclusion Based on the Eringen’s model, the stresses at the crack tip under tensile load and shear load at infinite of plate were approximated analytically by using singular stresses of the classical theory. Therefore, the estimation of nonlocal stresses at the crack tip can be obtained by using classical mixed mode stress intensity factors. A rectangular sheet with a central crack is investigated to demonstrate the estimation of stresses at the crack tip. Reference [1] Sudak LJ. Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics. J Appl Phys, 94:7281–7 (2003). [2] Wang Q, Varadan VK. Application of nonlocal elastic shell theory in wave propagation analysis of carbon nanotubes. Smart Mater Struct, 16: 178–90 (2008). [3] Filiz S, Aydogdu M. Axial vibration of carbon nanotube heterojunctions using nonlocal elasticity. Comp Mater Sci, 49:619–27 (2010).

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[4] Hu YG, Liew KM, Wang Q, He XQ, Yakobson BI. Nonlocal shell model for elastic wave propagation in single and double walled carbon nanotubes. J Mech Phys Solids, 56:3475–85 (2008). [5] Sladek J, Sladek V, Bazant ZP. Non-local boundary integral formulation for softening damage. Int. J. Num. Meth. Engn, 57: 103-116 (2003). [6] Eringen AC. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys, 54: 4703-4710 (1983). [7] Eringen AC. Nonlocal Continuum Field Theories, Springer-Verlag New York, Inc.,( 2002). [8] Altan SB. Existence in nonlocal elasticity, Archive Mechanics, 41: 25-36(1989). [9] Pisano AA, Sofi, A., Fuschi, P., Nonlocal integral elasticity: 2D finite element based solutions, Int. J. Solids and Struct. 46, 3838-3849 (2009).

[10] S.P. Timoshenko, J.N. Goodier, Theory of Elasticity, McGraw-Hill Book Company, (1951).


195

A Dual Boundary Element Method for Measurement of Electromechanical Impedance Fangxin Zou1, M. H. Aliabadi2 1

Department of Aeronautics, Imperial College London, South Kensington Campus, London, UK, [email protected]

2

Department of Aeronautics, Imperial College London, South Kensington Campus, London, UK, [email protected]

Keywords: Dual boundary element method, structural health monitoring, electromechanical impedance, piezoelectric transducer

Abstract. In this paper, a boundary element method (BEM) for the time-harmonic analysis of 3D solid structures with bonded piezoelectric actuators is presented. In particular, the characteristic of interest is the electromechanical impedance (EMI), which could be used for structural health monitoring (SHM) purposes. The complete formulation is obtained by coupling the semi-analytical finite element method (FEM), which is used for modelling the piezoelectric actuator, with the dual boundary element method (DBEM), which is employed for modelling the host structure. The analysis of the entire system is performed in Fourier domain. The results computed by the BEM are validated against those by an established FEM. Introduction In recent years, SHM has become a highly sought approach for ensuring the integrity of structures in engineering applications. The concept of smart structures has attracted much attention and research effort for its potential in SHM techniques. Among the available technologies, piezoelectric transduction patches are being used extensively due to some of their distinctive features. These transducers are not only capable of generating and receiving ultrasonic guided waves, but can also be used for examining the frequency responses of structures. The development of a feasible SHM technique is a systematic task. It requires knowledge from a variety of areas, ranging from structural mechanics to signal processing. Among these areas, a reliable and efficient mathematical model of the smart structure under investigation would offer valuable assistance in predicting the performance of the structure in service. Also, it could accelerate the development of techniques, which demand a large amount of data, such as artificial neural networks (ANNs), by generating this data numerically. In this paper, a piezoelectric smart structure for EMI based SHM techniques is modelled and analysed with BEM. The model of the piezoelectric actuator takes into account the full electromechanical behaviour, and is coupled with the host structure, which is formulated by DBEM, via BEM variables. The EMI values for the frequency range of interest are obtained by post-processing the solution of the boundary integral equations in Fourier domain. Expectedly, the results from BEM and FEM simulations show excellent agreement. Electromechanical Impedance When a piezoelectric transducer is bonded to a host structure, its electric impedance would depend not only on its own properties, but also on the properties of the host structure. The impedance of the coupled system, measured through the transducer, is referred to as the EMI. To obtain the EMI signature of a system, sinusoidal electric voltages of different frequencies are applied to the transducer, and the resultant electric currents are measured. The full complex expression of EMI is given by ܼሺ߱ሻ ൌ

ܸሺ߱ሻ ൌ ܴ ൅ ܺ݅ ‫ܫ‬ሺ߱ሻ

where ܴ is the resistance and ܺ is the reactance.

(1)

196


EMI based SHM techniques are generally very sensitive to the presence of defects, especially if they are local. When there is a defect in the system, the change in resonant frequencies would also alter the EMI signature. By using advanced processing algorithms, detailed characterisation of the defect is also achievable. Model of Piezoelectric Actuator An expression, which relates the variables of the top and the bottom surfaces of a piezoelectric transducer, is given by [1, 2] ෥௧ ෥௕ ࢛ ࡸ ሺ߱ǡ ݄ሻ ࡸ௨௏ ሺ߱ǡ ݄ሻ ࡸ௨ఙ ሺ߱ǡ ݄ሻ ࡸ௨஽ ሺ߱ǡ ݄ሻ ࢛ ‫ ۍ‬෩ ‫ ۍ ې‬௨௨ ‫ۍ ې‬෩ ‫ې‬ ሺ߱ǡ ሺ߱ǡ ሺ߱ǡ ሺ߱ǡ ࡸ ࢂ ࢂ ݄ሻ ࡸ ݄ሻ ࡸ ݄ሻ ࡸ ݄ሻ ௏௏ ௏ఙ ௏஽ ‫ ێ‬௧ ‫ ۑ‬ൌ ‫ ێ‬௏௨ ‫ ێ ۑ‬௕‫ۑ‬ ෥ ௧ ‫ࡸێ ۑ‬ఙ௨ ሺ߱ǡ ݄ሻ ࡸఙ௏ ሺ߱ǡ ݄ሻ ࡸఙఙ ሺ߱ǡ ݄ሻ ࡸఙ஽ ሺ߱ǡ ݄ሻ‫࣌ ێ ۑ‬ ෥௕ ‫ۑ‬ ‫࣌ێ‬ ෩ ௧ ‫ࡸۏ ے‬஽௨ ሺ߱ǡ ݄ሻ ࡸ஽௏ ሺ߱ǡ ݄ሻ ࡸ஽ఙ ሺ߱ǡ ݄ሻ ࡸ஽஽ ሺ߱ǡ ݄ሻ‫ࡰۏ ے‬ ෩ ௕‫ے‬ ‫ࡰۏ‬

(2)

෩ and ࡰ ෩ ෥ଶ ࢛ ෥ ଷ ሿ and ࣌ ෥ଵ ࢛ ෥ଵଷ ࣌ ෥ ଶଷ ࣌ ෥ ଷଷ ሿ are the mechanical displacement and stress, ࢂ ෥ ൌ ሾ࢛ ෥ ൌ ሾ࣌ where ࢛ are the electric potential and displacement in the x3-direction, ݄ is the thickness of the transducer, the subscripts ‫ ݐ‬and ܾ stand for the top and the bottom surfaces, and the tildes indicate nodal values. Through eq (2), it is not difficult to see that the model is essentially finite element based in the plane and analytical across the thickness. In order to derive the model of piezoelectric actuators, a couple of boundary conditions are taken into account. First of all, because the top surfaces of the actuators are not attached to anything, they are stress free. Secondly, because electric potential is a relative quantity, the bottom surfaces of the actuators are assigned with zero potential. By applying these two boundary conditions in eq (2), the model of piezoelectric actuators is found to be ෩௧ ෥ ௕ ൅ ࢶࢂ ࢚෤௕ ൌ ࢸ࢛

(3)

Dual Boundary Element Method The Navier-Cauchy equation, which governs the dynamics of an elastic isotropic body, in Fourier domain, is given by ܿଶ ଶ ‫ݑ‬௜ǡ௞௞ ሺ࢞ǡ ߱ሻ ൅ ሺܿଵ ଶ െ ܿଶ ଶ ሻ‫ݑ‬௞ǡ௞௜ ሺ࢞ǡ ߱ሻ ൌ ‫ݑ‬ሷ ௜ ሺ࢞ǡ ߱ሻ

(4)

The solution of eq (4), using BEM, is written as [3] ܿ௜௝ ൫࢞′ ൯‫ݑ‬௝ ൫࢞′ ǡ ߱൯ ൅ න ܶ௜௝ ൫࢞′ ǡ ࢞ǡ ߱൯‫ݑ‬௝ ሺ࢞ǡ ߱ሻ݀߁ ൌ න ܷ௜௝ ൫࢞′ ǡ ࢞ǡ ߱൯‫ݐ‬௝ ሺ࢞ǡ ߱ሻ݀߁

(5)

ͳ ‫ ݐ‬ሺ࢞′ǡ ߱ሻ ൅ ݊௜ ሺ࢞′ሻ න ܶ௞௜௝ ൫࢞′ ǡ ࢞ǡ ߱൯‫ݑ‬௞ ሺ࢞ǡ ߱ሻ݀߁ ൌ ݊௜ ሺ࢞′ሻ න ܷ௞௜௝ ൫࢞′ ǡ ࢞ǡ ߱൯‫ݐ‬௞ ሺ࢞ǡ ߱ሻ݀߁ ʹ ௝ ௰ ௰

(6)

௰

௰

After carrying out the collocations and the integrations in eq (5) and eq (6), a linear system of equations are resulted ෥ ሺ߱ሻ ൌ ࡳሺ߱ሻ࢚෤ሺ߱ሻ ࡴሺ߱ሻ࢛

(7)

By separating the known and the unknown nodal values, eq (7) can be rewritten as ෥ሺ߱ሻ ൌ ࢟ሺ߱ሻ ࡭ሺ߱ሻ࢞

(8)

In order to obtain the real part of EMI – resistance, damping needs to be included in the formulation. Among the established damping models, structural damping has been found to be the most appropriate one for modelling EMI based applications [4]. In FEM, the equation of motion, which takes structural damping into account, is given by ࡹ‫ݔ‬ሷ ሺ‫ݐ‬ሻ ൅ ࡷ‫ݔ‬ሺ‫ݐ‬ሻ ൅ ݅ߟࡷ‫ݔ‬ሺ‫ݐ‬ሻ ൌ ‫ܨ‬ሺ‫ݐ‬ሻ

(9)


197

where ߟ is a constant which can be found by model updating. For BEM, when structural damping is considered, an extra term is added to the right hand of eq (4) such that ܿଶ ଶ ‫ݑ‬௜ǡ௞௞ ሺ࢞ǡ ߱ሻ ൅ ሺܿଵ ଶ െ ܿଶ ଶ ሻ‫ݑ‬௞ǡ௜௞ ሺ࢞ǡ ߱ሻ ൌ ߱ଶ ‫ݑ‬௜ ሺ࢞ǡ ߱ሻ ൅ ݅ܿ

(10)

where ܿ is also a constant. Consequently, the frequency range to be solved becomes ߱௡௘௪ ൌ ඥ߱ ଶ ൅ ݅ܿ

(11)

It is worth noting that the constants ߟ and ܿ are not equivalent due to the difference in representation between eq (9) and eq (10). Coupling of Piezoelectric Actuator and Host Structure

Figure 1 Coupling of piezoelectric actuator and host structure [2] Fig. 1 shows a piezoelectric actuator that is perfectly bonded to a host structure. The boundary of the host structure is separated into two parts – the interface between the host structure and the piezoelectric actuator, and the rest of the boundary. As a result, eq (7) is expanded into ෥ ௛ ൅ ࡴ௜ ࢛ ෥ ௜ ൌ ࡳ௛ ࢚෤௛ ൅ ࡳ௜ ࢚෤௜ ࡴ௛ ࢛

(12)

where the subscripts ݅ and ݄ stand for the interface and the rest of the boundary. By substituting eq (3), which is written in terms of the BEM variables displacement and traction, into eq (12), and applying continuity conditions, the following expression is obtained ෩௧ ෥ ௛ ൅ ሺࡴ௜ ൅ ࡳ௜ ࢸሻ࢛ ෥ ௜ ൌ ࡳ௛ ࢚෤௛ െ ࡳ௜ ࢶࢂ ࡴ௛ ࢛

(13)

By separating the known and the unknown nodal values, eq (13) can be rearranged into ෩௧ ෥ ൌ ࢟ െ ࡳ௜ ࢶࢂ ࡭࢞

(14)

From eq (13) and eq (14), it can be seen that the presence of a piezoelectric actuator does not only add boundary conditions to the system of equations but also modifies the collocation matrix. Eq (14) is ready to be solved by an adaptive cross approximation algorithm [5]. Calculation of Electromechanical Impedance Value The solution of eq (14) contains the nodal displacements of the boundary of the host structure. Due to continuity condition, the nodal displacements of the interface are equivalent to those of the bottom surface of

198


the actuator. In order to obtain EMI values, the nodal stresses of the bottom surface of the actuator are firstly found by eq (3). Next, the following expressions are extracted from eq (2) for finding the nodal electric displacements of the top surface ෩ ௕ ൌ െࡸఙ஽ ିଵ ሺࡸఙ௨ ࢛ ෥ ௕ ൅ ࡸఙఙ ࣌ ෥௕ ሻ ࡰ ෩ ௧ ൌ ࡸ஽௨ ࢛ ෩௕ ෥ ௕ ൅ ࡸ஽ఙ ࣌ ෥ ௕ ൅ ࡸ஽஽ ࡰ ࡰ

(15) (16)

Then, the overall electric charge on the top surface is calculated by ܳ௧௢௣ ൌ න

஺೟೚೛

෩ ௧ ݀ߦ݀ߟ ࡺሺߦǡ ߟሻ‫ܬ‬ሺߦǡ ߟሻࡰ

(17)

where ࡺ contains the shape functions. Furthermore, the electric current is determined by ‫ܫ‬ሺ߱ሻ ൌ ݅߱ܳ௧௢௣ ሺ߱ሻ

(18)

Finally, the EMI is obtained using eq (1). Numerical Validation

Figure 2 BEM models of specimens (The piezoelectric actuators are coloured in red and the crack in green)


199

Figure 3 EMI signature of pristine specimen

Figure 4 EMI signature of cracked specimen Fig. 2 shows the BEM models of the specimens used for numerical validation. Fig. 3 and fig. 4 compare the EMI signatures of the pristine and the cracked specimens obtain from both BEM and FEM simulations.

200


Generally speaking, the agreement of the resistances is almost perfect while that of the reactances is reasonably well in terms of resonance frequencies and values of most of the peaks. In order to obtain the match of the resistances, several damping constants have been tested in both the BEM and the FEM models. However, because model updating is out of the scope of this paper, the damping constants, which have been considered, are merely 0.1, 0.01, 0.001 and 0.0001. Among these, the use of 0.01 in both models produces the best agreement. Nevertheless, for those who require more accurate models, the damping constant should be adjusted with smaller increments. On the other hand, for SHM applications, the more important feature is the difference in the EMI signatures between the pristine and the cracked specimens. As seen through fig. 3 and fig. 4, the existence of the crack yields more noticeable different in the resistances than in the reactances. This in turn explains the fact that resistances are generally used for the detection of defects. Conclusion In this paper, a BEM for the time-harmonic analysis of 3D solid structures with bonded piezoelectric actuators is introduced. The formulation includes a DBEM for modelling the host structure, and a semianalytical FEM for modelling the piezoelectric actuator. The actuator and the host structure are coupled via BEM variables and the analysis of the entire system is carried out in Fourier domain. The model is employed for the measurement of EMI values for SHM purposes. The results from BEM and FEM simulations demonstrate good coherence, and the existence of a crack is clearly identified. References 1. 2. 3. 4. 5.

Qing, G., J. Qiu, and Y. Liu, A semi-analytical solution for static and dynamic analysis of plates with piezoelectric patches. International journal of solids and structures, 2006. 43(6): p. 1388-1403. Zou, F., I. Benedetti, and M. Aliabadi, A boundary element model for structural health monitoring using piezoelectric transducers. Smart Materials and Structures, 2014. 23(1): p. 015022. Wen, P., M. Aliabadi, and D. Rooke, Cracks in three dimensions: A dynamic dual boundary element analysis. Computer methods in applied mechanics and engineering, 1998. 167(1): p. 139-151. Lim, Y.Y. and C.K. Soh, Towards more accurate numerical modeling of impedance based high frequency harmonic vibration. Smart Materials and Structures, 2014. 23(3): p. 035017. Benedetti, I. and M. Aliabadi, A fast hierarchical dual boundary element method for three-dimensional elastodynamic crack problems. International Journal for Numerical Methods in Engineering, 2010. 84(9): p. 1038-1067.


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Study on the Water Coning Phenomenon in Oil Wells Using the Boundary Element Method G. S. V. Gontijo1 , E. L. Albuquerque1 , E. L. F. Fortaleza1 1

University of Brasilia - Darcy Ribeiro Campus, North Wing - Brasilia - Brazil [email protected]

Keywords: Boundary Element Method, Water Coning, Oil Reservoir Simulation.

Abstract. This paper presents a study on the water coning phenomenon in oil wells. The scope of this work is the numerical modeling of the phenomenon, using the boundary element method. A review about the water coning phenomenon is presented, as well as the utilized mathematical model. A sub-regions boundary element method formulation with moving interface is implemented using continuous linear boundary elements. The simulation results are physically consistent, but they were not validated due to the lack of published analytical or experimental results.

Introduction An oil reservoir is composed by a porous medium which contains oil and, in most cases, water. In an oil production field, the water coning phenomenon is a limiting factor in the productivity of an oil well [6]. This phenomenon occurs due to the pressure gradient applied to the well in order to withdraw the oil from the reservoir. The pressure gradient reaches both fluids (oil and water) and since the water has greater mobility than the oil, it tends to flow toward the well, taking the shape of a cone. For certain values of flow, the water cone reaches the well, so that water will be produced instead of oil, reducing productivity. It is of interest to industry the knowledge of the behavior of the oil-water interface in order to prevent the water coning. Numerous works are devoted to study this phenomenon. Gas coning in free surface problems was studied by [3] and a BEM code for this class of problems was developed by [1]. Critical steady-state interface position and supercritical withdrawal was studied by [9] and [8]. Efforts to prevent the water coning was studied by [5]. Oil companies are always looking for ways to reduce the computational cost of their reservoir simulations. These processes consume massive computational resources due to the immense number of degrees of freedom of the usual problems. Dimensions of simple water coning simulations easily surpass the magnitude of 1 km in width and length. Nowadays, simpler models are gaining popularity among reservoir engineers [7] because of their efficiency regarding the simulation time. In this respect, the boundary element method (BEM) can be a powerful tool in the development of numerical reservoir simulators, whereas most current simulators were developed using the finite difference method. This work succeeded in the development of a numerical reservoir simulator specifically for the study of the water coning phenomenon, using the BEM. The developed simulator is classified as black oil, two-dimensional, vertical, two-phase, single-porosity type.

Problem Statement Mathematical Model. Consider a reservoir consisting of an homogeneous porous medium with constant and isotropic permeability k [darcy]. This reservoir is filled with two immiscible fluids, oil and water, each one with different density ρo,w [ML−3 ] and dynamic viscosity μo,w [ML−1 T −1 ]. Both fluids are considered incompressible and, at rest, are stratified by their density: the bottom layer consists of water, whilst the upper layer consists of oil. The interface between the fluids is considered sharp [4], i.e., under the interface the fluid is composed by 100% water and above it, the fluid is 100% oil. Oil withdrawal occurs permanently at a constant rate Q [L2 T −1 ] per unit width. The porous media flow is a potential problem governed by Darcy’s Law [4], which states that the velocity of the fluid is given by: qo,w = −Ko,w ∇φo,w [LT −1 ] (1)

202


where K and φ are the hydraulic conductivity of the porous medium and the velocity potential, respectively, given by: kρo,w g [LT −1 ] μo,w po,w φo,w = + zo,w [L] ρo,w g Ko,w =

(2) (3)

Applying the mass conservation principle at eq. (1), comes: ∇2 φo,w = 0

(4)

When the oil withdrawal is active, the well is represented by a punctual sink. The governing equation turns to be: Q (5) ∇2 φo = − δ (x − xs ) K where xs is the sink’s position vector. Therefore, eq. (5) must be satisfied in the oil zone and eq. (4) must be satisfied in the water zone of the reservoir. Boundary Conditions. The boundary conditions for the modeled reservoir are shown in the Fig. 1(a). The two vertical dotted lines show the limit of influence of the sink’s pressure gradient over the interface position. Beyond those lines, the interface stays in it’s non-disturbed position. This condition sets the limits of the simulated domain (Fig. 1(b)): the reservoir has a width such that the outer portion to its limits remains unchanged during the oil withdrawal.

(a)

(b)

Figure 1: Boundary conditions for the modeled reservoir. Potential: The potential at any portion of each fluid is given by eq. (3). When the sink is inactive, the potential at the interface is equal to the potential at any point of the same fluid. For example, the potential at a point immediately under the interface is equal to the potential of the water body at rest, which is the same to the whole sub-region that contains the water. In the same way, this is also valid for the oil. When the sink is active, however, the potential at the interface is unknown, and it also is for any inner point of the reservoir, with the exception of the lateral limits, which always have their potentials calculated by eq. (3) as: pw + zw ρw g po φ6 = φ8 = φo = + zo ρo g

φ2 = φ4 = φw =

(6) (7)

Flux: The upper and lower limits of the reservoir are impermeable walls, so that it doesn’t exist any normal flux to them, so: q1 · nˆ 1 = 0

(8)

q7 · nˆ 7 = 0

(9)


203

BEM Sub-regions Formulation In order to allow the analysis of the water coning phenomenon, it was necessary that the developed code could show the position of the oil-water interface along the withdrawal operation. Given the immiscible two-phase characteristic of the flow, it was implemented a BEM sub-regions formulation, in which each fluid was represented by one sub-region [2]. The boundary elements at the interface between the two sub-regions represent the oil-water sharp interface. It was used continuous linear boundary elements in the implementation of the code. The assembly of the influence matrices H and G used analytical integration procedure, in order to improve performance and accuracy.

Interface Equations Potentials Compatibility. The potential on a fluid depends on it’s density. Then, it is possible to deduce from eq. (3) that exists a potential gap at the interface. It is used, then, the continuity of the pressure field at the whole reservoir. By making po = pw = p and zo = zw = z in eq. (3), it is obtained: (φo − z)ρo g = (φw − z)ρw g Dividing eq. (10) by ρo g, results:

(10)

ρw ρo

(11)

φo − αφw = (1 − α)z

(12)

φo − z = (φw − z) Equation (11) can be rewritten as: where α=

ρw ρo

(13)

Equation (12) is the potential compatibility equation for an interface between two immiscible fluids. Therefore, for the reservoir illustrated in the Fig. 1(b), it results in: φ5 − αφ3 = (1 − α)z

(14)

Fluxes Balance. The interface is an abrupt boundary, where is considered that both fluids are always in contact with each other. There is a calculated flux in the normal direction to the interface. This flux must have the same magnitude in both sides of the interface, however, contrary signals. Thus, it is guaranteed that it won’t be either empty spaces or overlaps between the fluids. This leads to the fluxes balance equation for an interface between two immiscible fluids: q3 · nˆ 3 = −q5 · nˆ 5 (15) Equations (14) and (15) are the coupling equations for the oil-water interface. Interface Movement. The calculated flux at the interface is, actually, the velocity of the fluid that defines it. So, it is possible to deduce the equation that defines the interface position, due to the flow. Consider, for this, that a function F exists and is defined by: F = z − η(x,t) = 0

(16)

where η(x,t) is the function that express the height of the interface at a given x coordinate and a given time. Because the interface is a material stream, it is possible to do [10]: DF ∂F q = + · ∇F = 0 Dt ∂t θ

(17)

where θ is the medium porosity. Replacing eq. (16) at eq. (17), it is obtained: DF ∂η q =− + · (∇z − ∇η) = 0 Dt ∂t θ

(18)

204


Isolating the variation of η in time:

∂η 1 = (q · eˆ z − q · ∇η) ∂t θ Applying Darcy’s Law (eq. (1)), it is obtained: ∂η K ∂φ =− − ∇φ · ∇η ∂t θ ∂z

In order to implement in the BEM code, a relation between

∂η ∂t

and

∂φ ∂n

(19)

(20) is sought:

∂φ = ∇φ · eˆ n ∂n The unit normal vector may be written as: eˆ n = Replacing eq. (22) at eq. (21), leads to:

Expliciting

∂φ ∂z :

(21)

∇(z − η) |∇(z − η)|

(22)

∂φ − ∇φ · ∇η ∂φ = ∂z ∂n |∇(z − η)|

(23)

∂φ ∂φ = |∇(z − η)| + ∇φ · ∇η ∂z ∂n

(24)

K ∂φ ∂η =− |∇(z − η)| ∂t θ ∂n

(25)

Replacing eq. (24) at eq. (20), comes:

The quantity |∇(z − η)| in eq. (25) can be written as:

∂η |∇(z − η)| = 1 + ∂x

2 1/2 (26)

Replacing eq. (26) at eq. (25), it is obtained: ∂η K ∂φ =− ∂t θ ∂n

1+

∂η ∂x

2 1/2 (27)

Equation (27) defines the height variation of a certain node of the interface and was used in this study for the simulation of the interface movement. The implementation of this equation in the developed code, however, must consider the size of the time step used in the iterations. Writing eq. (27) in terms of finite differences, it is obtained: 1/2 ∂η 2 K zm+1 = zm − Δt qm 1 + (28) θ ∂x m where z is the interface’s height, Δt is the size of the time step and the sub-indexes m and m + 1 indicates the present time step and the immediately next one.

Numerical Results It was simulated the oil withdrawal from a fictitious reservoir with dimensions 5 × 4 m. Each fluid zone has 2 m height, as shown in Fig. 2(a). The permeability of the porous medium is k = 1 darcy and it’s porosity is θ = 0.20. The water density and viscosity are respectively ρw = 1025.18 kg/m3 and μw = 3 × 10−4 Pa.s. The oil density is ρo = 688.79 kg/m3 and it’s viscosity is μo = 3 × 10−3 Pa.s. The sink was located at coordinates x = 2.5 m and z = 3.5 m and it’s intensity is Q = 1.55 × 10−6 m3 /s per unit length. Fig. 2(b) shows the position of the interface in steady state. Fig. 3(a) shows the evolution of the interface central node’s height along the


(a)

205

(b)

Figure 2: (a) Discretized reservoir and (b) position of the interface in steady state. withdrawal time. Fig. 3(b) shows the flow rate that enters each sub-region of the reservoir by its lateral limits along time. It can be seen in Fig. 3(b) that the sum of both flow rates is equal to the sink’s intensity. In steady state, the water zone stays stationary and the oil flow rate is equal to the sink’s intensity. The final height of the central node is 3.12 m. In Fig. 2(a), the circle symbols represent the oil potential and the square symbols represent the water potential, both at the lateral boundaries of the simulated reservoir. The triangle symbols represents the impermeability condition at the bottom and top boundaries. In Fig. 2(b), the circle symbol inside the upper sub-region represents the position of the sink.

(a)

(b)

Figure 3: (a) Height of the central node of the interface and (b) flow entering the reservoir along time. It can be seen that, with the correct withdrawal parameters, the oil-water interface becomes stable and the water cone do not reach the well, i.e., only oil is produced.

Conclusions The obtained results are consistent with what physically occurs. A comparison between these results and analytical and experimental results is still needed in order to validate them. The developed BEM code is useful for transient analysis of the water coning phenomenon in oil wells and can be used in control applications which need to constantly know the position of the oil-water interface as well as its displacement rate.

206


Acknowledgments The authors would like to thank Chevron and PETROBRAS for the financial support of this work.

References [1] A. B. Dias Jr; E. L. Albuquerque and E. L. F. Fortaleza. The boundary element method applied to the analysis of fluid extraction from a reservoir. Advances in Boundary Element Techniques, 2013. [2] P. K. Banerjee. Boundary element methods in engineering. McGraw-Hill, 2 edition, 1981. [3] H. Zhang; D. A. Baray and G. C. Hocking. Analysis of continuous and pulsed pumping of a phreatic aquifer. Advances in Water Resources, 22(6):623–632, 1999. [4] J. Bear. Dynamics of fluids in porous media. American Elsevier, 1972. [5] S. K. Lucas; J. R. Blake and A. Kucera. A boundary integral method applied to water coning in oil reservoirs. The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 32:261– 283, 1991. [6] A. J. Rosa; R. S. Carvalho and J. A. D. Xavier. Oil reservoir engineering (in Portuguese). Interciência, 2006. [7] M. Crick. Reservoir simulation. Journal of Petroleum Technology, July 2013:76–89, 2013. [8] G. C. Hocking and H. Zhang. Coning during withdrawal from two fluids of different density in a porous medium. Journal of Engineering Mathematics, 65(2):101–109, 2009. [9] H. Zhang; G. C. Hocking and B. Seymour. Critical and supercritical withdrawal from a two-layer fluid through a line sink in a partially bounded aquifer. Advances in Water Resources, 32:1703–1710, 2009. [10] J. A. Ligget and P. L-F Liu. The boundary integral equation method for porous media flow. George Allen and Unwin, 1983.


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Investigations of dynamic interface crack problems in active bimaterials Felipe García-Sánchez1, Michael Wünsche2,3, Andrés Sáez2 and Chuanzeng Zhang3 1

Departamento de Ingeniería Civil, de Materiales y Fabricación, Universidad de Málaga, C/ Dr. Ortiz Ramos s/n, 29071 Málaga, Spain 2

3

Departamento de Mecánica de Medios Continuos y Teoría de Estructuras, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain

Department of Civil Engineering, University of Siegen, Paul-Bonatz-Str. 9-11, D-57076 Siegen, Germany

Keywords: Linear piezoelectric materials, electrically semi-permeable crack-face condition, crack-face contact condition, time-domain BEM, dynamic intensity factors.

Abstract. This paper presents a time-domain BEM for transient dynamic crack analysis in two-dimensional (2D) composites with passive non-piezoelectric and active piezoelectric components. A time-domain boundary element method is applied for numerical computations which uses the convolution quadrature of Lubich for temporal discretization. An explicit time-stepping scheme is obtained for computing the unknown boundary data. Non-linear boundary conditions are applied to take possible crack-face contact and the electric permeability of the medium inside the crack into account for realistic simulation. Numerical examples are presented to show the effects of the crack positions, the material combinations and the dynamic loading on the dynamic field intensity factors.

Introduction Piezoelectric materials are widely used in smart devices and structures by utilizing the coupling effects between the mechanical and electrical fields. Beside layered materials, fibre composites by combining a non-piezoelectric matrix and piezoelectric fibres can be optimized by taking advantages of the most beneficial properties of each constituent. Since piezoelectric composites are usually very brittle the static and dynamic crack analysis is an important task [1]. Beside cracks inside homogeneous domains, interface cracks may be induced under the in-service condition. Due to the mismatch of the mechanical and electrical properties of the material constituents, interface problems are significantly more complicated than the corresponding crack problems in homogeneous elastic or piezoelectric materials. Of considerable importance to fracture and damage mechanics, design and optimization as well as non-destructive testing of such composites and structures is the dynamic crack analysis. A key issue in the crack analysis of piezoelectric composites is to describe the mechanical and electrical boundary conditions on the crack-faces properly. The mechanical boundary conditions are usually assumed as self-equilibrated stresses and contact conditions are introduced to avoid physically unacceptable crack-face intersection. Several electrical crackface boundary conditions are proposed and discussed in the literature [1,2]. A non-linear semi-permeable electrical crack-face boundary condition has been presented by Hao and Shen [3] in order to take into account the influence of the medium inside a crack. In this paper transient dynamic crack analysis in two-dimensional (2D) composites with passive nonpiezoelectric and active piezoelectric components is presented. Both cracks inside homogeneous nonpiezoelectric or piezoelectric components and interface cracks between non-piezoelectric and piezoelectric layers or fibres and matrices are considered. A time-domain boundary element method is applied for numerical computations which uses the convolution quadrature of Lubich for temporal discretization. An explicit time-stepping scheme is obtained for computing the unknown boundary data. Non-linear boundary conditions are applied to take possible crack-face contact and the electric permeability of the medium inside

208


the crack into account for realistic simulation. Numerical examples are presented to show the effects of the crack positions, the material combinations and the dynamic loading on the dynamic field intensity factors.

Problem formulation and time-domain BIEs Let us consider a two-dimensional piecewise homogeneous and linear piezoelectric solid with interior or interface cracks of arbitrary shape. In the absence of body forces and free electrical charges and by using the quasi-electrostatic assumption the cracked piezoelectric solid satisfies the generalized equations of motion G , J, K 1,2

V iJ ,i (x, t ) U ] G JK uK (x, t ) , G JK ® JK , (1) ¯ 0, otherwise and the constitutive equations ] ViJ (x, t ) CiJKl u K ,l (x, t ) . (2) Here, ρ] is the mass density for a homogeneous domain ζ=1,…,N, GJK* is the generalized Kronecker delta, and uI, ViJ and CiJKl are the generalized displacements, the generalized stresses and the generalized elasticity tensor defined by V , J 1,2 u , I 1,2 , ViJ ® ij , uI ® i (3) M , I 4 ¯ ¯Di , J 4

CiJKl

cijkl , J, K 1,2 ° e , J 1,2, K 4 ° lij , ® °eikl , J 4, K 1,2 ¯° Hil , J, K 4

(4)

where ui and M are the displacements and the electrical potential, Vij and Di are the stresses and the electric displacements and cijkl, eijk and Hil are the elasticity tensor, the piezoelectric tensor and the dielectric permittivity tensor. A comma after a quantity represents spatial derivatives while a dot over a quantity denotes time differentiation. Lower case Latin indices take the values 1 and 2 (elastic), while capital Latin indices take the values 1, 2 (elastic) and 4 (electric). Unless otherwise stated, the conventional summation rule over repeated indices is implied. Further, the following initial conditions uI (x, t 0) u I (x, t 0) 0 , (5) the boundary conditions t I (x, t ) t I (x, t ) , x *t , (6) (7) u I (x, t ) u I (x, t ) , x *u with tI being the traction vector defined by t I (x, t ) V jI (x, t )e j (x) , (8) and the continuity conditions on the interface except the crack-faces t I (x, t ) t I (x, t ) , x *if , (9) (10) u I (x, t ) u I (x, t ) , x *if are applied. Here, ej is outward unit normal to the boundary, Γif is the interface between the homogenous domain Ωζ (ζ=1,2,…,N), Γt and Γu stand for the external boundaries where the tractions ti and the displacements ui are prescribed. On the upper and the lower crack-faces, Γc+ and Γc- respectively, selfequilibrated generalized tractions are considered and to avoid a physically meaningless material interpenetration between both crack-faces, the following constraint condition is introduced on the crack 'u 2 (x *c , t ) t 0 , (11) with Δu2(x,t) being the normal component of the generalized crack-opening-displacements defined by 'u I (x, t ) u I (x *c , t ) u I (x *c , t ) . (12) Further three different electrical boundary conditions are applied on the crack-faces. Taking into account the electrical permittivity κc inside the crack, the electrical semi-permeable crack-face boundary conditions are


Di (x *c , t )

Di (x *c , t ) , Di (x *c , t )

Nc

209

>M(x * >u (x * i

@ , t )@

c

, t ) M(x *c , t )

c

, t ) u i (x *c

,

(13)

where both opposite crack-faces are considered as a set of corresponding parallel capacitors. If both crackfaces are considered as electrically impermeable, eq. (10) simplifies to Di (x *c , t ) Di (x *c , t ) 0 . (14) In contrast, if both crack-faces are treated as electrical permeable, eq. (10) leads to Di (x *c , t ) Di (x *c , t ) , M(x *c , t ) M(x *c , t ) 0 . (15) The time-domain displacement BIEs for a cracked solid may be written as

³ >u

c IJ u I (x, t )

G IJ

@

(x, y, t ) t I (y, t ) t GIJ (x, y, t ) u I (y, t ) d*y

*b

(16)

³ t GIJ (x, y, t ) 'u I (y, t ) d*y , * c

where the free term cIJ is defined for a source point x on the smooth boundary by cIJ=1/2δIJ, Гb=Гu+Гt+Гif, an asterisk “*” denotes the Riemann convolution, uIJG(x,y,t) are the dynamic displacement fundamental solutions and tIJG(x,y,t) are the dynamic traction fundamental solutions defined by t GIJ (x, y, t ) CqIPr e q (y)u GPJ,r (x, y, t ) . (17) The time-domain traction BIEs are obtained by substituting eq. (16) into the constitutive equations (2) and taking the limit process x→Гc± to yield

³ >v

t J (x, t )

G IJ

@

(x, y, t ) t I (y, t ) w GIJ (x, y, t ) u I (y, t ) d*y

*b

(18)

³ w GIJ (x, y, t ) 'u I (y, t ) d*y , * c

with vIJG(x,y,t) and wIJG(x,y,t) being the traction and the higher-order traction fundamental solutions defined by v GIJ (x, y, t ) CpIKse p (x)u GKJ ,s (x, y, t ) , (19) (20) w G (x, y, t ) C e (x)C e (y)u G (x, y, t ) . IJ

pIKs p

qJLr q

KL, sr

The Riemann convolution integral is approximated by the convolution quadrature in this work, which requires the Laplace-domain fundamental solutions instead of the time domain fundamental solutions. The Laplace-domain fundamental solutions for homogeneous and linear piezoelectric solids can be represented in the 2D case by a line integral over the unit-circle as M · PIJm § p 1

@

(21) (22)

where n, cm and PIJm, p, Ei are the wave propagation vector, the phase velocities of the elastic waves, the projection operator, the Laplace parameter and the exponential integral, respectively [4]. It should be mentioned that the fundamental solutions can be divided into a static part plus a dynamic part u GIJ (x, y, p) u SIJ (x, y) u DIJ (x, y, p) . (23) While the static part can be simplified to an closed form and contains the singularities the dynamic part is regular and defined by a line integral over the unit-circle [4,5].

Numerical solution algorithm A spatial collocation method is used to solve the time-domain displacement BIEs (16) and the time-domain traction BIEs (18) numerically. The time-domain displacement BIEs are used on the external boundary and the interfaces while the time-domain traction BIEs are applied on the crack-face inside a homogenous domain. Quadratic elements are used for spatial discretization. On the crack-faces discontinuous quadratic

210


elements are adopted in order to fulfill the C1-continuity requirement of the CODs in the hypersingular traction BIEs. Quarter-point elements are used at the crack-tips inside the domain to describe the local behavior of the CODs properly. This ensures an accurate and direct calculation of the dynamic intensity factors from the numerically computed CODs. Despite the known oscillating behavior of the singularities for interfacial cracks [6], only quadratic elements have been implemented at crack tip for these preliminary results. The strongly singular and hypersingular boundary integrals are computed by a regularization technique based on a suitable change of variable as shown by García-Sánchez et al. [1]. The line integrals over the unit circle arising in the dynamic parts of the dynamic fundamental solutions are computed numerically by the standard Gaussian quadrature. More details about the solution algorithm can be found for example in García-Sánchez et al. [5]. After spatial and temporal discretizations and invoking the initial conditions (5), the boundary conditions (6), (7) and the continuity conditions (9), (10), an explicit time-stepping scheme can be obtained as K 1 ª º x K (C1 ) 1 «D1 y K (B K k 1 t k A K k 1 u k )» . (24) k 1 ¬ ¼ In eq. (24) yK is the vector of the prescribed boundary data, while xK represents the vector of the unknown boundary data, which can be computed time-step by time-step. The investigated initial-boundary value problem involves two different non-linear crack-face boundary conditions. At each time-step where positive crack-opening displacements are obtained an iteration algorithm is used to solve the semi-permeable electrical crack-face boundary conditions (13). In contrast, at time-steps where a physically unacceptable crack-face intersection occurs, an iterative crack-face contact analysis is applied to satisfy the mechanical constraint condition (11). Further information about both non-linear solution algorithms are shown in Wünsche et al. [2].

¦

Numerical results In this section, several numerical examples are presented and discussed. The following loading parameter χ is introduced to measure the intensity of the electrical loading e 22 D0 , F (25) H 22 V0 where e 22 and ε 22 are material properties, see (4), and σ0 and D0 are the mechanical and electrical loading amplitudes. For convenience, the mode-I, the mode-II and the mode-IV dynamic intensity factors for crack-tips inside a homogeneous layer or sub-domain are normalized by I

e 22 K IV ( t ) K I (t) K II ( t ) , K*II ( t ) , K *IV ( t ) . (26) K0 K0 H I22 K 0 The real part K1 and the imaginary part K2 of the complex dynamic stress intensity factors and the electrical displacement intensity factors K4 for interface cracks are normalized by K*I ( t )

K1* ( t )

K1 ( t ) , K*2 ( t ) K0

K 2 (t) , K*4 (t ) K0

I I with e 22 , ε 22 are properties of the material I and K 0 crack.

eI22 K 4 ( t ) H I22 K 0

(27)

V 0 Sa with being the half length of an internal

In the first example, we consider a square plate with a central fibre containing a crack of the length 2a as shown in Fig. 1. The plate is subjected to a tensile impact loading σ(t)=σ0H(t), where σ 0 is the loading amplitude and H(t) is the Heaviside step function. The geometrical dimensions h=20mm, r=h/2 and 2a=4.8mm are used. Epoxy is chosen as material for the passive non-piezoelectric matrix which has the elastic constants

C11 8.0 GPa , C12 4.4 GPa , C 22 8.0 GPa , C66 1.8 GPa , H11 0.0372 C /(GVm) , H 22 0.0372 C /(GVm)

(28)


211

and PZT-5H is used for the active piezoelectric fibre C11 126.0 GPa , C12 84.1GPa , C22 117.0 GPa , C66

23.0 GPa ,

(29) e21 6.5 C / m , e22 23.3C / m , e16 17.0 C / m , H11 15.04C /(GVm ) , H22 13.0 C /(GVm ) . The spatial discretization of the external boundary is done by an element-length of 4mm and the crack is divided into 6 elements and a time-step of cLΔt/h=0.04 is used in the computation. 2

2

2

Fig. 1: A cracked square plate with a central fibre under impact loading The normalized dynamic intensity factors obtained by the present time-domain BEM are presented in Fig. 2.

Fig. 2: Normalized dynamic intensity factors for Tip A and Tip B The normalized dynamic mode-I stress intensity factors are quite different for both crack-tips. Since the material for the domain I is non-piezoelectric the intensity factors are zero until the elastic waves reach the crack. Immediately after the wave arrival at the crack-tips, the dynamic stress intensity factors increase rapidly with increasing time and after reaching a peak they show a rather complex variation with increasing time. As expected, the electrical displacement intensity factor for the crack-tip inside the non-piezoelectric matrix is zero. In the second example as shown in Fig. 3, let us consider a rectangular piezoelectric plate with a central crack of length 2a subjected to a tensile impact loading σ(t)=σ0H(t). The geometry of the cracked plate is

212


determined by h=16mm, w=20mm, r=5mm and a=r. The same material properties as in the first example for the matrix and the fibre are applied. V( t )

I w r II

Tip A

Tip B 2a

h

Fig. 3: A cracked square plate with a central fibre under impact loading The boundary of the plate is divided into elements with a length of 4.0mm, the crack is discretized by 6 elements, and a time-step of cLΔt/h=0.06 is chosen. Plain strain condition is assumed. Three different configurations are considered. In the first case the material of the fibre is PZT-5H. In the second case the fibre has the same material properties than the matrix, which is identically to the homogenous cracked plate and in the third case a hole instead of a fibre is investigated. The normalized dynamic intensity factors of the present time-domain BEM for are shown in Fig. 4.

Fig. 4: Normalized dynamic IFs of both crack-tips for the different crack configurations The normalized dynamic mode-I stress intensity factors for both crack-tips show in generally a similar global behaviour for the three investigated configurations. Although the fibre has clearly a higher stiffness than the matrix the highest stress intensity factor is obtained at the right crack-tip for the fibre configuration. The lowest stress intensity factor is computed at the right crack-tip for the plate with the hole. This is expected since the hole shields this crack-tip. In contrast the stress intensity factors at left crack-tip are significantly higher.

Summary Transient dynamic analysis of cracks in two-dimensional composites with passive non-piezoelectric and active piezoelectric components is presented in this paper. Interior cracks as well as interface cracks are investigated. For the numerical computations a time-domain boundary element method is applied. Nonlinear boundary conditions are used to consider possible crack-face contact and the electric permeability of


213

the medium inside the crack for more realistic simulation. Numerical examples are presented and discussed to show the effects of the crack positions, the material combinations and the dynamic loading on the dynamic field intensity factors. The results indicate the suitability of the present BEM for the numerical simulation of this kind of modern smart materials. Acknowledgement This work was funded by the research project DPI2010-21590-C02-02 from the Spanish Ministerio de Economía y Competitividad and the research project P09-TEP-5054 from the Junta de Andalucía. The financial support is gratefully acknowledged.

References [1] Kuna M., Fracture mechanics of piezoelectric materials - where are we right now?,

Engineering Fracture Mechanics 2010; 77: 309-326. [2] Wünsche M., Zhang Ch., García-Sánchez F., Sáez A., Sladek J., Sladek V., Dynamic crack analysis in piezoelectric solids with non-linear electrical and mechanical boundary conditions by a time-domain BEM, Computer Methods in Applied Mechanics and Engineering 2011; 200: 2848-2858. [3] Hao T.H., Shen Z.Y., A new electric boundary condition of electric fracture mechanics and its applications, Engineering Fracture Mechanics 1994; 47: 793-802. [4] Wang C.-Y. and Zhang Ch., 3-D and 2-D dynamic Green’s functions and time-domain BIEs for piezoelectric solids, Engineering Analysis with Boundary Elements 2005; 29: 454-465. [5] García-Sánchez F., Zhang Ch., Sáez A., 2-D transient dynamic analysis of cracked piezoelectric solids by a time domain BEM, Computer Methods in Applied Mechanics and Engineering 2008; 197: 31083121. [6] Suo Z., Kuo C.M., Barnett D.M., Willis J.R., Fracture mechanics for piezoelectric ceramics, Journal of the Mechanics and Physics of Solids 1992; 40: 739-765.

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A (constrained) microstretch approach in living tissue modelling: a numerical investigation by the local point interpolation – boundary element method. Jean-Philippe Jehl1, Richard Kouitat Njiwa2 Université de Lorraine, Institut Jean Lamour - Dpt N2EV - UMR 7198 CNRS. Parc de Saurupt, CS 14234, 54042 Nancy Cedex France 1


Keywords: Micromorphic, Constrained microstretch, Isotropic BEM, Meshfree strong form. Abstract. Extended continuum mechanical approaches are nowadays increasingly adopted for the modelling of various types of materials with microstructure such as foams and porous solids. The potentialities of the microcontinuum approach, in the field of biomechanical modelling, are investigated. It is obvious that, in this area numerical investigation of the material response is of paramount importance. The work hereby is concerned with investigation of some potentialities of the (constrained) microstretch modelling. The fields equations of the problem are solved by a numerical approach combining the conventional isotropic boundary elements method with local radial point interpolation. Numerical examples show that the model is a good candidate for the mechanical modelling of living tissues.

Introduction It is more and more evident that the microstructure of the material must be taken into account in the establishment of its constitutive equation. Microstructural information is taken into account either implicitly or explicitly in the description of the continuum. In the case of biological materials it is not easy to apply a multi scale modelling or conventional mathematical approaches. Indeed, a living tissue such as heart tissue is non homogeneous with a complex micro organization. Following Rosenberg and Cimrman [1], it is appropriate to investigate the applicability of microcontinuum approaches in this field. The theory of micromorphic media of Eringen and Suhubi [2] is intended to capture the impact of the microstructure on the overall response of the material. In the full theory, a material point of a micromorphic medium has twelve degrees of freedom: the three traditional components of displacement and the nine components of a micro-deformation tensor. Usually it is specialized depending on the salient microscopic motions. When the material point of the medium can rotate and stretch, the medium is known as a microstretch material. In the constrained microstretch medium the material point can only dilate or contract. The latter is also known as a microdilatation medium and has already been applied to model foam and some porous media [3]. The literature on microstretch media is not dense. Indeed it is not easy to understand and qualify their response using conventional mechanical tests. We believe that numerical experiments are extremely useful in this field. Following, we have taken the party to investigate the response of a 3D (constrained) microstretch medium to various types of loading. The work presented is essentially numerical and based on a specifically developed simulation tool. First of all, we present the governing equations of a microstretch medium. Then, we describe the adopted numerical method called the “local point interpolation – boundary element method” (LPI-BEM). Finally, the ability of the model to represent the mechanical behaviour of living tissue such as heart tissue is demonstrated by the presented numerical examples

Governing Equations In the theory of microstretch medium, the material point is attached to a triad of directors that can rotate and stretch. The material point has seven degrees of freedom: the three components of the traditional displacement vector, the three components of a microrotation vector and the scalar microdilatation. The field equations of such a medium under quasi-static evolution without external body loads are [4]:


ߪ௝௜ǡ௝ ൌ Ͳ ݉௝௜ǡ௝ ൅ ߳௜௝௞ ߪ௝௞ ൌ Ͳ ‫ݏ‬௞ǡ௞ െ ‫ ݌‬ൌ Ͳ

215

(1) (2) (3)

In these equations, ߪ௜௝ is the stress tensor, ݉௜௝ is the couple stress tensor, ‫ݏ‬௞ the vector of internal hypertraction and the scalar ‫ ݌‬is the generalized internal body load. The latter can be viewed as an internal pressure. When the considered solid is homogeneous and isotropic the constitutive relations are as follows: ߪ௜௝ ൌ ߣߝ௥௥ ߜ௜௝ ൅ ʹሺߤ ൅ ߢሻߝ௜௝ െ ߢ‫ݑ‬௜ǡ௝ ൅ ߢ߳௝௜௞ ߮௞ ൅ ߟ߰ߜ௜௝ ݉௜௝ ൌ ߙ߱௥௥ ߜ௜௝ ൅ ሺߚ ൅ ߛሻ߱௜௝ ൅ ሺߚ െ ߛሻ൫߮௜ǡ௝ െ ߮௝ǡ௜ ൯Ȁʹ ‫ݏ‬௞ ൌ ܽ߰ǡ௞ ‫ ݌‬ൌ ߟߝ௥௥ ൅ ܾ߰

(4) (5) (6) (7)

߰ is the micro-stretch function, ‫ݑ‬௜ the macroscopic displacement vector and ߮௜ the microscopic rotation vector. ߝ௜௝ ൌ ൫‫ݑ‬௜ǡ௝ ൅ ‫ݑ‬௝ǡ௜ ൯Ȁʹ and ߱௜௝ ൌ ൫߮௜ǡ௝ ൅ ߮௝ǡ௜ ൯Ȁʹ. ߣ and ߤ are the Lame’s constants. ߙ, ߚ, ߛ and ߢ are the micropolar constants.ߟ, ܽ and ܾ are the microstretch elastic constants. With ݊௝ the outward normal vector on the boundary, the tractions acting at a regular point of the boundary are given by: (8) ‫ݐ‬௜ ൌ ߪ௝௜ ݊௝ , ݉௜ ൌ ݉௝௜ ݊௝ and ‫ ݏ‬ൌ ‫ݏ‬௝ ݊௝ The material parameters must fulfil the following constraints: ʹߤ ൅ ߢ ൒ Ͳ ߢ൒Ͳ ܽ൒Ͳ ܾሺ͵ߣ ൅ ʹߤ ൅ ߢሻ െ ͵ߟ ଶ ൒ Ͳ ܾ൒Ͳ ͵ߙ ൅ ߚ ൅ ߛ ൒ Ͳ ߚ൅ߛ ൒Ͳ ߛെߚ ൒Ͳ

Solution Method In the case of linear problems with well-established analytical fundamental solution, the boundary element method has already proved very efficient. As the fundamental solution of the field equations does not exist, the BEM loses its main appeal (reduction of the problem dimension by one) as traditional volume cell are needed in the so called “field boundary element method”. In order to alleviate this shortcoming, a number of strategies have been proposed. One can mention the dual reciprocity method (DRM) and the radial integration method (RIM) that allow to convert volume integrals into surface ones. In recent years, a large number of researchers have invested in the development of the so called meshless or meshfree methods. Among the various meshless approaches, the local point interpolation method is very attractive because it is simple to implement. The accuracy of this approach deteriorates in presence of Neumann type boundary conditions which are practically inevitable when solving solid mechanic problems. Liu et al [5] have proposed to circumvent this difficulty by adopting the so called weak-strong form local point interpolation method. In a recent paper Kouitat [6] proposed a strategy to take advantage of the conventional BEM method and the local point interpolation method. The LPI-BEM has proved efficient in the case of anisotropic elasticity [6], piezoelectric solid [7] and nonlocal elasticity [8]. It is adopted in this work and its main steps in the case of a microstretch medium are presented below. Assume that the kinematical primary variables are the sum of a complementary part and a particular term. That is : ‫ݑ‬௜ ൌ ‫ݑ‬௜ு ൅ ‫ݑ‬௜௉ , ߮௜ ൌ ߮௜ு ൅ ߮௜௉ and ߰ ൌ ߰ ு ൅ ߰ ௉ The complementary fields satisfy the following homogeneous equations: ଵ ு ு ு ҧ ൤ȟ߮௜ு ൅ ଵ ߮௝ǡ௝௜ ‫ܩ‬௨ҧ ቂȟ‫ݑ‬௜ு ൅ ‫ݑ‬௝ǡ௝௜ ൌͲ (9) ቃ ൌ Ͳ , ‫ܩ‬ఝ ൨ ൌ Ͳ , ܽ߰ǡ௞௞ ෥ೠ ଵିଶఔ

෥ക ଵିଶఔ

These equations are solved by the conventional boundary element method. Therefore, the following systems of equations are obtained: ሾ‫ܪ‬௨ ሿሼ‫ݑ‬ு ሽ ൌ ሾ‫ܩ‬௨ ሿሼ‫ ݐ‬ு ሽ , ൣ‫ܪ‬ఝ ൧ሼ߮ ு ሽ ൌ ൣ‫ܩ‬ఝ ൧ሼ݉ு ሽ and ൣ‫ܪ‬ట ൧ሼ߰ு ሽ ൌ ൣ‫ܩ‬ట ൧ሼ‫ ݏ‬ு ሽ (10)

216


The particular fields solve: ଵ ௉ ‫ܩ‬௨ҧ ቂȟ‫ݑ‬௜௉ ൅ ଵିଶఔ෥ ‫ݑ‬௝ǡ௝௜ ቃ െ ߢ‫ݑ‬௝ǡ௝௜ ൅ ߢ߳௜௝௞ ߮௞ǡ௝ ൅ ߟ߰ǡ௜ ൌ Ͳ

(11)

൫ఝೕǡ೔ೕ ିఝ೔ǡೕೕ ൯ ௉ ҧ ൤ȟ߮௜௉ ൅ ଵ ߮௝ǡ௝௜ ‫ܩ‬ఝ െ ߢ߮௜ െ ߢ߳௜௝௞ ‫ݑ‬௞ǡ௝ ൌ Ͳ ൨ ൅ ሺߚ െ ߛሻ ෥ ଵିଶఔ ଶ

(12)

௉ ܽ߰ǡ௞௞ െ ܾ߰ െ ߟ‫ݑ‬௥ǡ௥ ൌ Ͳ

(13)

ೠ

ക

The tractions at a regular point on the boundary are written as: ஺ ‫ݐ‬௜ ൌ ‫ݐ‬௜ு ൅ ‫ݐ‬௜௉ ൅ ߜ‫ݐ‬௜ with ‫ݐ‬௜஺ ൌ ൫ߣߝ௥௥ ߜ௜௝ ൅ ʹሺߤ ൅ ߢሻߝ௜௝஺ ൯݊௝ ሺ‫ ܣ‬ൌ ‫ܲݎ݋ܪ‬ሻ and ߜ‫ݐ‬௜ ൌ ൫െߢ‫ݑ‬௜ǡ௝ ൅ ߢ߳௝௜௞ ߮௞ ൅ ߟ߰ߜ௜௝ ൯݊௝ ஺ ஺ ݉௜ ൌ ݉௜ு ൅ ݉௜௉ ൅ ߜ݉௜ with ‫ݐ‬௜஺ ൌ ൫ߙ߱௥௥ ߜ௜௝ ൅ ሺߚ ൅ ߛሻ߱௜௝ ൯݊௝ ሺ‫ ܣ‬ൌ ‫ܲݎ݋ܪ‬ሻ and ߜ݉௜ ൌ ሺߚ െ ߛሻ൫߮௝ǡ௜ െ ߮௜ǡ௝ ൯݊௝ Ȁʹ ‫ ݏ‬ൌ ‫ ݏ‬ு ൅ ‫ ݏ‬௉ with ‫ ݏ‬஺ ൌ ܽ߰ǡ௝஺ ݊௝ ሺ‫ ܣ‬ൌ ‫ܲݎ݋ܪ‬ሻ

(14) (15) (16)

Let us now consider the solution of equations (11-13) by a local radial point collocation method. In this method [5], a field

Z x is approximated as: w(x) = N

with the constraint condition:

i=1

N i=1

Ri (r)ai +

M

Pj (x)bj

j=1

Pj (xi )ai = 0 j = 1-m

Ri r is the radial basis functions, N the number of nodes in the neighbourhood (support domain) of point

x and M is the number of monomial terms in the polynomial basis ܲ௝ ሺ‫ݔ‬ሻ. Coefficients ai and b j can be determined by enforcing the approximation to be satisfied at the ܰ nodes in the support domain of point x. After some algebraic manipulation the interpolation is written in the compact form: (x) = [ (x)] { / L } . For a given collocation centre, the followings are obtained: ௉ ෩ ൧൛‫ݑ‬Ȁ௅ ෩ ଵ ൧൛‫ݑ‬Ȁ௅ ൟ ൅ ߢൣȰ ෩ ଶ ൧൛߮Ȁ௅ ൟ ൅ ሼ‫׏‬ሽൣȰ ෩ ଷ ൧ሾߟሿ൛߰Ȁ௅ ൟ ൌ ሼͲሽ ሾ‫ܤ‬ሺ‫׏‬ሻሿ் ሾ‫ܥ‬௨ ሿሾ‫ܤ‬ሺ‫׏‬ሻሿൣȰ ൟ ൅ ߢൣȰ ௉ ෩ ൧൛߮Ȁ௅ ෡ ଴ ൧൛߮Ȁ௅ ൟ ൅ ሺߚ െ ߛሻൣȰ ෡ ଵ ൧൛߮Ȁ௅ ൟ ൅ ߢൣȰ ෡ ଷ ൧൛‫ݑ‬Ȁ௅ ൟ ൌ ሼͲሽ ሾ‫ܤ‬ሺ‫׏‬ሻሿ் ൣ‫ܥ‬ఝ ൧ሾ‫ܤ‬ሺ‫׏‬ሻሿൣȰ ൟ ൅ ߢൣȰ ்

்

௉ ෙ ൟ ൛߰Ȁ௅ ෙ ଵ ൟ ൛߰Ȁ௅ ൟ െ ሼ‫׏‬ሽ் ൣȰ ෩ ൧ሾߟሿ൛‫ݑ‬Ȁ௅ ൟ ൌ Ͳ ܽሼ‫׏‬ሽ் ሼ‫׏‬ሽ൛Ȱ ൟ െ ܾ൛Ȱ డ

In these equations, ሼ‫׏‬ሽ ൌ ቀడ௫

B( z )

ª z1 «0 « ¬« 0

0 z2 0

0 0 z3

z2 z1 0

z3 0 z1

భ

డ డ௫మ

డ ் ቁ డ௫య

and matrix B is given in terms of a vector z=(z1,z2,z3)T by

T

0º z3 » . The over matrices are properly constructed from the defined » z2 ¼»

interpolation. Now, consider all internal collocation points and considering that all particular integrals take zero value at boundary points, the particular integrals at internal collocation centres are written in the forms: ሼ‫ݑ‬௉ ሽ ൌ ሾ‫ܣ‬௨ ሿሼ‫ݑ‬ሽ ൅ ሾ‫ܤ‬௨ ሿሼ߮ሽ ൅ ሾ‫ܥ‬௨ ሿሼ߰ሽ ሼ߮ ௉ ሽ ൌ ൣ‫ܣ‬ఝ ൧ሼ߮ሽ ൅ ൣ‫ܤ‬ఝ ൧ሼ‫ݑ‬ሽ ሼɗ୔ ሽ ൌ ൣ‫ܣ‬ట ൧ሼ߰ሽ ൅ ൣ‫ܤ‬ట ൧ሼ‫ݑ‬ሽ Following a similar strategy, the tractions at the boundary points can be written in the forms:

(17) (18) (19)


217

ሼ‫ݐ‬ሽ ൌ ሼ‫ ݐ‬ு ሽ ൅ ሾ‫ܭܣ‬௨௨ ሿሼ‫ݑ‬ሽ ൅ ൣ‫ܭܣ‬௨ఝ ൧ሼ߮ሽ ൅ ൣ‫ܭܣ‬௨ట ൧ሼ߰ሽ ሼ݉ሽ ൌ ሼ݉ு ሽ ൅ ൣ‫ܭܣ‬ఝ௨ ൧ሼ‫ݑ‬ሽ ൅ ൣ‫ܭܣ‬ఝఝ ൧ሼ߮ሽ and ሼ‫ݏ‬ሽ ൌ ሼ‫ ݏ‬ு ሽ ൅ ൣ‫ܭܣ‬ట௨ ൧ሼ‫ݑ‬ሽ ൅ ൣ‫ܭܣ‬టట ൧ሼ߰ሽ After some algebraic manipulations the final coupled systems of equations are of the forms: ഥ௨ ሿሼ‫ݑ‬ሽ ൅ ൣ‫ܪ‬௨ఝ ൧ሼ߮ሽ ൅ ൣ‫ܪ‬௨ట ൧ሼ߰ሽ ൌ ሾ‫ܩ‬௨ ሿሼ‫ݐ‬ሽ ሾ‫ܪ‬

(20)

ഥఝ ൧ሼ߮ሽ ൅ ൣ‫ܪ‬ఝ௨ ൧ሼ‫ݑ‬ሽ ൌ ൣ‫ܩ‬ఝ ൧ሼ݉ሽ ൣ‫ܪ‬

(21)

ഥట ൧ሼ߰ሽ ൅ ൣ‫ܪ‬ట௨ ൧ሼ‫ݑ‬ሽ ൌ ൣ‫ܩ‬ట ൧ሼ‫ݏ‬ሽ ൣ‫ܪ‬

(22)

Remarkably, the final equations contain boundary primary variables and internal kinematical unknowns as in traditional BEM. Boundary conditions can be taken into account as usual and the resulting system of equations solved by a standard direct solver. Numerical examples ௤

In this work the multi-quadrics radial basis functions are adopted: ܴ௜ ሺ‫ݎ‬ሻ ൌ ൫‫ݎ‬௜ଶ ൅ ܿ ଶ ൯ where ‫ݎ‬௜ ൌ ԡ‫ ݔ‬െ ‫ݔ‬௜ ԡ, ܿ and ‫ ݍ‬are known as shape parameters. The shape parameter is taken proportional to a minimum distance ଴ defined as the maximum value among the minimum distances in the x, y and z directions between collocation points. The simulation tool, that has been developed, is general for a microstretch medium. It has been validated through comparison with analytical results in the case of uniform traction loading. Given the limited number of pages, we present the results of influence an infarction necrotic area relating to a tubular cylindrical geometry sample often adopted when drawing up a scheme of the left ventricle [9,10]. The cylinder has an outside radius of 0.75mm, a thickness of 0.5mm and measures 1.25mm in length. This specimen is loaded by 10-2GPa pressure on its outer boundary. Considering that, with respect to heart infarction, the most salient microscopic feature is the ability of the material point to “breath”, the model is specialized to microdilatation. For this case, all the microrotation parameters are set to zero. The results presented are obtained with the boundary of the specimen subdivided into 288 nine nodes boundary elements supplemented by 432 internal collocation centers. The material parameters adopted are: μ= 0.85 GPa, O = 3.4 GPa, a = 26 kN, b = 26 GPa and K=-10.15GPa. Figure 1 shows the deformation of a cross section of the cylinder. Let us remind the reader that, under this loading state, with regards to deformation, all cross section of the cylinder are identical. Note also that in this case the specimen behaves like an auxetic material, that is material with negative Poisson ratio.

FIG. 1 – Radial displacement in a cross section of the cylinder with microdilatation These results should be linked to the left ventricular ejection fraction (LVEF), i.e. the ratio between volume ejected by the left ventricle (end-systolic volume) and the initial volume (end-diastolic volume). This measurement holds clinical significance as it indicates the heart’s capacity to adequately pump blood (and therefore oxygen) around the body. Certain works ([11–14]) have used the LVEF as an indicator in prognosis for heart failure. P. Curtis et al. [15] state in their work that, as the LVEF rises from 15 to 45%, mortality

218


rates fall linearly. Once the 45% marker is overtaken, the mortality rate seems no longer affected by the EF. The ejection fraction calculated for the cases shown in Figure 1 is 84.7%. This criterion provides a solid base for considerations of this model’s capacity to analyze the mechanical behavior of the left ventricle. In the following, the material with the above parameters is called the safe specimen. Next we will consider a sample covering a small zone within which the parameter K values vary. This zone is intended as a representation of a post-infarction necrotic area. We first position it near to the outer boundary of the tube. Figure 2 displays a representation of the deformation in the middle of the sample. The deformation is no more uniform. In accordance with our expectations, the results indicate a significant regression in the LVEF (around 48%) when the parameter K of the necrotic zone is far removed from that of the healthy organ. This is due to a slight narrowing of the cylinder’s interior and a slight shortening of the tube itself. We also succeeded in reducing the axial deformation, as was achieved by Cho et al. in their clinical study [16] comprising heart failure patients with accompanied fall in LVEF. This constitutes yet further confirmation of this model’s great potential for modeling cardiac tissue behavior.

FIG. 2 – Axial deformation in a cross section of the cylinder, K=-10.15GPa in the “safe” zone and K=-2GPa in the affected zone. The necrotic zone is now positioned in proximity to the inner boundary of the tube, and our focus in this instance is particularly on the contractility of the neighboring points to this zone (Fig. 3). Unlike in previous cases, a large zone on the opposite side of the necrotic area shows a significant reduction in contractility. This result is in line with observations made by Reimer et al. in their series of experiments [17] who illustrated a phenomenon of tissue necrosis spreading from the endocardium to the epicardium or, in layman’s terms, from the interior to the exterior of the heart.

a)

b)

FIG. 3 – Loss of dilatation: affected zone initially in the proximity of the outer boundary a), or the inner boundary b).


219

4 Conclusion This primarily numerical work illustrates the great potential of the “local point interpolation – boundary element” method for addressing problems relating to the microstretch modelling of living tissue. The LPIBEM has already proved effective and accurate for anisotropic, nonlocal and piezoelectric elasticity. The method couples conventional isotropic BEM with local radial point interpolation applied to strong form differential equations. The solution procedure requires only little modifications of existing BEM code. The presented results prove the ability of the approach, to deal with localised non homogeneity such as a necrotic area. Considering heart infarction, we have observed a reduction in axial deformations following the introduction of an infarct zone, as well as the development of an infarction spreading from the endocardium to the epicardium; in layman’s terms: from the interior to the exterior of the heart. An affected zone positioned in proximity to the exterior of the heart has less of a reductive effect on the muscle’s capacity to contract than a zone positioned close to the inner wall. Following this study, in a future work, the conical geometry of the left ventricle will be taken into account. Also, the anisotropy of the physical parameters will be included in the model. References [1] J. Rosenberg et R. Cimrman, « Microcontinuum approach in biomechanical modeling », Mathematics and Computers in Simulation 61 (2003) 249-260 [2] A. C. Eringen et E. S. Suhubi, « Nonlinear theory of simple micro-elastic solids—I », International Journal of Engineering Science, vol. 2, no 2, p. 189̻ 203, mai 1964. [3] H. Ramézani, H. Steeb, et J. Jeong, « Analytical and numerical studies on Penalized Micro-Dilatation (PMD) theory: Macro-micro link concept », European Journal of Mechanics - A/Solids, vol. 34, p. 130̻ 148, juill. 2012. [4] D. Iesan et A. Pompei, « On the equilibrium theory of microstretch elastic solids », International Journal of Engineering Science, vol. 33, no 3, p. 399̻ 410, févr. 1995. [5] G. R. Liu et Y. T. Gu, « A local radial point interpolation method (LRPIM) for free vibration analyses of 2-D solids », Journal of Sound and Vibration, vol. 246, no 1, p. 29̻ 46, sept. 2001. [6] R. Kouitat Njiwa, « Isotropic-BEM coupled with a local point interpolation method for the solution of 3D-anisotropic elasticity problems », Engineering Analysis with Boundary Elements, vol. 35, no 4, p. 611̻ 615, avr. 2011. [7] N. Thurieau, R. Kouitat Njiwa, et M. Taghite, « A simple solution procedure to 3D-piezoelectric problems: Isotropic BEM coupled with a point collocation method », Engineering Analysis with Boundary Elements, vol. 36, no 11, p. 1513̻ 1521, nov. 2012. [8] M. Schwartz, N. T. Niane, et R. Kouitat Njiwa, « A simple solution method to 3D integral nonlocal elasticity: Isotropic-BEM coupled with strong form local radial point interpolation », Engineering Analysis with Boundary Elements, vol. 36, no 4, p. 606̻ 612, avr. 2012. [9] N. Kirchner et P. Steinmann, « Mechanics of extended continua: modeling and simulation of elastic microstretch materials », Computational Mechanics, vol. 40, no 4, p. 651̻ 666, sept. 2007. [10] Johnston P. R., 2003, “A cylindrical model for studying subendocardial ischaemia in the left ventricle,” Mathematical Biosciences, 186(1), pp. 43–61. [11] Nardinocchi P., Puddu P. E., Teresi L., and Varano V., 2012, “Advantages in the torsional performances of a simplified cylindrical geometry due to transmural differential contractile properties,” European Journal of Mechanics - A/Solids, 36(0), pp. 173–179. [12] Cohn J. N., Johnson G. R., Shabetai R., Loeb H., Tristani F., Rector T., Smith R., and Fletcher R., 1993, “Ejection fraction, peak exercise oxygen consumption, cardiothoracic ratio, ventricular arrhythmias, and plasma norepinephrine as determinants of prognosis in heart failure. The V-HeFT VA Cooperative Studies Group,” Circulation, 87(6 Suppl), pp. VI5–16. [13] Juillière Y., Barbier G., Feldmann L., Grentzinger A., Danchin N., and Cherrier F., 1997, “Additional predictive value of both left and right ventricular ejection fractions on long-term survival in idiopathic dilated cardiomyopathy,” Eur. Heart J., 18(2), pp. 276–280. [14] Hallstrom A., Pratt C. M., Greene H. L., Huther M., Gottlieb S., DeMaria A., and Young J. B., 1995, “Relations between heart failure, ejection fraction, arrhythmia suppression and mortality: analysis of the Cardiac Arrhythmia Suppression Trial,” J. Am. Coll. Cardiol., 25(6), pp. 1250–1257.

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[15] Curtis J. P., Sokol S. I., Wang Y., Rathore S. S., Ko D. T., Jadbabaie F., Portnay E. L., Marshalko S. J., Radford M. J., and Krumholz H. M., 2003, “The association of left ventricular ejection fraction, mortality, and cause of death in stable outpatients with heart failure,” Journal of the American College of Cardiology, 42(4), pp. 736–742. [16] Cho G.-Y., Marwick T. H., Kim H.-S., Kim M.-K., Hong K.-S., and Oh D.-J., 2009, “Global 2Dimensional Strain as a New Prognosticator in Patients With Heart Failure,” Journal of the American College of Cardiology, 54(7), pp. 618–624. [17] Reimer K. A., Lowe J. E., Rasmussen M. M., and Jennings R. B., 1977, “The wavefront phenomenon of ischemic cell death. 1. Myocardial infarct size vs duration of coronary occlusion in dogs,” Circulation, 56(5), pp. 786–794.


221

Boundary Element analysis of Mild Slope Equation problems with a monotonic bed profile A. Cerrato1 , J.A. González2 , L. Rodr´ıguez-Tembleque3

1

Escuela Técnica Superior de Ingenier´ıa, Universidad de Sevilla, Camino de los Descubrimientos s/n, E41092 Sevilla, SPAIN. [email protected], 2 [email protected], 3 [email protected]

Keywords: Fundamental Solution, Boundary Elements Method, Absorbing Boundary Conditions, Mild Slope Equation, Refraction, Diffraction.

Abstract. This work presents a fundamental solution for linear wave transmission problems in a non-homogeneous domain and its numerical implementation in a boundary element formulation. This problem is present in many fields of physics, for example in models of surface gravity waves governed by the Mild Slope Equation. The Boundary Element Method combined with this fundamental solution is a good approach for coastal problems with a bathymetric lines parallel to the coast. The numerical examples contained in this work present a comparison between the proposed fundamental solution and the analytical one for a constant wave number and its application in a problem with variable wave number. In both cases an excellent agreement between the proposed formulation and the theoretical solutions is obtained. Introduction The problem of linear wave transmission in a non-homogeneous medium is present in many physical problems. Physics like, for example, the travelling of surface gravity waves. The solution of the two dimensional Mild Slope Equation (MSE) is a well accepted approach to this problem: c ∇(ccg ∇φ(x, y)) + ω 2 φ(x, y) = 0 (1) cg In the above equation φ(x, y) is the velocity potential at the water surface, ω is the wave frequency, c is the phase velocity (c = ω/k) and cg is the group velocity (cg = ∂ω/∂k). The wave number k and the wave frequency are related by the well-know dispersion equation ω 2 = gk tanh(kd)

(2)

where d(x, y) is the local water depth. Equation (1) can be solved using the Finite Element Method (FEM), however the open boundaries and partial reflecting boundaries are difficult to be modelled and the result of those classical approaches gives a high degree of contamination in harbour models [6]. In case of a model bounded by a constant bathymetry, it is possible to use the Boundary Element Method (BEM) with the fundamental solution for the well-know Helmholtz equation. However, in a more realistic model, the water depth decreases as we approach to the coast. A typical approach of the wave number function for this case is shown in Figure 1. The implementation of a fundamental solution for a monotonic bed profile in a BEM can improve the solution of this kind of problems [4]. This work presents a Green function and their derivatives for slopping bottoms. Furthemore, these solutions are included in a boundary element formulation, solving some benchmark problems in order to demonstrate the accuracy of the BEM solution. Fundamental solution for the MSE on varying water depth In this section the fundamental solution for a variable water depth profile in one direction is going to be described. The first step in order to obtain this solution is to transform the MSE in the Helmholtz equation. This can be achieved with the change of variable: 1 φ = √ φ ccg

(3)

222


k(x) k = k3

k = k1 k = k2 (x)

x x=a x=b Figure 1: Wavenumber variation in x direction where φ is the original velocity potential, so the MSE becomes: k 2 φ = 0 ∇2 φ + where k is the modified wave number:

k2 =

(4)

√ ∇2 ( ccg ) 2 + k √ ccg

(5)

The Green’s function of the Helmholtz equation (4) ψ = ψ(x, y) satisfies k(x)2 ψ = −δ(x, y) ∇2 ψ +

(6)

being x the source point and being y the observed point. Applying the Fourier transform of the Green function ψ it is possible to reduce the Helmholtz equation in another one dimensional equation that depends the Fourier parameter [1]. ∞ ψ(r, r0 ; k)e−iyξ dy (7) ϕ(x, x0 ; ξ) = −∞ ∞ 1 ψ(r, r0 ; k) = ϕ(x, x0 ; ξ)e+iyξ dξ (8) 2π −∞ Introducing (8) in (4) it is obtained: ∂2ϕ + ( k(x)2 − ξ 2 )ϕ = −δ(x − x0 ) ∂x2

(9)

This equation can be solved numerically in the path a ≤ x ≤ b, where the boundary conditions in the limits correspond to a perfect absorption of the outgoing waves: ∂ϕ + i k12 − ξ 2 = 0, ∂x ∂ϕ − i k32 − ξ 2 = 0, ∂x

x = a;

(10)

x = b;

(11)

In the two remaining semi-infinite intervals (−∞, a] and [b, ∞] the solution of equation (9) is a plane wave (see [1]). The components of the gradient of ψ are: ∞ ∂ ∂ 1 ψ(r, r0 ; (ϕ(x, x0 ; ξ)) e+iyξ dξ k) = (12) ∂x 2π −∞ ∂x ∞ ∂ 1 ψ(r, r0 ; k) = iξϕ(x, x0 ; ξ)e+iyξ dξ (13) ∂y 2π −∞ Notice that the behaviour of (9) is asymptotic when ξ goes to infinity. ϕ(x, x0 ; ξ)

e−ξ|x−x0 | , 2ξ

ξ −→ ∞

(14)


223

To calculate numerically the Inverse Fourier Transform of ϕ(x, x0 ; ξ) in equation (8) avoiding the aliasing phenomena, it is necessary to integrate in the complex plane. The domain of integration is divided into three parts: the first path is in the ξ2 direction (complex part of ξ = ξ1 + iξ2 ), the second and third path is in the real and positive part of ξ (see [1]). Then, a well approach of (8) is 1 τy Ξ 1 e cosh(τ y)Re {E1 ((| x − x0 | +iy)Ξ)} (15) k) ϕ(x, x0 ; | ξ1 | −iτ )eiyξ1 dξ1 + ψ(r, r0 ; 2π 2π −Ξ The x derivative of the above expression can be obtained almost directly: ∂ 1 τy Ξ ∂ ψ(r, r0 ; e (ϕ(x, x0 ; | ξ1 | −iτ )) eiyξ1 dξ1 + k) ∂x 2π −Ξ ∂x +

(16)

e|x−x0 |Ξ 1 [− | x − x0 | cos((Ξ − iτ )y) + ysin((Ξ − iτ )y)] 2π (x − x0 )2 + y 2

The y derivative is more complicated because the terms that were multiplied by the hyperbolic sine function are now multiplied by the hyperbolic cosine and cannot be neglected. A proper approach is: ∂ 1 τy Ξ ψ(r, r0 ; e k) i(| ξ1 | −iτ )ϕ(x, x0 ; | ξ1 | −iτ )e(iyξ1 ) dξ1 − (17) ∂y 2π −Ξ Ξ 1 i(| ξ1 | −iτ )ϕ(x, x0 ; | ξ1 | −iτ )e−(iyξ1 ) dξ1 − − cosh(τ y) 2π −Ξ Ξ 1 i(ξ1 − iτ )ϕ(x, x0 ; | ξ1 | −iτ )e−(iyξ1 ) dξ1 + − cosh(τ y) 2π −Ξ +

e|x−x0 |Ξ 1 [− | x − x0 | sin((Ξ − iτ )y) − ycos((Ξ − iτ )y)] 2π (x − x0 )2 + y 2

In order to validate the presented fundamental solution, its behaviour for a constant depth is compared with the analytic fundamental solution of the Helmholtz problem. Then, a test for a variable depth profile is done. The Green function of the Helmholtz problem and its derivative are: 1 ∂ψH = ik H12 (k(|x − y|)) ∂r 4

1 ψH = −i H02 (k(|x − y|)), 4

(18)

In Figure 2 it is compared the solution of the proposed fundamental solution (ψ) and the analytic solution for a constant wave number (ψH ). The wave number is k = 11.21m−1 , and the wave period is T = 0.6s. Both solutions show perfect agreement. Figure 3 shows the fundamental solution for a variable wave number in x direction (k = k(x)). The depth function for this case is: d(x) = 0.00464x3 − 0.0348x2 + 0.3, x ∈ [0, 5]

(19)

where it can be observed how the depth, and consequently the wave length, is reduced in positive x-axis direction. Boundary integral equation Equation (4) can be rewritten in a weak form, multiplying by the fundamental solution (ψ(x, y)) and integrating in the domain: dΩ = 0 (∇2 φ + k 2 φ)ψ (20) Ω

Applying the divergence properties and the Gauss Theorem it leads to the dual reciprocity theorem as least: dΩ + ∇ψ · n dΓ (∇2 ψ + k 2 ψ)φ dΩ + ψ∇φ · n dΓ = (∇2 φ + k 2 φ)ψ (21) Ω

Γ

Ω

Γ

224


Figure 2: Values of the fundamental solution ψ and their derivatives for a domain with a constant wave number (k = 11.21m−1 ). The wave period is T = 0.6s. Left: Solution in x direction. Right: Solution in y direction.


225

Real(ψH)

0.25

Imag(ψ ) H

0.2

Real(ψ) Imag(ψ)

0.15 0.1

2

[m /s]

0.05 0 −0.05 −0.1 −0.15 −0.2 −0.25 0

0.5

1

1.5

2

2.5 x (m)

3

3.5

4

4.5

5

Figure 3: Values of the fundamental solution ψ for a domain with a variable wave number in x direction. In absence of internal loads the above expression can be write again only by boundary integrals: · n dΓ − (∇ψ(x, y) · n)φ(y) dΓ = 0 (22) C(x)φ(x) + ψ(x, y)∇φ(y) Γ

Γ

where C(x) is the free term, which values are: 1 for x in the interior of the domain Ω, 1/2 for a smooth boundary (x ∈ Γ) or, in general case C(x) = θ/(2π), where θ is the angle between boundary normals. Boundary Element Equation The Boundary Integral Equation (BIE) (22) can be solved numerically by discretizing the boundary into linear elements. Both variables (φ and q) are approximated by using linear shape functions. Note The fields of potential and flux become functions which depend of nodal values: that q = ∇φn. φ =

m

Ni φi ,

q =

i=1

m

Ni qi

(23)

i=1

and q are the nodal values of potential and flux respectively. N is the shape function where φ approximation matrix. Applying (23), the BIE (22) can be written as: Ci δij +

ne e=1

e ∂ψ(x, y) Nj φj dΓe = ∂n

n

Γe

e=1

Γe

ψ(x, y)Nj qj dΓe

(24)

where δij is the Kronecker δ-function. The above equation can be written in matrix form as: = G φ q H

(25)

Being H and G: ij = Ci δij + H ij = G

Γe

ne e=1

∂ψ(x, y) Nj φj dΓe ∂n

(26)

ψ(x, y)Nj qj dΓe

(27)

Γe

226


y φ(x = 0)

φ(x = 5)

k(x)

k(x = 5)

k(x) k(x = 0)

x

x=0 x=5 Figure 4: Numerical example description 1.5

Real(φin)

Imag(φin)

Real(φ)

Imag(φ)

1

φ

0.5 0 −0.5 −1 −1.5

0

0.5

1

1.5

2

2.5 x

3

3.5

4

4.5

5

Figure 5: Solution of channel problem along the x axis. Finally, the change of variable (3) should be undone, the above equation its now rewrite with the original potential φ. Hφ = Gq (28) where ij (√ccg )i δij Hij = H ij (√ccg )i δij Gij = G

(29) (30)

Numerical Example To validate model and test its performance, a numerical example with an analytic solution is solved. The validation test consist on a simulation of a plane wave in a channel with a variable depth profile along the x axis (Figure 4). The depth function of the channel is described in (19). The solution of the problem is the well-known incident wave [5]. Thus, the boundary condition is null flux on the walls of channel and potential is equal to incident potential in the start and in the end of the model channel. The incident potential (φin ) is: φin = A(x = 0)e−i

x 0

k(x)dx

(31)

where A is the wave amplitude. The wave period for the test is T = 0.6s so, the minimum and maximum values of wave number are k(x = 0) = 11.21m−1 and k(x = 5) = 34.07m−1 . The solution is shown in Figure 5 where a good accuracy of the methodology and the fundamental solution can be noticed.


227

Conclusions The accuracy of the Mild Slope fundamental solution for a variable depth profile in one direction has been demonstrated. This fundamental solution, implemented in a BEM formulation, allows to model more realistic boundary conditions for problems governed by the Mild Slope Equation (i.e. a typically coast bathymetry). Finally, the proposed BEM formulation can be combined with a FEM formulation in order to improve the solution of coastal problems with open boundaries. Acknowledgments This work was co-funded by the DGICYT of Ministerio de Ciencia y Tecnolog´ıa, Spain, research projects DPI2010-19331 and DPI2010-21590-C02-02, which were co-funded by the European Regional Development Fund (ERDF) (Fondo Europeo de Desarrollo Regional, FEDER). References [1] K.A. Belibassakis. The green’s function of the mild-slope equation: The case of a monotonic bed profile. Wave Motion, 2000. [2] R. P. Bonet. Refraction and diffraction of water waves using finite elements with a DNL boundary condition. Ocean Engineering, 2013. [3] Ken ichiro Hamanaka. Open, partial reflection and incident-absorbing boundary conditions in wave analysis with a boundary integral method. Coastal Engineering, 1997. [4] R. Naserizadeh, H. B. Bingham, and A. Noorzad. A coupled boundary element-finite difference solution of the elliptic modified mild slope equation. Engineering Analysis with Boundary Elements, 35(1):25 – 33, 2011. [5] A. C. Radder. On the parabolic equation method for water-wave propagation. Journal of Fluid Mechanics, 1979. [6] D. Steward and V. Panchang. Improved coastal boundary condition for surface water waves. Ocean Engineering, 28(1):139 – 157, 2001.

228


Analysis of wear on fiber-reinforced composites using boundary elements L. Rodr´ıguez-Tembleque1 and M.H. Aliabadi2 1

Escuela Técnica Superior de Ingenier´ıa, Universidad de Sevilla, Camino de los descubrimientos s/n, E41092 Sevilla, SPAIN. [email protected] 2 Department

of Aeronautics, Faculty of Engineering, Imperial College, University of London, South Kensington Campus, London SW7 2AZ, UK [email protected]

Keywords: Fiber Reinforced Composites, Anisotropic Wear, Anisotropic Friction, Contact Mechanics, Boundary Element Method.

Abstract. This work presents a numerical analysis of wear in fiber-reinforced plastics (FRP) that are subjected to different frictional contact conditions. New anisotropic wear and friction constitutive laws are also proposed to take into account the influence of the fiber’s orientation, on the tribological behavior of FRP. The formulation uses the Boundary Element Method (BEM) for computing the elastic influence coefficients, and contact operators over the augmented Lagrangian to enforce contact constraints. The new anisotropic wear and friction constitutive laws, and the boundary element formulation are applied to study how the fiber orientation, or sliding orientation affect the normal and tangential contact compliance, as well as wear evolution in a carbon FRP. Introduction Fiber-reinforced composite materials are being used increasingly for numerous applications in many different structural and mechanical components (i.e. in biomedical purposes like modern orthopaedic medicine and prosthetic devices [1]). Although fiber-reinforced plastics (FRP) are widely applied, there are not many numerical formulations that allow to analyze these polymer composites under different contact and wear conditions, especially due to the fact that particular contact and wear constitutive laws are required. Some experimental works have studied the significant influence of fiber orientation on the wear and frictional behavior of FRP composites. It has to be mentioned the works [2, 3, 4, 5, 6, 7], and more recently, [8]. Those experimental works showed that the coefficient of friction depends on several factors including the combination of materials, the surface roughness or the fiber orientation (i.e. the largest coefficient of friction was obtained when the sliding was normal to the fiber orientation, while the lowest one was obtained when the fiber orientation was transverse) (see Fig. 1). Even considering a sliding direction on a plane parallel to the direction of fibers, [2] observed that the coefficient of friction sliding in parallel direction was smaller than in the transverse direction. In summary, there is experimental evidence that it is not only important to consider anisotropy of the bulk material properties but also the anisotropy of the tribological properties, using proper contact and wear constitutive laws, and efficient numerical formulations. Contact problem formulation The contact problem between two linear anisotropic elastic bodies Ω α , α = 1, 2 with boundary ∂Ωα defined in the Cartesian coordinate system {xi } in R3 is considered. In order to know the relative position between both bodies at all times (τ ), a gap variable is defined for the pair I ≡ {P 1 , P 2 } of points (P α ∈ ∂Ωα , α = 1, 2), as g = BT (x2 − x1 ), where xα is the position of P α at every instant (xα = Xα + uαo + uα ), and matrix B = [e1 |e2 |n] is a base change matrix defined in [9, 10, 11, 12, 13], which expresses the pair I gap in relation to the local orthonormal base (see Fig. 2(a)).


(a)

(b)

229

(c)

Figure 1: Schematic diagram of a unidirectional FRP indicating the sliding directions: (a) Longitudinal, (b) transverse, and (c) Normal (μL ≤ μT ≤ μN ).

(a)

(b)

(c) Figure 2: (a) Contact pair I of points P α ∈ Ωα (α = 1, 2). (b) Elliptic friction law. (c) Orthotropic surface with parallel fibers.

230


The expression for the gap can be written as: g = ggo + BT (u2 − u1 ) BT (X2

(1)

− X1 )

being the geometric gap between two solids in the reference where ggo = gg + go , gg = configuration, and go = BT (u2o − u1o ) the gap originated due to the rigid body movements. In this work, the reference configuration for each solid (Xα ) that will be considered is the initial configuration (before applying load). Consequently, gg may also be termed initial geometric gap. In the expression (1) two components can be identified: the normal gap, gn = ggo,n + u2n − u1n , and the tangential gap or slip, gt = ggo,t + u2t − u1t , being uαn and uαt = [uαt1 , uαt2 ] the normal and tangential components of the displacements. Anisotropic contact law The unilateral contact law involves two conditions in the Contact Zone (Γ c ): impenetrability and no cohesion. Therefore for each pair I ≡ {P 1 , P 2 } ∈ Γc : gn ≥ 0 and tn ≤ 0. The variable tn is the normal contact traction defined as: tn = BTn t1 = −BTn t2 , where tα is the traction of point P α ∈ Γαc expressed in the global system of reference, and Bn = [n] is the third column of matrix B = [Bt |Bn ]. Tangential traction is defined as: tt = BTt t1 = −BTt t2 . Finally, the variables gn and tn are complementary: gn tn = 0, so this set of relations may be summarized on Γc by the so-called Signorini conditions: gn ≥ 0, tn ≤ 0, gn tn = 0. Friction constitutive laws for FRP can be accurately approximated by a convex elliptical friction cone, according to experimental works. The principal axes of the ellipse coincide with the orthotropic axes (Fig. 2(a)). The generic form of such anisotropic limit friction is given by (2) f (tt , tn ) = ||tt ||μ − |tn | = 0 where || • ||μ denotes the elliptic norm ||tt ||μ = (te1 /μ1 )2 + (te2 /μ2 )2 , and the coefficients μ1 and μ2 are the principal friction coefficients in the directions {e1 , e2 }. Eq. 2 constitutes an ellipse whose principal axes are: μ1 |tn | and μ2 |tn | (see Fig. 2(b)). The classical isotropic Coulomb’s friction criterion is recovered on Eq. 2 considering μ1 = μ2 = μ. The allowable contact tractions t must satisfy: f (tt , tn ) ≤ 0, defining an admissible convex region for t: the Friction Cone (Cf ). An associated sliding rule is considered, so the sliding direction is given by the gradient to the friction cone and its magnitude by the factor λ: g˙ e1 = −λ∂f /∂te1 and g˙ e2 = −λ∂f /∂te2 . To satisfy the complementarity relations: f (tt , tn ) ≤ 0, λ ≥ 0, λf (tt , tn ) = 0, the expression for λ factor is: λ = ||g˙ t ||∗μ , where the norm || • ||∗μ is dual of || • ||μ , so: ||g˙ t ||∗μ = (μ1 g˙ e1 )2 + (μ2 g˙ e2 )2 . Thus: te1 = −||tt ||μ μ21 g˙ e1 /||g˙ t ||∗μ and te2 = −||tt ||μ μ22 g˙ e2 /||g˙ t ||∗μ . To sum up, the unilateral contact condition and the elliptic friction law defined for any pair I ≡ {P 1 , P 2 } ∈ Γc of points in contact can be compiled as follows, according to their contact status: no contact (tn = 0, gn ≥ 0 and tt = 0), contact-adhesion (tn ≤ 0, gn = 0 and g˙ t = 0) and contact-slip (tn ≤ 0, gn = 0 and tt = −|tn |M2 g˙ t /||g˙ t ||∗μ ). The tangential slip velocity (g˙ t ) is expressed at time τk as: g˙ t Δgt /Δτ , where Δgt = gt (τk ) − gt (τk−1 ) and Δτ = τk − τk−1 , according to a standard backward Euler scheme. M is a diagonal matrix: μ1 0 (3) M= 0 μ2 whose coefficients are μ1 = μL + (μN − μL ) ϕˆ

μ2 = μT + (μN − μT ) ϕˆ

(4)

The expressions above establish a new constitutive friction law which can be applied to model friction in FRP. Parameter (0 ≤ ϕˆ ≤ 1) is the nondimensional fiber orientation constant (ϕˆ = 2ϕ/π ), and (0 ≤ ϕ ≤ π/2) is the fiber orientation relative to direction e1 (see Fig. 2(c)). The combined normal-tangential contact problem constraints can be formulated as [9, 10, 11]: t − PCf (t∗ ) = 0

(5)


231

Figure 3: Friction surface (f (tt , tn )/|tn |) as a function of the fiber orientation ϕ. ˆ

where the contact operator PCf was defined as PCf (t∗ ) = { PEρ (t∗t ) PR− (t∗n ) }T . The normal projection function, PR− (·), and the tangential projection function, PEρ were also defined in [9, 10, 11], as well as the augmented traction components (t∗ )T = [(t∗t )T t∗n ]: t∗t = tt − rt M2 gt and t∗n = tn + rn gn . rn and rt are the normal and tangential dimensional penalization parameters (r n ∈ R+ , rt ∈ R+ ), respectively. Anisotropic wear law The wear constitutive law is based on [9, 10]. In these works, wear evolution can be expressed in the following wear rate form: g˙ w = iw |tn |D˙ s , gw being the wear depth, D˙ s the tangential slip velocity module (D˙ s = ||g˙ t ||), and iw the dimensional wear coefficient or the specific wear rate. Assuming that the wear intensity iw is a function of the sliding direction parameter αv (iw = iw (αv )), wear velocity (g˙ w ) depends on the sliding direction. αv is the measure of the oriented angle between the given direction (e1 ) and the sliding velocity direction. Let us consider an orthotropic wear law, iw (αv ) = (i1 cos αv )2 + (i2 sin αv )2 , where: cos αv = g˙ e1 /||g˙ t ||, sin αv = g˙ e2 /||g˙ t ||, and i1 and i2 are the principal intensity coefficients: i1 = iL + (iN − iL ) ϕˆ

i2 = iT + (iN − iT ) ϕˆ

(6)

whose expressions (6) establish a new constitutive wear law which can be applied to model friction in FRP. Finally, postulating the wear rate to be proportional to the friction dissipation energy makes i L = kμL |tn |, iT = kμT |tn | and iN = kμN |tn |, so they are related to friction coefficients through the wear factor k. So the wear intensity can be written as iw = ||g˙ t ||i /||g˙ t ||, being ||g˙ t ||i = (i1 g˙ e1 )2 + (i2 g˙ e2 )2 . Finally, the anisotropic wear law can be defined by g˙ w = |tn | ||g˙ t ||i

(7)

For quasi-static contact problems, wear depth defined on instant τk , is computed as gw = gw (τk−1 ) + |tn | ||Δgt ||i

(8)

gw (τk−1 ) being the wear depth value on instant τk−1 . Due to the fact that the depth of removed material is computed for an instant τk , the normal contact gap (gn ) at the same time must be rewritten: gn = ggo,n + (u2n − u1n ) + gw .

232


Contact discrete variables and restrictions The contact tractions (tc ), the gap (g), and the displacements (uα , α = 1, 2), are discretized over the contact interface (Γc ). To that end, Γc is divided into N f elemental surfaces (Γec ). These elements (Γec ) constitute a contact frame. The contact tractions are discretized over the contact frame as: f tc ˆtc = N i = 1 δPi λi , where δPi is the Dirac delta on each contact frame node i, and λ i is the Lagrange f ˆ= N multiplier on the node (i = 1...N f ). In the same way, the gap is approximated as g g i = 1 δ Pi k i . In the expression above, ki is the nodal value. Therefore, taking into account the gap approximation, the discrete expression of Eq. 1 can be written as: (k)I = (kgo )I + (d2 )I − (d1 )I ,

(9)

for every contact pair I. In the expression above, k is the contact pairs gap vector and k go the initial geometrical gap and translation vector. Finally, the contact restrictions (Eq. 5) for every contact pair I can be expressed as: (Λt )I − PEρ ( (Λ∗t )I ) = 0

(Λn )I − PR− ( (Λ∗n )I ) = 0,

where augmented contact variables are defined as: (Λ∗t )I = (Λt )I rn (kn )I , and the value of ρ for the I pair: ρ = |PR− ( (Λ∗n)I )|.

− rt

M2 (k

(10) t )I

and

(Λ∗n )I

= (Λn )I +

Discrete boundary element coupling equations for solids ˜ − Gp ˜ = The boundary integral equations for a body Ω, can be written in a matrix form as: Hd F, where the vector d represents the nodal displacements, and F contains the applied boundary conditions. These equations can be written for contact problems as: A x x + Ap pc = F, being (x)T = [(xq )T (dd )T ] the nodal unknowns vector that collects the external unknowns (xq ), and the contact ˜ nodal displacements (dc ). pc is the nodal contact tractions. Ap is constructed with the columns of G ˜ and G, ˜ belonging to the contact nodal unknowns, and Ax = [Ax Ad ] with the columns matrices H corresponding to the exterior unknowns (Ax ), and the contact nodal displacements (Ad ). Considering a boundary element discretization for every solid Ωα (α = 1, 2), the resulting BEMBEM non-linear coupling equations set can be expressed according with [9], as ⎧ ⎫ ⎫ ⎡ ⎤ ⎪ x1 ⎪ ⎧ ˜1 ⎪ A1x 0 A1p C 0 ⎨ 2 ⎪ ⎬ ⎨ F1 ⎬ x 2 ⎣ 0 ˜2 0 ⎦ F (11) = −A2p C A2x ⎭ ⎪ Λ ⎪ ⎩ ⎪ Cg kgo ⎩ ⎭ 0 Cg ⎪ (C1 )T −(C2 )T k The first two rows in the expression above represent the equilibrium of each solid Ωα (α = 1, 2). The third row is the contact kinematics equations and the last row express the nodal contact restrictions. ˜ 1 Λ and p2c = −C ˜ 2 Λ. Expression 11 Vector Λ represents the nodal contact tractions, so that: p1c = C can be expressed, according to [9], as: ⎧ 1 ⎫ x ⎪ ⎪ ⎪ ⎬ ⎨ 2 ⎪ 1 x 2 ¯ R R Rλ Rg =F (12) Λ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ k xα being the solid Ωα (α = 1, 2) unknowns, vector Λ represents the nodal contact tractions, and ¯ the corresponding block matrices of these coupling the matrices R1 , R2 , Rλ and Rg , and vector F, systems.


(a)

233

(b)

(c)

Figure 4: (a) Sphere indentation over a FRP halfspace. (b) Boundary elements mesh details. (c) Cyclic tangential load.

Wear equations for contact problems The wear depth for every instant can be discretized over the contact frame, as a function of the nodal (k) (k) ˜ e , being N ˜ the shape functions matrix defined for the frame element Γe , values as gw gˆw = Nw c and we the nodal wear depth vector of element Γec . Therefore, the discrete form of kinematic equation for I pair, at instant k, is 2 (k(k) )I = (k(k) go )I + (d

(k)

)I − (d1

(k)

)I + (Cg n w(k) )I

(13)

where w(k) is a vector which contains the contact pairs wear depth, and matrix Cg n is constituted using the Cg columns which affect the normal gap of contact pairs [9, 10]. The discrete expression of Eq. 8 can be written for I pair as (k−1)

k (w(k) )I = (w(k−1) )I + |(Λ(k) n )I | ||(kt )I − (kt

where

(k) Λn

)I ||i

(14)

is a vector which contains the normal traction components of contact pairs at instant k.

Solution Scheme The quasi-static wear contact problem equations set: Eq. 10 to Eq. 14, allow to compute the variables on instant or load step (k), z(k) = [(x1 )T (x2 )T ΛT kT wT ]T , when the variables on previous instant are known. In this work z(k) is computed using the iterative Uzawa predictor-corrector scheme proposed in [9, 10, 11]. Numerical studies This example presents a steel sphere of radius R = 50 mm indented on a carbon FRP half-space (see Fig. 4(a)). The sphere is subjected to a normal displacement go,x3 = −0.02 mm and a tangential translational displacement of module: go,t = 0.008 mm, which forms an angle θ with axis x1 . The carbon FRP considered is IM7 Carbon/ 8551 − 7 with a volume fraction of 60 %, whose mechanical properties are: longitudinal Young modulus, E1 = 167.23 GP a, transverse Young modulus, E2 = E3 = 9.544 GP a, in-plane shear modulus, G12 = 5.292 GP a, transverse shear modulus ,G23 = 3.483 GP a, Poisson ratio ν12 = 0.272, and ν13 = 0.369. An anisotropic friction law is considered, being the friction coefficients: μL = 0.4, μT = 0.5 and μN = 0.55. For simplicity, due to the contact half-width (a) will be much less than the radius (R), the solids are approximated by elastic half-spaces, each one discretized using linear quadrilateral boundary elements. Fig. 4(b) shows the meshes details, where the half-space characteristic dimension is L = 1.2 mm. In this indentation problem, the influence of fiber orientation in the contact variables is considered. Figures 5(a) and (b), show the normal and tangential contact compliance variation with the fiber

234


orientation, relative to the load for the fiber alignment ϕ = 0o . For the normal load (Fig. 5(a)), the largest loads occur in the normal fiber orientation (ϕ = 90o ), and high differences can be observed for ϕ greater than 45o . For the tangential contact compliance (5(b)), with θ = 0o , the variation relative to the load Q(ϕ = 0) presents a different behavior. The largest discrepancies occur for a fiber orientation in the interval [0o , 45o ]. For ϕ = 90o , the tangential compliance is not affected by θ, because we recover the isotropic frictional behavior. Examining the Fig. 5(c), it is found that the variation of the orientation of the fibers has and important effect on the magnitude of the normal and tangential contact tractions.

(a)

(b)

(c)

Figure 5: Normal (a) and tangential (b) contact compliance variation with the fiber orientation. (c) Influence of fiber orientation on the contact tractions distribution. Wear coefficients: iL = 5 × 10−10 M P a−1 , iT = 6.25 × 10−10 M P a−1 and iN = 6.875 × 10−10 M P a−1 , are considered to study a fretting wear problem under gross slip conditions (see Fig. 4(c)). The wear volume evolutions after 100.000 cycles are presented in Fig.6(a), g o,t = 0.08 mm being the applied tangential load amplitude. Fig.6(b) shows the influence of ϕ and θ on the resulting wear volume. Examining the Fig. 6(b), it is found that the variation of the orientation of the fibers has and important effect on the magnitude of wear, as well as the sliding direction for a fiber orientation in the interval [0o , 45o ].

(a)

(b)

Figure 6: (a) Wear volume evolution considering different fiber orientations. (b) Influence of the fiber orientation and the sliding direction in the resulting wear volume.

Summary and conclusions This work presents new anisotropic wear and friction constitutive laws and its numerical implementation, to take into account the influence of the fiber orientation and the sliding direction, on the


235

tribological behavior of FRP. Some examples are presented to show the importance of considering this new constitutive tribological properties. In other case, we could over- or underestimate wear and contact magnitudes. Finally, the BEM reveals to be a very suitable numerical method for this kind of tribological problems, obtaining a good approximation on contact and wear variables with a low number of elements. Acknowledgment. The present research was supported by the was co-funded by the DGICYT of Ministerio de Ciencia y Tecnolog´ıa, Spain, by project DPI2010-19331 which were co-funded by the European Regional Development Fund (ERDF) (Fondo Europeo de Desarrollo Regional, FEDER). References [1] M.S. Scholz, J.P. Blanchfield, L.D. Bloom, B.H. Coburn, M. Elkington, J.D. Fuller, M.E. Gilbert, S.A. Muflahi, M.F. Pernice, S.I. Rae, J.A. Trevarthen, S.C. White, P.M. Weaver and I.P. Bond. The use of composite materials in modern orthopaedic medicine and prosthetic devices: A review. Composites Science and Technology 2011; 16: 1791–1803. [2] N. Ohmae, K. Kobayashi and T. Tsukizoe. Characteristics of fretting of carbon fibre reinforced plastics. Wear, volume(29): 345–353, 1974. [3] N.H. Sung and N.P. Suh. Effect of fiber orientation on friction and wear of fiber reinforced polymeric composites. Wear, volume(53): 129–141, 1979. [4] T. Tsukizoe and N. Ohmae. Friction and wear of advanced composite materials. Fibre Science and Technology, volume(18): 265–286, 1983. [5] M. Cirino, K. Friedrich, and R.B. Pipes. The effect of fiber orientation on the abrasive wear behavior of polymer composite materials. Wear, volume(121): 127–141, 1988. [6] O. Jacobs, K. Friedrich, G. Marom, K. Schulte and H.D. Wagner. Fretting wear performance of glass-, carbon-, and aramid-fibre/ epoxy and peek composites. Wear, volume(135): 207–216, 1990. [7] B. Vishwanath, A.P. Verma and V.S.K. Rao. Effect of reinforcement on friction and wear of fabric reinforced polymer composites. Wear, volume(167): 93–99, 1993. [8] T. Larsen, T.L. Andersen, B. Thorning, A. Horsewell and M. Vigild. Fretting wear performance of glass-, carbon-, and aramid-fibre/ epoxy and peek composites. Wear, volume(262): 1013–1020, 2007. [9] L. Rodr´ıguez-Tembleque, R. Abascal and M. H. Aliabadi. Anisotropic wear framework for 3D contact and rolling problems. Comput. Meth. Appl. Mech. Eng., volume(241): 1–19, 2012. [10] L. Rodr´ıguez-Tembleque, R. Abascal and M.H. Aliabadi. Anisotropic fretting wear simulation using the boundary element method. CMES–Computer Modeling in Engineering and Sciences, volum(87): 127–155, 2012. [11] L. Rodr´ıguez-Tembleque and R. Abascal. Fast FE-BEM algorithms for orthotropic frictional contact. Int. J. Numer. Methods Eng., volume(94): 687–707, 2013. [12] L. Rodr´ıguez-Tembleque, F.C. Buroni, R. Abascal and A. S´ aez. Analysis of FRP composites under frictional contact conditions. Int. J. Solids Struct., volume(50): 3947–3959, 2013. [13] L. Rodr´ıguez-Tembleque, A. Sáez and F.C. Buroni. Numerical study of polymer composites in contact. CMES–Computer Modeling in Engineering and Sciences, volume(96): 131–158, 2013.

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Analysis of crack onset and propagation at elastic interfaces by using Finite Fracture Mechanics M. Muñoz-Rejaa , L. Távaraa , V. Mantiˇca , P. Cornettib (a) Escuela Técnica Superior de Ingenier´ıa, Universidad de Sevilla Camino de los Descubrimientos s/n, 41092 Sevilla, Spain (b) Dipartimento di Ingegneria Strutturale, Edile e Geotecnica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy [email protected], [email protected], [email protected], [email protected]

Keywords: weak interface, LEBIM, BEM, FFM, coupled stress and energy criterion, composites

Abstract. A procedure able to predict crack onset and growth in an adhesive joint between two solids is presented. The procedure is based on the Linear Elastic-(Perfectly) Brittle Interface Model (LEBIM) combined with a Finite Fracture Mechanics (FFM) approach. The applied load necessary to produce an interface debond is predicted through the two coupled criteria based on: (i) the energy released due to the debond onset in the adhesive layer and (ii) the maximum stress produced in the adhesive layer. Each criterion, by itself, represents a necessary but not a sufficient condition to produce the debond. This procedure was implemented in a 2D Boundary Element Method (BEM) code. The adhesive layer is modelled by a distribution of elastic springs with an adequate stiffness. Although the procedure implemented allows a debond growth in mixed mode of fracture, a problem with mode II predominance is considered in the present work. This problem choice is due to the possibility to check the results with an available analytical solution. The results obtained with the coupled stress and energy criterion of FFM applied to the LEBIM are similar to those obtained by a Cohesive Zone Model. The advantage in using the present model is evident since a softening zone analysis is not necessary. Introduction The integrity of composite structures is determined by, among other conditions, durability and strength of their adhesive joints and interfaces in general. The adhesion must be guaranteed at several levels of the structure, namely at fibre-matrix interfaces, interfaces between unidirectional plies in laminates, joints between laminates and pieces, etc. Thus, one of the main concerns to design this kind of structures is the adequate characterization of the interfaces between solids on micro-, meso- and macro-scale. In several works, the behaviour of these joints/interfaces is modelled as a distribution of linear-elastic springs which represents the adhesive (see Lenci [1] for a review). Volkersen [2] and Goland with Reissner [3] were the first who used this model; since then many related works have been published, nevertheless in some problems the predictions do not coincide with the experimental evidence, that is why the model is still studied in order to find a way to improve it. Recently, Mantiˇc and Garc´ıa [4,5] studied the problem of fibre-matrix interface debond applying a failure criterion which characterize the crack onset and growth along the interface under remote transverse loading. Their criterion is based on the Finite Fracture Mechanics (FFM) and couples failure energy and stress criteria considering a perfect fibre-matrix interface. The aim of the present work is to study numerically a new criterion based on the FFM applied to LEBIM, to characterize the delamination produced between a composite layer and a concrete block. It should be mentioned that this criterion has already been successfully applied in [6] to the microscale problem of the fibre-matrix interface debond. Finite Fracture Mechanics applied to Linear Elastic-Brittle Interface The LEBIM, originally proposed in [7, 8], can be used to characterize the fracture of a weak interface between two solids, providing an accurate model of the interface damage problem. Nevertheless, when the interface under study is quite stiff, LEBIM may lead to inaccurate predictions. Lenci [1] compared the weak interface model with a perfect (strong) interface model showing that the latter predicts higher failure loads than the former. Mantiˇc and Garc´ıa [4, 5] and Cornetti et al. [9], respectively, applied the coupled criterion of FFM to the perfect and linear elastic interfaces, considering mixed


237

mode fracture in the former and pure fracture modes in the latter. In order to improve the original proposal of the LEBIM, the coupled criterion of FFM is included in the LEBIM in the present work, following the aforementioned previous studies, and implemented in a BEM code. The present novel approach to solve the interface damage problem is based on the interface strength and fracture toughness criteria, each of them representing a necessary but not sufficient condition for interface crack initiation and propagation. In the present model the interface is characterized by a spring distribution whose normal and shear stiffnesses are defined as kn and kt , respectively. So the normal and shear stresses σ and τ , at a point x on an undamaged part of the interface are proportional to the relative normal and shear displacements (δn and δt ): σ = kn δn , and τ = kt δt . (1) Therefore, the energy stored in a spring (per unit area) and which can be released is given as G = GI + GII ,

where

GI =

σ2+ 2kn

and

GII =

τ2 . 2kt

(2)

The stress and energy based fracture-mode-mixity angles are defined, respectively, as tan ψσ =

τ , σ

and

tan2 ψG =

GII kn = tan2 ψσ GI kt

(for σ ≥ 0).

(3)

In order to produce a crack onset and propagation the following incremental energy criterion must be fulfilled: Δa Δa G(a) da ≥ Gc (ψ(a)) da, (4) 0

0

where G(a) is the Energy Release Rate (ERR) associated to the crack tip at the position x = a, essentially it equals the energy (per unit area) stored at the spring located at the crack tip, cf. [1, 10], and is defined by (2) using the tractions σ(a) and τ (a) at the crack tip. Gc (ψ(a)) gives the fracture toughness (fracture energy) associated to the crack tip at the position x = a. Function Gc (ψ) is usually defined by a phenomenological law, as, e.g., the following one due to [11]: Gc (ψG ) = GIc (1 + tan2 (1 − λ)ψG ),

(5)

with GIc denoting the fracture toughness in pure mode I, λ is the fracture mode sensitivity parameter (0.2 ≤ λ ≤ 0.3 is the typical range for interfaces with moderately strong fracture mode dependence). Besides, for crack onset and propagation a stress criterion must be fulfilled too, Δa 1 t(x) dx ≥ 1, (6) Δa 0 tc (ψ(x)) where the traction vector modulus at a point x and its critical value are, t(x) = σ(x)2 + τ (x)2 and tc (ψ(x)) = σc (ψ(x))2 + τc (ψ(x))2 .

(7)

In the present work, following [8, 12], the energy fracture-mode-mixity angle is used defining kt ¯c 1 + tan2 [(1 − λ)ψG ] cos ψG and τc (ψG ) = σ ¯c 1 + tan2 [(1 − λ)ψG ] sin ψG , σc (ψG ) = σ kn

(8) with σ¯c being the critical stress for pure mode I. The present problem is governed by the following dimensionless parameter defined in [9]: 2kt GIIc μ= . (9) τ¯c2

2 /¯ According to Fig. 1, μ = τmax τc2 , with τmax and τ¯c , being respectively, the maximum and critical stresses associated to the energy and stress criteria for pure mode II. When μ = 1, the solution of the present model should revert to the solution of the original LEBIM. For increasing μ, keeping strength and fracture toughness constant, the interface becomes stiffer, so μ → ∞ leads to a perfect (yet softening) interface. As can be noticed, fracture toughness, strength and stiffness of the interface are independent in the present FFM+LEBIM model, in opposite to the original LEBIM, where these variables are related by an equation.

238


t

Figure 1: FFM+LEBIM law for pure mode II. The pull-push shear test In order to check the adequacy of the criterion proposed above, the well-known double pull-push shear test is studied by using the implemented BEM code, and the results obtained are compared with the analytical results obtained by Cornetti et al. [9]. The problem analysed is represented in Fig. 2. It consists of a concrete block with two overlaps of composite reinforcement. The test aims to measure the strength of the overlap joints, applying the loads shown in the figure. In the joint, the adhesive layer is mainly under shear strains. Thus, the fracture mode II prevails, although a rigorous elastic analysis shows that the fracture mode I also exists in the problem (see Suo and Hutchinson [13]). Nevertheless in the present work, the boundary conditions of the problem are designed to obtain normal stresses close to zero in the adhesive layer, this is because Cornetti et al. [9] assumed such a condition in their analytical studies. Besides, some other assumptions considered in [9] are not reproduced here, as uniform normal stress distribution in the solids and the absence of shear stresses. These assumptions on the problem solution, although close to reality, are a simplification of what really happens [14], thus when such an analytic solution is compared with numerical (FEM or BEM) or experimental results small differences may appear. Symmetry of the problem as shown in Fig. 3 Reinforcement

Adhesive

(a)

2h b

s

2s

Concrete block

Reinforcement

(b)

hr

Reinforcement

t b= t r Concrete block

s

s

l

Figure 2: Double pull-push shear test: (a) side view; (b) top view. is used in the numerical model of the double pull-push shear test studied. It is assumed that the adhesive is only able to transmit shear stresses using a null normal stiffness, kn = 0. In order to avoid large displacements, the normal displacements at the external face of the reinforcement are restrained. The two solids are considered linear, elastic and isotropic, their properties are presented in Table 1. A plane strain state is assumed in the system. Normal (σa ) and shear (τa ) stress distributions along the joint obtained by the present model are depicted in Fig. 4. It can be checked that shear stresses are in good agreement with the analytical results obtained by Cornetti et al. [9]. Although the similitude in the shear stresses obtained by both methods are noticeable, there are some differences produced by the previously described simplifications made in [9].


239

Table 1: Mechanical and geometrical properties (kt for μ = 4) Concrete block Reinforcement Adhesive

l(m) 0.72 0.48 kn (MPa/μm) 0.00

h(m) 0.12 0.8e-03 kt (MPa/μm) 0.18

E(GPa) 30.0 160.0 τ¯c (MPa) 1.94

ν 0.20 0.30 GIIc (Jm−2 ) 41.8

lr

y l hr

Reinforcement

x hb

s

Concrete block

kn (adhesive) = 0 lb

Figure 3: Numerical model used for double pull-push shear test.

0.04 τa Cornetti et al. (2012). τa Present Work.

0.03

0.02

a

a

τ /σ, σ /σ

σa Present Work.

0.01

0

−0.01

0

0.04

0.08 x(m)

0.12

0.16

Figure 4: The stress field along the interface. Fig. 5 shows the applied load in the reinforcement necessary to crack onset (Fc ) versus the dimenEr h r E b h b , sionless overlap longitude, scaled with the characteristic longitude of the problem lch = kt (E r hr +Eb hb ) for the different failure criteria used in the present work. In Fig. 5(a) the critical force is scaled with the maximum critical force obtained by means of the LEBIM in either analytical or numerical models, which is independent of the overlap longitude. The absolute differences between the analytical solution in [9] and the numerical results obtained in the present work are due to the differences in the initial hypotheses of both models, and the effect of the concrete block corner when the crack tip moves which it is not taken into account in the analytical model in [9]. In fact, it can be seen, comparing the plots in Fig. 5(a) and (b), that

240


(a) 1.2

1

Fc / F ∞ c

0.8

0.6

0.4

LEBIM. Cornetti et al. (2012) LEBIM. Present work. FFM. Cornetti et al. (2012) FFM+LEBIM. Present work. Stress criterion. Cornetti et al. (2012) Stress criterion. Present work.

0.2

0 0

1

2

3 l/l ch

4

5

6

(b) 120

100

F c (KN)

80

60

40

LEBIM. Cornetti et al. (2012) LEBIM. Present work. FFM. Cornetti et al. (2012) FFM+LEBIM. Present work. Stress criterion. Cornetti et al. (2012) Stress criterion. Present work.

20

0 0

1

2

3

l/l

4

5

6

ch

Figure 5: Failure load vs. dimensionless bond length according to different fracture criteria. (a) Dimensionless load. (b) Actual load. the represented dimensionless numerical results in Fig. 5(a) are closer to the dimensionless analytical solution, because the differences of the load obtained with the model (Fc ) are cancelled out by using the maximum critical load (Fc∞ ) obtained using the corresponding, either analytical or numerical, LEBIM. Conclusions A new computational procedure combining the FFM and LEBIM (corresponding to μ > 1) has been used to study a fracture mode II problem. It opens new possibilities to study the onset and propagation of cracks along interfaces and adhesive layers using realistic values of strength, fracture toughness and in particular layer stiffnesses, which can be significantly higher than in the original LEBIM (corresponding to μ = 1). It is interesting to observe that for the present double pull-push


241

shear test the predictions of the crack onset and propagation obtained by FFM and LEBIM differ only slightly from those obtained previously by an analytical model. It should be mentioned that, with the present model, normal stresses at the interface can be obtained if boundary conditions allow it, so the problem can be studied in a more general way in the future. Acknowledgements The work was supported by the Junta de Andaluc´ıa and European Social Fund (Projects of Excellence TEP-1207, TEP-2045, TEP-4051, P12-TEP-1050), the Spanish Ministry of Education and Science (Projects TRA2006-08077 and MAT2009-14022) and Spanish Ministry of Economy and Competitiveness (Projects MAT2012-37387 and DPI2012-37187). References [1] S. Lenci. Analysis of a crack at a weak interface. International Journal of Fracture, 108:275–290, 2001. [2] O. Volkersen. Die Nietkraftverteilung in Zugbeanspruchten Nietverbindungen mit Konstanten Laschen-querschnitten. Luftfahrtforschung, 15:4–47, 1938. [3] M. Goland and E. Reissner. The stresses in cemented joints. Journal of Applied Mechanics, 11:A17–A27, 1944. [4] V. Mantiˇc. Interface crack onset at a circular cylindrical inclusion under a remote transverse tension. Application of a coupled stress and energy criterion. International Journal of Solids and Structures, 46:1287–1304, 2009. [5] V. Mantiˇc and I.G. Garc´ıa. Crack onset and growth at the fibre–matrix interface under a remote biaxial transverse load. Application of a coupled stress and energy criterion. International Journal of Solids and Structures, 49:2273–2290, 2012. [6] M. Mu˜ noz-Reja, L. T´ avara, V. Mantiˇc and P. Cornetti. Crack onset and propagation in composite materials using finite fracture mechanics on elastic interfaces. Procedia Materials Science (In print), 2014. [7] L. Távara, V. Mantiˇc, E. Graciani, J. Ca˜ nas and F. Par´ıs. Analysis of a crack in a thin adhesive layer between orthotropic materials. An application to composite interlaminar fracture toughness test. CMES-Computer Modeling in Engineering and Sciences, 58(3):247–270, 2010. [8] L. Távara, V. Mantiˇc, E. Graciani and F. Par´ıs. BEM analysis of crack onset and propagation along fiber-matrix interface under transverse tension using a linear elastic-brittle interface model. Engineering Analysis with Boundary Elements, 35:207–222, 2011. [9] P. Cornetti, V. Mantiˇc and A. Carpinteri. Finite fracture mechanics at elastic interfaces. International Journal of Solids and Structures, 49:1022–1032, 2012. [10] A. Carpinteri, P. Cornetti and N. Pugno. Edge debonding in FRP strengthened beams: Stress versus energy failure criteria. Engineering Structures, 31:2436–2447, 2009. [11] J.W. Hutchinson and Z. Suo. Mixed mode cracking in layered materials. Advances in Applied Mechanics. 29:63–191, 1992. [12] V. Mantiˇc, L. Távara, A. Bl´ azquez, E. Graciani and F. Par´ıs. Application of a linear elasticbrittle interface model to the crack initiation and propagation at fibre-matrix interface under biaxial transverse loads. ArXiv preprint. arXiv:1311.4596, 2013. [13] Z. Suo and J.W. Hutchinson. Interface crack between two elastic layers. International Journal of Fracture, 43:118, 1990. [14] L.J. Hart-Smith. Adhesive-bonded double-lap joints. Technical report, NASA Contract Report 112235, 1973.

242


Simplified Assessement and Evaluation Procedure of Finite-Part Hypersingular Integrals Ney Augusto Dumont Civil Engineering Department, PUC-Rio, Rio de Janeiro, Brazil; email: [email protected] Keywords: Finite-part integrals, hypersingular integrals, boundary integral equations. Abstract: The present contribution, which was motivated by applications to strain gradient elasticity, revisits the concept of a finite-part integral, as originally conceived by Hadamard, and shows that, contrarily to precepts in the technical literature (H. R. Kutt, 1975), nonlinear coordinate transformations of finite-part integrals can be generally and consistently carried out without interfering with the evaluated results. Some confusion or undue differentiation between Hadamard finite part and Cauchy Principal Value representations (Mukherjee, 2000) is also clarified in passing, as well as it is shown that consistent coordinate transformations and numerical evaluations can be carried out without resorting to regularization techniques, which are just a welcome mathematical resource, but are unnecessary to understand and numerically deal with finite-part integrals of any order of singularity. The key features of the proposed developments are the use of integration by parts and the disclosure of a finite term that comes out as the singularity point is approached. The proposed transformation applies to a generally curved boundary segment (which does not need to be closed or part of a closed boundary), as a mathematical operation that dispenses with any mechanical interpretation to be built up and validated (although an accomplished implementation must end up physically meaningful). The general transformation rules for a finite-part integral are demonstrated for 2D problems (as 3D applications are straightforward developments).

Introduction The treatment of strongly singular integrals, to which the concept of a Cauchy Principal Value can be applied and a discontinuous (free) term subsequently evaluated, can in principle be considered as mastered in the literature on boundary integral equations since many decades ago. However, the case of hypersingular integrals, as sometimes required in fracture mechanics analysis, for instance, or in the implementation of the hypersingular boundary integral equations (HBIE), for curved boundaries in general, does not seem to have been satisfactorily understood up to the present days in spite of some not so recent seminal and mostly correct publications (see, for instance, Kutt [1], Mantic and Paris [2], Guiggiani [3] and Mukherjee [4]). The developments by Kutt [1] seem to have been the first to draw attention to Hadamard's concept of finitepart integrals [5] when the integrand contains an algebraic singularity. No criticism can be directed to Hadamard's proposition, which consisted on a too broad and too early set of statements that have ever since deserved some interpretation or refinement in order to be applicable to present day's challenging, practical numerical problems. However, it is to criticize that some particular interpretations have been widespread, such as by Kutt [1] himself, who states on page 51 that "We thus have the important fact that our kind of f.p. integral is invariant with respect to the general linear transformation (2.2.4) only if the exponent is a noninteger". Kutt's "our" finite-part integral has ever since become the standard one, with the result that a variable transformation is only feasible when translation or mirroring occurs, as cited by many authors [6]. However, a very important feature of a finite-part integral, which had been already enounced by Hadamard and is assumed by Kutt, is the fact that integration by parts can be applied. This key feature will be explored presently. The concept of a Cauchy Principal Value is also per se old enough and has been along the last several decades – and sometimes depending on a researcher's dwelling – discovered, enunciated or denominated in different ways, as a simple literature search indicates. The basic mathematical principle of this concept is too obvious to be repeated here. On the other hand, its relation with a finite-part integral is not so obvious – and sometimes controversial. Readers are referred to Mukherjee [4], for instance, who introduces some concepts – in order to differentiate between Hadamard Finite Part and Cauchy Principal Value representations – that are questionable or at least not exactly necessary for the understanding of the present subject matter. Mukherjee and several other authors (as cited therein, for example) use concepts of "regularization" of an integral, "limit


243

to the boundary" and smoothness of a boundary in their developments. Such concepts are very important – decisive, in fact, – to assess the ultimate mechanical meaning of a general singular integral over a closed curved boundary and the feasibility of its evaluation in terms of a Cauchy Principal Value plus a discontinuous term (when the singularity is locally addressed and eventually solved). It is worthwhile mentioning that Guiggiani (as given in the review paper [3]) adopts a conceptually different strategy than Mukherjee's by subdividing the evaluation of a hypersingular integral (which is assumed to have a sound mechanical meaning) into parts that, although involving in principle non-addressable singularities, must add up to a definite (finite) integral. The main feature of this strategy is that there is no need to explicitly assess a finite-part integral. On the other hand, it must be remarked that the regularization procedure adopted by Guiggiani (and collaborators) is actually not necessary, as shown in this paper. The concept of a finite-part integral is a welcome feature when dealing with singular or hypersingular integrals. The present paper proposes that a finite-part integral (with or without the preceding qualification "Hadamard", although always paying due tribute to this magnificent mathematician) is a mathematical entity to be evaluated, in a practical application, as part of a Cauchy Principal Value representation of a general singular integral. In fact, a finite-part integral also exists when a Cauchy Principal Value cannot be represented or evaluated (is non-existent or infinite, which is basically the same). Moreover, no geometrical interpretation and no regularization concepts are needed in the following developments, where it is also shown that general nonlinear variable transformations may be carried out to the user's best convenience without change of the numerical outcome. The developments are carried out for linear integrals, as required in a 2D boundary element implementation, initially for a singular integral, with a r −1 kernel, and then for a hypersingular integral, with a r −2 kernel. The generalization for a r − m kernel, with m > 2 , is conceptually straightforward, although possibly of difficult practical implementation (as well as of mechanical interpretation of the whole context of such an integral) and is presented in an extended version of this paper. Generalization to double integrals, as applied to 3D boundary element problems, is conceptually straightforward and should not differ from developments that are already classical in the technical literature [2, 3]. Given a linear singular or hypersingular finite-part integral over a curved boundary segment, the ultimate goal is to represent this integral in terms of some normalized interval, so that classical quadrature procedures can be applied.

Normalization rule for the r −1 singularity in the interval [ξ0 ,1] (singularity on the left) Preliminary developments. The finite-part integral x (1)

fp 0

1 f dx x

(1)

will undergo the nonlinear transformation x = (ξ − ξ 0 ) x (ξ ) , for a subsequent application to a curved

boundary segment with the singularity given in terms of a source-to-field distance r = ( ξ − ξ 0 ) r (ξ ) . As shown in the above equation, the singularity occurs on the left of the segment, for either ξ 0 = −1 or ξ 0 = 0 , as transformation rules of eq (1) are to be derived for either integration interval [ −1,1] or [0,1] in order to fit a user's convenience. The developments for a singularity on the right are not shown, but the results or interest are given in the Appendix. Both functions x (ξ ) and r (ξ ) must be positive in the integration interval. The generic function f ( x) ≡ f ( x(ξ ) ) ≡ f and its derivatives are assumed to be continuous in the interval

ξ 0 ≤ ξ ≤ 1 (more particularly, for the sake of practical numerical applications, it is required that f ( x(ξ ) ) be representable by a polynomial of a not too high order: in some cases it may be advisable to split the interval into subintervals). Only real functions of real variables are involved in the present developments, although it should not be difficult to generalize the concepts for a complex boundary element implementation. Then, for the finite part of the simplest case of what is known as a strong singular integral, the transformation rule from the interval 0 ≤ x ≤ x(1) to the normalized interval ξ 0 ≤ ξ ≤ 1 is derived as follows:

244


x (1)

fp 0

x (1) x (1) 1 d df fdx = fp ( ln( x) f ) dx − ln( x) dx 0 dx 0 x dx

= ln( x) f

ξ =1

− fp lim ln( x) f − ln ( x ) f 'dξ

= ln( x) f

ξ =1

− fp

= ln( x) f

ξ =1

= ln( x ) f

ξ =ξ0

1

ξ0

x →0 1

ξ0

( ln( x) f ) 'dξ + fp

1

1 x ' f dξ ξ0 x

− ln( x) f

ξ =1

(2)

1 1 + fp lim ln ( (ξ − ξ 0 ) x ) f + fp x ' f dξ ξ →ξ 0 ξ0 x

1

1 + fp f x 'dξ ξ0 x

According to Hadamard, a finite part integral can undergo integration by parts, which justifies the developments in the first row above. Observe that, for mathematical consistency, all terms on the right of a signal " = " either involve a finite part application or are the result of such an application. In the second row, use has been made of the identity df dx dx = f 'dξ . The normalization carried out from the first to the second rows leading from 0 ≤ x ≤ x(1) to ξ 0 ≤ ξ ≤ 1 is valid since it involves an improper integral. Moreover, the x (1) d evaluation fp ( ln( x) f ) dx = ln( x) f ξ =1 is justified because both operations of integration and derivation 0 dx refer to the same variable x , which is the same as establishing that fp lim ln( x) = 0 . This feature is also

x →0

observed in the evaluation of fp

1

ξ0

( ln ( (ξ − ξ ) x ) f ) 'dξ 0

in eq (2), with fp lim ln(ξ − ξ 0 ) = 0 . Equation (2) ξ →ξ 0

gives the final and complete normalization rule. The correction term involving ln( x ) ξ =ξ is void whenever 0

x ξ =ξ = 1 , such as in the cases of translation or mirroring, but also, for instance, in the particular non-linear 0

transformation x = ξ [1 + (A − 1)ξ ] for 0 ≤ x ≤ A and with ξ 0 = 0 . The scheme to be used for the numerical evaluation of the resulting normalized finite-part integral on the right of eq (2) is rather a matter of choice, as Kutt’s algorithm would work well [1], only with the known lack of elegance by dealing with a negative pair of abscissa and weight. A simpler and better solution would be to add and subtract a term, so that one ends up with the Gauss-Legendre quadrature, as shown in the Appendix.

Application to a curved boundary. The results of the above Section are now applied to the finite-part integral of a 2D boundary element problem defined in terms of the parametric variable ξ 0 ≤ ξ ≤ 1 that describes a smooth boundary segment Γ seg

1 1 fp f dΓ ≡ fp f (ξ 0 , ξ )dΓ(ξ ) Γ seg r Γseg r (ξ 0 , ξ )

(3)

where r is the absolute distance from a source point ξ 0 located on the left extremity of Γ seg to a general field point ξ and f is the regular part of the integrand, with mathematical properties in terms of ξ exactly as introduced before, whose definition depends on the problem that is being dealt with. As proposed, 0 ≤ r (ξ 0 , ξ ) ≤ r (ξ 0 ,1) with a strong singularity at ξ = ξ 0 . Moreover, there is for the moment no need to estipulate how successive boundary segments relate to each other, that is, whether there are corner points and which are the smoothness (or lack of) between adjacent boundary segments, since only the finite-part integral of eq (3) is of concern and it is only assumed that Γ seg is smoothly defined in terms of the parametric variable ξ .


245

Basic transformations. The Cartesian projections ( x, y ) ≡ ( x(ξ0 , ξ ), y (ξ0 , ξ ) ) of the distance r ≡ r (ξ0 , ξ ) from a field point to the source point along Γ seg are x = (ξ − ξ 0 ) x and y = (ξ − ξ 0 ) y , such that, for the distance between source and field points, r = ( ξ − ξ 0 ) x 2 + y 2 ≡ (ξ − ξ 0 ) r and r ' = r + (ξ − ξ0 ) r '

(4)

The expression of the Jacobian J in the geometric representation dΓ = J dξ is J = x '2 + y '2 = r 2 + 2 (ξ − ξ 0 ) r r '+ (ξ − ξ 0 ) ( x '2 + y '2 ) 2

(5)

Since

J r'

r 2 + 2 ( ξ − ξ 0 ) r r '+ (ξ − ξ 0 ) ( x '2 + y '2 ) 2

=

2

(6)

=0

(7)

r 2 + 2 (ξ − ξ 0 ) r r '+ ( ξ − ξ 0 ) r '2

then,

J r'

(

However, ∂ m−1 ( J r ') ∂ξ m −1

)

ξ =ξ 0

ξ =ξ0

J ′ = 1 and r'

≠ 0 for

ξ =ξ 0

m > 2 , in general.

Normalization rule. Given the above introductory transformations, the normalization of the integral in eq (3) becomes after substituting in eq (2) for the general integration variable x with r (as x is actually a 1 f J , dummy variable in the definition of the definite integral) and replacing the regular function f with r' r (1) 1 11 fp f dΓ = f J dr = ln J f Γ seg r r = 0 r r '

1

ξ =ξ 0

1 1 + fp f J r ' dξ ξ0 r r '

1

= ln J f

ξ =ξ 0

(8)

1 f J dξ + fp ξ0 r

The above transformation is justified because, according to eq (7), the behavior of the regular part of the integrand is not distorted as the field point ξ approaches the source and singularity point ξ 0 .

Normalization rule for the r −2 singularity in the interval [ξ0 ,1] (singularity on the left) Preliminary developments. The developments start in a similar way as to the nonlinear transformation x = (ξ − ξ 0 ) x (ξ ) for the finite-part integral of eq (2), this time for a x −2 kernel: x (1)

fp 0 =−

1 f x

1 =− f x 1 = − f x

x (1)

1 fdx = − fp x2 0

x (1) d 1 1 df dx f dx + 0 x dx dx x 1

+ ln( x )

df 1 df dx dξ + fp dx ξ =ξ0 ξ0 x dx d ξ

+ ln( x )

df d 1 1 dx + fp dξ f dξ + fp 2 f dx ξ =ξ0 dξ ξ0 x ξ0 d ξ x

+ ln( x )

df 1 + f dx ξ =ξ0 x

ξ =1

1

ξ =1

ξ =1

ξ =1

1

1 1 f 1 dx − fp lim dξ + fp 2 f x → 0 (ξ − ξ ) x dξ ξ0 x 0

(9)

246


Integration by parts is performed in the first row above, with subsequent evaluation of the finite part indicated for the first term on the right-hand side, as given in the second row, with a value only at ξ = 1 (which is justified, since both integral and differential operators refer to the variable x . The two rightmost terms in the second row come from the application of the normalization eq (2) to the last term in the first row. Integration by parts is then applied to the normalized finite-part integral in the second row, which results in the two corresponding terms in the third row. Evaluation of the first finite-part integral in this row leads to the results of the last row, with two terms that cancel out and a finite-part that must be carefully evaluated, since lim(ξ − ξ 0 )−1 ≠ lim x −1 . In fact, it is obtained by series expansion of ξ − ξ 0 as a function of x that x →0

x →0

1

ξ − ξ0

=

x (ξ ( x) ) x

=

x (ξ 0 ) x ' + + O( x) x x ' ξ =ξ0

(10)

with the result of the following finite-part definition: fp lim x→0

1

ξ − ξ0

=

x' x' ≡ x ' ξ =ξ0 x ξ =ξ0

(11)

The above limit necessarily exists if a valid variable transformation is being carried out in the interval ξ0 ≤ ξ ≤ 1 . It is finally obtained for eq (9) x (1)

fp 0

1

ln( x ) df 1 x' 1 dx f dx = dξ − 2 f + fp 2 f ξ x2 x d x dξ ξ0 x ξ =ξ0

(12) r (ξ =1)

Application to a curved boundary. It comes from r ' ξ =ξ = J ' ξ =ξ 0

1 fp 2 fdΓ = fp Γel r r = 0

1 f J dr r2 r '

and

2 , as obtained from eqs (4) and (5), the final expression for a curved boundary with 0

singularity at ξ = ξ 0 : r (ξ =1)

1 fp fdΓ = fp Γ seg r 2 r = 0

1 ln J df

J' 1 f J 1 dr = f − + fp 2 f J dξ 2 2 J dξ 2 J ξ0 r r r' ξ =ξ0

(13)

Summary Equations (8) and (13) are the normalization rules derived for the finite parts of singular and hypersingular integrals on a curved boundary segment that is smooth, but not necessarily smoothly connected to the adjacent boundary segments. They have been obtained using the basic properties of a finite-part integral, as laid out by Hadamard and independently from concepts such as regularization and limit to the boundary of a singular integral, also dispensing with any geometrical interpretation. Although the demonstrated nonlinear variable transformations are completely general, the intent was to arrive at an interval normalization, so that standard numerical quadrature schemes can be applied. The numerical evaluation of the normalized finite-part integrals can be in principle carried out in terms of Kutt's quadrature rules, although more convenient expressions for a Gauss-Legendre quadrature are given in the Appendix (and then coincide with the results obtained by Guiggiani [3], for instance). The complete developments for singularities of higher order as well as for some numerical illustrations are given in a companion full manuscript.

Acknowledgments The author acknowledges the financial support given by the Brazilian agencies CNPq and FAPERJ.


247

References [1] H.R. Kutt On the Numerical Evaluation of Finite Part Integrals Involving an Algebraic Singularity, Report WISK 179. The national Research Institute for mathematical Sciences, Pretoria, 1975. [2] V. Mantic and F. Paris Engineering Analysis with Boundary Elements, 16, 253-260 (1995). [3] M. Guiggiani Formulation and Numerical Treatment of Boundary Integral Equations with Hypersingular Kernels, in V. Sladek and J. Sladek (Eds), Singular Integrals in Boundary Element Methods, Computational Mechanics Publications, Southampton, 1998. [4] S. Mukherjee International Journal of Solids and Structures, 37, 6623-6634 (2000). [5] J. Hadamard Lectures on Cauchy's Problem in Linear Partial Differential Equations, Yale University Press (1923). [6] C.A. Brebbia, J.C.F. Telles and L.C. Wrobel Boundary Element Techniques – Theory and Applications in Engineering, Springer Verlag, Berlin (1984).

Appendix: Regularization and Gauss-Legendre quadrature The scheme to be used for the numerical evaluation of the resulting normalized finite-part integral on the right of eqs (2) or (13) is rather a matter of choice, as Kutt’s algorithm would work well, only with the known lack of elegance of dealing with either a negative or complex pair of abscissa and weight. A simpler and better solution seems to be the regularization of these integrals by adding and subtracting terms of a Taylor series expansion of f ( x(ξ )) , so as to arrive at a Gauss-Legendre quadrature. The developments are not shown, for space restrictions, but the results are given in the following for the integration intervals [ −1,1] and [0,1] . For a normalized integration interval [a,1] , where either a = 0 or a = −1 , the integration rules of the singular eq (8) for singularities ξ 0 = a and ξ 0 = 1 are, respectively, 1 n hi 1 1 fp fdΓ = GL f J dξ + f ln (1 − ξ 0 ) J − Γel r ξ0 r i =1 ξ i − ξ 0 ξ =ξ0

(14)

1 n h 1 1 fp fdΓ = GL f J dξ − f ln (1 − a ) J + i Γel r a r i =1 ξ i − 1 ξ =1

(15)

On the other hand, the integration rules for the hypersingular eq (13) for ξ 0 = a and ξ 0 = 1 are 1 n f 1 J' hi 1 1 fp + + 2 fdΓ = GL 2 f J dξ − 2 Γel r ξ0 r 2J J 1 − ξ 0 i =1 (ξi − ξ 0 )

ξ =ξ0

(16)

ξ =1

(17)

n 1 df hi + ln (1 − ξ 0 ) J − i =1 ξ i − ξ 0 J d ξ ξ =ξ0 1 n f 1 J' hi 1 1 fp + − 2 fdΓ = GL 2 f J dξ − 2 Γel r a r 2J J ξ 0 − a i =1 (ξ i − ξ 0 ) n 1 df hi − ln (ξ 0 − a ) J + i =1 ξ i − ξ 0 ξ =1 J d ξ

248


Fracture Analysis of Viscoelastic Nonhomogeneous Media Using Boundary Element Method Hugo Luiz Oliveira1 and Edson Denner Leonel2 University of São Paulo, School of Engineering of São Carlos, Department of Structural Engineering. Av. Trabalhador São Carlense, 400. São Carlos-SP, Brazil. 13.566-590. 1 2 [email protected] [email protected] Keywords: Viscoelasticity, Boundary Element Method, Fracture Mechanics.

Abstract. This paper addresses the mechanical analysis of two dimensional linear viscoelastic nonhomogeneous structures using the Boundary Element Method (BEM). This type of analysis has major importance in engineering field, as materials widely used in this domain exhibit mechanical response depending on time. The mechanical behaviour of viscoelastic bodies subjected to complex geometries and boundary conditions is accurately modelled by BEM. The mesh dimensionality reduction provided by BEM coupled to its intrinsic capability to represent high gradient fields make this method robust to deal fracture mechanics problems. Traditional BEM formulations deal viscoelastic problems using a convolutional relation between stress and strain tensors, generally leading to a formal space transformation. However, this problem may be also solved using alternative techniques to derive the time-marching process. Viscoelastic approaches are based on differential constitutive equations resulting from different associations among springs and dashpots, which origin models such as Maxwell, Kelvin-Voigt and Boltzmann. In this paper, these models are applied to analysis of fracture mechanics problems. The numerical viscoelastic approach is based on an explicit time procedure, in which one step Euler method approximation for time derivatives is applied. It leads to a linear system of equations that has to be solved at each time step. This approach allows imposing time-dependent boundary conditions, which represent structural loading phases. The Dual BEM formulation, in which crack faces are discretised using displacement and traction integral representations, is implemented in order to represent the mechanical structural behaviour along time of fractured solids. To validate the presented formulation one application is presented. The results obtained by the presented formulation show good agreement to reported data as well as numerical stability. Introduction The mechanical analysis of structures composed by materials that exhibit viscoelastic mechanical behaviour has major importance in engineering structures field. Many materials used in this field such as polymers, composites and concrete, for instance, exhibit noticeable time-effects. Due to these time-effects, such materials have, simultaneously, viscous and elastic properties, which make structures suffer from creep, relaxation and hysteresis problems [1]. In spite of many theoretical and experimental studies on viscoelastic fracture to be available in literature, numerical applications in this area are relatively limited [2]. [3] proposed a numerical solution, considering Laplace transform approach, in order to predict the crack opening displacement of a penny shaped crack in a viscoelastic material represented by Kelvin model. Stress and displacement fields were analysed in the neighbourhood of a crack filled with failed by [4]. The mechanical analysis of viscoelastic anisotropic bodies containing holes, inclusions and initial defects was performed by [5]. In this work the correspondence principle was used considering the sub-region BEM approach. The displacement discontinuity method was used by [6] in order to obtain a time integral formulation applied to analysis of quasi-static mechanical behaviour of asphalt. The strain energy release rate was studied by [7], which derived an expression from the functional related to the potential energy applying a direct time domain BEM. As presented above, few papers in literature reported the analysis of linear viscoelastic fracture using BEM. In this regard, the present study aims to contribute with this engineering structures field, by coupling Maxwell, Kelvin-Voigt and Boltzmann models to algebraic BEM equations in order to analyse linear viscoelastic fracture mechanics problems. The numerical viscoelastic approach is based on the explicit time procedure presented in [8], in which one step Euler method approximation for time derivatives is applied. To represent the mechanical structural behaviour along time of fractured solids, the dual version of BEM is adopted [9]. The nonhomogeneous structures are modelled, in this research, using the sub-region technique. This BEM technique enforces compatibility of displacements and equilibrium of forces along the interfaces of all multiple bodies that compose


249

the nonhomogeneous structure. One application is presented in order to validate the implemented formulation. The results obtained by the presented formulation show good agreement to reported data as well as numerical stability.

Viscoelastic Theory. Rheological models The solid mechanics theory uses simplified models to represent the real physical behaviour of bodies subjected to external loads and prescribed displacements. Regarding rheological approaches in the context of solid mechanics, the application of arrangements proposed by electrical circuit’s theory is extremely convenient to model this important mechanical phenomenon [10]. By using these arrangements, the physical phenomena involved are modelled by graphical elements (springs and dashpots) that can be connected together in order to represent the mechanical structural behaviour. The one-dimensional spring element, Fig. (1), is used to represent the mechanical behaviour of linear elastic materials, i.e., materials governed by Hooke's law. This element characterizes the material capability to recover its initial strain condition after the removal of external load. This element assumes that strain change occurs instantaneously. ߟ

‫ܧ‬

ߟ

E

ߪ

ߪ

ߪ

ߪ ߝ

ߪ

ߪ ߝ

Figure 1. Hooke’s model.

ߝ௘

ߝ௩ ߝ

Figure 2. Newton’s model.

Figure 3. Maxwell’s model.

The one-dimensional dashpot element, Fig. (2), is used to represent materials that have pure viscous mechanical behaviour. Materials governed by this mechanical behaviour present a dependence between the stress state into the body and the strain rate along time, not the strain state itself. The model composed only by a dashpot to represent the mechanical behaviour of a given material is known as Newton’s model. In this model, permanent strains are assumed to appear along time. The modelling of viscoelastic mechanical behaviour is performed by associating the elements presented in figures 1 and 2 in series and/or parallel sequences. This procedure leads to a convenient representation of both elastic and viscous mechanical behaviour of materials. One possible association of these elements concerns the direct association, in series sequence, of one spring and one dashpot, as presented in Fig. (3). This association represents the model know as viscoelastic Maxwell’s model. The following equations are used to describe mathematically the Maxwell’s model: Equilibrium Equation V t V e t V v t

Compatibility Equation H t H e t H v t

(1) .

Constitutive Equation V e t EH e t ; V v t K H v t in which V indicates normal stress, H normal strain, E represents Young’s modulus and K viscous coefficient. Superscripts e and v indicate elastic and viscous parts, respectively. The dot over each variable indicates variation along time. The differential equation that governs Maxwell’s model is obtained by differentiating the compatibility equation, Eq.(1), with respect to the time. Afterward, equilibrium and constitutive equations, Eq.(1), are applied in order to obtain the following representation: .

.

H t

V t V t E K

Equation 2 can be integrated along time leading to: V t 1 t H t ³ V W dW E K W0

(2)

(3)

The integration by parts of the kernel presented in Eq.(3) allows the determination of the following equation:

H t

t

ª1

³« W ¬E 0

t W º . V W dW K »¼

(4)

250


The integral kernel presented in Eq.(4) is known as creep function for Maxwell’s model, i.e.: 1 t W (5) D t W E K This integral kernel has to be evaluated using the linearity properties of hereditary functions, the invariance to temporal translation and the causality principle. As the effects are always posterior to the causes, Eq.(5) can be translated into temporal axis without interferences on previous actions. It is worth to stress that Eq.(5) is a linear equation with respect to the time. It means that if an external load be applied indefinitely, producing a constant stress level into the body, the material will flow indefinitely. This is a typical behaviour of fluids, which make this model, sometimes, be known as fluid Maxwell’s model. Similarly, the relaxation function for this model is written as follows:

G t W

Ee

E

K

t W

(6)

Another possible arrangement of spring and dashpot is illustrated in Fig. (4). This model is based on a parallel sequence of one spring and one dashpot, being known in literature as Kelvin-Voigt’s model. ‫ܧ‬

‫ܧ‬ଶ ‫ܧ‬ଵ

ߪ

ߪ

ߟ

ߪ

ߟ

ߝ௘

ߝ

Figure 4. Kelvin-Voigt’s model.

ߪ

ߝ ௩௘

Figure 5. Boltzmann’s model.

The governing equations for Kelvin-Voigt’s model are the following: Equilibrium Equation V t V e t V v t

Compatibility Equation H t H e t H v t

(7) .

Constitutive Equation V e t EH e t ; V v t K H v t The differential equation that governs this viscoelastic model is determined by introducing constitutive and compatibility equations into equilibrium equation. This procedure leads to: .

V t EH t K H t

(8)

The solution of Eq.(8), which is a first order linear differential equation, for a given history of applied stresses is obtained as follows:

H t

1

t

³ V t e

K f

E

K

t W

dW

(9)

The integration by parts of Eq.(9) leads to the determination of the creep function for Kelvin-Voigt’s model, which is equal to:

D t W

t W · 1§ ¨¨1 e K ¸¸ E© ¹ E

(10)

According to the kernels presented in Eq.(10), it is observed that when the time variable tends to infinity the solution obtained tends to an asymptotic elastic solution, H f V 0 E , where all stress is supported by the spring element. For a given history of structural strain it is possible to evaluate the structural stress state directly from Eq.(8). An experimental test of relaxation is physically impossible by Kelvin-Voigt’s model. It happens .

because H t H 0G t , i.e., an initial singular applied stress is required into experimental test. Consequently, there is not a relaxation function for this viscoelastic model. Finally, the arrangement involving a series and parallel sequence of springs and dashpot, as presented in Fig. (5), leads to the well-known viscoelastic Boltzmann’s model. This model is also known in literature as standard solid model. This viscoelastic model is governed by the following equations:


251

V t V e t V ve t

Equilibrium Equation

Compatibility Equation H t H e t H ve t

(11) .

Constitutive Equation V e t E1H e t ; V ve t E2H ve t K H ve t in which the superscript ve refers to the values calculated on the parallel sequence of one spring and one dashpot. The differential equation that governs the viscoelastic Boltzmann’s model is achieved by differentiating the compatibility equation with respect to the time. Afterwards, equilibrium and constitutive equations are applied, which allows the determination of the following linear differential representation: .

.

K E1 H t E1E2H t K V t > E1 E2 @V t

(12)

For a given history of applied stress, the strain history is determined from Eq.(12) as follows:

H t

E ª1 1 1 K2 t W º . e « » V W dW ³ E1 E2 E2 »¼ W0 « ¬ t

(13)

On the other hand, if a history of strain be applied into the structure, the history of stress is calculated as follows:

V t

E E ª 1 2 t W · º . E12 § K E e 1 « ¨ ¸ » H W dW 1 ³ ¸» E1 E2 ¨© W0 « ¹¼ ¬ t

(14)

The kernels presented into brackets on Eq.(14) represent the relaxation function for Boltzmann’s model. The relaxation function shows that, during the structural loading process, both elastic and viscous strains are developed.

BEM Integral Equations

Considering a two-dimensional homogeneous elastic domain, : , with a boundary, * , the equilibrium equation can be expressed in terms of displacements as follows:

ui , jj

b 1 u j , ji i 1 2X P

0

(15)

where P is the shear modulus, υ is the Poisson’s ratio, ui are components of the displacement field and

bi represents the body forces. Using Betti’s theorem, a singular integral for displacements can be obtained (without body forces), as follows: clk uk ³ Plk*uk d * *

³P u k

* lk

d*

(16)

*

in which Pk and uk are tractions and displacements on the boundary, respectively, the free term clk is equal to G lk for smooth contours and Plk* and ulk* are the fundamental solutions for tractions and displacements, [9]. It is worth to mention that for solids containing cracks, the use of only Eq. (16) for assembling the system of algebraic equations results in a singular matrix, because, both crack surfaces are located along the same geometrical path. In this regard, the Dual BEM, which is one of the most popular BEM formulations used to analyse random crack growth, is adopted in this work. In this formulation, the singular integral representation, Eq. (16), is used to determine the algebraic representation related to collocation points defined along the crack surface, whereas the hyper-singular integral representation is used to obtain the algebraic representation related to the collocation points located along the opposite crack surface. For external boundaries, the singular representation is sufficient to obtain the required algebraic relations. The hyper-singular integral (or traction integral) representation at the boundary or crack surface collocation points can be obtained from Eq. (16). First, this equation, which is written for an internal collocation, is differentiated in order to obtain the integral representation in terms of the strains. Then, using Hooke’s law, the stress integral representation is achieved. Finally, the integral representation of stresses for a boundary collocation is determined by carrying out the relevant limits. The Cauchy formula is applied to obtain the traction representation as follows:

252


1 Pj ] k 2

³

* ] k ³ Dkij Pk d *

* Skij uk (c ) d *

*

(17)

*

* * and Dkij contain the new kernels that were calculated from Plk* and ulk* , [9] and ] k indicates where the terms S kij

the cosines of the normal direction to the body’s boundary.

Algebraic BEM equations for Multi-Domain analysis To simulate the mechanical behaviour of solids composed by multi-domains, the sub-region BEM technique has to be applied. In the sub-region BEM approach, the body on analysis is divided into a finite amount of homogeneous sub-regions interconnected by interfaces. As previously presented, BEM analyses involving singular and hyper-singular integral representations are performed using Eq.(16) and Eq.(17). When multi-domains are considered, these equations have to be applied at each sub-domain individually. Then, the classical BEM system of algebraic equations is obtained for each subregion i of the entire solid as follows: (18) > Hi @Ûi ` >Gi @^Pi ` * in which matrix H contains the integration kernels Plk* and S kij whereas matrix G contains the integration

* kernels ulk* and Dkij . Vectors U and P contain the displacement and traction values on the body boundary,

respectively. Once the kernels

> Hi @

>Gi @

and

evaluated, different strategies can be performed in solving the

nonhomogeneous boundary value problem. The first step concerns the assembling of matrixes of each sub-region i into a global system of equations, as presented in Eq. (19).

ª H1 «0 « « « ¬0

º U1 ½ » °U ° » ®° 2 ¾° »° ° » H n ¼ ¯°U n ¿°

0 H2

0 0

0

ªG1 0 «0 G 2 « « « ¬0 0

º P1 ½ » °P ° » ®° 2 ¾° »° ° » Gn ¼ ¯° Pn ¿° 0 0

(19)

where n represents the number of sub-regions used into the discretisation of the entire solid. The global system of algebraic equations presented in Eq.(19) cannot be solved directly just by imposing the boundary conditions of the problem because along the interfaces neither tractions nor displacements values are known. Therefore it is necessary to enforce the compatibility of displacements and equilibrium of forces along all interfaces. These conditions can be written as follows: U side1 U side 2 Pside1 Pside 2 0 (20) The compatibilities conditions, Eq.(20), coupled to the boundary conditions have to be imposed on the global system of equations. By performing a convenient change on the columns of matrices H and G , all known variables are placed at the right hand side of this algebraic system whereas unknown variables, x , are placed at its left hand side. This system can be presented as follows:

> A@^x` > B @

^`

Once f

^`

f

(21)

is the vector of know boundary values, the system is solved and the unknowns variables

determined.

Viscoelastic BEM Formulations BEM formulations for analysis and modelling of viscoelastic problems are developed using the differential equations described in section 2. In the present section, these differential equations are used to determine the algebraic BEM equations for each viscoelastic model already presented. The BEM formulation for Maxwell’s model is presented in details in this section whereas for Kelvin-Voigt and Boltzmann models the algebraic equations are presented using inductive procedure in order to avoid repetitive expressions. Maxwell’s model. The formulation for Maxwell’s model is obtained by writing Eq.(2) in terms of stress:


253

.

.

V t K H t K

V t

(22)

E

Assuming that viscous and elastic material properties are proportional by a scalar J , K by [8], the above equation can be rewritten in the following form:

J E , as proposed

V t J ª« E H t V t º» ¬ ¼ .

.

(23)

It is worth to mention that Eq. (23) is also applicable for tri-dimensional problems. In this case, the above equation is rewritten as follows:

V ij t J ª«Cijkl H kl t V kl t º» ¬ ¼ .

.

(24)

in which Cijkl represents the forth order tensor of elastic material properties. The BEM integral representation for viscoelastic Maxwell’s model is obtained by using Eq.(24) to represent the stress state of the problem. Therefore, Eq.(24) is used in the equilibrium equation, V ij , j bi 0 , which is weighted by the fundamental solution of displacements, u * , leading to:

³u *

. . ª º P d * ³ uki* , j J «Cijkl H kl V kl » d : ³ uki* bi d : ¬ ¼ : :

* ki i

0

(25)

To facilitate the mathematical development, an auxiliary variable I has to be defined: . . ª º ³ uki* , j J «Cijkl H kl V kl » d : ¬ ¼ : * , the equation above can rewritten as follows: Bearing in mind that uki* , j Cijmn V kmn

I

.

.

.

(26)

.

* ³ J uki* , j Cijkl H kl d : ³ J uki* , j V kl d : ³ JV kmn H mn d : ³ J V kl uki* , j d :

I

:

:

:

(27)

:

The integration by parts of the right hand side of Eq.(27) leads to the following expression: .

.

.

.

.

* * * * ³ JV kij u i ] j d * ³ JV kij , j u i d : ³ J V ij uki ] j d * ³ J V ij , j uki d :

I

*

:

*

(28)

:

* Traditional BEM formulations considerer that V kmn ,n

.

' s, f G km and V ij , j

.

bi , which are the fundamental Kelvin equilibrium and the mechanical equilibrium conditions, respectively. In these conditions, ' and G indicates Dirac and Kroenecker operators, respectively, and s and f represents source and field points, respectively. Therefore, Eq.(28) is rewritten as follows: . . . ª . º ³ J Pki* u i d * ³ J ª¬' s, f G ki º¼ u i d : ³ J uki* Pi d * ³ J « bi » uki* d : ¼ * : * : ¬

I

(29)

The second integral term on the right hand side of Eq.(29) is simplified as follows:

³ J ª¬' s, f G

:

.

ki

.

º¼ u i d : J u k s

(30)

Therefore, Eq.(29) assumes the following form: .

.

.

.

J ³ Pki* u i d * J u k J ³ uki* Pi d * J ³ uki* bi d :

I

*

*

(31)

:

Equation 31 can used to rewrite Eq.(25). Then: .

.

.

.

* * J u k J ³ Pki* u i d * J ³ uki* Pi d * ³ uki* Pd i * J ³ uki b i d : ³ uki bi d : *

*

*

:

:

(32)

254


.

Assuming that body’s forces are null, as well as its variation along time, i.e., bi can be rewritten as follows: .

.

bi

0 , the equation above

.

J u k J ³ Pki* u i d * J ³ uki* Pi d * ³ uki* Pd i * *

*

(33)

*

It is worth to mention that Eq.(33) is written for points that belong to the body’s domain. To evaluate this equation for points positioned only at the body’s boundary, convenient limits have to be performed (as in classical BEM formulations). These limits lead to the following final integral equation: .

.

.

J cki u k J ³ Pki* u i d *

J ³ uki* Pi d * ³ uki* Pd i *

*

*

(34)

*

The values of all variables on the body’s boundary are approximated by shape functions. High order boundary elements are available in the computational code developed. The numerical evaluation of the kernels presented in Eq.(34) allows the determination of the algebraic BEM equations, which are equal to: .

.

JHU

J G P GP

(35) in which H and G result from the integration process, over the body’s boundary, of the fundamental kernels P* and u * , respectively. It is worth to mention that shape functions used on Eq.(35) are purely spatial, it does not account the variable variation along time. However, Eq. (35) represents a first order differential equation written into time domain. To solve this differential equation, a linear approximation into time domain is applied, using forward finite differences technique, aiming to determine the first derivative of this function. This procedure is performed by dividing the time of analysis into finite time steps, 't , in which the actual time is s, leading to the following:

U s 1 U s 't

.

U

.

P

P s 1 P s 't

(36)

Equation 34 can be rewritten using the results presented in Eq.(36) as follows:

§ U s 1 U s · § P s 1 P s · s 1 ¸ JG¨ ¸ GP 't 't © ¹ © ¹

JH¨

(37)

After some simple algebraic manipulations, the equation above assumes the following form:

§ 't · s 1 s s ¨1 ¸ GP HU GP J ¹ ©

HU s 1

(38)

Equation 38 represents the system of algebraic equations that must be solved, for finite time increments, in order to determine the unknown values on the body’s boundary using the viscoelastic Maxwell’s model. The stress state for all internal points is determined by deriving Eq.(33). This equation has to be derived with respect to the coordinates of source points, leading to: .

.

.

* * J u k ,l J ³ Pkil* u i d * J ³ ukil Pi d * ³ ukil Pd i * *

*

(39)

*

.

Bearing in mind that viscoelastic Maxwell’s model assures the following condition, V ij

. e

V ij

.

Cijkl u k ,l ,

then, Eq.(39) can be rewritten as follows:

1

.

V kl

J

_*

_*

_*

.

.

³ V kil Pdi * ³ V kil Pi d * ³ Pkil u i d * *

*

(40)

*

_*

in which V kil indicates the fundamental solution for stress [9]. The integral equation above is solved by using shape functions to approximate displacements and tractions on the body’s boundary. When this approximation is considered, Eq.(40) is algebraically represented as follows: .

V

1

J

.

.

G' P G' P H ' U

(41)


255

The equation above, as the differential equation for boundary values, is solved using a linear approximation into time domain. This procedure leads to the following algebraic equation:

V s 1

. s 1 · . s 1 §1 'tG ' ¨ P s 1 P ¸ 'tH ' U V s ©J ¹

(42)

As in this model elastic and viscous stresses are equal, as presented in Eq.(1), Eq.(42) allows determining the stress on the body for viscoelastic Maxwell’s model. Kelvin-Voigt’s model. The equation that allows the determination of unknown values of variables on the body’s boundary is obtained using similar procedures performed for Maxwell’s model. All mathematical manipulations are omitted in this sub-section for simplicity. For Kelvin-Voigt’s model this algebraic equation is written as follows: J · J § s 1 (43) GP s 1 HU s ¨1 ¸ HU 't © 't ¹ The total stresses on internal points are determined using the following algebraic equation: . s 1

V s 1 G ' P s 1 HU s 1 J H ' U

(44) The stresses determined on Eq. (44) are divided into elastic and viscous components using the next two algebraic equations. These portions are given by: J J · § · § (45) V es 1 ¨ V s 1 V es ¸ ¨1 ¸ V vs 1 V s 1 V es 1 't ¹ © 't ¹ © Boltzmann’s model. The algebraic equations that govern the Boltzmann’s model are obtained using similar procedure already used in Maxwell’s model. All mathematical manipulations are omitted in this sub-section just for simplicity. Based on this consideration, the unknown values for all variables on the body’s boundary are determined using the following algebraic equation:

J · § s 1 ¨1 ¸ HU t¹ ' ©

§ J E1 E2 · J HU s GP s ¨ ¸G t E t ' ' 2 © ¹

(46)

The total stresses are calculated based on the following algebraic equation:

§

V s 1 ¨ G ' P s 1 ©

. s 1 . s 1 · § E2 J E2 J E2 E2 E2 · J J H 'U s 1 H'U G' P V s ¸ ¨1 ¸ (47) E1 E2 E1 E2 E1 E2 't E1 E2 ¹ © 't E1 E2 ¹

The total stresses have to be divided into its elastic and viscous portions. These portions are given as follows: J J · § · § (48) V es 1 ¨ V s 1 V es ¸ ¨1 ¸ V vs 1 V s 1 V es 1 't ¹ © 't ¹ ©

Application This application refers to the structural analysis of the nonhomogeneous panel presented in Fig. (6). This sandwich panel is composed by three different materials, which have different mechanical behaviour. The structure is clamped at its left boundary and at its right end a parabolic distributed load is applied. Three cracks are presented in the structure, which are positioned at the middle span of each material, as presented by F1, F2 and F3 in Fig. (6). The mechanical properties for each material (domain) that compose the structure are presented in Table 1. Domain D1 D2 D3

Model Boltzmann Hooke Kelvin

E1(kN/cm²) 10000 22000 30000

E2(kN/cm²) 5000 -

X

0.22 0.20 0.15

J

(days) 30 45

Table 1. Material mechanical properties. To illustrate the mechanical time-effects, the crack mouth opening displacement (CMOD) was monitored along time for each crack into the structure. The total time considered in this analysis is 300 days, which was simulated with time intervals of 1 day. The variation of CMOD along time is illustrated in Fig. (7). It is observed that the CMOD grows along time for all cracks studied. This mechanical behaviour is observed independently of the model adopted to represent the mechanical structural behaviour along time. Figure 8 presents

256


the variation of the shear stress along time for a given point belonging to domain 3 (D3). The behaviour for total shear stress as well as for its elastic and viscous components is illustrated. According to this figure, it is observed that total stress does not depend on the viscoelastic material behaviour. As the viscous stress decreases the elastic stress grows in order to keep valuable the equilibrium requirements. Geometry Geometria ܽ ൌ ͳͲܿ݉

Load Carregame ܲ ൌ ͷ݇ܰ

Figure 6. Nonhomogeneous structure.

Figure 7. Variation of CMOD along time.

Figure 8. Variation of shear stress along time.

Conclusion This work presented a viscoelastic BEM formulation which was applied to analysis of fracture mechanics problems in nonhomogeneous bodies. The rheological models of Maxwell, Kelvin-Voigt and Boltzmann were presented as well as the algebraic BEM equations obtained from the differential equations the govern each of these viscoelastic models. The implemented BEM formulation was used to analysis of a nonhomogeneous domain, in which the obtained results were consistent with those observed in literature. It confirms the accuracy and robustness of the presented viscoelastic BEM formulation. The analysis of viscoelastic problems considering crack propagation is due in course and it is the next advance scheduled by the authors.

Acknowledgements Sponsorship of this research project by the São Paulo State Foundation for Research (FAPESP), project number 2012/24944-5 is greatly appreciated. This research is a part of the activities scheduled by the research project USP/COFECUB 2012.1.672.1.0.

References [1] X.Y. Zhu, W.Q. Chen, Z.Y. Huang, Y.J Liu, Eng. Analysis with Boundary Elements, 35, 170-178 (2011). [2] S. Syngellakis, Eng. Analysis with Boundary Elements, 27, 125-135 (2002). [3] J. Sladek, J. Sumec, V. Sladek, Ingenieur Archiv, 54, 275-282 (1984). [4] B.N. Sun, C.C. Hsiao, Computers & Structures, 30, 963–966 (1988). [5] Y.C. Chen, C. Hwu, Eng. Analysis with Boundary Elements, 35, 1010–1018 (2011). [6] J. Wang, B. Birgisson, Eng. Analysis with Boundary Elements, 31, 226-240 (2007). [7] S.S. Lee, Y.J. Kim, Eng. Fracture Mechanics, 51, 585-590 (1995). [8] Mesquita, A.D.; Coda, H.B. Eng. Analysis with Boundary Elements, 27, 885-895 (2003). [9] A. Portela, M.H. Aliabadi, D.P. Rooke, Int. J. of Numerical Methods in Engineering, 33, 1269-1287 (1992). [10] N.W. Tschoegl, The phenomenological theory of linear viscoelastic behavior – an introduction, Berlin: Springer-Verlag (1989).


257

Solution of time-domain problems using Convolution Quadrature methods and BEM++ T. Betcke [email protected]

N. Salles [email protected]

W. Smigaj [email protected]

Department of Mathematics, University College London 25 Gordon Street, London, WC1H 0AY, United-Kingdom

Keywords: boundary element method, time-domain problems, convolution quadrature. Abstract Convolution Quadrature methods are efficient techniques for the solution of time-domain wave problems in unbounded domains via Boundary Element Methods. In this proceeding we use the Convolution Quadrature approach to decouple the time-domain problems into a series of independent frequency-domain problems that can be solved efficiently in parallel. In contrast to previous approaches we solve many more frequency-domain problems than there are time steps. We demonstrate numerically that this approach approximates the underlying time-stepping scheme with exponential accuracy as the number of frequency problems is increased. The implementation of the method is done using BEM++, a modern C++ based boundary element library with an easy to use Python interface.

1

Introduction

Convolution Quadrature (CQ) is an efficient technique for the solution of time-domain wave problems via Boundary Element Methods. A well known interesting point is that CQ methods can be formulated in such a way as to obtain a number of independent frequency-domain problems, which can be easily solved in parallel. Typically, the number of frequency problems is chosen to be the same as the number of time steps. However, in this proceeding we demonstrate that by increasing the number of time steps one can achieve exponential convergence to the underlying time-stepping scheme that forms the basis of the Convolution Quadrature formulation. For simplicity, in this proceeding we focus on multistep schemes. But similar approaches are also possible for Runge-Kutta based Convolution Quadrature formulations. The proceeding is organised as follows. We first introduce parallel CQ schemes based on multistep methods and propose an approach in which the number of frequency domain solves is decoupled from the number of time steps. We then discuss the numerical implementation of the frequency problems with BEM++, an open source C++ based boundary element library. It offers the Galerkin discretisation of Laplace, Helmholtz and Maxwell kernels, and can be accessed via an easy to use Python interface. We conclude the proceeding with a numerical example that shows the exponential convergence of the proposed method. Let Ω ⊂ R3 be an obstacle and Γ = ∂Ω its boundary. We define the exterior domain where we solve the wave ¯ (see Figure 1). In this proceeding we consider the following acoustic problem: equation by Ωe = R3 \Ω ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

∂2u (x, t) − c2 Δx u(x, t) = 0, x ∈ Ωe , ∂t2 ∂u (x, 0) = 0, u(x, 0) = ∂t u(x, t) = g(x, t) for x ∈ Γ.

(1)

258


Ωe Ω

Γ

Figure 1: The exterior domain of interest Ωe and the boundary of the scatterer.

2

Convolution Quadrature Methods

In this section we present a simple approach for the solution of (1) based on Convolution Quadrature. The idea is based on a Z-Transform of the time steps of an underlying time-stepping scheme. This leads to a range of modified Helmholtz problems in the frequency domain, which can be solved in parallel. The time domain solution is then synthesised by an inverse Z-transform. The proposed approach is similar to previous CQ methods (see e.g. [1, 2, 8]). However, in contrast to previous presentations we decouple the number of frequency solves from the number of time steps.

2.1

The Convolution Quadrature method

First, we transform the wave equation(1) intoa first order system. To do this transformation, we introduce T 1 ∂u I 0 0 I , and B(x, t) = g(x, t), 0 . We obtain ,T = (x, t)]T , M = v(x, t) = [u(x, t), 0 0 Δx 0 c ∂t ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

1 ∂v(x, t) = M v(x, t), x ∈ Ωe , c ∂t v(x, 0) = 0, ∀x ∈ Ωe , T v(x, t) = B(x, t), x ∈ Γ.

(2)

Secondly, we apply a general multistep scheme of the form 1

γn−j vd (x, tj ) = M vd (x, tn ), cΔt n

(3)

j=0

to obtain the discrete time step values vd (x, tj ) that approximate the exact values v(x, t) at time steps tj , j = 0, . . .. We then apply the Z-transform: ⎡ ⎤ ∞ n ∞

⎣ 1 γn−j vd (x, tj )⎦ z n = M vd (x, tn )z n . (4) cΔt n=0

j=0

n=0

The convolution becomes a product after the Z-Transform. We hence obtain 1 γ(z)Vd (x, z) = M Vd (x, z), cΔt

(5)


259

∞ n n where γ(z) = ∞ n=0 γn z and Vd (x, z) = n=0 vd (x, tn )z . For Backward Euler it holds that γ(z) = 1 − z. We now rewrite (5) as a modified Helmholtz equation in the frequency domain: ⎧ ⎪ γ(z) 2 ⎨ Ud (x, z) − ΔUd (x, z) = 0, x ∈ Ωe , (6) cΔt ⎪ ⎩ U (x, z) = G(x, z), x ∈ Γ, d ∞ n n where Ud (x, z) = ∞ n=0 ud (x, tn )z and G(x, z) = n=0 g(x, tn )z . Hence, for each given z we can evaluate Ud (x, z) by solving the boundary value problem (6). The boundary conditions are obtained by a Z-transform of the time-domain boundary data. The time-domain solution is then obtained by the inverse Z-transform, which is given as a simple contour integral: Ud (x, z) 1 dz, (7) ud (x, tn ) = 2πi C z n+1 with C a contour around the origin in the region of convergence of the Z-transform. Hence, we use C = {z ∈ C : |z| = λ} for some appropriately chosen λ > 0. We approximate the integral using a trapezoidal rule with Nf frequencies at the points zk = e

−2πi Nk

f

with k = 1, . . . , Nf . This leads to (Nf )

ud (x, tn ) ≈ ud

(x, tn ) :=

Nf λ−n Ud (x, λzk ) . Nf zkn

(8)

k=1

Note that it is possible to reduce the number of frequency problems to solve by approximatively 2 since U (x, z) = U (x, z), as noticed in [5, Subsection 4.1]. (N )

The accuracy of computing ud f via (8) depends on the number Nf of frequency problems solved. In Section 4 we demonstrate that the rate of convergence is exponential.

3

Solving the frequency problems with BEM++

In this section we demonstrate how to implement the frequency-domain modified Helmholtz problems, using BEM++ [11]. BEM++ is an open-source boundary element library developed in C++ but with an extensive Python interface. It is fully object-oriented and can be easily extended. It makes heavy use of external projects, such as Trilinos [3] for the solution of linear systems, Dune [6, 7] for the grid management and implementation of basis functions on canonical elements, and TBB [4] for shared-memory parallelisation. Moreover, the library can make efficient use of AHMED 1.0 [10] for adaptive Cross Approximation (ACA) and Hierarchical Matrices (H-matrices) algebra. Interfaces to external FMM libraries are in development. The library currently supports Laplace, Helmholtz and Maxwell equations. Discretisation of kernels is done using a Galerkin approach, where the singular integrals are approximated by fully numerical singularity adapted quadrature rules (see e.g. [9]).

3.1

The frequency problem

In order to solve the wave equation, we have to be able to solve the modified Helmholtz equation: Δu(x) − k 2 u(x) = 0, x ∈ Ωe , u(x) = g(x), x ∈ Γ + Radiation condition when |x| → ∞.

(9)

260


Various methods are possible but since we solve in a unbounded domain, boundary element methods are well adapted. The kernel associated to this equation is Gω (x, y) =

e−ωx−y . 4πx − y

(10)

A possibility is to use the following indirect second kind formulation: Find φ ∈ L2 (Γ) such as: I + Kω φ(x) = g(x), x ∈ Γ. 2 where I is the identity operator and K is the double-layer boundary operator: ∂Gω (x, y) φ(y)dsy . Kω φ(x) = ∂ny Γ

3.2

(11)

(12)

Python implementation

For the implementation, we first load the BEM++ library in Python. i m p o r t numpy a s np import sys s y s . p a t h . a p p e n d ( ” ˜ / bempp / p y t h o n / ” ) from bempp i m p o r t l i b a s b e m p p l i b

We added in the path the directory containing the BEM++ python library. Second, we read the mesh and we define the numerical quadrature strategy. g r i d = b e m p p l i b . c r e a t e G r i d F a c t o r y ( ) . i m p o r t G m s h G r i d ( ” t r i a n g u l a r ” , ” . / s p h e r e −h − 0 . 1 . msh ” ) q u a d S t r a t e g y = b e m p p l i b . c r e a t e N u m e r i c a l Q u a d r a t u r e S t r a t e g y ( ” f l o a t 6 4 ” , ” complex128 ” ) o p t i o n s = bempplib . createAssemblyOptions ( ) c o n t e x t = bempplib . c r e a t e C o n t e x t ( q u a d S t r a t e g y , o p t i o n s )

There exist various options to modify the default accuracy of the numerical quadrature rule. Here, we will not go into detail of this but rather refer to [11]. Since we approximate the solution in L2 (Γ), we define piecewise constant basis functions and then the corresponding boundary layer operators. p c o n s t s = bempplib . c r e a t e P i e c e w i s e C o n s t a n t S c a l a r S p a c e ( c o n t e x t , g r i d ) mass matrix = bempplib . c r e a t e I d e n t i t y O p e r a t o r ( context , pconsts , pconsts , p c o n s t s ) dlOp = b e m p p l i b . c r e a t e M o d i f i e d H e l m h o l t z 3 d D o u b l e L a y e r B o u n d a r y O p e r a t o r ( c o n t e x t , p c o n s t s , p c o n s t s , p c o n s t s , wavenumber ) l h s O p = . 5 ∗ m a s s m a t r i x + dlOp

Operators created in BEM++ take three spaces as arguments, the domain space, the range space, and the space dual to the range space. The range space is not strictly necessary for Galerkin discretisations, but allows BEM++ to automatically implement routines for an operator algebra, including the product of boundary integral operators. At this level no matrix have been created. The following code creates a right hand side via user defined function, denoted by evalDirichletData. rhs = bempplib . c r e a t e G r i d F u n c t i o n ( context , pconsts , pconsts , e v a l D i r i c h l e t D a t a ) . coefficients ()

In what follows the iterative solver is setup, and the problem solved. The RealOperator class from BEM++ turns a complex operator into an equivalent real operator. This is to circumvent a bug in the handling of complex matrices via Gmres in some commercial Python distributions, and may not be necessary depending on how Scipy was compiled. The actual discretisation of the operator takes place via the weakForm method, which returns a discretised operator that is compatible to the Scipy Operator interface.


261

from s c i p y . s p a r s e . l i n a l g i m p o r t gmres n= l e n ( r h s ) A = R e a l O p e r a t o r ( ( . 5 ∗ m a s s m a t r i x + dlOp ) . weakForm ( ) ) b = np . h s t a c k ( [ np . r e a l ( r h s ) , np . imag ( r h s ) ] ) s o l r e a l , i n f o = gmres (A, b , t o l =1e −15 , m a x i t e r = 1 5 0 0 ) s o l =( s o l r e a l [ 0 : n ]+1 j ∗ s o l r e a l [ n : ] ) . r e s h a p e ( n , 1 ) np . s a v e ( ’ s o l u t i o n ’ , s o l )

To evaluate the solution in the domain, we first define a grid function by using the coefficients of sol in the space of piecewise constant basis functions. Then, a double layer potential is created to evaluate the solution at the evaluation_points: e v a l u a t i o n p o i n t s =np . a r r a y ( [ [ 0 . , 2 . , 0 . ] , [ 0 . , − 2 . , 0 . ] , [ 2 . , 0 . , 0 . ] ] ) gridFun=bempplib . c r e a t e G r i d F u n c t i o n ( context , pconsts , c o e f f i c i e n t s = s o l ) d o m a i n s o l u t i o n =bempplib . c r e a t e M o d i f i e d H e l m h o l t z 3 d D o u b l e L a y e r P o t e n t i a l O p e r a t o r ( context , wavenumber ) . e v a l u a t e A t P o i n t s ( g r i d F u n , e v a l u a t i o n p o i n t s ) np . s a v e ( ’ d o m a i n s o l u t i o n ’ , d o m a i n s o l u t i o n )

Here, we evaluate the domain solution at only 3 points but it’s possible to define a grid on a plan or in a volume to evaluate the domain solution.

4

Numerical result

In this section, we present a result for the acoustic scattering by a unit sphere, where the incoming wave is a plane wave localised in time by a Gaussian. The boundary condition in (1) writes as (t−t − k·x )2 p c k·x 2σ 2 )f e− , g(x, t) = cos 2π(t − c where f is the frequency, tp the time-of-arrival, σ the variance, and k is the wave vector defining the direction 3 for the computation. of the incident wave. We use f = 400, k = (1, 0, 0)T , tp = 0.01, c = 343 and σ = 1000π (N ) f (ref) (ref) − ud according to Nf . The reference solution ud is Figure 2 presents the absolute difference ud obtained by choosing Nf very large. We can observe exponential convergence with a rate that depends on the parameter λ of the contour for the inverse Z-Transform. In [12] a full analysis of the asymptotic exponential rate of convergence for Nf → ∞ is given. A particularly interesting point in Figure 2 is the point Nf = Nt , where the number of frequencies solved Nf is identical to the number of time steps Nt . This is the standard case in previous Convolution Quadrature approaches. We note that by increasing Nf a significantly better accuracy can be achieved.

Conclusion We have presented an easy-to-use Convolution Quadrature method. By choosing the number of frequencies Nf larger than the number of time steps we can obtain a significantly improved accuracy. The solution of the frequency-domain problems with BEM++ has been presented. An open source time-domain toolbox for BEM++ is in development, which will allow the parallel solution of time-domain problems using various CQ approaches. Acknowledgments. EP/K03829X/1.

This work was supported by Engineering and Physical Sciences Research Council Grant


Absolute difference:

|

ud

(Nf )

−

ud(ref) |

262

100 10-1 10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9 10-10 10-11 10-12 Nt=Nf 10-13 0 100

λ =0.95 λ =0.85 λ =0.75

200 300 400 N_f, Number of frequencies

500

600

Figure 2: Absolute difference of the solution for the scattering of the unit sphere using indirect second kind formulation for three different λ. The accuracy obtained usually is indicated by Nt = Nf .

References [1] L UBICH , C., Convolution quadrature and discretized operational calculus. I, Numerische Mathematik, (1988). [2] L UBICH , C., Convolution quadrature and discretized operational calculus. II, Numerische Mathematik, (1988). [3] H EROUX , M. AND

AL .,

An overview of the Trilinos project, ACM Trans. Math. Software, 2005.

[4] R EINDERS , J. , Intel threading building blocks: outfitting C++ for multi-core processor parallelism, O’Reilly Media, 2007. [5] BANJAI , L. AND S AUTER , S., Rapid Solution of the Wave Equation in Unbounded Domains, SIAM J. Numerical Analysis, 2008. [6] BASTIAN , P. AND Computing, 2008.

AL .,

A generic grid interface for parallel and adaptive scientific computing. Part I: abstract framework,

[7] BASTIAN , P. AND AL ., A generic grid interface for parallel and adaptive scienti?c computing. part II: Implementation and tests in DUNE, Computing, 2008. [8] BANJAI , L., Multistep and Multistage Convolution Quadrature for the Wave Equation: Algorithms and Experiments, SIAM SISC, (2010). [9] S AUTER , S. A.

AND

S CHWAB , C., Boundary element methods, Springer, 2011.

[10] B EBENDORF, M., Another software library on hierarchical matrices for elliptic differential equations, 2012, http:// bebendorf.ins.uni-bonn.de/AHMED.html. [11] S´ MIGAJ W. AND AL . , Solving Boundary Integral Problems with BEM++, ACM Trans. Math. Software, to appear. [12] B ETCKE , T., S ALLES N., AND S MIGAJ , W., Exponentially accurate evaluation of time-stepping schemes for the wave equation via Convolution Quadrature type Methods, To appear.


263

Sensitivity analysis by the fast multipole boundary element method V. Mallardo1 , M. H. Aliabadi2 1

2

Department of Architecture, University of Ferrara, via Quartieri 8, 44121 Ferrara (I) [email protected] Department of Aeronautics, Imperial College London, South Kensington Campus (UK) [email protected]

Keywords: optimization, identification, integral equation, Helmholtz equation. Abstract. The present paper intends to couple the Fast Multipole Method with the Boundary Element Method in 2D inverse acoustic problems. The procedure is aimed at simplifying the integrals involved in the governing Boundary Integral Equations on the basis of the multipole expansion. The approach makes the determination of the governing unknowns and of their sensitivities much faster if compared to the conventional approach. Introduction There has been much interest in the application of inverse analysis to engineering problems in the last twenty years. Geophysics, structural engineering, non-destructive testing are just a few examples in which inverse analysis has great importance. There are two major types of inverse problems: optimization and identification. Any numerical approach to these problems require repetitive resolution of a direct problem in which all the input data are known. Therefore the total CPU time grows linearly with the number of iterations that are necessary to reach the optimal solution. The optimal solution is generally the minimum of a suitable objective function written in terms of some either computed or experimentally measured data. In order to speed up the inverse analysis it could be convenient to determine the sensitivities at each iteration step. The sensitivities are the derivative of the governing field with respect to the design variables, i.e. the variables controlling the global shape in an optimization analysis or the target shape (for instance the shape of a flaw) in an identification analysis. Experiences of the Authors on identification procedures based on the sensitivity analyses are given in [1-3]. In [1] an internal flaw is identified on the basis of scattering measurements obtained by acoustic illumination of the flawed solid. In [2-3] the identification procedure is aimed at obtaining either an internal crack or an internal inclusion on the basis of boundary displacement measurements under different static loads. The numerical approach is based on the Boundary Element Method (BEM), enhanced by the implicit computation of the sensitivities. It is clear that CPU time and matrix storage can be a big issue in solving the inverse problems described above. There is no doubt that the BEM is more suited than the Finite Element Method (FEM) as it reduces the effort to iteratively re-generate the boundary at each optimization/identification step. A reference text dealing with the development of BEM in structural engineering is given by [4]. In 1983 Rokhlin [5] proposed an algorithm for rapid solution of classical boundary value problems for the Laplace equation based on iteratively solving integral equations of potential theory and commonly referred as Fast Multipole Method (FMM). The CPU time requirement obtained was proportional to N . The approach was afterward coupled to BEM with excellent results (see [6] as review) and the coupled procedure has showed to be able to cope problems of up to 106−8 unknowns with a desktop computer. An example of application of the Fast Multipole Boundary Element Method (FMBEM) for scalar wave propagation is given in [7]. It is, therefore, clear that the FMBEM can increase the performance of the inverse analysis by speeding up the necessary effort to compute the objective function and its derivative at each iteration step. However, the use of the FMM increases the complexity in the implementation with the BEM: the structure of the code changes completely and the pre-processor stage becomes more important than in the conventional approach.

264


The present paper intends to present a FMBEM inverse analysis for two-dimensional acoustics. The procedure may be used both for optimization, for instance to optimize the shape of passive noise control panels, and for identification, for instance in non-destructive testing by acoustic ultrawaves. The optimization algorithm An inverse problem can be set as minimization of a suitable objective function f (V) where V are the design variables, i.e. the variables governing the iteratively modified geometry: Find {V} which minimises f (V) In optimization the objective function is usually expressed in terms of the target to be minimized. For instance the optimization in acoustic and vibration involves a function that can be a measure of the maximum sound pressure level or of the scattering cross section in a fixed region. In identification the objective function is usually expressed in terms of the boundary field, as it is more commonly and more simply measurable by experimental techniques. In conclusion, in linear structural optimization/identification it is primary to obtain the boundary field. Such a task can be accomplished by the BEM rather than by the FEM as the discretization, that needs to be updated iteration by iteration, is reduced to the minimum effort. Afterwards, either the inverse problem can be carried out without further computations or simple postprocessing steps need to be performed. The objective function can be written as: f (V) = f (pΓ , qΓ ) (1) where pΓ and qΓ represent pressure and flux, respectively, on the external boundary Γ. The minimization can be carried out by updating the design variable vector V on the basis of a first order, non linear optimization technique. Such a technique requires the evaluation of the governing variables, pressure and flux in the present analysis, on the boundary as well as of their sensitivities. Both can be computed by the FMBEM approach that will be detailed in the next sections. The FMBEM direct problem The propagation of time-harmonic acoustic waves in a homogeneous isotropic acoustic medium (either finite or infinite) is described by the Helmholtz equation: ∇2 p(x) + k 2 p(x) = 0

(2)

under the boundary conditions: p(x) = p(x)

x ∈ Γ1

(3a)

q(x) = p(x),n = q(x)

x ∈ Γ2

(3b)

where p is the acoustic pressure, k = ω/c with ω =angular frequency and c =sound velocity, comma indicates partial derivative, Γ1 ∪ Γ2 = Γ, Γ is the boundary of the domain Ω under analysis, n = n(x) is the outward normal to the boundary in x, q is the flux and the barred quantities indicate given values. The boundary value problem described by the Eqs. (2-3) can be transformed into the following integral representation: c(ξ)p(ξ) + q ∗ (ξ, x)p(x)dΓ(x) − p∗ (ξ, x)q(x)dΓ(x) = 0 (4) Γ

Γ

where c(ξ) occurs in the limiting process from the internal point to the boundary point, being equal to 0.5 if the tangent line to the boundary at ξ is continuous. The fundamental solutions p∗ and q ∗ , in terms of the Hankel function of the first kind, can be found in any BEM book (see for instance [4]). As constant elements are here adopted, the procedure requires the evaluation of either the integral of p∗ or the integral of q ∗ on each boundary element.


265

For convenience, the complex notation is introduced, i.e. the collocation and field points are replaced by their complex representation. With such an assumption it is simple to show that the fundamental solutions in ξ, x coincide with their expression in complex notation: p∗ (ξ, x) = p∗ (z0 , z)

(5a)

q ∗ (ξ, x) = q ∗ (z0 , z)

(5b)

The FMM relations intervene in the evaluation of integrals involved in the eq. (4). The multipole expansion is the key point (see also [7] for details). If F (z0 , z)f indicates either p∗ (z0 , z)q or q ∗ (z0 , z)p, the following local expansion can be obtained: f Γj

where:

F (z0 , z)dΓ(z) =

∞ i (−1)p L−p (zL )Ip (z0 − zL ) 4 p=−∞

(6)

Ip (z) = (−i)p Jp (kr)eipθ

r, θ are the polar coordinates of z and Jp stands for the Bessel function of the The coefficients L−p are given by the following M2L translation: ∞

Ll (zL ) =

Ok+l (zL − zC )M−k (zC )

(7) pth

order.

(8)

k=−∞

where z0 − zL 0 Ni,p (ζ) = ζi+p − ζi ζi+p+1 − ζi+1 Eqs. 6 may yield the quotient 00 ; in such a case by definition such a quotient is set to zero. It is clear that Ni,0 is the well-known step function, zero everywhere except on the half open interval ζ ∈ [ζi , ζi+1 ), and that the computation of the function Ni,p requires the specification of the degree p and of a knot vector U . The above B-splines benefit from some properties that are revealed to be very useful in the context under analysis. Here are reported some useful ones. First, Ni,p (ζ) = 0 if ζ is outside the interval [ζi , ζi+p+1 ). Then, in any given knot span [ζj , ζj+1 ) at most p + 1 of the Ni,p are non zero. Furthermore, for an arbitrary knot span [ζi , ζi+1 ), ij=i−p Nj,p (ζ) = 1 for all ζ ∈ [ζi , ζi+1 ). It is worthy to underline that the last property does not imply that each B-spline assumes the unit value in one knot node and vanishes in the others, as it occurs with the classical polynomial shape functions. The derivative of B-spline is necessary to compute the stress in any internal point by eq. (5). Such derivatives are given by: Ni,p (ζ) =

p p Ni,p−1 (ζ) + Ni+1,p−1 (ζ) ζi+p − ζi ζi+p+1 − ζi+1

(6)

All derivatives of Ni,p (ζ) exist in the interior of a knot span. An important property is that at a knot, Ni,p (ζ) is p− l times continuously differentiable, where l is the multiplicity of the knot. In other words, the knot vector may contain repeated values depending on the type of continuity that is intended to be assigned to the geometry representation obtained by the B-splines. For instance, a double repeated knot implies a corner of the boundary, a triple repeated knot corresponds to either the start or the end of the represented curve. The geometry interpolation that is obtained by using the B-splines can be further improved by slightly transforming them. The new splines are called NURBS and their expression is given by: Ni,p (ζ)wi Ri,p (ζ) = n i=1 Ni,p (ζ)wi

(7)

where wi are the weights and n is the number of control points (to be discussed further). The superiority of the NURBS when compared to the B-spline is evident, for instance, to represent the circle arch.

284


A generic curve representing the boundary of an elastic domain subjected to generic static loads can be represented by the aid of the NURBS, that is: x(ζ) =

n

Ri,p (ζ)Pi

(8)

i=1

where P are the control points. Such a representation has some interesting features that are detailed in [5], for instance the strong convex hull property.

(a)

(b)

Figure 1: Example of boundary represented by NURBS. (a) geometry with control points (red circle) and some nodes (black filled circle). (b) geometry with element ends (cross) and collocation nodes (filled circle) An example of boundary representation by quadratic (p = 2) NURBS is given in Fig. 1. In Fig. 1a the control points are depicted by red circles and four indicative positions (ζ = 0, 0.25, 0.5, 0.75) are reported in black filled circles (ζ = 1 coincides with ζ = 0). The corresponding shape functions NURBS are drawn in Fig. 2 where the curves from left to right are N1,2 , · · · , N13,2 (i.e. the number of the control points with the first counted twice).

Figure 2: NURBS associated to the example in Fig. 1.

BEM plus NURBS When used for geometric representation, NURBS demonstrate to be much more powerful than all the shape functions nowadays known. The practical demonstration is that all the CAD softwares are NURBS based, but the simple representation of the circle is able to show the superior performance of the NURBS. Such a performance may be reproduced in modeling the displacements and the tractions involved in the BEM problem. The coupling is optimal as BEM, differently from FEM, at least in the


285

linear case, requires the boundary discretisation only. The NURBS representation of the elastostatic unknowns can be written as: u(ζ) = t(ζ) =

n i=1 n

Ri,p (ζ)di

(9)

Ri,p (ζ)qi

(10)

i=1

where it is worthy to underline that the unknowns di and qi do not represent, as it is common with other shape functions, the values of either displacement or traction in the nodes, but they are variables that have not any physical relation with u and t. The integral equation eq. (1) can be adapted to the new expression of the shape functions and it can be collocated in the collocation points in order to build the final system of equations. Some more details on the meaning of element and collocation point need to be provided. The element i must be intended as the part of curve going from ζi to ζi+1 of the knot vector, provided that ζi = ζi+1 . Different strategies are available to set the position of the collocation points ζ. The most used is given by the Greville abscissae, i.e: ζi+1 + · · · + ζi+p (11) ζi = p and it will be adopted in the present paper. Fig. 1b provides an example of subdivision of elements and location of collocation points as given by Eq. (11). On the basis of the discretization carried out by eqs. (8,10) the ith discretized integral equation can be written as: cij (ζ) =

p+1

l=1 p+1 NE qjl e=1 l=1

Rl,p (ζ)dlj + Γe

p+1 NE e=1 l=1

e dlj − Tij (ζ, x(ζ))Rl,p (ζ)J l (ζ)dζ = Γe

e Uij (ζ, x(ζ))Rl,p (ζ)J l (ζ)dζ

(12)

In analogous way can be obtained the expression providing the stress in any interior point. Collocating in each of the nodes located by eq. (11) provides the final system of equations similar to eq. (3). As it is clear looking at the approximation eq. (10), it is not straightforward to apply the BCs to the matrix eq. (3). The unknowns are parameter that have no physical meaning whereas the BCs are directly applied in the collocation nodes in terms of either displacement or traction. The procedure aimed at correctly applying the BCs in the procedure here presented are detailed in a separate paper under revision. Another issue raises with reference to the strongly singular integral involved in eq. (12). In a generic collocation node the p + 1 involved shape functions Ri,p may all result to be different from zero (see for instance ζ = 0.25 in Fig. 2). Therefore, the rigid body condition is not sufficient to determine the singular term in terms of the nonsingular terms belonging to the same row, that is, it must be computed directly (see [4] for details). Finally it is worthy to point out that the non-singular integrals can be computed on each element by Gaussian quadrature after a variable transformation ζ → η that allows Γe to be transform into +1 −1 . Numerical results In order to demonstrate the accuracy of the proposed procedure, a numerical example is presented. A cylinder (of radius Re = 1.0) containing a central cylinder hole (of radius Ri = 0.2) and subjected to the radial (compressive) pressure both on the external boundary (pe = 2) and on the internal boundary (pi = 1), is analyzed. In such a case an analytical solution is available and thus compared

286


to the numerical one. Young’s modulus and Possion’s coefficient are assumed equal to 100000 and 0.3, respectively. Geometry and loads of the example, along with control points, elements and collocation nodes, are depicted in Fig. 3. The two circles are generated as four nurbs each.

(a)

(b)

Figure 3: The numerical example. (a) geometry and load. (b) control points (red circle), collocation points (filled circle) and element extremes (cross) The results are listed in Table 1, where a comparison between analytical and numerical values is carried out. It is evident the excellent match with an error that is less than 0.04%. 4 |uext r |x10

4 |uint r |x10

analytic

0.11158

0.04832

BEM-NURBS

0.11160

0.04830

Table 1: Comparison of the radial displacement. It is worthy to underline that a classical BEM approach would require minimum 12 quadratic elements for circle in order to obtain the same accuracy. Conclusions A BEM approach involving the NURBS as shape functions has been presented. The procedure results to be more accurate than the classical BEM and it allows a simpler and more direct link with the common CAD softwares. References [1] T.J.R. Hughes, J.A. Cottrell, Y.Bazilev. Computer Methods in Applied Mechanics and Engineering, 194:4135–4195, (2005). [2] R.N. Simpson, S.P.A. Bordas, J. Trevelyan, T. Rabczuk, Computer Methods in Applied Mechanics and Engineering, 209-212:87–100, (2012) [3] L.C. Wrobel and M.H Aliabadi The Boundary Element Method, Vol2: Applications in Solids and Structures, Wiley (2002). [4] J. C. F. Telles. International Journal for Numerical Methods in Engineering, 24:959–973, (1987). [5] L. Piegl and W. Tiller The NURBS book, Springer (1997)


287

Boundary Element Analysis of Fibre-Reinforced Composites and Adhesion Joints with Bridged Cracks Mikhail Perelmuter Institute for Problems in Mechanics of RAS, pr. Vernadskogo 101-1, Moscow, 119526, Russia E-mail: [email protected] Keywords: boundary elements, composite, adhesion joints, bridged interfacial cracks Abstract. The multi-domain formulation of boundary element method is used to analyze the stress-strain state and fracture toughness of fibers-reinforced composites or adhesion joints with interface cracks or crack-like defects. It is assumed that unbroken fibers form bridged zones along the whole or part of a delamination region inside of the materials joint. For modeling purposes the bridged zone is considered as a part of the interface crack between two materials and it is assumed that distributed nonlinear spring-like bonds link and constrain the crack surfaces. The special crack tip boundary elements for asymptotic interfacial displacements and stresses modeling are used near the crack tips. Some results for structures of finite size, with the material junctions of different kinds and shapes, with variation of bridged zone size and fibers property, are presented. Introduction Analysis of cracks growth in composite materials and in adhesive joints using models of a crack process zone (cohesive or bridged) includes the problem of displacements and stresses computation in crack process zones and in vicinity of crack tips. In this paper the bridged crack approach will be considered as a crack process zone model. This approach it is assumed that the stress intensity factors do not vanish at a crack tip. The problem of displacements and stresses analysis for bridged cracks in homogeneous infinite media was considered in [1-2] (planar problem) and [3] (problem with axial symmetry). For bridged interfacial cracks between two semi-infinite plates the problems were solved in [4-5] by the singular integral-differential equation method (planar problem). In the last two decades a number of papers were devoted to the application of the boundary element method (BEM) to computation of the stress intensity factors (SIF) for cracks on bi-material interfaces in a finite size structures, see, for example, [6-8]. In all these papers interaction between crack surfaces was not assumed nor considered. Only few papers have been directed to analysis of bridged cracks in finite size structures, [910]. The procedure of the BEM application to interfacial bridged cracks had been developed in [11]. The main objective of this paper is the application of the approach proposed in [11] for curvilinear interfacial bridged cracks in finite size structures. Bridged interfacial crack To characterize the bridged interfacial crack model let us consider a crack (curvilinear in the general case) of length 2A on an interface of two dissimilar elastic materials, Fig.1a. The segments of length d1 and d 2 , adjacent to the crack tips, are the bridged zones. In these zones the crack surfaces interact with each other; this interaction restrains the crack opening. To describe mathematically the interaction between the crack surfaces, it is assumed that there are bonds between the crack surfaces at the bridging zone of the interfacial crack. The tractions in these bonds restrain the crack opening and decrease the SIF. The deformation law of these bonds, which is generally nonlinear, is given. The stresses (the bond tractions) Q( s ) appear under the external loading action in the bonds between the surfaces of the bridged interfacial crack placed at the boundary between different materials. These tractions Q( s ) have (in particular, in the simplest case of an uniaxial tension and a straight crack in the material interface [4]) the normal t y ( s ) and tangential tx ( s ) components ( s - is the local arch coordinate along the crack, in the case of Fig.1a , s { x , 0 - is the origin of the Cartesian coordinate system).

288


Q( s ) t y ( s ) it x ( s ),

i2

1

(1)

The crack surfaces in the bridged zone are loaded by the normal and tangential stresses, which are numerically equal to these components of tractions. The interfacial crack opening at the bridged zone, u s , can be written in the form similar to (1) u( s ) u y ( s ) iux ( s ), u y ( s ) u y ( s ) u y ( s ), u x ( s ) ux ( s ) u x ( s )

(2)

where ux , u y and ux , u y denote the displacements components of the upper and lower crack surfaces.

b) a) Fig.1 (a) Bi-material plate with bridged interfacial crack; (b) Plate with elastic inclusion and bridged circular interfacial crack under normal tension loading, W / R 10 .

The relation between the crack opening and the bond traction (the bond deformation law) depends on the physical nature of the bonds and their properties. The general form of the spring-like bond deformation law can be written as [4-5] u( s ) c y ( s,V )t y ( s ) ic x ( s,V )t x ( s ), c y , x ( s,V ) I1,2 ( s,V )

H , V Eb

t x2 t y2

(3)

where the functions c y and c x can be considered as the effective bond compliances in the OX and OY axes directions, I1,2 are dimensionless functions used for the description of non-uniform behavior of compliances over the bridged zone, H is the linear scale being proportional to the thickness of the bonding zone, Eb is the effective elasticity modulus of the bond, V is the modulus of the tractions vector of bonds. Note that the second formula in (3) presents the compliances of linear-elastic bonds if the functions Ii depend only on the coordinate s . In particular, the compliances have a uniform distribution over the crack bridged zone if the functions Ii are constants. The SIF for interfacial cracks can be defined on the basis of asymptotic relations for the crack opening and the stresses in the polar coordinates system U ,T with the origin at the crack tip (see Fig.1a). The crack opening is the difference between displacements of the upper ( T S ) and lower ( T S ) surfaces of the crack and according to [12] for an interfacial crack between two semi-infinite plates it can written as

u y ( U ) iu x ( U )

§ N1 1 N 2 1 · K iK II ¨ iE P 2 ¸¹ I U §U· © P1 ¨ ¸ , 2 1 2i E cosh(SE ) 2S © R ¹

E

ln D , 2S

D

P1 P 2N1 P 2 P1N 2

(4)


289

where K I and K II are the SIF for an interfacial crack, u y ( U ) and ux ( U ) are the components of the crack opening, D and E are the bielastic parameters, N1,2

3 4Q 1,2 or N1,2

(3 Q 1,2 ) / (1 Q 1,2 ) for plane strain

and plane stress, respectively, Q 1,2 and P1,2 are Poisson's ratios and the shear modulus of jointed materials 1 and 2, R is an arbitrary length to normalize the distance U in (4). Relation (4) for the crack opening has an oscillatory behavior as U o 0 and this means that the crack surfaces near its tips are overlapped. If the tension loading is predominating, the size of this zone is very small. Also within this model framework we assume that the lengths of the overlapping regions are much less than the crack bridged zone sizes. The asymptotic of the stress field in a neighborhood in front of the interfacial crack tip (T 0) is [12]

K I iK II § U · ¨ ¸ 2SU © R ¹

V yy ( U ) iV xy ( U )

iE

(5)

which also has an oscillatory behavior in a small zone in front of the crack tip. The SIF modulus is defined as

K

K I2 K II2

(6)

and it can be found from relation (4) for the crack opening behind the crack tip ( Ku ) and also from relation (5) for the stresses in front of the crack tip ( KV )

Ku ( U )

2cosh(SE ) 1 4 E 2 A

2S ux2 u 2y

U

, KV ( U )

2SU

V yy2 V xy2

(7)

In formulas (7) the values u x , u y and V xy ,V yy are defined at the position s behind or in front of the crack tip, respectively. These formulas will be used below for computation from numerical results of the SIF module. The crack opening along crack bridged zones and the stresses in front of crack tips, which are used in formulas (7), can be determined semi-analytically or numerically in the case of an infinite plate with bridged crack between two different materials under a uniform external loading. Detailed results for these problems are presented in [4-5]. For structures of finite size with bridged cracks of an arbitrary shape and with a spring-like bonds deformation law which is similar to (3) the crack opening along crack bridged zones, bonds stresses and stresses in front of crack tips can only be determined numerically. Boundary element formulation for bridged interfacial crack

The modeling of bridged interfacial cracks is based on the multi-domain BEM formulation, [13]. Within this approach, direct boundary integral equations (BIE) for elasticity problems are used for each homogeneous sub-region of the structure. The supplementary boundary conditions at the interfacial boundaries and at the bridged zones of cracks are introduced and used to eliminate additional variables on joint boundaries of subregions. For elasticity problems without body forces the direct BIE for any sub-region of the structure is given by [14]: (8) cij p ui p ³ ¬ªGij p, q ti q Fij p, q ui q ¼ºd * q , i, j 1,2 *

here cij p depends on the local geometry of boundary * (for a smooth boundary cij p 0.5G ij ), Gij p, q and Fij p, q are Kelvin's fundamental solutions for displacements and stresses, respectively, ui q and ti q are displacements and tractions over the boundary of the structure and the location of source and field points belonging to the sub-region boundary * are defined by the coordinates of the points q and p .

290


The displacements continuity and the tractions equilibrium supplementary conditions at the interfacial boundaries without cracks are the following uik p uif p , tik ( p )

ti f ( p )

(9)

where k and f are joint sub-regions numbers, ui p and ti ( p ) are displacements and tractions components at the boundary point p . The relationships between bond tractions and displacements difference of the upper and lower crack surfaces (the crack opening) at the crack bridged zone can be written in the following generalized form (the bonds deformation law) tn ,W ( p ) N n ,W ( p,V ) un ,W ( p ), N n ,W ( p,V )

1 cn ,W ( p,V )

J 1,2 ( p,V )

Eb H

(10)

where tn ,W p and un ,W p are the components of tractions vector and crack opening (see (2)) in the local coordinate system connected with the normal n and tangential W directions at the point p and belonging to the sub-region with number M , where M min( f , k ) , cn ,W ( p ,V ) are the compliance and N n ,W p,V are the stiffness of bonds depending on the distance from the crack tip, V is the tractions vector modulus at the current point p , and J 1,2 (I1,2 )1 , see (3). The boundaries of all sub-regions of the structure are subdivided into quadratic isoparametric elements for numerical solution of the BIE (8). For the interfacial crack displacements and stresses asymptotic modeling the special crack tip elements are used. For any point at the interface of joint sub-regions or at the interfacial crack bridged zone 4m nodal unknown functions ( m 2 for 2D problem) need to be defined. The multidomain BIE formulation provides 2m equations, and the auxiliary equations are generated for the remaining 2m unknown functions ( m from the continuity condition and m from the equilibrium condition across the jointed parts of subregions for cases without bridged zones, see (9), or m from the equilibrium conditions and m from the bond deformation law (10) for bridged zones of interfacial cracks). The displacements and tractions are regarded as nodal unknowns on jointed parts of sub-regions whereas the displacements of the upper and lower crack surfaces are regarded as nodal unknowns on interfacial crack bridged zones and the additional tractions are eliminated by substitution of the bond deformation law (10) into BIE (8). Computation of the stress intensity factors

The isoparametric formulation, which is used in this paper, provides the possibility to model the square-root asymptotic of displacements ( u ~ U ) for boundary elements adjacent to the crack tip, by moving the midpoint node of a quadratic boundary element to a quarter point position. This approach, proposed initially for finite element computation, has been extended to BEM [13] with correction for the traction asymptotic modeling in front of crack tips. This approach is also applied here to model the traction asymptotic for the bridged interfacial crack. The coordinates of the quarter-point boundary element at the crack bridged zone in the local coordinate system with the origin at the crack tip and with the axis U directed up over the boundary element on the crack surface ( T rS ) can be written as ( | [ |d 1 )

U ([ ) N k ([ )U k , U1 0, U 2 N1 [

0.25L, U3

0.5[ 1 [ , N 2 [ 1 [ 2 , N 3 [

L 0.5[ 1 [

(11) (12)

where [ is the parametric coordinate along the element, N k [ , ( k 1,2,3) are the shape functions of 3nodal quadratic element, L is the length of the boundary element jointed to the crack tip, U1,2,3 are the


291

coordinates of nodal points on this element. Substituting (12) into (11) yields that the variation of the local coordinate [ along the boundary element is

[ U 1 2

U

(13)

L

Displacements and traction variations along the boundary element with the quarter-point node have the form which is similar to (11) ui [

N k [ uik , ti [

N k [ tik , i 1,2, k 1,2,3

(14)

Substituting (13) into the first relation from (14) we get the proper asymptotic of the displacements near the crack tip U U (15) ui U Ai1 Ai 2 Ai 3 L L where the parameters Aik depend on the nodal displacements of the boundary element with quarter-point node [11]. Similar relations can be also written for the tractions distribution on the boundary element with quarter-point node in front of the crack tip, whereas the stresses should be varied according to 1 U . This singularity may be modeled by multiplying the equation for the traction on the adjoining quarter-point boundary element in front of the crack tip (which is similar to (15)) by L

2 1[

U

(16)

then, from the second relation of (14) and (16), it can be written as ti U

Bi1

L

U

Bi 2 Bi 3

U

(17)

L

where the parameters Bik depend on the nodal tractions of the element [11]. The module of SIF can be computed from the quarter-point node displacements on the boundary element adjoining to the crack tip. By letting U 0.25L in the first relation in (7) gives

K u ( L / 4)

4cosh(SE ) 1 4 E 2 A

2S ux2 u 2y L

where ux and u y are the components of the crack opening at the point U

(18)

0.25L , see formula (2). This is

the so-called one-point displacement (OPD) formula. Another equation for the computation of the SIF module can be obtained by the linear extrapolation to the crack tip of SIF (18) computed at the quarter-point node of the element and at the edge of the element (U L) 4 Ku L / 4 Ku L Ku U 0 (19) 3 where K ( L) is defined by (11) for U

L . This is the linear extrapolation displacements (LED) formula.

292


A different approach to the SIF computation is to use the nodal values of tractions on the boundary element adjacent to the crack tip, in front of it, as has been proposed in [15]. The SIF module according to this approach can be calculated as Kt

2S L

tx12 ty21 , tx1, y1

lim t x1, y1 U o0

U

(20)

L

where tx1 and ty1 are the tractions computed at the crack tip after the modification according to (16). This is the one-point traction (OPT) formula. Similarly to (19), it is possible to obtain the SIF module as linear extrapolation at the crack tip of the SIF module values computed at the quarter-point node and at the edge node on the boundary element adjacent to the crack tip in front of it (LET formula)

Kt

U 0

4 Kt L / 4 Kt L 3

(21)

where Kt ( L / 4) and K t L are obtained from the second relation in (7) by letting, respectively, U and U L 2 K t L / 4 0.5 2S L V xy2 ,2 V yy ,2 ,

Kt L

2 2S L V xy2 ,3 V yy ,3

0.25L

(22)

where the components of stresses on the boundary element are defined at the positions U 0.25L and U L , respectively. From relations (22) and taking into account the computed tractions at the quarter-point node of crack tip boundary element is a half of the physical tractions ([15]) we obtain K t L / 4 0.5T2 2S L ,

K t L T3 2S L

where T2 and T3 are the tractions vector modulus at the positions U boundary elements adjacent to the crack tip in front of it. Finally, formula (21) is transformed into Kt

U 0

4T2 T3 2S L 3

0.25L and U

(23)

L on the quarter-point

(24)

Numerical results

The algorithms of BIE system (8) solving for structures with bridged interfacial cracks and the SIF modulus computation were implemented into the computer code for 2D/3D and axisymmetric problems of elasticity and thermoelasticity. The quarter-point boundary elements with tractions modification according to (17) were used. The problem with cylindrical inclusion in a finite size plate (plane strain conditions) under a uniform tension was analyzed on the basis of the algorithm developed. It was assumed that there exists an arc-shaped zone of weakened adhesion bonds between the inclusion with circular cross-section and the matrix. This zone was considered as the bridged arc-shaped interfacial crack. Note, the problem of a bridged interfacial crack between a matrix and an inclusion taking into account the singularity at the crack tips was not considered and no solutions were published for this problem. In the presented paper numerical analysis the region was taken as a square plate with width 10 times larger as compare to the circular elastic inclusion diameter. Due to the symmetry of an uniaxial tension problem only half of the region with the proper boundary conditions was considered in the BEM analysis (see Fig.1b, where value of M 0 30 o is the half arc angle of the interfacial bridged crack configuration, W / R 10 , the


293

crack tip is at point 0 ). Point P of the model was fixed in the direction of the external tension loading to exclude a free body motion.

Fig.2 Circular bridged interfacial crack. The bridged stress along the bridged zone length.

Fig. 3 Fully bridged interfacial crack. The SIF modulus vs relative bonds stiffness.

The bridged stresses (the modulus of bonds tractions vector) distributions over the zone length for various relative sizes of circular crack bridged zone - / M0 are shown in Fig.2, where - is the size of the bridged zone in the angular measure and 0 - d M0 . The dimensionless angular parameter M / M0 defines a current point position along the zone. The structure symmetry line I I corresponds to the value M 0 and the angle measure of the bridged zone length starts at the crack tip (point 0 in Fig.1b). The stresses distribution tends to a uniform one with a small non-uniform region near the crack tip as the relative bridged zone length is increasing. The maximum value of bond stresses was observed at the edge of a bridged zone, and for - / M0 0.2 the value of maximal bond stresses is about V / V 0 | 4.3 . Note that the similar dependence of the bond stresses versus the bridged zone length was observed also for the straight bridged crack at the plate, [4]. The results of SIF module computation by formulas (18), (19), (20) and (21) (OPD, LED, OPT, LET, respectively) proposed in the previous section are presented in Fig.3 for fully bridged circular crack between the inclusion and the matrix. Differences between the SIF module values computed on the basis of these formulas are not too large with small increasing for the relatively hard bonds stiffness. The results obtained by the OPT formula (20) can be regarded as close to average values and this formula was used to compute the SIF module results presented below. The SIF module dependence on the relative stiffness bonds for several values of the inclusion and the matrix elastic modules ratio for fully bridged circular crack ( - / M0 1 ) is shown in Fig.4. The case N 0 0 on these graphs corresponds to crack without bridged zone. If the relative rigidity of the inclusion ( E1 / E2 ) is increasing, the SIF module is also increasing. The saturation of shielding influence of bonds initiates at relative soft bond stiffness in the cases of homogeneous body or the soft inclusion. For the case of a relatively hard bond, N 0 10 , the SIF module dependence on the relative bridged zone size (in the angular measure) is shown in Fig.5 for several values of the inclusion and the matrix elastic modules ratio. The shielding influence of bonds is increasing as the bridged zone size increases and the influence of the bridging zone length on the SIF module is stronger for the soft inclusion. If the relative size of the bridged zone is about - / M0 | 0.3 the shielding effect by bonds is close to saturation. The position of this point also slightly depends on the inclusion relative rigidity ( E1 / E2 ). Closing

The computation of the bridged stresses and the SIF module analysis is the first step in bridged cracks growth modeling. Analysis of bridged cracks growth can be implemented in the frame of the nonlocal criterion of bridged cracks growth [4,16] which can be easily incorporated into the developed boundary element code.

294


Acknowledgements: Financial support for this research was partly provided by Russian Foundation for Basic Research (Projects14-08-01163 and 14-01-00869). References

[1] Y. Weitsman Nonlinear analysis of crazes. Trans. ASME J. Appl. Mech. 53:97-102 (1986). [2] L.R.F. Rose Crack reinforcement by distributed springs, J. Mech. Phys. Solids 35(4):383-405 (1987) [3] N.V. Movchan and J.R. Willis Penny-shaped crack bridged by fibres, Quart. Appl. Math.56(2):327-340 (1998) [4] R.V. Goldstein and M.N. Perelmuter Modeling of bonding at an interface crack, Int. J. Fract. 99(12):53-79, (1999) [5] M.N. Perelmuter An interface crack with non-linear bonds in a bridged zone, J. Appl. Math. Mech. (PMM) 75(1):106-118 (2011) [6] R. Yuuki and S.B. Cho Efficient boundary element analysis of stress intensity factors for interface cracks in dissimilar materials, Eng. Fract. Mech. 34:179-188 (1989) [7] S.T. Raveendra and P.K. Banerjee Computation of stress intensity factors for interfacial cracks, Eng. Fract. Mech. 40:89-103 (1991) [8] A.R. Hadjesfandiari and G.F. Dargush Analysis of bi-material interface cracks with complex weighting functions and non-standard quadrature, Int. J. Solids Struct. 48:1499-1512 (2011) [9] Y.F. Liu, C. Masuda and R. Yuuki An efficient BEM to calculate weight functions and its application to bridging analysis in an orthotropic medium, Comput. Mech. 22:418-424 (1998) [10] A.P.S. Selvadurai Crack-bridging in a unidirectionally fibre-reinforced plate, J. Eng. Math. 68:5-14 (2010) [11] M. Perelmuter Boundary element analysis of structures with bridged interfacial cracks, Comput. Mech. 51:523-534 (2013) [12] J.R. Rice Elastic fracture mechanics concepts for interfacial cracks, Trans ASME J. Appl. Mech. 55:98103 (1988) [13] G. Balanford, A. Ingraffea, J. Liggett (1981) Two-dimensional stress intensity factor computations using the boundary element method, Int. J. for Numer. Meth. Eng. 17:387-404 [14] C.A. Brebbia, J.C.F. Telles and L.C. Wrobel Boundary Element Techniques - Theory and Applications in Engineering, Springer-Verlag, Berlin (1984) [15] J. Martinez, and J. Dominguez On the use of quarter-point boundary elements for stress intensity factor computations, Int J Numer. Meth. Eng. 20:1941-1950 (1984) [16] M. Perelmuter Nonlocal criterion of bridged cracks growth: Weak interface. J. Eur. Ceram. Soc. (2014), http://dx.doi.org/10.1016/j.jeurceramsoc.2014.01.033

Fig. 4 Circular fully bridged interfacial crack. The SIF modulus vs relative bonds stiffness. Variation of the elastic modules of materials.

Fig. 5 Circular bridged interfacial crack. The SIF modulus vs the bridged zone relative length.


295

Radon-Stroh formalism for 3D theory of anisotropic elasticity Federico C. Buroni1,† & Mitsunori Denda2,‡ †

School of Engineering, University of Seville, Camino de los descubrimientos s/n, E41092 Seville, Spain ‡ Mechanical and Aerospace Engineering Department, Rutgers University 98 Brett Road, Piscataway, New Jersey 08854-8058, U.S.A 1 [email protected], 2 [email protected]

Keywords:3D anisotropic elasticity; Stroh formalism; Radon transform; fundamental solution.

Abstract. In this paper we present a formalism for the solution of three-dimensional (3D) boundary value problems for general anisotropic solids combining the Radon transform and the Stroh formalism. The 3D problem is first reduced to a two-dimensional (2D) transformed space by the Radon transform. This resulting 2D problem is solved using the well-known Stroh formalism. Finally the 3D solution of the original problem is recovered by applying the inverse Radon transform. This Radon-Stroh formalism provides the general framework and solution technique for the most general boundary value problems in 3D anisotropic elasticity. As application example the solution for a point force in an infinite medium is considered; and the veracity of the approach is demonstrated with numerical results. Introduction In 1998, Wu [1] presents a very attractive scheme to treat general 3D problems by combining Radon transform and the Stroh formalism. This method has been applied to obtain the fundamental solution and a half-space Green’s function for a point force, and the solution for a planar dislocation loop in an infinite medium but no numerical result has been reported. Despite its tremendous potential, this formulation presented 15 years ago has remained very few explored. To the authors’ knowledge Wu’s approach was only used to solve the elliptical crack problem in [2]. The aim of this work is, starting from the constitutive law, to present an alternative derivation to [1] and validate the formulation with a numerical example. Application of this approach to the analysis of heat transfer problems has been presented by the authors in last BETEQ [3] (see also the extended version [4]). Reduction of the 3D physical space to a 2D Radon space Along the paper, it is assumed that Greek subindices range from 1 to 2 and Latin subindices from 1 to 3 unless otherwise is explicitly stated. Moreover, repeated indices imply summation. The differential ∂ ∂ ∂ is preferentially used; while the symbols ∂p and ∂s are reserved for ∂p and ∂s , operator ∂i := ∂x i respectively. Consider a function f (x) defined in a (xi ) (i = 1, 2, 3) Cartesian coordinate system in 3D. For the fixed value x3 , the 2D Radon transform in the plane with normal x3 is defined by [5] f (x) dSx , (1) fˇ(n, p; x3 ) = n·x=p

, 0)T ,

with T denoting transpose, is a unit vector and Sx is a line perpendicular to n where n = (n1 , n2 located at a (signed) distance p from the origin in the plane with normal x3 . The equilibrium equations in terms of the displacements um for a homogeneous linearly elastic solid in absence of body forces are (2) Ljm(∂x )um (x) = 0, where the differential operator Ljm (∂x ) := cijmn ∂i ∂n is defined in terms of the stiffness tensor cijmn which satisfies the usual symmetries and it is positive definite. Let now consider the constitutive law in terms of the displacement field such as the stress tensor σij is split as σij (x) = (cijkα ∂α + cijk3 ∂3 )uk (x).

(3)

296


Applying Radon transform (1) to equation (3) yields σ ˇij (n, p; x3 ) = (cijkα nα ∂p + cijk3 ∂3 )ˇ uk (n, p; x3 ).

(4)

Then, tractions on a surface with normals n = (cos θ, sin θ, 0)T and m = (0, 0, 1)T are

and

uk , σ ˇij ni = (Qjk ∂p + Rjk ∂3 )ˇ

(5)

uk , σ ˇij mi = (Rkj ∂p + Tjk ∂3 )ˇ

(6)

respectively, where the dependence of the functions in the transformed space has been omitted for simplicity, and matrices Qjm = cijmn ni nn ,

Rjm = cijmn ni mn ,

Tjm = cijmn mi mn ,

(7)

have been introduced. Note that Qjm and Tjm are symmetric and positive-definite guaranteeing the existence of their inverses. It can be deduced that the tractions on planes with normal vectors n and m can be obtained from σij ni = −mi ∂i χj and σij mi = ni ∂i χj , where χj are the components of the stress function. These are the directional derivatives of the stress functions, i.e., in the n- and m-directions. Hence, equations (5) and (6) can be rewritten as

and or in matrix form

−Q −RT

ˇj = (Qjk ∂p + Rjk ∂3 )ˇ uk , −∂3 χ

(8)

ˇj = (Rkj ∂p + Tjk ∂3 )ˇ uk , ∂p χ

(9)

0 I

ˇ ∂p u ˇ ∂p χ

=

R I T 0

ˇ ∂3 u ˇ ∂3 χ

,

where (I)ij = δij , and 0 the 3 × 3 null matrix. With the identity [6] R I I 0 0 T−1 = , I −RT−1 T 0 0 I equation (10) becomes

ˇ = ∂3 ω, ˇ N∂p ω

ˇ is defined by where vector ω

ˇ := ω

ˇ u ˇ χ

(10)

(11)

(12)

,

(13)

and N is a 6 × 6 matrix called the fundamental elasticity matrix [7] whose coefficients ∈ R and it is defined by N1 N2 , (14) N := N3 NT1 where

N1 = −T−1 RT ,

N2 = T−1 ,

N3 = RT−1 RT − Q.

(15)

Equation (12) is a matrix differential equation for the generalized plane strain problem, that is, the three components u ˇm depend on the coordinates p and x3 only (see Figure 1) [6]. This matrix transforms ˇ T with respect to the n-direction to obtain its derivatives in the derivatives of the dual vector (ˇ u, χ) ˇ, χ ˇ and the elastic constant cijkl for computing N the m-direction. It is important to point out that u are all referred to the global (xi ) Cartesian coordinate system. Next, consider the right-handed orthonormal triad (n, m, ˆ e) with associated coordinates p, x3 and s, respectively (see Figure 1). Stroh’s method seeks for displacement solutions, and hence strains and stresses, that are independent of x · ˆ e. That is, they depend on the projections of x on the plane


297

Figure 1: Sketch of the right-handed orthonormal triad (n, m, ˆ e) and the reference plane. spanned by the two orthogonal unit vectors n and m. Such a plane is called the reference plane. Eshelby et al. [8] note that the elastic field can be represented by means of an holomorphic function ψ(z) with suitable domain of definition, where the argument z → n · x + λm · x is a linear combination of a two-dimensional position on the reference plane. Thus, u ˇk = ak ψ(n · x + λm · x),

(16)

and the corresponding Airy stress function vector χ ˇk = bk ψ(n · x + λm · x),

(17)

are solutions of the two-dimensional elastic problem once ak , bk and λ are appropriately selected. It is easy to show that the stresses σ ˇij referred to the (xi ) Cartesian coordinate system are obtained as

and

˙ · x + λm · x), σ ˇ3j = bj ψ(n

(18)

˙ · x + λm · x), σ ˇ2j = cj ψ(n

(19)

˙ · x + λm · x), σ ˇ1j = dj ψ(n

(20)

where the dot over the function ψ denotes derivative with respect to its argument and the bj , cj and dj vectors are related to ak through

and

1 bj = (Rkj + λTjk )ak = − (Qjk + λRjk )ak , λ

(21)

cj = (c2jkα nα + λc2jk3 )ak ,

(22)

dj = (c1jkα nα + λc1jk3 )ak .

(23)

Note that aj and bj are the components of the well-known Stroh’s eigenvectors, instead, cj and dj are herein introduced in order to recover all the components of the stress tensor in the 3D solution. Substitution of the solutions (16) and (17) into (12) leads to the well-known six-dimensional eigenproblem [7] Nξα = λα ξα , (24) where vector ξα (α = 1, . . . , 6) is defined by ξα := (aα , bα )T . Eshelby et al. [8] have proved that λα (α = 1, . . . , 6) can not be real if the strain energy is positive. Then, the three complex λα (α = 1, 2, 3) with positive imaginary part are called the Stroh’s eigenvalues. The other three are the conjugate of ¯ α (α = 1, 2, 3) with overbar denoting complex conjugate. akα and bkα are the the remainder λα+3 = λ ¯kα , k-components of the Stroh’s eigenvectors corresponding to λα (α = 1, 2, 3). In addition, akα+3 = a bkα+3 = ¯bkα , ckα+3 = c¯kα and dkα+3 = d¯kα (α = 1, 2, 3). Therefore, assuming that λα (α = 1, 2, 3)

298


are distinct, the general solutions in real-form are obtained by superposition of the six solutions of the form (16) to (20) as 3 kα akα ψα (n · x + λα m · x) , (25) u ˇk = 2 α=1

χ ˇk = 2

3

kα bkα ψα (n · x + λα m · x) ,

(26)

α=1

and σ ˇik = 2

3 α=1

(1)

(2)

(i)

kα βkα ψ˙ α (n · x + λα m · x) ,

(27)

(3)

being βkα = dkα , βkα = ckα and βkα = bkα . For a particular problem all one has to do is to choose suitable holomorphic functions ψ and to determinate the complex unknowns kα in order to satisfy the boundary conditions of the problem under consideration. Of course, due to the symmetry of the stress tensor it is not necessary compute all the components. The potential and very promising feature of the presented formalism is indeed that the powerful Stroh formalism can be directly applied to solve the problem in the transformed space. Inversion procedure to the 3D physical domain The solution of the original 3D problem is obtained by applying the inverse 2D Radon transform. Let fˇ(n, p; x3 ) a general function in the Radon space, f (x) is recovered as [5] π +∞ ∂p fˇ(n, p; x3 ) 1 − dp dθ, (28) f (x) = − 2 2π 0 −∞ p−n·x where the inner π integral must be understood in the sense of Cauchy principal value as denoted by −. Let B(·) := 0 · dθ the Backprojection integral operator, and 1 +∞ f (p) H (f (p); p → t) := − dp, (29) π −∞ p − t the Hilbert transform of the function f (p); inversion formula (28) can be rewritten as π 1 ˇ(n, p; x3 )). H (∂p fˇ(n, p; x3 ); p → n · x) dθ = 2B(f¯ f (x) = − 2π 0 1 H (∂p f (p); p → t) has been introduced. where f˜(t) := − 4π ˜ ˇk (n, p; x3 )), where Consider now the inverse of the solution (25) as uk = 2B(u 3 +∞ ˙ ψα (ζα ) 1 ˜ dp , kα(θ) akα(θ) − u ˇk (n, p; x3 ) = − 2 2π −∞ p − n · x

(30)

(31)

α=1

where the dot over ψ denotes derivative with respect to its argument ζα → p + λα x3 . First the integration in (31) can be analytically carried out with some elementary techniques on Cauchy-type integrals [9]. Second step consists in perform the Backprojection where, in general, the integration must be numerically obtained. Thus, a solution for the k-component of the displacement is provided as 3 sgn(x3 ) ˙ B , (32) kα(θ) akα(θ) ψα(ζα )|ζα =nβ xβ +λα x3 uk (x) = π α=1

where the symbol (·) is used to denote imaginary part. In the same way, from equations (27) the solutions for the stress tensor are written as 3 sgn(x3 ) (i) kα(θ) βkα(θ) ψ¨α(ζα )|ζα =nβ xβ +λα x3 B . (33) σik (x) = π α=1 .


299

Table 1: Component u1 of the displacement evaluated at the point x = (1.6472193, 0.258202, −0.1414213) when a point force q = (1, 0, 0)T is applied at the point x = (0, 0, 0). 5.826441180703367 × 10−10 [m] Buroni et al. [10] 5.826441180703305 × 10−10 [m] Present work; 364 Gauss points 5.826441158773066 × 10−10 [m] Present work; 240 Gauss points 5.825911034132442 × 10−10 [m] Present work; 120 Gauss points 5.782000288924871 × 10−10 [m] Present work; 64 Gauss points 6.231541356975483 × 10−10 [m] Present work; 32 Gauss points

Application to the fundamental solution It is well-known that the 2D solution for a line force singularity δ(p)δ(x3 ) of strength q is obtained by 1 log(n · x + λx3 ) and kα = ajα qj in equations (25) and (27) [6]. Then, from the selecting ψ as ψ = 2πi previous section the corresponding 3D point force fundamental solutions for displacements and stresses are 3 −sgn(x3 ) 1 B ajα qj akα (34) uk (x) = 2π 2 n · x + λα x3 α=1

and

3 sgn(x3 ) 1 (i) B ajα qj βkα σik (x) = 2 2π (n · x + λα x3 )2

,

(35)

α=1

respectively, with q = (1, 0, 0)T , q = (0, 1, 0)T or q = (0, 0, 1)T . Consider a material with the following elastic constants [GPa]: c1111 = 1.1484, c1112 = −0.0514, c1113 = −0.1028, c1122 = 0.7578, c1123 = −0.1343, c1133 = 0.5562, c2222 = 1.2109, c1212 = 0.1828, c1213 = −0.0468, c1222 = −0.0027, c1223 = 0.0162, c1233 = −0.1407, c1333 = −0.1515, c1322 = −0.1353, c1323 = −0.0649, c2223 = −0.0031, c2233 = 0.7187, c2333 = −0.0875, c3333 = 1.0749, c2323 = 0.1875, c1313 = 0.1125. The proposed fundamental solutions have been checked with the implementation by Buroni et al. [10] and, in indeed, excellent agreement is obtained. Table 1 shows a comparison of the present work and the implementation by Buroni et al. [10] for the component u1 of the displacement evaluated at the point x = (1.6472193, 0.258202, −0.1414213) when a point force q = (1, 0, 0)T is applied at the point x = (0, 0, 0). Various Gauss points are considered in the standard numerical Gauss quadrature to perform the integral in the Backprojection operator. The stress tensor evaluated at the same point when a point force q = (1, 0, 0)T is applied at the origin is ⎛ ⎞ −0.22624622796011082 −0.027551525839378047 0.01879431837194627 ⎝ σ= −0.027551525839378147 −0.05922673495153985 0.02226097681318537 ⎠ . (36) 0.018794318371946252 0.022260976813185285 −0.00756968406806467 (1)

(2)

(3)

In this case 240 Gauss points have been used. Note that since βkα , βkα and βkα have been used for computing all components of the stress tensor an inevitable lack in the symmetry due to numerical errors is observed. Conclusions A derivation of the formulation proposed by [1] for the analysis of 3D anisotropic elastic materials has been presented. By applying the Radon transform to the constitutive law in term of the displacements and considering dual coordinate systems the 3D problem is reduced to a generalized plane strain one. This resulting 2D problem can be solved by using the well-known Stroh formalism. In order to recover the full 3D stress tensor two new vectors have herein proposed in equations (22) and (23). Then, the 3D solution of the original problem is recovered by applying the inverse Radon transform. A key advantage when compared with others methods is that the transformed space corresponds to a

300


physically meaningful 2D problem. Then, any existing Stroh solution can be pulled-forward to its corresponding 3D solution. As application example the fundamental solution has been considered. Numerical results validate the formulation. Application of this formalism to more general problems will be presented in a forthcoming paper. Acknowledgment This work was partially supported by the Ministerio de Ciencia e Innovación of Spain and the Consejería de Innovación, Ciencia y Empresa of Andalucia (Spain) under projects DPI2010-21590-C02-02 and P12-TEP-2546. F.C. Buroni would like to acknowledge to University of Seville the financial support through "Ayuda para la Movilidad de Profesores Ayudantes y Ayudantes Doctores del IV Plan Propio" for a two months stay at Rutgers University where this work has been done. References [1] K.C. Wu, Generalization of the Stroh formalism to 3-Dimensional anisotropic elasticity. Journal of Elasticity 51, 213-225 (1998). [2] K.C. Wu, On an elliptic crack embedded in an anisotropic material. International Journal of Solids and Structures, 37, 4841-4857 (2000). [3] F.C. Buroni, A. Sáez, M. Denda, Derivation of 3D anisotropic heat conduction Green’s functions using a 2D complex-variable method, In Proceedings of the 14th International Conference on Boundary Element & Meshless Techniques, BETEQ, A. Sellier, M.H. Aliabadi (eds). Paris, 2013 [4] F.C. Buroni, R.J. Marczak, M. Denda, A. Sáez, A formalism for anisotropic heat transfer phenomena: Foundations and Green’s functions. International Journal of Heat and Mass Transfer, (2014) http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.03.073 [5] I. M. Gel’fand, M. I. Graev, N. Ya. Vilenkin, Generalized Functions, Vol. 5, Academic Press, New York, 1966. [6] T.C.T. Ting, Anisotropic Elasticity. Theory and Applications. Oxford University Press, New York, 1996. [7] K. Ingebrigtsen, A. Tonning, Elastic surface waves in crystals, Physical Review, 184, 942-951 (1969). [8] J. D. Eshelby, W. T. Read, and W. Shockley, Anisotropic elasticity with applications to dislocation theory. Acta Metallurgica, 1, 251-259 (1953). [9] N. I. Muskhelishvili, Singular Integral Equations. Boundary Problems of Function Theory and their Application to Mathematical Physics. (J. R M. Radok, trans.) Noordhoff, Groningen, Holland, 1953. [10] F.C. Buroni, J.E. Ortiz and A. Sáez, Multiple pole residue approach for 3D BEM analysis of mathematical degenerate and non-degenerate materials. International Journal for Numerical Methods in Engineering 86, 1125-1143 (2011).


301

Three-dimensional linear elastic boundary element method with direct evaluation of singular integrals Cristiano J. Brizzi Ubessi1, Rogério José Marczak2 Universidade Federal do Rio Grande do Sul, Mechanical Engineering Department Rua Sarmento Leite 425, 90430-131, Porto Alegre, RS, Brasil 1

[email protected] 2

[email protected]

Keywords: Boundary element method, singular integrals, regularization, three-dimensional elasticity.

Abstract. This work presents the formulation and implementation of the boundary element method (BEM) to three dimensional linear elastostatics, with the direct evaluation of the strongly singular integral equations. The implementation follows with the traditional direct BEM formulation, and the discretization of the boundary is carried out with discontinuous elements, avoiding corner singularities, and enabling the use of unconnected meshes along the surfaces. The computation of the singular integral equations is accomplished by using the asymptotic expansions derived around a generic singular point. The analytical expressions for these expansions are presented in the article. The expansions are subtracted from the kernel to regularize it. This subtracted part is then added by computing a regular line integral along the boundary of the element. Both the integrals can be calculated with Gauss-type quadratures. It is observed that the present method need less points of integration for the same level of error when compared with other techniques. Several elasticity benchmarks are solved to demonstrate the efficiency and the accuracy of the present method. Introduction The three-dimensional boundary element method (BEM) has unique advantages for the mechanical engineering application, since the dimensionality the problem is reduced by one, and the accuracy of the solution is very high, due to it’s formulation. As a downside one should cite the fully populated matrices and the computational cost associated with computing the near- and strongly-singular integrals. Over the years all kinds of approaches has been done in order to compute BEM integrals with accuracy and efficiency. The less generalist methods rely on analytic integration, prohibiting the use of high order elements, and the most generalist use variable transformations over the Gauss-Legendre quadrature rule [1, 2, 3, 4], or special purpose quadratures [5, 6], both having to be computed during the preparation of matrices, and requiring significant number of integration points. [7], proposed a direct method based on singularity subtraction, more generalized and efficient than the last ones. Besides being very well explored on various fields of engineering, on 3D elasticity, [8] applied this method with continuous elements, didn’t presented convergence data, and examples were limited to infinite bodies. Recently, in the field of acoustics, [9], developed coordinate transformations that improved significantly the method’s efficiency, and with the benefit of curved high order discontinuous elements. On this paper the Direct Method described in [7] is implemented, in order to evaluate Its applicability and efficiency on linear 3D elasticity problems. To simplify the integration, discontinuous elements are used, which could also simplify the meshing. A standard implementation of BEM with cubic transformation (CT) [3] on the weakly singular integration and rigid body displacements (RBD) for the strongly singular terms is also implemented previously in order to compare with the Direct Method against it. The paper is organized as follows. The boundary for linear elastic is reviewed in section 2. On section 3 the Direct Method is then reviewed and the asymptotic expansions for 3D elasticity are described, as well as the method implementation, and some preliminary results obtained on the numerical integrations. Finally, on section 4, the benchmark tests are described and the results are presented. On section 5 the paper is closed with discussions and conclusion.

302


Boundary Integral Equations on Elasticity and BEM The BEM formulation is derived from the elasticity equilibrium equations, and a very didactic review could be found on [10]. The main boundary integral equation (BIE) from which the BEM comes is known as the 'Somigliana's identity':

(1) where y is a source point inside the domain, and x is the field point on the boundary, where the displacements and traction are already known. and are the fundamental solutions that give the displacement or traction in the direction k on point x on the boundary of an infinite body, when a unit load is applied in the direction l on point y, inside of this body. When the source point is taken to the boundary of the body, a limit has to be taken, which leads to a free term , that will depend on the shape of the boundary at y, resulting on the following BIE: (2) where, in the case of this work, where the boundary at y is always smooth, due to the use of discontinuous elements, . In order to obtain a numerical solution from Eq. (2), the body boundary must be discredited in a finite number of elements, to form a linear system of equations. The geometry of the body will be calculated by means of shape functions, so that the coordinates x lying inside an element j could be calculated in terms of the intrinsic coordinates : (3) where N are the number of nodes of the element, and is a vector containing the shape functions for each node evaluated at . Since discontinuous elements are used, the physical variables of the problem will be calculated in nodes recessed by a factor to the center of the element. Displacements and traction are interpolated with discontinuous interpolation functions, according to [11], resulting into: (4) Representing in matrix notation, and adding the Jacobian of the area transformation J (or transformation), the final equation for the boundary element method is then:

for the volume

(5)

where the summation over j states for all the elements, and over s states for all the domain cells, as already stated, when the node belongs to the element, otherwise it is equal to zero. Formulations for the free term coefficients , where the node is not on a smooth surface could be found on [12]. Further combination of the terms computed with Eq. (5), and boundary conditions, result in the system of equations that result in the BEM solution.


303

Direct Method One of the purposes of this work is to unify the evaluation of both weakly- and strongly-singular with the Direct Method, in order to compare the results with the ones obtained through CT and RBD method. All the procedure that arise on final formulation of this method is well described in [7,13,14]. The main procedure on this method is to isolate the singularity of the kernel containing the fundamental solution and interpolation function, by mean of an asymptotic expansion such that: (6) and then subtract it from the original kernel, integrate the singular part analytically performing a limit in the radial direction , resulting in the following integrals:

(7) where and are parameters of the exclusion region equation, where the limit was performed, and is the radius of the element edges as a function of (fig. 1a). To perform this integral numerically, one has to transform the local to a polar coordinate system, centered on , so that , and , resulting on a subdivision scheme in the square element, that could be seen on fig. 1b.

Figure 1: (a) variables of the direct method and (b) subdivision of the element in four triangles The complete formulation needed to transform the standard Gaussian quadrature to a polar coordinate system can be found on [15]. As stated by [7], for the displacement kernel, that contains , the asymptotic expansions are and , which means that this kernel is regular if evaluated over polar coordinates, otherwise it will be weakly singular. In the case of the traction kernel, containing , the asymptotic expansion are: , as already shown in [8], and is: (8) where is the vector that represent the first term of the Taylor series expansion of the radius, and of this vector:

is the norm

(9) and

, is the first term of the Taylor series expansion of the interpolation function at the a node.

304


Numerical Verification. In order to analyze the regularization procedure, one generic element, with a discontinuity factor of 15%, with the first node of the element as the source and field point, located at . The element used was a linear element with 4 nodes, lying on the plane with dimension 1x1m. In the case of RBD the mesh used to compute indirectly the tensor was a unit cube with 6 equal unit square elements. On fig. 2 convergence on the integration of the term is demonstrated with both methods implemented. The Direct Method proves to be faster on the convergence, and besides the considerable number of integration points, almost reaches the machine precision.

Figure 2: Results of numerical integration and error relative to the last result obtained with direct method.

According to [16], polar integration schemes need a larger number of integration points on elements with large aspect ratios, because singularities arise from the little distance between the source point and the edges of the element. It is proposed to verify the topological consistency of the Direct Method comparing it with the results presented by the author. To perform this analysis, the term of the element was evaluated varying one of the sides of the element. The integration was carried out with various sets of points from 2 to 64 points on both angular directions of each pair of opposite triangles of the element: is the number points on the two triangles that the angle were reduced, and is the number of points on the triangles where the local angle was raised. On table 1 the minimum number of points needed to obtain a relative error of are given, and some results obtained by [16] are also presented for a matter of comparison. It was observed that on the radial direction, even on the most stretched element, there was not considerable precision gain on raising the number of points.

Table 1: Minimum integration points to obtain an error minor than on numerical integration of tensor on a unitary element with both source and field points on the same node.

Results To evaluate the efficiency of the methods implemented on actual engineering problems, some benchmark tests were proposed. First, a cantilever beam under a constant shear tension, then, a plate under uni-axial stress with a round hole. The data used on both analysis were and .


305

Figure 3: Benchmarks analysed: (a) Cantilever beam with fixed end and under shear tension on opposite end and (b) rectangular plate with hole under uni-axial tension. Beam. The beam length was along the x axis and a square and constant cross section on the zy plane with side . The constant shear tension applied on the end of the beam was . Since this beam has a short length in terms of it's cross section height, the analytical solution given by the Euler-Bernoulli theory is not accurate enough, so in this case it was opted to use a finite element method (FEM) analysis as a reference for the error. On fig. 4 are plotted the results of maximum displacement, on the shear stressed end, and on fig. 5, the maximum stress, on the fixed end of the beam, for meshes from 22 to 530 elements, which represent from 264 to 6360 degrees of freedom (DOF). The program with CT and RBD is referred as CT&RBD, and the implementation with the direct method referred as Direct.

Figure

4:

Maximum displacements and error relative to the FEM solution for the studied beam.

Figure Maximum stress and error relative to the FEM solution for the studied beam.

5:

306


Plate. The thickness of the plate analyzed was , the bulk modulus and Poisson coefficient were same as the beam, as well as the value of the traction applied on the plate sides . As a first analysis with this problem, the recess of the elements was tested, since there is no standard value to be used, and both possibilities have it's own advantages, a lower recess should give less errors of extrapolation, and a larger recess could enhance the precision on the near-singular integration (between different elements) and polar coordinate integration. Considering those possibilities, four values of recess were suggested to be analyzed from 10% to 25\%. On table 2 the results of maximum stress and error relative to an analytical solution are given. The solution with a very fine mesh with FEM is also considered on this evaluation. The values obtained with the standard (CT&RBD) integration were closer to the analytical solution, while the Direct method suffers from the variation of the element recess. The FEM yielded the highest error in relation to the analytical solution. The Direct Method obtained larger error with less recess, what could be addressed to the near singularities arising when the source point is too close to the element edge. When the element recess is risen, the error of the Direct method approaches to the standard BEM error.

Table 2: Maximum stress and error relative to the analytical solution at , on the plate with thickness , for different BEM discontinuities, and 3D FEM.

In order to verify the distribution and the stress concentration on the symmetry plane, on fig. 6 are given the stress and error relative to the FEM solution, on the plate with thickness . The Direct method yielded a larger error to the FEM solution than CT&RBD, but with less variation between each element.

Figure Stresses and relative error to the FEM solution, at on the plate with thickness .

6:


307

Conclusion On this paper the implementation of the standard BEM formulation, with discontinuous elements, as well as the Direct method for evaluation of singular integrals was discussed. The results of the regularization and integration of the and kernels has shown its efficiency. The Direct Method performs an actual regularization of the BEM singularities, subtracting them. On the other hand, other kinds of difficulties appear on the numerical integration of the regularized kernel on the angular direction, suggesting to further researches on transformation techniques that could improve it's efficiency. The benchmark test results has proven that the BEM implementation with Direct Method is as accurate as the traditional BEM in some engineering problems, and that it yield consistent results as well. Acknowledgments The first author wish to express his thanks to CAPES for the financial support. References [1] N.I. Ioakimidis. One more approach to the computation of Cauchy principal value integrals. Journal of Computational and Applied Mathematics, 7(4):289 – 291, (1981). [2] (1981).

D.F. Paget. A quadrature rule for finite-part integrals. BIT Numerical Mathematics, 21(2):212–220,

[3] J. C. F. Telles. A self-adaptive co-ordinate transformation for efficient numerical evaluation of general boundary element integrals. International Journal for Numerical Methods in Engineering, 24(5):959–973, (1987). [4] K. Hayami and C. Brebbia. Quadrature methods for singular and nearly singular integrals in 3-d boundary element method, (invited paper). In Boundary Elements X, Proc. 10th Int.Conf. on Boundary Elements, (1988). [5] H.R. Kutt. The numerical evaluation of principal value integrals by finite-part integration. Numerische Mathematik 24(3):205–210, (1975). [6] P. Linz. On the approximate computation of certain strongly singular integrals. Computing, 35(34):345–353, (1985). [7] M. Guiggiani and P. Casalini. Direct computation of cauchy principal value integrals in advanced boundary elements. International Journal for Numerical Methods in Engineering, 24(9):1711–1720, (1987). [8] O.J.B. Almeida Pereira and P. Parreira. Direct evaluation of cauchy-principal-value integrals in boundary elements for infinite and semi-infinite three-dimensional domains. Engineering Analysis with Boundary Elements, 13(4):313 – 320, (1994). [9] J. Rong, L. Wen, and J. Xiao. An efficient method for evaluating bem singular integrals on curved elements with application in acoustic analysis. CoRR, abs/1306.0282, (2013). [10]

C. A. Brebbia and J. Dominguez. Boundary Elements - An Introductory Course. WIT Press, (1989).

[11] G. Beer, I. Smith, and C. Duenser. The Boundary Element Method with Programming. Springer Wien, New York, (2008). [12] V. Mantic. A new formula for the c-matrix in the somigliana identity. Journal of Elasticity, 33(3):191– 201, (1993). [13] M. Guiggiani and A. Gigante. A general algorithm for multidimensional cauchy principal value integrals in the boundary element method. Journal of Applied Mechanics, 57(4):906–915, December (1990).

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[14] M Guiggiani. Formulation and numerical treatment of boundary integral equations with hypersingular kernels. In J. Sladek V. Sladek, editor, Singular integrals in boundary element methods, volume 3 of Advances in Boundary Elements, pages 85–124. Computational Mechanics, (1998). [15] Koizumi and Utamura. A polar coordinate integration scheme with a hierarchical correction procedure to improve numerical accuracy on the boundary element method. Computational Mechanics, 7:183–194, (1991). [16] K. Hayami and H. Matsumoto. Improvement of quadrature for nearly singular integrals in 3d-bem. Boundary Element Method, XIV:201–210, (1994). [17]

W.D. Pilkey and D.F. Pilkey. Peterson’s Stress Concentration Factors. Wiley, (2008).


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Nk (x)

α

Fα (x)aαk

,

N N CT $ Ni 8 $ i ui aαk $ $ $

$) * aαk 9. $ . $ u1 u2

$ . $ %2') * $ ) %2' -.("#: ⎞ ⎛ −1 −1 {A11 B11 β1 + A12 B21 β2 } −1 −1 √ ⎜ {A11 B12 β1 + A12 B22 β2 } ⎟ ⎟ ⎜ Fl (r, θ) = r ⎝ −1 −1 β1 + A22 B21 β2 } ⎠ {A21 B11 −1 −1 β1 + A22 B22 β2 } {A21 B12

;


311

√

βi = cos θ + μi sin θ r θ A B μ

−L−1 M Z − MT L−1 M

−L−1 −MT L−1

Am Bm

= μm

Am Bm

m

!"# $ %&' ( $ !)# $ * +%,& $%,& Z := C1ij1 ;

cij (ξ)uj (ξ) + cij (ξ)pj (ξ) + Nr

Γ

Γ

M := C2ij1 ;

p∗ij (x, ξ)uj (x)dΓ(x) +

s∗rij (x, ξ)uj (x)dΓ(x) + Nr

Γc

Γc

L := C2ij2

p∗ij (x, ξ)Fα (x)aαk dΓ =

s∗rij (x, ξ)Fα (x)aαk dΓ = Nr

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u∗ij (x, ξ)pj (x)dΓ(x)

-.

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: h/w = 2 2a " 3 θ θ = 0 θ = 45 $

312


a

2h w

3.06

Reference [7] X−BEM Dual BEM quarter−point X−FEM (topologic) X−FEM (geometric)

3.04

√ KI / πa

3.02

3

2.98

2.96

2.94 30

40

50

60

70

80

90

100

2a

2h

2w


a/w = 0.5

313

! "# "# $

! % ! &'( ) $

$ "# )

"#

1.21


1.205

1.2

√ KI / πa

1.195

1.19

1.185

1.18

1.175

1.17

1.165 30

40

*

50

60

70

80

90

100

+ ! #

θ = 0

, ) ! - ) ./0 ./0

! &1(

$ )

θ = 45

!

"# !

! !

"# $ $ )

0.65

0.555


0.645

0.55

0.64

√ KII / πa

√ KI / πa

0.545

0.635

0.63

0.54

0.535

Reference [9]

0.625

X−BEM 0.53

Dual BEM quarter−point

0.62

X−FEM (topologic) X−FEM (geometric) 0.525

0.615

0.61 30

40

50

60

70

80

90

100

0.52 30

40

50

.* + ! #

60

70

80

90

100

θ = 45

2 ) 3 !

!

) )

- ! ! "# ) +

314


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4A5 0 ! & , & / 2 '((( 4+5 6 #

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# $" &

& G )+AE

3- &


315

The Method of Fundamental Solutions coupled with a Genetic Algorithm to Optimize Cathodic Protection Systems in Infinite Regions W. J. Santos1, J. A. F. Santiago2 and J. C. F. Telles3 Department of Civil Engineering, COPPE/UFRJ. Rio de Janeiro – Brazil 1

[email protected], 2 [email protected], 3 [email protected]

Keywords: Optimization, nonlinear least squares, cathodic protection, MFS, BEM.

Abstract. The method of fundamental solutions (MFS) is used for the solution of Laplace’s equation, with nonlinear boundary conditions, aiming at analyzing cathodic protection systems of external problems within infinite regions (e.g. seawater). In the MFS procedure, it is necessary to determine the intensities and the distribution of the virtual sources so that the boundary conditions of the problem are satisfied. The metallic surfaces, in contact with the electrolyte to be protected, are characterized by a nonlinear relationship between the electrochemical potential and the current density, called cathodic polarization curve. Thus, the calculation of the intensities of the virtual sources entails a nonlinear least squares problem. For external potential problems, a term representing a constant potential at infinity should be added to the MFS solution. The inclusion of this unknown potential value makes it possible to impose a self-equilibrated current density between the anodes and cathodes of the problem. Here, first, an example is presented to validate the standard MFS formulation as applied to the simulation of cathodic protection systems over external problems in comparison with the result of a direct boundary element (BEM) procedure. Second, a MFS methodology is presented, coupled with a genetic algorithm (GA), for the optimization of anodes positioning and their respective current intensity values. 1. Introduction The most common methods for modelling cathodic protection (CP) systems are the finite element method (FEM) and the boundary element method (BEM). Even though recent FEM publications have successfully been used by some researchers [1,2], the BEM is still the most appropriate technique to solve problems involving CP systems, mainly to solve large domain problems in homogeneous conductive media. The BEM requires only the discretization of the anode and cathode surfaces, which leads to better accuracy and reduction of computer run time when compared to FEM. Several applications of BEM to study CP systems have been reported in the literature, including reference to practical analyses performed by offshore oil companies [3,4,5]. Usually, the BEM implementation includes a Newton-Raphson solution algorithm to accommodate possible nonlinear boundary conditions [6]. Just like BEM, the method of fundamental solutions (MFS) is applicable when a fundamental solution of the differential equation in question is known, with the advantage of not requiring any integration procedure or specific treatment for the singularities of the fundamental solution. In the MFS, the approximate solution of the problem is represented in the form of a linear superposition of fundamental solutions with singular points located outside the domain of the problem. These singular points are called virtual sources and form a “pseudo-boundary” having no interception with the actual boundary of the region. The essence of the MFS is the use of a fundamental solution which satisfies the associated homogeneous differential equation in any point, except at the source point not located within the actual problem domain. The unknown source intensities producing the approximate solution are determined by imposing satisfaction of the boundary conditions at a set of boundary points. The basic ideas for the formulation of the MFS were first proposed by Kupradze and Aleksidze [7]. The MFS has successfully been applied for solving several problems. For example, potential problems [8], nonlinear problems [9], acoustic problems [10] and crack problems [11]. In Santos et al. [12] a formulation using GA was proposed with the MFS to simulate cathodic protection systems with nonlinear boundary conditions. The adopted GA was used to minimize a nonlinear error function, whose design variables were the coefficients of the linear superposition of fundamental solutions and the positions of the virtual sources, randomly distributed outside the problem domain.

316


In addition, coupled with the numerical method to solve the Laplace equation, optimization algorithms can be used to determine the optimum location and the corresponding current intensity values of the anodes in order to satisfy the corrosion protection criterion. Hence, the minimization of an objective function using, for example, genetic algorithms (GAs) and a penalty method for handling constraints can be adopted. This type of optimization can and has also been successfully performed using BEM [13,14]. The purpose of the present paper is to present a formulation using a GA and MFS to determine the optimum location and the optimum current intensity of the anodes, inserted in the electrolyte, leading to a practical optimized design procedure. Problems with nonlinear boundary conditions solved by MFS can be treated as nonlinear least square problems. Therefore, to determine the coefficients of the linear superposition of the fundamental solutions and the positions of the virtual sources, a nonlinear least squares algorithm is found necessary. In this paper, the minimization of the nonlinear functional is done using the MINPACK [15] routine LMDIF, which is a modified version of the Levenberg-Marquardt algorithm. Examples of application are presented considering infinite regions in ܴଶ . The results presented here include a comparison with a direct boundary element (BEM) solution procedure [16]. 2. The MFS formulation for Poisson equation with nonlinear boundary conditions In order to design the CP system, one needs to know the expected current density ሺ݅ሻ or the electrochemical potential (߶) over the metallic surfaces. The mathematical model of the problem, within the conducting domain π (electrolyte), is based on a Poisson equation for the electrochemical potential: ݇ߘ ଶ ߶ ሺ࢞ሻ ൌ ܾሺ࢞ሻǡ ࢞ ‫ א‬πǡ (1) where ܾ is a known function representing the anodes as external sources and ݇ is the conductivity of the electrolyte. Over the metal surfaces in direct contact with the electrolyte, the boundary conditions for Eq (1) are of the following form ݅ ሺ࢞ሻ ൌ ‫ ܨ‬ሺ߶ሻǡ ࢞ ‫߁ א‬ǡ (2) where ߁ is the boundary of π, ݅ ሺ࢞ሻ is the current density in the outward normal direction ࢔ and ‫ ܨ‬ሺ߶ሻ is a nonlinear function of ߶. The general solution ሺ߶௚ ሻ of Eq (1) is given by adding a particular solution ሺ߶௣ ሻ to the solution of the associated homogeneous equationሺ߶௛ ሻ, subjected to the corresponding homogeneous boundary conditions. One technique for obtaining a particular solution of Poisson's equation is based on the Newton potential [17], which is given by the integral

߶௣ ሺ࢞ሻ ൌ ‫׬‬π ‫ ܩ‬ሺࣈǡ ࢞ሻܾሺࣈሻ ݀πǤ

(3)

The function ‫ ܩ‬ሺࣈǡ ࢞ሻ is a fundamental solution of Laplace's equation given by ଵ

ଵ

ሺ૆ǡ ‫ܠ‬ሻ ൌ ቀୖቁ ǡ ଶ஠୩ where ܴ is the Euclidean distance between point ࣈ and the field point ࢞.

(4)

Considering an impressed current anode, the external sources can be treated as point sources and the term ܾሺࣈሻ becomes equal to ௡೛ೞ ௣௦ ܾሺࣈሻ ൌ ෌௠ୀଵ ܲ൫࢞௣௦ (5) ௠ ൯ߜ൫࢞௠ ǡ ࣈ൯ǡ where ࢞௣௦ represent the coordinates of the point sources, ܲሺ࢞௣௦ ሻ is the intensity of the source, ߜ is the Dirac delta "function" and ݊௣௦ is the number of point sources inserted in the electrolyte. Therefore, the particular solution can be written in the following form


317

௡೛ೞ

௣௦ ߶௣ ሺ࢞ሻ ൌ ෌௠ୀଵ ܲ൫࢞௣௦ ௠ ൯ ‫ܩ‬൫࢞௠ ǡ ࢞൯Ǥ

(6)

In addition, from Ohm's law, the particular solution for a current density is equal to ݅௣ ሺ࢞ሻ ൌ ݇

డథ೛ డ௡

ଵ

ൌ െ ଶగ ෍

௡೛ೞ ௠ୀଵ

௣௦

ଵ డோ

ܲ൫࢞௠ ൯ ோ డ௡ .

(7)

The approximate solution of the associated homogeneous problem by MFS is represented in the form of a linear superposition of fundamental solutions with singular pointsሺ࢞௦௣ ሻ located outside the domain of the problem. Thus, the electrochemical potential can be considered as the following summation ௡ೞ೛

߶௛ ሺ࢞ሻ ൌ ෍

௝ୀଵ

‫ܩ‬ሺ࢞ǡ ࢞௝௦௣ ሻ ܿ௝ ǡ

(8)

with ݊௦௣ being the total number of singular points (virtual sources) and the coefficients ܿ௝ present in the approximate solution are the unknown source intensities. Similarly, defining ‫ ܪ‬ൌ ݇

డீ డ௡

, the homogeneous solution for the current density ሺ݅௛ ሻ is given as ௡ೞ೛

݅௛ ሺ࢞ሻ ൌ ෍

௝ୀଵ

‫ܪ‬ሺ࢞ǡ ࢞௝௦௣ ሻ ܿ௝ Ǥ

(9)

The idea of MFS is to determine the coefficients ܿ௝ by imposing satisfaction of the boundary conditions at certain boundary nodes. In the present work, the polarization curve is given by the expression [18]: ݅ ൌ ‫ ܨ‬ሺ߶ሻ ൌ ݁

ഝశలవయǤవభ ഁభ

ଵ

െ ൤௜ ൅ ݁

ഝశఱమభǤల ഁమ

భ

ିଵ

൨

െ݁

షሺഝశళబళǤఱళሻ ഁయ

ି

ǡ

(10)

with ߶ and ݅ having units ܸ݉ and ߤ‫ܣ‬Ȁܿ݉ ଶ, respectively, and ߚଵ ǡ ߚଶ ǡ ߚଷ e ݅ଵ are given constant parameters: ߚଵ ൌ ʹͶܸ݉, ߚଶ ൌ ʹ͵ǤͶ͹ܸ݉,ߚଷ ൌ ͷͷܸ݉ e ݅ଵ ൌ ͺ͸ǤͲ͸ߤ‫ܣ‬Ȁܿ݉ ଶ. The conductivity of the electrolyte is equal to ݇ ൌ ͲǤͲͶ͹ͻπିଵ ܿ݉ ିଵ and the critical value of the electrochemical potential is ߶௖ ൌ െͺͷͲܸ݉ሺ‫ݏݒ‬Ǥ ሻ. The general solution of the problem must satisfy Eq (10), i.e., ݅௚ ൌ ݅௣ ൅ ݅௛ ൌ ‫ܨ‬൫߶௣ ൅ ߶௛ ൯ ൌ ‫ܨ‬൫߶௚ ൯.

(11)

The relationship given by Eq (11) generates a nonlinear least square problem in which the design variables are the coefficients ܿ௝ and the positions of the virtual sources. In least square problems, the objective function ݂ has the following special form ଵ

௡್೏ ଶ ‫ݎ‬௝ ሺࢉǡ ࢞ ௦௣ ሻ, ݂ሺࢉǡ ࢞ ௦௣ ሻ ൌ σ௝ୀଵ (12) ଶ where ݊௕ௗ is the number of boundary nodes, ࢉ is a vector containing the coefficients ܿ௝ Ԣ‫ ݏ‬and each ‫ݎ‬௝ is a smooth function referred to as a residual given by ௝ ௝ ‫ݎ‬௝ ൌ ݅௚ െ ‫ܨ‬൫߶௚ ൯ǡ ݆ ൌ ͳǡ ǥ ǡ ݊௕ௗ Ǥ

(13)

Considering the arrangement of the virtual sources on a contour geometrically similar to the actual boundary of the region under consideration, it is only necessary to search for the constant distance ߩ from the virtual sources to the problem boundary. Thus, design variables of Eq (12) are the coefficients ܿ௝ and a distance ߩ.

318


In this paper, the minimization of the functional of Eq (12) is carried out using the MINPACK routine LMDIF, which is a modified version of the Levenberg-Marquardt algorithm. In LMDIF, the Jacobian is evaluated internally by finite differences. LMDIF terminates when either a user-specified tolerance is achieved or the user-specified maximum number of function evaluations is reached. 3. MFS for external problems In external problems, the equation systems for BEM and MFS can be solved producing a solution that does not necessarily ensure conservation of current between the anodes and cathodes. This property can be included into the formulation through the satisfaction of ௡೛ೞ

௣௦

‫׬‬௰ ݅ ሺ‫ݔ‬ሻ݀߁ሺ‫ݔ‬ሻ ൌ െ ෌௠ୀଵ ܲ൫࢞௠ ൯ǡ for the Poisson equation with point sources of intensity ܲ.

(14)

Following [19], in simulations of cathodic protection systems for external problems using BEM, a constant potential at infinity ሺ߶ ஶ ሻ should be added in the boundary integral equation. Thus, the extra condition, Eq (14), can be employed to complete the system of equations of the BEM. In the case of MFS, the general solution is the contribution of the virtual sources plus the contribution of the real point sources. Therefore, with the inclusion of the extra unknown ߶ ஶ , the MFS general solution becomes ߶௚ ሺ࢞ሻ ൌ ෍

௡ೞ೛ ௝ୀଵ

௡

೛ೞ ௣௦ ஶ ‫ܩ‬ሺ࢞ǡ ࢞௝௦௣ ሻ ܿ௝ ൅ ෌௠ୀଵ ܲ൫࢞௣௦ ௠ ൯ ‫ܩ‬൫࢞௠ ǡ ࢞൯ ൅ ߶ Ǥ

(15)

To ensure the conservation of current between anodes and cathodes in the MFS formulation, the following relation needs to be considered ௡

௡

೛ೞ ௣௦ ೞ೛ σ௝ୀଵ ܿ௝ ൌ െ ෌௠ୀଵ ܲ൫࢞௠ ൯

(16)

In cathodic protection systems, Eq (16) will ensure that the structure to be protected becomes the cathode of the electrolytic cell while the virtual sources and point sources have an anodic behaviour. For external problems with nonlinear boundary conditions solved by MFS, it is necessary to include Eq (16) in the objective function ݂ of the least square problem, Eq (12). Furthermore, the design variables will be the coefficients ܿ௝ , the distance from the virtual to the problem boundary ߩ and the extra term ߶ ஶ . 4. GA for optimization of cathodic protection In order to guarantee effective cathodic protection the electrochemical potential distribution over the interface metal electrolyte should be kept, as uniform as possible, close to the critical potential (߶௖ ) value: ߶ ൑ ߶௖ Ǥ The optimum location and the optimum current intensity delivered by the anodes have to be determined in such a way as to satisfy the protection criterion. Hence, to satisfy the protection criterion with the minimum possible power input, it is necessary to minimize the following objective function: ܼ൫࢞௣௦ ǡ ܲሺ࢞௣௦ ሻ൯ ൌ ට

ଵ ௡್೏

௡್೏ σ௜ୀଵ ሾ߶ ௜ െ ߶௖ ሿଶ ൅ ‫ݒ‬ට

ଵ ௡್೏‫כ‬

ଶ ௣௦ ್೏ σ௡ ሺ ௣௦ ሻሻ ௜ୀଵ ݇௟ ሺ࢞ ǡ ܲ ࢞

(17)

where ܼ calculates the root mean square error (RMSE) between the electrochemical potential at each boundary node and the critical potential. Furthermore, ݊௕ௗ‫ כ‬is the number of boundary nodes that do not satisfy the protection criterion, the constant ‫ ݒ‬is a penalty number and function ݇௟ is equal to: ݇௟ ൫࢞௣௦ ǡ ܲ ሺ࢞௣௦ ሻ൯ ൌ ൫߶ ௜ െ ߶௖ ൯ ή ‫ݑ‬൫߶ ௜ െ ߶௖ ൯ǡ

(18)


319

where ‫ ݑ‬is the unit step function. Typical values of ‫ ݒ‬are within the range ͳͲଶ െ ͳͲହ . Similarly, the penalty number is also used to keep the potential over the interface metal electrolyte greater than a fixed potential value denoted ߶௟௣௟ (lower potential limit). The goal is to provide a potential distribution on the metal within a pre-established range (߶௟௣௟ ൏ ߶ ൑ ߶௖ ). In this paper, the minimization of Eq (17) is achieved using a genetic algorithm. The adopted GA used for the minimization has a binary representation and is inspired by the algorithm presented in [20]. However, some characteristics have been included as the two-point crossover and the elitism. Furthermore, the probabilities of mutation and crossover can vary linearly over the generations. 5. Numerical results Example 1 presents an initial simulation with the sole purpose of testing the standard MFS formulation to external problems, comparing results with a direct boundary element (BEM) solution procedure. The design variables of the nonlinear least squares problem solved by LMDIF are the coefficients ܿ௝ , the distance from the virtual to the problem boundary ߩ and the potential at infinity ߶ ஶ . In this simulation, the anodes are fixed in different positions within the electrolyte. Example 2 is selected to test the complete optimization model for cathodic protection systems using GA and MFS. The adopted GA design variables now include the anode coordinates and their respective current intensities. Example 1 In this example, the cathodic protection of a submerged metallic structure, with square shape, is analyzed. The dimensions of the structure are ͳͲͲܿ݉‫݉ܿͲͲͳݔ‬, each of the four anodes has a current intensity ofെͳͳͲͲͲǤͲߤ‫ ܣ‬and is located over the two coordinate axes at ± 100 cm; i.e. at 50 cm from the sides. The boundary was represented with ͳͷʹ boundary nodes and 76 virtual sources have been adopted. The initial solution to the coefficients ܿ௝ is considered equal to ሺͳͲǤͲǡ െͳͲǤͲǡ ǥ ǡͳͲǤͲǡ െͳͲǤͲሻ and the distance from the virtual to the problem boundary starts equal to ͳͲǤͲܿ݉. The initial value for the potential at infinity is ͲǤʹͷܸ݉. In Fig. 1 the virtual source arrangements of the starting distance and of the final optimum distance are illustrated. The optimum distance determined by routine LMDIF after 4982 function evaluations (NFEV) is ஶ ߩ ൌ ͵Ǥͻ͹͸͸ͻܿ݉. The unknown potential at infinity estimate obtained by the BEM procedure is ߶஻ாெ ൌ ஶ ൌ െͻ͸͵ǤͶͻܸ݉. The conservation of ͻ͸͵Ǥ͵ͷܸ݉ while the potential at infinity solved by LMDIF is ߶ெிௌ ௡ೞ೛ current between anodes and cathodes is approximately satisfied, as expected, since σ௝ୀଵ ܿ௝ ൌ Ͷ͵ͻͻͻǤͻ͸ and ௡

೛ೞ ෌௠ୀଵ ܲ൫࢞௣௦ ௠ ൯ ൌ െͶͶͲͲͲǤͲߤ‫ܣ‬. Fig. 2 presents the potential distribution on the metal surface determined by MFS and by BEM. The RMSE between the potential values on the boundary determined by BEM and MFS is െͲǤͲͺͲͷͻܸ݉.

Figure 1: Virtual sources arrangement.

Figure 2: Potential distribution on the metal.

320


The potential distribution in the electrolyte using BEM is presented in Fig. 3 whereas Fig. 4 indicates the MFS solution. The RMSE between the potential values at internal points determined by BEM and MFS isͲǤͲ͹ͻͺ͹ܸ݉.

Figure 3: Potential in the electrolyte using BEM.

Figure 4: Potential in the electrolyte using MFS.

As seen above, satisfactory coincidence of results between the MFS and BEM procedures has been obtained. The next example deal with a more elaborate practical cathodic protection system design application. Example 2 In this example the geometry of problem 1 is repeated considering four anodes randomly distributed within the electrolyte. The anodes have coordinates ‫ݔ‬௔௣௦ and ‫ݕ‬௔௣௦ with ranges within ሾെͳͷͲǤͲܿ݉ǡ ͳͷͲǤͲܿ݉ሿ and ሾെͳͷͲǤͲܿ݉ǡ ͳͷͲǤͲܿ݉ሿ, respectively. The range of current intensity is ሾെͳʹͲͲͲǤͲߤ‫ܣ‬ǡ െͳͲͲͲͲǤͲߤ‫ܣ‬ሿ. In addition, the following GA values have been used: population sizeൌ ͵Ͷ, maximum number generationsൌ ʹͲͲ, required precision ൌ ͲǤͳ, initial crossover probabilityൌ ͲǤ͸, final crossover probabilityൌ ͲǤͷ, initial mutation probability ൌ ͲǤͲͳ, final mutation probabilityൌ ͲǤͲ͵ and the penalty number ‫ ݒ‬ൌ ͳͲଷ . The goal of this optimization is provide the potential distribution on the metal within the range െͺ͹Ͳܸ݉ ൑ ߶ ൑ െͺͷͲܸ݉. The potential distribution over the metal surface after optimization is presented in Fig. 5. The optimum location determined by GA is presented in Fig. 6 and the optimum anode current intensity is equal toെͳͲͳʹͻǤͲ͵ߤ‫ܣ‬Ǥ

Figure 5: Boundary potential distribution.

Figure 6: Potential distribution in the electrolyte.

After optimization, the electrochemical potential over the structure is lower and close to the critical potential and the complete surface is protected from corrosion as desired.


321

6. Conclusions The main goal of this paper is the combination of a MFS formulation with an optimization approach to obtain a robust and efficient numerical technique for optimization of cathodic protection system design also valid for external problems. The preliminary analyzes performed using BEM and the proposed MFS indicated good agreement between the two techniques for electrochemical potential distributions on the metal surface and in the electrolyte. The adopted MINPACK routine LMDIF showed satisfactory estimates for the coefficients of the linear superposition of fundamental solutions, the positions of the virtual sources and the potential at infinity, satisfying the nonlinear boundary conditions (polarization curve) and ensuring conservation of current between anodes and cathodes in the MFS formulation. The complete design optimization results found also confirm GA as a robust procedure to determine the optimum locations and the minimum current intensity for the anodes located in the electrolyte. The proposed modelling strategy using GA and MFS reached the optimum protection criterion as expected, providing a potential distribution on the metal interface within a pre-established range ߶௟௣௟ ൑ ߶ ൑ ߶௖ , within which the structure is protected from corrosion with optimized anode locations and respective impressed currents. References [1] R. Montoya, W. Aperador and D. M. Bastidas Influence of conductivity on cathodic protection of reinforced alkali-activated slag mortar using the finite element method, Corrosion Science, 51, 2857-2862 (2009). [2] M. H. Parsa, S. R. Allahkaram and A. H. Ghobadi Simulation of cathodic protection potential distributions on oil well casings, Journal of Petroleum Science and Engineering, 72, 215-219 (2010). [3] J. C. F. Telles, W. J. Mansur, L. C. Wrobel and M. G. Marinho Numerical Simulation of a Cathodically Protected Semisubmersible Platform using PROCAT System, Corrosion, 46, 513-518 (1990). [4] J. A. F. Santiago and J. C. F. Telles On Boundary Elements for Simulation of Cathodic Protection Systems with Dynamic Polarization Curves, International Journal for Numerical Methods in Engineering, 40, 2611-2622 (1997). [5] BEASY User Guide Beasy, Computational Mechanics BEASY Ltd, Ashurst, Southampton, UK (2000). [6] J. P. S. Azevedo and L. C. Wrobel Nonlinear heat conduction in composite bodies: a boundary element formulation, International Journal for Numerical Methods in Engineering, 26, 19-38 (1988). [7] V. D. Kupradze and M. A. Aleksidze Aproximate method of solving certain boundary-value problems, Soobshch akad nauk Gruz SSR, 30, 529-536 (1963). [8] J. R. Berger and A. Karageorghis The method of fundamental solutions for heat conduction in layered materials, International Journal for Numerical Methods in Engineering, 45, 1681–1694 (1999). [9] A. Karageorghis and G. Fairweather The method of fundamental solutions for the solution of nonlinear plane potential problems, IMA Journal on Numerical Analysis, 9, 231–242 (1989). [10] E. G. A. Costa, L. Godinho, J. A. F. Santiago, A. Pereira and C Dors Efficient numerical models for the prediction of acoustic wave propagation in the vicinity of a wedge coastal region, Engineering Analysis with Boundary Elements, 35, 855–867 (2011). [11] E. F. Fontes Jr., J. A. F. Santiago and J. C. F. Telles On a regularized method of fundamental solutions coupled with the numerical Green’s function procedure to solve embedded crack problems, Engineering Analysis with Boundary Elements, 37, 1–7 (2013).

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[12] W. J. Santos, J. A. F. Santiago and J. C. F Telles An Application of Genetic Algorithms and the Method of Fundamental Solutions to Simulate Cathodic Protection Systems, Computer Modeling in Engineering & Sciences, 87, 23-40 (2012). [13] S. Aoki and K. Amaya Optimization of cathodic protection system by BEM, Engineering Analysis with Boundary Elements, 19, 147–156 (1997). [14] L. C. Wrobel and P. Miltiadou Genetic algorithms for inverse cathodic protection problems, Engineering Analysis with Boundary Elements, 28, 267-27 (2004). [15] B. S. Garbow, K. E. Hillstrom and J. J. Mor MINPACK Project, Argonne National Laboratory (1980). [16] C. A. Brebbia., J. C. F. Telles and L. C. Wrobel Boundary Element Techniques: Theory and Applications in Engineering, Springer, Berlin (1984). [17] O. D. Kellog, Foundations of Potential Theory, Dover, New York (1954). [18] J. F. Yan, S. N. R. Pakalapati, T. V. Nguyen and R. E. White Mathematical Modelling of Cathodic Protection using the Boundary Element Method with Nonlinear Polarisation Curves, Journal of the Electrochemical Society, 139, 1932-1936 (1992). [19] J. C. F. Telles, W. J. Mansur and L. C. Wrobel On boundary elements for external problems, Mechanics Research Communications, 11, 373-377 (1985). [20] Z. Michalewicz Genetic Algorithms + Data Structures = Evolution Programs, Spinger-Verlag (1996).


323

Stabilized FEM-BEM Solutions of MHD Flow in an Annular Pipe S. Han Aydın

∗

Department of Mathematics, Karadeniz Technical University, Trabzon, Turkey Keywords: Stabilized FEM, BEM, MHD pipe flow.

Abstract. In this study, we have considered the numerical solution of the magnetohydrodynamic (MHD) flow through a annular pipe with insulating pipe walls under the influence of a transverse magnetic field. The coupled equations are transformed into decoupled inhomogenous convection-diffusion type equations in order to apply stabilization in the finite element method (FEM) solution procedure. Further transformations are also applied to transform the equations first to homogenous form using a particular solution, and then to the homogenous modified Helmholtz equations. Boundary element method (BEM) application is performed in both cases by using corresponding fundamental solution of homogenous convection-diffusion and modified Helmholtz equations. Proposed stabilized FEM numerical scheme is efficient especially for the high values of the Hartmann number. Computations are carried in both circular and sqauare annular regions for several values of Hartmann number. Obtained numerical solutions display the well-known characteristics of the MHD pipe flow.

Introduction The magnetohydrodynamic (MHD) flow has great importance for the researchers since there are many applications in engineering, science and technology. Hartmann firstly investigated the MHD flow of viscous, incompressible flow between two plates [1]. Then, many number of authors have interested in the solutions of the MHD pipe flow equations both analytically and numerically (see [2, 3, 4] and references therein). Due to the coupled partial differential equations, most the solutions are obtained using some numerical methods. Using the advantages of boundary element method (BEM), TezerSezgin and Han Aydın have solved the MHD duct flow equations for several geometrical cross-sections having insulating walls [3]. Recently, the more general case of the problem, MHD pipe flow in a conducting medium for circular and square pipe cross-sections, is considered and solved again by using BEM [4]. Han Aydın has also solved the same problem for the very large values of the problem parameters using stabilized FEM and BEM coupling formulations [5]. ∗

Electronic address: [email protected];

1

324


Mathematical Model The governing MHD equations of steady, laminar, fully developed flow of a viscous, incompressible and electrically conducting fluid in an annular pipe domain (Ω) with insulated boundary walls (Γ1 and Γ2 ) subject to a constant and uniform applied magnetic field in the y -axis are given (in non-dimensional form) as [2]; ⎫ ∂B(x, y) ⎪ ⎪ = −1 ⎪ ∇2 V (x, y) + M ⎪ ⎪ ∂y ⎬ (x, y) ∈ Ω (1) ⎪ ⎪ ⎪ ∂V (x, y) ⎪ =0 ⎪ ∇2 B(x, y) + M ⎭ ∂y with the homogenous boundary conditions V (x, y) = B(x, y) = 0 on Γ1 and Γ2 where M is the Hartmann number [1]

Figure 1: Domain of the problem for circular and square annular regions

In order to apply FEM or BEM solution procedure, the coupled Eqns. (1) are decoupled first by defining new unknowns U1 (x, y) and U2 (x, y) as

U1 = V + B ,

U2 = V − B

(2)

which gives

∇ 2 U1 + M

∂U1 = −1 ∂y

∇ 2 U2 − M

∂U2 = −1. ∂y

(3)

with the boundary conditions

U1 = U2 = 0 on Γ1 and Γ2 .

FEM Formulation The weak formulation of the problem (3) can be stated as [5]: Find U1 , U2 ∈ V = H01 (Ω)2 such that

(∇U1 , ∇v) − M (

∂U1 , v) = (1, v) ∂y

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ∂U2 ⎪ (∇U2 , ∇v) + M ( , v) = (1, v) ⎪ ⎭ ∂y

2

,

∀v ∈ V.

(4)


325

Then, the Galerkin finite element formulation of the problem is; find U1h , U2h ∈ Vh such that

(∇U1h , ∇vh ) − M (

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

∂U2h , vh ) = (1, vh ) ∂y

⎪ ⎪ ⎪ ∂U2h ⎪ , vh ) = (1, vh ) ⎪ (∇U2h , ∇vh ) + M ( ⎭ ∂y

∀vh ∈ Vh .

,

(5)

Finally, if we insert the stabilization terms to the formulation, the SUPG type stabilized FEM variational formulation of the problem is written as ⎡

⎤

∂U2h

⎢ (∇U1h , ∇vh ) − M ( ∂y , vh ) − (1, vh ) ⎢ ⎢ ⎢ ⎢ ⎣ ∂U2h

(∇U2h , ∇vh ) + M (

, vh ) − (1, vh )

∂y

⎡ ∂vh ∂U1h ⎥ ⎢ (−M ∂y − 1)(−M ∂y ) ⎥ ⎢ ⎥ ⎢ ⎥ + τK ⎢ ⎥ ⎢ ⎦ ⎣ ∂U2h ∂vh

(M

∂y

− 1)(M

∂y

)

⎤ ⎥ ⎥ ⎥ ⎥=0 ⎥ ⎦

(6)

with the stabilization parameter [5]

τK =

⎧ hK ⎪ ⎪ if P ek ≥ 1 ⎪ ⎪ ⎨ 2M

(7)

⎪ ⎪ 2 ⎪ ⎪ ⎩ hK if P ek < 1

12

where hK is the diameter of the element K and P eK =

M hK 6

is the Peclet number.

BEM Formulation For obtaining homogeneous equations, we define w1 (x, y) and w2 (x, y)

w1 = U1 +

1 y, M

w 2 = U2 −

which results in

∇2 w1 + M

1 y M

(8)

∂w1 =0 ∂y (9)

∂w2 ∇ w2 − M =0 ∂y 2

with the new boundary conditions

w1 =

1 y, M

w2 = −

1 y on Γ1 and Γ2 . M

The fundamental solutions w1 and w2 satisfy the adjoint equations 9 are given by Pozrikidis [6] as;

w1 (ξ, η) =

1 ( M ry ) M e 2 K0 ( r) , 2π 2

w2 (ξ, η) =

1 (− M ry ) M e 2 K0 ( r) 2π 2

(10)

→ where − r = (rx , ry ) is the vector between the source and the field points respectively, and K0 is the zero order modified Bessel function of the second kind. After usual BEM procedure, we will get boundary

3

326


∂w1 ∂w2 ∂n , ∂n

element equations in matrix-vector form for w1 , w2 and

[H 1 ]{w1 } = [G1 ]{ where

G1ij =

Γj

e

M ry 2

K0 (

M r)dΓj , 2

Hij1 = πδij −

M ny 4π

Hij2

∂w1 }, ∂n

M = πδij + ny 4π

Γj

e

Γj

e

−

[H 2 ]{w2 } = [G2 ]{

G2ij =

M ry 2

K0 (

M ry 2

as

Γj

M r)dΓj + 2

e−

M ry 2

Γj

M K0 ( r)dΓj − 2

e

K0 (

M ry 2

Γj

e

−

∂w2 } ∂n

(11)

M r)dΓj 2

K1 (

M ry 2

M ∂r r) dΓj 2 ∂n

(12)

M ∂r K1 ( r) dΓj . 2 ∂n

A further transformation with

u2 = w2 e−ky

u1 = w1 eky ,

(13)

where k = M/2, reduces the equations to two homogeneous modified Helmholtz equations as

∇2 u1 − k 2 u1 = 0 (14) ∇2 u2 − k 2 u2 = 0 with the new boundary conditions

u1 =

1 ky ye , M

u2 = −

1 −ky ye . M

In this form, BEM formulation gives the following system as

∂u1 }, ∂n

[H]{u2 } = [G]{

∂ K0 (kr)dΓj , ∂n

Gi,j =

[H]{u1 } = [G]{ where

Hi,j = πδij +

Γj

Γj

∂u2 } ∂n

(15)

K0 (kr)dΓj .

(16)

Numerical Results We have tested proposed BEM and FEM formulations for both circular and square annular pipes. In Fig. (2), solutions obtained from all the formulations are compared for the small value of Hartmann number (M = 50). It is seen that, although all the solutions are almost the same for the induced current values, velocity values obtained with FEM and BEM using modified Helmholtz form formulation agree most with each other. Therefore, in the rest of the solutions, BEM formulation using convectiondiffusion form is not considered anymore. In Fig. (3) we have displayed the velocity and induced current behaviors for both circular and square annular pipes for the moderate value of the Hartmann number (M = 100). The boundary layers from bottom to top of the pipe are seen near the walls in the y -direction which is in the same direction of the applied magnetic field. Also, magnitudes of both fluid velocity and induced current

4


V 0.061 0.049 0.037 0.025 0.013 0.001

-3 -3

0

0

-3 -3

3

0

X

-3 -3

3

0

0

0

-3 -3

3

0

X

3

B 0.044 0.0264 0.0088 -0.0088 -0.0264 -0.044

3

Y

0

3

X B 0.045 0.027 0.009 -0.009 -0.027 -0.045

3

Y

3

Y

0

X B 0.045 0.027 0.009 -0.009 -0.027 -0.045

-3 -3

V 0.052 0.0418 0.0316 0.0214 0.0112 0.001

3

Y

0

V 0.053 0.0426 0.0322 0.0218 0.0114 0.001

3

Y

Y

3

327

0

-3 -3

0

X

3

X

Figure 2: Velocity of the fluid and induced currents for M = 50 and with BEM (convection-diffusion form) (left), BEM (modified Helmholtz form) (center), and FEM(right) are decrease when the Hartmann number increases.

0

-3 -3

3

X

0

-3 -3

0

X

-3 -3

3

3

0

-3 -3

0

Y 0

0

-3 -3

3

3

0

-3 -3

0

X

0

3

X V 0.029 0.0234 0.0178 0.0122 0.0066 0.001

3

X

B 0.028 0.0168 0.0056 -0.0056 -0.0168 -0.028

3

X B 0.024 0.0144 0.0048 -0.0048 -0.0144 -0.024

3

Y

Y

3

0

0

X V 0.032 0.0258 0.0196 0.0134 0.0072 0.001

Y

-3 -3

0

V 0.029 0.0234 0.0178 0.0122 0.0066 0.001

3

Y

Y

Y

0

B 0.023 0.0138 0.0046 -0.0046 -0.0138 -0.023

3

3

B 0.029 0.0174 0.0058 -0.0058 -0.0174 -0.029

3

Y

V 0.027 0.0231 0.0179 0.0127 0.0062 0.001

3

0

-3 -3

0

3

X

Figure 3: Velocity of the fluid and induced currents for M = 100 and with BEM (modified Helmholtz form) (first row) and FEM(second row) for circular and square annular pipes

5

328


Finally, stabilized FEM formulation is used to obtained solutions for very large values of the Hartmann number. In Fig. (4), the velocity and induced current solutions for both circular and square annular pipes are displayed for M = 1000. It is seen that, for the very large value of the Hartmann number the thickness of the boundary layers getting very small which is the well-known behaviour of the MHD duct flows.

V 0.0032 0.002735 0.002115 0.001495 0.00072 0.0001

0

-3 -3

0

B 0.0026 0.00156 0.00052 -0.00052 -0.00156 -0.0026

3

Y

Y

3

0

-3 -3

3

0

X

0

0

B 0.0028 0.00168 0.00056 -0.00056 -0.00168 -0.0028

3

Y

Y

3

-3 -3

3

X V 0.0029 0.00248 0.00192 0.00136 0.00066 0.0001

0

-3 -3

3

X

0

3

X

Figure 4: Velocity of the fluid and induced currents for M = 1000 and with stabilized FEM for circular (first row) and square (second row) annular pipes

References [1] J. Hartmann Math. Fys. Medd., 15(6) (1937). [2] L. Drago¸s, Magnetofluid Dynamics, Abacus Pres (1975). [3] M. Tezer-Sezgin, S. Han Aydın Eng Anal Bound Elem. 30 411–418 (2016). [4] M. Tezer-Sezgin, S. Han Aydın Computing 95(1) 751–770 (2013). [5] S. Han Aydın Open Journal of Fluid Dynamics 3 184–190 (2013). [6] C.A. Pozrikidis, Practical Guide to Boundary Element Methods with the Software Library BEMLIB, Chapman and Hall/CRC, (2002)

6


329

A new boundary approach for the 2D slow viscous MHD flow of a conducting liquid about a solid particle A. Sellier1 , M. Tezer-Sezgin2 and S. H. Aydin3 LadHyx. Ecole polytechnique, 91128 Palaiseau Cédex, France 2 Department of Mathematics, Middle East Technical University, 06531 Ankara, Turkey 3 Department of Mathematics, Karadeniz Technical University, 61080 Trabzon, Turkey 1 [email protected], 2 [email protected], 3 [email protected] 1

Keywords: MagnetoHydroDynamics, Two-dimensional flow, Stokes flow, Green tensor, Boundary-integral equation.

Abstract. This works examines the two-dimensional MagnetoHydrodynamic steady slow viscous flow of a conducting newtonian liquid about a solid insulating particle. Attention is paid not only to the flow, driven both by an imposed uniform ambient magnetic field B and a prescribed rigid-body motion of the particle, but also to the traction it exerts on the particle boundary. Symmetry shows that in the liquid domain the magnetic field is B whereas there is no electric field. The advocated procedure holds for a Stokes flow and consists in deriving a new suitable boundary formulation which rests on the analytical determination of the so-called free-space second-rank (two-dimensional) Green tensor and third-rank associated stress tensors obtained from the fundamental Stokes flow produced by a concentrated point force placed at a given point. As a result, relevant boundary-integral equations are proposed which permit one to get on the particle contour the required traction. Introduction As it is well know [1], a conducting liquid flows when subject to ambient magnetic and/or electric field(s). The resulting so-called MagnetoHydrodynamic (MHD) flow with pressure field p and velocity field u is driven by a non-uniform Lorentz body-force f . More precisely, designating by E and B the electric and magnetic fields prevailing and denoting by σ > 0 the liquid conductivity, one gets f = j∧B with j the current density expressed (via Ohm’s law) as j = σ(E + u∧ B). In general, bodies might also be immersed in the liquid and one then solves in the liquid domain the cumbersome incompressible Navier-Stokes equations with body force f for the flow (u, p) and the Maxwell equations for (E, B). In addition, proper far-field conditions and boundary conditions on each body surface are to be added. Clearly, the previous MHD problem is in general very involved since the governing equations are coupled through the Lorentz body force f . Fortunately, in some steady cases for which symmetry properties hold [2] such a coupling vanishes therefore yielding a much more tractable problem. For example, for a sphere translating in a quiescent liquid parallel with a prescribed uniform external magnetic field B in absence of far-field electric field it turns out [3] that (B, E) = (B, 0) in the entire liquid domain! The same property holds [4] for the quasi-steady two-dimensional MHD flow produced by the rigid-body motion (translation and/or rotation) of a solid particle in a conducting fluid which is quiescent and subject solely (no far-field electric field) to a uniform magnetic field B lying in the flow plane. Actually, [4] pays attention to the steady two-dimensional MHD flow about a translating disk with radius a and velocity U when inertial effects are negligible i. e., setting U = |U| > 0 and designating by ρ and μ the Newtonian conducting liquid uniform density and viscosity, for vanishing Reynolds number Re = ρ|U|/μ 1. The analytical treatment developed in [4] reveals that the MHD two-dimensional flow about the translating disk immersed in the ambient uniform magnetic field B with magnitude B = |B| > 0 deeply depends on the so-called Hartmann number M = aB σU/μ > 0. In addition and contrary to the case of the 2D Stokes flow about the disk in absence of magnetic field, there is no use to employ the Oseen approximation far from the body because the obtained 2D MHD Stokes flow adequately vanishes there. Note that [4] actually solved the problem of the translating disk by expanding the flow stream function as an infinite serie of terms. Unfortunately, such a procedure is not tractable in practice for other body shapes. Therefore, the present work introduces another approach to deal with the challenging case of a solid and arbitrary-shaped body experiencing

330


C

B = Be1 P

σ>0 O

Ωe3

μ, ρ U

D Figure 1: A solid plane particle P, with (closed) boundary C, immersed in the ambient uniform magnetic field B = Be1 and migrating with translational velocity U and/or angular velocity Ω = Ωe3 . a given rigid-body motion (translation and/or rotation). The advocated method consists of a new boundary formulation which makes it possible to reduce the task to the treatment of at the most three boundary-integral equations on the body boundary. Governing problem and fundamental Green and associated stress tensors This section introduces the governing problem and also analytically obtains the key associated free-space velocity Green tensor and associated stress tensor. Assumptions and resulting MHD equations As shown in Fig. 1, we consider a solid and plane particle P immersed in a Newtonian, conducting and unbounded liquid with uniform conductivity σ > 0, density ρ and viscosity μ. The particle has typical length a, attached point O and smooth boundary C with unit normal n directed into the liquid. Finally, the particle lies in a plane with Cartesian coordinates (0, x1 , x2 ) and we set

x = OM = x1 e1 + x2 e2 and also r = |x| = x21 + x22 . The conducting liquid two-dimensional flow, with pressure field p and velocity field u = u1 e1 +u2 e2 , is driven by the uniform magnetic field B = Be1 (with B > 0) imposed far from the particle and the particle rigid-body motion with typical velocity magnitude V > 0. More precisely, the liquid is quiescent far from the particle which experiences a translational velocity U = U1 e1 + U2 e2 (velocity of the attached point O) and/or angular velocity Ω = Ωe3 where e3 = e1 ∧ e2 (i. e. the rotation is normal to the particle). In addition, inertial effects are negligible i. e. Re = ρV /μ 1. Under those assumptions [4] there is no electric field in the liquid whereas the magnetic field is B = Be1 in the entire liquid domain D. Since the Lorentz body force then reads f = σ(u ∧ B) ∧ B, the steady liquid flow (u, p) obeys the following creeping flow equations and far-field behaviours μ∇2 u + σB 2 (u ∧ e1 ) ∧ e1 = ∇p and ∇.u = 0 in D,

(1)

(u, p) → (0, 0) as r → ∞

(2)

together with the no-slip boundary condition u = U + Ωe3 ∧ x on C.

(3)

The task consists in determining in the liquid domain the flow (u, p) and if necessary its stress tensor σ once the body shape and rigid-body motion (U, Ωe3 ) are prescribed. In practice, one is also often interested in getting the exerted hydrodynamic force F and torque C (about the point O) such that F=

C

σ.ndl, C =

C

x ∧ σ.ndl = [ (x1 e2 − x2 e1 ).(σ.n)dl]e3 . C

(4)

One can think about using a Finite Element Code in solving (1)-(3). Such a procedure would however require to adequately truncate the liquid domain and also mesh the still large resulting bounded domain. Here we instead develop a new boundary approach to efficiently deal with (1)-(3).


331

Free-space velocity second-rank Green tensor For further purposes it is fruitful to determine the fundamental flow (u, p) due to a concentrated point force of strength g = g1 e1 + g2 e2 placed at the point x0 . In other words, such a flow obeys μ∇2 u + σB 2 (u ∧ e1 ) ∧ e1 − ∇p = −δ(x − x0 )g and ∇.u = 0 for x = x0

(5)

(u, p) → (0, 0) as |x − x0 | → ∞

(6)

where δ is the two-dimensional delta pseudo-function. Clearly, the flow (u, p) and its stress tensor σ lineary depend upon the force g. As in [5], we thus adopt introduce a vector P and a so-called velocity second-rank Green tensor G such that 1 1 u(x) = G(x, x0 ).g, p(x) = P(x, x0 ).g. (7) 4πμ 4π Inspecting (5)-(6) immediately reveals that G(x, x0 ) = G(x − x0 ) and P(x, x0 ) = P(x − x0 ). Hence, one can restrict the attention to the special case of a concentrated force located at the origin O (case x0 = 0). In first getting the tensor G we micmick the treatment given in [6] for a similar threedimensional problem. First we apply the operator ∇ ∧ (∇∧) to each side of equation (5). Taking into account of the second equation (5) it follows that μ∇2 (∇2 u) − σB 2

∂2u = ∇ ∧ (∇ ∧ [δg]) ∂x21

(8)

with u = u(x), δ = δ(x) and x = x1 e1 + x2 e2 . Denoting by Δ the two-dimensional Laplacien operator, the required solution u then reads u=

σB 2 ∂ 2 H 1 ∇ ∧ (∇ ∧ [Hg]), Δ(ΔH) − =δ μ μ ∂x21

(9)

with function H to be determined. Once this is done, the velocity u = u1 e1 + u2 e2 is gained from (9) i. e. from the relations μu1 = [

∂2H ∂2H ∂2H ∂2H − ΔH]g1 + [ ]g2 , μu2 = [ ]g1 + [ 2 − ΔH]g2 . 2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2

(10)

Clearly, (10) reveals that it is sufficient to determine the functions ΔH and ∂H/∂x1 . In view of the second relation (9), this is achieved by introducing the so-called Hartmann layer thickness d [1,7] such that 1 μ (11) d= B σ and two auxiliary functions H− and H+ obeying [Δ +

1 ∂ H+ + H− ∂H d 1 ∂ , ]H− = [Δ − ]H+ = −δ, ΔH = − = − (H+ − H− ). d ∂x1 d ∂x1 2 ∂x1 2

(12)

From (6) both u1 and u2 are required to vanish far from the source point O and one therefore solely retains solutions H− and H+ going to zero as r = |x| becomes large. Denoting by K0 the usual modified Bessel function of order zero (which indeed vanishes at infinity; see [8]), the obtained solutions then read (see also [9]) H± =

1 ±x1 /(2d) r r d r 1 x1 ∂H x1 K0 ( ), ΔH = − cosh( )K0 ( ), = − sinh( )K0 ( ). e 2π 2d 2π 2d 2d ∂x1 2π 2d 2d

(13)

Appealing to (10), (13) and the definitions (7) then easily provides the Cartesian components of the second-rank velocity tensor G(x, x0 ) = Gij (x, x0 )ei ⊗ ej . Because K0 = −K1 , with K1 the usual modified Bessel function of order one, the results read x ˆ1 ˆ1 ˆ2 x ˆ1 rˆ rˆ x x ˆ1 rˆ x )K0 ( ) + sinh( )K1 ( ) , G12 (x, x0 ) = sinh( )K1 ( ) , (14) 2d 2d 2d 2d rˆ 2d 2d rˆ x ˆ1 ˆ1 x ˆ1 rˆ rˆ x (15) G21 (x, x0 ) = G12 (x, x0 ), G22 (x, x0 ) = cosh( )K0 ( ) − sinh( )K1 ( ) , 2d 2d 2d 2d rˆ

G11 (x, x0 ) = cosh(

332


ˆ = x − x0 , x ˆ .ei and rˆ = |ˆ with the notations x î = x x|. One should note the properties Gij (x, x0 ) = Gji (x0 , x) = Gij (x0 , x). Associated pressure and so-called third-rank stress tensor We now look at the pressure, i. e. the vector P(x, x0 ) occurring in (7). Again, we select x0 = 0 and go back to the first equation (5) which now reads ∇p = μ∇2 u + σB 2 (u ∧ e1 ) ∧ e1 + δ(x)g with u previously given by (9)-(10). Taking the divergence of this equation and using the property ∇.u = 0 gives Δp = ∇.(δg) + σB 2 ∇.[(u.e1 )e1 ]. Applying on each side the Laplacien operator further yields Δ(Δp) −

σB 2 [g.e1 ] ∂δ σB 2 ∂ 2 p = Δ[∇.(δg)] − . 2 μ ∂x1 μ ∂x1

(16)

Recalling that H satisfies the second equation (9) and using the definition (11) of the Hartmann layer thickness d then immediately shows that p = Δ[∇.(Hg)] − (

∂ΔH 1 ∂H ∂ g.e1 ∂H ) =[ − 2 ]g1 + [ (ΔH)]g2 . d2 ∂x1 ∂x1 d ∂x1 ∂x2

(17)

Accordingly, one arrives at the following components Pi = P.ei P1 (x, x0 ) =

1 rˆ rˆ x 1 rˆ x x ˆ1 x ˆ1 ˆ1 x ˆ1 ˆ2 [sinh( )K0 ( ) + cosh( )K1 ( ) ], P2 (x, x0 ) = cosh( )K1 ( ) . d 2d 2d 2d 2d rˆ d 2d 2d rˆ

(18)

Henceforth, we adopt the tensor summation notation for repeated indices with for instance g = gi ei . Let us also remind that the flow (u, p) solution to (5)-(6) has stress tensor σ(x) = σij (x, x0 )ei ⊗ ej . By linearity, one can introduce a so-called stress tensor T with cartesian components Tijk (x, x0 ) such that (see also [5]) 1 Tijk (x, x0 )gj . (19) σik (x, x0 ) = 4π As should be noticed by the reader, (19) does not read σ = T.g. Combining the definitions (7) and (19) immediately shows that Tijk (x, x0 ) = −δik Pj (x, x0 ) +

∂Gij ∂Gkj (x, x0 ) + (x, x0 ) ∂xk ∂xi

(20)

where the symbol δik designates the Kronecker delta. Because Tijk = Tkji the task reduces to the determination of six Cartesian components: T111 , T121 , T212 , T222 and the quantities T112 = T211 and T122 = T221 . After elementary manipulations one gets, using the results (14)-(15) and (18),

ˆ1 x ˆ1 x ˆ1 ˆ2 1 x ˆ1 rˆ x rˆ x rˆ 1 x ˆ1 + 2 sinh( )K1 ( )( − 31 ), sinh( )K1 ( ) − cosh( )K1 ( ) d 2d 2d rˆ 2d 2d rˆ 2d 2d rˆ rˆ x ˆ1 x ˆ1 2 1 2 rˆ xˆ2 rˆ rˆ x ˆ1 x ˆ2 , T222 (x, x0 ) = − cosh( )K1 ( ) + sinh( ) K1 ( ) − K1 ( ) d 2d 2d rˆ 2d rˆ 2d d 2d rˆ2 1 ˆ1 x ˆ2 x ˆ1 1 rˆ rˆ x K ( ) − K1 ( ) , T121 (x, x0 ) = 2 sinh( ) 2d 2d 1 2d rˆ 2d rˆ2 x ˆ1 ˆ2 1 rˆ x rˆ T212 (x, x0 ) = sinh( ) K1 ( ) 22 − K0 ( ) d 2d 2d rˆ 2d 1 rˆ x rˆ 1 x x ˆ1 ˆ1 x ˆ1 ˆ2 − cosh( )K1 ( ) + 2 sinh( )K1 ( )( − 32 ), d 2d 2d rˆ 2d 2d rˆ rˆ 2 x ˆ1 1 rˆ rˆ x ˆ1 x ˆ2 , T112 (x, x0 ) = T211 (x, x0 ) = sinh( ) K1 ( ) − K1 ( ) 2d d 2d rˆ 2d rˆ2 1 rˆ rˆ x x ˆ1 1 x ˆ1 ˆ1 sinh( )K0 ( ) − cosh( )K1 ( ) T122 (x, x0 ) = T221 (x, x0 ) = 2d 2d 2d d 2d 2d rˆ 2 1 rˆ rˆ x ˆ1 − x ˆ22 x ˆ1 1 + sinh( ) K1 ( ) − K1 ( ) 2 2d rˆ 2d 2d 2d rˆ

T111 (x, x0 ) =

(21) (22) (23)

(24) (25)

(26)


333

ˆ = x − x0 , xî = x ˆ .ei and rˆ = |ˆ where K1 designates the derivative of the function K1 and of course x x|. Note that Tijk (x, x0 ) = −Tijk (x0 , x). Advocated boundary formulation and boundary-integral equations This section briefly presents the new boundary formulation advocated to solve the MHD problem (1)-(3). Integral representations for the velocity Let us consider two MHD flows (u, p) and (u , p ) with stress tensors σ and σ obeying (1)-(2). Then, (1) rewrites ∇.σ = −σB 2 (u∧e1 )∧e1 and ∇.σ = −σB 2 (u ∧e1 )∧e1 in the liquid. Since each flow is divergence-free one thus gets u .(∇.σ) = u.(∇.σ ) in the liquid. Setting n = ni ei , f = σ.n = fi ei and exploiting the property Gij (y, x0 ) = Gji (x0 , y) one then arrives (see also [5]) at the following key integral representation for the Cartesian components of the velocity field uj (x0 ) = −

1 4πμ

C

Gji (x0 , y)fi (y)dl(y) +

1 4π

C

ui (y)Tijk (y, x0 )nk (y)dl(y)

for x0 in D.

(27)

In addition, for any MHD flow (u , p ) with stress tensor σ obeying (1) inside the particle P and exerting on its boundary C the traction −f with f = σ .n (since n is directed inside outside the particle, as illustrated in Fig. 1) one has the additional relation 0=−

1 4πμ

C

Gji (x0 , y)[σ .n](y).ei dl(y) +

1 4π

C

ui (y)Tijk (y, x0 )nk (y)dl(y).

(28)

The result (28) makes it possible to cast the integral (27) representation in another equivalent form. For x0 located in D we indeed select the following MHD flow inside the particle: u (y) = u(x0 ) and p = σB 2 [(u(x0 ) ∧ e1 ) ∧ e1 ].y. For such a flow σ .n = −p n and (28) holds. Combining this latter relation (28) with (27) then provides the integral representation uj (x0 ) = −

1 4πμ

C

Gji (x0 , y){fi (y) + σB 2 [(u(x0 ) ∧ e1 ) ∧ e1 ].yni (y)}dl(y)

+

1 4π

C

[ui (y) − ui (x0 )]Tijk (y, x0 )nk (y)dl(y)

for x0 in D.

(29)

The representation (27) is useful to obtain the velocity field u in the liquid domain once both u and σ.n are known on the particle boundary C. It however does not hold for x0 lying on the boundary C since, from (21)-(26), each component Tijk (y, x0 ) behaves as 1/|y − x0 | as x0 approaches y. By contrast, the equivalent representation (28) is regularized in the sense it also holds on the boundary C! Relevant key boundary-integral equation The MHD flow (u, p) governed by (1)-(3) has prescribed velocity u on the particle. The unknown surface traction f = σ.n is obtained by requiring (29) at the boundary. One has thus to invert for the vector s = si ei the following boundary-integral equation of the first kind −

1 4πμ

C

Gji (x, y)si (y)dl(y) = uj (x) −

1 4π

C

[ui (y) − ui (x)]Tijk (y, x)nk (y)dl(y), x on C.

(30)

From the knowledge of s one finally gets the traction as f (x) = s(x) − σB 2 {[(u(x) ∧ e1 ) ∧ e1 ].x}n(x). Observe that for a translation of the particle at the velocity U the vector s, f and the velocity field u satisfy −

1 4πμ

C

G(x, y).s(y)dl(y) = U for x on C, f = σ.n = s − σB 2 {[(U ∧ e1 ) ∧ e1 ].x}n(x),

u(x) = −

1 4πμ

C

G(x, y).s(y)dl(y) for x in D.

(31) (32)

334


Finally, for a general particle rigid-body motion (U, Ωe3 ) one has the following “shift” f (x) = s(x) + σB 2 [U.e2 + Ωx.e1 ](x.e2 )n(x). Conclusions A new boundary approach has been established to accurately compute the two-dimensional steady viscous MHD flow of a conducting liquid about a migrating solid plane particle when the liquid is subject far from the particle to a prescribed uniform magnetic field (parallel with the particle). Contrary to the available literature, the advocated approach is valid for arbitrary-shaped particles. It rests on the treatment of at the most three boundary-integral equations on the particle boundary and enables one to accurately compute at a reasonable cpu time cost the force and torque experienced by the moving particle. A boundary element numerical implementation together with benchmarks for a translating disk and new results for translating or rotating ellipses will be given and discussed at the oral presentation. References [1] R. J. Moreau Magnetohydrodynamics, Fluid Mechanics and its applications. Kluwer Academic Publisher. (1990). [2] K. Gotoh Stokes flow of an electrically conducting fluid in a uniform magnetic field. Journal of the Physical Society of Japan, 15 (4), 696-705 (1960). [3] W. Chester The effect of a magnetic field on Stokes flow in a conducting fluid. J. Fluid Mech., 3, 304-308 (1957). [4] H. Yosinobu and T. Kakutani Two-dimensional Stokes flow of an electrically conducting fluid in a uniform magnetic field. Journal of the Physical Society of Japan, 14 (10), 1433-1444 (1959). [5] C. Pozrikidis Boundary integral and singularity methods for linearized viscous flow, Cambridge University Press, (1992). [6] J. Priede Fundamental solutions of MHD Stokes flow arXiv: 1309.3886v1. Physics. fluid. Dynamics, (2013). [7] J. Hartmann Theory of the laminar flow of an electrically conductive liquid in a homogeneous magentic field. Det Kgl . Danske Videnskabernes Selskab . Mathematisk-fysiske Meddelelser, XV (6), 1-28 (1937). [8] M. Abramowitz and I. A. Stegun Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York. (1965). [9] C. Pozrikidis A practical guide to boundary element method with the software library BEMLIB. London: Chapman & Hall/CRC. (2002).


335

Direct Volume-to-Surface Integral Transformation for 2D BEM Analysis of Anisotropic Thermoelasticity Y.C. Shiah*, Chung-Lei Hsu, and Chyanbin Hwu Department of Aeronautics and Astronautics National Cheng Kung University, Tainan 701, Taiwan, R.O.C. (* Correspond to: [email protected]) Keywords: Direct volume-to-surface integral transformation, 2D anisotropic thermoelasticity. Abstract. As has been well documented for the boundary element method (BEM), a volume integral is present in the integral equation for thermoelastic analysis. Any attempt to directly integrate the integral shall inevitably involve internal discretisation that will destroy the BEM's notion as a true boundary solution technique. Among the schemes to overcome this difficulty, the exact transformation approach is the most elegant since neither further approximation nor internal treatments are involved. Such transformation for 2D anisotropic thermoelasticity has been achieved by Shiah and Tan [1] with the aid of domain mapping. This paper revisits this problem and presents a modified transformation process for 2D anisotropic thermoelasticity, where no domain distortion is involved. Being defined in the original Cartesian coordinate system, the volume integral is analytically transformed to the boundary, being derived using the Stroh formulism. This transformation is favorable especially when the corresponding anisotropic field is directly calculated using the anisotropic Green's function without resorting to domain mapping. In the end, numerical examples are provided to show the validity of such transformation. Introduction In engineering practice, thermoelastic analysis often plays an important role to ensure the integrity of structures when subjected to thermal loads. Being recognized as an efficient numerical tool, the BEM is characterized by its distinctive notion that only the boundary needs to be modeled. However, for treating thermal effects, an additional volume integral appears in the boundary integral equation (BIE) that shall destroy the BEM's notion as a truly boundary solution technique. This is because any means to directly integrate the extra volume integral will inevitably involve "cell" discretization throughout the whole domain. Over the years, there are several techniques proposed to avoid such domain discretization, such as the Monte Carlo method [2], the particular integral approach (e.g. [3-4]), the dual reciprocity method [5], the multiple reciprocity method [6], and the exact transformation method [7], abbreviated as ETM herein. Among these schemes mentioned above, the ETM is fundamentally the most appealing because it restores the BEM analysis as a purely boundary solution technique yet without requiring further numerical approximation and internal treatments. Although the ETM had been widely employed to treat equivalent body-force effects in isotropic elasticity, such transformation for 2D anisotropic elasticity was not achieved until Zhang et al. [8] first derived the transformed boundary integrals accounting for body forces in 2D anisotropic bodies. Following this success, Shiah and Tan [9] made use of a coordinate transformation that mapped the anisotropic thermal field to an equivalent isotropic one defined in the transformed coordinate system. Accounting for anisotropic thermal effects, this coordinate transformation enables the use of Green's 2nd Identity to exactly transform the volume integral into boundary ones [1], albeit defined in the new coordinate system. All kernel functions in their derivations are based upon the Lekhnitskii formulations [10]. For solving the anisotropic thermal field, it is quite expedient to treat the problem as if it is "isotropic" in the new coordinate system. It turns out that this volume integral transformation works well with the domain mapping technique, especially considering its expediency in solving the associated thermal field. However, when the anisotropic thermal field is to be solved directly using the anisotropic Green's function, such treatment appears to be less straightforward. In this paper, a direct transformation process is presented to treat the 2D anisotropic thermoelasticity such that no domain mapping is involved. All derivations follow the similar way but in a somewhat modified manner such that no coordinate transformation is involved. Also, all kernel functions, including the new one constructed accordingly to facilitate the transformation, are based upon the Stroh formalism [11]. For verifying the validity of the transformation, a few numerical examples are presented at the end.

336


BIE for 2D anisotropic thermoelasticity For generally anisotropic elastic media in two dimensions, the constitutive law between stresses Vij and strains Hij with thermal effects is governed by the well-known Duhamel-Neumann relation,

V ij

Cijkl H kl Eij 4 ,

(1)

where 4is the temperature change, Cijkl are the material stiffness coefficients, and Eij are the thermal moduli, given by Eij = CijkjDkl, Dkl being the coefficients of linear thermal expansion. The effective values of stiffness and thermal moduli depend on the corresponding condition of either plane stress or plane strain. For the steady-state condition without heat source, the anisotropic temperature field satisfies the following equation: kij 4 ,ij 0 , (2) where kij denote the heat conductivity coefficients. In a sequentially coupled manner, the temperature field is pre-calculated by the usual BEM analysis such that all thermal data on boundary nodes, including temperature and its gradients in all directions, are determined first for the subsequent thermoelastic analysis. As this is not the focus, the BEM analysis for the associated thermal field will not be discussed here. So, all the followings just take the thermal data as known values, determined via the usual BEM analysis as stated. For a linear elastic body with thermal effects in the domain :, the displacement u j and traction t j on the boundary surface *are cross-related with each other by the well known BIE as follows:

cij ( )u j ( ) ³ Tij* (, x)u j (x)d *(x)

³U

*

*

* ij

(, x)t j (x) d *(x) ³ 4(x) E jkU ij*, k (, x) d :(x) , :

(3)

where cij ( ) are the geometric coefficients of the source point ; U ij* ( , x) and Tij* (, x) are the fundamental

solutions of displacements and tractions. As the target of the present work, the last integral in eq. (3) is a volume integral that needs to be transformed to the boundary. In this study, the fundamental solutions were from the complex variable Stroh formalism for anisotropic elasticity, which can be written in matrix form as [2] [U ij* ] U* 2 Re{[AF( z )]T } , (4a)

2 Re{[BF,s ( z )]T } ,

(4b)

1 ln( zD zˆD ) ! AT . 2S i

(5)

[Tij* ] T* where

F( zD )

In eq. (4a) and (4b), A and B are the material eigenvector matrices, Re{} denotes the real part of a complex value, the superscript T represents the transpose of a matrix, the angular brackets stand for a 3 u 3 diagonal matrix, in which each element varies with its subscript index D , and F, s wF / ws gives the derivative of F along s, the tangential path of the body. In the above equations, the general complex variables zD and zˆD for the field point and the source point, located respectively at x ( x1 , x2 ) and

zD

( xˆ1 , xˆ 2 ) , are defined by

x1 PD x2 , zˆD

xˆ1 PD xˆ2

(6)

in which PD ( D =1, 2) are the material's eigenvalues. In a sequentially coupled manner, the temperature 4, determined independently via solving the BIE for the associated field problem, is treated as the known value in eq. (3). The work aims to transform the volume integral into surface ones, denoted by (7) Vi ³ 4(x) E jkU ij*, k (, x) d :(x) :

Consider the following identity:

³

:

( f i k jk 4, jk 4k jk fi , jk ) d :

³

:

[( fi k jk 4, j ), k (4k jk f i , k ), j ] d : ,

(8)


337

where fi is the component of an arbitrary function f. As a result of applying the Green's 2nd Identity to the right hand side of eq. (8), one immediately obtains

³

:

( f i k jk 4, jk 4k jk fi , jk ) d :

³ (fk *

i

jk

4, j nk 4k jk f i ,k n j ) d * .

(9)

From eq. (2), the first term in the integrand on the left hand side of eq. (9) vanishes and thus, eq. (9) becomes

³

:

4k jk f i , jk d:

³ ( f i k jk 4, j nk 4k jk f i ,k n j )d* . *

(10)

It immediately follows that, by making the following substitution in the above equation:

k jk f i , jk

E jkU ij*,k ,

(11)

one obtains

Vi

³ (4k *

jk

f i , k n j fi k jk 4, j nk ) d * .

(12)

Now, the task remains to determine the explicit expression of fi according to eq. (11). Let f be denoted by f(zD) in what follows for the derivations. Expansion of eq. (11) results in the following matrix form:

(k11 2k12 PD k22 PD2 ) fi cc( zD ) 1T u*i ,1 T2 u*i ,2 ,

(13)

where

1

ª E11 º « E » , 2 ¬ 21 ¼

ª E12 º * « E » , ui ¬ 22 ¼

ªU1*i º « * ». ¬U 2i ¼

(14)

For brevity, the terms preceding fi cc( zD ) in eq. (13) are denoted simply by kD , namely

kD

k11 2k12 PD k22 PD2 .

(15)

Substitution of eq. (4a) into eq. (13) leads to

f icc( zD )

1

S

ª ½ º 1 Im ® « (1T A T2 A PD ! ) !» A T ¾ i i , ˆ k ( z z ) D D D ¼ ¯¬ ¿

(16)

in which Im {} denotes the operation of taking imaginary part of the complex variable in the curly bracket; i i is the unit base vector for the unit load applied in the xi direction. Thus, direct integrations of eq. (16) yield

f i ( zD )

ª ½ º z zˆD Im ® « (1T A T2 A PD ! ) D [ln( zD zˆD ) 1] ! » A T ¾ i i . S ¯¬ kD ¼ ¿ 1

Consequentially, performing spatial differentiations upon f results in

(17)

338


f i ,1 ( zD )

ª ½ º 1 Im ® « (1T A T2 A PD ! ) ln( zD zˆD ) ! » A T ¾ i i , S ¯¬ kD ¼ ¿

(18a)

f i , 2 ( zD )

ª ½ º P Im ® « (1T A T2 A PD ! ) D ln( zD zˆD ) ! » A T ¾ i i . S ¯¬ kD ¼ ¿

(18b)

1

1

Up to this point, there is still one more issue that needs to be resolved, that is, the discontinuity along the branch cut of the logarithmic function in eq. (18a) and (18b). In the process of applying the Green's Theorem, the integrand must be continuous throughout the whole domain. As a consequence, the presence of the branch cut line inside the domain will invalidate such transformation. As has been discussed in [1], this issue can be avoided by simply re-orientating the branch cut line such that it is directed in the outward normal direction. However, such treatment does not provide the general resolution, especially when the boundary surface is concave or multiply connected in geometry. Consider the general case when the negative ] 1 -axis ( ] 1 x1 xˆ1 ) cuts through the

domain so that the intersected region is bounded by [a1, b1],…[am, bm]. As a consequence, taking the similar treatment as in [1] yields a series of extra line integrals added to the transformed BIE, expressed as cij u j ³ Tij*u j d * *

m

³Ut *

* ij j

d * ³ (4k jk f i , k n j fi k jk 4, j nk ) d * ¦ ³ Li (] 1 ) d ] 1 , *

n 1

bn

an

(19)

where

Li (] 1 ) qi qi ,1 qi , 2

qi k jk 4, j nk 4k jk qi , k n j ,

(20a)

ª ½ º ] 2 Re ® «(1T A T2 A PD ! ) D ! » A T ¾ i i , kD ¼ ¯¬ ¿

(20b)

ª 1 º T½ 2 Re ® «(1T A T2 A PD ! ) ! A ¾ ii , kD »¼ ¯¬ ¿ ª T PD º T ½ T 2 Re ® «(1 A 2 A PD ! ) ! A ¾ ii . kD »¼ ¯¬ ¿

(20c)

(20d)

In eq. (20a)-(20d), Re{} takes the real part of the complex values in the curly bracket. To this end, eq. (19) is a truly boundary integral equation that can be solved for boundary unknowns by the usual BEM analysis. Next, a few numerical examples are provided for illustrating the validity of the derived formulations. Numerical tests For verifying the veracity of the transformed BIE, a few tests were carried out, where all material properties as tabulated in Table 1 just followed those in [1]. Table 1. Material properties for the numerical examples E11 (GPa) 55

E22 (GPa) 21

Q 0.25

G22 (GPa) 9.7

D D 6.3×10 (/°C)

-6

-6

20×10 (/°C)

k11

k12

3.46 0.35 (W/m °C) (W/m°C)


The illustrations are separated into two parts. The former targets demonstration of the mathematical soundness of the derived formulations, whereas the latter is for showing the implementation in AEPH, a BEM code developed by the last author. For the first part, consider a square plate as Case I, where the approach of re-defining the branch cut is employed, and also consider a hollow disc as Case II, where the approach of adding an extra line integral is adopted. The transformed surface integrals were calculated using the 32-point Gauss quadrature scheme for integrating quadratic elements, whereas the original volume integral was evaluated directly using Mathematica, a commercial software package for mathematical calculations. For Case I, the example treats a square plate, dimensioned as shown in Fig. 1, for which eight quadratic elements are applied. The principal axes of the material are rotated by 200 counterclockwise so that the properties are generally anisotropic in the global Cartesian coordinates. Suppose the plate is subjected to a temperature change, described by

339

Figure 1. A square plate subjected to a thermal load- Case I

4 3x12 5.722255 x1 x2 5 x22 ,

(21)

which satisfies eq. (2) in the global coordinates. To resolve the discontinuity problem as aforementioned, the branch cut is directed in the outward normal direction to avoid intersections with the domain. Table 2 lists all results computed by the volume integral and the transformed boundary integrals as well. As can be seen from the comparison in Table 2, very Table 2. Numerical values of volume and boundary integral - Case I excellent agreements between the both are present, except for a few nodes when the calculated values V1 V2 Node are relatively small as compared Eq. (7) Eq. (12) % Diff. Eq. (7) Eq. (12) % Diff. with all the others. Indeed, the 1 5.707E-06 5.694E-06 0.23% 5.888E-06 5.884E-06 0.06% minor percentages of difference are 2 8.300E-06 8.298E-06 0.02% -1.427E-05 -1.427E-05 0.00% mainly derived from the numerical 3 2.700E-06 2.707E-06 0.26% -1.088E-05 -1.088E-05 0.03% integration and the mesh modeling 4 -9.607E-07 -9.553E-07 0.56% -1.285E-06 -1.282E-06 0.18% itself. 5 -5.442E-07 -5.418E-07 0.45% 4.752E-06 4.754E-06 0.04% To further prove the validity 6 6.556E-06 6.557E-06 0.01% -5.193E-06 -5.193E-06 0.00% of the transformation by adding an 7 7.398E-06 7.403E-06 0.07% -1.536E-05 -1.536E-05 0.00% extra line integral, Case II considers 8 3.478E-06 3.482E-06 0.10% -1.754E-05 -1.754E-05 0.00% a thin elastic hollow disk (Fig. 2) 9 -5.707E-06 -5.694E-06 0.23% -5.888E-06 -5.884E-06 0.06% with the inner radius ri=0.5 (m) and 10 -8.300E-06 -8.298E-06 0.02% 1.427E-05 1.427E-05 0.00% the outer radius ro=1 (m). For 11 -2.700E-06 -2.707E-06 0.26% 1.088E-05 1.088E-05 0.03% treating general anisotropy, the 12 9.607E-07 9.553E-07 0.56% 1.285E-06 1.282E-06 0.18% principal axes are arbitrarily rotated 13 5.442E-07 5.418E-07 0.45% -4.752E-06 -4.754E-06 0.04% counterclockwise by 55° with 14 -6.556E-06 -6.557E-06 0.01% 5.193E-06 5.193E-06 0.00% respect to the global Cartesian 15 -7.398E-06 -7.403E-06 0.07% 1.536E-05 1.536E-05 0.00% coordinate system. The temperature 16 -3.478E-06 -3.482E-06 0.10% 1.754E-05 1.754E-05 0.00% is assumed to be distributed by

4 3x12 6.154546x1 x2 2 x22 ,

(22)

which also conforms to eq. (2) with the conductivities defined in the global Cartesian coordinates. As shown in Fig. 2, 16 elements with a total of 32 nodes are employed to model the boundary. Table 3 lists the computed values of the volume integral obtained directly by eq. (7) and the transformed one in eq. (12). For removing the discontinuity, the branch cuts of the logarithmic function are chosen to be directed in the outward normal direction for the source nodes on the outside surface; otherwise, they are re-defined in the negative x1-axis for

340


those trapped inside. Once again, the discrepancies between the both results are insignificant indeed, generally falling within 0.5% except for those with relatively small magnitudes compared with all the others. Table 3. Numerical values of volume and boundary integral - Case II Node

Figure 2. A hollow disk subjected to thermal load- Case II

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

V1 Eq. (7) -1.012E-05 -1.566E-05 -1.453E-05 -7.089E-06 3.875E-07 2.777E-06 1.988E-06 3.837E-06 1.012E-05 1.566E-05 1.453E-05 7.092E-06 -3.875E-07 -2.777E-06 -1.988E-06 -3.837E-06 2.399E-06 4.232E-07 -1.618E-06 -3.412E-06 -4.687E-06 -5.248E-06 -5.011E-06 -4.010E-06 -2.399E-06 -4.232E-07 1.618E-06 3.412E-06 4.687E-06 5.248E-06 5.011E-06 4.010E-06

Eq. (12) % Diff. -1.011E-05 0.07% -1.565E-05 0.04% -1.453E-05 0.02% -7.089E-06 0.00% 3.789E-07 2.20% 2.774E-06 0.13% 1.987E-06 0.06% 3.833E-06 0.09% 1.011E-05 0.07% 1.565E-05 0.04% 1.453E-05 0.02% 7.089E-06 0.03% -3.789E-07 2.20% -2.774E-06 0.13% -1.987E-06 0.06% -3.833E-06 0.09% 2.396E-06 0.13% 4.237E-07 0.12% -1.615E-06 0.18% -3.408E-06 0.12% -4.683E-06 0.08% -5.242E-06 0.12% -5.004E-06 0.13% -4.005E-06 0.13% -2.396E-06 0.13% -4.204E-07 0.64% 1.619E-06 0.07% 3.412E-06 0.01% 4.686E-06 0.02% 5.248E-06 0.01% 5.012E-06 0.02% 4.012E-06 0.03%

V2 Eq. (7) 1.089E-06 7.616E-06 1.222E-05 1.089E-05 5.723E-06 1.847E-06 1.754E-06 2.201E-06 -1.089E-06 -7.616E-06 -1.222E-05 -1.090E-05 -5.723E-06 -1.847E-06 -1.754E-06 -2.201E-06 -3.720E-06 -2.909E-06 -1.655E-06 -1.486E-07 1.380E-06 2.699E-06 3.607E-06 3.965E-06 3.720E-06 2.909E-06 1.655E-06 1.486E-07 -1.380E-06 -2.699E-06 -3.607E-06 -3.965E-06

Eq. (12) % Diff. 1.097E-06 0.69% 7.616E-06 0.00% 1.221E-05 0.06% 1.099E-05 0.88% 5.718E-06 0.09% 1.845E-06 0.09% 1.752E-06 0.10% 2.198E-06 0.14% -1.097E-06 0.69% -7.616E-06 0.00% -1.221E-05 0.06% -1.099E-05 0.82% -5.718E-06 0.09% -1.845E-06 0.09% -1.752E-06 0.10% -2.198E-06 0.14% -3.718E-06 0.07% -2.911E-06 0.07% -1.654E-06 0.04% -1.492E-07 0.38% 1.378E-06 0.18% 2.695E-06 0.15% 3.601E-06 0.15% 3.961E-06 0.11% 3.718E-06 0.07% 2.906E-06 0.12% 1.652E-06 0.16% 1.475E-07 0.74% -1.380E-06 0.03% -2.698E-06 0.05% -3.606E-06 0.02% -3.965E-06 0.02%

The derived formulations have been implemented in the BEM analysis. For showing the implementation, the last example also treats Case II, while the inner and outer surfaces are prescribed with 1000C and 00C, respectively. For the elastic boundary conditions, the outside surface is fully constrained in all directions, while the inner surface is free of tractions by assumption. The general anisotropic properties used for the analysis correspond to a counterclockwise rotation of the principal axes by 300 with respect to the Cartesian coordinates. As usual for the sequentially coupled BEM analysis, the thermal field was first calculated using the same mesh (Fig. 2) to provide temperature data. As a result of solving the transformed BIE (eq. 19), the displacements on the inner surface are obtained. Simply for the purpose of verification, the same problem was also analyzed by ANSYS, finite element based software. Figure 3 shows the displacements computed by the both approaches. It is seen that most data are in satisfactory agreement. Principally, the discrepancy is mainly due to the mesh modeling for the anisotropic properties. Conclusions For the 2D thermoelastic BEM analysis, it is well known the thermal effect reveals itself as a domain integral. To restore the BEM's feature of boundary discretization, the domain integral needs to be transformed to the boundary. In this paper, a direct analytical transformation for 2D anisotropic thermoelasticity is presented that involves no coordinate transformation. All kernel functions, including the newly constructed one, are based on the Stroh formulism. Also, the derived formulations have been implemented in the BEM analysis. A few numerical tests were performed, showing the veracity of the transformed boundary integral.


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Figure 3. Computed displacements on the inner surface Acknowledgement The authors gratefully acknowledge the financial support from the National Science Council of Taiwan (NSC 102-2221-E-006-290-MY3 and NSC 101-2221-E-006-056-MY3).

References [1]

Y.C. Shiah and C.L. Tan, Computational Mechanics 23, 87-96 (1998).

[2]

C.V. Camp and G.S. Gipson, Boundary Element Analysis of Nonhomogeneous Biharmonic Phenomena, Springer-Verlag, Berlin (1992).

[3]

A. Deb and P.K. Banerjee, Commun. Appl. Num. Meth. 6, 111-119 (1990).

[4]

S. Ahmad and P.K. Banerjee, J. Engrg. Mech. Div. ASCE 112, 682-695 (1986).

[5]

D. Nardini and C.A. Brebbia, Boundary Element Methods in Engineering, Computational Mechanics Publications/Spring-Verlag, Berlin (1982).

[6]

A.J. Nowak and C.A. Brebbia, Eng. Anal. Bound. Elem. 6, 164-168 (1989).

[7]

F.J.Rizzo and D.J. Shippy, Int. J. Numer. Methods Engrg. 11, 1753-1768 (1977).

[8]

J.J. Zhang, C.L. Tan, and F.F. Afagh, Computational Mech. 19, 1-10 (1996).

[9]

Y.C. Shiah and C. L. Tan, Eng. Anal. Bound. Elem. 20, 347-351 (1997).

[10] S.G., Lekhnitskii, Theory of Elasticity of an Anisotropic Elastic Body, Holden-Day Inc., San Francisco (1963). [11] C. Hwu, Anisotropic Elastic Plates, Springer, NewYork (2010).

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Slow gravity-driven migration and interaction of a bubble and a solid particle near a free surface M. Guémas1,2 , A. Sellier1 and F. Pigeonneau2 Ecole polytechnique, 91128 Palaiseau Cédex, France 2 Surface du Verre et Interfaces, UMR125 CNRS St Gobain, 39 quai Lucien Lefranc, BP 135, 93303 Aubervilliers, Cedex, France [email protected], [email protected], [email protected] 1 LadHyx.

Keywords: Free surface, bubble, solid particle, surface tension, axisymmetric Stokes flow, Boundary-integral equation.

Abstract. The axisymmetric gravity-driven migration of two interacting bubble and solid particle near a free surface is examined. The solid particle location and the bubble and free surface shapes are numerically tracked in time. This is done by solving at each time step a steady Stokes problem (with mixed-type boundary conditions) owing to a boundary approach which makes it possible to reduce the task to the treatment of two boundary-integral equations on the unbounded liquid domain boundary. The theoretical material and the relevant numerical implementation, valid for a bubble and a free surface with either equal or unequal uniform surface tensions, are briefly described and preliminary numerical results for a nearly-neutrally buoyant solid sphere interacting with a bubble and a free surface with equal surface tensions are presented. Introduction In basic applications (geophysics, glass process,...) it is important to determine the gravity-driven migration of clusters of non-rigid bubbles immersed in a liquid near a free surface. Such a task is involved since the bubbles and free surface shapes are unknown and time-dependent. Moreover, the liquid flow is in general governed by the unsteady Navier-Stokes equations. Fortunately, for small “enough” bubbles one can neglect inertial effects therefore arriving at a much more tractable creeping flow quasi-steady problem. Within this convenient framework, [1] recently proposed a boundary approach to efficiently deal with axisymmetric configurations obtained when the free surface and the bubbles share the same axis of symmetry aligned with the imposed uniform gravity g. Later [2] extended [1] to the case of bubbles and free surface not necessarily having the same surface tension. However, one also encounters in practice clusters made of both bubbles and solid particles. For instance, in liquid glass nearly neutrally buoyant solid impurities coexist with bubbles and it is important to investigate to which extent such solid particles affect the migration of the bubbles. To deal with this issue, still in axisymmetric configurations, [3] proposed a theoretical boundary formulation for clusters made of M ≥ 1 bubble(s) and N ≥ 1 not-necessarily spherical solid particles consisting in (numerically) inverting N + 1 relevant boundary-integral equations on the liquid domain boundary (i. e. the free surface and the cluster’s boundary). This paper presents preliminary numerical results for a neutrally buoyant solid sphere interacting with one bubble and a free surface having identical surface tensions. Theoretical formulation and adopted boundary procedure This section briefly gives the governing problem and the relevant bondary-integral equations employed at each time step for one bubble interacting with one solid particle (case M = N = 1). For further details the reader is directed to [3] where the theory is presented for the general case of a collection of M ≥ 1 bubble(s) and N ≥ 1 solid particle(s). Axisymmetric governing Stokes flow problem We consider a solid particle P and a bubble B immersed in a Newtonian liquid, with uniform viscosity μ and density ρ, migrating under the action of a uniform gravity field g = −ge3 towards a


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free surface. At time t the solid has boundary Σ(t), the bubble has surface S1 (t) with uniform surface tension γ1 and the free surface with uniform surface tension γ0 is denoted by S0 (t). As sketched in Figure 1, all surfaces moreover admit the same axis of revolution (0, e3 ).

z

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n n

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Figure 1: Two interacting bubble B and solid sphere P ascending near the free surface S0 (t). At initial time t = 0 the bubble is spherical with radius a while the solid particle has radius a/2. The plotted shapes are the computed ones at normalized time t = 1.90 for Bo = ρga2 /(3γ1 ) = 2 (see also Figure 2 (c) in the section devoted to the numerical results). At initial time t = 0 the free surface is the x3 = 0 plane with pressure p0 , the bubble is spherical with radius a and all surface are separated. At each time t > 0 the axisymmetric flow in the liquid domain D(t) has pressure p+ρg.x+p0 and velocity u with typical magnitude V > 0. The solid particle, with uniform density ρs and length scale as ≤ O(a), translates (no rotation for symmetry reasons) at the velocity U (t)e3 whereas the bubble has constant volume V and constant pressure p1 . Assuming that Re = ρV a/μ 1, the flow (u, p) with stress tensor σ then obeys the following quasi-steady Stokes flow problem ∇ · u = 0 and μ∇2 u = gradp in D(t), (u, p) → (0, 0) as |x| → ∞,

(1)

σ · n = (ρg · x + γ0 ∇S · n) n on S0 (t), σ · n = (ρg · x − p1 + γ1 ∇S · n) n on S1 (t),

(2)

u = U (t)e3 on Σ(t)

(3)

where n denotes the unit normal on the liquid domain boundary directed into the liquid and H = [∇S · n]/2 designates the local average curvature. The solid particle with volume Vs has negligible inertia and therefore is to be force-free. Accordingly, one requires the additional condition e3 · σ · ndS = (ρs − ρ)Vs g. (4) Σ(t)

In a similar fashion, the bubble is also force-free. Such a property reads S1 (t) e3 · σ · ndS = −ρVg. It is however already satisfied by integrating on the bubble surface S1 (t) the boundary condition (2) there (since γ1 is uniform). Finally, since the bubble has time-independent volume, (1)-(4) is also supplemented with the key relation S1 (t)

u · n dS = 0.

(5)

In summary, one has to solve for the unknown flow (u, p) and solid particle velocity U (t)e3 the equations and boundary conditions (1)-(3) in conjunction with the requirements (4)-(5). Auxiliary Stokes flow and boundary method

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As shown in [3], a trick actually permits one to determine the velocity U (t)e3 without solving the entire problem! It appeals to the auxiliary Stokes flow (u , p ), with stress tensor σ , obeying (1) and the following boundary conditions and relation u · n dS = 0. (6) σ · n = 0 on S0 (t) ∪ S1 (t), u = e3on Σ(t), S1 (t)

σ · n

As seen in [3], once the resulting traction on Σ(t) and velocity u on the surface S0 (t) ∪ S1 (t) are gained, the velocity U (t) is then obtained from the relation (easily deduced from the usual reciprocal identity [4]) [ Σ(t)

e3 · σ · n dS]U (t) = (ρ − ρs )Vs g +

1 m=0

Sm (t)

u · (ρg · x + γm ∇S · n) n dS.

(7)

Accordingly, it is sufficient to successively obtain the unknown velocity on S0 (t)∪S1 (t) and the surface traction on the solid particle boundary Σ(t) for two similar problems consisting of (1)-(3) and (5): the first one for the auxiliary flow (u , p ) and the second one (after calculating U (t) from (5)) for the liquid flow (u, p). This is done by solving two coupled boundary-integral equations and the relation (5). More precisely [3], one has here to solve the [u(y) − u(x)] · T(y, x) · n(y)dS − G(y, x) · [σ · n](y)dS −8μπu(x) + μ S0 (t)∪S1 (t) Σ(t) = G(y, x) · [σ · n](y)dS for x on S0 (t) ∪ S1 (t), (8) S0 (t)∪S1 (t) u(y) · T(y, x) · n(y)dS − G(y, x) · [σ · n](y)dS μ S0 (t)∪S1 (t) Σ(t) G(y, x) · [σ · n](y)dS for x on Σ(t), (9) = 8μπu(x) + S0 (t)∪S1 (t) u · n dS = 0 (10) S1 (t)

where the second-rank velocity G and associated third-rank stress tensor T are defined in [4]. Implementation and preliminary numerical results This section gives a few informations on the employed numerical strategy and also provides numerical results for a neutrally buoyant solid sphere interacting with one bubble and a free surface having identical surface tensions. Implementation Since the problem is axisymmetric, cylindrical coordinates (r, φ, z) with r = x2 + y 2 , z = x3 and φ the azimuthal angle in the range [0, 2π] are employed. The traces Lm of Σm for m = 0, 1 and L of Σ in the φ = 0 half plane are also introduced. Then, performing an integration of the boundary problem (8)-(10) over φ yields another boundary problem involving the previous contours L0 , L1 and L. This latter problem is solved by a boundary element technique after truncating the unbounded contour L0 . As explained in [1], the numerical treatment of the resulting linear system appeals to a discrete Wielandt’s deflation method. In practice, we track in time the (truncated) free surface S0 (t), the bubble surface S1 (t) and the solid particle boundary Σ(t) by running a Kutta-Fehlberg algorithm. At time t, the knowledge of those surfaces makes it possible to calculate the curvature there, then the solid sphere velocity U (t) and finally the liquid velocity u on the liquid domain boundary. For a time step Δt the new surfaces at time t + Δt are obtained by advancing the fluid boundary with the displacement vector Δt(u.n)n.


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Numerical results for a neutrally-buoyant solid sphere We further present numerical results for a solid sphere with uniform density ρs and radius as interacting with a bubble and a free surface of identical surface tensions γ1 = γ. Morevover, as it appears in liquid glass, the sphere is neutrally buoyant with ρs /ρ = 0.94 (a value encountered in glass process). When distant from the sphere and the free surface the bubble B is spherical with radius a and migrates at the velocity V e3 (the one obtained in an unbounded liquid) given by V = ρga2 /(3μ). Note that because its volume is preserved as time evolves B has length scale a. Henceforth, we take 2a and 2a/V as length and velocity scale, respectively. Therefore, the normalized time t is t = ρgat/(6μ). Finally, the Bond number Bo which compares at the bubble surface the “gravity” term ρg.x with the capillary “term” γ1 ∇S · n is here defined as Bo = ρga2 /(3γ1 ). Let us first take as = a/2 and Bo = 2 and put at initial time t = 0 the solid sphere between the spherical bubble and the undisturbed free surface, the initial gaps between the sphere and each other surface being equal to a. 0.5

0.5

0

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Figure 2: Computed free surface, bubble and solid sphere locations and shapes for Bo = 2 at different normalized times: t = 0 (a), t = 0.120 (b), t = 1.90 (c) and t = 2.72 (d). As seen in Figure 2, two different regimes are found as time evolves. In a first “fast” regime the bubble ascends faster than the solid sphere and the sphere-bubble gap therefore decreases faster than the gap between the sphere and the free surface whereas the free surface is weakly deformed (compare Figure 2(a) with Figure 2(b)). In a second “slow” regime, illustrated in Figure 2(c) and Figure 2(d), the bubble and the sphere slowly migrate towards the free surface which now experiences a slight deformation due to the combined action of the sphere and the bubble. In this regime liquid films take place below and above the solid sphere with the film below being the thinner one.

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Not surprisingly, the computed shapes also depend upon the free-surface and bubble surface tensions γ1 = γ0 . This is illustrated by plotting in Figure 3 the obtained shapes for Bo = 1, 2 at different normalized times t starting with the same initial configuration.

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Figure 3: Compared shapes obtained at three different normalized times t0 = 0, t1 = 1.19 and t2 for as = a/2 at two different Bond numbers. (a) Bo = 1 and t2 = 2.58. (b) Bo = 2 and t2 = 2.72. When the surface tension is weaker (case of Bo = 1 depicted in Figure 3(a)) the bubble is less “flexible” and thus speeds up more the solid sphere (compared the sphere locations at time t = 1.20 in Figure 3(a) and Figure 3 (b)) during the previously-distinguished “fast” regime. In addition, for this ratio as = a/2 the deformation of the free surface at t = 1.20 appears to be larger for Bo = 1 than for Bo = 2 although the surface tension there is larger. Not surprisingly, during the second “slow” regime in which the drainage of both thin films really takes place the free surface deformation for Bo = 2 finally is more pronounced than for Bo = 1. In absence of sphere one would obtain (see [1]) a larger deformation of the free surface at each normalized time t for Bo = 2. This actually also occurs when the sphere is sufficiently small compared to the bubble. We illustrate this behaviour in Figure 4 for the case for as = a/4.

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Figure 4: Compared shapes obtained at three different normalized times t = 0, t = 1.19 and t = 2.03 for as = a/4 at two different Bond numbers. (a) Bo = 1. (b) Bo = 2. Conclusions Our preliminary results for a nearly neutrally buoyant sphere reveal that the obtained final configuration (free surface shape) is strongly sensitive to the sphere size (compared to the bubble) and


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to the Bond number. Additional results will be given and discussed at the oral presentation with attention also paid to the case of inequal free surface and bubble surface tensions γ0 and γ1 . References [1] F. Pigeonneau and A. Sellier Low-Reynolds-Number gravity-driven migration and deformation of bubbles near a free surface. Phys. Fluids, 23, 092302 (2011). [2] M. Guémas, F. Pigeonneau and A. Sellier Gravity-driven migration of one bubble near a free surface: surface tension effects. In Advances in Boundary Element & Meshless Techniques XIII. Editors P. Prochazca and M. H. Aliabadi. 115-120 (2012). [3] M. Guémas, F. Pigeonneau and A. Sellier Gravity-driven migration of bubbles and/or solid particles near a free surface. In Advances in Boundary Element & Meshless Techniques XIV. Editors A. Sellier and M. H. Aliabadi. 400-405 (2013). [4] S. Kim and S.J. Karrila Microhydrodynamics. Principles and selected applications. Martinus Nijhoff Publishers, The Hague, 1983.

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A Fast 2D-3D BEM Approach to Dynamic Ride-Sharing A. Brancati1,2 and S.M. Siniscalchi3,4 1

Visiting Researcher, Department of Aeronautics, Imperial College London, South Kensington, SW7 2AZ, UK 2

3

[email protected]

Facoltà di Ingegneria e Architettura, Università di Enna “Kore”, Cittadella Universitaria, 94100 Enna, IT 4

[email protected]

Keywords: Dynamic, Car-pooling, Ride-sharing, Matching Driver-Passenger, Cluster Tree, and Boundary Element Method.

Abstract. A new approach for solving the matching problem in a dynamic ride-sharing system is presented. The proposed approach suggests a set of alternative journeys involving a passenger and one, or more drivers without the need of optimising a cost function. The strategy is potentially able to reduce the computational burden of fetching alternative journeys than common algorithm based upon geo-localisation of starting and ending points only. Moreover, the strategy can be easily extended to multi-hop matching algorithm. The approach utilises tools inspired by the BEM communities, i.e. 2D and 3D pre-processing geometry discretisation and binary cluster tree tools. 1.

Introduction

Carpooling is a commuting mode based on sharing a ride inside a vehicle among a driver and one or more passengers in order to reach a specific destination [1]. The driver and the passengers are a priori aware that the ride begins at a predefined time. Carpooling usually requires long-term commitment among participants on their trips, and, it can be seen as a transport demand strategy for home-to-work transport of company workers (e.g. between home and work) [2]. Carpooling is a very popular in North America, where transportation authorities support it through a series of actions e.g., designing dedicated high occupancy vehicle (HOV) lane [3], enabling integrated mobility systems based on credits, etc. In North America and Europe, there are several carpooling agencies, such as Zimride, Ridejoy, Kangaride and Delaride, Mitfahrzentrale, Carpooling, Blablacar, Amovens, and Avacar. Although carpooling companies may (i) tackle the ride-matching problem adopting a proprietary algorithm, and (ii) provide different kinds of services to their users, they all adopt the “note board” conceptual schema to find drivers and passengers, namely: users provide information, about their planned trip (e.g., time to start the trip, origin and destination points, personal information, etc.) by posting an advertisement on the note board, the rideshare company identifies potential carpooling partners. In case of one-time short trips arranged on very short notice is better to refer as Ridesharing service (also known as real-time ridesharing, instant ridesharing, dynamic ridesharing, ad-hoc ridesharing, dynamic carpooling). Ridesharing is more flexible form of carpooling in which a one-time rideshare can be arranged on a very short notice or even en-route [4], and participants’ schedules are not rigid. Despite the number of benefits that a ridesharing service could provide to communities, only a few companies provide such a system, for instance in California by Transportation Network Companies, in Germany by Flinc.org, and in Ireland by Avego. An important role in ridesharing and carpooling services is played by the internal system to evaluate the matching between a passenger request and a driver’s ride. With the exception of a few services provided by start-ups (Flinc, Avego, and Carticipat), riders and drivers are matched when the starting and the ending points of the offered and required rides correspond, thus they do not allow riders to receive a ride when their journey starting points are in the middle of the driver’s journey. This limits the real potentiality of such services. In fact, according to [3,5-6], a successful dynamic ride-sharing system requires a multi-hop strategy, that is, riders may be matched with multiple drivers to reach their target destinations. So far, only Flinc, Avego, and Carticipat provide an automated system that matches single drivers and single riders. In this regards, Agatz et al. [8] presents a review of optimisation techniques for dynamic ride sharing. The assignment problem has been widely studied by several scientists as reported in [9]. Most of these


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studies on ride-share optimisation strategies focus on solving a specific problem: (i) minimise system-wide vehicle-miles, (ii) minimise the system-wide travel time, (iii) maximise the number of participants. All these approaches assume that participants’ schedules are fixed and regular, and travel points remain the same over time. In a practical dynamic ride-sharing applications, riders and drivers constantly leave and enter the system, and the optimisation strategy adopted have to be solved very rapidly and every few minutes. Hence, an optimisation approach is not a winning strategy. In this paper, a fast matching strategy inspired by the BEM approach is proposed to efficiently carry out the dynamic ridesharing task. The matching strategy is conceived to provide a list of possible driver lifts to riders, which are free to decide whatever of the proposed journeys match better their preferences (waiting time, walking distances, travel costs, feedbacks, possibility to smoke or to bring a bulky luggage, etc.). This strategy supports to reach the critical mass, i.e. the minimum number of people that are necessary for the service to work properly. The proposed technique is part of the Italian research project, City Free project [5] funded by the MIUR (Instruction, University and Research Ministry). The goal of the project is to introduce the rideshare service in Sicily, a Southern Italian Island, where this transportation modality is still at its infancy, for instance, only 9% of university students are acquainted with this concept. Carpooling/ridesharing is a sustainable transportation modality having the following advantages: 1) reduction in the number of vehicles on the route; 2) reduction in expenses for gas, toll road fees, etc.; 3) reduction in energy consumption and pollution, 4) reduction in demand for parking spaces at destination. Therefore, Ridesharing has the potential to solve many environmental, congestion, and energy problems, and can be a key asset for modern cities. 2.

BEM based Ridematching In this section, the computational complexity of the rideshare matching problem is analysed.

2.1 Single ride problem complexity In order to match a set of drivers’ rides with a single rider’s needs, each driver journey can be discretised, using a 2D pre-processing Boundary Element Method (BEM) procedure, with m middle points, thus with m-1 linear element. In the figure 1 the driver’s journey is discretised using seven middle points (A, B, C, D, E, F, and G), and six 2D linear elements, whereas the rider’s journey starting and ending points are clear visible with capital letters H and G. Let us consider that there are n journeys posted by drivers. In order to match a rider request, the system should check if any m points of each of the n lifts matches the starting location of the rider’s journey. Hence, there is the need to make mn comparisons to match the rider’s starting point. Later the system should check if any of the identified rides have a middle point where the rider needs to conclude his journey. The complexity of the problem is then O(mn). 2.1 Two-hop ride problem complexity In case of two-hope ride, i.e. a single rider receives a lift from two different drivers, the system need to operate mn comparisons for the starting as well as for the ending point. Once two possible pools g1 and g2 of rides have been identified (one for the starting point and one for the ending point), the system should compare if any of the middle points of one set of pools matches with one of the other set. Thus, there is the need of double comparisons, with respect to the previous case, plus an extra set of further comparisons between the two sets of pools, i.e. g1g2(m-1)2. The complexity of the problem is more then the double than the previous case.

Figure 1. Discretisation of the driver (in blue) and the rider (in red) journeys. 3.

Hierarchical matching strategy

The key idea of the proposed technique is to adopt the hierarchical matrix approach [10], a rather popular tool in the BEM community to speed up solving matrix population and matrix-vector

350


multiplication, to find the best match between drivers and passengers. The proposed strategy consists upon discretising the geographical region, where the rideshare service is operating, with linear quadrilateral element as those used in the 3D BEM pre-processing approach. The basic idea is to organize the quadrilateral elements into a binary tree, called cluster tree. The entire region is set as root of the tree, which is split into two subsets, called sons, according to a pre-defined geometrical criterion. Each cluster in the tree is referred to as tree node. Herein, the root is set to be rectangular and the middle point of its maximum extension is set to split the initial rectangle into two smaller rectangles with equal dimensions (see the first level of subdivision by the black line in the Fig. 2). The process is repeated till the maximum extension of rectangles is equal to or less then a certain value, called cardinality extension. The tree nodes that are not split any further are called leaves. The figure 2 shows the first three level of the cluster tree, whereas the figure 3 shows the application of this concept to Sicily. The Italian island is contained in a 350 km x 250 km rectangle. It is worth to notice that it can be discretised into 222 171 m x 122 m elements in just 22 subsequent subdivisions. Once a geographical region has been partitioned, it is necessary to discretise each ride available in the system (i.e., rides proposed by drivers as seen previously in the figure 1). The idea is to locate each middle point of proposed rides into separate elements, where the identifiers (IDs) of rides are stored, and to generate a list of contiguous elements that compose the initial journeys. To do so, each middle point is placed in a single element using the cluster tree approach. In case the cardinality extension is set to 130 m, and Sicily is set as the region where the ride is required, only 22 comparisons are required to locate each middle point inside the correct element, having dimension 171 m x 122 m.

Figure 2. Cluster tree: Root of the tree and 3-level of subdivisions.

Figure 3. Tree-based clustering of Sicily up to the forth level. Two alternative routes from Palermo to Catania are highlighted in light blue and grey, respectively. When a passenger searches for a possible ride, the starting and ending points of his/her journey are assigned to two separate elements. Then, it is simply necessary to verify whether any of the proposed journeys has a first middle point located in the same element of the starting point of rider’s journey and a second middle point at the same element where requested journey terminates. If so, a possible match between the offered and the required rides has been found. Moreover, the strategy is easily extended for the case of two-hop ride, i.e. a single rider receives a lift from two different drivers. In this case, it is only required to check if two rides, one from the pool contained


351

in the rider’s journey starting element and the other from the pool of the ending element, have a middle point located at the same element. 3.1 Single ride problem complexity The complexity of this problem is evaluated noticing that each proposed ride is included inside the cluster tree as soon as a driver put an advertisement. The geographical region is discretised using l levels of the cluster tree, thus with 2l elements. The procedure starts by locating both initial and final points into two separate elements of the discretised region using the cluster tree approach. At this stage only 2l comparisons are required. Two pools, g1 and g2, of the proposed journeys, are identified. If there exists a journey that has two middle points inside both elements containing the starting and ending points of the rider’s journey, then a possible match has been found. This procedure requires g1g2 comparisons. The whole procedure requires 2l+g1g2 comparisons. 3.2 Two-hop ride problem complexity The procedure starts, as in the previous case, by locating both initial and final points into two separate elements, and identifying two pools, g1 and g2 inside both elements. If there exists a common element that contains two middle points of two different journeys, coming from both pools, then a possible match has been found. This procedure requires mg1g2 comparisons. 4.

Numerical aspects

4.1 General consideration In the proposed strategy, three different discretised geometries are meant to be utilised, depending on the length of the proposed or requested journeys: (i) short journeys (below 5 km), (ii) medium journeys (between 5 km and 50 km), and (iii) long journeys (above 50 km). For the first case, Sicily is discretised with 222 elements and leaves are 122 m x 171 m, for the second with the 217 elements and leaves are 684 m x 977 m, and for the last with 210 elements and leaves are 7.813 m x 10.938 m (see table 1).

Table 1. Cluster tree subdivision of Sicily: level, dimension of two edge of the geometry of Sicily, number of elements. In the first case, the leaf dimensions have been chosen to let a rider reaches the meeting point simply by walking. The largest dimension of the last two cases is approximately 1/5 of the minimum journey length.

352


Driver deviation is not taken into account in the first case, whereas it is likely that a driver is disposed to deviate to get a rider in the other two cases. It should be noticed that the proposed approach suggests a driver’s ride to riders, and it does not arrange the meeting point. Driver and rider need to set up their meeting point. This is possible in case of medium and long journeys, where it is likely that the advertisement is posted at least a 12 hours before departure time and riders could take a different means of transport to meet a driver. Nevertheless, short trips are conceived to share a ride dynamically, i.e. with a very short notice, and with drivers lose no time in deviations. 4.2 Timing of matching rides Timing is a key constraint, and it complicates the matching problem even further. Most rideshare services require users to provide time preferences by a time window representation, e.g., earliest possible departure time and latest possible arrival time. In the proposed approach, different cluster trees should be generated for different time frames. For example, our strategy would be to generate a cluster tree every hour for short journeys, every day for medium journeys, and every tree days for long journeys. The matching algorithm can then be run using either one cluster tree or contiguous cluster trees with respect to the time for requested ride service. The latter helps riders obtaining a wider number of possible rides, which is a reasonable solution especially at service kick-off when only few rides are available. 4.3 Complexity of the problem With regards to the complexity of the problem, it should be notice that a good ride-sharing service should provide thousands of rides depending on the geographical region of the service. It has been evaluated [8] that a ride-sharing service is successful only if the 1% of the local population is an active user. Around 5 million people live in Sicily, which means that around 50 thousand users are needed. If we suppose that 1/3 of them are drivers and the others are passengers, the rideshare service should daily provide around 18 thousand rides in total. If Sicily is properly discretised, a single element should contain a pool of 20 rides on the average in case of long journeys, and much less in the remaining two abovementioned scenarios. In case of short journeys, ride requests should attain a peak during rush hours. The proposed approach can be easily scaled up, and if too many journeys are assigned to the same elements, a finer mesh can be used. The assignment of the driver’s routes to the leaves of the cluster tree is performed when the driver posts his/her journey. Therefore, the associated computational costs have not been considered for the complexity evaluation of the proposed matching algorithm. 5.

Conclusion and future work

A new approach for solving the matching problem in a ride-sharing/carpooling system, using tools coming from the BEM communities, i.e. 2D and 3D pre-processing geometry discretisation and binary cluster tree, has been presented. The goal is not to solve an assignment problem minimising a cost function, but to suggest to riders a set of possible journeys that drivers may have proposed, thus to facilitate the connection between ride-sharing/carpooling users. The strategy is potentially able to suggest much more journeys than algorithm utilised by common carpooling services, based upon geo-localisation of starting and ending points only, and to strongly reduce the matching solution time for multi-hop matching algorithm. Further research is required to implement the system and to test all the possible parameters. References [1] R.F. Teal. Carpooling: Who, how and why. Transp. Res. A, Gen., 21(3), 203–214, (1987). [2] R.F. Casey, L.N. Labell, R. Holmstrom. Advanced public transportation systems: The state of the art update ’96. Tech. rep., Transportation Research Board, Washington, D.C. (1996). [3] K.L. Kelley. Casual carpooling—enhanced. Journal of Public Transportation, 10(4), 119 – 130 (2007). [4] N. Agatz, A. Erera, M. Savelsbergh, X. Wang. Sustainable passenger transportation: Dynamic ridesharing. Tech. rep., Erasmus Research Inst. of Management (ERIM), Erasmus Uni, Rotterdam (2010). [5] www.cityfreeproject.it/en, Art. 8, DD 84/Ric Social Innovation, 02/03/2012. (2014).


[6]

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W. Herbawi, M. Weber. Ant Colony vs. Genetic Multiobjective Route Planning in Dynamic Multihop Ridesharing. 23rd IEEE International Conference on Tools with Artificial Intelligence, 282 – 288 (2011). [7] Philip A. Gruebele. Interactive System for Real Time Dynamic Multi-hop Carpooling. (2008). [8] N. Agatz, A. Erera, M. Savelsbergh, X. Wang. Optimization for dynamic ride-sharing: A review. European Journal of Operational Research, 223, 295–303 (2012). [9] D.W. Pentico. Assignment problems: a golden anniversary survey. European Journal of Operational Research, 176, 774-793 (2007). [10] A. Brancati, M.H. Aliabadi, I. Benedetti. Hierarchical Adaptive Cross Approximation GMRES Technique for Solution of Acoustic Problems Using the Boundary Element Method. CMES: Computer Modeling in Engineering & Sciences, 43(2), 149-172 (2009).

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Meshfree Modelling of Elastodynamic Response of Woven Fabric Composites Y. H. Chen1a, M.H. Aliabadi1b, and P. H. Wen2c 1

2

a

Department of Aeronautics, Imperial College, London, UK, SW7 2AZ

School of Engineering and Materials Science, Queen Mary, London, UK, E1 4NS

[email protected], b [email protected], c [email protected]

Keywords: Meshfree Methods, Radial Point Interpolation, Woven Fabric Composites, Unit Cell Model, Material Homogenisation, Elastodynamic Response

Abstract: The meshfree method based on the radial point basis function is used to model the elastodynamic response of plain-woven fabric composites. A smooth fabric cell model is employed for the homogenisation of material properties of woven fabric composites. After the homogenisation, the cell model is seen as a transversely isotropic continuum, based on which the central difference algorithm is implemented into the meshfree method to model the elastodynamic response of the composites. Case studies are carried out to verify the proposed method for the analysis of elastodynamic response of plain-woven fabric composites. 1. Introduction Woven fabric composites have been widely used in aerospace engineering owning to their high strength-toweight and stiffness-to-weight ratios, the ability of being tailored to produce required mechanical properties in specific directions, excellent corrosion resistance, and other desirable features. However, the design or development of structures and systems made of woven fabric composites is still largely based on semiempirical methods due to the lack of effective and efficient computational models for the prediction of mechanical response of woven fabric composites. As a result, the development of woven fabric structures is generally time consuming and financially prohibitive. The difficulty of developing computational models for woven fabric composites to predict their mechanical responses is mainly due to their highly complex geometry, namely, the interlacing nature of warps and wefts. In the past decades, extensive research has been conducted to predict mechanical responses of woven fabric composites. Among all the methods used for modelling woven fabric composites, the direct method and the homogenisation method are the most common. The direct method, used by Naik et al [1], Duan et al [2], Ji and Kim [3], and among many others, treats a woven composite as a three-dimension system and models each woven yarn and the matrix in the composite. On the other hand, the homogenisation method (also known as unit cell method), used by Bahei et al [4], Bogdanovich [5], Ji et al [6], and among many others, treats a woven composite as an anisotropic continuum by homogenising the mechanical properties (e.g. Young’s modulus) of the whole composite. The homogenised model is then used to predict the mechanical response of the composite subject to external loads. Based on the above two methods, various models have been developed for the prediction of mechanical responses of woven fabric composites, and these models are commonly developed based on the finite element method (FEM) due to its versatility and robustness in dealing with solids. However, because of the complex architecture of woven fabric composites and the need for maintaining reasonable aspect ratio of each element during the finite element analysis (FEA), these models are either oversimplified in order to maintain reasonable efficiency or computational prohibitive in order to maintain acceptable accuracy. Considering the limitation of the FEM in dealing with woven fabric composites, Wen and Aliabadi [7] and Li et al [8] applied meshfree methods and developed smooth unit-cell models for the homogenisation of


355

mechanical properties of woven fabric composites. In such a method, the complex architecture of woven fabric composites is mathematically implemented the constitutive equations, and the contradiction between efficiency and accuracy existing in FEM-based methods is effectively avoided. In view of the efficiency of meshfree methods in addressing woven fabric composites, the meshfree method that uses radial basis functions to construct the shape functions is employ to model the elastodynamic response of plain-woven fabric composites. With this method, the material properties of composite’s unit cell are firstly homogenised. Then, the unit cell is treated as an anisotropic continuum, based on which the elastodynamic response of the cell is derived using the central difference algorithm. Finally, case studies are carried out to verify the proposed modelling method. 2. Meshfree modelling of elastodynamic response of plain-woven fabric composites 2.1 Smooth fabric model In this paper, the smooth fabric model (SFM), proposed by Wen and Aliabadi [7], is used to conduct the homogenisation of plain-woven fabric composites. Such a model is obtained by firstly deriving the representative volume cell (RVE) and then dividing the RVE into four sub-cells. The following figure is a schematic representation of a typical plain-woven fabric composite, its RVE, and the SFM. As can be seen from the figure, the SFM is one quarter of the RVE.

Fig. 1: Schematic representation of a typical plain-woven composite (a), RVE (b), and SFM (c)

Fig. 2: The geometry of smooth fabric model

356


The geometry of the SFM proposed by Wen and Aliabadi is illustrated in Fig. 2. In such a model, the crosssections of both the warp and the weft yarns are assumed as half of an elliptic, while the undulations of the yarns are assumed to comply with cosine waves. Therefore, the top and bottom surfaces of both the warp and the weft yarns can be described using the following equations: ு ଶ

െ ସ

௪௔௥௣ ሺ‫ݔ‬ǡ ‫ݕ‬ሻ ൌ

ு ଶ

െ ସ

௕௢௧ ሺ‫ݔ‬ǡ ‫ݕ‬ሻ ൌ ௪௘௙௧

ு ଶ

൅ ସ

ு ଶ

൅ ସ

௕௢௧ ௪௔௥௣ ሺ‫ݔ‬ǡ ‫ݕ‬ሻ ൌ

௧௢௣

௧௢௣

௪௘௙௧ ሺ‫ݔ‬ǡ ‫ݕ‬ሻ ൌ

ு

ு

ு

ு

గ௬ ଶ

െ ସ ඨͳ െ

ு

గ௬ ଶ

൅ ସ ඨͳ െ

గ௫ ଶ

െ ସ ඨͳ െ

గ௫ ଶ

൅ ସ ඨͳ െ

ு

ு

ு

௫మ ௏೤ᇲ

మ

௫మ ௏೤ᇲ

మ

௬మ ௏೤ᇲ

మ

௬మ ௏೤ᇲ

మ

(1)

(2)

(3)

(4)

where the parameter ܸ௬ᇱ ൌ Ͷܸ௬ Ȁߨ. The undulation angles of the warp and weft yarns are defined as: గு గ௬ ଶ ሻ ଼

ߠ௪௔௥௣ ൌ ‫ି݊ܽݐ‬ଵ ሺ

ߠ௪௘௙௧ ൌ ‫ି݊ܽݐ‬ଵ ሺെ

గு గ௫ ଶ ሻ ଼

(5) (6)

2.2 Shape function construction for meshfree modelling In this paper, radial basis functions are used for the construction of shape functions, based on which the displacement of any point of interest A=(xA, yA, zA) in the problem domain can be approximated by: ሺ‫ۯ‬ሻ ൌ σ௡௜ୀ଴ ܴ௜ ሺ‫ۯ‬ሻܽ௜

(7)

where ݊ is the number of field nodes in the support domain (see Fig. 3) or the influence domain of the point A; and ܽ௜ are the unknown coefficients to be determined from ܴ௜ ሺ‫ۯ‬ሻ, which are the distances between the point A and its support nodes. According to Hardy [9], ܴ௜ ሺ‫ۯ‬ሻ is defined as: ܴ௜ ሺ‫ۯ‬ሻ ൌ ටܿ ଶ ൅ ܽ௫ଶ ሺ‫ݔ‬஺ െ ‫ݔ‬௜ ሻଶ ൅ ܽ௬ଶ ሺ‫ݕ‬஺ െ ‫ݕ‬௜ ሻଶ ൅ ܽ௭ଶ ሺ‫ݖ‬஺ െ ‫ݖ‬௜ ሻଶ where ܿ is a free parameter and chosen as 1; ܽ௫ , ܽ௬ , and ܽ௭ are scale factors. Fields Problem domain Support domain Support nodes Po Point of interest Fig. 3: Schematic representation of the support domain of point of interest in Meshfree methods

(8)


357

Since the point of interest is arbitrary and all the support nodes can be seen as the point of interest, Eq. (7) can be rewritten as: ሺ‫ۯ‬ሻ ൌ ࢶሺ‫ۯ‬ሻ࢛ ൌ σ௡௜ୀ଴ ߶௜ ሺ‫ۯ‬ሻ ‫ݑ‬௜

(9)

where ࢶሺ‫ۯ‬ሻ is the shape function, and defined as: ࢶሺ‫ۯ‬ሻ ൌ ࡾ் ሺ‫ۯ‬ሻࡾିଵ ଴

(10)

where ࡾ் ሺ‫ۯ‬ሻ ൌ ሼܴଵ ሺ‫ۯ‬ሻǡ ܴଶ ሺ‫ۯ‬ሻǡ ǥ ǡ ܴ௡ ሺ‫ۯ‬ሻሽ

ࡾିଵ ଴

ܴଵ ሺ‫ۯ‬ଵ ሻ ܴଶ ሺ‫ۯ‬ଵ ሻ ǥ ܴ௡ ሺ‫ۯ‬ଵ ሻ ିଵ ܴ ሺ‫ ۯ‬ሻ ܴଶ ሺ‫ۯ‬ଶ ሻ ǥ ܴ௡ ሺ‫ۯ‬ଶ ሻ ൌ൦ ଵ ଶ ൪ ‫ڭ‬ ‫ڭ‬ ‫ڰ‬ ‫ڭ‬ ܴଵ ሺ‫ۯ‬୬ ሻ ܴଶ ሺ‫ۯ‬୬ ሻ ǥ ܴ௡ ሺ‫ۯ‬୬ ሻ

2.3 Homogenisation of material properties ୘

For a static problem with a body force ࢈ ൌ ൛ܾ௫ ǡ ܾ௬ ǡ ܾ௭ ൟ on domain Ω and a prescribed traction force ୘

࢚ ൌ ൛‫ݐ‬௫ ǡ ‫ݐ‬௬ ǡ ‫ݐ‬௭ ൟ on boundary Γ, the global discrete system equations can be obtained using element free Galerkin method (EFG): ࡷࢁ ൌ ࡲ

(11)

where ࡷ ൌ න ሺࡸࢶሻ் ࡯ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݖ‬ሻሺࡸࢶሻ் ݀ȳ ஐ

ࡲ ൌ න ࢶ் ࢈݀ȳ ൅ න ࢶ் ࢚݀Ȟ ஐ

߲‫׎‬௜ ‫ۍ‬ ‫ݔ߲ ێ‬ ‫ێ‬ ࡸ் ൌ ‫Ͳ ێ‬ ‫ێ‬ ‫Ͳ ێ‬ ‫ۏ‬

୻

Ͳ

Ͳ

߲‫׎‬௜ ߲‫ݕ‬

Ͳ

Ͳ

߲‫׎‬௜ ߲‫ݖ‬

߲‫׎‬௜ ߲‫ݕ‬ ߲‫׎‬௜ ߲‫ݔ‬ Ͳ

Ͳ ߲‫׎‬௜ ߲‫ݖ‬ ߲‫׎‬௜ ߲‫ݕ‬

߲‫׎‬௜ ‫ې‬ ߲‫ۑ ݖ‬ ‫ۑ‬ Ͳ ‫ۑ‬ ‫ۑ‬ ߲‫׎‬௜ ‫ۑ‬ ߲‫ے ݔ‬

It is noted in Eq. (11) that the elasticity matrix ࡯ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݖ‬ሻ varies with the position of the material point in the plain-woven fabric composite. Since the matrix material is generally isotropic, the elasticity matrices of material points in the matrix are independent of coordinates. In terms of warp and weft yarns, they are generally treated as transversely isotropic material. Due to the undulation of the yarns, the global elasticity matrices of material points in the yarns are thus dependent of coordinates. Specific form of the elasticity matrix for either the warp or the weft yarns can be found at literature [7, 8]. Considering the geometry of the SFM, it should be homogenised as a transversely isotropic continuum with z-direction as the longitudinal direction. Thus, the elasticity matrix of the homogenised continuum can be written as:

358


‫ܥ‬ҧ ‫ ۍ‬ଵଵ ҧ ‫ܥێ‬ଵଶ ҧ ‫ێ‬ ‫ܥ‬ ത ‫ ێ ؠ‬ଵଷ Ͳ ‫ێ‬ ‫Ͳ ێ‬ ‫Ͳ ۏ‬

ҧ ‫ܥ‬ଵଶ ҧ ‫ܥ‬ଵଵ ҧ ‫ܥ‬ଵଷ

ҧ ‫ܥ‬ଵଷ ҧ ‫ܥ‬ଵଷ ҧ ‫ܥ‬ଷଷ

Ͳ Ͳ Ͳ

Ͳ Ͳ Ͳ

Ͳ Ͳ Ͳ ҧ ‫ܥ‬ସସ Ͳ Ͳ

Ͳ Ͳ Ͳ Ͳ ҧ ‫ܥ‬ହହ Ͳ

Ͳ ‫ې‬ Ͳ ‫ۑ‬ Ͳ ‫ۑ‬ Ͳ ‫ۑ‬ ‫ۑ‬ Ͳ ‫ۑ‬ ҧ ‫ے‬ ‫ܥ‬ହହ

(12)

Considering that the stress integration based on the un-homogenised SFM is equal to that based on the homogenised continuum, all the material constants in Eq. (12) can be averaged and approximated by applying special boundary conditions on the SFM. Specific boundary conditions applied for material homogenisation and the expressions of all the material constants in Eq. (12) can be found at literature [8]. 2.4 Elastodynamic response of plain-woven composites Since the SFM has been homogenised, it can be treated as a transversely isotropic continuum with average ୘

density ߩ. For a dynamic problem with a body force ࢈ ൌ ൛ܾ௫ ǡ ܾ௬ ǡ ܾ௭ ൟ on domain Ω and a prescribed traction ୘

force ࢚ ൌ ൛‫ݐ‬௫ ǡ ‫ݐ‬௬ ǡ ‫ݐ‬௭ ൟ on boundary Γ, the global discrete system equations can be obtained (the damping effect is neglected here): (13)

‫ܝۻ‬ሷ ሺ‫ݐ‬ሻ ൅ ۹‫ܝ‬ሺ‫ݐ‬ሻ ൌ ۴ ࢋ࢚࢞ ሺ‫ݐ‬ሻ where ‫ ۻ‬ൌ න ߩ૖் ૖݀ȳ ஐ

۹ ൌ න ሺ‫ۺ‬૖ሻ் ۱തሺ‫ۺ‬૖ሻ݀ȳ ஐ

۴ ࢋ࢚࢞ ൌ න ૖் ‫݀܊‬ȳ ൅ න ૖் ‫݀ܜ‬Ȟ ஐ

୻

Since the displacement vector ‫ܝ‬ሺ‫ݐ‬ሻ is time dependent, the central difference algorithm is to obtain the displacement history of the problem domain. Consider n+1, n, and n-1 time steps with ∆t as step size, According to Taylor series expansions of ‫ܝ‬௡ାଵ, and ‫ܝ‬௡ିଵ about time n∆t: ‫ܝ‬௡ାଵ ൌ ‫ܝ‬௡ ൅ ο‫ܝ‬ሶ ௡ ൅

ο ଶ ο ଷ ഺ ൅‫ڮ‬ ‫ܝ‬ሷ ൅ ‫ܝ‬ ʹ ௡ ͸ ௡

‫ܝ‬௡ିଵ ൌ ‫ܝ‬௡ െ ο‫ܝ‬ሶ ௡ ൅

ο ଶ ο ଷ ഺ ൅‫ڮ‬ ‫ܝ‬ሷ െ ‫ܝ‬ ʹ ௡ ͸ ௡

By carrying out addition and subtraction over the above two expansions, the first and second order of derivatives ‫ܝ‬ሶ ௡ , ‫ܝ‬ሷ ௡ can be obtained. Substituting the two derivatives into Eq. (13), the displacement vector at time step n+1 can be explicitly calculated from the former two steps: ଵ ‫ܝۻ‬௡ାଵ ο୲మ

ଶ

ଵ

ൌ ۴࢔ࢋ࢚࢞ ൅ ሺο୲మ ‫ ۻ‬െ ۹ሻ‫ܝ‬௡ െ ο୲మ ‫ܝۻ‬௡ିଵ

3. Results and Discussions 3.1 Verification of the SFM for material homogenisation

(14)


359

To verify the accuracy of the smooth fabric model for material homogenisation, both the matrix and the yarns are assigned with the same material. In order to obtain an analytical solution for comparison, the material is assumed as isotropic, and its Young’s modulus and Poisson’s ratio are 3500 MPa and 0.0 respectively. The elasticity matrix homogenised by the smooth fabric model is:

௛௢௠

͵ͶͻͺǤͻͷ ͲǤͲͳͲͶͻ ͲǤͲͲͶͳͲ Ͳ Ͳ Ͳ ‫Ͳۍ‬ǤͲͳͲͶͻ ͵ͶͻͺǤͻͷ ͲǤͲͲͶͳͲ Ͳ Ͳ Ͳ ‫ې‬ ‫ێ‬ ‫ۑ‬ ͲǤͲͲͶͳͲ ͲǤͲͲͶͳͲ ͵ͶͻͺǤͻͷ Ͳ Ͳ Ͳ ‫ۑ‬ ൌ‫ێ‬ Ͳ Ͳ ͳ͹ͶͻǤͲͶ Ͳ Ͳ ‫ۑ‬ ‫Ͳ ێ‬ ‫Ͳ ێ‬ Ͳ Ͳ Ͳ ͳ͹Ͷ͸Ǥͺͺ Ͳ ‫ۑ‬ ‫Ͳ ۏ‬ Ͳ Ͳ Ͳ Ͳ ͳ͹Ͷ͸Ǥͺͺ‫ے‬

Since the same material is applied on both the matrix and the yarns, this cell model is actually an isotropic block. Therefore, the analytical solution of the elasticity matrix should be:

௔௡௔

͵ͷͲͲ Ͳ Ͳ Ͳ Ͳ Ͳ ‫Ͳ ۍ‬ ͵ͷͲͲ Ͳ Ͳ Ͳ Ͳ ‫ې‬ ‫ێ‬ ‫ۑ‬ Ͳ Ͳ ͵ͷͲͲ Ͳ Ͳ Ͳ ‫ۑ‬ ൌ‫ێ‬ Ͳ Ͳ ͳ͹ͷͲ Ͳ Ͳ ‫ۑ‬ ‫Ͳ ێ‬ ‫Ͳ ێ‬ Ͳ Ͳ Ͳ ͳ͹ͷͲ Ͳ ‫ۑ‬ ‫Ͳ ۏ‬ Ͳ Ͳ Ͳ Ͳ ͳ͹ͷͲ‫ے‬

As can be seen from the above results, the homogenised material constants are in very good agreement with the analytical values, which justifies the smooth fabric model. 3.2 Verification of the meshfree model for elastodynamic analysis To verify the accuracy of the proposed meshfree modelling method for the prediction of elastodynamic response, a numerical simulation is also carried out on the smooth cell model using FEM software ABAQUS for comparison. Specific loading and boundary conditions are shown in Fig. 4, where both the bottom and the top surfaces are applied with a dynamic distribution load ɐ଴ ‫ܪ‬ሺ‫ݐ‬ሻ, and ‫ܪ‬ሺ‫ݐ‬ሻ is the Heaviside function. The volume fraction ௬ and height of the cell are 0.5385 and 0.3077 respectively. The materials parameters used are shown in Table 1.

Fig. 4: Loading and boundary conditions of homogenised continuum Materials ࡱࡸ ሺ‫܉۾ۻ‬ሻ ࡱࢀ ሺ‫܉۾ۻ‬ሻ ࡳࡸࢀ ሺ‫܉۾ۻ‬ሻ

࢜ࡸࢀ

࢜ࢀࢀ

ૉሺ܏Ȁ࢓࢓૜ ሻ

Matrix

3500

-

-

0.30

-

0.00128

Fibre

47770

18020

5494

0.249

0.314

0.00258

Table 1: Materials used for matrix and fibre yarns

360


The displacement histories at three different points (‫ݔ‬ଵ ൌ ͲǤͷǡ ‫ݔ‬ଶ ൌ ͲǤͷǡ ‫ݔ‬ଷ ൌ ݄ǡ ͲǤͻ݄ǡ ܽ݊݀ͲǤͺ݄) obtained by ABAQUS and the proposed meshfree method are shown in Fig. 5. As can be seen from the figure, the results obtained by the proposed method are in good agreement with that by ABAQUS. 0.018

displacement (mm)

0.016

Meshfree, x3=1.0h

0.014

Meshfree, x3=0.9h

0.012

Meshfree, x3=0.8h

0.01 0.008

Abaqus, x3=1.0h

0.006 0.004

Abaqus, x3=0.9h

0.002

Abaqus, x3=0.8h

0 0

0.0002

0.0004 0.0006 Time (s)

0.0008

0.001

Fig. 5: Displacement histories obtained by ABAQUS and meshfree method 5. Conclusions A meshfree model for the analysis of elastodynamic response of plain-woven fabric composites is developed. In this model, a smooth fabric cell is introduced and implemented into the constitutive equations for accurate homogenisation of material properties. As such, the complex geometry modelling typical existing in FEMbased homogenisation is avoided. Based on the homogenisation, the fabric cell is treated as a transversely isotropic continuum, and the central difference algorithm is used to obtain the displacement history of the cell under dynamic loading conditions. Case studies are carried out, and the results show that the proposed meshfree model is effective for elastodynamic analysis of plain-woven fabric composites. References [1] [2] [3] [4] [5] [6] [7] [8] [9]

N. K. Naik, Y. C. Sekher, and S. Meduri Composite Science and Technology, 60(5), 731-744 (2000). Y. Duan, M. Keefe, T. A. Bogetti, and B. Powers International Journal of Mechanical Sciences, 48(1), 33-43 (2006). K. H. Ji and S. J. Kim Journal of Composite Materials, 41, 175-200 (2007). Y. A. Bahei and M. A. Zikry Composite Science and Technology, 63(7), 923-942 (2003). A. E. Bogdanovich Journal of Materials Science, 41, 6547-6590 (2006). C. Ji, B. Sun, Y. Qiu, and B. Gu Applied Composite Materials, 14, 343-362 (2007). P. H. Wen and M. H. Aliabadi Journal of Multiscale modelling, 1(2), 303-319. L. Y. Li, P. H. Wen, and M.H. Aliabadi Composite Science and Technology, 71, 1777-1788 (2010). R. L. Hardy Journal of Geophysical Research, 76(8), 1905-1915 (1971).


361

Boundary element method applied for folded thick plates D. I. G. Costa1 , E. L. Albuquerque1, P. M. Baiz2 1

University of Brasilia - Darcy Ribeiro Campus, North Wing - Brasilia - Brazil [email protected]

2

Imperial College London - South Kensington Campus, London SW7 2AZ - U.K

Keywords: Boundary Element Method, Plate, Membrane, Drilling rotation.

Abstract. In this paper a boundary element formulation is developed for the analysis of structures formed by three-dimensional association of plates. A plate element is developed with six degrees of freedom per node given by three displacements and three rotations. It is built from the association of plane elasticity and shear deformable plate formulations. From these formulations we have two displacements from plane elasticity, one displacement and two rotations from the shear deformable plate formulation. In other to obtain three rotations, an equation for the in-plane rotation (drilling rotation) is included applying plane elasticity boundary integral equation in the expression for rotation. In the three-dimensional assembly, each plate element is defined as a sub-domain. After the necessary transformation of these equations to a common reference system, displacement-rotation compatibilities and traction-moment equilibrium conditions are taken into account. Quadratic discontinuous boundary elements are then used to discretize both geometry and unknown variables of the problem. Numerical examples are presented and their results are compared to those available in literature and the ones obtained with linear discontinuous elements.

Introduction Shells are very common engineering structures that are used in various applications such as car and aircraft frames, buildings, etc. For an analysis point of view, shells can be modelled based on some assumptions and restrictions due to their shape characteristics. Some of them are that one of its dimensions are small when compared to the others and that it carries loads via bending and membrane action [6]. This makes possible to model such structures as 2-dimensional, which simplify considerably the problem. There are some possibilities for describing shell behaviour, one of them is a combination of membrane and thick plates formulations, which results in a 5-degree of freedom model, also applicable for folded thick plates. As presented earlier in [3], membrane formulation has been modified in order to include a drilling degree of freedom, which makes the combined formulation complete, i.e, with six degrees of freedom. Such development avoid some restrictions experienced in previous works [4]. In this work we first present all boundary equations involved, both, membrane and thick plate parts. Second, drilling rotation boundary integral equation is derived and included in membrane standard formulation. After this, the complete system of equation is presented and numerical examples are used to evaluate the accuracy of the results.

Boundary element equations Boundary integral equation for plane elasticity. Boundary integral equations (BIE) for elastostatics twodimensional problems can be found in many BEM basic textbooks as [2] and [5]. It is obtained using the equilibrium equations from elasticity theory and Betti’s reciprocal work theorem. Boundary integral equation for plane elasticity in absence of body forces is given by: Cαβ (x )uα (x ) =

Γ

∗ ∗ Uαβ (x , x)tβ (x)dΓ(x) − − Tαβ (x , x)uβ (x)dΓ(x) Γ

(1)

where u and t are vectors of displacements and tractions, respectively. Integrals are solved at the boundary Γ, ∗ and T ∗ are displacement and traction fundamental solutions, respectively, given by: Uαβ αβ

362


∗ Uαβ ∗ Tαβ

1 1 δαβ + r,α r,β (3 − 4ν ) ln 8πμ (1 − ν ) r

1 ∂r = − (1 − 2ν )δαβ + 2r,α r,β + (1 − 2ν )(nα r,β − nβ r,α ) 4π (1 − ν )r ∂ n =

(2) (3)

where x and x are the source and field point coordinates, μ is the shear modulus, ν the Poisson ratio, r the distance from source to field point, and nα the unity normal vector. Cαβ (x ) is added to allow different positions for the collocation point, inside domain or exactly at the boundary. Boundary integral equation for shear deformable plates. For bending of plates, governing equations are given by: Mαβ ,β − Qα

= 0

Qα ,α + q = 0

(4) (α , β = 1, 2)

(5)

where Mαβ are bending moments and Qα are shearing forces. According to [7], the BIE can be derived by considering the integral representation of the governing equations (4) and (5) via the following integral identity: Ω

[(Mαβ ,β − Qα )Wα∗ + (Qα ,α + q)W3∗ ] dΩ = 0

(6)

where Wi∗ (i = α , 3) are weight functions. Following the standard procedure, applying Green’s second identity in Equation (6) and Betti’s reciprocal work theorem, we have, for a point on the boundary (Γ) [4]: ν ∗ Wi3∗ (x , x) − q(x)dΩ (7) W Ci j (x )wi (x ) + Pi∗j (x , x)w j (x)dΓ = Wi∗j (x , x)p j (x)dΓ + (1 − ν ) λ 2 iα ,α Γ Γ Γ where Ci j is included in order to allow different positions for collocation points and λ is the shear factor, wα are rotations, w3 is out of plane displacement. The terms p j are bending moment and shear tractions, respectively given by, pα = Mαβ nβ and p3 = Qα nα . The terms Pi∗j and Wi∗j represent the Reissner plate fundamental solutions and can be found in [7]. Equation (7) represents three boundary integrals, two for rotations and one for transversal displacements. Exact boundary integral equation for drilling rotations. Elastic solids will deform when subjected to loadings, and these deformations can be quantified by knowing the displacements field throughout the body. The continuum hypothesis establishes a displacement field at all points within an elastic solid. Applying linear theory of elasticity and some kinematic considerations, it is possible to find that displacement gradient tensor is given by ui,k [8]. This tensor can be divided into a symmetric and antisymmetric parts, which are strain (eik ) and rotation (wik ) tensors, respectively. A dual vector can be associated with the rotation tensor using wi = −1/2εik j wk j . Then for out of plane rotation, we have: w3 = w21 =

1 (u2,1 − u1,2 ) 2

(8)

which is the expression for drilling rotations in plane elasticity analysis. Equation (1) gives values for displacements at a specified collocation point (source point), that can be placed in any point. Derivatives to the source point of this expression need to be found in order to compute Equation (8). Taking into account that only fundamental solutions are function of source point position, the derivative of equation (1) with respect to x is given by: ∗ ∂T∗ ∂ Uαβ ∂ uα αβ (x ) = 2 − (x , x)tβ (x)dΓ(x) − = (x , x)uβ (x)dΓ(x) (9) ∂ xγ Γ ∂ xγ Γ ∂ xγ Derivatives of displacement and traction fundamental solutions, present in Equation (9), can be found in [2]. Substituting (9) into (8), we finally find the boundary integral equation for drilling rotation:


w3 (x ) = −=

∂ T2∗β (x , x) ∂ x1

Γ

−

∂ T1∗β (x , x) ∂ x2

uβ (x)dΓ(x) + −

∂ U2∗β (x , x) ∂ x1

Γ

363

−

∂ U1∗β (x , x) ∂ x2

tβ (x)dΓ(x) (10)

where x is the collocation point. When the source point belongs to the element being integrated, there are two types of singularities in expression (10). Those involving ∗ ∂ Uαβ

∗ ∂ Tαβ ∂ xγ

terms are hypersingular of order O(r−2 ) and

those involving ∂ xγ are strong singular of order O(r−1 ). The approach used in this work for solving these integrals is the same used in [1] based on an analytical treatment. Boundary integral system of equations. For each of the collocation nodes, Equation (1) and Equation (7) can be combined to give the linear system of equations in a matrix form:

p

p w p G b 0 0 H (11) = + 0 6×1 0 Hs 6×6 u 6×1 0 Gs 6×6 t 6×1 where u and w are generalized displacements, and t and p are generalized tractions. b is the domain load vector from plate formulation. Hp , Gp , Hs and Gs are boundary element influence matrices for plate bending and plane stress formulations, as in [4]. In order to solve the boundary integral equations numerically, the problem is discretized in boundary elements. In this work, quadratic discontinuous elements were used. Shape functions for this case are given by: 3 9 ξ− 8 4 9 2 N2 (ξ ) = 1 − ξ 4 3 9 ξ+ N3 (ξ ) = ξ 8 4

N1 (ξ ) = ξ

(12) (13) (14)

Shape functions (12), (13) and 14 are used to approximate geometry, generalized displacement, and traction fields using the parameter ξ that is used to map element domain to [−1, 1]. Combining Equations (1) and (10), we obtain: ⎫ ⎤⎧ η ⎫ ⎤⎧ ⎡ mη mη c c C12 0 ⎨ uc1 0 ⎨ um C11 U11 U12 ⎬ Ne 2 1 ⎬ 1 m η m η η ⎣ Cc Cc 0 ⎦ uc1 + ∑ ∑ ⎣ U21 U22 0 ⎦ um 21 22 2 2 ⎩ ⎭ ⎩ c1 ⎭ m=1 η =1 mη mη w3 U32 0 0 0 0 1 U31 ⎡ mη ⎤ ⎧ mη ⎫ mη T11 T12 0 ⎨ t1 ⎬ Ne 2 mη mη T22 0 ⎦ t2mη = ∑ ∑ ⎣ T21 ⎩ ⎭ mη mη m=1 η =1 T31 T32 0 0 ⎡

(15)

for the modified plane elasticity formulation, where Ne is the number of elements in the problem and 1

mη Uαβ

∗ = = Tαβ Φη J(ξ ) d ξ

mη Tαβ

∗ = − Uαβ Φη J(ξ ) d ξ −1 ∗ 1 ∗ ∂ T21 ∂ T11 J(ξ ) d ξ = = Nη − ∂ x1 ∂ x2 −1 ∗ 1 ∗ ∂ T22 ∂ T12 J(ξ ) d ξ = = Nη − ∂ x1 ∂ x2 −1 ∗ 1 ∗ ∂ U21 ∂ U11 = − Nη − J(ξ ) d ξ ∂ x1 ∂ x2 −1 ∗ 1 ∗ ∂ U22 ∂ U12 = − Nη − J(ξ ) d ξ ∂ x1 ∂ x2 −1

mη U31 mη U32 mη T31 mη T32

−1

(16)

1

(17) (18) (19) (20) (21)

364


and ⎡

Nη Φη = ⎣ 0 0

0 Nη 0

⎤ 0 0 ⎦ Nη

where η is the node number within an element. At this point, the formulation in this work differs from the standard one by the presence of terms in (18) to (21), related to the drilling rotation. New elemental influence matrices for the plane elasticity are then: ⎡

mη mη U12 U11 mη mη ⎣ H = U21 U22 mη mη U32 U31 s

⎤ 0 0 ⎦ 0

(22)

and ⎡

mη T11 mη G = ⎣ T21 mη T31 s

mη T12 mη T22 mη T32

⎤ 0 0 ⎦ 0

(23)

Now, for thick plates part, combining Equations (7) with shape functions approximation of generalized displacements and tractions, we have:

⎤ ⎧ c1 ⎫ ⎡ mη mη mη ⎤ ⎧ mη ⎫ c c c C12 C13 P13 P11 P12 C11 ⎨ w1 ⎬ Ne 2 ⎨ w1 ⎬ m η m η m η η c c c c1 ⎦ w ⎣ P ⎣ C P22 P23 ⎦ wm 21 C22 C23 2 ⎭+ ∑ ∑ 21 ⎩ c1 ⎩ m2 η ⎭ mη mη mη c c c m=1 η =1 u3 C31 C32 C33 P31 P32 P33 u3 ⎡ mη mη mη ⎤ ⎧ mη ⎫ W13 W11 W12 ⎨ p1 ⎬ Ne 2 m η m η mη ⎦ pmη = ∑ ∑ ⎣ W21 W22 W23 ⎩ m2 η ⎭ mη mη mη m=1 η =1 W31 W32 W33 t3 ⎡

(24)

where Pimj η Wimj η

1

= = Pi∗j Φη J(ξ ) d ξ −1

(25)

1

= − Wi∗j Φη J(ξ ) d ξ −1

(26)

Numerical results In this section, a problem for folded thick plates is solved and quadratic boundary element formulation is compared to the linear one and to analytical solution. The geometry depicted in Fig. 1 is formed by four plates joined in such a way to form a box structure in a cantilever configuration. For this case analytical solution, according to Bernoulli-Euler’s beam theory, is given by:

w(y) =

2F 3L2 y2 − y3 , I = 1/12 L1 L33 − (L1 − 2t) (L3 − 2t)3 6EI

(27)


365

z

F

x

L3

F L2

y L1

Figure 1: Geometry of the box cantilever. Dimensions are L1 = 0.8 m, L2 = 2 m and L3 = [0.025, 0.05, 0.1, 0.2] m. Mechanical properties are E = 70 GPa, ν = 0.3, thickness t = 2 mm and Force F = 5 kN. Solutions for this problem are presented in Fig. 2. Results were obtained considering four different configurations, varying dimension L3 to 0.025, 0.05, 0.1 and 0.2 m. In each one of them, comparison between BEM with quadratic and linear elements, Bernoulli-Euler’s analytical solution and finite element (Ansys, Shell281 elements) was performed. −3

7

x 10

0.03 Quadratic boundary elements (96) Linear boundary elements (160) Bernoulli−Euler beam theory Ansys FEM

6

Quadratic boundary elements (96) Linear boundary elements (160) Bernoulli−Euler beam theory Ansys FEM

0.025

5 0.02

z/L2

z/L2

4 0.015

3 0.01 2 0.005

1

0

0

0.2

0.4

0.6

0.8

0

1

0

0.2

0.4

y/L2

(a) L3 = 0.2 m

0.8

1

0.8

1

(b) L3 = 0.1 m 0.45

0.12 Quadratic boundary elements (96) Linear boundary elements (160) Bernoulli−Euler beam theory Ansys FEM

0.1

Quadratic boundary elements (96) Linear boundary elements (160) Bernoulli−Euler beam theory Ansys FEM

0.4 0.35 0.3

z/L2

0.08

z/L2

0.6 y/L2

0.06

0.25 0.2

0.04

0.15 0.1

0.02 0.05 0

0

0.2

0.4

0.6 y/L2

(c) L3 = 0.05 m

0.8

1

0

0

0.2

0.4

0.6 y/L2

(d) L3 = 0.025 m

Figure 2: Box cantilever z-displacements results for the side aligned with y axis at coordinates x = z = 0 m. Figure 2 shows a comparison of displacements in z-direction for the side aligned with y axis at coordinates x = z = 0 m, obtained with the above mentioned methods. Table 1 presents the relative error of boundary

366


element results when compared to Bernoulli-Euler’s beam’s theory. Table 1: Maximum displacements obtained via BEM for each size of L3 and relative error to Bernoulli-Euler’s solution. L3 0.2 m error (%) 0.1 m error (%) 0.05 m error (%) 0.025 m error (%) Linear 0.011892 9.1 0.049388 6.44 0.20426 3.48 0.66955 22.75 Quadratic 0.011716 7.49 0.046772 0.8 0.18463 6.47 0.69774 19.49 Bernouli 0.0109 0.0464 0.1974 0.8667

Conclusions This work has presented a boundary element formulation for folded thick plates which presents six degree of freedom per node. The drilling rotation just included allows the formulation to overcome restrictions of problems and geometries found in previous works. Also, an analysis of the element order was performed. It has indicated that quadratic discontinuous boundary elements haven’t shown a remarkably better performance when compared to linear ones. In some cases, results were even less accurate.

Acknowledgements The authors would like to thank CAPES for the financial support of this work.

References [1] E. L. Albuquerque. Analysis of dynamic problems in anisotropic materials using the boundary element method. PhD thesis, State University of Campinas, Campinas, Brazil, 2001. [2] M.H. Aliabadi. The Boundary Element Method, Applications in Solids and Structures. The Boundary Element Method. Wiley, 2002. [3] D.I.G Costa, E.L. Albuquerque, and P.M. Baiz. Exact computation of drilling rotations with the boundary element method. [4] T. Dirgantara and M. H. Aliabadi. Boundary element analysis of assembled plate structures. Communications in Numerical Methods in Engineering, 17(10):749–760, 2001. [5] J.H. Kane. Boundary Elements Analysis in Engineering Continuum Mechanics. Prentice Hall, New Jersey, 1994. [6] W. Kanok-Nukulchai. A simple and efficient finite element for general shell analysis. International Journal for Numerical Methods in Engineering, 14:179–200, 1979. [7] Y.F. Rashed. Boundary element formulations for thick plates. Topics in engineering. WIT Press, 2000. [8] M.H. Sadd. Elasticity: Theory, Applications, and Numerics. Elsevier Science, 2009.


367

Vertical Vibration of Rigid and Flexible Foundations Presenting Inertia Properties and Interacting with Transversely Isotropic Layered Media Josue Labaki1 and Euclides Mesquita2 1,2

200 Mendeleyev St, Department of Computational Mechanics, School of Mechanical Engineering, University of Campinas, Campinas, Brazil.

Keywords: Vertical Response, Circular Foundations, Flexible Foundations, Inertial Response.

Abstract. This article presents a numerical model to investigate the vibratory response of elastic and rigid circular plates embedded in viscoelastic, transversely isotropic, three-dimensional layered media. Inertia properties of the plates or foundations are taken into account. A boundary-value problem represents the stress and displacements response of the aforementioned medium due to axisymmetric vertical ring loads. The soil-foundation interaction problem is formulated as a constrained variational problem that considers the strain and kinetic energy of the plate and of its surrounding medium. The deflection profile of the embedded plate is shown for different governing parameters such as frequency and shape of excitation, inertia properties and configuration of the surrounding medium. Introduction In a previous paper, Labaki, Mesquita and Rajapakse [1] presented a model for the vertical vibration of rigid plates interacting with transversely isotropic, layered soil. Their model provided important insights into many characteristics of the dynamic interaction between foundations and their surrounding soil medium, but it was limited as it did not take into consideration the flexibility of the foundation and the effect of its mass in the vibration of the system. The present paper provides an extension of the model presented by Labaki, Mesquita and Rajapakse [1]. It describes a numerical scheme for the investigation of the inertial response of an elastic circular plate embedded in viscoelastic, transversely isotropic, three-dimensional layered media. The consideration of flexibility of the plate is based on a work by Rajapakse [2]. The model is built according to a variational formulation, in which an energy functional involving the strain and kinetic energy of the plate and of its surrounding medium is minimized. The minimization respects the constraint that its solution must satisfy the boundary conditions at the plate edge. By incorporating a term that corresponds to the kinetic energy of the plate in this energy functional, the resulting model enables investigating the effect of the inertia of the foundation on the response of the soil-foundation system. The model of the layered media in which the plate is considered to be embedded is described in detail by Labaki, Mesquita and Rajapakse [1] and Labaki [3]. The following section presents the formulation of the flexible plate possessing mass. Inertial Response of Embedded Elastic Plate This section presents a model for the inertial response of flexible circular plates. Consider an elastic, circular plate, with unit outer radius a, thickness h, Young’s modulus Ep, Poisson ratio Qp and mass density Up, embedded within an elastic media. The plate is under the effect of time-harmonic axisymmetric vertical loads. The model of elastic plate adopted in this work comes from the classical Kirchhoff plate theory. The model considers that the thickness h of the plate is small, compared with its radius a, and that the plate undergoes small deflections [4]. A trial solution for the deflection profile of the plate due to a uniformly distributed load of intensity q(r)=q0 can be written as [2]:

368


w r

N

¦ Dn r 2n ; 0drd1

(1)

n 0

The summation in Eq. (1) comprises an approximation for the deflection of the plate w(r) by power series. Each power profile r2n is weighted by a generalized coordinate Dn (n=0,N). According to Rajapakse [2], the appropriate boundary conditions to represent the present case of plate embedded within elastic media is that the bending moment and shear force at the plate edge are zero. In view of the deflection profile established by Eq. (1), these boundary conditions can be written as:

> B@^D` ^R`

(2)

In Eq. (2), [B] is a 2u(N+1) matrix and {R} is a 2u1 vector in which:

B11 B12

B22

0

(3)

B21

2 1 Q P

B1j

ª 4 j 1 2 2 1 Q p «¬

B2 j

4 j 1

R

^D`

0 0

2

(4)

j 1 º»¼ ; 3djd(N+1)

(5)

2 j 4 ; 3djd(N+1)

(6)

T

(7)

D0 D1 " D N

T

(8)

The strain and kinetic energy of the elastic plate and of its surrounding medium can be determined from the trial deflection profile established in Eq. (1). The strain energy of the plate, Up, can be expressed in a matrix form involving the generalized coordinates D [2]: Up

^D`T ª¬K p º¼ ^D`

(9)

In Eq. (9), [Kp] is an (N+1)u(N+1) matrix, the terms of which are: p K1j

K pj1

K ijp

4 i 1 j 1 SD ª 4 i 1 j 1 2 1 Q p ¬ 2i 2 j 6

(10)

0

2i 3 º¼ ; 2di,jd(N+1)

(11)

Analogously, the corresponding kinetic energy of an elastic plate of mass density Up and thickness h can be written as [5]: Tp

^D `T ª¬ M p º¼ ^D `

in which

(12)


Mijp

§ · 1 hUp S ¨¨ ¸¸ ; 1i,j(N+1) 2 i j 1 © ¹

369

(13)

The strain energy Uh of an elastic medium of volume V is given by [6]: Uh

³

1 V H 2 V ij ij

(14)

dV

Consider that vertical external loads are applied on a circular surface of radius a within the medium, and that these loads deflect the loading surface to the shape described by Eq. (1). Let tz(r) represent the resulting vertical traction field due to these loads. Equation (14) results in: Uh

1 1 2Sr t r w r dr z 2 0

³

(15)

Let tz(r) be composed of N+1 different terms of tnz(r) (n=0,N), each corresponding to a term Dnr2n of the power series inside the summation in Eq. (1):

tz r

N

¦ t nz r

(16)

n 0

A numerical approximation of Eq. (15) is obtained by considering that the loaded surface is made up of M concentric annular discs elements of inner and outer radii s1k and s2k (k=1,M). Is it assumed that the traction fields tnz(rk) acting on the annular disc element k is uniformly distributed. For the discretized surface, it holds: M

¦ uzz ri ,s1k ,s2k , Z t nz rk , Z

w n ri Dn ri2n ; i=1,M; n=0,N

(17)

k 1

Equation (17) results in the deflection profile Dnri2n of the annular disc element i (i=1,M) and it involves M unknown traction components tnz(rk) (k=1,M). This equation can be repeated for all annular disc elements i=1,M, resulting in a set of M linear equations: M º t r , ½ ª u1,1 u1,2 u1, nz 1 zz zz zz « 2,1 2,2 2, M » ° t r , ° ° ° u u u « zz zz zz » nz 2 ® ¾ « » # % ° ° « M,1 » M, M °t r , ° u M,2 u «¬u zz » nz M ¯ ¿ zz zz ¼

w r , r 2n ½ n1 ° n 1 ° ° w n r2 , n r22n ° ® ¾ # ° ° °w r , r 2n ° n M ¿ ¯ n M

(18)

in which u i,k zz

u zz ri , s1k , s 2k , Z , i,k=1,M

(19)

In the equations above, uZZ(ri, s1k, s2k, Z) designates the axisymmetric vertical displacements observed on a ring of radius ri upon application of an also axisymmetric vertical load of frequency Z, which is uniformly distributed on an annular area of radii s1k and s2k. The dependence of the traction

370


fields tnz(r) on the composition of the elastic medium is represented by this kernel function uzz(ri,s1k,s2k,Z). The derivation of these terms for the case of a layered, transversely isotropic half-space has been presented in [1] and [3]. The substitution of the traction field tz(r) from Eq. (16) into Eq. (15) yield the strain energy of the elastic medium Uh, which, in terms of generalized coordinates, is [2]: Uh

^D`T ª¬K h º¼ ^D`

(20)

In Eq. (20), [Kh] is an (N+1)u(N+1) matrix, the terms of which are given by:

Kijh

M

¦ t(i1)z rk S s2k s1k rk2j1 ; 1di,jd(N+1)

(21)

k 1

Now let q0 denote the intensity of a loading per unit area, that is uniformly distributed on a circular area of radius Rda. A concentrated force can be represented by this loading by making R small. The potential energy Eq of this loading, in view of the deflection profile w(r) from Eq. (1), can be written in terms of generalized coordinates: Eq

Fq ^`

(22)

in which ¢Fq² is a 1u(N+1) vector given by: Fiq

q0

R 2i ; 1did(N+1) i

(23)

Consider the strain and kinetic energy of a flexible plate, Up (Eq. 9) and Tp (Eq. 12), the strain energy of an elastic medium, Uh (Eq. 20), as well as the potential energy of the external loadings, Eq (Eq. 22). The Lagrangian function L of the system comprising the flexible plate and its surrounding medium, under the effect of q0 is given by [7]:

L

U p Tp U h E p Ô`

T

> B@^D` ^R`

(24)

in which {O} is a 2u1 vector of Lagrange multipliers given by:

Ô` Ô1

O2`

(25)

The term multiplying {O} in Eq. (24) come from the boundary conditions from Eq. (2). The inclusion of this term is necessary in order for the Lagrangian function to satisfy the boundary conditions at the plate edge. Expanding Eq. (24) yields:

L

T T T ^D` ª K p º ^D` ^D ` ª M p º ^D ` ^D` ª K h º ^D` Fq ¬ ¼ ¬ ¼ ¬ ¼

^D` Ô`T > B@^D` ^R`

The Lagrangian equation of motion for the plate problem can be written as:

(26)


d§ w · w L¸ L ¨ dt © wD i ¹ wD i

371

^0` , 0iN

(27)

and

w L wOi

^0` , i=1,2

(28)

The differentiation of Eq. (24) according to Eqs. (27) and (28) results in:

ª ª Ks º «¬ ¼ « > B@ ¬

> B@T º» °^D`°½ °^F` °½ ® ¾ ® ¾ >0@ »¼ ¯°Ô` ¿° ¯°^R`¿°

(29)

where ªKs º ¬ ¼

Tº T T ª Z2 « M p M p » K p K p K h K h ¬ ¼

(30)

The numerical solution of Eq. (29), which depends on the material properties and configuration of the system of elastic plate and its surrounding medium, results in the generalized coordinates Dn (n=0,N). The deflection profile of the plate can be obtained by substitution of Dn (n=0,N) into Eq. (1). Validation and Numerical Results The present implementation was used to reproduce classical results from the literature [8]. Figure 1 shows the case of a rigid plate at the interface of an isotropic half-space and two layers of unit thickness. These results are presented in terms of the normalized dynamic vertical compliance defined by: CZZ a 0

w r 0,a 0 Es Sq0a

(31)

in which a0 is the normalized frequency of excitation defined by a0=Za/cS, in which cS (c2S=c44/U) is the shear wave propagation speed in the homogeneous surrounding medium. 0.7

0.6 Pak and Gobert Present study

0.4 0.3 0.2 0.1

0.4 0.3 0.2 0.1

0 -0.1 0

Pak and Gobert Present study

0.5

0.5

-Im[C ZZ(a0)/C0ZZ(0)]

Re[C ZZ(a0)/C0ZZ(0)]

0.6

1

2 Frequency a0

3

4

0 0

1

2

3

4

Frequency a0

(a) (b) Figure 1: Normalized vertical dynamic compliance of a rigid plate between two isotropic layers of unit thickness and an isotropic half-space.

372


These results agree with the ones presented in [8] regarding the vertical vibration of a rigid circular plate embedded at a depth H/a=2 inside an isotropic half-space. The compliance shown in Fig. 1 is normalized by the vertical static compliance of a plate resting on the surface of the half-space, C0ZZ(a0=0), whose closed-form solution was derived in [8]. Deflection Profile. Figure 2 shows the normalized deflection profile w*(r), across the middle surface of a flexible plate (Kr=0.5) due to uniformly distributed static loads (a0=0). The normalization w*(r) and the relative flexibility Kr of the plate are defined as: w r Es

w* r

q 0 1 Qs2 a

and K r

1 Qs2 Eps §¨© ha ·¸¹ E

3

(32)

in which Es and Qs are the Young’s modulus and Poisson ratio of the surrounding medium, q0 is the intensity of the loading per unit area, and h is the thickness of the plate. The plate is embedded within an isotropic half-space and different numbers of isotropic layers of thickness hi=a. In Fig. 2a, the plate is under the effect of a load per unit area that is uniformly distributed on the entire surface of the plate (R/a=1). Figure 2b shows the corresponding results for the case of a load that is concentrated on an area of radius R/a=103, which approaches the case of a point load (Eq. 23). In these and in all subsequent results, the implementation considered M=20 concentric disc elements and N=6 generalized coordinates. 1

1 H/a=0 H/a=0.5 H/a=1.0 H/a=2.0 H/a=4.0 H/a=7.0

w*(r)

0.6

0.6

0.4

0.4

0.2

0.2

0 -1

-0.5

0 radius r/a

0.5

H/a=0 H/a=0.5 H/a=1.0 H/a=2.0 H/a=4.0 H/a=7.0

0.8

w*(r)

0.8

1

0 -1

-0.5

0

0.5

1

radius r/a

(a) (b) Figure 2: Normalized deflection profile of the flexible plate for different depths of embedment due to (a) uniformly distributed loads and (b) concentrated loads. In both cases of concentrated and distributed loads, deeper embedments result in smaller amplitudes of deflection of the plate (Fig. 2). This is physically consistent, since deeper embedments correspond to stiffer surrounding media. The size of the loaded area strongly influences the deflection profile across the plate. Influence of the Frequency of Excitation. The present model of embedded plates is capable of dealing with the case of time-harmonic loads. Figure 3 shows the influence of the frequency of excitation on the central deflection (r/a=0) of a flexible plate (Kr=0.5). Uniformly distributed loads are considered. Different depths of embedment H/a for the plate are considered. The plate is situated between the halfspace and the first layer above it. All layers and the half-space are homogeneous isotropic media. Figure 3 shows a consistent physical behavior. As the embedment ratio H/a increases, the stiffness of the medium also increases, and the static displacement decreases. The dynamic behavior, on the other hand, shows an increasing oscillating characteristic for larger values of the embedment parameter H/a. These oscillations on the displacement solutions denote the presence of a free surface at z=0.


373

0 H/a=0 H/a=0.5 H/a=1.0 H/a=2.0 H/a=4.0 H/a=7.0

Real[w*(0)]

0.4 0.3

-0.05 -0.1 Imag[w*(0)]

0.5

0.2 0.1

-0.2 H/a=0 H/a=0.5 H/a=1.0 H/a=2.0 H/a=4.0 H/a=7.0

-0.25 -0.3

0 -0.1 0

-0.15

-0.35 1

2

3

4

-0.4 0

1

Frequency a0

2

3

4

Frequency a0

(a) (b) Figure 3: Influence of the frequency of excitation on the central deflection of the flexible plate under uniformly distributed loads. Inertial Response. The present formulation enables the study of cases of foundations possessing mass. The inclusion of mass is done by considering the kinetic energy of the plate according to Eq. (12). In this section, a representative case of a relatively flexible plate (Kr=0.5) of outer radius a, thickness h/a=0.1 and Poisson’s ratio Q=0.3 is considered. The plate rests on the surface of a homogeneous halfspace and its surface is under uniformly distributed unit loads (R/a=1). UP=0

Real parts

UP=0

UP/U=1

0.4

0.2

0.3 0.2

0.1

-0.1 0

UP/U=1.5

0.4

0.3

0

UP/U=1

0.5

UP/U=1.5 Abs[w*(0)]

Re[w*(0)], -Im[w*(0)]

0.5

Imaginary parts

1

0.1

2 Frequency a

3 0

4

0 0

1

2 Frequency a

3

4

0

(a) (b) Figure 4: Central displacement of a flexible plate with different mass densities. The results from Fig. 4 show how the central displacement of the plate is affected as the mass density of the plate UP varies respectively to that of the underlying half-space, U. For these particular cases, there is an increase in the amplitude of the displacement of the plate as the relative mass density increases. Concluding Remarks In the present paper, a model of thin elastic plates under small axisymmetric deflections was presented. It was shown that the problem of embedded flexible plate can be accurately represented by establishing a variational problem, in which the solution of a constrained Lagrangian energy functional involving the strain and kinetic energy of the plate and of its surrounding medium results in the deflection profile of the embedded plate. Influence functions written according to an exact stiffness method were used to represent the transversely isotropic layered media in which the plate is considered to be embedded. The present implementation has been validated with a representative source from the literature. The deflection profile of a flexible plate due to distributed and concentrated static and timeharmonic loads were investigated. Different depths of embedment were considered. The results and observations in this paper show that the present numerical model of the interaction between elastic plates and layered media provide an important contribution for the understanding of the behavior of elastic foundations and anchors embedded within layered soils.

374


Acknowledgements The research leading to this article has been funded by the São Paulo Research Foundation (Fapesp) through grants 2012/17948-4 and 2013/23085-1. The support of grant 2013/08293-7 (CEPIDFapesp), Capes, CNPq and Faepex/Unicamp are also gratefully acknowledged. References [1] Labaki, J.; Mesquita, E. and Rajapakse, R. K. N. D. Vibrations of a Rigid Circular Foundation Embedded on a Transversely Isotropic Multilayered Soil. In: International Conference on Boundary Element Techniques. Advances in Boundary Element Techniques XIII, Prague, 2012. [2] Rajapakse, R. K. N. D. (1988): The Interaction Between a Circular Elastic Plate and a Transversely Isotropic Elastic Half-Space. International Journal for Numerical and Analytical Methods in Geomechanics, 12, 419-436. [3] Labaki, J. (2012): Vibration of Flexible and Rigid Plates on Transversely Isotropic Layered Media. Ph.D. Thesis, University of Campins, Campinas, Brazil. [4] Timoshenko, S. and Woinowsky-Krieger, S. (1964): Theory of Plates and Shells, McGraw-Hill Classic Textbook Series. [5] Rajapakse, R. K. N. D. (1989): Dynamic Response of Elastic Plates on Viscoelastic Half Space. Journal of Engineering Mechanics, 115(9), 1867-1881. [6]

Fung, Y. C. (1965): Foundations of Solid Mechanics, Prentice-Hall, Englewood Cliffs.

[7] Washizu, K. (1982): Variational Methods in Elasticity and Plasticity. 2nd Ed, Pergamon Press, New York. [8] Pak, R. Y. S and Gobert, A. T. (1991): Forced Vertical Vibration of Rigid Discs with Arbitrary Embedment, J. Eng. Mech, 117, issue 11, 2527.


375

A 2D BEM-FEM model of thin structures for time harmonic fluid-soil-structure interaction analysis including poroelastic media J.D.R. Bordón, J.J. Aznárez and O. Maeso Instituto Universitario SIANI, Universidad de Las Palmas de Gran Canaria. Edificio Central del Parque Científico y Tecnológico del Campus Universitario de Tafira, 35017. Spain. E-mail addresses: [email protected], [email protected], [email protected] Keywords: SBIE/HBIE dual boundary formulation, BEM-FEM coupling, fluid-soil-structure interaction, poroelasticity.

Abstract. This paper presents a two-dimensional BEM-FEM model of thin structures for time harmonic analysis when they are surrounded by inviscid fluid, viscoelastic and/or poroelastic media. The thin structures are considered as beams under the Euler-Bernoulli hypotheses, which are discretized by the FEM. The surrounding media are discretized by the BEM, where thin structures are seen as null thickness inclusions. The usage of the conventional SBIE for null thickness inclusions leads to a singular system of equations. To overcome this difficulty, the SBIE/HBIE dual formulation is used since it is the most direct approach. The HBIE and the SBIE/HBIE dual formulation for inviscid fluids and viscoelastic solids are well known, but not in the case of poroelastic solids. For this type of medium, a regularized form of the HBIE has been derived, which together with the SBIE/HBIE dual formulation are briefly presented. Also, appropriate equilibrium and compatibility conditions that couple the BEM equations (surrounding media) and the FEM equations (thin structures) are shown. This BEM-FEM model is validated against BEM-BEM models by studying a water reservoir where bottom sediments support a wall. Introduction This work presents a two-dimensional BEM-FEM dynamic model of thin structures surrounded by inviscid fluid, viscoelastic and/or poroelastic media. Thin structures buried or immersed in these types of media can be found in many applications: noise barriers, tunnels, retaining walls, sheet piles, slurry walls, etc.. The dynamic analysis of such structures can be performed using pure BEM or FEM models. However, these pure models face many difficulties. The BEM demands a mesh carefully executed, where the size of elements greatly depends on the thickness of the structure and the quasi-singular integration capabilities of the implementation. The FEM requires an important volume mesh for the surrounding media, and besides, it is unable to naturally incorporate the Sommerfeld radiation condition. This BEM-FEM model properly combines both methods and reduces these difficulties. It has been presented already for inviscid fluids [1]. The key of this approach is assuming that the thin structures are seen by the surrounding media as null thickness inclusions, being the surrounded media treated by the BEM and the thin structures by the structural FEM. It is well known that null thickness inclusions leads to a BEM degenerate system of equations if only SBIE (Singular Boundary Integral Equations) are used [2]. This can be avoided by the multiregion approach [3], which requires fictitious boundaries that increase the number of degrees of freedom. The most direct approach is the SBIE/HBIE (Hypersingular BIE) dual boundary formulation [4], whose difficulty lies on expressing the HBIE in an evaluable form. The HBIE can be treated by many ways [5], of which the reduction of the HBIE to a set of regular or weakly singular integrals is used in this work. The particular HBIE formulation has been already developed for elastostatics [6] and elastodynamics [7], and for inviscid fluids [1]. In this work, we briefly present the HBIE and the SBIE/HBIE dual formulation for the Biot's poroelastic media. The thin structures are considered as beams under the Euler-Bernoulli hypotheses with added rotational inertia. The FEM matrices have already been presented [8,1]. SBIE and HBIE for poroelastic media The poroelastic media is assumed to be a fluid-filled poroelastic material governed by Biot's equations. The following formulation uses the notation and procedures presented in [9], but for the two-dimensional problem.

376


Let Ω be a poroelastic region, and Γ its boundary ( Γ = ∂Ω ) with an orientation defined by its outward unit normal n . The SBIE for a collocation point xi ∈ Ω can be written as: * − (U n00 + JX '*j n j ) t0k* τ −τ * J 0 τ i dΓ = Γ 00*

0 δ i + Γ * * uk −U nl0 tlk uk

lk −τ l0

u0k* U n i i * * dΓ, C u + Γ T u dΓ = Γ U t dΓ ulk* tk (1)

The fluid equivalent stress τ and the displacements uk of the solid skeleton are gathered in the vector u of primary variables, while the fluid normal displacement U n and the tractions t k of the solid skeleton are gathered in the vector t of secondary variables. The fundamental solutions matrices can be divided into four submatrices: 00, 0k, l0, and lk; where the first index indicate where the load is applied, and the second index indicate where the response is being observed (0: fluid phase, l , k = 1,2 : solid skeleton). If xi ∈ Γ , then the integration domain is partitioned as Γ = limε →0+ [(Γ − ei ) + Γi ] , being ei the exclusion zone of Γ ,

and Γi an arc of radius ε that surrounds xi . Once the integration over Γi is done, the SBIE (1) turns into:

Ci u i + lim+ ε →0

Γ− ei

T*u dΓ = lim+ ε →0

Γ− ei

Jc i U*t dΓ, Ci = 00 0

0 clki

(2)

where the free-terms c00i and clki are similar to those of the potential and the elastostatic problems, respectively. The components of the U* fundamental solution matrix can be written as:

τ 00* =

1 1 μ η, η = 2 α1 K 0 ( ik1r ) − α 2 K 0 ( ik2 r ) , α j = k 2j − k2 λ + 2μ 3 2π k1 − k22

u0k* = −

τ l0* =

1 1 Q 1 Θr,k , Θ = ΘC ik1 K1 ( ik1r ) − ik2 K1 ( ik2 r ) , ΘC = − Z 2 2π 2 R λ + μ k − k22 ! " 1

1 Θr,l 2πJ

β β 1 (ψδ lk − χ r,l r,k ) , ψ = K 0 ( ik3 r ) + ik1 r K1 ( ik3r ) − k 2 −1 k 2 ik 1r K1 ( ik1r ) − ik 2r K1 ( ik2 r ) 2πμ 3 1 2 1 2 2 2 k k 1 μ χ = K 2 ( ik3 r ) − 2 β1 K 2 ( ik1r ) − β 2 K 2 ( ik2 r ) , β j = k 2 − 1 22 λ + 2μ j k1 − k22 k3

(3)

(4)

(5)

ulk* =

(6)

where K n ( z ) is the modified Bessel function of the second kind, order n , and argument z . The components of the T* fundamental solution matrix are obtained from: * * U n00 + JX '*j n j = ( − Zu0j* − Jτ 00 , j )nj

(7)

Q * * * * t0k* = λ u0m τ 00δ kj n j , mδ kj + μ ( u0k , j + u0j , k ) + R

(8)

* U nl0 = ( − Zulj* − Jτ l0* , j ) n j

(9)


377

Q * * * * tlk* = λ ulm τ l0δ kj n j , mδ kj + μ ( ulk , j + ulj , k ) + R

(10)

* and ulk* are written in a similar fashion to the scalar wave propagation and elastodynamic Since τ 00

problems, respectively, it is easy to identify the 00 and lk submatrices of U* and T* with those of these problems. All integrals are regular or weakly singular, except those associated with tlk* , which are strongly singular and must be regularized by interpreting them in the Cauchy Principal Value sense. Their solution is well known, see for example [6]. The HBIE of the poroelastic problem is obtained by differentiating the SBIE (1) with respect to the coordinates of the collocation point, and then applying: U ni = ( − Zu ij − Jτ ,i j ) nij

(11)

Q tli = λumi ,mδ lj + μ ( uli, j + u ij ,l ) + τ iδ lj nij R

(12)

where ni = ( n1i , n2i ) is the unit normal at the collocation point. Then, the HBIE can be written as: * − s00 1 0 U ni

0 δ i + Γ * tk lk − sl0

If

* −d00 s0k* τ dΓ = Γ * * slk uk −dl0

d0k* U n i i * * dΓ, C t + Γ S u dΓ = Γ D t dΓ d lk* tk

(13)

xi ∈ Γ , then the integration domain is partitioned in a similar way as the SBIE, i.e.

Γ = limε →0+ [(Γ − ei ) + Γi ] . However, some of the integrals associated with the matrix S* are hypersingular integrals, B

which

need

certain

continuity

conditions

in

order

to

be

solved.

Let

I = F ( x) /( x − x ) dx, A < x < B be a hypersingular integral, if F belongs to the Hölder function space i 2

A

C1,α , the I exists in the Hadamard Finite Part sense. In order to fulfill this condition, it is necessary to impose that τ ( xi ) ∈ C 1 and uk ( xi ) ∈ C1 . Once the integration over Γi is done, the HBIE (13) turns into:

− J i 1 1 0 U n 1 1 − lim

2 0 δ lk tki π ! ε →0+ ε " 0

0 i τ * * 2μ (λ + μ ) i + lim+ Γ− ei S u dΓ = lim+ Γ− ei D t dΓ ε →0 uk ε →0 λ + 2μ

(14)

where it has been assumed that Γ(xi ) ∈ C 1 , i.e. the boundary is smooth at the collocation point. The

integration over Γi has produced an unbounded term. However, it is cancelled by another unbounded term that emerges when the regularization process is performed to the integrals over Γ − ei associated with S* , resulting the Hadamard Finite Part of the original integral. Since the 00 and lk submatrices from S* have the same kind of singularity of the fundamental solutions of the scalar wave propagation and elastodynamic problems, respectively, their regularization process is similar to those of these problems. The regularization process for the scalar wave propagation problem can be found in [1], and for the elastodynamic problem in [6,7]. The 0k and l0 submatrices of S* contain strongly singular integrals similar to those that appear in the off-diagonal terms of the T* matrix of the elastodynamic problem. The D* fundamental solution matrix is similar to the T* matrix, except for some signs and that ni appears instead of n .

378


SBIE/HBIE dual boundary formulation for poroelastic media The SBIE/HBIE dual boundary formulation consists in the simultaneous collocation of the SBIE and the HBIE on the boundaries of null thickness inclusions, i.e. cracks, voids, or in our case, thin elastic bodies. It is the most direct and general approach to face problems with null thickness inclusions. Other techniques could be found in the introduction section of [4]. Let Γ be the boundary of a region Ω , resulting from the approaching of two identical boundaries, Γ + and Γ − , whose normal vectors are pointing at each other, until they are coincident. The face Γ + is the reference face, thus n and ni are defined on it. The variables of each face are indicated by ,+ or ,− . When the collocation point xi ∈ Γ , the integration domain for both the SBIE and the HBIE is (see Fig. 1):

{

Γ = lim+ Γ + − ei + + Γ i + + Γ − − ei − + Γ i − ε →0

}

(15)

Figure 1: Integration domain for the dual boundary formulation

If it is assumed that Γ(xi ) ∈ C 1 , then the regularized SBIE/HBIE dual boundary formulation for poroelastic media can be written as: 1 J 0 i+ i− * *

( u + u ) + CPV Γ T u dΓ = RPV Γ U t dΓ 2 0 δ lk 1 1 0 i + i − * *

( t − t ) + H PV Γ S u dΓ = CPV Γ D t dΓ 2 0 δ lk

(16)

where RPV (Riemann Principal Value), CPV (Cauchy Principal Value) and HFP (Hadamard Finite Part) before the integral operator sign is used to indicate that only the finite part of those integrals are considered. The integration and regularization process of those integrals is the similar to the previously explained. BEM-FEM coupling The BEM dual boundary formulation together with a FEM model for the thin structures and appropriate coupling conditions make possible to build a BEM-FEM model for thin structures immersed or buried in inviscid fluid, viscoelastic and/or poroelastic media. The FEM model of the thin structures is not explained in this paper, but it could be found in [8,1]. In this paper, the coupling conditions when the surroundings is a poroelastic region are presented. The coupling conditions for inviscid fluids [1] and viscoelastic solids can be obtained as particular cases of the poroelastic case. Let ϒ j be a BEM-FEM poroelastic soil – structure element composed by three sub-elements: ϒ +j and ϒ −j (boundary elements), and ϒ sj (finite element); see Fig. 2. The boundary elements are 3-noded quadratic

line elements, where each node is associated with four variables: fluid phase normal displacement U n , fluid phase equivalent stress τ , solid skeleton displacement vector u , and solid skeleton traction vector t . The


379

finite element is a 3-noded straight beam element, and has eight degrees of freedom. The vertex nodes i = 1, 2 have translation u ( i ) and rotation θ ( i ) , while the central node only has translation u (3) . The local axes are defined by the pair of vectors x '1 and x '2 . Each node is associated with axial s '1( i ) and lateral s '(2i ) forces due to axial and lateral load distributions, respectively.

Figure 2: Coupling between sub-elements ϒ +j , ϒ −j and ϒ sj (local numbering) The compatibility equations for a node i are: U n( i + ) = u ( i + ) ⋅ n, U n(i − ) = −u ( i − ) ⋅ n

(17)

u (i + ) = u (is ) , u ( i − ) = u (is )

(18)

where (25) establishes that both faces of the structure are impervious, and (26) establishes a perfect bonding between the soil and the structure. The equilibrium equation for a node i is:

τ ( i + ) n + t ( i + ) − τ ( i − ) n + t (i − ) + s '1( is ) x '1 + s '(2is ) x '2 = 0

(19)

which is in reality two equations, one for each coordinate. By examining Fig. 2, each vertex node has 17 unknowns in total, while the central node has 16 unknowns. The number of equations are: 3 (SBIE), 3 (HBIE), 3 for vertex nodes or 2 for the central node (FEM), 2 (impervious condition), 4 (perfect bonding), and 2 (equilibrium); in total 17 equations for vertex nodes, and 16 for the central node. Validation of the BEM-FEM model A modified problem from [9] is used to validate the model. The problem is a simplified water reservoir where bottom sediments support a wall (see Fig. 3). The upper part of the wall is immersed in the water and its base is buried in the bottom sediments. The reservoir has free tractions on the left and top boundaries, and horizontal displacements on the right and bottom boundaries. Four regions are present: dam wall Ω1 (viscoelastic solid), bottom sediments Ω 2 (poroelastic solid), water Ω3 (inviscid fluid), and the immersed/buried wall Ω 4 (viscoelastic solid). The viscoelastic regions Ω1 and Ω 4 have the following properties: density ρ = 2481.5 kg/m3 , shear modulus μ = 11500 MPa , Poisson’s ratio ν = 0.20 and damping coefficient ξ = 0.05 . The poroelastic region Ω 2 has the following properties: fluid phase density

ρ f = 1000 kg/m3 , solid skeleton density ρ s = 2640 kg/m3 , Lamé’s first constant λ = 17.9753 MPa , μ = 7.7037 MPa , damping coefficient ξ = 0.05 , porosity φ = 0.60 , null added density, Biot’s constants

380


R = 1.24416 ⋅ 109 N/m 2 and Q = 829.44 ⋅ 106 N/m 2 , and dissipation constant b = 3.5316 ⋅ 106 Ns/m 4 . The

fluid region Ω3 has a density ρ = 1000 kg/m3 and a wave propagation speed c = 1438 m/s .

Figure 3: Water reservoir with bottom sediments supporting a wall Three cases with three different widths of the immersed/buried wall are considered w = {1, 2,5} m , or in

terms of the slenderness L / w = {40, 20,8} . These cases are solved using the complete geometry using a BEM-BEM model, and using the developed BEM-FEM model for the immersed/buried wall. Note that the BEM-BEM model needs a different mesh for each case, while the BEM-FEM model only needs one mesh. The points B, C, DI, DD, EI and ED are selected points where several results are going to be plotted. In the plots, the normalized frequency ω / ω1 is used, where ω1 = 6.769 rad/s is the first natural frequency of the dam wall on rigid foundation. Fig. 4 and Fig. 5 show the u1 amplification factor abs[(u1 − 1) /1] at the point B and C, respectively. They show that the BEM-FEM model gets close to the BEM-BEM model as the slenderness of the wall increase. Even so, when the wall has slenderness L / w = 8 m , which is in the limit to consider it as a thin structure, the BEM-FEM model obtains a good reproduction of the response. Fig. 6 shows the pressure at the water at points DI and DD. Fig. 7 shows the fluid equivalent pressure at the bottom sediments at points EI and ED. Again, they show that the BEM-FEM model get close to the BEM-BEM model as wall thickness decrease.

Conclusions A two-dimensional BEM-FEM model of thin structures for time harmonic analysis when they are surrounded by inviscid fluid, viscoelastic and/or poroelastic media has been presented. The key of the model is using the BEM dual boundary formulation for the surroundings, and a beam finite element for the thin structures. The model has been validated through a simple water reservoir problem, showing excellent agreement.

Acknowledgments This work was supported by the Subdirección General de Proyectos de Investigación of the Ministerio de Economía y Competitividad (MINECO) of Spain and FEDER through research project BIA2010-21399C02-01. Also by the Agencia Canaria de Investigación, Innovación y Sociedad de la Información (ACIISI) of the Government of the Canary Islands and FEDER through research project ProID20100224. J.D.R. Bordón is a recipient of the fellowship TESIS20120051 from the Program of predoctoral fellowships of the ACIISI. The authors are grateful for this support.


Figure 4: u1 amplification factor at point B

Figure 5: u1 amplification factor at point C

Figure 6: pressure p at points DI and DD

381

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Figure 7: fluid equivalent stress τ at points EI and ED References [1] J.D.R. Bordón, J.J. Aznárez and O. Maeso. A 2D BEM-FEM approach for time harmonic fluidstructure interaction analysis of thin elastic bodies. Eng Anal Bound Elem, 43, 19-29 (2014). [2] G. Krishnasamy, F.J. Rizzo and Y. Liu. Boundary integral equations for thin bodies. Int J Numer Methods Eng, 37, 107-121 (1994). [3] G.E. Blandford, A.R. Ingraffea and J.A. Liggett. Two-dimensional stress intensity factor computations using the boundary element method. Int J Numer Methods Eng, 17, 387-404 (1981). [4] A. Portela, M.H. Aliabadi and D.P. Rooke. The dual boundary element method: effective implementation for crack problems. Int J Numer Methods Eng, 33, 1269-1287 (1992). [5] Tanaka M, Sladek V, Sladek J. Regularization techniques applied to boundary element methods. Appl Mech Rev, 47, 457-99 (1994). [6] A. Sáez, R. Gallego and J. Domínguez. Hypersingular quarter-point boundary elements for crack problems. Int J Numer Methods Eng, 38, 1681-1701 (1995). [7] F. Chirino and R. Abascal. Dynamic and static analysis of cracks using the hypersingular formulation of the boundary element method. Int J Numer Methods Eng, 43, 365-388 (1998). [8] L.A. Padrón, J.J. Aznárez and O. Maeso. BEM-FEM coupling model for the dynamic analysis of piles and pile groups. Eng Anal Bound Elem, 31, 473-484 (2007). [9] J.J. Aznárez, O. Maeso and J. Domínguez. BE analysis of bottom sediments in dynamic fluid-structure interaction problems. Eng Anal Bound Elem, 30, 124-136 (2006).

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