Advances in Boundary Element & Meshless ...

Advances in Boundary Element & Meshless Techniques XIX

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 as applied to the Boundary Element Method and Meshless Techniques. Previous conferences devoted to 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), Paris, France (2013), Florence, Italy (2014), Valencia, Spain (2015), Ankara, Turkey (2016) and Bucharest, Romania (2017).

Advances in Boundary Element & Meshless Techniques XIX

Edited by ISBN 978-0-9576731-5-1

EC ltd

F. Garcia-Sanchez L. Rodriguez-Tembleque M.H. Aliabadi

Advances In Boundary Element and Meshless Techniques XIX

Advances In Boundary Element and Meshless Techniques XIX

Edited by Felipe García Sánchez Luis Rodríguez-Tembleque Ferri M H Aliabadi

EC

ltd

Published by EC, Ltd, UK Copyright © 2018, Published by EC Ltd, 18 Sir Lancelot Close, Eastleigh, SO53 4HJ, UK Phone (+44) 2380 260334

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ISBN: 978-0-9576731-5-1

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International Conference on Boundary Element and Meshless Techniques XIX 9-11 July 2018, Malaga, Spain Organising Committee: Organising Committee: Professor Felipe García Sánchez UNIVERSIDAD DE MÁLAGA Escuela de Ingenierías. Despacho 2044.D Calle Dr. Ortiz Ramos s/n 29071 MÁLAGA [email protected] Professor Luis RodriguezTembleque Universidad de Sevilla, Dpto. Mecánica de Medios Continuos y Teoría de Estructuras Escuela Técnica Superior de Ingeniería, Avda. Camino de los Descubrimientos, s/n, 41092, Sevilla [email protected] Professor Ferri M H Aliabadi Department of Aeronautics Imperial College, London South Kensington Campus London SW7 2AZ Tel: +44 (0) 20759 45077 [email protected] International Scientific Advisory Committee Benedetti,I (Italy) Chen, Weiqiu (China)

Chen, Wen (China) Cisilino,A (Argentina) De Araujo, F C (Brazil) Delvare,F (France) Denda,M (USA) Dong,C (China) Dumont,N (Brazil) Gao,X.W. (China) Garcia-Sanchez,F (Spain) Hematiyan,M.R. (Iran) 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) Pineda,E (Mexico) Qin,Q (Australia) Saez,A (Spain) Sapountzakis E.J. (Greece) Sellier, A (France) Sharif Khodaei, Z (UK) Shiah,Y (Taiwan) Sladek,J (Slovakia) Saldek, V (Slovakia) Sollero.P. (Brazil) Taigbenu,A (South Africa) Tan,C.L (Canada) Telles,J.C.F. (Brazil) Wen,P.H. (UK) Wrobel,L.C. (UK) Yao,Z (China) Zhang, Ch (Germany))

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), Paris, France (2013), Florence, Italy (2014), Valencia, Spain (2015), Ankara, Turkey (2016) and Bucharest, Romania (2017). The present volume is a collection of edited papers that were accepted for presentation at the Boundary Element Techniques Conference held at the NH Hotel, Malaga, Spain, 9-11th July 2018. 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

CONTENTS A boundary element formulation for micromechanical modeling of piezoelectric ceramics I. Benedetti, V. Gulizzi and A. Milazzo

1

Identification of heat fluxes based on strain measurement and meshfree methods A. Khosravifard, A. Razavi, M.R. Hematiyan

7

Bimodular materials with finite block method in polar coordinate QX. Pan, JL. Zheng, PH. Wen and MH. Aliabadi

15

A Study on Spring-Dashpot Equivalency for the Dynamic Response of Piled Rafts: the Extent to which Energy Transmission through the Soil can be Disregarded PLA. Barros, J. Labaki, LF Lima and E. Mesquita

23

BEM Analysis of Thermoelastic Stresses in 3D Generally Anisotropic Solids Embedded with Non-uniform Heat Source YC. Shiah and Nguyen Anh Tuan

29

Complex-step assessment of shape sensitivity for the 3D indentation problem C Ubessi, RJ. Marczak and FC. Buroni

37

Hypersingular Boundary Integral Equation for harmonic acoustic problems in 2.5D domains J. Pizarro-Ruiz, E. Puertas and R.Gallego.

45

An IGABEM application in damage mechanics V. Mallardo, E. Ruocco, and G. Beer

51

Boundary Element Analysis of Damage Detection in Plates Using In-Plane Ultrasonic Waves Jun Li, Z. Sharif Khodaei, M.H. Aliabadi

57

Boundary element analysis of fractured CNT-polymer nanocomposites F. García-Sánchez, E. García-Macías, L. Rodríguez-Tembleque, and A. Sáez The boundary element method: a simple code for the arbitrarily high accurate analysis of 2D static and frequency-domain problems of general topology and shape NA. Dumont, Wellington T. de Carvalho and CA. Aguilar

61

An Efficient Fluid-Rigid Body Interaction Simulation of a 2 DOF Oscillating Wing Energy Harvester D. Talarico, A. Mazzeo, and M. Denda

75

The Application of BEM to MHD flow and heat transfer in a rectangular duct with temperature dependent viscosity E. Ebren Kaya and M. Tezer-Sezgin

83

High-cycle fatigue in polycrystalline materials by boundary elements I. Benedetti, V. Gulizzi

91

67

A Boundary Integral velocity representation of the steady Navier-Stokes equations for a body in an exterior domain uniform flow field E. Chadwick

97

A quasi-static delamination model with interface damage and friction for layered structures R. Vodička, F. Kšiňan

103

Inference of the Equivalent Initial Flaw Size Distribution with the Boundary Element Method using Maximum Likelihood Estimation L.Morse, Z.Sharif-Khodaei and M. H.Aliabadi

111

Fundamental coupled MHD viscous flow and electric potential produced by a point force located in a conducting liquid bounded by two parallel plane solid walls A. Sellier

115

Particle-wall interactions in axisymmetric MHD viscous flow A.Sellier and S. H. Aydin

123

The seismic site effects study of adjacent non-curved valleys based on boundary element method Z Khakzad, B.Gatmiri and D.Amini.

131

Green element and Shuffled Complex Evolution methods for groundwater contaminant source identification E.Onyari, A.Taigbenu and J. Ndiritu

139

Java application to solve thermoelastic contact problems using the Boundary Element Method J. Vallepuga-Espinosa, L. Sánchez-González and I. Ubero-Martínez

147

Methodology for automatic integration of nearly and weakly singular integrals arising in the boundary element method JDR. Bordón, JJ. Aznárez and O. Maeso

157

Approximated Fundamental Solutions Based on Levi’s Functions R.Gallego, E.Puertas

167

Somigliana Identity for Unbounded Cone-Shaped Domains V Mantič

173

Mesh generation in three-dimensional boundary element problems FM. Loyola, EL. Albuquerque

175

The Method of Fundamental Solutions for Two-Dimensional Stationary Thermoelastic Problems Involving Curved Line Heat Sources M. Mohammadi, MR. Hematiyan

183

New Approach with Domain Superposition to Solve Piecewise Homogeneous Elastic Problems LOC. Lara, C F. Loeffler, JP. Barbosa, WJ. Mansur

191

A couple-stress formulation for the boundary element method G. Hattori, P.A. Gourgiotis, J. Trevelyan

199

Boundary element contact modelling of CNT-polymer reinforced composites L. Rodrıguez-Tembleque, E. Garcıa-Macıas and A. Saez

205

Micro dynamic failure of 2D crystal aggregate structures using BEM and a hierarchical multiscale cohesive zone model JE. Alvarez, AF. Galvis and P. Sollero

213

Meshfree Analysis of Viscoplastic Problems Using Different Domain Integration Techniques Z. Kazemi, M.R. Hematiyan, A. Khosravifard

221

A hierarchic constitutive governed recursive methodology for obtaining threedimensional anisotropic fundamental solution: a theoretical approach Tales V. Lisbôa, Rogério J. Marczak

229

Application of SPR for Discontinuous Boundary Elements Results in 2D Elasticity and a New Method to Evaluate Tangential Stress Otávio A. A. da Silveira, Rogério J. Marczak

237

3D frictional contact analysis with Boundary Element Method and discontinuous elements using a Generalized Newton Method with line search Cristiano J.B. Ubessi , Rogério J. Marczak

245

Series expansion of the axi-symmetric fundamental solution for transversely isotropic elastic solids Enrique Graciani, Inmaculada Gómez-Rodríguez

251

Advances in Boundary Elements & Meshless Techniques XIX

1

A boundary element formulation for micromechanical modeling of piezoelectric ceramics I. Benedetti1,a, V. Gulizzi1,b and A. Milazzo1,c 1

Dipartimento di Ingegneria Civile, Ambientale, Aerospaziale e dei materiali,

Università degli Studi di Palermo, Viale delle Scienze, Edificio 8, 90128, Palermo, Italy a

[email protected], [email protected], [email protected]

Keywords: Boundary Element Method, Micromechanics, Piezoelectric ceramics, Polycrystalline materials

Abstract. In this work, we present a boundary element formulation for modelling polycrystalline piezoelectric ceramics at the scale of their constituent grains. The micro-morphologies are artificially generated using the Voronoi tessellation algorithm. The formulation is based on a multi-domain boundary element approach whereby each grain is modelled as a linear elastic generally anisotropic piezoelectric region. The fundamental solutions for piezoelectricity and their derivatives are obtained by means of a recently developed series representation in terms of spherical harmonics. Several periodic morphologies are tested to compute the effective properties of unpoled and partially poled piezoelectric polycrystalline materials. Introduction Piezoelectric ceramics are widely employed in engineering applications for their well-known capability of coupling the mechanical and the electrical fields. Bulk piezoelectric transducers for real-time Structural Health Monitoring (SHM) systems and for micro electro-mechanical systems (MEMS) represent typical examples [1,2]. However, such applications are limited by the brittleness of piezoelectric ceramic materials, whose performance can be deteriorated by the presence and the development of micro-cracks. For this reason, accurate and robust numerical tools for understanding and predicting their behavior are of relevant engineer interest. In the last decades, the advancements in materials microstructure characterization and the availability of high performance computing (HPC) facilities have allowed to study and model generally heterogeneous materials at the scales of their constituents, thus enhancing the understanding of the link between the microstructure properties and the macroscopic material behavior [3]. Polycrystalline materials, whose macroscopic properties are highly influenced by the size, the orientation and the deformation and failure mechanisms of the grains as well as by the properties and the failure mechanisms of the grain boundaries, are a typical example of heterogeneous materials. Within the field of computational mechanics, polycrystalline materials with elastic and elastoplastic constitutive behavior have been extensively studied by means of finite element method (FEM) approaches [4], whereas the literature on numerical models for piezoelectric polycrystals is not as rich [5]. In both cases, a drawback of FEM-based models is the high number of degrees of freedom of the resulting system, which may limit their applications, especially in the case of three-dimensional models. As an alternative to FEM models, the boundary element method (BEM) has proved effective in reducing the computational efforts in terms of the total degrees of freedom, given the appealing feature of expressing the problem in terms of boundary unknowns only [6]. In fact, BEM models have been successfully employed to study intergranular micro-cracking in 2D [7,8] and 3D [9,10] polycrystalline materials with elastic constitutive behavior. More recently, 3D BEM models have been employed to investigate specific deformation and failure mechanisms in polycrystals such as crystal plasticity [11], stress corrosion cracking [12] and inter- and trans-granular failure competition [13]. Concurrent two-scale models where macroscopic damage is extracted from explicit microscopic simulations of polycrystalline materials are available in the literature [14,15] and exemplify the effectiveness and capabilities of BEM approaches. BEM has also been employed to model piezoelectric bodies and for fracture mechanics problems [16,17].

2

Eds: F.García Sánchez, L.Rodríguez-Tembleque, M.H>Aliabadi

In this work, a boundary element formulation for polycrystalline materials with piezoelectric constitutive behavior is presented. The polycrystalline domain is generated using the Voronoi tessellation algorithm and the formulation is based on a multi-region approach where each Voronoi region represents a piezoelectric grain. To reproduce unpoled and partially poled piezoelectric polycrystals, the grains orientation is given in terms of a prescribed orientation distribution function. The grain boundaries are modelled as perfect interfaces where displacement and electric potential continuity and traction and surface charge equilibrium are enforced. The comparison of the presented numerical results with those available in the literature and with those provided by analytical bounds shows the accuracy of the formulation. Boundary Element formulation for polycrystalline piezoelectric materials In this section, the boundary element formulation for three-dimensional polycrystalline piezoelectric materials is presented. The morphology of each polycrystalline specimen is artificially generated using the Voronoi tessellation algorithm and numerically discretized as described in [18]. The reader interested in the details of the preprocessing stages of the present formulation, e.g. the features of the adopted meshing technique, is referred to Ref.[18]. For the scope of the present work, the constitutive relationships and the governing equations of linear piezoelectricity are first recalled; then, the model of the generic piezoelectric grain and of the overall aggregate is obtained by means of the boundary element formulation described next. In the following, u = {ui } and φ denote the displacements field and electric potential, respectively, σ = {σ ij } and γ = {γ ij } are the stress and strain tensors, respectively, and E = {Ei } and D = {Di } are the electric field and electric displacement vectors, respectively. Eventually, t = {ti } and ω denote the boundary tractions and the surface charge density, respectively, which are related to the stress tensor and the electric displacement vector as ti = σ ij n j and ω = − Di ni , respectively, being ni the outward unit normal of the considered piezoelectric domain. Piezoelectric constitutive behavior. The constitutive behavior of linear piezoelectricity can be expressed by the following relationships E σ ij = cijkl γ kl − ekij Ek

Di = eikl γ kl + κ ijγ E j

(1)

E where cijkl , eikl and κ ijγ are the elasticity tensor at constant electric field, the piezoelectric coupling tensor and the dielectric tensor at constant strain, respectively. In the previous equation and in the remaining part of the paper, summation is assumed over repeated subscripts.

Grain governing equations and boundary element formulation. In absence of body forces and volume free electric charges, the governing equations of piezoelectric media are the mechanical equilibrium

∂σ ij / ∂x j = 0

(2)

and the Gauss’ law for electrostatics

∂Di / ∂xi = 0,

(3)

which are coupled to the strain-displacement relationship

γ ij = (∂ui / ∂x j + ∂u j / ∂xi ) / 2

(4)

and the electric field-electric potential relationship

Ei = −∂φ / ∂xi .

(5)

It is possible to show [16] that Eqs.1-5 can be recast in a unified integral form by means of the following boundary integral equation


cij (y)U j (y) + ∫ Tij* (x, y)U j (x)dS (x) = ∫Uij* (x, y)Tj (x)dS (x), S

3

(6)

S

in which the generalized notation

⎧u Ui = ⎨ i ⎩φ

i≤3 ⎧t i ≤ 3 , and Ti = ⎨ i i=4 ⎩ω i = 4

(7)

has been introduced. In Eq.6, S denotes the boundary of the considered piezoelectric domain, x and y are the integration and collocation points, respectively, cij denotes the so-called free-term stemming from the collocation limiting process, and U ij* and Tij* are the fundamental solutions for piezoelectricity, which are computed using the spherical harmonics expansion presented in Ref.[19]. Eq.6 is valid for the generic piezoelectric grain of the polycrystalline morphology. Its discrete counterpart is obtained by suitably meshing the surface of each grains and by introducing suitable shape functions to approximate the generalized displacements U i and generalized tractions Ti over the grain surface. The interested reader is referred to Ref.[18] for further details on the employed meshing strategy. The interface equations linking neighboring grains and the boundary conditions enforced on the exterior of the aggregate are discussed next. Interface equations. In this work, the generic grain boundary between two adjacent grains is modelled as a perfect interface where displacement continuity, electric potential continuity, traction equilibrium and surface charge equilibrium are enforced. These hypotheses are implemented by means of the following interface equations

⎧ui〈 k 〉 − ui〈 l 〉 = 0 ⎧ ti〈 k 〉 + ti〈 l 〉 = 0 continuity ( ), and (equilibrium), ⎨ 〈k〉 ⎨ 〈k〉〈l 〉〈l 〉 ⎩φ − φ = 0 ⎩ω + ω = 0

(8)

where the superscript 〈 l 〉 denotes a quantity related to the generic face of the l -th grain. Periodic boundary conditions. Different types of boundary conditions can be enforced on the exterior of the polycrystalline morphology. In this work, since the numerical tests deal with the computation of the macroscopic properties of polycrystalline piezoelectric aggregates, periodic boundary conditions are considered as they have been shown to provide better results with respect to Dirichlet or Neumann boundary conditions. Considering two external faces of the aggregate, and more specifically, a master face m and a slave face s, the mechanical and electric periodic boundary conditions are implemented as follows

⎧⎪ui〈 s 〉 − ui〈 m〉 = γ ij ( x 〈j s 〉 − x 〈jm〉 ) ⎧φ 〈 s 〉 − φ 〈 m〉 = − E j ( x 〈j s 〉 − x 〈jm〉 ) ( ), a mechanica l nd (electrical ), ⎨ ⎨ ti〈 m〉 + ti〈 s 〉 = 0 ω 〈 m〉 + ω 〈 s 〉 = 0 ⎩ ⎩⎪

(9)

where γ ij and Ei are the components of the prescribed values of the strain field and the electric field, respectively. Polycrystalline system assembly and solution. The governing equations of the whole polycrystalline aggregate are obtained by writing the discrete counterpart of Eq.6 for each grain of the aggregate and by writing Eqs.8 and 9 for each couple of grain boundaries and coupled periodic faces, respectively. Considering a N g -grain aggregate, the resulting system can be written in the following matrix form

⎡ H 〈1〉 ⎢ ⎢ M ⎢ 0 ⎢ ⎣⎢ ←

L O L IU

⎡G 〈1〉 0 ⎤ 〈1〉 ⎧ ⎫ ⎥ U ⎢ M ⎥⎪ ⎪ ⎢ M M ⎬− 〈N 〉 ⎨ H g ⎥ ⎪ 〈 Ng 〉 ⎪ ⎢ 0 ⎥ U ⎭ ⎢ → ⎦⎥ ⎩ ⎣⎢ ←

L O L IT

0 ⎤ ⎧0 ⎫ 〈1〉 ⎥⎧ T ⎫ ⎪ ⎪ M ⎥⎪ ⎪ ⎪ M⎪ M ⎬=⎨ ⎬ 〈 Ng 〉 ⎥ ⎨ 0 G ⎥ ⎪⎩T〈 N g 〉 ⎪⎭ ⎪ ⎪ ⎪ → ⎦⎥ ⎩b ⎭⎪

(10)

where: the generic superscript 〈 g 〉 denotes a quantity related to the g-th grain, U 〈 g 〉 and T〈 g 〉 collect the nodal values of the generalized displacements and tractions, respectively; H 〈 g 〉 and G 〈 g 〉 stem from the numerical integration of Eq.6; IU and IT collect the coefficients of the generalized displacements and tractions appearing in the interface equations 8 and periodic boundary conditions 9; and b collects the prescribed


4

values appearing on the right-hand side of the periodic boundary conditions 9. The solution of the sparse system given in Eq.10 follows the same strategy adopted in [18] where special solvers were employed.

a)

b)

c) Fig.1: Apparent properties of BaTiO3 piezoelectric aggregates with isotropic orientation of the constituent grains: a) Young’s modulus, b) shear modulus and c) relative dielectric constant as functions of the considered number of grains. The dotted lines denote the single crystal constants. ε 0 = 8.854 ⋅ 10−12 Fm−1 denotes the vacuum permittivity constant. Constants are given in Voigt notation.

Computational tests In the present section, the presented formulation is employed for the computation of the macroscopic apparent properties of piezoelectric polycrystalline aggregates. Prescribed components of strain and electric fields are enforced according to Eqs.9 and the apparent properties are computed in terms of macroscopic stress field and macroscopic electric displacement field using the following volume averages

σ ij =

1 ti x j dS V ∫S

and

Di =

1 ω x j dS V ∫S

−

(11)

The selected material is a polycrystal made of BaTiO3 crystals, whose non-zero constitutive constants in the local reference system are taken from [20]. The orientation of the grains is defined by the three Euler angles α , β and γ according to the ZXZ convention, whose value is randomly chosen by means of three orientation distribution functions. More specifically, the angles α and γ , which represent rotation around the Z axis, are assumed to be uniformly distributed in the [0, 2π ) interval, whereas, in order to account for unpoled, partially poled and fully poled aggregates, the angle β is assumed to be distributed over the interval [0, βmax ) according to the following probability density function

pβ ( x ) =

sin x , x ∈ [0, β max ), 2 sin ( β max / 2) 2

(12)

where β max denotes the maximum angle between the global x3 direction of the aggregate and the local x3 axis of each grain. It is clear that βmax = π denotes an isotropic (completely random) orientation of piezoelectric


5

crystals, that in turn corresponds to a macroscopically unpoled state of the aggregate, whereas βmax = 0 denotes a situation in which the local x3 axes of the piezoelectric grains are aligned with the global x3 direction, i.e. a fully poled state.

a)

b)

c)

d)

e)

f)

Fig.2: Apparent macroscopic constitutive properties of a,c,e) unpoled and b,d,f) partially poled BaTiO3 piezoelectric aggregates: a,b) selected elastic constants, c,d) selected piezoelectric constants and e,f) selected dielectric constant as functions of the considered number of grains. The grey regions around the shown constants denote the corresponding Voigt and Reuss averages. The dotted lines denote the single crystal constants. ε 0 = 8.854 ⋅ 10−12 Fm−1 denotes the vacuum permittivity constant. Constants are given in Voigt notation.

The effect of the considered morphologies on the convergence of the macroscopic apparent properties is assessed by considering one hundred realizations of polycrystalline aggregates consisting of 10, 20, 50, 100 and 200 grains. First, the isotropic orientation of the grains is tested, i.e. βmax = π , and Fig.1a, 1b and 1c show the results of the numerical homogenization in terms of effective Young’s modulus, shear modulus and dielectric constant, respectively. The figures report the volume averages computed using Eq.11 and the ensemble averages over the analyzed realizations, from which it is possible to note that the convergence of the macroscopic properties is obtained with the 50-grain aggregates. Moreover, the results are compared with those available in the literature [20] and show the accuracy of the present formulation.

6


To gain further insight on the macroscopic behavior of polycrystalline piezoelectric aggregates, Fig.2 reports the values of the ensemble averages of a few selected apparent components of the elastic, piezoelectric and dielectric constitutive tensors of the aggregates as functions of the considered number of grains and for two cases of the maximum angle β max , i.e. βmax = π , and βmax = π / 4 . In both cases, it is interesting to note that aggregates consisting of 50 grains provide the converged values of the apparent properties. Moreover, it is worth highlight that the case βmax = π corresponds to a macroscopically isotropic solid, for which the piezoelectric effect disappears (see Fig.2c) and the components of elastic and dielectric tensors along different directions coincide (see Fig.2a and 2e). The piezoelectric effect is instead present when a partial poling is considered ( βmax = π / 4 ) as shown in Figs.2d. Finally, it is worth noting that the computed apparent properties fall within the Voigt and Reuss averages [21], which have been obtained analytically and are indicated in the figures by the shaded regions around the corresponding values of the apparent constitutive constants. Summary A boundary element formulation for polycrystalline materials with piezoelectric coupling has been presented. The formulation has the advantages of expressing the micro-mechanical problem in terms of intergranular generalized displacements and generalized tractions only, thus reducing the overall degrees of freedom with respect to FEM-based approaches. The numerical results in terms of apparent macroscopic properties of unpoled and partially poled polycrystalline aggregates show the accuracy and the capabilities of the method. References [1] W. Staszewski, C. Boller and G.R Tomlinson Health monitoring of aerospace structures: smart sensor technologies and signal processing, John Wiley & Sons, (2004). [2] I. Benedetti, M.H. Aliabadi and A. Milazzo Computer Methods in Applied Mechanics and Engineering, 199(9-12), 490-501 (2010). [3] S. Nemat-Nasser and M. Hori Micromechanics: Overall Properties of Heterogeneous Materials, North Holland, (1999). [4] F. Roters, P. Eisenlohr, L. Hantcherli, D.D. Tjahjanto, T.R. Bieler and D. Raabe. Acta Materialia, 58(4), 11521211 (2010). [5] C.V. Verhoosel and M.A. Gutiérrez Engineering Fracture Mechanics, 76(6), 742-760 (2009). [6] M.H. Aliabadi The boundary element method, applications in solids and structures Vol.2, John Wiley & Sons, (2002). [7] G.K. Sfantos and M.H. Aliabadi International Journal of Numerical Methods in Engineering, 69(8), 1590-1626 (2007). [8] G. Geraci and M.H. Aliabadi Engineering Fracture Mechanics, 176, 351-374 (2017). [9] I. Benedetti and M.H. Aliabadi Computational Materials Science, 67, 249-260 (2013). [10] I. Benedetti and M.H. Aliabadi Computer Methods in Applied Mechanics and Engineering, 265, 36-62 (2013). [11] I. Benedetti, V. Gulizzi and V. Mallardo International Journal of Plasticity, 83, 202-224 (2016). [12] I. Benedetti, V. Gulizzi and A. Milazzo Mechanics of Materials, 117, 137-151 (2018). [13] V. Gulizzi, C.H. Rycroft and I. Benedetti Computer Methods in Applied Mechanics and Engineering, 329, 168194 (2018). [14] G.K. Sfantos and M.H. Aliabadi Computer Methods in Applied Mechanics and Engineering, 196(7), 1310-1329 (2007). [15] I. Benedetti and M.H. Aliabadi Computer Methods in Applied Mechanics and Engineering, 289(1), 429-453 (2015). [16] E. Pan Engineering Analysis with Boundary Elements, 23(1), 67-76 (1999). [17] G. Davì and A. Milazzo International Journal of Solids and Structures, 38(40), 7065-7078 (2001). [18] V. Gulizzi, A. Milazzo and I. Benedetti Computational Mechanics, 56(4), 631-651 (2015). [19] V. Gulizzi, A. Milazzo and I. Benedetti International Journal of Solids and Structures, 100, 169-186 (2016). [20] A. Froehlich, A. Brueckner-Foit and S. Weyer, In Smart Structures and Materials 2000: Active Materials: Behavior and Mechanics, 3992, 279-288 (2000). [21] J.Y. Li, M.L. Dunn and H. Ledbetter, Journal of applied physics, 86(8), 4626-4634 (1999).


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Identification of heat fluxes based on strain measurement and meshfree methods A. Khosravifard1, A. Razavi2, M.R. Hematiyan3 Department of Mechanical Engineering, Shiraz University, Shiraz 71936, Iran 1

[email protected]

2

[email protected] 3

[email protected]

Keywords: Inverse analysis, Heat flux identification, Strain measurement, Meshfree methods

Abstract A new method for identification of unknown heat fluxes is proposed in this paper. The proposed identification method is based on the measured strains, rather than the usual temperature measurements. The basic idea of this work is that the thermal response of systems is much slower than their mechanical response. This means that when a heat flux is applied to a sufficiently constrained member, the strain field develops much faster than the temperature field. Therefore, the time variations of the applied heat flux can be better identified if the strain measurements are utilized. Especial attention is paid to problems in which large heat fluxes or thermal shocks are applied to a structure. In such problems, a large temperature gradient exists near the boundary and a dense nodal distribution should be used in the numerical analysis. In this work, an efficient meshfree method is used for fast and accurate analysis of the associated direct problems. Introduction Identification of unknown heat fluxes applied to bodies has long been an area of interest to researchers. Heat exchangers and furnaces, space shuttles, and slab surfaces are some examples of the situations in which the measurement of heat flux is crucial for proper design and functioning of a system [1–4]. Not only the direct measurement of heat fluxes is a cumbersome task, but also, heat fluxes are most often applied to inaccessible parts of a medium. For instance, in the furnaces or space vehicles during the re-entry to the atmosphere, large heat fluxes are applied to the boundaries of the medium which are quite impossible to measure directly. In such cases, the usual approach for determination of the heat flux is to perform an inverse analysis. This means that the heat fluxes are obtained based on measurement of a related quantity in a different part of the medium. For instance, in order to obtain the time variations of heat flux in a furnace, the temperature variations on the outer walls of the furnace can be measured. These measurements are then used in an inverse analysis to reconstruct the fluxes that have led to the measured variations of the temperature. Beck used the idea of inverse analysis for estimation of the heat flux density in a conducting solid, based on temperature measurements [5]. He introduced the concept of future time steps in order to obtain stable results. Woodbury proposed a thermocouple model in order to generate data necessary for determination of heat flux in a one-dimensional domain. For this purpose, he developed a computer code based on the inverse heat conduction problem [6]. Xu et al., proposed an inverse method for prediction of heat flux distribution in a cutting tool [7]. Chen et al., made use of the conjugate gradient method for estimation of unknown heat flux at the base of a pin fin. Their method requires no information on the functional form of the unknown heat flux [8]. Li and Yan formulated an inverse problem for identification of space and time variation of heat flux in turbulent forced convection between parallel flat plates [1]. They concluded that when the sensors are located closer to the position of the applied fluxes, more accurate results can be obtained. Hon and Wei used the method of fundamental solutions for the inverse analysis of heat conduction problems [9]. For the purpose of regularization, they used the Tikhonov algorithm along with the L-curve method. Loulou and Artioukhine solved the nonlinear unsteady inverse heat conduction problem in a finite hollow half cylinder. The problem formulation was based on the minimization of a residual function by the conjugate gradient method [10]. Taler et al. used the inverse analysis for identification of local heat fluxes in steam boilers [2]. They used the least squares method along with the

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Levenberg–Marquardt (LM) technique for obtaining unknown fluxes based on temperature measurements. Khorrami et al. utilized the system identification techniques for determination of unknown heat fluxes through temperature measurements. Their proposed identification algorithm was based on the linear artificial neural network [11]. Pacheco et al. presented a real-time identification method for determination of high magnitude heat fluxes. Their method is based on the steady-state Kalman filter [3]. Cui et al. proposed a modified LM algorithm for solution of inverse heat conduction problems. In this modified algorithm, sensitivity analysis is performed by means of complex variable differentiation technique [4]. By a rigorous review of the literature one can conclude that almost all of the methods that have been used for identification of heat fluxes make use of temperature measurements. To the best knowledge of the authors, there are only two works which use strain measurements for this purpose. Blanc and Raynaud proposed the use of temperature as well as strain measurements for reconstruction of unknown boundary conditions in inverse heat conduction problems [12]. Yang et al. presented a technique for solution of inverse thermoelasticity problem in an infinitely long annular cylinder [13]. They used the history of strain at one point of the domain to recover the applied heat flux. Both of the above mentioned works are applicable to simple one-dimensional problems. In the present work, a general formulation for identification of time history of heat fluxes based on the strain measurements is proposed. The main idea of this work is based on the fast response of strains in comparison to the temperature change in a system subjected to heat fluxes. In this way, heat fluxes can be reconstructed when the body is subjected to thermal shocks or when the measurements are taken far from the position of the applied flux. It will be shown that in such cases it is impossible to recover the heat flux history by temperature measurements. The numerical analysis of the associated thermoelasticity problem is performed by an improved meshless method [14, 15], which makes use of a robust integration technique [16]. The effectiveness of the proposed technique for identification of heat fluxes is assessed by a numerical example. Formulation of the inverse problem based on strain measurement Consider a domain which at some parts of its boundary ( q ) is subjected to a time and space varying heat flux. It is desired to obtain a reasonable estimation for the applied flux based on the measured values of an accessible quantity at some other parts of the boundary, i.e. the sampling points (see Fig. 1). In such problems, it is common to measure temperatures at the sampling points. However, if the heat flux is applied in a short time interval or if the sampling points cannot be selected near the position of the applied flux, the sampling points might not experience an observable temperature change during the sampling interval. On the other hand, if the domain has sufficient mechanical constraints, all parts of the domain, almost immediately, undergo a deformation field due to the application of the heat flux. Therefore, if strains are measured at the sampling points, there would be a better chance for identification of the applied heat flux.

Figure 1. Geometry and nomenclature of a typical heat flux identification problem.


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In order to obtain the spatial distribution of the unknown heat flux, the boundary portion q is divided into N parts. On each part, it is assumed that the heat flux is either constant or varies linearly. Also, we collect values of strain at M sampling points on the boundary. Based on the collected values of strain, and by using a proper inverse technique the unknown heat flux is estimated. If the estimation process is linear, i.e., if the thermo-physical properties of the medium does not vary with temperature, the whole domain approach can be adopted for the inverse analysis. In this approach, all the unknowns at all time steps are determined simultaneously. Herein, the whole domain process for estimation of the unknown heat flux is described. Suppose that we want to obtain the heat flux at N t time steps. Therefore, there would be a total of N  N t unknowns that can be collected in a vector X as follows:



(1)



(1)



(N ) T

(1) X  q1  qN q1  qN  q1 t  q N t Also, the measured values of strain at the M sampling points and at all time steps are collected in the vector Y; likewise the computed values of the strains based on the estimated heat fluxes are put in the vector V(X) :

Y  1



(1)

 M

V( X)  1

(1)

( 2)

( 2)

(N )

1( 2)   M ( 2)  1( N )   M ( N )

(1)

t

 M

(1)

t



T

1( 2)   M ( 2)  1( N )   M ( N ) t

t



(2)

T

(3) In Eqs (1) to (3) the superscripts refer to the time step, while the subscripts denote the boundary portion. Now using the Tikhonov regularization scheme, a cost function which includes a proper regularization term, is defined as follows:

f X  Y  VX Y  VX  XT X T

(4) The first term of the cost function guarantees that the computed and measured strains are as close as possible, while the goal of the second term is to reduce the unwanted oscillations of the solution. μ in this term is the socalled regularization parameter, the value of which should be selected appropriately. By minimization of the cost function, values of the unknown heat flux can be obtained; therefore, the partial derivative of the cost function with respect to the vector of unknowns should be set to zero, i.e.:

f  2ST Y  VX 2X  0 X

(5)

where S is the sensitivity matrix and is written as follows: (1) (1) (1) (1)  1(1) 1  1 1  1      (1) (1) (1) (N ) (N )  q2 q N q1 t q N t   q1              (1)  (1) (1) (1) (1)  M  M  M  M    M    (1) (1) (N ) (N )  q1(1) q2 q N q1 t q N t   ( 2) ( 2) ( 2) ( 2) ( 2)  1  1 1  1   1    (1) (1) (N ) (N )  q1(1) q2 q N q1 t q N t  S          ( 2) ( 2) ( 2) ( 2) ( 2)    M  M  M  M    M(1)    (1) (1) (N ) (N )  q1 q2 q N q1 t q N t            ( Nt ) ( Nt ) ( Nt ) (N )   ( Nt )  1 1  1 1 t   1 (1)     (1) (1) (N ) (N ) q2 q N q1 t q N t   q1           ( Nt ) ( Nt ) (N )   ( Nt )  ( Nt )  M  M  M t  M  M (1)     (1) (1) (N ) (N ) q2 q N q1 t q N t  (6)  q1 Values of the sensitivity matrix are partial derivatives of the strains with respect to the heat fluxes. For linear systems, one of the unknown fluxes at the first time step is set to unity, while all others are equal to zero. Also, all heat fluxes are set to zero for all other time steps:


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q1  1 , q2  q3    q N (1)

(1)

(1)

(1)

0

(7)

(8) q1  q2    q N  0 k  2,3,...N t Then by performing a direct analysis, values of the strains at sampling points for all the time steps are computed. These values comprise the first column of the sensitivity matrix, i.e.: (k ) (k ) (k )  1  2  M (9)  k  1,2,..., N t (1) (1) (1) q1 q1 q1 Other columns of the sensitivity matrix are calculated in a similar manner. After calculation of the sensitivity matrix and by performing some arithmetic manipulations, the vector of unknowns is obtained as follows: (k )

(k )

(k )



X  S TS  μI

 S Y 1

T

(10)

The meshfree RPIM for analysis of thermoelastic problems In an inverse analysis, several direct problems should be solved. In the present work, an improved version of the original meshfree radial point interpolation method (RPIM) is used for performing the direct analyses [14, 15]. In the first step, the temperature and displacement fields are approximated by the following relations: n

T x, t    Ti (t )i (x)

(11)

i 1

n

ux, t    u i (t )i (x)

(12)

i 1

where  i is the RPIM shape function, n is the number of nodes in the support domain of point x, and Ti and u i are the nodal values of the temperature and displacement vector, respectively. In an uncoupled thermoelasticity problem, the temperature field is obtained initially by solving the following discretized system of equations [14]:   K thT  F th (13) MT where: (14) M ij  ci j d





   j i  j   K ijth   k  i   y y   x x Fi (t )   q(x, t )i d

(15) (16)

q

where ρ is the mass density, c is the specific heat, k is the thermal conductivity of the material, and q is the applied heat flux to the boundary. After the temperature field is obtained by solution of Eq. (13), the displacement field at each time step is obtained by solving the following system of equations [15]: (17) K el Ut  Ftel where

0 i y   x K   B DB j d; Bi   i  i y i x   0 Ftel,i   BTi Dεth d el ij

T i



T

(18) (19)

In Eqs (18) and (19), D is the matrix of elastic constants, and ε th is the vector that collects the values of thermal strains, i.e.: T (20) εth   T  T0   T  T0  0 where α is the coefficient of thermal expansion, and T0 is the temperature at which the system is stress free. Solution of Eq. (17) at each time step gives the value of displacement field at any point of the domain, from which the strains can be calculated. In the present study, all of the domain integrals in Eqs (14) to (19) are evaluated by an effective technique, called the background decomposition method [16]. This domain integration method is


11

especially useful for computation of domain integrals in meshfree methods in which the nodal density varies drastically over the domain. Since in the presented example, the investigated body is subjected to large heat fluxes, a denser nodal distribution is selected near the respective boundary. Numerical results and discussion To investigate the effectiveness of the proposed technique for identification of large heat fluxes, a ceramic pipe which is subjected to a time and space varying heat flux at its inner surface is considered. The measured data are taken from strain gauges installed on the outer surface of the pipe. In order to simulate the experimental conditions, the strains are computed by a numerical analysis and then, random Gaussian errors are added to them. Fig. 2(a) depicts the schematic representation of this example problem. It is assumed that the loading condition on one quarter of the pipe repeats itself on other quarters. Therefore, only one quarter of the problem domain is modelled. In order to be able to predict the true temperature and strain distribution in the domain, the numerical model requires a dense nodal distribution near the inner surface, where there is a large temperature gradient. Fig. 2(b) shows the distribution of the nodal points in the numerical domain.

Figure 2. Representation of (a) the geometry of the inverse problem, and (b) the nodal distribution. The applied heat flux varies according to the relation q ( , t )  (100  400  ) f (t ) K W m . The space and time variations of the applied heat flux are plotted in Fig. 3. In the time period of the problem solution, the temperature of the outer surface of the pipe does not experience any change, while the strains develop almost immediately in all parts of the domain. Fig. 4 depicts the time variations of the temperature and strain at some points on the outer surface of the pipe. From this figure, one can conclude that the identification of the applied heat flux cannot be performed by temperature measurements at the outer surface of the pipe. However, the strains have a good sensitivity to the applied heat flux and can be therefore used in the estimation process. 2

Figure 3. The (a) space, (b) time variations of the applied heat flux.


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Figure 4. Variations of the (a) temperature, and (b) strain at some points on the outer surface of the pipe. For estimation of the applied heat flux, the inner boundary is divided into four parts, equal in length. On each boundary portion the spatial variation of the flux is assumed to be linear. Different levels of measurement error are considered in the inverse analysis. In Fig. 5 the time variations of the four identified heat fluxes are plotted for a measurement error of 2%. This figure clearly shows that the proposed inverse algorithm is capable of recovering the magnitude and time variations of the applied heat fluxes. It should be noted that the measured data are regularized prior to the inverse analysis of the problem in order to smooth the noisy data. The spatial variations of the estimated heat flux for some time instances are plotted in Fig. 6. It is seen that the present method has successfully reconstructed the spatial variations of the heat flux, as well. The accuracy of the results of the estimation problem with other levels of measurement error is reported in Table 1. Table 1. Average error of estimated heat fluxes for various levels of measurement error Measurement error

Average percent error of q1


Average percent error of q 3


No error

0.000021

0.000017

0.000011

0.000007

1%

1.31

1.08

0.55

0.56

2%

2.43

1.45

1.18

1.35

5%

7.39

4.31

2.75

2.61


13

Figure 5. Time variations of the estimated heat flux on the (a) 1st, (b) 2nd, (c) 3rd, and (d) 4th boundary portion.

Figure 6. Spatial variation of the estimated heat flux at (a) t  1 s , (b) t  4 s , (c) t  7 s , (d) t  15 s .

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Conclusion In the present work, an inverse methodology for estimation of heat fluxes based on strain measurements was proposed. The presented method is especially useful in cases where the temperature does not change at the measurement locations during the time period of the application of the heat flux. In such cases, one can benefit from the rapid development of the strains in the body. The effectiveness of the proposed algorithm was assessed by a numerical example and it was shown that in such cases the strain measurement is an excellent alternative for the temperature. The inverse problem was solved with different levels of measurement error and it was shown that the variations of the applied heat flux can be recovered by the proposed technique with acceptable accuracy. References [1] Li HY, Yan WM. Identification of wall heat flux for turbulent forced convection by inverse analysis. International Journal of Heat and Mass Transfer. 2003; 46(6):1041-1048. [2] Taler J, Duda P, Weglowski B, Zima W, Gradziel S, Sobota T, Taler D. Identification of local heat flux to membrane water-walls in steam boilers. Fuel. 2009; 88(2):305-311. [3] Pacheco CC, Orlande HRB, Colaco MJ, Dulikravich GS. Real-time identification of a high-magnitude boundary heat flux on a plate. Inverse Problems in Science and Engineering. 2016; 24(9):1661-1679. [4] Cui M, Yang K, Xu XL, Wang SD, Gao XW. A modified Levenberg–Marquardt algorithm for simultaneous estimation of multi-parameters of boundary heat flux by solving transient nonlinear inverse heat conduction problems. International Journal of Heat and Mass Transfer. 2016; 97:908-916. [5] Beck JV. Nonlinear estimation applied to the nonlinear inverse heat conduction problem. International Journal of Heat and Mass Transfer. 1970; 13(4):703-716. [6] Woodbury KA. Effect of thermocouple sensor dynamics on surface heat flux predictions obtained via inverse heat transfer analysis. International Journal of Heat and Mass Transfer. 1990;33(12), 2641-2649. [7] Xu W, Genin J, Dong Q. Inverse method to predict temperature and heat flux distribution in a cutting tool. Journal of Heat Transfer. 1997; 119(3):655-659. [8] Chen UC, Chang WJ. Hsu JC. Two-dimensional inverse problem in estimating heat flux of pin fins. International Communications in Heat and Mass Transfer, 2001; 28(6):793-801. [9] Hon YC, Wei T. A fundamental solution method for inverse heat conduction problem. Engineering Analysis with Boundary Elements, 2004; 28(5):489-495. [10] Loulou T, Artioukhine E. Numerical solution of 3D unsteady nonlinear inverse problem of estimating surface heat flux for cylindrical geometry. Inverse Problems in Science and Engineering. 2006; 14(1):39-52. [11] Khorrami M, Samadi F, Kowsary F, Mohammadzaheri M. Online estimation of multicomponent heat flux using a system identification technique. International Communications in Heat and Mass Transfer. 2013; 44:127134. [12] Blanc G, Raynaud M. Solution of the inverse heat conduction problem from thermal strain measurements. Journal of Heat Transfer. 1996; 118(4):842-849. [13] Yang YC, Chen UC, Chang WJ. An inverse problem of coupled thermoelasticity in predicting heat flux and thermal stresses by strain measurement. Journal of Thermal Stresses. 2002; 25(3):265-281. [14] Khosravifard A, Hematiyan MR, Marin L. Nonlinear transient heat conduction analysis of functionally graded materials in the presence of heat sources using an improved meshless radial point interpolation method. Applied Mathematical Modelling. 2011; 35:4157-4174. [15] Khosravifard A, Hematiyan MR. Nonlinear transient thermo-mechanical analysis of functionally graded materials by an improved meshless radial point interpolation method. 11th Int. Conference on Boundary Element and Meshless Techniques, 2010. [16] Hematiyan MR, Khosravifard A, Liu GR. A background decomposition method for domain integration in weak-form meshfree methods. Computers and Structures. 2014; 142:64-78.


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Bimodular materials with finite block method in polar coordinate Q.X. Pan1, J.L. Zheng1, P.H. Wen2 and M.H. Aliabadi3 1

School of Traffic and Transportation Engineering, Changsha University of Technology and Science, China 2 School of Engineering and Materials Science, Queen Mary, University of London, E1 4NS, UK 3 Department of Aeronautics, Imperial College London, SW7 2BY, UK

Keywords: Different Young's modulus, composites, asphalt-mixture materials, nonlinear partial differential equation, finite block method, mapping technique, differential matrix. Abstract. Due to the differences in Young's moduli under tensile and compressive loadings for the rock-like and asphalt-mixture materials, the numerical simulation becomes high nonlinear and more difficult. The meshless finite block method with polar coordinate is formulated to solve the deformation and stress associated with bimodular materials in this paper. With utilization of the Lagrange series, the first order partial differential matrices are derived firstly with the mapping technique. The numerical results are presented and comparisons have been made. 1. Introduction It is well known that in the classical elasticity theory, materials have the same elastic properties in tension and compression. However, this is only for a simplified modeling and does not account for material non-linearity. In the construction of traffic and transport engineering, some materials such as rock and concrete and asphalt-mixture materials, shown in Figure 1 demonstrate a significant difference between tensile and compressive strengths, as well as their mechanical properties, including elastic moduli E and Poisson ratios ν . The experimental data shows that the constitutive relationship between stress and strain has two straight lines with different slopes. It has been observed that most materials, including ceramics and some composites, have different tensile or compressive strain, subjected to the same tensile or compressive stress [1-3]. In this paper, the behavior of elastic bimodular materials is investigated by the meshless finite block method in polar coordinate system to deal the circular boundary. In the principal coordinates, the Hook's law in elasticity matrix related to principal stresses and strains is formulated for three dimensional problems with two kinds of regions in the field. Then, the stress-strain relations are transformed from principal coordinate system to global Cartesian coordinates. A set of nonlinear algebraic equation of partial differential equations in strong form are formulated in the polar coordinate system, in terms of the nodal values of displacement by using the first order partial differential matrices. Due to the


16

non-linearity of material coefficients, an iterative method is applied in the computations.

Figure 1. Constitutive relationship.

Figure 2. Principle stresses/directions.

2. Bimodular material constitute equation Two linear stress-strain relationships under either tension or compression loadings are shown in Figure 1. The following assumptions are taken into account for the meshless finite block method: (1) The media is elastic and continuous; (2) The deformation is small; (3) There is no initial stress. In the principal element, Hook's law, from assumption of elasticity, gives ε α = a11σ α + a12σ β + a13σ γ ,

ε β = a12σ α + a22σ β + a23σ γ , ε γ = a13σ α + a23σ β + a33σ γ , γ αβ = γ βγ = γ αγ = 0

(1)

where the coefficients aij depend on the principal stresses as following (1) If σ α > σ β > σ γ > 0 , a11 = a22 = a33 = 1 / E + , a12 = a13 = a23 = −ν + / E + .

(2)

(2) If σ γ < σ β < σ α < 0 , ` a11 = a22 = a33 = 1 / E − , a12 = a13 = a23 = −ν − / E − .

(3)

(3) If σ α > σ γ > 0 and σ β < 0 , a11 = a33 = 1 / E + , a22 = 1 / E − , a12 = a13 = a23 = −ν + / E + = −ν − / E − .

(4)

(4) If σ β > 0 and σ γ < σ α < 0 , a22 = 1 / E + , a11 = a33 = 1 / E − , a12 = a13 = a23 = −ν + / E + = −ν − / E − .

(5)

Cases (1) and (2) are defined as the first kind in which the classic linear elastic theory is valid. Cases (3) and (4) are the second kind and all coefficients of stress depend on


17

the signs of the principal stresses (tensile or compressive). In the Cartesian's coordinate system, the principal direction along principal stresses σ α , σ β and σ γ are defined, as shown in Figure 2 as α = (l1 , l2 , l3 ), β = (m1 , m2 , m3 ), γ = (n1 , n2 , n3 ) . For two dimensional plane stress case, all variables are independent of the z-axis. One has the principal directions l3 = m3 = n1 = n2 = 0, n3 = 1 and stresses σ z = σ γ = τ rz = τ θz = 0. The principal directions α = (l1 , l2 ) and β = (m1 , m2 ) are functions of stress state (σ r , σ θ ,τ rθ ) demonstrated in Figure 2. The Hook's law, from assumption of elasticity, gives ε r ` = a11σ r + a12 (σ θ + σ z ) + B3m12σ β , (6) ε = a σ + a (σ + σ ) + B m 2σ , θ

11 θ

12

z

r

3

2

β

γ rθ = 2 A1τ rθ + 2 B3m1 m2σ β .

where εr =

1 ∂v u 1 ∂u ∂v v ∂u , εθ = + , γ rθ = + − . r ∂θ r r ∂θ ∂r r ∂r

(7)

Figure 3. Principal directions in polar Figure 4. Continuous modes of Young's coordinate system. modulus. Therefore stresses can be written as σ r = 2 Aε r + Be + σ r* ,

σ r* = − B3 ( B + 2 Am12 )σ β ,

σ θ = 2 Aεθ + Be + σ θ* ,

σ θ* = − B3 ( B + 2 Am22 )σ β ,

τ rθ = Aγ rθ + τ r*θ

τ r*θ = −2 AB3m1m2σ β .

e=

(8)

∂u 1 ∂v u . + + ∂r r ∂θ r

It is obvious that all coefficients in the general elasticity law are dependent on the stress state in the field, which constructs the nonlinearity. In the general elastic theory,


18

Young's module is represented as a Heaviside function as show in Figure 3. Therefore, there is a jump between these different kinds of regions which is the main source and cause of the numerical computational divergence. In order to overcome this difficulty of computational divergence, two continuous modes are proposed as following Linear continuous model Young's modulus is approximated as  E+  E + + E − E + − E − + E (σ ) =  σ 2c  2 −  E

σ >c σ ≤c

(9)

σ < −c

where c is characteristic stress to be determined by the curve fitting technique with experiment data under uniaxial tensile/compressive loads. This parameter can be normalized to the applied load by c = βσ 0 , where β is the characteristic dimensionless factor and σ 0 is the applied pressure load in this paper. Or parameter c can be normalized to yielding stress of material, i.e. c = βσ s . Hyperbolic tangent continuous model Young's modulus is smoothed by E (σ ) =

E+ + E− E+ − E− tanh (σ / c) + 2 2

(10)

where tanh(σ / c) is a hyperbolic function. Although the linear mode is closer to bimodules mode than hyperbolic tangent mode, the hyperbolic model is smooth everywhere in the field and therefore it is easier to formulize all partial differential equations in mathematical modeling. 3. Partial differential equations for bimodular material The components of stress in polar coordinate system can be rewritten as

σ r = A[(ε r − λe) + B3 (λ − m12 )σ β ] = Aε r − Aλe + σ r* ,

σ θ = A[(εθ − λe) + B3 (λ − m22 )σ β ] = Aεθ − Aλe + σ θ* , τ rθ = A[γ rθ − 2 B3m1 m2σ β ] =

(11)

A γ rθ + τ r*θ , 2

a12 . Then, a11 + a12 substituting (11) into equilibrium equation, with zero body forces, yields

where σ r* = AB3 (λ − m12 )σ β ; σ θ* = AB3 (λ − m12 )σ β ; τ r*θ = − AB3m1 m2σ β , λ =

A

 ∂ 2u 1 ∂u u 1 ∂ 2v 1 ∂ 2 v  A  1 ∂ 2u ∂ 2v 3 ∂v ∂ 2u    A + + λ − + − + −  ∂r 2 r ∂r r 2 r ∂r∂θ r 2 ∂θ 2  2r  r ∂θ 2 ∂r∂θ − r ∂θ ∂r 2   

∂u 2u  ∂A ∂u ∂ (λA)  ∂u u 1 ∂v  1 ∂A  ∂v v 1 ∂u  − + −  + +  − + +  ∂r r  ∂r ∂r ∂r  ∂r r r ∂θ  2r ∂θ  ∂r r r ∂θ  ∂σ * σ * − σ θ* 1 ∂τ r*θ =− r − r − r r ∂θ ∂r

+2

(12)


19

3 ∂u ∂ 2v 1 ∂v v  A ∂ 2v A  1 ∂ 2u  − 2 + + − + 2  r ∂r∂θ r ∂θ ∂r 2 r ∂r r 2  r 2 ∂θ 2 −

Aλ  ∂ 2u 1 ∂u 1 ∂ 2v  1 ∂A  1 ∂u ∂v v  1 ∂A  u 1 ∂v  +  + − + + +  +   r  ∂r∂θ r ∂θ r ∂θ 2  2 ∂r  r ∂θ ∂r r  r ∂θ  r r ∂θ 

−

1 ∂ (λA)  ∂u u 1 ∂v  ∂τ * 2τ * 1 ∂σ θ* .  + +  = − rθ − rθ − ∂r r ∂θ  ∂r r r ∂θ  r r ∂θ

(13)

Boundary conditions are given u = u, v = v

(14) for the displacements, or (15) Tr = σ r nr + τ rθ nθ , Tθ = τ rθ nr + σ θ nθ for the tractions, where n is unit outward normal to the boundary shown in Figure 5. For the sake of convenience of analysis, we assume that σ α > σ β and hold m12 =

1 k2 k 2 , , m1m2 = , m = 2 1+ k2 1+ k2 1+ k2

where k = −t ± 1 + t 2 , t =

σ r − σθ . 2τ rθ

(16)

(17)

Figure 5. Polar coordinate system and Figure 6. Circular ring with bimodular normal to the boundary. composite media under pressure loads. A quadratic block with 8 seeds is mapped into a normalized domain (ξ ,η ) by using the shape functions as 1 N i = (1 + ξiξ )(1 + ηiη )(ξiξ + ηiη − 1) for i = 1,2,3,4 4 1 (18) N i = (1 − ξ 2 )(1 + ηiη ) for i = 5,7 2 1 N i = (1 − η 2 )(1 + ξiξ ) for i = 6,8 2


20

where (ξi ,ηi ) i = 1,2,...,8 are the seed coordinates. Therefore, the transformation of coordinate (mapping) for the real polar coordinate system can be written as 8

8

k =1

k =1

r = ∑ N k (ξ ,η )rk , θ = ∑ N k (ξ ,η )θ k

(19)

The relationship of partial differentials of function u ( x, y ) between the real Cartesian coordinate system and the normalized system is given by ∂u 1  ∂u ∂u  ∂u 1  ∂u ∂u  =  β11 + β12 , =  β 21 + β 22  ∂r J  ∂ξ ∂η  ∂θ J  ∂ξ ∂η 

,

(20)

where ∂r ∂r ∂ξ ∂η , β = ∂θ , β = − ∂θ , β = − ∂r , β = ∂r . (21) J= 11 12 21 22 ∂θ ∂θ ∂η ∂ξ ∂η ∂ξ ∂ξ ∂η Therefore, the first partial differential matrices in the real domain can be obtained by U r = Δ11Uξ + Δ12 Uη = Δ11Dξ + Δ12 Dη u = Dr u, (22) Uθ = Δ 21Uξ + Δ 22 Uη = Δ 21Dξ + Δ 22 Dη u = Dθ u

( (

) )

in which diagonal matrix Δ = diag{( β / J )i } and β (i ) / J (i ) is calculated from (21) at collocation point Qi (ξi ,ηi ) . It is apparent that the nodal values of any order partial differentials in vector form can be determined, in terms of the first order partial differential matrix in the normalized domain, where ξ ≤ 1 and η ≤ 1. 4. Iterative algorithm for nonlinear partial differential Meshless finite block method has different strategies, i.e. the numerical solutions of governing equation in strong form and in weak form. Both solutions are of different advantages which have been discussed by Wen and Cao [4], Li et al [5]. The meshless solution in strong form is employed in this paper and the equilibrium equations. They are non-linear partial differential equations due to the variables A, λ , br and bθ depend on the stress state (σ r , σ θ ,τ rθ ) in the field Q(r ,θ ) . Applying the partial differential matrices over Eq. (12) and (13) gives  2 2 ˆ ˆ2 1 ˆ ˆ 2 ˆ ˆ  ADr − A λ (Dr + RDr − R ) + 2 RA(RDθ + 2Dr − 2R ) + Dr A Dr − Dr A λ (Dr + R ) (23) 1 ˆ2   ˆ (D D − R ˆ D2 ) + 1 R ˆ A(D D − 3R ˆD )−D A R ˆ + R Dθ A Dθ u + − A λ R θ θ r θ r θ r λ Dθ 2 2   1 ˆ ˆ ) v = b + R Dθ A(Dr − R r  2


(

21

)

A ˆ 1 ˆ ˆ ˆ ˆ ˆ2  2 RDθ Dr − 3R − A λ RDθ (Dr + R ) + 2 Dr ARDθ + R Dθ A  ˆ D A (D + R ˆ ) u +  A D2 + R ˆD −R ˆ2 +R ˆ 2 AD2 − A R ˆ 2 D2 −R θ λ r r θ λ θ  2 r 

]

(

)

(24)

1 ˆ)+R ˆ 2 D AD − R ˆ 2D A D  v = b + Dr A ( Dr − R θ θ θ λ θr  θ θ 2  and b r = −Dr σ*r − Rˆ (σ*r − σθ* ) − Rˆ Dθ τ*rθ and bθ = −Dr τ*rθ − 2Rˆ τ*rθ − Rˆ Dθ σθ* , where stress

vectors of nodal value σ*r = {σ ri* } , σθ* = {σ θ*i } , τ*rθ = {τ ri* }T and diagonal matrices ˆ = diag{1/r } . By solving nonlinear governing A = diag{A i } , A λ = diag{λi Ai } , R i equations (23)(24) with boundary conditions, all nodal values of displacements can be obtained. Iterative algorithm by using the meshless finite block method can be realized with the following procedures Step 1: Set m = 0 and material is considered a single material, E = ( E + + E − ) / 2 , ν = (ν + + ν − ) / 2 and λ = −ν /(1 −ν ) . All non-linear coefficients including λ0 , A0 and force terms (br0 , bθ0 ) are set to be zero. The displacements (u 0 , v 0 ) and stresses (σ r0 , σ θ0 ,τ r0θ ) are obtained by solving Eqs (23) and (24) with boundary conditions; Step 2: Determine principal stresses (σ β0 ) and their directions (m10 , m20 ) , coefficients, T

T

including λ0 , A0 and force terms (br0 , bθ0 ) by stress state (σ r0 , σ θ0 ,τ r0θ ) at each collocation points P; Step 3: Set m = m + 1 ; Step 4: Determine displacements (u m , v m ) and stresses (σ rm , σ θm ,τ rmθ ) by solving the linear algebraic systems equation with boundary conditions, Step 5: Determine principal stress (σ βm ) and its direction (m1m , m2m ) ; Step 6: Check the average relative error ε . If ε < 106 go to Step 8 Step 7: set m = m + 1 and go to Step 4; Step 8: Print stresses and the computation terminated. 5. Numerical example Consider a ring of bimodular materials loaded by inner and outer pressures ( pa and pa ) as shown in Figure 6. the dimensionless parameters are defined κ = a / b, α = E − / E + . The node numbers in two directions are equal ( Nξ = Nη ). Poisson ratio ν + = 0.1 . There are two regions in the ring, i.e. (1) σ r < 0 and σ θ > 0 , when a ≤ r ≤ s and (2) σ r < 0 and σ θ < 0 when s ≤ r ≤ b . A quarter of ring is

considered in the numerical modeling. The normalized stresses with both linear continuous model and the exact solutions are shown in Figures 7 and 8 for different characteristic parameter β . The node number is selected as Nξ = 11 . In addition, the solutions with same modulus ( E − / E + = 1 ) are also presented to illustrate the influence of bimodular material property. It is clear that the effect on the hoop stresses σ θ is

22


much significant than that on radial direction stress σ r . Generally speaking, the numerical solution given by a linear model is slightly higher that that by a hyperbolic tangent model. Excellent agreement can be achieved when 0.05 < β < 0.25 . However, the oscillation of the results (divergence) in the iteration occurs when β < 0.01 caused by the discontinues of material property at the joint of two regions in the ring.

Figure 7. Normalized stress for different Figure 8. Normalized stress for different β β and comparison with exact solution when and comparison with exact solution when E− / E+ = 2 . E− / E+ = 2 . References [1] Ambartsumyan SA. The basic equations of the theory of elasticity for materials with different tensile and compressive strengths. Mekhanika Tverdogo Tela 1966; 2: 44-53. [2] Y. P. Tseng and Y. C. Jiang, Stress analysis of bimodulus laminates using hybrid stress plate elements, International Journal of Solids and Structures 1998; 35 (17): 2025-2028. [3] Ambartsumyan SA. Different Modulus Theory of Elasticity (in Russian) (Monograph), Moscow Science Publications, Physics and Mathematical Literature, 1982. [4] Wen PH, Cao P, Korakianitis T. Finite Block Method in elasticity. Engineering Analysis with Boundary Elements 2014; 46: 116-125. [5] Li M, Meng LX, Hinneh P, Wen PH. Finite block method for interface cracks. Engineering Fracture Mechanics 2016; 156: 25-40.


23

A Study on Spring-Dashpot Equivalency for the Dynamic Response of Piled Rafts: the Extent to which Energy Transmission through the Soil can be Disregarded Persio L A Barros1, Josue Labaki2, Luis F V Lima3 and Euclides Mesquita4 1

School of Civil Engineering, 224 Saturnino de Brito St, Campinas SP, Brazil, [email protected] 2,3,4

School of Mechanical Engineering, 200 Mendeleyev St, Campinas SP, Brazil, [email protected], [email protected], [email protected]

2

Keywords: Dynamic soil-foundation interaction, raft foundations, pile vibration, coupled foundations

Abstract. This article investigates the parameterization of a piled-raft foundation into a dynamic springdashpot equivalent. In this study, the piled-raft foundation consists of a rigid, circular surface plate in bonded contact with an embedded elastic pile. The system is under axisymmetric, vertical timeharmonic excitations. The coupled plate−pile−soil model is obtained in this work through the indirect boundary element method. The model accounts for the energy transference from the plate to the soil, from the plate to the pile head, and from the pile to the soil, as well as the energy transmission from the plate to the pile through the soil. The response of this fully coupled model is compared with an equivalent spring-dashpot system that disregards the energy transference from the surface plate to the embedded pile through the half-space. The purpose of this study is to verify the accuracy of the equivalent spring-dashpot model in representing the piled-raft foundation.

Introduction This article investigates the parameterization of a piled-raft foundation into a dynamic springdashpot equivalent. In this study, the piled-raft foundation consists of a rigid, circular surface plate [1] and an embedded elastic pile [2], which is connected to the center of the plate. The soil is modeled as a homogeneous, transversely isotropic half-space [3]. Relaxed bonded contact is considered at the plate−half-space and pile−half-space interfaces. The present analysis considers external, axisymmetric, vertical time-harmonic excitations. The fully coupled plate−pile−half-space model is obtained in this work through a superposition of Green’s functions, in the framework of the indirect boundary element method (I-BEM) [2]. The model accounts for the energy transference from the plate to the soil, from the plate to the pile head, and from the pile to the soil, as well as the energy transference from the plate to the pile through the soil. The response of the fully coupled model described above is compared with an equivalent spring-dashpot system. The equivalent system is obtained by measuring separately the dynamic impedance of a rigid circular plate at the surface of the soil and the dynamic impedance of an embedded elastic pile [4]. A spring-dashpot system comprising these two individual, independent impedances is taken to represent the piled-raft foundation. The resulting spring-dashpot system disregards the energy transference from the surface plate to the embedded pile through the half-space. The purpose of this analysis is to understand to which extent it is reasonable to disregard the energy transmission through the soil in such systems. The response of the equivalent system is compared for different constitutive parameters. The results provide insights into modeling other types of coupled foundations, such as pile groups.


24

Problem statement Consider a 3D, isotropic, elastic half-space, under axisymmetric, time-harmonic vertical excitations of frequency ω and amplitude Q0. Solutions for the displacement and stress fields in such media were presented by [3]. The half-space has Young’s modulus Es, Lamé constant µs, Poisson ratio νs, and mass density ρs. A rigid circular plate of radius ab and mass mb rests at the surface of the halfspace, and an elastic pile of radius ap, length lp, Young’s modulus Ep, and mass density ρp is embedded in the half-space. The contact surfaces between the plate and the soil and between the plate and the pile are bonded in the vertical direction.

Figure 1. Piled-raft foundation embedded in the half-space. Model of the fully-coupled foundation The Indirect-BEM model of the piled raft−half-space interaction proposed by the authors in [5] is used in the present work. A numerical approximation to the traction distribution at the plate−half-space interface can be obtained according to [6], in which the interface is discretized into nb concentric annular disc elements, in which the contact tractions are assumed to be constant. For the pile model, the pile is modeled as a 1D elastic bar element. The body of the bar is discretized into np two-noded elements with one degree-of-freedom (vertical) per node. The stiffness and mass matrices of the pile are assembled from its elemental stiffness matrices [7]. The pile−half-space interface is divided into np cylindrical segments along the length of the pile, and one disc element at the interface between the pile tip and the half-space. The contact tractions are constant in each of these elements. The relation between arbitrary, fictitious loads q applied at the discretized plate and pile elements and the resulting displacement and traction anywhere in the half-space is given by influence functions due to unit annular loads and cylindrical shell loads within those elements. These influence functions can be obtained by direct numerical integration of the Green’s functions for axisymmetric loads proposed by [3]. Kinematic compatibility and equilibrium conditions are imposed throughout the pile−half-space and plate−halfspace interfaces. This model, which is described mathematically in details by [5], yields:  K  −D   T  −1 ⋅ i 0   −m b ω2i 0T

ATpp

ATpb

Upp

U pb

Ubp

U bb

sb T Tbp

sb T Tbb

 0  −i 0  ( n p +1)×1   uf   0  q   0   p    = ( n p +1)×1 0  q b   0      n ×1  f b 1       Q0 

(1)


25

in which K = K − ω2M is the global, dynamic stiffness matrix of the classical finite-element 1D model of the pile [7],

 πa l(1)  pe  (1)  πa p le   0  A= 0   ⋮   0    0

⋯

0

0

0

πa p l(2) e

0

0

πa p l(2) e

0

0

0

0

⋱

0

(n p )

0

πa p le

0

πa p le

(n p )

πa 2p

              

(2)

transforms contact tractions in the half-space elements into nodal equivalents in the pile elements, Tij and Uij (i,j=p,b) are traction and displacement influence matrices, where the p and b indices correspond to the pile and plate elements,

1 2 0  D=⋮  0   0 

1 2 1 2

0

⋯

1 2

0  0    1 2  1 

0 0

⋱ 0

0

1 2

0

0

0

(3)

is a transformation matrix relating pile nodes with their corresponding half-space elements,

i0 = { 1

0

0 ⋯ 0} , T

(4)

and le is the length of each pile element. The solution of Eq. (1) yields the nodal displacements uf, the fictitious loads qb and qp at the plate and pile elements, and the magnitude f0 of the force at the plate−pile interface. The rigid-body displacement of the plate can be obtained by substituting uf into [5]:

u 0 = i 0T uf

(5)

Equivalent spring-dashpot model A spring-dashpot equivalent to the fully coupled piled-raft model described above has been proposed by [4]. The model consists of direct superposition of the models of the surface plate proposed by [1] and of the embedded pile proposed by [8]. Coupling between the two separate systems is obtained through direct kinematic compatibility and equilibrium at the plate-pile interface. The rigid-body displacement u0 of the plate in this model is obtained through

u0 =

Q0 αk p + (1 − α ) k b − ω2mb

,

(6)

26


in which kb is the vertical dynamic stiffness of the rigid surface plate interacting solely with the soil, without the presence of a pile, kp is the vertical dynamic stiffness of the plate interacting solely with the pile – no energy is transferred from the plate directly to the soil (even though energy is dissipated to the soil from the plate through the pile), and α (0≤α≤1) is a parameter indicating how much of the energy is transferred from the plate to the soil and from the plate to the pile. The bound α=0 is the case when all energy is transferred from the plate directly to the soil. The bound α=1 is the case when all energy is transferred from the plate to the pile, and then from the pile to the soil. It is expected that in practical applications, α is neither of these limits, but its actual value cannot be determined without some experimental measurement. For a full derivation of eq. (6), please refer to [4]. Numerical results

Re[u 0 ]

-Imag[u0 ]

Figure 2 compares the response of the piled-raft system obtained with the two models described in this paper. In these results, lp/ap=35, Ep=100Es, νs=νp=0.25, ab/ap=1, ρp/ρs=1, mb=0, and Es/µs=2.5. The results are presented in terms of the dynamic compliance c=u0/Q0 and the normalized frequency a0=ωap√(ρs/µs).

Figure 2. Comparison between the coupled piled-raft model and its spring-dashpot model. Figure 2 shows that, even though the equivalent spring-dashpot model provides a reasonable approximation to the response of the piled raft, the energy transference from the plate to the pile through the soil cannot be disregarded if a more detailed analysis is desired. Two remarkable observations from these results is that the response of the I-BEM model does not fall within the bounds predicted by the equivalent model (0≤α≤1) and that the static response of the system is also not predicted accurately by the equivalent model. The figure also shows that α is a value that is closer to 1 than to zero, which means that most of the energy is transferred from the plate to the soil through the pile. This is physically consistent, since the pile is much stiffer than the soil. Concluding remarks This article presented a study on equivalent spring-dashpot models of piled rafts interacting with soil media. The response of the equivalent model is compared with a fully coupled Indirect-BEM model of the system. The results show that the equivalent model gives a reasonable approximation to the response of the system, but fails to represent important characteristics of it.


27

References [1] J.Labaki, E.Mesquita and R.K.N.D.Rajapakse, Vertical Vibrations of an Elastic Foundation with Arbitrary Embedment within a Transversely Isotropic, Layered Soil. CMES: Computer Modeling in Engineering & Sciences, Vol. 103, No. 5, pp. 281-313 (2014). [2] P.L.A.Barros. Impedances of rigid cylindrical foundations embedded in transversely isotropic soils. International journal for numerical and analytical methods in geomechanics, 30(7), pp.683-702 (2006). [3] R.K.N.D.Rajapakse, and Y.Wang. Green’s Functions for Transversely Isotropic Elastic Half Space. Journal of Engineering Mechanics, 119, No. 9, pp. 1724-1746 (1993). [4] L.F.V.Lima, J.Labaki and E.Mesquita. Stationary Dynamic Response of a Rigid Circular Foundation Partially Supported by a Flexible Pile and Interacting with a Transversely Isotropic Soil. In: Iberian Latin American Congress on Computational Methods in Engineering, 2016, Brasília (2016). [5] P.L.A.Barros, J.Labaki and E.Mesquita, IBEM-FEM Model of a Piled Plate within a Transversely Isotropic Half-Space, unpublished (2018). [6] J.Lysmer. Vertical Motions of Rigid Footings, Ph.D. Thesis, University of Michigan, Ann Harbor, United States (1965). [7] T.J.R.Hughes. The finite element method: linear static and dynamic finite element analysis. Courier Corporation (2012). [8] J.Labaki, E.Mesquita and R.K.N.D.Rajakse. Vertical Vibrations of a Pile in a Transversely Isotropic Soil. ICGE International Conference on Geotechnical Engineering (2015).

28



29

BEM Analysis of Thermoelastic Stresses in 3D Generally Anisotropic Solids Embedded with Non-uniform Heat Source Y.C. Shiah*1 and Nguyen Anh Tuan1 1

Department of Aeronautics and Astronautics

National Cheng Kung University, Tainan 701, Taiwan, ROC. (* Corresponding to: [email protected]) Keywords: 3D anisotropic thermoelasticity, boundary element method (BEM), non-uniform heat source.

Abstract. As is well known in the BEM community, thermal effects reveal themselves as an additional volume integral that conventionally require domain discretisation for numerical computation of it. Considered as a major drawback, this domain discretisation will not only destroy the distinctive notion of boundary discretisation but also cause computational inefficiency. An ideal solution to avoid direct integration of this domain integral is to analytically transform the domain integral onto the boundary surface. In engineering practice, it is quite often to have anisotropic structures with non-uniform heat sources embedded inside due to internal heating of electricity or chemical reaction. In this article, the additional volume integral in the boundary integral equation due to the presence of heat generation rate in quadratic form is analytically transformed onto the surface without involving domain distortion. All formulations have been implemented in an existing BEM code, based on quadratic isoparametric elements. For illustration of this successful implementation, a numerical example is studied with numerical verifications by ANSYS, commercial software based on the finite element method. Introduction The boundary element method (BEM) has been recognized as an efficient numerical tool that is well known for its distinctive feature that only the boundary needs to be modeled. However, for treating thermal effects, the volume heat source reveals itself as a volume integral, which demands internal cell discretization throughout the whole domain and therefore destroy the distinctive feature of boundary modeling. To avoid direct integration of the volume integral, there are several schemes proposed over the years. Obviously, the exact transformation method, abbreviated here as ETM here, appears to be not only analytically elegant but also numerically efficient since no further approximation is involved except for the numerical integration itself. For treating 2D generally anisotropic thermoelasticity, Shiah and Tan [1] presented an analytically transformed boundary integral equation, considering the presence of constant volume heat source. As compared with 2D works, pertinent 3D works are extremely scarce indeed. An attempt to exactly transform the volume integral in the 3D thermoelastic BIE, was made by Shiah and Li [2], presenting solutions of an elliptic partial differential equation. Despite the success of applying the ETM to treat isotropic thermoelasticity for both 2D and 3D, its extension to 3D anisotropic thermoelasticity had not been so successful until Shiah [3] presented an exact volume-to-surface integral transformation for 3D. Following this work, Shiah and Chong [4] also implemented this approach to perform interior thermoelastic analysis of three-dimensional generally anisotropic bodies. The transformation previously presented in [3,4] still relies on domain distortion. Obviously, computations of the transformed boundary integrals defined in the mapped domain appear to be less direct and more complicated. The authors have accomplished the direct volume-to-surface integral transformation for 3D anisotropic thermoelasticity when the internal heat source is a constant; the work has been submitted elsewhere for consideration of possible publication. The present paper aims to extend the work to consider the volume heat source present in a quadratic form. This is because in engineering practice, it is more often to have applications with non-uniform heat source inside due to internal electrical heating or chemical reaction. Theoretically, distribution of the non-uniform heat source can always be simplified into quadratic form for simplicity. In this paper, the complete process to make the direct volume-to-surface integral transformation for 3D anisotropic thermoelasticity is presented when a quadratic distribution of volume heat source is involved. To illustrate the veracity of all formulations as well as our successful implementation, a benchmark example is investigated in the end.


30

BIE of Thermoelaticity As can be referred to many textbooks, the constitutive law between stresses ij and strains ij with thermal effects is governed by the well-known Duhamel-Neumann relation, that is

 ij  Cijkl  kl   ij , (i, j, k , l  1, 2,3)

(1)

where is the temperature change, Cijkl are the material stiffness coefficients, and ij represent the thermal modulii, given by ij= Cijkj kl with kl being the coefficients of linear thermal expansion. Under the steady-state condition, the anisotropic thermal field with non-uniform volume heat source is governed by

K jk , jk  H ( x1 , x2 , x3 ),

(2)

where Kij denote the heat conductivity coefficients and H ( x1 , x2 , x3 ) is the non-uniform heat generation rate. 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 [5]:

cij (ξ )u j (ξ )   Tij* (ξ, x)u j (x)d (x)   U ij* (ξ, x)t j (x) d (x)   (x) jkU ij*,k (ξ, x) d (x), 





(3)

where cij (ξ) are the geometric coefficients of the source point ξ , and U ij* (ξ, x) and Tij (ξ , x ) are the fundamental *

solutions of displacements and tractions, respectively. The three-dimensional Green's function of anisotropic elasticity has been derived by Lifshitz and Rosenzweig [6] to have an integral form. As the main issue of the present work, the last integral in Eq.(3), denoted here by Vj for brevity, is a volume integral that needs to be transformed to the boundary integral without any coordinate transformation. Before the complete 3D transformation process is elaborated, previous derivation of the direct transformation for 2D is reviewed first. Consider the following identity [7]:





( fi K jk , jk  K jk fi , jk ) d    [( f i K jk , j ), k  (K jk f i , k ), j ] d  , 

(4)

where f i is the component of an arbitrary function f. As a result of applying the Green's second identity to the right hand side of Eq.(4), one immediately obtains [7]





( fi K jk , jk  K jk f i , jk ) d    ( f i K jk , j nk  K jk f i ,k n j ) d  , 

(5)

For this, quadratic distribution of heat generation rates term H ( x1 , x2 , x3 ) in Eq.(2) is taken into account. By substituting Eq.(2) into Eq.(5), one obtains the following identity:





K jk fi , jk d    (K jk fi ,k n j  f i K jk , j nk ) d    fi H ( x1 , x2 , x3 ) d  . 



(6)

The last term in Eq.(6) is an additional volume integral that the previous work [7] does not consider. For transforming the additional volume integral, one may introduce a new fundamental solution Ri and Ki such that the followings are satisfied:

Ri ,kk  f i ,

(7)

K i ,kk  Ri ,

(8)


31

After the Green's second theorem is applied to Eq.(7), one immediately gets



( Ri ,kk H  H , kk Ri ) d    ( Ri ,k H  H k Ri ) nk d  ,

(9)

f i H d    ( Ri ,k H  H k Ri ) nk d    H , kk Ri d  ,

(10)













Due to the relation H ,kk  C0 (a constant) for arbitrary quadratic function, continually applying Green's second theorem to Eq.(10) yields





fi H d    ( Ri , k H  H k Ri ) nk d   C0  K i ,k nk d  , 

(11)



By use of Eq.(11), the identity of Eq.(6) is rewritten as





K jk f i , jk d    (K jk fi ,k n j  f i K jk , j nk ) d    ( Ri ,k H  H k Ri ) nk d   C0  K i ,k nk d  , (12) 





It immediately follows that by making the following substitution:

K jk fi , jk   jkU ij*,k ,

(13)

The last term of Eq.(3) can be rewritten as





   jkU ij*,k d    (K jk f i , k n j  fi K jk , j nk ) d    ( Ri ,k H  H k Ri ) nk d   C0  K i ,k nk d  , (14) 





Thus, substitution of Eq.(14) into Eq.(3) yields the transformed BIE,

cij u j   Tij*u j d    U ij*t j d    (K jk f i ,k n j  fi K jk , j nk ) d  





  ( Ri ,k H  H k Ri ) nk d   C0  K i , k nk d  

.

(15)



It is obvious that the transformed BIE has no volume integral at all. However, this transformation process is not completed yet unless the new functions fi , Ri and Ki are explicitly determined according to Eq.(7), Eq.(8) and Eq.(13), respectively. Derivations of the explicit expressions of the new fundamental solutions will be addressed next. Explicit Expressions of the New Functions As aforementioned, success of the direct and exact volume-to-surface integral transformation is considered complete only when the explicit expressions of the newly introduced new functions are explicitly determined. The explicit form of the Green’s function can be expressed by Ting and Lee [8], U * (x)=

1

1 4 r κ

4

q

n

Γˆ ( n )

,

(16)

n= 0

In terms of the spherical coordinates (r,,), the Green's function can be re-expressed as U * (r , , )=

H ( ,  ) , 4 r

(17)


32

where H ( ,  ) , referred to as the Barnett-Lothe tensor, only depends on the spherical angles ( ,  ) . Instead of computing the Barnett-Lothe tensor directly, Shiah et al. [9] have proposed to rewrite it as a double-Fourier-series as follows: H uv ( ,  ) 

a

a

 

m  a n  a

e

i m  n 

( m,n ) uv

 u , v  1, 2, 3  ,

,

(18)

( m,n ) where a is an integer sufficiently large to ensure convergence of the series; uv are unknown coefficients

determined by  u(vm , n ) 

1 4

2









    H  ,   e

 i  m   n 

uv

d d .

(19)

The first-order derivatives may be readily shown to have the following Fourier-series form:      m    1      4 r 2  m         m   

U u*v , l



  co s   sin   i n co s    e i m   n       i m sin  / sin     sin   sin   i n co s     u(vm , n ) e i  m   n       i m co s  / sin  

 

n





n

( m ,n ) uv



 

n

( m ,n ) uv

e i  m   n      co s   i n sin   

fo r l  1 fo r l  2

.

(20)

fo r l  3

Now, the task is to determine the new function fi by use of Eq.(13) and Eq.(20). In terms of the spherical coordinates, satisfaction of Eq.(13) implies fi must be independent of r and thus, all derivatives of fi taken with respect to r shall vanish. Since f i ( ,  ) should be a periodic function of the spherical angles, it can be expressed as a double-Fourier-series as follows: f i ( ,  ) 

a

a

 

ma na

C i( m , n ) e i ( m  n )

,

(21)

where Ci( m ,n ) are unknown coefficients to be determined and a is an integer guaranteeing convergence of the series. Thus, for determining the unknown coefficients Ci( m, n ) , one may perform the double integrations on Eq.(13) as follows:

1



2





     K 



jk

f i , jk sin 2    e-i( p  q ) d d 

1



2









    

jk

U ij*,k sin 2    e-i( p  q ) d d ,

(22)

The complete process to determine the unknown constant can be referred to [3] and thus, no further elaborations are provided here. After Ci( m, n ) are determined, fi can be calculated by Eq.(21) and its 1st-order derivatives are given by


f i ,t

33

 a a ( m , n ) i ( m  n ) e  i n cos  cos   i m sin  / sin      Ci ma n a  1 a a     C i( m , n ) e i ( m  n )  i n sin  cos   i m cos  / sin   r  ma n a  a a ( m , n ) i ( m  n ) e   i n sin      Ci  ma n a

( t  1) ( t  2) .

(23)

( t  3)

In Eq.(13), the other new function to be determined is R defined according to Eq.(7). By use of Eq.(21), Eq.(7) can be written in terms of the spherical coordinates as a a  2 Ri 2 Ri 1  2 Ri cot  Ri  2 Ri 1 + + 2 + 2 + 2 2 =   Ci( m , n ) ei( m  n ) . 2 2 2 r r r r  r  r sin   m  a n  a

(24)

Satisfaction of Eq.(24) implies R can be expressed in the following form: Ri ( r ,  ,  )  r 2

a

a

 

ma na

( m ,n ) i

e i ( m   n ) ,

(25)

where i( m , n ) are unknown coefficients to be determined. The process to determine i( m , n ) is pretty similar to that used in [3] and the details are omitted here for limited space. Once i( m , n ) are determined, the new function Ri can be calculated directly by Eq.(25) and their 1st-order derivatives are given by  a a ( m ,n ) i( m   n )   sin       i e  2sin  cos   i  n cos  cos   m   ( t  1) sin       m  a n  a  a a   cos     Ri ,t  r    i( m ,n ) ei( m   n  )  2sin  sin   i  n sin  cos   m   ( t  2) . sin       m  a n  a  a a    i( m ,n ) ei( m   n  )  2 cos   i n sin   ( t  3)  m  a n  a

(26)

By use of Eq.(25), Eq.(8) can be written in terms of the spherical coordinates as a a  2 K i 2  K i 1  2 K i cot   K i 2Ki 1 + + 2 + 2 + 2 =   C i( m , n ) e i ( m  n ) . 2 2 2 2 r r r r  r   r sin    ma na

(27)

Satisfaction of Eq.(27) implies K can be expressed in the following form: K i (r , , )  r 4

a

a

 

ma na

 i( m , n ) e i ( m   n ) ,

(28)

where i( m , n ) are unknown coefficients to be determined. Similar to the previous processes, i( m , n ) can be determined; the 1st-order derivatives of Eq.(28) are given by


34

K i ,t

 a a ( m , n ) i( m   n  )   sin  e    i  4 sin  cos   i  n cos  cos   m sin    m a na  a a   cos    r 3     i( m , n ) e i( m   n  )  4 sin  sin   i  n sin  cos   m sin    m a n a  a a     i( m , n ) e i( m   n  )  4 cos   i n sin    m   a n   a

   ( t  1)     ( t  2) . 

(29)

( t  3)

Up to this end, with the fully explicit expressions of fi, Ri and Ki and their 1st-order derivatives, the transformed BIE, Eq.(15), can now be solved via the conventional collocation process. As a secondary process, displacements at internal points can also be calculated using Eq.(15) by simply setting the free coefficient cij to be unity. By making spatial differentiations with respect to the locations of internal points for Eq.(12), one obtains the following BIE: u i ,t 





Tij*,t u j d    U ij* ,t t j d    (  K jk f i ,tk n j  f i ,t K jk  , j nk ) d  



  ( Ri , k H  H k Ri ) ,t n k d   C 0  K i ,tk n k d  

,

(30)



where calculations of U ij*,t and Tij*,t can be referred to [18] and thus, they are not elaborated here. In Eq.(30), the 2nd-order derivatives fi ,tk , Ri ,tk and K i ,tk can be derived by directly differentiations. Numerical Example For showing our implementation of the presented formulations, an example is studied that considers a doubly joined hollow sphere as shown in Fig. 1. Also, the geometric dimensions and the prescribed boundary conditions are indicated in Fig. 1. By assumption, the inner layer has an internal heat source, described arbitrarily by H  12x2  5 y2 10z2  4xy  8 yz  7xz  21x  3y  5z 10 . The outside surface is fully constrained in all directions and the inside surface of cavity is traction-free. For yielding generally anisotropic properties, rotations of the principal axes about the x/y/z-axis counterclockwise for each respective layer are prescribed as depicted in Fig. 1. Fig. 2 shows BEM mesh modeling with 1280 quadratic elements with 3592 nodes. For verifications, the problem was also analyzed using ANSYS, where 39936 SOLID227 elements with 180136 nodes were applied as displayed in Fig. 2.

Figure 1. Doubly joined hollow sphere subjected to thermal loads.


35

Figure 2. BEM and ANSYS mesh modeling. Alumina (Al2O3) is taken to be the inner material that has the following nonzero principal stiffness coefficients:

* C 11  465 GPa ,

C12*  124 GPa ,

C13*  117 GPa ,

* C 14  101 GPa ,

* C 33  563 GPa ,

* C 44  233 GPa .

(31)

The principal thermal properties of the material for analysis are * K11  18 (W / m 0 C),

* K 22  10 (W / m 0 C),

* K 33  25 (W / m 0 C),

* * 11*  8.1 10 6 (1 / 0 C),  22  5.4  10 6 (1 / 0 C),  33  9.2  10 6 (1 / 0 C).

(32)

Niobium (Nb) is taken to be the outside material with the following nonzero principal stiffness coefficients: * C11  246 GPa ,

C12*  134 GPa ,

* C14 

* C 33  246 GPa ,

0 GPa ,

C13*  134 GPa , * C 44  28.7 GPa .

(33)

Its principal thermal properties for analysis are * K 11  15 (W / m 0 C ),

* K 22  10 (W / m 0 C ),

* K 33  20 (W / m 0 C ),

* * *  11  7 . 2  10  6 (1 / 0 C ),  22  6 . 1  10  6 (1 / 0 C ),  33  8 . 5  10  6 (1 / 0 C ).

(34)

The von Mises stresses around the equators (on the x1-x2 plane) of the interface (r=1.5m), internal points (r=2.0m) and outside surface (r=3.0m) were calculated and plotted in Fig. 3. It can be seen that for perfect bonding, the analyses of the both approaches agree very well. Obviously, the equivalent stresses are relatively greater for places with smaller radius. Concluding Remarks As has been well know, the 3D thermoelastic BEM analysis problem involving arbitrary heat volume heat source that effects itself to an additional domain integral, internal treatments are still inevitable. In this research, present works adopt a BEM approach to treat that problem. A double Fourier series representation, the extra domain integral has been analytically transformed to boundary integral in Cartesian coordinate system. The value of numerical example illustrates the veracity of the process where the excellent results computed from domain integral corresponding with boundary integrals. The implemented BEM code yield reliable results demonstrate a

36


further development in BEM analysis, in restoring the BEM as a truly boundary solution technique in 3D thermoelasticity.

Figure 3. Distribution of the calculated equivalent stresses.

Acknowledgements:

The authors gratefully acknowledge the financial support of the Ministry of Science and Technology, Taiwan, ROC. (No: MOST 106-2221-E-006 -129) References [1] Y.C. Shiah and C.L. Tan Applied Mathematical Modelling, 23, 87-96 (1999). [2] Y.C. Shiah, Meng-Rong Li Engineering Analysis with Boundary Elements, 54, 13-18 (2015). [3] Y.C. Shiah Computer Methods in Applied Mechanics and Engineering, 278, 404-422 (2014). [4] Y.C. Shiah, Juin-Yu Chong Journal of Mechanic, 32, 725-735 (2016). [5] V. Sladek, J. Sladek Applied Mathematical Modelling, 8, 413-418 (1984). [6] I.M. Lifshitz, L.N. Rosenzweig Zh. Eksp. Teor. Fiz, 17, 783-791 (1947). [7] Y.C. Shiah, Chung-Lei Hsu, Chyanbin Hwu Computer Modeling in Engineering and Sciences, 102, 257-270 (2014). [8] T.C.T. Ting, V.G. Lee The Quarterly Journal of Mechanics and Applied Mathematics, 50, 407-426 (1997). [9] Y.C. Shiah, C.L. Tan, C.Y. Wang Engineering Analysis with Boundary Elements, 36, 1746-1755(2012).


37

Complex-step assessment of shape sensitivity for the 3D indentation problem Cristiano Ubessi1,† , Rogério J. Marczak2,† & Federico C. Buroni3,‡ † †

Departamento de Engenharia Mecânica - DEMEC, Universidade Federal do Rio Grande do Sul, Sarmento Leite 425, 90050-160, Porto Alegre, Brasil ‡ Department of Mechanical Engineering and Manufacturing, University of Seville, Camino de los descubrimientos s/n, E41092 Sevilla, Spain 1 [email protected], 2 [email protected],3 [email protected]

Keywords: Shape Sensitivity, Anisotropy, Fourier Series, Contact,

Abstract. This paper presents a shape sensitivity study of a 3D indentation problem involving crystalline anisotropic material. The BEM is used to solve the elastic contact problem. In this paper we use the Complex Step Method to obtain the shape sensitivity at the interest regions. This complex variable method requires robust and efficient real-valued fundamental solutions. For this purpose a Fourier series representation of the anisotropic fundamental solution, proposed by other authors, is employed. Whenever possible, results are compared with analytical solutions or previous works from the literature. Introduction Shape sensitivity is one of the key tasks inside of a correponding optimization algorithm. Too few are the cases where it could be calculated analytically, and very often the numerical approach is the only option, due to problem complexity, or to the non availability of closed form expressions for the underlying problem. The increasing rate of anisotropic material applications in engineering on the last few decades, leads to the necessity of structural optimization algorithms adapted to these technologies. The integration of these materials very often is through contact conditions, what leads to a more profound analysis of such problems. Since many years, single crystal turbine blades are a recurring example of such conditions. Those blades are commonly manufactured using a unique face centered cubic crystalline to avoid fluence phenomena at the high temperatures which they operate. The Boundary Element Method is well-known for its accuracy on the solution of contact problems, since its formulation intrinsically treats the displacements and tractions with same order of approximation allowing for the use of load-incremental techniques such as in Refs. [1, 1, 2, 3, 4, 5, 6, 7]. Ref.[8] suggests that the augmented Lagrangian formulation of the contact restrictions using a Newton method to solve the resulting non linear problem avoids trial and error contact state estimations. This formulation is extensively used in other recent works, [9, 10], and particularly has been applied to solve anisotropic contact problems in [11, 12, 13]. Closed form or explicit fundamental solutions for 3D anisotropic elasticity are a very recent topic under closure. Among other works, [14, 15] has presented explicit formulas for their evaluation, which only depends on the numerical evaluation of the 6th order eigenvalue problem. Recent developments in the efficient evaluation of 3D anisotropic fundamental solutions could be found in refs. [16, 17]. This approximated Green functions increase the generality of the resulting BEM code, and allows for simple adaptation of the program. In refs. [18], [19, 20, 21, 22],the shape sensitivity in contact problems were studied using direct or implicit differentiation of the integral equations. In Ref. [23], the complex step method (CSM) was evaluated and compared with finite differences and other works using direct differentiation of boundary integral equations. Recently this method gained some attention in the algorithmic differentiation literature, because of its special properties: it does not involve a difference, so no round off errors are included and the method is insensitive to the step size. The size related error can be proved by a Taylor series to be on the order of O(h2 ). This results in a safe step sizes ranging from 1 × 10−10 to 1 × 10−300 .


38

This paper presents a study on numerical shape sensitivity procedures on contact problems using anisotropic BEM formulation. The paper organized as follows: First the complex step method is introduced. The BEM is briefly presented considering the complex part added by the CSM, and how the sensitivity is obtained from the problem solutinon. The contact problem is described, along with the restrictions it imposes. The resulting non linear system of equations is presented with an analogy to problems with multiple regions. On the results section, the classical Hertz contact problem, considering two elastic regions, is analyzed using a BEM mesh and the solution obtained is compared with the analytical one. Final considerations are done, which closes this work. Complex-step method and boundary integral equations The complex-step method (CSM) was first proposed by [24], based on the works of [25], which used complex variables to compute derivatives of functions through Cauchy-Riemann theorem. Since then, this method has been widely used and it is well-known for its simplicity and the capability to obtain the numerically exact derivative of any real function. [24] presents essential Taylor series expansion which accounts for the error order of the method. The approximated derivative, including its error, is ∂f Im [f (x0 + ih)] + O(h2 ). (1) = ∂x x=x0 h As could be seen, only one function evaluation is needed. As the step size h is generally lower than the machine precision, the term O(h2 ) tends to vanish by the round-off, resulting thus in a numerically exact derivative. CSM and BEM The application of the CSM to BEM is straightforward and the only precautions are relative to the programming language used. As the approaches vary, in C++ and Fortran 90 is possible to use operator overloading to adapt existing code, other languages may have built in complex math, but some standard functions not, so care must be taken [26]. A practical example is that in Matlab the transpose operator should be the .’ and not ’, since the latter is the Conjugate transpose. In the evaluation of shape sensitivities in a BEM formulation, the end variables receiving the complex step are node positions. The step could be performed point to point, evaluating then each individual node sensitivity, or in a more general design approach, e.g. placing the complex perturbation at spline nodes or varying the rounding radius of a corner. The latter is generally faster and results that various nodes of the problem are incremented with varying step sizes and axial directions. To write the Somigliana identity on the boundary, lets consider the complex step was added to a design variable γ which now is γˆ = γ + i.h. This will result that, possibly, many of the positions on the boundary are now complex, with imaginary parts correponding to the effect of that increment on the design variable γ. One can write then that either source x and field y points could are complex, Z Z Cu(ˆ y) + P∗ (ˆ x, y ˆ) u(ˆ y)dΓ = U∗ (ˆ x, y ˆ) p(ˆ y)dΓ, (2) Γ x

Γ y

where, i.e.: x ˆ = x + ih , and y ˆ = y + ih , where the superscript on the step means that the step is particular to that specific position and should not be thought as a constant step in every boundary position. U∗ and P∗ are the displacement and traction fundamental solution tensors, which will be further detailed in the next section. If the boundary at y ˆ is smooth, C = 12 I. With the element discretization geometric variables are calculated PN in termsj of the parametric coordinates ξ = (ξ1 , ξ2 ), by means of standard shape functions: x = n=1 φn (ξ)xn . To simplify the formulation and implementation, the collocation points are placed inside the elements with an offset to the geometrical nodes, in a discontinuous element scheme. Displacements and PN ¯ ¯ tractions are then interpolated with the discontinuous interpolation functions, φ, u = n=1 φn (ξ)ujn , P j ¯ p= N n=1 φn (ξ)pn . The linear and quadratic interpolation functions for the could be found on ref. [27]. After the discretization Eq. (2) the integral is divided resulting in the following sum for each element,


Ci ui +

39

˜e e j=1 Hi u

PN

=

˜e e j=1 Gi p ,

PN

(3)

R R ¯ are obtained for each element e and field point ¯ and G ˜ ei = e U∗ φ, ˜ ei = e P∗ φ, where the matrices H Γ Γ i. Eq. (2) in matrix notation, and further combination of the terms computed after a collocation process over all boundary nodes will result in the algebraic system of equations that leads to the BEM solution, i.e., Hu = Gt. (4) The sensitivity of displacements, tractions, or even postprocessed stress, with respect to the design variable γ, could be obtained as in the following example ∂u Im [u] = , ∂γ h

∂p Im [p] = . ∂γ h

(5)

Fundamental solutions for general anisotropic elasticity The displacement fundamental solution can be expressed as a singular term by a modulation function H as 1 Hjk (x) (6) Ujk (x) = 4πr where x = rˆ e with r = |x| 6= 0. The modulation function Hjk (x) depends on the direction of x but not on its modulus, so Hjk (x) = Hjk (ˆ e) and that is one of the three Barnett-Lothe tensors which is symmetric and H(ˆ e) = H(−ˆ e). The tensor HJK can be evaluated as [14, 28], Z 1 +∞ −1 (7) Γ (p)dp, HJK (ˆ e) = π −∞ JK with ΓJK (p) = QJK + (RJK + RKJ )p + TJK p2 , being QJK = CiJKm ni nm , RJK = CiJKm ni mm , TJK = CiJKm mi mm . Vectors ni and mi are the components of any two mutually orthogonal unit vectors such that (n, m, ˆ e) is a right-handed triad. Moreover, H and U are independent of the choice of the unit vectors m and n on the oblique plane. As mentioned in the previous section, equation (6) shows that the displacement fundamental solution can be expressed via separation of variables as a singular function just depending on the radial distance r, and the modulation function Hjk (ˆ e) which depends only on ˆ e. This vector can be expressed in terms of spherical coordinates θ and φ as ˆ e = (sin φ cos θ, sin θ sin φ, cos φ) (−π ≤ θ < π, 0 ≤ φ < π), and therefore the Barnett-Lothe tensor Hjk (θ, φ), which is a periodic function in both θ and φ with a period of 2π in each one of them. By virtue of this periodic nature, Hjk (θ, φ) admits a double Fourier series representation, which, for the purposes of this work, is written in its trigonometric form according to [17], α X

Hjk (θ, φ) =

(m,n)

Rjk

(m,n)

cos(mθ + nφ) + Ijk

sin(mθ + nφ).

(8)

m,n=−α (m,n)

The Fourier expansion coefficients λjk (m,n) λjk

1 = 2 4π

Z

π

Z

(m,n)

= Rjk

(m,n)

+iIjk

, are obtained through the following equation

π

HJK (θ, φ) (cos (mθ + nφ) − i sin (mθ + nφ)) . −π

(9)

−π

This integration can be numerically performed by standard Gauss quadrature. It should be noted that the period of φ is π as mentioned, H(ˆ e) = H(−ˆ e), although the angle has been above defined in the range [0, 2π). This Fourier series could be further simplified, as could be more easily seen by Eq. (9), (m,n) (−m,−n) λjk and λjk are complex conjugates. It could be made function of the positive terms only, (m,n) (m,n) ¯ λ =λ . jk

jk


40

The fundamental solution for the tractions Tjk , on a surface with normal ni , which represents the j-component, of the generalized traction vector produced by a generalized xk -direction point force, is obtained by Tjk = Cijml Umk,l ni

(10)

where Ujk,l , the derivative of Ujk , easily obtained using the chain rule, Ujk,l (x) =

∂Ujk ∂r ∂Ujk ∂θ ∂Ujk ∂φ + + , ∂r ∂xl ∂θ ∂xl ∂φ ∂xl

(11)

∂(r, θ, φ) are simple trigonometric derivatives, and calculated only once ∂xl for the entire tensor. Detailed simplifications and explicit equations for the Fourier series representation of the displacement fundamental solution and its derivative is found in [17].

where the partial derivatives

Contact Kinematic variables In this work the BEM formulation is assuming small strains and displacements, and node on node contact. The nodes are assumed to be positioned in a conforming scheme. The contact variables in the discrete form will then be related to the possible contact node pairs. The contact frame is based on the master nodes, and the gap variable g is obtained through the following relation g = BT (x2 − x1 ) + BT (u2 − u1 ),

(12)

where x1 and x2 are master and slave nodal coordinates, u1 and u2 are the respective nodal displacement vectors, on the global Cartesian coordinate system, and B is a change of base matrix constructed with the three unit vectors which form a local coordinate system with origin at x1 , the master node position, i.e., B = [t1 , t2 , n] . Contact restrictions The well known frictional contact restrictions are:  g˙ t = 0; Contact - Stick,  tn ≤ 0; gn = 0;



tn ≤ 0; gn = 0; ktt k = µ |tn | ; tn = 0; gn ≥ 0; tt = 0;

g˙ t · t˙ t = − kg˙ t k t˙ t ;

Contact - Slip, Separated.

(13)

The frictional contact law is fulfilled by means of projection operators, i.e. functions which project the augmented version of the contact variables in to the admissible solution region. They are well documented in [10]. Contact treatment with Boundary Element Discretization One of the well known advantages of BEM in contact problems is that the contact tractions are already part of the unknowns. The SLE for the contact problem will be formed by in a similar mode as it is done for multiple-region problems, the following system arise, nc nc nc nc c c c c nc nc ¯ Γnc ¯ Γnc 1 p 1 u HΓ1 uΓ1 − GΓ1 pΓ1 + HΓ1 uΓ1 − GΓ1 pΓ1 = G ¯ Γ1 − H ¯ Γ1 nc nc nc nc c c c c nc nc ¯ Γnc ¯ Γnc 2 p 2 u HΓ2 uΓ2 − GΓ2 pΓ2 + HΓ2 uΓ1 + GΓ2 pΓ1 = G ¯ Γ2 − H ¯ Γ2

(14)

where the bonded connections on the interface between the two regions was set by the displacement and traction compatibility conditions. Equation (14) is sufficient to calculate a bonded problem, where is assumed the interface region remains the same at all times and support tractions in all directions. To incorporate contact restrictions on a similar equation system one have to write Eq. (14), along with two additional sets of equations: The kinematic relations which arise from the gap, i.e., Eq. (12), and the projection operators, which represent the contact restrictions, depending on the contact state of the node pair, resulting in a system Θ(z) = Rz − f , composed by the following system of non-linear equations,


41

t¯0z

r2 ko

r1

(b)

(a)

Figure 1: (a) Problem description and (b) mesh used on the Hertzian contact example.

 nc AΓ1   0 Θ(z) =   0 0 n

Γc1

H 0 1 ˜ (C )T 0

0 nc AΓ2 0 0

0 c HΓ2 ˜ 2 )T −(C 0

1

Γc1

˜ G C ˜ 2 GΓc2 −C 0 Pλ

 Γnc  x 1     Γnc  c    0  uΓ1    b 1       Γnc    Γnc  2 b 2 0 x c −  uΓ2  k      Cg      go       Λ 0  Pg    k

(15)

nc

where AΓα c are the columns from HΓα or −GΓα (Eq. (14)) relative to the independent unknowns xΓα for the α region, which are presented in a more compact manner. Anp are the matrices relative to the traction unknowns on the possible contact region. The contact tractions (Λ) as well as the gap (k) are considered on their local coordinate system, by the incorporation of the rotation matrices on the system of equations, i.e., B, which are assembled for each contact pair on the main rotation matrix ˜ α on Eq. (15). Matrix Cg = I and kgo vector is the sum of the initial gap minus and presented as C the rigid body displacements. The resulting non-linear system is solved using the gereralized Newton method (GNMls) from [29]. Results The problem analyzed in this work consists of two spheres in contact (Fig. 1), allowing the direct comparison with the classical Hertz solution and its derivative with respect to the incremented variables. The mesh used for each sphere on this problem has 264 rectangular elements, and is illustrated on Fig. 1(b) for the quadratic case. The linear and quadratic cases resulted in 6936 and 14400 degrees of freedom respectively, 9 percent of it belonging to the kinematic relations (gap) and the contact restrictions. The offset used on the discontinuous elements was a = 12.5%, and all material and geometrical properties considered on this example are shown on Table 1. Prescribed tractions t¯z were applied on the upper face of the half sphere. A symmetry boundary condition was applied at the faces coincident with xy,zx and zy, restricting the displacements solely at the outward normal direction. The friction coefficient used on this problem was µ = 0.1, as this particular value was used in ref. [10] for a similar example. Also are shown the maximum contact pressure p0 Hertz and the contact area radius predicted for the displacements applied uz , from [30]. The boundary conditions were applied on a single load step and the initial parameters used on the GNMls were q = 1.33, β = 0.75, rt = rn = 0.70. On figure 2a, the normal component of the contact tractions λn normalized with the maximum


42

Table 1: Sphere on Sphere contact problem parameters and results from the hertz solution. ESt νSt

2.0 0.3

Pa m/m

Sphere 1

Young’s Modulus Poisson Ratio

ENi GNi νNi

1.0 0.89377 0.3762

Pa Pa m/m

Sphere 2

Young’s Modulus Shear Modulus Poisson Ratio

µ r1 , r 2

0.1 1.0

N/N m

t¯z p0 a

1.6 × 10−4 2.546 × 10−2 0.020

Pa Pa m

Coulomb friction Sphere radius Pressure applied on the top sphere Maximum hertz contact pressure (Ref.[30]) Radius of the contact area

Hertz pressure p0 are shown. These values were extracted from a line of points close to plane of symmetry xz, with constant y. As can be seen, the normal tractions agree with the analytical solution with a considerable difference. The norm of tangential tractions kλt k/p0 µ divided by the friction coefficient are also plotted in the same figure. Although the maximum normal tractions does not present a pronounced difference between the linear and quadratic elements, at the tangential direction is possible to see a greater difference between the two. The quadratic elements also predict a larger slip region, as one of the additional nodes of the Q8 appears as the last one in the slip region. At this transition region the normal tractions are in much better agreement with the closed form solution, what points out to a better approximation of the quadratic elements. Sensitivity analysis The complex step was appended the radial position of the sphere nodes, i.e., ˆ = X/R · ∆, where ∆ = 1e − 30. The contact normal and tangential traction sensitivities were h analyzed. The comparison for the normal components is done with the analytical solution found in ref. [21], which could be obtained differentiating the Hertz solution for the pressure distribution in relation to the variable R∗ , the combined sphere radius. The results obtained for linear (Q4) and quadratic (Q8) elements are plotted on figure 2b. Although the tangential tractions do not present a closed form solution, the derivative for the normal components agree very well with the analytical, especially for the quadratic elements. Near the end of the contact area where a singularity appears, it is very difficult to the linear elements to match the results provided by the quadratic. Conclusion This work evaluated the use of the complex step method (CSM) to obtain parameters sensitivity on contact problems with friction involving anisotropic materials. A BEM-BEM contact formulation along with a Fourier series representation of the anisotropic elastic fundamental solutions was employed. A brief literature review was presented and discussed. The basic mehthodology for the analysis was also presented, and principally, the integration of the CSM with the BEM. A direct comparison between the results from linear and quadratic (discontinuous) elements show that the former were in fact accurate for contact problems, mostly due to the superior number of nodes in the contact region, if one compares to a continuous scheme. Its also possible to point out that the quadratic elements resulted on a result in much higher agreement with respect to the normal component of the contact tractions. The normal tractions sensitivity was close enough to the analytical solution, representing its general behaviour with an acceptable error. The results of sensitivity for the tangential tractions are revealing, as they are apparently do not scale with the friction coefficient. More detailed investigations on the results presented in this work will be published in a forthcoming paper. Acknowledgments The first author wish to express his thanks to Brazilian CNPq and CAPES/PDSE for the doctoral scolarship.


43

Advances in Bundary Element and Meshless Techniques XIX

43

1.2

8

1

6

0.8

4

0.6

2

0.4

0

0.2

−2

0

0

0.2

0.4

0.6 0.8 x/a (m/m)

tn /p0 Hertz λn /p0 α = 20 Q4 λn /p0 α = 20 Q8 (a)

1

1.2

Ref. [10]. kλt k p0 µ kλt k p0 µ

α = 20 Q4 α = 20 Q8

−4

·10−2

0

0.2

∂tn /∂R∗

0.4

0.6 0.8 x/a (m/m)

1

1.2

kIm[λt ]k/h α = 20 Q4 Im[λn ]/h α = 20 Q4 kIm[λt ]k/h α = 20 Q8 Im[λn ]/h α = 20 Q8 Hertz

(b)

Figure 2: (a) Normal and tangential tractions over the possible contact region (b) Sensitivity of the contact tractions to variation of combined sphere radius R∗ .

References [1] Torbjörn Andersson. The boundary element method applied to two-dimensional contact problems with friction. In Boundary element methods, pages 239–258. Springer, 1981. [2] F. Paris and J.A. Garrido. An incremental procedure for friction contact problems with the boundary element method. Engineering Analysis with Boundary Elements, 6(4):202 – 213, 1989. [3] JA Garrido, A Foces, and F Paris. Bem applied to receding contact problems with friction. Mathematical and Computer Modelling, 15(3):143–153, 1991. [4] KW Man, MH Aliabadi, and DP Rooke. Bem frictional contact analysis: load incremental technique. Computers & structures, 47(6):893–905, 1993. [5] KW Man, MH Aliabadi, and DP Rooke. Bem frictional contact analysis: modelling considerations. Engineering analysis with boundary elements, 11(1):77–85, 1993. [6] F Paris, A Blazquez, and J Canas. Contact problems with nonconforming discretizations using boundary element method. Computers & structures, 57(5):829–839, 1995. [7] J.A. Garrido and A. Lorenzana. Receding contact problem involving large displacements using the bem. Engineering Analysis with Boundary Elements, 21(4):295 – 303, 1998. Contact Mechanics. [8] Luís Rodríguez-Tembleque, José Ángel González, and Ramón Abascal. A formulation based on the localized lagrange multipliers for solving 3D frictional contact problems using the BEM . Numerical Modeling of Coupled Phenomena in Science and Engineering: Practical Use and Examples, page 359, 2008. [9] José A González, KC Park, Carlos A Felippa, and Ramon Abascal. A formulation based on localized lagrange multipliers for bem–fem coupling in contact problems. Computer Methods in Applied Mechanics and Engineering, 197(6):623–640, 2008. [10] L. Rodríguez-Tembleque and R. Abascal. A fem-bem fast methodology for 3d frictional contact problems. Comput. Struct., 88(15-16):924–937, August 2010. [11] L Rodriguez-Tembleque, FC Buroni, R Abascal, and A Sáez. 3d frictional contact of anisotropic solids using bem. European Journal of Mechanics-A/Solids, 30(2):95–104, 2011.

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[12] L Rodríguez-Tembleque, A Sáez, and FC Buroni. Numerical study of polymer composites in contact. Comput. Model. Eng. Sci, 96(2):131–158, 2013. [13] L. Rodríguez-Tembleque, F.C. Buroni, R. Abascal, and A. Sáez. Analysis of frp composites under frictional contact conditions. International Journal of Solids and Structures, 50(24):3947 – 3959, 2013. [14] Federico C Buroni and Andrés Sáez. Three-dimensional green’s function and its derivative for materials with general anisotropic magneto-electro-elastic coupling. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, volume 466, pages 515– 537. The Royal Society, 2010. [15] Federico C Buroni and Andrés Sáez. Unique and explicit formulas for green’s function in threedimensional anisotropic linear elasticity. Journal of Applied Mechanics, 80(5):051018, 2013. [16] YC Shiah, CL Tan, and CY Wang. Efficient computation of the green’s function and its derivatives for three-dimensional anisotropic elasticity in bem analysis. Engineering Analysis with Boundary Elements, 36(12):1746–1755, 2012. [17] CL Tan, YC Shiah, and CY Wang. Boundary element elastic stress analysis of 3d generally anisotropic solids using fundamental solutions based on fourier series. International Journal of Solids and Structures, 50(16):2701–2711, 2013. [18] Zeki Erman and Roger T. Fenner. Three-dimensional design sensitivity analysis using a boundary integral approach. International Journal for Numerical Methods in Engineering, 40(4):637–654, 1997. [19] G.K. Sfantos and M.H. Aliabadi. A boundary element sensitivity formulation for contact problems using the implicit differentiation method. Engineering Analysis with Boundary Elements, 30(1):22 – 30, 2006. [20] Azam Tafreshi. Shape design sensitivity analysis with respect to the positioning of features in composite structures using the boundary element method. Engineering Analysis with Boundary Elements, 30(1):1–13, January 2006. [21] Azam Tafreshi. Shape sensitivity analysis of composites in contact using the boundary element method. Engineering Analysis with Boundary Elements, 33(2):215 – 224, 2009. [22] Azam Tafreshi. Simulation of crack propagation in anisotropic structures using the boundary element shape sensitivities and optimisation techniques. Engineering Analysis with Boundary Elements, 35(8):984 – 995, 2011. [23] Daniel Contreras Mundstock and Rogério José Marczak. Boundary element sensitivity evaluation for elasticity problems using complex variable method. Structural and Multidisciplinary Optimization, 38(4):423–428, 2009. [24] William Squire and George Trapp. Using complex variables to estimate derivatives of real functions. SIAM Review, 40(1):110–112, 1998. [25] J. N. Lyness and C. B. Moler. Numerical differentiation of analytic functions. SIAM Journal on Numerical Analysis, 4(2):202–210, 1967. [26] Joaquim RRA Martins, Peter Sturdza, and Juan J Alonso. The complex-step derivative approximation. ACM Transactions on Mathematical Software (TOMS), 29(3):245–262, 2003. [27] G. Beer, I. Smith, and C. Duenser. The Boundary Element Method with Programming. Springer Wien, New York, 2008. [28] Federico C Buroni, Jhonny E Ortiz, and Andrés Sáez. Multiple pole residue approach for 3d bem analysis of mathematical degenerate and non-degenerate materials. International Journal for Numerical Methods in Engineering, 86(9):1125–1143, 2011. [29] P. Alart and A. Curnier. A mixed formulation for frictional contact problems prone to newton like solution methods. Comput. Methods Appl. Mech. Eng., 92(3):353–375, November 1991. [30] Kenneth Langstreth Johnson and Kenneth Langstreth Johnson. Contact mechanics. Cambridge university press, 1987.


45

Hypersingular Boundary Integral Equation for harmonic acoustic problems in 2.5D domains Pizarro-Ruiz J.1, Puertas E.2*, Gallego R.3 Departamento de Mecánica de Estructuras e Ingeniería Hidráulica, ETS de Ingenieros de Caminos, Canales y Puertos, Universidad de Granada, Avenida Fuentenueva s/n, 18002 Granada, Spain, 1

[email protected], [email protected] [email protected]

*

Corresponding author.

Keywords: BEM, hypersingular, 2.5D, coincident contour, potential problem, Helmholtz equation, acoustic barrier, wave propagation

Abstract. A challenge faced when modeling crack-like contours by the Boundary Element Method, is to obtain an accurate approximation of integrals which have singular and/or hypersingular kernels. In this paper, we apply a procedure based on regularization of the kernels to compute singular and hypersingular integrals for time harmonic wave problems in two and a half dimensional (2.5D) domains. For this purpose, the Fundamental Solution is described and the study of the terms that arise taking its limit as r→0 is carried out. These terms lead to Singular Boundary Integral Equation (SBIE) and Hypersingular Boundary Integral Equation (HBIE). Regular terms of SBIE and HBIE are integrated by applying numerical quadratures, whereas singular and hypersingular terms are calculated analytically. Numerical results are compared to analytical ones for potential and acoustic problems with crack-like contours to demonstrate the accuracy of the proposed formulation. Introduction Wave propagation analysis over crack-like contours, is used to model different kind of problems such as acoustic barriers to reduce or eliminate rail or car traffic noise or antiplane shear problems over cracks. Boundary element method (BEM) is a good tool to tackle it because it shows strong performance in wave propagation analysis; but modeling these special contours using BEM leads to the integration of functions with hypersingular kernels, which often are not easy to evaluate. For this purpose, many authors have proposed different techniques to face this problem. In [2] the authors use a linear function to represent the integrand function over the domain; [4] present a regularization of the kernels by using simple solutions; in [3] polar transformation and Lauren expansion series are used to derivate the hypersingular integral; [6] employ image source technique to avoid hypersingular terms. In this work, we apply the regularization procedure presented in [5] in order to compute singular and hypersingular integrals in two and a half dimensional (2.5D) domains. For a better understanding of the reader, this work shows the formulation for the particular case of acoustic barriers; but this procedure can be applied to any wave propagation problem in 2.5D with crack-like contours by just changing the meaning of the variables and constants, provided that the problem follows Helmholtz equation. Boundary Element Formulation The scalar Helmholtz equation can be written as:

1 𝜕 2𝑝 1 − ∇2 𝑝 − 2 𝑏 = 0 2 2 𝑐 𝜕𝑡 𝑐

(1)

In the particular case of 3D acoustics, c is the wave propagation speed inside the fluid, b are the external forces, 𝜕

𝜕

𝜕

p is the fluid pressure and ∇= (𝜕𝑥 , 𝜕𝑦 , 𝜕𝑧). For time harmonic problems, we can make a Fourier transform along time-axis to convert the problem from time-domain to frequency-domain. Also, if the geometry of the


46

acoustic domain does not vary along z-coordinate, we can define a parameter kz called wave number, and make another Fourier transform along z-axis: ∞

∞

𝑃(𝑥, 𝑦, 𝑘𝑧 , 𝜔) = ∫ ∫ 𝑝(𝑥, 𝑦, 𝑧, 𝑡)𝑒 −𝑖(𝜔𝑡−𝑘𝑧 𝑧) 𝑑𝑧𝑑𝑡

(2)

−∞ −∞

This formulation leads to find the solution of eq (1) as an infinite sum of waves with different kz and ω. Substitution of eq (2) into eq (1), assuming there is no external forces (B=0) and with some rearrangement, gives the homogeneous Helmoltz equation in 2.5D:

∇2 𝑃 + 𝑘𝑓 𝑃 = 0 𝜕

𝜕

where ∇= (𝜕𝑥 , 𝜕𝑦) and 𝑘𝑓 =

𝜔2

(3)

− 𝑘𝑧2. Let Γ be the 2D contour of the acoustic domain, X={xi} be a point over

𝑐2

Γ, and let Y={yi} be the collocation point of the fundamental solution of eq (3); standard boundary integral equation (PBIES) of this equation is:

𝑐(𝒀)𝑃(𝒀) + ∫ 𝑄 ∗ (𝑿; 𝒀)𝑃(𝑿)𝑑Γ(𝐗) − ∫ 𝑃∗ (𝑿; 𝒀)𝑄(𝑿)𝑑Γ(𝐗) = 0 Γ

(4)

Γ

where 𝑄, the pressure gradient and c(𝒀) is the so called “free term”. 𝑃∗ and 𝑄 ∗ are, respectively, the fundamental solutions for the pressure and the pressure gradient: 1

(2)

(2)

𝑃∗ (𝑿; 𝒀) = 𝑃∗ (𝑟) = 4𝑖 𝐻0 (𝑘𝑓 𝑟) ; 𝑄 ∗ (𝑿; 𝒀) = 𝑄 ∗ (𝑟) = (2)

𝑖𝑘𝑓 𝜕𝑟 4 𝜕𝒏

(2)

𝐻0 (𝑘𝑓 𝑟)

(5)

where 𝐻0 , 𝐻1 are Hankel functions of second kind and r, the Euclidean distance between the point over the domain X and the collocation point Y:

𝒓 = 𝑿 − 𝒀 ; 𝑟 = |𝑿 − 𝒀|

(6)

Figure 1: a) Example of a domain with a barrier on it inside b)Both faces of the barrier being infinitely close to each other.

When there is a rigid barrier inside the domain, its contour can be expressed as a sum of three parts: the exterior contour Γ𝐶 , and two coincident contours, one for each face of the barrier, Γ + and Γ − [1]. As shown in Fig. 1, both Γ + and Γ − are infinitely close, so if Y is over one of these contours, eq (4) leads to an incompatible system of equations. Let n={ni} be the normal vector of Γ + , we can have –n as the normal vector of contour Γ − . Then, with some rearrangement, eq (4) when Y is over the barrier is:

𝑐(𝒀)𝑃(𝒀) + ∫ 𝑄 ∗ (𝑿; 𝒀)𝑃(𝑿)𝑑Γ(𝐗) + ∫ 𝑄 ∗ (𝑿; 𝒀)Δ𝑃(𝑿)𝑑Γ(𝐗) Γ+

Γ𝑐

−∫ 𝑃 Γ𝑐

∗ (𝑿;

𝒀)𝑄(𝑿)𝑑Γ(𝐗) − ∫ 𝑃 Γ+

∗ (𝑿;

(7) 𝒀)Δ𝑄(𝑿)𝑑Γ(𝐗) = 0


47

where ∆𝑃 = 𝑃+ − 𝑃− and ∆𝑄 = 𝑄 + − 𝑄 − . Using eq (7), instead of two separate equations for each face of the barrier, make the system of equation indeterminate. Thus, in order to have a determinate system, a new equation is needed. For this purpose, eq (7) can be derived respect Y over the direction defined by the vector N={Ni}, which is the normal vector in the collocation point. This leads to:

𝑄 + (𝒀) + ∫ 𝑆 (𝑿; 𝒀)𝑃(𝑿)𝑑Γ(𝐗) + ∫ 𝑆 (𝑿; 𝒀)Δ𝑃(𝑿)𝑑Γ(𝐗) − ∫ 𝐷 (𝑿; 𝒀)𝑄(𝑿)𝑑Γ(𝐗) Γ+

Γ𝑐

Γ𝑐

(8)

− ∫ 𝐷 (𝑿; 𝒀)Δ𝑄(𝑿)𝑑Γ(𝐗) = 0 Γ+

where 𝜕

𝜕

𝐚) 𝑆(𝑿; 𝒀) = 𝜕𝑦 𝑄 ∗ (𝑿; 𝒀)𝑁𝑖 ; b) 𝐷(𝑿; 𝒀) = 𝜕𝑦 𝑃∗ (𝑿; 𝒀)𝑁𝑖 𝑖

𝑖

(9)

and the summation convention for repeated indices is adopted. Eq (8) is called Gradient Boundary Integral Equation (QBIES). Singular and Hypersingular Integrals Formulation described in the previous section leads to integrals with singular and hypersingular kernels. Analyzing 𝑆(𝑿; 𝒀) equation, eq (9a):

−𝑖𝑘𝑓 1 𝜕 ∗ 𝜕𝑟 𝜕𝑟 (2) (2) 𝑄 (𝑿; 𝒀)𝑁𝑖 = [ (𝑛𝑖 − 2 𝑟,𝑖 ) 𝐻1 (𝑘𝑓 𝑟) + 𝑘𝑓 𝑟,𝑖 𝐻0 (𝑘𝑓 𝑟)] 𝑁𝑖 𝜕𝑦𝑖 4 𝑟 𝜕𝒏 𝜕𝒏

(10)

where 𝑟,𝑖 = 𝜕𝑟⁄𝜕𝑥 . In order to get non-regular terms of eq (9), Lauren expansion of Hankel functions is done. 𝑖

Two non-regular terms are obtained: 



1 1 2𝜋 𝑟 2 −𝑘𝑓2 4𝜋

𝑛𝑖 𝑁𝑖

𝑛𝑖 𝑁𝑖 log(𝑟)

which is hypersingular

which is singular

Both terms are integrated following the procedure described in [5]. Analyzing 𝐷(𝑿; 𝒀) equation, eq (9b):

𝑖𝑘𝑓 (2) 𝜕 ∗ 𝑃 (𝑿; 𝒀)𝑁𝑖 = 𝐻 (𝑘𝑓 𝑟)𝑟,𝑖 𝑁𝑖 𝜕𝑦𝑖 4 1

(11)

After making a Lauren expansion of this function, all terms obtained are regular. Thus, no special integration is needed. Numerical Examples In order to show the accuracy of the previous method, we have calculate a theoretical example. Let have a 2.5D domain where there is a rigid barrier. We are going to compare results obtained after an analysis with both PBIES and QBIES. In the case of PBIES, barrier can be model as an ellipse contour where its semi-minor axe b →0. On the other hand, for a QBIES formulation, barrier will be model as a pair of coincident contours, as shown previously. Assuming 𝑘𝑧 ad c given, boundary conditions expressed in Fig. (2), and all units according to the International System of Units:

48


(a)

(b) Figure 2 a) Model with a pair of coincident contours and its boundary conditions. b) Model with ellipse barrier. Crosses are internal points, where P will be calculate.


49

After both analysis, we show the results for the base contour, and for the barrier. Also, we have calculate the pressure in some internal points showed in Fig. (2):

Figure 3 Comparison between P calculated with QBIES (Coincident Contours), and PBIES (ellipse) obtained in the base contour for different b values. Dimension of the contour has been normalized from 0 on the left to 1 on the right.

Figure 4 Comparison between P increment obtained with QBIES formulation (Coincident Contours), and PBIES (ellipse) obtained in the barrier for different b values. Dimension of the contour has been normalized from 0 on the left down to 1 on the right and up.

50


Figure 5 Comparison between P obtained with QBIES formulation (Coincident Contours), and PBIES (ellipse) obtained in internal points, for different b values. Distance from the barrier has been normalized from 0 on the left and down to 1 on the right and up.

Conclusions As we can see in Fig. (3-5), as long as semi-minor axe b tends to 0, PBIES result tends to the solution with QBIES formulation. When b is too small, the system of equations obtained with PBIES formulation is not consistent. In contrast, coincident contour formulation have a greater performing, as the barrier is already infinitely thin. Thus, the accuracy of the results is better.

Bibliography [1] Bordón, J.D.R., Aznárez, J.J., Maeso, O. A 2D BEM-FEM approach for time harmonic fluid-structure interaction analysis of thin elastic bodies. Engineering Analysis with Boundary Elements 43, 19-29 (2014) [2] Gray, L. J. Evaluation of hypersingular integrals in the boundary element method. Mathematical and Computer Modelling, 15, (3–5), 165–174 (1991). [3] Guiggiani, M., Krishnasamy, G., Rudolphi, T. J., and Rizzo, F. J. A General Algorithm for the Numerical Solution of Hypersingular Boundary Integral Equations. Journal of Applied Mechanics, 59 (1992) [4] Rudolphi, T. The use of simple solutions in the regularization of hypersingular boundary integral equations. Mathematical and Computer Modelling, 15, 269–278, (1991). [5] Sáez, A., Gallego, R., and Dominguez, J. Hypersingular quarter‐point boundary elements for crack problems. International Journal for Numerical Methods in Engineering, 38 (1995). [6] Tadeu, A., António, J., Mendes, P. A., and Godinho, L. Sound pressure level attenuation provided by thin rigid screens coupled to tall buildings. Journal of Sound and Vibration. (2007).


51

An IGABEM application in damage mechanics Vincenzo Mallardo1,a , Eugenio Ruocco,2,b , and Gernot Beer3,c 1 Department 2 Department

of Architecture, University of Ferrara, Ferrara (Italy)

of Engineering, University of Campania, Caserta (Italy)

3 Institute

a

of Structural Analysis, Graz University of Technology, Graz (Austria) [email protected] (Corresponding author), b [email protected], c [email protected]

Keywords: NURBS, Integral equation, nonlinear, mapping, Isogeometric, BEM.

Abstract. In this paper an application coupling the Isogeometric Boundary Element Method with the continuum damage mechanics is proposed in three-dimensional geometry. The novelty of the approach stands both in the discretization phase in which independent NURBS functions are used to model geometry plus governing variables and in the computation step where a special mapping procedure is applied to compute the arising volume integrals. The the use of the internal cells is avoided. The implementation is verified on a simple test case. Introduction Design in engineering and architecture is usually carried out by following two main steps, the generation of the geometry and the analysis of the structure. This two-step approach is quite common both in architecture and in engineering, from civil to aeronautic to mechanic engineering. The first phase encompasses the design of the shape along with all the technological details whereas the latter is carried out both by usually running some structural analyses with the aid of numerical software and by defining the main structural details. Both steps proceed simultaneously and require a repetitive structural analysis to update the continuous modifications that are carried out to achieve the ”best” solution. Computer-Aided Design (CAD) was developed as a tool for representing the geometry. On the other hand, Computer-Aided Engineering (CAE) was created to model and analyze the structural behavior. Since their origin, that occurred in different times, there has been a gap between the representation of the geometry and its structural modeling. One difference is related to the fact the ”structural geometry” does not coincide with the ”architectural” one as the former contains the structural components only and it does not give prominence to many decorative and technological features that, on the other hand, are included in the latter. This is still an open field of research that has become overriding for instance in laser scanning data acquisition systems research [1]. Another important difference is related to the way the model is described by shape functions, that is usually performed by B-Spline and Non Uniform Rational B-Splines (NURBS) in CAD, by polynomial functions commonly in CAE. In the late 2000 a novel approach [2] was proposed in the Finite Element context: the structural unknowns are discretized with the same functions adopted to represent the geometry, i.e. the NURBS. The approach, commonly referred as isogeometric, has rapidly gained significant popularity because the geometry data can be directly taken from CAD programs and thus the need for structural meshing is eliminated. Besides, it allows easy mesh refinements without influencing the imported geometrical data. A natural application of the isogeometric approach is represented by the Boundary Element Method (BEM) [3]. Such a method, at least in the linear case, allows the discretization of the boundary only, thus optimizing the advantages. In the nonlinear context, such as plasticity or damage for instance, it is necessary to discretize the volume involved by the dissipation. Commonly, such a discretization is carried out with polynomial cells. A novel approach [4] was recently presented in threedimensional (3D) plasticity to overcome the generation of the cells in the volume. An inclusion of different material or a region involving plasticity is discretized by a special mapping procedure that transforms the inclusion/plastic-zone into a unit cube volume with the aid of NURBS surfaces. In such


52

a unite cube all the integration can be carried out more easily. The procedure was then successfully extended to two-dimensional [5] and 3D [6] steady incompressible viscous flow. A natural continuation of the approach is in continuum damage mechanics. Quasibrittle materials, such as concrete, rock, tough ceramics or ice, are characterized by the development of nonlinear fracture process zones, which can be macroscopically described as regions of highly localized strains. The classical linear elastic fracture mechanics cannot be directly applied to such a phenomenon because of the existence of narrow fracture process zone containing a large number of distributed microcracks at the fracture front. In the last thirty years Continuum Damage Mechanics (CDM) has grown as a link between the classical Continuum Mechanics and the Fracture Mechanics. Damage is mainly localized in relatively small zones, that is, the nonlinear volume is limited in comparison with the overall size of the finite domain. The smallness of the nonlinear zone amplifies the advantages of the BEM [7]-[10]. CDM models the development, the growth and finally the coalescence of microdefects which could lead to the occurring of macrocracks and eventually of rupture. One of the most commonly used damage models introduces an arbitrary variable tensor to model the growth and the diffusion of the microcracks inside the solid. Such a tensor is introduced in the constitutive equations in order to describe the region of the body in which a degradation of the material elastic properties due to the microcracking phenomenon occurs. Interesting developments of CDM are discussed in [11] and [12]. In the former the application of the Object-Oriented Paradigm to the constitutive modeling of a wide range of materials of engineering interest, among which brittle materials, is investigated with the intent of creating a fully independent computational framework. In the latter a new solution strategy for the non-linear implicit formulation of the BEM is presented and applied to the CDM context. In the present paper a BEM approach is used to model the continuum damage problem. Geometry and boundary unknowns are discretized by NURBS functions in independent way. Such an independence allows to refine the unknown field without involving the geometry. The damaged zone is not discretized into cells but dealt with a special mapping procedure that reduces it to a unit cube. The consequent integral computations are computed straightforward. The final procedure is tested on a single example for which an analytical solution is available. The constitutive model Following an idea by [13], the nonlinear behaviour of quasi-brittle materials can be phenomenologically represented by continuum damage models in which a tensor d gives, for any plane containing the point, the measure of the micro-crack diffusion. The value of such a tensor in any point is related to the ratio between the effective area of the intersection of all microcracks lying in the plane and the area of the intersection of the plane with the representative volume element (RVE). in the present contribution the damage is assumed to be isotropic, i.e. a scalar variable d rather than a tensor is used, and the formulation is confined to the case of small induced strains. The stress-strain relation can be written in the following way: el σij = (1 − d)Cijkl ϵkl

(1)

where Cel is the fourth order elastic moduli tensor and σ and ε are the stress and the strain tensor, respectively. In order to set the model consistently with thermodynamic principles [10], it is necessary to introduce the energy per unit of volume Y : Y := −

1 1 ∂f (d) el el ϵij Cijkl ϵkl = ϵij Cijkl ϵkl 2 ∂d 2

(2)

It is also necessary to introduce the kinematic internal variable γ and the force X whose definition is the following: X := hγ (3)


53

It can be proved that such a formulation is consistent with thermodynamic principles assuming the existence of an Helmholtz free energy of the type: 1 1 el ψ(ε, d, γ) = ϵij f (d)Cijkl ϵkl + hγ 2 . 2 2

(4)

Equivalently with plasticity, a damage activation function g(Y, X) can be assumed. Under the hypothesis of generalised associative damage behaviour we have: g(Y, X) = Y − Y0 − X

(5)

The evolution equations read: .

.

.

.

.

g(Y, X) ≤ 0 d = λd = γ with λd ≥ 0 and λd g = 0

(6)

in any point of the subvolume Vd . At the generic iteration and at the points where the damage activation function is zero (points y ∈ Vd ⊆ V ), the response is either elastic or damaging and the following relations must hold (in Vd ): .

g≤0

.

λd ≥ 0

.

.

λd g = 0

(7)

The isogeometric BEM The Boundary Integral Equations (BIEs) involved in continuum damage are similar to those governing the classical elasto-plasticity. Following the procedure shown in [4], the BIE can be written in incremental form as: ∫ ∫ . . . cij (ξ)uj (ξ) + − Tij (ξ, x)uj (x)dΓ(x) = Uij (ξ, x)tj (x)dΓ(x) + Γ

∫

Γ

∫

.d

Uij (ξ, X)tj (X)dSd (X) +

+ Sd

.d

Uij (ξ, X)bj (X)dVd (X)

(8)

Vd

where Γ is the boundary of the domain, the integral on the left hand side is to be interpreted in the sense of Cauchy principal value, Uij and Tij are the well-known fundamental solutions, the coefficient cij is related to the shape of the boundary around ξ ∈ Γ, x and X are boundary and internal points, . d .d

respectively, and bj , tj can be expressed in terms of the damaged stress increment in the subvolume Vd , with boundary Sd , involved by the damage process. The procedure requires the computation of the strain and stress increments in the subvolume Vd . The integral relation providing the strain increment at any internal point y can be written as: ∫ ∫ . . . ϵij (y) = Sijk (y, x)tk (x)dΓ(x) + Rijk (y, x)uk (x)dΓ(x) + Γ

Γ

∫

.d

∫

Sd

.d

Sijk (y, X)bk (X)dVd (X)

Sijk (y, X)tk (X)dSd (X) +

(9)

Vd

where Sijk (y, X) and Rijk (y, X) are the strain tensor at point X due to a unit point load and a unit point displacement, respectively, at point y in the direction k, whose expressions can be found in any BEM book (see for instance [3], [14]).


54

.d

.d

The terms bk and tk involved in eqs (8-9) can be written in terms of the damaged stress tensor .d increment σ ij , i.e.: .d

∂ σ kl bk = ∂xl .d

.d

.d

tk = σ kl nl

(10) .d

It can be noticed that both are written in terms of the tensor increment σ . This is usually named as . ”damaged stress” increment and it can be obtained in terms of the ”actual stress” increment σ and of . el the ”elastic stress” increment σ by the following relation: .

. el

.d

.

.d

el el σ ij := σ ij − σ ij = Cijlm εlm − Cijlm εlm

(11)

In the case under analysis, it is easy to show that: .

. el

σ ij = (1 − d)σ ij

(12)

The integral equations eqs (8-9) can be discretized to compute, at each iteration,. both the boundary unknowns and the strain tensor in any internal point (necessary to determine Y ). The collocation method is used to build the governing system of equations. Boundary discretization. The usual approach in BEM is to divide the boundary Γ into NΓ continuous (usually quadratic) elements in order that geometry, displacement and tractions can be approximated by products of polynomial shape functions MΓn with nodal values. In the present paper a different discretization scheme is carried out. The boundary surface is divided into patches, and geometry plus unknowns are approximated by independent NURBS-based functions, that is: x

e

=

.e

uj =

K ∑ k=1 Ku ∑

Rk (s, t) xek

(13)

.e

Rku (s, t) ujk

(14)

k=1 K ∑ t

.e

tj =

.e

Rkt (s, t) tjk

(15)

k=1

where e refers to the number of the patch, Rk , Rku and Rkt are NURBS basis functions with local .e .e coordinates (s, t), xe locates the control point, uk and tk are related to boundary displacement and traction increment vectors. Discretization and collocation approach allow to transform the integral equation eq (8) into a system of equations. The imposition of the boundary conditions is not straightforward as the unknowns are parameters that have no physical meaning. Applying the boundary conditions at non-interpolatory control points may introduce an error. A correction procedure is proposed in [15], [16]. Sub-volume integration. Some integrals in equations eqs (8-9) are to be computed in the damaged volume Vd and on its boundary Sd . No cells are adopted in the present work. The damaged zone is defined by two NURBS surfaces and a linear interpolation between them: x(s, t, r) = (1 − r) xI (s, t) + r xII (s, t)

(16)

where: K ∑

x (s, t) =

k=1

K ∑ II

I

I

RkI (s, t) xIk

II

and x (s, t) =

RkII (s, t) xII k .

(17)

k=1

and s = (s, t, r) is a local coordinate system in a unit cube (see Fig. 1). The superscripts I and II refer to the bottom (dark-violet) and top (sea-green) surfaces respectively. With the present mapping procedure, all the numerical computations can be carried out in the unit cube and then mapped back to the global coordinate system.


55

Fig. 1: Geometry definition of the damaged subvolume Vd with the proposed approach. Left: the subvolume in the global coordinate system. Right: the same subvolume after mapping.

Numerical example The example tests the presented procedure for the case of a damage that is spread over a 1 x 1 x H bar. The geometry of the bar is defined by six NURBS surfaces (patches) with basis functions of order 1. The beam is loaded with a uniform displacement utop on the top and fixed at the bottom. The NURBS representation of the boundary unknowns is refined to the second order. The damaged subvolume is coincident with the entire domain and a very coarse 2 x 2 x 1 grid of internal point is set. In such a case 1

ttop

0.75

0.5

0.25

Analytic results Numeric results 0 0

0.25

0.5

0.75

1

1.25

1.5

1.75

utop

Fig. 2: Test example: top z−traction versus imposed top z−displacement.

it is possible to obtain an analytical solution with ν = 0. By setting Y = Y0 when utop = 1 on the top, we obtain the following solution for utop > 1: d=

1 E u2top − 1 2 H2 h

σz = (1 − d)E

utop H

(18)

where E is the Young’s modulus. The numerical results are compared with the analytical ones in Fig. 2 being H = 10, E = 10, Y0 = 0.05 and h = 0.1. As it is clear from the figure, the error is negligible (less than 1.5%).

56


Summary An application of the isogeometric BEM was presented in the CDM context. Geometry, boundary unknowns, boundary conditions and volume integrals were discretized by NURBS. A novel mapping procedure that avoids the use of 3D cells was adopted in the damaged subvolume. The procedure was tested on a simple example. Acknowledgement Italian Prin 2015 project ”Advanced mechanical modeling of new materials and structures for the solution of 2020 Horizon challenges” (2015JW9NJT) is acknowledged for the support. References [1] C. Alessandri, M. Garutti, V. Mallardo, G. Milani: Int. J. Archit. Herit. Vol. 9(2) (2015), p. 111129. [2] T. Hughes, J. Cottrell, Y. Bazilevs: Comput. Meth. Appl. Mech. Eng. Vol. 194 (39-41) (2005), p. 4135-4195. [3] M.H. Aliabadi: The boundary element method, vol. 2: applications in solids and structures (Wiley, London 2002). [4] G. Beer, V. Mallardo, E. Ruocco, B. Marussig, J. Zechner, C. Duenser, T. P. Fries: Comput. Meth. Appl. Mech. Eng. Vol. 315 (2017), p. 418-433. [5] G. Beer, V. Mallardo, E. Ruocco, C. Duenser: Comput. Meth. Appl. Mech. Eng. Vol. 326 (2017), p. 51-69. [6] G. Beer, V. Mallardo, E. Ruocco, C. Duenser: Comput. Meth. Appl. Mech. Eng. Vol. 332 (2018), p. 440-461. [7] C. Alessandri, V. Mallardo: Comput. Mech. Vol. 24 (1999), p. 100-109. [8] V. Mallardo, C. Alessandri: Comput. Mech. Vol. 26 (2000), p. 571-581. [9] V. Mallardo, C. Alessandri: Eng. Anal. Bound. Elem. Vol. 28 (2004), p. 547-559. [10] V. Mallardo: Int. J. Fracture Vol. 157 (2009), p. 13-32. [11] L. Gori, S.S. Penna, R. L. S. Pitangueira: Comput. Struct. Vol. 187 (2017), p. 1-23 [12] R.G. Peixoto, F.E.S. Anacleto, G.O. Ribeiro, R.L.S. Pitangueira, S.S. Penna: Eng. Anal. Bound: Elem. Vol. 64 (2016), p. 295-310. [13] D. Krajcinovic: Damage mechanics (North-Holland series in Applied Mathematics and Mechanics 1996). [14] G. Beer: Advanced numerical simulation methods - From CAD Data directly to simulaton results (CRC Press/Balkema 2015). [15] V. Mallardo, E. Ruocco: CMES-Comp. Model. Eng. Vol. 102 (2014), p. 373-391. [16] V. Mallardo, E. Ruocco: European Journal of Computational Mechanics Vol. 25 (2016), p. 71-90


57

Boundary Element Analysis of Damage Detection in Plates Using InPlane Ultrasonic Waves Jun Li 1, a, Zahra Sharif Khodaei 1, b, M. H. Aliabadi 1, c 1

Department of Aeronautics, Imperial College London, South Kensington Campus, City and Guilds Building, Exhibition Road, SW7 2AZ, London, UK.

a

[email protected], b [email protected], c [email protected]

Keywords: boundary element method (BEM); dual boundary element method; wave propagation; damage detection; structural health monitoring (SHM); piezoelectric transducer

Abstract The aim of this paper was to conduct numerical simulations of damage detection process for plate structures using the boundary element method (BEM). A boundary element formulation based on the plane stress theory is used to model the propagation of the fundamental symmetric Lamb mode (S0) and the fundamental shear horizontal mode (SH0) in an isotropic plate. Crack problems are modelled with the dual boundary element method. Piezoelectric (PZT) actuators are used to generate diagnostic signals. The coupling relationship between the PZT actuators and the host plate are described using a semi-analytical model. Numerical examples show that BEM results are accurate and promising in modelling the damage detection processes. 1. Introduction Lamb waves have been widely used in health monitoring of plate structures because they can travel long distances with less dissipation and are sensitive to small flaws. In structural health monitoring (SHM) applications, numerical modelling plays a very important role because it can help to understand the interaction of Lamb waves with damage, and help to conduct experiments. Finite element method (FEM) is commonly used for this numerical modelling because commercial FEM software can provide easy access for researchers and engineers to carry out the simulation [1, 2]. However, since Lamb waves used for SHM applications are ultrasonic waves, FEM becomes computationally inefficient in the modelling of such high-frequency waves. Boundary element method (BEM), in which only boundaries, rather than entire domain, need to be discretized, has been shown to be a more efficient and numerically stable method than FEM in the modelling of SHM applications for a three-dimensional beam [3]. The boundary element formulation used in Ref. [3] is not suitable for plate structures because of the large surface/volume ratio. Therefore, in order to model Lamb waves in plate structures, other formulations should be used. There are two types of in-plane waves in an isotropic plate within low frequency range: the fundamental symmetric Lamb mode (S0) and the fundamental shear horizontal mode (SH0). SH0 is a non-dispersive mode, and S0 mode can be considered to be nondispersive at low frequencies. It has been shown that these two waves can be captured accurately using the plane stress theory if the frequency is not too high [4]. Therefore, a boundary element formulation based on the plane stress theory was used in our study to model both S0 mode and SH0 mode. In this study, Laplace transform method is used to solve the elastodynamic equation. A through-thickness crack is chosen as the form of damage and the crack problem is modelled using dual bondary element method. Piezoelectric (PZT) actuators are used to generate in-plane waves and an equivalent-traction model is used to couple the actuators with the host plate. Numerical results show that BEM can be used as an alternative numerical method in the damage detection applications for plate structures. 2. Numerical Modelling The Laplace transform of governing equation for an isotropic elastic plate is:

c

2 1

 c22  ui ,ij (x, s)  c22u j ,ii (x, s)  s 2u j (x, s)

(1)


58

where s is the Laplace transform parameter; c1  E   1   2  is the longitudinal wave velocity and

c2  E 2 1    is the shear wave velocity; E,  and  denote Young’s modulus, Poisson’s ratio and material density, respectively. The boundary integral equations and numerical implementation strategies can be found in Ref. [5]. In our study, dual PZT disks are used to excite pure S0 mode and the configurations can be found in Ref. [6]. Since the size of the PZT disks can be negligible in comparison to the size of the host plate, according to Ref. [7], the influence of PZT actuators on the plate can be simplified into an equivalent surface traction. The traction is normal to the disk circumference and can be written as:

Fr  

E pzt 1   pzt

d 31V ,

(2)

where E pzt and  pzt are Young’s modulus and the Poisson’s ratio of the PZT actuator; d 31 is the piezoelectric strain constant and V represents the voltage applied to the actuator. This surface traction can be considered to be body force in the boundary integral equations. 3. Numerical examples As shown in Figure 1, an aluminium plate with a central through-thickness crack is used for the numerical analyses. The thickness of the plate is 1.6 mm. The diameter and the thickness of the disk actuators are 6.9 mm and 0.5 mm, respectively. The material properties of the plate and the PZT actuators are the same as those in Ref. [8]. The displacement responses are measured at the point P1.

Figure 1. Configuration for numerical modelling

Figure 2. Dispersion curves based on different theories

Hanning-windowed sinusoidal tonebursts with five-cycle are used as diagnostic signals and their expressions can be written as:  1 2 f c t   5  V (t )  Va sin(2 f ct ) 1  cos( ) H   t  , 2 5   fc  

(3)

where Va and f c denote the peak voltage and the central frequency; H stands for the Heaviside function. Figure 2 compares the Lamb-Rayleigh dispersion curve for the S0 mode with the dispersion curve obtained using the plane stress theory. It can be seen that the plane stress theory can provide satisfactory approximations in the frequency range between 0 to 200 kHz. In the following calculations, Va is set to 80V and the central frequency is 200 kHz. At such frequency, the group speed of the S0 mode based on the plane stress theory is 5432 m/s, which is fairly close to 5409 m/s, the speed from the Lamb-Rayleigh dispersion curve.


59

Figure 3 compares the displacements at P1 in the intact plate with those in the cracked plate. Three wave packets can be observed in Figure 3 (a). The first wave packet is caused by the S0 mode starting from the actuator. The second one represents the reflected S0 mode wave from the lower plate side, and the reflected SH0 mode generates the third packet. When the crack exists, the signal from the actuator will be reflected from the crack surface and diffracted at the crack tips. The reflected and diffracted waves affect the signals measured at P1. (a)

(b)

Figure 3. Displacements at point P1: (a) intact plate, (b) cracked plate It can be seen how the crack influences the displacement responses at P1 from Figure 3 (b). The wave packet W1 does not change because it comes from the actuator directly. The wave packet W2 is composed of four waves: the reflected S0 mode waves from both the crack surface and the plate side, and the diffracted S0 waves from the two crack tips. Thus, W2 is enhanced and its amplitude is higher than the signal from the intact plate. Because of symmetry, the two reflected S0 waves cannot cause displacement u2 at P1, so the packet W3 contains only the two diffracted S0 waves. For the same reason, although W4 is composed of the two reflected SH0 waves and two diffracted SH0, W5 only reflects the information of the diffracted SH0 mode. It is worth noting that there is a phase lag between W3 and W2. This is because more time are required for the diffracted S0 waves to arrive at P1 than the reflected waves. The dual actuators excite the incident S0 wave and then this wave reaches the left crack tip. After that, a diffracted S0 wave is generated and travels to P1. The total wave-propagation distance in this process is 226.47 mm, which is the same for the diffracted S0 wave from the right crack tip because of symmetry. According to Snell’s law, the total wave propagation distance for the incident S0 wave and the reflected S0 waves is 223.61 mm. It means that the incident and the diffracted S0 waves has a longer distance to travel, and therefore more time is used. Since the signal of W3 only contains the diffracted S0 waves, it can be used, combined with the signal of W1, to estimate the group speed of the S0 wave. The envelopes of W1 and W3 are obtained using the Hilbert transform. The time difference between the peaks of the two envelopes is 23.55 𝜇𝑠, and the corresponding difference of propagation distance is 126.47 mm. Thus, the estimated group speed of S0 mode is 5370.3 m/s. The analytical group speed from the plane stress theory is 5432 m/s. The relative error is only 1.1%, which results from the size of PZT actuators and numerical errors. This small relative error indicates that BEM results can provide satisfactory accuracy in the modelling of damage detection applications. 4. Conclusions In this paper, the damage detection process for an isotropic plate is simulated using the boundary element method. The boundary element formulation based on the plane stress theory is adopted to model in-plane waves within the low-frequency range. Crack problems are modelled using the dual boundary element method. The interaction of an incident S0 Lamb wave, excited by piezoelectric disks, with a throughthickness crack is captured with the proposed numerical method. BEM simulations have been shown accurate through comparing the estimated group speed of S0 mode from numerical results and the analytical solutions. Future work is planned to apply BEM, combined with damage detection algorithms, to the identification of crack locations and sizes.

60


Acknowledgments This research was supported by a grant provided by the China Scholarship Council (CSC). References [1] Sharif-Khodaei Z, Ghajari M, Aliabadi M. Determination of impact location on composite stiffened panels. Smart Materials and Structures. 2012;21:105026. [2] Katsikeros CE, Labeas G. Development and validation of a strain-based structural health monitoring system. Mechanical Systems and Signal Processing. 2009;23:372-83. [3] Zou F, Benedetti I, Aliabadi M. A boundary element model for structural health monitoring using piezoelectric transducers. Smart Materials and Structures. 2013;23:015022. [4] Zak A, Krawczuk M, Ostachowicz W. Propagation of in-plane waves in an isotropic panel with a crack. Finite Elements in Analysis and Design. 2006;42:929-41. [5] Fedelinski P, Aliabadi M, Rooke D. The Laplace transform DBEM for mixed-mode dynamic crack analysis. Computers & structures. 1996;59:1021-31. [6] Yang C, Ye L, Su Z, Bannister M. Some aspects of numerical simulation for Lamb wave propagation in composite laminates. Composite structures. 2006;75:267-75. [7] Banks HT, Smith RC, Wang Y. Smart material structures: modeling, estimation, and control: John Wiley & Son Ltd; 1996. [8] Lu Y, Ye L, Su Z, Yang C. Quantitative assessment of through-thickness crack size based on Lamb wave scattering in aluminium plates. NDT & e International. 2008;41:59-68.


61

Boundary element analysis of fractured CNT-polymer nanocomposites F. García-Sánchez1, E. García-Macías2a, L. Rodríguez-Tembleque2b, and A. Sáez2c 1

2

Escuela de Ingenierías Industriales, Universidad de Málaga, Doctor Ortiz Ramos s/n, 29071 Málaga, SPAIN, [email protected]

Escuela Técnica Superior de Ingeniería, Universidad de Sevilla, Camino de los descubrimientos s/n, 41092 Sevilla, SPAIN, [email protected], [email protected], [email protected]

Keywords: Cracks, stress intensity factors, carbon nanotubes, traction boundary integral equation. Abstract. In this work, we apply the dual BEM formulation to analyze the fracture behavior of Carbon NanoTube (CNT) Reinforced polymer Composites (CNTRC). The macroscopic elastic moduli of the CNTRC are computed by using the Eshelby-Mori-Tanaka approach. To this end, the CNT are modelled as equivalent fibers, in order to properly capture the scale difference between micro- and nano-scales. Parametric studies are then conducted to illustrate the effect on the fracture parameters (stress intensity factors and energy release rate) of different CNT volume fractions as well as CNT distributions in the polymer matrix. Namely, the cases of: (i) randomly distributed CNT; (ii) aligned CNT; and (iii) agglomerated CNT are studied. Introduction Carbon-based nanomaterials have garnered the attention of the scientific community in recent years, due to the possibilities they offer to develop novel multifunctional nanocomposites [1,2]. One example of carbon-based nanomaterials are the Carbon NanoTubes (CNTs), which inherently exhibit high mechanical strength and electrical conductivity. Because of the interesting properties of these carbon nanomaterials, new functionalities can be added to the reinforced material, including strength increase or strain sensing. For instance, the piezoresistivity of CNT-polymer composites is very promising because it opens an alternative to the conventional piezoelectric materials used for sensing in structural health monitoring. The study of fracture mechanics in this class of materials is crucial to predict their structural integrity or to avoid monitoring malfunction. In this work, the dual boundary element formulation developed by the authors [3] is applied to study the fracture behavior of CNT-polymer nanocomposites. The effective elastic properties of the composites are estimated by an Eshelby-Mori-Tanaka micromechanical constitutive model [4,5], which has been extended in order to account for agglomeration effects [6]. The CNT are modelled in terms of equivalent fibers [7], in order to capture the scale difference between micro- and nano-scales. Detailed parametric analyses are presented to illustrate the influence on the fracture parameters of micromechanical features of the CNTRC such as CNT content, orientation and dispersion. Modeling of carbon nanotube reinforced polymer composites (CNTRC) Consider a linear elastic and isotropic polymer matrix reinforced with dispersed long straight CNT. In this work, the macroscopic elastic moduli of the resulting CNTRC are computed using Mori-Tanaka’s (MT) mean-field homogenization approach [5]. Three different filler configurations are considered, namely: polymers doped with randomly oriented CNTs, uniaxially aligned CNTs and randomly dispersed CNTs with agglomeration effects. Composites reinforced with straight CNTs. The MT method extends the theory of Eshelby [4], limited to dilute dispersions of inclusions in an elastic, homogeneous and isotropic medium, to the case of multiple inhomogeneities. If we assume that all the fibers are equal and aligned, the effective stiffness of the two-phase composite can be expressed as [8]

𝐂 = (𝑉𝑚 𝐂m + 𝑉𝑟 𝐂r : 𝐀): (𝑉𝑚 𝐈 + 𝑉𝑟 𝐀)−1

(1)

with 𝐂m and 𝐂r being the stiffness tensor of the matrix and the equivalent fiber, respectively. 𝐈 stands for the fourth rank identity tensor, while 𝑉𝑚 denotes the volume fraction of the matrix and 𝑉𝑟 = 1 − 𝑉𝑚 is the corresponding CNT volume fraction. 𝐀 is the so-called dilute strain concentration tensor, given by


62

𝐀 = {𝐈 + 𝐒: (𝐂m )−1 : (𝐂r −𝐂m )}−𝟏

(2)

where 𝐒 is the well-known Eshelby's tensor [4]. The matrix is assumed to be elastic and isotropic, whilst each straight CNT is modeled as an equivalent long fiber with transversely isotropic elastic properties. In the case that all the fibers were uniaxially aligned, the resulting composite would also be transversely isotropic. In order to extend the model and account for any general fiber orientation distribution, the approach proposed by Schjødt-Thomsen and Pyrz [9] is considered herein. This approach is based in the direct integration of the stiffness tensor in Eq. (1) weighted by an Orientation Distribution Function (ODF), Λ(𝜃, 𝜙), with 𝜃 and 𝜙 being the Euler angles that define each CNT orientation, as sketched in Fig. 1(a), to yield 2𝜋

〈𝐂〉 =

𝜋/2

∫0 ∫0

2𝜋

𝐂(𝜃, 𝜙)Λ(𝜃, 𝜙)𝑠𝑖𝑛𝜃𝑑𝜙𝑑𝜃 𝜋/2

∫0 ∫0

(3)

Λ(𝜃, 𝜙)𝑠𝑖𝑛𝜃𝑑𝜙𝑑𝜃

Agglomeration of CNTs. Due to their low bending stiffness and high aspect ratio, CNTs tend to agglomerate in bundles inside the polymer matrix. Such agglomerations may substantially decrease the resulting mechanical properties of the CNTRC. To account for this phenomenon, the two-parameter agglomeration model proposed by Shi et al. [6] is implemented in this work. In this way, the regions with higher concentrations of CNTs are assumed to adopt spherical shapes and are considered as inclusions with different elastic properties from the surrounding material, as Fig. 1(b) illustrates. The total volume 𝑉𝑟 of CNTs is subsequently divided in two parts: 𝑉𝑟 = 𝑉𝑟𝑖𝑛𝑐𝑙 + 𝑉𝑟𝑚 , with 𝑉𝑟𝑖𝑛𝑐𝑙 being the volume of CNT dispersed in the concentrated regions and 𝑉𝑟𝑚 standing for the volume of CNTs dispersed in the rest of the matrix. The agglomeration of CNTs is described by introducing the following two parameters 𝜉=

𝑉 𝑖𝑛𝑐𝑙 𝑉𝑟𝑖𝑛𝑐𝑙 ; 𝜁= 𝑉 𝑉𝑟

(4)

where 𝑉 𝑖𝑛𝑐𝑙 and 𝑉 denote the volume of the spherical inclusions and the total volume of the composite, respectively. Note that when all the CNTs are uniformly dispersed in the surrounding matrix, 𝜁 = 0 , while a value of 𝜁 = 1 indicates that all the CNTs are located in the inclusions. In the case where all the CNTs are uniformly dispersed, 𝜉 = 𝜁. Bigger values of 𝜁, with 𝜁 > 𝜉, indicate a more heterogeneous spatial distribution of CNTs. On this basis, the previously presented micromechanics model can be readily extended to the case of nonuniform distributions by considering bundles as sphere inclusions and first estimate the effective elastic stiffness of the inclusions and the matrix to, subsequently, calculate the overall properties of the resulting CNTRC. CNT equivalent fiber. The above micromechanics equations fail to capture the scale difference between the nano-and micro- levels, as it does not take into account the van der Waals interactions between the CNTs and the surrounding polymer matrix. In order to overcome this difficulty, in this work we adopt the equivalent fiber approach developed by Shokrieh and Rafiee [7]. In this manner, the computed effective stiffness of the fiber equivalent to the CNT and its interphase (Fig. 1(c)) may be efficiently used in combination with the micromechanics equations to predict the actual CNTRC behavior.

(a)

(b)

(c)

Figure 1: (a) Euler angles defining straight CNT orientation; (b) Schematic representation of agglomeration; (c) definition of CNT equivalent fiber.


63

Dual BEM formulation The dual BEM formulation for fracture mechanics problems is based on both the displacement and traction boundary integral equations. For a solid Ω with boundary Γ containing a crack, in the absence of body forces, it may be written as ∗ ∗ 𝑐𝑖𝑗 𝑢𝑗 + ∫ 𝑝𝑖𝑗 𝑢𝑗 𝑑Γ = ∫ 𝑢𝑖𝑗 𝑝𝑗 𝑑Γ

(5)

∗ ∗ 𝑐𝑖𝑗 𝑝𝑗 + 𝑁𝑟 ∫ 𝑠𝑟𝑖𝑗 𝑢𝑗 𝑑Γ = 𝑁𝑟 ∫ 𝑑𝑟𝑖𝑗 𝑝𝑗 𝑑Γ

(6)

Γ

Γ

Γ

Γ

where 𝑢𝑗 and 𝑝𝑗 are the elastic displacements and tractions at the boundary; 𝑐𝑖𝑗 is the so-called free term that ∗ ∗ depends on the geometry of the boundary at the collocation point; 𝑝𝑖𝑗 and 𝑢𝑖𝑗 are the fundamental solution tractions and displacements, respectively; Nr stands for the outward unit normal at the collocation point; and ∗ ∗ the kernels 𝑠𝑟𝑖𝑗 and 𝑑𝑟𝑖𝑗 are obtained by differentiation of the fundamental solution as ∗ ∗ ∗ ∗ 𝑠𝑟𝑖𝑗 = 𝐶𝑟𝑖𝑚𝑛 𝑝𝑚𝑗,𝑛 ; 𝑑𝑟𝑖𝑗 = 𝐶𝑟𝑖𝑚𝑛 𝑢𝑚𝑗,𝑛

(7)

with 𝐶𝑟𝑖𝑚𝑛 being the elastic stiffness tensor of the CNTRC. In this work, we apply the formulation previously developed by García-Sánchez et al. [3] for cracked anisotropic materials, so that the interested reader may refer to [3] for details on the regularization procedure implemented to evaluate the singular and hypersingular integrals in Eqs. (5) and (6), the implemented fundamental solutions or the BE discretization. In particular, continuous quadratic elements are employed for the non-cracked boundary, whilst discontinuous quadratic elements with the two extreme collocation nodes shifted towards the element interior are used to mesh the cracks in order to ensure the continuity of the displacements and their derivatives. The asymptotic displacement behavior near the crack-tip is simulated by placing discontinuous quarter-point elements at the crack-tips (Fig. 2(a)). Mode-I and mode-II (KI, KII) stress intensity factors (SIF) are directly extrapolated from the computed displacements at the closest collocation node to the crack-tip (NC1) from 𝐊={

Δ𝑢𝑁𝐶1 𝐾𝐼𝐼 } = 2√2𝜋⁄𝐿 [𝑅𝑒(𝐁)]−1 { 1𝑁𝐶1 } 𝐾𝐼 Δ𝑢2

(8)

where 𝐁 is defined in [3] in terms of the elastic moduli of the CNTRC at the macro-level and Δ𝑢𝑖𝑁𝐶1 denote the crack opening displacements. Subsequently, the energy release rate GM may be computed from 1

𝐺𝑀 = 2 𝐊 𝑇 𝑅𝑒(𝐁)𝐊

(9)

Numerical results The fracture response of a rectangular plate with an inclined central crack, as illustrated in Fig. 2(b), is next analyzed for both uniformly distributed and agglomerated CNTs. The plate is loaded by a uniform traction at two opposite sides. The height to width ratio of the plate is ℎ/𝑤 = 2, with the crack length being 2𝑎 = 0.4𝑤. The BE discretization consists of six discontinuous quadratic elements on the crack line, with the two at the crack-tips being quarter-point, plus 24 continuous quadratic elements on the external boundary. The material consists of a polymer matrix with isotropic properties (𝐸𝑚 = 10 𝐺𝑃𝑎; 𝜈𝑚 = 0.3) reinforced with straight CNTs. The CNT properties are taken from reference [7], where the material properties of the CNT equivalent fiber defined to obtain the macro-elastic properties of the CNTRC (Fig. 1(c)) are obtained from an atomisticbased FEM analysis as 𝐶𝑁𝑇−𝑒𝑞

𝐸11

𝐶𝑁𝑇−𝑒𝑞



= 649.37 𝐺𝑃𝑎; 𝐸22 = 11.27 𝐺𝑃𝑎; 𝐺12 = 5.13 𝐺𝑃𝑎; 𝜈12 CNT tube thickness: 𝑡 𝑐 = 0.34 𝑛𝑚; CNT diameter: 𝐷 𝑐 = 2.034 𝑛𝑚

= 0.28

Effect of volume fraction and CNT orientation. The modes I and II SIF are analyzed for different filler contents, with CNT volume fractions ranging from 𝑉𝑟 = 1% to 𝑉𝑟 = 5%, as Fig. 3 depicts. This figure also illustrates the effect of the CNT orientation, defined by angle 𝜙 (Fig. 2(b)), for the case of aligned CNT


64

distributions. The values of the computed SIF are normalized with respect to those obtained for the CNTRC with randomly oriented CNTs and the same volume fraction, KIr and KIIr. Fig. 3 illustrates the importance of conveniently defining the orientation of the fillers to maximize the mechanical response of the CNTRC. Table 1 summarizes the elastic properties corresponding to both the cases of: (i) aligned CNTs (transversely isotropic CNTRC, with the local 𝑥1 -axis coinciding with the CNT-axis); and (ii) randomly oriented CNTs (isotropic CNTRC).

Figure 2: (a) Geometry of rectangular CNTRC plate with central slant crack and CNTs aligned along 𝐸1 direction. (b) Discontinuous quarter-point element with collocation nodes at 𝜂 = −0.75, 𝜂 = 0, 𝜂 = +0.75.

𝑽𝒓 (%) 𝑬𝟏𝟏 (𝑴𝑷𝒂) 0

10000,00

𝑬𝟐𝟐 (𝑴𝑷𝒂)

𝑮𝟏𝟐 (𝑴𝑷𝒂)

𝝂𝟏𝟐

10000,00

3846,15

0,3

𝑬𝟏𝟏 (𝑴𝑷𝒂)

aligned CNT

𝑬𝟐𝟐 (𝑴𝑷𝒂)

𝑮𝟏𝟐 (𝑴𝑷𝒂)

𝝂𝟏𝟐

randomly oriented CNT

1

23026,43

10550,32

3868,63

0,300

12211,19

12211,23

4731,54

0,290

2

36052,86

10723,28

3891,25

0,299

14416,65

14416,73

5617,05

0,283

4

62105,71

10871,26

3936,87

0,299

18818,07

18818,22

7388,45

0,273

5

75132,14

10913,71

3959,89

0,298

21016,07

21016,27

8274,34

0,270

Table 1. Elastic properties of the CNTRC: (i) aligned CNTs; (ii) randomly oriented CNTs. Effect of CNTs agglomeration. Finally, the effect of agglomeration of CNTs is analyzed. To this end, randomly oriented straight CNTs are considered and one of the agglomeration parameters is fixed, at 𝜉 = 0.2, so that the energy release rate GM is obtained for varying 𝜁, as Figure 4 illustrates for several CNT volume fractions. As in the previous case, the obtained energy release rate values are normalized with respect to those obtained for a uniform distribution (without agglomerations) of randomly oriented CNTs and the same volume fractions. Fig. 4 illustrates the detrimental effect that agglomeration implies on the fracture response of the CNTRC. Table 2 shows the elastic properties corresponding to several agglomeration parameters. As randomly oriented CNTs are assumed, the resulting CNTRC behavior is isotropic.


65

(a)

(b)

(c)

(d)

Figure 3. Modes I and II SIF for a central slant crack in a rectangular CNTRC plate with aligned CNTs: (a) 𝑉𝑟 = 1%; (b) 𝑉𝑟 = 2%; (c) 𝑉𝑟 = 4%; (d) 𝑉𝑟 = 5%.

Figure 4. Influence of CNTs agglomeration on energy release rate for a central slant crack in a rectangular CNTRC plate with randomly oriented CNTs: (a) 𝑉𝑟 = 1%; (b) 𝑉𝑟 = 2%; (c) 𝑉𝑟 = 4%; (d) 𝑉𝑟 = 5%.


66

Agglomeration parameters

𝝃 = 𝟎, 𝟐 ; 𝜻 = 𝟎, 𝟐

𝝃 = 𝟎, 𝟐 ; 𝜻 = 𝟎, 𝟓

𝝃 = 𝟎, 𝟐 ; 𝜻 = 𝟎, 𝟖

𝑽𝒓 (%)

𝑬(𝑴𝑷𝒂)

𝑮(𝑴𝑷𝒂)

𝝂

𝑬(𝑴𝑷𝒂)

𝑮(𝑴𝑷𝒂)

𝝂

𝑬(𝑴𝑷𝒂)

𝑮(𝑴𝑷𝒂)

𝝂

1

12211,18

4731,55

0,290

12107,38

4689,71

0,291

11820,11

4573,93

0,292

2

14416,64

5617,07

0,283

14080,08

5481,34

0,284

13193,08

5123,57

0,287

4

18818,04

7388,48

0,273

17841,13

6994,37

0,275

15408,4

6012,31

0,281

5

21016,04

8274,38

0,270

19673,44

7732,77

0,272

16389,81

6406,81

0,279

Table 2: Elastic properties of the CNTRC with randomly oriented CNTs and agglomeration effects. Conclusions A dual boundary element approach based on both the displacement and traction boundary integral equations has been implemented for the analysis of cracked CNT reinforced polymer composites (CNTRC). The mechanical properties of the CNTRC have been determined by applying the Eshelby-Mori-Tanaka homogenization approach, which has been further extended to account for agglomeration effects of the CNT spatial distribution inside the matrix. The hypersingular integrals appearing in the formulation are transformed into regular ones, which can be numerically evaluated, plus simple singular integrals with known analytical solution, following the authors’ previous formulation for anisotropic materials. Discontinuous quarter-point elements are used to capture the crack-tip behavior. Stress intensity factors and energy release rates are accurately computed from the nodal crack opening displacements at the crack tip element. The results show the key influence of the micromechanical features of the CNTRC (such as CNT content, orientation and dispersion) on the resulting fracture parameters. Acknowledgements This work was supported by the Ministerio de Economía y Competitividad of Spain and the European Regional Development Fund under projects DPI2014-53947-R and DPI2017-89162-R. The financial support is gratefully acknowledged. References [1] Baughman, R.H., Zakhidov, A.A., De Heer, W.A. Science, 297, 787-792 (2002). [2] Pandey, G., Thostenson, E.T. Polymer Reviews, 52, 355-416 (2012). [3] García-Sánchez, F., Sáez, A., Domínguez, J. Computers and Structures, 83, 804-820 (2005). [4] Eshelby, J.D. Proc. R. Soc. A, 241, 376-396 (1957). [5] Mori, T., Tanaka, K. Acta Metallurgica, 21, 571-574 (1973). [6] Shi, D.-L., Feng, X.-Q., Huang, Y.Y., Hwang, K.-C., Gao, H. Journal of Engineering Materials and Technology, Transactions of the ASME, 126, 250-257 (2004). [7] Shokrieh, M.M., Rafiee, R. Composite Structures, 92, 647-652 (2010). [8] Benveniste, Y. Mechanics of Materials, 6, 147-157 (1987). [9] Schjødt-Thomsen, J., Pyrz, R. Mechanics of Materials, 33, 531-544 (2001).


67

The boundary element method: a simple code for the arbitrarily high accurate analysis of 2D static and frequency-domain problems of general topology and shape Ney A. Dumont*1, Wellington T. de Carvalho2 and Carlos A. Aguilar1 1

Department of Civil and Environmental Engineering, Pontifical Catholic University of Rio de Janeiro, Brazil, [email protected] 2

Coordenação de Matemática, CEFET-RJ, Brazil, [email protected]

Keywords: Boundary elements, numerical integration, quasi-singularity, time-dependent problems.

Abstract. This paper proposes the application to two-dimensional time-dependent problems of a boundary element code that carries out all numerical integrations highly accurately and is combined with a generalized modal analysis framework that relies on the frequency power series expansion of the problem’s fundamental solution. The numerical integration developments apply to the evaluation of the matrices of both the conventional, collocation boundary element method and its variational counterpart, called the simplified hybrid boundary element method. The paper displays the key developments related to the precise numerical evaluation of the integrals obtained from the frequency expansion of the fundamental solution, as particularized to potential problems. A simple, although very elucidating, numerical example is displayed.

Introduction A revisit of the collocation boundary element method was proposed by Dumont [1] on the basis of two previous key developments [2 – 5], to include, as for elasticity problems: a) the conceptual reformulation of the ensuing single-layer potential matrix, with the correct approximation of boundary tractions (today’s concepts seem to have been unduly borrowed from the finite element literature), which should be required to be in balance independently from mesh discretization [2, 3]; b) a convergence theorem for general curved boundaries and high order elements, for 2D and 3D problems; c) a unified numerical evaluation of regular, improper, quasi-singular, singular and hypersingular integrals that relies exclusively on mathematical concepts and only uses Gauss-Legendre quadrature plus adequate corrections for general curved boundaries – thus leading to arbitrarily high accurate results independently from the singularity or quasi-singularity intensity. This integration procedure is extended to the accurate numerical evaluation of matrices related to the formulation of transient problems in the frequency domain, which includes expansions in frequency series [6, 7]. The later subject is also approached in the frame of the variational boundary element method. A numerical example illustrates the applicability of the proposed procedure to problems of complicated topology as well as with high stress gradients including the evaluation of results at internal points. For the sake of simplicity, the following developments are applied to potential problems, which present exactly the same mathematical issues as for elasticity but are simpler to manipulate. Readers are referred to [6, 7] for specific developments that deal with the time-dependent problem including evaluation of results at internal points in the frame of the proposed hybrid boundary element method (see also [8]). Details mainly concerning the proposed quadrature techniques are given by Carvalho [9]. The economical and accurate evaluation of the matrix coefficients G j and H j of a frequency power series expansion, as developed in a subsequent section, comprises the whole scope of the present paper. A thorough development is being prepared for publication.

Basic equations As proposed by Dumont, the Hellinger-Reissner potential has led to the hybrid boundary element method [10], a generalization of Pian's earlier achievements in the finite element method. The hybrid boundary


68

element method can be applied in a simplified (no longer fully variational, however still consistent) version [6, 7], H T p*

= p − p p , U * p*

= d − dp

 H T U * −1 ( d − d p ) = p − p p

(1)

where H mn ≡ H happens to be the same double layer potential matrix of the conventional, collocation ∗ ∗ boundary element method, U mn = U nm ≡ U* is the problem's underlying displacement fundamental solution (or Green's function, as to be used in the present context) obtained at a boundary node n in terms of force parameters pm∗ ≡ p* applied at a boundary node m (and referring to its adjacent segments) by means of which

* pm* + σ ijp . In the above equation, d m ≡ d are boundary an elastic body's stress field can be expressed: σ ijs = σ ijm

nodal displacements, pm ≡ p are equivalent boundary nodal forces and the quantities with the superscript ( ) p correspond to some given particular solution that eventually takes into account body actions. As indicated above, one may arrive at a stiffness-type matrix to solve either static or time-dependent problems [6, 7]. The proposed numerical integration scheme also applies to the conventional, collocation boundary element method in terms of the single layer potential matrix Gmℓ ≡ G and the vector of traction forces tℓ ≡ t , also considering a term t p fort the domain actions [1, 2]: G (t − t p ) = H (d − d p )

(2)

Frequency-dependent fundamental solution for a 2D potential problem The frequency-domain Helmholtz equation for a 2D potential problem, in terms of the wave number ω and solely dependent on the radial distance r to a singular point source, in polar coordinates, d 2u ∗ (r , ω ) 1 du ∗ (r , ω ) + + ω 2u ∗ (r ) = 0 dr 2 r dr has as real-variable, convenient fundamental solution [6, 7, 9] (with γ as the Euler's constant)

u∗ (r , ω ) =

γ + ln ω − ln 2 1 BesselJ (0, ω r ) − BesselY (0, ω r ) 2π k 4k

(3)

(4)

in such a way that, when expanded in a frequency power series (here truncated after the sixth term), 2 2 4 2 6 2 − ln r 2 r ( ln r − 2 ) 2 r ( ln r − 3) 4 r ( 3ln r − 11) 6 u (r , ω ) = + ω − ω + ω 4π k 24 π k 28 π k 21033 π k (5) r 8 ( 6ln r 2 − 25 ) 8 r10 ( 30ln r 2 − 137 ) 10 ω + ω + O (ω12 ) − 217 33 π k 2193353 π k the leading term turns out to be the fundamental solution for steady state problems, already normalized for a unit point source and using k as (say) the constant conductivity parameter of an isotropic, homogenous medium. At least for a bounded domain, this expression it easier to conceptually grasp and implement than when using the Hankel function that would come out after explicit imposition of the Sommerfeld radiation condition (as proposed above, the radiation condition at infinity is undefined). A simplified, compact (2 n ) expression of this equation is given in terms of polynomial terms pu,(2lnn ) and pu,reg as ∗

u∗ (r , ω ) =

− ln r 2 1 5 (2 j ) + )ω 2 j + O(ω12 )  ( ln r 2 pu,(2lnj ) + pu,reg 4π k π k j =1

(6)

The body’s boundary geometry is described in Cartesian coordinates ( x(ξ ), y (ξ ) ) in terms of a parametric variable ξ . The outward unit normal n has Cartesian projections ( y ′, − x′ ) J , where a prime denotes derivative with respect to ξ and J is the Jacobian of the coordinate transformation, in such a way that an infinitesimal boundary segment of ( x(ξ ), y (ξ ) ) becomes represented as d Γ = J d ξ . The normal flux q along the boundary is expressed from eq. (5) as


q∗ (r , ω ) =

+

69

ln r 2 − 1 2 r xy′ − yx′  1 − ω +  23 π J  2r 2  r 6 (12 ln r 2 − 47 ) 15 3

2 3

ω8 −

2

( 2ln r

2

27

r 8 ( 30ln r 2 − 131) 2183352

− 5)

ω4 −

r 4 ( 3ln r 2 − 10 ) 2932

ω6

 ω10  + O (ω12 ) 

(7)

(2 j ) or, more compactly, in terms of polynomial terms pq,(2lnj ) and pq,reg ,

q∗ (r , ω ) =

5 xy′ − yx′  1 2 (2 j − 2) (2 j − 2) 2j 12  2 +  ( ln r pq, ln + pq,reg ) ω  + O(ω ) π J  2r j =1 

(8)

We are presently interested in the development and evaluation of the single-layer and double-layer potential matrices of the collocation boundary element method, as truncated after O(ω10 ) :

G (ω ) ≡ Gsℓ (ω ) = J

 ∗ N ℓoe ( ) dΓ u ω s at ℓ  J Γ

5   − ln r 2 1 5 oe 2 (2 j ) (2 j ) 2j = J at ℓ   + ln r p + p ω N d ≡ + ω 2 jG j G ξ (  ℓ   u, ln u,reg ) 0 π π 4 k k j =1 j =1 s Γ 

(9)

H (ω ) ≡ H sf (ω ) =  qs∗ N of e dΓ Γ

5 5  xy′ − yx′  1 o 2 (2 j − 2) (2 j − 2) 2j 2j =  2 +  ( ln r pq, ln + pq,reg ) ω  N f e dξ ≡ H 0 +  ω H j j =1 j =1 Γ π  2r 

(10)

In these expressions, the boundary is discretized with ne elements of order oe (= 1, 2, 3 for linear, quadratic, cubic, etc), so that there are nn = ne oe nodal points distributed along the boundary, to which s = 1… nn source points for the fundamental solution of eqs. (5) and (7) are applied, and coinciding with f = 1⋯ nn field points for the potential interpolation functions N ofe (partition of unit), as denoted by the subscripts. Moreover, as for 2D problems, there are ℓ = 1… (oe + 1) ne (also field) reference points for the interpolation of the boundary normal flux in terms of the functions N ℓoe J

at ℓ

J (not partition of unit). The number of reference points ℓ

outnumbers by nn (for 2D problems) the number of nodal points because of the (in general) lack of uniqueness of the outward unit normal n on the left and right of a node belonging to two adjacent elements. (As remarked in references [1] and [2], the dichotomy into continuous and discontinuous elements is just one of some unfortunate misconceptions to be wiped out from the technical literature on boundary elements.) The accurate evaluation of G 0 in eq. (9) has already been adequately dealt with in References [1, 4, 5]: it involves an improper integral, when the points referring to s and ℓ geometrically coincide, or complex quasi-singularities, when the distance from s to ℓ is small as compared to an element’s length. The same references above also show how to adequately evaluate H 0 , both for the case of s and f coinciding, when finite part integrals and a local jump term must be obtained, and for different types of quasi-singularities. The expression of the remaining matrices, as developed for up to ω10 , is shown in the following. It must be pointed out that, in the consistent formulation of the collocation boundary element method, as proposed by Dumont [1, 2], only polynomial terms – whether or not multiplying ln r 2 – have to be dealt with in the evaluation of the matrices G (ω ) and H(ω ) even in the case of curved elements. However, although there are no singularities involved in the evaluations of the high order matrices shown above, the presence of the term ln r 2 makes the direct application of the Gauss-Legendre quadrature inaccurate, which requires the addition of a correction term, as proposed next.


70

Complex quasi-singularity issues Let p j ≡ p j (ξ ) be part of a generic integrand indicated in either eq. (9) or (10) that multiplies ln r 2 and is related to the multiplying term r 2 j , j = 0,1, 2… , for the case of a generic quasi-singularity at a point

ξ 0 = a ± bi, where i = −1 , that is for r (ξ 0 ) = 0 . Different cases of singularities or quasi-singularities are implied in this statement depending on the values of a and b . Writing r (ξ ) = w(ξ ) ρ (ξ ) , where ρ (ξ ) ≠ 0 in the vicinity of the integration interval (which may not be guaranteed if one is dealing with a very distorted boundary segment) and w(ξ0 ) = (ξ − a − bi )(ξ − a + bi ) = (ξ − a) 2 + b 2 . The evaluation of the double pole a ± bi is straightforward (to shown in the extended version of this paper). As indicated next, the explicit evaluation of the (so to say) regular part ρ (ξ ) of r (ξ ) is actually not required. One may write [1, 4] for the following generic integral: 1  1 r2 2 p ln r d ξ = p ln dξ +  p j ln w dξ    j 0 j 0 w  0  1

1

≈ GL  p j ln r dξ + 2

0

(11)

(  p ln w dξ − GL p ln w dξ ) 1

0

1

j

j

0

where the indicated Gauss-Legendre quadrature with ng points is accurately applicable to a polynomial of order 2ng − 1 that approximates p j ≡ p j (ξ ) . This may be eventually improved, as developed in the following, taking advantage of the fact that there is a term w j (ξ ), j = 0,1, 2… , implicit in 2j

j

p j (ξ ) , since

2 j −2

r (ξ ) = w (ξ ) ρ (ξ ) , as defined before. The proposition shown in eq. (11) is just a generalization of the procedure proposed for the evaluation of G 0 [1, 2, 4, 5]. However, while the analytical evaluation of the term 1

 ln(w) udξ 0

of G 0 is relatively simple,



1

0

p j ln( w)dξ requires some manipulation to be obtained. A

numerical quadrature using ln( w) as a weighting function is not feasible, as the evaluation of different abscissas related to different values of ξ 0 = a ± bi is prohibitively time-consuming. There are basically three possibilities for the accurate evaluation of

First possibility of evaluating



1

0



1

0

p j ln w dξ .

p j ln w dξ . The first possibility deals with the direct, analytical

evaluation of the integral. The following expression was proposed in Appendix 1 of Reference [4]: j  ln wξ dξ = ln w

ξ j +1 − A−1 j +1

+

2  c 2 A0 − aA−1 ξ −a j ξ k +1  arctan −  Ak  , b≠0 j +1 b b k + 1 k =0

(12)

The terms Ak , k = 0,… , j are the coefficients of the polynomial obtained as the quotient between ξ j +1 (ξ − a ) and

w:

Aj = 1,

Aj −1 = a,

Ak − 2 = 2aAk −1 − c 2 Ak , k = j , j − 1,…,1 .

This

recurrence

formula

is

of

straightforward implementation, although the development of the polynomial p j (ξ ) in canonical form is not a simple matter, as it may be of very high order, as shown in Table 1 and discussed latter on.

Second possibility of evaluating



1

0

p j ln w dξ . One may apply n successive integration by parts to arrive

at a sufficiently low term that multiplies ln w ,



1

0

1

 d k pj  d n pj n p j ln w dξ =  (−1)  w + (−1)  ln w dξ n  d ξ k  k =0 0 d ξ  0 n −1

1

k

as for G 0 [1]. However, this also presupposes an efficient manipulation of p j (ξ ) in its canonical form.

(13)




Third possibility of evaluating

1

0

71

p j ln w dξ . A further development of eq. (11) is proposed in connection

with a Gauss-Legendre quadrature with ng points,

1 pj  1 j p ln( r )d ξ ≈ GL p ln r d ξ + w ln w d ξ − GL p ln w d ξ ( )    j j j j 0 0 0  0 w   j ng ng  ng p j   hw  1 1 j ≈ GL  p j ln r 2 dξ +   j hiw −  ( p j ln w ) hi  = GL  p j ln r 2 dξ +  p j  i j − ln whi  0 0 w  ξì  i =1 w ξ  i =1 i =1   ξì ì   1

1

2

2

(14)

j

where the quadrature introduced for the term p j w j is carried out with weights hiw obtained in such a way that the generic integral



1

0

p

ng −1

(w

ln( w) ) dξ , where p

j

ng −1

≡p

ng −1

(ξ ) is an arbitrary polynomial of order

ng − 1, is accurately evaluated. Observe that the lower accuracy of order ng − 1 affects the term p j w j , which on the other hand can be approximated by a low-order polynomial. Table 1 shows the expressions of the polynomial degrees of the terms involved in the evaluation of G j and H j in eqs. (9) and (10) for an element of order oe that spans either curved or straight segments. The evaluation of such degree expressions for different element orders up to quartic elements and up to expansion terms G 5 and H 5 is also shown in Table 2. The term p j w j is still of very high degree, for some ultimate applications, although, as indicated, the problem becomes always easily tractable when a problem’s geometry is sufficiently accurate if approximated by straight segments: at most six abscissas for an accurate evaluation. Table 1. Degrees of the polynomials p j and p j w j for the evaluation of G j and H j in eqs. (9) and (10). For a curved segment Expression of p j for

For a straight segment

Degree of p j

Degree of p j w j

Degree of p j

Degree of p j w j

G j : r 2 j N oe

(2 j + 1)oe

(2 j + 1)oe − 2 j

2 j + oe

oe

H j : r 2 j − 2 ( xy ′ − yx′ ) N oe

(2 j + 1)oe − 1

(2 j + 1)oe − 2 j + 1

2 j + oe − 1

oe + 1

Table 2. Evaluations of Table 1 for p j and p j w j , the latter both for curved and straight segments. oe

G1

H1

G2

H2

G3

H3

G4

H4

G5

H5

1

3, 1, 1

2, 2, 2

5, 1, 1

4, 2, 2

7, 1, 1

6, 2, 2

9, 1, 1

8, 2, 2

11, 1, 1

10, 2, 2

2

6, 4, 2

5, 5, 3

10, 6, 2

9, 7, 3

14, 8, 2

13, 9, 3

18, 10, 2

17, 11, 3

22, 12, 2

21, 13, 3

3

9, 7, 3

8, 8, 4

15, 11, 3

14, 12, 4

21, 15, 3

20, 16, 4

27, 19, 3

26, 20, 4

33, 23, 3

32, 24, 4

4

12, 10, 4

11, 11, 5

20, 16, 4

19, 17, 5

28, 22, 4

27, 23, 5

36, 28, 4

35, 29, 5

44, 34, 4

43, 35, 5

In this proposition, one evaluates a term G j or H j in eqs. (9) and (10) by first carrying out the GaussLegendre quadrature of the indicated integrals, as if dealing with solely regular terms, according to eq. (14), j

and then adds a correction term. For this sake, the weights hiw must be first evaluated, in such a way that, if p

ng −1

≡p

ng −1

(ξ ) is an arbitrary polynomial of order ng − 1 and ξi are the abscissas of the Gauss-Legendre

quadrature in the interval [0, 1],



1

0

p

ng −1

(w

ng

j

ln( w) ) dξ =  p i =1

(

ng −1

)

ξì

hiw

j

(15)


72

n

The weights hiw are obtained directly by applying the above expression to the evaluation of the Lagrange polynomial of order ng − 1 , which can be accomplished provided that the analytical expression of the integral represented above is known in terms of the singularity point ξ 0 = a ± bi . This requires writing the Lagrange n −1

polynomial Li g in canonical form, according to an implemented algorithm to be given in the expanded version of the present paper. The particular case of a real quasi-singularity, for ξ 0 = a , is directly obtained from the above developments. This third, and apparently more feasible possibility of dealing with the present quasisingularity issues, is already implemented for a potential problem [9].

Computational implementations The displacement u ( x, t ) of a fixed-free truss element of length L , cross sectional area A , elasticity modulus E and mass density ρ is given by n

u ( x, t ) = − a  j =1

2( −1) j ( sin(bω1t ) − bω1 sin(ω j t ) ω j ) 2 j

2

2 1

(ω − b ω ) Aρ L

(2 j − 1)π  (2 j − 1)π x  sin   , with ω j = 2L 2L  

E

ρ

(16)

when submitted to a periodic load p (t ) = a sin(bω1t ) at its free end, where ω j are the natural vibration frequencies. For the particular case of Poisson’s ratio ν = 0 , one may simulate this phenomenon as a twodimensional potential problem. Moreover, the very simple case of eq. (16) may be used as the target solution of geometrically as well as topologically complicated cases [6, 7, 11]. In fact, as illustrated on the left of Figure 1, the fixed-free truss element may be modeled as a two-dimensional figure, in the case with length L = 3m and cross sectional area A = 3.5 × 1m2 , out of which an irregularly shaped figure is cut, as shown: a polygon with vertices (0,0) , (1,1) , (3,1.5) , (1.5,3.5) , (0, 2) . In the present case, the cut-out figure has its boundary discretized with 30 quadratic elements. Owing to the applied boundary condition u (0, t ) = 0 , as indicated, there are 47 nodes and degrees of freedom to be considered. One is also using E = 210GPa and

ρ = 7850 Kg m3 , and, for the applied load, a = −100MN and b = 1.35 . The problem is discretized with 30 quadratic elements. The code is an improvement of a Maple code developed by Chaves [7]. More details are given by Carvalho [9]. Since the segments are all straight, in the present illustration, the integrand related to p j w j , j = 1,2,3 , for evaluations according to the third possibility discussed in the previous Section, is actually simple to deal with. Examples with curved elements are being prepared for a more extensive publication. The graphics on the right of Figure 1 show the displacements evaluated along the segment BC on the left for different time instants, as the result of the problem formulated with 3 mass matrices. The graphics in Figure 2 show on the top and bottom displacement results at the depicted boundary point Q (2.2, 2.5) and internal point A (1.5,1.5) , for simulations run with one, two and three mass matrices [6, 7, 9]. As reported by Carvalho [9], the approximations introduced to solve the timedependent problem given by eq. (1) lead to round-off errors in the nonlinear eigenvalue solution [8], so that many key eigenpairs may get lost in the evaluation process. This is why the solution with three mass matrices is not overwhelmingly better than the one with two mass matrices. This actually deserves a long and detailed analysis for different configurations of loading, geometry and topology as well as machine precision. It is nevertheless always worth admiring the high stability of the employed generalized modal analysis [7-9].

Summary Owing to space restrictions, only a brief introduction of the proposed method could be dealt with in the present paper. The advantages of accurately evaluating the singular and quasi-singular integrals of both the


73

collocation and the hybrid boundary element method are extended to the more challenging case of the matrices comprised by a previously proposed generalized modal analysis of time-dependent problems. A full manuscript with more details – particularly concerning the depicted possibilities of evaluating the analytical integral in eq. (11) – and several illustrative examples including high order, curved elements is being prepared. p (t ) = a sin(bω1t )

Figure 1. Left: irregular cut-out of a fixed-free truss element submitted to a periodic load at the extremity, with displacements given by eq. (16); right: displacements measured along segment BC for different time instants.

References [1] N.A.Dumont Int. J. Comp. Meth. and Exp. Meas. 6(6), 965-875 (2018). [2] N.A.Dumont In C.A.Brebbia, ed., Boundary Elements and Other Mesh Reduction Methods XXXII, pp 227-238, WITPress, Southampton (2010). [3] N.A.Dumont Computational Mechanics 22(1), 32-41 (1998). [4] N.A.Dumont Engineering Analysis with Boundary Elements, 13, 155-168 (1994). [5] N.A.Dumont and M.Noronha Computational Mechanics 22(1), 42-49 (1998). [6] N.A.Dumont and R.A.P.Chaves Computer Assisted Mechanics and Engineering Sciences (CAMES) 10, 431-452 (2003). [7] R.A.P.Chaves The Simplified Hybrid Boundary Element Method Applied to Time-Dependent Problems, Ph.D. Thesis (in Portuguese), Pontifical Catholic University of Rio de Janeiro, Brazil (2003). [8] N.A.Dumont International Journal for Numerical Methods in Engineering 71, 1534-1568 (2007). [9] W.T.Carvalho, Assessment of some conceptual and numerical issues in the analysis of generalized nonlinear eigenvalue problems in the simplified hybrid boundary element method, Ph.D. Thesis (in Portuguese), Pontifical Catholic University of Rio de Janeiro, Brazil (2017). [10] N.A.Dumont Applied Mechanics Reviews, 42(11), part 2, S54-S63 (1989). [11] N.A.Dumont, R.A.P.Chaves and G.H.Paulino International Journal of Computational Engineering Science (IJCES) 5(4), 863-891 (2004).

74


Figure 2. Displacements at boundary point Q (top figure) and at the internal point A shown on the left of Figure 1 for a simulation with 30 quadratic elements using 1, 2 and 3 mass matrices.


75

An Efficient Fluid-Rigid Body Interaction Simulation of a 2 DOF Oscillating Wing Energy Harvester David Talarico1, Aaron Mazzeo2, and Mitsunori Denda3 1,2,3

Rutgers University Department of Mechanical and Aerospace Engineering Piscataway, NJ, United States

1

Email: [email protected] 2Email: [email protected] 3Email: [email protected]

Keywords: energy harvesting, computational fluid dynamics, CFD, parallel processing, renewable energy, unsteady fluid dynamics, fluid-rigid body interaction

Abstract. In recent years, oscillating wing energy harvesters have been increasingly investigated as a possible alternative to traditional wind and tidal turbines. Several numerical studies have demonstrated that relatively high efficiencies are achievable within a specific range of prescribed kinematic parameters. In practice, however, it has proven challenging to create a cost effective apparatus which can replicate these precise kinematics. Typically a bulky mechanism with a single degree of freedom is implemented in order to couple the relative phases of the wing pitch and heaving motions. In order to overcome these shortcomings, one must develop an adaptive, scalable means by which one may achieve the kinematics required for optimal fluid-structure coupling across a range of incident velocities. This paper assesses the dynamic response of a structurally simple, two degree of freedom flapping-wing power generation apparatus using a fast potential flow solver derived using a vortex method coupled to a multi-phase rigid body solver. The low computation time of the vortex flow solver is further reduced using a parallel implementation on an inexpensive graphics processing unit (GPU) which allows for the unsteady fluid solver to run on a consumer-level PC in a fraction of time required by traditional solvers with exceptional levels of accuracy across a wide range of input parameters. The results indicate a certain range of feasible operating conditions which can be adjusted using certain control parameters. A discussion of possible strategies for control and successful implementation is provided. Introduction The idea for flapping wing energy harvesting was first presented by McKinney and Delaurier in 1981 [1]. In their numerical study, they constrained an oscillating wing to a sinusoidal pitching and plunging motion. They kept the maximum pitch amplitude to within 15 degrees to prevent flow separation, and they only simulated laminar flow cases. Unfortunately, due to the lack of computational power at the time, these necessary constraints forced them to simulate very low efficiency cases of flapping wing energy harvesting. Nearly three decades later, with the help of an extraordinary decrease in the cost of computational power, Kinsey and Dumas released a landmark paper [2] which showed how the leading edge vortex of a pitching and heaving airfoil can enhance aerodynamic forces under certain optimal conditions. This enhancement occurred far beyond the stall angle, resulting in a highly separated flow the McKinney and Delaurier had never been able to study with their limited computational resources. Kinsey and Dumas further showed that the same wing may produce propulsion or harvest energy, depending on the maximum pitching angle ( ) and a dimensionless parameter called the reduced frequency ( ), defined by

,

(1)

where f is the frequency of oscillation of the airfoil, c is the chord length, and U is the oncoming flow velocity. Moreover, an observer can distinguish between a propulsive and an energy harvesting flapping wing simply by noting from which side of the wing the flow separates. If the flow separates on the side of the wing leading the heaving motion, the wing is energy harvesting. If the flow separates from the side of the wing that is trailing the heaving motion, the wing is propulsive. Using the commercial CFD software FLUENT, Kinsey and Dumas conducted a parametric study revealing promising efficiency levels of 34%. Subsequently, there have been several designs proposed to achieve the kinematics required for optimal flapping-wing power generation. Kinsey et al. [3] built and tested a single-degree of freedom dual four bar linkage mechanism which allowed them to prescribe the sinusoidal pitching and plunging motions used in their numerical studies. Surprisingly, this experiment achieved efficiencies as high as 40%, exceeding predictions made by their numerical study. Other theoretical kinematic

76


models [4,5] allowed for the pitching and plunging of the wing to occur independently, thus creating a two-degree of freedom system. Although numerical simulations of such systems assume that power can be extracted from both the pitching and plunging motions, there is little mention of how this may be achieved in practice. One practical implementation of a two-degree-of-freedom oscillating wing energy harvester from Talarico and Hynes [6] is shown in Fig. 1. This experimental apparatus consists of a flapping wing, a transmission system, a flywheel, and a generator. The driving wing is a NACA 0012 airfoil that is constrained to rotate about its half chord inside of a block which, in turn, is permitted to slide along a pair of rails. A timing belt is fixed at one point along its length to the block, and is routed around a timing pulley which is, in turn, coupled to a two-way-to-one-way rotation converter. The one way rotational output of the converter is coupled through the use of a one way locking bearing to the shaft that is, in turn, coupled to the flywheel and generator. The airfoil is constrained by a pair of identical adjustable stops to Figure 1 – Two DOF Apparatus from [6] rotate symmetrically about a zero degree angle of attack. An elastic tube (not shown) is fixed on one end to the frame upstream from the wing near the center of the rails and on its other end to the leading edge of the airfoil. At the end of each stroke the tube is pulled taught, causing the airfoil to rotate and the direction of lift to be reversed. An optional damper (not shown) may be included to slow the rotation of the wing after rotating past a zero degree angle of attack. Even with the built in adjustments, the apparatus in [6] was only able to achieve a limited range of reduced frequencies given the small force to mass ratio which resulted from the available parts and a limited wind tunnel test section size. A scaled up version operating outdoors in a wind or tidal application would not suffer these limitations. It is therefore the goal of the present study to create an efficient computational solver which can determine the power extraction efficiency and stability of the system in Fig. 1 across a broad range of parameters. Fluid-Rigid Body Interaction Methods For the fluid dynamics portion of the code, the apparatus in Fig. 1 is modelled as a 2D cross section of an infinite span flat plate wing undergoing simultaneous heaving and pitching motions. A flat plate model was chosen due to its computational simplicity, noting that in [2], it was shown that wing thickness has a negligible effect on power extraction efficiency. For the rigid body equations of motion the wing is modelled as a finite span plate with a finite mass and moment of inertia sliding on a frictionless track. A simplified transmission model is employed where the linear motion of the cart is numerically converted into one way rotation and a torque is only transmitted to the flywheel (and generator) using a timing pulley when the magnitude of the angular velocity of the pulley is equal to or greater than the magnitude of the angular velocity of the flywheel. This arrangement simulates a two way to one way rotation converter connected to a one way locking bearing. The radius of the timing pulley ( ) determines the mechanical advantage from the wing to the flywheel, and the flywheel is assumed to be rigidly coupled to the generator input shaft. Vortex Method. An in house vortex method code is used to calculate the forces and moments on the wing at each time step. Vortices are placed on the wing surface (a line in 2D) to model the effect of viscosity in the boundary layer as in thin airfoil theory. Collocation points are spaced at the midpoint of the chord segment connecting adjacent vortices. At each time step, vortices are shed from the leading and trailing edges and convected based on the local fluid velocity. In order to solve for the strength of the bound vortices at each time step, the nonpenetration condition and Kelvins theorem provide a set of linear equations:


77

∑ ∑

, ,

(2) (3)

where, and are the wing-normal component of the velocity at collocation point i induced by the bound vortex and the entire wake vortices , respectively. In addition, is the normal velocity component of the wing at collocation point i, and m is the total number of bound vorticies. For m bound vortecies with unknown circulations , there will be (m-1) collocation points and thus (m-1) equations from (2). Therefore equation (3) allows for a unique solution to be determined for the ’s by defining the equation. Impulse-momentum theory is then used to calculate forces and moments on the wing in terms of the time derivatives of the linear and angular impulses. When simulating multiples oscillation cycles, one may find that the amplitude of the moment on the wing tends to slowly increase over time because the vortices in the model exert increasing moments on the wing as they are convected downstream. This is due to the lack of viscosity in the model, which in nature would tend to cause the vortices to decay over time. In order to simulate vortex decay, one can simply remove the oldest vortices after some critical threshold time. This method is most suitable for vortex flows where the dominant form of decay is a result of Crow instability, as this type of instability produces rapid decay. This method of vortex removal also has the added benefit of reducing the number of calculations in the far field wake, where the additional computational effort would provide little value. Since the method of force calculation relies upon impulse momentum theory, the forces and moments are calculated as the time derivative of the momentum. Therefore, simply removing vortices will result in large spikes in the force and moment plots, as the removal process represents a step change in momentum. In order to overcome this issue, two “far field” vortices are created once the oldest vortex has exceeded the time threshold for removal. The leading edge and trailing edge far field vortex has the average circulation and position of the removed vortices which originated from the leading and trailing edges, respectively. In this way, the vortices can be removed without creating step changes in momentum.

Figure 2 - Wake vortex plot of a power extracting wing after several oscillation cycles


78

Fig. 2 shows an example of the wake developed by an oscilating flat plate wing in a power extraction regime after several cycles. The thin black line on the left is the wing (translating down), the fluid velocity is from left to right, and the black and red circles represent vortices shed from the leading and trailing edges of the wing, respectively. The starting vortex can be seen at the far right of the figure, and the von Kármán vortex street can be clearly identified. Denda et al. [7] demonstrate good agreement between this vortex method solver and the open source full Navier Stokes solver OpenFOAM over a wide range of kinematic parameters. Massively Parallel Implementation. Given that one of the goals of the present study is to create a highly efficient solver, the code was analyzed to determine whether or not it could benefit from the highly parallel processing enabled by the GPU. Some of the functions did not take more time to run as the simulation progressed (i.e. with more wake vortices). Others only increased approximately linearly with the number of wake vortices . However, one function in particular, the velocity calculation at each wake vortex point (for convection), increased with the square of the number of wake vortices , which greatly reduced the efficiency of the solver after more than three cycles. Since the calculation of each velocity contribution is independent of the others, this function is a prime candidate for parallel implementation on the GPU. First, a pre-processor creates three arrays; it creates one for the location of the contributing vortex, one for the circulation of the contributing vortex, and one for the vortex location at which the velocity is to be determined. Therefore, the array indices represent one of the permutations required to calculate the velocity contribution from every vortex at every vortex point. The results of the GPU implementation were very encouraging. For simulations over 3 oscillation cycles the same output from the GPU runs over 80 times faster than on the CPU. Fig. 3 shows the results of a simulation of only 1 cycle (55 times), because for larger numbers of cycles the time plot for the GPU is difficult to see. For example, the largest number of cycles tested on the GPU was 16.5 cycles, which took 1 hour and 15 minutes to run. By comparison, the same simulation would have taken just over 5 days on the CPU! Fig. 4 illustrates the relationship between the number of cycles simulated and the ratio of the CPU time to the GPU time. Based on a regression analysis of the run time data , the ratio seems to asymptotically approach .

120 CPU

100 Wall Clock Time (s)

GPU 80 60 40 20 0 0

50

100 Time Step Index

150

Figure 3 - Comparison of CPU vs. GPU Implementation

200


79

100 95

CPU/GPU Time Ratio

90 85 80 75 70

65 60 55 50 0

5

10 Number of Cycles

15

Figure 4- Ratio of CPU to GPU Run Time vs. Number of Cycles Simulated

Rigid Body Motion of the Wing System. The basic motion of the wing is defined in terms of rotation (“pitching”) about the half chord and translation (“heaving”) of the center of rotation in a direction substantially perpendicular to the flow velocity. A top-view diagram of the system is provided in Fig. 5. The origin of spacefixed Cartesian coordinate system defined by basis vectors ̂ ̂ is located at the midpoint of the track, with the ̂axis antiparallel to the wind velocity. For convenience, a second non-inertial wing-fixed coordinate system is defined by the basis vectors ̂ ̂ is placed at the center of rotation of the wing, with the ̂ directed along the chord of the wing. The angle between ̂ and ̂ is denoted as . Other important parameters in Fig. 5 include the free stream velocity , the stiffness of the elastic tube k, the heaving velocity v, the pitching velocity , the mass of the bearing block , the mass of the wing , and the mass moment of inertia of the wing about its center of mass . In order to simplify the equations of motions and reduce the number of input parameters, it is assumed that the center of mass coincides with the center of rotation.

Figure 5 - Diagram of the 2 DOF System (Top View)

Equations of Motion. The equations of motion represent an indefinitely repeated cycle of motion given that the input parameters of the simulation are chosen such that the wing does not get “stuck” at any point in the cycle. The system is said to be in “stable oscillation” if it is able to continuously oscillate when subjected to a given set of input parameters.


80

All of the forces and moments that affect the motion of the wing are shown in Fig. 6. is the force of the elastic tube, is the lift force, is the force of the timing belt, is the moment created by the fluid, is the moment provided by the rotational damper, and is the estimated generator torque at a given rpm. From this, one may find the equations of motion to be the following, ⃑ (⃑

̂

(4)

,

⃑ ) ̂

(5)

where is the moment of inertia of the flywheel/generator rotational assembly, is the angular acceleration of the generator’s rotor, is the radius of the pulley that drives the flywheel, ⃑ is the vector from the center of the wing to the tube attachment point on the wing, and is the linear acceleration of the pivot point of the wing. The following relations further define the force of the tube and the moment of the damper, , ⃑

|⃑ |

(6) ⃑ , |⃑ |

(7)

where is the damping coefficient of the damper, ⃑ is the vector from the tube attachment point on the wing to the tube attachment point on the frame, and is the unstretched length of the tube. During a given oscillation cycle the wing goes through several motion phases, each which requires different adjustments to the equations of motion. For example, in the phase depicted in Fig. 6, let us assume that the tube has been pulled taught. Therefore, all of the terms in equations (4) and (5) are required, unless , where is the angular velocity of the flywheel/generator. In this case the

and

terms would be neglected because

the one way locking bearing does not permit any force to be transferred to from the flywheel to the wing when . Therefore, the angular velocity of the flywheel during these times is determined independently from the motion of the wing. This is done by simply incrementing the angular velocity of the flywheel by

(8)

for each time step . Once the wing pitches to reverse the direction of lift, accelerates in the other direction, and “catches up” to the flywheel again, then the terms and are once again included in the equations of motion.

Figure 6 – Force/Moment Diagram


81

Simulation Procedure Overview. Due to the fact that the vortex method uses a time derivative to calculate its forces and moments, the two solvers interact iteratively instead of concurrently. Moreover, the central difference scheme produces much more accurate results than a backward difference scheme. The iterative solver proceeds as follows: 1. The oscillation frequency and heaving amplitude are estimated using a quasi-steady lift approximation. 2. A prescribed sinusoidal motion allows for the calculation of the initial force and moment arrays using the vortex method. 3. The solid body solver begins. 4. The forces and moments are interpolated to a given solid body time step depending on the current state of the wing. This effectively scales the time axis of the force and moment arrays such that the solid body solver and the vortex solver are always synchronized. 5. The solid body solver continues to calculate the motion for one full cycle after the frequency and heaving amplitude remain within a small tolerance from cycle to cycle. 6. The resulting motion is input into the vortex solver to obtain new force and moment vectors. 7. The I/O exchange between the vortex solver and the solid body solver iterates until the resulting power, frequency and heaving amplitude by the solid body solver remains within a certain tolerance from one iteration to the next. Results Based on a qualitative assessment of the preliminary results, it appears that one primary factor in determining whether or not a given set of input parameters will achieve stable oscillation is if there is enough kinetic energy in the wing/block translating mass by the time the tube pulls taught to overcome the minimum energy requirement to move from the stable pitched configuration to a point past a zero degree angle of attack. Once the wing has pitched a small amount past this point, the reversal in the direction of lift tends to carry the wing to the opposite fully pitched configuration. One factor in determining the efficiency of power extraction is the ratio of the inertia of the flywheel to the inertia of the translating mass. If this ratio is too low, the angular velocity of the generator rotor fluctuates a large amount. This causes poor efficiency due to the steep drop off in generator efficiency outside its rated speed range. Further details of the results will be discussed at the conference. References [1] M. William, and J. DeLaurier Wingmill: an oscillating-wing windmill, Journal of Energy, 5(2), 109-115 (1981). [2] K. Thomas, and G. Dumas Parametric study of an oscillating airfoil in a power-extraction regime, AIAA journal, 46(6), 1318-1330 (1981). [3] T. Kinsey et al Prototype testing of a hydrokinetic turbine based on oscillating hydrofoils, Renewable energy, 36(6), 1710-1718 (2011). [4] Z. Peng and Q. Zhu Energy harvesting through flow-induced oscillations of a foil, Physics of Fluids, 21(12), 123602 (2011). [5] Q. Zhu Energy harvesting by a purely passive flapping foil from shear flows, Journal of Fluids and Structures, 34, 157-169 (2012). [6] D. Talarico and K. Hynes Bio-Inspired Wind Energy Harvester, Proc. ASME. 44816, ASME 2012 6th International Conference on Energy Sustainability, Parts A and B. [7] M. Denda, P. K. Jujjavarapu and B. C. Jones A vortex approach for unsteady insect flight analysis in 2D, European Journal of Computational Mechanics, 25(1-2), 218-247 (2016).

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83

The Application of BEM to MHD flow and heat transfer in a rectangular duct with temperature dependent viscosity E. Ebren Kaya1 and M. Tezer-Sezgin2 1,2

Department of Mathematics, Middle East Technical University, 06800, Ankara, Turkey, e-mail: 1 [email protected], 2 [email protected]

Keywords: BEM, DRBEM, MHD duct flow, variable viscosity.

Abstract. The steady, laminar, fully developed MHD flow of an incompressible, electrically conducting fluid with temperature dependent viscosity is studied in a rectangular duct together with its heat transfer. Although the induced magnetic field is neglected due to the small Reynolds number, the Hall effect, viscous and Joule dissipations are taken into consideration. The momentum equation for the pipe-axis velocity and the energy equation are solved iteratively. Firstly, the momentum equation is solved by using the boundary element method (BEM) with a parametrix (Levi function) since the diffusion term contains variable viscosity parameter depending on the temperature exponentially. Then, the energy equation is solved by using the dual reciprocity boundary element method (DRBEM). The temperature and the velocity behaviours are examined for several values of Hartmann number 0 ≤ Ha ≤ 10, dimensionless viscosity parameter B = 0, 1, 2, Brinkmann number Br = 0, 1, 2 and the Hall parameter m = 0, 3, 8. As Ha is increasing, the velocity magnitude drops which is a well known property of the MHD duct flow. Increasing B reduces both the flow and the temperature magnitudes whereas the increase in the Hall parameter accelerates the flow and increases the fluid temperature.

Introduction For many areas such as designing heat transfer, cooling of electronic systems, chemical reactors, nuclear reactor, combustion systems, the problems of MHD flow and heat transfer via rectangular ducts have very important position. Therefore, there have been many theoretical and experimental studies for these problems. Some researchers and their methods are the followings. Türk and Tezer-Sezgin [1] have studied natural convection flow in square enclosures under magnetic field using the finite element method (FEM). In their study, the momentum equations include the magnetic effect and the induced magnetic field due to the small magnetic Reynolds number is neglected. They showed that, the FEM with quadratic elements enables one to solve MHD natural convection flow under the effect of a magnetic field for large values of Rayleigh and Hartmann numbers. Alsoy-Akgün and Tezer-Sezgin [2] have studied natural convection MHD flow equations in cavities by using the dual reciprocity boundary element (DRBEM) and the differential quadrature method (DQM). They have compared these two methods in terms of the flow and temperature behaviors concluding that the DRBEM and the DQM give almost the same accuracy. Although these studies differ from each other in their numerical implementations, they have a common feature which is the constant viscosity in the momentum equations considered in all of them. However, the momentum equations can contain a variable viscosity depending on space and time variables and even on the temperature. Sayed-Ahmed and Attia [3] have solved MHD flow and heat transfer with variable viscosity for Newtonian fluids in a rectangular duct with the Hall effect by using the finite difference method (FDM). Their results are confirmed by an experimental study of laminar flow and heat transfer in 2:1 rectangular duct filled with mineral oil, conducted by Xie and Hartnett [4] since the viscosity of the mineral oil changes dramatically. They obtained good agreement between the experimental data and the analytical solution. Shin, et al. [5] also investigated the influence of variable viscosity of temperature-dependent fluids on the laminar heat transfer and friction factor in a 2:1 rectangular duct. The governing mass, momentum and energy equations were solved using a finite volume method (FVM). They get the excellent agreement with the experimental results conducted by Xie and Hartnett [4]. There are quite a number of fluid dynamics problems in which the fluid viscosity is varying with space, time and even with temperature. In this case, the diffusion operator contains the viscosity function in its outer derivative terms. Thus, if the weighted residual


84

statement is required for an equation containing a diffusion operator with a variable function, the corresponding fundamental solution or a weight function is required. For this purpose, AL-Jawary and Wrobel [6] derived such a weight function for variable coefficient diffusion equation. They have also considered the general partial differential equation with variable coefficients and they have used a parametrix (Levi function) which is usually available to solve the problem containing variable coefficients directly and accurately. In their study, with the help of the parametrix, the differential equation is reduced to a boundary-domain integral or integro-differential equation (BDIE or BDIDE). In this paper, the steady, laminar, fully developed MHD flow of an incompressible, electrically conducting fluid with temperature dependent viscosity is studied in a rectangular duct together with its heat transfer as in the reference study [3]. Although the induced magnetic field is neglected by taking small magnetic Reynolds number, the Hall effect, viscous and Joule dissipations are taken into consideration. First of all, the momentum equation containing variable viscosity parameter depending on temperature is solved by using the boundary element method (BEM). The BEM procedure given in this paper uses the fundamental solution (parametrix) which treats the variable viscosity in the diffusion term directly, [6]. That is, it is capable of addressing to the diffusion operator of the equations in its original form. Then, the energy equation is solved by using the dual reciprocity boundary element method (DRBEM) leaving the Laplacian term alone and taking the rest of the terms as inhomogeneity. The velocity and the temperature profiles are obtained for several values of Hartmann number (Ha), Brinkmann number (Br), viscosity parameter (B) and the Hall parameter (m). The BEM implementation discretizing only the boundary with constant elements captures the well known behavior of the MHD flow and the temperature of the fluid. Therefore, its computational cost is very small comparing with the other numerical methods.

Mathematical formulation The laminar, steady flow of a viscous, incompressible, electrically conducting fluid is considered in a long pipe with the pipe-axis velocity w = w(x, y) and the temperature T = T (x, y) varying only in the cross-section (duct) Ω = [0, a] × [0, b] of the pipe. The governing non-dimensional momentum and energy equations with viscosity coefficient µ = e−BT are ∂w ∂w ∂ ∂ Ha2 w (µ ) + (µ ) = −1 + ∂x ∂x ∂y ∂y 1 + m2

(1) in Ω

∇2 T + Brµ

∂w ∂x

2

+

∂w ∂y

2 +

Ha2 Br 2 w w = 2 1+m wm

(2)

where 2

B = (blρc p wm a (dTm /dz))/k,

Z bZ a

m = σ β B0 , wm =

wdxdy, 0

0

√ aB0 σ , Ha = √ µ0

Br =

kµ02 (−d p/dz) . ρc p wm (dTm /dz)

Here, B0 , σ and β are the applied magnetic field intensity, electric conductivity of the fluid and the Hall factor, respectively. ρ, c p , k, µ0 and l, a, b are the density, the specific heat capacity, the thermal conductivity, the dp viscosity at T = Tm of the fluid and the characteristic lengths in the z- and x-, y-directions, respectively. and dz dTm represent the pressure and temperature gradients, respectively. The boundary conditions w = 0 and T = 0 dz are imposed on the walls Γ of the duct indicating no-slip velocity and the cold wall conditions.

The BEM and DRBEM formulation The velocity equation (1) is solved by using the BEM with the fundamental solution of diffusion equation ln|r − ri | containing a variable coefficient which is called as a parametrix P(x, xi ; y, yi ) = derived in [6], and 2π µ(xi , yi )


85

the energy equation (2) is solved by using the DRBEM with the fundamental solution of Laplace’s equation 1 u∗ = ln|r − ri | given in [7], where r = (x, y) and ri = (xi , yi ) are the variable and fixed points in Γ ∪ Ω, 2π respectively. These two equations are solved iteratively. The boundary of the duct is discretized by N number of constant elements. Multiplying the velocity equation (1) with the parametrix P(x, xi ; y, yi ) and applying Green’s second identity two times, the following discretized boundary-domain integral equation is obtained.

ci wi −

N

1 2π

∑

Z

w

j=1 Γ j

2

(N/4) 1 + ∑ 2π µ(xi , yi ) j=1

1 (r − ri )~n µ(x, y) dΓ j + 2 |r − ri | µ(xi , yi ) 2π

N

Z

∑

µ(x, y)

j=1 Γ j

∂w 1 ln|r − ri | dΓ j ∂n µ(xi , yi )

(3)

(N/4)2 Z − 1 (r − ri )→ n ∂ µ(x, y) Ha2 w)dΩ j dΩ j = w ln|r − ri |(−1 + ∑ 2 |r − ri | ∂n 2π µ(xi , yi ) j=1 Ω j 1 + m2 Ωj

Z

where 0 i0 denotes the boundary point, Γj and Ω j are constant boundary element of length l and domain cell, respectively. The constant ci is 1/2 and 1 when the source(fixed) point is on the boundary and in the interior of the domain, respectively. The vector ~n is the outward normal vector to Γ. The equation (3) can be written in the matrix-vector form as Hw − Gq = I 1 (w) − I 2 (w)

(4)

where the matrices H and G are given as 1 2π

Hi j = ci δi j −

1 Gi j = 2π

Z Γj

Z

(r − ri )~n µ(x, y)dΓ j , |r − ri |2

ln|r − ri |dΓ j ,

Gii =

Γj

N

Hii = −

∑

Hi j

j=1, j6=i

l l (ln( ) − 1). 2π 2

The vectors I 1 (w) and I 2 (w) have the entries as Ii1 (w)

− (r − ri )→ n ∂ µ(x, y) = dΩ j wi 2 |r − r | ∂n Ωj i Z

Ii2 (w) =

Z

ln|r − ri |(−1 +

Ωj

i, j=1,...,N

Ha2 wi )dΩ j 1 + m2

with r = (x, y) a variable point. As it is mentioned, the energy equation (2) is solved by using the DRBEM with the fundamental solution 1 of Laplace’s equation u∗ = ln|r − ri |. Multiplying with u∗ both sides of the equation and applying Green’s 2π second identity two times on the left hand side of the equation, the following boundary-domain integral equation is obtained Z Z Z ∂w 2 ∂w 2 Ha2 Br 2 w ∗ ∗∂T −ci Ti + q T dΓ − u dΓ = − Brµ + − w + u∗ dΩ (5) ∂n ∂x ∂y 1 + m2 wm Γ Γ Ω ∂ u∗ , Γ is the boundary and Ω is the domain. ∂n The integrand on the right hand side of the equation (5) is considered as inhomogeneity b(x, y). This inhomogeneity term can be expanded by a series of radial basis functions as

where q∗ =

b(x, y) = −Brµ

∂w ∂x

2

∂w + ∂y

2 −

N+L Ha2 Br 2 w w + = ∑ α j f j (x, y) 1 + m2 wm j=1

(6)


86

where f j (x, y) are the radial basis functions which are connected with the particular solutions uˆ j of the eqution ∇2 uˆ j = f j , the unknown coefficients α j ’s are undetermined contants and N and L denote the number of boundary and interior points, respectively. Substituting f j = ∇2 u j in (6) and therefore in (5), and applying Green’s second identity two times again for the right hand side of the equation (5), the following boundary integral equation is obtained

−ci Ti +

Z

∗

q T dΓ −

∗∂T

Z

u

Γ

∂n

Γ

N+L

dΓ =

∑

Z Z ∗ ∗ α j ci uî j − q uˆ j dΓ + u qˆ j dΓ Γ

j=1

(7)

Γ

∂ uˆ j ∂ x ∂ uˆ j ∂ y ∂ uˆ j , and it can be written as qˆ j = + . ∂n ∂x ∂n ∂y ∂n The discretization of the boundary Γ with N constant elements and taking L interior points, we obtain the matrix-vector equations (since Fα = b and b = F −1 b) where the term qˆ j is defined as qˆ j =

∂T ˆ −1 b = (HUˆ − GQ)F ∂n

HT − G

(8)

where the vector b has entries

bi =

∂w ∂x

− Brµ

2 i

∂w + ∂y

2 i

Ha2 Br 2 wi w + − . 1 + m2 i wm

ˆ Qˆ and F are constructed by taking each of the vectors uˆ j , qˆ j and fi j = 1 + ri j , ri j being the The matrices U, distance from the point i to the point j, as columns, respectively. The components of the H and G matrices are given for constant elements as H i j = ci δi j −

Gi j =

1 2π

1 2π

Z

Z

Γj

∂ (ln|r − ri |)dΓ j , ∂n

ln|r − ri |dΓ j ,

Γj

Gii =

N

H ii = −

∑

Hi j

j=1, j6=i

l l (ln( ) − 1). 2π 2

The space derivatives of the velocity w in the vector c are computed by using the coordinate matrix F as ∂ w ∂ F −1 = F w and ∂x ∂x

∂ w ∂ F −1 = F w. ∂y ∂y

Numerical results and discussions The discretized matrix-vector equations for the flow and the energy equations (4) and (8), respectively are solved iteratively by taking w = 0 and T = 0 initially. The domain of the problem, the cross-section of the pipe is taken as a square [0, 1] × [0, 1] and discretized by using N = 100 constant boundary elements and L = 625 interior nodes. The equavelocity lines and isolines of the problem are simulated for several values of the Brinkman number as Br = 0, 1, 2, the Hartmann number range 0 ≤ Ha ≤ 10, the viscosity parameter B = 0, 1, 2 and the Hall parameter m = 3, 5, 8. For some values of the problem parameters, the velocity is relaxed wn+1 = αwn+1 + (1 − α)wn with a relaxation parameter 0 < α < 1 to slow down the decrease of velocity magnitude. Fig. 1 shows the velocity and temperature behaviors as Ha is increasing. It is observed that as Ha increases the velocity magnitude drops due to the damping effect of the magnetic field, this is as expected property for MHD duct flow. It is observed that as Ha increases the temperature magnitude also drops. From Fig. 2, it is also observed that, as the viscosity parameter B is increasing the magnitudes of the velocity and the temperature drop due to the increment in the viscosity. That is, as B is increasing the viscosity term µ is exponentially approaching to zero.


Ha = 5

Ha = 10

87

Temperature

Velocity

3 Advances in Bundary Element Ha and = Meshless Techniques XIX

87

max w = 0.03364 min T = −0.10979

max w = 0.02136 min T = −0.10628

max w = 0.010511 min T = −0.09539

Temperature

Velocity

Figure 1: Equavelocity lines and isolines for m = 0, Br = 0, B = 1 and α = 0, 04. B=0 B=1 B=2

max w = 0.06407 min T = −0.11326

max w = 0.05815 min T = −0.11282

max w = 0.05266 min T = −0.11211

Figure 2: Equavelocity lines and isolines for Ha = 3, m = 3, Br = 1. From Fig. 3, it is seen that increasing the Hall parameter m increases the magnitude of the velocity since it reduces the effective conductivity σ /(1 + m2 ) which decreases the damping magnetic force. It is also observed that as the Hall parameter m increases the magnitude of the temperature increases. The effects of the Hall parameter and Hartmann number on the Nusselt number Nu are given on Table 1 Z Z 1 1 1 1 where Nu = − and θm = wθ dxdy. It is observed that as Ha increases, the values of Nusselt 4θm wm 0 0 number are increasing. However, as the Hall parameter m is increasing, the values of Nusselt number are decreasing. But, it can be seen that, the effect of m on Nu may be neglected for small Hartmann number. In this table, the results for m = 0 and Ha = 3, 4, 5 are obtained by using a relaxation parameter taken as α = 0.04.


88

m=3

m=8

88

Temperature

Velocity

m =Meshless 0 Advances in Bundary Element and Techniques XIX

max w = 0.03364 min T = −0.10900

max w = 0.05815 min T = −0.11282

max w = 0.06407 min T = −0.11365

Figure 3: Equavelocity lines and isolines for Ha = 3, Br = 1, B = 1 (α = 0.04 for m = 0 case). Table 1: The effect of the Hall parameter m and Ha on Nu (Br=0, B=1). m Ha 0.0 1.0 2.0 3.0 4.0 5.0 0.0 3.0 5.0 8.0

3.6418 3.6418 3.6418 3.6418

3.6904 3.6471 3.6438 3.6426

3.8456 3.6668 3.6500 3.6450

3.8719 3.6870 3.6644 3.6492

3.9616 3.7049 3.6761 3.6597

4.0449 3.7628 3.6891 3.6661

Conclusion The steady, laminar, fully developed MHD flow of an incompressible, electrically conducting fluid with temperature dependent viscosity is studied in a rectangular duct together with its heat transfer. The momentum equation for the pipe-axis velocity and the energy equation are solved iteratively. Since the diffusion term contains variable viscosity parameter depending on the temperature exponentially, the momentum equation is solved by using the boundary element method (BEM) with a fundamental solution (parametrix) [6] which treats the variable viscosity in the diffusion term directly. Thus, the use of this fundamental solution makes it possible to discretize the diffusion operator of the equation in its original form. Then, the energy equation is solved by using the dual reciprocity boundary element method (DRBEM). The obtained results are in accordance with the well known property of the MHD duct flow. As the value of Hartmann number is increasing, the velocity magnitude drops, however increasing the Hall parameter m increases the velocity and temperature magnitude. Also, increasing the viscosity parameter B reduces the magnitudes of the velocity and the temperature due to the exponentially decaying viscosity term inside the diffusion term.

References [1] Ö. T ÜRK , M. T EZER -S EZGIN, FEM solution of natural convection flow in square enclosures under magnetic field, International Journal of Numerical Methods for Heat & Fluid Flow, 23, 844-866 (2013). [2] N. A LSOY-A KGÜN , M. T EZER -S EZGIN, DRBEM and DQM solutions of natural convection flow in a cavity under a magnetic field, Progress in Computational Fluid Dynamics, 13, 270-284 (2013).


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[3] M. E. S AYED -A HMED , H. A. ATTIA, MHD Flow and Heat Transfer in a Rectangular Duct with Temperature Dependent Viscosity and Hall Effect, International Communication of Heat Transfer, 27, 1177-1187 (2000). [4] C. X IE , J. P. H ARTNETT, Influence of variable viscosity of mineral oil on laminar heat transfer in a 2:1 rectangular duct, International Journal of Heat and Mass Transfer, 35, 641-648 (1992). [5] S. S HIN , Y. I. C HO , W. K. G INGRICH , W. S HYY, Numerical study of laminar heat transfer with temperature dependent fluid viscosity in a 2:1 rectangular duct, International Journal of Heat and Mass Transfer, 36, 4365-4373 (1993). [6] M. A. A L -JAWARY, L. C. W ROBEL, Numerical solution of two-dimensional mixed problems with variable coefficients by the boundary-domain integral and integro-differential equation methods, Engineering Analysis with Boundary Elements, 35, 1279-1287 (2011) . [7] P. W. PARTRIDGE , C. A. B REBBIA , L. C. W ROBEL, The Dual Reciprocity Boundary Element Method, Computational Mechanics Publications, Southampton Boston (1992).

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91

High-cycle fatigue in polycrystalline materials by boundary elements Ivano Benedetti1,a, Vincenzo Gulizzi2,b 1

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

[email protected], b [email protected]

Keywords: Boundary Element Method, Micromechanics, High-cycle fatigue, Cohesive Zone Modeling Polycrystalline materials

Abstract. In this work, we present a boundary element formulation for grain-scale modelling of polycrystalline materials subjected to high-cycle fatigue loads. Polycrystalline aggregates are artificially generated by means of Laguerre-Voronoi tessellations, suitably employed to generate either convex or nonconvex morphologies. The formulation is based on a multi-region boundary element approach and each grain is modelled as a generally anisotropic domain randomly oriented in the three-dimensional space. The fundamental solutions for anisotropic elasticity and their derivatives are obtained in terms of spherical harmonics expansions. To model the deterioration of the grain boundaries under the action of cyclic loads, the interface equations implement specific cohesive laws, which simultaneously account for the static and the cyclic contributions to the total damage, whose consistency is assessed at each fatigue cycle jump. Few pseudo and fully three-dimensional polycrystalline micro-morphologies are tested using the presented formulation, which can be used in the multiscale modelling of polycrystalline materials as well as for the prediction of the fatigue life of micro devices subject to high-cycle loads.

Introduction Fatigue represents one of the most common failure mechanisms in engineering components subjected to cycling loads and it characterized by levels of applied failure stress far below the strength of the employed material [1]. Fatigue failure mechanisms are articulated in different stages: damage initiates and develops at a microscopic scale and subsequently emerges at the macroscopic scale leading the component rupture. Understanding and predicting damage initiation and evolution at the microscopic scale represents an extremely complex task, due to the different mechanisms simultaneously involved and to the wide heterogeneity of materials microstructures. For such reasons, numerical tools to model the micromechanics of high-cycle fatigue are of high engineer interest. In this work, a model for the analysis of degradation and failure in polycrystalline materials subjected to cyclic loads is described. Polycrystalline materials, either metals or ceramics, are extensively used in structural applications. Polycrystalline silicon (polysilicon) is used in the manufacturing of Micro-ElectroMechanical Systems (MEMS), often subjected to cyclic loads [2]. At a microscopic level, polycrystalline materials are characterized by the morphology and crystallography of the grains and by the complex physical and chemical properties of the inter-granular interfaces, which play an important role in their micro-mechanics. The polycrystalline micromechanics, including damage and failure, has been investigated employing several different experimental and computational techniques. Grain-scale models have been developed for material homogenization and for the analysis of quasi-static failure and stress corrosion cracking [3,4]. Crystal plasticity finite elements with fatigue crack initiation criteria based on accumulation of slip have been used to assess the fatigue degradation of polycrystalline aggregates [5]: such studies often consider microstructures in which the initiation and evolution of fatigue damage is driven by the cyclic accumulation of plastic. On the other hand, in some polycrystalline systems plastic slip may be highly confined and some models, not based on slip accumulation criteria, have been proposed to study their fatigue degradation, taking into account the effects of microstructure randomness on the fatigue of MEMS [6]. In this work, we describe a 3D grain-scale boundary integral formulation for the analysis of polycrystalline aggregates under cyclic loading. Only inter-granular degradation is considered: the method is then suitable when plastic slip is absent or highly confined, e.g. in fine-grain MEMS [6]. The formulation represents a development of a previous grain-boundary framework successfully used for computational


92

homogenization [7,8], for the analysis of quasi-static inter/trans-granular degradation [9-13], for hydrogen embrittlement [14] and multiscale analysis of polycrystalline components [15,16].

Grain-boundary integral formulation The high-cycle fatigue formulation for polycrystalline aggregates is based on the following items: a) Representation of the polycrystalline morphology through Voronoi-Laguerre tessellations; b) Boundary integral representation of the bulk grains mechanics; c) Decomposition of inter-granular damage into cyclic and quasi-static contributions; d) Cohesive zone modelling of the inter-granular interfaces e) Polycrystalline system assembly; f) High-cycle fatigue algorithm: Envelope load representation and cycle jump solution strategy. Such items are described in the following paragraphs, before the section reporting some numerical results obtained by applying the proposed method. Artificial polycrystalline microstructures. The first step in the construction of the proposed computational model is a suitable representation of the material microstructure. Three-dimensional (3D) Voronoi-Laguerre tessellations are employed here: it has been recently demonstrated that such tessellations offer a flexible tool for the generation of satisfying first order morphological approximations of real microstructures [17]. Each tessellation's cell represents an individual crystal with random crystallographic orientation in space. It is worth mentioning that a special algorithm for the generation of tessellations of non-convex domains has been specifically implemented: this constitutes a useful tool for the generation of components of MEMS. Boundary integral representation of the bulk crystals micromechanics. Each grain is represented as a linear elastic anisotropic domain and the Boundary Element Method (BEM) for 3D anisotropic elasticity is used as numerical formulation [18]. The boundary integral equations employed for a generic grain k are k %k c% ij  x  u j  x  



BI  BE

k k %k T% ij  x, y  u j  y  dB  y  



k k U%ijk  x, y  t% j  y  dB  y 

(1)

BI  BE

k where u%kj and t% j are components of displacements and tractions of points belonging to the surface of the grain k, the tilde refers to quantities expressed in local reference systems set on the grain faces, U%k and T%k ij

ij

k ij

are the 3D displacement and traction anisotropic fundamental solutions [19] and c% are coefficient stemming from the limiting boundary collocation procedure. Eq.1 is used for each grain, by direct collocation at the nodes of a suitably generated grain-boundary mesh, which takes into account the specific features of the polycrystalline morphology [7,11]; the integrals are evaluated over all the grain faces, which include inter-granular interfaces BI and external non-contact faces BE , over which the external boundary conditions are assigned. Inter-granular damage decomposition. In the proposed framework, a Continuum Damage Mechanics (CDM) based cohesive law for high-cycle inter-granular fatigue degradation is adopted. The total damage d * is decomposed into damage induced by quasi-static overloads and damage induced by cyclic loads [20], so that the damage rate can be expressed as d * d s* d c*   N N N

(2)

where N represents the number of load cycles, considered as a continuous and differentiable time-like variable. The above expression can also be expressed in discrete terms, considering the accumulation of damage over a finite number of cycles N. Interface quasi-static degradation: cohesive zone modelling. The inter-granular interfaces are represented through cohesive traction-separation laws embodying a damage parameter tracking the


93

irreversible interface degradation. For a generic initially pristine interface, kinematical continuity holds. When, for a certain interface pair, the local stress state fulfills the threshold condition 1

teff

2 2  2   %    t%  t n  t    Tmax     

(3)

then an extrinsic cohesive law of the form

t% u%,d *   Tmax K  d *    u%

(4)

is introduced to link the interfacial traction with the local displacement jumps. In Eqs.3-4: teff is the % effective traction, expressed in terms of normal and tangential tractions t% n and t t components;  and  give different weights to mode I and II loadings, Tmax is the interface cohesive strength of an interface; d *  [0,1] expresses the current level of total irreversible damage, such that the interface is pristine for d *  0 , while it is failed for d *  1 . It is important to realize that, in Eq.4, d * is the current total damage, which comprises both the accumulated quasi-static and cyclic damage contributions. The accumulation of quasi-static damage is governed by the evolution rule

d s*  N  N   max d  N  N  , d s*  N  ,

  u% n with d   cr  u  n

1

2

  u%    2  crs    us 

2

2  

(5)

where d is the current effective displacement. In the proposed formulation, the interface components of displacements and tractions are the primary variables, so that the computation of effective tractions and displacements can be straightforwardly accomplished once the solution of the polycrystalline system is available at a step of the adopted cycle jump strategy. Upon failure, Eq.(4) is replaced by the laws of frictional contact mechanics, that address separation, stick and slip of the micro-crack surfaces. For further details about the adopted cohesive-frictional model, the interested readers are referred to Refs.[9,10]. Interface cyclic degradation. In this model, inter-granular degradation is modelled in a CDM background. According to Ref.[6], where fatigue degradation in conditions of confined plasticity is studied, the law employed to express the fatigue damage rate, at a generic interface pair, is assumed as  r  tˆ  dc*    N   r  1  d *    

m

(6)

where d * is the current total damage,  r and m are material parameters and r   is a function returning the

range of a cyclically evolving function, so that r  tˆ  is an effective inter-granular traction range, which should be suitably identified in relation to the considered load case [6]. Polycrystalline system assembly. The collocation of Eq.1 at the mesh nodes of each grain and the subsequent integration, performed within the framework of BEM, produce a system of algebraic equation that, complemented by the sets of boundary and interface conditions, taking into account the fatigueinduced mechanical degradation, allow the incremental-iterative solution of the polycrystalline problem. The polycrystalline system assumes the form 0 0   A1  y1  BCs   0   x1    O 0 M      M    y  BCs   0 0 A Ng     x N   Ng  {... Ψ  d *  ...}   g   ψ   

(7)


94

where the blocks A k contain columns of the BE matrices corresponding to the unknown displacements and tractions of the k-th grain, contained in the vector x k , while y k derives from the enforcement of the known boundary conditions (BCs). The block Ψ  d *  implements the interface equations, evolving as a function of

the local values of inter-granular damage, whose components are collected in the vector d * . System (7) must be coupled with the aforementioned rules for updating the components of damage.

High-cycle fatigue solution algorithm: envelope load and cycle jump strategy. In high-cycle problems the direct computation of individual cycles is often too expensive and then unfeasible. For such a reason, an envelope load representation of the external loads [20] and a cycle jump strategy [21] are used to tackle the high-cycle analysis. The envelope load representation consists in loading the aggregate considering the envelope of the load cycles maxima, justified by the observation that most cyclic damage is induced during the loading phase of each cycle. On the other hand, in the cycle jump strategy the system is assessed at discrete cycle intervals  N1 , N 2 ,..., N k ,... , with N k 1  N k  N k , and it is assumed that between N k and N k 1 the system is subjected to the cyclic action of N k cycles. If the system status at N k is known, the system is sampled at N k 1 , after N k cycles during which it has experienced cyclic degradation. At N k 1 the system will be more compliant and the quasi-static damage evolution rules will be assessed. The specimen is thus alternately loaded by quasi-static and cyclic steps. In a quasi-static step, the structure is subjected to the corresponding maximum value of load (envelope load) and damage is evolved according to the quasi-static cohesive rules, upon numerical solution of the underlying structural problem. In a cyclic step, cyclic damage is evolved according to rate laws in Eq.(6). Due to the structure of the damage update rules, whose functional expressions depend on the current value of accumulated damage, quasi-static and cyclic contributions are inherently coupled.

Computational tests The described framework has been applied for the analysis of the fatigue behavior of the polycrystalline aggregate shown in Fig.1a. The aggregate has been generated using the algorithm developed for the tessellation of non-convex domains. It contains 452 grains whose material properties are given in Table 1. No texture is assigned to the grains, so that their orientation is random in the 3D space. The interface properties are also reported in Table 1. The aggregate is subjected to the fully alternated cyclic load   t  whose maximum amplitude (used as envelope load) is  . The simulations are then run in load control: the load amplitude is kept constant, so that the displacement cycles amplitude grows as N grows. Fig.1b shows the obtained -N curve: as the load amplitude  decreases, the fatigue life of the analyzed component becomes longer. The obtained results agree well with those reported in Ref.[2]. Property Elastic constants [GPa] Interface strength [MPa] Cohesive law constants Critical displacement jumps [µm] Cyclic damage rate resistance stress [MPa] Cyclic damage rate exponent

Component c11 c12 c44 Tmax

   uncr  uscr r m

Value 166 64 79.6 550 1 2 1.89 102

( 2 /  ) uncr 2668 8.93

Table 1: Elastic and interface properties of the tested polycrystalline aggregate


95

 t 

 t  a

b

Figure 1: High cycle fatigue of a 3D polycrystalline specimen: a) analyzed tessellation; b) -N curve obtained by applying the developed formulation.

Summary A three-dimensional boundary element method for grain level modelling of high-cycle fatigue degradation in polycrystalline materials has been presented. In the boundary element model the micro-mechanical problem is expressed in terms of inter-granular displacements and tractions, which are then directly used in the traction-separation laws, thus providing a natural implementation of the intergranular degradation processes. Additionally, the need of meshing only the boundaries of the crystals provides simplification in data preparation and reduction in the number of DoFs. The discussed test shows how the model may be employed to investigate the degradation and failure mechanisms in polycrystalline systems, providing a valuable tool for the analysis and design of MEMS.

References [1] Schijve J, Fatigue of structures and materials. Springer Science & Business Media, 2001 [2] Jalalahmadi B, Sadeghi F, Peroulis D. A numerical fatigue damage model for life scatter of MEMS devices, Journal of Micro electro mechanical Systems, 18 (5), 1016-1031, 2009 [3] Roters F, Eisenlohr P, Hantcherli L, Tjahjanto DD, Bieler TR, Raabe D. Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: Theory, experiments, applications. Acta Materialia, 58(4), 1152-1211, 2010 [4] Benedetti I, Barbe F. Modelling polycrystalline materials: an overview of three-dimensional grain-scale mechanical models. Journal of Multiscale Modelling, 5(01), 1350002, 2013 [5] McDowell D, Dunne F. Microstructure-sensitive computational modeling of fatigue crack formation, International Journal of Fatigue, 32 (9), 1521 – 1542, 2010 [6] Bomidi JAR, Weinzapfel N, Sadeghi F. Three‐dimensional modelling of intergranular fatigue failure of fine grain polycrystalline metallic MEMS devices. Fatigue & Fracture of Engineering Materials & Structures, 35(11), 1007-1021, 2012 [7] Benedetti I, Aliabadi MH. A three-dimensional grain-boundary formulation for microstructural modeling of polycrystalline materials. Computational Materials Science, 67, 249-260, 2013 [8] Benedetti I, Gulizzi V, Mallardo V. A grain boundary formulation for crystal plasticity. International Journal of Plasticity, 83, 202-224, 2016

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[9] Sfantos GK, Aliabadi MH. A boundary cohesive grain element formulation for modelling intergranular microfracture in polycrystalline brittle materials. International Journal of Numerical Methods in Engineering, 69(8), 1590-1626, 2007 [10] Benedetti I, Aliabadi MH. 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 [11] Gulizzi V, Milazzo A, Benedetti I. An enhanced grain-boundary framework for computational homogenization and micro-cracking simulations of polycrystalline materials. Computational Mechanics, 56(4), 631-651, 2015 [12] Geraci G, Aliabadi MH. Micromechanical boundary element modelling of transgranular and intergranular cohesive cracking in polycrystalline materials. Engineering Fracture Mechanics, 176, 351-374, 2017 [13] Gulizzi V, Rycroft CH, Benedetti I. Modelling intergranular and transgranular micro-cracking in polycrystalline materials. Computer Methods in Applied Mechanics and Engineering, 329, 168-194, 2018 [14] Benedetti I, Gulizzi V, Milazzo A. Grain-boundary modelling of hydrogen assisted intergranular stress corrosion cracking. Mechanics of Materials, 117, 137-151, 2018 [15] Sfantos GK, Aliabadi MH. Multi-scale boundary element modelling of material degradation and fracture. Computer Methods in Applied Mechanics and Engineering, 196(7), 1310-1329, 2007 [16] Benedetti I, Aliabadi MH. Multiscale modeling of polycrystalline materials: A boundary element approach to material degradation and fracture. Computer Methods in Applied Mechanics and Engineering, 289, 429-453, 2014 [17] Quey R, Renversade L. Optimal polyhedral description of 3D polycrystals: method and application to statistical and synchrotron X-ray diffraction data. Computer Methods in Applied Mechanics and Engineering, 330, 308333, 2018 [18] Aliabadi MH. The boundary element method, applications in solids and structures, Vol.2, John Wiley & Sons, 2002 [19] Gulizzi V, Milazzo A, Benedetti I. Fundamental solutions for general anisotropic multi-field materials based on spherical harmonics expansions. International Journal of Solids and Structures, 100, 169-186, 2016 [20] Robinson P, Galvanetto U, Tumino D, Bellucci G, Violeau D. Numerical simulation of fatigue-driven delamination using interface elements, International Journal for Numerical Methods in Engineering, 63(13), 1824–1848, 2005 [21] Paepegem WV, Degrieck J. Fatigue degradation modelling of plain woven glass/epoxy composites, Composites Part A: Applied Science and Manufacturing, 32 (10), 1433 – 1441, 2001


97

A Boundary Integral velocity representation of the steady Navier-Stokes equations for a body in an exterior domain uniform flow field E. Chadwick School of Computing, Science and Engineering, University of Salford, Salford M5 4WT, UK Keywords: Navier-Stokes equations, NSlets, Oseen flow, Stokes flow.

Abstract. Consider uniform flow past a fixed, closed body in an unbounded domain governed by the incompressible Navier-Stokes equations. A velocity representation is given as an integral distribution of Green’s functions of the Navier-Stokes equations which we shall call NSlets. The strength of the NSlets is the same as the force distribution over the body surface. Introduction. A Boundary Integral velocity representation is given for the Navier-Stokes equations in terms of the Green’s functions called NSlets for a body in an exterior domain uniform flow field. The result is obtained by generalizing existing results in the literature for Stokes [7] [6], Oseen [4] and Euler flow [1]. These represent the flow by an integral distribution over the body boundary of their respective Green’s functions which have a strength given by the force distribution. For the Stokes and Oseen flow this is well-known in the literature and unremarkable as the equations are linear [7] [6] [4]. However, in last year’s BETEQ 2017 conference the same result was obtained for Euler flow by using eulerlets, and this governing differential equation is non-linear [1]. Given that the same simple representation occurs for Stokes, Oseen and Euler flow, the natural question to ask addressed here is whether this is an underlying representation that is also satisfied by Navier-Stokes flow? A theoretical argument is presented which suggests this. We start by introducing a new method and showing that it gives the expected existing results in Oseen, Stokes and Euler flow. We then apply this method to the Navier-Stokes equations and get the new result that the Navier-Stokes velocity is given by an integral distribution of NSlets over the body boundary. An Oseen representation of the velocity by an integral distribution of oseenlets. Applying the Oseen linearisation u†i = δi1 + ǫui (1) with dimensionless variables to the steady incompressible Navier-Stokes equations with body force −fi acting on the fluid u†j u†i,j = −p†i + (1/Re)u†i,jj − fi , u†i,i = 0 (2) where ǫ Aliabadi

98

Advances in Bundary Element andZMeshless Techniques XIX (m) (m) (m) {[Oi + fi ]ui − [Oi + fi ]ui }dΣ′ = 0, (5) Σ where Σ is a region of space, and dΣ′ = dx′1 dx′2 ...dx′n is an infinitesimal element of n−dimensional (m) (m) integrated space. Following Oseen, the term Oi ui − Oi ui in the integrand in (5) can be reformulated

with an outer differential because (m)

Oi

(m)

ui − Oi ui

(m)

(m)

= ui ui,1 + ui p,i (m)

(m)

= −ui ui;1 − ui p;i

(m)

(m)

− (1/Re)ui ui,jj − ui;1 ui (m)

(m)

− (1/Re)ui ui;jj − ui;1 ui

(m)

(m)

− p;i ui

(m)

(m)

− p;i ui

(m)

+ (1/Re)ui;jj ui

(m)

+ (1/Re)ui;jj ui

(m)

(m)

= [−ui ui ];1 − [ui p(m) ];i − [pui ];i − (1/Re)[ui ui;j − ui;j ui ];j h i (m) (m) (m) (m) = −ui ui δj1 − uj p(m) − puj − (1/Re)[ui ui;j − ui;j ui ] ;j

= oj;j ,

(6)

where the outer differential applies to the relation (m)

oj = −ui ui

(m)

δj1 − uj p(m) − puj

(m)

(m)

− (1/Re)[ui ui;j − ui;j ui

].

(7)

Let us define the following boundaries: ∂Σ0 which defines the boundary of the body and is sufficiently smooth that a normal exterior to the body n exists; the boundary ∂Σ∞ which defines the surface a radius R∞ distance away from the body in the limit as R∞ → ∞, see figure 1.

∂Σ∞

R∞ → ∞ n

∂Σ0

Figure 1: The boundaries


99

Let Advances us definein Bundary the following spaces: Σǫ Techniques which defines Element and Meshless XIX a space within a radius ǫ of the point x; Σ0 which defines a space within a distance ǫ from the body boundary ∂Σ0 , Σ∞ which defines the space exterior to the boundary ∂Σ∞ , and Σ which defines all space including the space exterior and interior to the body surface ∂Σ0 , see figure 2.

Σ∞ Σ − Σ∞ − Σǫ − Σ0

Σǫ ǫ Σ0

ǫ ǫ

Figure 2: The spaces Then the integral (5) reduces to Z

0 =

(m)

ZΣ

=

Σ∞

(m)

(m)

]ui − [Oi + fi ]ui }dΣ′ Z Z (m) (m) (m) {Oi ui − Oi ui }dΣ′ + fi ui dΣ′ +

{[Oi

+ fi

Σǫ

(m)

Σ0

f i ui

dΣ′ . (8)

Taking each integral term in (8) in turn, from (6) and applying the divergence theorem the first term becomes Z Z Z (m) (m) ′ {Oi ui − Oi ui }dΣ = oj;j = − oi ni d∂Σ′ = 0, (9) Σ∞

Σ∞

∂Σ∞

for the outward pointing normal n to the boundary ∂Σ∞ , and for the velocity and pressure perturbations going to zero at infinity, ui , p → 0, see for example [3]. The second term in (8) is Z Z (m) ′ fi ui dΣ = δδim ui dΣ′ = um , (10) Σǫ

Σǫ

and the third term is Z Z (m) ′ fi ui dΣ = Σ0

∂Σ0

Z

ǫ −ǫ

(m) fi ui dn′ d∂Σ′

=

Z

∂Σ0

Z

ǫ −ǫ

(m) Fi δui dn′ d∂Σ′

=

Z

(m)

∂Σ0

Fi ui

d∂Σ′ ,

(11)

where the Rforce is considered as a distribution of Dirac delta functions fi = Fi δ over the body boundary ǫ such that −ǫ δdn′ = 1 where dn′ is a length element normal to the surface.


100

Oseen representation. Therefore, bringing together (9), (10) and (11) in (8) gives the Oseen velocity representation Z Z (m) ′ um = Fi ui d∂Σ′ = Fi u(i) (12) m d∂Σ ∂Σ0

(m) ui

∂Σ0

(i) um .

since for the oseenlet = This is the Oseen representation used, for example, by Olmstead and Gautesen [4], amongst many others. Stokes representation. Considering the Oseen equations as a far-field asymptotically matched to a near-field Stokes flow for low Reynolds number, we see that in the near-field the oseenlets tend towards the stokeslets, see for example, Chadwick [2]. Therefore, for Stokes flow the same representation of the velocity holds Z Z (m)

um =

∂Σ0

(m)

Fi ui

d∂Σ′ =

∂Σ0

′ Fi u(i) m d∂Σ

(13)

(i)

since for the stokeslet ui Pozrikidis [6].

= um . This is the representation given in, for example Shankar [7] and

Euler representation. We can also consider the Oseen equations as a far-field asymptotically matched to a near-field Euler flow by Chadwick [1]. Therefore, for Euler flow the same representation of the velocity holds Z Z (m) ′ ′ um = Fi ui d∂Σ = Fi u(i) (14) m d∂Σ ∂Σ0

(m) ui

∂Σ0

(i) um .

since for the eulerlet = This is given by Chadwick [1]. So, this new method gives all the existing representations in the literature for Oseen, Stokes and Euler flow. Navier-Stokes representation. Let us now apply the method to Navier-Stokes flow. Letting Ni = ui;1 + uj ui;j − p;i − (1/Re)ui;jj and

(m)

Ni

(m)

(m) (m) ui;j

= ui;1 + uj

(m)

− p;i

(15) (m)

− (1/Re)ui;jj

then following the same analysis for the Oseen representation (8), we get Z (m) (m) (m) 0 = {[Ni + fi ]ui − [Ni + fi ]ui }dΣ′ ZΣ Z Z (m) (m) (m) ′ ′ = {Ni ui − Ni ui }dΣ + fi ui dΣ + Σ∞

Σǫ

(16)

(m)

Σ0

f i ui

dΣ′ . (17)

In the far-field the flow becomes Oseen flow and so from (9) we get Z Z Z Z (m) (m) (m) (m) {Ni ui − Ni ui }dΣ′ = {Oi ui − Oi ui }dΣ′ = oj;j = − Σ∞

Σ∞

Σ∞

oi ni d∂Σ′ = 0.

(18)

∂Σ∞

So again, substituting (18) into (17) gives um =

Z

∂Σ0

(m)

Fi ui

d∂Σ′ .

(19)


101

Conclusion. First, a new method is given confirming known results in the literature that show the velocity for Oseen, Stokes and Euler flow is given by an integral distribution of Green’s functions over the body whose strength is the body force distribution. Then, the same method is used to show that the NavierStokes velocity is also given by the same integral distribution of Green’s functions over the body boundary. The method is the same for internal as well as external flows, and the extension to time-dependent flows is trivial. This would then appear to provide a general solution to the Navier-Stokes equation, one of the millenium problems, that can also be solved numerically in a straightforward way by an appropriate numerical approximation to give a boundary element formulation. Currently, a boundary element formulation for (19) is being developed, with the aim of testing on two Navier-Stokes flow problems: two-dimensional steady uniform flow past a circular cylinder for Reynolds number up to 40, and uniform flow past a two-dimensional semi-infinite flat plate.

References [1] E. Chadwick, J.Christian, and K. Chalasani. Using eulerlets to model steady uniform flow past a circular cylinder. European Journal of Computational Mechanics, to appear, 2018. [2] E.A. Chadwick. The far-field Green’s integral in Stokes flow from the boundary integral formulation. Computer Modeling in Engineering Sciences, 96(3):177–184, 2013. [3] N. Fishwick and E. Chadwick. The evaluation of the far-field integral in the Green’s function representation for steady Oseen flow. Physics of Fluids, 18:113101: 1–5, 2006. [4] W. Olmstead and A. Gautesen. Integral representations and the Oseen flow problem. Mechanics Today, 3:125–189, 1976. [5] C.W. Oseen. Neure Methoden und Ergebnisse in der Hydrodynamik. Akad. Verlagsgesellschaft, Leipzig, 1927. [6] C. Pozrikidis. Fluid Dynamics. Springer, USA, 2009. [7] P.N. Shankar. Slow viscous flows. Imperial College Press, London, UK, 2007.

102



103

A quasi-static delamination model with interface damage and friction for layered structures Roman Vodička, Filip Kšiňan Technical University of Košice, Civil Engineering Faculty, Vysokoškolská 9, 042 00 Košice, Slovakia, [email protected],[email protected] Keywords: interface damage, anisotropic friction; quasi-static delamination, cohesive interface, anisotropic elasticity, energy formulation, quadratic programming, BEM.

Abstract A quasi-static model for numerical analysis of interface crack initiation and propagation is proposed. It also considers friction between layers of a structure made of anisotropic materials if the crack evolves in the shear mode under compression. The approach is based on balance of stored energy, work of external forces, and dissipation, looking for a kind of weak solution. Numerically, the solution is approximated by a time stepping procedure to express the evolution in a variational sense, non-linear programming to solve pertinent minimisations, and by the SGBEM to provide the elastic solutions for the solids. Numerical results are also presented, demonstrating properties of the presented model and showing its usefulness in engineering problems. Introduction A model for numerical analysis of interface damage which leads to interface crack initiation and propagation in layered structures made of anisotropic materials presented in [15] is upgraded to cover the assumptions of frictional contact problems arising after crack initiation between the layers remaining in contact. Damage and subsequent cracking of an interface can be modelled by an additional internal variable pertinent to damage distributed on this interface [4,11]. Such damage variable enables to avoid stress singularities and also to describe the interface relations between displacement gaps and stresses as in cohesive zone models (CZM) in terms of damage as described in [8, 13]. Nevertheless, practical calculations with anisotropic materials show how complicated the stress strain relations could be, see [2, 5], e.g. the process of deterioration of the material near the crack tip may develop in several stages. Roughness of the interface surfaces together with the anisotropy of the adjacent layers require to consider also a frictional contact problem in a damageable interface as in [10]. The broken interface, if kept under pressure, represent a typical contact problem treated here as normal compliance contact in order to facilitate mathematical interpretations for friction [19, 20]. Additionally, the solids also involve a viscous rheology, which is used here along with the normal compliance contact to guarantee that the friction model has a unique solution, cf. [6]. In relation to intended anisotropy of solids in contact, it is also worthy to consider anisotropic friction [21]. The orientation of material axes does not have to be the same as, say, load orientation and friction can be direction dependent at the interface, see also [14]. In the model, various types of layered materials can be used, especially those made of laminated and fibrous composites. The layers, considered as homogenised anisotropic material, may have independently defined material symmetries. Modelling of damage takes into account various relations between interface stresses and displacement gaps providing the structural response as in known cohesive zone models [13]. The proposed mathematical approach is physically represented by energy evolution and provides a kind of weak solution. The solution evolution is approximated by a semi-implicit time stepping algorithm (a staggered scheme). For obtaining an elastic state variables, the algorithm implements the Symmetric Galerkin BEM into (sequential-)quadratic programing solvers based on [3]. Description of the interface model First, we describe a model for interface damage with friction based on that from [15]. A simplified situation is considered in the sense that no variable is dependent on x3 coordinate, see Figure 1. Therefore, the geometrical representation by a 2D section Ω of the cylindrical 3D body Ω×[−h, h] is allowed supposing the state of generalized plane strain in the analyzed element. All displacement or traction variables have than three components


104

(geometrically independent of x3 ). Only two such bodies will be considered for the sake of simplicity of explanation and notation, though the theory can be generalized to consider several bodies, even in a full 3D context. The bodies are supposed to occupy domains Ωη ⊂R2 (η=A, B) with bounded Lipschitz boundaries ∂Ωη =Γη , Figure 1(right). Let ⃗nη denote the unit outward normal vector defined along smooth parts of Γη , let ⃗sη denote ~nA ~s A ~z

gA ΓA D

Ω

fA ΓA N

A

ΓC ΓB N

Ω

B

ΓB D

fB

x2 x3

B 0 x1 ~s ~z ~nB

gB

Figure 1: Section of a 3D cylindrical body and the used notation for two bonded domains in such a section. the unit tangential vector such that it defines anti-clockwise orientation of Γη . The cohesive contact zone in the considered section-domain Ω denoted by Γc is defined as the common part of ΓA and ΓB , i.e. Γc =ΓA ∩ΓB . The Dirichlet and Neumann boundary conditions are defined on the outer boundary η η η η parts, respectively, prescribing displacements as uη =gD (t) on ΓD and tractions as pη =fN (t) on ΓN at a time instant η η η η t. It is naturally assumed that ΓD and ΓN are disjoint parts of Γη , disjoint also from Γc , and Γη =ΓD ∪ΓN ∪Γc . η Additionally, the Dirichlet part is considered to lie far from the contact boundary, i.e. ΓD ∩Γc =∅. The difference across Γc of the functions defined in ΩA and ΩB will be denoted by [[·]]. In particular, the displacement gap on the contact boundary Γc means [[u]]=uA |Γc −uB |Γc . Further, we will need the gaps of the normal and tangential components. The normal component reads [[ ]] [[ ]] [[ ]] u n = u ·⃗nB = − u ·⃗nA = −uB |Γc ·⃗nB − uA |Γc ·⃗nA , (1) Similarly, the tangential component (in the(plane of Ω) . . . ) [[u]]s and the out of plane component [[u]]z can be written. Total tangential component is then [[u]]t := [[u]]s , [[u]]z . We also use the convention that the ‘dot’ will stand for ∂ the partial time derivative ∂t . The initial-boundary value problem takes a form of a nonlinear evolution governed by a stored energy functional E and a nonsmooth potential of dissipative forces R. In the Biot-equation form, it reads as a system of differential inclusions

.. .. ∂ . R(u, ζ; u, ζ) + ∂ E (g (t); u, ζ) ∋ 0,

∂u. R(u, ζ; u, ζ) + ∂u E (gD (t); u, ζ) ∋ F (t), ζ

ζ

D

(2a) (2b)

for u=(uA ,uB ) and gD =(gAD ,gBD ), where the symbol ∂ refers to partial subdifferentials relying on the convexity of . . R(u, ζ; ·, ζ), R(u, ζ; u, ·), E (gD ; ·, ζ), and E (gD ; u, ·). It involves the functionals E , R in the specific form:  ∑ ∫ 1 η η   ε(u ):C :ε(uη ) dΩ  η  η 2  if uη |Γη = gD ,  η=A,B Ω D ∫ (3a) E (gD ; u, ζ) = [[ ]]) [[ ]] 1 ([[ ]]− )2 and 0 ≤ ζ ≤ 1 on Γc 1(  + K(ζ) u · u + k u dΓ  G n  2  Γc 2   +∞ else, ∫ √ ([[ ]]) . [[ .]] [[ ]]− [[ .]]⊤   G u · | ζ| + k u M(ζ) u t dΓ c G u n  t   Γc ∫  ∑ . .. 1 .η η .η R(u, ζ; u, ζ) = (3b) + ε(u ):D :ε(u ) dΩ, if ζ ≤ 0 on Γc ,  η 2  Ω  η=A,B    +∞ else,


105

and a linear functional F (t) acting on u as ∑ ∫ η ⟨F (t), u⟩ = fN (t) · uη dΓ.

(3c)

η

η=A,B ΓN

The evolution system Eq. (2) holds for the instants t∈[0, T] within a fixed time range T. We will further consider the initial conditions at time t=0 for the displacement and the damage parameter: η

uη (0, ·) = u0

on Ωη ,

ζ(0, ·)=1

and

on Γc .

(4)

In Eq. (3), Cη and Dη are the positively definite symmetric 4th-order tensors of elastic and viscosity moduli, respectively, in the domain Ωη . The parameter kG is the compression stiffness of the normal-compliance contact allowing for interpenetration of the bodies (we introduced the notation x− = min(0, x) for any real x), but for large values of kG being a realistic replacement of the Signorini contact efficiently implementable by QP. The interface is considered as an infinitesimally thin layer of an adhesive, whose elastic moduli are introduced by the matrix K. Then, K is the positively definite 2nd-order tensor of interface stiffness depending on the actual state of damage expressed by the value of the damage parameter ζ. Each component of K is defined by a function, increasing in ζ, with K(1) being the initial (undamaged) stiffness of the adhesive. Also, it is supposed that (K(ζ)v) ·v is a convex function of ζ for any v. The problem in Γc is usually expressed in a local coordinate system defined by normal and tangential components in the plane defined by Ω and the z coordinate as in Eq. (1) and below it. Then, K is simplified to a diagonal matrix of the following form K(ζ)=diag (kn , ks , kz ) Φ(ζ) with a prescribed convex function Φ according to the considered cohesive zone model representing the tractiondisplacement-gap relation, e.g. for the common bilinear CZM: Φ(ζ)=βζ/(1+β−ζ), with a parameter β>0 defining the slope of the softening branch in the relation between displacement gap and contact traction, see [16]. The interface fracture energy Gc , required to break a unit length of the interface (reduced to the section Ω), is considered to depend on the current state of displacement gap [[u]]. Any phenomenological law describing its dependence on current displacement gap can be used, being a representation of crack-mode sensitivity of such energy, say that of Hutchinson [7], or Benzeggagh [1], e.g. Gc

([[ ]]) kn [[u]]2n + ks [[u]]2s + kz [[u]]2z u = k [[u]]2 k [[u]]2 k [[u]]2 n n + sGII s + zGIII z GI c

I

II

c

(5)

c

III

The parameters Gc , Gc and Gc express the partial fracture energies in the pure Mode I, in the pure Mode II and in the pure Mode III, respectively. Finally, the friction matrix M is made dependent on the current state of damage M(ζ) = (1 − ζ)θ M0 ,

θ ≥ 1,

(6)

where the symmetric positive definite matrix M0 reflects the frictional relations in the totally damaged contact zone. Numerical solution and computer implementation The numerical procedures proposed for the solution of the above problem consider the time discretisation and the spatial discretisation separately. The procedures are expressed in terms of the boundary data only by making use of the Symmetric Galerkin BEM (SGBEM). The time discretisation scheme is defined by a semi-implicit algorithm with a fixed time step size τ such that τ k =kτ for k=1, . . . Tδ . In order to obtain such an algorithm from Eq. (2), the rates are approximated by the

.

finite difference: for damage ζ≈ ζ −ζτ , where ζ k denotes the solution at the instant τ k , and for displacements . k k−1 u≈ u −uτ . The differentiation with respect to the rates can be replaced by the differentiation with respect to ζ k or uk , so that from Eqs. (2) we obtain ( ) uk −uk−1 τ∂uk R uk−1 , ζ k−1 ; , 0 + ∂uk E (gD (τ k ); uk , ζ k−1 ) ∋ F (τ k ), (7a) τ ( ) k k−1 ζ −ζ τ∂ζ k R uk−1 , ζ k−1 ; 0, + ∂ζ k E (gD (τ k ); uk , ζ k ) ∋ 0, (7b) τ k

k−1


106

if we solve the inclusions separately in u and ζ, respectively. Such a separation provides a variational structure to the solved problem, with two minimizations in each time step. First, the minimization of ( ) ⟨ ⟩ k−1 k k k−1 k k−1 k−1 u−u Hu (u) = E (gD (τ ); u, ζ ) − F (τ ), u + τR u , ζ ; ,0 . (8a) τ provides uk , and then the minimization of ( Hζ (ζ) = E (gD (τ ); u , ζ) + τR k

k

k

k−1

u

,ζ

k−1

ζ−ζ k−1 ; 0, τ

) (8b)

provides ζ k . The role of the SGBEM in the present computational procedure is to provide a complete boundary-value solution from the given boundary data for each domain in order to calculate the elastic strain energy in these domains. Thus, the SGBEM code calculates unknown tractions along Γc ∪ΓD , assuming the displacements jump at Γc to be known from the used minimization procedure, in the same way as proposed and tested in [17]. Once all the boundary data (displacements and tractions) are obtained from the solution of the SGBEM code, the pertinent energy functionals H k in Eq. (8) can be calculated. As an SGBEM code is used for the solution of these BVPs, it is convenient to change the domaink integrals with u in Eq. (3) to boundary based integrals ∫ ∫ η η η ε(u ):C :ε(u )dΩ = pη (uη ) · uη dΓ. (9) Ωη

Γη

As our model includes also viscosity, a simple transformation of the displacement variable is used for reformulation of the viscoelastic problem in the solids in terms of recursive elastostatic problems which are solved by the elastostatic SGBEM, similarly as in [9,19]. The transformation is based on a restriction of general viscosity tensor Dη introduced in (3b) to the form Dη =τ r Cη , with the prescribed relaxation time τ r >0 being the same for all subdomains. With this assumption, let us introduce a new ’displacement’ v at the time step τ k , as vk := uk + τ r

uk − uk−1 . τ

(10)

The transformed boundary displacement v and tractions p are obtained by the standard SGBEM for multidomain problems, [17], using the boundary integral equations ∫ ∫ 1 η η η η v (ξ) = U (ξ, η) · p (η)dΓ(η) − − Tη (ξ, η) · vη (η)dΓ(η), for a.a. ξ ∈ ΓD ∪ Γc , (11a) η η 2 ∫Γ ∫Γ 1 η η p (ξ) = − Tη∗ (ξ, η) · pη (η)dΓ(η)− = Qη (ξ, η) · vη (η)dΓ(η), for a.a. ξ ∈ ΓN . (11b) η η 2 Γ Γ in a anisotropic media. Stated in the previous section, the restriction to generalized plane strain state is consider and the fundamental solutions U, T, Q are supposed according to [14, 15]. Remarks on the minimisation algorithms The problem is numerically solved by recursive minimisation of the functionals (8a) (expressed in terms of v) and (8b). Here, we generally have convex functionals for whose minimisation sequentially quadratic programming algorithms are applied. Nevertheless, under certain conditions the algorithm reduces to mere QP with constraints. For (8b), it is true if the function K(ζ) is quadratic. For the other functional, the situation is more complicated as there are two non-smooth terms: one related to the normal contact term containing (·)− expression, the other one from the frictional term. The former can be reduced to a quadratic term with an additional constraint introducing a new variable υ satisfying relations υ ≥ 0,

τr υ + wn ≥ − zk−1 . τ n

(12)

This is classical trick, known also as a Mosco-type transformation, implemented also in [13, 19]. The friction term can also be treated similarly if the friction were isotropic as shown in 2D case in [19]. For aniostropic friction, this non-smooth term is regularised by a smooth one [20]. The regularisation of the function


107

√ J(w)= w⊤ M(ζ)w, which is a cone, eliminates its vertex, being the only non-smooth point of the function J. There are several possibilities how to do it, e.g.: { √ J(w), if J(w) > ε, Jε1 (w)= J 2 (w) + ε 2 , or Jε2 (w)= 3ε (13) 3 2 1 4 8 + 4ε J (w) − 8ε 3 J (w), otherwise, with a small positive parameter ε. After the discretisation, all variables are approximated by the pertinent boundary element mesh introduced by SGBEM, see [12, 17]. The approximation formula can be written in the schematic form ∑ ωh (x) = Nn (x)ωn , (14) n

where ω can be v, w, p, ζ, or υ and ωn are the nodal unknowns associated to the node xn . In what follows, the nodal values ωn are grouped into column vector ω. In minimisation of the functional (8a) with the constraints (12), it is useful to reformulate the constraints as bound constraints. First, the constraint can be written in terms of the nodal variables as ( )( ) ( ) I 0 υ 0 ≥ , (15) I I wn − ττr zn with the identity matrix I. The inequality is defined by a full rank matrix, therefore the following relation holds: ) ( ) ( )( ) ( ) ( υ I 0 y1 y1 0 . = , with ≥ wn −I I y2 y2 − ττr zn

(16)

This implies the same number of bound constraints as provided by the more general conditions (15). The functional (8a) can be expressed after discretisation in a general matrix form as 1 k Hv,h (y) = y⊤ Akh y + Kh (y) − (bkh )⊤ y + ckh , y ≥ ξ, (17) 2 where the term Kh pertains to the discretised version of the non-quadratic friction term in (3b) with the square root regularised by one of the functions proposed in (13). The bound ξ is in fact determined by the constraints applied on y1 , y2 in (16). The constrained minimum is denoted by yk . Similarly, the discrete form of the functional (8b) can be written as k Hζ,h (ζ) = Hh (ζ) − (Gkh )⊤ ζ + ckh ,

0 ≤ ζ ≤ ζk−1 h .

(18)

The function Hh can be quadratic as the first term in (17), if the damage dependent stiffness matrix K depends quadratically on ζ [13]. Otherwise, it is generally convex as the Kh term in (17). Anyhow, the constrained minimum is denoted by ζk . In the general case, we apply the QP algorithm sequentially, see [13]. Therefore, in the case of K for example, we first approximate it by a quadratic form using the Taylor polynomial 1 (y − y0 )⊤ K′′ (y0 ) (y − y0 ) , (19) 2 where y0 is known. It should be noted that the second derivative is positive due to required convexity of both K and H functions. The methods of QP can be implemented numerically by various algorithms. Here, we use Conjugate Gradient based algorithms with bound constraints [3]. The pertinent algorithms are not described explicitly as they can be found in the aforementioned reference. Summing it up, in each time step k we first start with m=1 and y0 =yk−1 and iteratively find the minimum of h k,m Hv,h (v), the minimiser being denoted ym until a convergence criterion is met for ym1 . Then, we put ykh =ym1 . K(y) ≈ K(y0 ) + K′⊤ (y0 ) (y − y0 ) +

k,m Then again we first start with m=1 and now ζ0 =ζk−1 and iteratively find the minimum of Hζ,h (ζ), the minh k imiser being denoted ζm until a convergence criterion is met for ζm2 . Then, we put ζh =ζm2 . This we apply recursively up to prescribed time T.


108

Numerical example The present formulation of crack initiation and growth at an interface between two layers of a laminate has been tested numerically by a computer code, which was implemented in M . The developed numerical algorithm exploits the variationally based SGBEM to calculate the elastic solution at the interface and in each subdomain and a conjugate gradient based method for constrained minimization leading to the contact solution as in [13, 14, 18]. A square plate consisting of two layers of thicknesses h1 =0.5mm, h2 =1mm is considered in the example. The geometry of a plate cross-section, loading conditions and constraints are shown in Figure 2. The load is y 50 0.22

−g

0.5 1

−f

x2 x3

f

f(t) [MPa]

g(t) [mm]

g

x1 2

18

2 0.25 t [s]

0 0.02 0.03

Figure 2: The two-layer plate cross-section geometry and the loading conditions. There are no external forces applied in x3 direction, though the fixed constraint includes also this component. applied in two steps. First, a vertical increasing pressure f is applied at the unconstrained end of the plate until the maximum value of fmax =50 MPa is reached. Thereafter, horizontal displacements are prescribed so that they increase from 0 to gmax =0.22 mm. In this kind of loading, the influence of friction can be observed at the interface close to the applied load. The elastic properties of the solids are: EL =134 GPa, ET =7.7 GPa, GTT =2.76 GPa, GLT =4.3 GPa, ν LT =0.3, ν TT =0.4 for the bottom layer, and EL =136 GPa, ET =9.8 GPa, GTT =5.2 GPa, GLT = 4.7 GPa, ν LT =0.28, ν TT =0.15 for the top layer. Additionally for the top layer, the fibres are oriented along the x1 axis. The other layer contains the fibres rotated by the angle π/4 about the x2 axis. This skew orientation of fibres causes the out-of-plane deformation and also interface forces. The viscous properties are neglected, i.e. τ r =0. Based on that, we may compare behaviour of the structure for isotropic and anisotropic friction by taking M0 =μ 20 M in (6) as follows: isotropic friction: anisotropic friction:

μ 0 = 0.4, M = I, θ = 4, ) ( (1 cos π8 0 ⊤ Q , Q= μ 0 = 0.4, M = Q 4 − sin π8 0 1

sin π8 cos π8

(20a)

) , θ = 4,

(20b)

where in the anisotropic case the rotation matrix Q determines principal directions of friction in this orthotropic description. The choice of θ causes the friction to become significant only if the damage variable ζ is close to zero, i.e. for an almost corrupted interface. The interface stiffnesses contained in K are kn =ks =kz =10 TPam−1 . The material characteristics which −2 govern the crack propagation in (5) are: GIc =0.26k Jm−2 , GIIc =GIII c =1.002 kJm , α=1. It can also be noted that under these conditions the maximal interface stresses are pn =30MPa, ps =pz =60MPa. The solution used the time step τ=0.1ms with a non-uniform boundary element mesh refined especially at the vicinity of the edge where the plate is loaded as shown in Figure 3. ℓmin = 0.05mm ℓmax = 0.5mm

Figure 3: The boundary element mesh. The evolution of interface damage can be observed in Figure 4 for both cases of friction. The extent of the interface crack can be read from the distribution of ζ as ζ close to zero naturally means the crack. The first used instant corresponds to the instant when ζ reaches zero initially. The second instant is the point of minimum reaction force calculated at the place of prescirbed displacement for the lower layer. The third instant is taken the same for both friction options and it just documents how the damage and contact traction develop. Generally, three zones in the interface can be distinguished, the right end of the interface is not so interesting

1 (b) (a) 0.8 0.6 0.4 0.2 0 60

109

1 [-]

t [s] 0:2043 0:2054 0:2300 pn ps pz

40 20

pn ; ps ; pz [MPa]

pn ; ps ; pz [MPa]

1 [-]


0 -20 -40 -60

0

5

10

15

20 22

1 (c) (b) 0.8 0.6 0.4 0.2 0 60

t [s] 0:2042 0:2054 0:2300 pn ps pz

40 20 0 -20 -40 -60

0

5

10

x1 [mm]

15

20 22

x1 [mm]

Figure 4: Distributions of the interface variables: tractions p and damage ζ at selected time instants t, (a), (b) denote the choice of friction according to (20). (The line thickness determined by the time instant depending on the friction mode is valid both for damage and traction graphs.) as it only reflects the fixed constraints. At the left edge, the stress distribution according to friction conditions appears: both tangential components are present due to skew orientation of the fibres in the layers of the plate. The ratio between these two components varies for two friction options prescribed. In the central part of the interface, if the crack propagates, the tangential stress distribution can be seen. The special shape of the traction distribution in the zone where ζ changes from 1 to 0 appears due to prescribed interface stiffness function K(ζ), which pertains to well known bilinear CZM. Finally, the deformation at the last calcuated instant is presented in Figure 5. The shape of deformation is x2 [mm]

2 0 -2

t = 0:2300s

-4 0

5

10

15

20

x1 [mm] 0

0.01

0.02

0.03 0.04 u3 [mm]

0.05

0.06

Figure 5: Deformations in the plate cross-section plane (the hue intensity according to the displacement in the out-of-plane direction). the same for both friction options, therefore only the anisotropic one is used. The out of plane displacement evolves in the same way as increases the vertical displacement. If there is a substantial crack, as it is for the used instant, the shown results document that the skew orientation of the principal material directions gives rise also to a tearing deformation of the plate layers. Conclusions The approach for quasi-static delamination in layered anisotropic structure, intended to be understood as layers in a fibre-composite laminate, presented previously in [15] was reconsidered with friction if a crack propagates in a shear mode under pressure. Though in the numerical implementation it considers only two dimensional crosssection of such a structures, the deformation state takes also the out-of-plane direction into account, provided that the orientation of the fibres is not restricted, say, to parallel or perpendicular orientation of fibres in the adjacent layers. The numerical solution has even demonstrated that such a situation is applicable and significant influence of the out-of-plane Mode III crack has been detected. The results also show that considering a form of anisotropy of friction is relevant in calculations. Acknowledgement Authors acknowledge support from the Slovak Ministry of Education by the grants VEGA 1/0078/16 and VEGA 1/0477/15.

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References [1] M.L. Benzeggagh and M. Kenane. Measurement of mixed-mode delamination fracture toughness of unidirectional glass/epoxy composites with mixed-mode bending apparatus. Compos. Sci. Technol., 56:439– 449, 1996. [2] C.G. Dávila, C.A. Rose, and E.V. Iarve. Modeling fracture and complex crack networks in laminated composites. In V. Mantič, editor, Mathematical Methods and Models in Composites, pages 297 – 347. Imperial College Press, 2013. [3] Z. Dostál. Optimal Quadratic Programming Algorithms, volume 23 of Springer Optimization and Its Applications. Springer, Berlin, 2009. [4] M. Frémond. Dissipation dans l’adhérence des solides. C.R. Acad. Sci., Paris, Sér.II, 300:709 – 714, 1985. [5] R. Gutkin, M.L. Laffan, S.T. Pinho, P. Robinson, and P.T. Curtis. Modelling the R-curve effect and its specimen-dependence. Int. J. of Solids and Structures, 48(11–12):1767 – 1777, 2011. [6] W. Han, M. Shillor, and M. Sofonea. Variational and numerical analysis of a quasistatic viscoelastic problem with normal compliance, friction and damage. J. Comput. Appl. Math., 137:377–398, 2001. [7] J.W. Hutchinson and Z. Suo. Mixed mode cracking in layered materials. Advances in Applied Mechanics, 29:63 – 191, 1991. [8] A. Mielke and T. Roubíček. Rate-Independent Systems – Theory and Application. Springer, 2015. [9] C.G. Panagiotopoulos, V. Mantič, and T. Roubíček. A simple and efficient BEM implementation of quasistatic linear visco-elasticity. Int. J. Solid Struct., 51:2261 – 2271, 2014. [10] M. Raous, L. Cangemi, and M. Cocu. A consistent model coupling adhesion, friction and unilateral contact. Comput. Meth. Appl. Mech. Eng., 177(6):383–399, 1999. [11] T. Roubíček, M. Kružík, and J. Zeman. Delamination and adhesive contact models and their mathematical analysis and numerical treatment. In V. Mantič, editor, Mathematical Methods and Models in Composites, pages 349 – 400. Imperial College Press, 2013. [12] A. Sutradhar, G.H. Paulino, and L.J. Gray. The symmetric Galerkin boundary element method. SpringerVerlag, Berlin, 2008. [13] R. Vodička. A quasi-static interface damage model with cohesive cracks: SQP–SGBEM implementation. Eng. Anal. Bound. Elem., 62:123–140, 2016. [14] R. Vodička, E. Kormaníková, and F. Kšiňan. Interfacial debonds of layered anisotropic materials using a quasi-static interface damage model with coulomb friction. Int. J. Frac., 2018. (submitted). [15] R. Vodička and F. Kšiňan. A quasi-static model of delamination for structures made of anisotropic layers implemented by SGBEM and nonlinear programming. In L.Marin and M.H. Aliabadi, editors, Advances in Boundary Element & Meshless Techniques XVIII, pages 154–161, Eastleigh, 2017. EC ltd. [16] R. Vodička and V. Mantič. An energy based formulation of a quasi-static interface damage model with a multilinear cohesive law. Discrete and Cont. Dynam. Syst.–S, 10(6):1539–1561, 2017. [17] R. Vodička, V. Mantič, and 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., 17(3):173–203, 2007. [18] R. Vodička, V. Mantič, and T. Roubíček. Energetic versus maximally-dissipative local solutions of a quasi-static rate-independent mixed-mode delamination model. Meccanica, 49(12):2933–296, 2014. [19] R. Vodička, V. Mantič, and T. Roubíček. Quasistatic normal-compliance contact problem of visco-elastic bodies with Coulomb friction implemented by QP and SGBEM. J. Comp. Appl. Math., 315:249–272, 2017. [20] P. Wriggers. Computational Contact Mechanics. Springer, Berlin, 2006. [21] A. Zmitrowicz. Models of kinematics dependent anisotropic and heterogeneous friction. Int. J. Solids Struct., 43:4407–4451, 2006.


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Inference of the Equivalent Initial Flaw Size Distribution with the Boundary Element Method using Maximum Likelihood Estimation Llewellyn Morse*, Zahra Sharif-Khodaei and M. H. Aliabadi Department of Aeronautics, Imperial College London, South Kensington Campus, City and Guilds building, Exhibition Road, SW7 2AZ, London, UK *[email protected] Keywords: Boundary Element Method (BEM), Equivalent Initial Flaw Size (EIFS), Uncertainty quantification, Fatigue, Crack growth, Maximum Likelihood Estimation (MLE) Abstract A method is proposed in this work for the inference of the Equivalent Initial Flaw Size (EIFS) distribution using the Boundary Element Method (BEM). The EIFS distribution for a cracked stiffened panel is determined using Maximum Likelihood Estimation (MLE). Various sources of uncertainty are considered, such as uncertainty in loading conditions, measurement of crack size during inspections, and in fatigue crack growth model parameters. Results suggest that MLE is effective at estimating the statistics of an EIFS distribution in the absence of prior information. 1. Introduction To accurately determine the fatigue life of a structure, it is necessary to grow the cracks in the structure from their actual initial flaw size (IFS) to a maximum permissible crack size. Current damage tolerance guidelines specify that the designer should assume that these IFSs are equal to the lower limit of detectability of the Non-Destructive Inspection (NDI) techniques used during manufacture [1], however, this can lead to conservative estimates of the structure’s fatigue life [2]. Due to their small size (of the order of several µm [3]), the use of crack growth models that accurately simulate small crack growth behaviour is necessary. However, this is a difficult task considering that such small cracks have been found to behave in a very anomalous manner [4] due to significant interaction between the crack and the material’s microstructure [5]. The idea of the Equivalent Initial Flaw Size (EIFS) can be used to avoid the above difficulties. The EIFS is not a physical quantity, but can be thought of as a model calibration parameter. One of the main advantages of the EIFS is that a long-crack growth model (e.g. Paris’ law) can be used. Many methods have been used to establish the EIFS, these include back-extrapolation [6-11], equivalent pre-crack sizes (EPS) [5, 10, 11], and the Kitagawa-Takahashi diagram [12-14]. Statistical approaches, involving the use of Bayesian updating and MLE have also been undertaken [15-17]. In the approaches detailed in [15-17], all of the uncertainty in the fatigue crack growth procedure is represented by a single noise term. By instead quantifying the individual sources of error, the EIFS distribution could be more accurately determined [18]. The method proposed in this work for determining the EIFS distribution involves the consideration of multiple sources of uncertainty, such as uncertainty in loading conditions, crack size measurement, and in the crack growth model parameters. The Dual Boundary Element Method (DBEM) is used to model fatigue crack growth, and enables the incremental growth of cracks in a simple and automatic manner with no remeshing [19]. 2. Methodology In this work, an automatic iterative crack growth procedure is used with the DBEM. The direction of growth for each increment is determined using the maximum principal stress criterion. The J-integral is used to calculate the mode-I and mode-II stress intensity factors at the new crack tip of each increment. In contrast to FEM techniques which require the remeshing of the structure as the crack grows, the use of the DBEM requires no remeshing. As the crack grows, new elements are added to 1


112

the existing crack tip, resulting in new rows and columns being added to the already existing system of equations. The DBEM is therefore very effective when used for fatigue crack growth. A comprehensive description of the crack growth procedure used in this work can be found in [19]. MLE is used in this work to determine the EIFS distribution for a structure. MLE involves determining the most likely parameters of a statistical model, given a series of observations. In this work, the parameters of interest are the statistics (mean 𝜇 and standard deviation 𝜎) of the EIFS distribution. Observations are in the form of routine inspections of an aircraft structural component amongst an aircraft fleet of the same model, it is therefore assumed that the component’s geometry will be identical between inspections and that the loading conditions will be similar. In each inspection, the size of any cracks and the number of cycles at which the inspection was carried out are recorded. A comprehensive description of the MLE procedure used in this work can be found in [20]. 3. Example – Stiffened aircraft inspection panel An example featuring a stiffened aircraft inspection panel was investigated to demonstrate the procedures detailed in section 2 for estimating the EIFS distribution. The BEM model seen in Figure 3.1 was created using 44 quadratic elements. Two stiffeners are attached near the central hole, as seen by the vertical lines with crossed markers. Cracks are assumed to most likely form in region A. The panel is under tension along its top and bottom edges. The Paris law constants 𝐶 and 𝑚, maximum stress 𝜎𝑚𝑎𝑥 , and stress ratio 𝑅 are treated as random variables.

A

Figure 3.1. BEM model of the aircraft inspection panel. Boundary nodes are shown as dots, stiffener nodes are shown as crosses. A crack is assumed to most likely form in region A. The data from 20 inspections was used in MLE. It can be seen from Table 1 that the estimates for the mean and standard deviation are very close to their actual values, showing percentage errors of only 0.3% and 3.2% respectively. From Figure 3.2 it can be seen that the estimate of the overall EIFS distribution matched the actual EIFS distribution very well.

Table 1. True values for the parameters of the EIFS distribution, and those estimated from MLE. Parameter

True

Estimated

𝜇

3.00 mm

2.99 mm

Error (%) 0.33

2


𝜎

113

0.500 mm

0.516 mm

3.2

Figure 3.2. Probability distributions for the mean and standard deviation of the EIFS distribution. The EIFS distribution is also shown. The data from 20 inspections are used. 4. Conclusions A method is proposed in this work for the statistical inference of the EIFS distribution using the BEM. The proposed method was applied to an example featuring a stiffened aircraft inspection panel. MLE was able to provide estimates for the mean and standard deviation of the EIFS distribution of 2.99 mm and 0.516 mm respectively. When compared to the actual values of 3 mm and 0.5 mm, this represents a small amount of error. Overall, results suggest that MLE is effective at estimating the statistics of an EIFS distribution in the absence of prior information, even when the data from only 20 inspections is used. Using this EIFS distribution, it is possible to estimate the fatigue life of the structure and of other similar structures under similar conditions. Acknowledgements This research was supported by a grant provided by the Engineering and Physical Sciences Research Council (EPSRC). References [1] Gallagher JP, Giessler FJ, Berens AP, Engle RM. USAF damage tolerant design handbook Guidelines for the Analysis and Design of Damage Tolerant Aicraft Structures. Wright-Patterson Air Force Base: USAF, 1984. [2] Forth SC, Everett RA, Newman JA, editors. A Novel Approach to Rotorcraft Damage Tolerance. 6th Joint FAA/DoD/NASA Aging Aircraft Conference; 2002. [3] Merati A, Eastaugh G. Determination of fatigue related discontinuity state of 7000 series of aerospace aluminum alloys. Engineering Failure Analysis. 2007;14(4):673-85. [4] Lankford J, Hudak SJ. Relevance of the small crack problem to lifetime prediction in gas turbines. International Journal of Fatigue. 1987;9(2). [5] Barter SA, Sharp PK, Holden G, Clark G. Initiation and early growth of fatigue cracks in an aerosapce aluminium alloy. Fatigue Fract Eng Mater Struct. 2001;25(2):111-25. [6] Fawaz SA. Equivalent initial flaw size testing and analysis. Air force research laboratory, Wright-Patterson AFB, 2000 Contract No.: AFRLVA-WP-TR-2000–3024. [7] Fawaz SA. Equivalent inital flaw size testing and analysis of transport aircraft skin splices. Fatigue & Fracture of Engineering Materials & Structures. 2003. [8] Johnson WS. The history, logic and uses of the Equivalent Initial Flaw Size approach to total fatigue life prediction. Procedia Engineering. 2010;2(1):47-58.

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[9] Moreira PMGP, de Matos PFP, de Castro PMST. Fatigue striation spacing and equivalent initial flaw size in Al 2024-T3 riveted specimens. Theoretical and Applied Fracture Mechanics. 2005;43(1):89-99. [10] Molent L, Sun Q, Green AJ. Characterisation of equivalent initial flaw sizes in 7050 aluminium alloy. Fatigue & Fracture of Engineering Materials & Structures. 2006;29(11):916-37. [11] White P, Molent L, Barter S. Interpreting fatigue test results using a probabilistic fracture approach. International Journal of Fatigue. 2005;27(7):752-67. [12] Liu Y, Mahadevan S. Probabilistic fatigue life prediction using an equivalent initial flaw size distribution. International Journal of Fatigue. 2009;31(3):476-87. [13] Xiang Y, Lu Z, Liu Y. Crack growth-based fatigue life prediction using an equivalent initial flaw model. Part I: Uniaxial loading. International Journal of Fatigue. 2010;32(2):341-9. [14] Lu Z, Xiang Y, Liu Y. Crack growth-based fatigue-life prediction using an equivalent initial flaw model. Part II: Multiaxial loading. International Journal of Fatigue. 2010;32(2):376-81. [15] Makeev A, Nikishkov Y, Armanios E. A concept for quantifying equivalent initial flaw size distribution in fracture mechanics based life prediction models. International Journal of Fatigue. 2007;29(1):141-5. [16] Cross R, Makeev A, Armanios E. Simultaneous uncertainty quantification of fracture mechanics based life prediction model parameters. International Journal of Fatigue. 2007;29(8):15105. [17] Sankararaman S, Ling Y, Mahadevan S. Statistical inference of equivalent initial flaw size with complicated structural geometry and multi-axial variable amplitude loading. International Journal of Fatigue. 2010;32(10):1689-700. [18] Sankararaman S, Ling Y, Shantz C, Mahadevan S. Inference of equivalent initial flaw size under multiple sources of uncertainty. International Journal of Fatigue. 2011;33(2):75-89. [19] Aliabadi MH. The Boundary Element Method: Applications in solids and structures: John Wiley and Sons; 2002. [20] Morse L, Sharif-Khodaei Z, Aliabadi MH. Statistical inference of the Equivalent Initial Flaw Size Distribution using the Boundary Element Method under multiple sources of uncertainty. Key Engineering Materials. 2018;(Submitted).

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Fundamental coupled MHD viscous flow and electric potential produced by a point force located in a conducting liquid bounded by two parallel plane solid walls A. Sellier LadHyx. Ecole polytechnique, 91128 Palaiseau Cédex, France Keywords: 3D MHD Stokes flow, Fundamental flow, Walls.

Abstract. This work deals with the steady coupled 3D fundamental MHD creeping flow and electric potential induced by a force point, with given unit strength e, in a conducting Newtonian liquid bounded by two solid, no-slip and parallel motionless plane walls. In the liquid the magnetic fied is uniform (vanishing magnetic Reynolds number) and taken to be normal to the walls which may be either perfectly conducting or insulating surfaces and not necessarily of the same natures. This paper gives the governing problem and solves it in the axisymmetric case of a unit force e normal to the walls while the more-involved case of e parallel with the walls will be addressed at the oral presentation. Introduction The dertermination of the coupled flow (velocity u and pressure p), electric potential φ and magnetic field B about a solid body moving in a conducting Newtonian liquid, with uniform conductivity σ > 0 and viscosity µ, is one tremendously-involved problem encountered in MagnetoHydrodynamics [1,2]. Indeed, one gains (B, u, p, φ) by simultaneously solving the non-linear Navier-Stokes equations with Lorentz body force fL and the Maxwell equations. Those equations are coupled by the current density j which usually adopts the widely-employed Ohm’s law j = σ(u ∧ B − ∇φ) and drives B while fL = j ∧ B. Fortunately, simplifications might occur depending upon the range of this problem dimensionless Reynolds magnetic number Rem , Reynolds number Re and Hartmann number Ha. For a liquid with uniform magnetic permeability µm > 0 and density ρf , a magnetic field with typical magnitude B > 0, a body with length scale a and a flow with p velocity typical magnitude V > 0 one has Rem = µm σV a, Re = ρV a/µ and Ha = a/d with d = ( µ/σ)/B the so-called Hartmann layer thickness [3]. In practice, Rem ≪ Re. Thus, upon neglecting inertial effects, i.e. assuming that Re ≪ 1, one gets Rem ≪ 1. In such circumstances the magnetic field is undisturbed by the body and therefore adopts in the entire liquid domain the uniform ambient value it takes far from the body. Moreover, (u, p) is a steady creeping flow driven by the non-uniform body force fL = σ(u ∧ B − ∇φ) ∧ B with B given and uniform. From the this latter property the charge conservation property ∇.j = 0 yields for φ the equation ∆φ = ∇.(u ∧ B). The associated fundamental solution (u, p, φ) produced by a force point, with unit strength e, in an unbounded liquid has been analytically obtained in [4]. As shown in [5], whenever (u, p) has axis of revolution parallel with B then ∇φ = 0 and the problem is even more tractable. This property permitted [5,6] to obtain the axisymmetric flow about a solid sphere, with radius a, translating parallel with B for Ha either small or large. The fundamental solution (u, p, φ) derived in [4] has been employed in [7] to give the axisymmetric fundamental Stokes produced by distributing force points on a circular ring normal to the magnetic field B. The material obtained in [7] made then it possible in [8] to build a new boundary formulation to accurately obtain the axisymmetric MHD creeping flow (u, p) about a solid axisymmetric body translating parallel with both B and its axis of revolution. The case of a bounded liquid domain, also encountered in applications, has attracted less attention. However, one can cite [9,10] for the axisymmetric flow of liquid bounded by a plane solid wall. Recently, [11] derived the fundamental MHD solution (u, p, φ) produced by a force point force located in a conducting liquid bounded by a plane, no-slip and motionless solid wall normal to the applied ambient uniform magnetic field B. In [11] both cases of a perfectly conducting wall and of an insulating wall have been handled. In addition, the solution obtained in [11] enabled the authors to gain in [12] the axisymmetric flow (not potential) produced by point forces located on a circular ring immersed in the liquid and parallel with the wall. The present work looks at the fundamental MHD solution (u, p, φ) produced by a force point force, with given arbitrary unit strength e, located in a conducting Newtonian liquid bounded by two solid, no-slip


116

and parallel motionless plane walls normal to the ambient uniform magnetic field B. Each wall may be either perfectly conducting or insulating. Governing fundamental MHD problem and advocated decomposition This section presents the MHD problem governing the fundamental coupled and singular viscous flow and electric potential produced by a point force located in a conducting liquid bounded by two parallel no-slip motionless walls. It also gives a suitable decomposition by which the task requires to solve another regular equivalent MHD problem. Addressed fundamental MHD problem As illustrated in Fig. 1, we consider a point force, with arbitrary unit strength e, located at point x0 in a conducting liquid domain with uniform viscosity µ and conductivity σ > 0 occupying the 0 < z < h domain D. Σh (z = h)

z σ, µ D

e • x0

B = Bez O

•

z0 > 0 Σ0 (z = 0) x

Figure 1: A concentrated force with unit strength e located at point x0 in the 0 < z < h liquid domain D bounded by two motionless, plane and parallel no-slip walls Σ0 (z = 0) and Σh (z = h > 0). In other words, D is bounded by the parallel Σ0 (z = 0) and Σh (z = h > 0) plane walls. We use Cartesian coordinates (O, x, y, z) (with origin O on Σ0 ) and designate by x0 (x0 , y0 , z0 ) the point force location with 0 < z0 < h and by x(x, y, z) a point in the liquid domain. We neglect inertial effects so that, as seen in the introduction, the magnetic Reynolds number vanishes and the magnetic field B in the liquid is uniform. Here, B is taken to be normal to the walls, i.e. B = Bez with B a prescribed constant. The point force produces not only a creeping flow, with velocity u and pressure p, but also a coupled electric potential φ. Taking for the current density f the Ohm’s law j = σ(−∇φ + u ∧ B) and enforcing the charge conservation property ∇.j = 0 then yields the following fundamental equations and far-field behaviour µ∇2 u = ∇p + σB∇φ ∧ ez − σB 2 (u ∧ ez ) ∧ ez − δ(x − x0 )e for x 6= x0 ∈ D,

(1)

∇.u = 0 and ∆φ = B∇.(u ∧ ez ) for x 6= x0 ∈ D, (u, ∇φ, p) → (0, 0, 0) far from x0

(2)

with ∆ and δ the three-dimensional Laplacian operator and Dirac delta pseudo-function, respectively. Each wall has unit normal n directed into the liquid and is no-slip, motionless and either insulating (j.n = 0) or perfectly conducting (j ∧ n = 0). Moreover, the no-slip walls are not necessarily of the same nature. Thus, we supplement (1)-(2) with the boundary conditions u = 0 and φ = 0 (conducting) or ∇φ.ez = 0 (insulating) on Σ0 ,

(3)

u = 0 and φ = 0 (conducting) or ∇φ.ez = 0 (insulating) on Σh .

(4)

In summary, we look at the fundamental MHD quantities (u, p, φ) solution to (1)-(4). As mentionned in p the introduction, a basic length for this latter problem is the Hartmann layer thickness d = ( µ/σ)/|B|. As the magnetic field B vanishes d → ∞ and φ = 0 while (u, p) becomes the fundamental pure (no body


117

force) Stokes flow nicely obtained in [13]. Because the solution derived in [13] is intricated we expect the determination of the more involved fundamental quantities (u, p, φ) to be cumbersome. Free-space analytical solution and advocated decomposition It has been possible in [4] to analytically derive the solution to (1)-(2) in absence of walls, i.e. for the unbounded liquid case. At point x the associated velocity field v and electrostatic potential ψ read v(x, x0 ) =

1 B {∇ ∧ (∇ ∧ [He])}, ψ(x, x0 ) = ∇.[H(ez ∧ e)] µ µ

(5)

with function H = H(x, x0 ), depending upon the Hartmann layer thickness d, solution to the problem 1 ∂2H = δ(x − x0 ) for x 6= x0 , H → 0 far from x0 . (6) d2 ∂z 2 The function H is available in [7] together with the flow associated pressure q. Now, the solution (u, p, φ) to (1)-(4) is written as follows ∆(∆H) −

u(x, x0 ) = v(x, x0 ) + U(x, x0 ), p(x, x0 ) = q(x, x0 ) + P (x, x0 ), φ(x, x0 ) = ψ(x, x0 ) + Φ(x, x0 ). (7) From the previous decomposition and governing equations, the regular MHD quantities (U, P, Φ) are immediately found to satisfy the auxiliary problem µ∇2 U = ∇P + σB∇φ ∧ ez − σB 2 (u ∧ ez ) ∧ ez in D,

(8)

∇.U = 0 and ∆Φ = B∇.(U ∧ ez ) in D, (u, ∇Φ, P ) → (0, 0, 0) far from x0 ,

(9)

U = −v and Φ = −φ or ∇Φ.ez = −∇φ.ez on Σ0 or Σh .

(10)

Solution for a force strength normal to the wall In this section we solve the problem (8)-(10) for the axisymmetric case e = ez . Advocated form of the auxiliary MHD viscous flow The point force is located at M0 (x0 , y0 , z0 ) with x0 = OM0 and 0 < z0 < h. For e = ez , the fundamental free-space flow (v, q) and the flow (u, p) are axisymmetric about the (M0 , ez ) axis and without swirl. Moreover, the associated electrical potential ψ and φ vanish. Accordingly, the auxiliary flow (U, Q) has no swirl and is axisymmetric with associated electric potential Φ = 0. It is then (see, for instance, [1]) expressed in terms of a single function F (x, x0 ) as ∂2F ∂2F ](x, x0 ), µUy (x) = [ ](x, x0 ), ∂x∂z ∂y∂z ∂2F ∂2F ∂ F µUz (x) = −[ 2 + ](x, x0 ), Q(x) = [ ∆F − 2 ](x, x0 ). 2 ∂x ∂y ∂z d

µUx (x) = [

(11) (12)

As the reader may easily check, the previous flow (U, Q) indeed satisfies (8)-(9) for Φ = 0 as soon as the function F obeys 1 ∂2F L(F ) := ∆(∆F ) − 2 2 = 0 for 0 < z < h. (13) d ∂z By virtue of (11)-(12), the far-field requirements (10) are ∂2F ∂2F ∂2F ∂ F → 0, + → 0, ∆F − 2 → 0 for |x − x0 | → ∞; t = x, y. 2 2 ∂x∂z ∂x ∂y ∂z d

(14)

Taking (5) for e = ez shows that the free-space velocity v and pressure q are obtained from the free-space function H(x, x0 ), solution to (6), by relations similar to (11)-(12). Therefore, the velocity boundary conditions (10) require that ∂2 ∂2 ∂2 [F + H](x, x0 ) = [ 2 + 2 ][F + H](x, x0 ) = 0 for z = 0, h; t = x, y. ∂t∂z ∂x ∂y

(15)


118

Clearly, the work for e = ez then consists in obtaining the new generating function F solution to (13)-(15). Solution in two-dimensional Fourier space Inspecting (13)-(15) shows that F (x, x0 ) = F (t1 , t2 , z, z0 ) with t1 = x − x0 and t2 = y − y0 . Hence, we use the two-dimensional Fourier tranform fˆ of a function f (t1 , t2 ) such that 1 fˆ(q) = 2π

Z

∞

−∞

Z

∞

f (t1 , t2 )eiq.t dt1 dt2 ,

−∞

c c ∂f ∂f = −iq1 fˆ, = −iq2 fˆ ∂x ∂y

(16)

where q = q1 ex + q2 ey denotes the vector in the two-dimensional Fourier space and i is the usual complex number obeying i2 = −1. Setting q = |q| and taking the Fourier transform of (13) provides for Fˆ (q, z) the equation 1 ∂ 2 Fˆ ∂ 4 Fˆ q 4 Fˆ − (2q 2 + 2 ) 2 + = 0 for 0 < z < h and q ≥ 0. (17) d ∂ z ∂z 4 Seeking solutions of (17) of the form A(q)eαz gives four different real values for α. These values are α1 < α2 < 0 < −α2 < −α1 with 1 1 1 α1 = −α0 − , α2 = −α0 + , α0 = (q 2 + 2 )1/2 2d 2d 4d α1 α2 2 2 2 2 α1 + − q = 0, α2 − − q = 0, α1 α2 = q 2 . d d

(18) (19)

Hence, the general solution Fˆ to (17) is Fˆ (q, z) = A1 (q)e−α1 z + A2 (q)eα1 z + A3 (q)e−α2 z + A4 (q)eα2 z .

(20)

The occurring functions Al (l = 1, ..., 4) are dictated by the two-dimensional Fourier transform of each veˆ locity boundary conditions (15). Designating by H(q, z) the Fourier transform of the free-space generating functionH(x, x0 ) = H(t1 , t2 , z − z0 ), one arives at the equations ˆ A1 + A2 + A3 + A4 = −H(q, 0),

(21)

ˆ A1 e + A2 e + A3 e + A4 e = −H(q, h), ˆ ∂H −α1 A1 + α1 A2 − α2 A3 + α2 A4 = −[ ](q, 0), ∂z −α1 h

α1 h

−α2 h

α2 h

−α1 A1 e−α1 h + α1 A2 eα1 h − α2 A3 e−α2 h + α2 A4 eα2 h = −[

(22) (23) ˆ ∂H ](q, 0). ∂z

(24)

The right-hand sides of (21)-(24) have been analytically obtained from [7]. Curtailing the details and recalling the definition (18) of α0 , it is found that d[α2 eα1 z0 − α1 eα2 z0 ] ˆ d[α2 eα1 (h−z0 ) − α1 eα2 (h−z0 ) ] ˆ H(q, 0) = , H(q, h) = , 8πq 2 α0 8πq 2 α0 ˆ ˆ ∂H d[eα2 z0 − eα1 z0 ] ∂ H d[eα1 (h−z0 ) − eα2 (h−z0 ) ] [ ](q, 0) = , [ ](q, h) = . ∂z 8πα0 ∂z 8πα0

(25) (26)

The problem (21)-(26) is solved, using the identity α1 α2 = q 2 , by the Maple Software. It is the found that 3

3

2

f11 α1 + f21 α2 + (f13 α1 + f23 α2 )/q Fˆ (q) = Fˆ (q) = d[ ], 8πα0 q 2 D(α1 , α2 , q, h) d ∂F ∂ Fˆ f00 q 2 + f12 α21 + f22 α22 [ ](q) = [ ](q) = d[ ], ∂z ∂z 8πα0 q 2 D(α1 , α2 , q, h) {d(p11 α1 + p21 α2 ) − f00 }q 2 + d(p13 α31 + p23 α32 ) − f12 α21 − f22 α22 ˆ Q(q) = 8πα0 dq 2 D(α1 , α2 , q, h)

(27) (28) (29)


119

with arising functions coefficients fkl and pkl analytically obtained. More precisely, D(α1 , α2 , q, h) = cosh(α1 h) cosh(α2 h) − sinh(α1 h) sinh(α2 h)[

α21 + α22 ] − 1, 2q 2

(30)

f11 (α1 , α2 , h, z, z0 ) = − cosh[α2 (z − z0 )]{1 − eα2 h cosh(α1 h)} + sinh(α2 h){cosh[α1 (h − z − z0 )] + eα1 h cosh[α1 (z − z0 )]/2} − sinh(α2 z0 ) cosh(α1 z) − sinh(α2 z) cosh(α1 z0 ) − cosh[α1 (h − z0 )] sinh[α2 (h − z)] − sinh[α2 (h − z0 )] cosh[α1 (h − z)], sinh(α1 h) cosh[α2 (z − z0 )] f13 (α1 , α2 , h, z, z0 ) = cosh[α2 (h − z − z0 )] − 2 e−α2 h

(31) (32)

and also f21 (α1 , α2 , h, z, z0 ) = −f11 (α2 , α1 , h, z, z0 ) and f23 (α1 , α2 , h, z, z0 ) = −f13 (α2 , α1 , h, z, z0 ). In addition f00 (α1 , α2 , h, z, z0 ) = cosh(α2 z0 ) cosh(α1 z) − cosh(α1 z0 ) cosh(α2 z) + sinh[α1 (z − z0 )]{1 − eα1 h cosh(α2 h)} − sinh[α2 (z − z0 )]{1 − eα2 h cosh(α1 h)} + cosh[α1 (h − z0 )] cosh[α2 (h − z)] − cosh[α2 (h − z0 )] cosh[α1 (h − z)],

(33)

f12 (α1 , α2 , h, z, z0 ) = sinh[α2 (h − z0 )] sinh[α1 (h − z)] − sinh(α2 z0 ) sinh(α1 z) − sinh(α2 h){sinh[α1 (h − z − z0 )] − eα1 h sinh[α1 (z − z0 )]}/2 − sinh(α1 h){sinh[α2 (h − z − z0 )] − eα2 h sinh[α2 (z − z0 )]}/2

(34)

and f22 (α1 , α2 , h, z, z0 ) = −f12 (α2 , α1 , h, z, z0 ). Finally, the functions pkl are defined as p11 (α1 , α2 , h, z, z0 ) = − sinh[α1 (z − z0 )]{1 − eα1 h cosh(α2 h)} − sinh(α1 h){sinh[α2 (h − z − z0 )] + eα2 h sinh[α2 (z − z0 )]}/2 + cosh[α2 (h − z0 )] cosh[α1 (h − z)] − cosh(α2 z0 ) cosh(α1 z),

(35)

p13 (α1 , α2 , h, z, z0 ) = sinh(α2 z0 ) sinh(α1 z) − sinh[α2 (h − z0 )] sinh[α1 (h − z)] + sinh(α2 h){sinh[α1 (h − z − z0 )] − eα1 h sinh[α1 (z − z0 )]}/2

(36)

and p2k (α1 , α2 , h, z, z0 ) = p1k (α2 , α1 , h, z, z0 ) for k = 1, 3. Resulting auxiliary flow The flow (U, Q) is gained from the two-dimensional Fourier transforms of Ux , Uy , Uz and Q which, by virtue of (11)-(12), are ˆx = −iq1 µU

3ˆ ˆ ˆ ∂ Fˆ ˆy = −iq2 ∂ F , µU ˆz = q 2 Fˆ , Q ˆ = ∂ F − (q 2 + 1 ) ∂ F . , µU ∂z ∂z ∂z 3 d2 ∂z

(37)

Note that Fˆ , ∂ Fˆ /∂z and Pˆ solely depend upon q = |q|. Setting ρ = {(x − x0 )2 + (y − y0 )2 }1/2 , one then easily arrives at Ux =

A x − x0 A y − y0 B C [ ], Uy = [ ], Uz = , Q= 8πµd ρ 8πµd ρ 8πµd 8πd2

with the following definitions Z ∞ [f00 q 2 + f12 α21 + f22 α2 2 ]J1 (ρ q)dq A= , α0 D(α1 , α2 , q, h) 0 Z ∞ f11 α1 + f21 α2 + (f13 α31 + f23 α32 )/q 2 B= [ ]qJ0 (ρ q)dq, α0 D(α1 , α2 , q, h) 0 Z ∞ (p11 α1 + p21 α2 − f00 )q 2 + p13 α31 + p23 α32 − f12 α21 − f22 α22 C= [ ]J0 (ρ q)dq α0 qD(α1 , α2 , q, h) 0

(38)

(39) (40) (41)

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in which J0 and J1 designate the usual Bessel functions, ρ = ρ/d, h = h/d, α0 = dα, α1 = dα1 and α2 = dα2 . Furthermore, in (39)-(41) the functions D, fkl and pkl are expressed by replacing in the previous definitions (30)-(36) the quantities h, z, z0 , q, α1 and α2 with overlined ones. Conclusions The MHD problem governing the coupled creeping flow (u, p) and electric potential φ produced by a point force located at M0 (x0 ) in a conducting liquid bounded by two parallel solid plane walls normal to the magnetic field has been presented. Its solution is analytically obtained for a force strength normal to the walls, a case for which (u, p) is axisymmetric about the (M0 , B) axis whereas φ = 0. The more tricky case of a force strength parallel with the walls require to determine coupled flow (u, p) and electric potential φ. It will be addressed at the oral presentation. [1] A. B. Tsinober MHD flow around bodies. Fluid Mechanics and its Applications (Kluwer Academic Publisher, 1970). [2] R. J. Moreau Magnetohydrodynamics, Fluid Mechanics and its applications. Kluwer Academic Publisher. (1990). [3] J. Hartmann Theory of the laminar flow of an electrically conductive liquid in a homogeneous magnetic field. Det Kgl . Danske Videnskabernes Selskab . Mathematisk-fysiske Meddelelser, XV (6), 1-28 (1937). [4] J. Priede Fundamental solutions of MHD Stokes flow arXiv: 1309.3886v1. Physics. fluid. Dynamics, (2013). [5] W. Chester The effect of a magnetic field on Stokes flow in a conducting fluid. J. Fluid Mech.”, vol 3, 304-308 (1957). [6] W. Chester The effect of a magnetic field on the flow of a conducting fluid past a body of revolution. J. Fluid Mech.”, vol 10, 459-465 (1961). [7] A. Sellier and S. H. Aydin Fundamental free-space solutions of a steady axisymmetric MHD viscous flow. European Journal of Computational Mechanics, vol 25, issue 1-2; 194-217 (2016). [8] A. Sellier and S. H. Aydin Creeping axisymmetric MHD flow about a sphere translating parallel with a uniform ambient magnetic field. MagnetoHydrodynamics, vol 53, 1; 5-11 (2017). [9] A. B. Tsinober Axisymmetric MagnetoHydrodynamic Stokes flow in a half-space. MagnetoHydrodynamics, 4, 450-461 (1973). [10] A. B. Tsinober Green’s function for axisymmetric MHD Stokes flow in a half-space. MagnetoHydrodynamics, 4, 559-562 (1973). [11] A. Sellier Fundamental MHD creeping flow bounded by a motionless plane solid. European Journal of Computational Mechanics, vol 26, issue 4; 411-429 (2017). [12] A. Sellier and S. H. Aydin Axisymmetric fundamental MHD viscous flows bounded by a solid plane wall normal to a uniform ambient magnetic field. Submitted to European Journal of Computational Mechanics. [13] R. B. Jones Spherical particle in Poiseuille flow between planar walls. The Journal of Chemical


Physics, vol 121, 483-500 (2004).

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Particle-wall interactions in axisymmetric MHD viscous flow A. Sellier 1 and S. H. Aydin2 LadHyX. Ecole polytechnique, 91128 Palaiseau Cédex, France Department of Mathematics, Karadeniz Technical University, 61080 Trabzon, Turkey 1

2

Keywords: Particle-wall interactions, MagnetoHydroDynamics, Axisymmetric MHD flow, Boundary-integral equation.

Abstract. This work examines the interactions between a solid sphere translating, normal to a close no-slip plane wall, in a conducting liquid normal in which prevails a uniform ambient magnetic field normal to the wall. Neglecting inertial effects and appealing to fundamental flows, recently obtained elsewhere by the authors, makes it possible to gain the flow about the moving sphere by a boundary approach. The task is actually reduced to the treatment of a boundary-integral equation for the surface traction arising on the sphere surface and, eventually, on the plane wall. A numerical implementation is achieved and the hydrodynamic force experienced by the translating sphere is calculated. As a consequence, the sensitivy of this force to the wall-sphere gap and to the Hartmann number Ha = a/d (comparing the sphere radius a with the so-called Hartmann layer thickness d) is investigated. For comparisons, the liquid pressure and velocity fields are also finally computed, for given sphere location and Ha, both with or without (unbounded liquid case) the wall. Introduction When a solid particle migrates in a conducting and unbounded Newtonian liquid subject to uniform ambient (i.e. far-field) magnetic and electric fields, one has to simultaneously determine in the liquid: the flow pressure p, the flow velocity u, the electrical field E′ and the magnetic field B′ . Taking the usual Ohm’s law, the current density j in the liquid reads j = σ(E′ + u ∧ B′ ) with σ > 0 the uniform liquid conductivity. As a result, (p, u, E′ , B′ ) is obtained by solving the non-linear Navier-Stokes and Maxwell equations [1,2] which are coupled through the Lorentz body force fL = j ∧ B′ and j, respectively. Such a task is tremendously involved and the solution depends upon three dimensionless numbers: the magnetic Reynolds number Rem , the Reynolds number Re and the Hartmann number Ha. For a body length scale a, a flow typical velocity magnitude V > 0 and uniform liquid density ρf , viscosity µ > 0 and magnetic permeability µm > 0 one takes Rem = µm σV a and Re = ρf V a/µ. For applications Rem ≪ Re. Thus, for negligible inertial effects Re ≪ 1 and also Rem ≪ 1 so that the magnetic field B′ adopts in the entire liquid its prescribed uniform far-field value B. Moreover, (u, p) is a creeping flow obeying the linear Stokes equations with a non-uniform body force fL . The resulting MHD p linear problem for (p, u, E′ ) still depends upon the Hartmann number Ha = a/d, with d = ( µ/σ)/|B| the so-called Hartmann layer thickness [3]. Again, it is pretty involved since there is a coupling between E′ and u through the charge conservation property ∇.j = 0 which reads ∇.E′ = −∇.(u ∧ B). However, as soon as the flow (u, p) has axis of revolution parallel with the uniform magnetic field B and no swirl the charge conservation property becomes ∇.E′ = 0. In absence of ambient electric field one arrives at E′ = 0 in the entire liquid domain [1,4]. The axisymmetric flow (u, p), without swirl, thus satisfies the Stokes equations with body force σ(u ∧ B) ∧ B and the previous motivating problem of the migration of a solid body becomes much more tractable! It has been solved for a sphere, with radius a, translating parallel with B for Ha either small [5] or large [6]. Recently, a boundary approach has been proposed in [7] to accurately compute the drag on and the flow about the sphere whatever Ha. This method appeals to two fundamental axisymmetric viscous flows, without swirls, obtained in [8]. Although bounded liquid domains are also encountered in applications quite a very few papers deal with the case of a bounded MHD flow. Within our creeping flow framework one can cite [9] for a viscous axisymmetric flow bounded by a solid plane motionless wall and with axis of revolution normal to the wall and parallel to the magnetic field. Note also that [9] actually considers either a no-slip or a slip wall and allows the flow to have a swirl. In the present work we investigate to which extent the solution obtained in [7] for a sphere translating parallel with the magnetic field B in an unbounded liquid is affected when the


124

liquid is bounded by Element a planeand solid no-slip and motionless wall normal to B. This is achieved by employing Advances in Bundary Meshless Techniques XIX (and also comparing) two different boundary approaches based on fundamental axisymmetric viscous MHD flows without swirl obtained in [8] and [10]. Governing MHD problem and adopted boundary approaches This section presents the considered MHD problem and also the two adocated boundary formulations employed to numerically solve it. Governing axisymmetric MHD problem and related issues As announced in the introduction and illustrated in Fig.1, we consider a solid sphere, with radius a and center O ′ , translating at the prescribed velocity U = U ez parallel with the magnetic field B = Bez in a conducting Newtonian liquid. The liquid, with uniform conductivity σ > 0 and viscosity µ, is bounded z

S

n

D

a

(σ, ρl , µ)

O ′•

ez

r

x

•

er z

B = Bez O

n

Σ(z = 0)

• r

Figure 1: A solid sphere immersed in a bounded conducting Newtonian liquid and translating normal to the no-slip z = 0 plane wall Σ. by the solid and motionless no-slip z = 0 plane wall Σ. Its flow is axisymmetric and without swirl with pressure field p and velocity field u. Neglecting inertial effects, the flow (u, p) then obeys µ∇2 u = ∇p − σB 2 (u ∧ ez ) ∧ ez and ∇.u = 0 in D , (u, p) → (0, 0)as

|x′ |

→ ∞, u = U ez on S, u = 0 on Σ

(1) (2)

with x′ = O′ M for a point M in the liquid, S the sphere boundary and D the bounded liquid domain. The sphere center distance to the wall is l = h + a with h > 0 the sphere-wall gap. The origin point O is located on Σ with, as shown in Fig. 1, OO′ = lez . For each point M of the domain D ∪ S ∪ Σ we note x = OM and employ cylindrical coordinates (r, z, θ) with θ ∈ [0, 2π], z = x.ez and r = {|x|2 − z 2 }1/2 ≥ 0. Therefore, x = rer + zez with local unit vector er = er (θ) shown in Fig. 1. Since the axisymmetric flow is without swirl u(x) = ur (r, z)er + uz (r, z)ez while p(x) = p(r, z). Moreover, (u, p) has stress tensor σ and exerts on the liquid boundary S ∪ Σ a surface traction f = σ.n which reads f = fr (r, z)er + fz (r, z)ez . Invoking the problem symmetries, the sphere experiences a zero torque about its center O ′ and a force F given by Z Z F=

f dS = [2π

S

fz (P )r(P )dl(P )]ez = −6πµaλU ez

(3)

C

where C is the half-circle trace of S in the θ = 0 half plane and λ > 0 the so-called drag coefficient. Note that in (3) each point P lies on C and has cylindrical coordinates r(P ), z(P ) and θ(P ) = 0. Advocated boundary representations and related boundary-integral equations As done in [7] for the unbounded liquid case, we can tackle the linear MHD axisymmetric problem (1)-(2) by using a boundary formulation. Actually, two different boundary formulations are here proposed.


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The first one, further denoted M1, appeals to the fundamental axisymmetric MHD flow produced in an unbounded liquid by distributing on the ring with radius rP > 0 and location z = zP point forces with strength Fr er + Fz ez and (Fr , Fz ) constant. This basic flow is without swirl and its pressure q(x) = q(r, z) and velocity v(x) = vr (r, z)er + vz (r, z)ez have been analytically determined in [8]. Introducing points M (r, z) and P (rP , zP ) in the half θ = 0 plane, taking indices α and β in {r, z} and adopting henceforth the usual tensor summation convention, yields vα (x) = [

1 1 ]G∞ (M, P )Fβ , q(x) = [ ]Pβ∞ (M, P )Fβ for M 6= P 8πµ αβ 8π

(4)

with so-called free-space Green velocity tensor components G∞ αβ (M, P ) and Green pressure vector components Pβ∞ (M, P ) expressed versus (z − zP , r, rP , d) in [8]. Mimicking the procedure worked out in [11] for the Ha = 0 Stokes flow case then provides for the flow (u, p), solution to (1)-(2), the following key integral representations Z 1 G∞ (M, P )fβ (P )r(P )dl(P ) for x ∈ D ∪ S ∪ Σ, uα (x) = − (5) 8πµ C∪L αβ Z 1 P ∞ (M, P )fβ (P )r(P )dl(P ) for x ∈ D (6) p(x) = − 8π C∪L β where the unbounded straight line L denotes the trace of Σ in the θ = 0 half plane. In a similar fashion, [10] recently obtained the fundamental axisymmetric MHD flow having a vanishing velocity on the z = 0 wall Σ and produced in the bounded z > 0 liquid domain by distributing on the ring with radius rP > 0 and location z = zP > 0 point forces with strength Fr er + Fz ez and (Fr , Fz ) constant. This second flow velocity and pressure read as in (4) with previous free-space components ∞ wall wall (M, P ) available in [10]. As shown in G∞ αβ and Pβ (M, P ) replaced by components Gαβ and Pβ wall wall [10], Gwall αβ (M, P ) = Gβα (P, M ). Moreover, by essence Gαβ (M, P ) = 0 for z = 0. Consequently, the counter-parts of (5)-(6) for this second boundary-approach, termed M2, are Z 1 uα (x) = − Gwall (M, P )fβ (P )r(P )dl(P ) for x ∈ D ∪ S ∪ Σ, (7) 8πµ C αβ Z 1 p(x) = − (8) P wall (M, P )fβ (P )r(P )dl(P ) for x ∈ D. 8π C β In summary, (5)-(6) and (7)-(8) are single-layer boundary integral representations of the required MHD flow (u, p) for M1 (free-space fundamental flow) and M2 (wall bounded fundamental flow), respectively. Enforcing the velocity boundary conditions (2) on S ∪ Σ now results in boundary-integral equations for the unknown traction f = fr (r, z)er + fz (r, z)ez on C ∪ L for M1 and on C for M2 (for which the condition u = 0 at z = 0 is already satisfied from (7) in virtue of the previously-noticed property Gwall αβ (M, P ) = 0 for z = 0). Denoting by δ the usual Kronecker symbol, these boundary-integral equations are Z G∞ (9) αβ (M, P )fβ (P )r(P )dl(P ) = −8πµU δαz for M ∈ C ∪ L in method M1, C∪L Z Gwall (10) αβ (M, P )fβ (P )r(P )dl(P ) = −8πµU δαz for M ∈ C in method M2. C

In summary, for M1 or M2 one first obtains the traction f solution to (9) or (10) and subsequently gains the flow (u, p) in the entire liquid domain by exploiting the relevant boundary intregral representations (5)-(6) or (7)-(8). Numerical results This section briefly describes the achieved numerical implementation and presents numerical results for the drag coefficient experienced by the sphere and the flow about the sphere.


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Numerical treatment and benchmarks As in [8], quadratic 3-node boundary elements are used to discretize the half-circle contour C and also for M1 the truncated line L (it is truncated beyond r = L). Equally-sized elements are used on C while both elements of equal or uneqal lengths have been employed in M1 for the truncated line L. Details regarding wall the accurate computation of the key components G∞ αβ and Gαβ are given in [7] and [10], respectively. For vanishing Ha one must approach the results available in the literature (see, for instance, [12]) for the pure Stokes flow case. Note that [12] makes use of bipolar coordinates with seemingly computations run in simple precision accuracy. Thus, we also performed double-precision computations in bipolar coordinates to check [12]. Comparisons are made, taking Ha = 0.01 and several sphere center normalized locations l/a, for drag coefficient computed either with M1 or M2. We put equal-sized boundary elements on C and L in M1 with line L truncated at L = 5a. It means that for N elements on C we spread N ′ ∼ LN/π elements on L. For M2 we then take again N elements on C. l/a 1.02 1.05 1.1 1.2 1.5 2 3 6

M2(20) 57.6 21.77 11.49 6.344 3.205 2.125 1.569 1.227

M1(20) 62.86 21.96 11.46 6.345 3.203 2.120 1.557 1.195

M2(40) 52.96 21.65 11.47 6.342 3.205 2.125 1.569 1.227

M1(40) 54.26 21.72 11.48 6.342 3.203 2.120 1.557 1.195

M2(60) 52.46 21.63 11.47 6.341 3.205 2.126 1.569 1.228

M1(60) 53.22 21.67 11.47 6.342 3.202 2.121 1.557 1.195

M2(80) 52.27 21.62 11.47 6.341 3.205 2.126 1.569 1.228

M1(80) 52.81 21.65 11.47 6.341 3.202 2.121 1.557 1.195

[A] 51.76 21.59 11.46 6.341 3.205 2.126 1.569 1.228

[12] 51.76 21.59 11.47 6.341 3.206 2.126 1.57 1.228

Table 1: Computed drag coefficients λ at Ha = 0.01 versus l/a and the number N of elements (given in parenthesis) spread on C (for M2 L = 5a and N ′ ∼ LN/π elements are used on the truncated line L). The values obtained at Ha = 0, using bipolar coordinates with either double precision [A] or single precision [12] accuracy level, are given for comparison purposes. As shown in Table 1, the results reveal a good convergence of both methods versus N. Not surprisingly, N is required to increase as the normalized sphere-wall gap l/a − 1 drops. It is also observed that M2 is more accurate than M1. Additional investigations are reported in Table 2 for the convergence of the computed drag versus the method and N for different Ha and a close sphere with l/a = 1.1. Again, for M1 the line L is truncated beyond L = 5a. Other values 2 ≤ L/a ≤ 5 have been tested and found to provide very close results. Ha 0.1 1 10

M2(20) 11.483 11.742 24.818

M1(20) 11.502 11.762 24.915

M2(40) 11.472 11.730 24.791

M1(40) 11.480 11.740 24.855

M2(60) 11.469 11.727 24.783

M1(60) 11.473 11.733 24.837

M2(80) 11.468 11.726 24.779

M1(80) 11.470 11.730 24.828

Table 2: Computed drag coefficients λ versus Ha, N and the employed method for a close sphere with l/a = 1.1. For M 1 the truncature length is L = 5a whatever Ha. As illustrated in Table 2, for a given value of Ha the predictions of M1 and M2 converge as N increases. Moreover, taking M2 with N = 60 is sufficient to ascertain a good accuracy level in the domain l/a ≥ 1.1 and Ha ≤ 10. Finally, M2 has been found to run faster than M1 for each calculation. This trend is illustrated for Ha = 0.01 in Table 3. Drag coefficient sensitivity to the sphere location and to the Hartmann number In view of the previous results, the drag coefficient has been computed using M2 with N = 60 for different sphere normalized location l/a ≥ 1.1 and Hartmann number Ha ≤ 10.


Method M1 M2

N =5 70 54

127

N = 10 164 70

N = 20 702 147

N = 40 1574 375

Table 3: Cpu time (in second) versus the employed method and the number N of quadratic boundary elements used on the half-circle contour C. Here Ha = 0.01. 26

λ

24 22 20 18 16 14 12 10 8 6 4 2 0

1

1.5

2

2.5

3

3.5

4

4.5

5

l/a

Figure 2: Computed drag coefficient λ versus l/a ≥ 1 for Ha = 0.01 (solid line), Ha = 0.5(◦), Ha = 1 (dashed line), Ha = 3(•), Ha = 5() and Ha = 10(). The results are displayed in Fig. 2. Not surprisingly and as for the Ha = 0 pure Stokes flow case, the drag coefficient increases slightly for a given Hartmann number as the sphere approaches the wall. Moreover, for a given sphere location increasing Ha (i.e. for a given liquid increasing the magnitude of the ambient magnetic field) results beyond Ha ∼ 1 in a large increase of the drag experienced by the translating sphere. Illustrating flow patterns As explained in a previous section, both advocated boundary approaches make also it possible to compute the flow (u, p) in the liquid domain. Here we plot in the half θ = 0 plane the isolevel contour lines of the following normalized quantities p′ = ap/(µU ), u′r = ur /U, u′z = uz /U.

(11)

This has been done, introducing the normalized variables r ′ = r/a and z ′ = z/a, for several sphere normalized location l/a and a few values of the Hartmann number Ha. For a sake of conciseness, we confine the attention to l/a = 2 and Ha = 1. We also plot, for comparisons purposes, the results obtained with the wall (present work) and also without the wall (unbounded liquid case, see also [7]). It has also been numerically checked at some points in the bounded liquid domain that M1 and M2 predict (taking the same mesh on the half-circle contour C) close values for u′r , u′z and p′ . Finally, the results presented below for the wall case are obtained using M2. As seen in Fig.3, the no-slip wall strongly affects in its vicinity both radial and axial velocity components


128

5

0.0

z′

z′

5 0.1

8

0.1

4

4 0.12

0.0

3

086 00..0

2

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-0 .1

-0.2 -0.25

6 .0 -0 4 1 0 0 . -0-0.

-0 .1 -0. 14

0 0

0.4

6

-0 .1

2

1

2

0. 4

z′

z′

5

-0 .0 6

r′

3

0.3

5

-0.02

3

-0. 1

-0.12

-0.4 -0.35

r′

0

-0.0 4

1

2

-0.01

-0.02

3 -0.

1

0.04

5

-0.2-0.25 -0.15 -0.1 -0.05

0 0

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0 0..14 12

-0.1

1 0.

05 0.

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4

0

15 0.

0. 06

0

0. 08

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0.05

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05 0.

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0 0

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75

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0 -0.

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0.7

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5

0.15

0

0.1

3

0.8

0.7

3

2

r′

3

0 0

1

2

r′

3

Figure 3: Isolevel curves of the normalized velocity components u′r (top left), u′r in absence of wall (top right), u′z (bottom left) and u′z in absence of wall (top right) for l/a = 2 and Ha = 1. u′r and u′z . The same trend are observed for the streamlines and the pressure pattern in Fig. 4. Conclusions


129

z′

5

z′

5

4

4

3

3

2

2

1

1

0 0

1

2

r′

3

1

2

r′

3

0.2

z′

5

z′

5

0 0

5

0.5

4

4 0.6

0.8 1

3

3

1.8 1.2

1.5

1 -0.

5 .2 -0

-0.5 -1

1

0 -0. 4

-0. 2

.1 -0

-2

1

2

-1. 2

2

0.1

4 0. 0.2

0

.4 -1

-3

0 0

1

2

r′

3

0 0

1

2

r′

3

Figure 4: Flow Streamlines with (top left) or without (top right) the wall and isolevel curves of the normalized pressure p′ with (bottom left) or without the wall (bottom right) for l/a = 2 and Ha = 1.

Two boundary approaches have been proposed to determine the MHD axisymmetric viscous flow about a sphere experiencing in a liquid metal bounded by a plane no-slip wall Σ a translation normal to the

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wall and parallel to a prescribed uniform ambient magnetic field. The first method, M1, appeals to two free-space fundamental axisymmetric creeping flows obtained in [7] and involves both Σ and the sphere boundary S. In contrast, the second method, M2, solely involves S. However, it makes use of two other and more complicated fundamental axisymmetric viscous flows recently derived in [10]. One then finally has to invert a boundary-integral equation governing the flow surface traction on S for M 2 and on Σ ∪ S for M1. As revealed by our computations, M2 turns out to be more accurate and less cpu time requiring that M1. Moreover, the drag experienced by the sphere is seen to deeply depend upon the sphere location and the Hartmann number Ha. Finally, as illustrated for a sphere-wall gap equal to the sphere radius and Ha = 1, the wall-sphere interactions strongly affect the MHD flow prevailing in absence of wall (i.e. in the unbounded liquid case studied in [7]). Additional flow patterns for other different values of l/a and Ha will be shown and discussed at the oral presentation. References [1] A. B. Tsinober MHD flow around bodies. Fluid Mechanics and its Applications (Kluwer Academic Publisher, 1970). [2] R. J. Moreau Magnetohydrodynamics, Fluid Mechanics and its applications. Kluwer Academic Publisher. (1990). [3] J. Hartmann Theory of the laminar flow of an electrically conductive liquid in a homogeneous magnetic field. Det Kgl . Danske Videnskabernes Selskab . Mathematisk-fysiske Meddelelser, XV (6), 1-28 (1937). [4] K. Gotoh Magnetohydrodynamic flow past a sphere. Journal of the Physical Society of Japan, 15 (1), 189-196 (1960). [5] W. Chester The effect of a magnetic field on Stokes flow in a conducting fluid. J. Fluid Mech., vol 3, 304-308 (1957). [6] W. Chester The effect of a magnetic field on the flow of a conducting fluid past a body of revolution. J. Fluid Mech., vol 10, 459-465 (1961). [7] A. Sellier and S. H. Aydin Creeping axisymmetric MHD flow about a sphere translating parallel with a uniform ambient magnetic field. MagnetoHydrodynamics, vol 53, 1; 5-11 (2017). [8] A. Sellier and S. H. Aydin Fundamental free-space solutions of a steady axisymmetric MHD viscous flow. European Journal of Computational Mechanics, vol 25, issue 1-2; 194-217 (2016). [9] A. B. Tsinober Axisymmetric magnetohydrodynamic Stokes flow in a half-space. MagnetoHydrodynamics, vol 4; 450-461 (1973). [10] A. Sellier and S. H. Aydin Axisymmetric fundamental MHD viscous flows bounded by a solid plane wall normal to a uniform ambient magnetic field. Submitted to European Journal of Computational Mechanics. [11] C. Pozrikidis Boundary integral and singularity methods for linearized viscous flow, Cambridge University Press, (1992). [12] S. L. Goren The Hydrodynamic Force Resisting the Approach of a Sphere to a Plane Wall in Slip Flow Journal of Colloid and Interface Science, vol 44, issue 2; 356-360 (1973).


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The seismic site effects study of adjacent non-curved valleys based on boundary element method 1

Zahra Khakzad1, Behrouz Gatmiri2 and Dana Amini2 Department of Structural Civil Engineering, Islamic Azad University of Roudehen, Tehran, Iran, [email protected]

2

Department of Civil Engineering, University of Tehran, Tehran, Iran, [email protected]

Keywords: Non-curved Adjacent Valleys, Site Effect, Seismic response, Hybrid Numerical Method, Topographic effects.

Abstract. A comprehensive numerical analysis on the seismic site effects due to ground irregularities is performed. Two dimensional (2D) non-curved adjacent configurations valleys under incidence of vertically propagating SV waves is modelled with HYBRID program which combining finite elements in the near field and boundary elements in the far field in various topographical conditions. This paper focuses on the modelling of non-curved adjacent valleys and influence of shapes, depth ratios and distance between the valleys on the ground amplification at various points across the valleys have been studied. In this study, the valleys are characterized by their depth (H), half width at the surface(L) and distance between two adjacent valleys (D) and the calculations are made for different depth ratios (H/L=0.2, 0.4, 0.6, 1) and distance between two adjacent valleys (D=L,2L,3L). Finally, some criteria are proposed in terms of engineering applications to assess the spectral response at the surface of non-curved adjacent valleys.

1. Introduction It has been recognized that effects of geometrical of a site can significantly affect the nature of strong ground motion during earthquakes. The modification of the seismic movement due to local topographical and geotechnical conditions is called site effect. Certainly in the recent past, there have been numerous cases of recorded motions and observed earthquake damage pointing towards geometrical and geotechnical amplification as an important effect. Thus study of site effects is one of the most important topics in earthquake engineering. Geometrical of a site modify the nature of seismic waves in transition from depth to the surface. The majority of seismic codes rest on seismic site effects by using one-dimensional (1D) model. The purpose of this paper is study of site effects in two-dimensional (2D) non-curve adjacent valleys in a building code. The 2D wave scattering is studied with a hybrid numerical method, combining finite elements in the near field and boundary elements in the far field (FEM/BEM). This program has been developed by Gatmiri and his coworkers (Gatmiri, B. & Kamalian, M, 2002; Gatmiri, B. & Arson, C. Nguyen, K.H, 2007 and 2008; Gatmiri, B. & Dehghan, K, 2005).

2. Summary of previous works Gatmiri et al have performed several parametric analyses of site effects. In order to better clarify the usage of HYBRID program, some of these studies are mentioned in the following. It should be noted, that sediments are modeled by finite elements and substratum is represented by boundary elements, which is adapted to the study in the far field. Gatmiri et al., (2007, 2008) studied various configurations and considered the influence of configuration of irregularities, slope angle of irregularities and dimensionless frequency of incident wave. The several salient features of topographic effects obtained are as follows: The seismic ground motion was amplified at the crest of ridges, at the upper corner of slopes and at the edges of canyons; it was systematically attenuated at the base of these reliefs. This conclusion was normally verified for the cases of low dimensionless frequency. The ground motion as not homogeneous as in case of the half-space, but it strongly varied on the free field. There were

132


successive regions that movements of round ere attenuated. The magnitude of response at a location on the top surface was dependent on the distance from this location on the relief. This distance was a function of the frequency content of the relief itself. The effects of topography were also influenced by the slope angle of the relief. Generally, the stiffer the slope of the relief was, the more the effects of topography due to this relief were accentuated. The topographic effects of a relief on the seismic response of that relief strongly depended on the frequency content of the excitation. In general, the higher the excited frequency was the more significant and complex were the site effects due to relief, and the wider the region influenced by the presence of the relief was, especially for the wavelengths comparable to or lower than the characteristic dimension of the relief. Gatmiri and Arson, (2008) studied several parametric analysis in order to characterize the combined effects of topographical irregularities and sedimentary filling on ground motion under seismic solicitation due to vertically incident SV Ricker wave. Indeed, the horizontal displacement in a canyon tend to be attenuated at the centre and slightly amplified at the edge but in an alluvial basin, horizontal displacements are amplified at the centre and can be locally attenuated near the edge if depth is large enough. A qualitative comparison between seismic response of the filled and empty was carried out suggesting that 2D geotechnical effects increase with depth and steep sidedness. Gatmiri et al., (2009, 2011) studied acceleration response spectra of different empty valleys. Curves were collected on a unique figure, which characterized topographical effects in a quantitative and qualitative way in the spectral domain. The results showed that maximum amplification was reached at the edge point of valleys. The spectral acceleration responses were classified according to a unique geometrical criterion except for elliptical valleys: the “S/A” ratio (where S is the area of the valley opening, and A indicates the angle between the horizontal line and slope in the above corner) (Fig. 1). The spectral response increased by increasing the parameter of S/A, in elliptical valleys for each depth ratio.

Figure 1. Definition of parameters S, A Sedimentary aspect of alluvial valleys was underlined by Gatmiri and Foroutan, (2012). New criteria were offered in order to develop simple methods to incorporate 2D combined site effects in building codes. Filling ratio effects of Non-curved alluvial valleys and the influence of the changes in impedance ratio between sediments and the bedrock were investigated. The derived conclusions are presented briefly as follows: Existence of sediments could smooth valley’s response at the edge and amplify it at the centre. When combining the depth and shape effects, two geometrical parameters S/A and sin (A) were presented; by increasing S/A, SR*sin (A) increased (S and A are similar to prior work).In order to combine filling ratio and depth ratio effects, the two geometrical parameters S1/A and H/L were considered. As increasing the S1/A, SR*H/L increased (S1, the area which was occupied by sediment, and H/L was the valley’s depth ratio) (Fig. 2). Spectral ratio had an inverse relation to impedance ratio. By sediment softening in comparison to rocky bed, the spectral ratio increased and the seismic response of a configuration became more and more complicated and the data sequencing became more and more difficult. Finally, variation S1/A n1/ß as a function of dimensionless parameter SR*sin (A)* H1/L (H1 was sediments depth) was plotted as a linear trend.


133

Figure 2. Definition of parameters S1, H/L

3. PROBLEM PARAMETERS 3.1. Geometrical parameters In order to evaluation of influence of shapes of empty non-curved adjacent valleys on the site effects the different shapes of valleys include rectangular, triangular and trapezoidal was modelled. Valleys are characterized by their depth, H and their half width at the surface, L (Fig. 3). Simulations are carried out with depth ratios, H/L, equal to 0.2, 0.4, 0.6, 1 and distance between two adjacent valleys (D=L,2L,3L).. The value of L for all of the valleys is kept equal to100m.

Y

Y X

X

H

L

H

H

L

Y

H

L

X

H

L

H

H

L

Y

H

L

X

H

H

H

L

Y

H

L

L

Y

X

X

H

L

L

Y

X

L

L

Y

X

H

H

L

H

L

L Y X

H

L

H

L

Y X

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L

H

L

Y X

H

L

H

L

Y X

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L

H

L

Figure 3. The shematic shape of valleys with different distance


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3.2. Mechanical parameters of the materials In adopted models, the rocky bed are assumed to be homogeneous linear elastic materials. The main parameters of the bedrock are given in Table 1. The data used as input in this research are the digits and numbers considered for simulation in this program and their practical application calls for assessment of the extent by which they are factual and statistical as well as their sensitivity of results to these parameters, an assessment which is beyond the scope of objectives of this study. E(MPa)

ν

ρ(kg/m³)

C(m/s)

6720

0.4

2450

1000

Table 1. Mechanical characteristics of bedrock

3.3. Incident wave characteristics The main focus of this work is the study of the effect of 2D geometrical irregularities on modification of seismic response and this study relies on simplified geometrical conditions as seismic loading is considered to be the simplest one; vertically incident SV Ricker wave. Imposed displacements are therefore expressed as;

Where Amplification A0 is constant value of 1; predominant frequency (f) is thus equal to 2 Hz; and TP = TS = 0.5s. The incident signal lasts 3s, but it can be seen from Fig. 2 that amplitude is nearly zero as soon as it reaches t = 1s. That is why the window has been defined from t=0 to t=3s (Fig. 4). It should be noted, in all the models above, vibration is applied to the base of the left valley (Fig. 5).

Figure 4. Incident Ricker signal

Figure 5. Point of the wave diffusion


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4. THE EFFECT OF AMOUNT ADJACENT ON SEISMIC BEHAVIOR OF NONCURVED VALLEYS 4.1. 2D Site effects in non-curved adjacent valleys The aim of this section is compare of influence of non-curved adjacent valleys on the seismic response of valleys with different ratios of H/L. The geometrical characteristics of valleys are displayed in Figure 1. In all of the shapes of valley, L is half of the width of the valley and is equals to 100 m for all the valleys, as well as the distance between the adjacent valleys was determined 3L and the depth of valleys are H. In the present work, simulations are carried out with a depth (H) equal to 20, 40, 60, and 100 m and for different ratios (H/L) equal to 0.2, 0.4, 0.6, and 1. According to the following graph, the results of different models show a general trend that spectral ratio is increased with increasing H/L ratio, and this increase is more evident in the inner edge of the valleys. The spectral ratio at the inner edge of rectangular valleys are more critical than spectral ratio at the inner edge of trapezoidal and triangular valleys (Fig. 6).

2.0

Spectral Ratio

1.5

1.0

0.5

H/L=0.2

H/L=0.4

H/L=0.6

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0.0 -1.5

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a) Rectangular 2.0

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c) Triangular

-0.5

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1.5

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X/L

Figure 6. The results of Different models

H/L=1 2.5

3.0

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4.2. The effect of distance between adjacent valleys on seismic response In order to evaluate the behaviour of adjacent valleys, one of the parameters to be considered is the distance between the valleys. Therefore, in this section the models are considered with different distance (D=L, D=2L, D=3L) and the results of them are presented in (Fig. 7). According to this figure, we can be said that the spectral ratio is decrease at constant depth by increasing the distance between two adjacent valleys. The amount of this reduction in the rectangular valleys is more evident and the amount of this magnification is too less in the triangular valleys.

)A(

)B(

)C(

Figure 7. The results of different models in different distance, (A)Rectangular, (B)Trapezoidal, (C)Triangular


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5. Conclusion Site effects in non-curve adjacent valleys are studied by means of a hybrid numerical technique. The main results of this study are:  Spectral ratio is increased with increasing H/L ratio.  In the constant H/L ratio, spectral ratio at the adjacent rectangular valleys is more than spectral ratio at the trapezoidal and triangular valleys.  According to the results, the spectral ratio at the inner edge of all valleys is more critical and the value of that is nearly uniform, between two adjacent valleys.  Spectral ratio is increased with decreasing D distance ratio, It means that by increasing the distance between two adjacent valleys, The magnitude of the magnification in different points decreases.  In the different distance ratio, spectral ratio at the adjacent rectangular valleys is more than spectral ratio at the trapezoidal and triangular valleys.

6. References B.Gatmiri and M. Kamalian, Two-dimensional transient wave propagation in an elastic saturated porous media by a hybrid FE/BE method, Proceedings of the fifth European conference of numerical methods in geotechnical engineering, Paris, France, 947–56 (2002). B.Gatmiri and K.Dehghan, Applying a new fast numerical method to elasto-dynamic transient kernels in HYBRID wave propagation analysis, Proceedings of the sixth conference on structural dynamics (EURODYN2005), Paris, France, p.1879–84 (2005). KV. Nguyen and B. Gatmiri, Evaluation of seismic ground motion induced by topographic irregularity. Int J Soil Dyn Earthquake Eng 27:183–8 (2007). B. Gatmiri, C. Arson and KV. Nguyen, Seismic site effects by an optimized 2D BE/FE method I. Theory, numerical optimization and application to topographical irregularities. Int J Soil Dyn Earthquake Eng 28:632–45 (2008). B. Gatmiri and C. Arson, Seismic site effects by an optimized 2D BE/FE method II. Quantification of site effects in two-dimensional sedimentary valleys. Int J Soil Dyn Earthquake Eng 28:646–61 (2008). B.Gatmiri and C.Arson and K.H.Nguyen, Seismic site effects by an optimized 2D BE/FE method I. Theory, numerical optimization and application to topographical irregularities, Int J Soil Dyn Earthquake Eng 28, 632–645 (2008). B. Gatmiri, P. Maghoul and C. Arson, Site-specific spectral response of seismic movement due to geometrical and geotechnical characteristics of sites. Int J Soil Dyn Earthquake Eng 29:51–70 (2009). B.Gatmiri and P.Maghoul, Site-specific spectral response of seismic movement due to geometrical and geotechnical characteristics of sites, Int J Soil Dyn Earthquake Eng 29, 51–70 (2009). B. Gatmiri, S. Lepense and P. Maghul, A multi-scale seismic response of two dimensional sedimentary valleys due to the combined effects of topography and geology. J Multi scale Model 03:133–49 (2011).

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B. Gatmiri and T. Foroutan, New criteria on the filling ratio and impedance ratio effects in seismic response evaluation of the partial filled alluvial valleys. Int J Soil Dyn Earthquake Eng 41:89–101 (2012). B.Gatmiri and D.Amini, Practical Recommendations of Spectral Response Analysis in NonCurved Alluvial Valleys Using Hybrid FE/BE Method, Journal of Multiscale Modelling Vol. 5, No. 2 (2013) 1350006 (2013). B.Gatmiri and D.Amini, Impact of geometrical and mechanical characteristics on the spectral response of sediment-filled valleys, Int J Soil Dyn Earthquake Eng 67, 233-250 (2014).


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Green element and Shuffled Complex Evolution methods for groundwater contaminant source identification Ednah Onyari1, Akpofure Taigbenu2 and John Ndiritu2 1

Civil & Chemical engineering, University of South Africa, P. Bag 6, Florida, South Africa. [email protected] 2 School of Civil & Environmental Engineering, University of the Witwatersrand, P Bag 3, Wits 2050, South Africa. [email protected] & john.ndiritu @wits.ac.za Keywords: Green element method, Shuffled complex evolution algorithm, groundwater source identification

Abstract. Despite extensive efforts in understanding the movement of contaminants in the subsurface, its associated inverse problem remains a challenge. Inverse problems that seek to evaluate pollution source characteristics or aquifer parameters from scanty observation data at monitoring wells present computational challenges of solution non-existence and non-uniqueness and numerical instability. In this paper a numerical method with an optimization approach are used to solve the pollution source identification problem. The Green element method (GEM) is used to solve the contaminant transport equation, while the Shuffled complex evolution (SCE) approach is utilized to search for the unknown parameters. From evaluating the computational burden of the SCE, it is found that as the number of complexes in the SCE increases the computational resources also increase without necessarily improving the accuracy of the inverse solutions. Time discretization has huge implications for both the computation effort and the inverse solutions; smaller time steps yield better solutions at a high computation effort and vice versa. The influence of the observation site on the computational resources is trivial, but very significant on the recovered source solutions.

Introduction The fate of contaminants in the environment particularly water resources has been an issue of great concern for decades. The direct movement of various contaminants in the subsurface environment has been extensively studied with mathematical equations of the phenomenon presented in various works [1,2]. Modelling and simulation of these mathematical equations result in unique solutions of the spatial and temporal distributions of the contaminant. On the other hand, the presence of a contaminant in an aquifer may be known from observations of its concentration in monitoring wells which, if reformulated to determine the pollution source characteristics, becomes an inverse problem. Its solution presents numerical challenges of numerical instability and a solution that might be non-unique and non-existent [3]. Direct approaches cannot satisfactorily solve these problems, and thus various methods have been proposed for inverse groundwater contaminant transport problems: optimization approaches, geostatistical and probabilistic, and regularization methods among others. Optimization techniques when combined with numerical or analytical solutions of the governing equations, unique solutions of the inverse problems can be obtained. Various optimization techniques that have been used in inverse groundwater contaminant transport include regression and linear programming [4],


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non-linear maximum likelihood [5] embedded optimization [6], genetic algorithms [7] heuristic harmony [8], simulated annealing [9] and shuffled complex evolution [10]. In this paper, we implement the Green element method (GEM) [11] in conjunction with the shuffled complex evolution algorithm (SCE) [12] to recover the source strengths of pollution sources. The SCE method had been extensively and successfully used in various applications, as in the calibration of various rainfall-runoff models, groundwater flow calibration, and inverse groundwater contaminant problems [10]. The SCE has been found to be computationally expensive when used to solve inverse problems due to the large number of evaluations of the fitness function, which require repeated solution of the simulation model [13]. This work seeks to evaluate the computational features of the GEM-SCE method when applied to the recovery of the release history of boundary pollution sources in a 2-D groundwater system.

Contaminant transport equation The processes of diffusion, dispersion and advection of a contaminant in a 2-D groundwater system, Ω, can be described mathematically as [1-2]

D2C  S

C  V C t

(1)

where  is the 2-D gradient operator in x and y directions, S is the retardation factor, C is the contaminant concentration, V=iu+jv is the velocity vector, t is time, D is the hydrodynamic dispersion coefficient. The solution of eq (1) is subject to the initial condition in Ω: C(x,y,t=0) = C0(x,y) and the boundary conditions: (2a) C ( x, y, t )  f1 along Γ1,

 DC.n  f 2

along Γ2

(2b)

in which n is the outward pointing normal vector on the boundary. The inverse problem addressed in this paper has an additional boundary segment Γ3 from which pollutant (whose strength is unknown) is released into the aquifer. On this boundary neither the concentration nor the flux is known. In essence the domain Ω has a boundary Γ that comprises three parts Γ1, Γ2 and Γ3 (Figure 1). To determine the release history of the unknown boundary pollution source, concentration data are available at observation wells Zj=(xj,yj), j=1,2 …, P, within the aquifer. In many real life circumstances, these observed data have measurement errors and the actual measured concentrations can be expressed as (3) C (Z j )  C (Z j ) 1    RAN (Z j )  j  1, 2, ..., P where σ is the error or noise level on the observed concentration C(Zj) and RAN are random numbers that are generated from a standard normal distribution. GEM-SCE Model The GEM numerical simulator solves eq (1) directly. It uses Green’s second identity to transform eq (1) into an integral one, employing the fundamental solution of 2G    r  ri  . The integral equation is

 C    G  C  D C( ri )    C  G ds    G  S  V C dA  0 n n     t   

(4)


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in which ri=(xi,yi) is the source point, and  is the nodal angle at the source point. Eq (4) is implemented by discretizing the computational domain into rectangular elements and interpolating the quantities C, V and P by basis functions of the Langrange family (C ≈ NjCj). The discrete element equation for each element Ωe with boundary Γe is dC j (5) Rij C j  Lij q j  Wij S  U ikj uk C j  Y ikj vk C j  0 dt Where Rij  D[  N j Gi ds   ij  ], Lij   N jGi ds, Wij  e

e

 N G dA, j

i

e

U ikj 

N

e

N j k

x

Gi dA, Yikj 

N

e

N j k

y

Gi dA

(6)

in which Gi=ln(r-ri) and q  DC / n is the normal contaminant flux. Aggregating the discrete eq (5) for all elements, and using the difference approximation: dC/dt≈[C(2)–C(1)]/∆t at the time t=t1 + θ∆t, where 0≤ θ ≤1 and ∆t is the time step between the current time t2 and the previous time t1, eq (5) becomes Wij    Eij  S t 

 (2)  Wij  (2) (1)   Eij  C (1)  C j   Lij q j   S j   Lij q j  t   

(7)

where Eij  Rij  Uikj uk  Yikj vk , ω = 1-θ, and the superscripts represent the times at which the quantities are evaluated. The global matrix equation of (7) is achieved by retaining the concentration, C, and

contaminant flux, q, at the external nodes, and expressing the latter in terms of the former at the internal nodes so that only C is calculated at the internal nodes [11].

D2C  S

C  V C t

C( x, y, t )  f1

 DC.n  f 2

Fig. 1: Schematic of the problem statement The SCE algorithm [12] is a global optimizer that combines the best features of controlled random search algorithms with the competitive evolution concept and that of complex shuffling. The SCE generates a population of possible solutions to the problem and divides this population into a number of sub-populations (complexes). Based on a statistical ‘reproduction’ process the simplex geometric shape is used to direct the


142

search. The downhill simplex method is used to evolve each complex for a set number of evolutions. The improved solutions from the complexes are then shuffled to enable the exchange of ‘good traits’ among the complexes. The ability of this technique to escape local regions of attraction in a search space is taken care of by the process of shuffling and reformulation of new complexes. The resulting complexes are evolved and shuffled repeatedly until the set criterion for terminating the search is met. The process of complex shuffling and competitive evolution embodied in the algorithm ensures that the information contained in the sample is efficiently and thoroughly exploited and does not become degenerate [14]. In this work, the SCE technique is implemented with the objective to search for a feasible set of the unknown variables that minimize deviations between the estimated and the observed concentrations of the pollutant at selected observation locations and from known boundary conditions. The optimization model therefore can be written as:

Minimize J 

2 1 Nt  Ni cal meas  C  xm , ym , tk   C  xm , ym , tk    Nt k 1  m1 

(8)

Subject to the following constraints: C ( L)  C  C (U ) and q( L)  q  q(U )

(9)

where the superscripts cal and meas refer to the calculated and the measured values, respectively, Nt is the number of time steps, and Ni is the number of nodes at which concentration values are available. The constraints in eq (9) are imposed on the lower and upper bounds of the concentration and fluxes to ensure consideration of only practically acceptable ranges. Thus, the SCE approach solves eq (8) subject to eq (9) when linked to the direct GEM solver of the contaminant transport equation. In order to exploit the SCE algorithm a number of parameters have to be determined. Suitable values are presented for the user-specified parameters as a function of the number of parameters (no) to be optimized [15]. These parameters include the number of points in each complex (mo) = (2no+1), number of points to select in complex (qo) = (no+1), number of consecutive offspring generated by each sub-complex (opt) =1 and the number of evolution steps taken by each complex (opt) =mo. The only parameter that remains to be specified is the number of complexes (po). 4. Results & Discussion A test case of pollution source identification is used to evaluate the performance of the GEM-SCE method. This test case is a 1-D contaminant transport problem with an exponentially decaying boundary source along x=0 that is continuously injected into a uniform flow field. The aquifer parameters are: D = 6km2/yr, u=6km/yr, and S =1. Initially the concentration in the aquifer is zero everywhere and while the boundary conditions are:

C ( x  0, t )  Cs expt and C ( x  25km, t )  0

(10)

The analytical solution is given by Mariño [16]:

C ( x, t ) 

Cs et  x  t  x(u   )  exp  erfc   2  2D   2 Dt

  x  t   x(u   )    exp  2 D  erfc       2 Dt 

(11)


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in which,    u 2  4 D  2 . The GEM simulation is carried out in a rectangular domain with dimension of 1

25km in the x-direction, and the y-dimension is taken large enough so that the top and bottom boundaries do not have any influence on an essentially one dimensional problem. The domain is discretized into 25 rectangular elements with the size of each element in the x-direction, Δx = 1km. The top and bottom boundaries are no-flux boundaries, while the flux and concentration are not specified along x=0 and at x=25km the concentration is zero. Two observation points, denoted as Z1 and Z2, are placed on the top and bottom boundaries at x=1km. The sensitivity of the inverse solutions with respect to the computational parameters in the SCE code, such as the number of complexes (po) and the sample size (so) is evaluated. The optimization and search termination parameters that are used for the simulations are given in Table 1. Table 1 : SCE-UA optimization and search termination parameters Description of parameter Notation Values Number of decision variables no 50 (2 unknowns and 25 timesteps) Number of complexes po Varied (2,5,8,10,15,and 20) Number of points in each complex mo 101 Sample size so po×mo Number of points to select in complex qo 51 Optimization parameter αopt 1 Optimization parameter βopt 101 Fig. 2 shows the variation of the objective function with the corresponding number of simulations for the different number of complexes. It is observed that the objective function decreases at a faster rate with fewer complexes, but the starting value of the objective function is slightly lower with higher number of complexes. From Fig. 3, the computation time and the number of simulations increases almost linearly with increase in the number of complexes. The recovered release history of the pollutant at the boundary x=0 is presented in Fig. 4 which shows equal performance for the different number of complexes. This implies that an increase in the number of complexes does not necessarily improve the inverse solution but increases the burden on computational resources.

Fig. 2: Objective function versus number of simulations for various numbers of complexes.

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Fig. 3: Computation time and number of simulations for various number of complexes.

Fig. 4: Recovered release history considering various number of complexes. The influence of time discretization on the GEM-SCE model’s performance is evaluated. For various values of time step ∆t, the variation of the mean error of the inverse concentration distribution with time is presented in Fig. 5 when the number of complexes is set at po=5. Whereas there is significant improvement in the solution accuracy with reduced time step, it is at the prize of a huge increase in the computation time. Using a time step of ∆t = 0.05, the influence of the location Z1 of the two observation wells on the numerical solution of the pollution source release is assessed. When Z1 is 1km, the source recovery is well reproduced but when Z1 is 3km, the solution oscillates about the exact solution (Figure 7). This is an agreement with our earlier finding [17] that adequate information must be available at the observation point in order to have meaningful numerical results, and this is usually the case when the observation point is located close to the pollution source.


Fig. 5: Mean error for the different time discretizations.

Fig. 6: Recovered release history considering different time discretization.

Fig. 7: Recovered release history considering different observation locations.

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146


Conclusion This paper has evaluated the capability and performance of the GEM-SCE model for pollution source recovery, taking into account its computing resource requirements. SCE, as an evolution algorithm, has proven successful for solving inverse problems, but it is notorious for its high computational burden, since the fitness function has to be evaluated repeatedly, requiring several runs of the numerical model. This work sought to evaluate the computational parameters that influence the performance of the method. It is found that with increase in the number of complexes in the SCE, which consequently increases the sample size, the search space increases resulting in the increase of the computation time, but not necessarily resulting in better prediction of the pollution source release history. The computation burden increases significantly when small time steps are used in the calculation, due to the increased number of decision variables.

References [1] J. Bear Dynamics of fluids in porous media, Elsevier Science publishing Co., New York, USA (1972). [2] R.A. Freeze, and J. A. Cherry Groundwater. Prentice-Hall, New Jersey (1979). [3] N-Z. Sun, Inverse problems in Groundwater Modelling:Theory and applications of transport in porous media. The Netherlands: Kluwer Academic publishers (1999). [4] S. M. Gorelick, B.Evans and I. Remson Water Resources Research, 19(3),779–790 (1983). [5] B. J. Wagner Journal of Hydrology, 135, 275–303 (1992). [6] P. S. Mahar, and B. Datta Journal of Water Resource Planning and Management. ASCE, 123(4), 199– 207 (1997). [7] R. M. Singh, and B. Datta Journal of Hydrologic Engineering, 11(2) 101–109, 2006. [8] M. T. Ayvaz Journal of Contaminant Hydrology, 117, 46–59 (2010). [9] O. Prakash and B. Datta Journal of Water Resource and Protection, 6, 337–350 (2014). [10] E. K. Onyari, A.E. Taigbenu and J. Ndiritu EWRI World Environmental & Water Resources Congress 22-26 May 2016, Florida, USA. 2016. [11] A.E. Taigbenu Engineering Analysis Boundary Elements, 36, 125–136 (2012). [12] Q. Duan, S. Sorooshian, and V. K.Gupta Water Resources Research , 28, 1015–1031 (1992). [13] E. Onyari Green element solutions for inverse groundwater contaminant problems, PhD Thesis, Wits, 2016. [14] Q. Duan, V. K. Gupta and S. Sorooshian Journal of Optimisation Theory Application, 76(3), 501–521 (1993). [15] Q. A. Duan, S. Sorooshian, and V. K. Gupta Journal of Hydrology, 158, 265–284 (1994). [16] M. A. Mariño Journal of the Hydraulics Division, ASCE, 100, 151–157 (1974). [17] E. K. Onyari, A. E.Taigbenu Journal of HydroInformatics , 19(1), 2017.


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Java application to solve thermoelastic contact problems using the Boundary Element Method José Vallepuga-Espinosa1, Lidia Sánchez-González2 and Iván Ubero-Martínez1 1

Universidad de León, Departamento de Tecnología Minera, Topográfica y de Estructuras, Campus de Vegazana s/n, 24071, León, Spain ([email protected], [email protected]) 2

Universidad de León, Departamento de Ingenierías Mecánica, Informática y Aeroespacial, Campus de Vegazana s/n, 24071, León, Spain ([email protected])

Keywords: 3D Thermoelastic Contact Problem, Boundary Element Method, Object Oriented Programming, Human Computer Interaction

Abstract In this paper we propose a user friendly application (BEMAPEC) that solves 3D thermoelastic contact problems besides elastic and heat transfer problems. The programmed algorithm is based on the Boundary Element Method. First, we define the geometry of the solids, then solids are meshed being possible to use different ratios and divisions. Boundary and contact conditions are specified by clicking on the graphics and choosing a solid. The problem is solved by using an iterative procedure being possible to save the partial results as well as re-mesh certain areas to guarantee convergence. Finally, results are showed on the solids using different color scales allowing us to choose only certain surfaces of any solid. Results can be exported to a csv file. This program is an alternative to other Finite Element Method software and unique among the existing programs.

Introduction The Boundary Element Method (BEM) has been used in several applications in different scientific fields, for example in thermoelasticity such as the dynamic coupled thermoelastic problems under thermal shock loading [1], thermo-elastoplastic problems of thick graded plates [2]. Other authors used the BEM to solve either thermoelastic [3] contact problems in 2D and elastic [4] and thermoelastic contact problems in 3D [5]. The iterative method using the BEM developed by Vallepuga and Foces [5] to solve 3D thermoelastic contact problem is implemented in BEMAPEC. Imperfect contact problems with no constant thermal resistance can also be solved by means of BEMAPEC. Libraries to solve linear equation systems related to BEM can be found but the entire contact problem cannot be solved. For this reason, in this work it is presented BEMAPEC (Boundary Element Method Applied to Problems of thermal, elastic and thermoElastic contact) which is a java application that solves thermoelastic contact problems between three dimensional solids by means of the BEM. Elastic and heat transfer problems are also solved. BEMAPEC can be run in any operating system, so, it is the unique multi-platform application that solves 3D thermoelastic contact problem with a variable resistance and it is an alternative to other Finite Element Method software. This work is structured as follows: First of all, it is described briefly the iterative method proposed by Vallepuga and Foces [5] to solve 3D thermoelastic contact problems. Afterwards, an application herein developed is presented by solving an application example. Finally, the conclusions and future works are detailed.

3D thermoelastic contact problem Two elastic bodies A and B defined by their boundaries ( ) are in contact and boundary conditions are imposed. This problem is solved applying the boundary integral eq (1) and eq (4) that is obtained applying the Somigliana’s identity to any point of the boundary of each solid. The resulting linear equations are coupled imposing the contact conditions [6]. The equation becomes as follows for potential problems:


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(1) the free term of the integral equation and the fundamental solutions for the potential problem:

Being

(2) (3) For the 3D thermoelastic problem, considering small deformations due to static loads and stationary conduction thermal, the integral equations are as follows:

(4) Where

is the free term of the elastic problem,

and

are the Kelvin singular

solutions. (5) (6) Moreover, and to the behavior elastic law.

are the vectors derived from the consideration of thermal deformations

(7) (8) is the Kronecker´s delta [7], are the normalized eigenvectors of the vector that links the application point of the load with the considered point is the component of the normal vector to the boundary and are the local axis directions. In order to apply these equations, both boundaries are discretized in with planar and constant triangular elements and subdivided in two main zones: initial contact zone ( ) common to both solids and a free zone for each solid ( , ). After doing the integrations over the elements and imposing the boundary and contact conditions, eq (1) and eq (4) become into two linear systems of equations as follows: Where



Thermal problem:

Where is a square matrix, independent terms vector. 

Elastic problem:

is the vector (

thermal unknowns and

is the


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Where is a square matrix, is the vector ( thermal unknowns and is the independent terms vector and is a vector whose components are the results of the integrals that appear in the second member of equation (4). The procedure proposed by Vallepuga and Foces [5] solves the system of equations by adding elastic and thermal problems as it is shown in Fig. 1.

Fig. 1: Schema of the problem

Application example The example solved by means of BEMAPEC corresponds to the elastic punch over an elastic foundation, see Fig. 2, solved by Foces [4] but including thermal boundary conditions. Mechanical and thermal properties are gathered in Table 1. Table 1: Mechanical and thermal properties

E [GPa] ν





Elastic punch Elastic foundation 210 210 0.3 0.3

Elastic boundary conditions o The perpendicular displacements are avoided ( . o On the rest of the faces . Thermal boundary conditions o The temperature is constant at and respectively.

and at

to the faces

with value


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o

On the rest of the faces,

The heat flows from the elastic punch to the elastic foundation that has less distorsionability, so, an imperfect contact problem appears. Then, the correlation used proposed by [8], the value of the thermal resistance is obtained. (9) This problem is firstly solved for a thermal resistance equal to zero ( thermal resistance given by contact surface roughness with and .

. Then, it is solved for a parameters: ,

Fig. 2: Problem geometry

In order to solve the problem by means of BEMAPEC, it is needed to start introducing the geometry of the solids of the problem, which is defined by means of their surfaces. Thus, the coordinate points that shape each surface. The geometry of the problem is plotted in BEMAPEC, so it can be easily check as it is shown in Fig. 3. Once the geometry has been defined, the material properties (Table 1) of each solid and the type of contact can be easily introduced in BEMAPEC.

Fig. 3: Geometry of the problem in BEMAPEC


151

After that, meshing is done by planar triangular elements whose node is located in the barycentre as it is shown in Fig. 4. The mesh is created by entering the number of divisions and the ratio in each axis. Moreover, BEMAPEC also plots the normal vectors to the surface.

Fig. 4: Problem mesh

Finally, the boundary conditions described previously are introduced in BEMAPEC. These boundary conditions can be introduced by surfaces of by elements. The boundary conditions to be introduced can be divided in elastic boundary conditions (displacements and tractions) and thermal boundary conditions (temperatures, force convection and temperature gradients).

Fig. 5: Elastic boundary conditions

152


Fig. 6: Thermal boundary conditions

The problem is solved obtaining elastic and thermal results. The results are displayed using an OpenGL environment (Fig. 7) and they can be obtained in a tabular output or save them in a csv file. Moreover, they can be plotted using a grey or a color scale respectively and they can also be interpolated. Furthermore, a surface of the solid can be selected and zoom can be applied on it in order to analyse the results in more detail.

Fig. 7: Temperature distribution

The temperature distribution along the z axis is shown in Fig. 7. For thermal resistance and , the temperature distribution is lineal along the z axis but for the case of , a small disturbance appears in the contact zone due to the increase of the thermal resistance. This small disturbance can be seen in following Fig. 8, in which the temperature distribution in the contact zone between both solids is shown. The temperature values are higher for than for , since the thermal resistance value is higher. For , the higher values of temperature are located where the thermal resistance values are higher (Fig. 10).


153

Fig. 8: Temperature distribution at the contact zone

As the resistance increases, the temperature variations in the contact zone are smaller, since the traction distribution in the mentioned contact zone is more uniform.

Fig. 9: Traction distribution at the contact zone

Fig. 9 shows the traction distribution in the contact zone. In which can be seen that the traction values increase as we move away from the centre producing a punching effect at the edges. In addition, the tractions values are higher for than for , since the thermal resistance value is higher. Finally, Fig. 10 shows the thermal resistance values at the contact zone for the solved cases.

Fig. 10: Thermal resistance at the contact zone

Conclusions In this work it is presented BEMAPEC, which is a Java program to solve elastic, thermal and thermoelastic contact problems in three-dimensions. BEMAPEC solves an entire contact problem between 3D solids unlike the other existing programs like BEASY. Moreover, BEMAPEC also solves thermoelastic imperfect contact problems with a thermal resistance in function of the contact tractions. These reasons make BEMAPEC an alternative to other FEM software programs. The algorithm developed by Vallepuga [5] was developed in Fortran, leading to a very arduous process of data definition and result interpretation. Thus, BEMAPEC offers several advantages. Firstly, the fact of

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having developed it in an object-oriented programming language facilitates the maintenance of the program and its possible extensions in the future, allowing the modular addition of new functionalities such as interaction with other applications in fortran or another language. Secondly, the fact of having a friendly graphical interface offers the advantage of easily defining solids as well as their properties and the characteristics of the problem. Moreover, once the problem is solved, the application allows the treatment of the results, both in tabular and graphical form by means of the gray scale or color display of the variations of tractions, movements, thermal resistances, temperatures and thermal gradients along the surfaces of 3D solids. Thus, for a given problem, the user can analyze the output produced by the program by visually checking the behavior of the solids.

Future works Future works are focused in two lines of work: - Incorporation of physical phenomena to those already treated in this work. - Improvement of the algorithms and techniques used in the resolution of physical problems. In the first line, the following can be cited: Consider the effects of friction on the interface of both bodies (incremental procedure). - The effect of heat transfer by radiation on the contact interface can also be included. The inclusion of radiation conditions introduces a new non-linearity in the problem since the equations that govern this type of heat transmission depend on the 4th power of the temperature. A method, such as that of Newton-Rhapson, should be used to solve the resulting non-linear system of equations. - The type of thermal loads can be extended, such as point, linear or evenly distributed thermal bulbs without excessive difficulty. - The method of analysis could be extended to the study of contact problems between several solids. Regarding the second possible line of research, it could be included a library of elements different from the triangular constant of a node, such as, for example, non-continuous quadrilateral elements with a greater order of approximation.

References [1] X.-W. Gao, B.-J. Zheng, K. Yang, and C. Zhang. Radial integration BEM for dynamic coupled thermoelastic analysis under thermal shock loading. Computers & Structures, 158:140-147, oct 2015. [2] R. Vaghefi, M. Hematiyan, and A. Nayebi. Three-dimensional thermos-elastoplastic analysis of thick functionally graded plates using the meshless local PetrovGarlekin method. Engineering Analysis with Boundary Elements, 71:34-49, oct 2016. [3] P. Alonso and J. Garrido García. BEM applied to 2D thermoelastic contact problems including conduction and forced convection in interstitial zones. Engineering Analysis with Boundary Elements, 15(3):249259, jan 1995. [4] Paris, F., Foces, A., Garrido, J. Application of boundary element method to solve three dimensional elastic contact problems without friction. Computers and Structures 43, 19–30 (1992). [5] J. Vallepuga Espinosa and A. Foces Mediavilla. Boundary element method applied to three dimensional thermoelastic contact. Engineering Analysis with Boundary Elements, 36(6):928-933, jun 2012. [6] C. A. Brebbia, J. C. F. Telles, and L. C. Wrobel. Boundary Element Techniques. In Boundary Element Techniques, pages 177-236. Springer Berlin Heidelberg, Berlin, Heidelberg, 1984.


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[7] F. Hartmann. Computing the C-matrix in non-smooth boundary points. New developments in boundary element methods, pages 367-379, 1980. [8] M. Cooper, B. Mikic, and M. Yovanovich. Thermal contact conductance. Journal Heat Mass Transfer, 12:279-300, 1969.

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Advances in Boundary Elements & Meshless Techniques XIX Advances in Bundary Element and Meshless Techniques XIX

157 157

Methodology for automatic integration of nearly and weakly singular integrals arising in the boundary element method J. D. R. Bordón1, J. J. Aznárez1 and O. Maeso1

1

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: {jacobdavid.rodriguezbordon,juanjose.aznarez,orlando.maeso}@ulpgc.es

Keywords: adaptive integration, nearly singular integrals, weakly singular integrals, boundary element method

Abstract. This paper presents a simple and efficient methodology for the automatic integration of nearly and weakly singular integrals. Its main feature is the use of formulas for estimating the required quadrature order which take into account the desired error, the main components of the fundamental solution, the element geometry and order, and the collocation point position. For distantly and nearly singular integrals, it uses a set of formulas depending on the desired error, singularity order and collocation point position. For weakly singular integrals, it uses formulas obtained from error analysis of the integrand. Its efficacy is shown through several examples.

Introduction The implementation of codes using the boundary element method requires effective and efficient evaluation of the integrals required by the method. Preferably, these should be evaluated analytically, but this can only be done in certain cases, usually for planar and low-order elements, and simple fundamental solutions. However, boundary element integrals are generally evaluated by numerical integration due to its versatility. Before performing the numerical integration, some analytical transformation (subdivision, change of variables, integration by parts) may be applied in order facilitate the computation, or even being able to evaluate the integral at all. The number of developed mathematical and computational techniques for handling boundary element integrals is quite large, see e.g. [1], and these focus mainly on two of the most problematic ones: nearly (or quasi) singular integrals and singular integrals, which can contain a weak, strong or hyper singularity. Strongly singular and hypersingular integrals can be reduced to weakly singular integrals, other non-singular (possibly nearly singular) integrals and analytical terms via regularization procedures. For the most common problems and fundamental solutions there are regularization procedures available, see e.g. [2–4]. Thus, the present methodology only considers up to weakly singular integrals. The aim of this work is to present the mentioned methodology and its ingredients, where, despite most of the ingredients are known, some contributions in the form of reliable a priori selection of quadrature rules for nearly and weakly singular integrals are presented. Methodology The methodology is comprised of four steps: pre-processing, collocation point projection distance and position on the element, distant or nearly singular integration if the collocation point does not lie on the element, and singular integration if the collocation point lies on the element. Pre-processing. At this stage, all elements in a given mesh are studied in order to verify their validity, and several auxiliary calculations are performed for later needs. The first auxiliary calculation consists on building a bounding ball with center c and radius R for each element. Then, characteristic lengths Lc (maximum element edge length) are calculated, although it is also

158 Advances in Bundary Element and Meshless Techniques XIX


158

possible to conservatively take Lc =2 R . Finally, the required quadrature ( N ϕ J ) for integrating the shape functions over the element ( ∫Γ ϕd Γ ) with a prescribed relative error ϵ is determined. Collocation point projection distance and position on the element. At processing (building) and post-processing (internal point calculation for example) stages, boundary integrals have to be evaluated for a given collocation point x i and element Γ . The first step in this calculation is determining where and how far is the singularity with respect to the element. The accuracy of the calculation becomes less important as the collocation point moves away from the element. This is due to the fact that the integrand becomes smoother from the integration domain point of view, and no treatment of the integrand singularity is required. On the other hand, the accuracy is central for nearly singular integrals, as their treatment requires the projection distance and position in local coordinates of the nearest element point to the collocation point. For weakly singular integrals, obviously only the position of the collocation point in local coordinates is needed. Hence, four different algorithms are considered: 1) Bounding ball. In order to cheaply identify cases of distant collocation point – element configurations where only a crude approximation of the distance is needed, the element bounding ball is considered. If ‖c−xi‖> γ R , then the dimensionless distance d is simply d =‖c−xi‖/ Lc , otherwise a more accurate algorithm is required. Note that γ R establishes the distance beyond which a rough estimate of d is enough. A value of γ =4 is taken on the basis that N v ( d )≈ N c (d ) and the variation of N v ( d ) is small for d > 2 (see next section). 2) Distance minimization via element nodes sampling. The minimum distance and the local coordinates of the nearest element point between collocation point and element is taken by using element nodes coordinates. Thus, d =‖x(k)−x i‖/ L c and ¯ξ i =ξ (k) , where k is the node index giving minimum d . If the resulting d 2 , the distance minimization based only on element nodes sampling is used for d ≤2 , which is used as the starting point of a gradient-based minimization algorithm if d 0 ). These integrals are regular and thus can be evaluated via numerical integration using conventional Gauss-Legendre (GL) quadrature. The required quadrature order vary widely depending on element order, collocation point – element relative position and the actual type fundamental solution. General-purpose integration routines using a posteriori error estimates [5] can be used, but they are too inefficient to be massively used. Therefore, ad hoc strategies have to be considered for them, where the singularity ln r or 1/r n is considered dominant for small d →0 and the rest of the integrand becomes dominant as d →∞ . The case of d →0 correspond to the so-called nearly or quasi- singular integrals, and it is probably the most problematic. Since the pioneering work of Lachat and Watson [6], a priori error estimates have been used to select the appropriate quadrature rule, i.e. adaptive quadrature order, and/or subdivide the integration domain, i.e. adaptive subdivision. Another relevant work using adaptive quadrature order with subdivision is that of Jun, Beer and Meek [7]. In both cases, error analysis is used in order to provide N ( d ) curves as a function of singularity order and desired relative error, although these are very conservative [9]. The landmark work of Telles [8] solves the


159


159

problem from a different perspective, and, instead of using a divide and conquer technique, it seeks for a coordinate transformation which smooths out the singularity. Telles’ transformation greatly reduces the required quadrature order, especially for d 10−6 and 2≤N ≤30 , which is the range explored by the numerical experiments. Denoting ¯ξ i as the nearest point of the element to the collocation point, the required quadrature rule as a function of the location of the collocation point, the desired relative error and the singularity order is: i i 2 (3) N qs (d , ξ¯ , ϵ , n)=nint {[ ξ¯ ] [ N v (d )− N c ( d )]+ N v (d )} (line) i i i 2 ¯ (4) N qs (d , ξ , ϵ , n)=nint {[ ¯ξ1 ¯ξ2 ] [ N v (d )−N c (d )]+ N v (d )} (quadrilateral) i i i i i i N qs (d , ξ¯ , ϵ , n)=nint {4[ ξ¯ 1( ¯ ξ1−1)+ ¯ξ 2 ( ¯ξ1 + ¯ξ 2−1)][ N v (d )− N c ( d )]+ N v (d )} (triangular) (5) These final equations interpolate N v (d ) and N c (d ) curves which are assigned to vertices and centers of elements in order to take into account the location of the collocation point projection. Despite that N v (d ) and N c (d ) have been obtained for a line element, they are conservative for quadrilateral and triangular elements. In order to treat cases with small d requiring N qs >30 , a simple recursive uniform subdivision of the integration domain is performed. This classical strategy is robust and powerful, but computationally very expensive for arbitrarily small d , where other techniques are more appropriate. The case of d →∞ (say d > 2 ) corresponds to what it is here called distant singular integrals, and it is straightforward to tackle. In these cases, the singularity is smooth, and the products of shape functions and Jacobians become dominant. Since these depend only on the element itself (geometry and field variable interpolation), it is possible to determine the required quadrature at a pre-processing stage by using the integrand with singular terms removed. At that stage, a simple adaptive integration with a posteriori error estimation gives the required N ϕ J for a prescribed relative error. The transition between distant and nearly singular dominance can not be determined beforehand, thus the final quadrature to be used is simply: N =max {N qs , N ϕ J } (6) Func. r

−n

Ch. Coef. path Nc

c 00

c 10

c 01

c 11

c 20

c 02

p0

2.44394E-1 -7.84513E-2 3.89533E-2 8.78312E-4 -1.71917E-3 -1.57591E-3

p1

-8.42122E-1 -8.77826E-2 1.67950E-2 6.52270E-4 -2.64981E-3 -8.47201E-4



Nv

160

q1

-2.69303E-2 -6.13831E-2 2.27392E-2 9.78975E-4 -2.03038E-3 -9.05929E-4

p0

1.85587E-1 -8.44178E-2 3.99252E-2 1.00303E-3 -2.01246E-3 -1.59668E-3

p1

-8.01290E-1 -7.06056E-2 5.71330E-3 -1.95225E-4 -1.98446E-3 -6.68082E-4

q1

-2.35352E-1 -5.14186E-2 1.69310E-2 -1.49138E-5 -1.33037E-3 -1.26470E-3

Table 1: Coefficients for estimating N ( d ) curves when using plain GL quadrature Func.

Ch. Coef. path Nc

r

−n

Nv

c 00

c 10

c 01

c 11

c 20

c 02

p0

2.93790E-1 -7.65184E-2 3.00113E-2 5.89008E-4 -1.76427E-3 -1.36342E-3

p1

-9.83391E-1 -8.67614E-2 2.60288E-2 7.76775E-4 -2.36193E-3 -1.28914E-3

q1

-4.54535E-1 -7.39415E-2 3.76722E-2 9.82768E-4 -1.88978E-3 -1.80796E-3

p0

1.11552E-1 -9.00432E-2 3.96056E-2 1.06697E-3 -2.24281E-3 -1.60303E-3

p1

-6.58563E-1 -4.73546E-2 1.94236E-3 -2.90874E-4 -1.18598E-3 -4.59887E-4

q1

-2.66100E-1 -3.60484E-2 1.32881E-2 -1.80992E-4 -7.60155E-4 -1.22473E-3

Table 2: Coefficients for estimating N ( d ) curves when using GL quadrature with Telles’ transformation. Weakly singular integrals ( d =0 ). Despite containing a singularity, weakly singular integrals can be understood in the Riemann sense, i.e. the integrand is absolutely integrable. If no treatment of the weak singularity is performed, then a prohibitive quadrature order is required in order to achieve a reasonable low error. For weakly singular integrals over line elements (singularity ln r ), there are many approaches including the use of the Telles’ transformation. In the present methodology, the use of the GaussAnderson quadrature [10] is considered because it includes ln r as weighting function, i.e. it perfectly deals with the weak singularity. The only drawback is the need for the expansion of r in terms of the local coordinate. Since in the pre-processing stage N ϕ J has been obtained for each element, this is precisely the required Gauss-Anderson quadrature. For weakly singular integrals over surface elements (singularity 1/r ), many approaches have been devised, from which the classical quadrilateral-to-triangle degenerated mapping and the use of polar coordinates properly deals with the problem. In both cases, the element is subdivided into triangular regions with vertices at the element vertices and at the collocation point. They suffer from two serious defects which affect their efficiency: dependency on the location of the collocation point and element shape (aspect ratio and skewness). The first defect is analogous to a near singularity but along the angular coordinate, and it is treated with similar techniques. The second defect is also a near strong singularity, which appears when tangent vectors along local coordinates are neither equal nor orthogonal at the collocation point, i.e. not conformal. Rong et al. [11] proposed a polar transformation which imposes exactly this condition for triangular elements. They use a sigmoidal transformation in order to treat the near strong singularity. However, there is an exact transformation which cancels out this strong singularity, and it was proposed by Khayat et al. [12]. In the present work, we use Rong’s idea but generalized also for quadrilateral elements, and Khayat’s transformation for exactly removing the near strong singularity. The resulting methodology is free from any source of near singularities, and leads to a scheme with manageable influence of collocation point position and element shape. This allows to propose formulas for a priori selection of quadrature rules.


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161

Elements are subdivided at the collocation point into triangular regions with right angles at edges, see Fig. 1 and Fig. 2, allowing the use of different quadrature rules along angular and radial coordinates for each subdivision.

Figure 1: Triangular subdivision pattern of triangular elements.

Figure 2: Triangular subdivision pattern of quadrilateral elements.

The complete scheme is depicted in Fig. 3. The contribution of each subdivision d is obtained after the following transformations: 1) Transformation from the element reference space to the real space T g :ξ →x : N nodes

x (ξ )= ∑ ϕ(k ) (ξ ) x(k) k =1

‖ξ

(7)

‖

∂x ∂x × (8) ∂ 1 ∂ ξ2 2) Transformation from subdivision reference space to element reference space T S d : ζ →ξ : i Sd (9) ξ=ξ +S ζ (1)−Sd i (2)−Sd i ξ − ξ ξ1 − ξ1 Sd (10) S = 1(1)−S d 1i (2)−Sd ξ2 − ξ2 ξ 2 −ξi2 J=

(

)

d d (11) J Sd =( ξ(1)−S −ξ i1)( ξ(2)−S − ξi2 )−( ξ(2)−Sd −ξi1 )( ξ(1)−Sd −ξ i2) 1 2 1 2 3) Transformation from subdivision space to subdivision reference space T C d : η→ζ : (12) ζ =C Sd (η−ηi −S d ) i−Sd i−Sd −1 ( η1 −1)/ η2 CS d = (13) 1 −η1i−Sd / ηi−Sd 2

(

)

i−S d (14) J Cd =1/ η2 i−S d , is such that conformality of where the location of the collocation point in the η space, i.e. η the subsequent polar coordinates is present in the real space. For each subdivision d , it is necessary to calculate ηi−S d that fulfill the conformality conditions at the collocation point: η η η η η η Condition I (equal length) T1 · T1 =T2 ·T 2 , and Condition II (orthogonality) T1 · T2 =0 , where: ∂ x i ∂ x i ∂ ξ ∂ζ Sd (15) Tηj = ∂ η = ∂ ξ ∂ ζ l ∂ ηk =Tξl S Sd lk C kj j l k j and where j , k ,l=1,2 and Einstein summation convention is implied. The solution is: ξ ξ ξ ξ a l b k (Tl · Tk ) i −S d b l b k (Tl · Tk ) i−Sd i−Sd 2 (16) η1 =− = ) ξ ξ , η2 ξ ξ −( η1 al a k (Tl · Tk ) a l a k (Tl ·T k ) (1)−Sd i where a j =ξ(2)−Sd and b j =ξ(1)−Sd −ξ j −ξ j , and the notation is such that it expands as j j a l b k (Tlξ · Tξk )=a1 b1 (T1ξ · Tξ1 )+ a 1 b 2 (Tξ1 · T2ξ)+a 2 b1 (T2ξ · Tξ1 )+ a 2 b 2 (Tξ2 · Tξ2) . etcetera. 4) Transformation from polar coordinates space to subdivision space T P d :( ρ , θ )→η :

( ) ( )

√


162 Advances in Bundary Element and Meshless Techniques XIX i −S d 1

162

i−Sd 2

(17) η1 =η + ρ · cos θ , η2 =η + ρ · sin θ J P d =ρ (18) Sd (1 )−Sd (2)−S d where ( ρ , θ )∈[0 , ρ ,θ ]: ¯ ]×[ θ i−S d η2 Sd (19) ¯ρ =− sin θ η1i−Sd 1− ηi−Sd (1 )−Sd (2)−Sd 1 θ = π +arccos =2 π −arccos , θ (20) i−Sd 2 i−S d 2 i−Sd 2 d 2 ) √(η1 ) +( η2 ) √(1− η1 ) +( ηi−S 2 ~ 5) Transformation from altered polar coordinates to polar coordinates T A d :( ρ , θ )→(ρ , θ ) : ~ θ =π + 2 arctan exp iθ−S d (21) η2 sin θ J A d =− i −S d (22) η2 ~ ~ ~ where θ ∈[ θ (1 )−S d , θ (2)−Sd ] : (1)−Sd ~(1)−Sd i−Sd −π , ~ θ(2)−S d − π θ =η2 ln tan θ θ (2)−S d = ηi−Sd ln tan (23) 2 2 2 ~ 6) Transformation from the unit square to altered polar coordinates T UA d :( ρ ' , θ ' )→( ρ , θ ) :

}(

{

){ }

ρ 0 ρS d 0 ¯ ρ' = + (24) ~ ~ (1)−S d ~ Sd θ' θ θ 0 Δθ d ηi−S S d ~Sd 2 J UA d =¯ρ Δ θ =− Δ~ θ Sd (25) sin θ ~ ~ ~ where Δ θ Sd = θ (2)−S d − θ (1)−S d . A GL product rule of N ρ ' × N θ ' points is used in the unit square ( ρ ' , θ ' )∈[0,1]×[0,1] . The numerical integration of a weakly singular integrand f is becomes:

{}

NS

I =∫Γ f d Γ= ∑

N ρ'

N θ'

∑∑

f (ρ '

(k ρ ' )

,θ'

(k ρ' )

) J J Sd J C d J P d J A d J UA d w

(k ρ' )

d =1 k ρ ' =1 k θ' =1 (k ρ ' )

w

(k θ' )

(26)

(k ) where ρ ' and w are the quadrature points and weights along coordinate ρ ' , and θ ' (k ) and w(k ) are the quadrature points and weights along coordinate θ ' . The resulting product of jacobians (except J ) is constant for each triangular subdivision and proportional to ρ : (1)−Sd i (2)−S d i (2 )−Sd i (1)−Sd i (ξ 1 −ξ1 )( ξ 2 − ξ2 )−( ξ1 −ξ1 )( ξ2 −ξ2 ) ~S d J Sd J Cd J P d J A d J UA d =ρ Δθ (27) d ηi−S 2 which ensure no near singularity as the collocation point approaches the edges. Furthermore, thanks to the imposed conformality conditions at the collocation point: r =ρ √ c 0 +c 1 ( θ ) ρ +O( ρ2) (28) where c 0 is a constant (see Eq. (33)), which guarantees a local shape independent cancellation of the weak singularity ( r →ρ √ c0 as r →0 ). Since this scheme produces an exact mapping from polar coordinates to the real space as r →0 , and, as a consequence, also a good mapping in the surroundings of the collocation point, it is possible to find under certain assumptions error estimators for angular and radial coordinates. The integral form considered for error estimators is the following: 1 1 ϕ I =∫ ∫ J J Sd J Cd J P d J A d J UA d d ρ ' d θ ' (29) 0 0 r The shape function term ϕ represents the leading monomial, which after transformation leads to: ~(1 )−S d +Δ ~ θS d θ ' d 2p 2p p θ ϕ∼K ϕ( ρ cos θ )p ( ρ sin θ ) p= K ϕ ( ηi−S ( ρ ' ) sinh (30) ) 2 ηi2−S d ρ'

θ'

θ'

(

)


163


163

where for linear elements p=1 , for quadratic elements p=2 , etcetera. The Jacobian J is the square root of a polynomial for curved elements, but reduces to a simple polynomial for planar elements. For moderately curved elements, J can be considered as: ~(1 )−Sd ~S d 2( p+q ) +Δ θ θ ' 2( p +q) p+q θ J ∼K J ( ρ cos θ )q ( ρ sin θ )q → ϕ J =K J ( ηi−Sd ( ρ ' ) sinh (31) ) 2 ηi2−S d The distance expansion from Eq. (28) is approximated as: r ∼ρ √ c 0 √1+ c1 ( θ )/c 0 ρ=ρ √ c0 √ 1+~ c ρ' (32) c where due to conformality at the collocation point 0 is a constant: ∂x i ∂x i ∂x i ∂x i (33) c 0= ∂ η · ∂ η = ∂ η · ∂ η 1 1 2 2 c depends on the angular coordinate: and ~

(

)

( )( ) ( )( ) i−Sd

[(

)( )

i

( ( ) ( ) ( ) ( )) ( ( ) ( ) ( ) ( )) ] ( )( )

i

i

2 η2 ∂x ∂ x cos θ ∂ x ∂ x sin θ ∂x ∂ x ∂x ∂ x ~ · + · + 2 · · c =− + 2 2 2 ∂ η2 ∂ η1 ∂ η1 η2 ∂ η2 c 0 sin θ ∂ η1 2 2 ∂ η1 ∂ η2 ∂ η1 i

2

i

3

i

+ 2

2

i

3

i

2

i

2

∂x ∂ x ∂x ∂ x + · · 2 ∂ η2 ∂ η1 η2 ∂ η1 ∂ η2

i

2

i

2

i

2

sin θ cos θ 2

(34)

2

sin θ cos θ 2

From Eq. (27), it follows that J Sd J Cd J P d J A d J UA d =ρ K T . Therefore, the final integral to be considered for error estimation becomes: ~(1)−Sd + Δ ~ θS d θ ' 2( p +q ) p+ q θ ( ρ ' ) sinh 2 ( p+q ) 1 1 ηi−Sd K ϕ K J K T ( ηi−Sd ) 2 2 I= dρ ' d θ' (35) ∫ ∫ ~ √ 1+ c ρ ' √c 0 0

(

)

0

The classical error estimator based on 2 N -derivatives [6] of the integrand can be used under certain assumptions, which will be described next. In terms of relative error, it can be written as:

|E I( I )|≤2( e

ϵ=

n

ρ'

(36)

H ρ ' +e θ' H θ ' )

c . It can be neglected where the normalization integral I n is calculated analytically assuming null ~ because its small value for moderately curved elements has no significant impact for the purpose of normalization: 2( p +q) 1 1 ~(1)−S d ~Sd K ϕ K J K T ( ηi−Sd ) 2 θ +Δ θ θ ' 2( p+q ) I ∼I n= (ρ ' ) d ρ ' ∫ sinh p +q d θ ' =K I ρ ' I θ ' (37) ∫ d ηi−S √c 0 0 0 2

(

The error term related to the radial coordinate is considered to be: 2( N ρ ' !)

4

|

2 N ρ'

1 d 1 2N I I 1+ (2 N ρ ' + 1)[(2 N ρ ' ) !] ρ ' θ ' d ρ ' √ ~c ρ ' 4 2N ~ ( N ρ ' !) [(4 N ρ ' −1)! ! ] 2 c = 2 (1+~ c ρ' ) |I ρ ' I θ '| (2 N ρ ' +1)[(2 N ρ ' ) !]3 √1+~c ρ '

ϵρ '= 2 eρ ' H ρ ' =

3

ρ'

(

)

|

)

(38)

ρ'

which, after applying Stirling‘s formulas for approximating factorials and double factorials, it can conservatively be approximated as: 2N ~ 1 1 c (39) ϵρ ' ≈ c ρ' ) |I ρ ' I θ '| √ 1+~ c ρ ' 4 (1+ ~

(

)

ρ'

2 ( p+ q) which is invertible with respect to N ρ ' . We neglect the presence of ( ρ ' ) for the calculation of 2 N derivatives, but we do take into account its influence by taking N ρ ' as the maximum between using Eq. (39) and considering N ρ ' = p+ q+1 . We sample Eq. (39) at mid-point ρ ' =0.5 , and at



164

~ ~(1)−S d ~(2 )−Sd } , and then the most unfavourable case is taken. The error term related to the ,θ θ ={ θ angular coordinate is considered to be:

|

)| )

2N ~(1)−S d ~Sd 1 d + Δ θ θ' p +q θ sinh ( 2 N θ ' +1)[( 2 N θ ' )! ]3 I ρ ' I θ ' d θ ' 2 N ηi−Sd 2 4 ~S d 2 N ~(1)−S d ~(1)−Sd ( N !) ( p +q )Δ θ ( p+ q )( θ +θ ) 2 θ' = cosh i−S d i −S d |I ρ ' I θ '| (2 N θ ' + 1)[(2 N θ ' ) !]3 η2 2 η2 4

2 ( N θ' ! )

ϵθ ' =2 e θ ' H θ '=

(

θ'

θ'

)

(

θ'

(

(40)

c is considered null, the radial coordinate is considered to be ρ ' =1 , and the angular where ~ coordenate is sampled at the mid-point θ ' =0.5 . It can be conservatively be approximated as: 2N ( p+q)(~ θ (1 )−Sd +~ θ (1)−S d ) e( p+ q)Δ ~ θS d 1 ϵθ ' ≈ cosh (41) d |I ρ ' I θ '| 2 ηi−S 8 N θ ' ηi−Sd 2 2

(

)(

)

θ'

which is not invertible with respect to N θ ' , but it is easily solvable numerically. Numerical examples For the sake of brevity, we are going to show only results regarding accuracy of error estimators for weakly singular integrals, i.e. Eqs. (39) and (41), which are probably the most relevant contribution of the present work. In order to show the efficacy of the methodology, a set of seven quadratic quadrilateral elements with different aspect ratios, skewnesses and curvatures are considered (see Fig. 3).

Figure 3: Quadratic quadrilateral elements considered for the numerical examples The accuracy of estimating N ρ ' and N θ ' for these elements is measured by considering the following canonical weakly singular integral: ϕ (42) I k =∫Γ−T 1 k d Γ r which considers only the integration domain of the subtriangle T1 (see Fig. 2). Since these elements are quadratic quadrilaterals, we consider p=2 (shape functions) and q=3 (planar element). A grid of 50×50 collocation points ( ξ i ∈[−0.99,0 .99]×[−0.99,0 .99] ) are tested. Fig. 4 shows the estimation accuracy by showing a monotonic curve of Δ N ρ ' =N estimated and −N required ρ' ρ' estimated required . It shows that Δ N ρ ' and Δ N θ ' is mostly near or moderately greater than Δ N θ ' =N θ ' − N θ' 0, which means that the estimated quadratures are reasonable conservative and accurate.


165 165

Figure 4: Monotonic curves of differences between estimated and actual required N ρ ' and N θ ' when integrating subtriangle T1 for collocation points in ξ i ∈[−0.99,0 .99]×[−0.99,0 .99] .


Acknowledgments


166

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 BIA2017-88770-R. The authors are grateful for this support. References [1] M. Tanaka, V. Sladek, and J. Sladek. Regularization techniques applied to boundary element methods. Applied Mechanics Reviews, 47, 457–499 (1994). [2] R. Gallego and J. Domínguez. Hypersingular BEM for transient elastodynamics. International Journal for Numerical Methods in Engineering, 39, 1681–1705 (1996). [3] J. Domínguez, M. P. Ariza, and R. Gallego. Flux and traction boundary elements without hypersingular or strongly singular integrals. International Journal for Numerical Methods in Engineering, 48,111–135 (2000). [4] J. D. R. Bordón, J. J. Aznárez and O. Maeso. Dynamic model of open shell structures buried in poroelastic soils. Computational Mechanics, 60, 269–288 (2017). [5] P. J. Davis and P. Rabinowitz. Methods of Numerical Integration, Academic Press (1984). [6] J.C. Lachat and J.O. Watson. Effective numerical treatment of boundary integral equations: a formulation for three-dimensional elastostatics. International Journal for Numerical Methods in Engineering, 10, 991–1005 (1976). [7] L. Jun, G. Beer, and J.L. Meek. Efficient evaluation of integrals of order 1/r, 1/r^2 and 1/r^3 using Gauss quadrature. Engineering Analysis, 2, 118–123 (1985). [8] 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, 959–973 (1987). [9] S. Bu and T. G. Davies. Effective evaluation of non-singular integrals in 3D BEM. Advances in Engineering Software, 23, 121-128 (1995). [10] D. G. Anderson. Gaussian quadrature formula for −∫ 10 ln(x ) f ( x)dx . Mathematics of Computation, 19(91), 477–481 (1965). [11] J. Rong, L. Wen, and J. Xiao. Efficiency improvement of the polar coordinate transformation for evaluating bem singular integrals on curved elements. Engineering Analysis with Boundary Elements, 38, 83–93 (2014). [12] M. A. Khayat and D. R. Wilton. An improved transformation and optimized sampling scheme for the numerical evaluation of singular and near-singular potentials. IEEE Antennas and Wireless Propagation Letters, 7,377–380 (2008).


167

Approximated Fundamental Solutions Based on Levi’s Functions Gallego R.1∗ , Puertas E.2 Department of Structural Mechanics and Hydraulic Engineering, School of Civil Engineering, University of Granada, Fuentenueva Campus, 18071 Granada, Spain. 1 [email protected]; 1 [email protected] Keywords: Fundamental Solution, Levi Function, Functionally Graded Materials

Abstract. Fundamental Solutions (FS) are useful for numerical methods, specifically in applications such as the Boundary Element Method (BEM) or the Method of Fundamental Solutions (MFS). Obtaining analytically the FS for a given problem is frequently unfeasible since it entails the solution of a complex system of differential equations. In this paper, a novel method for the computation of approximate FS based in enhanced Levi’s functions is presented. The first terms of the enhanced Levi’s functions are computed analytically until the residual is continuous at the collocation point. The last step in the computation of the FS is completed approximating the residual by Modified Radial Base Functions that are particular solutions of the problem equations. The method is applied to potential problems with variable coefficients, and validated comparing to available analytically computed FS’s. Introduction Fundamental Solution are well known for many Partial Differential Equations (PDEs) with constant coefficients [1,2]. But when the coefficients are variable the solution is available in explicit form only for some special cases [3]. Several techniques have been proposed to solve problems with variable coefficients that involve changes in the Boundary Element Method formulation. The parametrix (Levi function [4]) is used as a substitute of a Fundamental Solution in order to reduce a boundary value problem into a system of boundary-domain integral equations [5,6,7]. Furthermore, Radial Basis Functions (RBFs) have been applied to solve many problems in science and engineering [8]. RBFs are employed in the development of methods for solving partial differential equations [9,10,11,12,13]. The goal of this paper is to prove the feasibility of obtaining Fundamental Solution by combining Levi Functions and Radial Basis Functions in problems with variable coefficients. Definition of problem Let’s consider a general linear partial differential operator A(x) of order n. Its Fundamental Solution is given by the solution U(x; y) of the system of equations A(x)U(x; y) = δ(x; y)

(1)

where x, y ∈ Ω ⊂ Rn are the “observation” and “collocation” point, respectively. A Levi function of the operator A(x) is any function UL(x; y) such that A(x)UL(x; y) = δ(x; y) + BL(x; y)

(2)

where BL(x; y) is a regular function in x and y, which is called, the “residual”. Defining ∆U(x; y) = U(x; y) + UL(x; y)

(3)

which is termed the “remainder”, is obvious that, A(x)∆U(x; y) = BL(x; y)

(4)

The method for calculate the Fundamental Solution U(x; y) is divided in two steps: firstly a Levi Function UL(x; y) of A(x) is computed using an iterative procedure [5]; secondly, the remainder ∆U(x; y) is approximated using Modified Radial Basis Functions. The complete Fundamental Solution will be given by, U(x; y) ' UL(x; y) + ∆U(x; y)

1

(5)


168

Calculation of a Levi Function Following [5] the differential operator A(x) is decomposed in two terms, A(x) = A0 (y) − T(x; y)

(6)

where A0 (y) is the principal part of A(x) with frozen coefficients, i.e. only the derivatives order n are kept, and x is substituted in the remaining coefficients by the fixed value y. Here T(x; y) is simply T(x; y) = A0 (y) − A(x). Formally this decomposition can be written as, A(x) = A0 (y){I − A−1 0 (y)T(x; y)} and therefor the inverse of A(x) is given by, −1 −1 −1 A−1 (x) = [A0 (y){I − A−1 = {I − A−1 0 (y)T(x; y)}] 0 (y)T(x; y)} A0 (y) −1 Taking the (formal) expansion of {I − A−1 0 (y)T(x; y)} −1 {I − A−1 = I + ∆ + ∆2 + ∆3 + · · · 0 (y)T(x; y)}

and therefore, A−1 (x) = (I + ∆ + ∆2 + ∆3 + · · · )A−1 0 (y) where ∆ = A−1 0 (y)T(x; y). From this expansion an iterative process can be implemented U(x; y) = A−1 δ(x; y) =

∞ X

U (i)(x; y)

i=0

where −1 (i−1) i (x; y) U (i)(x; y) = (A−1 0 T ) U0(x; y) = A0 T U

and U (0)(x; y) = U0(x; y) = A−1 0 δ(x; y) is the Fundamental Solution for A0 (y). Provided the expansion above converges, a Levi Function of the operator A(x) will be obtained truncating the expansion after N terms, N X −1 UL(x; y) = A δ(x; y) = U (i)(x; y) i=0

The number of terms N is such that BL(x; y) = A(x)(U(x; y) − UL(x; y)) is regular. An alternative expansion can be obtained writing the operator A(x) as, A(x) = (I − T(x; y)A−1 0 (y))A0 (y) and therefore,

2

3

−1 −1 0 0 0 A−1 = A−1 = A−1 0 (I − T A0 ) 0 (I + ∆ + ∆ + ∆ + · · · )

Here ∆0 = T(x; y)A−1 0 (y). From this formal expansion the Fundamental Solution is obtained as, U(x; y) = A−1 δ(x; y) = A−1 0

∞ X

B (i)(x; y)

i=0

with, B (0)(x; y) = δ(x; y) (i−1) B (i)(x; y) = T A−1 (x; y) 0 B

With this second alternative, the computation of the Fundamental Solution is reduced to these recursive expressions, (i) U (i)(x; y) = A−1 0 B (x; y)

(7)

B (i)(x; y) = T(x; y)U (i)(x; y)

(8)

2


169

B (0)(x; y)

with = δ(x; y). Formally, equations eqs. (7) and (8) permits the calculation of the Fundamental Solution to the desired level of accuracy; however, for differential operators of practical interest the first step in the iteration eq. (7) is not feasible for i > 0. Even for cases when the iteration can be applied, the ensuing expressions become unwieldy in a few iterations and therefore the expansion is not suitable for practical computation of the Fundamental Solution for a given level of accuracy. However, for the computation of a Levi function the iteration can be modify defining the following scheme, (i) U (i)(x; y) = A−1 0 B (x; y)

B

(9)

0 (i+1)

(x; y) = T(x; y)U (i)(x; y)

B

(i+1)

(x; y) = P S[B

(10)

0 (i+1)

(x; y)]

(11)

where P S[f ] = means singular part of f and is obtained retaining only the most singular terms in f at r = |x− 0 y| = 0 and then freezing the coefficients at x = y. The iteration stops at i = N , when P S[B (N +1)(x; y)] = 0, and it depends on the order of A(x). Calculation of the remainder ∆U(x; y) using Modified Radial Basis Functions In the previous section a procedure for obtaining a Levi function of A(x) is presented. In this section a method to obtain the remainder ∆U(x; y) using an approximation of the residual BL(x; y) is developed. Firstly, it is easily shown that the residual BL(x; y) is given by, BL(x; y) =

N X

(i)

BR (x; y) + B (N +1)(x; y)

i=1 (i)

0 (i)

0

0

0

where BR (x; y) = B (x; y) − B (i)(x; y) = B (i)(x; y) − P S[B (i)(x; y)], i.e., the regular part of B (i)(x; y). The problem to be solved is given by, A(x)∆U(x; y) = BL(x; y) where BL(x; y) is regular. The solution of this problem can be found using standard procedures based in Radial Basis Functions (RBF) as shown in [11-12]. However, since in most applications one is interested in the solution of this problem for many points y, a procedure that combines RBF and the Method of Particular Solutions (MPS) is developed. Let zj ∈ Ω, and let f (x; zj ) be a Radial Function (RF) whose value at any point x ∈ Ω depends only on the distance from the point zj . The first step is to compute a particular solution up (x, zj ) from the equation (12) Ap (x)up (x; zj ) = f (x; zj )

(12)

where Ap is the principle part of A(x) at a point x = zp . The point zp can be zj or a fixed point zp ∀j. From this particular solution a Modified Radial Basis Function (MRBF) is computed by, A(x)up (x; zj ) = fp (x; zj )

(13)

and the residual BL(x; y) is approximated by a linear combination of these MFBRs in the following form, BL(x; y) '

M X

λj (y)fp (x; zj )

(14)

j=1

Collocating this equation at x = zi the following system of linear equations is obtained, Fλ(y) = BL where λ(y) = (λ1 (y), λ2 (y), . . . , λL (y))T

(15)

BL = (B(z1 ; y), B(z2 ; y), , . . .)T

(16)

F = (Fij ) = (fp (zi ; zj )) 3

(17)


170

The matrix F is called the kernel matrix. This matrix is independent from the collocation point y. So, it is necessary to calculate and factorize it just once. Solving this linear system of equations, and considering equations eqs. (12) to (14), the remainder ∆U(x; y) is obtained by, M X ∆U(x; y) ' λj (y)up (x; zj ) j=1

Finally, the Fundamental Solution is given by, U(x; y) ' UL(x; y) +

N X

λj (y)up (x; zj )

j=1

where UL(x; y) is the Levi Function from the previous section. Fundamental Solution of Functionally Graded Materials The proposed methodology is tested in the equation ∇ · (k(x)∇u(x)) = δ(x, y) In this case, A(x) = ∇ · (k(x)∇) and therefore, A0 = A(y) = ∇ · (k(y)∇) = k0 ∇2 and T(x; y) = A0 − A = −∇ · ({k(x) − k0 }∇) = −k0 ∇ · (s(x, y)∇) 0 where s(x, y) = k(x)−k , and k0 = k(y). With these definition, s(y, y) = 0. k0 2 (0) It is easy to obtain U (x; y) = A−1 0 δ(x, y) since A0 = k0 ∇ ,

U (0)(x; y) = −

1 log r 2πk0

where r = |x − y|. Applying the recursive iteration given by eqns.(9) to (11) the next terms of the Levi function are obtained, 1 (r1 s10 + r2 s01 ) log r 4πk0 1 (2) U (2)(x; y) = − a + b(2) log r 96πk0

U (1)(x; y) = −

where, a(2) = 3r2 (s201 + s210 ) + (7s10 s20 + 5s02 s10 + 2s01 s11 )r13 + (7s01 s02 + 5s01 s20 + 2s10 s11 )r23 b(2) = (15s210 − 3s201 )r12 + (15s201 − 3s210 )r22 + 36s01 s10 r1 r2 + + (4s10 s20 − 4s02 s10 − 2s01 s11 )r13 + (4s01 s02 − 4s01 s20 − 2s10 s11 )r23 + + 6(3s10 s11 − s01 s02 + s01 s20 )r12 r2 + 6(3s01 s11 − s10 s20 + s02 s10 )r1 r22 ∂s ∂s ∂2s ∂2s and ri = xi − yi , s10 = ∂x , s = , s = , s = and s02 = 01 11 20 2 ∂x2 ∂x1 ∂x2 ∂x 1 x=y

0(3) Bs (x; y)

The function = the Levi Function is given by,

x=y T(x; y)U (2)(x; y)

x=y

1

x=y

∂2s . ∂x22 x=y

is regular, since the operator A(x) is order 2, and therefore

UL(x; y) = U (0)(x; y) + U (1)(x; y) + U (2)(x; y) The value of BL(x; y) is long, an it is not reproduced in this communication. 4

(18)


171

-1.0 -0.5 0.0

-2

-1

0

1

2 0.5 1.0 0.00 0.0 -0.01 -0.1 -0.02 -0.2 -0.03 -1.0

-2

-0.5

-1

0.0

0

0.5

1

1.0

2

Figure 1: Levi Function (left) and remainder (right) 0.004

0.4

0.002

0.3

0.000

0.2

-0.002

0.1

-0.004

Π 2

-0.006

Π

3Π 2

2Π

-0.1 0

1

2

3

4

5

6

Figure 2: Remainder (left) and residual (right) around the collocation point (θ = 0 . . . 2π) for different values of r (0.01, 0.02, . . . , 0.1)) Numerical example A numerical example is presented to validate de accuracy of the method. We consider an specific case described in [14]. We define a time-independent problem with material properties varying in the direction x2 . For a potential φ(x), the flux equation is qi = −k(x2 )φ,i (19) where k(x2 ) is the material physical parameter, whose variation is, −2 2 k(x2 ) = ke2βx2 αe2βx2 + 1 e−2βx2 αe2βx2 + 1

(20)

The Fundamental Solution is given in this case by U(x; y) = p(x2 )h(x; y) where −1 −1 eβx2 αe2βy0 + 1 p(x2 ) = eβy0 k0−1 αe2βy0 + 1

(21)

1 {K0 (βr) + [log (β/2) + γ] I0 (βr)} 2π

(22)

h(x; y) =

Let us consider a potential problem with exponentially graded material properties: α = 0.215, β = 1.5, k = 5, y0 = 0.2. The Levi Function expressed in equation 18 is represented in Figure 1. The remainder BL(x; y) is approximated using the Euclidean MRBFs and it is represented in Figure1. In Figure 2 the remainder (left) and the residual (right) around the collocation point y is represented at diferent distances r.

5

172


Conclusions In this paper, a method is developed and implemented for obtaining Fundamental Solution in problems with variable coefficients. The following remarks apply to the present approach: • This technique is easy to implement numerically. • Levi function is available for obtain Fundamental Solution . • Modified Radial Basis Functions provide very high rates of convergence. References [1] C. A. Brebbia and J. Domínguez. Boundary elements: An introductory course. McGraw Hill Book Co., (1989). [2] E. Kausel. Fundamental Solutions in Elastodynamics. A compendium. Cambridge University Press (2006). [3] D.L. Clements. Fundamental solutions for second order linear elliptic partial differential equations. Computational Mechanics, 22, 26–31 (1998). [4] E.E. Levi. I problemi dei valori al contorno per le equazioni lineari totalmente ellittiche alle derivate parziali. Memorie della Societa Italiana di Scienze XL, 16, 1–112 (1909). [5] A. Pomp. Levi functions for linear elliptic systems with variable coefficients including shell equations. Computational Mechanics, 22, 93–99 (1998). [6] M.A. AL-Jawary and L.C. Wrobel. Numerical solution of two-dimensional mixed problems with variable coefficients by the boundary-domain integral and integro-differential equation methods. Engineering Analysis with Boundary Elements, 35, 1279-1287 (2011). [7] S.E. Mikhailov. Analysis of Segregated Boundary-Domain Integral Equations for Variable-Coefficient Dirichlet and Neumann Problems with General Data. arXiv:1509.03501, (2015). [8] M.D. Buhmann. Radial Basis Functions, Theory and Implementations. Cambridge University Press (2003). [9] W. Chen and M. Tanaka. A meshless, exponential convergence, integration-free, and boundary only RBF technique. Computers and Mathematics with Applications, 43, 379–391 (2002). [10] M.A. Golberg and C.S. Chen. The method of fundamental solutions for potential, Helmholtz and diffusion problems. In M.A. Golberg, editor, Boundary Integral Methods: Numerical and Mathematical Aspects, 103–176. WIT Press (1998). [11] E.J. Kansa. Multiquadrics - a scattered data approximation scheme with applications to computational fluid dynamics - I. Computers and Mathematics with Application, 19 (8/9), 127–145 (1990). [12] E.J. Kansa. Multiquadrics - a scattered data approximation scheme with applications to computational fluid dynamics - II. Computers and Mathematics with Application, 19 (8/9), 147–161 (1990). [13] V.D. Kupradze and 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 Physics, 4, 82–126 (1964). [14] N.A. Dumont, R.A.P. Chaves and G.H. Paulino. The hybrid boundary element method applied to functionally graded materials. In C.A. Brebbia, A. Tadeu, V. Popov, editors, Boudary Elements XXIV (2002).

6


173

Somigliana Identity for Unbounded Cone-Shaped Domains

1

Vladislav Mantič1 Escuela Técnica Superior de Ingeniería, Universidad de Sevilla Camino de los Descubrimientos s/n, Sevilla, 41092, Spain [email protected]

Keywords: boundary integral equation, linear elasticity, coefficient tensor, free term, infinite domain, half space, dihedron, cone

Abstract. Somigliana displacement identity for unbounded homogeneous linear elastic domains with unbounded boundary in 3D is studied. We assume that sufficiently far from the origin the domain coincides with a cone. The results presented can easily be generalized to unbounded boundaries asymptotically approaching a cone boundary. Without loss of generality we can assume that the cone vertex is located at the origin of the coordinate system. Then, this cone is defined by a subdomain of the unite sphere S2, this definition covering examples of cones like half-space, dihedron, pyramid (polyhedral cone), circular cone, etc. The domain boundary is assumed to be sufficiently smooth to guarantee that all the integrals in the Somigliana displacement identity are well-defined. Specifically, we consider piecewise smooth and uniformly Lipschitz boundary. First the integral of the fundamental tractions Tij(x,y), i.e. tractions of the Kelvin elastic fundamental solution for displacements Uij(x,y), on this unbounded cone-shaped boundary is calculated for the collocation point situated inside the domain, outside the domain, at a smooth or non-smooth boundary point. It is shown that this integral is given by the difference of two symmetric tensors, the first one independent of the collocation point given by the cone shape and orientation, and the second one dependent on the position of the collocation point with respect to the domain and its boundary. Then, the Somigliana displacement identity for such cone-shaped domain and elastic solutions satisfying usual conditions at infinity, vanishing displacements ui=o(1) and σij=o(R-1), is deduced for any collocation point in both strongly singular and regularized forms. In the case of the elastic solution behavior at infinity is prescribed by an elastic field, the deduced Somigliana identity form should be applied to the difference between the problem solution and the given elastic field, as otherwise the integrals appearing in this identity could become infinite. The present theoretical results will be useful to solve many problems including this type of unbounded domains with unbounded cone-shaped boundaries, e.g., in geotechnics and fracture mechanics, applying the substructuring method in presence of several materials.

174



175

Mesh generation in three-dimensional boundary element problems F. M. Loyola1, E. L. Albuquerque2 Department of Mechanical Engineering, Faculty of Technology, University of Brasília 70910-900, Brasília, Brazil 1

[email protected] 2

[email protected]

Keywords: Boundary Element Method, Mesh generation, Delaunay triangulation, Non uniform rational Bsplines.

Abstract. This work presents a complete approach for the analysis of heat conduction problems in solids using the boundary element method. The analysis begins with the drawing of the solid in computer aided design programs (CAD programs). This drawing is then exported in the IGES (Initial Graphics Exchange Specification) format which is read in the developed program. To read the file with the .igs extension, an open source package available on the internet is used. Then, the geometry is decomposed into threedimensional surfaces in space that are parametrized in a parametric plan using non-uniform rational Bsplines, i.e., NURBS. Each of these surfaces is discretized in the parametric plane using the Delaunay triangulation. This triangulation is done using a two-dimensional triangular finite element method mesh generator which is also an open source package. The triangular mesh in the plane is then transported into space. The boundary conditions of the problem are then imposed, the influence matrices of the boundary element method are assembled and a linear system is solved to compute the unknown variables in the boundary. Finally, the temperature in the surface of the solid is shown on a heat map. The program is applied in problems with high geometric complexity in order to demonstrate the developed formulation ability to analyze complex geometry problems. Introduction. Various areas of engineering seek to find results accurate enough to model real problems. They usually look for analytical solutions, which are accurate but difficult to find for more complex problems. Defining an analytical solution requires simplifications, in order to reduce the complexity of the equations that govern the problem, which may end up generating a model that does not correspond to reality. Faced with this scope, the importance of numerical methods has been growing, as they approximate solutions of problems, even complex ones. Numerical methods have the advantage of not requiring many of these simplifications, allowing more realistic analysis. When it comes to engineering problems solving, some methods stand out: Finite Differences Method (FDM); Finite Element Method (FEM); Boundary Element Method (BEM). Besides these, it is worth mentioning the growth of the meshless methods. BEM has advantages over FEM, although the latter is the most commercially used method for engineering problems solving. According to [1], the Boundary Element Method reduces the dimension of the problem in one order – a 3D solid may be modeled as a 2D surface – and has the feature of being simple in geometric data preparation. In FEM, the entire domain must be discretized to get the solutions, increasing the number of equations and burdening the user with generating a mesh both on the surface and inside the domain [1, 2]. Numerical simulations requires a large amount of time for sending data from CAD (Computer Aided Design) generated models to solvers, as seen in [3, 4, 5]. Some reasons for that are the translation from CAD files to a geometry compatible with the analyzing software and the mesh generation. Among the huge variety of numerical analysis, three-dimensional are the ones that present the greatest challenges, mainly due to the geometry of the problem that can be highly complex. In this work we will use the convenience of importing CAD files into Matlab/Octave and then analyzing with BEM. The program will analyze temperature and heat flow in solids. To do this, IGES toolbox [6] package will be used, which, as the name suggests, imports files in the iges format for Matlab/Octave. This process streamlines the work of the engineer, who can use a CAD software of his choice to generate the geometry, export it in IGES format and then read it and generate the mesh in a program of boundary elements. After this, another toolbox was used in order to create the mesh. MESH2D [7] can create triangular unstructured meshes on trimmed or untrimmed surfaces. In FEM, it can be used only in 2D problems, but in


176

those using BEM, it is possible to solve 3D problems. This can happen because, as said before, one of the main features of BEM is to reduce the dimensions of the problem in one order. Geometric modelling. This section gives a brief description of the Bézier curves and NURBS curves and surfaces, which can be seen in details in [8, 9, 10, 11]. Next, it exemplifies IGES file format and briefly explains which are its main entities. Some relevant references are [12, 13, 14], and the latter also addresses the generation of triangular mesh directly on the 3D surface. B-Spline. The origin of the B-splines is in Bernstein's polynomials which are defined by Equation. Bernstein's polynomials of order 1 are identical to the Lagrange functions. Note that B-splines are generalizations of Bézier curves – those are formed by one or more of these, with continuity between the segments. The mathematical description of a B-spline is given by: 𝑁𝑖,0 (ξ) =

𝑁𝑖,𝑝 ξ =

1 if ξi ≤ ξ ≤ ξi+1 0 otherwise

ξi+p+1 − ξ ξ − ξi 𝑁𝑖,𝑝−1 ξ + 𝑁 ξ ξ𝑖+𝑝 − ξ𝑖 ξ𝑖+𝑝+1 − ξ𝑖+1 𝑖+1,𝑝−1

(1)

(2)

A B-spline curve is given by: 𝑛

𝐂(ξ) =

𝑁𝑖,𝑝 ξ 𝐁i

(3)

𝑖=1

where 𝐁𝑖 is the control point.

Figure 1: Quadratic B-spline curve in two dimensions [17]. NURBS. Non Uniform Rational Basis Spline (NURBS) are based on the B-splines, but with the difference of controlling the curve using weights. One of the most important advantages of NURBS when compared to B-splines is the ability to exactly represent a wide array of objects that cannot be exactly represented by polynomials [5, 8]. For instance, a quarter circle as in Figure, which shows the exact representation of the quarter circle by a quadratic NURBS in blue compared to a non-rational B-spline curve.


177

Figure 2: Quadratic NURBS exactly representing the quarter circle, in contrast to a non-rational quadratic B-spline curve [17].

We can define the weighting function as: 𝑛

W(ξ) =

𝑁𝑖,𝑝 ξ wi

(4)

𝑖=1

where N𝑖,𝑝 (ξ) is the standard B-spline basis function. The NURBS basis is given by: 𝑝

𝑅𝑖 ξ =

𝑁𝑖,𝑝 ξ 𝑤𝑖 = 𝑊(ξ)

𝑁𝑖,𝑝 ξ 𝑛 î=1 𝑁î,𝑝

𝑤𝑖 ξ 𝑤î

(5)

Finally, the NURBS curve is: 𝑛 𝑝

𝐂(ξ) =

𝑅𝑖 ξ 𝐁i

(6)

𝑖=1

which is form identical to a B-spline curve. B-spline or NURBS surfaces are obtained by a tensor product between two curves. Rational surfaces are defined analogously in terms of the rational basis function: 𝑝,𝑞

𝑅𝑖,𝑗 ξ, η =

𝑛 î=1

𝑁𝑖,𝑝 𝑛 𝑗 =1 𝑁î,𝑝

ξ 𝑀𝑗 ,𝑞 η 𝑤𝑖,𝑗 ξ 𝑤î 𝑁î,𝑝 ξ 𝑀𝑗 ,𝑞 η 𝑤î,𝑗

(7)

When a NURBS surface is created, there are two sides, front and back. Figure 3 shows an example of a surface with its normal vectors pointing outward. Usually, when normal vectors are considered on closed surfaces, the inward-pointing (pointing towards the interior of the surface) and outward-pointing are usually distinguished. This information is essencial in the scope of this study, because it has direct impact on the results – if the normals are pointing to the wrong side, the results will not be accurate.


178

Figure 3: Surface normals. IGES file format. IGES was an initiative of the United States Air Force through the Integrated Computer Aided Manufacturing project between 1976 and 1984. The focus was on developing procedures, processes and integration between CAD/CAM softwares required for the aerospace industry and to reduce costs. At the time, the data generated by different CAD systems were incompatible with each other, hampering the development of new technologies. IGES is a text file, with information divided by sectors. The first part contains the general information of the file, such as directory, date, author, among others. It should be noted that only the second part is relevant for the simulation with the BEM, because it is where the model information is. This type of file has two types of entities - geometric ones and non-geometric ones. The former are points, lines, profiles, surfaces, among others. The latter are data of colors or explanatory texts, for example. By default, IGES has 80 character columns. Three-dimensional BEM formulation. Potential problems, the kind that we are aiming to solve in this study, are governed by Laplace’s equation, which in three-dimensional problems can be written as: 𝛻²𝑇 =

𝑞 𝑘

(8)

where 𝑇 represents the temperature in a given point, 𝑞 is a source if 𝑞 > 0 and a sink if 𝑞 < 0. Figure 4 represents a solid with volume and surface 𝑉 and 𝑆, respectively. Heat transfer in this body is governed by Equation (8).

Figure 4: Solid with volume V and surface S. Developing the integral equation for Laplace’s equation, making Equation (8) times a weight function 𝜔(𝑥, 𝑦), considering 𝑞 = 0 and integrating over domain 𝑉, it is assumed that the result is zero (weighted residuals method):


179

𝛻 2 𝑇 𝜔𝑑𝑉 = 0 𝑉

𝑉

𝜕²𝑇 𝜔𝑑𝑉 + 𝜕𝑥²

𝜕²𝑇 𝜕²𝑇 𝜕²𝑇 + + 𝜔𝑑𝑉 = 0 (9) 𝜕𝑥² 𝜕𝑦² 𝜕𝑧² 𝑉 𝜕²𝑇 𝜕²𝑇 𝜔𝑑𝑉 + 𝜔𝑑𝑉 = 0 𝜕𝑦² 𝜕𝑧² 𝑉

𝑉

The step-by-step of the development of the equation can be seen in [1,15]. In the end, the integral can be used to represent an intern or a boundary point as follows: 𝑇𝑞 ∗ 𝑑𝑆 −

𝑐𝑇 𝑃 =

𝑇 ∗ 𝑞𝑑𝑆

𝑆

(10)

𝑆

where 𝑐 is a constant depending on the source point. Discretization of integral boundary equations. Basically, the BEM formulation transforms differential equations into integral boundary equations, thus eliminating the discretization of the domain. These integrals can be solved numerically or analytically, integrating over the boundary. A strategy for solving integrals over surface S is depicted in Figure 5. The 𝑆 surface is divided in a number of smaller surfaces 𝑆1 , 𝑆2 , … , 𝑆𝑛 , 𝑆 = 𝑛𝑖=1 𝑆𝑖 , where 𝑛 is the number of surfaces in which the boundary has been divided. After that, is possible to approximate each of the smaller surfaces in more simple surfaces that can be easily described by mathematical functions. Each surface 𝑆1 , 𝑆2 , … , 𝑆𝑛 is approximated to 𝛤1 , 𝛤2,…, 𝛤𝑛 , which can be described by polynomials, the so called boundary elements. Yields: 𝑛

𝛤=

𝛤𝑖

(11)

𝛤𝑖 = 𝑆

(12)

𝑖=1

and

𝑛

𝑙𝑖𝑚 𝛤 =

𝑛→∞

𝑖=1

Equation (10) can be rewritten as: 𝑛

𝑛 ∗

𝑐𝑇 𝑃 =

𝑇 ∗ 𝑞𝑑Γ

𝑇𝑞 𝑑Γ − 𝑗 =1

Γ𝑗

𝑗 =1

(13)

Γ𝑗

In this study, elements 𝛤𝑖 are described by plane triangles. They have got constant Jacobian over the element as seen in [15]. Constant triangular elements. In discretization using constant elements, the geometry is approximated by plane triangles with a node in the centroid of each element. In addition, both temperature and flux are assumed to be constant. Since the 𝑗 node will always be in the centroid of the 𝑗 element (always in a smooth boundary 1 region), 𝑐 = 2 . Thus, the integral equation is approximated by: 1 𝑇 = 2 𝑖

𝑛

𝑛 ∗

𝑇𝑗 𝑗 =1

𝑞 𝑑Γ − Γ𝑗

𝑇 ∗ 𝑑Γ

𝑞𝑗 𝑗 =1

Γ𝑗

(14)


180

1 − 𝑇𝑖 + 2

𝑛

𝑛 ∗

𝑇𝑗

𝑞 𝑑Γ = Γ𝑗

𝑗 =1

(15)

Γ𝑗

𝑗 =1

𝑞 ∗ 𝑑Γ, 𝐻𝑖𝑗 =

𝑇 ∗ 𝑑Γ

𝑞𝑗 if 𝑖 ≠ 𝑗

Γ𝑗

1 − + 2

𝑞 ∗ 𝑑Γ, Γ𝑗

if 𝑖 = 𝑗

(16)

𝑇 ∗ 𝑑Γ

𝐺𝑖𝑗 = Γ𝑗

The matrix equation can be written as follows: 𝑛

𝑛

𝐻𝑖𝑗 𝑇𝑗 = 𝑗 =1

𝐺𝑖𝑗 𝑞𝑗

(17)

𝑗 =1

Packages. As mentioned before, two Matlab toolboxes (or packages) has been used. The idea is to use available codes to create, as much as possible, an integrated heat transfer analysis tool. The first package used, IGES toolbox, has the ability to make Matlab open and read IGES entities (such as point, line, shell, surface etc). With the solid’s data in the numerical computing environment, it is possible to edit and use its properties. Secondly, with the model’s data already available, the other package used, MESH2D, can play its role – it creates unstructured triangular meshes on 2D planes. As we are using BEM, only the boundaries need to be discretized, making possible to apply the method to solids using surfaces discretization only. After this process, the idea is to use parametric meshing as seen in [16]. The strategy is to represent the surface to be discretised by a bivariate analytical function, in which all points on the 3D surface are mapped to the 2D parametric space. The mesh generation process is carried out entirely on the parametric space by MESH2D. The final surface mesh is obtained by proper transformation of the mesh generated on the parametric space back to the 3D space. Results. In this study, the example is a bracket (Figure 5) with 16 surfaces (Figure 6). The mesh generation process is shown in Figure 7. For ease of understanding, the mesh generation process for a surface is detailed in Figure 7(a)-(c). It starts creating the boundary points of the surface in the parametric 2D space. After that, internal points are generated with Delaunay triangulation. Then, it proceeds to the next step – transferring from 2D to 3D. Afterwards, the result is detailed in Figure 5(c). As the elements are discontinuous, their nodes do not have to match the two faces.

(a) (b) (c) Figure 5: (a) bracket; (b) exploded-view; (c) detail of bracket’s edge.


181

Figure 6: Surfaces numbering.

(a) (b) (c) Figure 7: (a) Generation of boundary points; (b) Delaunay triangulation; (c) transferring mesh from 2D to 3D. Finally, for analyzing the heat transfer phenomena, let the flux be assumed as known and equals zero in all surfaces but numbers 1 and 16, where the temperature is 10; and 4, where the temperature is zero. This mesh was created using 8422 constant elements. The heat map can be seen in Figure 8.

Figure 8: Heat map for bracket.

182


Conclusions. The focus of this work was to study a way to easily create a model using a regular CAD software, read the data in a numerical computing environment, generate a mesh and then solve potential problems. From this work, it can be concluded that using packages available online for free can save time from the user, demanding less effort in order to create the mesh and solve the problem. This kind of approach that integrates and automates process of modeling and analyzing deserves more research, as diminishes production costs, as seen in [5]. References [1] L. C. Wrobel. The Boundary Element Method, Vol1: Applications in Thermo-Fluids and Acoustics, Wiley (2002). [2] T. Belytschko et al. Element free galerkin methods. International journal for numerical methods in engineering, John Wiley & Sons, Ltd, v. 37, n. 2, p. 229–256 (1994). [3] Y. Bazilevs et al. Isogeometric analysis using T-splines. Computer Methods in Applied Mechanics and Engineering 199.5-8 (2010): 229-263. [4] G. Beer. Advanced Numerical Simulation Methods: From CAD Data Directly to Simulation Results. CRC Press (2015). [5] J. A. Cottrell, T. JR Hughes, Y. Bazilevs. Isogeometric analysis: toward integration of CAD and FEA. John Wiley & Sons (2009). [6] P. Bergström. IGES Toolbox - File Exchange - MATLAB Central. [online] Mathworks.com. Available at: http://www.mathworks.com/matlabcentral/fileexchange/13253-iges-toolbox [Accessed Jan. 19th 2017]. [7] D. Engwirda. MESH2D - Delaunay-based unstructured mesh generation. [online] Mathworks.com. Available at: http://www.mathworks.com/matlabcentral/fileexchange/25555-mesh2d-delaunay-basedunstructured-mesh-generation [Accessed Jan. 19th 2017]. [8] D. F. Rogers. An introduction to NURBS: with historical perspective. Elsevier (2000). [9] L. Piegl; W. Tiller. The NURBS book. Springer Science & Business Media (2012). [10] G. E. Farin. NURB curves and surfaces: from projective geometry to practical use. AK Peters, Ltd. (1995). [11] G. E. Farin, J. Hoschek; M.-S. Kim. Handbook of computer aided geometric design. Elsevier (2002). [12] B. M. Smith. Iges: A key to CAD/CAM systemns integration. IEEE Comp. Graphics Applic., v. 3, n. 8, p. 78–83 (1983). [13] B. Smith, J. Wellington. Initial graphics exchange specification (IGES); version 3.0. (1986). [14] Y. Ito, K. Nakahashi. Direct surface triangulation using stereolithography data. AIAA journal, v. 40, n. 3, p. 490–496 (2002). [15] F. M Loyola. Modelagem tridimensional de problemas potenciais usando o método dos elementos de contorno. Master thesis, University of Brasília (2017). [16] D. S. H. Lo. Finite Element Mesh Generation. CRC Press (2015). [17] J. Valentin. Bsplines.org Available at: https://bsplines.org/flavors-and-types-of-b-splines/ [Accessed Fev. 17th 2018].


183

The Method of Fundamental Solutions for Two-Dimensional Stationary Thermoelastic Problems Involving Curved Line Heat Sources M. Mohammadi1, M.R. Hematiyan2 1

Department of Mechanical Engineering, College of Engineering, Shiraz Branch, Islamic Azad University, Shiraz, Iran [email protected] 2

Department of Mechanical Engineering, Shiraz University, Shiraz 71936, Iran [email protected]

Keywords: Method of fundamental solutions; Curved line heat source; Thermoelasticity, Meshfree

Abstract. In this work, the method of fundamental solutions (MFS) is used to solve two dimensional steady-state thermoelastic problems involving curved line heat sources. The geometrical shape of the curved heat sources can be arbitrary and sufficiently complicated. Each curved line heat source is modelled by putting together several simple sources with quadratic variations. The analyses for temperature and stress are carried out without considering any internal points or internal cells. Two numerical examples are presented to verify the validity and efficiency of the proposed formulation. It is concluded that the presented meshfree formulation is very efficient and useful in comparison with the finite element method in which accurate results are obtained only by considering condensed nodes near the curved line heat source. 1. Introduction In practical thermoelasticity, it is quite often to have internal concentrated heat sources distributed over curved paths. Boundary methods such as the boundary element method (BEM) and the MFS are powerful approaches for solving these problems. To date, the BEM has been used effectively to solve direct and inverse problems containing concentrated heat sources. Shiah et al. [1] used the BEM for thermoelastic analysis of two-dimensional anisotropic media containing point sources. They could solve the problem with a boundary-only discretization. Mohammadi et al. [2] used the BEM for direct analysis of two- and threedimensional thermoelastic problems involving arbitrary curved line heat sources. They effectively solved the problem without considering any internal points/cells. Similar to the BEM, the MFS is applicable when a fundamental solution of the problem is known. In the MFS, the solution is approximated as a linear combination of fundamental solutions. After the first numerical formulation of the MFS by Mathon and Johnston [3], various applications of this method have been presented in the literature. An overview can be found in the survey papers [4,5]. Although the MFS in conjunction with the dual reciprocity method (DRM) and in combination with the so called method of particular solutions (MPS) has been used in [6] and [7,8], respectively, for analysis of thermoelastic problems; but, to the authors' knowledge, no formulation has been presented for analysis of thermoelastic problems involving heat sources concentrated on curved lines. The present work uses the MFS-MPS to analyse the two dimensional stationary thermoelastic problems involving internal curved line heat sources. The method presented here, can be simply employed without considering any internal points or internal cells and therefore it preserves the attractiveness of the MFS as a boundary-type mesh-free method. Two numerical examples are presented to demonstrate the effectiveness of the proposed method in comparison with the BEM and the finite element method (FEM). 2. Basic equations and formulation Consider the medium Ω with its boundary Γ (Fig. 1). In the presence of heat sources, the governing equation of steady-state heat conduction can be expressed as: s (x) ∇ 2τ (x) = − in Ω (1) k The boundary conditions can be written as: ∂τ (xb ) f1 (xb )τ (xb ) + f 2 (xb ) = f3 (xb ) on Γ (2) ∂n


184

where f1 , f 2 and f 3 are given functions on the boundary, ∇ 2 is the Laplace operator, n is the direction normal to the boundary, x is a point in the domain, x b is a point on the boundary, τ is the temperature, s(x) is a known function describing the heat source distribution in the domain and k is the thermal conductivity.

Fig. 1: Domain, main boundary, pseudo boundary, and a curved line heat source Navier equations for two-dimensional problems with consideration of the thermal effects are given as [8]: 2 G 2  ∂ u j (x)  ∂τ (x) G∇ 2ui (x) + −γ = 0 i = 1,2 in Ω (3) ∑ 1 − 2ν j =1  ∂xi ∂x j  ∂xi Mechanical boundary conditions can be expressed as follows: ui ( xb ) = gi ( xb ) on Γ1 and ti ( xb ) = hi ( xb ) on Γ2 i = 1,2 (4) Γ1 U Γ2 = Γ

where ui and t i are the components of displacement vector and traction vector, respectively, G is the modulus of rigidity and g i and hi are prescribed functions on the boundary. In eq (3) γ is:

γ = 2Gα (1 + ν ) (1 − 2ν ) where plane strain  ν , υ = plane stress ν (1 + ν )

α  α = α (1 + ν ) (1 + 2ν )

plane strain plane stress

(5)

α represents the coefficient of thermal expansion and ν is the Poisson’s ratio. The solution of eq (1) in the MFS is represented as follows: N

τ ( x ) = ∑ a lτ * ( x, ξ l )) + τ p ( x )

(6)

l =1

where ξ l and al are location and intensity of l th source point on the pseudo boundary Γ′ , respectively, and

N is the number of these source points. τ * is the fundamental solution of the Laplace operator and is expressed as: −1 (7) τ * ( x, ξ l ) = ln (r ( x,ξ l ) ) 2π where r is the magnitude of the vector r = (r1 , r2 ) , which connects the source point ξ l to the point X.

τ p (x) is the particular solution and it can be obtained by constructing the associated Newton potential in the following domain integral form [9]: 1 τ p (x) = ∫ s (ξ ) τ * (x, ξ ) dV(ξ ) kΩ

(8)


185

Considering N collocation points on the main boundary and satisfying the thermal boundary conditions at these points, the coefficients al can be determined. Considering thermal effects in the Navier equation, the solution of this equation can be represented as follows: ui ( x) = uih ( x) + uip (x ) i = 1,2 (9) The components of the homogeneous displacement vector (solution of the Navier equation without thermal terms) are expressed as follows [8]: N

[

uih (x ) = ∑ clU1*i ( x, ξ l ) + cl + NU 2*i ( x, ξ l ) l =1

]

i = 1,2

(10)

where: U ij* =

 1 1 (3 - 4ν ) ln  δ ij + r,i r, j 8π (1 − ν )G  r

]

i , j = 1,2

(11)

The coefficients cl in eq (10) are unknowns which can be determined by applying boundary conditions. The components of the particular displacement vector (solution of the Navier equations with thermal terms) can be represented by: 1 N 1 uip ( x ) = ∑ al ri ( x, ξ l )τ * ( x, ξ l ) + s (ξ ) ri ( x, ξ )τ * ( x, ξ ) dV(ξ ) i = 1,2 (12) ∫ A l =1 kA Ω where: 4G (1 − v ) A= (13) γ (1 − 2ν ) The components of the traction vector at boundary points can be calculated as follows: ti ( x b ) = tih (x b ) + tip ( x b ) i = 1,2 (14)

[

]

p

where tih and ti are components of the homogeneous traction vector and the particular traction vector, respectively. The components of homogeneous traction vector can be derived from the following equation [8]: N

[

]

tih ( x b ) = ∑ cl P1*i ( x b , ξ l ) + cl + N P2*i ( x b , ξ l ) i = 1,2 l =1

(15)

where:

)]

−1 ∂r  (1 - 2ν )δ ij + 2 r, j r,i − (1 - 2ν ) r,i n j − r, j ni i , j = 1,2  4π (1 − ν ) r  ∂n in which, ni are the components of the unit vector n normal to the boundary. The components of the particular traction vector can be calculated as follows: tip ( x b ) = σ ijp ( x b ) n j ( x b ) i = 1,2

(

Pij* =

)

(

In order to calculate the components of the particular stress tensor σ relation can be used:

 

σ ijp ( x) = 2G ε ijp ( x) +

ν 1 − 2ν

p ij

(16)

(17)

at a point, the following stress-strain

 

ε kkp ( x )δ ij  − γ τ ( x )δ ij i , j = 1,2

(18)

The components of the particular strain tensor ε ijp in a point can be determined using the straindisplacement relation as:

ε ijp ( x ) =

ri ( x, ξ l ) r j ( x, ξ l )   1 p 1 N    + ui , j ( x ) + u jp,i ( x ) = ∑  al τ * ( x, ξ l )δ ij − 2 A l =1   2πr 2 ( x, ξ l )  

(

)

(19) ri ( x, ξ ) r j ( x, ξ )   * 1  dV( ξ ) i, j = 1, 2 s (ξ ) τ ( x, ξ )δ ij − kA Ω∫ 2πr 2 ( x, ξ )   For calculation of stress tensor components in a point of the domain, the following equation can be used:


186

σ ij ( x) = σ ijh ( x) + σ ijp (x ) i, j = 1,2

(20)

In which the components of the homogeneous stress tensor σ can be calculated as follows [8]: h ij

N

[

]

σ ijh ( x ) = ∑ cl Ψ1*ij ( x, ξ l ) + cl + N Ψ2*ij ( x, ξ l ) i , j = 1, 2 l =1

where:

[

(21)

]

−1 (22) ( 1- 2ν ) r, j δik + r, k δij − r,i δ jk + 2 r,i r, j r, k i , j , k = 1,2 4π (1 − ν ) r As it can be seen in equations (8), (12) and (19); the particular solutions associated with the heat source effect are represented by domain integrals. In the next section, these integrals are calculated for curved line heat sources without any need to internal cells or internal points. * Ψijk =

(

)

3. Computations for curved line heat sources Each arbitrary curved line heat source can be considered as a combination of several simple segments with quadratic variations of intensity and geometry. The domain integrals in equations (8), (12) and (19) are converted to integrals on the path of quadratic segment as follows: −1 IT ( x) = ∫ s (ξ )ln[r (x, ξ )]dl 2πk l source

I u (x) =

−1 ∫ s(ξ ) ri ( x, ξ ) ln[r (x, ξ )] dl 2πkA l source

I ε ( x) =

ri ( x, ξ ) rj ( x, ξ )   −1  dl s (ξ )  ln [r (x, ξ )]δ ij + ∫ 2πkA l source r 2 ( x, ξ )  

i = 1,2

(23)

i, j = 1,2

where dl is an infinitesimal element of the quadratic segment. Each quadratic segment is discretized by three points, i.e. ( x1, y1 ) at the starting point, ( x2 , y2 ) at the middle point, and ( x3 , y3 ) at the end point. The intensity per unit length of the heat source at these points are represented by g 1 , g 2 and g 3 , respectively [2]. The coordinates of a point on the quadratic segment can be modelled as: xs = (N1 x1 + N 2 x2 + N 3 x3 ) y s = ( N1 y1 + N 2 y2 + N 3 y3 )

(24)

where N 1 , N 2 and N 3 are the quadratic shape functions as follows: 1 1 (25) N1 = η (η − 1) N 2 = −(η + 1)(η − 1) N 3 = η (η + 1) 2 2 η is a dimensionless local coordinate aligned with the quadratic segment that varies from -1 to 1. The infinitesimal element dl can be represented as: dl = dx s + dy s = Jdη where J is the Jacobian and expressed as follows: 2

2

2

(26) 2

 dN dN 2 dN 3   dN1 dN 2 dN 3  J =  x1 1 + x2 + x3  +  y1 + y2 + y3  d η d η d η d η d η dη     The intensity function of the heat source along the quadratic segment can be written as follows: s(ξ ) = N1g1 + N 2 g 2 + N3 g3 substituting eq (26) and eq (28) into equations (23) results in:

(27)

(28)


187

 −1 1  3  ∑ N m g m  ln [r (η )]Jdη IT ( x) = ∫ 2πk −1 m =1   −1 1  3  ∑ N m g m (ri (η )ln[r (η )])Jdη I u (x) = i = 1,2 (29) ∫ 2πkA −1 m =1  r (η )r (η )   −1 1  3  ∑ N m g m  ln[r (η )]δ ij + i 2 j  Jdη i , j = 1,2 I ε ( x) = ∫ 2πkA −1 m =1 r (η )   The integrals in equations (29) can be computed using conventional numerical integration methods. It should be noted that the integrals in equations (29) will be weakly singular in the cases that the field point x is exactly on the curved line heat source and therefore, they can be calculated by various numerical methods. In other words, stress, displacement and temperature have finite values at points on the curved line heat source. 4. Numerical examples In the following examples, the state of plane strain is considered and it is assumed k = 60 W m oC ,

α = 11 .7 × 10 −6 1 o C , E = 200 GPa and ν = 0.3 . 4.1. Two semi-circular heat sources in a circular domain In the first example a circular domain which is centered at ( 0,0) with radius R = 0.5 m is considered.

Boundary conditions are τ B = 10 oC and u x = u y = 0 . According to Fig. 3a, two semi-circular heat sources which are centered at ( 0.125,0) and ( −0.125,0) with radius of r = 0.125m are considered in the domain. Their intensity functions are as follows: s1 = 16000 x + 4000 , s2 = −16000 x + 4000 (30)

Fig 3: a) The boundary conditions and the curved line heat sources in a circular domain, b) The finite element mesh In the MFS modelling of the problem, only eight boundary points and eight source points are used. Pseudo boundary is considered as a circle with radius R′ = 5 m and each semi-circular heat source is modelled with two quadratic segments. The obtained results by the present MFS, along the vertical diameter of the circle, are compared with those of the BEM and the FEM in Fig. 4. In the BEM, 8 linear boundary elements are used and each semi-circular heat source is modelled with 2 quadratic heat sources [2]. In the FEM, 3463 quadratic quadrilateral elements and 10502 nodes are used and each semi-circular segment is modelled as an area heat source over a semi-circular strip with the thickness 0.01 m. The finite element mesh is shown in Fig. 3b.

188


Fig. 4: Variations of field variables along the vertical diameter: a) temperature, b) displacement in y direction, c) stress in y direction As can be seen in Fig. 4, the presented MFS formulation yields very accurate results in comparison with the BEM and the FEM 4.2. A domain with a large number of circular heat sources In this example, a 1 × 1 m square domain is considered. In this domain, 30 circular heat sources in different locations are considered. The radius of each heat source is 0.05 m and the intensity of each heat source is 2500 W/m. The boundary conditions and the layout of the heat sources are shown in Fig. 5a.

Fig. 5: a) A square with 30 circular heat sources, b) The finite element mesh


189

In the proposed MFS, each circular heat source is modelled with 4 quadratic segments and 36 collocation points and 36 source points are considered on the main and pseudo boundaries, respectively. The layout of these points is selected according to the procedure presented in the reference [10] and is shown in Fig. 6.

Fig. 6: The MFS modelling of the square with 30 circular heat sources The results of the present MFS for different variables, along the line AB in comparison with the FEM are shown in Fig. 7. In the FEM, 7113 quadratic elements and 21214 nodes are used and each circular source is modelled as an area heat source on a ring with thickness of 0.005 m. The finite element mesh is shown in Fig. 5b.

Fig. 7: Variations of field variables along the line AB of the square: a) temperature, b) displacement in y direction, c) stress in x direction

190


5. Conclusions A formulation based on the MFS for 2D analysis of thermo-elastic problems involving curved line heat sources was presented. The shape and the intensity function of the curved line heat source can be arbitrary and sufficiently complicated. For modeling of a curved line heat source in the FEM, an area source over a narrow part of the domain must be considered and the analysis accuracy depends on the mesh density especially near the concentrated heat source. On the other hand, concentrated heat sources can be efficiently modelled in the proposed MFS without considering any internal cells or internal points. By presenting two examples, the effectiveness and efficiency of the presented MFS were demonstrated. The obtained results show that the proposed MFS is very efficient in comparison with the BEM and FEM and gives accurate results even with a small number of collocation points. References

[1] Y. C. Shiah, T. L. Guao and C. L. Tan, Two-dimensional BEM thermoelastic analysis of anisotropic media with concentrated heat sources, CMES Computer Modeling in Engineering and Sciences, 7, 321-338 (2005) [2] M. Mohammadi, M. R. Hematiyan and A. Khosravifard, Boundary element analysis of 2D and 3D thermoelastic problems containing curved line heat sources, European Journal of Computational Mechanics, 25, 147-164 (2016) [3] R. Mathon and R. L. Johnston, The approximate solution of elliptic boundary value problems by fundamental solutions, SIAM Journal of Numerical Analysis, 14, 638-650 (1977) [4] G. Fairweather and A. Karageorghis, The method of fundamental solutions for elliptic boundary value problems, Advances in Computational Mathematics, 9, 69–95 (1998) [5] A. Karageorghis, D. Lesnic and L. Marin, A survey of applications of the MFS to inverse problems, Inverse Problems in Science and Engineering,19, 309–36 (2011) [6] C.C. Tsai, The method of fundamental solutions with dual reciprocity for three dimensional thermoelasticity under arbitrary forces, Engineering Computations, 26, 229-244 (2009) [7] A. Karageorghis and Y. S. Smyrlis, Matrix decomposition MFS algorithms for elasticity and thermoelasticity problems in axisymmetric domains. Journal of Applied and Computational Mathematics, 206, 774–795 (2007) [8] L. Marin and A. Karageorghis, The MFS–MPS for two-dimensional steady-state thermoelasticity problems, Engineering Analysis with Boundary Elements, 37, 1004-1020 (2013) [9] K.E. Atkinson, The numerical evaluation of particular solutions for Poisson's equation, IMA Journal of Numerical Analysis, 5, 319-338 (1985) [10] M.R. Hematiyan, A. Haghighi and A. Khosravifard, A two-constrained method for appropriate determination of the configuration of source and collocation points in the method of fundamental solutions for 2D Laplace equation, Advances in Applied Mathematics and Mechanics, In press (2018)


191

New Approach with Domain Superposition to Solve Piecewise Homogeneous Elastic Problems L. O. C. Lara1, C. F. Loeffler1,2, J. P. Barbosa2,3, W. J. Mansur4 1

Mechanical Engineering Department, Federal University of Espírito Santo, UFES, Centro Tecnológico, Av. Fernando Ferrari, 540-Bairro Goiabeiras, 29075-910 Vitoria, ES, Brazil, [email protected] 2

Mechanical Engineering Post-Graduate Program, Federal University of Espírito Santo, UFES, Centro Tecnológico, Av. Fernando Ferrari, 540-Bairro Goiabeiras, 29075-910 Vitoria, ES, Brazil, [email protected] 3

Federal Institute of Espírito Santo, IFES, Campus São Mateus, Rodovia BR 101 Norte, Km 58 Litorâneo, 29932540, São Mateus, ES – Brazil, [email protected] 4

Civil Enginneering Department,COPPE, Federal University of Rio de Janeiro, Centro de Tecnologia, Bloco B, Ilha do Fundão, 21945-970, P.B. 68506, Rio de Janeiro, RJ, Brazil, [email protected] Keywords: Boundary Element Method, Piecewise Homogeneous Elastic Problems, Domain Superposition.

Abstract. In this work, an alternative methodology is presented for solution of piecewise homogeneous elastic problems; previously it was successfully tested on Laplace Problems, adapted here to be applied to static cases of linear elasticity. It is based on the sum of elastic energy retained in each different sector. Two examples are then solved by the proposed technique, to evaluate its applicability in this class of problems, a heterogeneous elastic sheet subjected to bending moment and a heterogeneous elastic hollow cylinder submitted to uniform pressure on the inner surface; in both cases simulated the robustness and adequacy of the proposed technique were checked. The results obtained in the present paper are compared with Finite Element Method solutions. Introduction Numerical solution of piecewise homogeneous elastic problems is preferably performed by methods that discretize the domain, such as the Finite Element Method [1,2] the Finite Difference Method [3], etc., since different values of the properties are easily introduced inside each sector or sub-domain. The use of sub-regions technique is still the most well known BEM technique to deal with sector located heterogeneity [4]. However in certain complex situations the insertion of many internal boundaries together with other phenomena produces harmful effects, such as the loss of accuracy, increase in computational cost and more elaborate programming [5]. A previous work has presented an alternative BEM approach for solving heterogeneous scalar potential problems [6] with relative simplicity. This work presents the extension of that new approach, that showed encouraging results solving scalar potential problems, which hereinafter is named Domain Superposition Technique (DST). Its idea is substantially different from the classic sub regions technique, since energy principles support the proposed procedure. Using the DST, the complete problem is modeled as a superposition of a homogeneous surrounding domain and other sub-domains with different properties. The sum of elastic energies in each sector is computed suitably generating a consistent mathematical model given in a usual form of BEM integrals. It must be highlighted that the same DST idea concerning the energy contribution of each homogeneous sector exists in problems with domain actions. Loeffler and Mansur [7] used this approach to account for sectorial loads with the Dual Reciprocity technique [8]. With this new approach (DST), all sectors are mathematically connected by means of the influence coefficients, which in the BEM standard procedure are generated by integrations performed on the sub-domains boundary, with the source points located at all nodal points, either internal or on the boundaries. The final DST H matrix is full, but its order is lower than the order of standard


192

sub region H matrix, since the tractions on internal boundaries do not appear in the final DST system. Researchers are increasingly trying to propose new formulations to solve problems with sector located heterogeneity. For example, approach concerning sectorially homogeneous scalar and vectorial problems were studied using the mathematical formalism of the Potential Theory [9-11] and of soil-structures interaction problems when FEM and BEM are employed together [12]. Cases of integration between zoned plate bending [13,14] in which the common nodal points of different systems are assembled in a general matrix, among others [15-19], were also studied. It must be highlighted that there are meaningful theoretical differences concerning the aforementioned methods and the DST technique of this work: all mentioned works do not employ the concept of energy superposition to connect suitably different sub-domains where a surrounding homogeneous domain is taken as reference. Just the idea of linkage of distinct domains using internal points is employed in the above mentioned papers. The good performance achieved in Laplace problems [6] accredits the DST for other more elaborate applications without great difficulties of implementation, such as elasticity problems. The extension of DST to this kind of problem opens an opportunity to examine later more complex cases such as plasticity and fracture mechanics problems. In this work, to evaluate the applicability the DST in piecewise homogeneous elastic problems, two examples are then solved by the proposed technique, a heterogeneous elastic sheet subjected to bending moment and a heterogeneous elastic hollow cylinder submitted just to uniform pressure on the inner surface. The results obtained in the present paper are compared with Finite Element Method solutions. Domain Superposition Technique (DST) Consider a domain consisting of two regions with different physical properties, as shown in Fig. 1, in which the complete domain Ω(X) is composed of the sum of Ωe and Ωi; both λe, λi, µe and µi are physical properties, constant inside each sub-domain. In this formulation, unlike what is done in the traditional sub-regions approach, a complete or surrounding domain with homogeneous properties is elected and the other sub domains are correlated with it.

Ωe µe, λe

i

i

i

Ω, µ, λ

µe, λe Ωe+Ω Ωi=Ω Ω

=

+

Ωi

µi-µ µe λi- λe

Figure 1. Complete and sectorial domains with homogeneous properties. Considering that the kernel of the integrals is comprised by integrable functions, the following integral equation can be exposed based on the domain division concept exposed above:

µ e ∫ u e j ( X),ii u *j (ξ ; X)d Ωe ( X) + (λ e + µ e ) ∫ u ei ( X),ij u *j (ξ ; X)d Ωe ( X) + Ωe

Ωe

+ µ i ∫ u i j ( X),ii u *j (ξ ; X) d Ωi ( X) + (λ i + µ i ) ∫ u i i ( X),ij u *j (ξ ; X)d Ωi ( X) = 0 Ωi

(1)

Ωi

Performing some simple mathematical operations considering that λi=λe+λ* and µi=µe+µ*, the following integral equation can be achieved:

µ e ∫ u e j ( X),ii u*j (ξ ; X)d Ωe ( X) + (λ e + µ e ) ∫ u ei ( X),ij u*j (ξ ; X)d Ωe ( X) + Ω

+µ

*

Ω i i * ∫ u j (X),ii u (ξ ; X)d Ω (X) + (λ + µ ) ∫ u i (X),ij u j (ξ ; X)d Ω (X) = 0 i

Ωi

* j

i

*

*

Ωi

(2)


193

In Eq. (2),the auxiliary function uj* (ξ;X) is the Kelvin fundamental solution [4] and ui(X) represents the vector component of the displacement field in “i” direction and X = X (x1, x2). Possible body forces are not considered here. Equation (2) synthesizes the DST aim: the problem as a whole can be analyzed as a superposition of a contribution related to a surrounding homogeneous domain and other sectors that are also homogeneous. Due to the BEM features, this contribution is given in term of balance of elastic energy. Equation (2) can be rewritten, after application of the Divergence Theorem [4,20] and, for convenience, the adoption of dyadic structure for the fundamental solution and its associated traction derivative, denoted respectively uij* and pij* to represent displacements and tractions generated in the direction j at field point X, as results of a unit load acting in the direction i applied at source point . When a dyadic coefficient Cij is introduced as a function of the positioning of the source point (if it lies within the domain, outside it or exactly on the boundary), the complete inverse boundary integral term takes the following form:





µ e  − C ij (ξ ) u j (ξ ) + ∫ p j ( X ) u ij* (ξ ; X ) d Γ ( X ) − ∫ u j ( X ) p ij* (ξ ; X )d Γ ( X )  + 

Γ



Γ

  + µ *  − C iji (ξ ) u ij (ξ ) + ∫ p ij ( X ) u ij* (ξ ; X ) d Γ i ( X ) − ∫ u ij ( X ) p ij* (ξ ; X )d Γ i ( X )  = 0  Γi Γi 

(3)

Thus, since the different properties µ are given explicitly in Eq. (3), the dyadics u ij* and p ij* in two-dimensional problems are given by [4]: u ij* =

p ij* =

1 8πµ (1 − ν

  1  ( 3 − 4ν ) ln  r  δ ij + r, i r, j  )   

−1  ∂r  ((1 − 2ν )δ ij + 2 r, j r,i ) − (1 − 2ν )( r,i n j − r, j n i )   4π (1 − ν ) r  ∂ n 

(4)

(5)

BEM inverse integral equation can be interpreted according energy principles: there is equilibrium of elastic energy stored and the work done by the applied tractions. In the proposed method energy present in each subdomain is accounted only by elastic energies contained in each sub-domain, in an approach to that concerning that problems in which there are body forces or any other domain loading. The main difference is related to the values of body forces which are usually known while the displacements on the internal boundaries are obtained after the discretization and complete solution of the BEM system. Notice that the work of tractions on internal boundaries is not null, but the energy contribution of each sub domain in the energy balance in the whole system - which is not yet known – is done just advantageously by the elastic energy, since it is given as a function of displacements. Thus, being the energy stored in the internal sector given just by the elastic energy, which is represented in terms of the values of displacement u ij at internal source points, Eq. (3) can be simplified so that:

  * *  − C ij (ξ ) u j (ξ ) + ∫ p j ( X ) u ij (ξ ; X ) d Γ ( X ) − ∫ u j ( X ) p ij (ξ ; X )d Γ ( X )  = Γ Γ    µ*  + e  ∫ u ij ( X ) p ij* (ξ ; X )d Γ i ( X )  µ  Γ i 

(6)

In this numerical procedure, the displacement at internal points should be calculated simultaneously with boundary nodal unknowns appearing explicitly in the final BEM system, as shown in Eq. (7):

 H cc  H  ic

u  H ci   c   G cc    = H ii   u   G ic  i

p  0 ci   c     0 ii   p   i 

(7)


194

Where coupling submatrix Hic and Hci coefficients are all non-null. In Eq. (7), ui are the displacement at internal points that define the sectorial boundary Γi(X) of the region Ωi(X); uc and pc are values at nodal points on the boundary Γ(X). Such as occur with the sectorial body loads, it is necessary to transmit information from the sectorial domain Ωi(X) to the complete domain Ω(X); thus, the submatrix Hci represents coefficients generated by integration on internal boundaries Γi(X) with source points based on boundary nodes belonging to surrounding boundary Γ(X). It must be highlighted a particular aspect related to stress calculations, both at internal points and at the sectorial nodal points as well: These values must be found solving a new BEM problem that is geometrically defined by each sub domain. Numerical Experiments Two examples are then solved by the proposed technique. Sectors with different physical properties were considered. To validate the results obtained by means of the new approach (DST), results of both examples were compared with the results obtained by the Finite Element Method (FEM). Heterogeneous elastic sheet. The first example considers a rectangular geometry with physical property µ1, in which a circle with distinct property exists inside it. Figure 2 shows the geometry of the problem, as well as the boundary conditions and physical properties values of each sector.

L=12

A

P=0.12

P=0.12

y

µ1=1

B R=1.2

W=6

µ2=2

x

P

P

Figure 2. Geometric characteristics and boundary conditions for the heterogeneous elastic sheet. Figures 3 and 4 shows the results of displacements obtained by FEM (10000 elements) and BEM methods, where two meshes were used for simulation (92 and 224 elements), the displacements were obtained along the region indicated in Figure 2 by the letters A and B. 8.00E-01

8.00E-01

6.00E-01

FEM

FEM

DST (92 elements)

4.00E-01

4.00E-01

2.00E-01

2.00E-01

Ux

Ux

6.00E-01

0.00E+00

DST (224 elements)

0.00E+00

-2.00E-01

-2.00E-01

-4.00E-01

-4.00E-01

-6.00E-01

-6.00E-01 -8.00E-01

-8.00E-01 -6

-4

-2

0 x

2

4

6

-6

-4

-2

0 x

2

4

6

Figure 3. Comparison between displacements, obtained by FEM and BEM (DST), along the boundary A for the heterogeneous elastic sheet.

2.50E-02 2.00E-02 1.50E-02 1.00E-02 5.00E-03 0.00E+00 -5.00E-03 -1.00E-02 -1.50E-02 -2.00E-02 -2.50E-02

FEM

195

DST (92 elements)

Ux

Ux


-1.5

-1

-0.5

0 y

0.5

1

1.5

2.50E-02 2.00E-02 1.50E-02 1.00E-02 5.00E-03 0.00E+00 -5.00E-03 -1.00E-02 -1.50E-02 -2.00E-02 -2.50E-02

FEM

-1.5

-1

DST (224 elements)

-0.5

0 y

0.5

1

1.5

Figure 4. Comparison between displacements, obtained by FEM and BEM (DST), along the internal interface B for the heterogeneous elastic sheet. It must be highlighted that, as expected, the extension of the DST to elastic problems shows encouraging results, since it is possible to verify a very good agreement between the DST and the results obtained by the Finite Element Method. It can be seen very accurate results obtained by using both meshes, 92 and 224 elements, even with the BEM having much less nodal points than the FEM. Heterogeneous elastic hollow cylinder. The second example considers a thick hollow cylinder that has the domain composed of three sectors submitted just to uniform pressure on the inner surface. Figure 5 shows the geometry of the problem, as well as the boundary conditions and property values of each sector.

Figure 5. Geometric characteristics and boundary conditions for the heterogeneous elastic hollow cylinder. Figure 6 detects the results of displacements by FEM (10000 elements) and BEM methods, where two meshes were used for simulation (62 and 155 elements). The displacements were calculated along the horizontal line at y=0. Confirming the good level of numerical results obtained in previous simulations [6], in which just Laplace Problems were solved, the DST approach in the simulation of piecewise homogeneous elastic problems, in particular heterogeneous elastic hollow cylinder, have shown excellent performance. Similarly to first example, it can be seen very accurate results obtained by using both meshes, 62 and 155 elements.



5.00E-01

4.50E-01

4.50E-01 FEM

4.00E-01

DST (62 elements)

FEM

4.00E-01

DST (155 elements)

3.50E-01 Ux

3.50E-01 Ux

196

5.00E-01

3.00E-01

3.00E-01

2.50E-01

2.50E-01

2.00E-01

2.00E-01 1.50E-01

1.50E-01 0.4

0.5

0.6

0.7 x

0.8

0.9

1

0.4

0.5

0.6

0.7 x

0.8

0.9

1

Figure 6. Comparison between displacements, obtained by FEM and BEM (DST), along y=0 for the heterogeneous elastic hollow cylinder. Results obtained with the solution of the second example confirm again the performance of the new approach (DST), considering the FEM method as reference relative to the BEM. Conclusions The results of the examples tested show that the proposed approach presents itself as an important alternative to the standard sub-regions technique in the context of the boundary element method. DST performance confirmed that the basic principles of the DST can be extended to many other kind of elastic problems. It is possible to consider that this technique can be successfully extended to the solution of Elastoplastic and Fracture Mechanics problems. The promising potential of the technique also extends to other special cases of mechanics of great importance, such as seismic analysis. In addition to good accuracy, the DST shows a huge advantage in terms of computational implementation and data input. Despite the model construction of a full boundary element H matrix, its size is reduced in comparison with the size of H matrix achieved through the classical sub-regions technique.

Acknowledgements This research was supported by the National Council for Scientific and Technological Development – CNPq (Brazil).

References [1] J. Reddy An Introduction to the Finite Element Method, McGraw-Hill, New York (2005). [2] O. Zienkiewcz, R. Taylor, J.Z. Zhu The Finite Element Method: Its Basis and Fundamentals, 7th Ed. (2013). [3] R. LeVeque Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems, SIAM, Philadelphia (2007). [4] C.A. Brebbia, J.C.F. Telles, L.C. Wrobel Boundary Element Techniques, Springer-Verlag, Berlin (1984). [5] L. Xiaoping, W. Wei-Liang A new sub-region Boundary Element Technique based on the Domain Decomposition Method. Engineering Analysis with Boundary Elements, v. 29, p. 944–952 (2005). [6] C.F. Loeffler, W.J. Mansur Sub-regions without subdomain partition with boundary elements. Engineering Analysis with Boundary Elements, v. 71, p.169-173 (2016). [7] C.F. Loeffler, W.J. Mansur Analysis of time integration schemes for boundary element applications to transient wave propagation problems, in: C.A. Brebbia (Ed.), Boundary Element Techniques: Applications in Stress Analysis and Heat Transfer, Computational Mechanics Publishing, UK, p. 105-124 (1987).


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[8] P.W. Partridge, C.A. Brebbia, L.C. Wrobel The Dual Reciprocity, Boundary Element Method, Computational Mechanics Publications and Elsevier, London (1992). [9] O.D. Kellog Foundations of Potential Theory, Dove, New York (1953). [10] C. Pozrikidis Bodundary Integral and Singularity Methods for Linearized Viscous Flow, Cambirdge University Press, New York (1992). [11] C. Pozrikidis Reciprocal identities and integral formulations for diffusive scalar transport and Stokes flow with position-dependent diffusivity or viscosity, Journal of Engineering Mathematics, v. 96, p. 95–114 (2016). [12] D. Soares, O. Von Estorff, W. J. Mansur Iterative coupling of BEM and FEM for nonlinear dynamic analyses, Computational Mechanics, 34, v. 1, p. 67-73 (2004). [13] J. B. Paiva, W. S. Venturini Plate Bending analysis by the boundary element method considering zoned thickness domain, Soft. Eng. Workstations, v. 4, 183-185 (1988). [14] J. B. Paiva, W. S. Venturini Alternative boundary element approach to compute efforts along zoned domain interfaces. Eng. Analysis with Boundary Elements, v. 12, pp. 143-148 (1993). [15] J. B. Paiva, W. S. Venturini Analysis of building structures considering plat-beam-column interactions, in Boundary Element Technique: application in stress Analysis and Heat Transfer, ed. C. A. Brebbia and W. S. Venturini, CML Publisher (1987). [16] A. Reis, E. L. Albuquerque, L. Torsani, L. Palermo Jr., P. Sollero Computation of moments and stresses in laminated composite plates by the Boundary Element Method, Engineering Analysis with Boundary Elements, v. 35, p. 105-113 (2011). [17] W. S. Venturini Alternative Formulations of the Boundary Element Method for Potential and Elastic Zoned Problems, Engineering Analysis with Boundary Elements, 9, p. 203–207 (1992). [18] L. G. S. Leite, W. S. Venturini Boundary element formulation for 2D solids with stiff and soft thin inclusions, Analysis with Boundary Elements, v. 29, p. 257–267 (2005). [19] M. Wagdy, Y. F. Rashed Boundary Element analysis of multi-thickness shear-deformable slabs without subregions, Engineering Analysis with Boundary Elements, v. 43, p. 86–94 (2014). [20] J. T. Katsikadelis Boundary elements: theory and applications, Elsevier (2002).

198



199

A ouple-stress formulation for the boundary element method 1,2∗

G. Hattori

1

1

1

, P.A. Gourgiotis , J. Trevelyan

Department of Engineering, Durham University, DH1 3LE, Durham, UK.

2 Department

of Engineering, University of Cambridge, CB2 1PZ, Cambridge, UK

gh465 am.a .uk, panagiotis.gourgiotisdurham.a .uk, jon.trevelyandurham.a .uk

Keywords: ouple-stress, size ee ts, mi rome hani s. Abstra t. A boundary element methodology is developed for the analysis of two-dimensional linear elasti bodies within the ontext of ouple-stress theory. This theory is the simplest generalised ontinuum theory that an ee tively model size ee ts in elasti solids. The ouple-stress theory, whose origins go ba k to the beginning of the last entury, has attra ted a renewed and growing interest during re ent years. The ouple-stress fundamental solutions are expli itly derived and used to onstru t the boundary integral equations of the problem with the aid of the re ipro al theorem. A new boundary integral equation is also required to model the rotations and moments. We validate the formulation using an analyti al solution from the literature. Introdu tion It is well known that the ma ros opi al behaviour of most mi rostru tured materials with non-homogeneous mi rostru ture, like erami s, omposites, bones and foams is strongly inuen ed by their material

hara teristi lengths, espe ially in the presen e of large stress (or strain) gradients [1℄. Relevant size ee ts have been found when the representative s ale of the deformation eld be omes omparable to the length s ale of the mi rostru ture [2℄. Unfortunately, the lassi al theory of elasti ity is inherently size independent and annot model properly su h situations. To ir umvent this di ulty, a generalised ontinuum approa h is proposed in the present study based on the theory of ouple-stress elasti ity (also known as onstrained Cosserat theory) [3, 4℄. This theory may be viewed as a rst step generalisation of lassi al elasti ity theory where the strain-energy density and the resulting onstitutive relations depend, besides the usual innitesimal strains, on rotation gradients. Chara teristi material lengths are introdu ed through the onstitutive formulation showing that the ouple-stress theory en ompasses the analyti al possibility of size ee ts, whi h are absent in the lassi al theory. In the present work, a boundary element method (BEM) is developed for plane-strain problems in isotropi Cosserat materials. The Green's fun tions for a on entrated for e and moment are derived in losed form. A new boundary integral equation arises sin e the rotation and the displa ements are independent on the boundary of the body. To validate our formulation, the ben hmark problem of an innite plate with a hole is solved and the ee ts of the pertinent Cosserat lengths upon the ma ros opi response of the material are investigated.

Governing equations For a body that o

upies a domain in the (x1 ,x2 )-plane under onditions of plane-strain, the equations of equilibrium assume the following form:

σ11,1 + σ21,2 + b1 = 0

(1)

σ12,1 + σ22,2 + b2 = 0

(2)

σ12 − σ21 + m13,1 + m23,2 + Y3 = 0

(3)

where σ represents the stress tensor, m13 and m23 stand for the non-vanishing omponents of the

ouple-stress tensor, bj are the body for es and Y3 is the body moment, and (, i) represents the partial derivative with respe t to xi . Remark that the variables Y3 and mi3 ) have the subs ript index 3 sin e all rotation in a plane strain problem are taken with respe t to the x3 -axis.


200

The kinemati relations are given by:

ε11 = u1,1 ; ε22 = u2,2 ; ε12 = ε21 = ω=

1 (u1,2 + u2,1 ) 2

(4)

1 (u2,1 − u1,2 ) ; κ13 = ω,1 ; κ23 = ω,2 2

(5)

where εij is the strain tensor, ui are the displa ements, ω represents the rotation with respe t to the x3 -axis and κ13 and κ23 are the non-vanishing omponents of the urvature tensor. The onstitutive equations in the ouple-stress theory are [5℄:

σ11 = (λ + 2µ)ε11 + λε22

(6)

σ22 = (λ + 2µ)ε22 + λε11

(7) (8)

σ12 + σ21 = 4µε12 2

(9)

2

(10)

m13 = 4µℓ κ13 m23 = 4µℓ κ23

where λ and µ are the Lamé moduli, and ℓ is the hara teristi material length. Combining Eqs. (1-10), we obtain the following system of fourth order partial dierential equations for the displa ements

1 µ∇2 u1 + (λ + µ)d,1 + 2µℓ2 (∇2 ω),2 + b1 + Y3,2 = 0 2

(11)

1 µ∇2 u2 + (λ + µ)d,2 + 2µℓ2 (∇2 ω),1 + b2 − Y3,1 = 0 2 where d is the dilatation as is dened as

(12)

d=

∂u1 ∂u2 + ∂x1 ∂x2

(13)

The boundary element method (BEM) for the ouple-stress theory The boundary element method (BEM) has be ome a popular method sin e the dis retisation is only required at the boundaries. This feature redu es the size of the problem to be solved, ompared to domain dis retisation methods su h as the nite element method (FEM). The displa ement boundary integral equation (DBIE) for the ouple-stress formulation is given by Z Z Z Z F F F cij (ξ)ui (ξ)+− Tij (x, ξ)ui (x) dΓ(x)+ Mj (x, ξ)ω(x) dΓ(x) = Uij (x, ξ)ti (x) dΓ+ ΩFj (x, ξ)m(x) dΓ(x) Γ

Γ

Γ

Γ

(14) where Γ stands for the boundaries of the domain Ω; − denotes the Cau hy Prin ipal Value (CPV) integration; ti are the tra tions; m represents the moment tra tion; cij is the free term deriving from the CPV integration of the strongly singular kernels TijF ; UijF and TijF are the displa ement and tra tion fundamental solutions, respe tively, while MjF and ΩFj are the moment and rotation fundamental solutions. Note that the supers ript F relates to a on entrated unit for e load a ting in an innite domain for the fundamental solutions present in the DBIE. A new boundary integral equation is ne essary to model the rotation of the problem. The rotation boundary integral equation (RBIE) is dened as Z Z Z Z C C C c(ξ)ω(ξ)+ Ti (x, ξ)ui (x) dΓ(x)+− M (x, ξ)ω(x) dΓ(x) = Ui (x, ξ)ti (x) dΓ+ ΩC (x, ξ)m(x) dΓ(x)

R

Γ

Γ

Γ

Γ

(15) where TiC , UiC , M C and ΩC are the tra tion, displa ement, moment and rotation fundamental solutions arising from an out-of-plane on entrated unit moment about the x3 axis in an innite domain. The additional free term c(ξ) arises from the strongly singular kernel M C . The fundamental solutions for the ouple-stress theory are detailed in the next se tion.


201

Fundamental solutions The fundamental solutions were obtained using the plane-strain assumptions and an isotropi ouplestress material. As mentioned previously, the Green's fun tions are al ulated for two dierent loading

onditions: one for the in-plane on entrated unit for e (as in lassi al elasti ity) and one for the out-of-plane on entrated unit moment. The fundamental solutions arising from the unit for e are given as: 2 h r i 1 hri 1 1 1 2ℓ − K2 Uik = [r,i r,k − (3 − 4ν) log rδik ] − r,i r,k − δik + K0 δik 8πµ(1 − ν) 2πµ 2 r2 ℓ 2 ℓ (16) h i eqk r,q ℓ r Ωk = − − K1 (17) 4πµℓ r ℓ 1 1 Σijk = [(1 − 2ν)(r,j δik + r,i δkj − r,k δij ) + 2r,i r,j r,k ] − (r,j δik + r,j δkj + r,k δij − 4r,i r,j r,k ) 4π(1 − ν)r πr 2 h r i hri 2ℓ 1 − K (r r r − r δ )K (18) − 2 ,i ,j ,j 1 ,k ik r2 ℓ πℓ ℓ 2 h r i e ℓ ℓ h r i eqk 2ℓ ik Mi3k = r,i r,q − K2 − − K1 (19) π r2 ℓ πr r ℓ where Kn [x] is the modied Bessel fun tion of n-th order, δij is the Krone ker delta, ri = xi − ξi , |r| = |x − ξ| is the distan e between ollo ation and observation points, and eij stands for the 2D permutation tensor (e11 = e22 = 0, e12 = −e21 = 1). In the same way, the fundamental solutions for the unit moment applied in the x3 -axis are dened by: h r i eiq r,q r Ui3 = − 1 − K1 (20) 4πµr ℓ ℓ hri 1 Ω3 = K (21) 0 8πµℓ2 ℓ 2 hri hr i eij 2ℓ 1 (e r r + e r r ) − K − K Σij3 = (22) iq ,j ,q jq ,q ,i 2 0 4πℓ2 r2 ℓ 4πℓ2 ℓ hri r,i Mi33 = − K1 (23) 2πℓ ℓ The tra tions TijF and T C and moments MjF and M C are obtained by multiplying the stress tensor Σijk and moment tensor Mi3k by the unit outward normal n.

Regularisation The DBIE and RBIE present strong and weak singularities that must be dealt before a numeri al integration s heme an be applied. In the ase of weak singularities, the Telles transformation [6℄ provides a simple regularisation pro edure to remove the O(ln r) singularity, making possible to use a regular Gauss-Legendre quadrature. For the strongly singular kernels, there is a singularity of O(1/r). In this ase, we employ the rigid body motion and rotation assumptions. These pro edures avoid the al ulation of the singular integral in an expli it form, as performed by Guiggiani [7℄ using the singularity subtra tion te hnique. The rigid body motion is explained in detail in [8℄. The rigid body rotation follows the same idea for a xed rotation with respe t to a given point in spa e. In this work we assume a rotation with respe t to the origin of the oordinate system.

Hole in an innite domain subje ted to unixial/bi-axial loading We validate our BEM formulation for ouple-stress using the analyti al solution proposed by Mindlin [9℄, who has investigated the ouple-stress inuen e in the stress eld for an innite plate ontaining a hole of radius a for uniaxial and biaxial loading onditions. The problem is illustrated in Figure 1.


σq

a σp

Figure 1: Innite plate with a hole.

ngential stress -

0

2.9

−0.1

σθθ

3

2.8

Min tangential stress -

σθθ

We model the innite plate as a square plate of dimension w where w = 500a. The material properties are E = 10000 (Young's modulus) and ν = 0.33 (Poisson's ratio) with a = 1. Initially we investigate the uniaxial problem (σp = 0, σq = 1) where we al ulate the maximum and minimum tangential stress at the hole and ompare with the analyti al solution of [9℄ for dierent values of the length s ale parameter ℓ. The results are shown in Figures 2(a) and 2(b).

Max tangential stress -

gential stress -

σp

σq

ag repla ements

gential stress -


202

2.7

2.6

PSfrag repla ements

2.5


2.4

2.3

Min tangential stress 2.2


2.1

2 0 10

1

2

a/ℓ

10

(a) Maximum stress

−0.2

−0.3

−0.4

−0.5

−0.6

−0.7

−0.8

−0.9

BEM Analyti al [9℄ 10

BEM Analyti al [9℄

3

10

−1 0 10

1

10

2

a/ℓ

10

3

10

(b) Minimum stress

Figure 2: Tangential stress on the hole due to an axial loading. For a/ℓ ≥ 100, the ouple-stress solution oin ides with the lassi al elasti ity one (σmax = 3, σmin = −1), and when the length s ale size is similar to the hole radius, one an verify a redu tion of the stresses in the hole due to the stiening of the material. An ex ellent agreement is a hieved between the BEM ouple-stress and the analyti al solution. Next, we study the biaxial problem, assuming σp = 1 and σq = 2. The results are illustrated in Figures 3(a) and 3(b). Again, an ex ellent agreement is obtained between the BEM ouple-stress and


the analyti al solution. 5

4.8

Min tangential stress -

gential stress -

σm

σn Max tangential stress -

gential stress -

2

BEM Analyti al [9℄

4.9

ag repla ements

gential stress -

203

4.7

4.6

PSfrag repla ements

4.5


4.4

Min tangential stress 4.3

4.2


1.8

1.6

1.4

1.2

1

4.1

4 0 10

BEM Analyti al [9℄ 1

10

2

a/ℓ

10

(a) Maximum stress

3

10

0.8 0 10

1

10

2

a/ℓ

10

3

10

(b) Minimum stress

Figure 3: Tangential stress on the hole due to a biaxial loading.

Con lusions We presented a formulation for ouple-stress elasti ity in the BEM framework. New fundamental solutions have been obtained in losed form. The formulation has been validated against an analyti al solution, illustrating the a

ura y of the proposed approa h. Future work will involve the development of an isogeometri formulation to obtain a smoother representation of the variables.

A knowledgements The rst author a knowledges the Fa ulty of S ien e, Durham University, for his Postdo toral Resear h Asso iate funding.

Referen es [1℄ R. Maranganti and P. Sharma. Length s ales at whi h lassi al elasti ity breaks down for various materials. Physi al Review Letters, 98:195504, 2007. [2℄ N. A. Fle k and J. W. Hut hinson. Strain gradient plasti ity. Advan es in Applied Me hani s, 33:295361, 1997. [3℄ R. D. Mindlin and H. F. Tiersten. Ee ts of ouple-stresses in linear elasti ity. Ar hive for Rational Me hani s and Analysis, 11:415448, 1962. [4℄ R. A. Toupin. Elasti materials with ouple-stresses. Ar hive for Rational Me hani s and Analysis, 11:385414, 1962. [5℄ M. Sokolowski. Theory of Couple-Stresses in Bodies with Constrained Rotations. Springer, 1970. [6℄ J. C. F. Telles. A self-adaptive o-ordinate transformation for e ient numeri al evaluation of general boundary element integrals. International Journal for Numeri al Methods in Engineering, 24(5):959973, 1987. [7℄ M. Guiggiani. Formulation and numeri al treatment of boundary integral equations with hypersingular kernels. Singular integrals in boundary element methods, pages 85124, 1998.

204


[8℄ C. A. Brebbia and J. Domínguez. Boundary Elements: An Introdu tory Course (se ond edition). Computational Me hani s Publi ations, 1992. [9℄ R. D. Mindlin. Inuen e of ouple-stresses on stress on entrations. Experimental Me hani s, 3(1):17, 1963.


205

Boundary element contact modelling of CNT-polymer reinforced composites L. Rodr´ıguez-Tembleque1 , E. Garc´ıa-Mac´ıas2 and A. Sáez3 Escuela Técnica Superior de Ingenier´ıa, Universidad de Sevilla, Camino de los descubrimientos s/n, E41092 Sevilla, SPAIN 1 [email protected], 2 [email protected], 3 [email protected]

Keywords: Carbon Nanotube, Nanocomposites, Indentation, Frictional Contact, Boundary Element Method.

Abstract. The unique intrinsic physical properties, particularly rigidity and strength-to-weight ratio, of carbon nanotubes (CNTs) suggest that they are ideal fillers for high performance composites. Although some studies have revealed the promising potential of these nanoparticles to tailor the tribological properties of polymer-based composites, the number of theoretical studies on the characterization of the frictional behavior of carbon nanotube reinforced composites is still very scant. In this work, a 3D boundary element formulation for contact modeling is applied to study the indentation response of CNT-polymer nanocomposite films. The effective elastic properties of the composites are estimated by an Eshelby-Mori-Tanaka micromechanical constitutive model. In addition, the homogenization framework has been extended in order to account for waviness and agglomeration effects. Detailed parametric analyses are presented to illustrate the influence of micromechanical features such as fiber content, orientation, waviness and dispersion. The simulation results demonstrate that the indentation response of CNT-polymer nanocomposites can be optimized by controlling the micromechanical characteristics. Introduction. Governing equations This work studies the indentation response of a CNTRC halfspace (HS) domain. So, as Fig. 1 shows, two elastic solids Ωl ⊂ R3 (l = 1, 2) in contact are considered. Two kind of CNTs distributions are considerec on Ω1 : random distribution of CNTs (RCNT) or aligned distribution of CNTs (ACNT). The boundary of Ωl , ∂Ωl , is divided into three disjoint parts: ∂Ωlt with prescribed tractions t˜li (i = 1, 2, 3), ∂Ωlu with imposed displacements u ˜li and ∂Ωlc represents the potential contact surfaces, which have l outward unit normal vectors ni . Under small displacement and strain assumption, these boundaries are almost coincident (i.e. ∂Ω1c ≃ ∂Ω2c ) so we can define a common contact surface ∂Ωc with a normal vector nc,i ≃ n1i ≃ −n2i . Moreover, the infinitesimal strain tensor εij can be obtained from derivatives of the displacements field ui in Ω1 ∪ Ω2 as: εij = (ui,j + uj,i )/2 in Ω1 ∪ Ω2 . Assuming static loading conditions, the mechanical equilibrium equations on the domains Ωl , in the absence of body forces, are σij,j = 0 σij nlj = t˜i 1n 2 σij c,j = −σij nc,j = pi

in Ω1 ∪ Ω2 , on ∂Ω1t ∪ ∂Ω2t , on ∂Ωc ,

(1)

σij being the components of Cauchy stress tensor, nli the unit normal on ∂Ω1t ∪ ∂Ω2t , pi is the contact 1 and σ 2 are restrictions of σ to a particular domain Ωl . Finally, the stress and strain traction and σij ij t ij tensors for a general anisotropic linear elastic material are coupled through the linear constitutive law σij = Cijkl εkl ,

(2)

where Cijkl denotes the components of the elastic stiffness tensor, which satisfies the following symmetries: Cijkl = Cjikl = Cijlk = Cklij and it is positive definite. The effective stiffness tensor for Ω1 is provided by an Eshelby-Mori-Tanaka micromechanical constitutive model and which is going to be explained later on.

206


Figure 1: Indentation response of a CNT reinforced composite halfspace (HS) domain. Two kind of CNTs distributions are considered on Ω1 : random distribution of CNTs (RCNT) or aligned distribution of CNTs (ACNT). Contact boundary conditions Unilateral contact law. The unilateral contact conditions in ∂Ωc can be defined as: gn ≥ 0, pn ≤ 0, gn pn = 0, where pn = p · nc is the normal contact pressure and gn = (go + un ) is the contact gap, go being the initial gap between the bodies and un = (u2 − u1 ) · nc being the relative displacements. These normal contact constraints can be reformulated in a more compact form as: pn − PR− (p∗n ) = 0,

(3)

where PR− (•) is the normal projection function (PR− (•) = min(0, •)), p∗n = pn +rn gn is the augmented normal traction, where rn is a penalization parameter (rn ∈ R+ ). Frictional contact law. The experimental works [1, 2] shows that ACNT reinforced composites exhibit a frictional response which is clearly affected by the fiber orientation relative to the sliding direction. Consequently, an anisotropic friction law for fiber-reinforced composite materials [3] should be considered. The ACNT fibers can be oriented with any angle (0 ≤ φ ≤ π/2) relative to direction e1 , as defined in Fig.2 (a). So the principal friction coefficients in the directions {e1 , e2 } can be defined as: ˆ µ1 = µL + (µN − µL ) φ, (4) ˆ µ2 = µT + (µN − µT ) φ,

(5)

where µL , µT and µN are the friction coefficients in longitudinal, transverse and normal direction, respectively (see Fig.2 (b)), and φˆ = 2φ/π is the nondimensional fiber orientation angle (0 ≤ φˆ ≤ 1). The friction coefficients {µL , µT , µN } can be obtained from experimental analysis, in consonance with [1, 2]. Consequently, the frictional response of the CNTRC depends on the sliding direction (ψ) and ˆ So, an orthotropic friction law, whose principal friction coefficients the CNT fiber orientation (φ). are µ1 = µL and µ2 = µT , is obtained when the fibers are parallel to the sliding plane (φˆ = 0). An isotropic friction law (i.e. µ1 = µ2 = µN ) is obtained when the fibers are normal to the sliding plane (φˆ = 1). So the Coulomb friction restriction defines the admissible region for tangential contact tractions as: ˆ = ||pt || ˆ − |pn | = 0, f (pt , pn , φ) (6) µ(φ)


207

(a)

(b) Figure 2: (a) Arbitrary ACNT fibers direction. (b) Schematic diagram of ACNT reinforced composite indicating the longitudinal, transverse and normal sliding directions. where pt = p − pn nc and || • ||µ denotes the elliptic norm ||pt ||µ = restrictions can be summarized as: ||pt ||µ < |pn | ⇒ g˙ t = 0 on

q (pe1 /µ1 )2 + (pe2 /µ2 )2 . These

∂Ωc ,

||pt ||µ = |pn | ⇒ pt = −|pn |M2 g˙ t /||g˙ t ||∗µ on

(7) ∂Ωc .

(8)

For quasi-static frictional contact problems, the tangential slip velocity g˙ t defined in the above expressions can be expressed at time τk as follows: g˙ t ≈ ∆gt /∆τ , where ∆gt = gt (τk ) − gt (τk−1 ), ∆τ = τk − τk−1 , and gt = (u2 − u1 ) − un nc , being zero the initial tangential slip. In Eq. (8), the value for the tangential contact traction was presented in [3] assuming an associated sliding rule, the p norm || • ||∗µ is dual of || • ||µ i.e. ||g˙ t ||∗µ = (µ1 g˙ e1 )2 + (µ2 g˙ e2 )2 , and M is a diagonal matrix: M = diag(µ1 , µ2 ). Finally, proceeding in the same way as in the normal contact law, the frictional contact constraints (7–8) are rewritten in a more compact form as: pt − PEρ (p∗t ) = 0,

(9)

where p∗t = pt − rt M2 g˙ t (rt ∈ R+ ) is the augmented tangential traction and PEρ (•) : R2 −→ R2 is the tangential projection function: PEρ (p∗t ) = p∗t , if ||p∗t ||µ < |PR− (p∗n )|, and PEρ (p∗t ) = ρ p∗t /||p∗t ||µ if ||p∗t ||µ ≥ |PR− (p∗n )|, being ρ = |PR− (p∗n )|. Micromechanics effective moduli of CNT In this work, the macroscopic elastic moduli of CNT-reinfoced polymer composites are computed by mean-field homogenization approaches. In particular, two different filler arrangements are considered, namely composites doped with uniformly distributed straight CNTs and composites doped with heterogeneous dispersions of CNTs. Throughout this section, a boldface letter stands for fourth-order tensor, and a colon between two tensors denotes inner product, A : B ≡ Aijkl Bklmn . Composite reinforced with aligned straight CNTs. Let us define the Representative Volume Element (RVE) of a polymer matrix doped with a sufficient number of CNTs in such a way that, from a statistical perspective, the RVE represents the composite as a whole [4]. The matrix is defined as isotropic with Young’s modulus Em and Poisson’s ratio νm , and perfect bonding between phases is assumed. In order to define the orientation of the inclusions, a local coordinate system {1′ , 2′ , 3′ } is



x3

Matrix x1

2

,

,

2

α1

1

0 α2

3

x2

1

3

,

CNT

Figure 3: RVE of polymer matrix doped with straight CNTs. fixed at each filler as represented in Fig. 3. The constitutive matrix for inclusions with transversely isotropic properties (1’ is the axis of material symmetry) in the local coordinate system takes the form:   nr lr kr − mr 0 0 0  lr kr + mr lr 0 0 0   kr − mr lr kr + mr 0 0 0    Cr =  (10) 0 0 0 mr 0 0     0 0 0 0 pr 0  0

0

0

0

0 pr

where kr , lr , mr , nr and pr are fiber Hill’s elastic moduli [5]. In this context, the effective stiffness tensor provided by the Eshelby-Mori-Tanaka model can be written as: C = Cm + fr h(Cr − Cm ) : Ai : (fm I + fr hAi)−1

(11)

where fr and fm denote the filler and matrix volume fractions, respectively, I the fourth-order identity tensor, and Cm the stiffness tensor of the matrix. The tensor A stands for the dilute mechanical h i−1 strain concentration tensor defined as: A = I + S : (Cm )−1 : (Cr − Cm ) , with S being the Eshelby’s tensor. The terms enclosed with angle brackets in Eq. (11) represent the average value of the corresponding term over all orientations. In order to conduct these orientational averages, the orientation of a filler aligned in the local axis 1’ is defined with respect to the material coordinate system by two Euler angles, α1 and α2 , as indicated in Fig. 3. The base vectors e and e′ of the global {1, 2, 3} and local {1′ , 2′ , 3′ } coordinate systems are related via the transformation matrix g as e′i = gij ej . Thus, the coordinate transformation of a fourthorder tensor P into the local coordinate system is explicitly represented in terms of the transformation ′ matrix as Pijkl = gip gjq gkr gls Ppqrs . Due to the high number of inclusions contained in the RVE, the description of their orientation field is of statistical nature. The probability of finding a filler in an infinitesimal range of angles [α1 , α1 + dα1 ] × [α2 , α2 + dα2 ] is given by Ω(α1 , α2 ) sin α1 dα1 dα2 , with Ω(α1 , α2 ) being the so-called Orientation Distribution Function (ODF). Let us note that, in the case of randomly oriented fillers, the ODF is defined as a constant of value Ω = 1/2π. Hence, the orientational average of any tensorial function F(α1 , α2 ) is defined through: hFi =

Z

2π 0

Z

π/2

F(α1 , α2 )Ω(α1 , α2 ) sin α1 dα1 dα2

(12)

0

Some inconsistencies in the Eshelby-Mori-Tanaka method have been reported in the literature, i.e., diagonally asymmetric stiffness tensors for arbitrary filler orientation distributions and violation of the Hashin-Shtrikman-Walpole bounds for randomly oriented filler configurations [6]. To tackle these inconsistencies, Schjødt-Thomsen and R. Pyrz [7] proposed an extended Eshelby-Mori-Tanaka method. This approach utilizes a direct orientational average of the Eshelby-Mori-Tanaka stiffness


209

tensor for the case of perfectly aligned fillers as: D E C = Cm + fr (Cr − Cm ) : A : (fm I + fr A)−1 .

(13)

Agglomeration of CNTs. An important phenomenon to be taken into account in the simulation of CNT-based composites is the appearance of non-uniform spatial distributions of nanoinclusions. The tendency of nanotubes to form agglomerates is ascribed to the electronic configuration of the tube walls and their high specific surface, what favors the appearance of large van der Waals (vdW) attraction forces among nanotubes [8]. It has been reported in the literature that CNT agglomerates act as defects in the microstructure, exhibiting considerable weakening effects on the macroscopic mechanical properties [9]. In order to incorporate agglomeration effects in the numerical simulations, the twoparameter agglomeration model introduced by Shi et al. [10] is adopted in this work. This approach distinguishes two different regions, one with high filler concentration, corresponding to clusters, and another with low filler concentration, that is to say, the surrounding composite. Hence, the total volume of CNTs, Vr , dispersed in V , can be divided into the following two parts: Vr = Vrbundles + Vrm

(14)

where Vrbundles and Vrm denote the volumes of CNTs dispersed within the bundles and in the surrounding matrix, respectively. In order to characterize the agglomeration degree, two parameters, ξ and ζ, are introduced as follows: ξ=

Vbundles , V

ζ=

Vrbundles Vr

(15)

where Vbundles is the volume occupied by the bundles in the RVE. The agglomeration parameter ξ represents the volume ratio of bundles with respect to the total volume of the RVE. On the other hand, ζ stands for the volume ratio of CNTs within the bundles with respect to the total volume of fillers. After some straightforward manipulations, the CNT volume fractions in the bundles and the surrounding composite, c1 and c2 , respectively, can be expressed as: ζ c 1 = fr , ξ

c 2 = fr

1−ζ 1−ξ

(16)

It is extracted from Eq. (16) that ζ ≥ ξ must be fulfilled in order to impose a higher filler concentration in the bundles. The limit case ζ = ξ represents an uniform distribution of fillers (c1 =c2 ), whilst the heterogeneity degree grows (c1 ≥ c2 ) for larger values of ζ up to ζ = min (1, ξ/fr ). Hence, the homogenization process is conducted through two separate steps. Firstly, the overall constitutive tensors of the bundles, Cin , and the surrounding composite, Cout , are obtained with polymer and CNTs as matrix and reinforcing phases, considering filler volume fractions of c1 and c2 , respectively. To this aim, the previously overviewed homogenization approach (Eq. (13)) can be applied by considering fr =ci , i=1,2. Secondly, the overall constitutive tensor of the composite, C∗ , is computed considering the surrounding composite as matrix and bundles as ellipsoidal inclusions (fr =ξ). In this work, two different agglomeration configurations are investigated, including fully aligned and randomly oriented fillers. Boundary element equations After a collocation procedure on each boundary ∂Ωl , the boundary integral equations can be written as Hd = Gp, where d and p contains the values of all nodal displacements and tractions, respectively. That expression can be rearranged according to the boundary conditions as: Ax = F, passing all the unknowns to vector x on the left-hand side. For contact problems, the interface discretization on ∂Ωc is performed such that node to node contact is considered. Consequently, each node on ∂Ω1c forms a contact pair I with one almost coincident node on ∂Ω2c . So, equation Ax = F can be written for solid


210

Ωl (l = 1, 2) as: Alx xl + Alp plc = Fl , where (xl )T = [(xle )T (dlc )T ] is the nodal unknowns vector that collects the contact nodal displacements (dlc ) and the external unknowns (xle ). Alx is constructed with the columns of matrices Hl and Gl , and Alp with the columns of Gl belonging to the contact nodal unknowns. Therefore, the discrete equilibrium equations can be formulated as: ˜ 1 λ = F1 , A2x x2 − A2p C ˜ 2 λ = F2 . A1x x1 + A1p C

(17)

In (17), vector λ contains the normal and tangential contact tractions of every contact pair I, which are ˜ l (i.e. p1c = C ˜ 1λ related with the nodal boundary element nodal tractions vector through the matrix C ˜ 2 λ). and p2c = −C The discrete gap for every contact pair I is approximated as: g = Cgn go + (C2 )T x2 − (C1 )T x1 , where g contains the normal and tangential gap of every contact pair I and go contains the initial gap. Cgn is an assembling matrix. Finally, the contact restrictions (3) and (9) are applied to every contact pair I: (λn )I − PR− ( (λn )I + rn (gn )I ) = 0 , (λt )I − PEρ ( (λt )I − rt M2 (gt )I ) = 0,

(18)

where λn and λt contain the normal and tangential contact tractions of every contact pair I, respectively. In this work, the nonlinear system (17-18) is solved using an iterative Uzawa’s method. Numerical example: Indentation response of CNTs A steel sphere of radius R = 100×10−3 m is indented on the resulting CNTRC halfspace (HS) domain. For that purpose, the halfspace region is discretized by linear quadrilateral boundary elements, using 16 × 16 elements on the potential contact zone. The influence of the fiber volume fraction on the indentation response of RCNT and ACNT is studied, where different ACNTs orientations (φ) are also considered (see Fig. 1). The elastic properties of the matrix are Em = 10 GPa, νm = 0.3, and the elastic properties of the equivalent fiber are computed based on the concept of equivalent fiber [11]. The spherical indenter is subjected to a normal displacement go,x3 = −80 µm and the indentation force P is computed by integration of the normal contact pressures. Fig. 4 (a) shows the indentation forces as a function of the volume fraction of CNT and CNT distributions, i.e., RCNT and ACNT. In both cases, the normal contact compliance increases with the fiber volume fraction. However, it has been noted that for an orientation φ greater than 45o , the ACNTs lead to a greater increments. Fig. 4 (b) shows the influence of the fiber orientation (φ) in the tangential contact force under gross and partial slip conditions. For a vr = 1% ACNT-RC, the friction coefficients for this ACNT reinforced composite are: µL = 0.77, µT = 0.82, µN = 0.67, being µ = 0.9 the friction coefficients for the epoxy (vr = 0%). These values have been considered from [1]. We can observe the great influence of the fiber orientation in the tangential contact force under, both, gross and partial slip conditions. For gross slip conditions (go,t = 100 µm), the tangential contact force is even more affected by the CNTs orientation, with differences close to 30 % between tangential forces observed for φ = 45o and φ = 90o . Finally, Fig. 4 (c) shows the effect of agglomeration on the indentation response of RCNT and ACNT for vr = 5%. We can see that the normal contact compliance of RCNT is affected by the agglomerations effects. However, the agglomeration effects on ACNT are highly affected by the fiber orientations. The normal contact compliance of ACNT is almost not affected by the agglomerations effects for fiber orientations within the interval [0◦ , 45◦ ]. Nevertheless, the indentation force presents differences up to 30 % between the CNTRC with no agglomerations (ζ = 0.2) and the CNTRC with a high number of agglomerates ζ = 0.8, when the filler orientation is greater than 60◦ . Conclusions The influence of micromechanical properties on the indentation response of CNTRCs has been studied using a 3D boundary element analysis. The boundary elements, which prove to be very robust and accurate for this kind of contact simulations, provide the elastic influence coefficients for the CNT


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(a)

(b)

(c) Figure 4: (a) Indentation forces as a function of the volume fraction of CNT and CNT distributions, i.e., RCNTACNT. (b) Influence of the fiber orientation (φ) in the tangential contact force relative to the epoxy configuration, under partial slip and gross slip conditions. (c) Effect of agglomeration on the indentation response of RCNT and ACNT. reinforced halfspace. The effective constitutive properties of the composites have been predicted by an Eshelby-Mori-Tanaka micromechanical model. This model allows us to consider aligned CNTs distributions or to simulate more realistic features typically reported for CNT-reinforced polymers, i.e., agglomeration effects. Moreover, in order to study uniaxially-aligned CNT reinforced composites under frictional indentation conditions, an anisotropic friction constitutive law has been considered. This contact constitutive law allows us to assess the influence of the fiber orientation (φ) on the friction coefficients. The presented studies show that, not only the CNT volume fraction, but also the fibers orientation or the CNT dispersion (i.e. non-uniform spatial distribution of nanoinclusions due to agglomeration effects) have a fundamental effect on normal and tangential contact compliances or on contact traction magnitude and distribution. In this way, we may maximize the effectiveness of the reinforcement and therefore the properties of the CNTRC, by controlling the CNT fiber alignment, dispersion or volume fraction. Finally, it should be noted that the proposed methodology can be useful to design new CNT reinforced mechanical systems with better resistance under frictional contact conditions.

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Acknowledgments This work is supported by the Ministerio de Ciencia e Innovaci´ on, Spain, and by the Consejer´ıa de Innovaci´ on, Ciencia y Empresa, Junta de Andaluc´ıa (Spain), through the research projects: DPI201453947-R, DPI2017-89162-R and P12-TEP-2546, which were co-funded by the European Regional Development Fund (ERDF). E. G-M was also supported by a FPU contract-fellowship from the Spanish Ministry of Education Ref: FPU13/04892. The financial support is gratefully acknowledged. References [1] Huaiyuan Wang, Li Chang, Xiaoshuang Yang, Lixiang Yuan, Lin Ye, Yanji Zhu, Andrew T. Harris, Andrew I. Minett, Patrick Trimby, and Klaus Friedrich. Anisotropy in tribological performances of long aligned carbon nanotubes/polymer composites. Carbon, 67:38–47, feb 2014. [2] Hui Zhang, Longbin Qiu, Houpu Li, Zhitao Zhang, Zhibin Yang, and Huisheng Peng. Aligned carbon nanotube/polymer composite film with anisotropic tribological behavior. Journal of Colloid and Interface Science, 395:322–325, apr 2013. [3] Luis Rodr´ıguez-Tembleque and M.H. Aliabadi. Numerical simulation of fretting wear in fiberreinforced composite materials. Engineering Fracture Mechanics, 168:13–27, 2016. [4] Sia Nemat-Nasser and Muneo Hori. Micromechanics: overall properties of heterogeneous materials, volume 37. Elsevier, 2013. [5] R. Hill. A self-consistent mechanics of composite materials. Journal of the Mechanics and Physics of Solids, 13(4):213–222, aug 1965. [6] Y. Benveniste, G.J. Dvorak, and T. Chen. On diagonal and elastic symmetry of the approximate effective stiffness tensor of heterogeneous media. Journal of the Mechanics and Physics of Solids, 39(7):927–946, jan 1991. [7] J. Schjødt-Thomsen and R. Pyrz. The Mori–Tanaka stiffness tensor: diagonal symmetry, complex fibre orientations and non-dilute volume fractions. Mechanics of Materials, 33(10):531–544, oct 2001. [8] A Allaoui. Mechanical and electrical properties of a MWNT/epoxy composite. Composites Science and Technology, 62(15):1993–1998, nov 2002. [9] Anastasia Sobolkina, Viktor Mechtcherine, Vyacheslav Khavrus, Diana Maier, Mandy Mende, Manfred Ritschel, and Albrecht Leonhardt. Dispersion of carbon nanotubes and its influence on the mechanical properties of the cement matrix. Cement and Concrete Composites, 34(10):1104– 1113, 2012. [10] D. L. Shi, X. Q. Feng, Yonggang Y. Huang, K. C. Hwang, and Huajian Gao. The effect of nanotube waviness and agglomeration on the elastic property of carbon nanotube-reinforced composites. Journal of Engineering Materials and Technology, 126(3):250, 2004. [11] Mahmood M. Shokrieh and Roham Rafiee. On the tensile behavior of an embedded carbon nanotube in polymer matrix with non-bonded interphase region. Composite Structures, 92(3):647– 652, February 2010.


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Micro dynamic failure of 2D crystal aggregate structures using BEM and a hierarchical multiscale cohesive zone model J. E. Alvarez, A F. Galvis and P. Sollero Faculty of Mechanical Engineering, University of Campinas, Brazil, {jealvarez, andres.galvis, sollero}@fem.unicamp.br

Keywords: Dynamic Analysis; Boundary Element Method; Crystal Aggregate Structures; Cohesive Zone Model; EAM Potential.

Abstract. In this work, the dynamic failure of 2D crystal aggregate structures is analyzed using the boundary element method (BEM) and the multiscale cohesive zone model (MCZM). To describe the transient approach, the dual reciprocity method (DRM) is considered. Regarding the microscopic modeling, anisotropic properties are randomly fitted. Furthermore, the crystal morphology is reproduced using Voronoi Tessellations. In order to model the grain interfaces, cohesive elements (CIEs) are constructed as new regions to be analyzed. Thus, deformation gradients are evaluated at the CIEs during the strain wave propagation. Thereby, the strain energy is transferred to a local lattice arrangement using the Cauchy-Born rule. Finally, the bond energy is computed using the EAM potential. At the same time that a cut-off parameter reaches the breaking bond, the corresponding CIE disappears. The numerical results are provided by an example. Introduction. Nowadays, some of the primary goals of science and technology are to understand and to control the behavior of materials at different space and time scales. The reason can be understood as macroscopic domains are affected by microscopic conditions. In metallic materials, widely used in manufacturing processes, crystal aggregate structures characterize the microscopic domain. These geometries present random distributions that affect the elastic behavior of the aforementioned materials. In view of increasing need to control failure variables, and to improve the modeling based on smaller scales, the material analysis using enhanced approaches need to be considered. Recently, research works of crystal aggregate structures have been developed in order to obtain the mechanical response in a multiscale framework. The finite element method (FEM) is widely found in simulation analysis. This method has been used to compute the influence of micro defects over a macroscopic scale by static boundary conditions [1]. Similarly, this computational technique is considered to model heterogeneous materials [2], and for application involving crystal aggregate domains with multiscale cohesive zone models (MCZM) [3, 4]. Despite the FEM requires a straightforward implementation, the resolution of response gradients tied to volume mesh refinement. Owing to the high gradients and internal force concentration that crystal materials present, the BEM has been applied as an alternative computational method for the multiscale analysis. Sfantos and Aliabadi [5] introduced a 2D macro and micro analysis of polycrystalline materials under static loads; the authors considered the representative volume elements (RVEs) to transfer the mechanical response between scales. Galvis and Sollero [6] presented a dynamic analysis of 2D polycrystalline materials coupling the BEM and MCZM, where the micro and the atomistic scale were modeled based on the Lennard Jones potential. Referring to the application of the non-linear and the time-dependent problem, the DRM is effectively used to solve the transient analysis [7]. Wang et al. [8] analyzed scattering of elastic waves due to a crack in infinite bi-dimensional anisotropic media. Saez and Dominguez [9] presented a formulation of the scattering of harmonic waves due to a crack in transversely isotropic media using a numerical fundamental solution. Regarding an atomistic scale, several works have developed using molecular dynamics (MD) and quantum mechanics (QM) [10, 11]. However, it is impossible to conduct such simulations of large scales for


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practical applications [12]. As an alternative, the MCZM is presented as a simulation technique to couple the continuum and the discrete domain [13, 14, 15]. The MCZM was introduced by Lyu et al. [12] at the last version of multiscale analysis in order to model fracture in metals; the authors used the third-order Cauchy-Born rule and a barycentric FEM formulation to build shape functions for hexagonal cohesive zones. Furthermore, the MCZM formulation is based on both atomistic potentials and the information related to deformation gradients. To model the atomistic behavior, the Lennard-Jones potential has been extensively implemented [6, 13, 15]. Thereby, it is an attractive task to implement an enhanced potential to obtain reliable results, particularly, in simulations with metallic materials [16]. The EAM potential was introduced to analyze the bulk, surface and cluster properties of metals. This potential, based on a second-moment approximation to the tight-binding density of states, has a simple analytic form showing satisfactory approximations in simulations of point defects, grain boundaries, surface and amorphization transition for metals and alloys [17]. This work presents a 2D dynamic analysis coupling the microscopic and the atomistic scales using BEM and MCZM, respectively. In order to simulate a crystal aggregate material, anisotropic properties are considered. Furthermore, a parallelization on a shared memory architecture using Fortran-OpenMP is developed to evaluate the BEM matrices and to increase the number of regions inside the domain. The MCZM is implemented based on the EAM potential. Finally, simulation results are presented in order to describe the dispacement wave at time scale. Modelling of crystal aggregate structures. In order to reproduce the heterogeneous domain, the Voronoi tessellation algorithm is implemented [18]. This algorithm has been extensively used to represent a grain morphology, e.g. [6, 19, 20]. Figure 1 a) illustrates a 2D polycrystalline structure characterized by random elastic properties for each region and generated within an area of 1.0 mm2 . This process is similarly developed by Sfantos and Aliabadi [5]. Each region is considered as a single crystal with orthotropic elastic behavior and specific material orientation. Considering as xyz the geometry coordinate system and 123 the material coordinate system, three cases emerge in view of which of the three material axes coincide with the z-axis (out of plane) of the geometry; thus Case 1: 1 ≡ z, Case 2: 2 ≡ z and Case 3: 3 ≡ z (working plane is assumed the xy). Subsequently, every generated crystal is characterized by one of the aforementioned cases in a completely random manner.

y 2

1 x

a)

b)

Figure 1: Modeling of the crystal aggregate structure: a) anisotropic properties, b) boundary discretization. It is useful to model each crystal as a continuum body applying displacement compatibility and traction


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equilibrium at the grain interfaces. Regarding the mesh discretization, boundary discontinuous elements of three nodes are used to represent the domain, avoiding the approximation among corners. The Figure 1 b) shows the boundary discretization for a crystal aggregate structure. BEM and MCZM implementations. After the representation of the crystal aggregate domain, the evaluation of mechanical behavior using BEM and MCZM is implemented. A general flowchart of the computational code is illustrated by Figure 2.

Input BEM implementation threads

OpenMP

End Figure 2: Flowchart for the dynamic simulation. First, the input parameters such as anisotropic properties, boundary element discretization, load conditions and time integration are introduced. Subsequently, the BEM implementation is computed using DRM and the Houbolt’s algorithm. The Equation (1) describes the matrix form to compute the dynamic simulation based on a multizone approach.

Ai Hij −Gij 0 Hij Gij

   bc  xi  k        j   i   0 u B 0 0 0 0 i i = + 0  0 0 0 Bj  Aj  tij         bc  kj x j τ+∆τ τ+∆τ (1)

Mi Mij 0 Mij

 uαi    j 0 0 uαi  0 0 Mj   uα j

   

,

  

where Hi and Gi are the influence matrices that are used during the BEM computation. Mi is the mass matrix. Ai , Bi are the blocks of influences matrices that belong to the boundary of the polycrystalline


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material. Vector xi represent all the traction and displacement unknowns to be evaluated with the elements corresponding to the boundaries, and kbc i are known boundary conditions applied respectively at the instant τ + ∆τ. In the interfaces, displacements uij and traction tij are evaluated. The terms uαi and uα j are defined j for the block belonging to the boundaries and uαi for the interfaces. The algorithm proposed by Kane [21] is used to assemble the system of equations. Regarding the RAM and computational processing, this work implements a treatment for the sparse matrix, previously developed in [22]. Owing to the large number of regions to be evaluated, the critical sections of the BEM code are parallelized. This is based on a Fortran-OpenMP platform. Furthermore, the solution for the system is carried out using the PARDISO solver [23, 24]. During the simulation process, the MCZM is implemented after knows the BEM solution. A cut-off parameter is fitted indicating whether the CIE disappears or not. In addition, the CIE can be represented as a bulk medium with elastic anisotropic properties, see Figure 3.

Figure 3: Cohesive Interface Element (CIE). The number of CIEs corresponds to the number of elements in each interface, and the average orientations (between two grains) αAB characterize the corresponding elastic behavior. Furthermore, each CIE presents compatibility with the nodes on the grain boundary. By modeling the interfaces with CIEs, and knowing its respective deformation, the elastic energy can be transferred to an atomistic lattice. For this, the Cauchy-Born rule (CBR) is used [25]. The Equation (2) describes the positions ri of two particles based on deformation gradients hFc i, hGc i and a reference position Ri evaluated from the lattice configuration. 1 ri = hFc i · Ri + hGc i : (Ri ⊗ Ri ) 2

(2)

where “( : )” expresses the inner product of a third-order tensor with a second-order tensor. The result is a tensor of the first order. The symbol “(⊗)” denotes the standard tensor product. After disrupting the atomic position between two particles using Equation (2), the EAM formulation is considered to evaluate the bond energy. The Equation (3) contains both the attractive and repulsive energy of two atoms. The repulsive part is evaluated by pair potential and the attractive part by a many-body potential [16]. UT =

1 V (ri j ) − ∑ f (ρi ), 2∑ ij i

(3)


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The first term in Eq. (3) is the conventional central pair-potential summation, which is expressed by a quartic polynomial equation as (r − c)2 (c0 + c1 r + c2 r2 + c3 r3 + c4 r4 ), r ≤ c , (4) V (r) = 0, r>c where c is a cut-off parameter assumed to lie between the second and third neighbor atoms. co , c1 , c2 , c3 , and c4 are the potential parameters to be fitted. The second term in Eq. (3) is the n-body term. The embedding function f can be expressed by f (ρi ) =

√ ρi ,

(5)

where, according to the linear superposition approximation, the host electronic density ρi is written as ρi = ∑ A2 φ (ri j ),

(6)

j6=i

And the electronic function is expressed by (r − d)2 + B2 (r − d)4 r ≤ d φ (r) = , 0, r>d

(7)

where d is assumed to lie between the second and third neighbor atoms. B is a potential parameter. The parameters to computed the EAM potential are available in [17]. Numerical results. The aggregate structure contains 1,000 grains, 9,034 cohesive zones, it is discretized by 60,438 discontinuous boundary elements and 181,314 nodes. Iron anisotropic properties are selected: C11 = 230 GPa, C12 = 135, C44 = 117 GPa, the constants are available in [26]. Iron is represented at the atomistic scale by a body-centered cubic (BCC) lattice. Thus, the parameters to compute the EAM ˚ and c = 2.96 A, ˚ B=0A ˚ −2 , A = 0.931312 eVA ˚ −1 , c0 = 26.27034 eV A ˚ −2 , potential are: d = 4.05 A ˚ −3 , c2 = 6.957871 eV A ˚ −4 , c3 = −0.303077 eV A ˚ −5 , and c4 = −0.085092 eV A ˚ −6 . c1 = −24.40109 eV A The boundary conditions to simulate the speciment are shown in Figure 4.

Figure 4: Boundary conditions. A ramp load is imposed at the lower and upper surfaces. The time step is fixed in ∆τ = 3.8µs. In addition, a pre-crack is created in the right middle part of the crystal aggregate structure. The Figure 5 describes the displacement wave under mentioned conditions.


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0e+00 0.000

4.5e-8 0.010

0.015 8.9e-8

1.3e-7 0.020

0.025 2e-07 [m]

Figure 5: The displacement wave and the crack propagation path. Figure 5 shows the crack propagation along grain boundaries. The simulation starts with the crack being propagated horizontally. Then it bifurcates due to perturbation of the displacement wave reflected from the boundary conditions. Conclusions. In this work, the simulation of a crystal aggregate structure is shown due to a dynamic load. The BEM was applied in order to evaluate the displacement wave through the material. Furthermore, the MCZM was included to model a homogenized atomistic scale. A path crack propagation was presented, and it was the consequence of cohesive forces between interfaces. This work was based on the EAM potential in order to improve the metallic approximation. Further, a large number of regions were possible to simulate due to the parallelism implementation. Acknowledgement. The authors would like to thank the Brazilian National Council for the Scientific and Technological Development (CNPq) for their financial support of this research. Grant numbers: 312493/2013 − 4, 54283/2014 − 2, 134625/2016 − 1. Furthermore, the authors would like to thank the Centre for Computational Engineering and Sciences (CCES-CEPID/UNICAMP) for providing access to computational facilities.


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G. Liu, D. Zhou, Y. Bao, J. Ma, Z. Han, Theoretical and Applied Fracture Mechanics 90, 65 (2017). P. Fu, H. Liu, X. Chu, Y. Xu, International Journal of Computational Methods 13, 1650030 (2016). X. Zeng, S. Li, Computer Methods in Applied Mechanics and Engineering 199, 547 (2010). J. Qian, S. Li, Journal of Engineering Materials and Technology 133, 011010 (2011). G. Sfantos, M. Aliabadi, Computer Methods in Applied Mechanics and Engineering 196, 1310 (2007). A. F. Galvis, P. Sollero, Computers & Structures 164, 1 (2016). P. Partridge, C. Brebbia, L. Wrobel, Computational Mechanics Publication pp. 1–10 (1992). C.-Y. Wang, J. Achenbach, S. Hirose, International Journal of Solids and Structures 33, 3843 (1996). A. Saez, J. Dominguez, International Journal for Numerical Methods in Engineering 44, 1283 (1999). J. S.-L. Gibson, et al., Materials Research Letters 6, 142 (2018). Y. Zhang, S. Jiang, Metals 7, 432 (2017). D. Lyu, H. Fan, S. Li, Engineering Fracture Mechanics 163, 327 (2016). H. Fan, S. Li, GAMM-Mitteilungen 38, 268 (2015). S. Li, et al., Computer Methods in Applied Mechanics and Engineering 229, 87 (2012). X. Zeng, S. Li, International Journal of Multiscale Computational Engineering 10, 391 (2012). M. Griebel, S. Knapek, G. Zumbusch, Numerical Simulation in Molecular Dynamics (Springer, 2007). X. Dai, Y. Kong, J. Li, B. Liu, Journal of Physics: Condensed Matter 18, 4527 (2006). A. Okabe, B. Boots, K. Sugihara, S. N. Chiu, Spatial tessellations: concepts and applications of Voronoi diagrams, vol. 501 (John Wiley & Sons, 2009). H. D. Espinosa, P. D. Zavattieri, Mechanics of Materials 35, 333 (2003). S. Ghosh, Y. Liu, International Journal for Numerical Methods in Engineering 38, 1361 (1995). J. H. Kane, Boundary element analysis in engineering continuum mechanics (1994). A. F. Galvis, R. Q. Rodr´ıguez, P. Sollero, Computers & Structures 200, 11 (2018). C. G. Petra, O. Schenk, M. Lubin, K. Gärtner, SIAM Journal on Scientific Computing 36, C139 (2014). C. G. Petra, O. Schenk, M. Anitescu, IEEE Computing in Science & Engineering 16, 32 (2014). J. Ericksen, The Cauchy-Born hypothesis for crystals in phase transformation and material instabilities in solid (1984). H. Kiewel, H. Bunge, L. Fritsche, Texture, Stress, and Microstructure 28, 17 (1996).

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Meshfree Analysis of Viscoplastic Problems Using Different Domain Integration Techniques Z. Kazemi1, M.R. Hematiyan2, A. Khosravifard3 Department of Mechanical Engineering, Shiraz University, Shiraz 71936, Iran 1

[email protected]

2 3

[email protected]

[email protected]

Keywords: Viscoplastic, Meshfree radial point interpolation method; Background mesh; Cartesian transformation method

Abstract. In this work, the meshfree radial point interpolation method (RPIM) is used for analysis of viscoplastic problems. In the formulation of the RPIM for viscoplastic problems, some domain integrals appear that should be accurately evaluated. The domain integrals are usually evaluated using the standard Gaussian quadrature method with a background mesh. The background mesh integration (BMI) does not lead to a truly meshfree technique. On the other hand, truly meshfree integration methods such as the Cartesian transformation method (CTM) can also be used for evaluation of domain integrals. The objective of this work is to investigate the accuracy and efficiency of the BMI and the CTM for evaluation of domain integrals in the RPIM for viscoplastic problems. For this end, a viscoplastic problem with a relatively complicated geometry is considered. The problem is analyzed with three different configurations of nodes using both BMI and CTM and accuracy and convergence of each method are investigated. Introduction In Galerkin meshfree methods, which are based on global weak formulation, it is necessary to compute some integrals over the domain of the problem. Many important meshfree methods such as the radial point interpolation method (RPIM) [1,2], the element free Galerkin (EFG) method [3], the reproducing kernel particle method (RKPM) [4], and the partition of unity (PU) method [5] are based on global weak formulation. Success in these meshfree methods depends on the total number of nodes, the size of support domain, the shape parameters, and the accuracy of integration technique. Usually, the background mesh integration (BMI) is used for evaluation of the domain integrals and therefore, these methods cannot be considered as truly meshfree methods. In the BMI method, the standard Gaussian quadrature (GQ) method is utilized. The BMI isn’t accurately performed when the background mesh doesn’t conform to the integration domain [6]. To overcome this difficulty, it is necessary to use relatively large number of integration points, which increases the computational cost. Due to this important difficulty to achieve accurate domain integration in the BMI method, some researchers have attempted to propose new efficient integration techniques or to enhance the existing integration methods for evaluation of domain integrals in meshfree methods. Beissel and Belytschko [7] have suggested a stabilized nodal integration procedure to obviate the necessary use of background mesh. Dolbow and Belytschko [6], proposed a numerical integration technique for the EFG method that reduced the errors due to the varying nodes density. In their study, different sizes of integration cells were used for different parts of the domain with different nodal density. In the study of Zhou et al. [8], a stabilization term was added to the potential energy functional to enhance the weak form formulation. The CTM for evaluation of domain integrals in meshfree methods was presented in [9]. In this method, the domain integrals are transformed into a boundary integral and a 1D integral. After some mathematical manipulations, the domain integral can be reconstructed by a matrix-vector multiplication of nodal values and integration weights. Yavuz and Kanber [10], used tetrahedral cells to obtain more accurate solution of the RPIM for analysis of elasto-static problems with a background mesh. Hillman and Chen [11] suggested a naturally stabilized nodal integration scheme based on gradients of the strains at the nodes. Wang and Wu [12] used a triangular shaped background


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mesh to improve the precision and efficiency of the stabilized conforming nodal integration method proposed in [13]. Accurate and efficient evaluation of domain integrals is more important in nonlinear structural mechanics, where we are confronted with inherently long duration processes. In viscoplastic problems, combined consideration of rheological and plastic behaviour of materials causes the problem to be severely nonlinear. This characteristic necessitates suitable numerical methods for meshfree analysis of viscoplastic problems. Obviously, choosing a proper integration technique, can significantly improve the accuracy and efficiency of viscoplastic meshfree analyses. The objective of this work is to investigate the accuracy and efficiency of the BMI method and the CTM for evaluation of domain integrals in the RPIM for viscoplastic problems. For this purpose, a viscoplastic problem with three different configurations of nodes using both BMI and CTM is analyzed and accuracy and convergence of each method are investigated. Basic equations of elasto/viscoplasticity The knowledge of viscoplasticity is important in understanding behaviour of materials under high strain rate or subjected to stress at high temperature. We consider a general two dimensional domain  with the boundary   t  u , where t and u are the natural and the essential boundaries. Ignoring the inertia effects, the incremental form of the governing equation, boundary conditions, strain-displacement relation, and the constitutive relation for a viscoplastic problem can be expressed as follows: (1) .dσ  db  0 in  (2) (dσ)n  d t on t (3) du  d u on u 1 (4) dε  du  du  in  2 dσ  Dt dε (5) where σ , ε , u and b are the stress tensor, the strain tensor, the displacement vector and the body force vector, respectively. t , and u , are the prescribed vectors of traction and displacement, respectively. n is the outward unit vector normal to the natural boundary.  and . represent the gradient and the divergence operators, respectively. D t in eq. (5), in the elastic range, is the elastic stiffness matrix De , and in the viscoplastic range, is the viscoplastic tangent stiffness matrix Dvp . More details about computation of Dvp can be found in [14]. The strain rate can be decomposed into the elastic part ε e and viscoplastic part ε vp as follows:

ε  ε e  ε vp

(6) The associated viscoplastic flow rule obtained using the Perzyna von-Mises model [15] can be expressed as follows:



ijvp

where

  eff     1    y



f  ij

 is the viscoplastic parameter and the sign

In eq. (7),  eff 

 y   y  H

vp

 and

(7) is the Macaulay brackets notation.

3 1 Sij : Sij , where Sij   ij   kk ij are the deviatoric stress components. Also, 3 2

f   eff   y ; where H is the isotropic hardening parameter and 

t

vp

 0

is the accumulated viscoplastic strain.

2 vp vp  ij :  ij dt 3


223

The meshfree RPIM formulation for viscoplastic problems In this section, the main equations of the RPIM formulation for elasto/viscoplastic problems are described. In order to interpolate the incremental displacement vector u(x) at any point of interest x , a total number of m nodes is considered over the domain and for each node a support domain including n node is constructed. The approximation function u x  is expressed as follows: h

n

u h x    i u i  φT x  u

(8)

i 1

where u is the incremental nodal displacement vector, and φ x  is the vector of RPIM shape functions. The incremental form of strain-displacement and constitutive equations can be expressed as follows: T

n

ev  Bu   Bi u i

(9)

i

n

Δσ  Dt Δe v  Dt B Δu   Dt i Bi Δu i

(10)

i

where

i , x 0    Bi   0 i , y  i , y i , x   

(11)

Constructing the weighted integral of eq. (1) and equating it to zero and after some mathematical manipulation, the discritized system of equations is botained as follows [2]: m

K

ij

u j  Fi

i  1, 2, , m

(12)

j

where

 kij11 K ij   12 kij

kij12   kij22 

(13)

 f 1   f 1  Fi   i2    i2   f i   f i 

(14)

j K ij is the nodal stiffness matrix and Fi is the nodal force vector. In eq. (14) f i is a part of the force vector due to

domain effects and f i j is another part due to boundary effects. To calculate the elements of these matrices, it is nessesary to evaluate the following integrals: (15) K ij   BTi Dt B j d 

Fi   φTi bd   φTi t d 

(16)

t

In most meshfree studies, the BMI method has been used as a standard tool for evaluation of domain integrals. In this work, the BMI method as well as the CTM, which is a well performed method for evaluation of domain integrals in the BEM [16,17] and meshfree methods [9] are used and compared for computation of domain integrals. The CTM has been applied for evaluation of domain integrals in different mechanical problems such as nonlinear transient heat conduction analysis [18], vibrational analysis of sandwich beams [19], solidification problems [20] and viscoplastic analysis [14,21]. For problems with high stress concentration, where nodes are dense in a part of the domain, e.g. crack propagation problems, a modified version of the CTM can be used [22, 23]. In the next section, the CTM is described briefly.


224

Evaluation of domain integrals using the CTM The integral of an arbitrary function z ( x, y ) over a 2D region  with the boundary  , can be expressed as follows: (17) I   z ( x, y) d 

With the help of a rectangle ( X 1  x  X 2 , Y1  y  Y2 ) that surrounds the domain, (see Fig. 1), the integral in eq. (17) is changed into two separate 1D integrals [9]. These 1D integrals are:

I

Y2

 r ( y) dy

(18)

Y1

r ( y) 

X2

 z ( x, y) dx

(19)

X1

By evaluating these 1D integrals using the composite Gaussian quadrature (CGQ) method, the domain integral in eq. (17) is expressed as follows [9]: a

b

I   J jy wi R(i )

(20)

j 1 i 1

where a is the number of sub-divisions in the vertical direction, b is the number of Gauss points in each subdivision, wi are the Gaussian quadrature weights, and i are Gauss points. In eq. (20), R( )  r ( y( )) where y (i ) is the so-called integration ray and J jy  dy d  ( y j 1  y j ) / 2 is the Jacobin of transformation.

The function r (i ) can be expressed as follows: c

x2i

i 1

x 2 i 1

r( y j )   (

 z ( x , y )dx) j

j

(21)

where c is the number of intersection parts of each integration ray with the domain. The integral on the right hand side of eq. (21) can be computed as follows: x2i

g

h

x  z ( x j , y j )dx   J i wl Z ( l )

(22)

k 1 l 1

x 2 i 1

where g and h denote, the number of integration sub-divisions in the interval x2 j 1  x  x2 j and the number of Gauss points in each sub-division, respectively. wl are the Gaussian quadrature weights and  l are the Gauss points. In this equation Z ( l )  z ( x( l ), y j ) and J ix  ( x2 j  x2 j 1 ) / 2 g is the Jacobin. Now, the domain integral in eq. (17) can be computed by summation of product of two vectors, the vector of the CTM integration weights, W CTM , and the vector of the integrand at the CTM integration points, Z , i.e.:

I

N CTM

 WiCTM z( x i , yi )  W CTM Z i 1

(23)


225

Figure 1. The integration domain in an auxiliary rectangle The following equation can be used to evaluate the field variable at the CTM integration points, based on the values of the field variable at the nodes: z int  Sz nodal (24)

where, z int ( N CTM 1 ) and z nodal ( m  1 ) are the vectors containing the values of the field variable at the CTM integration points and at the nodes, respectively. The matrix S ( N CTM  m ) contains the values of the interpolating shape functions i.e., Sij   j (xi ) . Using the CTM, the domain integrals in eqs. (15) and (16) can be computed as follows:

kijrs 

NCTM

 WlCTM ( Dlt ,rs Silx S xjl  Dlt ,33 Sily S jly )

fi r 

l 1 N CTM

W l 1

CTM

l

blr Sil

r, s  1: 2

r  1: 2

(25) (26)

rs where kij is the (r , s ) element of the stiffness matrix K ij and Dtl ,ij is the l th component of the vector Dt,ij ,

which contains the value of the (i, j ) element of the matrix Dt . Also, the two matrices S respectively, i.e.:

Sijx 

 j (xi ) x

x

; Sijy 

and S

y

 j (xi ) y

contain the x-derivative and the y-derivative of the shape functions,

; i  1: NCTM , j  1: m .

(27)

Numerical example A numerical example is considered in order to demonstrate the effectiveness of the CTM compared to the BMI method for evaluation of domain integrals in the RPIM formulation of viscoplasticity. The geometry depicted in Fig. (2) is considered as the domain of the problem. With the assumption of the plane strain condition, the upper edge of the domain is stretched by a traction that is progressively increased to 360 MPa in 30 seconds and then decreased back to zero in the next 30 seconds. The results of the proposed method are verified with an FEM reference solution obtained by ANSYS with a very fine mesh (3321 nodes and 3200 quadrilateral 4-node elements). The mechanical properties of the material are considered as follows: E  210 GPa,   0.3,  y  240 MPa, H iso  5.8 GPa ,   104 and   3 For each integration method, the problem is solved with three different total numbers of integration points, i.e. 4000, 6000 and 8000 points. The problem domain is represented by 756 nodes, configuration of which is shown in Fig. 2.


226

Figure 2. The geometry and meshfree nodes configuration The results are presented for the point at the upper left corner of the domain. The displacement component in ydirection, i.e. u y , viscoplastic strain  y , the stress in the y-direction, i.e.  y , and the accumulated viscoplastic vp

strain  are depicted in Figs. 3-6, respectively. These figures illustrate the convergence of the RPIM for both the CTM and the BMI method for different numbers of integration points. It is clearly observed that for the cases with 6000 and 8000 integration points, the results obtained by the CTM, show excellent agreement with those obtained from the reference FEM solution. It is also observed that the results obtained by the CTM with 4000 integration points are almost as accurate as those obtained by the BMI method with 8000 integration points. vp

Figure 3. u y at the upper left corner of the domain for different numbers of integration points using the CTM and BMI method

Figure 3.  y at the upper left corner of the domain for different numbers of integration points using the CTM and vp

BMI method


Figure 3. 

vp

227

at the upper left corner of the domain for different numbers of integration points using the CTM and BMI method

Figure 3. The stress at the upper left corner of the domain for different numbers of integration points using the CTM and BMI method Conclusion In this work, the conventional integration method with a background mesh and the CTM, which is a truly meshfree integration method were employed for evaluation of domain integrals appearing in the formulation of the RPIM for viscoplastic problems. The accuracy of the computations was assessed by considering three different numbers of integration points. The effectiveness of the CTM as compared to the BMI method was clearly observed. In the numerical example presented in this work, it was observed that the CTM with half the number of integration points gives the results as accurate as the BMI method. References [1] Liu GR, Gu YT. A local radial point interpolation method (LRPIM) for free vibration analyses of 2-D solids. Journal of Sound and Vibration. 2001; 246(1):29-46. [2] Liu GR, Gu YT. An Introduction to Meshfree Methods and Their Programming. Berlin Heidelberg New York, Springer, 2005. [3] Belytschko T, Lu YY, Gu L. Element-free Galerkin methods. International Journal for Numerical Methods in Engineering. 1994; 37(2):229-56. [4] Liu WK, Jun S, Zhang YF. Reproducing kernel particle methods. International Journal for Numerical Methods in Fluids. 1995; 20(8-9):1081-106. [5] Melenk JM, Babuška I. The partition of unity finite element method: basic theory and applications. Computer methods in applied mechanics and engineering. 1996; 139(1-4):289-314.

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[6] Dolbow J, Belytschko T. Numerical integration of the Galerkin weak form in meshfree methods. Computational Mechanics. 1999; 23(3):219-30. [7] Beissel S, Belytschko T. Nodal integration of the element-free Galerkin method. Computer methods in applied mechanics and engineering. 1996;1 39(1-4):49-74. [8] Zhou JX, Wen JB, Zhang HY, Zhang L. A nodal integration and post-processing technique based on Voronoi diagram for Galerkin meshless methods. Computer methods in applied mechanics and engineering. 2003; 192(35):3831-43. [9] Khosravifard A, Hematiyan MR. A new method for meshless integration in 2D and 3D Galerkin meshfree methods. Engineering Analysis with Boundary Elements. 2010; 34(1):30-40. [10] Yavuz MM, Kanber B. On the usage of tetrahedral background cells in nodal integration of RPIM for 3D elasto-static problems. International Journal of Computational Methods. 2015; 12(06):1550036. [11] Hillman M, Chen JS. An accelerated, convergent, and stable nodal integration in Galerkin meshfree methods for linear and nonlinear mechanics. International Journal for Numerical Methods in Engineering. 2016; 107(7):603-30. [12] Wang D, Wu J. An efficient nesting sub-domain gradient smoothing integration algorithm with quadratic exactness for Galerkin meshfree methods. Computer Methods in Applied Mechanics and Engineering. 2016; 298:485-519. [13] Chen JS, Wu CT, Yoon S, You Y. A stabilized conforming nodal integration for Galerkin meshfree methods. International journal for numerical methods in engineering. 2001; 50(2):435-66. [14] Kazemi Z, Hematiyan MR, Vaghefi R. Meshfree radial point interpolation method for analysis of viscoplastic problems. Engineering Analysis with Boundary Elements. 2017; 82:172-84. [15] Perzyna P. Fundamental Problems in Viscoplasticity. Advances in Applied Mechanics. 1966; 9:243-377. [16] Hematiyan MR. A general method for evaluation of 2D and 3D domain integrals without domain discretization and its application in BEM. Computational Mechanics. 2007; 39(4):509-20. [17] Hematiyan MR. Exact transformation of a wide variety of domain integrals into boundary integrals in boundary element method. Communications in Numerical Methods in Engineering. 2008;24(11):1497-521. [18] Mohammadi M, Hematiyan MR, Marin L. Boundary element analysis of nonlinear transient heat conduction problems involving non-homogenous and nonlinear heat sources using time-dependent fundamental solutions. Engineering Analysis with Boundary Elements. 2010; 34(7):655-65. [19] Bui TQ, Khosravifard A, Zhang C, Hematiyan MR, Golub MV. Dynamic analysis of sandwich beams with functionally graded core using a truly meshfree radial point interpolation method. Engineering structures. 2013; 47:90-104. [20] Khosravifard A, Hematiyan MR. Meshless analysis of casting process considering non-Fourier heat transfer. Iranian Journal of Materials Forming. 2016 Oct 1;3(2):13-25. [21] Kazemi Z, Hematiyan MR, Shiah YC. An Efficient Load Identification for Viscoplastic Materials by an Inverse Meshfree Analysis. International Journal of Mechanical Sciences. 2018. [22] Hematiyan MR, Khosravifard A, Liu GR. A background decomposition method for domain integration in weak-form meshfree methods. Computers & Structures. 2014 Sep 30; 142:64-78. [23] Khosravifard A, Hematiyan MR, Bui TQ, Do TV. Accurate and efficient analysis of stationary and propagating crack problems by meshless methods. Theoretical and Applied Fracture Mechanics. 2017; 87:21-34.


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A hierarchic constitutive governed recursive methodology for obtaining three-dimensional anisotropic fundamental solution: a theoretical approach Tales V. Lisbôa1, Rogério J. Marczak1 1

Mechanical Engineering Department, Universidade Federal do Rio Grande do Sul Rua Sarmento Leite, 435, Centro, 90050-170, Porto Alegre, Brazil [email protected], [email protected]

Keywords: Anisotropy, Fundamental Solutions, Constitutive Hierarchy, Homotopy-like method

Abstract. The paper’s aim is to present a recursive methodology to determine the three-dimensional anisotropic fundamental solution. This methodology is based on a homotopy-like method which determines the solution of differential equations in a recursive manner by superposing the differential operator and expanding its solution in infinite terms. Each solution part is then derived by a correlation between the operator’s terms and previous solutions. The operator’s decomposition is related to a constitutive hierarchy. Being the isotropic symmetry the irreducible one, or the lowest one in the hierarchy, the methodology is essentially the isotropic solution enhanced recursively by anisotropic terms. Aspects such as convergence and degeneracy of the solution are analysed and discussed. Introduction

Fundamental Solutions are essential to numerical methods such as the Boundary Elements Method (BEM) [1] and Method of Fundamental Solutions (MFS) [2]. Characteristics such as the property of transforming domain integrals into boundary ones come from these distributions, which can be described as the response of an infinite or semi-infinite domain which an infinitesimal loading is applied and which is submitted to Sommerfeld boundary conditions. In considering the linear elastic behavior of three-dimensional solids, the Hooke’s Law is used so as to determine de relationship between stresses and strains. The crystalline symmetry has strong influence on such relation, modifying inner symmetries of the constitutive tensor [3]. It is well-known that, for some material symmetries, there are no fundamental solutions, in their analytical (close) form. Shiah et al. [4] have produced a fundamental solution and its first and second derivatives using the modular tensor in spherical coordinates and double Fourier Series. Buroni et al. [5] have utilised the residual theorem integration scheme, as Wang [6], and the second Barnett-Lothe tensor, as Távara et al. [7], consequently, the three-dimensional fundamental solution can be written then as a unique close form solution. Although the good results obtained by both approaches, in a way or another, the solution are approximations. Indeed, by Abel-Rufini theorem it can be shown that the fundamental solutions in the three-dimensional elasticity cannot be exact, since the characteristic equation (polynomial equation) of this problem has degree six and no algebraic solution can be obtained, unless the polynomial roots belong to a Galois group. That is exactly the case of isotropic and transversally isotropic symmetries: both have analytic fundamental solutions [3, 7, 8]. Moreover, concerning fundamental solutions and constitutive hierarchy [9, 10, 11], some considerations relative to non-removable spurious singularities should be addressed. It is well-known that fundamental solutions derived for higher symmetries (considering the hierarchic tree) fails when considered for lower ones. This is called mathematical degeneracy and the solution can either fall into a singularity or be different to its respective, when defined directly by the same symmetry [12, 13]. To the best of the


230

authors’ knowledge, no fundamental solution (or procedure to obtain it) has presented a way to avoid such problem of having specific fundamental solutions for each material symmetry. Furthermore, for as no polynomial equation with degree equal or greater than five can be written in terms of its coefficients (or its roots) unless they belong to specific groups or number symmetry, for slight changes of material properties, a new solution must be determined. The proposition herein presented has as objective the demonstration of a theoretical procedure that could determine fundamental solutions in an approximate way, similarly as Shiah et al. [4] and Buroni et al. [5], however, eliminating the degeneracy problem. Moreover, this solution is written in terms of the spatial coordinates and the constitutive properties, in an explicit manner. Consequently, a “simple” solution can define the elastic response of all material symmetries. In being defined as a recursive procedure, convergence parameters are placed and suggestions for cases in which these criteria are not satisfied are discussed. It is important to state that, being a theoretical approach, no solution is presented. Based on the Adomian Decomposition Method [14], the procedure utilises a constitutive hierarchy [9, 10, 11] and a constitutive decomposition [15] to provide a physical meaning to the operator’s superposition. The decomposition basically transforms any constitutive tensor (with any symmetry) into two new ones: one with a specific lower (with respect to the original one) symmetry and a remainder one. So that one can always satisfy such criteria, the isotropic symmetry is considered to be the first one. This choice also eases the procedure first step, since the isotropic fundamental solution is available. This paper is arranged as follows: in the chapter 2, the three-dimensional elasticity differential operator is presented and operated by the Fourier Transform; in the chapter 3, the constitutive hierarchy as well as the decomposition are demonstrated; in chapter 4, the recursive system to determine the fundamental solutions is presented and discussed; in chapter 5, the errors of the methodology along with the convergence parameters are shown. Matrix notation is used in this paper, where bold low-case letters denote vectors (or first-order tensors) and bold high-case letters correspond to matrices (or second-order tensors). Three-dimensional Elasticity differential equations

The differential equations that govern the three-dimensional elasticity problem can be written as: (1.a) (1.b)

in which and correspond to the elastic response and the font term, respectively, and denotes the differential operator. Equation (1) must have a set of boundary conditions in order to develop a well-posed problem. Moreover:

(2.a)

0 0 0

0 0

0

0 0

0

(2.b)


231

The Fourier Transform is applied to eqs. (1) over the three dimensions, due to the property that converts partial differentials into algebraic-complex functions and to the fact that elliptic differential equations, as eq. (1.a), result in real-value functions. Thus: (3.a) (3.b)

knowing that groups the transformed differentials, maps the spatial into the frequency domain and the hat over functions denotes they are defined in the frequency domain. Being the matrix operator of eq. (3.b) positive definite, the eq. (3.a) can be algebraic manipulated, resulting in: (4)

can be found applying the inverse transform. It results in: 1 2

(5)

where Ω is the integral’s domain ( ∞, ∞ for each variable) and . . . On the eq. (5), the boundary conditions should be applied and the solution can be developed by analytic or numeric solution method. In Fundamental Solutions, the source term can be changed to Dirac’s delta function to simulate the infinitesimal load. Its Fourier Transform is equal to the unity ( 1) and, therefore, the eq. (5) results in: 1 2

(6)

in which is an 3 3 identity matrix, a unit vector which represents the direction of the infinitesimal load – the direction of the Dirac’s delta and is the fundamental solution of the operator . One should observe that the Dirac’s delta is applied in the coordinate’s system origin, which should not be viewed as a simplification: this is made for sake of simplicity. The eq. (6) can be also found in Ting [3] for three-dimensional elasticity and it is submitted to the Sommerfeld boundary conditions. Equation (6) is modified so as to present the tensorial form of the fundamental solution as: ∴

in which,

1 2

(7)

is the tensor fundamental solution.

Constitutive Tensor’s Decomposition

The constitutive hierarchy, as posed by Cowin and Mehrabadi [9], Chadwick et al. [10] and Ting [11], is presented in Figure 1. The arrows system shows how the hierarchy goes from the triclinic to the isotropic symmetry.


232

In order to understand the idea of constitutive hierarchy, the concept of constitutive symmetry must be presented: two different materials, i. e., with different elastic properties, belong to the same constitutive symmetry when they share the same symmetric planes. As a result, each symmetry has its unique set of planes. A hierarchic relationship is then developed when a higher symmetry is a subset of a lower one. By observing Figure 1, the triclinic symmetry, which has none symmetric plane, is the highest symmetry in the scheme while the isotropic symmetry, which has all symmetric planes, is the lowest one.

Figure 1. Hierarchy tree containing the eight crystalline symmetries [9, 10, 11] The presented hierarchy has its importance to the constitutive decomposition. Essentially, it guides the decomposition, when more than two symmetries are considered to superpose the original tensor. Moreover, it presents the possible decompositions of a specific symmetry. Browayes & Chevrot [15] have developed a constitutive decomposition based on projections of an anisotropic constitutive tensor on the isotropic tensor. This idea simplifies the process of decomposition due to the fact that the constitutive properties are grouped into a vector while the projections are matrix operators. Furthermore, these projections are independent of the anisotropic tensor symmetry: it must be only a higher symmetry than the projected one. Consequently, one can define a decomposition of any anisotropic tensor into an isotropic one as follow: 1 2 15 1 2 15

(8.a) (8.b)

in which and are the Voigt’s and Dilatational tensors, respectively, defined via fourth-order and . and are the elastic properties of an isotropic constitutive tensor as material. These constants have relationship neither with natural nor synthetic material. The superposition of the original tensor is developed as:

(9)

where and correspond to the superposed tensors with isotropic and anisotropic symmetries as well as is the original one, normally with the same symmetry as the anisotropic tensor. In evaluating the decomposition defined by eqs. (8)-(9), one may note that does not maintain the positivity, a necessary property in constitutive tensor in order not to have either negative or zero strain energy, when a non-trivial strain field is observed. Nonetheless, for the recursive methodology, it is required that only the isotropic tensor to be positive definite.


233

Recursive Methodology

The procedure used to obtain the fundamental solution is based on Adomian’s Decomposition Method [14]. Developed so as to derive solution for non-linear differential equations, when applied to linear problems, it is defined, essentially, by three steps: 1. Superposition of the differential operator into two terms (Linear and Remainder terms); 2. Expansion of the solution into an infinite series; 3. Evaluation of each expanded solution’s term by a relationship between the differential operator terms and previous solution’s parts. Basically, the recursive procedure is described by the third step, while the first two are related to algebraic preparation of the recursive system. The constitutive decomposition is herein utilised to superpose the differential operator into two terms with physical meaning. Thus, in applying eq. (9) to eq. (1.b), one finds: (10)

Both operators in the right side of eq. (10) have the same structure. The second step can be written as: ⋯

⋯

(11)

and, in assembling both eqs .(10)-(11) together, one develops the base for the recursive system as: ⋯ in which

. In inserting all the terms relative to

(12) to the font term, one assembles

recursive procedure as: (13.a) (13.b) (13.c)

⋮

(13.d)

⋮ In being constructed by , eq. (13.a) solution reassembles the isotropic fundamental solution. This is the main reason of using the isotropic symmetry in the decomposition: its fundamental solution is readily available for three-dimensional elasticity (and for many other physical problems). Moreover, one shall observe that the remaining equations have exactly the same differential equation, differencing each other in the font term (which is determined by the previous solution and ). As consequence, one requires only the particular solution since the homogeneous one is determined by eq. (13.a). The solution is truncated when a desired accuracy is achieved. The mathematical degeneracy is pathology in anisotropic fundamental solutions related to singularities concerning elastic properties. Reducing the material’s complexity, relationship between such constants may lead to division per zero, mostly due to the characteristic equations of the differential equations. Nevertheless, as observed in eqs. (13), the anisotropic properties are in the numerator part of the system while in the denominator, only the isotropic terms. As being obtained by a positive definite constitutive tensor, no poles related to constitutive properties should be found in solutions obtained by such


234

procedure. In spite of being a simple qualitative approach, it presents strong arguments to demonstrate that the degeneracy should not be observed. Convergence Criteria

In order to describe the convergence criteria of the recursive methodology, the Fourier Transform is used. An error between the theoretical and the approximate solutions can be evaluated in the frequency domain via Plancherel Theorem [16]. First of all, the norm of a second-order tensor can be defined as: | |

⋯

tr

(14)

The inner product between two equal tensors which depend on the spatial coordinates is defined as: 〈

〉

,

|

tr

Now, one considers Thus:

(15)

|

the error between the theoretical (close) and the approximate fundamental solution. (16)

By eq. (15), results in: |

|

|

(17)

|

The difference between both solutions, in the frequency domain, is defined as: (18)

1

where: 1

1

(19)

2

The tensor of eq. (19) is decomposed into its principal values (eigenvalues) as:

(20)

By considering eq. (17) and eq. (19), one defines the error norm as: ∙

∙

(21)


235

Via eq. (21), the quadratic error norm can be determined given a recursive iteration quantity . It can be observed that, on order to obtain a smaller error in each approximation step, eqs. (20)-(21) impose that: (22) One may reconstruct the operators using eq. (22) by pre- and post-multiplying eq. (22) by , resulting in: 1

2

and (23)

1

Further manipulating eq. (23), one achieves the convergence criterion dependent only on the constitutive tensor, as: 0 ∴ 2

(24)

0

where the inequalities are related to the positivity sense of matrix:

0 for all non-trivial .

Summary and Conclusion

The paper’s aim was the development of an anisotropic fundamental solution based on a crystalline class hierarchy. An additive decomposition of the constitutive tensor was proposed to simplify the calculus of an anisotropic solution. A known solution (isotropic) is used and the remainder term influence is inserted in a recursive manner. The procedure convergence is presented in terms of the error norm. It is shown that the convergence depends only on the constitutive decomposition and the anisotropic degree of the original constitutive tensor. One may observe that the procedure can be applied several times and the convergence criterion is more related to a maximum step size than requirement of using the procedure. Even though the solutions obtained through this methodology do not have an analytical close form, they do not degenerate and, due to the solutions’ superposed form, the first and second derivatives can be straightforward determined. Nevertheless, the solutions reduce them self to other more symmetric given the fact that the materials’ singularities stays in the base solution. The next steps are the development of a numerical procedure to determine anisotropic solutions via an isotropic response. Using the results, semi-analytical procedures will be pursuit to put the material properties in evidence. Acknowledgements

Lisbôa would like to acknowledge DAAD (Deutscher Akademischer Austauchdienst) and CAPES (Coordenação de Aperfeiçoamento Pessoal de Nível Superior) for the granted doctorate scholarship. Marczak would like to acknowledge CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) for funding this reseach project. Lisbôa also like to thank Prof. Dr. –Ing. Habil. Ch. Zhang for the discussions about this theme and the idea of solution.

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References [1] Cheng, A. H.-D., & Cheng, D. T. (2005). Heritage and Early History of the Boundary Element Method. Engineering Analysis with Boundary Elements, 29(3), 268 - 302. [2] Fairweather, G., Karageorghis, A. (1998). The method of fundamental solutions for elliptic boundary value problems. Advances in Computational Mechanics, 9(69), 69-95. [3] Ting, T. -C. T. (1996). Anisotropic Elasticity: Theory and Applications. New York, NY: Oxford University Press, Inc. [4] Shiah, Y. C., Tan, C. L., & Wang, C. Y. (2012). Efficient computation of the Green's function and its derivatives for three-dimensional anisotropic elasticity in BEM analysis. Engineering Analysis with Boundary Elements, 36(12), 1746 - 1755. [5] Buroni, F. C., Ortiz, J. E., & Saez, A. (2010). Multiple Pole Residue Approach dor 3D BEM Analysis of Mathematical Degenerate and Non-Degenerate Materials. Numerical Methods in Engineering, 86(9), 1125 - 1143. [6] Wang, C. -Y. (1997). Elastic Fields Produced by a Point Source in Solids of General Anisotropy. Journal of Engineering Mathematics, 32(1), 41 - 52. [7] Távara, L., Ortiz, J. E., Mantič, V., & París, F. (2008). Unique Real-Variable Expressions of Displacement and Traction Fundamental Solutions Covering all Transversely Isotropic Elastic Materials for 3D BEM. International Journal for Numerical Methods in Engineering, 74(5), 776 - 798. [8] Nakamura, G., & Tanuma, K. (1997). A Formula for the Fundamental Solution of Anisotropic Elasticity. The Quartely Joournal of Mechanics and Applied Mathematics, 50(2), 179 - 194. [9] Cowin, S. C., & Mehrabadi, M. M. (1995). Anisotropic Symmetries of Linear Elasticity. Applied Mechanics Reviews, 48(5), 247 - 285. [10] Chadwick, P., Vianello, M., & Cowin, S. C. (2001). A New Proof that the Number of Linear Elastic Symmetries is Eight. Journal of the Mechanics and Physics of Solids, 49(11), 2471 - 2492. [11] Ting, T. –C. T. (2003). Generalized Cowin-Mehrabadi Theorems and a direct proof that the number of elastic symmetries is eight. International Journal od Solids and Structures, 40(25), 7219 – 7142. [12] Shi, G., & Bezine, G. (1988). A General Boudnary Integral Formulation for Anisotropic Plate Bending. Journal of Composite Materials, 22(8), 694 - 716. [13] Paiva, W. P., Sollero, P., & Albuquerque, E. L. (2002). Analysis of the Fundamental Solution for Anisotropic Thin Plates. 15th ASCE Engineering Mechanics Conference. New York, NY. [14] Adomian, G. (1994). Solving Frontier Problems of Physics: The Decomposition Method. Dordrecht: Kluwer Academic Publishers. [15] Browayes, J. T., & Chevrot, S. (2004). Decomposition fo the Elastic Tensor and Geophysical Applications. Geophysical Journal International, 159(2), 667 - 678. [16] Mirsky, L. (1955). An Introduction to Linear Algebra (3rd Edition ed.). London: Oxford University Press.


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Application of SPR for Discontinuous Boundary Elements Results in 2D Elasticity and a New Method to Evaluate Tangential Stress Otávio A. A. da Silveira1, Rogério J. Marczak2 1

Civil Engineering Department, UFSC

Campus Reitor João David Ferreira Lima, Rua João Pio Duarte da Silva, 205, Bairro Córrego Grande 88037-000, Florianópolis, Brazil [email protected] 2

Mechanical Engineering Department, Universidade Federal do Rio Grande do Sul Rua Sarmento Leite, 435, Centro, 90050-170, Porto Alegre, Brazil [email protected]

Keywords: Boundary element methods, boundary stress, stress recovery, stress smoothing.

Abstract. The use of discontinuous elements in the boundary element method (BEM) does not provide continuous results across de boundary mesh, i.e. variables are not single valued across element interfaces. The implementation of a smoothing technique, able to retrieve continuous results for isoparametric discontinuous boundary elements in two-dimensional elasticity is proposed. The methodology is based on the recovery of smoothed values at the geometric nodes shared by two elements, using least squares fit of the physical nodes values in the neighbourhood. New solutions with the same degree of interpolation of the original ones are obtained in each element from the recovered values and, consequently, a continuous solution can be achieved. Moreover, continuous as well as discontinuous boundary elements generate discontinuous, low accuracy results for the tangential component of stress, which is usually obtained by post processing. This paper presents a new proposal for computing that stress component based on the use of a lower number of sampling for the evaluation of displacement derivatives prior the application of the Hooke’s law. The efficiency of the proposed techniques is verified by solving static elasticity problems using linear and quadratic elements.

Introduction The relaxation of continuity conditions in discretization-based methods has gained impulse among several numerical methods in the last decade. The use of discontinuous elements in boundary element methods (BEM) is somewhat old, but discontinuous Galerkin finite element methods (FEM) are prime examples of the developments in this field. These approaches greatly simplify the computational implementation of the solution methods, and may increase their efficiency, particularly in nonlinear problems or problems containing discontinuous fields. In the BEM context, there are a number of advantages in the use of discontinuous boundary elements in spite of the characteristic interelement discontinuities. Discontinuous interpolation presents C1 continuity on all physical nodes, which simplifies the computation of strongly singular integrals. It also avoids the need of double nodes in cases containing corners and discontinuities in the boundary conditions. In addition, the use of discontinuous elements has already proved its efficiency in the solution of multidomain BEM formulations and FEM-BEM couplings (Zhang and Zhang, 2002). On the other hand, the

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recovery of variables at the ends of the elements by simple extrapolation or by averaging the extrapolated results of two or more elements is usually inadequate. The issue approached by the present work is related to the boundary stress components not directly evaluated from BEM boundary solution. It is well known that in 2D elasticity problems only of two stress components are directly given by the traction components along the boundary. The remaining component of the stress tensor must be computed by mixing the known stress components and another term, evaluated by differentiating the shape functions in order to estimate the normal strain in the tangential direction. Regardless the elements are continuous or not, lower accuracy is generally found for these post-processed stress components due to the reduction by one degree in the approximation polynomial. Therefore, one can expect problems similar to those found in FEM for Mindlin plates, where the shear strain is evaluated by mixing p polynomials for the plate rotations with p-1 polynomials for the derivatives of the transverse displacement. This is not a robust method because of two reasons: (a) mixing primal variables with dual ones (obtained by numerical differentiation) may lead to ill-conditioned equations (Guiggiani, 1994); (b) nodes are not the optimal ordinates to recover derivative (dual) variables. Although it is a viable technique for many applications, the tangential stress component may present significant errors when coarse meshes are employed. Aiming the evaluation of more reliable values for the stress components on the boundary, a low-cost alternative technique for computing the normal tangential stress component is presented and tested in this work. The proposed alternative technique for evaluation of the tangential stress is implemented for linear and quadratic discontinuous boundary elements, and used to solve 2-D elasticity benchmarks. The results obtained are compared with the conventional BEM results.

Alternative Tangencial Stress Calculation in Discontinuous Elements In numerical analysis, the computation of quantities by combining interpolated values and its derivatives must be done with care, as the optimal sampling points of the derivatives are not coincident with the interpolation points themselves. This issue is relatively common in many branches of computational mechanics. Prime examples can be found in FEM, for instance, in the calculation of stress in twodimensional elasticity elements, in the evaluation of shear strains in structural elements (plate/shell), or in the pressure-velocity coupling in fluid mechanics. This is essentially the very same problem that causes the locking phenomenon of in low order thick plate finite elements (Oñate et al. 1992, Zienkiewicz et al. 1993). A similar problem occurs in the standard evaluation of the tangential boundary stress components for elasticity in BEM, although not characterized by the same consequences as in FEM. As aforementioned, the missing boundary stress components in the conventional BEM are obtained using shape functions derivatives (tangential strain) and boundary tractions (Brebbia et al., 1984); however, it is known that this technique not necessarily provides good results along the whole element (Guiggiani, 1994). This work suggests a small change in the use of Hooke’s law in order to obtain a more reliable estimate of the tangential stress component for boundary elements without any significant increase in the computational cost. Basically, the tangential strain is sampled at optimal locations, instead of the nodes. It is important to note that the ideas presented herein are implemented and tested for 2D elasticity discontinuous boundary elements, but they can be used in a fairly broad class of problems, regardless the continuity of the interpolation. Standard technique for tangential stress calculation: In 2D elasticity problems, the normal (σnn) and shear (σnt) boundary stress are directly related to the boundary tractions (pn, pt) in a local coordinate system (n, t). Assuming that the tractions are written in


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the global coordinate system, the boundary stress components are easily obtained by rotating the tractions according to the local system: σ nn   p1    = R (α )   σ  nt   p2 

(1)

where R(α) is a rotation marix and α is the angle between global and the local coordinate systems (Fig. 1). The tangential strain (εtt) is obtained by using the interpolated displacements (Zhao, 1996): ε tt =

dut du1 du = t1 + 2 t2 dt dt dt

ε tt =

∴

dφ (ξ )  1  dφi (ξ ) i ∑  dξ u1t1 + di ξ u2i t2  J  

(2)

where t1 and t2 are components of the unit tangential vector in x1 and x2 directions, respectively, u1i and u2i are the nodal displacements in the i-th node in x1 and x2 directions, respectively; and φi(ξ) are the physical interpolation functions (J is the Jacobian of the element transformation to the normalized space).

n

t

x2

α P Ω

x1

Γ

Figure 1. Coordinate system over the boundary. The tangential stress component (σtt), can be obtained by Hooke’s law for plane-strain: σ tt (ξ ) =

1 ν σ nn (ξ ) + 2Gε tt (ξ ) 1 −ν 

(3)

where ν is the Poisson’s ratio, and G is the shear modulus. Therefore, when Eq.(3) is used in the standard BEM, it sums two polynomial terms of different orders. Depending on where σtt is evaluated, this procedure process may lead to unreliable results unless the ξ coordinate is known to be an optimal point to retrieve derivative quantities (present in the εtt term). Alternative technique for tangential stress calculation: The existence of points able to represent optimally the derivative of an interpolated function is well known and can be proved mathematically. In the FEM, these points are known as Barlow points, and they are used to evaluate stress fields from differentiation of interpolated displacements (Barlow, 1976 and Prathap, 1996). When the interpolation function is of the polynomial type, these points are located at the Gauss-Legendre stations corresponding to one order less than the minimum order necessary to integrate the interpolation function exactly. The underlying idea of the scheme proposed here is to use these points


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to evaluate Eq.(3). To the best of the authors’ knowledge, there are no similar studies correlating these aspects in the BEM context. In the case of linear boundary elements, the normal stress is obtained directly from the traction forces, and therefore it is a linear function (as well as the displacements). The tangential deformation is represented by a constant function in each element since it is obtained by the displacement derivative. The combined use of these two functions, through the Hooke’s law, is the origin of often unsatisfactory results (Guiggiani, 1994). The present work suggests the use of the central point of the element – Gauss point for a linear function integration (ξ = 0) – to sample the differentiation of the interpolated displacement. The coordinate ξ = 0 delivers the best estimate for the tangential strain along the element. It is worth to note that the evaluation of this strain at nodal locations (ξ = ±1) will overestimates or underestimates the strain value. In summary, it is proposed that both, the normal stress and tangential strain should be evaluated at the center of the element, thus obtaining a constant function for tangential stress over each element. For clarity, Table 1 compares both ways for the evaluation of the tangential stress in linear elements. For quadratic discontinuous elements where, a priori, the normal stress and tangential strain are represented by quadratic and linear functions respectively, it is suggested that the Gauss points for a cubic quadrature (ξ = ±1/√3) should be used to represent the tangential strain field along the element. Therefore, replacing the standard technique, the calculation of the tangential stress is performed using the values for normal stress and tangential strain just at two points. A linear interpolation of the values obtained in the Gauss points is made in order to obtain the nodal stress values of the quadratic element. Table 2 shows the two methods for tangential stress calculation on quadratic elements. In summary, the tangential stress calculation is made with one degree less than the other stress components. This method may initially seems less sound than the conventional procedure, but later it will be shown that when used with the smoothing technique described by Silveira (2007), the proposed scheme leads to better results. In many cases, it was found that the conventional scheme will produce wrong signs to the εtt term in Eq.(3), a direct consequence of it being sampled at non-optimal points. Table 1. Tangential stress calculation for linear discontinuous boundary elements*. STANDARD TECHNIQUE

1 (νσ nn1 + 2Gε tt1 ) 1 −ν 1 σ tt2 = (νσ nn2 + 2Gε tt2 ) 1 −ν

σ tt1 =

ALTERNATIVE TECHNIQUE 1 σ tt (ξ1 ) = (νσ nn (ξ1 ) + 2Gε tt (ξ1 ) ) 1 −ν σ tt1 = σ tt2 refers to constant interpolation of σ tt ( ξ1 )

* ξ1 = 0 , and the superscript represent the associated nodal value of the variable.

Table 2. Tangential stress calculation for quadratic discontinuous boundary elements*. STANDARD TECHNIQUE 1 (νσ nn1 + 2Gε tt1 ) 1 −ν 1 σ tt2 = (νσ nn2 + 2Gε tt2 ) 1 −ν 1 σ tt3 = (νσ nn3 + 2Gε tt3 ) 1 −ν

σ tt1 =

* ξ1 = −1

3 , ξ2 = 1

ALTERNATIVE TECHNIQUE 1 σ tt (ξ1 ) = (νσ nn (ξ1 ) + 2Gε tt (ξ1 ) ) 1 −ν 1 σ tt (ξ 2 ) = (νσ nn (ξ 2 ) + 2Gε tt (ξ 2 ) ) 1 −ν σ tt1 , σ tt2 e σ tt3 are obtained by linear interpolation of σ tt ( ξ1 ) e σ tt (ξ 2 )

3 , and the superscript represent the associated nodal value of the variable.


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Numerical Results In order to investigate the performance of the proposed technique, this section shows some results. Numerical integration was carried out using 16 Gauss points, in order to minimize the influence of quadrature errors. Dimensions, material properties, and other physical data are given without units, but they were specified to represent a compatible system of units. The material properties used in all case are: E = 210e9 and ν = 0.3 Plane-stress condition is assumed throughout this section. Square-plate with a central hole under traction: A 100×100 square plate with a central hole of radius R = 5 was analyzed. Due to symmetry, only one quarter of the plate was considered (Fig. 2). The traction loading along the upper side was set to P = 1. The offset of all boundary elements used in the mesh is 15% of the element length. Linear and quadratic elements were used with two different meshes for each type of element. Mesh 1 used an element size of 2.5 along the straight boundaries and four elements along the quarter-circle. Mesh 2 used an element size of 1.25 and eight elements along the quarter-circle. P

50

r

θ B

A

50

Figure 2. Squared plate with a central hole under uniform traction. Regarding the application of the alternative method for tangential stress calculation, it can be used with or without a smoothing procedure (Silveira, 2007), leading to four possibilities for postprocessing the results: • Method A: Discontinuous BEM without smoothing – the raw results of discontinuous elements are considered with standard tangential stress calculation (section 2.1). • Method B: Discontinuous BEM with smoothing – same as Method A, but the results are smoothed. • Method C: Modified discontinuous BEM without smoothing – raw results of discontinuous elements with alternative tangential stress calculation as outlined in section 2.2. • Method D: Modified discontinuous BEM with smoothing – same as Method C, but the results are smoothed. These methods were used to post-process the normal radial stress along the edge AB, which is the tangential component along that piece of boundary. Figure 3 compares graphically these results for linear elements with meshes 1 and 2.

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The graphs depicted in Fig. 3 shows that the response which agrees more closely to the analytical solution is the smoothed solution considering the alternative tangential stress calculation. As expected, it can be seen that the alternative tangential solution without smoothing is simply an element average value from discontinuous BEM without smoothing. Figure 4 shows the recovery of radial stress on the same edge, this time using quadratic elements. As in Fig. 3, these graphs show the four types of post-processed results against the analytical solution. Although the differences between the four methods are not as drastic as in the case of linear elements, it is evident that the smoothed results obtained with alternative tangential stress calculation agree more closely to the analytical solution, particularly at point A. Another important aspect is that the differences between all methods tend to vanish where analytical solution is less oscillatory (away from stress concentration areas). Interestingly, it is also evident from Figs.3-4 that none of the methods provided very good results near the hole, although the modified stress calculation seems to recover the better ones. This is direct consequence of the different signals of the terms in Eq. (3), i.e. the high gradients of the tangential strain near the hole are miscalculated when the displacements are differentiated at nonoptimal locations. Of course, this effect becomes more conspicuous when coarse meshes are used.

(a)

(b) Figure 3. Radial stress recovery. Linear elements: (a) mesh 1, (b) mesh 2.


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(a)

(b) Figure 4. Radial stress recovery. Quadratic elements: (a) mesh 1, (b) mesh 2.

Conclusions This work presented an alternative technique for tangential stress component calculation in BEM methods, which estimates more reliable results when compared to the standard boundary stress technique. Moreover, it was used a method to obtain smoothed results from the results of tangential stress component, since it is now evaluated at different positions than the nodes. This method of smoothing appears to work very well with the alternative technique for tangential stress calculation. The increase in the computational cost is negligible. The scheme proposed presents potential to be used with other types of elements or different governing equations without hurdles.

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References [1] Barlow, J., Optimal Stress Locations in Finite Element Models, International Journal For Numerical Methods in Engineering, vol. 10, pp. 243-251, (1976). [2] Brebbia, C.A., Telles, J.C.F., Wrobel, L.C., Boundary Element Techniques, Springer Verlag, Berlin Heidelberg (1984). [3] Guiggiani, M., Hypersingular Formulation for Boundary Stress Evaluation, Engineering Analysis with Boundary Elements, vol. 13, pp. 169-179 (1994). [4] Silveira, O.A.A., Implementação de Técnicas de Suavização de Resultados para Elementos de Contorno Descontínuos. Dissertação de Mestrado, UFRGS/Porto Alegre. (2007) [5] Prathap, G., Barlow Points and Gauss Points and the Aliasing and Best Fit Paradigms, Computer & Structures, vol. 58, No. 2, pp. 321-325 (1996) [6] Zhang, Xiaosong, Zhang, Xiaoxian, Coupling FEM and Discontinuous BEM for Elastostatics and Fluid–structure Interaction, Engineering Analysis with Boundary Elements, vol. 26, pp. 719-725 (2002). [7] Zhao, Z.Y., Interelement Stress Evaluation by Boundary Elements, International Journal for Numerical Methods in Engineering, vol. 39, pp. 2399-2415 (1996). [8] Zienkiewicz, O.C., Zhu, J.Z., Wu, J., Superconvergent Patch Recovery Techniques – Some Further Tests, vol. 9, pp. 251-258 (1993).


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3D frictional contact analysis with Boundary Element Method and discontinuous elements using a Generalized Newton Method with line search Cristiano J.B. Ubessi1 , Rogério J. Marczak2 1

Mechanical Engineering Post Graduation Program - PROMEC, UFRGS, Brazil, [email protected] 2

Mechanical Engineering Department - DEMEC, UFRGS, Brazil, [email protected]

Keywords: Frictional Contact, Boundary Element Method, Discontinuous Elements, Generalized Newton Method.

Abstract: This paper descibes the implementation of an algorithm for the solution of 3D elastic contact problems with friction using the Boundary Element Method (BEM) with discontinuous elements. A standard BEM formulation is used, and the coupling of the potential contact zone is imposed through a projection function which treats each region independently, being updated along with the changes to the contact state. Contact restrictions are fulfilled through the augmented Lagrangian, and the non-linear solution is found using the Generalized Newton Method with line search. With this method is possible to skip the calculation of the non-linear derivatives, allowing for a fast solution of the problem. Hertz type contact problems are solved to demonstrate the accuracy of the method and to provide a comparison case with analytical solutions found in the literature. Introduction Contact type problems are too often found in engineering applications. While some could be simplified or even assumed to be irrelevant, there exists cases where it is the reason of existence of the engineering problem by itself. Wear, tear, fatigue, friction, among others, are all problems which could arise by the contact occurrence. With the fast growth on the use of advanced and high performance materials in engineering, rises the need to predict the contact conditions when using these materials. The Boundary Element Method is well known for its ability to solving contact problems, since its formulation intrinsically treats the displacements and tractions with same order of approximation. This enables the direct application of the contact constraints without the need of penalty parameter or Lagrangian multipliers. Since the pioneer work of [4], which used that property to develop an incremental loading technique to solve contact problems on 2D, a few other works are also found on the literature using the same principles, such as [4], [5], [6], [7], [8] and [9]. Another works are found such as [10], [11], which use the Lagrangian multiplier or the penalty parameter methods, which are mandatory to treat contact problems with FEM. Though they could be used, are not needed to treat contact with pure BEM discretization. More recently, motivated by [12], which treated FEM-BEM coupled problems, [13], based on the works of [14], [3], used the Augmented Lagrangian formulation, which circumvents some weaknesses existing on Lagrangian multiplier and penalty methods. The contact restrictions are imposed in the form of projection functions, resulting on a very robust framework to both FEM and BEM frictional contact analysis. The resulting non linear system of equations is then solved by the Generalized Newton Method with line search (GNMls), which is simply a generalization of the standard Newton method to B-differentiable functions, and which convergence is independent of the penalization parameter used on it. With an unconstrained optimization between each step of the newton method is also possible to accelerate its convergence. The resulting equations can be further simplified with the complementarity properties of the normal tractions and gap, reducing the number of DOFs needed on the SLE solution. The method was also used by [15] to study 3D frictional contact on anisotropic media using BEM. This paper describes part of our research on contact analysis using BEM, where the GNMls was implemented on the resolution of contact problems using discontinuous elements. The paper is presented on the following methodology: First the contact problem is described along with the restrictions it imposes. The boundary integral formulation leading to BEM equations is presented. The contact variables on the discrete form, along with the projection functions and the Augmented Lagrangian variables which imposes the restrictions on the discrete form. The resulting non linear system of equations is presented with an analogy to problems with multiple regions. The GNMls is described with the linearized Jacobian. On the results section, the classical Hertz contact problem, considering two elastic regions, is analyzed using a BEM mesh and the solution obtained is compared with the analytical one. Final considerations are done, which closes this work.

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Methodology The elastic contact problem. Consider the simple problem of two separated elastic bodies, which may come in to contact, as illustrated on Fig. 1. Treating this as a classical Boundary Value Problem (BVP), one knows by anticipation the prescribed conditions at the bodies boundaries: Tractions are null on the contours which are free to displace in any direction, and are unknowns on this regions ( ); Displacement are generally prescribed as null in some directions, also called restrictions, but can also be different from zero, and in both cases the tractions will be unknowns on those regions. The contact boundary, have conditions which depends on the contact state, defined by the distance between the two bodies, which cannot be negative. When it is positive, the surfaces are free and the tractions are null. Compatibility conditions must be set when the distance is zero.

Figure 1: Solid under consideration The conditions relate the displacement and the tractions on the contact surface, and will depend on the existence of friction or not. Kinematic variables. In this work the BEM formulation is assuming small strains and displacements, and node on node contact. The nodes are assumed to be positioned in a conforming scheme, i.e., the slave nodes are positioned as closely as possible to the master, or matching the displacement path performed by the contact pair. The contact variables in the discrete form will then be related to the possible contact node pairs. The contact frame is based on the master nodes, and the gap variable is obtained through the following relation (1) where and are master and slave nodal coordinates, and are the respective nodal displacement vectors, on the global Cartesian coordinate system, and is a change of base matrix constructed with the three unit vectors which form a local coordinate system with origin at , the master node position, i.e., (2) Contact laws. The unilateral contact condition and the friction law could be summarized for any pair of points, The contact problem is modeled as a variyng boundary condition, which has restrictions according to the contact state of the contact node pair. The well known frictional contact restrictions are

(3) The time rate appearing in equation (3) must be approximated. As done in [13], a finite differences scheme approximates the at time as follows (4) Boundary Element Method formulation. The BEM formulation of this work is folowing the Somigliana’s well known identity on the boundary brebbia2012:

(5)


247

where, in the case of this work, the boundary at is always smooth due to the use of discontinuous elements, i.e., . To obtain a numerical solution from Eq. (5), the boundary is discretized in to a finite number of elements, forming an algebraic system of equations. The geometry of the body will be calculated by means of shape functions in terms of the parametric coordinates . The physical variables of the problem will be calculated in physical nodes that are offset relative to the geometrical nodes. Displacements and tractions will be interpolated with the discontinuous interpolation functions, :

(6) In this work, discontinuous rectangular elements were used, and the interpolation functions could be found on [17]. Writing Eq. (5) in matrix notation, and further combination of the terms computed after a collocation process over all boundary nodes will result in the algebraic system of equations that leads to the BEM solution, i.e., (7) Contact constraints. The well known frictional contact restrictions are:

(8) The frictional contact law is fulfilled by means of projection operators, i.e. functions which project the the contact variables in to the admissible solution region [13]. The normal tractions projector function takes the form: variable, augmented normal traction is defined as: parameter. The tangential projector function takes the form:

, , where

. The mixed is a positive penalization

,

(9) where

. The tangential contact restriction then is written as (10)

and the augmented tangential traction is defined as: could differ from the normal one.

where the positive penalization parameter

The constraints of the combined normal-tangential contact problem can be formulated as: contact operator is then defined as

. The

(11) where the region

is the augmented friction circle with radius

.

Contact treatment with Boundary Element Discretization. One of the well known advantages of BEM in contact problems is that the contact tractions are already part of the unknowns. The SLE for the contact problem will be formed by in a similar mode as it is done for multiple-region problems (e.g., [16]), i.e., considering the geometry illustrated on Fig. 1, the following system arise, (12)

248

Eds: F.García Sánchez, L.Rodríguez-Tembleque, M.H.Aliabadi

where the bonded connections on the interface between the two regions was set by the displacement and traction compatibility conditions. Equation (12) is sufficient to calculate a bonded problem, where is assumed the interface region remains the same at all times and support tractions in all directions. To incorporate contact restrictions on a similar equation system one have to write Eq. (12), along with two additional sets of equations: The kinematic relations which arise from the gap, i.e., Eq. (1), and the projection operators, which represent the , contact restrictions, depending on the contact state of the node pair, resulting in a system composed by the following system of Non linear equations,

(13) where are the matrices relative to the independent unknowns (mixed tractions and displacements) for the th region, including the displacement unknowns on the possible contact region, are the matrices relative to the traction unknowns on the possible contact region. The tractions on the contact interface ( ) as well as the gap ( ) are considered on their local coordinate system, by the incorporation of the rotation matrices on the system of on equations, i.e., Eq. (2), which are assembled for each contact pair on the main rotation matrix presented as Eq. (13). The matrices are the same rotation matrices, but assembled only on the positions of corresponding to displacement unknowns of the contact regions. the matrix , and is the initial gap. The projector matrices are also assembled for each contact pair, and will depend on the contact pair state, following the formulation from [13]. Results The problem analyzed in this work is the Hertz classical one, with the particularity of both solids being elastic, not so commonly seen on the literature, since generally one of the bodies is considered rigid and a half space on the plane region with BEM, in favor of a simpler solution and a closer geometry to that assumed on the closed form solution. The half space representation is numerically simpler to solve, and the use of a rigid indenter is closer to a pure prescribed displacements problem, than to an elastic contact problem, as the final displacements of the contact regions are known a priori. This classical problem also permits the evaluation of the algorithm when a small region of the solids are transferring the load, concentrating the contact stress in a small area. The mesh used for the the sphere and the cube on this problem has 888 and 468 8-node quadratic elements applied on the lower face of the half respectively, and is illustrated on Fig. 2. The prescribed displacements sphere and the cube upper face was restricted on the direction. On all elements at the and plane a symmetry boundary condition was applied, restricting the displacements on the outward normal direction, and freeing it on the tangential direction. The offset used on the discontinuous elements was , and all material and geometrical properties considered on this example are shown on Table 1. Also is brought the predicted maximum contact pressure from [18], which do not consider frictional effects on its formulation, so the coefficient of friction was set to .

Figure 2: Mesh used on the Hertzian contact example


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The load was applied on a single step, and the algorithm convergence is brought on Fig. 3. The initial parameters used on the GNMls were , . To accelerate the solution time, an iterative sover (GMRES) was used, and a lower initial tolerance was set, along with an adaptive tolerance reduction lower than the last solution residual. As can be seen on the graph, although the solver could not converge to the desired tolerance on the final iterations, the final GNMls residual was lower than the desired value.

Figure 3: Newton method convergence: Residual for each iteration( ), solver tolerance (tol), relative solver residual (res), and obtained on line search. On Fig. 5a, the obtained displacements on the contact region are plotted as a function of the radius from the center of the sphere along with the analytical solution. Both numerical and analytical displacements were normalized with the prescribed displacements . The mesh nodes were disposed on a rectangular grid, so the results are scattered along their relative radius . On Fig. 5b, the obtained normal tractions on the contact region are plotted in the same way as the displacements, and normalized with the maximum analytic contact pressure . As can be seen, the normal displacements and tractions agree with the analytical solution. Conclusion On this work it was possible to evaluate the use of discontinuous BEM to resolve a classical contact problem. A brief review of BEM contact literature was presented. The GNMls presents a good convergence rate, needing a few iterations to start converging at a logarithmic rate. Results obtained for displacements and tractions on the contact regions show a good agreement with the analytical solution. Acknowledgements The first author wish to express his thanks to CNPq for the doctoral scholarship. References [1] Fichera, G. (1963). “Sul problema elastostatico di Signorini con ambigue condizioni al contorno.” Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat., 34:138–142. ISSN 0392-7881.

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Figure 4: Traction and displacements along the possible contact region selected for the problem [2] Fichera, G. (1973). “Boundary value problems of elasticity with unilateral constraints.” In Linear Theories of Elasticity and Thermoelasticity, pages 391–424. Springer. [3] Alart, P. and Curnier, A. (1991). “A Mixed Formulation for Frictional Contact Problems Prone to Newton Like Solution Methods.” Comput. Methods Appl. Mech. Eng., 92(3):353–375. ISSN 0045-7825. 10.1016/00457825(91)90022-X. [4] Andersson, T. (1981). “The boundary element method applied to two-dimensional contact problems with friction.” In Boundary element methods, pages 239–258. Springer. [5] Paris, F. and Garrido, J. (1989). “An incremental procedure for friction contact problems with the boundary element method.” Engineering Analysis with Boundary Elements, 6(4):202 – 213. ISSN 0955-7997. http://dx.doi.org/10.1016/0955-7997(89)90019-2. [6] Garrido, J., Foces, A., and Paris, F. (1991). “BEM applied to receding contact problems with friction.” Mathematical and Computer Modelling, 15(3):143–153. [7] Man, K., Aliabadi, M., and Rooke, D. (1993). “BEM frictional contact analysis: load incremental technique.” Computers & structures, 47(6):893–905. [8] Man, K., Aliabadi, M., and Rooke, D. (1993). “BEM frictional contact analysis: modelling considerations.” Engineering analysis with boundary elements, 11(1):77–85. [9] Paris, F., Blazquez, A., and Canas, J. (1995). “Contact problems with nonconforming discretizations using boundary element method.” Computers & structures, 57(5):829–839. [10] Yamazaki, K., Sakamoto, J., and Takumi, S. (1994). “Penalty method for three-dimensional elastic contact problems by boundary element method.” Computers & structures, 52(5):895–903. [11] Rodríguez-Tembleque, L., González, J. Á., and Abascal, R. (2008). “A formulation based on the localized Lagrange multipliers for solving 3D frictional contact problems using the BEM .” Numerical Modeling of Coupled Phenomena in Science and Engineering: Practical Use and Examples, page 359. [12] González, J. A., Park, K., Felippa, C. A., and Abascal, R. (2008). “A formulation based on localized Lagrange multipliers for BEM–FEM coupling in contact problems.” Computer Methods in Applied Mechanics and Engineering, 197(6):623–640. [13] Rodríguez-Tembleque, L. and Abascal, R. (2010). “A FEM-BEM Fast Methodology for 3D Frictional Contact Problems.” Comput. Struct., 88(15-16):924–937. ISSN 0045-7949. 10.1016/j.compstruc.2010.04.010. [14] Pang, J.-S. (1990). “Newton’s method for B-differentiable equations.” Mathematics of Operations Research, 15(2):311–341. [15] Rodriguez-Tembleque, L., Buroni, F., Abascal, R., and Sáez, A. (2011). “3D frictional contact of anisotropic solids using BEM.” European Journal of Mechanics-A/Solids, 30(2):95–104. [16] Brebbia, C., Telles, J., and Wrobel, L. (2012). Boundary Element Techniques: Theory and Applications in Engineering. Springer Berlin Heidelberg. ISBN 9783642488603. [17] Beer, G., Smith, I., and Duenser, C. (2008). The Boundary Element Method with Programming. Springer Wien, New York. [18] Johnson, K. L. and Johnson, K. L. (1987). Contact mechanics. Cambridge university press.


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Series expansion of the axi-symmetric fundamental solution for transversely isotropic elastic solids Enrique Graciani1,a, Inmaculada Gómez-Rodríguez1 1

Escuela Técnica Superior de Ingeniería, Universidad de Sevilla Camino de los Descubrimientos s/n, Sevilla, 41092, Spain a [email protected]

Keywords: elasticity, fundamental solution, singularities, transverse isotropy, axial symmetry

Abstract. A series expansion of the fundamental solution for axisymmetric problems in transversely isotropic elastic solids in the vicinity of the collocation point, not located at the symmetry axis, has been carried out. To this end, the fundamental solution for displacements, 𝑈 𝒙, 𝒚 , and tractions, 𝑇 𝒙, 𝒚 , has been written in terms of the complete elliptic integrals of the first, second and third kind. In the vicinity of the collocation point, 𝒙, the integration point, 𝒚, can be expressed as 𝒚 𝒙 𝜀 𝒓, with 𝒓 being a unit vector and 𝜀 being the distance between the collocation and the integration point. Introducing the series expansion of the complete elliptic integrals of the first and second kind in the expressions of the components of the fundamental solution, the singular terms have been isolated. The series expansion of the complete elliptic integral of the third kind is not required, because the terms containing it are not singular in the vicinity of the collocation point. As a result of the series expansion, the fundamental tractions are expressed as the sum of a strongly singular term, proportional to 𝜀 , a weakly singular term, proportional to ln 𝜀 , a bounded term with infinite slope at the collocation point, proportional to 𝜀ln 𝜀 , and a regular term. Analogously, the fundamental displacements are expressed as the sum of a weakly singular term, a bounded term with infinite slope at the collocation point and a regular term. The implementation of the fundamental solution has been validated applying the Somigliana Indentity to problems with known solution (including the fundamental solution itself as the solution of one of these problems). Moreover, to check that the singular terms are correctly identified, they have been successively subtracted to the original kernels and represented in the vicinity of the collocation point. Finally, it has been demonstrated that if quasi-isotropic material properties are employed, the expressions obtained for transversely isotropic materials reproduce those corresponding to isotropic materials.