Advances in Boundary Element & Meshless ...

ISBN 978‐0‐9547783‐9‐2

Advances in Boundary Element & Meshless Techniques XIII

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

EC ltd

Advances in Boundary Element & Meshless Techniques XIII

Edited by P Prochazca M H Aliabadi

Advances In Boundary Element and Meshless Techniques XIII

Advances In Boundary Element and Meshless Techniques XIII

Edited by P Prochazka M H Aliabadi

EC

ltd

Published by EC, Ltd, UK Copyright © 2012, Published by EC Ltd, P.O.Box 425, Eastleigh, SO53 4YQ, England Phone (+44) 2380 260334

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ISBN: 978-0-9547783-9-2

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

International Conference on Boundary Element and Meshless Techniques XIII 3-5 September 2012, Prague, Czech Republic Organising Steering Committee: Prof. Petr Prochazka Czech Technical University in Prague Faculty of Civil Engineering Department of Mechanics Thakurova 7 166 29 Prague 6 Czech Republic [email protected] Prof. Ferri M H Aliabadi Department of Aeronautics Imperial College, South Kensington Campus London SW7 2AZ [email protected] International Scientific Advisory Committee Abascal,R (Spain) Abe,K (Japan) Baker,G (USA) Beskos,D (Greece) Blasquez,A (Spain) Chen, W (China) Cisilino,A (Argentina) Denda,M (USA) Dong,C (China) Dumont,N (Brazil) Estorff, O.v (Germany) Gao,X.W. (China)

Garcia-Sanchez,F (Spain) Gatmiri,B (France) Hartmann,F (Germany) Hematiyan,M.R. (Iran) Hirose, S (Japan) Kinnas,S (USA) Liu,G-R (Singapore) Mallardo,V (Italy) Mansur, W. J (Brazil) Mantic,V (Spain) Marin, L (Romania)) Matsumoto, T (Japan) Millazo, A (Italy) Minutolo,V (Italy) Ochiai,Y (Japan) Panzeca,T (Italy) Perez Gavilan, J J (Mexico) Pineda,E (Mexico) Qin,Q (Australia) Saez,A (Spain) Sapountzakis E.J. (Greece) Sellier, A (France) Shiah,Y (Taiwan) Sladek,J (Slovakia) Saldek, V (Slovakia) Sollero.P. (Brazil) Stephan, E.P (Germany) Taigbenu,A (South Africa) Tan,C.L (Canada) Telles,J.C.F. (Brazil) Wen,P.H. (UK)

Yao,Z (China)

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) and Brasilia, Brazil (2011). The present volume is a collection of edited papers that were accepted for presentation at the Boundary Element Techniques Conference held at the Diplomat Hotel, Prague, Czech Republic during 3th-5th September 2012. Research papers received from 18 counties 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 September 2012 CONTENT

A three-dimensional boundary element model for the analysis of polycrystalline materials at the microscale I. Benedetti, M.H. Aliabadi Three-dimensional unsteady thermal stress analysis by triplereciprocity boundary element method Y OchaiI, V Sladek and J Sladek Green element solutions to steady inverse heat conduction problems A E. Taigbenu and J Ndiritu Anisotropic wear simulation using the boundary element method L. Rodrıguez-Tembleque, R. Abascal, M.H. Aliabadi Non-Local plastic deformation gradients for localization phenomenon and regularization of strain-softening behavior using the meshless finite point method L. Pérez Pozo, A. Campos Rodríguez An inverse multi-loading boundary element method for identification of elastic constants of 2D anisotropic bodies M.R. Hematiyan, A. Khosravifard, Y.C. Shiah, L. Tan Analysis of phase-change heat conduction problems by an improved CTM based A. Khosravifard, M. R. Hematiyan BIE reduction for long cylindrical shapes – the Laplace, Poisson and Helmholtz equation P Jablonski

1

7

15

23

29

39

47

55

Fiber shape optimization in linear elasticity P P. Prochazka

61

Optimal shape of fibers in composites exposed to combustion P P Prochazka, M J Valek

67

Meshless method in analytical formulation and application to elastodynamics P.H. Wen and M.H. Aliabadi Symmetric BEM as discretization tool for FEM analysis of thin plates M. Mazza General framework for localization and regularization for local and non-local damage model using the meshless finite point method L. Pérez Pozo, F. Chacana Yorda Genetic algorithms and the method of fundamental solutions for simulations of cathodic protection systems W. J. Santos, J. A. F. Santiago and J. C. F. Telles Slipping stokes flow about a solid particle experiencing a rigidbody motion A. Sellier Slow viscous migration of a solid particle near a plane wall with a general non-isotropic slip condition N. Ghalia and A. Sellier Gravity-driven migration of one bubble near a free surface: surface tension effects M. Guemas, A. Sellier and F.Pigeonneau Dual BEM analysis of semipermeable cracks in magnetoelectroelastic solids under time-harmonic loading G Hattori, R.Rojas-Diaz, M Denda, F.Garcia-Sanchez, A Saez A BEM formulation for the dynamic analysis of shear deformable symmetric composite plates J. Useche, H. Alvarez Nonlinear dynamic analysis of plates stiffened by parallel beams

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E.J.Sapountzakis1 and J.A. Dourakopoulos MFS solution of inverse boundary value problems in twodimensional linear thermoelasticity Liviu Marin and Andreas Karageorghis Finding hidden cavities in a three-dimensional layer based on combined Genetic Algorithms and the Optimal Linearized Cost Function A E Martínez-Castro, I H. Faris and R Gallego On the transient response analysis of thick elastic plates by the boundary element method W.L.A. Pereira V.J. Karam, J.A.M. Carrer, W.J. Mansur Regularization of the divergent integrals using generalized function based approach V.V. Zozulya Alternative forms of Green’s function as a source for infinite product representation of elementary functions Y A Melnikov Stability investigation of radial basis functions and time evolution E J Kansa BEM study of the symmetry of the debond onset at the fibrematrix interface under transverse load I.G. Garcıa, V. Mantic, E. Graciani An efficient numerical scheme for the evaluation of the fundamental solution and its derivatives in 3D generally anisotropic elasticity Y.C. Shiah, C.L. Tan and C.Y. Wang Hyper-singular dual reciprocity boundary element formulation applied to diffusive-advective problems C F Loeffler, F Costalonga

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Hierarchical-ACA DBEM for anisotropic three-dimensional timedomain fracture mechanics A Milazzo, I Benedetti, M H Aliabadi 3D boundary element analysis of delamination crack using the modified crack closure integral I. Benedetti A. Milazzo, C. Orlando

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212

New traction fundamental solutions for anisotropic materials F C Buroni, A Saez

218

A meshless method for the Reissner-Mindlin Plate Model based on stabilised weak form P M Baiz, J S Hale

224

BEM for the second problems of the Stokes system D Medkova A Multiregion Technique for the Boundary Element Method with Drilling Rotation D. I. G. Costa, E. L. Albuquerque, P. M. Baiz A large deflection analysis of laminate composite thin plates by the boundary element method L. S. Campos, E. L. Albuquerque Computation of stress in orthotropic thick plates by the boundary element method A. P. Santana, E. L. Albuquerque, A. Reis and J. F. Useche Geometric and creeping material nonlinearities in shear deformable plate with the boundary element method E. Pineda, M.H. Aliabadi, J. Núñez-Farfán, A. RodríguezCastellanos Scattering of elastic waves in fluid-layered solid interfaces by the indirect boundary element method

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A. Rodríguez-Castellanos, E. Pineda León, J. Nuñez-Farfán, E. Olivera-Villaseñor, Andrei Kryvko Advances in numerical modeling of cohesive crack using a meshless Finite Point Method L. Pérez Pozo, F. Valdivia Bugueño Vibrations of a rigid circular foundation embedded on a transversely isotropic multilayered soil J Labaki, E Mesquita and N Rajapakse

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A new truly meshless method in the near-incompressible linear elasticity analysis J. Belinha, L M J S Dinis, R M Natal Jorge

282

The elasto-static analysis of structures using a new developed meshless method J. Belinha, L M J S Dinis, R M Natal Jorge

289

On a strategy to implement a fast and expedite boundary element method N A Dumont and C Andrés A Marón

295

Analysis of anisotropic symmetric plates by the adaptive cross approximation R. Q. Rodrıguez, P. Sollero and E. L. Albuquerque The method of fundamental solutions applied to crack problems using Tikhonov's regularization and the numerical Green's function procedure E.F. Fontes Jr. J.A.F. Santiago. J.C.F. Telles An SGBEM implementation of an energetic approach for mixed mode delamination R Vodicka, V Mantic Multidomain SBEM analysis of two dimensional elastoplastic contact problems T. Panzeca, F. Cucco, M. Salerno

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How to use the SGBEM in the practical engineering? Panzeca T, Terravecchia S. and Zito L.

331

Vibration analysis of vehicle cab by BEM with block SS method H.F. Gao, T. Matsumoto, T. Takahashi, T. Yamada

337

Curvature singularities on surfaces of water waves G Baker and Jeong-Soon Im

343

An Isogeometric Boundary Element Method for Interior Acoustic Analysis using T-splines R N. Simpson, M A. Scott, H Lian A novel local active noise control strategy using a control volume formulated by a fast boundary element approach A. Brancati, M.H. Aliabadi and V. Mallardo A fast 3D-BEM approach to local active noise control: a sensitivity analysis A. Brancati, M.H. Aliabadi Effect of aspect ratio on the stability analysis of uniaxially loaded plates with eccentric holes by the boundary element method P. C. M. Doval, E. L. Albuquerque and P. Sollero Structural Health Monitoring of Sensorised Panels using Z Sharif Khodaei, S Leme and M H Aliabadi Structural Health Monitoring of Delaminated Composite Structures by the Boundary Element Method A. Alaimo, A. Milazzo, C. Orlando

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Advances in Boundary Element and Meshless Techniques

A three-dimensional boundary element model for the analysis of polycrystalline materials at the microscale I. Benedetti1,2,a, M.H. Aliabadi1,b 1

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

2

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: Polycrystalline materials, Microstructure Modelling, Material Homogenization, Multi-region formulation, Anisotropic Boundary Element Method.

Abstract. A three-dimensional multi-domain anisotropic boundary element formulation is presented for the analysis of polycrystalline microstructures. The formulation is naturally expressed in terms of intergranular displacements and tractions that play an important role in polycrystalline micromechanics, micro-damage and micro-cracking. The artificial morphology is generated by Hardcore Voronoi tessellation, which embodies the main statistical features of polycrystalline microstructures. Each crystal is modeled as an anisotropic elastic region and the integrity of the aggregate is restored by enforcing interface continuity and equilibrium between contiguous grains. The developed technique has been applied to the numerical homogenization of SiC and the obtained results agree very well with available data. Introduction Macroscopic material properties depend on the material microstructure. Understanding the link between micro- and macro-properties is an important and technologically relevant task of modern Materials Science. The estimation of the effective material properties can be carried out at different levels [1]. A modern approach to material homogenization is the use of numerical models for the simulation of the material behavior at the microstructural scale [2]. Polycrystalline materials constitute an important class of heterogeneous materials [3]. Many engineering materials (metals, ceramics) present a polycrystalline microstructure. The internal structure of polycrystals is determined by the size and shape of the grains, by their crystallographic orientation and by different types of defects. A crucial role in the determination of the polycrystalline aggregate properties is played by the intergranular interfaces and their defects [4]. The polycrystalline microstructure can be investigated by using different experimental techniques [5,6]. These provide fundamental information but require sophisticated equipment, material manufacturing and preparation and complicated post-processing, resulting then generally expensive and time consuming. A viable alternative, or complement, to the experimental characterization is offered by Computational Micromechanics [2]. The dramatic increase in computational power and the formulation of reliable mathematical models allow to simulate the response of complex microstructures at little cost, thus complementing and accelerating the experimental campaigns when, for example, the design of a new material is pursued. In the present study, a three-dimensional boundary integral formulation for the analysis of polycrystalline microstructures is presented. The technique is alternative to the more used FEM and its typical features are: a) the simplification in the artificial microstructure generation and modelling, especially in relation to the meshing of the artificial microstructure, since only the discretization of the grains surface is required; b) the microstructural problem is formulated

1

2

Eds: P Prochazka and M H Aliabadi

directly in terms of intergranular displacements and tractions, which play an important role in polycrystalline micromechanics, especially when damage and micro-cracking are involved [7]. Artificial microstructure The artificial microstructure must retain the main topological, morphological and crystallographic features of the aggregate. For polycrystalline materials, Voronoi tesselations are widely used for the generation of the microstructural models [4,8,9]. The Voronoi cells are convex polyhedra bounded by flat polygonal convex faces. Voroni tessellations have the advantage of being analytically defined, relatively simple to generate and possess some features that make them suitable for numerical treatment, (straight edges and flat faces). Here the Hardcore Voronoi tessellation is adopted for generating the microstructure: the additional hardcore constraint produces more regular grains and tessellations. The assignation of a specific orientation to each crystal of the aggregate completes the microstructure representation. In this work, each grain is assigned a random orientation from a uniform distribution in the group of rotations in the three-dimensional space. Microstructure boundary element model Material modelling. Each grain is modeled as a three-dimensional linear elastic orthotropic domain with arbitrary spatial orientation. This is not restrictive, as the majority of single metallic and ceramic crystals present general orthotropic behavior. Grain boundary element formulation. Each crystal is modeled by using the Boundary Element Method (BEM) for 3D anisotropic elasticity [10]. The polycrystalline aggregate is seen as a multiregion problem, so that different elastic properties and spatial orientation can be assigned to each grain [11]. Given a volume bounded by an external surface and containing N g grains, two kinds of grains can be distinguished: the boundary grains, intersecting the external boundary, and the internal grains, completely surrounded by other grains. Boundary conditions are prescribed on the surface of the boundary grains lying on the external boundary, while interface continuity and equilibrium conditions are forced on interfaces between adjacent grains, to restore the integrity of the aggregate. In general, the boundary integral equation for a generic grain G k is written

cijk x u kj x

³

Tijk x, y u kj y dB k y

BC BNC

³

U ijk x, y t kj y dB k y ,

(1)

BC BNC

where uik and tik denote boundary displacements and tractions of the grain G k , and U ijk and Tijk represent the components of the 3D anisotropic displacement and traction fundamental solutions. The integrals appearing in Eq. 1 are extended over the entire surface of the grain, given by the union of contact surfaces BC , in common with other grains, where interface conditions apply, and

non-contact surfaces BNC , where boundary conditions apply. Eq. 1 is complemented by the k k °ui ui ®k k °¯ti ti

Boundary conditions

k j °ui ui ® k j °¯ti ti

G uikj

0

Interface conditions ,

(2)

where the overbar denotes prescribed quantities, while the tilde represents quantities expressed in an interface local reference system, more suitable for the interface conditions. The interface conditions involve surface displacements and tractions from two different grains, G k and G j . After discretization and integration of Eq. 1 for each grain, the final system of equations for the aggregate can be written

Advances in Boundary Element and Meshless Techniques 0 º ª A1 0 ª x º «0 % 0 »» « 1 » « # » « 0 0 A Ng » « « » « xNg » ¼ >I@ «¬ »¼ ¬

ª y1 º « # » « », « yN g » « » «¬ 0 »¼

3

(3)

where the vectors xk contain the unknown components of displacements and tractions, the matrix blocks A k are the grain boundary element matrices, the matrix I contain the coefficients of the interface conditions and the terms yk stem from the boundary conditions. System 3 is highly sparse and the use of specialized sparse solvers is then desirable to speed up the numerical solution. Grain boundary element discretization. The presented formulation has the remarkable advantage that only meshing of the grains surfaces is required. The artificial microstructure is, in this context, a collection of flat convex polygonal faces. Plane triangular linear elements are used to discretize such faces. Constant or linear discontinuous triangular elements are implemented for representing the unknown boundary fields. The mesh generator Triangle (http://www.cs.cmu .edu/~quake/ triangle.html, [12]) is used for the creation of a two-dimensional high-quality mesh of each plane cell face. Since the Voronoi tessellations used for microstructure modelling have stochastic nature, care must be taken to ensure mesh consistency and homogeneity to the greatest extent. This is achieved by introducing a discretization parameter d m governing the mesh density, so to create meshes as homogeneous as possible: d m sets the number of segments in which the average length edge in the tessellation is subdivided into. Fig. 1 shows the mesh of a tessellation with 150 grains and the representative mesh of few single grains taken from the same tessellation.

Figure 1: Mesh of a tessellation containing 150 grains and representative mesh of few grains.

Numerical estimation of elastic properties of SiC Before proceeding with the determination of the effective properties of the considered polycrystals, the developed technique has been tuned in terms of mesh density and boundary element type. As already described, the mesh density is controlled by the parameter d m . Moreover, in this work, two different types of elements have been implemented: constant elements and discontinuous linear

4


elements. To tune the method, a copper polycrystal with N g 10 grains is first analyzed. The material properties for the copper crystals are given in [8]. The aggregate is subjected to two different sets of linear displacement boundary conditions, the first enforcing a macro elongation *33 and the second enforcing a macro shear strain *13 . The convergence of the computed stress volume averages is checked, in order to set both mesh density d m and element type for the subsequent set of computations. Enforcing *33 by means of linear boundary displacements allows ˆ for the aggregate, and in particular to compute the third column of the apparent stiffness matrix C ˆ ˆ ˆ C , C and C . In the same way, by enforcing * , the fifth column of the apparent stiffness 13

23

33

13

matrix can be evaluated, and in particular Cˆ 55 . The numerical convergence of such quantities is shown in Fig. 2. In the plot, the apparent quantities are normalized with respect to the homologous value obtained by using the most refined scheme, i.e. the finest mesh with linear discontinuous elements, so that the trend of the quantities Cˆ ij / Cˆ ijref is shown. The linear scheme does not show a remarkable dependence on the mesh refinement for any considered apparent quantities, and the computed values can be considered converged even for the coarser mesh. The same behavior is noticed if constant traction boundary conditions are enforced and the components of the compliance matrix are evaluated, instead of the stiffness matrix. As a consequence, linear discontinuous elements with d m 1 will be used in the following computations.

Figure 2: Convergence of apparent elastic properties with element type and mesh density.

After numerical tuning, the macroscopic effective properties of silicon carbide (SiC) are estimated. The performed analysis takes into account the stochastic nature of the microstructure, in terms of grain size, morphology and orientation. Aggregates with N g 10 , 20, 50, 100 and 150 grains have been simulated and for each number of grains N R 100 realizations have been generated. Each realization differs from the others in terms of both geometry and crystallographic orientation. Given a polycrystalline realization, consisting of N g grains and subjected to a given set of consistent boundary conditions, since the material is supposed to not develop microcracks, stress and strain volume averages can be used to extract the apparent elastic modula, see for example [1,2]. Kinematic uniform boundary conditions, i.e. linear displacement boundary conditions

Advances in Boundary Element and Meshless Techniques corresponding to prescribed macro-strains, have been enforced on each simulated realization. Table 1 lists the elastic constants for hexagonal single crystal SiC, as measured by Arlt and Schodder [14]: these constants define the material of the single grains, which are then given a random orientation in the three-dimensional space. Fig. 3 shows the mean values and the scatter of the apparent Young’s modulus and shear modulus over the considered number of realizations: since linear displacement BCs are applied, the apparent properties approach the effective ones from higher values; moreover, it is worth noting how the scatter decreases when an higher number of grains is considered. The values of the effective elastic modula E and G for polycrystalline SiC have been reported by various authors, see Lambrecht et al. [15] and references therein. In this study, the average values, calculated over N R 100 realizations of aggregates with N g 150 grains, are E=456 GPa and G=193 GPa, which are in very good agreement with the values E=448 GPa and G=192 GPa, reported by Carnahan [16], who used low porosity samples and extrapolated the values to zero porosity. The average computed value of the Poisson ratio was Q 0.181 , close to the value Q 0.168 yielded by Carnahan estimations.

C11 502

C12 95

C13 96

C33 C44 C66 565 169 203.5

Table 1: Elastic constants for hexagonal single-crystal SiC [Gpa].

Figure 3: Apparent properties of SiC against number of grains in the simulated aggregate.

Summary A three-dimensional boundary element formulation for the analysis of polycrystalline microstructures has been developed. The technique allows a remarkable simplification in the generation of the artificial microstructure model and it is directly formulated in terms of intergranular displacements and tractions, which play an important role in polycrystalline micromechanics. The developed method has been applied to the determination of the effective properties of silicon carbide and the results have shown remarkable agreement with literature data in the framework of numerical homogenization.

Acknowledgements This research was supported by a Marie Curie Intra European Fellowship within the 7th European Community Framework Programme.

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References [1] S. Nemat-Nasser, M. Hori: Micromechanics: overall properties of heterogeneous materials (North-Holland, Elsevier, The Netherlands, 1999). [2] T.I. Zohdi and P. Wriggers: An introduction to computational micromechanics, (Springer, Berlin, 2005). [3] B.L. Adams and T. Olson: Prog Mater Sci, Vol. 43, (1998), p. 1. [4] A.G. Crocker, P.E.J. Flewitt, and G.E. Smith: Int Mater Rev, Vol. 50(2), (2005), p. 99. [5] K.M. Döbrich, C. Rau, and C.E. Krill III: Metall Mater Trans A, Vol. 35A, (2004), p. 1953. [6] M.A. Groeber et al.: Mater Charac, Vol. 57, (2006), p. 259. [7] G.K. Sfantos and M.H. Aliabadi: Int J Numer Meth Eng, Vol. 69, (2007), p. 1590. [8] F. Fritzen, T. Böhlke and E. Schnack: Comput Mech, Vol. 43, (2009), p. 701. [9] R. Quey, P.R. Dawson, and F. Barbe: Comput Meth App Mech Eng, Vol. 200, (2011), p. 1729. [10] N.A. Schclar: Anisotropic analysis using boundary elements, (Computational Mechanics Publications, Southampton, 1994). [11] M.H. Aliabadi: The boundary element method: applications in solids and structures, (John Wiley & Sons Ltd, England, 2002). [12] J.R. Shewchuk: in Applied Computational Geometry: Towards Geometric Engineering, ed. M.C. Lin and D. Manocha, vol. 1148 of Lecture Notes in Computer Science, Springer-Verlag, Berlin, (1996). [13] T. Kanit et. al: Int J Solids Struct, Vol. 40, (2003), p. 3647. [14] . Arlt and G.R. Schodder, Journal of the Acoustical Society of America, Vol. 37(2), (1965) p.384. [15] W.R.L. Lambrecht, B. Segall, M. Methfessel, M. van Schilfgaarde. Physical Review B, Vol. 44(8), (1991), p.3685. [16] R.D. Carnahan, Journal of the American Ceramic Society, Vol. 51(4), (1968), p.223.


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Three-Dimensional Unsteady Thermal Stress Analysis by Triple-Reciprocity Boundary Element Method Yoshihiro OCHIAI1, Vladimir Sladek2 and Jan Sladek2 1

Kinki University, Dept. of Mechanical Engineering, 3-4-1 Kowakae, Higashi-Osaka, Osaka, 577-8502 Japan, E-mail: [email protected] 2 Institute of Construction and Architecture, Slovak Academy of Science, 845 03 Bratislava, Slovakia Key Words: Thermal Stress, Boundary Element Method, Heat Conduction, Meshless Method

Abstract. The conventional boundary element method (BEM) requires a domain integral in unsteady thermal stress analysis with heat generation or an initial temperature distribution. In this paper it is shown that the three-dimensional unsteady thermal stress problem can be solved effectively using the triple-reciprocity boundary element method without internal cells. In this method, the distributions of heat generation and initial temperature are interpolated using integral equations and time-dependent fundamental solutions are used. A new computer program was developed and applied to solving several problems. Introduction The unsteady thermal stress problems cannot be solved easily, without using internal cells, by the conventional boundary element method (BEM), in general. Only special cases of ploblems, such as unsteady thermal stress problems with constant heat generation and uniform initial temperature distribution can be solved by the standard BEM without the need for internal cells. When an analysis of thermal stress under arbitrary heat generation or a non-uniform initial temperature distribution within the domain is carried out by the BEM, a domain integral is involved in general [1,2]. However, by including the domain integral, the merit of BEM is lost, since the unknowns are not localized on the boundary alone like in pure BEM. Thus, several other methods have been considered. Nowak and Neves proposed a multiple-reciprocity method [3]. Tanaka et al. have proposed a dual-reciprocity BEM for transient heat conduction problems [4], and V. Sladek and J. Sladek proposed a local boundary integral equation for unsteady heat conduction problems [5, 6]. However, these methods do not employ a time-dependent fundamental solution, which gives an accurate result. A Laplace transformation can remove the time dependence of the problems, however, it is not suitable under complicated time-dependent boundary conditions. The Laplace transformation method requires internal cells for the initial temperature distribution. Recently, the efficient treatment of domain integrals has been proposed by the triple-reciprocity BEM or improved multi-reciprocity BEM for steady heat conduction, steady thermal stress and elastoplastic problems [7-10]. The triple-reciprocity BEM for two-dimensional heat conduction and thermal stress analysis for an unsteady state has also been proposed [11-17]. In this paper, the triple-reciprocity BEM is developed for three-dimensional unsteady heat conduction problems. In this method, the heat generation and initial temperature distributions are interpolated using the boundary integral equations. Since the domain integrals are eliminated, no internal cells are required in the present triple-reciprocity method and the time-dependent solution is employed. A new computer program was developed and applied to solving several problems. Theory Unsteady heat conduction In unsteady heat conduction problems with heat generation W1S (q, t ) , a temperature T is obtained by solving 2T

W1S

O

N 1

wT , wt

(1)

where N , O and t are the thermal diffusivity, heat conductivity and time, respectively. Denoting an

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DUELWUDU\WLPHDQGWKHLQLWLDOWHPSHUDWXUHE\ĲDnd T10 S (q,0) , respectively, the boundary integral equation for the temperature in the case of unsteady heat conduction problems is expressed by [1, 2] cT ( P, t )

t

N ³0 ³ * [T (Q,W )

wT1* ( P, Q, t ,W ) wT (Q, W ) * T1 ( P, Q, t , W )]d*dW N wn wn

³

:

t

³ ³T 0

:

* 1 ( P, q, t ,

W)

W1S (q,W )

O

T1* ( P, q, t ,0)T10 S (q,0)d: ,

d:dW

(2)

where c=0.5 on the smooth boundary and c=1 in the domain. ī and ȍ represent the boundary and the domain, respectively, p and q are respectively an observation point and a loading point, and r is the distance between p and q. The notations p and q are written as P and Q on the boundary, respectively. In the case of three-dimensional problems, the time-dependent fundamental solution T1* ( p , q ,t , W ) in Eq. (2) for the unsteady temperature analysis problem and its normal derivative are given by [1, 2] T1* ( p, q, t ,W ) wT1* ( p, q, t ,W ) wn

1 exp[ a ] [4S N (t W )]3 / 2 2r

wr

S 3 / 2 [4N (t W )]5 / 2 wn

exp( a )

(3) (4)

where r2 (5) 4N (t W ) As shown in Eq. (2), when there is an arbitrary initial temperature or heat generation distribution, a domain integral becomes necessary. Interpolation An interpolation method for a distribution of heat generation W1S (q,W ) is shown using the boundary integral equations to avoid the use of internal cells. The polyharmonic function a

T1[ f ] ( p, q ) for the steady state is given by

T [ f ] ( p, q )

with 2T [ f 1]

T [ f ] for ( f

r 2 f 3 , r 4S 2 f 2 !

pq ,

(6)

1, 2,... ) and 2T [1] ( p, q ) G ( p q ) .

It is appropriate to utilize the following equations for the three-dimensional interpolation [16,17]: 2W1S (q,W ) W2S (q,W ) W ( q, W ) 2

S 2

(7)

M

¦ W (qm ,W )G (q qm ) , PA 3

(8)

m 1

where M is the number of internal points for interpolation. Assuming the spatial distribution of W2s (q,W ) to be governed by Eq. (8) with point sources, it is known that W2s (q,W ) will be divergent at these source points as M

the particular solution

¦ T [1] ( p, qm )W3PA (qm ,W ) . Nevertheless, we can evaluate

W2s (q,W ) on the boundary.

m 1

The term W2S of Eq. (7) corresponds to the sum of the curvatures w 2W1S / wx 2 , w 2W1S / wy 2 and w 2W1S / wz 2 . The term W3PA is the unknown strength of a Dirac function distribution. From Eqs. (7) and (8), the following equation can be obtained. 4W1S (q,W )

M

¦W

PA 3

m 1

(qm ,W )G (q qm )

(9)


9

This equation corresponds to equation for the deformation of a fictitious thin plate with M point loads. The “deformation” W1S (q,W ) is given, but the “force of the point load” W3PA (q,W ) is unknown. W3PA (q,W ) is obtained inversely from the “deformation” W1S (q,W ) of the fictitious thin plate. W2S

corresponds to the moment of the thin plate. The “moment” W2S on the boundary is assumed to be 0, which is the same as that in a natural spline. This indicates that the thin plate is simply supported. Similarly, the distribution of the initial temperature can be interpolated as follows. 2T10 S (q,0)

T20 S (q,0) M

¦ T

T (q,0) 2

0S 2

0 PA 3

(qm ,0)G (q qm )

(10)

(11)

m 1

Furthermore, the polyharmonic function T f* ( p , q ,t , W ) in the unsteady heat conduction problem are defined by 1 w· * § 2T f*1 ( p, q, t ,W ) T f* ( p, q, t ,W ) , ¨ 2 T ( p, q, t ,W ) G ( p q )G (t W ) . N wt ¸¹ 1 ©

(12)

Using Green's theorem twice, and Eqs. (7)-(12), Eq. (2) becomes t

wT1* ( P, Q, t ,W ) wT (Q,W ) * T1 ( P, Q, t ,W )]d *dW wn wn S wW f (Q,W ) wT f* 1 ( P, Q, t ,W ) S [T f*1 ( P, Q, t ,W ) W f (Q,W )]d *dW

N ³0 ³* [T (Q,W )

cT ( P, t )

t N 2 ¦ (1) f ³0 O f 1

N Om

M

³

*

wn

wn

2

¦ ³0 W3PA (qm ,W )T3* ( P, qm , t ,W )dW ¦ (1) ³ [T t

f

f 1

1

*

* f 1

( P, Q, t ,0)

wT f0 S (Q,0) wn

wT f* 1 ( P, Q, t ,0) 0 S T f (Q,0)]d * wn

M

¦ T30 PA (qm , 0)T3* ( P, qm , t , 0) .

(13)

m 1

Similarly, starting from the governing equations (7) and (8), we obtain the integral equation constraints for W1S and W2S [9-11] 2

¦ (1) f ³ *{T [ f ] (P, Q)

cW1S ( P,W )

wW fS (Q,W )

f 1

³ *{T

cW2S ( P,W )

[1]

wn

M

wT [ f ] ( P, Q) S W f (Q,W )}d * T [2] ( P, qm )W3PA (q m ,W ) (14) wn m 1

¦

M

[1]

(P, Q)

wW2S (Q,W ) wT ( P, Q ) S W2 (Q,W )}d * T [1] ( P, qm )W3PA (qm ,W ) wn wn m 1

¦

(15)

Eventually, from the governing equations (10) and (11) for the initial temperature T10 S and T20 S , we obtain

2

cT10 ( P,0)

¦ (1) ³ {T f

*

f 1

cT20 S ( P,0)

[f]

³ {T *

(P, Q)

wT f0 S (Q,0)

wn

M

wT [ f ] ( P, Q) 0 S T f (Q,0)}d * T [2] ( P, qm )T30 PA (qm ,0) (16) wn m 1

¦

M

[1]

(P, Q)

wT20 S (Q,0) wT [1] ( P, Q) 0 S T2 (Q,0)}d * T [1] ( P, qm )T30 PA (qm ,0) (17) wn wn m 1

¦

U nsteady polyharmonic function

The three-dimensional unsteady polyharmonic function

T f*1 ( p, q, t ,W ) in Eq. (12) is determined as

T f*1 ( p, q, t ,W )

since 2 f ( r )

1 w §

¨r

r 2 wr ©

2

w wr

· ¹

f (r ) ¸ .

1

³ r ³r T 2

2

* f

( p, q, t ,W )drdr ,

(18)

10


Thus, the polyharmonic function T f* ( P, q, t ,W ) in the unsteady state and its normal derivative are explicitly given by 1 J (1/ 2, a) , a r 2 / E , E 4N (t W ) 4S 3/ 2 r E ª§ 1 · a º T3* q, p, t ,W a ¸ J (1/ 2, a ) e » , 3/ 2 «¨ 2 a¹ 8S ¬© ¼

T2* q, p, t ,W

x

³t

with J being the incomplete gamma function defined as J (D , x)

(19) (20)

D 1 t

e dt .

0

Unsteady thermal stress Next, in order to obtain the thermal stresses in uncoupled quasi-static thermoelasticity, let us consider the thermoelastic displacement potential )( P, t ) for unsteady problems given by [4] N

c) ( P , t )

N³

wI1* ( P, Q, t ,W )

wT (Q,W ) * I1 ( P, Q, t ,W )]d*dW wn S W1 (q,W ) * d:dW ³ I1* ( P, q, t ,0)T10 S (q,0)d: ³: I1 ( P, q, t ,W ) :

t

³0 ³ * [T (Q,W ) t

wn

O

0

(21)

Denoting Poisson's ratio by Q and the coefficient of linear thermal expansion by D , m0 is given by m0=(1+Q)D/(1-Q). Now, let us introduce the high-order function I f ( p, q, t ,W ) defined by I *f ( p, q, t ,W ) T f*1 ( p, q, t ,W )

(22)

Using Green's theorem twice, and Eqs. (7), (8), (10), (11), Eq. (21) becomes N

c) ( P , t )

t

³0 ³ * [T (Q,W )

2

NO 1 ¦ ( 1) f f 1 M

N O 1 ¦

³

t

m 1 0

t

wI1* ( P, Q, t ,W ) wn

ª wW fS (Q,W )

³ 0 ³ * «« ¬

wn

wT (Q, W ) * I1 ( P, Q, t , W )]d*dW wn

I *f 1 ( P, Q, t ,W ) W fS (Q,W ) 2

W3PA ( qm ,W )I3* ( P, qm , t ,W ) dW ¦ ( 1) f f 1

wI *f 1 ( P, Q, t ,W ) º wn

» d * dW ¼»

ª wT f0 S (Q, 0) wI *f 1 ( P, Q, t , 0) º * 0S « » d* I ( P , Q , t , 0) T ( Q , 0) f f 1 ³* « wn wn ¬ ¼» M

¦ T30 PA ( qm , 0)I3* ( P, qm , t , 0) (23) m 1

Using the relationship between the thermoelastic displacement potential ) ( P, t ) and the displacement, the boundary integral representation for the displacement is obtained as [4, 15] cij ( P )u j ( P, t )

³*

[uij ( P, Q ) p j (Q, t ) pij ( P, Q )u j (Q, t )]d * N

2 wT (Q,W ) >1@ º ui ( P, Q, t ,W ) » d * dW NO 1 ¦ ( 1) f wn f 1 ¼ M

N O 1 ¦

t

³ 0 ³ * [W f (Q,W ) ³

t

m 1 0

2

¦ ( 1) f f 1

³ * [T f

0S

(Q, 0)

S

>1@ ª w ui ( P, Q, t ,W ) « T ( Q , W ) ³0 ³* « wn ¬ t

> f 1@ ( P, Q, t ,W ) wW S (Q,W ) f > f 1@ ( P, Q, t ,W )]d * dW u

w ui

wn

wn

i

>3@

W3PA ( qm ,W )ui ( P, qm , t ,W ) dW

> f 1@ ( P, Q, t ,W ) wT 0 S (Q, 0) f >3@ > f 1@ ( P, Q, t ,W )]d * M ¦ T30 PA ( qm , 0)ui ( P, qm , t , 0) ui

w ui

wn

wn

m 1

and cij is the free coefficient. Moreover, u j and p j are the j-th components of the displacement and

(24)


11

surface traction, respectively. Kelvin's solutions, namely, uij ( p, q ) and pij ( p, q ) , are given by

1 [(3 4Q )G ij r ,i r , j ] 16S (1 Q )Gr wr 1 {[(1 2Q )G ij 3r ,i r , j ] (1 2Q )(r ,i n j r , j ni )} wn 8S (1 Q )Gr 2

uij ( P, Q)

pij ( P, Q)

(25) (26)

and ni is the unit normal component. where Q is Poisson's ratio, and G is the shear modulus. The i-th component of a unit normal vector is denoted by ni . Moreover, r ,i w r / w xi .

q, p, t ,W

m0T2* ,i

m0 r ,i J (1.5, a ) 2S 3 / 2 r 2

ui[ 2] (q, p, t , W )

m0T3* ,i

m0 r ,i 1 [J (0.5, a ) J (1.5, a )] a 8S 3 / 2

m0 r ,i r 2

^>J (2.5, a) J (1.5, a)@ a2 (1 a1)J (0.5, a)`

ui

[1]

ui[3] ( p, q, t ,W )

32S 3/ 2

tf

³

8NS

t f 1 tf

32NS

t f 1 tf

3/ 2

4

³

128NS

t f 1

af

, a f 1

af

r2 4N (t t f )

(29)

(30)

f 1 § 1· 1/ 2 z ½ e ¾ ® 2 J (3 / 2, z ) ¨ 1 ¸ J (1/ 2, z ) z © z¹ ¯ 2z ¿ a f 1

(31)

a

m0 r r,i

[3]

ui ( p , q , t , W ) dW

z

(28)

a

m0 r r,i

[2]

ui ( p , q , t , W ) dW

`

1

J (1/ 2, z ) J (3 / 2, z )

3/ 2

2

³

^

m0 r,i

ui[1] ( p, q, t ,W ) dW

(27)

3/ 2

1 · 1§ 5 · 1/ 2 z ½ f § 2 1 e ¾ ®¨ 2 ¸ J (1/ 2, z ) ¨ 2 ¸ z z¹ 9© ¯© 9 z 2 z ¹ ¿ a f 1

(32)

Internal stress In the same manner, internal stress can be obtained. V ij ( p, t ) N

³ * [V kij ( p, Q) pk (Q, t ) Skij ( p, Q)uk (Q, t )]d *

ª

t

³ 0 ³ * ««T (Q,W ) ¬

2

NO 1 ¦ ( 1) f f 1

t

³ 0 ³ * [W f (Q,W ) S

M

N O 1 ¦ 2

¦ ( 1) f f 1

³ * [T f

0S

(Q, 0)

>1@

wV ij ( P, Q, t ,W ) wT (Q,W ) >1@ º V ij ( P, Q, t ,W ) » d * dW wn wn ¼

> f 1@ ( P, Q, t , 0)

wV ij

wn

³

t

m 1 0

> f 1@ ( P, Q, t ,W ) wW S (Q,W ) f > f 1@ ( P, Q, t ,W )]d * dW V ij

wV ij

wn

wn

>3@

W3PA ( qm ,W )V ij ( P, qm , t ,W ) dW

M wT f0 S (Q, 0) > f 1@ >3@ V ij ( P, Q, t , 0)]d * ¦ T30 PA ( qm , 0)V ij ( P, qm , t , 0) (33) wn m 1

where

V ij[1] ( p, q, t ,W ) V ij[2] ( q, p, t ,W )

V ij[3] ( p, q, t ,W )

½° Gm0 ° G ij [(1 Q )J (3/ 2, a) 2QJ (5 / 2, a)] r ,i r , j 2J (5 / 2, a) ¾ 3/ 2 3 ®1 2Q S r ¯° ¿° Gm0

1 ª1 º ª3 º½ J (1/ 2, a ) » r ,i r , j « J (3 / 2, a ) J (1/ 2, a ) » ¾ ®G ij « J (3 / 2, a ) 1 2Q ¬a ¼ ¬a ¼¿

4S 3 / 2 r ¯

Gm0 r

ª 2 ®G ij « ¬1 2Q

32S 3 / 2 ¯

1 2Q a · 1 º ½ § J (3 / 2, a ) » ¾ e ¸ ¨ (1 2Q 1/ a )J (1/ 2, a ) a © ¹ a2 ¼ ¿

(34) (35)

12

Eds: P Prochazka and M H Aliabadi ª

3

¬

2

r,i r, j « 2(1 1/ a )J (1/ 2, a )

a

J (3 / 2, a )

2 a

º

ea »

¼

(36)

Numerical Examples To verify the efficiency of this method, an unsteady thermal stress distribution in a sphere is analyzed. The initial temperature of the sphere is T0 10qC , and the temperature on the surface suddenly becomes 0D at the time t 0 . It is assumed that the thermal diffusivity is N 16 mm2s-1 and the radius of the sphere is b 10 mm. Figure 1 shows the boundary elements. In this example, internal points are not necessary. Figure 2 shows the radial distributions of the temperature field at several time instants. The solid lines in Fig.2 show the exact solutions, which are given by 2bT0 f (1) n 1 nS r N n 2S 2t T (r , t ) ¦ n sin b exp( 2 ) (37) Sr n 1 b Young's modulus E, Poisson's ratio Q and the coefficient of linear thermal expansion D are assumed to be 210 GPa, 0.3 and 11u10-6 K-1, respectively. The sphere is not loaded mechanically. Figure 3 shows the numerical and exact results for the radial and circumferential thermal stress distributions. Next, an unsteady thermal stress distribution in the sphere with the constant heat generation W0 / O 10 K mm 2 is obtained. The initial temperature of the sphere is T0 0qC , and the temperature on the surface is 0qC . The other specifications are the same as in Figs.2 and 3. In this case again, internal nodes are not employed. Figures 4 and 5 show the radial distributions of the temperature and stress fields at several time instants. The solid lines show the exact solutions. Finally, the unsteady temperature distribution in a solid circular cylinder with an initial temperature distribution is analyzed. The outer diameter is 2b, the outer temperature is 0 DC , and the upper and lower

Fig.1

Boundary elements of sphere region

Fig.3

Fig.2

Temperature distributions in sphere

Stress distributions in sphere ( n 150 )


Fig.4 Temperature distributions in sphere with heat generation Fig.5 Stress distributions in sphere with heat generation

surfaces are adiabatic isolated. Figure 6 shows the boundary elements and internal points. The distribution of initial temperature is given by T (r ,0) T0

b2 r 2 , b2

(41)

and step heating is assumed. The thermal diffusivity is N 16 mm2s-1, b=10 mm and T0 100 DC . Figure 7 shows the exact and numerical results for the temperatures at t =0.2, 0.5, 1 and 2 s obtained by present method. Young's modulus E, Poisson's ratio Q and the coefficient of linear thermal expansion D are assumed to be 210 GPa, 0.3 and 11u10-6 K-1, respectively. Figure 8 shows the radial and circumferential thermal stress distributions.

Fig.6

Circular cylinder with initial temperature

Fig.7 Temperature distributions in cylinder Fig.8 Stress distributions in cylinder

13

14


Conclusion The triple reciprocity boundary element method has been developed for unsteady thermoelastic 3D problems within quasi-static uncoupled thermoelasticity. The well known BEM dimensionality reduction is achieved since the unknowns are localized on the boundary alone. Arbitrary heat sources as well as initial temperature distributions are allowed with prescribing their values at certain interior and boundary points. The triple reciprocity formulation utilizes only low order polyharmonic fundamental solutions. The formulation as well as the developed computer code and the efficiency of the proposed method have been verified in several numerical test examples for which the exact solutions are available. Acknowlwdgements This work was partially supported by the Slovak Research and Development Agency under the contract No. APVV-0032-10. References [1] C. A. Brebbia, J. C. F. Telles and L. C. Wrobel, Boundary Element Techniques㸫Theory and Applications in Engineering, pp. 47-107, Berlin, Springer-Verlag, 1984. [2] L. C. Wrobel, The Boundary Element Method, Volume 1, John Wiley & Sons, West Sussex, pp.97-117 (2002). [3] Nowak, A. J., and Neves, A. C., The Multiple Reciprocity Boundary Element Method, Computational Mechanics Publication, Southampton, Boston, (1994). [4] Tanaka, M., Matsumoto, T. and Takakuwa, S., Dual Reciprocity BEM Based on Time-Stepping Scheme for the Solution of Transient Heat Conduction Problems, Boundary Elements XXV, WIT Press, pp. 299-308, (2003). [5] Sladek, J. and Sladek, V., Local Boundary Integral Equation Method for Heat Conduction Problem in an Anisotropic Medium, Proceeding of ICCES2003, Chap. 5, (2003). [6] Sladek, V., Sladek, J., Tanaka, M. and Zhang, Ch., Transient heat conduction in anisotropic and functionally graded media by local integral equations, Engineering Analysis with Boundary Elements, Vol. 29, pp. 1047-1065, (2005). [7] Ochiai, Y. and Sekiya, T., Steady Heat Conduction Analysis by Improved Multiple-Reciprocity Boundary Element Method, Engineering Analysis with Boundary Elements, Vol. 18, pp. 111-117, (1996). [8] Ochiai, Y. and Sekiya, T., Steady Thermal Stress Analysis by Improved Multiple-Reciprocity Boundary Element Method, Journal of Thermal Stresses, Vol. 18, No. 6, pp. 603-620, (1995). [9] Ochiai, Y. and Kobayashi, T., Initial Stress Formulation for Elastoplastic Analysis by Improved Multiple-Reciprocity Boundary Element Method, Engineering Analysis with Boundary Elements, Vol. 23, pp. 167-173, (1999). [10] Y. Ochiai, Two-Dimensional Unsteady Heat Conduction Analysis with Heat Generation by Triple-Reciprocity BEM, International Journal of Numerical Methods in Engineering, Vol. 51, No. 2, pp. 143-157(2001). [11] Y. Ochiai, V. Sladek and J. Sladek, Transient Heat Conduction Analysis by Triple-Reciprocity Boundary Element Method, Engineering Analysis with Boundary Elements, Vol. 30, pp. 194-204 (2006). [12] Ochiai, Y., Two-Dimensional Unsteady Thermal Stress Analysis by Triple-Reciprocity Boundary Element Method, Journal of Thermal Stresses, Vol. 24, No. 3, pp. 233-253, (2001). [13] Y. Ochiai and V. Sladek, Numerical Treatment of Domain Integrals without Internal Cells in Three-Dimensional BIEM Formulations, CMES (Computer Modeling in Engineering & Sciences), Vol. 6, No. 6, pp. 525-536, (2004). [14]Y. Ochiai, Multidimensional Numerical Integration for Meshless BEM, Engineering Analysis with Boundary Elements, Vol. 27, No. 3, pp. 241-249, (2003). [15] Ochiai Y., Two-Dimensional Unsteady Heat Conduction Analysis with Heat Generation by Triple-Reciprocity BEM, International Journal of Numerical Methods for Engineering, Vol.51, No.2, pp.143-157(2001). [16] Ochiai, Y., Nishitani, H., and Sekiya, T., Two-dimensional thermal stress analysis under unsteady state by improved boundary element method, Journal of Thermal Stresses, Vol.19, pp.107-121, (1996). [17] Ochiai, Y., Kitayama, Y., T h r e e - d i m e n s i o n a l U n s t e a d y H e a t C o n d u c t i o n A n a l y s i s b y T r i p l e - R e c i p r o c i t y Boundary Element Method, Engineering Analysis with Boundary Elements, Vol. 33, No.6, pp.789-795, (2009). [18] Sharp, S. and Crouch, S.L., Boundary integral method for thermoelasticity problem, Transaction of the ASME, Journal of Applied Mechanics, Vol.53, pp.298-302, (1986).


Green element solutions to steady inverse heat conduction problems Akpofure E. Taigbenu1 and John Ndiritu School of Civil and Environmental Engineering, University of the Witwatersrand. P. Bag 3, Johannesburg, WITS 2050. South Africa. 1 [email protected] Keywords: Green element method, steady inverse heat conduction, regularized singular value decomposition, evolutionary algorithm.

Abstract The recent formulation of the Green element method (GEM), which approximates the internal normal fluxes with a second-order difference approximation, is used to solve steady inverse heat conduction problems in 2D spatial dimensions. The new formulation is more amenable for such problems than the flux-based one because the data for retrieving the temperature within an inaccessible part of the computational domain and/or the heat flux on a part of the boundary are only supported by the former formulation. Noting the comparable accuracy of the new formulation to the flux-based one [1], the application of the former to inverse problems further elicits its computational robustness. Two solution strategies are used in conjunction the GEM in addressing the inverse problem. The first uses a Tikhonov regularization with the singular value decomposition (SVD) to solve the ill-conditioned discrete equations arising from the element-by-element implementation of the singular integral equations. The second uses an evolutionary optimization (EO) technique that incorporated into the GEM solution of the direct problem in predicting the unknown field variables and constrained by the functional of the available data. Two examples of steady inverse heat conduction problems in homogeneous and heterogeneous media are simulated by the current GEM formulation. Both approaches give good prediction of the temperature and heat fluxes. Although the results show that the SVD sometimes outperforms the EO, the latter tends to be consistently more accurate but at the expense of higher computing times.

Introduction The solution of the inverse heat conduction problem (IHCP) continues to generate considerable interest in mathematical and computational circles because of the challenges it presents and its ubiquitous practical applications in fields of science and engineering. Various facets of IHCP are encountered, ranging from estimation of medium parameters (thermal conductivity and specific heat capacity) [2,3], recovery of field variables of temperature and heat flux [4-7], and recovery of the spatial and temporal distributions of heat sources/sinks [8,9]. Typically, the problems arise when sensor measurements of temperature at some points and heat fluxes along a part of the boundary are available, and there is the need to predict medium parameters or temperature and/or fluxes where measurements are unavailable, or heat source distributions. Unlike the direct problem which gives unique solution and whose numerical solution gives rise to a coefficient matrix that is well conditioned, the inverse problem might yield non-unique solutions and the coefficient matrix, arising from its numerical discretization, is usually ill-conditioned. The degree of illconditioning depends to a large extent on the distribution of the sensors, and Blanc et al. [10] presents a guide on how the sensors can be distributed in order reduce the degree of ill-conditioning and thereby enhance the stability of the solutions for IHCP. In this paper, the Green element method is applied to the steady 2-D IHCP in which temperature and heat fluxes are predicted at parts of the computational domain using available measured data. The formulation of the GEM that is used in this work is that recently presented in Taigbenu [1] in which a second-order approximation is carried out for the internal normal fluxes so that only solution for the temperature or the flux is obtained at external nodes and for the temperature at internal nodes. This formulation was shown to exhibit comparable accuracy as the flux-based formulation, and also readily lends itself to solving inverse problems. The resultant coefficient matrix from applying this formulation to the IHCP is ill-conditioned, and it is handled by the SVD method with Tikhonov regularization. Using a different solution strategy for the IHCP, the GEM solves the direct heat conduction problem with unspecified boundary data that are predicted by an evolutionary optimization (EO) technique that minimizes a functional that is based on the available data. Two examples (one in a homogeneous domain and the other in a heterogeneous domain) of steady IHCP are solved by these two solution strategies. There are instances when the SVD gives more accurate prediction of field variables than the EO, but the EO

15

16


solutions are generally consistently more accurate but at the price of increased computing time because of the repeated solution of the direct problem by GEM in minimizing the functional. Governing Equation The IHCP addressed in this paper is governed by the differential equation

>KT @ Q ( x, y ) (1) where is the 2-D gradient operator with the spatial variables x and y, T is the temperature field, K is the thermal conductivity of the medium and Q represents heat sources and sinks. The specified boundary conditions are: T ( x, y ) T1 on ī1 (2a)

KT n q2 on ī2 T ( x, y ) T3 and KT n

(2b)

q3 on ī3

(2c) where n is the unit outward pointing normal on the boundary. The differential equation applies to a domain ȍ with boundary ī= ī1 ī2 ī3 ī4. The IHCP problem requires the solution for the temperature T and heat flux q KT n on the boundary ī4 when a combination of appropriate boundary conditions presented in eqs (2a, b, c) are specified and measurements of the temperature are available at internal points in the domain. The measured temperature value at any internal point (xm,ym) is denoted as Tm=T(xm,ym), with Ni available sensor measurements. When the boundary ī4 does not exist, the problem reduces to the direct one which is solved without requiring information on the temperature at internal points. To be able to solve the inverse problem, the number of discrete equations that are generated by GEM has to be equal to or greater than the number of unknowns. Eq. (1) is expressed as a Poisson equation that allows the use of the fundamental solution to the Laplace differential operator. 2T < T Q / K (3) where < ln K . The integral equation of eq. (3) is obtained by employing Green’s theorem

q· § OTi ³ ¨ TG n G ¸dS ³³ G> < h Q / K @ dA 0 K © ¹ * :

(4)

Where G is the fundamental solution of the Laplacian which is the logarithmic function, the subscript i denotes the source or collocation node ri=(xi,yi) and O is the nodal angle at ri obtained from the Cauchy part of the integration of the Dirac delta function at the source node. The boundary and domain integrals in eq (4) are implemented in the Green element sense over sub-domains or elements that are used to discretize the computational domain. On these elements, Lagrange-type interpolations are prescribed for T, q/K and Ȍ, that is T§NjTj (Nj is linear interpolation function). Introducing the interpolation into eq (3) results in the discrete element equations applicable to each sub-domain or element denoted as ȍe. That is §q· §Q· (5) RijT j Lij ¨ ¸ U imj ... ıN > 0. N is the rank of the matrix A, and ui and vi are the ith column of the matrices U and V, respectively. In the least square solution of eq (8), the Euclidian norm ŒAp-bŒ2 is minimized to give the solution for the unknowns p p B 1s (10) Where B=AtA is a N×N matrix, and s=Atb is a N×1 vector. Introducing the expression for A in eq (9) into eq (10) gives

uit s

N

¦V

p

i 1

vi

(11)

i

The small singular values cause instability in the solution for p, and for that reason, regularization can be carried out or singular values less than a prescribed threshold are truncated and considered to have zero value. The former approach is the regularized SVD approach, while the latter is the truncated SVD. The former is followed in this work. The Tikhonov regularization is a smoothening technique that attempts to stabilize the numerical results from solving the ill-conditioned system of equations of the IHCP problem [12]. The zeroth-order Tikhonov regularization technique minimizes ŒAp-bŒ2+ĮŒpŒ2, resulting in the solution for p that is given by

Vi

N

¦D V

p(D )

i 1

2 i

uit svi

(12)

where Į is the regulation parameter. The factor ıi/(Į+ıi ) in eq (12) serves to dampen the contribution of the small singular values. The choice of Į has to be carefully done so that it is not too small to retain the instability in the numerical solution or too large to have smooth solutions that do not reflect the physics of the problem being addressed. The second solution strategy for the IHCP use GEM to solve the direct problem and an evolutionary optimization to predict the unspecified boundary data in the direct GEM simulation. The direct problem is governed by eq. (1) but subject to the conditions T ( x, y ) T1 on ī1 (13a) 2

KT n q2 on ī2 (13b) On the boundary segment ī3, either T or q (not both) is prescribed so that at the node where the segment intersects another part of the boundary, only one value of T or q is required. In case q is specified, then the calculated T is compared with its given value. On boundary segment ī4 where no data on T and q are available, the EO is used to predict one of the variables in order to calculate the other. Essentially, the EO solution is obtained by minimizing the functional

¦ >T Ni

J

m 1

cal

@

N3

>

( xm , ym ) T meas ( xm , ym ) ¦ Ti cal Ti meas 2

@

2

(14)

i 1

The first part of the functional refers to the Ni internal points where temperature measurements are available and the second part refers to the boundary segment ī3 where there exist N3 nodes at which the flux is prescribed. It should be noted that if the temperature is prescribed on ī3, the second part of the functional then evaluates the deviation between the calculated and prescribed fluxes. The superscripts cal and meas refer to calculated values from the Green element simulation and the measured or given values. A brief overview EO algorithm is herein presented. The EO algorithm that is applied here is the Shuffled Complex Evolution (SCE-UA) method originally developed by Duan et al. [13] for catchment model calibration. The SCE-UA generates a population of

18


possible solutions to the problem (nodal variable T or q that should be predicted on ī4) and divides this population into a number of sub-populations (complexes). The downhill simplex method is then used to evolve each complex for a set number of evolutions [14]. The improved solutions from the complexes are then shuffled to enable the exchange of good traits among the complexes. The resulting complexes are evolved and shuffled repeatedly until the set criterion for terminating the search is met. The criterion applied here was the reduction of the functional to a positive value very close to zero. The SCE-UA optimizer requires the values of several parameters to be set and the guidelines proposed by Duan et al. [15] were applied in achieving this. Numerical Examples Two examples are used to assess the accuracy of the inverse formulations of the GEM earlier presented. The medium of the first example is homogeneous, while the second is non-homogeneous. Both examples have exact solutions which are benchmarked with the numerical results. These examples had previously been addressed by the local integral equation (LIE) formulation [5]. Example 1 The spatial domain is a square [0.01×0.01] with medium material properties that are homogeneous. The thermal conductivity K=0.01, and there is no heat source in the medium, Q = 0. The exact solution for this test example that satisfies the governing equation with stationary boundary conditions is

T

y a

(15)

Where a is the length of the domain. Three cases of boundary conditions, presented in Figures 1a, 1b and 1c, are evaluated. In case (i), zero flux and a linear temperature variation with y are prescribed on the right boundary segment, while no data on T and q are available on the other three boundaries. Temperature measurements are available at 19 internal points (along x=1.25×10-3, y=1.25×10-3 and y=8.75×10-3) indicated by the empty circles in Figure 1a. In case (ii), a zero flux is prescribed on the left and right boundaries, while unit temperature and unit flux are specified on the top boundary. The bottom boundary is a ī4 boundary where T and q are not known. No temperature measurements are available within the domain (Figure 1b). Case (iii) has unit temperature specified on the top boundary, a linear temperature distribution and zero flux along the right boundary, and no boundary data on the left and bottom boundaries. 14 internal temperature measurements are available along y=1.25×10-3 and y=8.75×10-3 (Figure 1c). Using a grid of 8×8 linear rectangular elements with 32 boundary nodes and 49 internal nodes, numerical simulations are carried out with the GEM using the regularized SVD method to evaluate the overdetermined system of equations, and the EO to predict the unknowns of the direct model till the functional given by eq (14) is less than 10-5. The relative error for the simulated temperature field is evaluated by the relationship

H

T cal T exact T exact

(16)

Where 1/ 2

T

§ · ¨ ³³ T 2 dA ¸ ¨ ¸ ©: ¹

(17)

The relative errors for the predicted temperature field by the SVD and EO approaches incorporated into the GEM are presented in Table 1 for the three cases of example 1. The values of the relative error that are available for the LIE formulation of Sladek et al. [5] are also presented in Table 1. The results for the temperature and the flux along the boundary ī4 are presented in Figures 2a and 2b for case (i), Figures 3a and 3b for case (ii) and Figures 4a and 4b for case (iii). The results indicate that both solution approaches give good prediction of the temperature and flux on the boundaries where they are not specified for the three cases. Except for case (iii), the EO gives a better prediction than the SVD. The results for the relative error of the LIE simulations are comparable to the GEM simulations, despite the fact that LIE used twice the number of boundary nodes of the GEM.

Advances in Boundary Element and Meshless Techniques *

19

T=1, q=-1

q=0

*

q=0

q=0

T=y/a

x

*

x

*

(a) Case (i)

(b) case (ii) T =1

q=0 T =y/a

*

x

*

(c) Case (ii) Figure 1: Problem domain for three cases of example 1

Table 1: Errors from the numerical simulations of Example 1 Case (i) Case (ii) RegulariRegulariSolution zation, Į Relative zation, Į Relative approach parameter error, İ parameter error, İ SVD 10-8 7.09×10-4 10-10 3.64 ×10-3 EO 3.21×10-4 9.43×10-3 LIE (SVD)* 7.60×10-4 2.50×10-3 * Source [5] 1.2

Case (iii) Regularization, Į Relative error, parameter İ 10-4 3.03 ×10-3 2.14 ×10-3 Not available

1.2 1 0.8

T(x,y=0.01) 1 0.8

q

T

0.6 0.4

Exact

q(x=0,y)

-0.2

SVD

Exact SVD EO

-0.4 -0.6 -0.8 -1

EO

0.2

q(x,y=0)

0.6 0.4 0.2 0

0 T(x,y=0)

q(x,y=0.01)

-1.2

-0.2 0

0x10

-3

2x10

-3

4x10

-3

6x10 x, y

-3

8x10

-2

1x10

0

0x10

-3

2x10

-3

4x10

-3

6x10

-3

8x10

1x10

-2

x, y

(a) (b) Figure 2: Numerical solutions of example 1 case (i) for temperature and flux along the boundaries ī4 where they are both not specified

20

Eds: P Prochazka and M H Aliabadi 1.5 Exact SVD EO

1

q(x,y=0)

T(x,y=0)

0.1

0

0.5 Exact SVD EO

0 0

0x10

-0.1 0

-3

0x10

2x10

4x10

-3

-3

-3

6x10

8x10

1x10

-3

2x10

-3

-3

4x10

6x10

-3

8x10

-2

1x10

x

-2

x

(a) (b) Figure 3: Numerical solutions of example 1 case (ii) for temperature and flux along the boundary ī4 where they are both not specified

1

1.5

0.8

q(x,y=0) 1

0.6

q

T

Exact

0.4

SVD

0.5

Exact

EO

SVD EO

0.2

0 q(x=0,y)

0 T(x,y=0) -0.5

-0.2 0

0x10

-3

2x10

-3

4x10

-3

6x10 x, y

-3

8x10

-2

1x10

0

0x10

-3

2x10

-3

4x10

-3

6x10 x, y

-3

8x10

-2

1x10

(a) (b) Figure 4: Numerical solutions of example 1 case (iii) for temperature and flux along the boundaries ī4 where they are both not specified

Example 2 In this example, the boundary conditions presented for case (iii) of the first example are considered. The thermal conductivity K behaves exponentially with respect to the spatial variable y. That is (18) K K 0 e Oy where K0=0.01, and two values of the parameter Ȝ are used in the numerical simulations: Ȝ=0.2 and Ȝ=0.5. The exact solution is

T

e Oy 1 e O a 1

(19)

The value of the regularization used in the SVD method is 10 . The values of the relative error, İ of the temperature field from the GEM simulations and those of LIE formulation of Sladek et al. [5] for this example are presented in Table 2 for two values of the exponents Ȝ=0.2 and Ȝ=0.5. From the relative errors of the numerical predictions of the temperature field, the SVD performed better than the EO. Both GEM solutions from the SVD and EO techniques are superior to those of the LIE formulation, in spite of the latter using twice the number of boundary nodes of the GEM. The numerical solutions along the left and bottom boundaries, where both T and q are unspecified, are presented in Figures 5a and 5b for the case of Ȝ=0.2 and -6


Figure 6a and 6b for the case Ȝ=0.5. q.

21

Both techniques (SVD and EO) give equally good prediction of T and 1.2

1

q(x,y=0) 1

0.8

0.8 0.6

٣=0.2 Exact SVD EO

q

T

0.6 =0.2

0.4

0.4

Exact SVD

0.2

EO

0.2 q(x=0,y)

0

0 T(x,y=0)

-0.2

-0.2 0

0x10

-3

2x10

-3

4x10

-3

6x10 x, y

-3

8x10

-2

1x10

0

-3

0x10

2x10

-3

4x10

-3

6x10 x, y

-3

8x10

-2

1x10

(a) (b) Figure 5: Numerical solutions for temperature and flux along the boundaries ī4 where they are both not specified (Example 2, Ȝ=0.2) 1

1.2

0.8

1

q(x,y=0) 0.8

0.6

٣=0.5 Exact SVD EO

q

T

0.6

٣=0.5

0.4

0.4

Exact SVD

0.2

0.2

EO

q(x=0,y) 0

0 T(x,y=0)

-0.2

-0.2 0x100

2x10-3

4x10-3 6x10-3 8x10-3 x, y

1x10-2

0x100

2x10-3

4x10-3 6x10-3 8x10-3 x, y

(a) Figure 6: Numerical solutions for temperature and flux along the boundaries ī4 where they are both not specified (Example 2, Ȝ=0.5)

Table 2: Errors from the numerical simulations of example 2 Ȝ=0.2 Ȝ=0.5 RegulariRegulariSolution zation, Į Relative zation, Į Relative approach parameter error, İ parameter error, İ SVD 10-6 6.59×10-4 10-6 8.31×10-4 EO 2.2×10-3 2.83×10-3 -3 LIE (SVD)* 6.40×10 1.30×10-2 * Source [5]

Conclusion

The recent formulation of the GEM has been used to solve the steady IHCP in 2-D homogeneous and heterogeneous domains. Two fundamentally different solution strategies have been incorporated into GEM in achieving the results presented in this paper. One solution strategy uses the regularized SVD technique in solving the over-determined ill-condition system of discrete equations, while the other uses the EO to predict the unspecified boundary data for the direct GEM

1x10-2

22


simulation of the problem. Both strategies exhibit good predictive capabilities of the IHCP, but the EO uses considerably more computing times because of the repeated GEM solution of the direct problem as the functional is minimized. The GEM solutions are superior to those of the LIE despite using a finer discretization of the numerical examples. Acknowledgements Special thanks go to the National Research Foundation which provided the financial support for this research work. References [1] A.E. Taigbenu, Engineering Analysis with Boundary Elements, 35, 125-136 (2012). [2] C-H. Huang and J-Y Yan Int. J. Heat Mass Transfer, 38, 3433-3441 (1995). [3] B. Sawaf and M.N. Özisik Int. J. Heat Mass Transfer, 38, 3005-3010 (1995). [4] T.T.M. Onyango, D.B. Ingham and D. Lesnic J. Engineering Mathematics, 62 85-101 (2008) [5] J. Sladek, V. Sladek and Y.C. Hon Engrg. Anal. with Boundary Elements, 30, 650-661 (2006). [6] D. Lesnic, L. Elliot and D.B. Ingham Int. J. Heat Mass Transfer, 39, 1503-1517 (1996). [7] Y. Jarny, M.N. Özisik and J.P. Bardon Int. J. Heat Mass Transfer, 34, 2911-2919 (1991). [8] C-H. Huang, J-X. Li and S. Kim Applied Mathematical Modelling, 32 417-431 (2008) [9] L. Yan, C-L Fu and F-L. Yang Engrg. Anal. with Boundary Elements, 32, 216-222 (2008). [10] G. Blanc, M. Raynaud and T.H. Chau Rev Gén Therm, 37, 17-30 (1998). [11] G.H. Golub and V.F. Van Loan, Matrix Computations, John Hopkins Univ. Press (1996). [12] A.N. Tikhonov and V.Y. Arsenin, Solutions of ill-posed Problems, Winston-Wiley (1977). [13] Q. Y. Duan, S. Sorooshian and V. Gupta, Water Resources Research 28, 1015-1031 (1992). [14] J. A. Nelder and R. A. Mead, Comput J., 7, 308-313 (1965). [15] Q. Y. Duan, S. Sorooshian and V. Gupta, J. Hydrology, 158, 265-284 (1994).


Anisotropic wear simulation using the boundary element method L. Rodr´ıguez-Tembleque1 , R. Abascal1 , M.H. Aliabadi2 1

Escuela Técnica Superior de Ingenier´ıa, Universidad de Sevilla, Camino de los descubrimientos s/n, E41092 Sevilla, SPAIN. [email protected] [email protected]

2 Department

of Aeronautics, Faculty of Engineering, Imperial College, University of London, South Kensington Campus, London SW7 2AZ, UK [email protected]

Keywords: Anisotropic Friction, Anisotropic Wear, Contact Mechanics, Boundary Element Method.

Abstract. This paper presents a new methodology to compute anisotropic wear on 3D frictional contact problems. The formulation is based on the Boundary Element Method for computing the elastic influence coefficients, and it uses projection operators over the augmented Lagrangian to enforce contact constraints. Anisotropic wear and frictional models are considered. Along with the proposed methodology, an accelerated algorithm is proposed. The present formulation is illustrated with a simple example, in which some studies about the influence of anisotropy on wear are presented. Introduction Wear is present in most of mechanical interface interaction problems: contact, fretting, or rollingcontact, and it is one of the main reasons for inoperability in mechanical components. Its estimation allows engineers to predict the useful life of a mechanical element, to reduce costs of inoperability, or obtain an optimum design (i.e., selecting proper materials, shapes and surface finishing according to the mechanical conditions and durability). So the economic implication of wear prediction can be of enormous value to the industry. In the contact models, the tribological properties (friction coefficient and wear intensity) are considered isotropic: friction is assumed to be constant and isotropic, as well as wear intensity (i.e., the isotropic Holm-Archard wear law [1, 2]). This is true when the contact surfaces present an isotropic roughness. However, in a great number of engineering applications, the distribution of the asperities and hollows on the surfaces are not identical on every point. For example, after most machining operations, surfaces present some particular striation patterns, and highs and hollows are clearly oriented in the surface. In these cases, an anisotropic friction and wear laws have to be considered. Particularly, in a large number of machining processes, the striations are mutually orthogonal. For such cases, a specific friction and wear model will be considered: the orthotropic friction and wear law, which provide a better description of the frictional behavior. This work presents a new methodology to compute anisotropic wear on 3D frictional contact problems. This formulation, based on previous works [3, 4], using the Boundary Element Method (BEM) for computing the elastic influence coefficients, and contact operators over the augmented Lagrangian, to enforce contact constraints. An anisotropic wear model based on [5] is considered, together with an orthotropic friction law [6, 7]. Finally, the proposed methodology and algorithm are illustrated with a simple example, in which the influence of anisotropy on wear is important. Contact discrete variables and restrictions To consider the contact between two solids Ωα (α = 1, 2), we have to compute the contact gap for each pair I of nodes in contact (1) (k)I = (kgo )I + (d2 )I − (d1 )I k being the contact pairs gap vector and kgo the initial geometrical gap and rigid body displacement vector [8].

23

24


Contact restrictions for every contact pair I are summarized in: the Non-penetration condition, the Coulomb friction law and the Principle of maximum energy dissipation. The mathematical expressions for these contact restrictions, can be classified into two groups: normal and tangential. • Normal direction: The unilateral contact conditions can be written, in the form of a complementarity relation, as: (kn )I ≥ 0 ;

(pn )I ≤ 0 ;

(pn )I (kn )I = 0

(2)

• Tangential direction: For tangential direction, the fulfilment of friction law and principle of maximum dissipation is guaranteed by: (pt )I ≤ µ |(pn )I | ; (kt )I = −λ(pt )I

; λ ≥ 0 ; (kt )I ((pt )I − µ |(pn )I |) = 0

(3)

where || • ||µ denotes the elliptic norm ||(pt )I ||µ =

(pte1 )I µ1

2

+

(pte2 )I µ2

2 (4)

and the coefficients µ1 and µ2 are the principal friction coefficients in the directions {e1 , e2 }. The contact restrictions (2) and (3) for every contact pair I can be expressed in a discrete form as: (Λt )I − PEρ ( (Λ∗t )I ) = 0

(Λn )I − PR− ( (Λ∗n )I ) = 0

(5)

where augmented contact variables were defined in [9, 10] as: (Λ∗t )I = (Λt )I − rt M2 (kt )I and (Λ∗n )I = (Λn )I + rn (kn )I , as well as the projection functions: PR− and PEρ , with ρ = |PR− ( (Λ∗n)I )|. BE-BE contact equations The boundary integral equations for a body Ω, can be written in a matrix form as: Hd − Gp = F

(6)

where the vector d represents the nodal displacements, and F contains the applied boundary conditions. These equations are well known and can be found in many books like [11] or [12]. Equation (6) can be written for contact problems as: Ax x + Ap pc = F, being x the nodal unknowns vector that collects the external unknowns and the contact displacements; pc is the nodal contact tractions; Ap is constructed with the columns of G belonging to the contact nodal unknowns, and A x , with the columns of matrices H and G, corresponding to the exterior unknowns and the contact nodal displacements. So the BE-BE contact system can be expressed, according to [9, 10], as: ⎡

A1x ⎢ 0 ⎢ ⎣ (C1 )T 0

0 A2x −(C2 )T 0

⎤⎧ 1 ˜1 A1p C 0 x ⎪ ⎪ ⎨ 2 2 2 ˜ x −Ap C 0 ⎥ ⎥ ⎪ Λ 0 Cg ⎦ ⎪ ⎩ k Pλ Pg

⎫ ⎪ ⎪ ⎬

⎧ ⎪ ⎪ ⎨

F1 F2 = ⎪ Cg kgo ⎪ ⎪ ⎭ ⎪ ⎩ 0

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

(7)

˜ 1 Λ and p2c = −C ˜ 2 Λ. where vector Λ represents the nodal contact tractions, so that: p1c = C The first two rows in the expression above represent the equilibrium of each solid Ωα (α = 1, 2), the third row is the contact kinematics equations, and the last row expresses the nodal contact restrictions. Matrices Pλ and Pg are the non-linear terms obtained by assembling the matrices (Pλ )I and (Pg )I , associated to the I pair of nodes in contact. The expression of the matrices depends on the I pair contact state:


25

• No-Contact: (Λ∗n )I ≥ 0 ⎤ ⎤ ⎡ 1 0 0 0 0 0 (Pλ )I = ⎣ 0 1 0 ⎦ , (Pg )I = ⎣ 0 0 0 ⎦ 0 0 1 I 0 0 0 I ⎡

• Contact-Adhesion: (Λ∗n )I < 0 ⎡ 0 (Pλ )I = ⎣ 0 0

and (Λ∗t )I µ < |(Λ∗n )I | ⎤ ⎤ ⎡ 0 0 0 rt 0 0 ⎦ 0 0 ⎦ , (Pg )I = ⎣ 0 rt 0 0 −rn 0 0 I I

• Contact-Slip: (Λ∗n )I < 0 and (Λ∗t )I µ ≥ |(Λ∗n )I | ⎡ ⎡ ⎤ ⎤ 1 0 ωe∗1 0 0 0 ∗ 0 ⎦ (Pλ )I = ⎣ 0 1 ωe2 ⎦ , (Pg )I = ⎣ 0 0 0 0 −rn 0 0 0 I I

(8)

(9)

(10)

being: (ω ∗t )I = (Λ∗t )I /(Λ∗t )I µ . To describe more clearly the algorithms of resolution in the following section, the system (7) can be rewritten in a more compact form as: ⎧ 1 ⎫ d ⎪ ⎪ ⎪ 1 ⎬ ⎨ 2 ⎪ ¯ x F R R2 Rλ Rg (11) = 0 0 0 P λ Pg ⎪ ⎪ ⎪ Λ ⎪ ⎭ ⎩ k ¯ the corresponding block matrices of these being the matrices R1 , R2 , Rλ and Rg , and vector F, systems. Anisotropic wear discrete equations Wear modeling on the contact surfaces of two bodies requires to include the wear depth variable on the contact kinematic equations. Consequently, the contact gap for the pair I of nodes in contact (1), can be expressed for the instant (k) as: (k) )I + (d2 (k(k) )I = (kgo

(k)

)I − (d1

(k)

)I + (Cg n w(k) )I

(12)

where w(k) is a vector which contains the contact pairs wear depth, and matrix Cg n is constituted using the Cg columns which affect the normal gap of contact pairs. The discrete wear depth evolution for every load step (k) can be computed as (w(k) )I = (w(k−1) )I + (∆w(k) )I (k)

(k−1)

(∆w(k) )I = |(Λn )I | ||(kkt )I − (kt (k)

(13) )I ||i

where Λn is a vector which contains the normal traction components of contact pairs at instant k, and ∆w(k) is the wear depth increment. The wear intensity norm || • ||i were defined in [10] as ||f ||i = (i1 fe1 )2 + (i2 fe2 )2 (14) being i1 and i2 the principal intensity coefficients.

26


Solution method The nonlinear equations set for quasi-static anisotropic contact problems (11) can be solved using the following iterative Uzawa scheme predictor-corrector: (1) Initialization: z(0) = z(k−1) .

(2) Predictor step, solve:

R1

R2

Rg

⎤(n+1) d1 2 ¯ (k) . ⎣ x ⎦ = −Rλ Λ(n) + F k ⎡

(3) Corrector step, update the contact tractions Λ(n+1) for every contact pair I : (Λn(n+1) )I = PR− ( (Λn(n) )I + rn (kn(n+1) )I ) (n+1)

(Λt (n+1)

being ρ = |(Λn

(n)

(n+1)

)I = PEρ ( (Λt )I − rt M2 [(kt

(15)

(k−1)

)I − (kt

)I ] )

(16)

)I |.

(4) Finish when: Ψ z(n+1) ≤ ε, otherwise return to step 2. Finally, we obtain the solution for the load step (k): z(k) = z(n+1) and the resulting accumulated wear: (w(k) )I = (w(k−1) )I + (∆w(k) )I (k)

(k)

(17)

(k−1)

)I ||i . being (∆w(k) )I = |( Λn )I | ||( kt − kt The algorithm can be accelerated using the Steffesen’s method. For the normal contact tractions point of view, corrector step 3 (15) can be viewed as a fixed point iteration. So the convergent sequence (n+1) ¯ (n+1) )I }: {(Λn )I } can be accelerated as {(Λ n (n+1)

¯ (Λ n (n+2)

where (Λn

)I = (Λ(n) n )I +

(n+1) (n) )I − (Λn )I ]2 [(Λn (n+2) (n+1) (n) [(Λn )I − 2(Λn )I + (Λn )I ]

(18)

)I is computed with a second predictor-corrector loop as (Λn(n+2) )I = PR− ( (Λ(n+1) )I + rn (k(n+2) )I ) n n

This Aitken’s ∆2 method is not applied for fixed-point tangential sequence convergence is slower than the normal problem.

(19) (n+1) {(Λt )I },

because its

Application This example considers the indentation on an elastic block by a rigid spherical punch. The punch radii is R = 100 m, and the block dimensions are: Lo = 0.1 m and L1 = L2 = 1 m (see Fig. 1(a)). The material properties considered are: Young module E = 210 GPa and Poisson coefficient ν = 0.3. The domain is discretized by linear quadrilateral boundary elements, using 20 × 20 elements on the potential contact zone. A rigid body displacement go,x3 = −0.07 mm is applied on 40 loads steps, while three different cases of anisotropic friction limits and wear laws are considered (see Fig.1(b)): (a) isotropic (µ1 =µ2 = 0.3 and i1 =i2 = 10 × 10−12 P a−1 ), (b) orthotropic (µ1 = 0.2, µ2 = 0.3, i1 = 6.66×10−12 P a−1 and i2 = 10×10−12 P a−1 ), and (c) orthotropic (µ1 = 0.1, µ2 = 0.3, i1 = 3.33×10−12 P a−1 and i2 = 10 × 10−12 P a−1 ). Figures 2(a-c) display the resulting wear depth distribution in every case. We can see how the level of anisotropy on the tribological properties affects the wear intensity and its distribution over the contact zone. For example,it can be observed in Fig.2(c) that wear depth in x1 = 0 is four times bigger than the one in x2 = 0.


(a)

27

(b)

Figure 1: (a) Indentation of a rigid sphere over an elastic block. (b) Wear intensity functions as a polar diagrams: (a) isotropic (i1 =i2 = 10 × 10−12 P a−1 ); (b) orthotropic (i1 = 6.66 × 10−12 P a−1 , i2 = 10 × 10−12 P a−1 ); (c) orthotropic (i1 = 3.33 × 10−12 P a−1 , i2 = 10 × 10−12 P a−1 ).

(a)

(b)

(c)

10−12

P a−1 );

Figure 2: Wear depth distributions: (a) isotropic (i1 =i2 = 10 × (b) orthotropic (i1 = 6.66 × 10−12 P a−1 , i2 = 10 × 10−12 P a−1 ); (c) orthotropic (i1 = 3.33 × 10−12 P a−1 , i2 = 10 × 10−12 P a−1 ).

The examples have been solved using the proposed algorithm, considering rn = 1012 and rt = 1013 for the augmented Lagrangian, and ε = 10−2 as a termination limit. Parameters rn and rt cannot be chosen larger, because the algorithms might become unstable. All the examples have been solved using the accelerated Uzawa scheme proposed, which allows to obtain an important CPU time reduction compared with the classical one (Fig.3). Conclusions This work presents a novel numerical methodology for solving 3D anisotropic frictional contact problems, including anisotropic wear. The BE contact methodology is based on an augmented Lagrangian formulation, using a fast predictor-corrector Uzawa scheme, which allows to significantly reduce the number of iterations compare with the classical Uzawa. The present formulation has been applied to compute wear on an incremental contact problem under partial slip. All the studies presented show the importance of considering wear in the contact process because of its influence in the contact variables. Furthermore, when the distribution of the asperities and hollows on the surfaces are not identical on every point it has to be considered an anisotropic friction and wear laws. In other case, we could over- or underestimate wear magnitudes and its distribution over the contact zone, as it was shown in the numerical examples. In this kind of problems, the BEM reveals to be a very suitable

28


Figure 3: CPU time reduction on every step due to the accelerated Uzawa scheme.

numerical method, considering only the degrees of freedom involved on the problem, and obtaining a very good accuracy on anisotropic contact variables with a reduced number of elements. Acknowledgments This work was co-funded by the DGICYT of Ministerio de Ciencia y Tecnolog´ıa, Spain, research project DPI2010-19331, and by the Consejer´ıa de Innovaci´ on Ciencia y Empresa de la Junta de Andaluc´ıa, Spain, research project P08-TEP-03804. References [1] Holm R. Electric Contacts. Almquist and Wiksells Akademiska Handb¨ ocker. Stockholm, 1946. [2] Archard JF. Contact and rubbing of flat surfaces, J. Appl. Phys. 1953; 24:981–988. [3] Rodr´ıguez-Tembleque, L., Abascal, R., Aliabadi, M.H. A boundary element formulation for wear modeling on 3D contact and rolling-contact problems. Int. J. Solids Struct., 2010; 47: 2600–2612. [4] Rodr´ıguez-Tembleque, L., Abascal, R., Aliabadi, M.H. A boundary element formulation for 3D fretting-wear problems. Eng. Anal. Bound. Elem., 2011; 35: 935–943. [5] Zimitrowicz, A. Constitutive equations for anisotropic wear, Int. J. Engng. Sci., 1993; 31: 509–528. [6] Zimitrowicz, A. Models of kinematics dependent anisotropic and heterogeneous friction, Int. J. Solids Struct., 2006; 43: 4407–4451. [7] Zimitrowicz, A. Contact stress: a short survey of models and methods of computations, Arch. Comput. Methods, 2010; 80: 1407–1428. [8] Rodr´ıguez-Tembleque, L., Abascal, R. A FEM-BEM fast methodology for 3D frictional contact problems. Comput. Struct. 2010; 88: 924-937. [9] Rodr´ıguez-Tembleque, L., Abascal, R. Accelerated FE-BEM algorithms for 3D anisotropic frictional contact problems. Int. J. Numer. Methods Eng., 2012; submitted. [10] Rodr´ıguez-Tembleque, L., Abascal, R., Aliabadi, M.H. Anisotropic wear framework for 3D contact and rolling problems. Comput. Meth. Appl. Mech. Eng., 2012; submitted. [11] Brebbia, C.A., Dominguez J. Boundary Elements: An Introductory Course (second edition). Computatinal Mechanics Publications. John Wiley & Sons, 1992. [12] Aliabadi, M.H. The Boundary Element Method Vol2: Applications in Solids and Structures. John Wiley & Sons, 2002.


Non-Local plastic deformation gradients for localization phenomenon and regularization of strain-softening behavior using the meshless Finite Point Method L. Pérez Pozo1, A. Campos Rodríguez2 1 2

email: [email protected]

email: [email protected]

Aula UTFSM-CIMNE, Departamento de Ingeniería Mecánica, Universidad Técnica Federico Santa María, Avenida España 1680, Valparaiso, Chile. Keywords: Meshless, softening, localization, non-local deformations, gradients plasticity.

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Advances in Boundary Element and Meshless Techniques An inverse multi-loading boundary element method for identification of elastic constants of 2D anisotropic bodies M.R. Hematiyan1, A. Khosravifard2, Y.C. Shiah3, L. Tan4 1

Department of Mechanical Engineering, Shiraz University, Shiraz, Iran, [email protected]

2

Department of Mechanical Engineering, Shiraz University, Shiraz, Iran, [email protected]

3

Department of Aerospace and Systems Eng., Feng Chia University, 100 Wenhwa Road, Seatwen 407, Taichung, Taiwan, ROC, [email protected]

4

Department of Mechanical and Aerospace Engineering, Carleton University, Ottawa, ON, Canada K1S 5B6, [email protected]

Keywords: Inverse method, boundary element method, anisotropic elasticity. Abstract. An inverse technique based on the boundary element method (BEM) and elastostatic experiments for identification of elastic constants of orthotropic and general anisotropic 2D bodies is presented. Displacements at several points on the boundary of the body, obtained by a few known load cases are used in the inverse analysis to find unknown elastic constants of the body. Using data from more than one elastostatic experiment results in a more accurate and stable solution for the identification problem. In the inverse analysis, sensitivities of displacements of only boundary points with respect to the elastic constants are needed. Therefore, the BEM is a very useful and effective method for this purpose. An iterative Tikhonov regularization method is used for the inverse analysis. A method for appropriate selection of the regularization parameter appearing in the inverse analysis is also proposed. Convergence and accuracy of the presented method with respect to measurement errors and number of load cases are investigated by presenting several examples. 1. Introduction Anisotropic composite materials are widely used in engineering structures. Some natural materials and biostructures show anisotropy too. Identification of elastic constants of human-made or natural anisotropic materials and structures is very important for predicting their behaviour. The most important methods for identification of elastic constants are based on either static or dynamic measurements. Some researchers have presented inverse methods based on the finite element method (FEM) and static measurements; see for example [1]. Cunha and Piranda [2] presented a method based on the FEM and dynamic measurements for identification of stiffness properties of composite tubes. Rikards et al. [3] presented a method for identification of elastic constants of laminates based on dynamic measurements. They used the FEM in their formulation. A few researchers have used the BEM for identification of elastic constants of orthotropic or anisotropic materials. Ohkami et al. [4] presented an identification method based on static measurements and the BEM for a 2D orthotropic medium. They used the Gauss-Newton method in their formulation. Huang et al. [5] presented an inverse BEM based on displacement measurements and Levenberg-Marquardt method for identification of elastic parameters of 2D orthotropic bodies. Comino and Gallego [6] presented an inverse method based on the BEM for identification of elastic constants of 2D anisotropic materials. They used static measurements and LevenbergMarquardt method in their inverse technique. In the above-mentioned researches on properties identification of anisotropic materials, the unknowns have been computed using only one static load case. To obtain an acceptable solution using only one static experiment, a complicated load case should be considered. Often, it is impossible to make a sample with a simple standard geometry for performing the required measurements. In such cases, the original body should be used for the measurements. On the other hand, setting up a single experiment with a load case that efficiently includes effects of all elastic constants may be impossible or difficult. Therefore, it is reasonable to carry out several experiments with different simple load cases to find out the unknowns. This is the main idea of the present paper. In this work, an inverse method based on static experiments for identification of elastic constants of orthotropic and general anisotropic 2D bodies is proposed. Displacements at several boundary points, obtained by a few known load cases

39

40


are considered as measured data. Equations generated from inverse formulation of all load cases are coupled and solved simultaneously. The BEM is used for the sensitivity analysis in the inverse method. The BEM is a very useful and effective method for this purpose, because a sensitivity analysis of displacements at only boundary points with respect to the elastic constants is needed in the inverse analysis. An iterative Tikhonov regularization method, including a regularization parameter, is used for the inverse analysis. A procedure for appropriate selection of the regularization parameter is also proposed. By presenting several examples, the convergence and accuracy of the presented method are investigated. 2. The BEM for elastostatic analysis of 2D anisotropic bodies Before dealing the core issue of the inverse analysis, it is useful to review some fundamental equations in twodimensional anisotropic elasticity and the corresponding formulation in the BEM. For a homogeneous generally anisotropic elastic material in plane stress, the constitutive relations can be expressed in matrix form as H 11 ½ ª a11 a12 a16 º V 11 ½ V 11 ½ ª c11 c12 c16 º H11 ½ ° ° « ° ° « ° ° »° »° (1) ®H 22 ¾ «a12 a 22 a 26 » ®V 22 ¾ , ®V 22 ¾ «c12 c22 c26 » ®H 22 ¾ , °2H ° «a °W ° °W ° «c °2H ° » » c c a a 26 66 ¼ ¯ 12 ¿ ¯ 12 ¿ ¬ 16 ¯ 12 ¿ ¬ 16 26 66 ¼ ¯ 12 ¿ where Vij and Hij (i, j=1, 2) represent the stresses and strains, respectively, and the coefficients cmn and amn are the elastic stiffness and compliance constants of the material, respectively. These compliances may be given in terms of engineering constants as follows [7]:

a11 1 / E1 ,

a22

1 / E2 , a12

Q 12 / E1

Q 21 / E2 ,

a16 K12,1 / E1 K1,12 / G12 , a26 K12, 2 / E2 K 2,12 / G12 , a66

1 / G12 ,

(2)

where Ek is the Young’s modulus in the direction of the xk-axis and G12 is the shear modulus on the x1-x2 plane; Qij is the Poisson’s ratio, and Ki,jl, Kij,l are the coefficients of mutual influence of the first and second kind, respectively. Equation (2) is also applicable to the case of plane strain, provided bjk is substituted for ajk by (3) b jk a jk a j 3 a k 3 / a 33 , ( j , k 1, 2) ,

where, with the index 3 referring to the x3-axis, and am3 are given by a j 3 Q j 3 / E j Q 3 j / E3 , a33 1 / E3 , a 63 K12,3 / E3 K 3,12 / G12 .

(4)

By introducing Airy’s stress functions, Lekhnitskii [7] has shown that the characteristic equation for an anisotropic material in stable equilibrium is a11 P 4 a16 P 3 2a12 a 66 P 2 a 26 P a 22 0 . (5) It has further been shown that the roots of this characteristic equation must be complex, and are given by two distinct pairs of complex conjugates. They are denoted by P j D j i E j , ( j 1, 2) , (6) where i= 1 and Ej must be positive from thermodynamic considerations. By following the above notation for material properties, the position of a general field point at (x1, x2) can be described by z j x1 P j x 2 , ( j 1, 2) . (7) The analytical basis of the BEM is the boundary integral equation (BIE). Two key requirements are necessary for its derivation, namely, (a) the fundamental solution to the governing differential equations, and (b) a reciprocal theorem relating the displacements and the tractions on the elastic body. These are provided, respectively, by the unit load solutions for an infinite body, and Betti-Rayleigh’s reciprocal work theorem. Carrying out the usual limiting process results in the BIE, relating the displacements ui and the tractions ti on the boundary S of the domain : , as follows: (8) C ij ( P) u i ( P) = t i (Q ) U ij ( P , Q) dS u i (Q) T ij ( P , Q ) dS ,

³ S

³ S

in which P and Q represent the source point and the field point on S, respectively, and C ij are coefficients associated with the boundary geometry at P. In the boundary integral equation, U ij ( P , Q) and T ij ( P , Q) are the fundamental solutions for displacements and tractions at Q in the xi-direction, respectively, when a unit load is applied at P in the xj-direction. Their explicit forms are given by [8]: (9a) U ij ( P , Q) = 2 Re { r i1 A j1 log z1 + r i2 A j2 log z 2 } ,

Advances in Boundary Element and Meshless Techniques T1 j ( P , Q) = 2 n1 Re { P12 A j1 / z1 + P 22 A j 2 / z 2 } - 2 n2 Re { P1 A j1 / z1 + P 2 A j 2 / z 2 }, T2 j ( P , Q) = 2 n1 Re{ P1 A j1 / z1 + P 2 A j 2 / z 2 } 2 n2 Re{ A j1 / z1 + A j 2 / z 2 },

41

(9b) (9c)

where rij and Aji are complex quantities associated with the material properties, Re{} is the operator which takes the real part of the quantities in the parentheses, and zi is a generalized complex variable. This variable for the field point Q at (x1, x2), with reference to the source point P at (xp1, xp2), is defined as follows: (10) zi = ( x1 - x p1 ) + P i ( x2 - x p 2 ) ] 1 P i ] 2 , where ]i represent the local coordinates with the origin located at P. In Eqs. (9b) and (9c), ni are components of the unit outward normal vector at Q. The BIE, Eq. (8), can generally be solved only by numerical means. It involves, discretizing the boundary into a mesh with, say, M elements and N distinct nodes. When using quadratic isoparametric elements, the geometry and all the primary solution variables are assumed to vary in a quadratic manner over each element. With the use of interpolation by shape functions Nc(]) expressed in terms of the intrinsic coordinate ] ( 1 d ] d 1 ), the coordinates and solution variables at a general field point can then be expressed as

xi (] )

3

¦N

c

3

¦N

(] ) xic , u i (] )

c 1

c

(] ) u ic , t i (] )

c 1

where N c (] ) are given by

3

¦N

c

(] ) t ic ,

(11)

c 1

N 1 (] ) ] 1 ] / 2, N 2 (] ) 1 ] 2 , N 3 (] ) ] 1 ] / 2 . (12) By substituting Eqs. (11) and (12) into the BIE, Eq. (8), the discretized form of the boundary integral equation is obtained, as follows: M

a a C ij ( P ) ui ( P ) =

3

¦¦ t ³ U b c i

b 1 c 1

M

3

¦¦ b 1 c 1

ij ( P

a

, Q ) N c (] ) J ] dS

S

b

uic

,

³

(13)

a c T ij ( P , Q ) N (] ) J ] dS

S

where Pa stands for the a-th node of the mesh (Pa=1~N) and the superscripts b and c represent the b-th element and the c-th node of each element, respectively. J ] is the Jacobian of coordinate transformation. Equation (13) represents a set of 2N linear algebraic equations for the unknown displacements/tractions at the boundary nodes. It can be solved by, for example, Gaussian elimination. 3. Inverse analysis

A direct anisotropic elastic problem in which material properties, boundary conditions, and loadings are known is a well-posed problem, i.e. it possesses a unique and stable solution. However, an inverse problem with unknown elastic constants is an ill-posed problem. In the inverse problem, the unknowns are to be found by using some additional information obtained by measurements. In an inverse analysis, an optimization method including a sensitivity analysis and a regularization technique should be employed. 3.1 The inverse problem statement and formulation A general 2D anisotropic body with unknown elastic constants is considered. The vector of elastic constants which contains these unknowns is defined as follows T c >c1 c2 c6 @ , (14) where c1 c11 , c2 c22 , c3 c66 , c4 c12 , c5 c16 , c6 c26 . (15) To find these material constants, a few elastostatic experiments with different loadings and constraints are performed. The number of experiments may be two, three, or more. We consider three load cases to formulate the problem. The formulation with a different number of load cases is similarly accomplished. An anisotropic body under three different loadings and constraints is shown in Fig. 1. In each load case, displacements at some selected boundary points are measured. Location of measurements can be different in each load case. Assume there are N1 , N 2 , and N 3 measurement data in the load cases 1, 2, and 3, respectively.

42


Figure 1: A body subjected to three different load cases The vectors of measured data are represented by Y (1) , Y ( 2) , and Y (3) , where, for example, Y (1) contains measured data obtained from the load case 1 and can be expressed as follows:

>Y

Y (1)

(1) 1

@

T

Y2(1) YN(11) .

(16)

Vector of displacements at the sampling points in the load case 1 computed by considering a set of elastic constants is represented by U (1) , which can be expressed as follows:

>U

U (1) ( 2)

(1) 1

U 2(1) U N(11)

@. T

(17)

( 3)

Vectors U and U are similarly defined. To find the vector of elastic constants, the Tikhonov regularization method is used. In this method, the following function S is formed: S (Y U )T (Y U) PcT c , (18) where the vectors Y and U are expressed as follows

>

@

T

>

@

T

(19) Y Y (1) Y ( 2 ) Y ( 3) , U U (1) U ( 2) U (3) . In Eq. (18), P is a regularization parameter. The first term in Eq. (18) is used to make sure that the difference between the vectors Y and U is small. The second term in Eq. (18) is used to prevent the elastic constants vector having a large norm. Small values of P lead to oscillatory solutions in some cases. Increasing the value of the regularization parameter damps the oscillations; however, the difference between the measured and computed values of displacements at the sampling points increases. The vector c is found by minimizing S, which leads to wS 2 XT (Y U) 2 Pc 0 . (20) wc The matrix X in Eq. (20) is the sensitivity matrix of all load cases, which can be expressed as follows:

>X

X

where X

( L)

X( L)

(1)

X ( 2)

X ( 3)

@

T

is the sensitivity matrix of the load case L and is expressed as ª X 11( L ) X 12( L ) X 16( L ) º « ( L) ( L) (L) » X 22 X 26 « X 21 » L 1, 2, and 3 « » « (L) » (L) (L) ¬« X N L1 X N L 2 X N L 6 ¼»

Components of the sensitivity matrix X ( L ) can be expressed as follows wU i( L ) X ij( L ) . wc j

(21)

(22)

(23)

In order to compute the components of the sensitivity matrix given in Eq. (22), the derivative of boundary displacements with respect to the elastic constants should be computed. Two main approaches are usually used for this purpose. One is the use of finite differences, and the second is carried out by differentiating the integral or matrix equations, representing the problem. The second approach is much more complicated. In this work, the approach based on the finite difference method is used.


43

The unknown vector c should be found using Eq. (20) by an iterative procedure. Suppose that the vector ~c is a ~ ~ ~ suggestion for the elastic constants vector, and U (1) , U ( 2) , and U (3) are the corresponding displacement vectors for the load cases 1, 2, and 3, respectively. The displacement vector can be approximately represented as follows ~ U U X(c ~c ) , (24) where ~ ~ ~ ~ T U U (1) U ( 2 ) U ( 3) . (25) Substituting Eq. (24) into Eq. (20) and after some mathematical manipulations, the following relationship is obtained ~ c [ XT X PI ]1[ XT (Y U) XT X~c ] . (26) Eq. (26) should be used in an iterative procedure, and therefore, it is better to be written in the following form c k 1 [( X k )T Xk P k I ]1[( X k )T (Y U k ) ( X k )T Xk c k ] , (27) where k and k+1 represent iteration numbers. The convergence criterion is defined as c k 1 c k d e . (28)

>

@

where e is a specified tolerance. The existence and uniqueness of the solution for the inverse problem may be assured by physical reasoning. If a sufficient number of measured data (greater than the number of unknowns) are used in the inverse analysis, a solution very close to the exact solution can be obtained even in case of noisy input data.

3.2. Selection of the regularization parameter The regularization parameter P k in Eq. (27) should be carefully selected at each iteration. Suppose that the measured data have a Gaussian noise, and each measured displacement can be expressed as follows: Yi Yi exact eiYi exact i 1, 2, ..., N t (29) where Nt is the total number of measured data and ei is the relative error. ei is a random number from a Gaussian distribution with a zero mean and the standard deviation Vˆ . We usually have sufficient information about the measurement error and there exists a suitable estimation for the standard deviation. Assuming the vector E contains errors of computed displacements with respect to measured data, and using Eq. (24), we can write: E U(c k 1 ) Y U k X k (c k 1 c k ) Y (30) When we use a small number of measurement data, for example, equal to the number of unknowns, with P k the computed vector c STD(E) Vˆ

k 1

0,

will be very noisy; however, the value of E will be very small and (31)

where STD stands for the standard deviation. In this case, selecting a positive value for P k will result in a much better solution for c k 1 with a larger STD(E) . The regularization parameter P k is selected in a such way that (32) STD (E) | Vˆ When we use a sufficient number of measurement data (considerably more than the number of unknowns) we will have STD (E) ! Vˆ even with P k 0 . In this case, satisfactory results are obtained with P k 0 .

4. Numerical examples In this section, the proposed inverse technique is applied for identification of elastic constants of bodies. Two different cases are considered. In the first case, the body is made of an orthotropic material; a generally anisotropic material is treated in the second case. In each case, a direct analysis is performed and the displacements at several boundary points obtained using this direct analysis, are used in place of experimental measurements. To account for the inherent experimental errors, a vector of errors with Gaussian distribution is added to the results of the direct analysis. The effect of the standard deviation of the errors on the identified elastic constants is studied. The shape of the body for which the elastic constants are sought is shown in Fig. 2. For the sake of sensitivity analysis by the BEM, the boundary of the body is discretized by 34 quadratic boundary elements.

44


Fig. 3 depicts the three load cases used for identification of the material constants. In this figure, the value of the applied traction q, for each case is equal to 104 N/m. 20 sampling points are selected on the boundaries of the body in each load case. The dots in the figure represent the sampling points at which measurements are made. The elastic constants of the body are obtained in three different situations. First, the measurements of only the first load case are used for identification of the material constants. Then, the measurements of the first and the second load cases are used together, and another set of constants are identified. Finally, the measurements of all the three load cases are used together to obtain material constants. The results of these three situations are compared to a case in which all loadings are applied to the body simultaneously (see Fig. 4).

Figure2: The geometry of the problem along with 34 boundary elements

Figure 3: The three load cases used for determination of material constants

Figure 4: The load case with simultaneous application of various loadings

4.1. Case I: Identification of elastic constants of an orthotropic body In this case, it is assumed that the body shown in Fig. 2 is made of an orthotropic material for which the vector of elastic constants is: c [10 5 3.5 1.5]T GPa (33) The relative error of measured data is assumed 10%, i.e. a relative error with a Gaussian distribution and the standard deviation of Vˆ 0.033 is assumed. Table 1, gives the values of elastic constants obtained by load cases of Fig. 3. This table also presents the values of elastic constant obtained by application of all loadings in one experiment (see Fig. 4). Table 1 suggests that the most reliable results are obtained in the case that three different experiments are performed and the results of all three experiments are used together for identification of the elastic constants.

Advances in Boundary Element and Meshless Techniques Another benefit of using more than one test is that relatively simple tests can be performed in each case for identification of constants. If only one test is to be used, the loadings should be much more complicated than the ones used in this paper. The effects of measurement errors on the identified elastic constants are also investigated. Table 2, reports the elastic constants predicted for three different cases with 3, 5, 10, and 20 percent error. The results of this table are obtained with the three loadings of Fig. 3. Fig. 5 depicts the values of the elastic constants versus the iteration number for the case with three tests and 10% measurement error. This figure clearly shows that the convergence of the method is very fast. Table 1: The identified elastic constants (in GPa) of the orthotropic material with various load cases (with 10% measurement error) c11 (error) c22 (error) c66 (error) c12 (error) Exact value 10 5 3.5 1.5 1-test 10.41 (4.1%) 4.95 (1.0%) 3.53 (0.9%) 1.17 (22%) 2-test 11.32 (13%) 5.02 (0.3%) 3.61 (3.1%) 1.69 (12%) 3-test 10.38 (3.8%) 5.02 (0.3%) 3.43 (2.0%) 1.41 (5.8%) 1-test (all-in-one loading) 10.06 (0.6%) 5.13 (2.5%) 2.99 (15%) 1.42 (5.6%) Table 2: Effect of measurement error on the identified elastic constants of the orthotropic body c11 (error) c22 (error) c66 (error) c12 (error) Exact value (GPa) 10 5 3.5 1.5 3% measurement error 10.11 (1.1%) 5.00 (0.1%) 3.48 (0.6%) 1.48 (1.6%) 5% measurement error 10.19 (1.9%) 5.01 (0.2%) 3.45 (1.0%) 1.46 (2.8%) 10% measurement error 10.38 (3.8%) 5.02 (0.3%) 3.43 (2.0%) 1.41 (5.8%) 20% measurement error 10.80 (8.0%) 5.03 (0.7%) 3.36 (3.9%) 1.32 (13%)

Figure 5: Convergence of the proposed method for identification of elastic constants of the orthotropic body

4.2. Case II: Identification of elastic constants of an anisotropic body In this case, it is assumed that the body shown in Fig. 2 is made of an anisotropic material for which the vector of elastic constants is as follows: c [544.8 531.1 243.5 153.6 81.2 89.7]T GPa (34) These values of elastic constants for the anisotropic material are reported in a paper by Tan et al. [9]. Table 3, lists the identified elastic constants based on the load cases of Fig. 3. The results are also compared with those obtained by the load case of Fig. 4. A close review of the table suggests that the most reliable results are obtained in the case that three tests with the three load cases are conducted and the measurements are used together. The values reported in Table 3, are obtained when the tolerance of measurement errors is considered to be 10%. To investigate the effect of measurement errors on the identified elastic constants, a vector of errors with Gaussian distribution is generated. The standard deviation of the errors for the same vector is so chosen as to result in 3, 5,

45

46


10, and 20 percent tolerance of measurement error. Table 4, reports the elastic constants predicted for each case. The results of this table are obtained with the three loadings of Fig. 3. Table 3: The identified elastic constants of the anisotropic body with various load cases (with 10% measurement error) c11 c22 c66 c12 c16 c26 (error) (error) (error) (error) (error) (error) Exact value (GPa) 544.8 531.1 243.5 153.6 -81.2 89.7 386.4 524.0 231.8 85.4 -20.5 122.7 1-test (29%) (1.3%) (4.8%) (44%) (75%) (37%) 542.3 533.3 236.3 158.9 -64.4 93.3 2-test (0.5%) (0.4%) (3.0%) (3.4%) (21%) (4.0%) 527.1 526.3 245.0 140.0 -82.7 92.5 3-test (3.2%) (0.9%) (0.6%) (8.9%) (1.8%) (3.1%) 1-test 559.6 533.7 287.5 136.1 -110.0 111.7 (all-in-one loading) (2.7%) (0.4%) (18%) (11%) (35%) (25%) Table 4: Effect of measurement error on the identified elastic constants of the anisotropic body c11 c22 c66 c12 c16 c26 (error) (error) (error) (error) (error) (error) Exact value (GPa) 544.8 531.1 243.5 153.6 -81.2 89.7 539.3 529.6 243.9 149.4 -81.6 90.5 3% measurement error (1.0%) (0.3%) (0.2%) (2.7%) (0.5%) (0.9%) 535.8 528.6 244.2 146.7 -81.9 91.1 5% measurement error (1.7%) (0.5%) (0.3%) (4.5%) (0.9%) (1.5%) 527.1 526.3 245.0 140.0 -82.7 92.5 10% measurement error (3.2%) (0.9%) (0.6%) (8.9%) (1.8%) (3.1%) 510.9 522.01 246.5 127.1 -84.3 95.3 20% measurement error (6.2%) (1.7%) (1.3%) (17%) (3.8%) (6.2%)

5. Conclusions An inverse method for identification of the elastic constants of orthotropic and anisotropic 2D materials was presented. The proposed method is based on the BEM and static measurements. This method uses measured data from more than one experiment. In the numerical examples, it was observed that using two or three experiments instead of one experiment results in solutions that are more accurate. Since the number of measured data obtained from several experiments is considerably larger than the number of unknowns, the inverse analysis can be carried out simply even without any regularization performed.

References [1] D. Lecompte, A. Smits, H. Sol, J. Vantomme, D. Van Hemelrijck International Journal of Solids and Structures, 44 (5), 1643-1656 (2007). [2] J. Cunha, J. Piranda Experimental Mechanics, 40, 211–218 (2000). [3] R. Rikards, A. Chate, G. Gailis International Journal of Solids and Structures, 38, 5097–5115 (2001). [4] T. Ohkami, Y. Ichikawa, T. Kawamoto International Journal for Numerical and Analytical Methods in Geomechanics, 15, 609-625 (1991). [5] L.X. Huang, X.S. Sun, Y.H. Liu, Z.Z. Cen Engineering Analysis with Boundary Elements, 28, 109–121 (2004). [6] L. Comino, R. Gallego Inverse Problems in Science and Engineering, 13 (6), 635-654 (2005). [7] S.G. Lekhnitskii Anisotropic Plates, Gordon & Breach Science Publisher, New York (1968). [8] T.A. Cruse Boundary Element Analysis in Computational Fracture Mechanics. Kluwer Academic Publisher, Dordrecht, The Netherlands (1988). [9] C.L. Tan, Y.C. Shiah, C.W. Lin CMES: Computer Modelling in Engineering and Sciences, 41 (3), 195–214 (2009).


47

Analysis of phase-change heat conduction problems by an improved CTMbased RPIM A. Khosravifard1, M. R. Hematiyan2 1

Department of Mechanical Engineering, Shiraz University, Shiraz, Iran, [email protected] 2 Department of Mechanical Engineering, Shiraz University, Shiraz, Iran, [email protected] Keywords: Phase-change, Cartesian transformation method, meshless radial point interpolation method, solidification Abstract. A truly meshless method for the analysis of phase-change heat conduction problems is presented. The method is based on the global Galerkin weak formulation, and the RBFs are used for the construction of the shape functions. A modified version of the Cartesian transformation method (CTM) is developed for the efficient and accurate evaluation of the domain integrals. In the presented technique, a denser arrangement of integration points is selected only near the solidification front. As a result, without selecting a large number of integration points, the discontinuity in the material properties across the solidification front can be handled accurately. The effectiveness of the proposed method for prediction of the solidification front position and temperature distribution in the computational domain, are assessed through numerical examples. 1. Introduction Transient heat conduction involving solidification or melting is usually known as a phase change or moving boundary problem. Phase change problems are important to many engineering applications such as materials processing, purification of metals, growth of pure crystals from melts and solutions, solidification of castings and ingots, welding, electroslag melting, zone melting, thermal energy storage, ice making, aerodynamic ablation, and numerous other applications. Because of the complexity and nonlinearity of the governing equation of solidification, numerical methods are most often used for the analysis of such problems. Classically, the numerical analysis of the phase change heat transfer problems were performed using the finite difference and finite element methods. Voller has done an excellent survey on the fixed grid techniques available for the analysis of solidification problems by the FDM and FEM [1]. Mackerle presented a bibliographic list of the FEM and BEM techniques for the analysis of phase change heat transfer problems [2]. Furthermore, the mesh-free methods have shown great potential in the field of moving and free boundary problems. Being independent of a pre-defined meshing of elements, mesh-free methods are best suited for the analysis of moving boundary problems. Vertnik and Sarler developed a new local radial basis function collocation method for the analysis of solid-liquid phase change systems [3]. Zhang et al. used the finite point method (FPM) for modeling metal solidification processes in continuous casting [4]. Zhang et al. used the FPM along with the meshless local Petrov-Galerkin method for numerical simulation of solidification process and evaluation of thermal stresses of continuous casting billet in mold [5]. Zhihua et al. utilized the element free Galerkin (EFG) method for analysis of heat transfer problems with phase change [6]. Smoothed particle hydrodynamics (SPH) method has also been used for the simulation of solidification process by Fang et al. [7]. Yang and He presented a new smoothing method for modeling the effective heat capacity in the EFG method. They used the proposed method for the analysis of solidification problems [8]. In the present work, the meshless radial point interpolation method (RPIM) along with a modified version of the Cartesian transformation method (CTM) is utilized for the numerical analysis of solidification problems. The phase change problem is accompanied by the liberation or absorption of energy. A moving interface exists between the two phases of the material, with an abrupt difference in thermo-physical properties of the matter on the sides of the interface. Therefore, the integrals in the formulation of the computational method should be carried out with an especial care. If a large number of integration points are used, the accuracy of the

48


computational method is guaranteed, however, the overall efficiency of the method will deteriorate. In this paper, a modified version of the CTM is introduced, which automatically selects a denser arrangement of integration points near the moving interface. In this way, without a considerable increase in the number of integration points, accurate results are obtained. Some numerical examples are provided to demonstrate the efficiency and accuracy of the proposed technique. 2. Formulation of the solidification problem The solidification problem involves considering a domain which is originally occupied by a liquid, having a temperature T0(x) equal to or above the melting point Tm. As the liquid is cooled, it starts to solidify and an interface between the liquid and solid is formed (see Fig. 1). The solid-liquid interface at any instance of time is represented by *I , and the solid and liquid phases are denoted by : s and : l , respectively. If the solidifying material is not pure, there would not be a sharp interface between the solid and liquid phases. Instead, there exists a so-called mushy zone ( : m ) between the two phases. In such cases, the interface between the solid phase and the mushy zone is named the solidus line ( *s ). Likewise, the interface between the liquid phase and the mushy zone is termed the liquidus line ( *l ).

Figure 1: The geometry and terminology of the solidification problem. The energy balance equation can be applied to the solidification problem by two distinct approaches. In the first approach, the energy balance is written for each phase separately, and some further compatibility and balance equations are considered on the moving interface. When this approach is implemented in numerical methods, a so-called moving grid should be utilized. The grid should be updated to match the solid and liquid phases, as the solidification process progresses. In the second approach, the energy balance equation is written for the whole domain, and the latent heat effects are implicitly accounted for. The numerical methods that utilize the latter approach are usually referred to as the fixed grid techniques. In fixed grid techniques, the energy balance equation is written in terms of the enthalpy function, as follows:

>k (x, T )T (x, t )@

wH , wt

(1)

where k is the thermal conductivity, and H is the enthalpy function or the total heat content, which for the isothermal phase change is defined as:

H (T )

H (T )

³

T

³

Tm

Tr

Tr

Ucs (T ) dT T

Ucs (T ) dT UL ³ Ucl (T ) dT Tm

T Tm ;

(2-a)

T t Tm .

(2-b)

For phase change occurring over a finite temperature range, the enthalpy function is written as:

H (T )

³

T

Tr

Ucs (T ) dT

T Ts ;

(3-a)


H (T ) H (T )

³

Ts

³

Ts

Tr

Tr

T

ª dL

º

Uc f (T )» dT Ucs (T ) dT ³ « U T ¬ dT ¼

Ts d T Tl ;

49

(3-b)

s

Tl

T

Ts

Tl

Ucs (T ) dT UL ³ Uc f (T )dT ³ Ucl (T )dT

T t Tl .

(3-c)

In the preceding equations, cs, cl, and cf are the specific heats corresponding to the solid phase, liquid phase, and the mushy zone, respectively. ȡ is the material density, and L is the latent heat of fusion. Ts and Tl are the solidus and liquidus temperatures, respectively. One of the earliest methods in the context of the fixed grid techniques is the effective heat capacity method, in which the governing equation of the solidification process is written as a nonlinear heat conduction problem. To do so, the time derivative of the enthalpy function is written as follows:

wH wt

wH wT . wT wt

(4)

Consequently, the energy balance equation, Eq. (1), can be recast as:

wT >k (x, T )T (x, t )@ , (5) wt where Ceff wH wT is termed the effective heat capacity, and for the alloy materials is obtained by direct Ceff

differentiation of Eq. (3):

Ceff

U cs UL °° ® Uc f T lTs ° °¯ Ucl

T Ts Ts T Tl .

(6)

T ! Tl

The variation of the enthalpy and the effective heat capacity are shown schematically in Fig. 2.

Figure 2: Typical variation of the enthalpy function and the effective heat capacity with temperature. From the figure, it can be inferred that for narrow temperature range of the phase change, the effective heat capacity varies sharply. In the limit, for isothermal phase change the effective heat capacity will become infinite. The step-like behavior of the effective heat capacity in the phase-change interval may cause some oscillations in the numerical solution. The remedy is to smooth the variation of C eff with respect to the temperature. In this paper, the sigmoid functions are used to smooth the effective heat capacity [8]. 3. A Truly meshless formulation of the solidification problem One major benefit of expressing the governing equation of the solidification process in the form given by Eq. (5) is the exact resemblance of the equation with that of the heat conduction. In this way, the discretized form of the

50


heat conduction equation can be readily used for the solidification process. The discretized form of the phasechange heat conduction equation can be written as follows [9]: K (T )T F(T , t ) , M (T )T (7) where:

M ij

³

K ij

³

Fi

:

Ceff (x, T )IiI j d: ,

ª wI wI j wIi wI j º k (x, T ) « i d: , wy wy »¼ ¬ wx wx g (x, T , t )Ii d: ³ q Ii d* .

:

³

:

*Nat

(8) (9) (10)

In the preceding equations, Ii is the RPIM shape function of the node i, the heat source function is represented by g, and q is the prescribed heat flux on the boundary of the problem domain. Although the above-mentioned equations can be used for the analysis of the solidification problems, however, because of abrupt change of the thermo-physical properties of the matter across the solidification front, care should be taken when dealing with numerical integrations appearing in the weak form. In this paper, the CTM is used for meshless evaluation of the domain integrals. Khosravifard et al. have shown that the CTM is a versatile tool for evaluation of domain integrals in transient nonlinear problems [9]. 3.1. A conforming CTM for evaluation of domain integrals in phase-change problems Fig. 2 clearly shows how the effective heat capacity varies with temperature. The step jumps in this function can be captured, only if an adequately large number of integrations points are used. Nevertheless, apart from the regions near the solidification front, the variations of the thermo-physical properties are usually smooth. As a result, fewer number of integration points is needed in such regions. To make a compromise between these two parts of the domain, a conforming CTM is proposed, which automatically selects a denser distribution of integration points near the solidification front. In this way, without considerably increasing the number of integration points, the sharp variations of the effective heat capacity can be captured. In the conventional CTM, a domain integral is evaluated by the following expression [9]: N

I

³ f x d: ¦ w :

CTM i

f (x i ) ,

(11)

i 1

where, N is the number of CTM integration points, and wiCTM and xi are the CTM integration weight and point, respectively. In the conventional CTM, the arrangement of the integration points is selected based on the local as well as global geometry of the integration domain. No particular care is taken to the specific function being integrated. Herein, a methodology is introduced for selection of the CTM integration points, based not only on the local geometry of the integration domain, but also, on the local behavior of the integrand. To describe the methodology of the proposed conforming CTM, it is assumed that the local behavior of the integrand is known. This is the case in the analysis of the solidification problems, in which the effective heat capacity is known to have sharp variations near the solidification front. Fig. 3 shows a typical integration domain, along with the location of the solidification front, i.e. the solidus and liquidus lines. At first, the conventional CTM is used and the initial CTM integration points and weights for the domain are evaluated. Later, as a solidification front forms, a denser arrangement of the integration points is used near the solidification front. The criterion that is used to check if an integration interval should be refined or not, is the temperature of the nodes near that interval. If an integration interval is near a node for which the temperature is in the range of (1 G1 )Ts d T d (1 G 2 )Tl , the interval is refined. In the previous expression, į1 and į2 are relatively small numbers, say 0.01. The flowchart of Fig. 4 explains the procedure used for selection of the integration points of the conforming CTM.


Figure 3: Typical integration points of the conforming CTM.

Figure 4: A Flowchart for the algorithm used for selection of conforming CTM integration points.

51

52


4. Numerical example, results, and discussion In this section, a solidification problem, associated with the continuous casting process, is analyzed by the proposed meshless methods. For evaluation of the domain integrals, both the conventional and conforming CTM are used, and the results are compared. In addition, convergence of the proposed method with respect to the number of nodes of the meshless method is investigated. To assess the accuracy of the results, the problem is also analyzed by ANSYS using a fine mesh. The problem geometry and boundary conditions are depicted in Fig. 5. It should be mentioned that because of the symmetry of the continuous casting model, only a quarter of the problem domain is modeled and analyzed. The material is initially at T 1845 K , and in the time duration of 300 seconds, heat fluxes with intensities of 4×105 W/m2 are applied to the boundaries. The thermo-physical properties of the material used are reported in Table 1.

Figure 5: The problem geometry and boundary conditions. Table 1: Thermo-physical properties of the casting ks kl cs cl 36.6 W/mK 256.2 W/mK 682 J/kg K 710 J/kg K

ȡ 7400 kg/m3

L 272 KJ/kg

Ts 1775 K

Tl 1790 K

Three different nodal arrangements are used for investigating the convergence characteristic of the proposed method with respect to the number of nodes. These three nodal arrangements are shown in Fig. 6.

Figure 6: The nodal arrangements used in the meshless method. Fig.7 depicts the position of the solidus line at t=300 sec. In this figure, the positions of the solidus line obtained by the conventional and conforming CTM are compared with that of ANSYS. The results of this figure are obtained using the arrangement with 176 nodes. Number of CTM integration points in both cases is almost the same. The number of integration points utilized by the conventional and conforming CTM is 560 and 592, respectively. This means that without the need to increase the number of integration points, accurate results can be obtained by the use of the conforming CTM. Fig. 8 plots the integration points of the conventional and conforming CTM. This figure shows how the density of the integration points varies according to the position of


53

the solidus and liquidus lines. The solid lines in the plot of conforming CTM are the solidus and liquidus line positions.

Figure 7: Solidification front position (1775 K isotherm) at t=300 sec.

Figure 8: The conventional and conforming CTM integration points. To visualize the progress of the solidification process with time, the position of the solidification front, i.e. the 1775 K isotherms, at some instances of time are plotted in Fig. 9.

Figure 9: The progress of the solidus line during the casting process. In Fig. 10, the temperature of a point at (0.075 0.0833) during the casting process is plotted. The results are obtained by the RPIM with 475 nodes. The domain integral is carried out by the conforming CTM. Furthermore, in order to investigate the convergence characteristic of the proposed technique, the problem is analyzed by the nodal arrangements of Fig. 6. The position of the liquidus line obtained by the three nodal configurations is compared in Fig. 11.

54


Figure 10: the temporal variation of temperature at (0.075 0.0833).

Figure 11: Liquidus line position (1790 K isotherm) at t=300 sec.

5. Conclusions A conforming Cartesian transformation method, for the efficient evaluation of the domain integrals in meshless analysis of the phase-change heat conduction problems was introduced. In this method, without increasing the total number of integration points, the sharp variations of the thermo-physical properties can be captured. The accuracy of the proposed technique in prediction of the solidification front position and temperature distribution was assessed through some numerical examples. References [1] V.R. Voller, C.R. Swaminathan and B.G. Thomas International Journal for Numerical Methods in Engineering, 30, 875–898, (1990). [2] J. Mackerle Finite Elements in Design and Analysis, 32, 203–211, (1999). [3] R. Vertnik and B. Sarler International Journal of Numerical Methods for Heat & Fluid Flow, 16, 617–640, (2006). [4] L. Zhang, Y.M. Rong, H.F. Shen and T.Y. Huang Journal of Materials Processing Technology, 192–193, 511–517, (2007). [5] L. Zhang, H.F. Shen, Y.M. Rong and T.Y. Huang Materials Science and Engineering A, 466, 71–78, (2007). [6] G. Zhihua, L. Yuanming, Z. Mingyi, Q. Jilin and Z. Shujua, Cold Regions Science and Technology, 48, 15– 23, (2007). [7] H.S. Fang, K. Bao, J.A. Wei, H. Zhang, E.H. Wu and L. Zheng Numerical Heat Transfer, Part A: Applications, 55, 124–143, (2009). [8] H. Yang and Y. He International Communication in Heat and Mass Transfer, 37, 385–392, (2010). [9] A. Khosravifard, M.R. Hematiyan and L. Marin, Applied Mathematical Modelling, 35, 4157–4174, (2011).


55

BIE reduction for long cylindrical shapes – the Laplace, Poisson and Helmholtz equation Paweá JabáoĔski Electrical Engineering Faculty, CzĊstochowa University of Technology, Al. Armii Krajowej 17, 42-200 CzĊstochowa, Poland, [email protected]

Keywords: BIE, BEM, cylindrical symmetry, Laplace equation, Helmholtz equation.

Abstract. A use of boundary integral equation (BIE) in solving the boundary problems in such 2D domains like circles and rings is considered in the paper. By expanding the excitation and the solution into the Fourier series, the problem is reduced to 1D. The fundamental solutions for each spatial harmonic is determined. Examples of use of the reduced BIE are given. One of the consequences is explaining the reason why the conventional boundary element method (BEM) crashes sometimes for circles of unit radius. As a kind of by-product, some definite integrals were found. Introduction Boundary integral formulation has been proved to be an efficient tool in describing and solving many field problems, and resulted in the boundary element method (BEM) [1]. The original inspiration of this paper was a strange crash during solving a boundary problem for a unit circle by means of BEM [2, 3], although theoretical considerations led to ‘good’ results. The motivation was also the will of reducing the dimension of a 2D problem in a region of axial symmetry. For such problems, BIE becomes an algebraic equation and requires no numerical implementation in the form of BEM. General considerations BIE in a ring. Consider annular domain ȍ of radii a and b > a in which function u satisfies the following equation: ∇ 2 u − ț 2u = − f ,

(1)

where f is a known function and ț is a known constant (including zero). The meaning of function u depends on the physical phenomena under consideration. The corresponding boundary integral equation written for point P has the following form [1]: c( P )u ( P) +

³

S1 ∪ S 2

∂u ∂G u d S = ³ G d S + ³ fG d ȍ , ∂n ∂n S ∪S ȍ 1

(2)

2

where c(P) – the geometric coefficient (1/2 for P on smooth boundary), and G – the fundamental solution for eq. (1). Since boundaries S1 and S2 are circles, eq. (2) can be rewritten in a more expanded form: c( P )u ( P ) −

∂G ∂u ∂u ∂G u ( ȡ) ȡ dș + ³ u ( ȡ) ȡ dș = − ³ G ȡ dș + ³ G ȡ dș + F ( P) , ∂ ȡ ∂ ȡ ∂ ȡ ∂ȡ ȡ=a ȡ =b ȡ=a ȡ =b

³

(3)

where (ȡ, ș) are polar coordinates of point Q on circle S1 or S2, and for brevity F ( P) = ³ fG d ȍ . ȍ

(4)

56


Reduced BIE. Suppose that f can be expanded into Fourier series. Since eq. (1) is linear, u can be expanded into Fourier series, too. Therefore, ∞

f (r , ĳ) =

¦ f k ( r ) e j kĳ ,

u (r , ĳ) =

k = −∞

∞

¦ uk (r )e j kĳ .

(5)

k = −∞

Using this expansions in eq. (3) leads us to the following equation for kth angular harmonic: c(r )uk (r ) − hk (r , a)u k (a) + hk (r , b)uk (b) = − g k (r , a)qk (a ) + g k (r , b)qk (b) + Fk (r ) ,

(6)

where qk(r) = uk(r)/r, and 2ʌ

g k (r , ȡ) = e − j kĳ ȡ ³ Ge j kș d ș ,

(7)

0 2ʌ

· ∂ §1 ∂G j kș e dș = ȡ ¨¨ g k (r , ȡ) ¸¸ , ∂ ȡ ȡ ȡ ∂ © ¹ 0

hk (r , ȡ) = e − j kĳ ȡ ³

(8)

R2

Fk (r ) =

³ f k ( ȡ) g k (r , ȡ) d ȡ .

(9)

R1

In the subsequent paragraphs, the explicit formulas for gk and hk are derived for G corresponding to the Laplace or Helmholtz equations. The second form of eq. (8) allows hk to be found if gk is already known. Special cases. If a = 0, the annulus degenerates to a punctured disk, and eq. (6) becomes c(r )u k (r ) + hk (r , b)u k (b) = g k (r , b)qk (b) + Fk (r ) ,

(10)

and if b ĺ , then by suitable selection of reference point, eq. (6) can be written as c(r )u k (r ) − hk (r , a )u k (a ) = − g k (r , a )qk (a ) + Fk (r ) .

(11)

The Laplace, Poisson, and Helmholtz equations Laplace equation. For Laplace equation ț = 0, f = 0, and

G=

1 1 1 ln = ln[r 2 + ȡ 2 − 2rȡ cos(ș − ĳ)] , 2ʌ P − Q 4ʌ

(12)

where (r, ĳ) and (ȡ, ș) are polar co-ordinates of point P and Q, respectively. Then g0 can be found with formula (4.226.2) in [4], which after some transformation gives g 0 (r , ȡ) = − ȡ ln

ȡ+r+ ȡ−r 2

− ȡ ln ȡ for r ≤ ȡ, = − ȡ ln max( ȡ, r ) = ® ¯− ȡ ln r for r ≥ ȡ.

(13)

Consequently, h0 can be easily found by the second form of the eq. (8):

h0 (r , ȡ) = − ȡ

− 1 1 + sgn( ȡ − r ) ° 1 = ®− 2 ȡ+r+ ȡ−r ° ¯0

for r < ȡ, for r = ȡ, for r > ȡ.

(14)


57

If k 0, then the integrals defining gk and hk are hard to evaluate (e.g. Mathematica 7.0 fails). They cannot be found in tables of integrals like [4], too. The idea to find gk is to use the method of separation of variables, which gives the general solution as follows: ∞

u (r , ĳ) = c0 + d 0 ln r +

¦ (ck r |k| + d k r −|k| )e j kĳ .

(15)

k = −∞ , k ≠ 0

where ck, dk are constants depending on boundary conditions. In a circle of radius R, to keep u finite at r = 0, constant d0 = dk = 0. Using then eq. (15) in eq. (10) with Fk(r) = 0, one obtains for r < R k

k

r + hk (r , R) R = g k (r , R) k R

k −1

,

(16)

Since hk is defined in terms of gk, eq. (16) is a differential equation for gk. Its specific solution can be find quite easily so that

g k (r , R ) =

R §r· ¨ ¸ 2k © R ¹

k

for r < R .

(17)

A similar procedure can be carried out for r > R. The final results for k 0 can be written as follows:

ȡ g k (r , ȡ) = 2k

|k |

§ ȡ+r− ȡ−r · ¸ = ȡ ¨ ¨ ȡ+r + ȡ−r ¸ 2k ¹ ©

|k |

§ min( ȡ, r ) · ȡ ¨¨ max( ȡ, r ) ¸¸ = 2 k © ¹

r |k | ȡ −|k | ® −|k | |k | ¯r ȡ

for r < ȡ, for r > ȡ.

(18)

Note that gk is continuous at r = ȡ, as it should be according to eq. (7). Thus, gk(r, r) = r/2|k|. Functions gk and hk are depicted in Fig. 1. A careful differentiation of the first form of eq. (18) in eq. (8) gives

− 12 r |k | ȡ −|k | ° hk (r , ȡ) = ®0 ° 1 r −| k | ȡ | k | ¯2

for r < ȡ, for r = ȡ,

(19)

for r > ȡ.

a)

b)

Fig. 1. Plots of gk(r, 1) and hk(r, 1) for Laplace equation and k = 0, 1, 2, 3

Helmholtz equation. If ț 0, the fundamental solution can be chosen as G=

(

)

1 1 K 0 (ț P − Q ) = K 0 ț r 2 + ȡ 2 − 2rȡ cos(ș − ĳ) , 2ʌ 2ʌ

where Kn(z) is the modified Bessel function of the second kind of order n. Then by eq. (7)

(20)

58


g k (r , ȡ) =

ȡ 2ʌ

³ K 0 (ț

2ʌ

)

r 2 + ȡ 2 − 2rȡ cos ȥ e j kȥ dȥ ,

0

(21)

This integral can be found in [4] as formula 6.681.13, but only for k = 0 and r = ȡ. Integral hk can be found by Mathematica 7.0, only for k = 0 and r = ȡ, too, but it involves special function Meijer G. The general case requires special methods, similar to those used in the previous section. It can be shown that

g k (r , ȡ) = ȡI k [ 12 ț ( ȡ + r − ȡ − r )]K k [ 12 ț ( ȡ + r + ȡ − r )] ,

(22)

for r < ȡ, țȡI k (țr ) K k′ (țȡ) ° hk (r , ȡ) = ® 12 țr[ I k′ (țr ) K k (țr ) + I k (țr ) K k′ (țr )] for r = ȡ, °țȡI ′ (țȡ ) K (țr ) for r > ȡ, k ¯ k

(23)

where In(z) is the modified Bessel function of the first kind of order n. Since I'n(z)Kn(z) í In(z)K'n(z) = zí1 [5], the case of r = ȡ in eq. (23) can be simplified as follows

hk (r , r ) = 12 + țrI k (țr ) K k′ (țr ) = − 12 + țrI k′ (țr ) K k (țr ) .

(24)

Non-homogeneous equation (f 0). To evaluate Fk(r) according to eq. (9), it is convenient to split the integral as follows: Fk (r ) =

r

R2

R1

r

³ f k ( ȡ) g k (r , ȡ) d ȡ + ³ f k ( ȡ) g k ( r , ȡ) d ȡ .

(25)

Then the integrals can be found using different forms for gk(r, ȡ) in intervals R1 ȡ r and r ȡ R2. Examples and consequences Crash in circle of unit radius. As a benchmark problem, let us find the electrostatic potential V in a long cylinder of radius R the boundary of which has potential U. With fringing neglected, this simple electrostatic problem has a solution V(r) = U. Surprisingly, the standard BEM procedure crashes during solving the system of equations if R = 1. To understand what really happens, let us consider the problem with use of the reduced polar BIE. Eq. (10) with r = b = R yields 1 u ( R) − 12 u ( R) 2

= − R ln Rq( R)

( 12 − 12 )U = − R ln Rq( R)

R ln Rq( R) = 0 .

(26)

Hence, q(R) = 0 in agreement with theory. If R = 1, however, the equation becomes 0q(R) = 0. Nothing strange that numerical calculation of q(R) fails in this case due to a division by zero. Similar results can be obtained for domain r R. Eq. (11) takes the following form: 1 u ( R ) + 12 u ( R) 2

= R ln Rq ( R)

u ( R) = R ln Rq( R) ,

(27)

from which it follows that u(R) can be always found for a given q(R), whereas numerical evaluation of q(R) will fail if R = 1 (division by RlnR). These examples show that R = 1 is a kind of ‘critical’ value. Dirichlet/Neumann problem in a ring. This example concerns a more general case. Eq. (6) written for boundary points r = a and r = b of annulus a r b gives the following system of equations: °[ 12 − hk (a, a)]uk (a ) + hk (a, b)u k (b) + g k (a, a)qk (a) − g k (a, b)qk (b) = Fk (a ), . ® °¯− hk (b, a )uk (a ) + [ 12 + hk (b, b)]u k (b) + g k (b, a )qk (a ) − g k (b, b)qk (b) = Fk (b).

(28)


59

Depending on the boundary conditions (Neumann or Dirichlet) for the ring, two of the values uk(a), uk(b), qk(a), qk(b) for each k are known. The two other values can be usually found from eq. (28). Table 1 shows the main matrix A and its determinant for each of four possible cases. Unknowns 1

qk(a) qk(b)

2

qk(a) uk(b)

3

uk(a) qk(b)

Main matrix A

ț = 0, k = 0

ª g k ( a , a ) − g k ( a, b ) º « g (b, a) − g (b, b) » k ¬ k ¼ hk (a, b) º ª g k ( a, a ) « g (b, a) 1 + h (b, b)» k k ¬ ¼ 2

b ab ln b ln a

detA ț = 0, k 0 Ĳ

2k

−1

4k 2

ab

abI a K b ( I a K b − I b K a )

a

abțI a K b ( I b′ K a − I a K b′ )

2k

íalnb

ª 12 − hk (a, a) − g k (a, b)º « » ¬ − hk (b, a) − g k (b, b) ¼ ª 12 − hk (a, a ) hk (a, b) º « » 1 ¬ − hk (b, a ) 2 + hk (b, b)¼

1+ Ĳ 4k

2k

blnb

−

ț0

1+ Ĳ 4k

b

abțI a K b ( I b K a′ − I a′ K b )

2k

1− Ĳ abț 2 I a K b ( I a′ K b′ − I b′ K a′ ) 4 Table 1. Determinants of main matrices corresponding to the four cases of Neumann/Dirichlet problem in ring a r b (0 < Ĳ = a/b < 1, Ia = Ik(ța), I'a = I'k(ța), Ka = Kk(ța), K'a = K'k(ța), Ib = Ik(țb), I'b = I'k(țb), Kb = Kk(țb), K'b = K'k(țb)) 4

uk(a) uk(b)

0

As shown in Table 1, detA 0 if ț 0 (argț < ʌ/2), or k 0, whereas detA can be zero if ț = 0 and k = 0. In case 4, which corresponds to Neumann problem, detA = 0 explicitly (what is connected with non-uniqueness of solution of the internal Neumann problem). Note that if b = 1, detA = 0 also in cases 1-3. It is worth observing, too, that if argț = ʌ/2, which can be obtained for undamped time-harmonic waves, detA can be zero for specific values of parameters a, b, ț (eigenvalues and modes). This does not mean that there are no solutions; rather numerical computations can fail due to singular matrix. Conductive cylinder in electroconductive field. This example shows the methodology in using the reduced BIE. Consider a long, homogeneous cylinder of radius R and electrical conductivity pȖ0 placed in an open conductive medium of conductivity Ȗ0 in which an externally applied potential, Vs(x, y), exists. This potential can be expanded into Fourier series so that Vs (r , ĳ) =

∞

¦Vsk (r )e j kĳ ,

k = −∞

2ʌ

Vsk (r ) =

1 Vs (r , ĳ)e − j kĳ d ĳ . 2ʌ ³0

(29)

Eq. (10) for the cylinder and eq. (11) for its exterior, with r = R, give the equations as follows:

§ 12 u k ( R) + hk ( R, R)u k ( R) = g k ( R, R)qkint ( R), ¨ ¨ 1 u ( R) − h ( R, R)u ( R) = − g ( R, R)q ext ( R) + V ( R), k k k k sk ©2 k

(30)

where ‘int’ and ‘ext’ refer to ‘internal’ and ‘external’ domain, respectively. The continuity of current across the boundary leads to relationship pqkint(R) = qkext(R). Thus, uk ( R ) =

1 2

qkint ( R) =

k = 0, ° Vs 0 ( R ) Vsk ( R ) =® 2 1 ≠ 0, V ( R ) k p + phk ( R, R ) + 2 − hk ( R, R ) °¯ p +1 sk 1 ext qk ( R ) = p

1 2

0 k = 0, ° + hk ( R, R ) uk ( R ) = ® 2 k V ( R ) k ≠ 0. g k ( R, R ) °¯ p +1 R sk

(31)

(32)

60


Eq. (10) written for r < R, and eq. (11) written for r > R, give finally: Vs 0 ( R) k = 0, ° ukint (r ) = g k (r , R)qkint ( R) − hk (r , R)uk ( R) = ® 2 r k °¯ p +1 ( R ) Vsk ( R) k ≠ 0, ° ukext (r ) = − g k (r , R)qkext ( R) + hk (r , R)uk ( R) + Vsk ( R) = ® °¯[1 −

Vs 0 ( R) p −1 R k ]Vsk ( R) p +1 r

( )

(33) k = 0, k ≠ 0.

(34)

The final expressions for potential inside and outside the cylinder are given by the second of eqs. (5). Formulas for integrals. One of the consequences of the carried out considerations is finding formulas for some integrals that are hard to evaluate in a conventional way. Gathering formulas (13) and (18), by eq. (7) together with eq. (12), we obtain the following formula

ȡ+r+ ȡ−r °4ʌ ln 2 ° 2 2 ³ ln( ȡ + r − 2rȡ cos ȥ ) cos kȥ dȥ = ® 2ʌ § ȡ + r − ȡ − r °− 0 ¨ °¯ k ¨© ȡ + r + ȡ − r

for k = 0,

2ʌ

(35)

|k |

· ¸ ¸ ¹

for k = 1, 2, 3,

Similarly, eq. (22) yields

³ K 0 (ț

2ʌ

)

r 2 + ȡ 2 − 2rȡ cos ȥ cos kȥ d ĳ = 2ʌI k [ 12 ț ( ȡ + r − ȡ − r )]K k [ 12 ț ( ȡ + r + ȡ − r )] .

0

(36)

for k being an integer. Differentiating these expressions with respect to r or ȡ, one can obtain more interesting formulas.

Concluding remarks The polar BIE reduces the dimension of the problem to 1D and makes the boundary integral equation an algebraic one. Its use in solving a boundary problem is equivalent to the method of separation of variables in polar coordinates. In addition, it shows that numerical calculations by means of BEM can fail if the boundary is a circle of unit radius. Some remedial procedures are given in [2, 3]. It seems that in case ț = 0, k = 0 avoiding circles of unit radius, by using other units of length (e.g. 1 cm ĺ 10 mm), is a good idea.

References [1] C.A.Brebbia The boundary element method for engineers, Pentech Press (1978). [2] J.T.Chen, S.R.Lin and K.H.Chen Degenerate scale problem when solving Laplace’s equation by BEM and its treatment, Int. J. Numer. Meth. Engng, 62, 233–261 (2005). [3] W.J.He, H.J.Ding and H.C.Hu Degenerate scales and boundary element analysis of two dimensional potential and elasticity problems, Computers and Structures 60(1), 155–158 (1996). [4] I.S.Gradshteyn and I.M.Ryzhik Table of Integrals, Series, and Products, Elsevier (2007). [5] M.Abramowitz and I.A.Stegun Handbook of mathematical functions with formulas, graphs, and mathematical tables (1972).


61

Fiber shape optimization in linear elasticity Petr P. Prochazka CTU in Prague, Civil Engineering, Thakurova 7, Prague, Czech Republic, e-mail: [email protected] Keywords: Shape optimization, composite structure, minimum Lagrangian, minimum stress and strain

Abstract. It appears that in comparison to classical theories, where the shape of fibers in composite structures do not play any important role, this shape can basically influence the distribution of stresses and displacements of the entire composite at macro-level. A closer sight at the microstructure reveals a dependency of the fiber shape and the mechanical properties of the macro-structure, i.e. also the overall stresses and strains. It is proved that the different optimal shape is attained for different fiber and matrix ratios and for the ration of material properties of the phases. In the study put forward an approach to homogenization of composites and to optimization being formulated in a special, advantageous, way is suggested.

Preliminary considerations and denotations Optimization of composites based on homogenization is studied in this paper. The homogenization suggested by Suquet in [1] is adopted here using boundary element technique and the main idea of optimization follows the methodology of [2]. Similar problem is solved in [3] for heat transfer problem. Let us denote the mechanical quantities to be computed: S, E macroscopic stress and strain tensors being independent of position and valid for the given RVE, microscopic stresses and strains related to point y . ı ( y ), İ ( y ) The average < f >V of function f defined on domain V is the Lebesque measure of the function over V: 1 (1) F= f ( y ) dV meas V

³

V

Hence, the relations between the stresses ı and strains İ at micro-level and the same tensors S and E at macro-level on unit cell ȍ are given as, S =< ı ( y ) > ȍ , E =< İ ( y ) > ȍ

(2)

Before tackling the optimization the overall stiffness is calculated using localization and homogenization procedures for arbitrary shape of the phases. Since these procedures are well known only brief description of them is presented, as the quantities obtained are used in formulating the optimization problem. Note that the formulation fulfils the one-dimensionally oriented fibers and only a cross-section perpendicular to the fiber direction. It reduces generally three-dimensional problem to two-dimensional. Localization using BEM As said above the problem is solved in 2D, i.e. the third coordinate disappears in specification of points and the tensors of the second order are of the type ( 2 × 2) , for example. The unit cell is supposed to be put into a periodic composite structure. In that follows its shape is square. Let us suppose only two phases ȍ = ȍf ∪ ȍ m with distinctive material properties Lf and Lm , i.e. stiffness matrices of fiber ȍ f (the first phase) and that of the second phase ȍm , the matrix. The problem of

62


homogenization consists in finding such an overall stiffness matrix L* , or compliance matrix M * , both belonging to the shape of phases and their given material properties, which are formulated as, ı = L* : İ

or

İ = M * : ı , L* = ( M * ) −1

(3)

There is a wide scale of approaches leading to the solution of problem of localization. The approach selected in this paper consists in splitting the micro-strain İ into the sum of the overall strain E and the fluctuating strain ~ İ:

1 ~ ~≡u ~( y) , ~ ~) , İ= E+~ İ , ~ İ ≡ İ (u~ ) , u İ = (䌛u + 䌛T u 2

(4)

~ = {u~ , u~ } is the vector of fluctuating displacements, related to the fluctuating strain by the where u 1 2 kinematical equations (44). Note that from the kinematical equations it follows that providing that the fluctuating strain is known the fluctuating displacements are unique but the movement of the rigid body, which can be disregarded. The problem of localization in linear elasticity is formulated as: for given E find the fluctuation displacement tensor u~ obeying the Navier equations: div ( L( y ) : ~ İ ( y )) = −div ( L( y ) : E ) = − pC . į īC ,

and the periodic boundary conditions,

(5)

where

p C = [ Lf - Lm ] : E . nf The stiffness matrix L is a step tensor-function of position y in 2D, i.e. if split it is uniform on both the fiber and matrix, it means that L( y ) = Lf for y ∈ ȍf and L( y ) = Lm for y ∈ ȍm . Vector pC is created by interfacial tractions and įīC is the distribution of Dirac’s function along the interface between the phases:

pC = [ Lf - Lm ] : E . nf . The formula is in compliance with [1]. For this reason the right hand side of the latter equation must be taken in the sense of distributions. In the next text axisymmetry of the problem considered can be assumed without loss of generality. Moreover, star-shaped fibers are supposed, i.e. there is a point (origin of the coordinate system) the rays from which cross the interfacial segment only and only once. For all these reasons only the first quadrant is taken into account in what follows. In order to complete the localization it remains to formulate the boundary conditions being valid for a periodic unit cell which itself is symmetric, see [1]. Let us successively introduce the unit impulses of the overall strain tensor, i.e.

ª1 0 º E11 = « », ¬0 0 ¼

ª0 0 º E 22 = « », ¬0 1 ¼

E12 =

1 ª0 1º , 2 «¬1 0»¼

(6)

where the first tensor describes a unit normal extension in direction y1 , the second in y 2 and the third the shear strain. Since the linear elastic problem is solved (neither debond of phases nor any plasticity are taken into account, for example), superposition of the distinctive overall unit strains, eq. (22), can be carried out ~: and tensors ȕ of fourth order are obtained as an impact of the solution of eq. (5) for u ~ ( y )) = ȕ f ( y ) : E for y ∈ ȍ f , İ f (u

~ ( y )) = ȕ m ( y ) : E for y ∈ ȍ m , İ m (u

(7)


63

As eq. (41) holds valid the process leads to fourth-order "concentration factor tensor" H defined as ( I is the fourth-order unit tensor), İ f (u( y )) = ( I + ȕ f ( y )) : E = H f ( y ) : E for y ∈ ȍ f

(8)

İ m (u( y )) = ( I + ȕ m ( y )) : E = H m ( y ) : E for y ∈ ȍ m

Since the shape optimization closely related to moving boundary problem the boundary element formulation seems to be extremely advantageous. Coming back to eq. (5) and applying the well-known procedure in terms of linear boundary elements to both phases leads to a system of equations of the type: ~ =B :p , Af : u f f f

~ =B : p Am : u m m m

(9)

where A,B are square matrices, u,p are vectors their components are values of displacements and tractions, respectively, on the boundary of constituents. The subscripts denote fiber (f) and matrix (m), so that the equations in eq. (9) hold valid separately. Splitting the boundaries into that falling in with īC and the remaining parts yields: f ª K 11 « f ¬« K 21

f º ~ out ½ out ½ K 12 °u f ° ° p f ° , = f » ® ~ in ¾ ® in ¾ K 22 ¼» °¯ uf °¿ °¯ pf °¿

m ª K 11 « m ¬« K 21

m º ~ in ½ in ½ K 12 ° um ° ° p m ° , = m » ® ~ out ¾ ® out ¾ K 22 ¼» °¯u m °¿ °¯ p m °¿

K = B −1 A

(10)

where the quantities with in are assigned to the nodal points at īC and that with the superscript out are ~ in = u ~ in and p in + p in = p , one connected with the values outside of īC . Since on īC it holds u C f m f m eventually gets: f ª K 11 « f « K 21 « 0 ¬

f K 12 f K 22

m + K 11 m K 21

~ out ½ p out ½ 0 º u f f ° m » ° ~ in ° ° K 12 » ® uf ¾ = ® p C ¾ ~ out ° ° p out ° K 2m » °u ¼¯ m ¿ ¯ m ¿

(11)

where the matrix of the system is banded but generally not symmetric. Using periodic boundary conditions ~ and ʌ follows from the previous equation for successive unit impulses E . u

Homogenization Knowing the concentration factor tensors H f and H m one is capable of formulating equations solving the problem of getting the overall material stiffness tensor L* according to eq. (3). The overall strain E is assumed to be given independently of the shape of the unit cell and of the shape of the fiber. The loading of the unit cell will be given by unit impulses of the three choices of components of E ≡ Eij , i.e. we successively select E i0 j0 = E j0i0 = 1; E ij = 0 for either i 0 ≠ i or j 0 ≠ j , cf. eq. (6). From eqs. (3) and (11) one gets a sequence of relations: ı ( y ) = L( y ) : İ ( y ) = (ț f ( y ) Lf : A f ( y ) + ț m ( y ) Lm : A m ( y ) ) : E

(12)

where țf , ț m are the characteristic functions of domains of fiber and matrix, respectively, i.e. ț f ( y ) = 1 for y ∈ ȍ f and is 0 otherwise; similarly for ț m . Using eq. (3) and Hooke’s law for the overall properties it immediately follows:

64


Ȉ=

³ ı ( y) dȍ = ³

ȍ

³

L( y ) : İ ( y ) dȍ = [ Lf :

ȍ

ȍ

A f ( y ) dȍ f + Lm :

f

³ ȍ

A m ( y ) dȍ m ] : E = L* : E

(13)

m

and then the overall properties L* are given as

³

L* = Lf :

ȍ

H f ( y ) dȍ f + Lm :

f

³

ȍ

H m ( y ) dȍ m = meas ȍ f Lf : < H f > ȍ f +

m

+ meas ȍ m Lm : < H m >

ȍm f

= meas ȍ (c f Lf : < H f >

= cf L : < H > f

+ cm L : < H m

ȍf

ȍf m

+ c m Lm : < H m > >

ȍm

) = (14)

ȍm

As seen below the concentration tensors are dependent on the shape of fiber and serve as an inevitable tool for obtaining the overall properties of the composite.

Optimization In this section adjustment of the above defined problem is put forward, taking into account special nature of composite structures. In the previous sections formulated relations are considered and in the sense of this reformulation of the functional can be carried out. Under the above circumstances Hill's energy condition holds valid, and the constrained internal energy is established as: Ȇ=

³

³

³

³

1 1 1 ı : İ dȍ − Ȝ( dȍf − C ) = < ı : İ > − Ȝ( dȍf − C ) = Ȉ : E − Ȝ ( dȍf − C ) 2 2 2 ȍ

ȍf

ȍf

(15)

ȍf

and Ȉ = [c f ( Lf − Lm ) : < A f > ȍf + Lm ]E , and only the concentration factors are dependant of the values

of ps , s = 1,2,…,n. Since the problem remains linear elastic, superposition of loadings due to successively given by components of the overall strain tensor can be used. Without lack of generality, let us consider a symmetric unit cell. The overall strain Eij is assumed to be given independently of the shape of the unit cell and of the shape of the fiber. The loading of this unit cell will be given by unit impulses of E ij , i.e. we successively select Ei0 j0 = E j0i0 = 1; Eij for either i0 ≠ i or j0 ≠ j . It remains to specify the domain ȍ f by means of its corresponding boundary. Suppose the approximate polygonal shape of the fiber under study. Connect the origin with each node at this polygonal boundary; the distance of the i-th node from the origin of the coordinate system is denoted as pi. In this way we obtain n triangles Ts, s = 1,...,n. It obviously holds:

³

ȍf

n

dȍ = meas ȍ f =

¦

meas Ts .

(16)

s =1

Euler's equations The stationary requirement leads to differentiation of the functional by the shape (design) parameters p s ∂Aklf Įȕ ( p) ∂AklmĮȕ ( p) ∂Ȇ (u, ȍ ) 1 f ∂ = [ Lijkl < > f + Lm > m ]Eij E Įȕ + Ȝ ijkl < 2 ∂p s ∂p s ∂p s ∂p s

³ dȍ = 0

ȍf

(17)


65

which can be rewritten as: ∂Aklf Įȕ ( p) ∂AklmĮȕ ( p) 1 f > f + Lm > m ]E ij E Įȕ [ Lijkl < ijkl < ∂p s ∂p s 2 for each s =1,…,n E s + Ȝ = 0, Ȝ = − ∂ dȍ ∂p s f

³

(18)

ȍ

The quantities Es corresponds to the strain energy density at the point of the interfacial boundary, in our case at the nodal point ȟ s . The equation (18) requires Es to have the same value for any s. In other words, if the strain energy density were the same at any point on the "moving" part of the boundary, the optimal shape of the trial body would be reached. For this reason the body of the structure should increase its area (in 3D its volume) at the nodal point ȟ s of the boundary, if Es is larger than the true value of − Ȝ , while it should decrease its value when Es is smaller than the correct − Ȝ . As, most probably, we will not know the real value − Ȝ in advance, we estimate it from the average of the current values at the nodal points. Differentiation by Ȝ completes the system of Euler's equations: n

¦

meas Ts = C

(19)

s =1

Example First, a square symmetric unit cell is considered with fiber volume ratio equal to 0.75. Since we compare energy densities at nodal points of the interfacial boundary, the relative energy density may be regarded as the comparative quantity influencing the movement of the boundary īC . As said in the previous section, the higher value of this energy, the larger movement of the nodal point of īC should aim at the optimum. In both cases of volume ratios we used the following material properties of phases: Young's modulus of fiber Ef= 210 MPa, Poisson's ratio Ȟ f = 0.16; on the matrix Em= 17 MPa, and Ȟ m = 0.3. We started with the radius r = 0.696 of a circle and "unit moves" of the parameters p s where given by the change of radius by 2.2 %. In Fig. 1 the optimal shape is depicted and in Fig. 2 the same distribution is considered for weak fibers and stiff matrix, i.e. the material properties are switched.

Fig. 1 Optimal shape for stiff fiber

Fig. 2 Optimal shape for stiff matrix

Next the same material properties are used with stiff matrix. The variation is applied to the volume fractions as: first meas ȍf = 0.28 and second meas ȍf =0.21. In both latter cases an additional constraint

66


is implied as: the length of rays cannot be less then 0.2 . The optimal shapes seem to be more interesting then that in the previous two cases.

Fig. 3 Stiff fiber and fiber ratio = 0.28

Fig. 4 Stiff fiber and fiber ratio = 0.22

Conclusions In this paper constrained variational principle has been applied to the solution of optimal fiber shape design in a unit cell of periodic composite structure. When searching for optimal shape of fibers in composite structures, many formulations have been used in the past. They often started with minimum energy defined on admissible spaces of fiber domains. In this new formulation on of a typical restriction of the fiber domain is involved directly in the functional, which is established on definition of the strain energy function. Any composite structure possesses a pleasant property: the material in such a structure is step tensor function. As such it enables one to fully utilize the Green theorem and simplify the quantities defined over domains of phases to their values on hyperplane. A natural requirement is the restriction to the constant volume or area in many methods of solution of optimal shape design of composites, say, when solving a periodic distribution of fibers. The requirement of the constant volume or area seams to be restrictive, particularly when expecting application of the variational principles to larger range of problems. Actually, it is not so. The constant C may change, too. Thus the formulation has to be extended in such a way that C is involved into the problem as a new variable and may be variated (differentiated) in some reasonable way. It is necessary to point out that the extreme of the functional Ȇ is found as neither the minimum nor the maximum, but the functional should be minimum with respect to the displacements and maximum with respect to the Lagrangian multiplier Ȝ . It is no surprise than other constraints have to be sometimes applied for realistic reasons, such as requirement on limited rays defining the fiber shape to avoid possible penetration of the rays through the unit cell boundary or to avoid possible negative values of the rays, which can occur due to the integral definition of the volume (area) of the fiber. In [2] an example of nonrealistic solution is discussed for various types of beams (plates). It is also proved there that the optimal solution leads to the minimum stress in the macroscopic composite and an also minimum displacement is attained.

Acknowledgment: Financial support of the Grant agency of the Czech Republic, grant number P105/00/0266 is gratefully acknowledged. References [1] P.M. Suquet Homogenization techniques for composite media: Lecture Notes in Physics 272 (eds. E. SanchesPalencia and A. Zaoi) Part IV, Sprinter Verlag Berlin, 194-278 (1985) [2] P.P. Prochazka, V. Dolezel and T.S. Lok Optimal shape design for minimum Lagrangian: Eng. Anal. with Bound. Elem. 33, 447–455 (2009) [3] P.P. Prochazka, M. Valek Optimal Shape of Fibers in Composites Exposed to Combustion: BETEQ 2012. Prague


Optimal Shape of Fibers in Composites Exposed to Combustion Petr P Prochazka, Martin J Valek CTU in Prague, Civil Engineering, Thakurova 7, Prague, Czech Republic, e-mail: [email protected] Keywords: Shape optimization, composites, heat transfer, minimum energy

Abstract. In classical theories of homogenization and localization of composites the effect of shape of inclusions is not taken into account. Applying more precise theoretical and numerical tools it appears that the classical theories desire corrections in this direction. Today there are many types of materials which enable producers to fabricate the fiber cross-sections and model them in various shapes, so that it is meaningful to carry out the optimization. In the paper optimal shape of fibers is sought to admit as small/large amount of heat energy to pass through a composite as the constituents allow. Introduction The optimal shape of fiber is characterized in a symmetric unit cell, which is positioned in a composite structure. Conductivity or harmonic problem is to solve so that the linear conductivity equation mediate a representative formulation. The coefficients of conductivity are different on phases and are given in advance. Hence, the design parameters are connected with the shape of fiber, which is assumed to be star-shaped. Since a contact problem is basically discussed, boundary element method appears to be the most appropriate in this case. This task appears not to be solvable uniquely and even can exceed the realistic situation. This is why additional constraints, or side conditions, have to be put forward. First, a reasonable condition is the restriction of volume (in 2D area) of the fiber. Still, this cannot be sufficient to meet a realistic situation, since the side conditions are mostly formulated in integral form, i.e. positive and negative signs can lead to nonrealistic geometry. For that, restrictions on the shape characteristics should be added, such as the diameters or tangential slope of certain directions of the interfacial fiber-matrix boundary. The mathematical formulation and subsequent numerical treatment provide a reasonable, fully usable in practice, layout. As mentioned in many publications, e.g. [1], plenty of approaches are available on how to solve this problem. Hereinafter similar procedure as that used in [2] is applied for formulating and calculating the proper shape of fibers. Classical approach in localization and homogenization of elastic composites belongs to Suquet, [3] and that of steady state heat transfer can be found also in [4], in which periodic composites are studied. More or less this procedure is applied in this paper to unit cell concept. Optimization of the fiber shape in a composite structure due to heat load is discussed in [5], where the problem of a variance between given overall properties and calculated from the given material properties of phases is as small as possible. In this text the application of the steady-state problem can be extended heat and mass density transfer, filtration of the Newtonian liquid, etc. First, homogenization approach will be suggested and a variational formulation will characterize the optimization problem, so that the formulation in terms of boundary elements can easily be derived. Second, cost functional will be created and provide designers with a range of possible conditions according to their request.

Basic considerations and equations In this paper 2D problem of steady state heat transfer in composite structure is to be treated. Two-phase composite is taken into account with one phase denoted as fiber and the other as matrix, both in a periodic unit cell, which is cut out of a representative volume element V describing the neighborhood of a typical point of macrostructure.

67

68


Let the body representing the composite is denoted as ȍ ⊂ V ∈ R 2 and its boundary ∂ȍ is supposed to be Lipschitz continuous such as in case the shape of the unit cell ȍ is square (0,1) × (0,1) , for example. V is the macroscopic body. Isotropic phases ȍ f ⊂ ȍ and ȍ m ⊂ ȍ represent the fiber and the matrix, respectively. Fig. 1 offers a layout used in what follows. Note that more general shapes are mentioned in [5], where a special treatment on how to simplify complicated unit cells is also discussed based on body transformations. The transformations create a group of base bodies. In RVE coordinate system 0 x1 x 2 is introduced while the unit cells are equipped by local coordinate system 0 y1 y 2 , as 2D problem is discussed.

Fig. 1: Unit cell Now the periodic conditions will be précised. The unit cells in the RVE are homothetic, i.e. there is a constant, say " , and to any point P ∈ V is always P′ , which is identified by the law: P′( y ) = P( y + ") = " PP′ and the function to be considered as periodic has exactly the same value at P and P′ , while the gradient posses the same norm but the direction is opposite. Hence the homothetic property is applicable in both y1 and y2 directions. Denote L the macroscopic length of RVE. The periodicity can also be defined in the following way: take a small number İ = " / L , and consequently, ȍ = İV .

The conservation law is assumed in the standard divergence form applied to temperature u İ (x ) for arbitrary İ as, ∂ ∂ İ (1) (c İ u )=0 ∂xi ∂xi where c İ ≡ c İ ( x) = c( x / İ ) = c( y ) is dependent on the position in ȍ , consisting of two subdomains, fiber and matrix, which are equipped by different conductivity values cf (fiber) and cm (matrix), where cf and cm are constants. This means that the coefficient of conductivity c İ (x ) is defined as: c( y ) = cf

for

y ∈ ȍf

and

c( y ) = c m

(2)

otherwise

The partial equations are written as ∇q İ = 0 ,

q İ = c İ ∇u İ

(3)

where u İ is still the, ∇ is the nabla operator, and q İ is the flux vector, gradient of u İ . For statistically isotropic material with the periodic boundary conditions an analog of the well known Hill condition holds valid as: < q İ ∇u İ >=

1 meas ȍ

³

ȍ

q İ ∇u İ dȍ ( x) =< q İ >< ∇u İ >=

1 meas ȍ

³

ȍ

q İ dȍ ( x) ×

1 meas ȍ

where meas ȍ is the volume in 3D or area in 2D, mostly considered equal to unit.

³ ∇u

ȍ

İ

dȍ ( x)

(4)


69

Homogenization In order to get relations between local and overall properties of the composite an asymptotic expansion of u İ and q İ are considered for small enough İ : u İ ( x ) = u0 ( x, y ) + İ u1 ( x,y ) + ... q İ ( x ) = q0 ( x, y ) + İ q1 ( x, y ) + ...

y = x/İ

,

(5)

where u i and qi are ȍ − periodic in y . In what follows coordinates x and y are first taken as independent and afterwards y is substituted by x / İ . Differentiation of the first order is applied to eq. (1), so that ∂ ∂ 1 ∂ is now read as . Substituting eq. (5) to eq. (1) yields at O( İ −2 ) : operator + ∂xi ∂x i İ ∂y i ∂ ∂ (c u 0 ) = 0 u 0 ( x, y ) = u 0 ( x ) ∂y i ∂y i

(6)

and the implication in Eq. (7) is valid but a constant, which can be disregarded. From Eq. (1) at 0( İ −1 ) and with respect to Eq. (7) yields:

§ ∂u1 ∂u 0 ∂ ª + «c( y )¨¨ ∂y i ¬« © ∂y i ∂xi

·º ∂u1 º ∂ ª ∂ ¸¸» = 0 «c ( y ) »=− ∂y i ¬ ∂y i ¼ ∂y i ¹¼»

ª ∂u 0 º «c ( y ) » ∂xi ¼ ¬

(7)

in the sense of distributions. Starting from the definition of distributions the latter equation can be rewritten as: ∂ ∂y i

ª ∂u1 º «c ( y ) » = − qC į īC ∂y i ¼ ¬

and the periodic boundary conditions,

(8)

where q C are interfacial flows and į īC is the distribution of Dirac’s function along the interface between the phases and q C ( y ) = [c f − c m ]

as

∂u 0 f n ( y ) = [c f − c m ] n f ( y ) ∂xi

(9)

∂u0 are considered as a unit impulses. ∂xi

Boundary element formulation Since the shape optimization closely related with moving boundary problem the boundary element formulation seems to be extremely advantageous. Starting equation will be eq. (8) with the right hand side equal to Dirac’s function, which being multiplied by a function u1* , integrated successively over ȍf and ȍ m applying linear approximations over boundary elements and splitting the boundaries into that lying on īC and the remaining parts finally yields:

70

Eds: P Prochazka and M H Aliabadi m º in ½ in ½ K12 ° u m ° ° qm ° = , K ij = Bik−1 Akj where Aij u j = Bij q j m » ® out ¾ ® out ¾ K 22 ¼» °¯u m °¿ °¯q m °¿

f º out ½ out ½ m K12 °u f ° °qf ° ª K11 = , « m f » ® in ¾ ® in ¾ K 22 ¼» °¯ u f °¿ °¯ qf °¿ ¬« K 21

f ª K11 « f ¬« K 21

(10)

where u and q are vectors of temperature and fluxes, respectively, their components are values at nodal points of the corresponding boundaries, A and B are square, generally not symmetric matrices of approximations, and quantities with superscript in are assigned to the nodal points at īC and that with the superscript

q fin

in + qm

out

in are connected with the values outside of īC . Since on īC it holds u fin = u m and

= qC , one eventually gets: f ª K11 « f « K 21 « 0 ¬

0 º u fout ½ qfout ½ ° m » ° in ° ° K12 » ® u f ¾ = ® qC ¾ out ° ° out ° q K 2m » °u m ¼¯ ¿ ¯ m ¿

f K12 m + K11 m K 21

f K 22

(11)

where the matrix of the system is banded but generally not symmetric. Using periodic boundary conditions u and q follows from the previous equation for unit impulse ∂u 0 / ∂x . Moreover, q ( y ) = c(

ci* =

where

∂u1

³ c( y)(1 + ∂y

ȍ

∂u1 ∂u 0 ) < q >= c * < ∇u > , + ∂yi ∂xi

) dȍ ( y ) = c f

i

³

(1 +

ȍf

∂u1 ) dȍ ( y ) + c m ∂y i

³

(1 +

ȍm

∂u1 ) dȍ ( y ), i = 1,2 ∂y i

(12)

Now the main advantage of the boundary element formulation appears: applying the Green theorem leads us to interface integrals as:

ci* = c f meas ȍf + c m meas ȍ m + c f

³ u1ni dȍ( y) + cm ȍ³ u1ni f

m

∂ȍf

= c f meas ȍf + c m meas ȍ m + c f

³ u1ni dȍ( y) + cm ȍ ³− ī u1ni ∂ȍ − ī f

m

f C

+ cf

³ u1ni dȍ( y) + cm ī³ u1ni ī f

m

C

dȍ ( y ) =

m

dȍ ( y ) +

(13)

m C

dȍ ( y )

C

so that the unpleasant volume integrals in (13) are avoided. This is again a very important advantage of the boundary element formulation. Note that c1* = c 2* = c * because of the symmetry considered.

Optimization Similarly to the optimization of elastic beams, [2], the energy functional is formulated using Lagrangian multiplier Ȝ constraining the given area of the fiber. Hence, the problem can be established as: =

1 2

³

ȍ

³

q∇u dȍ ( y ) − Ȝ ( dȍ − meas ȍ f ) = ȍf

³

1 < q >< ∇u > − Ȝ( dȍ − meas ȍf ) → stationary 2 f ȍ

which means that the above functional is minimum with respect to u but maximum in Ȝ .

(14)


71

The shape of the fiber is identified by radii ps , s = 1,2,…,n of nodes located at the interface īC . Because of the considered symmetry only the first quarter of unit cell (shaded) is observed and the origin is located at the lower left vertex of the unit cell, Fig. 1. In this way we obtain n triangles Ts, s = 1,...,n, which approximate the domain ȍ f . It obviously holds:

³ dȍ = meas ȍ

n

f

= ¦ meas Ts .

(15)

s =1

ȍf

In certain cases of fiber volume ratio with combination of the given phase conductivities restrictive conditions have to be applied to the admissible beams of nodes at the interfacial boundary. This can be done in various ways. A typical lowest value of the length of any node at īC is bounded from below by a given value 0 < p < p s and the highest length is constrained by the conditions as y i < h < 1 , p, h are reals selected in advance. If the above bounds on the beams are attained a special procedure needs to be used, see [3]. It requires an internal iteration, as the improvement of the boundary using collinear mapping to ensure the condition about constant fiber volume fraction has to be carried out.

Euler's equations The stationary requirement leads to differentiation of the functional by the shape (design) parameters ps 1 ∂ < ∇u > 2 ∂p s , s =1,…,n Ȝ = Ȝs ∂ dȍ ∂p s f

³

(16)

ȍ

The equation (21) requires Ȝ having the same value for any s. In other words, if this requirement were attained at any point on the "moving" part of the interfacial boundary the optimal shape of the trial body would be reached. For this reason the body of the composite structure should increase its area (in 3D its volume) at the nodal point of the boundary identified by ps if Ȝ is larger than the true value of the target, while it should decrease its value when Ȝ is smaller than the correct Lagrangian multiplier. As, most probably, real value of the target is not known a priori, its estimate is done by averaging the current values at the nodal points. So, approximation of Ȝ will be expressed as: Ȝapprox =

1 n

n

¦=1 Ȝ

s

(17)

s

Differentiation by Ȝ completes the system of Euler's equations. It remains to ensure that the fiber volume friction is constant with the value given a priori. For this aim a collinear mapping is applied after completing the shift of nodes at the interface.

Examples Unit cell is considered with various fibers volume ratios. Since we compare energy densities at nodal points of the interfacial boundary, the relative energy density may be regarded as the comparative quantity influencing the movement of the boundary īC in a proper direction. As said in the previous section, the higher value of this energy, the larger movement of the nodal point of īC should aim at the optimum. The process of iterations will end if the Euclidean distance between current and previous energies be less then given admissible error.

72


In the following examples fiber and matrix ratios are given as: meas ȍf , meas ȍ m and also the conductivities c f and c m are prescribed. In the two tests considered here c f = 5 and c m = 1 , meas ȍf = 0.4 and meas ȍ m = 0.6 with resulting optimal shape presented in Fig. 2 and being attained with relative error 1.8e-04 after twenty seven iterations using the step of iteration 0.1 to 0.005. Restriction on the length of rays to nodes is applied as: the length of any ray cannot exceed the value of 0.45. For the next case c f = 1 and c m = 5 , meas ȍf = 0.4 and meas ȍ m = 0.6 , the relative error is 2.5e-04 after eighteen iterations using the same self adopting steps as before. The result is depicted in Fig. 3.

Fig. 2: First case

Fig. 3: Second case

Conclusions New optimization procedure is put forward in this paper based on homogenization technique. The problem which has been solved deals with homogenization of coefficients of the linear harmonic equation. The optimization is formulated in term of energy. A special constraint is adopted, which is involved in the formulation of optimal shape by Lagrangian multiplier, enabling us to show that the stationary point is attained for energy density being equal at each nodal point of the interfacial boundary. This condition leads us to an elegant and efficient numerical approach. The computer program enables now the user to get various optimal shapes according to his requirements.

Acknowledgment: Financial support of the Grant agency of the Czech Republic, grant number P105/00/0266 is gratefully acknowledged. References [1] V.J. Challis, A.P. Roberts and A.H. Wilkins Design of three dimensional isotropic microstructures for maximized stiffness and conductivity.: Int. J. Solids and Structures, 45, (14-15), 4130-4146 (2008) [2] P.P. Prochazka, V. Dolezel and T.S. Lok Optimal shape design for minimum Lagrangian: Eng. Anal. with Bound. Elem. 33, 447–455 (2009) [3] P.M. Suquet Homogenization techniques for composite media: Lecture Notes in Physics 272 (eds. E. Sanches-Palencia and A. Zaoi) Part IV, Sprinter Verlag Berlin, 194-278 (1985) [4] T. Lévy Fluids in porous media and suspensions: Lecture Notes in Physics 272 (eds. E. SanchesPalencia and A. Zaoi) Part II, Sprinter Verlag Berlin, 64-119 (1985) [5] J. Dvorak Optimization of composite materials, PhD thesis, Charles University, June 1996


Meshless Method in Analytical Formulation and Application to Elastodynamics P.H. Wen1a and M.H. Aliabadi2b 1

Department of Engineering, Queen Mary, University of London, London, UK, E1 4NS 2

Department of Aeronautics, Imperial College, London, UK, SW7 2BY a

[email protected], [email protected]

Keywords: Meshless method, radial basis function, analytical solutions, elastodynamics. Abstract. The analytical forms of integrals in the meshless local integral equation method in the Laplace space are derived and implemented for elastodynamic problems in this paper. The meshless approximation based on the radial basis function (RBF) is employed for the implementation of displacements. A weak form for a set of governing equations with a unit test function is transformed into local integral equations. A completed set of closed forms of the local boundary integrals are obtained. As the closed forms of the local boundary integrals are obtained, there are no any domain or boundary integrals to be calculated numerically in this approach. Introduction The finite element method is nowadays routinely used for linear, nonlinear and large scale problems (Zienkiewicz[1] ). However, the finite element method suffers from drawbacks such as the generation of a finite element mesh with thousands of nodes. Problems such as crack growth are particularly different, although XFEM has made much progress in this respect. The boundary element method (BEM) is also a well-established technique for the analysis of certain engineering problem such as fracture mechanics (see Aliabadi [2]) and acoustics (see Wrobel[3]). In recent years, the computational mechanics community has turned its attention to so-called mesh reduction methods. These mesh reduction methods (commonly referred to as Meshless or Meshfree) have received much interest since Nayroles et al [4] proposed the diffuse element method. Later, Belyschko et al [5] and Liu et al [6] proposed element-free Galerkin method and reproducing kernel particle methods, respectively. MLPG is reported to provide a rational basis for constructing meshless methods with a greater degree of flexibility. Local Boundary Integral Equation method (LBIE) with moving least square and polynomial radial basis function (RBF) has been developed by Sladek et al [7]. However, Galerkin-base meshless methods, except MLGP presented by Atluri[8] still include several awkward implementation features such as numerical integrations in the local domain. In this paper, the meshless local integral method is presented for two dimensional dynamic problems. With the use of radial basis functions, the analytical solutions for all domain integrals in the weak form of are derived. The weak formulations for the governing equations with a unit test function are obtained exactly for the local domain integrals. As the closed form of the local boundary integrals are obtained, the computational time are reduced significantly. A numerical inversion technique, Durbin’s inversion method, is applied to determine each variable in the time domain. The accuracy of the proposed method is illustrated by comparing the numerical results with available analytical solution and results with boundary element method.

Meshless local integral equation method

Consider a linear elastic body in three dimensional domain : with boundary * . The governing equations of motion can be written as V ij , j f i U ui (1)

where V ij are stresses, f i is the body force,

(i 1, 2 j has

1, 2 for 2D) and ( i 1, 2, 3 j

U is the density of material and ui is the acceleration

1, 2, 3 for 3D) . By Hook’s law for plane stress problem, one

73

74


§ wu 2 wu · E § wu1 wu 2 · ¸ ¨¨ ¨ (2) Q 1 ¸¸ , V 12 2 ( 1 x x w Q ) ¨© wx 2 wx1 ¸¹ w 1 ¹ © 2 For two dimensional problem, where E is Young’s modulus, Q is Possion ratio and P E / 2(1 Q ) the E 1 Q 2

V 11

§ wu1 wu ¨¨ Q 2 x w wx 2 © 1

· ¸¸ , V 22 ¹

E 1 Q 2

shear modulus. The boundary conditions are given as

ui

u i0

on *u

ti

V ij n j

t i0

(3)

on *t

in which u i0 and t i0 are the prescribed displacements and tractions respectively on the displacement boundary *D and on the traction boundary *T , and ni is the unit normal outward to the boundary * . In the local boundary integral equation approach, the weak form of differential equation over a local integral domain : s can be written as

³ (V

ij , j

f i U ui )u i* d:

0

(4)

:s

where u i* is a test function. By use of divergence theorem, (4) can be rewritten in a symmetric weak form as

³V

ij

*s

n j u i* d* ³ (V ij u i*, j f i u i* U ui )d:

0

(5)

:s

If there is an intersection between the local boundary and the global boundary, a local symmetric weak form in linear elasticity may be written as

³V

:s

ij

u i*, j d: ³ t i u i* d* ³ t i u i* d* *D

Ls

³t

0 i

*T

u i* d* ³ ( f i U ui )u i* d:

(6)

:s

in which, Ls is the other part of the local boundary inside the local integral domain : s ; *D is the intersection between the local boundary *s and the global displacement boundary; *T is a part of the traction boundary as shown in Figure 1. The local weak forms in (5) and (6) are a starting point to derive local boundary integral equations if appropriate test functions are selected. A step functions can be used as the test functions u i* in each integral domain

support domain of x

node x

ȍ Local integral domain ȍs ȍ

Node in support domain ȟk

Ls

*s=*D+*T

Figure 1. Arbitrary distributed node, support domain of x, local integral domain for weak formulation.

u i* (x)

M i (x) at x (: s *s ) . ® at x : s ¯ 0

(7)


75

where M i (x) is arbitrary function. In the Laplace transform domain, the Laplace transform of function

f (t ) is defined as f

~ f (s)

³ f (t )e

pt

dt

(8)

0

where p is a Laplace parameter. Equation (6) becomes

³ V~ n u d* ³ (V~ u ij

* i

j

ij

*s

* i, j

~ f i u i* p 2 U u~i )d:

0

(9)

:s

The approximation scheme

Consider a local domain w: s shown in Figure 1, which is the neighbourhood of a point x

( ^x1 , x 2 ` ) and is considered as the domain of definition of the RBF approximation for the trail function at x and also called as support domain to an arbitrary point x. Generally the support domain is chosen as a circle centred at x. To interpolate the distribution of function u in the local domain w: s over a number of randomly distributed nodes ȟ [= ^ȟ 1 , ȟ 2 ,..., ȟ K `, ȟ k function u at the point x can be expressed by

([ k1 , [ k 2 ), k

1,2,..., K ], the approximation of

K

u ( x)

¦R

k

(x,ȟ k )a k

R (x)a(x)

k 1

(10) where R (x)

^R1 (x,ȟ), R2 (x,ȟ ),..., RK (x,ȟ )` is the set of radial basis functions centred around the point x K ( x1 , x 2 ) ], â k `k 1 are the unknown coefficients to be determined. The radial basis function selected

[ multi-quadrics

c 2 ( x1 [ k1 ) 2 ( x 2 [ k 2 ) 2

Rk (x,ȟ k )

(11) where c is a free parameter. In order to guarantee unique solution of the interpolation problem, the displacement field can be interpolated by K

u i ( x)

¦R k 1

T

k

(x,ȟ )a k ¦ Pt (x)bt

R (x)a P(x)b

t 1

(12) along with the constraints K

¦ P (ȟ t

k

)a k

0,

1d t dT

k 1

(13)

where ^Pt `t 1 is a basis for PT 1 , the set of d-variate polynomials of degree d T 1 . In this paper, following polynomials are considered (T=6) T

P

^1, x , x , x 1

2

2 1

, x1 x 2 , x 22 `

(14) A set of linear equations can be written, in the matrix form, as

R 0 a P0 b (15) where matrix

ui ,

P0 a

0

76


ª R1 (ȟ 1 ) R2 (ȟ 1 ) « R (ȟ ) R (ȟ ) 2 2 « 1 2 « . . « . « . « . . « «¬ R1 (ȟ K ) R2 (ȟ K )

R 0 (ȟ )

... ... ... ... ... ...

RK (ȟ 1 ) º R K (ȟ 2 ) »» . » » . » . » » RK (ȟ K )»¼ K uK

and

ª P1 (ȟ 1 ) P2 (ȟ 1 ) « P (ȟ ) P (ȟ ) 2 2 « 1 2 « . . « . « . « . . « ¬« P1 (ȟ K ) P2 (ȟ K )

P0 (ȟ )

... ... ... ... ... ...

PT (ȟ 1 ) º PT (ȟ 2 ) »» . » » . . » . » » PT (ȟ K )¼» K uT

(16) Solving these equations in Eq.(15) gives

b

ȕu, a

Įu

ȕ

P

T 0

1 0

R P0

1

P0 R 01 , Į T

>

R 01 I P0 P0 R 01P0 T

1

P0 R 01 T

@

(17) where I denotes the diagonal unit matrix. Substituting the coefficients a and b from Eq.(18) into Eq.(12), we can obtain the approximation of the field function in terms of the nodal values u (x) ĭ(x)u, ĭ(x) R (x)Į P(x)ȕ . (18) in which ĭ(x) is defined as shape function, matrix R (x) and P(x) are scale (1 u K ) and (1 u 6) matrix respectively. It is worth noting that the shape function depends uniquely on the distribution of scattered nodes within the support domain and it has the Kronecker Delta property.

Analytical solutions for test functions Consider a unit test function, i.e. M i (x) 1 and the local domain is enclosed by several straight lines as shown in Figure 2, therefore, the local boundary integral equation becomes L

~

³ V~ n d* ¦ n ³ V~ d* ³ ( f ij

j

lj

*s

ij

*l

l 1

i

p 2 Uu~i )d:

0

(19)

:s

where L is number of straight line. Suppose there are nodes both in the domain and on the boundary, M M : M T M D , where M : indicates the number of nodes collocated in domain, M T and

M D are numbers of nodes on the traction/displacement boundaries and consider the radial basis function interpolation in (18) and relationship between stress and strain in (2), then (4) becomes K

L

ª

K

§

E

¦¦ «¦ ¨© 1 Q j 1 l 1

¬i

1

2

T · · º § E F1il n1l PF2il n2l ¸D ij ¦ ¨ G1tl n1l PG2tl n 2l ¸E tj » u~1( j ) 2 ¹ ¹ ¼ t 1 © 1 Q

T º ~ ( j) ª § EQ § EQ l l · l l · ¦¦ «¦ ¨ 1 Q 2 F2il n1 PF1il n2 ¸D ij ¦ ¨ 1 Q 2 G2tl n1 PG1tl n 2 ¸ E tj » u 2 ¹ ¼ ¹ t 1© j 1 l 1 ¬i 1 © K

L

(20a)

K

I1


K

ª

L

§ EQ

K

¦¦ «¦ ¨© 1 Q j 1 l 1

¬i

2

1

77

T · § EQ · º F1il n 2l PF2il n1l ¸D ij ¦ ¨ G1tl n2l PG2tl n1l ¸ E tj » u~1( j ) 2 1 Q ¹ ¹ ¼ t 1©

(20b)

T º ~ ( j) ªK § E § E l l · l l · I2 ¦¦ «¦ ¨ 1 Q 2 F2il n 2 PF1il n1 ¸D ij ¦ ¨ 1 Q 2 G2tl n 2 PG1tl n1 ¸E tj » u 2 © ¹ © ¹ j 1 l 1 ¬i 1 t 1 ¼ ~ for x k , k 1,2,...M : , where the integral functions I i ³ ( f i p 2 U u~i )d: and K

L

:s

sl

wRi

0

j

³ wx

F jil

ds, G jtl

sl

wPt

0

j

³ wx

sin E l , n2l

Consider n1l

ds

(21)

cos E l , we have solutions in closed form

F1il

(r2 r1 ) cos E l [( x al 1 [ i1 ) sin E l ( x al 2 [ i 2 ) cos E l ] sin E l ln(d1 / d 2 )

F2il

(r2 r1 ) sin E l [( x al 2 [ i 2 ) cos E l ( x al 1 [ i1 ) sin E l ] cos E l ln(d1 / d 2 ) L

b ( xbl 1 , xbl 2 )

1

xk

x2 2

xk

x2

sl

'2

( x1l , x 2l ) '1

El

l

a

n

l

x1

( x al 1 , x al 2 )

x1

Figure 2. Local integral domain with straight boundary lines.

r1

c 2 ( x al 1 [ i1 ) 2 ( x al 2 [ i 2 ) 2

r2

c 2 ( xbl 2 [ i1 ) 2 ( xbl 2 [ i 2 ) 2

Figure 3. Rectangular local integral domain.

(22)

d1

( x al 1 [ i1 ) cos E l ( x al 2 [ i 2 ) sin E l r1

d2

( xbl 1 [ i1 ) cos E l ( xbl 2 [ i 2 ) sin E l r2

If the area of local integral domain is small, the domain integral can be approximated as

Ii

p 2 U u~i : s

(23)

However, for a rectangular local integral domain as shown in Figure 3, we have its analytical form as K

In

K

i

j 1

where

ª

¦ «¬¦ J D i 1

ij

T º ~ ¦ H t E tj » f n( j ) u~n( j ) t 1 ¼

(24)

78


1 2 1 c x 2 ln x1 c 2 ( x1 [ i1 ) 2 ( x 2 [ i 2 ) 2 c 2 x1 ln x 2 c 2 ( x1 [ i1 ) 2 ( x 2 [ i 2 ) 2 2 2

Ji

1 1 x1 x 2 c 2 ( x1 [ i1 ) 2 ( x 2 [ i 2 ) 2 x 23 ln x1 c 2 ( x1 [ i1 ) 2 ( x 2 [ i 2 ) 2 3 6 1 3 2 2 x1 ln x 2 c ( x1 [ i1 ) ( x 2 [ i 2 ) 2 6

§ cx1 c3 tan 1 ¨ ¨ c 2 x 2 x c 2 (x [ )2 (x [ )2 3 i1 i2 2 2 1 2 ©

· ¸ ¸ ¹

x1 '1 / 2

x2 ' 2 / 2

x1 '1 / 2

x2 ' 2 / 2

and

H1

x1 x 2

x1 '1 / 2

x2 ' 2 / 2

x1 '1 / 2

x2 ' 2 / 2

x1 '1 / 2 1 x1 x 22 x1 '1 / 2 2 1 2 2 x1 '1 / 2 x1 x 2 x1 '1 / 2 4

H3 H5

1 2 x1 '1 / 2 x x2 2 1 x1 '1 / 2 x2 ' 2 / 2 1 3 x1 '1 / 2 x1 x 2 , H4 x2 ' 2 / 2 x1 '1 / 2 3 x2 ' 2 / 2 1 3 x1 '1 / 2 x1 x 2 , H6 x2 ' 2 / 2 x1 '1 / 2 3 , H2

x2 ' 2 / 2 x2 ' 2 / 2 x2 ' 2 / 2 x2 ' 2 / 2 x2 ' 2 / 2 x2 ' 2 / 2

For the nodes on the traction boundary, (9) should be introduced

~

~0

³ t d* ³ t i

* *T

i

d*

Ii

for x k k

1,2,..., M T

(25)

*T

For the displacement boundary nodes, we can introduce the displacement equation directly, i.e. u~i (ȟ k ) u~i0 , k 1,2,...M D . Therefore, there are total 2 u ( M : M T M D ) linear algebraic equations which are used to determine the same number unknowns of displacements either in the domain or on the traction boundary. In the Laplace transform domain, a total number of L+1 samples in the transformation space s k ,

k 1, 2,..., L, are selected. Physical values are calculated for these transform parameters and the real value at time t must be obtained by an inverse transform. Here, the method given by Durbin [9] is used. L ~§ 2e K t ª 1 ~ 2kS · § 2kS t ·½º i ¸ exp¨ (26) ¸ ¾» « f (K ) ¦ Re® f ¨K T ¹ T ¬ 2 © T ¹ ¿¼ k 0 ¯ © where f ( p k ) stands for the transformed variables in the Laplace space for parameters

f (t )

pk

K 2kS i / T ( i

1) . The selection of parameters K and T only slightly affects the accuracy of 5 / t0 and T / t0 20 in the following b / c1 is the unit of time, b is specified length and longitudinal wave speed

the numerical solution. In our computations, we have chosen K examples, where t 0

E (1 Q ) /[(1 Q )(1 2Q ) U ] ) . In addition, the dynamic stress intensity factor is evaluated by crack opening displacement (COD) in the Laplace transform domain. For mode I fracture, the stress intensity factor for plan stress problem is related to crack opening displacement, in the transformed domain, as following c1 (

~ KI

2S ~ E 'u 2 , 'u~2 8(1 Q 2 ) r0

u~2 u~2 .

(27)

where r0 indicates the distance between the collocation point and crack tip, 'u~2 is the crack opening displacement (COD) in the Laplace transform domain.

Numerical example: A Single central crack in rectangular plate under tension


79

Consider a rectangular plate of width 2b and length 2h with a centrally located crack of length 2a. It is loaded dynamically in the direction perpendicular to the crack by a uniform tension V 0 H (t ) on the top and the bottom. Due to the symmetry, a quarter of plate is considered as shown in Figure 4. Here Poisson’s ratio Q 0.3 and Young’s modulus is unit. V 0 H (t )

Firstly we observe the accuracy of stress intensity factor with the density of collocation point. The analytical static solution for a square plate b / h 1 containing a central crack, if a / b

h

0.5 , is K I

1.325V 0 Sa

[10]

and the

result by the second nodal value with 21u 21 node distribution is K I 1.331V 0 Sa . Therefore, the relative error can be expected to be less than 1% for elastostatic problems. Therefore, we use the second nodal value with 21 u 21 node distribution to evaluate stress intensity factor in the following examples. .

x2 crack tip

a x1 b

Figure 4. Rectangular plate with a central crack of length 2a under tension V 0 H (t ) . Three geometries of rectangular plate are considered in this example, i.e. b / h 0.5 , b / h 1 and b / h 2 while a / b 0.5 . The total numbers of nodes for each case are selected as 21u 11 , 21u 21 and 21u 41 respectively. Figures 5, 6 and 7 show the normalized stress intensity factors various against the normalized time c1t / b for different geometry of plate. In addition, the results given by Wen et al [11] using the mesh free Perov-Galerkin method and the indirect boundary element method (fictitious load method [12]) are presented in the same figures for comparison. Apparently before the arrival time of dilatation wave traveling from the top of plate, the stress intensity factor should remain to be zero. The agreement between these solutions is considered to be good.

6HULHV This paper

KI/V0Sa

MFPG [11] 6HULHV BEM [12] 6HULHV

c1t / b

Figure 5. Normalized stress intensity factor K I / V 0 Sa against the normalized time c1t / b for b / h

0 .5 .

80


This paper 6HULHV

MFPG [11] 6HULHV

BEM [12] 6HULHV

KI/V0Sa

c1t / b


1 .0 .

This paper 6HULHV

KI/V0Sa

MFPG [11] 6HULHV 6HULHV BEM [12]

c1t / b


2.

Conclusion This paper demonstrated the availability of meshless local integral method to two dimensional elastodynamic fracture problems by the Laplace transform technique. Considering a local integral domain with local support domain, the analytical formulations were derived in the Laplace transform domain. Durbin’s inversion method was applied to determine all variable in the time domain. The dynamic stress intensity factor of mode I was evaluated by using the COD technique. We can conclude with the following observations: (1) meshless local integral method is valid to deal with elasto-dynamic problem in Laplace space; (2) analytical formulations of all integrals save CPU time; (3) numerical solutions are stable and convergent for suitable selection of free parameters; (4) this method can be extended to elastoplastic, plate/shell bending and nonlinear problems directly.


References [1] O.C. Zienkiewicz The finite element method, McGraw Hill, London 1977. [2] M.H. Aliabadi The boundary element method, applications in solids and structures, Wiley, Chicester, 2002. [3] L.C.Wrobel The boundary element method, vol1: application to thermo-fluids, Wiley Chicesater, 2002. [4] B. Nayroles, G. Touzot & P. Villon, Generalizing the finite element method: diffuse approximation and diffuse elements, Computational Mechanics, 10, 307-318, 1992. [5] T. Belytschko, Y.Y. Lu & L. Gu, Element-free Galerkin method, Int. J. Numerical Methods in Engineering, 37, 229-256, 1994. [6] W.K. Liu, S. Jun & Y. Zhang, Reproducing kernel particle methods, Int. J. Numerical Methods in Engineering, 20, 1081-1106, 1995. [7] V. Sladek, J. Sladek., Ch. Zhang, Comparative study of meshless approximations in local integral equation method, CMC: Computers, Materials, & Continua, 4, 177-188, 2006. [8] S.N. Atluri, The Meshless Method (MLPG) for Domain and BIE Discretizations, Forsyth, GA, USA, Tech Science Press, 2004. [9] F. Durbin, Numerical inversion of Laplace transforms: an efficient improvement to Dubner and Abate’s method, The Computer J., 17, 371-376, 1975. [10] D.P. Rooke, D.J. Cartwright, Compendium of Stress Intensity Factors, London, Her Majesty’s Stationery Office, 1976. [11] P.H. Wen, Dynamic Fracture Mechanics: Displacement Discontinuity Method, Computational Mechanics Publications, Southampton UK and Boston USA, 1996.

81

82


Symmetric BEM as discretization tool for FEM analysis of thin plates M. Mazza1 1

Dipartimento di Strutture, Universita´ della Calabria 87030 - Rende (Cosenza), Italy [email protected]

Keywords: Boundary integral equations, finite elements, analytical integration, complex variables.

Abstract. A thin plate finite element, able to overcome the typical drawbacks of traditional finite elements, is presented. It is designed by replacing the usual polynomial interpolations, which ensure displacement continuity but do not always respect the normal derivative continuity, with a symmetric boundary integral description of the static and kinematic fields. The entries of the stiffness matrix are evaluated by a complex variable integration technique which provides compact analytical results easy to use in a computer code, improving the accuracy of the numerical results.

Introduction The polynomial shape functions used in standard finite element models limit number and location of interpolation nodes, element shapes and quality of approximation. Moreover a large number of variables with a high computational cost is required when complex structural problems have to be analyzed. These drawbacks can be limited by interfacing finite and boundary element models. These approaches compel to combine the well established finite element computer codes with more complicate programs based on boundary elements. An interesting alternative consists of using boundary integral equations for the design of finite elements having flexible shape and arbitrary number of nodes. In this approach the usual polynomial interpolation of the domain fields is replaced by the boundary integral description of the boundary fields. So doing the typical accuracy of the boundary element approximation could be combined with the flexibility of the finite element methods. So far this possibility has been investigated only in few works where boundary integral equations associated with point [1] or distributed [2] sources have been used for the finite element analysis of plane elasticity problems. In the present work the symmetric boundary integral approach is used as a discretization tool to design a new thin plate finite element. The discrete forms of the boundary integral equations associated with static and kinematic sources, weighted in the Galerkin sense, are used to define the strain energy on the boundary of the proposed finite element. This procedure ensures the interelement continuity of both displacement and normal derivatives and allows a direct evaluation of the corner reactions differently from finite elements where they are obtained a posteriori. Similarly to a classic symmetric boundary element model, the entries of the stiffness matrix descend from the double (single) boundary integration of the product between a singular fundamental solution and two (one) polynomial shape functions. The accurate and efficient evaluation of these coefficients entails an high computational cost which can be minimized by means of an analytical integration technique based on complex variables and a specific integration rule. In this way the heavy symbolic computations, necessary for the integrations in real variables in presence of high order shape functions and articulated fundamental solutions, are strongly reduced. Moreover, the Gauss transformations, necessary for evaluating singular contributions, allow to reduce the number of types of prime integrals, making easier the entire integration process and the development of the finite element computer code. Boundary integral formulation The bending of a thin plate, defined over a domain Ω, delimited by the boundary Γ, subjected to the transversal load p and having flexural rigidity D, is usually described in differential form by the field equation D∆∆w = p. The boundary integral approach represents an alternative description of this problem.


83

In the standard formulation the above field equationis is weighted by a fundamental solution associated to a unit point source, integrated over the domain and then transferred to the boundary by the divergence theorem. Considering the fundamental solution ws∗ associated to a unit point static source s ∈ {F, C}, the generic kinematic field f ∈ {w, θ} is described by the following boundary integral equation nc (Rs∗ (j) w(j) − R(j) ws∗ (j) ) − pws∗ dΩ = 0 (1) c f (i) + (t∗s w + m∗s θ − t ws∗ − m θs∗ )dΓ + Γ

Ω

j=1

w being the transversal displacement, θ the normal slope, m the bending moment, t the equivalent shear and R (j) being the corner reaction at the generic singular point j of the boundary Γ. Specifically R (j) is − defined by the difference m+ t − mt between the twisting moments applied at the common end j of the two sides, marked by + and − and merging at this point. Here the quantities marked with an asterisk are the fundamental solutions and the factor c distinguishes sources located on the boundary (c = 1/2) from sources inside the domain (c = 1). In a similar way, adopting a fundamental solution corresponding to a unit point kinematic source s ∈ {∆θ, ∆w, ∆θt }, the generic static field f ∈ {m, q, mt } is defined by the boundary integral equation nc (R(j) ws∗ (j) − Rs∗ (j) w(j) ) + pws∗ dΩ = 0 (2) c f (i) + (t ws∗ + m θs∗ − t∗s w − m∗s θ)dΓ + Γ

Ω

j=1

The same equation (2) allows also the boundary integral description of both the equivalent shear t, obtained adding the tangent derivative of the boundary integral equation of twisting moment mt (equation 2 written for s = ∆θ) to the boundary integral equation of shear q (equation 2 written for s = ∆w), and the corner reaction R(j) , obtained writing the boundary integral equation of twisting moment mt for point sources s = ∆θt+ and s = ∆θt− applied at the common end of two contiguous sides. The weighted boundary integral equations used in a symmetric boundary element model are obtained integrating on Γ the boundary integral equations (1) and (2) weighted by unit distributed sources F , C, −∆θ, −∆w and interpolating sources and boundary fields by identical shape functions according to the Galerkin approach. So doing the following discrete equations are obtained in the case of sources located on the boundary ∗ (j) ∗ (j) ¯ (j) W ∗F t¯ + Θ∗F m ¯ − M ∗F θ¯ − T ∗F w ¯ − RF w ¯ (j) + W F R = Ψwt w/2 ¯ ∗ (j) ∗ (j) ¯ (j) ∗ ¯ ∗ ∗ ¯ ∗ (j) ¯ ¯ − MC θ − T C w ¯ − RC w ¯ + W C R = Ψθm θ/2 W C t + ΘC m

∗ (j) ∗ (j) ¯ (j) t¯ − Θ∗∆θ m ¯ + M ∗∆θ θ¯ + T ∗∆θ w ¯ + R∆θ w ¯ (j) − W ∆θ R = Ψmθ m/2 ¯ ∗ (j) ∗ (j) ¯ (j) − W ∗∆w t¯ − Θ∗∆w m ¯ + M ∗∆w θ¯ + T ∗∆w w ¯+R w ¯ (j) − W R = Ψtw t¯/2

− W ∗∆θ

∆w

∆w

(3) (4) (5) (6)

where the transversal load p has been assumed equal to zero for the sake of brevity. In presence of corners the set of equations has to be completed considering the boundary integral equations of transversal (i) displacement and corner reaction at the corner j, weighted by unit point sources −∆θt and F (i) ∗ (i) ∗ (i) ∗ (i) ∗ (i) ∗ (ij) ∗ (ij) ¯ (j) ¯ (j) /2 −W ∆θt t¯ − Θ∆θt m ¯ + M ∆θt θ¯ + T ∆θt w ¯ + R∆θt w ¯ (j) − W ∆θt R =R

(7)

∗ (i) ∗ (ij) ∗ (ij) ¯ (j) θ¯ − T F w ¯ − RF w ¯ (j) + W F R =w ¯ (j) /2

(8)

∗ (i) WF

t¯ +

∗ (i) ΘF

m ¯ +

∗ (i) MF

In equations (3-8) the barred symbols are used to mark vectors collecting the interpolation parameters ¯ (j) ). In the same equations the bold located at boundary regular points (see t¯) and at corner points (see R ∗ upper-case letters without superscripts (e.g. W F ) represent matrices collecting coefficients deriving from the double boundary integration of the product between two shape functions and a fundamental solution. ∗ (j) In a similar way the bold upper-case letters with superscripts (e.g. RF ) represent vectors containing coefficients obtained by the single boundary integration of the product between a shape function and a ∗(j) fundamental solution. As an example two generic entries of W ∗F and RF are given here (h) (k) ∗ (j) (h) ∗ (j) W ∗F [h, k] = ψt ψt wF∗ dΓ(k) dΓ(h) , RF [h] = ψt RF dΓ(h) (9) Γ(h)

Γ(k)

Γ(h)

84


As for the matrices Ψ, they contain the entries deriving from the single boundary integration of two shape functions. For instance the entry corresponding to the hth source interpolation parameter and to the k th field interpolation parameter is given by Ψwt [h, k] =

(h)

Γ(h)

ψt ψw(k) dΓ(h)

(10)

Finally, the symmetry of the boundary system is achieved selecting the equations on the basis of the boundary conditions, to exploit both the presence of identical shape functions (i.e. ψw = ψ∆w , ψθ = ψ∆θ , ψm = ψC , ψt = ψF ) and the reciprocity relationships linking pairs of fundamental solutions (i.e. wC∗ = θF∗ , ∗ ∗ ∗ ∗ ∗ ∗ w∆θ = m∗F , w∆w = t∗F , w∆θ = m∗t F , θ∆θ = m∗C , θ∆w = t∗C , θ∆θ = m∗t C , m∗∆w = t∗∆θ , t∗∆θt = m∗t ∆w ). t t Stiffness matrix by weighted boundary integral equations Finite element models are usually developed by interpolating the mechanical fields involved in the domain integral of the strain energy. For thin plates the choice of polynomial shape functions is quite problematic owing to the high order of continuity required along the interfaces between the finite elements. This trouble can be removed defining the strain energy in terms of boundary fields and making use of the weighted boundary integral equations. Strain energy in terms of boundary variables The bending strain energy of a thin plate finite element can be defined on the boundary as follows 1 1 Φ= mij w,ij dΩ = mij nj w,i dΓ − mij,j nj w dΓ + mij,ij w dΩ = 2 Ω 2 Γ Ω Γ 1 = (mθ + mt θt − tw) dΓ + mt,t w dΓ − pw dΩ 2 Γ Γ Ω

(11)

if the Green formula is applied to the original domain integral. Integrating by parts the second term on the right-hand side of the above equation nc nc (j + ) (j − ) mt,t w dΓ = (mt − mt )w(j) − mt w,t dΓ = R(j) w(j) − mt θt dΓ (12) Γ

Γ

j=1

Γ

j=1

the bending strain energy (11) takes the form nc 1 (j) (j) (m θ − t w)dΓ + R w − pwdΩ Φ= 2 Γ Ω j=1

(13)

Assuming p = 0, interpolating the boundary fields by polynomial shape functions and denoting by bold letters denote the vectors which collect interpolation parameters and the matrices of the shape functions, the above equation can be rewritten as nc 1 1 (j) T T T T T (j) ¯ ¯ ¯ Φ= R = tT Ψut u ¯ dΓ + w ¯ (14) m ¯ Ψm Ψθ θ − t Ψt Ψw w 2 2 Γ j=1 where

⎡

⎤ t¯ ¯ ⎦ t=⎣m ¯ (j) R

⎡ Ψut = ⎣

−

Γ

⎤ ΨTt Ψw 0 0 ΨTm Ψθ 0 ⎦ 0 Γ 0 0 I

⎤ w ¯ u = ⎣ θ¯ ⎦ w ¯ (j) ⎡

,

(15)

Manipulation of the weighted boundary integral equations The weighted boundary integral equations (3-6) and (7-8) allow to evaluate the stiffness matrix of the thin plate finite element by using equation (14) instead of the domain integral of the strain energy. To achieve T T this result equations (3-6) are multiplied for t¯ , m ¯ T , −θ¯ , −w ¯ T leading to

Advances in Boundary Element and Meshless Techniques T T ∗ (j) ∗ (j) ¯ (j) t¯ W ∗F t¯ + Θ∗F m ¯ − M ∗F θ¯ − T ∗F w ¯ − RF w ¯ (j) + W F R ¯ = (t¯ Ψwt w)/2 ∗ (j) ∗ (j) ¯ (j) ∗ ¯ ∗ ∗ ¯ ∗ T (j) T ¯ m ¯ ¯ − MC θ − T C w ¯ − RC w ¯ + WC R W C t + ΘC m = (m ¯ Ψθm θ)/2 T T ∗ (j) ∗ (j) ¯ (j) θ¯ W ∗∆θ t¯ + Θ∗∆θ m = −(θ¯ Ψmθ m)/2 ¯ − M ∗∆θ θ¯ − T ∗∆θ w ¯ − R∆θ w ¯ (j) + W ∆θ R ¯ (j) ∗ (j) ∗ (j) ¯ ¯ T W ∗∆w t¯ + Θ∗∆w m w = −(w ¯ T Ψtw t¯)/2 ¯ − M ∗∆w θ¯ − T ∗∆w w ¯ − R∆w w ¯ (j) + W ∆w R

85

(16) (17) (18) (19)

¯ (j) T as follows while equations (7-8) are multiplied for −w ¯ (j) T , R ∗ (i) ∗ (i) ∗ (i) ∗ (i) ∗ (ij) (j) ∗ (ij) ¯ (j) ¯ (j) )/2 (20) w ¯ (j) T W ∆θt t¯ + Θ∆θt m ¯ − M ∆θt θ¯ − T ∆θt w ¯ − R∆θt w ¯ + W ∆θt R = −(w ¯ (j) T R ∗ (i) ∗ (i) ∗ (ij) (j) ∗ (ij) ¯ (j) ¯ (j) T W ∗ (i) t¯ + Θ∗ (i) m ¯ (j) T w(j) )/2 (21) R ¯ − M F θ¯ − T F w ¯ − RF w ¯ + WF R = (R F F Equations (16,17,21) and (18,19,20) can be expressed, respectively, by the compact formulas tT Guu t − tT Gut u = tT Ψut u uT Gtu t − uT Gtt u = uT Ψtu t where Guu

⎡

⎤ ⎡ ∗ (j) T∗ WF ⎢ ∆w ∗ (j) ⎥ T W C ⎦ = Guu , Gtt = ⎣ T ∗∆θ ∗ (i) ∗ (ij) T ∆θt WF ⎤ ⎡ ∗ (j) W ∗∆w Θ∗∆w M ∗F RF ⎢ ∗ (j) ⎥ ∗ M C RC ⎦ = ⎣ W ∗∆θ Θ∗∆θ ∗ (i) ∗ (i) ∗ (i) ∗ (ij) W ∆θt Θ∆θt MF RF ⎤ ⎡ 0 0 Ψwt Ψmθ Ψut = ⎣ 0 Ψθm 0 ⎦ Ψtu = ⎣ 0 0 0 I 0

W ∗F Θ∗F ⎢ ∗ = ⎣ WC Θ∗C ∗ (i) ∗ (i) WF ΘF ⎡ T ∗F ⎢ Gut = ⎣ T ∗C ∗ (i) TF ⎡

⎤ ∗ (j) M ∗∆w R∆w ∗ (j) ⎥ M ∗∆θ R∆θ ⎦ = GTtt ∗ (i) ∗ (ij) M ∆θt R∆θt ⎤ ∗ (j) W ∆w ∗ (j) ⎥ W ∆θ ⎦ = GTtu ∗ (ij) W ∆θt ⎤ 0 0 Ψtw 0 ⎦ 0 I

(22) (23)

(24)

(25)

(26)

I being the identity matrix. Adding member to member equations (22) and (23) most of their terms cancel out each other due to the presence of the same interpolation parameters and shape functions and due to the presence of reciprocal fundamental solutions. This algebraic manipulation leads to the formula tT Guu t − uT Gtt u = 2 tT Ψut u

(27)

where the right-hand side of equation (27) is twice the bending strain energy (14). Expressing now ¯w ¯ (j) ] T in terms of u = [w, ¯ θ, ¯ (j) ] T by means of the equation (22) and replacing this result in t = [m, ¯ t¯, R equation (27), the bending strain energy takes the final form t = G−1 uu (Gut + Ψut )u

→

1 Φ = uT (Gut + Ψut )T G−1 uu (Gut + Ψut ) − Gtt u 2 K

(28)

It allows the definition of the stiffness matrix K of a triangular, quadrilateral or polygonal thin plate finite element. The equivalent shear, the bending moment and the normal slope are interpolated by C 0 linear shape functions while the transversal displacement is described using C 1 two cubic hat-defined shape functions to approximate the contributions of pure transversal displacement (u) and of tangent rotation (θt ). All the functions are hat defined on a support made of two contiguous boundary elements. So doing the degree of freedom of each intermediate node of the proposed finite element are u, θ and θt while the degree of freedom of each corner node are the normal slopes on the two sides θ and θ merging at the corner (Figure 1).

86


yq = ym = yt

+1

x

-1

yu

q,m,t

x

-1

yqt

u

qt

+1

x

+1 -1

Figure 1: Shape functions of the boundary fields of the finite element. Efficient evaluation of the stiffness matrix In the proposed approach most of the entries of the stiffness matrix K descend from the double (single) boundary integration of the product between two (one) shape functions and a fundamental solution (see Guu , Gtt , Gut ) while only few terms are obtained by single boundary integration of shape functions (see Ψut ). More specifically the articulated expressions of the fundamental solutions and their singular behaviours give rise to different computational problems which affect the performance of the proposed thin plate finite element. Standard quadrature rules are usually considered an simple and efficient tool to integrate the large number of terms arising from the product between shape functions and fundamental solution when the singularity of the fundamental solution is not active. However these numerical integration techniques become cumbersome for large problems and inaccurate in the presence of nearly singular situations. Analytical integration techniques in the real plane [?, Mazza1]llow these drawbacks to be overcome, saving computer time and obtaining accurate numerical results but paying the penalty for heavy algebraic manipulations in the presence of generically oriented integration domains and high order interpolation functions. Moreover it is worth noting that special integration techniques have to be used to compute the boundary integrals when the singular behaviour of the fundamental solution is actived. The complex variable integration technique proposed in [5] represents an efficient tool to overcome the above mentioned drawbacks. When the orders of singularity are not greater than O(1/r) (see some submatrices of Guu ), the complex variable technique can be directly applied without regularizing the kernels. Denoting by n and ν the unit vectors normal to the boundary elements where the field point x = (x1 , x2 ) and the source point ξ = (ξ1 , ξ2 ) are located, the fundamental solutions can be defined in the complex plane by replacing the real expression of the distance r = |x − ξ| and its derivatives r1 , r2 by the formulas r = |z|

,

r1 =

z + z¯ 2|z|

,

r2 =

z − z¯ 2 i |z|

,

m1 =

m+m ¯ 2

,

m2 =

m−m ¯ 2

m ∈ {n, ν}

(29)

being z = x + iξ and z¯ = x − iξ. The shape functions are described instead in the complex plane as ⎧ ⎨ a = 1/ (ν n ¯ − n¯ ν) A = i a(b + 2(¯ ν z)) h1 = x A 1 − ξ1 b = 2 ((ν)h2 − (ν)h1 ) (30) C λ = i a(c + 2(¯ nz)) h2 = x2 − ξ2C ⎩ c = 2 ((n)h − (n)h ) 2 1 A C C where (xA 1 , x2 ) and (ξ1 − ξ2 ) are the global cartesian coordinates of the initial ends A and C of the two boundary elements where are located the field and the source points. Each indefinite boundary integration on the boundary Γ or Γλ is performed by using the rule p dj d0 p i n2j+1 → Γ = Γ z p h[¯ z ]dΓ = βj j (z p ) h[¯ z ]d¯ z βj = (z ) = z p (31) 2j+1 0 −i ν → Γ = Γλ dz dz Γ j=0 j+1

When the orders of singularity are greater than O(1/r) (see submatrices of Gtt and Gut ), the complex variable integration technique has to include a regularization procedure based on integration by parts. As an example, the regularization of the entry Gtt [u(h) , u(k) ], corresponding to the u(h) and u(k) interpolation parameters of the source and field distributions, is considered below (Figure 2). It is initially defined as (k) (h) ∗ (k) (h) (k) ∗ (k) Gtt [u(h), u(k) ] = ψu(h) [λ] ψu(k) []t∗∆w dΓ dΓλ + ψu(h) [λ]R∆w dΓλ + ψu(k) []t∗∆θ(h) dΓ +R (h) (h)

Γλ

(k)

Γ

(h)

Γλ

(k)

Γ

t

∆θt

(32)


87

c (h)

(h)

y c [l]

=

b

=

d=a

(k)

(h)

y d [l]

(k)

y b [l ]

(k)

y a [l ]

Figure 2: Entry Gtt [u(h), u(k)]. The regularization of this expression requires different integration by parts depending on the involved (h) fundamental solution. More specifically four integrations by parts of t∗∆w , two on the support Γλ = (c, d) (h) (k) (k) of the cubic shape function ψu and two on the support Γ = (a, b) of the cubic shape function ψu (h) (k) ∗ ∗ while two integration by parts on Γλ and Γ are necessary for the kernels mt ∆w and t∆θt , respectively. It is worth noting that the integration by parts of the similar coefficients of the fundamental solutions, here marked by the same latin capital letters (for instance A.1 is similar to A.2) 3 3 z nν nν 1 (6c (6c 24c1 k22 (33) k k ) −2 k k ) −2 O(1/r4 ) : t∗∆w = −2 1 2 3 1 2 3 z4 z4 n3 ν 3 z¯5

O(1/r3 ) : m∗t ∆w

A.1

B.1

z n2 ν 1 = −2 6 i c1 k22 (2 i c1 k2 k3 ) +2 z3 n2 ν 3 z¯4

O(1/r3 ) : t∗∆θt = +2

2

nν z3

A.2

(2 i c1 k2 k3 ) −2 B.2

O(1/r2 ) : m∗t ∆θt = −2

z 1 2c1 k22 n2 ν 2 z¯3

1 n3 ν 2

C.1

(34)

C.2

z 6 i c1 k22 4 z¯

(35)

C.3

(36)

C.4

provide singular boundary coefficients which cancel out each other thanks to both the continuity of shape functions and their derivatives on the integration domains and their zero values at the ends of the same domains. In the above expressions n and ν denote the normal versors to the boundary elements of the (k) (h) supports Γ and Γλ , z and z¯ are the complex and conjugate complex variables connecting source and field points while k1 , k2 and k3 are constant coefficients of the involved fundamental solutions. Completed the regularization the entry (32) takes the new form (h) (k) d 2 ψu [λ] d 2 ψu [] ∗ λλ Gtt, r [u(h) , u(k) ] = t ddλ (37) (h) (k) dλ 2 d 2 Γλ Γ where

2 1 3n2 n 1 z ν2 ln(z) + k2 t∗ λλ = 2 k1 k2 − 2 + k3 + 2 2ν ν n 2ν 2 z¯

(38)

It is worth noting that expression (37) proves to be more compact than (32) as the two linear shape functions d2 ψu [λ]/dλ2 , d2 ψu []/d2 and the regularized fundamental solution t∗ λλ have taken the place of the two

88


cubic shape functions ψu [λ] and ψu [] and of the highly singular fundamental solutions t∗∆w , m∗t ∆w , t∗∆θt and m∗t,∆θt , respectively. The restricted number of terms appearing in (37) can be now integrated representing the shape functions in the complex plane by (30) and applying the integration rule (31). Regularizing the other entries in a similar way, the matrices Gtt and Gtu are replaced respectively by the regularized matrices ⎤ ⎡ ∗ ∗ T F, r M ∗F T ∆w, r M ∗∆w, r ∗ ∗ , Gut, r = ⎣ T C, r M C ⎦ = GTtu, r Gtt, r = (39) T ∗∆θ, r M ∗∆θ, r ∗ (i) ∗ (i) T F, r M F where the absence of the subscript r indicates that no regularization has been carried out. The above regulirization approach provides complex variable expressions not containing the specific integration extremes which are added only when the stiffness matrix of each finite element of the mesh has to be evaluted. For instance, the final value of an entry of Gtt, r is obtained specifying the real location of the boundary ele(h) (k) ments p ∈ (c, d) of the support Γλ and q ∈ (a, b) of the support Γ in order to evaluate the contributions gac , gad , gbc and gbd . To this end, the real extremes λ1 , λ2 of p and 1 , 2 of q of the two boundary elements are replaced by the complex quantities z11 , z22 , z12 , z21 and conjugate complex z¯11 , z¯22 , z¯12 , z¯21 which denote the distance between the ends of the considered boundary elements taken into account, obtaining Gtt, r [u(h) , u(k) ] = (gpq [z11 , z¯11 ] − gpq [z12 , z¯12 ] − gpq [z21 , z¯21 ] + gpq [z22 , z¯22 ]) (40) p=c,d q=a,b ∗, (i)

The final value of an entry of the submatrix M F, r , belonging to the matrix Gut, r , is evaluted by Gut, r [u(h) , θ(i) ] = (gp [z1S , z¯1S ] − gp [z2S , z¯2S ])

(41)

p=c,d

where the complex quantities z1S , z2S represent the distance between the ends of the boundary elements of the support and the corner point S. z12

l1

n (a)

l2 l n l1

z11 z22 z21

z1S l1

n

l2 l

(b)

z2S l2 l

Figure 3: Complex integration bounds for double (a) and single (b) boundary integrals. References [1] Bulgakov, V. E., Bulgakova, M. V.. Multinode finite element based on boundary integral equations. Int. J. Numer. Meth. Engng., 43, 533–548, 1998. [2] Aristodemo, M., Leone, L. and Mazza, M., “Energy based boundary elements”, Meccanica, 36, 463– 477, (2001). [3] Fraeijs De Veubeke, B., “A conforming finite element for plate bending”, Int. J. Solids Struct., 4(1), 95–108 (1968). [4] Leonetti, L., Mazza, M. and Aristodemo, M., “A symmetric boundary element model for the analysis of Kirchhoff plates”, Eng. Anal. with Bound. Elem., 33, 1–11 (2009). [5] Mazza, M., and Aristodemo, M., “Costruzione efficiente di modelli BEM simmetrici di lastre di Kirchhoff”, GIMC 2008 – XVII Convegno Italiano di Meccanica Computazionale.


89

General framework for localization and regularization for local and non-local damage model using the meshless finite point method L. Pérez Pozo, F. Chacana Yorda Aula UTFSM-CIMNE, Departamento de Ingeniería Mecánica. Universidad Técnica Federico Santa María, Avenida España 1680, Valparaiso, Chile. [email protected], [email protected] Keywords: Meshless, damage, softening, strain localization, fracture energy, non-local deformation. Abstract. The application of elastic-plastic strain softening/damage in the theory of continuity has been largely development in the finite element method (FEM) context, here are some problems whit the size and orientation of the mesh in the results. As to the local model the FPM posses intrinsic non-local properties, one of this given by the weighting function is used for incorporate a intrinsic length scale which regularizes the problem. For the second model with gradient type regularization the strong formulation of the FPM allows the use of high-order differentially shape functions with which we can approximate directly the fields of locals and non-locals displacement. Consequently this work develop the numerical implementation of both damage models and the solution for the nonlinear problem by Newton Raphson iterative scheme. The validation of the obtained results is made starting from typical benchmark problems and available results on associated literature.

Introduction The finite points method (FPM) was proposed by Oñate et al [1, 2] initially with the purpose of solving convective transport and fluid flow problems. Later, its application was extended to advection-diffusion transport [3] and incompressible flow problems [4]. In the context of solid mechanics, FPM has been applied successfully in elasticity [5, 6, 7, 8], solid dynamics [9] and non-linear material behavior problems [10]. The non-dependence on a mesh or integration procedures is an important aspect which transform the FPM in a truly meshless method. The continuous damage models may be employed to describe the evolution of failure processes between the undamaged state and macroscopic crack initiation [11]. Regarding numerical simulations, this presents pathological mesh sensitivity, in that way different solutions have been proposed in the literature to remedy this physically unrealistic behavior by means cohesive crack models [12], Crack bands model [13] and regularized models [14]. In this work we focus on the regularized models via non-local effects, incorporated a material characteristic length and the implicit gradient-enhanced continuum model based on non-local displacements [15]. For this damage model the FPM posses intrinsic non-local properties [16], one of this is used to incorporate a internal length scale and for the gradient-enhanced model the strong formulation of the FPM allows the use of highorder differentially shape functions with which we can approximate directly the fields of local and non-local displacements [17].

90


The finite points method. Let ȍூ be teh interpolation sub-domain, called cloud, and let ‫ݏ‬௝ with ݆ ൌ ͳǡʹǡ ǥǡ ݊be a collection of ݊ points with coordinates ‫ݔ‬௝ ‫ א‬ȍூ . The unknow function ‫ݑ‬ሺ‫ݔ‬ሻ may be approximated within ȍூ by ் (1) ‫ݑ‬ሺ‫ݔ‬ሻ ؆ ‫ݑ‬෤ሺ‫ݔ‬ሻ ൌ σ௠ ௜ ‫݌‬௜ ሺ‫ݔ‬ሻߙ௜ ൌ ࢖ሺ‫ݔ‬ሻ ࢻ where ߙ ൌ ሾߙଵ ǡ ߙଶ ǡ ǥ ǡ ߙ௠ ሿ் and the vector ࢖ሺ‫ݔ‬ሻ, called base interpolating functions, contains typically monomial in the space coordinates ensuring that basis is complete. For a 2D problem, ࢖ ൌ ሾͳǡ ‫ݔ‬ǡ ‫ݕ‬ሿ் ࢌ࢕࢘࢓ ൌ ૜ and ࢖ ൌ ሾͳǡ ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݕݔ‬ǡ ‫ ݔ‬ଶ ǡ ‫ ݕ‬ଶ ሿ் ࢌ࢕࢘࢓ ൌ6

(2) (3)

Can be used. The function ‫ݑ‬ሺ‫ݔ‬ሻ can now be sampled at the ݊ points belonging to ȍூ giving ‫ݑ‬௛ ࢖் ‫ݑ‬ොଵ ‫ ۓ‬ଵ௛ ۗ ‫ ۓ‬ଵ் ۗ ‫ݑ‬ො ࢛௛ ൌ ‫ݑ‬ଶ ؆ ൞ ଶ ൢ ൌ ࢖ଶ ߙ ൌ ۱ࢻ ‫ڭ‬ ‫ۘ ڭ۔‬ ‫ۘ ڭ ۔‬ ‫ݑ‬ො௡ ‫்࢖ە‬௡ ۙ ‫ݑە‬௡௛ ۙ

(4)

Where ‫ݑ‬௝௛ ൌ ‫ݑ‬ሺ‫ݔ‬௝ ሻ are the unknow values of ‫ ݑ‬at point, ‫ݑ‬ො௝ ൌ ‫ݑ‬ොሺ‫ݔ‬௝ ሻare the approximate values, and ࢖௝ ൌ ࢖ሺ‫ݔ‬௝ ). In the case of FEM, the number of points is chosen so that ݉ ൌ ݊ . In this case ۱ is a square matrix [18]. If ݊ ൐ ݉ ,۱ is nor a square matrix and the approximation cannot fit all the ‫ݑ‬௝௛ values. This problema can be solved determining the ‫ݑ‬ො values by minimizing with respect to the ߙ parameters the sum of the square of the error ateach point weighted wiht a function ߱ሺ‫ݔ‬ሻ as ‫ ܬ‬ൌ σ௡௝ୀଵ ߱൫‫ݔ‬௝ ൯ሺ‫ݑ‬௝௛ െ ‫ݑ‬ොሺ‫ݔ‬௝ ሻሻଶ ൌ σ௡௝ୀଵ ߱൫‫ݔ‬௝ ൯ሺ‫ݑ‬௝௛ െ ࢖்௝ ߙሻଶ

(5)

This approximation is called weighted least square (WLS) interpolation. Note that for ߱ሺ‫ݔ‬ሻ ൌ ͳ the standard least square (LSQ) method is reproduced. Function ߱ሺ‫ݔ‬ሻ is usually built in such way that it takes a unit vale in thevicinity of the point ‫ ܫ‬typically called star node where the function (or its derivatives) are to be computed and vanishes outside a region ȍூ surrounding the point. The región ȍூ can be used to define the number of sampling points ݊ in the interpolation región. In this work, he normalized Gaussian weight function ߱ሺ‫ݔ‬ሻ is used, thus ݊ ൒ ݉ is always required. Several possibilities for selecting the weighting function ߱ሺ‫ݔ‬ሻ can be found in Oñate et al., Taylor et al. And Perazzo [1, 2, 19, 20]. Minimization of eq. (5) with respect to ߙ gives ߙ ൌ ۱ ିଵ ‫ܝ‬୦ ǡ ۱ ିଵ ൌ ‫ିۯ‬ଵ ۰ǡ (6) Where


‫ ܣ‬ൌ σ௡௝ୀଵ ߱ሺ‫ݔ‬௝ ሻ࢖ሺ‫ݔ‬௝ ሻ࢖் ሺ‫ݔ‬௝ ሻ ‫ ܤ‬ൌ ߱ሺ‫ݔ‬ଵ ሻ࢖ሺ‫ݔ‬ଵ ሻǡ ߱ሺ‫ݔ‬ଶ ሻ࢖ሺ‫ݔ‬ଶ ሻǡ ǥ ǡ ߱ሺ‫ݔ‬௡ ሻ࢖ሺ‫ݔ‬௡ ሻ

(7)

The final approximation is obtained by substituting ߙ from eq. (6) into eq. (1) giving ‫ݑ‬ොሺ‫ݔ‬ሻ ൌ ࢖் ۱ ିଵ ࢛௛ ൌ ĭ் ࢛௛ ൌ σ௡௝ୀଵ ߶௝ூ ‫ݑ‬௝௛

(8)

Where ିଵ ் ିଵ ߶௝ூ ሺ‫ݔ‬ሻ ൌ σ௠ ூୀଵ ‫݌‬ூ ሺ‫ݔ‬ሻூ௝ ൌ ࢖ ሺ‫ݔ‬ሻ۱

(9)

Are the shape functions. It must be noted that because of least square approximation‫ݑ‬ሺ‫ݔ‬௝ ሻ ؄ ‫ݑ‬ොሺ‫ݔ‬௝ ሻ ് ‫ݑ‬௝௛ . That is, the local values of the approximating function do not fit the nodal unknown values. The WLS approximation described above depends strongly on the shape and the way in which the weighting function is applied. The simplest way is to define a fixed unction ߱ሺ‫ݔ‬ሻ for each of the ȍூ interpolation subdomains [1, 2, 19 ]. Let ߱ሺ‫ݔ‬ሻ be a wighting function satisfying ߱௜ ሺ‫ݔ‬୧ ሻ ൌ ͳ ߱௜ ሺ‫ݔ‬ሻ ് Ͳ‫ א ݔ‬ȍூ ߱௜ ሺ‫ݔ‬ሻ ൌ Ͳ‫ ב ݔ‬ȍூ

(10)

Then the minimization of ‫ ܬ‬in eq (5) becomes ‫ܬ‬ூ ൌ σ௡௝ୀଵ ߱ூ ൫‫ݔ‬௝ ൯ሺ‫ݑ‬௝௛ െ ‫ݑ‬ොሺ‫ݔ‬௝ ሻሻଶ minimun

(11)

The Gaussean weight function used in this work is given by ୣ୶୮൫ିௗೕ Τ௖ ൯ିୣ୶୮ሺିሺ௥ Τ௖ሻሻ

߱ூ ൫‫ݔ‬௝ ൯ ൌ ൝


ǡ ݂݅݀௝ ൑ Ͳ

Ͳǡ ݂݅݀௝ ൐ Ͳ

(12)

Where ݀௝ ൌ‫ݔ צ‬௝ െ ‫ݔ‬ூ ‫צ‬ǡ ‫ ݎ‬ൌ ‫ݍ‬൛‫ݔ‬௝ ‫ א‬ȍூ ห ‫ݔ צ‬௝ െ ‫ݔ‬ூ ‫צ‬ሽ and ܿ ൌ ߚ‫ݎ‬. The support of this function is isotropic, circular and spherical in two and three-spatial, respectively. A detailed description of the effects of the parameters ‫ ݍ‬and ߚ on the numerical approximation and some guidelines for their setting have been presented in [20]. Note that according to eq (1), the approximate function ‫ݑ‬ොሺ‫ݔ‬ሻ is defined in each interpolation domain ȍூ . In fact, different interpolation sub-domains can yield different shape function ߶௝ூ . As a consequence of this, a point belonging to two or more overlapping clouds has different values of the shape functions which means that ߶௝௜ ് ߶௝௞ . The interpolation is now multivalued within ȍூ and, therefore for the approximation to be useful, a decision must be made in order to limit the choice to a single value. Indeed, the approximate function ‫ݑ‬ොሺ‫ݔ‬ሻ will be

91

92


typically used to provide the value of the unknown function ‫ݑ‬ሺ‫ݔ‬ሻ and its derivatives only in specific regions within each interpolation sub-domain. For instance by using point collocation the validity of the interpolation is limited to a single point ‫ݔ‬ூ . Damage models. The so called continuum damage model have been used thoroughly to simulate the behavior of materials that present degradation of the mechanical properties due to small fissures that appears during the loading process. To characterize this, the concept of effective stress ߪത is introduced. In one dimension we can writte ߪ ൌ ሺͳ െ ݀ሻߪ ഥ (13) where ݀ is the damage parameter which range from 0 to 1. The effective stress and strain are related by the Hooke´s law ߪത ൌ ‫ߝܧ‬ (14) where E is the elastic modulus. Thus, substituting eq (13) into eq (14) yields ߪ ൌ ሺͳ െ ݀ሻ‫( ߝܧ‬15) the properties of both damage models are summarized in Table 1 and Table 2. TABLE 1: Generic equation of a local damage model Constitutive equation. ߪ ൌ ሺͳ െ ݀ሻ‫ߝܥ‬ (16) Strain. ߝሺ‫ݔ‬ǡ ‫ݐ‬ሻ ൌ ‫ݑ׏‬ሺ‫ݔ‬ǡ ‫ݐ‬ሻ (17) Local variable. ܻሺ‫ݔ‬ǡ ‫ݐ‬ሻ ൌ ܻሺߝሺ‫ݔ‬ǡ ‫ݐ‬ሻሻ (18) Damage evolution. ݀ሺ‫ݔ‬ǡ ‫ݐ‬ሻ ൌ ݀ሺܻሺ‫ݔ‬ǡ ‫ݐ‬ሻሻ (19)

TABLE 2: Damage model based on non-local displacement, gradient version Constitutive equation. ߪ ൌ ሺͳ െ ݀ሻ‫ߝܥ‬ (20) Local strain. ߝሺ‫ݔ‬ǡ ‫ݐ‬ሻ ൌ ‫ݑ׏‬௔ ሺ‫ݔ‬ǡ ‫ݐ‬ሻ (21) Non-local displacement ‫ݑ‬௚ ሺ‫ݔ‬ǡ ‫ݐ‬ሻ െ ݈ ଶ ‫׏‬ଶ ‫ݑ‬௚ ሺ‫ݔ‬ǡ ‫ݐ‬ሻ ൌ ‫ݑ‬௔ ሺ‫ݔ‬ǡ ‫ݐ‬ሻ (22) Non-local strain ߝ௚ ሺ‫ݔ‬ǡ ‫ݐ‬ሻ ൌ ‫ݑ׏‬௚ ሺ‫ݔ‬ǡ ‫ݐ‬ሻ (23) Non-local state variable ܻ௚ ሺ‫ݔ‬ǡ ‫ݐ‬ሻ ൌ ܻሺߝ௚ ሺ‫ݔ‬ǡ ‫ݐ‬ሻሻ (24) Damage evolution ݀ሺ‫ݔ‬ǡ ‫ݐ‬ሻ ൌ ݀ሺܻ௚ ሺ‫ݔ‬ǡ ‫ݐ‬ሻሻ (25)

Numerical implementation. Local damage model in FPM. Considerer the system of differential equations which governs the behavior of a solid ‫ ߪ׏‬൅ ߩܾ ൌ Ͳ‫ א ݔ׊‬π (26) ߪ ȉ ݊ ൌ ‫ݐ‬ҧ‫ א ݔ׊‬Ȟ୲ ሺʹ͹ሻ ‫ ݑ‬ൌ ‫ݑ‬௣ ‫ א ݔ׊‬Ȟ୳ (28) using the point collocation method, we can obtain a discrete system of equations[6] ‫ܮ‬୘ ߪ ൅ ߩܾ ൌ Ͳ‫ א ݔ׊‬π (29) ܰ ் ߪ ൌ ‫ݐ‬ҧ‫ א ݔ׊‬Ȟ୲ ሺ͵Ͳሻ ‫ ݑ‬ൌ ‫ݑ‬௣ ‫ א ݔ׊‬Ȟ୳ ሺ͵ͳሻ where ‫ ܮ‬is a operator that defines the differential equation an N contains the normal direction on the external contour.


93

To obtain an equivalent system in term of displacement, use the stress-strain relation and the strain-displacement relationships as follows [22]. ߪ ൌ ሺͳ െ ݀ሻ‫ߝܥ‬, where ߝ ൌ ‫( ݑܮ‬32) ሺ‫ ܫ‬െ ݀ሻ‫ ݑ߶ܮܥ ்ܮ‬௛ ൅ ሺ‫ ܫ‬െ ݀ሻሺ‫߶ܮ‬ሻ் ‫ ݑ߶ܮܥ‬௛ െ ሺ‫݀߶ܮ‬ሻ் ‫ ݑ߶ܮܥ‬௛ ൌ െߩܾ‫ א ݔ׊‬π (33) ܰ ் ‫ ݑ߶ܮܥ‬௛ ൌ ‫ݐ‬ҧ‫ א ݔ׊‬Ȟ୲ ሺ͵Ͷሻ ߶‫ ݑ‬௛ ൌ ‫ݑ‬௣ ‫ א ݔ׊‬Ȟ୳ (35) with its compact form ‫ܭ‬ௗூ ‫ݑ‬௛ ൌ ݂ூ ‫ ܫ‬ൌ ͳǡ ǥ Ǥ ݊ (36) Where ‫ܭ‬ௗ is the stiffness matrix in FPM and ݂ூ contain the equilibrium and boundary conditions. Non-local damage model in FPM. Consider the stiffness matrix and the model properties of Table 2, to approximate the diffusion-reaction eq (22) we use the second-order shape function directly [17] and then we obtain the new stiffness matrix for the non-local problem. ‫ܭ‬ே௅ூ ൌ ൤

‫ܭ‬ௗூ െ߶

૙ ൨ (37) ሺ߶ െ ݈ ଶ ‫ܦ‬ሻ

Where D is the diffusivity matrix defined as, ‫ ܦ‬ൌ σଷ௞ୀଵ ߶௞௞ (38) Here ߶௞௞ represent the spatial differentially shape functions. The compact form is now ‫ܭ‬ே௅ூ ‫ݑ‬௛ ൌ ݂ூ (39) Where ‫ݑ‬௛ ൌ ሾ‫ݑ‬௔௛ ǡ ‫ݑ‬௚௛ ሿ and ݂ூ contain the equilibrium condition and the zeros of the reaction-diffusion equation. One-dimensional elastic damage example. The example consists of a rod subjected to uniaxial tensile load by displacement control as show fig. 1. The central tenth of the bar is weakened 10% by a reduction in Young´s modulus to force the localization . The dimensionless geometrical and mechanical parameters are summarized in Table 3. Table 3: Uniaxial tension test Description Length of the bar Length of weaker part Cross-section Young´s modulus Damage Threshold Final strain

Symbol ‫ܮ‬଴ ݈௜ ‫ܣ‬ ‫ܧ‬ ߝ଴ ߝ௙

Value 100 10 1 20000 10-4 1.25* 10-2

In Fig. 2 is plotted the behavior of the internal variable of damage for the multiple discretizations and the variation of the specific energy for the regularized local-damage model via internal length.

94


Figure 1: One dimensional rod model

Figure 2: Results with a various discretizations. a) Final damage profile. b) Stress-Strain profile central node. In Fig.3 is plotted the evolution for load-unload for 320 nodes and, showing an elastic response for the specific load step.

Figure 3: Load- unload with partial damage. a) Damage evolution. b) Partial load-unload process. In Fig. 4 is plotted the behavior of the internal variable of damage for multliple discretizations and the reaction force for the regularized non-local damage model via non-local displacements. These results are according to [15].


95

Figure 4: Full loading process for different discretizations and fixed internal length ݈ ଶ ൌ ͷ. a) Damage evolution. b) Reaction force vs displacements. In Fig. 5 is plotted the behavior of the non-local model for a fixed number of nodes and multiplies internal lengths into the diffusion-reaction equation. The results are similar as in [15].

Figure 5: Full loading process for 81 nodes and multiplies internal length a) Damage evolution. b) Reaction force vs displacements. Conclusion. Has been implemented two models of isotropic damage in strong form by the meshless finite points method to simulate the non-linear materials behavior. As the results demonstrate the FPM is able to approximate the localization phenomena with fracture energy regularization by the consideration of characteristic material length and the diffusion-reaction equation. This works concludes the opening of a new alternative viewpoint to the classical weak formulation based on MEF. For a computational aspect, the use of the shape function to approximate all the fields and the simplicity of the development algorithm can be highly attractive.

96


As future lines, the FPM should be advanced to optimize the clouds formulation criterion at the borders and for the damage side the extension to 2D and 3D and a coupling technique FPM-FEM for the application of the non-local gradient enhanced models. Acknowledgements. The authors acknowledge the financial support from the Chilean agency CONICYT (FONDECYT Project 11100253) and UTFSM-DGIP ( PIIC 2011 studentship). References [1] Oñate E, Idelsohn S, Zienkiewicz O, Taylor R, Sacco C. A stabilized finite point method for for analysis of fluid mechanics problems. Comput Methods Appl Mech Eng 1996; 139:315-46. [2] Oñate E, Idelsohn S, Zienkiewicz O, Taylor R. A finite point methods in computational mechanics, application to convective transport and fluid flow. Int J Numer Methods Eng 1996;39:3839-66. [3]Oñate E, Idelsohn S. A mesh free finite point method for advective-diffusive transport and fluid flow problems. Computat Mesch 1988;21_283-92. [4] Oñate E, Sacco C, Idelsohn S. A finite point method for incompressible flow problems. Comput Vis Sci 2000; [5] Oñate E, PerazzoF, Miquel J. A finite point method for elasticity problems. Comput Struct 2001;79:2152-63 [6] Perazzo F. Una metodología numerica sin malla para la resolución de las ecuaciones de elasticidad mediante el método de puntos finites. Univeritat Politécnica de Cataluña, Barcelona España. Tesis doctoral; 2002. [7] Perazzo F, Oller S, Miquel J, Oñate E. Avances en el método de puntos finites para la mecanica de solidos. Revista internacional de métodos numéricos en ingeniería 2006;22:153-68. [8] Martin A. Análisis y formulación de un estimador del error en el método sin malla de punto finites. Universidad Técnica Federico Santa María, Valparaíso Chile. Trabajo de titulo; 2006. [9] Perazzo F, Miquel J, Oñate E. EL método de puntos finites para problemas de la dinámica de sólidos. Revista internacional de métodos numerícos en ingenieria 2004;20:235-46. [10] Pérez-Pozo L, Perazzo F. Non-linear material behavior analysis using the meshless finite point method In: 2nd ECCOMAS thematic conference on meshless methods, Porto, Portugal ; 2007. P. 251-68. [11] Lemaitre J. and J. L. Chaboche. Mechanics of solid materials. Cambridge University Press. [12] Hilleborg A., M. Modeer, and P.A. Petterson. Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement and concrete research.6, 773-782. [13] Oliver J. A. E. Huespe, M. D. Pulido, and E. Chaves . From continuum mechanics to fracture mechanics: the strong discontinuity approach. Engineering Fracture Mechanics 69(2), 113-136. [14] Rodríguez-Ferran A., I. Morata , and A. Huerta. A new damage model based on non-local displacements. International Journal for numerical and analytical methods in geomechanics 29(5), 473-493. [15] Bazant, Z. P. and B. H. Oh. Crack band theory for fracture in concrete. Materials and structures 16(3),155-177. [16] Pérez-Pozo L, Chacana F. Avances en la reguarización de la energía de gracira en un modelo de daño isotropic mediante un método sin malla. CILAMCE 2011. [17] Pérez-Pozo L, Campos A. Regularización de la localización por medio de gradients de deformación plastic no local z el método sin malla de puntos finites. [18] Zienkiewicz O. & Taylor R. El método de los elementos finitos, vol. 1. Centro internacional de métodos numéricos en ingeniería, Barcelona España; 2000. [19] Taylor R, Idelsohn S, Zienkiewics O, Oñate E. Moving least square approximations for solution of differential equations. CIMNE research report, 1995; 74. [20] Ortega E, Oñate E, Idelsohn S. An improved finite point method for trhee-dimensional potential flows, Computat Mech 2007;40:949-63.


Genetic Algorithms and the Method of Fundamental Solutions for Simulations of Cathodic Protection Systems W. J. Santos1, J. A. F. Santiago2 and J. C. F. Telles3 Programa de Engenharia Civil, COPPE/UFRJ. Rio de Janeiro – Brazil 1

[email protected], 2 [email protected], 3 [email protected]

Keywords: Optimization, MFS BEM.

Abstract. The aim of this paper is to present numerical simulations of Cathodic Protection (CP) Systems using a Genetic Algorithm (GA) and the Method of Fundamental Solutions (MFS). MFS is used to obtain the solution of the associated homogeneous equation with the non-homogeneous equation subject to nonlinear boundary conditions defined as polarization curves. The adopted GA will minimize a nonlinear error function, whose optimization variables are the coefficients of the linear superposition of fundamental solutions and the positions of the source points, located outside the problem domain. In this work, the anodes added to the CP system are considered as point sources and therefore the integral that represents the particular solution can be obtained analytically. The results presented here include a comparison with a direct boundary element (BEM) solution procedure. Simulations are performed considering finite regions in ܴଶ .

Introduction Cathodic Protection (CP) is a corrosion prevention technique which uses the electrochemical properties of metals to insure that the structure to be protected becomes the cathode of an electrolytic cell. The technique is commonly used for protecting metallic structures placed in aggressive environments, e.g. ship hulls, offshore structures and underground pipelines. In a CP system, the location and the impressed current of the anodes have to be determined with the goal of providing a uniform potential distribution on the metal surface, below a critical potential (߶௖ ) at which the structure is protected from corrosion. The electrochemical potential problem is governed by the Poisson equation with boundary conditions given by a polarization curve, which is a non-linear relationship between the electrochemical potential (߶) and current density (݅). The BEM is one of the most appropriate techniques to solve problems involving CP systems. Several applications of BEM to study CP systems have been reported in the literature, including reference to practical analyses performed by oil companies [1,2,3]. The MFS belongs to the class of meshless methods and is a technique which can also be applied to CP problems, even though not many references can be found in the literature. In the MFS, the approximate solution of the problem is represented in the form of a linear superposition of fundamental solutions with singular points located outside the domain of the problem. These singular points are called source points and form a “pseudo-boundary” having no common points with the actual boundary of the region. The essence of the MFS is the use of a fundamental solution which satisfies the associated homogeneous differential equation in any point except at the source point. The unknown source intensities producing the approximate solution are determined by imposing satisfaction of the boundary conditions at a set of boundary points (collocation points). Just like BEM, MFS is applicable when a fundamental solution of the differential equation in question is known, with the advantage of not requiring any integration procedure or specific treatment for the singularities of the fundamental solution. The basic problem for the application of the MFS is the determination of the positions of the source points. Generally, in 2-D applications the arrangement of the source points is on a circular contour or on a contour

97

98


geometrically similar to the actual boundary of the region under consideration. Nevertheless, the accuracy of the numerical solution depend on the radius of such a circle or on the distance from the source points over the geometrically similar boundary contour to the problem boundary, especially due to possible illconditioning of the algebraic system of equations formed. In [4], for instance, a genetic algorithm (GA) is proposed for the optimal positioning of source points. The singular value decomposition (SVD) idea can also provide means to obtain acceptable solutions to the ill-conditioned equations system and has been successfully applied to MFS [5]. The present work uses MFS to obtain the numerical solution of the associated homogeneous equation which, added to the particular solution, represents the electrochemical potential of metal surfaces immersed in electrolytes (domains). The aim is to simulate CP systems capable of providing a potential distribution on the metallic structure surface below the critical potential. The physical behaviour of metal surfaces is modelled by a polarization curve, which describes the nonlinear relation between potential and current densities. Thus, the unknown coefficients of the linear superposition of fundamental solutions and the positions of source points, located outside the problem domain, are determined by minimizing a nonlinear error function. This is here accomplished using a GA. Examples of application are presented considering finite regions in ܴଶ for different geometries.

The MFS for CP The mathematical model of the problem, within this conducting domain π (electrolyte), is based on a Poisson equation for the electrochemical potential: ݇ߘ ଶ ߶ ሺ࢞ሻ ൌ ܾሺ࢞ሻǡ ࢞ ‫ א‬πǡ (1) where ܾ is a known function representing the anodes as external sources and ݇ is the conductivity of the electrolyte. In the present work, the metal surfaces are considered to be in direct contact with the electrolyte and therefore the boundary conditions related to eq (1) are given in the following form ݅ ሺ࢞ሻ ൌ ‫ ܨ‬ሺ߶ሻǡ ࢞ ‫߁ א‬ǡ (2) where ߁ is the boundary of π, ݅ ሺ࢞ሻ is the current density in the outward normal direction ࢔ and ‫ ܨ‬ሺ߶ሻ is a nonlinear function of the ߶. The general solution ሺ߶௚ ሻ of eq (1) is given by adding a particular solution ሺ߶௣ ሻ to the solution of the associated homogeneous equationሺ߶௛ ሻ, subjected the corresponding homogeneous boundary conditions. Any particular solutions of eq (1) can be written integral form as follows

߶௣ ሺ࢞ሻ ൌ ‫׬‬π ‫ ܩ‬ሺࣈǡ ࢞ሻܾሺࣈሻ ݀Ǥ

(3)

The function ‫ ܩ‬ሺࣈǡ ࢞ሻ is a fundamental solution of Laplace's equation given by ଵ

ଵ

ሺ૆ǡ ‫ܠ‬ሻ ൌ ଶ஠୩ ቀ ୰ ቁ ǡ where ‫ ݎ‬is the Euclidean distance between point ࣈ and the field point ࢞.

(4)

Treating the anodes as point sources, the term ܾሺ࢞ሻ is equal to ௡

೛ೞ ௣௦ ܾ ሺ࢞ሻ ൌ ෌௠ୀଵ ܲ൫࢞௣௦ (5) ௠ ൯ߜ൫࢞௠ ǡ ࢞൯ǡ ௣௦ where ࢞ are the coordinates of the point sources, ܲሺ࢞ ሻ is the intensity of the source given in amps (A), ߜሺࣈǡ ࢞ሻ is the Dirac delta function and ݊௣௦ is the number of point sources inserted in the electrolyte. Therefore

௣௦


௡೛ೞ

99

௣௦ ߶௣ ሺ࢞ሻ ൌ ෍ ܲ൫࢞௣௦ ௠ ൯ න ‫ ܩ‬ሺࣈǡ ࢞ሻߜ൫࢞௠ ǡ ࣈ൯ ݀ߦ

ൌ

௠ୀଵ ଵ ଶ஠୩

π

௡

ଵ

೛ೞ ෌௠ୀଵ ܲ൫࢞௣௦ ௠ ൯ ቀ ቁ ǡ

where now ‫ ݎ‬is the Euclidean distance between point

௣௦ ࢞௠

(6)

୰

and the point ࢞.

In addition, from Ohm's law, the particular solution for a current density is equal to ݅௣ ሺ࢞ሻ ൌ ݇

డథ೛

ൌെ

డ௡

ଵ ଶగ

෍

௡೛ೞ ௠ୀଵ

ଵ డ௥

ܲ൫࢞௣௦ ௠ ൯

௥ డ௡

.

(7)

The numerical solution ߶௛ can be obtained by BEM or MFS. This paper presents the formulation of MFS. The approximate solution of the problem by MFS is represented in the form of a linear superposition of fundamental solutions with singular pointsሺ࢞௦௣ ሻ located outside the domain of the problem. Thus, the electrochemical potential may be written by the summation ߶௛ ሺ࢞ሻ ൌ ෍

௡ೞ೛

௦௣

௝ୀଵ

‫ܩ‬ሺ࢞ǡ ࢞௝ ሻ ܿ௝ ǡ

(8)

with ݊௦௣ being the number of source points and the coefficients that occur in the approximate solution are the unknown constants. Similarly, defining ‫ ܪ‬ൌ ݇

డீ డ௡

, the homogeneous solution for the current density ሺ݅௛ ሻ is given as ݅௛ ሺ࢞ሻ ൌ ෍

௡ೞ೛ ௝ୀଵ

௦௣

‫ܪ‬ሺ࢞ǡ ࢞௝ ሻ ܿ௝ Ǥ

(9)

The aim in MFS is to determine the coefficients ܿ௝ Ԣ‫ ݏ‬by satisfaction of the boundary condition at collocation points. The polarization curve of the structure describes data obtained by a series of experiments in a standard corrosion cell using the dc-potentiodynamic technique [6], and is given by the expression: ݅ ൌ ‫ ܨ‬ሺ߶ሻ ൌ ݁

ഝశలవయǤవభ ഁభ

ଵ

െ൤ ൅݁

ଶ

௜భ

ഝశఱమభǤల ഁమ

ିଵ

൨

െ݁

ି

ഝశళబళǤఱళ ഁయ

ǡ

(10)

with ߶ and ݅ having units ܸ݉ and ߤ‫ܣ‬Ȁܿ݉ , respectively, and ߚଵ ǡ ߚଶ ǡ ߚଷ e ݅ଵ are given constant parameters: ߚଵ ൌ ʹͶܸ݉, ߚଶ ൌ ʹ͵ǤͶ͹ܸ݉,ߚଷ ൌ ͷͷܸ݉ e ݅ଵ ൌ ͺ͸ǤͲ͸ߤ‫ܣ‬Ȁܿ݉ ଶ. The conductivity of the electrolyte is equal to ݇ ൌ ͲǤͲͶ͹ͻπିଵ ܿ݉ ିଵ and the critical value of the electrochemical potential is ߶௖ ൌ െͺͷͲܸ݉. The general solution of the problem must satisfy eq (10), i.e., ݅௚ ൌ ݅௣ ൅ ݅௛ ൌ ‫ܨ‬൫߶௣ ൅ ߶௛ ൯ ൌ ‫ܨ‬൫߶௚ ൯. This relationship results in a problem of nonlinear least squares with optimization variables defined as the coefficients ܿ௝ Ԣ‫ ݏ‬and the positions of the source points. The optimization is solved using a GA, which will minimize the following objective function ଵ

௡

ଶ

ೞ೛ ൣ݅௚௡ െ ‫ܨ‬൫߶௚௡ ൯൧ ǡ ܼሺࢉǡ ࢞ ௦௣ ሻ ൌ ට௡ σ௡ୀଵ ೎೛

(11)

where Z calculates the difference between the current density ሺ݅௣ ൅ ݅௛ ሻ calculated values and eq (10), applied in the general solution of the electrochemical potential, at each collocation point. Also, ݊௖௣ is the number of collocation points and ࢉ is a vector containing the coefficients ܿ௝ Ԣ‫ݏ‬. The adopted GA used for the minimization of eq (11) has a binary representation and is inspired by the algorithm presented in [7]. However, the two-point crossover, elitism, and the probabilities of mutation were included and crossover vary linearly over the generations.

100


Applications Simulations are performed considering metal surfaces in contact with an electrolyte and modelled by a polarization curve, defined in eq (10). An anode is placed in the electrolyte with the goal of providing a potential distribution on the metal surface below the critical potential߶௖ ൌ െͺͷͲܸ݉. ௡೎೛

ଷ௡೎೛

In these applications, ݊௦௣ ൌ ଶ . The number of the optimization variables is equal to ଶ , because includes the coefficients ܿ௝ Ԣ‫ ݏ‬and the positions of source points, ࢞ ௦௣ ൌ ሺ‫ݔ‬௦௣ ǡ ‫ݕ‬௦௣ ሻ. It is also assumed that all source points should be located in a certain area; i.e., the distant between any source point and geometry should not be greater than a particular value. The range of coefficients is assumed to be ሾെͷͲͲͲǤͲǡ ͷͲͲͲǤͲሿǤ Example1. In the first simulation a metallic structure in the form of a rectangle with a localized reentrance in the upper right corner is studied. The dimensions of the structure are ͳͲͲܿ݉‫ݔ‬ͷͲܿ݉. An anode is placed in the electrolyte with a current intensity ofെͷͲͲͲͲǤͲͲߤ‫ܣ‬, sufficient to maintain the potential on the metal surface below the critical potential. The calculations were performed for ݊௖௣ ൌ ͳͷͲ collocation points, ݊௦௣ ൌ ͹ͷ source points and 225 optimization variables. The distance between any source point and the geometry should not be greater than͵Ͳܿ݉. The plot of the source points arrangement determined by GA after 10000 generations is presented in Fig. 1. Fig. 2 presents the potential distribution on the metal determined by MFS and by BEM with 150 constant boundary elements. The error, defined in eq (11), between the potential values on the boundary determined by BEM and MFS is ͲǤͻͶͺͲͻ͹ܸ݉. The computational time for convergence was approximately 12min.

Figure 1: Source points arrangement by GA.

Figure 2: Potential distribution on the metal.


In Fig. 3 the potential distribution in the electrolyte using BEM is presented and in Fig. 4 the MFS counterpart result is presented. The error between the potential values in the internal points determined by BEM and the MFS isͲǤʹͷ͹ͳͶͻܸ݉.

Figure 3: Potential in the electrolyte using BEM.

Figure 4: Potential in the electrolyte using MFS.

Example2. The next simulation this a metallic structure in the form of a square cross. The dimensions of the structure are ͳͷͲܿ݉‫ͳݔ‬ͷͲܿ݉. Similarly to the first example, an anode is placed in the electrolyte with current intensity given byെͻͳͲͲͲǤͲͲߤ‫ܣ‬. The calculations were performed for ݊௖௣ ൌ ͳ͸ͺ collocation points, ݊௦௣ ൌ ͺͶ source points and 252 optimization variables. The distance between any source point and geometry should not be greater thanʹͷܿ݉. The plot of source points arrangement determined by GA after 20000 generations is presented in Fig. 5. In Fig. 6 the potential distribution on the metal determined by MFS and by BEM with 168 constant boundary elements is presented. The error between the potential values on the boundary determined by BEM and the MFS is ͲǤͻͻͺ͵͸ʹܸ݉. The computational time for convergence was approximately 35min.

Figure 5: Source points arrangement by GA.

Figure 6: Potential distribution on the metal.

101

102


In Fig. 7 is presented the potential distribution in the electrolyte using BEM and Fig. 8 presents the MFS solution. The error between the potential values in the internal points determined by BEM and MFS isͲǤ͹͵ͻʹͶͶܸ݉.

Figure 7: Potential in the electrolyte using BEM.

Figure 8: Potential in the electrolyte using MFS.

Conclusions The analyzes performed using constant elements BEM and the proposed GA MFS indicate a good agreement between the electrochemical potential distribution on the metal surface and in the electrolyte. The results found confirm GA as a robust optimization procedure to work on such problems. In the first example were determined 225 optimization variables and in the second example there were 252 optimization variables.

References

[1] J. C. F. Telles, W. J. Mansur, L. C. Wrobel and M. G. Marinho Numerical Simulation of a Cathodically Protected Semisubmersible Platform using PROCAT System, Corrosion, 46, 513-518 (1990). [2] J. A. F. Santiago and J. C. F. Telles On Boundary Elements for Simulation of Cathodic Protection Systems with Dynamic Polarization Curves, International Journal for Numerical Methods in Engineering, 40, 2611-2622 (1997). [3] P. Miltiadou and L. C. Wrobel A BEM-Based Genetic Algorithm for Identification of Polarization Curves in the Cathodic Protection System, International Journal for Numerical Methods in Engineering, 54, 159174 (2002). [4] R. Nishimura, M. Nishihara, K. Nishimori and N. Ishihara Automatic Arrangement of Fictitious Charges and Contour Points in Charge Simulation Method for two Spherical Electrodes, J Electr, 57, 337-346 (2003). [5] P. A. Ramachandran Method of Fundamental Solutions: Singular Value Decomposition Analysis, Communication in Numerical Methods in Engineering, 18, 789-891 (2002). [6] J. F. Yan, S. N. R. Pakalapati, T. V. Nguyen and R. E. White Mathematical Modelling of Cathodic Protection using the Boundary Element Method with Nonlinear Polarisation Curves, Journal of the Electrochemical Society, 139, 1932-1936 (1992). [7] Z. Michalewicz Genetic Algorithms + Data Structures = Evolution Programs, Spinger-Verlag (1996).


Slipping Stokes flow about a solid particle experiencing a rigid-body motion A. Sellier LadHyx. Ecole polytechnique, 91128 Palaiseau Cédex, France e-mail: [email protected] Keywords: Stokes flow, Navier slip condition, Boundary-integral equation, Green tensor.

Abstract. The net hydrodynamic force and torque exerted on a slipping and arbitrarily-shaped solid particle, experiencing a prescribed slow rigid-body migration in a quiescent Newtonian liquid, are determined by appealing to a new approach. The advocated method rests on a boundary formulation which makes it possible to easily calculate the flow about the particle and actually reduces the task to the treatment of six relevant boundary-integral equations on the particle surface. Those integral equations are numerically inverted by implementing a boundary element collocation method. The whole strategy is then tested against analytical and numerical results obtained elsewhere for a spherical and a spheroidal particle. Introduction As it is well known, the flow about a solid body deeply depends upon the prescribed boundary conditions on the body surface. Usually one requires the widely-employed no-slip condition which specifies that the fluid velocity is equal to the body velocity at each point of the surface. However, this condition breaks down for some surfaces such as, for instance, hydrophobic ones. In such circumstances, one then usually applies the famous Navier [1] slip condition (see equation (2) below) which introduces on the body surface a slip velocity proportional to the tangential viscous stress. More precisely, the constant of proportionality is λ/µ with µ the liquid kinematic viscosity and λ ≥ 0 the so-called slip length characterizing the surface slipping properties. Whenever inertial effects are negligible the flow about the moving solid and slipping particle is a Stokes flow driven by the adopted slip Navier condition on the body surface. Within this framework, the determination of this flow for a particle with arbitrary shape and slip lentgh λ is unfortunately a very challenging task! This explains why the available analytical, asymptotic or numerical treatments solely apply to a sphere (analytical solution in [2]), a slighty deformed sphere (asymptotic treatments in [3-6]) and a spheroidal or axisymmetric particle (numerical approach in [7-10]). This work presents a new method which, in contrast to the previous literature, is valid for arbitrarily-shaped particles. Addressed problem and analytical solution for a spherical particle. This section presents the problem for an arbitrarily-shaped slip particle. Governing equations and basic issues As sketched in Fig. 1, we consider a solid particle P with smooth surface S and attached point O

µ, ρ

S

P O •

n U Ω

Figure 1. A solid particle P experiencing a prescribed rigid-body migration.

103

104


immersed in a quiescent and unbounded Newtonian liquid with uniform density ρ and viscosity µ. This particle experiences a prescribed rigid-body migration with translational velocity U, which is the velocity of its point O, and angular velocity Ω. Morevover, the particle has typical length scale a and the flow about it has pressure p and velocity u with typical magnitude V. Assuming that Re = ρV a/µ vanishes all inertial effects are negligible and (u, p) obeys in the liquid domain D the steady Stokes equations and far-field behaviour µ∇2 u = ∇p and ∇.u = 0 in D, (u, p) → (0, 0) at ∞.

(1)

At the smooth particle surface S, with unit normal n directed into the liquid, the slipping of the flow (u, p) with stress tensor σ is here taken into account by enforcing the so-called and widely-employed Navier [1] condition u(M ) = U + Ω ∧ OM + λ{σ.n − (n.σ.n)σ.n}/µ on S

(2)

where λ ≥ 0 designates the slip length. Solving (1)-(2) for prescribed rigid-body motion (U, Ω) and slip length λ then provides the flow (u, p) and the surface traction f = σ.n prevailing on the particle surface. One is then able to subsequently calculate the following net hydrodynamic force F and torque Γ, about the attached point O, experienced by the moving particle

F=

S

σ.ndS, Γ =

S

x ∧ σ.ndS.

(3)

At a first glance such a problem seems simple. This is true for a sphere [2] while it already becomes rather involved for a translating spheroidal particle (see [3-10]). In practice, solving (1)-(2) for a particle with arbitrary shape for prescribed rigid-body motion (U, Ω) and slip length λ ≥ 0 is still a very challenging issue for which there is, to the author’s very best knowledge, not yet a general treatment! Analytical solution for a sphere. As mentioned earlier, it is indeed possible to analytically solve the problem (1)-(2) for a spherical particle with center O and radius a. Here we successively distinguish two cases: (i) The translating sphere. When the sphere translates, without rotating, at the velocity U it is possible to seek the Stokes flow (u, p) about the sphere under the following form u=

s (s.x)x (d.x)x d s.x +3 − 3 and p = 2µ( 3 ) for r = |x| > a. + r r3 r5 r r

(4)

In other words (see, for instance, [11]) the flow (u, p) is produced by a Stokeslet and a potentiel dipole located at the sphere center having unknown strength s and d, respectively. Note that (1) is satisfied whatever (s, d). Enforcing the Navier boundary condition (2) requires that a3 s − ad = a4 U + 6λd, a3 s + 3ad = −6λd. Accordingly, one gets s=

3a(1 + 2λ/a)U a3 U , d=− 4(1 + 3λ/a) 4(1 + 3λ/a)

(5)

(6)

and the surface force f = σ.n, the net force F and the net torque Γ exerted on the sphere then read

f =−

3µa3 (U.x)x 1 + 2λ/a U + 6λ , F = −6πµa[ ]U, Γ = 0. 2(1 + 3λ/a) a3 1 + 3λ/a

(7)

(ii) The rotating sphere. When the sphere rotatates, without translating, at the velocity Ω we put at its center a rotlet with unknown strength γ. Hence, we take this time u=

γ∧x and p = 0 for r = |x| > a. r3

(8)


105

The flow (u, p) given by (8) fulfills (1) and the vector γ is obtained from the Navier boundary condition (2). One easily gets a3 Ω (9) γ= 1 + 3λ/a and the resulting surface force f = σ.n, net force F and net torque Γ f =−

Ω 3µ[Ω ∧ x] , F = 0, Γ = −8πµa3 [ ]. a(1 + 3λ/a) 1 + 3λ/a

(10)

Boundary formulation. This basic section presents a new boundary approach to efficiently solve the problem (1)-(2) for arbitrarily-shaped particles. Velocity representation. Henceforth, we shall employ the usual tensor summation convention with, for instance, x = xi ei and n = ni ei . It is well known that the velocity field u gouverned by (1) receives in the entire liquid domain the following integral representation [12] u(x).ej =

1 8π

S

[

ei .σ.n ](y)Gij (y, x) − [u(y).ei ]Tijk (y, x)nk (y) dS(y) for x in D µ

(11)

with, denoting by δ the Kronecker delta symbol, the following definitions δij [(y − x).ei ][(y − x).ej ] , + |x − y| |x − y|3 6[(y − x).ei ][(y − x).ej ][(y − x).ek ] . Tijk (y, x) = − |x − y|5 Gij (x, y) =

(12) (13)

Setting u = uj ej and exploiting the identity

Tijk (y, x)nk (y)dS(y) = 0 for x in D,

Sn

(14)

makes it possible to cast (11) into the equivalent and fruitful form

8πuj (x) =

S

[ui (y) − ui (x)]Tijk (y, x)nk (y)dS(y) −

1 µ

S

[ei .σ.n](y)Gij (y, x)dS(y) for x in D. (15)

The representation (15) clearly shows that it is possible to calculate the velocity field u = uj ej in the entire liquid domain D from the knowledge of only two surface quantities on the particule boundary: the vectors σ.n and u. Relevant boundary-integral equation. Let us introduce on the particule surface S the unknown quantity a and vector a tangent to S as a = n.σ.n/µ, a = [σ.n − (n.σ.n)n]/µ = ai ei .

(16)

Inspecting (12)-(13) clearly shows that (15) also holds for x located on the particule surface S! Injecting in the obtained relation the Navier boundary condition (2) then easily provides the following regularized boundary-integral equation Li [a, a] = [U + Ω ∧ OM].ei for x on S and i = 1, 2, 3

(17)

with the linear operators Li [a, a] = −8πλai (x) − −

S

S

Gki (y, x)nk (y)a(y)dS(y)

Gki (y, x)ak (y)dS(y) + λ

S

[ak (y) − ak (x)]Tkil (y, x)nl (y)dS(y).

(18)

106


Inverting (17) with a.n = 0 provides on the particle surface the required traction σ.n = µ[an + a]. Invoking (2) one then immediately gains the velocity u on the surface and also, if needed, the velocity u in the entire liquid domain by appealing to (15). Numerical implementation and comparisons. As previously-explained, the proposed approach reduces to the accurate treatment of the boundaryintegral equation (17). We briefly present below the implemented numerical procedure and also report benchmark tests and preliminary new results. Numerical strategy The boundary-integral equation (17) is numerically inverted by resorting to a collocation point method. More precisely, we use on the particle surface S a N −node mesh obtained by appealing, for a sake of accuracy, to 6-node curvilinear and triangular boundary elements. At each nodal point we introduce not only the unit normal n but also two additional unit vectors t1 and t2 tangent to the particule surface S and such that t1 .t2 = 0. By definition (recall (16)) the vector a satisfies a.n = 0 and therefore writes a = at1 t1 +at2 t2 . Accordingly, one ends up for a given N −node mesh with 3N unknown quantities at the nodal points: the normal components a (again, recall (16)) and the new components at1 and at2 . Of course, when handling (17) one has to employ the relation ak = at1 t1 .ek + at2 t2 .ek . Once discretized, (17) then yields a linear system AX = Y with a 3N × 3N, non-symmetric and fullypopulated influence matrix A (the reader is directed to [13] for additional details regarding the proper way to calculate the entries of this matrix A). Finally, the system AX = Y is solved by Gaussian elimination. Comparisons The advocated boundary approach and associated numerical implementation have been benchmarked against the analytical results available for spherical and spheroidal particles. For further purposes we actually consider a general ellipsoidal particle with center O and semi-axis bi > 0. The surface S then admits the equation (x1 /b1 )2 + (x2 /b2 )2 + (x3 /b3 )2 = 1 and symmetries easily show that for such an orthotropic particle F = −6πµb1 fi U and Γ = 0 for U ∧ ei = Ω = 0,

(19)

Γ = −8πµb31 ci Ω and F = 0 for Ω ∧ ei = U = 0

(20) (21)

where the occurring (dimensionless) friction coefficients f1 , f2 , f3 , c1 , c2 and c3 depend upon the ellipsoid normalized slip length λ/b1 ≥ 0 and geometry through the two parameters b2 /b1 and b3 /b1 . By virtue of (7) and (10), one gets for a sphere with radius a (take b1 = b2 = b3 = a) the identities f1 = f2 = f3 = fsphere and c1 = c2 = c3 = csphere with fsphere = (1 + 2λ/a)csphere and csphere = (1 + 3λ/a)−1 . As seen in Table 1, our numerical computations nicely retrieve those analytical results as the number N of nodal points spread on the sphere boundary increases. Table 1: Computed friction coefficients fi and ci for a sphere with radius a versus the number N of nodal points for λ/a = 0.5, 2. fi ci fi ci

λ/a 0.5 0.5 2 2

N = 74 0.80287 0.39817 0.72045 0.14188

N = 242 0.80018 0.39988 0.71483 0.14278

N = 1058 0.80001 0.39999 0.71433 0.14285

analytical 0.8 0.4 0.71429 0.14286

Additional comparisons for the force friction coefficients fi have been achieved against the results given in [7,11] for a translating (no rotation) prolate or oblate spheroid such that b2 = b1 and b3 > b1

Advances in Boundary Element and Meshless Techniques or b3 < b1 , respectively. Here f1 = f2 and, as illustrated in Table 2, the computed values of f1 and f3 perfectly agree with the ones obtained in [7,11] using another procedure. Table 2: Computed force friction coefficients f1 = f2 and f3 for a spheroidal particle with λ/b1 = 0.1, 1 using N1 , N2 or N3 collocation points on the particle surface. One takes N1 = 74, N2 = 242 and N3 = 1058 for the oblate spheroid (b3 /b1 = 0.5) and N1 = 170, N2 = 530 and N3 = 2210 for the prolate spheroid (b3 /b1 = 2). f1 f3 f1 f3 f1 f3 f1 f3

λ/b1 0.1 0.1 1 1 0.1 0.1 1 1

b3 /b1 0.5 0.5 0.5 0.5 2 2 2 2

N1 0.7131 0.8479 0.5444 0.7748 1.3003 1.1153 1.1254 0.8159

N2 0.7144 0.8453 0.5416 0.7704 1.2994 1.1162 1.1234 0.8142

N3 0.7143 0.8448 0.5403 0.7696 1.2994 1.1163 1.1233 0.8141

[7, 11] 0.7142 0.8448 0.5402 0.7696 1.2994 1.1163 1.1233 0.8141

Conclusions A new boundary formulation has been established to accurately calculate at a reasonable cpu time cost the net force and torque experienced by a migrating, slipping and arbitrarily-shaped solid particle. The approach, which makes it also possible to easily compute the flow about the particule at any desired location in the fluid domain, has been nicely tested against the results obtained elsewhere for a spherical or spheroidal particle by other authors using quite different techniques. In order to illustrate the ability of the method to cope with non-necessarily axisymmetric particles, new results for non-spheroidal ellipsoids will be also presented and discussed at the oral presentation. Finally, one should note that the advocated treatment might be extended to the challenging problem of a slipping particle immersed in a prescribed and arbitrary ambient Stokes flow for which the available results are unfortunately restricted to the case of a spherical particle (see, for instance, [14]). References [1] C. L. M. H. Navier Mémoire sur les lois du mouvement des fluides. Mémoire de l’Académie Royale des Sciences de l’Institut de France, VI, 389-440 (1823). [2] A. B. Basset A treatise on Hydrodynamics, Vol. 2. Dover, New York, (1961). [3] D. Palaniappan Creeping flow about a slighty deformed sphere. Z. Angew. Math. Phys., 45, 832–838 (1994). [4] H. Ramkissoon Slip flow past an approximate spheroid. Acta Mech., 123, 227–233 (1997). [5] S. Deo and S. Datta Slip flow past a prolate spheroid. Indian J. Pure Appl. Math., 33, 903–909 (2002). [6] S. Senchenko and H. J. Keh Slipping Stokes flow around a slightly deformed sphere. Physics of Fluids, 18, 088104. (2006). [7] H. J. Keh and Y. C. Chang Slow motion of a slip spheroid along its axis of revolution. International Journal of Multiphase Flow, 34, 713–722 (2008).

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[8] H. J. Keh and C. C. Huang Slow motion of axisymmetric slip particles along their axes of revolution. Keh, H. J. & Huang, C. H. 2004 Slow motion of axisymmetric slip particles along International Journal of Engineering Science, 42, 1621–1644 (2004). [9] Y. C. Chang and H. J. Keh Translation and rotation of slightly deformed colloidal spheres experiencing slip. Journal of Colloid and Interface Science, 330, 201–210 (2009). [10] Y. C. Chang and H. J. Keh Theoretical study of the creeing motion of axially and fore-and-aft symmetric slip particles in an arbitrary directions. European Journal of Mechanics B/Fluids, 30, 236–244 (2011). [11] J. Happel, H. Brenner Low Reynolds number hydrodynamics, Martinus Nijhoff, (1973). [12] C. Pozrikidis Boundary integral and singularity methods for linearized viscous flow, Cambridge University Press, (1992). [13] M. Bonnet Boundary Integral Equation Methods for Solids and Fluids, John Wiley & Sons Ltd, (1999). [14] H. J. Keh and S. H. Chen The motion of a slip spherical particle in an arbitrary Stokes flow. European Journal of Mechanics B/Fluids, 15, 791–807 (1996).


109

Slow viscous migration of a solid particle near a plane wall with a general non-isotropic slip condition 1

N. Ghalia1 and A. Sellier2 Laboratory of Engineering Mathematics, Polytechnic School of Tunisia, University of Carthage. BP 743, La Marsa, Tunisia e-mail: [email protected] 2 LadHyx. Ecole polytechnique, 91128 Palaiseau C´ edex, France e-mail: [email protected]

Keywords: Stokes flow, Anisotropic Navier slip condition, Boundary-integral equation, Green tensor.

Abstract. A boundary formulation is proposed to investigate the slow migration of a solid particle suspended in a quiescent Newtonian liquid bounded by a plane, motionless and impermeable wall where a recently-introduced anisotropic Navier slip condition holds. The approach rests on the determination of a Green tensor complying with this new slip condition and reduces the task to the treatment of one Fredholm boundary-integral equation of the first kind on the particle surface. Introduction The sedimentation of a suspension bounded by a plane wall Σ is driven both by particle-particle and particle-wall interactions and is therefore in general very difficult to predict. For dilute suspensions one can fortunately ignore, at least at a first level of approximation in terms of the weak volume fraction, particle-particle interactions and thus confine the analysis to the still-challenging problem of one solid particle located near the plane, impermeable and motionless x3 = 0 boundary Σ. Usually, one applies a no-slip condition on both the particle surface and the bounding wall which means that the fluid velocity u vanishes for x3 = 0. However, for a slipping wall (for instance for a rough or hydrophobic wall) such a no-slip condition breaks down and is usually replaced with the so-called Navier [1] (isotropic) slip condition u1 = λ

∂u1 ∂u2 , u2 = λ and u3 = 0 at Σ(x3 = 0) ∂x3 ∂x3

(1)

where λ designates the wall effective slip length. Clearly, the above condition (1) assumes that the wall behaves in a similar fashion in all tangent direction. Because anisotropic patterned surfaces are now encountered in applications a different and anisotropic extended slip Navier condition has been recently proposed [2,3]. Such a condition introduces two othogonal tangential directions on the wall: the fastest one e1 with slip length λ1 ≥ 0 and the slowest one e2 with slip length λ2 such that 0 ≤ λ2 ≤ λ1 . It reads u1 = λ 1

∂u1 ∂u2 , u2 = λ2 and u3 = 0 at Σ(x3 = 0) ∂x3 ∂x3

(2)

and retrieves for λ2 = λ1 the usual slip condition (1). Several studies (see, for instance, [4-8]) deal with the Navier condition (1) by employing the bipolar coordinates and therefore unfortunately restricting the analysis to a spherical particle. Moreover, there is currently no available work coping with the anisotropic boundary condition (2) even for a sphere. Actually, it turns out that determining the flow about and the net hydrodynamic force and torque exerted on a non-spherical and arbitrarily-shaped particle experiencing a prescribed rigid-body migration is a key and tremendously-involved issue. This challenging task is handled in the present paper by appealing to a boundary formulation which, by judiciously selecting and caclulating a suitable Green tensor complying with the anisotropic Navier condition (2), ends up with the treatment of one Fredholm boundary-integral equation on the particle surface (i. e. not also on the unbounded slipping wall Σ!). This paper introduces this boundary approach whereas the associated numerical

110


implementation and new results will be detailed and discussed at the oral presentation. Governing problem As illustrated in Fig. 1, we consider a solid and arbitrarily-shaped particle P immersed in a Newtonian liquid, with uniform viscosity µ and density ρ, occupying the x3 > 0 semi-infinite domain D above the x3 = 0 plane, impermeable and slipping wall Σ.

x3 S O

P n

D σ, µ

n

Σ(x3 = 0)

O

x1 Figure 1. A solid particle P experiencing a rigid-body migration (U, Ω) near the x3 = 0 impermeable and slipping plane boundary Σ. The particle has length scale a, attached point O and smooth surface S with unit outward normal n directed into the liquid domain D. By translating at the velocity U (the velocity of the point O ) and rotating at the angular velocity Ω it induces a liquid flow with pressure field p and velocity field u with typical magnitude V. Assuming that Re = ρV a/µ inertial effects are negligible and the flow (u, p) then obeys the following quasi-steady Stokes equations and far-field and boundary conditions µ∇2 u = ∇p and ∇.u = 0 in D, (u, p) → (0, 0) at ∞,

(3)

u = U + Ω ∧ O M on S.

(4)

Of course, one must supplement the problem (3)-(4) with relevant boundary conditions on the x3 = 0 plane wall Σ. Here we assume that this slipping surface admits two orthogonal slowest and fastest tangential directions e2 and e1 with slip length λ2 ≥ 0 and λ1 ≥ λ2 . Adopting henceforth Cartesian coordinates (O, x1 , x2 , x3 ) and the usual tensor summation convention with, for instance, u = ui ei , we prescribe on the wall the following Navier slip condition u3 = 0, u1 = λ1

∂u1 ∂u2 , u2 = λ2 on Σ(x3 = 0). ∂x3 ∂x3

(5)

Assuming known particle geometry and location, our problem then consists in solving (3)-(5) for prescribed rigid-body motion (U, Ω) and wall slip lengths λ1 and λ2 . In addition to the flow (u, p), having a stress tensor σ, we are also interested in calculating the net hydrodynamic force F and torque Γ (about the attached point O ) exerted on the particle and such that

F=

S

σ.ndS, Γ =

S

O M ∧ σ.ndS.

(6)

Boundary formulation This section shows how it is possible to reduce the problem (3)-(5) to a boundary formulation solely involving the particle surface (and thus not also the unbounded slipping wall).


111

Green tensors and integral representation Let us consider a Stokes flow (u, p) with stress tensor σ satisfying (3). As it is well known [9], it is both possible and fruitful to express in the entire liquid domain D the velocity field u solely in terms of the resulting surface traction σ.n and velocity u on the liquid boundary S ∪ Σ. To do so, we select, as illustrated in Fig. 2, a so-called pole y in the y3 > 0 half-space and introduce for k = 1, 2, 3 three Stokes flows with pressure p(k) and velocity v(k) due to a concentrated point force of strength ek placed at the selected pole y.

x3

x y

•

•

h = y3 > 0

O •

Σ(x3 = 0)

y

x1

Figure 2. A pole y and its symmetric y with respect to the plane wall Σ(x3 = 0). Here x is the so-called observation point. In other words, those flows obey µ∇2 v(k) = ∇p(k) − δ3d (x − y)ek , ∇.v(k) = 0 in the x3 > 0 half-space (u(k) , p(k) ) → (0, 0) as |OM| → ∞,

(7) (8)

with δ3d (x− y) = δd (x1 − y1 )δd (x2 − y2 )δd (x3 − y3 ) where δd designates the Dirac pseudo-function. One should note that these flows are not unique since no boundary conditions are imposed on the wall Σ at that stage! The knowledge of the selected flows (v(k) , p(k) ) provides a so-called second-rank velocity Green tensor G = Gjk (x, y)ej ⊗ ek and associated third-rank stress tensor T = Tijk (y, x)ei ⊗ ej ⊗ ek with Cartesian components defined as Gjk (x, y) = v(k) (x, y).ej . Under these definitions it then easily appears that any Stokes flow (u, p), with stress tensor σ, obeying (3) satisfies in the entire fluid domain the following velocity integral representation [9,10] u(x).ej = −

S∪Σ

[ei .σ.n](y)Gij (y, x) − [u(y).ei ]Tijk (y, x)[n(y).ek ] dS(y), x in D.

(9)

Selected Green tensor and relevant boundary-integral equations The basic representation (9) deserves two remarks: (i) It holds whatever the boundary conditions satisfied on the particle surface S and on the slipping wall Σ by the Stokes flow (u, p). Hence, it does not take into account any boundary condition on S ∪ Σ. (ii) It involves the unbounded wall Σ. Fortunately, it is possible to circumvent these drawbacks by adequately selecting the Green tensor G. Here, we choose a specific Green tensor Gc (and associated stress tensor Tc ) so that the integrals over the boundary Σ vanish in (9). As the reader may easily check by applying the usual reciprocal identity [11], this is obtained by requiring the Green tensor Gc to fulfill the anisotropic Navier slip condition (5). Accordingly, the retained Green tensor Gc is given by the (unicity of the solution) Stokes flows (v(k) , p(k) ) subejct by (7)-(8) and the additional slip boundary conditions v(k) .e3 = 0, v(k) .e1 = λ1

∂v(k) .e2 ∂v(k) .e1 and v(k) .e2 = λ2 ∂x3 ∂x3

on Σ.

(10)

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Injecting (5) and (10) then makes it possible to cast the representation (9) into the new form u(x).ej = −

S

c [ei .σ.n](y)Gcij (y, x) − [u(y).ei ]Tijk (y, x)[n(y).ek ] dS(y), x in D.

(11)

Finally, one has to enforce the boundary condition (4) on the particle surface S for the Stokes flow (u, p). By virtue of this boundary condition u is a rigid-body velocity on S and therefore, invoking [10], one easily shows that the second integral on the right-hand side of (11) actually vanishes. Consequently, the velocity representation becomes u(x).ej = −

S

[ei .σ.n](y)Gcij (y, x)dS(y), x in D.

(12)

It is possible to show (see the determination of the selected Green tensor in [12]) that Gc (y, x) behaves as 1/|y − x| as points y and x approach. Accordingly, the identity (12) also holds for x located on S. Hence, one ends up with the following Fredholm boundary-integral equation of the first kind [U + Ω ∧ O M].ej = −

S

[f .ei ](y)Gcij (y, x)dS(y) for x = OM on S

(13)

for the traction σ.n arising on the particle surface when the particle migrates with velocities (U, Ω). One should also note that (10) and (12)-(13) easily show that u given by (12) satisfies the slip condition (5). In view of the above results the advocated procedure to solve (3)-(5) consists in the following steps: (i) Obtain the pecific Green tensor Gc . (ii) Solve the boundary-integral equation (13) to gain the force f exerted by the flow (u, p) on the particle surface. (iii) Calculate the net force and torque F and Γ given by (6) from the knowledge of f . (iv) Compute, if needed, the velocity field u in the entire liquid domain D by exploiting the integral representation (11). Determination of the selected Green tensor and numerical strategy In this section we explain the adopoted method to determine the Green tensor Gc and direct for further details the reader to [12]. Determination of the selected Green tensor As already pointed out, the Green tensor G is not unique. For example, in absence of boundary conditions on the wall Σ one ends up with the so-called free-space Oseen-Burgers Green tensor Gfree with Cartesian components and associated pressure field pf ree,(k) given by 8πµGfree jk (x, y) =

δjk [(x − y).ej ][(x − y).ek ] , + |x − y| |x − y|3

4πpf ree,(k) (x, y) = (x − y).ek .

(14) (15)

We then write the required selected Green tensor Gc , obtained for Stokes flows (v(k) , p(k) ) satisfing the additional boundary conditions (10), as G = Gfree + R with R a tensor regular in the entire x3 > 0 half-space. More precisely, denoting by y the symmetric of the selected pole y with respect to the plane wall Σ(x3 = 0) we adopt the following decomposition for the selected Stokes flows (vc,(k) , pc,(k) ) (k)

Gcjk (x, y) = Gfjkree (x, y) − Gfjkree (x, y ) + wj (x, y), p

c,(k)

(x, y) = p

f ree,(k)

(x, y) − p

f ree,(k)

(k) wj (x, y)

(x, y ) + s

(k)

(x, y).

(16) (17)

and pressure s(k) (x, y) has been recently The determination of the velocity components achieved in [12]. It makes use of a two-dimensional Fourier transform upon the variables R1 =

Advances in Boundary Element and Meshless Techniques (x − y).e1 and R2 = (x − y).e2 The solution is analytically obtained in the Fourier space and the (k) quantities wj and s(k) are finally gained by performing a two-dimensional inverse Fourier transform. Details are available in [12]. Numerical strategy The boundary-integral equation (13) is numerically inverted by resorting to a collocation point method [13]. This is achieved by using on the particle surface S a N −node mesh made of 6 − node curved triangular boundary elements. The discretized counter-part of (13) then becomes a linear system AX = Y with 3N × 3N sqaure fully-populated and non-symmetric influence matrix A. Such a system is solved by running a LU factorization algorithm. Conclusions A boundary approach has been proposed to efficiently compute the net hydrodynamic force and torque exerted on a solid and arbitrarily-shaped particle experiencing a prescribed slow rigid-body migration in a Newtonian liquid bounded by a plane, motionless, impermeable and slipping wall where a recently-introduced anisotropic Navier condition is imposed. Upon determining a specific Green tensor complying with such a slip condition on the wall, it has been possible to reduce the task to the treatment of one Fredholm boundary-integral equation of the first kind on the particle surface. Numerical results and comparisons with the ones obtained for the isotropic usual slip Navier condition will be presented for spherical and ellipsoidal particle at the oral presentation. References [1] C. L. M. H. Navier Mémoire sur les lois du mouvement des fluides. Mémoire de l’Académie Royale des Sciences de l’Institut de France, VI, 389-440 (1823). [2] H.A. Stone, A. D. Stroock and A. Adjari Slippage of liquids over lyophobic solid surfaces. Annu. Rev. Fluid Mech., 36, 381-411 (2004). [3] M.Z. Bazant and O. I. Vinagradova Tensorial hydrodynamic slip. J. Fluid Mech.”, 613, 125-134 (2008). [4] H. Luo and C. Pozrikidis Effect of slip on the motion of a spherical particle in infinite flow and near a plane wall. J. Eng. Math.,62(1), 1-21. (2008). [5] M. E. O’Neill and B. S. Bhatt Slow motion of a solid sphere in the presence of a naturally permeable surface. Q. J. Mech. Appl. Math.,44, 91-104 (1991). [6] A.M.J Davis, M.T. Kezirian and H. Brenner On the Stokes-Einstein model of surface diffusion along solid surfaces: Slip boundary conditions J.Colloid Interface Sci.,1065, 129-140 (1994). [7] H. Loussaief Ecoulement de suspensions avec condition de glissement sur la paroi. PHD Thesis. Laboratory of Engineering Mathematics, Polytechnic School of Tunisia, BP 743, La Marsa, Tunisia. (2008). [8] F. Feuillebois, H. Loussaief and L. Pasol Particles in Creeping Flow Near a Slip Wall. CP1186, Applications of Mathematics in Technical and Natural Sciences Editors M. D. Todorov and C.I. Christov. American Institute of Physics, 3-14 (2009). [9] S. Kim and S. J. Karrila Microdydrodynamics: Princples and Selected Applications, Butterworth, (1991).

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[10] C. Pozrikidis Boundary integral and singularity methods for linearized viscous flow, Cambridge University Press, (1992). [11] J. Happel and H. Brenner Low Reynolds number hydrodynamics, Martinus Nijhoff, (1973). [12] A. Sellier and N. Ghalia Green tensor for a general non-isotropic slip condition. CMES, 78, No.1, 25–50 (2011). [13] M. Bonnet Boundary Integral Equation Methods for Solids and Fluids, John Wiley & Sons Ltd, (1999).


115

Gravity-driven migration of one bubble near a free surface: surface tension effects M. Guémas1 , A. Sellier1 and F.Pigeonneau2 Ecole polytechnique, 91128 Palaiseau Cédex, France 2 Surface du Verre et Interfaces, UMR125 CNRS St Gobain, 39 quai Lucien Lefranc, BP 135, 93303 Aubervilliers, Cedex, France e-mail: [email protected] e-mail: [email protected] 1 LadHyx.

Keywords: Bubble, free surface, surface tension, Stokes flow, Boundary-integral equation, film drainage.

Abstract We pay attention not only to the time dependant film thickness of the liquid film (so-called drainage) between a bubble and a free surface but also to the bubble and free surface shapes when allowing different uniform surface tensions on these surfaces. This is achieved by extending the recent boundary approach recently employed in [1] for surfaces having identical surface tension. Preliminary numerical results clearly show the sensitivity of the drainage and time-dependent surfaces to the surface tension ratio. 1. Introduction Interactions between bubbles and a free surface in a gravity field play a key role in many chemical and geophysical applications. Indeed, as one bubble approaches a free surface, a complex interplay arises between the squeezing of the liquid flow (and film) surrounded by the bubble and the free surface, and the deformations of the aforementioned surfaces shapes. A recent numerical investigation by Pigeonneau and Sellier [1] examined the sensitivity to Bond number (see definition in 2.1) of both the time dependent free surface and bubble(s) shapes and the time dependent film thickness between the free surface and the closest bubble. This was performed for axisymmetric free surface and bubble having the same axis of revolution parallel with the applied gravity field and the same uniform surface tension. However, as suggested by a recent paper [2] dealing with the surface tension interaction at the glass-liquid-tin-gas phase interface, the obtained results are very likely to be modified if we consider the case of a free surface and a bubble having different (uniform) surface tension γ0 > 0 and γ1 > 0, respectively. In this work, we therefore extend the study achieved in [1] by considering the case γ0 = γ1 still resorting to a boundary-integral implementation. 2. Governing problem 2.1 Axisymmetric problem As sketched in Fig. 1, we consider a bubble B1 immersed in a Newtonian Fluid with uniform density ρ and viscosity µ bounded by a free surface subject to a uniform gravity g = −gez , with the magnitude g. The ambient fluid above the free surface is a gas with a uniform pressure p0 and both the temperature T1 and the pressure p1 inside the bubble are assumed uniform and constant in time. The bubble surface S1 (t) and the free surface S0 (t) have uniform surface tension γ1 > 0 and γ0 > 0, respectively. As buoyancy effects drive the bubble toward the free surface, the shape of each surface evolves in time. At initial time, the bubble is taken spherical with radius a and the free surface is the z = 0 plane. At any time t, the two deformed bubble surface S1 (t) and free surface S0 (t) are axisymmetric having identical axis of revolution parallel with the gravity g, and the flow in the liquid domain D(t) has pressure p + ρg.x and velocity u with typical magnitude U = ρga2 /(3µ). All inertial effect are neglected, i.e the Reynolds number Re obeys Re = ρU a/µ 1. Assuming quasi-steady bubble and free surface deformations, the flow (u, p) then satisfies the following far-field behavior and Stokes equations µ∇2 u = grad[p] and ∇ · u = 0

in D(t), (u, p) → (0, 0) as |x| → ∞

(1)

116


where x = OM. The flow (u, p) has stress tensor σ and, denoting (see Fig.1) by n the unit normal on S0 ∪ S1 directed z

γ0

po

S0 (t)

O

n

r

h1 γ1 n D(t)

p 1 B1 T1

S1 (t)

g = −gez

Figure 1: One bubble B1 ascending near a free surface S0 (t). into the liquid, one also requires the boundary conditions σ · n = (ρg · x − pm + γm ∇S · n) n on Sm for m = 0, 1

(2)

where [∇S · n]/2 = H is the local average curvature with [∇S · n], the surface divergence of the unit normal n. Moreover, there is no mass transfer accross the liquid boundary which implies that V.n = u.n on Sm for m = 0, 1

(3)

with V the material velocity on each surface Sm . Finally, since the bubble volume is time-independent, one also requires Sm

u · ndS = 0 for m=0,1.

(4)

To summerize, we represent the time-dependent shape of the free surface and of the bubble by successively running at each time t the following key steps: i) First, from the knowledge of the surface traction σ · n computed using (2), we solve (1) in conjonction with (4) to obtain the unique solution u on the surfaces S0 and S1 . ii) Then, one calculates the component V · n exploiting the relation (3) which allows us to move the surfaces S1 (t) and S0 (t) between time t and time t + dt. 2.1 Relevant boundary-integral equations By virtue of (1)-(2) and (4) the velocity field u may be computed at any point x0 located in the liquid domain D(t) by solely appealing to two surface quantities u and σ · n on the entire liquid boundary S = S0 ∪ S1 . One requires the surface traction σ · n given by the boundary condition (2) to gain the unknown velocity u on S. This is achieved by letting x0 tend onto this surface S. Since pm and γm are uniform on each surface Sm , the following boundary-integral equation for the unknown velocity u on the liquid boundary is then expressed as

u(x0 ) −

1 1 [(ρg · x + γ0 ∇S · n)n](x) · G(x, x0 )dS − u(x) · T(x, x0 ) · n(x)dS = 4π S 4πµ S0 1 [(ρg · x + γ1 ∇S · n)n](x) · G(x, x0 )dS for x0 on S + 4πµ S1

(5)


117

where the symbol − means a weakly-singular integration in the principal value sense of Cauchy[3] and, setting xi = x.ei and x0,i = x0 .ei , the tensors G and T have Cartesian components given by δij (xi − x0,i )(xj − x0,j ) , + |x − x0 | |x − x0 |3 (xi − x0,i )(xj − x0,j )(xk − x0,k ) Tijk (x, x0 ) = −6 |x − x0 |5 .

Gij (x, x0 ) =

(6) (7)

with δij the Kronecker symbol. Since we restrict the analysis to theaxisymmetric configuration depicted in Fig.1, we adopt cylindrical coordinates (r, φ, z) with r = x2 + y 2 , z = x3 and φ the azimuthal angle in the range [0, 2π]. Setting u = ur er + uz ez = uα eα (with α = r, z) and n = nr e + nz ez = nα eα on the entire contour L = L0 ∪ L1 , with Lm the trace of the surface Sm integrated over φ, then makes it possible to transform (5) as 1 Bαβ (x, x0 )[−ρgz + γ0 ∇S · n]nβ (x)dl 4πuα (x0 ) − − Cαβ (x, x0 )uβ (x)dl = − µ L0 L 1 Bαβ (x, x0 )[−ρgz + γ1 ∇S · n]nβ (x)dl for x0 on L − µ L1

(8)

with β = r or β = z, the differential arc length dl in the φ = 0 plane and the so-called single-layer and double-layer 2 × 2 square matrices Bαβ (x, x0 ) and Cαβ (x, x0 ) given in Pozrikidis [4]. 3. Numerical method In this section, we will briefly introduce the numerical procedure based on a collocation method and a discrete Wiedlandt deflation technique and direct for further details the reader to [1]. The boundary-integral equation (8) is numerically inverted by appealing to the following key steps: (i) First, a T truncated free surface contour and the bubble contour are divided into Ne curved boundary elements with the preserved x → −x symmetry. Each boundary element has Nc collocation points spread by a Gauss or a uniform distribution. The associated velocity u and the surface traction f = σ ·n are then approximated on each element using a isoparammetric interpolation. By introducing the components f .eα and u.eα at our Ne Nc nodal points, we end up with two given 2Ne Nc stress vector F and unknown 2Ne Nc velocity vector U. Discretizing the boundary-integral equation (8) then shows that these vectors satisfy the 2Ne Nc -equation linear system U − C.U = B.F.

(9)

The two matrices B and C are related to the quantities Bαβ and Cαβ introduced in 2.2 which are integrated on each boundary element by regularizing the weakly-singular terms of Bαα when the node x0 belongs to the selected boundary element. (ii) By combining (8) and (5), one finds a unique solution U. This solution is here obtained by performing a so-called Wiedlandt’s deflation technique to solve (9). (iii) Note that the major issue for the present work is to precisely calculate the quantity ∇S · n on each discretized surface Sm . An adequate approximation of this quantity indeed dictates the accuracy of the velocity u calculated through (2) on the fluid boundary. This is achieved by putting enough nodes on each boundary element. (iv) The shape of each surface Sm is tracked in time using the boundary condition (3) and solving the equation dx/dt = u(x, t) for each nodal point. A Runge-Kutta-Fehlberg method performs this

118


task using a time-step selected by controlling the errors for the second and third-order schemes. Furthermore, as one shape Sm become nearly time-independent, the adjusted time step is then very small and the computations is stopped. 4. Numerical results This section presents numerical results for one bubble acsending toward a free surface. First, we look at the dependence of the film thickness h on the z-axis and its sensitivity to the surface tension ratio r = γ0 /γ1 for a given Bond number Bo = ρga2 /γ1 based on the bubble surface tension. At initial time, the bubble is spherical with radius a and the distance between its center and the flat z = 0 free surface is equal to 3a. Adopting 2a as the length scale, the normalized distance between the bubble surface and the free surface is therefore hN = h/2a. Henceforth, t denotes the time normalized by 6µ/ρga. The numerical implementation is performed taking 45 boundary elements uniformaly spread on the entire contour L1 ∪ T and computations are run using 1000 iterations in time. Calculations are stopped whenever the time step selected as explained in section 3. becomes too small.

1

log hN

r=1 r = 0.2 r = 0.3 r = 0.5 r=2 r=3 r=5

0,1

0,01

0

1

0,5

1,5

2

t

1

log hN

r = 0.2 r = 0.5 r = 0.3 r=5 r=3 r=5 r=1

0,1

0,01

0

0,5

1

1,5

2

t

Figure 2: log hN versus t at (a) Bo = 0.3 and (b) Bo = 1 for different values of the surface tension ratio r = γ0 /γ1 . In order to illustrate the film thickness sensitivity to the surface tension ratio, we plot the evolution in time of the film thickness hN , for a large range of surface tension ratio γ0 /γ1 and a given Bond number. As seen in Fig. 2, the film thickness exhibits an exponential decay as time increases whatever


119

the ratio r = γ0 /γ1 . Observe that at Bo = 0.3 the drainage is significantly enhanced (when compared to the case γ0 = γ1 ) while γ0 > γ1 or reduced while γ0 < γ1 . In contrast, at Bo = 1 the drainage is less sensitive to γ0 /γ1 .

1

1

(a)

(b)

0,5

z

0,5

z

0

0

-0,5

-0,5

-1

-1

-1,5

-1,5

-2 -1,5

-1

-0,5

0

0,5

1

-2 -1,5

1,5

-1

-0,5

x

0,5

1

1,5

0,5

1

1,5

x

1

1 (d)

(c) 0,5

z

0

0,5

z

0

0

-0,5

-0,5

-1

-1

-1,5

-1,5

-2 -1,5

-1

-0,5

0

x

0,5

1

1,5

-2 -1,5

-1

-0,5

0

x

Figure 3: Bubble and free surface shape at Bo = 1 for (a) γ0 /γ1 = 1, (b) γ0 /γ1 = 2, (c) γ0 /γ1 = 0.5 and (d) γ0 /γ1 = 0.2. Dashed lines indicate the bubble and free surface shapes for t = 0.59 while t = 1.746 at finale stage. Fig. 3 depicts the time-dependent bubble and free surface shapes sensitivity to the surface tension ratio. Note that the free surface is slightly less deformed when γ0 /γ1 = 2 (Fig. 3 (b)) than when γ0 = γ1 (Fig. 3 (a)). This explains why, in Fig. 2 the drainage is indeed enhanced for γ0 /γ1 > 1. When γ0 < γ1 , though the drainage observed in Fig. 2 (b) are closely identical for γ0 /γ1 = 0.5 and γ0 /γ1 = 0.2, the final free surface shapes shown Fig. 3 (c) and Fig. 3 (d) are slightly different. 5. Conclusions Our investigations reveal that both the drainage and the time-dependent shapes of the bubble and free surface depend upon the surface tension ratio γ0 /γ1 . Additional numerical results will be reported and discussed at the oral presentation. Finally, one should note that the proposed boundary-integral

120


approache is also able to deal with several bubbles.

References [1] F. Pigeonneau and A. Sellier. Low-Reynolds-Number gravity-driven migration and deformation of bubbles near a free surface. Phys. Fluids, 23:092302, 2011. [2] V. I. Nizhenko and Yu. I. Smirnov Surface Phenomena and interfacial interaction at the glass-liquid tin-gas phase interface. Powder Metallurgy and Metal Ceramics, 42:075084, 2003. [3] J. Hadamard Lecture on Cauchy’s problem in linear differential equations, Dover Publications, Inc., New York, 1932. [4] C. Pozrikidis Boundary integral and singularity methods for linearized viscous flow. Cambridge University Press, 1992.


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! " # $ !

! % $ ⎛

⎞ ⎛ KII ⎜ KI ⎟ ⎜ π −1 ⎜ ⎟ K=⎜ ⎝ KIV ⎠ = 8r Y ⎝ KV

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p

(iω cy )

; ϕ = ϕ0 e

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u0 ϕ0 φ0 cp # # , cp =

κ1 =

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γ22 e22 − β22 h22 22 h22 − β22 e22 ; κ2 = 2 2 γ22 22 − β22 γ22 22 − β22

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C12 u0 + e21 ϕ0 + h21 φ0 ⎥ ⎢ C22 u0 + e22 ϕ0 + h22 φ0 ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎥=⎢ 0 ⎥ ⎢ ⎥ ⎢ e22 u0 − 22 ϕ0 − β22 φ0 ⎥ ⎢ ⎦ ⎣ 0

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D2+ = D2− = 0 ; B2+ = B2− = 0 ,

D2+ = D2− ; φ+ = φ− , B2+ = B2− ; ϕ+ = ϕ−

!"# !$# %&& ' (

D2c = −0

φ+

D2+ = D2− ; B2+ = B2−

− φ−

= −0

− u+ 2 − u2

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− ϕ−

∆u4 ∆u5 ; B2c = −γ0 + = γ0 ∆u2 ∆u2 u2 − u− 2

)

D2+ = D2− = D2c B2+ = B2− = B2c 0 γ0

'

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0 = 0

γ0 = 0

*

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(

+ ),- . + '

∆u4 = 0

∆u5 = 0

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p4 = 0

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p5 = 0

+ '

( , +' 1 , %&& ' ( , . ' 2 + *

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125

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ka[j]

(ω) = hka m ∆u5 (ω); ∆u5

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k[0]

kb[j]

(ω) = hkb e ∆u4 (ω) .

k[0]

k[0]

kb[j]

(ω) = hkb m ∆u5 (ω) .

k[0]

∆u2 (ω) Dn (ω) Bn (ω)

ϕ φ

ki hki e , hm ξm ki ki m (ω) = Dn (ξm , ω) ki γm (ω) = Bnki (ξm , ω)

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Uh

3

12

@

³ uD ( X , t ) u D ([ , X ) d: ( X ) Uh:³ u ( X , t ) u * i

:

3

³

[ , X ) d: ( X ) kUh ³ u3 ( X , t ) ui*D ,D ([ , X ) d: ( X )

* i3 (

(7)

:

where k Q / 6(1 Q )O2 . In this expression, [ and X are source point and field point, respectively; cij G ij / 2 for points [ in a smooth boundary and cij G ij for internal points.


157

Note that the last term in eq. (7) refers to the translational inertia term, being an additional part of the integral equations for dynamic analysis of thick elastic plates, not included by Providakis and Beskos [16]. In the expression above the load is a constant factor multiplying the first domain integral on the right-hand side of eq. (7), because of the consideration of constant loading over the entire domain : . The expressions for moments and shear forces at an internal point [ are written as [22, 23]

³ p ( X , t )uDE ([ , X ) d* ( X ) *³ u *

M DE ([ , t )

Uh 3 12

*

k

k

* ([ , X ) d* ( X ) [ , X ) d* ( X ) p ³ wDE

* k ( X , t ) pDEk (

*

³ uT ( X , t ) uDET ([ , X ) d: ( X ) Uh:³ u ( X , t ) uDE *

*

3

³ p ( X , t )u E ([ , X ) d* ( X ) *³ u * 3 k

k

12

³ :

uT ( X , t ) u3*ET

([ , X ) d: ( X ) Uh

³ :

pG DE (8a)

:

[ , X ) d* ( X ) p ³ w3*E ([ , X ) d* ( X )

* k ( X , t ) p3 E k (

*

Uh 3

* [ , X ) d: ( X ) kUh ³ u3 ( X , t ) zDE ([ , X ) d: ( X )

3(

:

and QE ([ , t )

Q (1 Q )O2

*

[ , X ) d: ( X ) kUh ³ u3 ( X , t ) z3*E ([ , X ) d: ( X )

u3 ( X , t ) u3*E 3 (

(8b)

:

The tensors that appear with the asterisk in eqs. (7) and (8) represent the static fundamental solution, see Refs. [16, 22]. Numerical procedure Figure 1 consider the boundary * discretized by isoparametric quadratic straight one-dimensional elements, each one denoted by * j , and the domain : discretized by constant triangular three nodes cell,

the domain of a cell l being denoted by : l .

*j

:

x2

*

:l

x1 Figure 1: Domain discretized with boundary elements and internal cells. BEM guidelines consider boundary element and internal cell approximations as follows: Displacements U ( j ) and surface forces P ( j ) within an element j are computed from nodal values, U (n ) and P (n ) , according to U ( j ) NU ( n ) and P ( j ) NP ( n ) (9a) while the translational inertia U (l ) at internal points are approximated by (9b) U (l ) NU ( m ) Substituting (9) into (7), these resulting equations are written for all boundary nodes and for all cell nodes, with nodes being collocation points [ . Then, the following system of equations is formed [22, 23]: HU (t ) GP (t ) B (t ) MU (t ) (10) When source and field points are located on the same element, with the source point fixed on the boundary, the integrals corresponding to the submatrices H and G , and to the vector B present singularities. The submatrices G and the vector B have singularities of kind ln r and to solve this problem the co-ordinate transformation of the second degree presented by Telles [24] and used by Karam [25] is used. Houbolt’s scheme [13] is employed to integrated eq. (10) on time; for each time step a non-symmetric system of equations AX b is solved. To calculate the moments and shear forces at internal points, the expressions (2) are utilized, where the differentiation of displacements are substituted by derivatives of the integral equation (7). According to eq. (8) and approximations given by eq. (9), the responses for internal forces can be calculated at each time step, see Ref. [22, 23].

158


Example Consider a simply supported square plate subjected to a suddenly applied uniform load p0 1.0 , where the following material and geometric parameters are considered: modulus of elasticity E 1.0 , Poisson’s ratio Q 0.3 , mass density U 1.0 , side length a 1.0 . Some analyses are made to various thicknesses of plates, where are considered four relations h / a equal to 0.10 , 0.20 , 0.30 and 0.40 . Fig. 2 shows the type of discretization and load function, and the boundary conditions. Due to the symmetry, only one quarter of the plate is discretized by 32 boundary elements and 128 domain cells. The total time of analysis t 0 adopted is equal to two times the first fundamental period. The first fundamental period for the four relations h / a used here are, respectively, equal to 10.8871 , 5.9432 , 4.4384 and 3.7538 , whereas the time step adopted is equal to 1.25 x 10 2 .

y Tx M y w 0 Tx

0

Qx

0

Mx

M xy

Ty

0 0

p( t )

0

w 0

p0

t (s)

x A M xy T y Q y 0 B

t0

Figure 2: Mesh with boundary condition due to the symmetry and load function. The numerical results, with and without additional translational inertia terms, are compared with the thick plate analytical solution presented by Pereira [23] that contains these additional inertia terms in the formulation. The results presented here are obtained from series expansions considering 9 terms for deflection, rotations, bending moments and shear forces. In this work, the superscript ‘a’ indicates consideration of additional translational inertia terms in the present formulation. The relative percentage error is evaluated by the following expression: 'A error x100 % with 'A ' ' 0 and A ' 0max (11) A The error is computed at each discrete time until the final time of the analyses. ' 0 is the analytical response with the additional translational inertia term while ' is the value for the proposed numerical formulation; ' 0max is the maximum amplitude obtained from analytical solution for thick elastic plate that contains additional translational inertia terms in the formulation, which can be either that of the first or of the second analysis peak depending on what item of the computed time response is being considered ( D : deflection, R : rotation, B : bending moment, S : shear force). Table 1 shows ' 0max values for the four relations h / a , where the values correspond to the right and left peaks. Table 1: Maximum amplitude ' 0max for the four relations h / a . h/a D R B S

0.10 94.9779 288.9210 0.1090 0.6302

94.0323 291.3678 0.1084 0.6351

0.20 13.7039 35.4183 0.1095 0.5846

13.2699 35.8404 0.1058 0.6289

0.30 4.7476 10.3459 0.1133 0.6272

4.9283 10.2079 0.1119 0.5997

0.40 2.4425 4.2327 0.1031 0.6625

2.4500 4.2298 0.1052 0.6071

Figs. 3-6 shows the time history of the responses. The values of the deflection and bending moment are calculated at point A and the values of the rotation and shear force at point B , see Fig. 2. It can be observed from Fig. 3 that for h/a=0.10, the influence of the new terms over deflection and rotation responses is hard to note. However, for internal forces a small difference between the two approaches can be easily detected; for the bending moment and shear force, respectively, whose maximum error of 0.93% and 2.20% occurs. Fig. 4 shows for h/a=0.20 errors of 1.65%, 1.15%, 3.84% and 4.17% for the deflection, rotation, bending moment and shear force, respectively. According to Fig. 5 for h/a=0.30 errors for the variables are respectively equal


159

to 1.74%, 2.85%, 5.32% and 5.64%. Thus, the responses of the internal forces already show an important contribution due to the inclusion of translational inertial terms. Fig. 6 presents time results for h/a=0.40. The maximum errors for deflection and rotation are respectively 3.18% and 2.84%, and for bending moment and shear force are 11.57% and 8.70%. Thus, it is confirmed the sensibility of the response to the consideration of the additional translational inertia terms and its importance in the evaluation of internal forces. 0.12

100

Bending moment

Deflection

75 50 25

0.08

0.04

0

0 -25

-0.04

5.5

11 Time

16.5

22

0

300

0.8

200

0.6 Shear force

Rotation

0

100

0

5.5

11 Time

16.5

22

16.5

22

0.4

0.2

-100

0

0

5.5

11 Time

16.5

22

0

5.5

11 Time

Figure 3: Time history of the responses for h / a 0.10 . Thick plate theory with additional translational inertia terms; present approach; presenta approach. 16

0.12

Bending moment

Deflection

12 8 4

0.08

0.04

0

0 -4

-0.04

0

3

6 Time

9

12

0

40

6 Time

9

12

9

12

0.8

30

0.6 Shear force

Rotation

3

20 10

0.4

0.2

0 -10

0 0

3

6 Time

9

12

0

3

6 Time

Figure 4: Time history of the responses for h / a 0.20 . Thick plate theory with additional translational inertia terms; present approach; presenta approach.


6

0.12

4

0.08

Bending moment

Deflection

160

2

0

-2

0

-0.04

0

2.25

4.5 Time

6.75

9

0

12

0.8

8

0.6 Shear force

Rotation

0.04

4

0

2.25

4.5 Time

6.75

9

6.75

9

0.4

0.2

-4

0

0

2.25

4.5 Time

6.75

9

0

2.25

4.5 Time

3

0.12

2

0.08

Bending moment

Deflection

Figure 5: Time history of the responses for h / a 0.30 . Thick plate theory with additional translational inertia terms; present approach; presenta approach.

1

0

-1

0

-0.04 0

1.9

3.8 Time

5.7

7.6

0

1.9

3.8 Time

5.7

7.6

5.7

7.6

0.8

5 3.75

0.6 Shear force

Rotation

0.04

2.5 1.25

0.4

0.2

0

0

-1.25 0

1.9

3.8 Time

5.7

7.6

0

1.9

3.8 Time

Figure 6: Time history of the responses for h / a 0.40 . Thick plate theory with additional translational inertia terms; present approach; presenta approach. In this work, the boundary element method is used to discretize the space employing quadratic continuous and discontinuous elements for boundary discretization, and constant three nodes triangular cells for domain discretization. According to Figs. 5 and 6, note that the responses already show the loss of accuracy due to the consideration of element of the domain that occurs with increasing thickness of the plate.


Conclusions In this paper, an analysis of the transient response of thick elastic plates based on Reissner’s theory is presented, the contribution of additional translational inertia terms to the integral equation of displacements and internal forces being computed. Moreover, the BEM approach described here employs quadratic continuous and discontinuous elements for the boundary, and constant three nodes triangular cells for the domain, while for time integration Houbolt’s acceleration operator is considered. Thus, boundary and domain unknowns are determined simultaneously for each time step. The numerical simulations carried out with the additional term considered by the present formulation modified results obtained without this term and its contribution in the analyses carried out here being more relevant for relation h/a>0.20, i.e., for moderately thick or thick plates. Acknowledgments The authors are grateful for the financial support from CNPq and special grateful for the partnership Capes-FAPERJ, Pos-Doctorate Support Program in Rio de Janeiro–2009, registered under nº. 10/2009. References [1] I.H. Shames, C.L. Dym. Energy and finite element methods in structural mechanics. US: McGraw-Hill (1985). [2] S.P. Timoshenko, S. Woinowsky-Krieger. Theory of plates and shells. New York: McGraw-Hill (1959). [3] A.W. Leissa. Vibration of plates. Washington: NASA SP-160 (1969). [4] J.M. Biggs. Introduction to structural dynamics. US: McGraw-Hill Book Company (1964). [5] E. Reissner. On the theory of bending of elastic plates. Journal of Mathematics and Physics, 23: 184-191 (1944). [6] E. Reissner. The effect of transverse-shear deformation on the bending of elastic plates. Journal of Applied Mechanics, 12 (2): 69-77 (1945). [7] E. Reissner. On bending of elastic plates. Quarterly of Applied Mathematics, 5 (1): 55-68 (1947). [8] R.D. Mindlin. Influence of rotatory inertial and shear on flexural motions of isotropic elastic plates. Journal of Applied Mechanics, 18: 1031-1036 (1951). [9] R.D. Mindlin, A. Schacknow, H. Deresiewicz. Flexural vibrations of rectangular plates. Journal of Applied Mechanics, 23: 430-436 (1956). [10] K.M. Liew, C.M. Wang, Y. Xiang, S. Kitipornchai. Vibration of Mindlin plates. Programming the pVersion Ritz Method. Netherlands: Elsevier Science Ltd (1998). [11] Y.K. Cheung, S. Chakrabarti. Free vibration of thick, layered rectangular plates by a finite layer method. Journal of Sound and Vibration, 21: 277-284 (1972). [12] T. Mikami, J. Yoshimura. Application of the collocation method to vibration analysis of rectangular Mindlin plates. Computers and Structures, 18: 425-431 (1984). [13] K.J. Bath. Finite element procedures. New Jersey: Prentice Hall (1996). [14] T. Rock, E. Hinton. Free vibration and transient response of thick and thin plates using the finite element method. Earthquake Engineering and Structural Dynamics, 3: 51-63 (1974). [15] C.P. Providakis, D.E. Beskos. Dynamic analysis of plates by boundary elements. Applied Mechanic Review ASME, 52: 213-236 (1999). [16] C.P. Providakis, D.E. Beskos. Inelastic transient dynamic analysis of Reissner-Mindlin plates by the D/BEM. International Journal for Numerical Methods in Engineering, 49: 383-397 (2000). [17] F.R. Mittelbach. The energetic finite difference method in the analysis of axisymmetric problems of thin and thick plates. D.Sc. thesis (in Portuguese), RJ, Brazil: COPPE/UFRJ (2007). [18] O.L. Roufaeil, D.J. Dawe. Vibration analysis of rectangular Mindlin plates by finite strip method. Computers and Structures, 12: 833-842 (1980). [19] J. Sladek, V. Sladek, H.A. Mang. Meshless LIBIE formulations for simply supported and clamped plates under dynamic load. Computers and Structures, 81: 1643-1651 (2003). [20] M. Batista. Refined Mindlin-Reissner theory of forced vibrations of shear deformable plates. Engineering Structures, 33: 265-272 (2011). [21] W.L.A. Pereira, W.J. Mansur, V.J. Karam, J.A.M. Carrer. A formulation for free vibration analysis of thick elastic plates by the boundary element method. In: Proceedings of the XXXII Ibero-Latin American congress of computational methods and engineering (XXXII CILAMCE), CD-ROM, Brazil (2011). [22] W.L.A. Pereira. A general formulation for dynamic analysis of thick plates by the boundary element method. D.Sc. thesis (in Portuguese), RJ, Brazil: COPPE/UFRJ (2009).

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[23] W.L.A. Pereira, V.J. Karam, J.A.M. Carrer, W.J. Mansur. A dynamic formulation for the analysis of thick elastic plates by the boundary element method. Engineering Analysis with Boundary Elements, 36: 1138-1150 (2012). [24] J.C.F. Telles. A self-adaptive coordinate transformation for efficient numerical evaluation of general boundary element integrals. International Journal for Numerical Methods in Engineering, 24: 959-973 (1987). [25] V.J. Karam. Plate bending analysis by the BEM including physical nonlinearity. D.Sc. thesis (in Portuguese), RJ, Brazil: COPPE/UFRJ (1992).


Regularization of the Divergent Integrals Using Generalized Function Based Approach V.V. Zozulya Centro de Investigacion Cientifica de Yucatan A.C., Calle 43, No 130, Colonia: Chuburná de Hidalgo, C.P. 97200, Mérida, Yucatán, México. E-mail: [email protected] Keywords: weakly singular, singular, hypersingular integrals, regularization, boundary integral equations. Abstract. This article considers weakly singular, hypersingular integrals, which arise when the boundary integral equation (BIE) methods are used for 3-D potential theory problem solution. For regularization of the divergent integrals, an approach based on the theory of distribution and application of the second Green theorem has been explored. The equations, that allow easy calculation of the weakly singular and hypersingular integrals for piecewise constant approximation have been considered for arbitrary convex polygon and for piecewise linear approximation for rectangular and triangular BE have been considered. In presented equations all calculations can be done analytically, no numerical integration is needed. Calculation of the divergent integrals for circular, rectangular and triangular area has been done.

1. Introduction. Classical approach for divergent integrals treatment has one significant disadvantage. Divergent integrals with different singularity need different definition, different theoretical justification and different methods for their calculation [4, 8, 9]. For example, the weakly singular integrals are considered as improper integrals, the singular integrals are considered in the sense of Cauchy principal value (PV) [7], the hypersingular integrals had been considered by Hadamard as finite part integrals (FP) [6]. In modern mathematics divergent integrals have well established theoretical foundation. It based on the theory of generalized functions (distributions) which permit to apply the same approach for the divergent integrals with different type of singularities. According to this theory divergent integrals with any type of singularity can be considered as functional (generalized functions) defined in special functional spaces and on specially defined test functions [3]. In our previous publications [5, 10-14] approach based on the theory of distributions has been developed for regularization of the divergent integrals with different singularities with arises in applications of the BEM. We explore the presented in [2] approach, which permit interprets definite integrals as distributions and apply it to solution of the problems of fracture mechanics firstly in [10]. Then it was further developed for regularization of the 2-D hypersingular integrals which appear in static and dynamic problems of fracture mechanics in [14]. Regularized formulas for different type of divergent integrals have been reported in [12, 14]. More applications of the developed regularization method can be found in review article [5]. Further development of that approach and application of the second Green’s theorems in the sense of theory of distribution has been done in [11]. The developed approach can be applied for regularization of a wide class of divergent integral regularization and not only for hypersingular integrals regularization but also for a wide class of divergent integral regularizations and any polynomial approximation. In this paper, the approach which is based on the theory of distributions and the second Green’s theorem is developed and applied for the divergent integral regularization. Generalized GaussOctrogradskii and Green theorems which are applicable for the case of singular functions has been

163

164


obtained using methods presented in [2]. Then the generalized second Green theorem has been used for the development of the regularized formulas for divergent integrals calculation. Special case of 2-D divergent integral regularization has been done for the weakly singular and hypersingular integrals and regular formulas for their calculation have been obtained. The weakly singular and hypersingular integrals for piecewise constant approximation have been considered for arbitrary convex polygon and for piecewise linear approximation have been considered for rectangular and triangular BE. Using obtained here regularized formulas divergent integrals have been calculated for circular, quadratic and triangular arias. It is important to mention that in presented equations all calculations can be done analytically, no numerical integration is needed. 2. n-D divergent integrals. Let as consider function f (x) which is defined in the finite region with : R n such that all its singularities are concentrated in the sub region :H : . In the region _

: \ :H including boundary function is regular and possesses all necessary derivatives. Let as consider a definite integral over finite region

I0

³ f (x)dx

(1)

:

What does the I 0 symbol means for such singular function? Classical approach cannot answer this question in general. Only for special type of singularity answer exists and each type of singularity needs its specific consideration. In order to consider this integral in the sense of the distribution, we introduce the function g (x) , such that

f ( x ) ' k g ( x) where ' k

(2)

' ' ' ... ' ' k is called the k – dimensional Laplace operator. k

This representation of the function f (x) can be considered in the classical sense in the region :0 , but in the region : it has to be considered in the sense of distribution. In the region ___

:0 R n / : function is regular and smooth up to boundary, that means f (x) C k (: 0 ) . The boundary w: is satisfied usual conditions of smoothness, which are discussed in any standard courses of analysis. We also introduce test function I ( x) C f ( R n ) , such that M ( x) 1 , x : . Function M (x) is finite and extended smoothly to the region :0 . In this case, its derivatives are equal to zero at the region : including its boundary w: i.e., _

' kI (x) 0, x :

(3)

Let us consider the scalar product, which is a definition of the singular function f (x) in the sense of distributions

( f ,I )

³

f (x)I ( x)dx

(4)

Rn

Taking into account that test function is finite

³ I ( x) '

Rn

k

g (x) dx

³

::0

I ( x ) ' k g ( x) d :

(5)


165

_

Since the derivatives of the test function M (x) are equal zero on : , the integration by parts using the second Green theorem in the sense of distribution gives

³

³

I (x)' k g (x)dx (1) k

::0

g (x)' kI (x)d : (1) k

::0

³ g (x)' I (x)dV k

(6)

:0

The integration by parts in reverse order for the last integrals above leads to the result

³ I ( x) '

k

k 1

¦ (1) ³ [I (x)w '

g ( x) d :

i 1

k i 1

n

i 0

:0

g (x) g (x)w n ' k i 1I (x)]dS (1) k

w:0

³ g ( x) ' I ( x) d : k

(7)

:0

Here, w n ni w i is the normal derivative on the surface with respect to x and ni (x) is a unit normal to the surface. Taking into account that

³

³

f (x)I ( x)d :

f (x)I ( x)d :

::0

Rn

³

f (x)I (x)d :

(8)

:0

with considering (6) and (7) we find the formula for calculation of the divergent integral for functions of the type (2) with any singularity F .P.³ f (x)dV

F .P.³ ' k g ( x)dV

:

³w '

k 1

n

:

g (x)dS

(9)

w:

For example weakly singular, singular and hypersingular integrals of any dimension can be calculated in the same way using formula (9). In specific case of k 1 , we get generalization of the well-known Gauss-Octrogradskii theorem for the case of functions of the type (2) with any singularity. F .P.³ f (x)d :

F .P.³ 'g (x)d : ³ w n g (x)dS

:

:

(10)

w:

Thus formulas (9) and (10) generalized Gauss-Octrogradskii theorem for defined above singular functions and give powerful tool for their calculation. Using this formula one can easily calculate the 2-D divergent integrals which also can be calculated using standard technique and compare results. Let us consider integrals of the type Ik

1

³r

k

d :, k ! 0, k z 2

(11)

:

The integrals of this type may be regularized using the Ostrogradskii-Gauss theorem (10). In this case f (x)

1 and (k 2) 2 r k 2

1 and g (x) rk

Ik

F .P.³

:

dS rk

1 (k 2) 2

³w

w:

n

1 dS r k 2

rn 1 ³ r k dS (k 2) w:

(12)

Here rn ( xD yD )nD and D 1, 2 . This integral can be calculated analytically over circular area. Introducing polar coordinates we will get Ik

rn 1 ³ r k dS ( k 2) w:

1 (k 2)

2S

r

³r

k

rdI

0

For specific cases of k 1 and k 3 we get well known results I1 One more example, integrals of the type

1 2S ( k 2) r k 2

2S r and I 3

(13) 2S r 1 respectively.

166


³

I k1,1

:

In this case f (x)

x1 x2 d :, k ! 3,5 rk

(14)

x1 x2 1 . Regularization formula in this case has the (k 2)(k 6) r k 2

x1 x2 and 'g (x) rk

form

³

I k1,1

:

x1 x2 rn · 1 § r* ³ ¨© r k 2 r k ¸¹ dS ( k 2)(k 6) w:

x1 x2 d: rk

(15)

where r* x1n2 x2 n1 . As it is shown in [26, 29], because of symmetry these integrals are equal to zero over circular area. We can extend presented approach and regularize functions of the type

f ( x ) M ( x) ' k g ( x)

(16)

First we consider simple the case k 1 and functions of the type

f ( x ) M ( x ) 'g ( x )

(17)

where M ( x) C (:) is regular differentiable function. All other assumptions are the same as in the previous case. In the same way as in (4) consider functional in the sense of distributions

( f ,I )

³ M (x) f (x)I (x)dx ³ I (x)M (x)'g (x)dx

Rn

(18)

Rn

Using second Green theorem we obtain

³ I (x)M (x)'g (x)dx

Rn

³

I (x)M (x)'g (x)dx

::0

³

g (x) I (x)M (x) dx

(19)

::0

Last integral can be represented in the form

³

g (x) I (x)M (x) dx

::0

³ M (x)g (x)I (x)dx ³

:0

I (x)g ( x)M ( x)dx

(20)

::0

Integrating integral over domain :0 by path in inverse order using first Green theorem we obtain

³ I (x)M (x)'g (x)dx ³ I (x)M (x)w w:

:0

n

g (x)dS ³ I (x)M (x)g ( x)dx ³ M (x)M (x)g (x)dx :0

(21)

:0

From the above equations we obtain the extension of the first Green theorem for the case of singular functions in the form I0

F .P.³ M (x)'g (x)d :

³ M ( x)w

:

w:

n

g (x) dS ³ g (x)M (x)d :

(22)

:

In the same way as in classical analysis one can find extension of the second Green theorem for the case of singular functions in the form I0

F .P.³ f (x)d : :

³ M ( x )w

w:

n

g (x) g (x)w nM (x) dS ³ g (x)'M (x)d :

(23)

:

In order to consider regularization of the divergent integrals with singular function of the type (16) we can just apply second Green theorem k times. In this case the following formula takes place


F .P.³ f (x)d :

I0

:

k 1

¦ (1) ³ M (x)w i 1

n

w:

i 0

g (x) g (x)w nM ( x) dS (1) k ³ g (x)' k M (x)d :

167

(24)

:

With this formula divergent integral with any type of singularity can be represented as summa of boundary term and regular integral. For the BEM applications it is very important to consider functions with singularities of the type f ( x)

1 rm

(25)

This case has been already analyzed using generalized function approach in [22]. Regularization formulae formula for this specific case of singularity has the form I0

F .P.³ V

M ( x) rm

where g (x)

k 1

dV

¦ (1) ³ [' i 1

i 0

Pk , and Pk r m2k

M ( x )w n

k i 1

wV

(1) k i

k 1 0

Pi P 1 i w n ' k i 1M ( x)]dS (1) k ³ m 2 k ' k M ( x)]dV r m 2i r m 2i V r

(26)

1 for k , m ! 1 . (m 2i ) 2

In next sections we will demonstrate application of the developed approach to regularization and calculation of the divergent integrals that appear in BEM. 2. Piecewise constant approximation. The piecewise constant approximation is the simplest one. Interpolation functions in this case do not depend on the FE form and dimension of the domain. They have the form 1 ¯0

Iq ( x ) ®

x S n ,

(27)

x S n .

In order to simplify situation we transform global system of coordinates such that the origin is located at the nodal point, where y 0 0 , the coordinate axes x1 and x2 are located in the plane of the element, while the axis x3 is perpendicular to that plane. In this case x3 0 and n1 0 n2 0 n3 1 and fundamental solutions have the following simple form U (x y )

1 , W (x, y ) 4S r

0, K (x, y )

0, F ( x, y )

1 4S r 3

(28)

Regular representations for integrals with these kernels can be easy obtained from the general formula (26). They can be presented in the form. J1

W .S . ³

dS r

F .P. ³

dS r3

Sn

J3

Sn

³

wSn

rn dl , r

³

wSn

rn dl r3

(29)

Now corresponding integrals (29) in can be calculated over polygon using the formulas J1

1 K ¦ rn (k ) I1,0 2'1 nˆ1 (k )nˆ2 (k ) I1,1 , 2k 1

J3

¦ rn (k ) I 3,0 2'1 nˆ1 (k )nˆ2 ( k ) I 3,1 K

k 1

Here we use the following notation for the integrals

(30)

168


[l

1

I p ,l

(' k )l 1 ³

1

r p ([ )

d[

(31)

These integrals may be calculated analytically 1

I1,0

'k ³

1

1

I 3,0

'k ³

1

1 d[ r ([ )

1 1

1

, I1,1

1

1 r ([ )

ln r (k ) ' k [ r ([ )

3

d[

' k [ r (k ) , I 3,1 (r (k ) r2 (k ))r ([ ) 1 2

(' k ) 2 ³

1

[ r (t )

d[

[ d[ r ( [ )3 1

r ([ ) 1 r (k ) I1,0 , 1

1

1

(' k ) 2 ³

r (k )' k [ r 2 (k ) (r 2 (k ) r2 (k ))r ([ ) 1

(32)

It is important to mention that above formulas can be applied for calculation of regular integrals, which appear when collocation point y q is situated outside of the area of integration Sn . Obtained formulas valid for any collocation point situated inside or outside of the BE. Only for points situated at the vertexes of the boundary element special consideration is needed. It will be done in next section. In the Table 1 are presented results of the divergent and regular integrals calculations for the unit side square with coordinates of the vertexes 1 {0.5, 0.5}, 2 {0.5, 0.5}, 3 {0.5, 0.5} , 4 {0.5, 0.5} and for the unit side triangle coordinates of the vertexes 1 {0.5, 0.289} , 2 {0, 0.577} , 3 {0.5, 0.289} respectively. Calculations have been done for collocation points with coordinate y10 0.0 and different coordinates y20 which correspond to points situated inside and outside of the integration area. Table 1. Points y20

0.0 0.1 0.2 2.0 3.0

J 30,0

J 10,0

J 30, 0

Unit

J 10,0

square

Unit

triangle

3.525 3.496 3.407 0.505 0.335

-11.31 -11.61 -12.69 0.137 0.039

2.281 2.239 2.123 0.218 0.145

-18.00 -19.08 -22.31 0.057 0.016

In order to validate the regularized equations (30) with integrals (31) and (32) we compare our calculations of the hypersingular integrals with the results reported in [14], and for weakly singular and regular integrals with results obtained using regular 2-D numerical Gauss quadrature. Our calculations show that results obtained using presented here regularized equations agree with the results obtained by other methods. Also it is important to mention that there are two possibilities for calculation of the integrals in regular representations (30). The first one is to calculate corresponding integrals using formulas (31) and numerical integration and the second one is to calculate corresponding integrals using analytical formulas (32). Our calculations show that with analytical formulas (32) results are more accurate and time of calculation is 5-7 times faster in comparison with numerical integration of the integrals (31) and 8-12 times faster than obtained with 2-D numerical calculation using Gaussian quadrature. As it was mentioned before regularized equations (30) valid for collocation points situated inside or outside of the integration area Sn , but they do not valid in the vicinity of boundary of the integration area including the boundary wSn . In the Fig. 1 and Fig. 1 are presented results of calculation of the integrals (29) for collocation point situated on the vertical line that pass through the point y 0 0 for unit square and triangle respectively. Our calculations show that regularized formula (30) for weakly singular integrals valid everywhere in the area Sn including boundary wSn .

Advances in Boundary Element and Meshless Techniques For hypersingular integrals regularized formula (30) gives good results for collocation point situated inside or outside of the integration area Sn , but in the vicinity of boundary wSn corresponding integrals are divergent. For regular integrals formula (30) has very good coincidence with calculation using Gaussian quadrature in both cases weakly singular and hypersingular integrals. Also one can see that Gaussian quadrature cannot be used for calculation of the divergent integrals.

Fig.1. Calculations of weakly singular and hypersingular integrals for unit square versus y20

Fig.2. Calculations of weakly singular and hypersingular integrals for unit triangle versus y20 It is important to mention that we can calculate direct value of the hypersingular integrals when collocation point belong to the contour of integration y 0 wS n . If collocation point situated inside of the side k of polygon we have to exclude that side from summation in (30) side with number k and calculate the rest. For side k we have 1-D hypersingular integral. which have to be taken into account in the in the final result. It the same way can be calculated integrals when collocation point belong to the vertex for the polygon (see [28 ] for details). Final results for rectangle with collocation point situated at the middle of the down side y10 0, y20 0.5 are J1 2.406 , J 3 6.471 , for triangle with collocation point situated at the middle of the down side y10 0, y20 0.289 are J1 1.616 , J 3 8.309 and for vertex y10 0, y20 0.577 are J1 0.951 , J 3 3.154 . It is important to mention that all calculations here can be done analytically, no numerical integration is needed. Conclusions. It has been shown that method of the divergent integral regularization, which is based on the theory of distribution gives not only good mathematical foundation, but also very efficient tool for their calculations. Distributions based approach consider divergent integrals with different

169

170


singularities as functional defined in special functional spaces. Using methods developed in [2] we obtain generalized Gauss-Octrogradskii and Green theorems with are applicable for the case of singular functions Obtained regularization formulas can be used also for regular integrals calculation. Such integrals appear when collocation point move to the BE that not belong to the area of integration. For verification of the developed equations and investigation their behavior for various collocation points computer algebra software Mathematica has been used. We compare our calculations with reported in the literature and with numerical calculations of the regular integrals using Gaussian quadrature. In all considered cases good agreement is observed. Our study shows that calculation of the regular integrals with our regularized formulas take much less time than with Gaussian quadrature. Therefore presented here and in other our publications regularized formulas can be recommended not only for divergent integrals calculations but also for regular one.

References [1] [2] [3] [4]

[5]

[6] [7] [8] [9] [10]

[11] [12]

[13] [14]

Balas, J., Sladek, J. and Sladek, V. Stress analysis by boundary element methods, Elsevier, Amsterdam, 1989. Courant R. and Hilbert D. Methods of mathematical physics, Vol.II. Jonh Wiley&Sons, New York, 1968. Gel'fand I.M. and Shilov G.E. Generalized functions, Vol.1, Academic Press, New York, 1964. Chen J.T. and Hong H.-K. Review of dual boundary element methods with emphasis on hypersingular integrals and divergent series. Applied Mechanics Review, 52(1), pp. 17-33, 1999. Guz A.N. and Zozulya V.V. Fracture dynamics with allowance for a crack edges contact interaction, International Journal of Nonlinear Sciences and Numerical Simulation, 2(3), 173-233, 2001. Hadamard J., Le probleme de Cauchy et les eguations aux devivees partielles lineaires hyperboliques, Herman, Paris, 1923. Michlin S. G. Multidimensional singular integrals and integral equations, Pergamon Press, Oxford, 1965. Sladek V. and Sladek J. (eds.) Singular Integrals in Boundary Element Methods, WIT Press, Southampton, 1998. Tanaka, M., Sladek, V. and Sladek, J. Regularization techniques applied to boundary element methods. Applied Mechanics Reviews, 47(10), 457-499, 1994. Zozulya V.V., Integrals of Hadamard type in dynamic problem of the crack theory. Doklady Academii Nauk. UkrSSR, Ser. A. Physical Mathematical & Technical Sciences, 2, 1991, 1922, (in Russian). Zozulya V.V. Regularization of the divergent integrals. I. General consideration. Electronic Journal of Boundary Elements, 4(2), 49-57, 2006. Zozulya V.V. Regularization of the Divergent Integrals in Boundary Integral Equations. In: Advances in Boundary Element Techniques. (Eds. Ch. Zhang, M.H. Aliabadi and M. Schanz), Published by EC, Ltd, UK, 2010, 561-568. Zozulya V.V. Divergent Integrals in Elastostatics: Regularization in 3-D Case, CMES: Computer Modeling in Engineering & Sciences, 70(3), 253-349, 2010. Zozulya V.V. and Gonzalez-Chi P.I. Weakly singular, singular and hypersingular integrals in elasticity and fracture mechanics, Journal of the Chinese Institute of Engineers, 22(6), 763775, 1999.


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0

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2000

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4000

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6000

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8000

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1.4

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183

184


BEM study of the symmetry of the debond onset at the fibre-matrix interface under transverse load I.G. Garc´ıa, V. Mantiˇc, E. Graciani ´ Escuela Tecnica Superior de Ingenier´ıa, Universidad de Sevilla Camino de los Descubrimientos s/n, 41092 Sevilla, Spain [email protected], [email protected], [email protected]

Keywords: interface crack, crack initiation, coupled stress and energy criterion, finite fracture mechanics, VCCT,

composite, BEM

Abstract. A Boundary Element Method code is used in combination with the coupled stress ad energy criterion of the Finite Fracture Mechanics (FFM) to solve the controversy about if the initiation of failure at the fibre-matrix interface under transverse loading is symmetric or not. The coupled criterion allows predicting the critical remote stress leading to the failure for the two failure configurations: symmetric and asymmetric. The comparison of both results shows that the asymmetric failure requires a lower critical remote stress which is in accordance with experimental results. Introduction The failure mechanism in Fibre Reinforced Composites (FRC) under transverse loads typically initiates as partial debonds at the fibre-matrix interface (or very close to this interface) [1]. A simplified model composed by a unique fibre bonded to an infinite matrix has been intensively used to study the onset and growth of the crack along the fibre-matrix interface assuming plain strain at the plane perpendicular to the fibre-axis. Several works analyzed the problem of the initiation by modelling the interface behaviour employing Cohesive Zone Models (CZM) [2,3], and Linear Elactic-Brittle Interface Model (LEBIM) [4] or the coupled stress and energy criterion of the Finite Fracture Mechanics (FFM) [5, 6]. These works agree in some predictions as the size effect of the fibre-radius. However, there is a controversy about the symmetry of the partially debonded configuration after the crack initiation. Two main geometries are observed after the onset in numerical results: two symmetric and separated debonds, typically in CZM with some exceptions, see [2, 3], or an asymmetric configuration with one debond in LEBIM models [4], which is however implementation dependent. In view of this, the aim of this work is to study the two failure scenarios, symmetric and asymmetric applying the coupled criterion of the FFM. This criterion, proposed by Leguillon [7], assumes that a crack of a finite length initiates when two conditions are fulfilled simultaneously: a stress condition which imposes that normal tractions along the expected crack path have to exceed a critical value, and an energy condition expressing the first law of thermodynamics applied to the energetic balance between the state before and after the crack onset. Fig. 1 shows a schema of the problem under study. The initial situation is given by a circular section of the fibre with radius a perfectly bonded to an unbounded matrix with a remote transverse load σ∞ x applied. A monotonic increase of σ∞ x leads to a situation where both conditions for the debond onset imposed by the coupled criterion are fulfilled. The critical value of σ∞ x predicted by the criterion depends on the failure configuration assumed. Two different failure situations showed in Fig. 1 and denoted as “symmetric” and “asymmetric” configurations are considered in the present work. The elastic solution in both failure configurations will be calculated with the aid of a Boundary Element Method (BEM) code developed by one of the authors [8]. Stress criterion The tensile stress criterion applied to this problem imposes that a crack at the interface can appear at those points where normal tractions σ exceed a critical value σc identified with the tensile strength of the interface. In view of Bimaterial glass/epoxy

E1 (GPa) 70.8

ν1 0.22

E2 (GPa) 2.79

ν2 0.33

α 0.919

β 0.229

k 1.44

m 1.56

Table 1: Elastic properties of the composite studied (1.- fibre, 2.- matrix).


185

Figure 1: Schema of the problem: Two possible configurations of the debonds along the fibre-matrix interface under transverse load. the initial symmetries, this criterion can be studied just for y ≥ 0 and x ≥ 0, see Fig. 1. This implies that θ ≥ 0. Thus, this condition can be written as, (1) σ(θ) ≥ σc , ∀θ ∈ [0, ∆θ], where ∆θ is the debond semiangle after its onset (or debonds in the symmetric configuration). Values of normal tractions along the interface can be extracted from the analytic solution of Goodier [9] as shown in [5]. Normal tractions are decreasing for θ ≥ 0, see Fig. 2(a) and [1, 5, 6], then, the condition in Eq. (1) is fulfilled if it is verified for θ = ∆θ. As a consequece, the final expression of this criterion can be written as, σ∞ 1 x ≥ = s(∆θ) σc k − m sin2 ∆θ

(2)

where k and m are elastic parameters of the bimaterial given in terms of Dundurs’ parameters α and β, see [5] for a detailed discussion and Table 1 for their values for glass/epoxy. This expression plotted in Fig. 2(b) defines a minimal load as a function of the debond semiangle ∆θ. A more exhaustive analysis of the consequences of this criterion can be found in [5, 6]. Note that the situation before the crack onset is identical for both failure configurations. Energy criterion A debond of a finite semiangle ∆θ is allowed if the incremental variation of energy between the state before and after the crack onset fulfills the first law of thermodynamics, which can be expressed as, ∆Π + ∆Ek + ∆Γ = 0,

(3)

where ∆Π and ∆Ek are the change in the potential elastic and kinetic energy, respectively, and ∆Γ is the energy dissipated in irreversible processes during the crack onset. Under the assumption of a quasistatic initial state: ∆Ek ≥ 0. The change in potential energy ∆Π can be related to the Energy Release Rate (ERR) G by the classical expression G = −dΠ/d(aθd ) where θd is the debond semiangle. Finally, the energy dissipated Ed can be approximated by the integration of the “instantaneous” interface fracture toughness Gc . Thus, taking into account these assumptions, the balance in Eq. (3) leads to [5, 6], ∆θd ∆θd G(θd )dθd ≥ Gc (ψ(θd ))dθd , (4) 0

0

186


6 2.0

k

5

1.5 Σ, Τ Σx

Σ

1.0

Com p ression s Σx

Traction s

0.5

Σc

3

0.0

2

0.5

1

1.0

Τ

1 or 2 d eb on d s

4 NOT SAFE 1/ k SAFE

0 0

20

40

60

80

0

20

40

Θ º

60

80

Θº

(a)

(b)

Figure 2: Plot of (a) normal tractions along the fibre-matrix interface and (b) the condition imposed by the stress criterion for the semiangle of debond ∆θ and the remote stress σ∞ x where θd is an intermediate debond semiangle and the angle ψ(θd ) is a measure of the fracture mode mixity. The dependence of Gc on the fracture mode mixity is approximated by the phenomenological law of Hutchinson & Suo [10]: Gc (ψ) = G1c (1 + tan2 ((1 − λ)ψ(θd ))) where G1c is the interface fracture toughness in pure mode 1 and λ is a mode mixity sensivity parameter. G and ψ will be evaluated by postprocessing numerical results from a BEM analysis. These magnitudes depend, in general, on the debond semiangle and debond configuration (one or two debonds), the fibre radius, the remote load and the elastic parameters of fibre and matrix. As a result, the amount of numerical solutions required would be enormous since it is necessary to compute, a priori, an elastic solution for each combination of values for these variables. In order to reduce the number of elastic solutions to compute, an adequate application of the dimensional analysis allows us to rewrite Eq. (4), see [11] for a detailed discussion, in the following manner: ∆θd 2 ∆θd (σ∞ x ) a ˆ d )dθd ≥ G1c G(θ Gˆ c (ψ(θd ))dθd (5) ∗ E 0 0 where Gˆ =

E∗ G σ2x a

and Gˆ c = Gc /G1c are the dimensionless energy release rate and interface fracture toughness, 2 1−ν 1−ν2 respectively, and E ∗ = 2/ E1 1 + E2 2 is the harmonic mean of the effective Young’s moduli. By a rearrangement in Eq. (5), a form of the energy criterion, analogous to Eq. (2), can be obtained as

∆θd ˆ ∗ Gc (ψ(θd ))dθd σ∞ G E 1 1c 0 x ≥ , (6) ∆θd σc σc a ˆ d )dθd G(θ 0 γ √ g(∆θd )

where γ is a dimensionless brittleness parameter defined in [5] and g a ratio of the dimensionless dissipated energy ˆ d ) and to the released energy at the debond onset. The function g(∆θ) is evaluated numerically. In particular, G(θ ψ(θd ) are obtained by postprocessing BEM results. BEM analysis BEM is an excellent computational tool for the evaluation of Gˆ and ψ for the interface cracks in the present problem. In particular, the BEM code developed in [8] is very suitable for this kind of analysis since it solves interface cracks problems in presence of contact according to the Comminou model [12]. A plain strain analysis is carried out for a relatively large finite square cell of matrix with a small circular inclusion placed in its centre, see Fig. 3(a). In order to approximate the solution for an infinite matrix, the cell side length is 400/3 times larger than the circular inclusion radius. A horizontal traction is imposed on the vertical sides of the cell, whereas free-stress condition is applied at the horizontal sides as can be seen in Fig. 3(a). A uniform mesh of 10 elements is used along the cell sides. The fibre-matrix interface is modelled by two conforming meshes, for the inclusion and matrix side. Two types of boundary conditions are imposed at nodes on this interface


(a)

187

(b)

Figure 3: (a) Model and mesh used in the BEM analysis to compute Gˆ and ψ, and (b) angular size of the boundary elements ∆ϑelem situated at the interface as a function of the angle θ for the debond semiangle θd = 30◦ . depending on the debond semiangle and failure configuration studied: perfect adhesion or frictionless contact, respectively, for nodes situated outside or inside the debonded zone. The mesh is strongly refined at the debond crack tips because of the strong stress gradient herein. Thus, nodes at the interface are divided into two types: • Nodes situated closer to a crack tip than a certain distance where a mesh refinement takes place. The lengths of boundary elements are given by a geometric progression, ϑn+1 = ϑn + δϑhom · q−n

(7)

where ϑn > 0 is the polar coordinate along the interface of the node n with the origin at the crack tip (where ϑ = 0). δϑhom is the length of the elements at the zones of the interface with homogeneous mesh. q = 1.2 is the common ratio of the geometric progression. The minimum element length is δϑmin = 0.00004◦ corresponding to elements contiguous to the crack tip. • The interface situated outside the neartip zone is modelled with a uniform mesh with elements of length δϑhom = 2◦ is employed. An algorithm is developed in order to generate automatically the mesh as a function of θd , number of debonds, q, δϑhom and δϑmin . Fig. 3(b) shows the angular size of elements for a debond semiangle θd = 30◦ as a function of the polar coordinate θ of the element center at the interface. The generation of the input files is automatic and a sequence of numerical analysis is carried out for ∆θd = 2◦ and the symmetric and asymmetric case. The application of the Virtual Crack Closure Technique (VCCT) [13] to the BEM results of tractions and displacement near the crack tip gives an excellent approximation of G by, δθa δθa 1 1 G(δθa ) = σ(ϑ)+ ∆un (δθa − ϑ)− dϑ + τ(ϑ)+ ∆ut (δθa − ϑ)− dϑ (8) 2δθa 0 2δθa 0 GI

G II

where σ and τ are, respectively, the near-tip normal and shear tractions, ∆un and ∆ut the normal and tangential gaps between the crack faces, respectively, and δθa is the angle of the virtual crack. The value of the dimensionless ERR associated to mode 1, Gˆ I , and mode 2, Gˆ II , does not tend to a fixed value for δθa → 0+ for interface cracks, see [12, 14], whereas Gˆ = Gˆ I + Gˆ II has a limit for δθa → 0+ . A Gauss-Chebyshev quadrature is applied to compute the integrals in Eq. (8) following [15]. Fig. 4(a) shows the results for Gˆ as a function of θd for the symmetric and asymmetric cases and the excellent agreement of this with the classical Toya’s solution when the open model of interface cracks is valid. It is interesting to remark that both cases have almost identical values of Gˆ for small θd . However, for moderate and large θd , Gˆ is larger for the asymmetric than for the symmetric case because of the shielding effect between the two debonds.

188


5

100

Toya' s solu tion

80

4 G

3

1 d eb on d

60

Ψº

2 d eb on d s

40

2

20

2 d eb on d s

1

1 d eb on d

0

0 0

20

40

60

0

80

20

40

60

80

Θ d º

Θ d º

(a)

(b)

ˆ and (b) interface stress-based fracture mode mixity ψ as functions of the Figure 4: (a) Dimensionless ERR G, debond semiangle θd for the asymmetric (1 debond) and asymmetric case (2 debonds). 6

3.0 2.5

2 debonds

5 scen ario A

2.0 Σ c Σc

4

scenario A Σc 3

1.5 1 debond

1.0

2

0.5 scenario B 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

scen ario B

1 0 0

1

2

Γ

3

4

Γ

(a)

(b)

σ∞ c

Figure 5: (a) Critical remote stress and (b) difference in % of of γ, (c) example of single-fibre test results [16].

(c)

σ∞ c

for both failure configurations as a function

The stress-based fracture mode mixity ψ is evaluated from the near-tip interface tractions by tan ψ(θl ) = τ(θl )+ /σ(θl )+ , where θl is a reference angle [5, 12, 14]. Following the recomendations in [5], we set θl = 0.1◦ . Results for ψ are plotted in Fig. 4(b). This plot shows that the fracture mode mixity is very similar for the symmetric and asymmetric case, and consequently will not have a large influence on the competition between the symmetric or asymmetric debond onset. Coupled criterion and results Stress criterion in Eq. (2) and energy criterion in Eq. (6) impose two conditions for two unknowns: the dimensionσ∞ less critical remote stress σxc and the debond semiangle ∆θ at the onset. Following the hypothesis of Leguillon in [7], we assume that the onset occurs for the minimum remote stress when both conditions are fulfilled. The analysis and evaluation of this remote critical stress is not simple because of the complexity of functions s in Eq. (2) and g in Eq. (6). Two scenarios are possible depending on the γ value. A computational algorithm solving the present nonlinear problem is necessary to obtain the critical remote stress σ∞ c , see [11] for details. The two scenarios found, named scenarios A and B, correspond to critical remote stresses essentially governed by the stress and energy criteria, respectively, see [5, 11] for a detailed discussion. Following the algorithm presented in [5], results of combining both criteria are plotted in Fig. 5 as a function of γ for glass/epoxy. Fig. 5(a) shows that the predicted remote stresses for both cases are very similar but being always smaller for the asymmetric case as is confirmed in Fig. 5(b). This means that the asymmetric failure is the failure mode predicted by the coupled criterion. This agrees with experimental results, see for example Fig. 5(c) where the debonds have grown from the free border to the centre.


Concluding remarks The Boundary Elements Method has been used to clarify a controversy on the initiation of debonds at the fibrematrix interface under uniaxial transverse loads by applying the coupled criterion of the Finite Fracture Mechanics. The controversy is if the failure configuration is characterized by two debonds as is predicted by the majority of Cohesive Zone Models simulations or by one debond. The present analysis predicts that the asymmetric configuration requires a smaller critical remote stress, so it is the failure configuration prefered by the coupled criterion. This result agrees with experimental results. This work has demonstrated that the combination of the coupled criterion and the BEM is an excellent tool to predict the failure initiation in the cases where numerical results are necessary to evaluate the governing parameters of the coupled stress and energy criterion. Acknowledgments V. M. thanks to Prof. Federico Par´ıs (University of Seville) and Prof. Janis Varna (Luleå University of Technology) who presented him the problem studied in this work in the middle of 1990s. This work was supported by the Junta de Andaluc´ıa and the Spanish Ministry of Science and Innovation, through the Projects TEP4051 and MAT200914022, respectively, and the FPU Grant of the Spanish Ministry of Education corresponding to I.G. Garc´ıa. References [1] F. Par´ıs, E. Correa, and V. Mantiˇc. Kinking of transversal interface cracks between fiber and matrix. Journal of Applied Mechanics, 74(4):703–716, 2007. [2] A. Carpinteri, M. Paggi, and G. Zavarise. Snap-back instability in micro-structured composites and its connection with superplasticity. Strength, Fracture and Complexity, 3(2):61 – 72, 2005. [3] R. Han, M.S. Ingber, and H.L. Schreyer. Progression of failure in fiber-reinforced materials. Computers, Materials and Continua, 4(3):163–176, 2006. [4] L. Távara, V. Mantiˇc, E. Graciani, and F. Par´ıs. BEM analysis of crack onset and propagation along fiber-matrix interface under transverse tension using a linear elastic-brittle interface model. Engineering Analysis with Boundary Elements, 35(2):207–222, 2011. [5] V. Mantiˇc. Interface crack onset at a circular cylindrical inclusion under a remote transverse tension. Application of a coupled stress and energy criterion. International Journal of Solids and Structures, 46(6):1287 – 1304, 2009. [6] V. Mantiˇc and I. G. Garc´ıa. Crack onset at the fibre-matrix interface under a remote transverse biaxial load. Application of a coupled stress and energy criterion. International Journal of Solids and Structures, doi: 10.1016/j.ijsolstr.2012.04.023, 2012. [7] D. Leguillon. Strength or toughness? A criterion for crack onset at a notch. European Journal of Mechanics, A/Solids, 21(1):61–72, 2002. [8] E. Graciani. Formulation and implementation of the Boundary Elements Method to axisymmetric problems with contact. Application to the charaterization of the fibre-matrix interface in composites (In Spanish). PhD thesis, School of Engineering, University of Seville, 2006. [9] J.N. Goodier. Concentration of stress around spherical and cylindrical inclusions and flaws. J. of Applied Mechanics, 55:39–44, 1933. [10] J.W. Hutchinson and Z. Suo. Mixed mode cracking in layered materials. Advances in Applied Mechanics, 29:63–191, 1992. [11] I. G. Garc´ıa, V. Mantiˇc, and E. Graciani. Debonding at the fibre-matrix interface under remote tranverse tension. one or two symmetric debonds? (to be submitted), 2012. [12] V. Mantiˇc, A. Blázquez, E. Correa, and F. Par´ıs. Analysis of interface cracks with contact in composites by 2D BEM. In M. Guagliano and M.H. Aliabadi, editors, Fracture and Damage of Composites, chapter 8, pages 189–248. WIT Press, Southampton, 2006. [13] G. R. Irwin. Analysis of stresses and strains near the end of a crack traversing a plate. J. of Applied Mechanics, 24(3):361–364, 1957. [14] V. Mantiˇc and F. Par´ıs. Relation between SIF and ERR based measures of fracture mode mixity in interface cracks. International Journal of Fracture, 130(2):557–569, 2004. [15] E. Graciani, V. Mantiˇc, and F. Par´ıs. A BEM analysis of a penny-shaped interface crack using the open and the frictionless contact models: Range of validity of various asymptotic solutions. Engineering Analysis with Boundary Elements, 34(1):66–78, 2010. [16] E. Correa. Micromechanical analysis of the matrix failure in fibre reinforced composites (In Spanish). PhD thesis, School of Engineering, University of Seville, 2008.

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190


An Efficient Numerical Scheme for the Evaluation of the Fundamental Solution and its Derivatives in 3D Generally Anisotropic Elasticity Y.C. Shiah1, C.L. Tan2 and C.Y. Wang1 1. 2

Department of Aerospace Engineering and Systems Engineering Feng Chia University, Taichung, Taiwan, R.O.C. Department of Mechanical & Aerospace Engineering Carleton University, Ottawa, Canada K1S 5B6 (* Corresponding author; Email: [email protected])

Keywords: Anisotropic elasticity, fundamental solutions, Green’s functions, Stroh’s eigenvalues, Fourier series.

Abstract. This paper presents an alternative numerical scheme to compute the fundamental solution by Ting and Lee [1], and its derivatives, for 3D general anisotropic elasticity. By taking advantage of the periodic nature of the angles when it is expressed in the spherical coordinate system, the Green’s function is represented as a double Fourier series. The Fourier coefficients are determined numerically only once, irrespective of the total number of field points involved in the BEM analysis of a problem. The derivatives of the fundamental solution can also be obtained simply by direct spatial differentiation of the double Fourier series without further numerical or significant analytical steps. Some minor issues that arise and ways to resolve them are discussed. Sample numerical results are presented to demonstrate the significantly superior efficiency of computing these quantities, while achieving the same accuracy, when compared to other direct analytical formulations based on the same Green’s function. This is especially so when large number of field points are involved, as is typically the case when modelling practical engineering problems. Introduction The fundamental solution or Green’s function to the governing differential equation of the physical problem, and its derivatives are essential items in the direct formulation of the boundary element method (BEM) and some meshless methods. In elastic stress analysis, the Green’s function for displacements and its first derivatives are used in the derivation of the conventional displacement-boundary integral equation (BIE); higher order derivatives of this fundamental solution are required for evaluating the stresses at interior points via Somigliana’s identity, and in, e.g., the formulation of the traction-BIE. The fundamental solution for displacements in a 3D generally anisotropic solid that was first derived by Lifschitz and Rozentsweig [2] was not of closed-form. It was expressed as a line integral around a unit circle; its integrand consists of the Christoffel matrix, defined in terms of the elastic constants of the material. Much attention was paid over the past few decades (see, e.g., [3]-[8]) to simplify the above-mentioned line integral into more explicit analytical forms, as well as on the development of efficient algorithms for their accurate and stable numerical evaluation since it was first implemented in BEM by Wilson and Cruse [9]. An algebraic, real variable form of the Green’s function for displacements in a 3D generally anisotropic body, and its derivatives, was derived by Ting and Lee [1] and Lee [10], respectively. They are expressed in terms of Stroh’s eigenvalues. Because of their explicit forms, they could be implemented into existing BEM codes in a relatively simple manner. This was, however, only recognised quite recently; its introduction into BEM formulations was first carried out by Tavara et al [11] for the special case of transverse isotropy, and by the present lead authors for full general anisotropy [12, 13], as well as by Buroni and Saez [14] more recently. The BEM implementation in [12, 13], although fairly straightforward, revealed the relative inefficiency of computing the higher order derivatives of the Green’s function because of the presence of very high order tensor terms (up to 10th order for 2nd order derivatives). This prompted Lee [15] to re-examine the problem; new general forms of the Green’s function derivatives, expressed in terms of spherical coordinates, were obtained without the need to introduce very high order tensors. Using this revised approach and the residue theorem for high-order poles, the lead authors derived the explicit algebraic expressions of the 1st and 2nd order derivatives for implementation in the BEM for computing internal point stresses in 3D generally anisotropic solids [16-18] recently; the better computational performance over the previous formulation is also demonstrated. These explicit forms of the derivatives are, however, tedious and their implementation, even if it is relatively quite straightforward, is quite involved. It


191

is expected that for even higher-order derivatives of the Green function which are required for hypersingular BEM formulations, it will likely be even more so. In this paper, an alternative approach to evaluate Ting and Lee’s [1] fundamental solution and its derivatives is proposed. Instead of directly evaluating the explicit algebraic expressions as derived in [1], [13] or [16], the fundamental solution is expressed as a double Fourier series. This is feasible by virtue of the periodic nature of the angles in the spherical coordinate system used in Ting and Lee’s fundamental solution. The Fourier coefficients can be numerically obtained, and is done only once irrespective of the number of field points in the numerical solution domain. Furthermore, the derivatives of the fundamental solution can be directly performed by spatial differentiation on the Fourier series. In what follows, the fundamental solution of Ting and Lee [1] will first be reviewed. The conversion of this solution into the form of a double Fourier series, the subsequent determination of its derivatives, and the numerical evaluation of these quantities will then be described. Due to space limitations, the full expressions for many of the terms that appear in the equations will not be presented; the reader will be referred to the relevant references where they be found. Numerical examples are given to demonstrate the significant computational advantage of this scheme, particularly when the number of field points is relatively large, as is typically the case for practical engineering problems. Fundamental solution for displacements The Green’s function for displacements in a generally anisotropic 3D body that was derived by Ting and Lee [1] can be expressed in simple closed-form as 1 (1a) U( x ) H[x ] , 4S r

or in spherical coordinates as

H(T , I ) , 4S r

U( r ,T , I )

(1b)

In eq. (1), r is the radial distance between the load and field point, and H (T , I ) , the Barnett-Lothe tensor, depends only on the spherical angles (T, I) defined in the usual sense. It can be expressed in terms of Stroh’s eigenvalues as 1 4 ˆ (n) H(T ,I ) (2) ¦ qn ī , T n= 0

ˆ ( n ) and q may be found in [1], [12]. Since H (T , I ) is periodic, where the explicit expressions for T , ī n with an interval width of 2S, for the both ș andI, it can be represented by a double Fourier series as follows, H uv (T ,I )

Ouv( m ,n )

f

f

¦ ¦O

m f n f

1 4S 2

( m ,n ) uv

S

S

e

i mT n I

u, v

,

³ S ³ S H T ,I e

i m T nI

uv

1, 2, 3 , dT d I ,

(3a) (3b)

where “i” is used to denote 1 . Equation (3b) may be numerically integrated, say by Gaussian quadrature; if k abscissa points are employed, it may be re-written as

Ouv( m ,n )

1 k k ¦¦ wp wq fuv( m,n ) S [ p ,S [q , 4 p 1q 1

(4)

where w and [ are the weights and the Gauss points, respectively; and f uv( m ,n ) T , I represents the integrand in eq. (3b), i.e.

f uv( m ,n ) T , I

H uv T ,I e i m T nI .

(5)

192


To ensure good accuracy of the numerical integration, a relatively large number for k may be required. This is especially so when m and n may also be large (§ 20) for a highly general anisotropic material, giving rise to rapidly varying values of the integrand, f ij( m ,n ) T , I . Figure 1 shows the variations of the real parts of f ij( m ,n ) T , I for an example anisotropic material when m=n=20; the plots for the imaginary parts are similar. Extensive numerical tests conducted suggest that 64-point Gauss quadrature may evaluate eq.(4) very satisfactorily for m=n=20. Thus, k=64 is employed for the present study, the Gauss abscissae and H12

H11

I

I

T

T H13 H22

I I T

T

H23 H33

I

I T T

Figure 1: Variations of f

(20,20) ij

T , I for a generally anisotropic material

weights for which may be found in, e.g., Stroud and Secrest [19]. It is worth emphasizing that determination of the coefficients Ouv( m ,n ) using eq. (4) is only done once, no matter how many field points there are in the solution domain for which the evaluations of the Green’s function and its derivatives are required. For practical engineering problems in 3D BEM analysis, the number of evaluations of these quantities is in the order of 106 and higher. The CPU-time for processing the one-time calculation of the Fourier coefficients using k=64 is therefore insignificant in practice. Thus for the purpose of numerical evaluation, the fundamental solution can also be written as


U uv ( r,T , I )=

193

1 D D ( m ,n ) i m T n I , ¦ ¦ Ouv e 4S r m D n D

(5)

where D is an integer sufficiently large to yield accurate results.

Derivatives of the displacement fundamental solution

In the earlier work by Lee [10], the differentiation of Green’s function, eq. (1a), is carried out directly, in the Cartesian coordinate system and the 1st order derivatives so obtained are expressed as follows U ij ,l

1 ª S yl H ij C pqrs ys M lqiprj yq M sliprj º , ¼ 4S 2 r 2 ¬

(6)

where C pqrs are the stiffness coefficients of an anisotropic body; and, yi are the components of a unit position vector y = x/r in the spherical coordinate system. The explicit algebraic expressions of the terms in the 6th order tensor M ijklmn , which are in terms of Stroh’s eigenvalues, is fairly elaborate and may be found in [10]. The 2nd order derivatives were also derived in [10], for which 10th order tensors need to be introduced. Although quite straightforward to implement into a BEM code, it became clear from the work in [12, 13] that the task of numerically evaluating these very high order tensors leave much to be desired. By differentiating with respect to the spherical coordinates as an intermediate step, separating the terms associated with the radial distance, and then applying the chain rule for the total derivative, Lee [15] re-derived Uij,l into a different algebraic form. It eliminates the use of the very high order tensors seen previously. Performing partial differentiations of the Green’s function in the spherical coordinate system, the displacement derivatives can be written as wU ij wr wU ij wT wU ij wI . wr wxl wT wxl wI wxl

U ij ,l

(7)

The partial derivatives of Uij with respect to r, T, and I are obtained as wU ij

U ij

wr

r

,

wU ij

I1 J 1 , 4S 2 r

wT

wU ij wI

I2 J 2 , 4S 2 r

(8)

where the explicit expressions of I1, I2, J1, J2 are given in [15]. Using the same approach with residue theorem for high-order poles, the present lead authors have derived the explicit expressions for the 2nd order derivatives [17]. This involves, first, making spatial differentiation of Uij,l, viz, w 2U ij

wU ij , k wr wU ij , k wT wU ij , k wI , wr wxl wT wxl wI wxl

wxk wxl

(9)

followed by the partial differentiation with respect to the spherical coordinates.This yields the following: w 2U ij wr 2

w Uij 2

wT 2

=

U ij r2

wU ij wr

1 § wI 'ij wJ 'ij · w Uij ¨ ¸, 4S 2 r © wT wT ¹ wI 2 2

,

w 2U ij wr wT

1 wU ij , r 2 wT

w 2U ij wr wI

1 § wI "ij wJ "ij · w Uij ¨ ¸, 4S 2 r © wI wI ¹ wTwI 2

1 wU ij r 2 wI

1 § wI "ij wJ "ij · ¨ ¸ 4S 2 r © wT wT ¹

(10)

(11)

194


The component terms in eq (11) are again expressed in terms of Stroh’s eigenvalues; their explicit expressions are also fairly elaborate and are given in [17]. Instead of formulating the partial derivatives in an exact analytical manner above, the partial differentiations are carried out on the Fourier series form of the Green’s function here. Substitution of eq. (5) into eq. (7) and carrying out the indicated differentiations yields the following:

U ij ,l

D D ( m, n ) i mT nI ª cosT sin I i n cos I º ° ¦ ¦ Oij e « » for l 1 ° m D n D ¬ i m sin T / sin I ¼ ° 1 ° D D ( m, n ) i mT nI ª sin T sin I i n cos I º ® ¦ ¦ Oij e « » for l 2 4S r 2 ° m D n D ¬ i m cos T / sin I ¼ ° D D ( m , n ) i mT n I ° ¦ ¦ Oij e for l 3 ¬ª cos I i n sin I º¼ ° m D n D ¯

(12)

Thus, the first derivatives of the Green’s function can be evaluated from eq. (12) directly; the Fourier coefficients, Ouv( m ,n ) , have been previously determined. Equation (12) is much more concise algebraically and can also be computed more efficiently when compared to the previous approaches mentioned above. However, due care should be taken when I=0 or S for l 1 or l 2 as these conditions give rise to a numerical singularity. These conditions correspond to cases where the load and field points are both on the x3-axis; the spherical angle T and its derivative, being functions of the Cartesian coordinates, become multivalued and are ill-defined. Physically, the spatial differentiation w / wx1 for points along the x1-axis implies the rate of change along T=0; similarly, w / wx2 implies the rate of change in the direction T=S/2. This singularity may be easily removed by introducing a small perturbation forI, say I= 10-6, and selecting T 0 for l 1 , and similarly, selecting T S / 2 for l 2 . For convenience, T 0 can be selected for l =3. Thus, for field points along the x3-axis, the first derivatives of the fundamental solution may be written as

U ij ,l

D D ( m,n ) i nI for l 1 ° ¦ ¦ J ij i n e cos I m D n D ° § mS · nI ¸ i¨ 1 °° D D ( m, n ) O i n e © 2 ¹ cos I for l 2 . 2 ® ¦ ¦ ij 4S r ° m D n D ° D D ( m,n ) i nI for l 3 ° ¦ ¦ Oij e cos I °¯ m D n D

(13)

In eq. (13), I=0 and I=S are chosen when the field point lies along the positive x3-axis and along the negative x3-axis, respectively. Thus, the issue of the numerical singularity can be easily overcome by either using eq.(13) or simply introducing a small perturbation to I and properly selecting T as described above. In a similar manner, the 2nd order derivatives of the Green’s function can be obtained by chain rule again as follows U ij ,lk

wU ij ,l wr wU ij ,l wT wU ij ,l wI wr wxk wT wxk wI wxk

(14)

Substitution of eq. (12) into eq. (14) and carrying out the indicated differentiations gives the six components of the 2nd order derivatives. Only the first component is shown below in eq. (15), due to space limitations. The other components have similar forms; they are evidently more concise than those derived in [10], [17].


U ij ,11

½ ° ° ° ° § cos T sin I i n cos I · °2sin I cos T ¨ ° ¸ ° ° © i m sin T / sin I ¹ ° ° D D ª º i m i n sin T cos T sin I cos I 1 sin T ° ° J ( m , n ) ei m T n I ® « » ¾ 3 ¦ ¦ ij I 4S r m D n D sin i m i m T T I cos sin / sin ° ° ¬« ¼» ° ° º° ª § cos I i n sin I · ° »° ° cos I cos T « cos T ¨¨ i n sin I i n cos I ¸¸ ¹ »° « © ° « 2 »° ° ¬« i m sin T cos I i n sin I / sin I »¼ ¿ ¯

195

(15)

Again, due care should be exercised when the load and field point lie simultaneously on the x3-axis; similar schemes as described above can be used to avoid the numerical singularity in the computations. Numerical examples

The accuracy and computational efficiency of the Fourier series scheme proposed is illustrated here. To this end, numerical values of the fundamental solution and its 1st and 2nd order derivatives are obtained for an increasing number of field points in an anisotropic medium. These results are compared with those computed using two previous approaches which have also been previously implemented by the lead authors [12,13], [16-18] as listed below:

Approach 1: Approach 2:

Green’s function eq. (1) eq. (1)

1st order derivatives eq. (6) eq. (7)

2nd order derivatives finite difference of eq.(6) eqs.(9) – (11)

The finite difference scheme is adopted in Approach 1 for the evaluation of the 2nd order derivatives as the effort involved in computing the 10th order tensors present in these derivatives in [10] is quite prohibitive. The material chosen here is an alumina crystal which has the following non-zero elastic stiffness coefficients: C11=465 GPa; C22=563 GPa; C33=233 GPa; C12=124 GPa; C13=117 GPa; C14=101 GPa. The principal material axes are deliberately rotated clockwise with respect to the three Cartesian axes in sequence by 30o, 45o and 60o, respectively to demonstrate the capability of the present scheme for full anisotropy; the resulting fully populated stiffness matrix of the elastic constants is as follows: ª544.8 153.6 57.3 10.5 «153.6 531.1 28.4 -14.7 « « 57.3 28.4 654.4 19.8 [C ] « 10.5 -14.7 19.8 106 .4 « « 65.7 -18.1 -6.4 24.8 « 89.7 10.4 13.3 ¬«-81.2

65.7 -18.1 -6.4 24.8 167.9 22.5

-81.2 º 89.7 »» 10.4 » » GPa 13.3 » 22.5 » » 243.5¼»

(17)

Table 1 lists the numerical values of the fundamental solution for an arbitrarily chosen field point (r,T, I)Ł (1, ʌ/3, ʌ/4) computed directly using eq. (1) and that using the present double Fourier series representation for different values of Į in eq. (5). It can be seen that the percentage errors are very small even with Į=10 and are generally of the order of 10-3 when Į=20. The computed numerical values of the 1st and 2nd order derivatives at the same field point using the three approaches described above are shown in Tables 2 and 3, respectively, with Į=20 chosen for the double Fourier series scheme. As can be seen in these tables, they are in excellent agreement with one another, the corresponding numerical values being identical up to sixth decimal place in most cases.

196


Figure 2 shows variation of the CPU times of the present scheme and Approach 2, relative to those of Approach 1, to compute all the components of the fundamental solution, its 1st order and the 2nd order derivatives together when the number of field points is increased from N=5 to N=106. The computations were carried out using a PC with an Intel quad-core processor. In these computations, Approach 2 is generally about 50% more efficient than Approach 1; but when N exceeds N=103, the present scheme with Į =10 would take less CPU time than either of these two approaches. The superior is seen when N exceeds about 104 if Į =20 is employed, noting again that these values of N are several orders of magnitude less than what would normally be encountered in practical problems.

Approach 1 Approach 2 Fourier Į=10 Fourier Į=20

1.6 Normalized CPU-time

1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0

103

104

105

106

N Figure 2: Variation of the normalized CPU-time with the number of field points

Conclusions

The fundamental solution of displacements of Ting and Lee [1] for a 3D generally anisotropic solid and the corresponding exact expressions for its derivatives have been implemented in BEM formulations in recent years. Although these implementations have been straightforward because of their algebraic forms, they are nevertheless still quite involved as its derivatives, in particular, are mathematically very elaborate. In this study, a very efficient alternative scheme has been proposed to compute the fundamental solution and its derivatives. The feasibility of this scheme lies in the form of Ting and Lee’s Green’s function that readily permits its representation as a double Fourier series. The Fourier coefficients need to be obtained numerically for a given anisotropic material only once, and all subsequent derivatives of the Green’s function can be obtained by performing spatial differentiation of the double Fourier series. Numerical results have been presented to demonstrate the accuracy of the solutions and the superior computational efficiency of this scheme. Of significance to note too is that the its implementation into BEM codes is made even easier because of its simpler and less elaborate analytical forms as compared to previous formulations.


197

Table 1: Comparison of the computed Green’s function at r=1.0,T=S/3,I=S/4 Uij U11 (10-12m) U12 (10-12m)

Approach1 0.336396 0.021313

Fourier

D=10

D=12

D=14

D=16

D=18

D=20

Series Sum 0.336431 0.336381 0.336389 0.336397 0.336396 0.336396 |% Error| 0.010423 0.004450 0.002033 0.000425 0.000096 0.000134 Series Sum 0.021433 0.021329 0.021300 0.021311 0.021315 0.021314 |% Error| 0.562254 0.073902 0.061398 0.011528 0.008529 0.001478

Series Sum -0.006627 -0.006501 -0.006522 -0.006540 -0.006536 -0.006535 U13 -0.006535 (10-12m) |% Error| 1.401946 0.528032 0.194557 0.073778 0.021024 0.000581 U22 (10-12m)

0.364307

Series Sum 0.364747 0.364262 0.364253 0.364311 0.364314 0.364307 |% Error| 0.120616 0.012416 0.014993 0.001174 0.001853 0.000177

Series Sum -0.008409 -0.008435 -0.008468 -0.008464 -0.008460 -0.008460 U23 -0.008461 (10-12m) |% Error| 0.610457 0.308514 0.089202 0.040721 0.008300 0.009228 U33 (10-12m)

0.375787

Series Sum 0.376262 0.375773 0.375729 0.375788 0.375794 0.375791 |% Error| 0.126303 0.003882 0.015380 0.000078 0.001785 0.000864

Table 2: Comparison of the computed first-order derivatives at a sample field point (1.0, S/3, S/4) Uij,l U11,1 U11,2 U11,3 U12,1 U12,2 U12,3 U13,1 U13,2 U13,3 U22,1 U22,2 U22,3 U23,1 U23,2 U23,3 U33,1 U33,2 U33,3

Approach 1 -0.046012 -0.216088 -0.265592 0.066941 0.021507 -0.082238 0.075717 -0.019233 -0.011961 -0.014921 -0.116213 -0.407104 0.052837 -0.034652 0.015556 -0.141946 -0.429518 -0.088497

Approach 2 -0.046012 -0.216088 -0.265592 0.066941 0.021507 -0.082238 0.075717 -0.019233 -0.011961 -0.014921 -0.116213 -0.407104 0.052837 -0.034652 0.015556 -0.141946 -0.429517 -0.088497

Fourier -0.046003 -0.216084 -0.265599 0.066954 0.021511 -0.082248 0.075717 -0.019238 -0.011956 -0.014878 -0.116196 -0.407140 0.052830 -0.034654 0.015561 -0.141984 -0.429531 -0.088471

|% Diff.| 0.019461 0.001786 0.002705 0.019963 0.017136 0.012548 0.000281 0.026329 0.035329 0.292022 0.015287 0.008907 0.013168 0.007534 0.029799 0.027016 0.003259 0.030176

198


Table 3: Comparison of the computed second-order derivatives at a sample field point (1.0, S/3, S/4) Uij,lk U11,11 U11,12 U11,13 U11,22 U11,23 U11,33 U12,11 U12,12 U12,13 U12,22 U12,23 U12,33 U13,11 U13,12 U13,13 U13,22 U13,23 U13,33 U22,11 U22,12 U22,13 U22,22 U22,23 U22,33 U23,11 U23,12 U23,13 U23,22 U23,23 U23,33 U33,11 U33,12 U33,13 U33,22 U33,23 U33,33

Approach 1 -0.151482 0.047583 0.164674 -0.003451 0.590386 0.157583 -0.172021 0.045261 -0.142523 -0.082709 -0.011834 0.314114 -0.092932 -0.173575 -0.017374 0.045166 0.102070 -0.045879 -0.390608 0.155377 0.102947 -0.080658 0.320864 0.822114 0.019137 -0.065790 -0.102038 0.095531 0.048173 -0.034699 -0.339161 0.538232 0.104942 0.728906 0.314492 -0.074520

Approach 2 -0.151482 0.047583 0.164674 -0.003451 0.590386 0.157583 -0.172021 0.045261 -0.142523 -0.082709 -0.011834 0.314114 -0.092932 -0.173575 -0.017374 0.045166 0.102070 -0.045879 -0.390608 0.155377 0.102947 -0.080658 0.320864 0.822114 0.019137 -0.065790 -0.102038 0.095531 0.048173 -0.034699 -0.339161 0.538232 0.104942 0.728906 0.314492 -0.074520

Fourier -0.151511 0.047548 0.164693 -0.003365 0.590318 0.157652 -0.172278 0.045218 -0.142395 -0.082685 -0.011844 0.314087 -0.092992 -0.173608 -0.017316 0.045196 0.102076 -0.045924 -0.391084 0.155318 0.103112 -0.080367 0.320592 0.822369 0.019072 -0.065840 -0.101943 0.095401 0.048318 -0.034885 -0.338994 0.538167 0.105023 0.728187 0.315187 -0.075238

|% Diff.| 0.019187 0.073714 0.011892 2.481185 0.011439 0.043795 0.149554 0.095061 0.089877 0.028680 0.079892 0.008492 0.063760 0.019263 0.333724 0.065538 0.005295 0.099442 0.121790 0.038010 0.161019 0.360180 0.084869 0.031079 0.341625 0.075540 0.093503 0.136092 0.300636 0.536725 0.049436 0.012062 0.077047 0.098731 0.221085 0.963638

Acknowledgement

The authors gratefully acknowledge the financial support from the National Science and Engineering Research Council of Canada and the National Science Council of Taiwan (NSC 96-2221-E035-011-MY3). References


[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

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229

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230


BEM for the second problems of the Stokes system D. Medková ˇ a 25, 115 67 Praha 1, Institute of Mathematics, Zitn´ CZECH REPUBLIC, [email protected]

Keywords:Stokes system; first problem; second problem; single layer potential; double layer potential; integral equation method; successive approximation Abstract: A weak solution of the second problem for the Stokes system in Sobolev space is studied in a bounded Lipschitz domain with connected boundary. A solution is looked for in the form of a hydrodynamical single layer potential. It leads to an integral equation on the boundary of the domain. Necessary and sufficient conditions for the solvability of the problem are given. Moreover, it is shown that we can obtain a solution of this integral equation using the successive approximation method. Then the consequences for the direct boundary integral equation method are treated. A solution of the second problem for the Stokes system is the sum of the hydrodynamical single layer potential corresponding to the boundary condition and the hydrodynamical double layer potential corresponding to the trace of the velocity part of the solution. Using boundary behavior of potentials we get an integral equation on the boundary of the domain where the trace of the velocity part of the solution is unknown. It is shown that we can obtain a solution of this integral equation using the successive approximation method.

1

Introduction.

There are two relevant second boundary value problems for the Stokes system ∆u = ∇p

in G,

∇·u=0

in G.

(1)

One with the boundary condition ∂u − pn = g ∂n

on ∂G

(2)

and the second one with the boundary condition T (u, p)nG = g

on ∂G.

(3)

G

Here n = n is the outward unit normal vector of G, u = (u1 , . . . , um ) is a velocity field, p is a pressure and ˆ − pI T (u, p) = 2∇u

(4)

is the corresponding stress tensor. Here I denotes the identity matrix and ˆ = 1 [∇u + (∇u)T ] ∇u 2

(5)


231

is the strain tensor, with (∇u)T as the matrix transposed to ∇u = (∂j uk ), (k, j = 1, . . . , m). Remark that ∇ · u = ∂1 u1 + . . . + ∂m um is the divergence of u. We shall study a weak solution of these problems in H 1 (G) on a bounded domain G with connected Lipschitz boundary in Rm both by indirect and direct BEM. For the proofs see [2] and [3].

2

Indirect BEM

We shall look for a solution of the second boundary value problem for the Stokes system in the form of a hydrodynamical single layer potential. Denote by ωm the surface of the unit sphere in Rm . For x ∈ Rm and j, k = 1, . . . , m define ⎧ 2−m xj xk 1 ⎨ δjk |x| , m > 2, 2ωm m−2 + |x|m Ejk (x) = (6) xj xk 1 1 ⎩ δ , m = 2, ln + 2 jk 4π |x| |x| Qk (x) =

xk . ωm |x|m

(7)

For Ψ = [Ψ1 , . . . , Ψm ] ∈ H −1/2 (∂G) define the hydrodynamical single layer potential with density Ψ by (EG Ψ)(x) = E(x − y)Ψ(y) dy (8) ∂G

whenever it makes sense and the corresponding pressure (QG Ψ)(x) = Q(x − y)Ψ(y) dy, x ∈ Rm \ ∂G.

(9)

∂G

Fix y ∈ ∂G such that there is the unit outward normal nG (y) of G at y. For x ∈ Rm \ {y}, j, k ∈ {1, . . . , m} set m (yj − xj )(yk − xk )(y − x) · nG (y) , ωm |x − y|m+2 1 (y − x) · nG (y) ˜ G (x, y) = K δjk jk 2Hm−1 (∂B(0; 1)) |y − x|m

G Kjk (x, y) =

(yj − xj )(yk − xk )(y − x) · nG (y) (yk − xk )nG (yj − xj )nG j (y) k (y) . − + m+2 m m |y − x| |y − x| |y − x| ˜ G (x, y) = −nG (y) · ∇y Ejk (y − x) + Qj (y − x)nG (y).) For (Remark that K +m

jk

k

Ψ ∈ L2 (∂G), x ∈ ∂G define

Ψ(x) = lim KG

δ0 ∂G\B(x;δ)

K G (y, x)Ψ(y) dy,

232


˜ Ψ(x) = lim K G

δ0 ∂G\B(x;δ)

˜ G (y, x)Ψ(y) dy K

˜ can be extended as bounded operwhenever this integral exists. Then KG ,K G ators on H −1/2 (∂G). Let g, Ψ ∈ H −1/2 (∂G). Put u = EG Ψ, p = QG Ψ. Then u, p is a weak solution of the problem (1), (3) if and only if 12 Ψ − KG Ψ = g; ˜ Ψ = g. u, p is a weak solution of the problem (1), (2) if and only if 12 Ψ − K G

Theorem 1. Denote Rm = {Ax + b; AT = −A} the space of rigid body motions. Fix g ∈ H −1/2 (∂G, Rm ). Then there is a weak solution of the problem (1), (3) if and only if g, w = 0 ∀w ∈ Rm . (10) Suppose now that g satisfies (10) and Ψ0 ∈ H −1/2 (∂G, Rm ). For a nonnegative integer k put Ψk+1 = [(1/2)I + KG ]Ψk + g. (11) Then there is Ψ ∈ H −1/2 (∂G, Rm ) such that Ψk → Ψ in H −1/2 (∂G, Rm ) as k → ∞. Moreover, there are constants 0 < q < 1, C > 0 depending only on G such that

Ψk − ΨH −1/2 (∂G) ≤ Cq k gH −1/2 (∂G) + Ψ0 H −1/2 (∂G) . (12) If we put u = EG Ψ, p = QG Ψ then u, p is a weak solution of the problem (1), (3). Theorem 2. Fix g ∈ H −1/2 (∂G, Rm ). Then there is a weak solution of the problem (1), (2) if and only if g = 0. (13) g, c = 0 ∀c ∈ Rm , i.e ∂G −1/2

(∂G, Rm ). For a nonnegative Suppose now that g satisfies (13) and Ψ0 ∈ H integer k put ˜ ]Ψk + g. Ψk+1 = [(1/2)I + K (14) G Then there is Ψ ∈ H −1/2 (∂G, Rm ) such that Ψk → Ψ in H −1/2 (∂G, Rm ) as k → ∞. Moreover, there are constants 0 < q < 1, C > 0 depending only on G such that

Ψk − ΨH −1/2 (∂G) ≤ Cq k gH −1/2 (∂G) + Ψ0 H −1/2 (∂G) . (15) If we put u = EG Ψ, p = QG Ψ then u, p is a weak solution of the problem (1), (2). ˜− In the numerical practice we approximate g, so we solve the equations 12 Ψ ˜ ˜ −K ˜ =g ˜ Ψ ˜ , 12 Ψ ˜ , where g ˜ is close to g. Since the operators 12 I − KG Ψ=g , KG G


233

1 ˜ 2 I − KG are not invertible these equations might not be solvable. To overcome this difficulty we define modified operators.

Theorem 3. Put 1 ˜ Ψ = K ˜G M ψ− c

Ψ dy,

∂G

c=

1 dy.

(16)

∂G

˜ ≤ q < 1. Then there is an equivalent norm on H −1/2 (∂G) such that 12 I + M Let now g ∈ H −1/2 (∂G, C m ), g = 0. Fix Ψ0 ∈ H −1/2 (∂G, C m ). For a nonnegative integer k put

1 ˜ Ψk + g. I +M Ψk+1 = 2 ˜ Ψ = 1 Ψ − K ˜ Ψ = g and Ψ − Ψj ≤ Then Ψk → Ψ in H −1/2 (∂G), 12 Ψ − M 2 q j [g + Ψ0 ] for arbitrary j. If we put u = EG Ψ, p = QG Ψ then u, p is a weak solution of the problem (1), (2). Theorem 4. Let f1 , . . . fm(m+1)/2 form a basis of the space of rigid body motions Rm and 1 for j = k, fj (y) · fk (y) dy = 0 for j =

k ∂G

Put

m(m+1)/2 Ψ − M Ψ = KG

fj Ψ, fj .

j=1

Then there is an equivalent norm on H −1/2 (∂G) such that 12 I + M ≤ q < 1. Let now g ∈ H −1/2 (∂G, C m ) be such that g, w = 0 for all w ∈ Rm . Fix Ψ0 ∈ H −1/2 (∂G). For a nonnegative integer k put

1 Ψk+1 = I + M Ψk + g. 2 Then Ψk → Ψ in H −1/2 (∂G, C m ), 12 Ψ − M Ψ = 12 Ψ − K Ψ = g and Ψ − Ψj ≤ q j [g + Ψ0 ] for arbitrary j. If we put u = EG Ψ, p = QG Ψ then u, p is a weak solution of the problem (1), (3).

3

The direct BEM

First we define two types of hydrodynamical double layer potentials. For Ψ = [Ψ1 , . . . , Ψm ] ∈ L2 (∂G) define (DG Ψ)(x) = K G (x, y)Ψ(y) dy (17) ∂G

234


the hydrodynamical double layer potential corresponding to the problem (1), (3) and the corresponding pressure ΠG (x, y)Ψ(y) dy, (18) (ΠG Ψ)(x) = ∂G

ΠG k (x, y) =

2 ωm

−m

(yk − xk )(y − x) · nG (y) nG (y) + k m |y − x|m+2 |y − x|

and by

.

˜ G (x, y)Ψ(y) dHm−1 (y) K

(WG Ψ)(x) =

(19)

∂G

the hydrodynamical double layer potential corresponding to the problem (1), (2) and the corresponding pressure (RG Ψ)(x) = RG (x, y)Ψ(y) dHm−1 (y), ∂G

RkG (x, y) =

nG 1 m(yk − xk )(y − x) · nG (y) k (y) . − Hm−1 (∂B(0; 1)) |x − y|m |x − y|m+2

If x ∈ ∂G define

KG Ψ(x) = lim

0 ∂G\B(x;)

˜ G Ψ(x) = lim K

0 ∂G\B(x;)

K G (x, y)Ψ(y) dy.

˜ G (x, y)Ψ(y) dy. K

˜ G are bounded linear operators whenever these limits exist. Then KG and K on H 1/2 (∂G). Moreover, If Ψ ∈ H 1/2 (∂G) then DG Ψ, WG Ψ ∈ H 1 (G) and ˜ G ]Ψ is the trace of WG Ψ. [(1/2)I + KG ]Ψ is the trace of DG Ψ; [(1/2)I + K Let u ∈ H 1 (G), p ∈ L2 (G) is a solution of the Stokes system (1). Denote by ˜ the trace of u. If u, p is a weak solution of the problem (1), (2) then u ˜ (x), u(x) = EG g(x) + WG u

˜ (x) p(x) = QG g(x) + RG u

x ∈ G.

(20)

Using boundary behavior of hydrodynamical potentials we get 1 ˜ Gu ˜ −K ˜ = EG g u 2

on ∂G.

(21)

If u, p is a weak solution of the problem (1), (3) then ˜ (x), u(x) = EG g(x) + DG u

˜ (x) p(x) = QG g(x) + ΠG u

x ∈ G.

(22)


235

Using boundary behavior of hydrodynamical potentials we get 1 ˜ = EG g ˜ − KG u u 2 Theorem 5. Let g ∈ H −1/2 (∂G, Rm ), nonnegative integer k put

on ∂G.

(23)

˜ 0 ∈ H 1/2 (∂G). For a g = 0. Fix u

˜ G ]˜ ˜ k+1 = [(1/2)I + K u uk + EG g. 1/2

(24) 1/2

˜ ∈ H (∂G) such that u ˜k → u ˜ in H (∂G) as k → ∞. Then there is u Moreover, there are constants 0 < q < 1, C > 0 depending only on G such that

˜ H 1/2 (∂G) ≤ Cq k gH −1/2 (∂G) + ˜ ˜ uk − u (25) u0 H 1/2 (∂G) . ˜ is a solution of the equation (21). If u, p are given by (20) in The function u ˜ is the trace of u G, then u, p is a weak solution of the problem (1), (2) and u on ∂G. Theorem 6. Put ˜Ψ = K ˜ Gψ − 1 M c

Ψ dy, ∂G

c=

1 dy. ∂G

Then there is an equivalent norm on H 1/2 (∂G) such that 12 I + M ≤ q < 1. ˜ 0 ∈ H 1/2 (∂G). For a nonnegative Let g ∈ H −1/2 (∂G, Rm ), g = 0. Fix u integer k put ˜ ]˜ ˜ k+1 = [(1/2)I + M uk + EG g. u ˜k → u ˜ in H 1/2 (∂G) as k → ∞. ˜ ∈ H 1/2 (∂G) such that u Then there is u ˜u ˜ Gu ˜ j ≤ q j [EG g + ˜ ˜ −M ˜ = 12 u ˜ −K ˜ = EG g and ˜ u−u u0 ] for Moreover, 12 u arbitrary j. If u, p are given by (20) in G, then u, p is a weak solution of the ˜ is the trace of u on ∂G. problem (1), (2) and u Theorem 7. Let g ∈ H −1/2 (∂G) be such that g, w = 0 for all w ∈ Rm . ˜ 0 ∈ H 1/2 (∂G). For a nonnegative integer k put Fix u ˜ k+1 = [(1/2)I + KG ]˜ u uk + EG g.

(26)

˜ ∈ H 1/2 (∂G) such that u ˜k → u ˜ in H 1/2 (∂G) as k → ∞. Then there is u Moreover, there are constants 0 < q < 1, C > 0 depending only on G such that

˜ H 1/2 (∂G) ≤ Cq k gH −1/2 (∂G) + ˜ ˜ uk − u (27) u0 H 1/2 (∂G) . ˜ is a solution of the equation (23). If u, p are given by (22) in The function u ˜ is the trace of u G, then u, p is a weak solution of the problem (1), (3) and u on ∂G.

236


Theorem 8. Let f1 , . . . fm(m+1)/2 form a basis of the space of rigid body motions Rm . Suppose that 1 for j = k, fj (y) · fk (y) dy = 0 for j =

k. ∂G

Put

m(m+1)/2

M Ψ = KG Ψ −

fj Ψ, fj .

j=1

Then there is an equivalent norm on H 1/2 (∂G, C m ) such that 12 I +M ≤ q < 1. ˜ 0 ∈ H 1/2 (∂G). Let g ∈ H −1/2 (∂G) be such that g, w = 0 for all w ∈ Rm . Fix u For a nonnegative integer k put ˜ k+1 = [(1/2)I + M ]˜ uk + EG g. u ˜ in H (∂G), 12 u ˜j ≤ ˜ − Mu ˜ = 12 u ˜ − KG u ˜ = EG g and ˜ ˜k → u Then u u−u q j [EG g + ˜ u0 ] for arbitrary j. If u, p are given by (22) in G, then u, p is a ˜ is the trace of u on ∂G. weak solution of the problem (1), (3) and u 1/2

Acknowledgements The work was supported by the Academy of Sciences of the Czech Republic, Institutional Research Plan No. AVOZ10190503 and grant No. P201/11/1304.

References [1] G.C.Hsiao and W.L.WendlandBoundary Integral Equations, Springer(2008) [2] D.Medkov´ aConvergence of the Neumann series in BEM for the Neumann problem of the Stokes system. Acta Applicandae Mathematicae,116, 281– 304(2011) [3] D.Medkov´ aBEM for the first and second problems of the Stokes system, ´ AVCR ˇ no. IM-2012-1, Praha 2012 Preprint MU [4] C.PozrikidisBoundary integral and singularity methods for linearized viscous flow. Cambridge texts in Applied Mathematics, Cambridge University Press (1992) [5] O.SteinbachNumerical Approximation Methods for Elliptic Boundary Value Problems. Finite and Boundary Elements, Springer(2008)


A Multiregion Technique for the Boundary Element Method with Drilling Rotation D. I. G. Costa1 , E. L. Albuquerque1 , P. M. Baiz2 1

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

2

Imperial College London - South Kensington Campus, London SW7 2AZ - U.K

Keywords: Boundary Element Method, Multiregion BEM, Membrane, Drilling Rotation.

Abstract. In this paper, a boundary element formulation is developed for the analysis of structures formed by multiregions, considering drilling rotation. The plane elasticity boundary integrals are enriched with the inclusion of an extra degree of freedom, that is the drilling rotation. In order to have the same number of equations and unknowns, an extra equation is created based on consideration that the drilling rotation is the true rotation of plane elasticity theory. In the multiregion assembly, each region is defined as a sub-domain. In order to carry out the assembly, displacement-rotation compatibilities and traction equilibrium conditions are taken into account. Numerical examples are presented and their results are compared to results available in literature. Introduction. Shells are elements that differ from plates for the possibility of being curved and for being able to carry out flexural and membrane loads. This type of structure can tolerate high levels of loads if membrane stresses are predominant. In finite elements technology, shell elements can be obtained in two main ways, through the combination of the plates and membrane formulations and through a degenerated solid approach. Plates are considered, in general, three dimensional structures and their thickness is small when compared to its other spatial dimensions. Loads are applied out of plane, that is because in plane loads are treated with 2-dimensional elasticity. The first plate theory was proposed by [1], widely known as the thin plate theory. Another theories were presented by Reissner and Mindlin [2, 3], which takes in to account the shear deformation and the transverse normal stresses. Shear deformable plate theories are called also thick plate theories [4]. Theory related to membrane stresses can be directly obtained from the 2-D elasticity theory. Shear deformable plate formulations provide three degrees of freedom, a displacement and two rotations, while membranes contribute with two more displacements. To accomplish the task of producing an initial formulation for shells, it is necessary to include one more degree of freedom in the membrane formulation, a in-plane rotation known as the drilling degree of freedom or drilling rotation. Many works in finite element methods have debated this inclusion, the first of them was [5], what lead many other researchers to give contributions like [6, 7, 8, 9]. Recently, [10] have found that the formulation of Allman can take the form of a simple partition of unity, which allows the extension of the concept to many other types of elements, including boundary element method and meshless techniques. This concept has been applied successfully by [15] within the boundary element method to analyze composite materials. This paper presents a multiregion technique to assemble plane structures in which drilling rotation is included in the BEM formulation. This is a first step to obtain a formulation to the analysis of thin walled 3D structures, in which the association of flat macro elements is necessary.

237

238


Shape Functions Enriched with Drilling Rotation. To make the inclusion of the drilling degree of freedom, enriched shape functions are used. The nodal drilling parameter is included and, after some analysis, it is imposed as the true rotation of elasticity theory. Following the derivation presented in Allman [5] and the simplification made by [10], for linear elements one can find:

ue1 ue2

=

2

1 0 λ ξˆβ (ξ )(−n1 ) 0 1 λ ξˆβ (ξ )(−n2 )

∑ Nβ (ξ )

β =1

⎫ ⎧ ⎪ uβ ⎪ ⎨ 1 ⎬ β u2 ⎪ ⎭ ⎩ β ⎪ ω

(1)

where Nβ (ξ ) are shape functions, ξˆβ = ξ − ξβ /2 for the two nodes, n1 and n2 are components of the normal vector and λ is an arbitrary non-zero value that, according to [10], do not have an influence on displacement numerical results, but affects the drilling degree of freedom. This constant can also be used to help making this degree of freedom equal to the real rotation of elasticity theory [9, 15]. Inclusion of eq (1) will change the typical system of equations normally found in the BEM. After some observation, one can conclude that this new system has more unknowns (because of the rotations) than the number of equations available. To overcome this problem more equations must be found. Relating drilling rotations to the true rotations. Because of the inclusion of drilling degree of freedom in the formulation, it is necessary to find more equations in order to obtain a solvable equations system. These equations can be obtained by creating a functional that can be used to enforce that the computed rotations will be the true rotations of elasticity:

γ (ψ0 − ω¯)2 dΓ (2) 2 Γ where ψ0 is the true rotation field and γ is a penalty parameter that, in this case, does not affect the solution. To include this equation in the system, we have to obtain its discretized form. Taking the first variation of eq (2) with respect to the drilling rotation field and making it equal to zero in order to minimize the functional, we obtain: Π (uα0 , ω¯) =

δ Π (u, ω¯) = δ ω¯

Γ

δ ω¯h ψ0h − ω¯h dΓ = 0

(3)

After substituting the discretized forms of ω¯h and ψ0h , transforming the integral to local coordinates and performing some simplification, we obtain: [δ dω ]T2×1

+1 −1

[Kω ×0ω ]2×6 J m (ξ ) d ξ δ dm 0ω 6×1 = 0

(4)

where [Kw×0w ] =

N1 (ξ ) J(nξ1 ) dNd1ξ(ξ )

N2 (ξ ) J(nξ1 ) dNd1ξ(ξ ) N1 (ξ ) J(nξ1 ) dNd2ξ(ξ ) N2 (ξ ) J(nξ1 ) dNd2ξ(ξ )

N1 (ξ ) J(nξ2 ) dNd1ξ(ξ ) N2 (ξ ) J(nξ2 ) dNd1ξ(ξ ) N1 (ξ ) J(nξ2 ) dNd2ξ(ξ ) N2 (ξ ) J(nξ2 ) dNd2ξ(ξ )

N1 (ξ )N1 (ξ ) N2 (ξ )N1 (ξ ) N1 (ξ )N2 (ξ ) N2 (ξ )N2 (ξ )

(5)

and [δ dwm ]T6×1 =

u110

u120

ω 1 u210 u220 ω 2

These equation are only used in the collocation element and each line is added to its both nodes, depending on the position of source point, if it is located in the first or second node.


239

Boundary Element Equations. For plane elasticity and by Betti’s reciprocal theorem, displacements in domain Ω in the absence of body forces, can be written as: uα (X ) =

Γ

s Uαβ (X , x)tβ (x)dΓ(x) −

Γ

s Tαβ (X , x)uβ (x)dΓ(x)

(6)

s and T s are displacement and traction fundamental solutions for plane stress problem, rewhere Uαβ αβ spectively [11]. Taking X’ to the boundary in eq (6), we obtain:

csαβ (x )uα (x ) =

Γ

s s Uαβ (x , x)tβ (x)dΓ(x) − − Tαβ (x , x)uβ (x)dΓ(x)

(7)

Γ

Using the enriched shape functions and the weak form of the rotational residual given in eqs (1) and (4), respectively, the system of equations become: ⎫ ⎡ ⎫ ⎤⎧ ⎤⎧ c1 c1 c c ∗mc1 H ∗mc1 H ∗mc1 ⎪ C11 C12 0 ⎪ H11 ⎨ u10 ⎪ ⎬ ⎬ ⎨ u10 ⎪ 12 13 ⎢ c ⎥ ⎢ ∗mc1 c ∗mc1 H ∗mc1 ⎥ 0 ⎦ uc1 H22 uc1 ⎣ C21 C22 ⎦ 20 ⎪ + ⎣ H21 23 20 ⎪ + ⎪ ⎪ ⎩ ω c1 ⎭ Kα 1 Kα 2 Kα 3 0 0 0 ⎩ ω c1 ⎭ ⎫ ⎤ ⎧ mβ ⎤⎧ ⎡ ∗mβ ∗mβ ∗mβ c2 ∗mc2 H ∗mc2 ⎪ ⎪ H11 H12 H12 H13 ⎬ ⎨ u10 ⎪ ⎨ u1 0 2 Ne 13 ⎥ ⎢ ∗mβ ∗mβ ∗mβ ⎥ mβ ∗mc2 H ∗mc2 c2 + H22 u ⎦ ⎦ ⎣ H21 H22 H23 u2 ∑ ∑ 23 2 0 ⎪ ⎪ ⎭ m=1, m=c β =1 ⎩ ω c2 ⎪ ⎩ m0β Kα 5 Kα 6 0 0 0 ω ⎡ mβ ⎤ ⎧ mβ ⎫ mβ G11 G12 0 ⎪ ⎬ ⎨ t1 ⎪ Ne 2 ⎢ ⎥ β mβ mβ = ∑ ∑ ⎣ Gm G22 0 ⎦ t2 21 ⎪ ⎪ ⎭ ⎩ m=1 β =1 0 0 0 1 ⎡

⎡

H ∗mc2 ⎢ 11 ∗mc2 ⎣ H21 Kα 4

⎫ ⎪ ⎬ ⎪ ⎭

(8)

where c is the collocation node, the second superscript in displacements and tractions refers to the element node. The subscript α assume values 1 or 2 according to the collocation point in the element. Multiregion BEM. For a multiregion association routine using the boundary element method, it is necessary to find a way to impose compatibility and equilibrium conditions at interfaces between different regions. This task can be done in many ways. In this work, the “Stiffness” Matrix method described by [12] is used. This approach has been chosen to allow the use of an analogous procedure already used in finite element method when dealing with shells as an assembly of flat elements [13]. The problem must be divided in two steps for each subregion N. The first step is to solve the problem with zero displacements at the interface: AN xN0 = bN0

(9)

where the vector of unknowns xN0 is formed for tractions at the interface and depends on the boundary conditions at the free nodes of the subregion. Matrix AN is the assembled left hand side, bN0 contains the right hand side due to known boundary conditions values at the free nodes and due to zero displacements at the interface nodes. The second step is basically solve the same problem but applying unit displacement at each node of the interface at a time. Therefore we have to solve nint (number of nodes × number of dof per node) systems of equations. AN xNn = bNn

(10)

contains tractions where n = 1, 2..., nint is the interface degree of freedom under consideration. at interface nodes and displacements or tractions depending on boundary conditions at the free nodes. Vector xN can be separated in an interface part (i) and a free nodes part ( f ). Vector xNn

240


{x}N =

tNi xNf

(11)

After solving Equations (9) and (10), the solution can be expressed in terms of the displacements of the interface ui as:

tNi xNf

=

tNi0 xNf0

+

KNi BNf

uNi

(12)

where matrices KN (“Stiffness” Matrix) and BN are defined by: KN = tNi1 · · · tNiint BN = xNf1 · · · xNfint

(13) (14)

After all region “stiffness” matrices are computed, they are assembled using compatibility and equilibrium conditions. In the case of only two regions connected, displacements at the interface can be determined by:

I K + KII ui = − tIi0 + tII i0

(15)

Solving this system and substituting ui in (12), unknown tractions and displacements can be found for each subregion. Numerical results. At this point, plane elasticity formulation with enriched shape functions has been implemented in a multiregion code that uses “Stiffness” Matrix approach. The implementation in an assembled plate structures code is yet to be done. Because of this, only plane elasticity results, including drilling rotations and subregions are presented in this paper. In order to show the accuracy of the method and that inclusion of drilling rotation do not affect the solution, a thick cantilever beam analysis is presented. Its dimensions are shown in Fig. 1. Properties are E = 30000 Pa and ν = 0.25. There is a uniformly distributed shear load of resultant of 40 N on right edge while the left edge is clamped. The beam was discretized with 24 discontinuous linear elements for each subregion.

y(m)

10 5 0 0

10

20

30

40

50

x(m)

Figure 1: Thick cantilever beam representation showing its dimensions 48 × 12, the 2 subregions, discretization in linear discontinous elements, and internal points. The thickness of the beam is 1. Deformed shape (with enlarged displacements) of the beam is showed in Fig. 2. The color map u21 + u22 . The analytical value of tip vertical

results are indicating the total displacements given by ut =


241

Total displacement 20 0.3 15 y(m)

0.2 10 0.1

5 0

0 0

10

20

30

40

50

x(m)

Figure 2: Deformed geometry obtained with BEM analysis.

0.9

0.85

BEM

utip /utip

analytical

0.95

0.8

0.75 10

20

30

40

50 60 70 Number of elements

80

90

100

110

Figure 3: Convergence study of the cantilever beam. displacement is utip y = 0.3558 m [8], while in this work it was obtained u2 = 0.3293 m, what gives a relative error of 7.31%. Drilling rotation values in the same position were calculated as ω = 0.0100144, what gives an relative error of 4.9% when compared to the result of [8]. The accuracy of the formulation, in theory can be improved using more Gauss points and a scheme of integration with subdivisions, or using quadratic elements. A convergence study was also made and is shown in Fig. 3. Conclusions. This paper has shown the results of an enriched BEM formulation for plane elasticity that includes drilling rotations. To make possible to assemble flat macroelements without relating the rotations obtained to any rotational stiffness, “Stiffness” matrices were used to solve the subregions problems in an attempt to use the procedure described by FEM researchers. The work showed here is only the first part of the plate assembly implementation. Next steps will be the coupling of the formulation just presented to plates formulation, to extend the code to work with any number of subregions and interfaces, and to include anisotropic fundamental solutions. Acknowledgments. The authors would like to thank CAPES for the financial support of this work.

References [1] G. Kirchhoff Uber das gleichgewicht und die bewegung einer elastischen scheibe. J. Reine Angew. Math., 40:51-88, 1850.

242


[2] E. Reissner On bending of elastic plates. Quart. Applied Mathematics, 5:55-68, 1947. [3] R. D. Mindlin. Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. J. Applied Mechanics 18:31-38, 1951. [4] Y. F. Rashed. Boundary Element Formulation of Thick Plates. WIT press, 2000. [5] D. J. Allman. A compatible triangular element including vertex rotations for plane elasticity analysis. Computer and Structures, 19:1-8, 1984. [6] P. G. Bergan , C. A. Felippa. A triangular membrane element with rotational degrees of freedom. Computer Methods in applied mechanics and engineering, 50:25-69, 1985. [7] T. R. J.Bergan, F. Brezzi. On drilling degrees of freedom. Computer Methods in applied mechanics and engineering, 72:105-121, 1989. [8] E. Turska, K. Wisniewski, Enhanced Allman quadrilateral for finite drilling rotations. Comput. Methods Applied Mech. Engrg, 195:6086-6109, 2006. [9] M. Huang, Z. Zhao, C. Shen An effective planar triangular element with drilling rotation. Finite elements in analysis and Design, 46:1031-1036, 2010. [10] R. Tian, G. Yagawa, Allman’s triangle, rotational DOF and partition of unity. Int. J. Numer. Meth. Engng, 69:837-858, 2007. [11] P. H. Wen, M. H. Aliabadi, A. Young. Plane stress and plate bending coupling in BEM analysis of shallow shells. Int. J. Numer. Meth. Engng, 48:1107-1125, 2000. [12] G. Beer, I. Smith, C. Duenser. The Boundary Element Method with Programming. Springer Wien, Germany, 2008. [13] O. C. Zienkiewicz, R. L. Taylor. The Finite Element Method Volume 2: Solid Mechanics. Springer Wien, Germany, 2008. [14] A. Ibrahimbegovic, E. L. Wilson. A unified formulation for triangular and quadrilateral flat shell finite elements with six nodal degrees of freedom. Communications in Applied Numerical Methods, 7:1-9, 1991. [15] P. M. Baiz, E. L. Albuquerque, P. Sollero. Drilling Rotations and Partition of Unity in Composite Plates by the Boundary Element Method. Advances in Boundary Element and Meshless Techniques XII, 12:324-329, 2011.


A large deflection analysis of laminate composite thin plates by the boundary element method L. S. Campos∗, E. L. Albuquerque† Faculty of Technology University of Brasilia, Brazil

Abstract Boundary-integral equations for large deflections of composite laminate thin plates are presented. Quadratic boundary elements are used to discretise the boundary. Domain integrals that arise from nonlinear terms are transformed into boundary integrals using the radial integration method. As a result, the obtained formulation does not demand domain discretisation. For the solution of the non-linear system, the total incremental method is used. A numerical example is presented and comparisons with other numerical results are made to demonstrate the accuracy of the proposed method. The formulation illustrates the adaptability of the boundary element methods to non-linear problems of anisotropic materials. Keywords: Large deflections, boundary element method, radial integration method, thin plates, anisotropic materials, composite materials.

1 Introduction The boundary element method has already been established as an extremely efficient and effective technique in the analysis of linear and non-linear structural problems, as can be seen in the book of Aliabadi [3]. For linear problems, it is always possible to find a fundamental solution considering all terms of governing equations, producing formulations with pure boundary discretization. However, when nonlinear differential equations are considered, fundamental solutions are found only to the linear portion of the equations. As a result, non-linear formulations have not only boundary integrals but also domain integrals. Thus, the advantage in reducing the dimensionality, as is the case in linear boundary element method, is lost in the non-linear case. Some procedures has been developed to avoid domain discretization, as can be seen in the works of Purbolaksono and Aliabadi [6] or Wen et al. [10, 11]. Apart from numerous articles where the boundary element method is applied to non-linear isotropic problems, to the best of authors knowledge, it has not been found in literature a paper where the boundary element method is applied to a non-linear analysis of anisotropic materials. The linear theory of Kirchhoff for thin plates is valid only when the transverse deflections are approximately less than the thickness of the plate. The extension to large deflections, where the non-linear terms are retained in the kinematic relationships, leads to a pair of coupled non-linear fourth order equations for the transverse displacement and the stress function for the in-plane stress resultants. Purbolaksono and Aliabadi [7] discuss and compare some approaches to compute large deflections of isotropic plates using the boundary element method. Non-linear formulations demand iterative solutions that, depending on the chosen approach, is necessary to rebuild influence matrices for each load step. This kind of procedure gives better results. However, they are very time consuming. This work presents a boundary element formulation for the analysis of large deflections in laminate composite thin plates. The formulation presented by O’Donoghue e Atluri [5] is extended for anisotropic materials. ∗ [email protected] † [email protected]

243

244


2

Boundary element formulation

The hypothesis of large deflections used in this work is: • Deflections u3 (displacements in the direction of axis x3 , transversal to the plate) are large (many times bigger than the thickness h of the plate) but strains are still small when compared to the unity (changes in geometry can be disregarded). Membrane stress resultants Ni j (i, j = 1, 2), i.e., stress due to extension, compression or shear on the medium plane of the laminate, integrated along the thickness, for a composite laminated plate, considering large deflections, are given by (Chia [4]): ⎧ ⎫ ⎡ A11 ⎨ N11 ⎬ = ⎣ A12 N22 ⎩ ⎭ N12 A16

A12 A22 A26

⎧ ⎫⎞ ⎧ ⎫ ⎫(l) ⎧ ⎫(nl) ⎤ ⎛⎧ 2 ⎪ A16 ⎨ N11 ⎬ ⎨ u3,1 ⎬ ⎨ N11 ⎬ ⎨ u1,1 ⎬ 1 ⎪ ⎟ ⎜ + , + = A26 ⎦ ⎝ u2,2 u23,2 N N ⎠ ⎩ 22 ⎭ ⎭ 2⎪ ⎩ 22 ⎭ ⎩ ⎩ 2u u ⎪ ⎭ A66 u1,2 N12 N12 3,1 3,2 (1)

where: ⎧ ⎫(l) ⎡ A11 ⎨ N11 ⎬ = ⎣ A12 N22 ⎭ ⎩ N12 A16

⎫ ⎤⎧ A16 ⎨ u1,1 ⎬ ⎦ A26 u ⎩ 2,2 ⎭ A66 u1,2

A12 A22 A26

(2)

and ⎧ ⎫(nl) ⎡ A11 ⎨ N11 ⎬ = ⎣ A12 N22 ⎭ ⎩ N12 A16

A12 A22 A26

⎫ ⎤⎧ ⎪ u23,1 ⎪ A16 ⎨ ⎬ , A26 ⎦ u23,2 ⎪ ⎪ A66 ⎩ 2u3,1 u3,2 ⎭

(3)

indices (l) and (nl) stand for linear and non-linear components, respectively, Ai j (i, j = 1, 2, 6) are terms of the laminate extension stiffness matrix; ui (i = 1, 2) are displacement in directions of axes x1 and x2 , u3 is the deflection, i.e., displacement in the direction x3 , normal to the plate surface. In the absence of body forces, governing equations of an anisotropic plate under large deflections are written as (Shi [8]): Ni j, j = 0,

(4)

D11 u3,1111 + 4D16 u3,1112 + 2(D12 + D66 )u3,1122 + 4D26 u3,1222 + D22 u3,2222 = Ni j u3,i j , (5) where i, j, k = 1, 2; D11 , D22 , D66 , D12 , D16 , and D26 are terms of the laminate bending stiffness matrix. Integral equations for in-plane displacements, obtained by the reciprocity and Green theorems and equations (4) and (5), are given by (O’Donoghue e Atluri [5]): ci j u j (Q) + + (l)

(l)

Ω

Γ

ti∗j (Q, P)u j (P)dΓ(P) = (nl)

Ni j (P)u∗ki, j (P, Q)dΩ,

Γ

(l)

u∗i j (Q, P)t j (P)dΓ(P) (6)

where ti = Ni j n j stand for tractions on the boundary of the plate, parallel to the plane defined by x1 − x2 ; n j is the unity vector normal to the boundary at the source point Q; P is the field point; asterisks


245

stands for fundamental solutions. Fundamental solutions of anisotropic plane elasticity can be found in Sollero and Aliabadi [9]. The constant ci j is introduced in order to take into account that the source point Q can be in the domain, on the boundary, or outside the domain. Resultant stresses for a point Q ∈ Ω are given by: (l)

cik Nk j (Q) + +

Ω

Γ

∗ Ski j (Q, P)uk (P)dΓ(P) =

Γ

(l)

D∗ki j (Q, P)tk (P)dΓ(P)

(nl)

D∗ki j (P, Q)Nk,ll (P)dΩ

(7)

where Dik j and Sik j are linear combinations of plane elasticity fundamental solutions. The integral equation for anisotropic thin plate, obtained by the reciprocity and Green theorems, and by equation (5), is given by:

Ku3 (Q) +

Γ

Vn∗ (Q, P)w(P) − Mn∗ (Q, P)

Nc

i=1

Γ

= ∑ Rci (P)u∗3ci (Q, P) + +

Ω

Ni j (P)u∗3,i j (Q, P) dΩ +

Nc ∂ w(P) dΓ(P) + ∑ R∗ci (Q, P)u3ci (P) ∂n i=1

Vn (P)u∗3 (Q, P) − Mn (P)

Γ

∂ u∗3 (Q, P) dΓ(P) ∂n

[ti (P)u∗3 (Q, P)u3,i (P)

−ti (P)u3 (P)u∗3,i (Q, P) dΓ,

(8)

where ∂∂()n stands for derivative in the direction of the vector n that is normal to the boundary Γ in the outward direction of domain Ω; Mn and Vn are bending moment and equivalent shear force, respectively; Rc is the thin plate corner reaction; u∗3ci is the deflection of the corner i; constant K is introduced in order to take into account that the source point Q can be in the domain, on the boundary, or outside the domain. Fundamental solutions of anisotropic thin plate can be found in Albuquerque et al. [2]. An additional equation is necessary in order to have an equal number of unknown variables and equations. This equation is given by:

K

∂ u3 (Q) + ∂m

Γ

∂ Vn∗ ∂ Mn∗ ∂ w(P) (Q, P)w(P) − (Q, P) dΓ(P) ∂m ∂m ∂n

Nc ∂ u∗3ci ∂ R∗ci (Q, P)u3ci (P) = ∑ Rci (P) (Q, P) ∂m i=1 ∂ m i=1 ∂ u∗ (Q, P) ∂ 2 u∗3 − mn (P) (Q, P) dΓ(P) + Vn (P) 3 ∂m ∂ n∂ m Γ ∗ ∂ u3,i j (Q, P) ∂ u3,i (P) dΩ + + u3 (Q, P)Ni j (Q, P) ti (P)u∗3 (Q, P) ∂m ∂m Ω Γ ∗ ∂ u3,i (Q, P) −ti (P)u3 (P) dΓ, ∂m Nc

+∑

where

∂ () ∂m

(9)

is the derivative in the direction of the vector m that is normal to the boundary Γ on the source (nl)

point Q in the outward direction of the domain Ω, ti = Ni j n j and ti

(nl)

= Ni j n j .

246


As can be seen, domain integrals arise in all integral equations. These domain integrals are given in the form: P(Q) = b(P)v∗ (Q, P)dΩ, (10) Ω

where b and v∗ are generic body force and fundamental solution, respectively. The body force is approximated over the domain Ω as a sum of M products between approximation functions fm and unknown coefficients γm , that is: b(P) =

M

∑ γm fm + ax + by + c

(11)

m=1

with M

M

M

m=1

m=1

m=1

∑ γm xm = ∑ γm ym = ∑ γm = 0

(12)

Domain integrals in the form of equation (10) approximated by the series given by (11) and (12) are transformed into boundary integrals by the radial integration method. Details of this method was given by Albuquerque and Aliabadi in [1] and will not be repeated here.

3

Numerical Results

Consider a clamped square plate, with side length a and thickness h, subjected to a uniform transverse load q. This plate is discretized by 20 discontinuous boundary elements of equal length and 25 uniformly distributed internal points. Two situations are analysed, quasi-isotropic (E1 /E2 ∼ = 1) and orthotropic (E1 /E2 = 2). Other material constants are given by ν12 = 0.316 and G12 = E2 /(1 + ν12 ). Results for both situations are shown in Figure 1 together with results obtained by Purbolaksono and Aliabadi [7] for the isotropic situation. In Figure 1, the following normalizations are applied: Q=

qa4 E2 h4

(13)

and u3c , (14) h where u3c is the transversal displacement at the centre of the plate. As it can be seen, there is a good agreement between the isotropic results of Purbolaksono and Aliabadi [7] and the quasi-isotropic results obtained in this work. Besides, the orthotropic deflections are, as expected, smaller than the isotropic and quasi-isotropic case. Z=

4

Conclusions

This paper presented a boundary element formulation for the analysis of large deflections in anisotropic thin plates. The formulation has a pair of coupled non-linear fourth order equations for the transverse displacement and the stress function for the in-plane stress resultants. Domain integrals that arise from non-linear terms are transformed into boundary integrals using the radial integration method. The numerical results presented good agreement with literature, which demonstrates the adaptability of the boundary element methods to non-linear problems of anisotropic materials.

Acknowledgment The first author would like to thank the CNPq (The National Council for Scientific and Technological Development, Brazil), AFOSR (Air Force Office of Scientific Research, USA), and FAPESP (the State of São Paulo Research Foundation, Brazil) for financial support for this work.


1.4

1.2

1

Z

0.8

0.6 Present work, quasiíisotropic Present work, orthotropic Reference [5], isotropic

0.4

0.2

0

0

20

40

60

80

100

120

140

160

180

Q

Figure 1: Normalized central deflection versus normalized load for the clamped square plate under uniformly distributed load.

References [1] E. L. Albuquerque and M. H. Aliabadi. A boundary element formulation for boundary only analysis of thin shallow shells. CMES - Computer Modeling in Engineering and Sciences, 29:63–73, 2008. [2] E. L. Albuquerque, P. Sollero, W. Venturini, and M. H. Aliabadi. Boundary element analysis of anisotropic kirchhoff plates. International Journal of Solids and Structures, 43:4029–4046, 2006. [3] M. H. Aliabadi. Boundary element method, the application in solids and structures. John Wiley and Sons Ltd, New York, 2002. [4] C. Y. Chia. Nonlinear analysis of plates. MacGraw-Hill, New York, 1980. [5] P. E. O’Donoghue e S. N. Atluri. Field/boundary element method approach to the large deflexion of thin flat plates. Computers and Structures, pages 427–435, 1987. [6] J. Purbolaksono and M. H. Aliabadi. Buckling analysis of shear deformable plates by boundary element method. International Journal for Numerical Methods in Engineering, 62:537–563, 2005. [7] J. Purbolaksono and M. H. Aliabadi. Large deformation of shear-deformable plates by the boundary-element method. Journal of Engineering Mathematics, 51:211–230, 2005. [8] G. Shi. Flexural vibration and buckling analysis of orthotropic plates by the boundary element method. J. of Solids and Structures, 26:1351–1370, 1990. [9] P. Sollero and M. H. Aliabadi. Fracture mechanics analysis of anisotropic plates by the boundary element method. Int. J. of Fracture, 64:269–284, 1993. [10] P. H. Wen, M. H. Aliabadi, and A. Young. Large deflection analysis of reissner plate by boundary element method. Computers and Structures, 83:870–879, 2005. [11] P. H. Wen, M. H. Aliabadi, and A. Young. A post buckling analysis of reissner plates by the boundary element method. Journal of Strain Analysis for Engineering Design, 41:239–252, 2006.

247

248


Computation of Stress in Orthotropic Thick Plates by the Boundary Element Method A. P. Santanaa , E. L. Albuquerqueb , A. Reisb and J. F. Usechec a Federal Institute of Maranhao ˜ Department of Mechanics and Materials ˜ Luis, MA, Brazil Sao

[email protected] b University of Bras´ılia - UnB Faculty of Mechanical Engineering Bras´ılia, DF, Brazil

[email protected] [email protected] c University

Technological of Bolivar - UTB Faculty of Mechanical Engineering ´ Park Industrial and Technological Carlos Velez Pombo Cartagena, Colombia [email protected]

Keywords: Boundary Element Method, Plates and Composite Materials.

Abstract. This work presents a formulation of the boundary element method for the computation of stresses on the boundary of orthotropic thick plates. Fundamental solutions are obtained using Hormander operator and Radon transform. Domain integrals that arise in the formulation due to distributed transversal loads are transformed into boundary integrals by the radial integration method. Only the boundary is discretized in the formulation. Numerical results show good agreement with results available in literature.

1

Introduction

In recent years, the boundary element method (BEM) has become an attractive tool for resolution of complex problems which have the formulation described by partial differential equations. Analysis of plate bending problems using the BEM has attracted the attention of many researchers during the past years, proving to be a particularly adequate field of applications for that technique. The first work about thick Reissner and Mindlin plates using boundary method element has been proposed by [1]. Stress computation by the BEM has been addressed by some works in literature. [15] and [16] have discussed how to compute stresses in plane elastic, [17] presented the integral formulation to compute moments and stresses in laminated composites and [18] presented the formulation of the BEM to thin plates under dynamic loads. This present work proposes a numerical procedure to compute stress at internal points of orthotropic plates using a static boundary element plate formulation that follows the Mindlin hypotheses. It uses the fundamental solution proposed in [8] that takes into account the effects of shear deformation and was derived by means of Hörmander operator and the Radon transform. Some numerical examples concerning orthotropic plate bending problems are analyzed with the BEM.


2

249

Mindlin Plate Theory

The Mindlin’s theory assumes displacement distribution through the thickness. Using the assumptions of the classical theory, he removed the hypothesis of the transverse shear deformation equal zero in the mid plane, but considering that distortion variation is null. Thus:

∂ γ13 = 0. ∂ x3 ∂ γ23 = 0. ∂ x3 So, equations of equilibrium for the plate are given by:

(1) (2)

Mαβ ,β − Qα = 0.

(3)

Qα ,α + q = 0.

(4)

where q is the distributed transverse load per unit area in the x3 direction. The bending moments Mαβ and she the shear forces Qα for orthotropic plates are expressed in terms of the rotations and the lateral displacement as: Mαβ = Dαβ (wα ,β + wβ ,α ) +Cαβ wγ ,γ .

(5)

Qα = Cα (wα + w3,α ).

(6)

where no summation is assumed in eq (5) and eq (6) with respect to the indices α , β and the material parameters are given as [11]: D11 =

D1 2 (1 − ν21 ),

C11 = D1 ν21 , D1 =

D12 = D21 = Dk =

C22 = D2 ν12 ,

E1 h3 12(1−ν12 ν21 ) ,

C1 = G13 kh

D2 2 (1 − ν12 ),

D22 = D2 =

G12 h3 12 ,

C12C21 = 0,

E2 h3 12(1−ν12 ν21 ) ,

D1 ν21 = D2 ν12 ,

C2 = G23 kh.

,

in which k = 5/6 in the Reissner plate theory, E1 and E2 represent Young’s moduli, G12 , G13 and G23 are shear moduli, ν12 and ν21 are Poisson’s ratios, respectively.

3

Differential Equations of Equilibrium

The differential equation of equilibrium is give by [8]: Li jU j + bi = 0.

(7)

where bi represent body forces and Li j are Navier differential operators, which can be written as: L11 = D1 ∂∂x2 + Dk ∂∂x2 −C1 ,

L22 = Dk ∂∂x2 + D2 ∂∂x2 −C2 ,

L12 = L21 = (D1 µyx + Dk ) ∂ x∂1 ∂ x2 ,

L13 = −L31 = −C1 ∂∂x1 ,

L23 = −L32 = −C2 ∂∂x2 ,

∂2

∂2

1

2

2

2

1

2

2

2

2

1

2

L33 = C1 ∂ x2 +C2 ∂ x2 .

(8)

250


The values of the constants are found to be: C11 = D1 µyx ,

C12 = C21 = 0, D1 =

Ex h3 12(1−µxy µyx ) ,

D2 =

D1 µyx = D2 µxy ,

C22 = D2 µxy ,

Ey h3 12(1−µxy µyx ) ,

Dk =

C1 = Gzx kh,

Gxy h3 12 ,

C2 = Gzy kh.

where Ex and Ey are elastic moduli; µxy and µyx are Poisson ratios; Gxy , Gzx , and Gzy are shear moduli; h is the thickness of the plate, and k = 5/6.

4

Fundamental Solution

The fundamental solutions of the orthotropic thick plate taking into account the transverse shear deformation are a set of particular solutions of the differential eq (7) under a unit concentrated load, i.e., the solutions satisfy the following inhomogeneous differential equations: Liadj jUk∗j (ζ , x) = −δ (ζ , x)δki .

(9)

in which δ (ζ , x) denotes the Dirac delta function, ζ represents the source point, x is a field point, and Liadj j is the adjoint operator. Following Hörmander’s operator method, the solutions of eq (9) can be written as: j Uk∗j (ζ , x) =co Lad jk φ (ζ , x). j where φ (ζ , x) is a unknown scalar function and co Lad jk

(10)

j is the cofactor matrix of the operator Lad jk

that is given

by: co ad j Lαβ

= Eαβ ∇2 ∇2k − Bαβ

co ad j L3α

= −co Lαad3j =

co ad j L33

= D1 Dk

∂2 ∂2 −C1C2 . ∂ xα ∂ xβ ∂ xα ∂ xβ

(11)

∂ ∂2 ∂2 (Eα 3 2 + Bα 3 2 −C1C2 ). ∂ xα ∂ x2 ∂ x1

(12)

∂4 ∂4 2 + (D1 D2 − D21 µyx − 2D1 Dk µyx ) 2 2 . ∂ x14 ∂ x1 ∂ x2

(13)

+D2 Dk

∂4 ∂2 ∂2 − (D1C2 +C1 Dk ) 2 − (C1 D2 +C2 Dk ) 2 +C1C2 . ∂ x24 ∂ x1 ∂ x2

The following symbols have been introduced: E11 = D2 ,

E22 = D1 ,

E12 = E21 = 0,

B11 = D2 − Dk ,

B22 = D1 − Dk ,

B12 = B21 = (D1 µyx + Dk ),

E13 = C1 D2 −C2 (D1 µyx + Dk ),

B13 = C1 Dk ,

E23 = C2 Dk ,

(14)

B23 = C2 D1 −C1 (D1 µyx + Dk ),

∇2k = C1 ∂∂x2 +C2 ∂∂x2 , 2

2

1

2

∇2 =

∂2 ∂ x12

+ ∂∂x2 . 2

2


251

By substituting eq (10) into eq (9), we obtain the following equation:

∂4 ∂4 2 2 + (D D − D µ − 2D D µ ) + 1 2 1 yx k 1 yx ∂ x14 ∂ x12 ∂ x22 ∂4 ∂2 ∂4 +D2 Dk 4 −C1C2 D1 2 + 2(2Dk + D1 µyx ) 2 2 ∂ x2 ∂ x1 ∂ x1 ∂ x2 ∂4 +D2 4 Φ(ζ , x) = −δ (ζ , x) ∂ x2

∇2k D1 Dk

(15)

The derivation of the fundamental solution of eq (9) is reduced to that of eq (15). As soon as the solution of eq (15) is obtained, substituting it into eq (10) and by differentiation we can get the solutions of eq (9). The eq (15) is a sixth order partial differential equation. Using the plane wave decomposition method also known Radon transformed, the partial differential eq (15) can be reduced to an ordinary differential equation, which simplifies the treatment of the problem. We first expand δ (ζ , x) into a plane wave (see, for example, [8]):

δ (ζ , x) = −

1 4π 2

2π 0

| ω1 (x − ζ ) + ω2 (y − η ) |−2 d θ ,

(16)

in which (ω1 , ω2 ) are the coordinates of a point on the unit circle, i.e., ω1 = cos(θ ), ω2 = sin(θ ), (x, y) and (ζ , η ) are the coordinates of a field point and a source point, respectively. Similarly, φ (ζ , x) can be written as: Φ(ζ , x) =

2π 0

ϕ (ρ )d θ ,

where ρ = ω1 (x − ζ ) + ω2 (y − η ), ϕ (ρ ) is a function depending only on ρ . By substituting eq (16) and eq (17) into eq (15), and considering differential relationship we obtain the following equation: d4 dρ 4

d2 1 2 − p ϕ (ρ ) = 2 2 | ρ |−2 , dρ 2 4π a

(17) ∂ ∂ xα

= ωα ddρ ,

(18)

in which 2 − 2D D µ )ω 4 ω 2 +C D D ω 2 ω 4 +C D D ω 4 ω 2 +C D D ω 6 a2 = C1 D1 Dk ω16 +C1 (D1 D2 − D21 µyx 1 k yx 1 2 k 1 2 2 1 k 1 2 2 2 k 2 1 2 2 − 2D D µ )ω 2 ω 4 , +C2 (D1 D2 − D21 µyx 1 k yx 1 2 b2 = C1C2 [D1 ω14 + 2(2Dk + D1 µyx )ω12 ω22 + D2 ω24 ], p2 = b2 /a2 . The solution of eq (15) is now reduced to solve the ordinary differential eq (18). After four times integration of eq (18) and leaving out the constants of integration, we obtain: 1 d 2 ϕ (ρ ) − p2 ϕ (ρ ) = − 2 2 p2 ln | ρ | . dρ 2 8π a

(19)

The solution of Eq. (19) can be written as follows:

ϕ (ρ ) = f1 (ρ )exp(pρ ) + f2 (ρ )exp(−pρ ).

(20)

252


By the method of variation of parameters, the coefficients f1 (ρ ) and f2 (ρ ) can be obtained. By substituting f1 (ρ ) and f2 (ρ ) into eq (20), we obtain: 1 [p2 ρ 2 ln | ρ | +2ln | ρ | +3 + exp(pρ ) 8π 2 p4 a2

ϕ (ρ ) =

−exp(−pρ )

∞ exp(−pσ ) ρ

σ

−∞

σ

ρ exp(pσ )

dσ

d σ ].

(21)

Substituting eq (21) into eq (17) and integrating, we can obtain the function Φ(ζ , x). The generalized displacement and boundary tractions can be expressed in the following forms: Ui∗j (ζ , x) =

2π

Pi∗j (ζ , x) =

0

U˜ i∗j (ρ )d θ ,

2π 0

(22)

P˜i∗j (ρ )d θ .

(23)

Details of the implementation of eq (22) and eq (23) can be found in [8].

5

Boundary Integral Equations

The boundary integral equation of the orthotropic thick plates taking into account the transverse shear deformation is given by:

ci j (ζ )U j (ζ ) + − Pi∗j (ζ , x)U j (x)dΓ = Γ

Γ

Ui∗j (ζ , x)Pj (x)dΓ +

Ω

q(x)Ui3∗ (ζ , x)dΩ,

(24)

where ζ , x ∈ Γ are source point and field point, respectively. The value of ci j (x) is equal to δi j /2 when x is located on a smooth boundary. The eq (24) represents three integral equations, two (i = α = 1, 2) for rotations and one (i = 3) for deflection. The last integral on the right hand side of eq (24), that is a domain integral is transformed into boundary integral by the radial integrations method. Bending moments at any internal point ζ can be computed by differentiating eq (24) with respect to the coordinate of the source point ζ and then substituting in eq (5) and eq (6) to give: Mαβ (ζ ) =

Qβ (ζ ) =

Γ

Γ

kernels Ui∗jk ,

∗ Uαβ k (ζ , x)pk dΓ(x) −

U3∗β k (ζ , x)pk dΓ(x) − Pi∗jk

Γ

Γ

∗ Pαβ k (ζ , x)uk (x)dΓ(x) + q

P3∗β k (ζ , x)uk (x)dΓ(x) + q

Γ

Γ

∗ Wαβ (ζ , x)dΓ(x)

W3∗β (ζ , x)dΓ(x)

(25)

(26)

and Wi∗jk

are found in [12] for isotropic plates. where the [14] proposed that the bending stresses vary linearly and transverse shear stresses vary parabolically over the thickness via the following relationships: 12x3 σαβ = 3 Mαβ h 2x3 2 3 σα 3 = 1− Qα 2h h

(27) (28) (29)


253

These relationships satisfy the three-dimensional differential equations of equilibrium provided that the bending moment and the shearing force satisfy the equilibrium eq (3) and eq (4). From the consideration of equilibrium in the three-dimensions, the transverse normal stresses can be obtained as follows:

σ33 =

6

2x3 2x3 2 q 3− 4h h

(30)

Numerical Results

Consider a square clamped-plate under uniformly distributed load with amplitude q = −2, 07 × 106 N/m2 . The plate is orthotropic with the following material properties: Ey = 0, 6895 × 1010 Pa, Ex = 2 × Ey , Gxy = Gxz = Gyz =299,78×106 Pa and νxy = 0, 25. The edges of plate is a = 0, 254 m and thickness h = 0, 0127 m. Constant boundary elements with equal length were used in the discretization. The results for moments and stresses are given in Table 1. Moments are compared with values calculated using the Meshless Petrov Galerking Method (MLPG) by [11]. As it can be seen, our work present a good agreement with the results of [11]. Table 1: Moments and Stresses at the center of plate. NE 20 40 60 [11]

Mxx (N.m) 4182,1 4119,3 4098,5 4089,5

σxx (N/m2 ) 1, 5557 × 108 1, 5324 × 108 1, 5246 × 108

As can be seen in Table 1, results are in good compared with results of [11]. The results to moments in direction x presented a difference of 0.002% in the results in comparasion with the literature. Wasn’t found in the literature any work to compute stresses in orthotropic thick plates.

7

Conclusions

This paper has discussed the use of the boundary method element for analysis of orthotropic plates. The formulation is applicable to thick plates to compute moments and stresses in internal points. Was used constants elements to discretization of the boundary. Only one problem was discussed to demonstrate the accuracy and efficiency of the formulation implemented.

Acknowledgment The authors would like to thank the State of Maranhão Research Foundation (FAPEMA) for the financial support of this work.

254

8


References

[1]

Weeën, F.V., Other B.B., ”Application of the boundary integral equation method to Reissner’s plate model”. Journal. Num. Meth. Eng. v.18, n.1, 1-10, 1982.

[2]

Long, S.Y., Brebbia, C.A., Telles, J.C.F., ”Boundary element bending analysis of moderately thick plates”. Comp. Mech. Publications v.5, n.8, 64-74, 1988.

[3]

Karam, V.J., Telles, J.C.F., ”On boundary elements for Reissner’s plate theory”. Eng. Analysis v.5, n.1, 21-27, 1988.

[4]

Ribeiro, G.O., ”Sobe a formulao do mtodo dos elementos de contorno para a flexo de placas usando as hipteses de Reissner.”. Escola de Engenharia de So Carlos, Universidade de So Paulo - USP, Tese (Doutorado) p.266, 1992.

[5]

Hörmander, H., ”Linear partial differential operators”. Berlin: Springer Verlag, 1963.

[6]

Abramowitz, M., Stegun, I.A., ”Handbook of mathematical functions”. New York: Dover Publications, 1965.

[7]

De Barcellos, C.S., Silva, L.H.M., ”A boundary element formulation for the Mindlin’s plate model”. Boundary Element Techniques: Applications in Stress Analysis and Heat Transfer, Southampton, 1987.

[8]

Wang J., Huang M., ”Boundary element method for orthotropic thick plates”.Acta Mechanica Sinica 7, 258-266, 1991.

[9]

Sanches, L.C.F., ”Uma resoluo de placas com a teoria de Mindlin atravs do mtodo dos elementos de contorno”.Faculdade de Engenharia Civil, Universidade de Campinas, Dissertao (Mestrado) p.207, 1998.

[10]

Useche, J.F., ”Anlise pelo mtodo dos elementos de contorno de placas de Reissner trincadas e reparadas com compsitos colados”.Faculdade de Engenharia Mecnica, Universidade de Campinas, Tese (Doutorado) p.165, 2007.

[11]

Sladek J., Sladek V., Zhang Ch., Krivacek J. and Wen P.H., ”Analysis of orthotropic thick plates by meshless local Petrov-Galerkin (MLPG) method”. Int. J. Numer. Meth. Engn. 67, 1830-1850, 2006.

[12]

Rashed Y.F., ”Boundary method formulations for thick plates”.Topics in Engineering, W.I.T. V.35, Publications Inc., 1999.

[13]

Timoshenko S., Woinowski-Krieger S., ”Theory of plates and shells”.McGraw-Hill Book Company, Second Edition, 1959.

[14]

Reissner, E., ”On bending of plastic plates”.Quart. Applied Mathematics, 5, 55-68, 1947.

[15]

Zhao, Z., ”On the calculation of boundary stress in boundary elements”.Engineering Analysis with Boundary Elements, 16, 317-22, 1995.

[16]

Zhao, Z. and Lan, S., ”Boundary stress calculation-comparison study”.Computers & Structures, 71, 77-85, 1999.

[17]

Reis, A., Albuquerque, E. L., Torsani, L. F., Palermo, L. J., Sollero, P., ”Computation of moments and stresses in laminated composite plates by the boundary elements method,”.Engineering Analysis with Boundary Elements, 35, 105-113, 2011.

[18]

Campos, L. S., Sousa, K. R. P. ; Santana, A. P., Reis, A. ; Albuquerque, E. L. and Sollero, P., ”Stress analysis of thin plate composite materials under dynamic loads using the boundary element,”.In: XII International Conference on Boundary Element and Meshless Techniques, 2011, Braslia. Advances in Boundary Element and Meshless Techniques, 190-195, 2011.


255

Geometric and Creeping Material Nonlinearities in Shear Deformable Plates with the Boundary Element Method E. Pineda1, M.H. Aliabadi2, J. Núñez-Farfán3, A. Rodríguez-Castellanos3 1

Escuela Superior de Ingeniería y Arquitectura, Instituto Politécnico Nacional, México D. F., email: [email protected]. 2

Department of Aeronautical Engineering, Imperial College London, London SW72AZ.

3Instituto Mexicano del Petróleo, México D F, e-mail: [email protected]; [email protected]

Keywords: Boundary Element Method, Large Deflections, Creep analysis.

Abstract. In this work a new boundary element formulation is developed for the analysis of shear deformable plates with combined geometric and creeping material nonlinearity. The new formulation includes a time dependent analysis (creep) for shear deformable plates. Also this formulation allows for material experiencing large deflections and small creep strains. The standard power law creep equation is considered and an initial stress formulation is employed to formulate the boundary integral equations due to creep. Secondary creep analysis for a long period of time is applied in this research. Domain integrals due to external load, large deflection and creep appear in the formulation. In dealing with domain integrals due to large deflection and creep, the approach considered here is cellcell, where both domain integrals due to large deflection and creep are analyzed using domain discretization technique. However, the discretization is in the whole domain due to the nature of creep. In the domain discretization technique 9-nodes of quadrilateral cells are implemented. An example is presented and comparisons are made with other numerical solution to demonstrate the validity and accuracy of the proposed formulation. Introduction Plate structures are widely used in engineering applications. There are two widely used plate theories. The first plate theory was developed by Kirchhof [1] and is commonly referred to as the classical theory. The other was developed by Reissner [2], and is known as the shear deformable theory. The classical theory is adequate for analyzing certain applications, however, for problems involving stress concentrations and cracks the theory has been shown not to be in agreement with experimental measurements. Recent advances in boundary element method for plate bending can be found in Aliabadi [3]. Most of the above papers deal with linear analysis of plate in which the change in deflection is proportional to the change of load. For nonlinear analysis, the applications of boundary element method can be found in the works by Tanaka [4]. Tanaka [4] studied the application of BEM to the problem of elastic thin plate bending with large deflection, and presented an incremental integral formulation, which is equivalent to the Von Karman equation. Kamiya and Sawaki [5] investigated the large deflection of elastic plate based on the Berger’s equation. General nonlinear differential equations for the Mindlin plate theory was introduced by Pica, Wood and Hinton [6] for the analysis of nonlinear plate bending in the application of finite element method. Mukherjee [7] has been working in the application of the BEM to nonlinear problems time-dependent inelastic deformation. A BEM formulation for viscoplastic problems [8] was followed by a numerical implementation for two dimensional problems. Other applications of the BEM include elastic and very recently inelastic fracture mechanics. Furthermore, several boundary element formulations for the solution of time-dependent inelastic problems arising in creeping metallic structures subjected to high temperature gradients were investigated by Providakis [9-11]. The accuracy and efficiency of such methods were demonstrated by obtaining stress distribution for crack specimens subjected to pure bending. Governing equations

256


Consider an element in the deformed state as shown in Fig. 1. In order to define a general formulation for combined geometrical and material nonlinearities (creep) of plate bending, it is considered that creep strains are only due to bending and membrane, hence total strain rates can be defined as

ሶ and ሶ

ሶ ሶ

ɖ஑ஒ ൌ ɖୣ஑ஒ ൅ ɖୟ஑ஒ

(1)

ሶ ɂ஑ஒ ൌ ɂୣ஑ஒ ൅ ɂୟ஑ஒ

(2)

ሶ

ሶ

ɀ஑ଷ ൌ ɀୣ஑ଷ

(3)

Where, ɖ஑ஒ are the total bending strain rates, ɂ஑ஒ are the total in plane strain rates, and ɀ஑ଷ are the shear strain rates, respectively. The total bendingሶ strainሶ rates consist of linear ɖୣ஑ஒ parts and nonlinear parts ɖୟ஑ஒ . Similarly total inplane strain rates consist of linearሶ parts ɂୣ஑ஒ and nonlinear parts ɂୟ஑ஒ . ሶ ሶ ሶ

ሶ ሶ

Fig. 1. Stress resultant equilibrium in nonlinear plate element The nonlinear parts of eq. (1) and (2) are due to large deflection (nl) and creep (c), and can be expressed as: ୡ ɖୟ஑ஒ ൌ ɖ୬୪ ஑ஒ ൅ ɖ஑ஒ

and ሶ

ሶ

(4)

ሶ ୡ ɂୟ஑ஒ ൌ ɂ୬୪ ஑ஒ ൅ ɂ஑ஒ

(5)

The generalized strain rates above can be expressed in terms of generalized displacement rates as: ሶ ሶ ሶ ɖ஑ஒ ൌ

ሶ ሶ ൫஑Ǥஒ ൅ ஑Ǥஒ ൯ ൅ ଷǤஒ ଷǤ஑ ɂ஑ஒ ൌ Ǣ ʹ ሶ

and ሶ

஑ǡஒ ൅ ஒǡ஑ ʹ

ሶ ሶ

ሶ

(6)

(7)


ሶ ሶ

ሶ

ɀ஑ଷ ൌ ஑ ൅ ଷǤ஑

257

(8)

ሶ rotations ஑ are translations and w3 is the deflection. Nahdi [12] derived the relationships between Where, ஑ are stress resultants and strains by using the Reissner’s variational theorem of elasticity [2]. Displacement integral equations The governing equations together with the Betti’s reciprocal theorem are used to derive the displacement integral equations:

ሶ

‫כ‬ ሺȱ ᇱ ǡ ሻ஑ ሺሻȞ ஘஑ ஘ ሺȱ ᇱ ሻ ൅ නሶ ஘஑ ୻ ‫כ‬ ‫כ‬ ୬୪ ሺȱ ᇱ ǡ ሻ ஑ ሺሻȞ െ න ሶ ஘஑ǡஒ ሺȱ ᇱ ǡ ȱሻ஑ஒ ሺȱሻπ ൌ නሶ ஘஑ ୻

(9)

୕

‫כ‬ ୔ ሺȱ ᇱ ǡ ȱሻ஑ஒ ሺȱሻπ ൅ න ሶ ɂ஘஑ஒ ୕

where r denotes a Cauchy principal value integral, and ୧୨ are the jump terms. eq. (9) constitutes the boundary displacement integral equations for plate bending problem. Stress integral equations

ሶ

The stress resultants can be expressed as: ‫כ‬ ‫כ‬ ‫כ‬ ‫כ‬ ୬୪ ሺȱ ᇱ ሻ ൌ නሶ ஑ஒஓ ሺȱ ᇱ ǡ ሻ ஓ ሺሻȞ െ නሶ ஑ஒஓ ሺȱ ᇱ ǡ ሻஓ ሺሻȞ ൅ න ሶ ஑ஒஓǡ஘ ሺȱ ᇱ ǡ ȱሻஓ஘ ሺȱሻπ ஑ஒ ୻

୻

୕

୔ ሶ ୔ ሶ ൣʹሺͳ ൅ ɋሻ஑ஒ ൅ ሺͳ െ ͵ɋሻ஘஘ Ɂ஑ஒ ൧ ୔ ሺȱሻπ െ െ න ɂሶ ‫כ‬஑ஒஓ஘ ሺȱ ᇱ ǡ ȱሻஓ஘ ͺ

(10)

୕

‫כ‬ ‫כ‬ ‫כ‬ ‫כ‬ where the kernels ୧ஒ୩ , ୧ஒ୩ are linear combination of the first derivatives of ୧୨‫ כ‬, and ୧୨‫ כ‬. The kernels ஑ஒஓ , ஑ஒஓ ‫כ‬ ‫כ‬ are the linear combination of the first derivatives of ஑ஒ and ஑ஒ . The kernels ɖ‫כ‬୧ஒஓ஘ , ɂ‫כ‬୧ஒஓ஘ are the linear ‫כ‬ . combination of the first derivatives of ɖ‫כ‬୧஑ஒ and ɂ஑ஒஓ

Multiaxial Creep Stress States In practice, it is found that the multiaxial characteristics of creep are very similar to non-linear formulations, and are commonly based on the Prandtl-Reuss flow rule and the von Mises effective stress criterion. So, the multiaxial case of the time hardening approach is obtained as follows: ͵ (11) ሶ ɂୡ ൌ ሺɐୣ୯ ሻ୬ିଵ ୧୨ ୫ିଵ ʹ B is a material constant that depends on the temperature, n and m are material properties that indicates the creep stage, ɐୣ୯ and ୧୨ are the equivalent and deviatoric stresses, respectively. Discretization and system of equations In order to analyze the problem by the boundary element method, the integral equations are discretized. The boundary Ȟ is divided into boundary elements and the domain π where the existence of creep deformation is

258


expected is divided into cells. The boundary is discretized using quadratic isoparametric elements both continuous and semi discontinuous elements are used and the domain is discretized using 9 nodes quadrilateral (continuous and semi discontinuous) cells. The semi discontinuous boundary elements are used to avoid corner problems in the boundary and the semi discontinuous cells to avoid the coincident nodes between boundary and cell. The internal values are taken at the cell nodes, which will avoid calculation of deflection derivative on the boundary nodes. The deflection derivative is used to calculate the nonlinear terms due to large deflection. After discretization and point collocation on the boundary as well as in the domain, the equations can be written in the matrix form as: ୵ Ͳ ୡ ୵ Ͳ ஓஒ ଷǡஒ Ͳ ቃ ൜ ൠ ൅ ൜ ൠ െ ቂ ቃ൜ ൠ ቋ൅ቂ ୳ቃ ቊ ୬୪ (12)

୳ Ͳ Ͳ ୳ ୡ Ͳ ஓஒ ሶ ሶ ሶ ሶ ሶ ሶ ሶ ሶ ሶ ሶ where [H] and [G] are the well-known boundary element influence matrices, [B] and [T] are the influence matrices for large deflection and creep, respectively. The superscript w and u show the plate and the in-plane mode, respectively. ሼሽ, ሼሽሼሽ, ሼሽ are the displacement and the traction rate vectors. ൛ൟ is the load rate vectors. ൛ ୡ ൟ, ൛ ୡ ൟare the bending and membrane stress resultant term, respectively. ሶ ሶ ሶ ሶ ሶ ሶ Numerical Example ሶ

ቂ

୵ Ͳ

୵ Ͳ ቃቄ ቅ ൌ ቂ ୳ Ͳ

The bending simply-supported circular plate, shown in Fig. 2. under constant uniform pressure is considered. A linear creep law given by 1.491x10-15 ɐଷ . The material 2024-T3 aluminum alloy at 600°F with the properties E=7.4x106 psi and Poisson ratio of 0.3. Results are presented in Fig. 3, good agreements is observed for all strains calculated at the central node.

Fig. 2.- BEM mesh for a circular plate under uniform pressure.


EL+LD+CRE

259

EL+CRE

8.00EͲ10

CreepStrains

7.00EͲ10 6.00EͲ10 5.00EͲ10 4.00EͲ10 3.00EͲ10 2.00EͲ10 1.00EͲ10 0.00E+00 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (hours)

Fig. 3.- Large deformation and secondary creep strains in the center of a circular plate. Fig. 3 shows the comparison of two cases: the blue line is the combination of the elastic, large deformation and secondary stage of creep strains. On the other hand, the red line corresponds to the elastic and creep strains only. Conclusions The application of BEM to combined geometric and creep material nonlinearities for shear deformable platebending analysis was presented. An overview of the formulations of the boundary integral equations is presented including the integral equations used to calculate the internal values. The numerical implementation of the formulation was performed. From the results obtained it can be concluded that our preliminary results mach well with respect to those obtained in the literature. Therefore, BEM method seems to be a good tool to deal with non linear behavior considering large deformations and creep.

References [1] Lei XY, Huang MK, Wang XX. Geometrically nonlinear analysis of a reissner’s type plate by boundary element method. Comput Struct 1990; 37(6):911–6. [2] Ribeiro GO, Venturini WS. Elastoplastic analysis of Reissner’s plate using the boundary element method Plate bending analysis with boundary element. Southampton: Computational Mechanics Publications; 1998 p. 101–25. [3] Aliabadi MH. Plate bending analysis with boundary element. Southampton: Computational Mechanics Publications; 1998. [4] Vander WeeeÞn F. Application of boundary integral equation method to reissner’s plate model. Int J Numer Methods Eng 1982; 18:1–10. [5] Karam VJ, Telles JCF. On boundary elements for reissner’s plate theory. Eng Anal 1988;5:21–7. [6] Purbolaksono J, Aliabadi MH. Large deformation of shear deformable [7] S.A. Mukherjee, Boundary Element Methods in Creep and Fracture. Applied Science Publishers LTD, Barking Essex, England, 1982. [8] V. Kumar, S.A. Mukherjee, Boundary-integral equation formulation for time-dependent inelastic deformation in metals, International Journal of Mechanical Sciences 19 (1977) 713-724. [9] C.P. Providakis, Boundary element analysis of creeping V-notched metallic plates in bending, Engineering Fracture Mechanics 64 (1999) 129-140. [10] C.P. Providakis, Boundary element approach to creep deformation of edge notched and cracked specimens, Theoretical and Applied Fracture Mechanics 38 (2002) 191-202. [11] C.P. Providakis, D/BEM implementation of Robinson's viscoplastic model in creep analysis of metals using a complex variable numerical technique, Advances in Engineering Software 33 (2002) 805-816. [12] Naghdi PM. On the theory of thin elastic shells. Q Appl Math 1956;14: 369–80. [13] J.L. Sanders, G.H. McComb, Jr and F.R. Schlechte, A variational theorem for creep with application to plates and columns, NACA Techn. Note 4003 (1957).

260


Scattering of Elastic Waves in Fluid-layered Solid Interfaces by the Indirect Boundary Element Method. A. Rodríguez-Castellanos1, E. Pineda León2, J. Nuñez-Farfán1, E. OliveraVillaseñor1, Andrei Kryvko3 1

Instituto Mexicano del Petróleo, Eje Central Lázaro Cárdenas 152, Gustavo A Madero, México D F, México, [email protected] 2

Instituto Politécnico Nacional, ESIA, Unidad Profesional Zacatenco, México D. F. México, [email protected]

3

Instituto Politécnico Nacional, ESIME, Unidad Profesional Zacatenco, México D. F. México.

Keywords: Indirect boundary element method, fluid-solid interface, scattering of elastic waves, discrete wave number, layered solid, Green’s function.

Abstract. This work deals with the scattering of elastic waves by fluid-layered solid interfaces. The Indirect Boundary Element Method (IBEM) is used to study this wave propagation phenomenon in 2D fluid-layered solid models. The source is represented by a Hankel’s function of second kind and is always placed in the fluid. The IBEM is based upon an integral representation for scattered elastic waves using single-layer boundary sources. This approach is called indirect because the sources’ strengths are calculated as an intermediate step. Moreover, this formulation is regarded as a realization of Huygens’ Principle. The results are presented in frequency and time domain. In order to demonstrate the accuracy of the method, results were validated with those obtained by the Discrete Wave Number method applied to a fluid solid interface joining two half-spaces, one fluid and the other an elastic solid. Various aspects related to the different wave types that emerge from this kind of problems are emphasized. A near interface pulse generates changes in the pressure field and can be registered by receivers located in the fluid. Among the phases that can be detected we have interface Scholte’s waves and trapped waves within the interface layer. The IBEM can be used with confidence to model irregular interfaces. Introduction In many areas of physics and engineering, the study of fluid-solid interfaces has always attracted interest. Important developments to study the dynamic behavior of an ocean layer over an elastic solid by means of analytical solutions can be found in the original work of Biot [1] and Ewing et al. [2], where attention to Stoneley and Rayleigh waves was paid. Other applications have been used to understand the behavior of interface waves directed to the ocean floor [3,4]. Carcione and Helle [5] studied the physics related with these interfaces in a variety of seabed mechanical properties, from soft sediments to crustal rocks. Analytical results regarding the characteristics of Rayleigh waves in a marine environment, excited by deep earthquakes have been presented in Yoshida [6,7]. In this work we used the Indirect Boundary Element Method (IBEM) in 2D to study the interactions between acoustic and elastic waves, near a fluid and an elastic layered solid interface. This formulation could be considered as a numerical realization of Huygens’ principle, in which the diffracted waves are built at the boundary from which they are radiated. Mathematically speaking, this is completely equivalent to the well-known Somigliana’s representation theorem. Formulation of the problem by means of the Indirect Boundary Element Method For the IBEM, the source pulse (incident wave) in the fluid is represented by a Hankel function and applied as shown in Fig. 1.


261

Fig. 1 a) Fluid-layered solid interface excited by a source in the fluid; b) Boundary element mesh used to solve the problem. Considering that Newton’s 2nd law governs the motion within the fluid we can write:

wV ij x

w 2ui x wt 2 , i , j 1,3 .

UF

wx j

(1)

The stresses in the fluid can be linked to the pressure generated by the incident pulse:

V ij x p 0 x G ij i, j 1,3 , . F

(2)

As a consequence of eqs (1) and (2), the displacement field in a given direction, in frequency domain, is proportional to the gradient of pressure in the same direction:

wp 0 (x) U F Z 2 wn . F

1

F

u n0 (x)

(3)

To express the diffracted wave field (for pressure and displacement, respectively) in the fluid due to the source effect upon the elastic medium, we use a “single-layer” integral representation for pressures:

³G

F

p d ( x)

wF

F

(x, [ ) < ([ ) dS[

(4)

where

G F (x, [ )

U FZ 2 4i

H 0( 2) (Zr / c F )

(5)

and

u nd (x) c1< (x)

1

F

U FZ

2

wG F (x, [ ) u < ([ )dS[ wF wn

³

(6)

full space Green’s function for the fluid, and < x boundary displacement density for the 0.5 and defines the region orientation. U and Z are the mass density and circular frequency,

F Where; G (x)

fluid. c1

respectively. r represents the distance between the source and the receiver. c F is the wave velocity of the fluid. The pressure and displacement field in the fluid, can be written, respectively:

p F x

u nF x

p 0 x p d x , F F u n0 x u nd x F

F

(7)

(8) . Since the source is only applied in the fluid, only diffracted waves appear in the solid and they can be established as follows.

262


Consider a domain V, with boundary wS . If the domain is occupied by an elastic solid, the displacement field under harmonic excitation can be expressed, neglecting body forces, by means of the single-layer boundary integral equation: (9) u id x Gij x, [ I j [ dS[ wS , and (10) t id x c2Ii x Tij x, [ I j [ dS [ wS , where ui x = ith component of the displacement at point x , Gij x; ȟ = Green’s function, I j ȟ = force density

³

³

in the direction j at point ȟ . tid x i th component of traction, c2

0.5 , Tij x; ȟ traction Green’s function.

The 2D Green’s functions for full spaces can be seen in [8,9]. Boundary Conditions From the configuration depicted in Fig. 1b, it is convenient to partition the domain in three regions (R, E and F), for which proper boundary conditions should be established. These conditions for interfaces can be written as follows: For the fluid-solid interface

u3R x u3F x , x w1 R wF ,

(11)

t1R x 0 , x w 1 R ,

(12)

t 3R x p F x , x w 1 R .

(13)

For the continuous solid interface

t iR x t iE x , x w 2 R uiR x uiE x , x w 2 R

wE , i=1, 3

wE

(14) (15)

,i=1, 3

Writing eq (11) as a function of the diffracted field eq (9) in the solid and the incident and diffracted fields (3 and 6) in the fluid, one obtains: (16) wG F x, [ 1 wp o x wr R R G dS c < x , x [ I [ j j 1 [ 3 2 ³wR ³wF wx3 < [ dS[ x wF w1 R wr wx3 UFZ , . The tangential traction-free condition (12) is expressed from the integral form (10): (17) 1 R R R

c2 Ii x ³ T1 j x, [ I j [ dS [ wR 2

0

, x w 1 R .

Eq (13) can be written by: c2IiR x ³ T3Rj x, [ I jR [ dS[ wR

cZ H 02 Zr c F ³ G F x, [ < [ dS[ x w F wF 1 ,

w1 R .

(18)

Eqs (14) and (15) represent the continuity condition that must exist between the interface of regions R and E (boundaries w 2 R and wE ). These are defined as follows:


c 2I iR x ³ TijR x, [ I jR [ dS [ wR

³

wR

GijR x, [ I jR [ dS [

³

wE

c 2I iE x ³ TijE x, [ I jE [ dS [ x w R wE 2 ,

GijE x, [ I jE [ dS [ x w R 2 ,

263

(19)

wE ,

(20)

wE .

The discretization of the eqs (16) to (20) leads to the system of Fredholm integral equations to be solved. Assuming that surface densities I x and < x are constant on each boundary element that forms the surfaces of regions R, E and F (see Fig. 1b) and Gaussian integration (or analytical integration, when the Green’s function is singular) is performed. Validation and application of the IBEM To verify the accuracy of IBEM, results from DWN formulation we considered for various models with fluidsolid interfaces. Borejko [10] developed theoretical and experimental studies in order to show the emergence of interface waves. The interfaces studied are: Case 1 (Fluid-Pitch), Case 2 (Fluid-Limestone), Case 3 (FluidGranite), Case 4 (Fluid-Sandstone) and Case 5 (Fluid-Sandstone-Granite). The material properties for the models used are described in Table 1. Table 1 Material properties for numerical examples Material Pitch Limestone Granite Sandstone Fluid for all cases

D

E

U

(m/sec) 2443 4810 6100 2670 1500

(m/sec) 1000 2195 2977 1090 -----

(kg/m3) 1270 2500 2700 2200 1000

For comparison purposes an interface model joining two half-spaces was chosen, one fluid and an elastic solid, for the Cases 1 and 2. As shown in Fig. 1, one receiver located at the distances a=0.05m and b=1.00m was considered. In Fig. 2, the amplitudes of the pressure for this receiver are displayed for the Cases 1 and 2. Results from IBEM are plotted with solid and dashed lines to represent Limestone and Pitch, respectively. Calculations from DWN are depicted with circles for Case 1 and asterisks for Case 2. Good agreement can be appreciated between IBEM and DWN results. It is clear from this figure that the response for both materials vary significantly and can be associated with the relative value of the shear wave velocity of the solid in comparison with the fluid wave velocity [10]. For shear wave velocities higher than the fluid’s compressional wave velocity, the pressure spectrum shows more simple patterns in comparison with the opposite case, where some conspicuous peaks are observed.

Fig. 2 Narrow band spectrum of the pressure at the receiver located at a=0.05 m and b=1.0 m for case 1 (dashed line for IBEM and circles for DWN method) and case 2 (solid line for IBEM and asterisks for DWN method).

264


In Fig. 3, pressure fields in time domain are shown for the Cases 3, 4 and 5. To this end, the Fast Fourier Transform (FFT) algorithm was used a Ricker wavelet as source. All models were analyzed for frequency increments of 150 Hz, reaching a final frequency of 19200 Hz. By means of the FFT, it is possible to observe the different kinds of waves that emerge. In all cases, 25 receivers were located in the fluid. The first one was placed at a distance of b=1.0 m from the source, and the rest of the receivers were placed using a distance increment of 0.05 m. The distance a=0.05 m (see Fig. 1) for all cases. For Case 5 (Fluid-Sandstone-granite interface) two layer thicknesses were considered, which are h=0.20 m (Fig. 3c) and h=0.40 m (Fig. 3d). In Fig. 3a, the wave propagation phenomenon for Case 4 (Fluid-Sandstone interface) is shown. Here, it is possible to observe the influence of the compressional wave velocity D of the Sandstone, represented as t ps , the direct wave that travels in the fluid and that is perceived by the receivers, is shown with t df , and the Scholte's interface s . The superscript s represents "Sandstone", while f is for "Fluid". Borejko [10] pointed wave is illustrated using t Sc out these of waves in his theoretical and experimental studies. For this case, the velocities measured were t ps | 2600 ms-1, t df 1500 ms-1 and t Scs 937.5 ms-1.

In Fig. 3b, the wave fronts that emerge from the Fluid-Granite interface interactions are depicted. Here, it is possible to identify wave velocities associated with pseudo Rayleigh, direct and Scholte's waves, which propagate g at t pR 3076.9 ms-1, t df 1500 ms-1 and t Scg 1500 ms-1, respectively. The superscript g refers to "Granite". The results for these last two cases agree with those obtained by Borejko [10]. The Scholte's wave travels at a velocity close to the direct wave in the fluid and, therefore, only one wave front is seen. This was also reported by [10]. For the interface model (Case 5), two layer thicknesses were studied, as mentioned above. In this case, the influence of the Granite is evident. In Fig. 3c (for h=0.20 m), the direct wave represented by tdf and the four fronts associated with Scholte's wave velocity ( t Scg ) are clearly observed. Moreover, two wave fronts that travel at a velocity of t ss 1090 ms-1 are identified. This velocity coincides with the shear wave velocity E of the Sandstone. The subscript s stands for S wave velocity. In Fig. 3d, these same wave fronts appear, but less interactions are evidently appreciated due to the considered layer thickness (h=0.40 m).


Fig. 3 Synthetic seismograms of pressures by IBEM; a) for case 4; b) case 3; c) case 5 (h=0.20 m); and d) for case 5 (h=0.40 m). Conclusions We used the Indirect Boundary Element Method to model wave propagation phenomenon at 2D fluid-layered solid interfaces. This indirect formulation can give a deep physical insight on the generated diffracted waves because it is closer to the physical reality and can be regarded as a realization of Huygens’ principle. In any event, it is fully equivalent to the classical Somigliana’s representation theorem. In order to verify accuracy we tested the method by comparing results with the semi-analytical solution known as the Discrete Wave Number method. A near interface source generates scattered waves that can be registered by receivers located in the fluid. Results were presented in the frequency and time domain, where some aspects related to the different wave types that emerge from this kind of problems were pointed out. The results between IBEM and DWN show a very good agreement. The corresponding results for the two solid cases show the emergence of trapped waves within the surface layer and possible resonances. References [1] M.A.Biot Bull. Seism. Soc. Am., 42, 81-93(1952). [2] W.M.Ewing, W.S.Jardetzky, F.Press Elastic waves in layered earth, McGraw-Hill Book Co. (1957). [3] W.L.Roever, J.H.Rosenbaum, T.F.Vining Journal of the Acoustical Society of America, 55, 1144-1157 (1974). [4] R.vanVossen, J.O.A.Robertsson, C.H.Chapman Geophysics, 67, 618-624 (2002). [5] J.M.Carcione, H.B.Helle Geophysics, 69, 825-839 (2004). [6] M.Yoshida Bull. Earthq. Res. Inst. 53, 319-338 (1978). [7] M.Yoshida Bull. Earthq. Res. Inst. 53, 1135-1150 (1978). [8] F.J.Sánchez-Sesma, M.Campillo Bull. Seism. Soc. Am., 81, 1-20 (1991). [9] A.Rodríguez-Castellanos, F.J.Sánchez-Sesma, F.Luzón, R.Martin Bull. Seism. Soc. Am., 96, 1359-1374 (2006). [10] P.Borejko International Symposium on Mechanical Waves in Solids, Zhejiang University, Hangzhou, China, 15-16 (2006).

265

266


Advances in numerical modeling of cohesive crack using a meshless Finite Point Method L. Pérez Pozo, F. Valdivia Bugueño Aula UTFSM-CIMNE, Departamento de Ingeniería Mecánica, Universidad Técnica Federico Santa María, Avenida España 1680, Valparaíso, Chile. [email protected], [email protected]

Keywords: Meshless, fracture, cohesive crack, strain softening. Abstract. Fracture in quasi-brittle heterogeneous materials has been treated extensively with numerical modeling using the Finite Element Method (FEM), showing a dependence on the size and orientation of the mesh in the results, for this reason the use of meshless techniques in the fracture models appears as an alternative to this dependences. In this work, a cohesive crack model using a meshless Finite Point Method (FPM) is presented. In the formulation of the stress-strain constitutive relationship the characteristic length is replaced by a geometric length given by the direction of the principal stress, which is based on the Hillerborg’s fictitious crack model where the fracture energy is considered as a material property. Rankine’s yield criterion and an incremental iterative NewtonRaphson scheme are used to predict the inelastic behavior of the cracked zone in the material. Introduction There are different points of view to classify the constitutive model based on classical mechanics, including models are the cohesive crack models and the models based on the theory of damage and plasticity. Each one of these groups includes constitutive models that reproduce numerically the fracture behavior of brittle and quasibrittle materials [1]. The damage mechanics, based on the irreversible thermodynamics process, has been introduced to simulate various materials that exhibit, in the meaning of loss of stiffness, an alteration of its elastic properties during the load process due a reduce of the resistant effective area [2]. In [3] the concept of effective stress was defined. In his work, Kachanov discussed the damage variable as a scalar, whose value ranged from 0 to 1. Later, several researchers extended this theory by approaching the damage variable as a tensor. On the other hand, the cohesive crack models are based on the continuum mechanics using hypothesis and parameters of the fracture mechanic [4]. The Fictitious Crack Model [5] has the advantage of avoiding the problems of objectivity due the dependence on the size and orientation of the mesh in FEM, and an approximation of this formulation based on a stress-strain relationship is presented in [6], where a characteristic length is replaced by a geometric length, but using FEM the dependence on the mesh still remains an important factor on the results. The non-dependence on a mesh or an integration procedure, makes that the Finite Point Method (FPM) [7,8,9,10] becomes in a meshless method, and the formulation of the constitutive equation based on the damage theory and related with the cohesive crack model, depends only on a geometric length given by the direction of the principal stresses.


267

The Finite Points Method Let πூ be the interpolation sub-domain, also called cloud, and let ܵ௝ with ݆ ൌ ͳǡʹǡ ǥ ǡ ݊ be a collection of ݊ points with coordinates ‫ݔ‬௝ ‫ א‬πூ . The unknown function ‫ݑ‬ሺ‫ݔ‬ሻ may be approximated within πூ by: ் ‫ݑ‬ሺ‫ݔ‬ሻ ؆ ‫ݑ‬෤ሺ‫ݔ‬ሻ ൌ σ௠ ௜ ‫݌‬௜ ሺ‫ݔ‬ሻߙ௜ ൌ ࢖ሺ‫ݔ‬ሻ ࢻ

(1)

where ߙ ൌ ሾߙଵ ǡ ߙଶ ǡ ǥ ǡ ߙ௠ ሿ் and the vector ࢖ሺ‫ݔ‬ሻ, called base interpolating functions, contains typically monomials in the space coordinates ensuring that the basis is completed. In general, ࢖ ൌ ሾͳǡ ‫ݔ‬ǡ ‫ݕ‬ሿ் ݂‫ ݉ݎ݋‬ൌ ͵ ࢖ ൌ ሾͳǡ ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݕݔ‬ǡ ‫ ݔ‬ଶ ǡ ‫ ݕ‬ଶ ሿ் ݂‫ ݉ݎ݋‬ൌ6 ࢖ ൌ ሾͳǡ ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݖ‬ǡ ‫ ݔ‬ଶ ǡ ‫ݕݔ‬ǡ ‫ݖݔ‬ǡ ‫ ݕ‬ଶ ǡ ‫ݖݕ‬ǡ ‫ ݖ‬ଶ ሿࢀ ݂‫ ݉ݎ݋‬ൌ ͳͲǢ ݁‫ܿݐ‬

(2)

(3) (4)

can be used. The function ‫ݑ‬ሺ‫ݔ‬ሻ can now be sampled at the ݊ points belonging to ȍூ giving ‫ݑ‬௛ ࢖் ‫ݑ‬ොଵ ‫ ۓ‬ଵ௛ ۗ ‫ ۓ‬ଵ் ۗ ‫ݑ‬ ො ‫ݑ‬ ଶ ؆ ൞ ଶ ൢ ൌ ࢖ଶ ߙ ൌ ۱ࢻ ࢛ ൌ ‫ڭ‬ ‫ۘ ڭ۔‬ ‫ۘ ڭ ۔‬ ‫ݑ‬ො௡ ‫்࢖ە‬௡ ۙ ‫ݑە‬௡௛ ۙ ௛

(5)

where ‫ݑ‬௝௛ ൌ ‫ݑ‬ሺ‫ݔ‬௝ ሻ are the unknow values of ‫ ݑ‬at point, ‫ݑ‬ො௝ ൌ ‫ݑ‬ොሺ‫ݔ‬௝ ሻ are the approximate values, and ࢖௝ ൌ ࢖ሺ‫ݔ‬௝ ). In the case of FEM, the number of points is chosen so that ݉ ൌ ݊ . In this case ۱ is a square matrix [11]. If ݊ ൐ ݉, ۱ is not a square matrix and the approximation cannot fit all the ‫ݑ‬௝௛ values. This problem can be solved determining the ‫ݑ‬ො values by minimizing with respect to the ߙ parameters the sum of the square of the error at each point weighted with a function ߱ሺ‫ݔ‬ሻ as ‫ ܬ‬ൌ σ௡௝ୀଵ ߱൫‫ݔ‬௝ ൯ሺ‫ݑ‬௝௛ െ ‫ݑ‬ොሺ‫ݔ‬௝ ሻሻଶ ൌ σ௡௝ୀଵ ߱൫‫ݔ‬௝ ൯ሺ‫ݑ‬௝௛ െ ࢖்௝ ߙሻଶ

(6)

This approximation is called weighted least square (WLS) interpolation. Note that for ߱ሺ‫ݔ‬ሻ ൌ ͳ the standard least square (LSQ) method is reproduced. Function ߱ሺ‫ݔ‬ሻ is usually built in such way that it takes a unit vale in the vicinity of the point ‫ ܫ‬typically called star node where the function (or its derivatives) is computed and vanishes outside a region πூ surrounding the point. The región πூ can be used to define the number of sampling points ݊ in the interpolation region. In this work, the normalized Gaussian weight function ߱ሺ‫ݔ‬ሻ is used, thus ݊ ൒ ݉ is always required. Several possibilities for selecting the weighting function ߱ሺ‫ݔ‬ሻ can be found in [12,13,14,15].

Minimization of eq. (6) with respect to ߙ gives ߙ ൌ ۱ ିଵ ‫ܝ‬୦ ǡ ۱ ିଵ ൌ ‫ିۯ‬ଵ ۰

(7)

where ‫ ܣ‬ൌ σ௡௝ୀଵ ߱ሺ‫ݔ‬௝ ሻ࢖ሺ‫ݔ‬௝ ሻ࢖் ሺ‫ݔ‬௝ ሻ ‫ ܤ‬ൌ ߱ሺ‫ݔ‬ଵ ሻ࢖ሺ‫ݔ‬ଵ ሻǡ ߱ሺ‫ݔ‬ଶ ሻ࢖ሺ‫ݔ‬ଶ ሻǡ ǥ ǡ ߱ሺ‫ݔ‬௡ ሻ࢖ሺ‫ݔ‬௡ ሻ

(8)

The final approximation is obtained by substituting ߙ from eq. (7) into eq. (1) giving ‫ݑ‬ොሺ‫ݔ‬ሻ ൌ ࢖் ۱ ିଵ ࢛௛ ൌ Ȱ் ࢛௛ ൌ σ௡௝ୀଵ ߶௝ூ ‫ݑ‬௝௛ where ିଵ ் ିଵ ߶௝ூ ሺ‫ݔ‬ሻ ൌ σ௠ ூୀଵ ‫݌‬ூ ሺ‫ݔ‬ሻூ௝ ൌ ࢖ ሺ‫ݔ‬ሻ۱ are the shape functions.

(9) (10)

268


It must be noted that due least square approximation‫ݑ‬ሺ‫ݔ‬௝ ሻ ؄ ‫ݑ‬ොሺ‫ݔ‬௝ ሻ ് ‫ݑ‬௝௛ . That is, the local values of the approximating function do not fit the nodal unknown values. The WLS approximation described above depends strongly on the shape and the way in which the weighting function is applied. The simplest way is to define a fixed unction ߱ሺ‫ݔ‬ሻ for each one of the πூ interpolation subdomains [16]. Let ߱ሺ‫ݔ‬ሻ be a wighting function satisfying ߱௜ ሺ‫ݔ‬୧ ሻ ൌ ͳ ߱௜ ሺ‫ݔ‬ሻ ് Ͳ‫ א ݔ‬πூ ߱௜ ሺ‫ݔ‬ሻ ൌ Ͳ‫ ב ݔ‬πூ

(11)

The Gaussian weight function used in this work is given by ୣ୶୮൫ିௗೕ Τ௖൯ିୣ୶୮ሺିሺ௥ Τ௖ሻሻ

߱ூ ൫‫ݔ‬௝ ൯ ൌ ൝


ǡ ݂݅݀௝ ൑ Ͳ

Ͳǡ ݂݅݀௝ ൐ Ͳ

(12)

where ݀௝ ൌ‫ݔ צ‬௝ െ ‫ݔ‬ூ ‫צ‬ǡ ‫ ݎ‬ൌ ‫ݍ‬൛‫ݔ‬௝ ‫ א‬πூ ห ‫ݔ צ‬௝ െ ‫ݔ‬ூ ‫צ‬ሽ and ܿ ൌ ߚ‫ݎ‬. The support of this function is isotropic, circular and spherical in two and three-spatial, respectively. A detailed description of the effects of the parameters ‫ ݍ‬and ߚ on the numerical approximation and some guidelines for their setting have been presented in [15,16]. Note that according to eq (1), the approximate function ‫ݑ‬ොሺ‫ݔ‬ሻ is defined in each interpolation domain πூ . In fact, different interpolation sub-domains can yield different shape function ߶௝ூ . As a consequence of this, a point belonging to two or more overlapping clouds has different values of the shape functions which means that ߶௝௜ ് ߶௝௞ . The interpolation is now multivalued within πூ and, therefore to the approximation be useful, a decision must be made in order to limit the choice to a single value. Indeed, the approximate function ‫ݑ‬ොሺ‫ݔ‬ሻ will be typically used to provide the value of the unknown function ‫ݑ‬ሺ‫ݔ‬ሻ and its derivatives only in specific regions within each interpolation sub-domain. For instance by using point collocation the validity of the interpolation is limited to a single point ‫ݔ‬ூ . Isotropic Damage Model In order to characterize the isotropic damage model, the concept of effective stress ߪത is introduced [1]. In one dimension the following relation is defined ߪ ൌ ሺͳ െ ݀ሻߪ ഥ (13) where ݀ is the scalar damage parameter, which ranges from 0 to 1. The effective stress ߪത and the strain ߝ are related by the Hook’s law ߪത ൌ ‫ߝܧ‬ (14) ߝ ൌ ‫ݑ׏‬ሺ‫ݔ‬ǡ ‫ݐ‬ሻ (15) where ‫ ܧ‬is the elastic modulus of the material and ‫ ׏‬denotes the gradient operator. Substituting eq(14) into eq(13) yields ߪ ൌ ሺͳ െ ݀ሻ‫ Ͳߝܧ‬൑ ݀ ൑ ͳ (16) The damage evolution ݀ሺ‫ݔ‬ǡ ‫ݐ‬ሻ ൌ ݀ሺܻሺ‫ݔ‬ǡ ‫ݐ‬ሻሻ (17) depends on an internal variable, the inelastic strain ߝ ௜௡௘௟ , which is part of the proposed model ܻሺ‫ݔ‬ǡ ‫ݐ‬ሻ ൌ ߝ ௜௡௘௟ ሺ‫ݔ‬ǡ ‫ݐ‬ሻ (18) Fictitious Crack Model [5] This model defines its behavior as a function of the principal stresses. The material has an elastic behavior before reaches the rupture limit, which is defined by the Rankine’s yield criterion. Once reached this limit, a failure occurs in the material, and it adopts an orthotropic solid structure. In the cracked zone is considered a loss of stiffness in the normal direction of the crack, directly related with the displacements as follows ீ೑ ఙ (19) ‫ ݑ‬ൌ ݈ െ ʹ మ ሺߪ െ ݂‫ݐ‬ሻ ா

௙೟


269

where ݈ is a geometric length, ‫ܩ‬௙ is the fracture energy and ݂௧ is the yield stress of the material (both material properties). In the same way, this relation can be formulated for the total strain [6] ߝ ൌ ߝ ௘௟ ൅ ߝ ௜௡௘௟ (20) ଶீ೑ ఙ (21) ߝ ൌ ൅ మ ሺ݂‫ ݐ‬െ ߪሻ ா

௟௙೟

The inelastic strain is defined as follow ଶீ೑ ߝ ௜௡௘௟ ൌ మ ሺ݂‫ ݐ‬െ ߪሻ

(22)

௟௙೟

and the stress as a function of the total strain ߪ ൌ ߝ‫ܥ‬ଵ െ ‫ܥ‬ଶ where ଶீ೑ ଵ ‫ܥ‬ଶ ൌ ‫݂ܪ‬௧ ‫ܥ‬ଵ and ‫ܪ‬ൌ మ ‫ܥ‬ଵ ൌ భ ; ಶ

௟௙೟

ିு

Now, the inelastic strain ߝ ௜௡௘௟ as a function of the total strain ߝ ஼ ߝ ௜௡௘௟ ൌ ߝ ቀͳ െ భ ቁ ൅ ‫ܥ‬ଶ Ȁ‫ܧ‬ ா The crack evolution is governed by the yielding function as follows ఌ ೔೙೐೗

ߪ ൌ ݂௧ െ ு Substituting eq(25) into eq(13) ߝ ௜௡௘௟ ሺͳ െ ݀ሻߪ ഥ ൌ ݂௧ െ ‫ܪ‬ then the damage variable ݀ can be related with the inelastic strain computed from eq(24) ݀ ൌͳെ

(23)

൬௙೟ ି

ഄ೔೙೐೗ ൰ ಹ

ഥ ఙ

(24) (25)

(26)

Numerical implementation Consider the following system of differential equations which governs the behavior of a solid ‫ ߪ׏‬൅ ߩܾ ൌ Ͳ‫ א ݔ׊‬π (27) ߪ ȉ ݊ ൌ ‫ݐ‬ҧ‫ א ݔ׊‬Ȟ୲ (28) ‫ ݑ‬ൌ ‫ݑ‬௣ ‫ א ݔ׊‬Ȟ୳ (29) Using the point collocation scheme [11], is obtained a discrete system of equations [15] ‫ܮ‬୘ ߪ ൅ ߩܾ ൌ Ͳ‫ א ݔ׊‬π (30) ܰ ் ߪ ൌ ‫ݐ‬ҧ‫ א ݔ׊‬Ȟ୲ (31) (32) ‫ ݑ‬ൌ ‫ݑ‬௣ ‫ א ݔ׊‬Ȟ୳ where ‫ ܮ‬is a operator that defines the differential equation and ܰ contains the normal direction on the contour. An equivalent system of equations in terms of displacements [17] and using the final FPM approximation defined in eq(9), is obtained ሺ‫ ܫ‬െ ݀ሻ‫ ݑ߶ܮܥ ்ܮ‬௛ ൅ ሺ‫ ܫ‬െ ݀ሻሺ‫߶ܮ‬ሻ் ‫ ݑ߶ܮܥ‬௛ െ ሺ‫݀߶ܮ‬ሻ் ‫ ݑ߶ܮܥ‬௛ ൌ െߩܾ‫ א ݔ׊‬π (33) (34) ܰ ் ‫ ݑ߶ܮܥ‬௛ ൌ ‫ݐ‬ҧ‫ א ݔ׊‬Ȟ୲ (35) ߶‫ݑ‬௛ ൌ ‫ݑ‬௣ ‫ א ݔ׊‬Ȟ୳ With its compact form (36) ‫ܭ‬ௗூ ‫ݑ‬௛ ൌ ݂ூ Ǣ ‫ ܫ‬ൌ ͳǡ ǥ Ǥ ݊ where ‫ܭ‬ௗ is the stiffness matrix in FPM and ݂ூ contains the equilibrium and boundary conditions. This equation is computed using an incremental iterative Newton-Raphson scheme. Numerical results One-dimensional damage example The central tenth of the road (Figure 1) is weakened (10% reduction in Young’s modulus), the geometrical and material parameters are summarized in table 1.

270


Length of the rod, ‫ܮ‬ Idem of weakened part Young’s modulus, ‫ܧ‬ Tensile strength, ݂௧ Fracture Energy, ‫ܩ‬௙

100 [mm] 10 [mm] 20000 [Mpa] 2 [Mpa] 1.25 [N/mm] Tabla 1

Figure 1 Uniaxial tensile test

The results are compared with [17] and are shown below (Figure (2) (3)), which were computed with 320 points and clouds with 5 points:

Figure 1 Stress-Strain curve of a totally damaged point

The figure(2) show the behavior of the central point in a stress-strain curve and the figure(3) the behavior of the same point in a stress-displacement curve.


Figure 2 Stress-displacement curve of a totally damaged point

Figure 3 a) semi-damaged point, b) undamaged point

In the figure(4) are shown the behavior of a semi-damaged (left) and a undamaged (right) point in a stress-strain diagram. It can be seen that the undamaged point has an elastic response during loading-unloading, while the semi-damaged point has an elastic and damaged behavior during load, to finally unload with an elastic response. These results are according to [17]. In the figure(5a) is shown the damage, and the inelastic strain is shown in the figure(5b).

271

272


Figure 4 a) Damage Evolution, b) Inelastic strain evolution

Conclusion. Has been proposed a model for isotropic damage, which makes use of the classical cohesive crack model to simulate the inelastic behavior of the quasi-brittle materials. The Rankine criterion initiated the crack at the node and the non-linearity behavior was computed using an incremental iterative Newton-Raphson scheme. The development of a meshless Finite Point Method and the implementation of the cohesive crack model provide an independence of a mesh and a characteristic length in the results. The Finite Point-calculations show that the isotropic damage model with the cohesive crack model can properly reproduce the literature stress-displacement and stress-strain diagrams in quasi-brittle materials during tensile load in one dimensional analysis. The proposed model can be extended to two and three dimensions and an application of an algorithm tangent operator by Perturbation Method to ensure convergence of the iterative method are applications that will be investigated in our future research. Acknowledgments. The authors wish to acknowledgment the support by the CONICYT-Chile (FONDECYT 11100253) and DGIP-UTFSM (Internal Project 251150). References [1] Oller S. Fractura mecánica: Un enfoque global. Ediciones UPC, Universidad Politécnica de Cataluña, Barcelona, España, (2001). [2] Maugin, G. A. The thermodynamics of plasticity and fracture. Cambridge University Press, (1992). [3] Kachanov L. M. Time of rupture process under creep conditions. Inzvestia Akademii Nauk. Otd Tech Nauk. 8; 26-31, (1958). [4] Rashid, Y. R. Analysis of prestressed concret pressure vessels. Nuclear Engineering and Design – Vol.7 – Nr.4, (1968). [5] Hillerborg A., Modeer M., Petersson P. Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement and Concrete Research – Vol. 6 – Nr.6, 773-782, (1976). [6] R. de Borst. Some recent issues in computational failure mechanics. International Journal for Numerical Methods in Engineering, 52:63-95, (2001). [7] Oñate E., Perazzo F., and J.Miquel. A finite point method for elasticity problems. Computer and Structures, 79:2151–2163, (2001).


[8] Perazzo F., Oller S., Miquel J., and Oñate E. Avances en el método de puntos finitos para la mecánica de sólidos. Revista Internacional de Métodos Numéricos en Ingeniería, 22:153–168, (2006). [9] Pérez-Pozo L. and Perazzo F. Non-linear material behaviour analysis using meshless finite point method. In 2nd ECCOMAS Thematic Conference on Meshless Methods, 251–268. Porto, Portugal, (2007). [10] Pérez-Pozo L., Perazzo F., and Angulo A. A meshless FPM model for solving nonlinear material problems with proportional loading based on deformation theory. Advances in Engineering Software, 40:1148–1154, (2009). [11] Zienkiewicz O. & Taylor R. El método de los elementos finitos, vol. 1. Centro internacional de métodos numéricos en ingeniería, Barcelona España, (2000). [12] Oñate E, Idelsohn S, Zienkiewicz O, Taylor R. and Sacco C. A stabilized finite point method for analysis of fluid mechanics problems. Comput Methods Appl Mech Eng, 139:315-346, (1996). [13] Oñate E, Idelsohn S, Zienkiewicz O. and Taylor R. A finite point methods in computational mechanics, application to convective transport and fluid flow. Int J Numer Methods Eng, 39:3839-3866, (1996). [14] Taylor R, Idelsohn S, Zienkiewics O. and Oñate E. Moving least square approximations for solution of differential equations. CIMNE research report, p.74, (1995). [15] Perazzo F. Una metodología numérica sin malla para la resolución de las ecuaciones de elasticidad mediante el método de puntos finitos. Univeritat Politécnica de Cataluña, Barcelona España. Tesis doctoral, (2002). [16] Pérez L. Simulación numérica del comportamiento no-lineal de materiales utilizando aproximaciones elásticas y el método sin malla de puntos finitos. Universidad Técnica Federico Santa María, Valparaíso, Chile. Tesis doctoral, (2008). [17] Pérez L., Chacana F. and Quelín J. Regularización de la energía de fractura para el análisis de daño isotrópico mediante el método sin malla de puntos finitos. Mecánica Computacional, Vol XXX: 755-772, (2011).

273

274


Vibrations of a Rigid Circular Foundation Embedded on a Transversely Isotropic Multilayered Soil Josué Labaki1, Euclides Mesquita2 and Nimal Rajapakse3 1,2

3

Department of Computational Mechanics, School of Mechanical Engineering, State University of Campinas. 200 Mendeleyev St., 13083-970, Campinas-SP, Brazil.

Faculty of Applied Sciences, Simon Fraser University. 8888 University Road, V5A 1S6, Burnaby-BC, Canada. 1

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

Keywords: circular foundations, multilayered media, transverse isotropy.

Abstract. This paper examines the steady-state response of a rigid circular plate embedded in a multilayered viscoelastic, transversely isotropic, three-dimensional medium. A boundary-value problem corresponding to the case of distributed vertical ring loads in an arbitrary interface inside the multilayered medium is introduced. The model of embedded disc is formulated in terms of a discretized integral equation, which couples the rigid displacements of the disc with the tractions acting over its surface. The results in this paper are presented in the form of dynamic compliance of the disc. Introduction The study of interaction between rigid foundations with transversely isotropic geomaterials has important practical applications in earthquake engineering and seismology. The case of a rigid foundation embedded in the interface of different transversely isotropic materials is of particular interest to the study of foundations and anchors. The present study is concerned with the steady-state vertical vibratory response of a rigid annular or solid circular disc embedded within viscoelastic, transversely isotropic, heterogeneous multilayered soil. Figure 1 illustrates the geometry of the system considered, for an example of two layers plus an underlying half-space.

Figure 1. A rigid circular foundation at the interface of two different materials. The Cauchy-Navier equations, which describe the behavior of the aforementioned medium, are solved by using Hankel integral transforms. Hankel transforms are the most suitable transforms in the


275

present case because cylindrical coordinates are used. The viscoelastic behavior of the medium is introduced by Christensen’s elastic-viscoelastic correspondence principle. A boundary-value problem corresponding to the case of distributed vertical ring loads in an arbitrary interface inside the multilayered medium is introduced. The model of embedded disc is formulated in terms of a discretized integral equation, which couples the rigid displacements of the disc with the tractions acting over its surface. The disc is discretized by a number of annular discs, and over each of these elements the traction is considered to be constant. The system of the resulting discretized integral equations is solved numerically, which gives the tractions over each elementary disc. The weighted summation of these results by the respective area of the elements gives the total force applied over the plate corresponding to a unitary rigid displacement. The results in this paper are presented in the form of dynamic compliance of the plate for different compositions of materials of the multilayered medium. The present solutions contribute to the study of the dynamic response of foundations and anchors embedded in the soil. Three-Dimensional Transversely Isotropic Full-Space Consider a three-dimensional transversely isotropic full-space with density ρ and elastic constants cij, with a cylindrical coordinate system O(r, θ, z) positioned in such a way that its z axis is orthogonal to the material’s plane of symmetry. Rajapakse and Wang [1993] derived a solution for the motion of such a medium. The axisymmetric vertical displacements and stresses are given by: ∞

uim = ³ u*imλdλ, i = r, θ, z

(1)

∞ σijm = σ*ijmλdλ, 0

i, j = r, θ, z

(2)

u*rm = a1 A m + a1 Bm + a 2 C m + a 2 D m + a 3 E m + a 3 Fm

(3)

u*θm

(4)

0

³

In which:

( ) * u zm = − ( a 7 A m − a 7 Bm + a 8 C m − a 8 D m ) σ*rzm = −c 44 ( b51 A m − b51 Bm + b52 C m − b52 D m + b53 E m − b53 Fm ) σ*θzm = c 44 ( b 41 A m − b 41 Bm + b 42 Cm − b 42 D m + b 43 E m − b 43 Fm ) σ*zzm = c 44 ( b 21 A m + b 21 Bm + b 22 C m + b 22 D m ) = − a 4 A m + a 4 Bm + a 5 C m + a 5 D m + a 6 E m + a 6 Fm

Am = A me−δξ1z , Bm = Bme+δξ1z , Cm = Cme−δξ2 z Dm = Dm e

+δξ2 z

, E m = Eme

−δξ3 z

and Fm = Fme

+δξ3 z

(5) (6) (7) (8) (9) (10)

The index m corresponds to the Fourier expansion with respect to the ș-coordinate. In the case of axisymmetric vertical loading, m=0 [Rajapakse and Wang, 1993; Wang, 1992]. The other parameters presented in Eqs. (3) to (8) depend on the material properties and are described in details in the Appendix. The coefficients Am, Bm, Cm, Dm, Em and Fm are arbitrary functions that can be determined from the boundary and continuity conditions of a given problem.

276


Multilayered Medium Consider the multilayered soil model shown in Fig. 2. Each of the N layers and the underlying half-space is made of a homogeneous transversely isotropic elastic material, the behavior of which is described by Eqs. 1 and 2. The material constants, mass density and thickness of the nth layer are denoted by c(n)ij, ρ(n) and hn, respectively.

Figure 2 Geometry and notation of a multilayered soil system. Let u*(n)im1 denote the ith displacement component at the top surface of the nth layer (z=zn), and denote the ith displacement component at the bottom surface of the nth layer (z=zn+1). Analogously, let σ*(n)ijm1 denote the ijth stress component at the top surface of the nth layer (z=zn), and σ*(n)ijm2 denote the ijth stress component at the bottom surface of the nth layer (z=zn+1). The * superscript indicate a transformed domain. The three displacement components from Eqs. 3 to 5 and stresses from Eqs. 6 to 8 can be written in the following matrix form: u*(n)im2

(n) (n) u*(n) m = Gm a m

(11)

(n) (n) σ*(n) m = Fm a m

(12)

where *(n) u*(n) m = u rm1

u*(n) θm1

u*(n) zm1

u*(n) rm2

u*(n) θm2

*(n) *(n) *(n) *(n) σ*(n) m = −σ rzm1 −σθzm1 −σ zzm1 σrzm2

(n) a (n) m = Am

B(n) m

C(n) m

D(n) m

E(n) m

u*(n) zm2

T

(13)

σ*(n) θzm2

(n) Fm

σ*(n) zzm2

T

(14)

T

(15)

The traction vector, on the other hand, is expressed by: *(n) p*(n) m = p rm1

p*(n) θm1

p*(n) zm1

p*(n) rm2

p*(n) θm2

p*(n) zm2

T

= σ*(n) m

(16)


277

where p*(n)im1 denote the traction acting at the top surface of the nth layer in the ith direction, and p*(n)im2 denote the traction acting at the bottom surface of the nth layer in the ith direction. Notice that in the underlying half-space (N+1), only the terms A(N+1)m, C(N+1)m and E(N+1)m are involved in the formulation, in order to satisfy Sommerfeld’s radiation condition [Sommerfeld, 1949]. Since the vectors a(n)m are common to the expressions of displacements and stresses, Eqs. 11 and 12 can be combined into:

( )

(n) (n) (n) (n) σ*(n) m = Fm a m = Fm G m

−1 *(n) *(n) u m = K(n) m u m ; n=1,2,…,N,N+1 (half-space)

(17)

In Eq. 17, the matrix K(n)m is the stiffness matrix of the layer n. It depends on the material properties of the layer and its thickness, on the frequency of excitation and on the Hankel space variable λ. The expression of K(n)m is rather long and has to be determined numerically. Let ℘*nim denote the Hankel transform of an external concentrated or distributed load applied at th the n interface of two layers, in the direction of i (i=r,θ,z). ℘*nim=℘*nim(ω) is a time-harmonic load with circular frequency ω. The following equation describes the traction discontinuity at that interface: *(n −1) *(n) *(n −1) *(n) ℘*n im = pim2 + pim1 = σim2 − σim1 ; i=r,θ,z

(18)

Equation 18 is applied together with Eq. 17 for all the layers n=1,2,…,N,N+1 to form the following global stiffness matrix of the multilayered medium:

℘m = K m u*m

(19)

In Eq. 19, ℘m=℘m(λ) is the vector of external loads applied at the layer interfaces, given by Eq. 20; u*m=u*m(λ) is the vector of resulting displacements of points of the interfaces, given by Eq. 21, and Km=Km(λ) is the global stiffness matrix of the medium, given by Eq. 22. All these terms are in the Hankel transformed domain, and depend on the Hankel space parameter λ. N +1 N +1 ℘m = ℘1rm ℘1θm ℘1zm ! ℘rm ℘θNm+1 ℘zm

T

T u*m = u*rm ( r, z1 ) u*θm ( r, z1 ) u*zm ( r, z1 ) ! u*rm ( r, z N +1 ) u*θm ( r, z N +1 ) u*zm ( r, z N +1 )

(20) (21)

(22) When Eq. 19 is assembled, the continuity of displacements at the interfaces, which states that u*(n−1)im2=u*(n)im1, is implicitly guaranteed. The solution of displacements from Eq. 19 must be

278


integrated along λ as described in Eq. 1 to obtain the displacements at the layer interfaces in the physical domain.

Rigid Circular Foundation Consider time-harmonic vertical excitation, of circular frequency ω, of a rigid massless disc of zero thickness and radius a embedded in an elastic interface as shown in Fig. 1. The vertical displacements of the disc can be related to the interface contact traction through the following integral equation: a

³0 u zz ( r, z = 0, ω) t z ( r, z = 0, ω) dr = ∆z ( r, z = 0, ω)

(23)

In Eq. 23, tz denotes the component of interface traction jump under the disc in the vertical direction; ∆z denotes the vertical displacement of the disc and uzz denotes the vertical displacement, as described in Eq. 1, due to a unit vertical ring load. A numerical solution to Eq. 23 is obtained by discretizing the disc into M concentric annular discs of inner and outer radii s1k and s2k (k=1,M). The tractions tz acting on each disc element are assumed to be constant. The discretized version of Eq. 23 is given by: M

¦ u zz ( ri ,s1k , s2k , ω) t z ( rk , ω) = ∆ z ( ri ) ; i=1,M

(24)

k =1

In Eq. 24, uzz(ri, s1k, s2k, ω) is the vertical displacement of a rigid annular disc element, represented by the radius ri=(s1i+s2i)/2 (i=1,M), due to a unit vertical load applied in an annular disc element k of inner and outer radii s1k and s2k (k=1,M). The derivation of this displacement component is described in the previous section. For every disc element k, this term is multiplied by the unknown constant vertical traction jump tz(rk) acting on it. This yields the actual vertical displacement ∆z(ri) that each ring i experiences. Equation 24, which presents M unknown tractions tz(rk) is repeated for all discs i=1,M to allow a set of M equations. The vertical displacement ∆z(ri) of each disc is set to a unit value ∆0z=1. This set of equations is solved for the tractions acting on each disc element. The total vertical force acting on the disc is given by Eq. 25, in which Ak is the area of the disc element k. Fz ( ω) =

2 2 − s1k ¦ A k t z ( rk , ω) = ¦ π ( s2k ) t z ( rk , ω) M

M

k =1

k =1

(24)

The dynamic vertical compliance of the system comprising the buried rigid plate and its surrounding medium, for each frequency ω, is defined by Eq. 25, in which E(m) is the Young’s modulus of one of the two materials of the interface. m C Z ( ω) = ∆ 0z aE ( ) / Fz ( ω)

(25)

Numerical Results The presented solution is used to study the dynamic vertical compliance of a rigid plate embedded in different configurations of multilayered media.


279

The normalized vertical dynamic compliance of a rigid plate at a depth H/a=2 within an isotropic half-space (E=2.5, ν=0.25) is shown in Fig. 3. These results match exactly the ones presented by Pak and Gobert [1992]. The compliance shown in Fig. 3 is normalized by the vertical static compliance of a plate resting on the surface of the half-space, C0Z(ω=0), whose closed-form solution was derived by Pak and Gobert [1992]. 0.8

1

0.7

0.8 -Im[C (ω )/C0(0)]

Z

Z

Re[C (ω )/C0(0)]

0.6

0.6

0.4

Z

Z

0.4

0.5

0.2

0.3 0.2

0 -0.2 0

0.1

1

2 Frequency ω

3

4

0 0

1

2

3

4

Frequency ω

Figure 3 Normalized vertical dynamic compliance of a rigid plate embedded at a depth H/a=2 within an isotropic half-space. Three models of multilayered soils composed of transversely isotropic layers are considered. The models are shown in Table 1. In all three configurations the plate is placed on the surface of the underlying half-space. The material properties used in each case are shown in Table 2. In the results of Fig. 4, the dynamic compliance is normalized by the static compliance of the plate in the same configuration.

Case A B C

Table 1. Multilayered soil configurations used in the present study. Material and thickness of Material and thickness of Material and thickness of layer 1 layer 2 layer 3 Beryl rock, h1=1.0 Beryl rock, h3=∞ Silty Clay, h1=0.5 Layered soil, h2=0.5 Beryl rock, h3=∞ Silty Clay, h1=0.3 Layered soil, h2=0.7 Beryl rock, h3=∞

Figure 4 Three different layered soil configurations with an embedded rigid plate. Table 2. Material constants of some transversely isotropic materials, with c’ij=cij/c44 (Wang, 1992). Material c’11 c’12 c’13 c’33 c44 (104 MN/m2) Beryl Rock 4.13 1.47 1.01 3.62 1.00 Silty Clay 2.11 0.43 0.47 2.58 2.70 Layered Soil 4.46 1.56 1.24 3.26 1.40 Ice (257K) 4.22 2.03 1.62 4.53 0.32

280

Eds: P Prochazka and M H Aliabadi 1 Case A Case B Case C

0.2

Z

Z

0.4

0.4

Z

-Im[C (ω )/C (0)]

0.5

0.6

0.3 0.2 0.1

0 0

Case A Case B Case C

0.6

Z

Re[C (ω )/C (0)]

0.8

1

2

3

0 0

4

1

Frequency ω

2

3

4

Frequency ω

Figure 4 Normalized vertical compliance of a rigid plate embedded in different configurations of transversely isotropic multilayered soil. In the second example, the effect of an additional layer over a foundation is studied. In the first case, a rigid foundation rests on the surface of a half-space of Beryl Rock (Fig. 5). An ice sheet of thickness a forms on top of the foundation. The effect of this additional layer on the vertical compliance of the foundation in shown in Fig. 6. The material properties of the transversely isotropic ice are given in Table 2.

Figure 5 Configuration of a plate resting on the surface of a rock and after the deposit of a thick layer of ice. 1 Without ice sheet With ice sheet

0 -0.2 -0.4 0

Z

0.5 0.4

Z

0.2

-Im[C (ω )/C (0)]

Z

0.4

Without ice sheet With ice sheet

0.6

0.6

Z

Re[C (ω )/C (0)]

0.8

0.3 0.2 0.1

1

2 Frequency ω

3

4

0 0

1

2

3

4

Frequency ω

Figure 4 Normalized vertical compliance of a rigid plate embedded in different configurations of transversely isotropic multilayered soil.

Concluding remarks This work has presented a study of vertical vibrations of a solid rigid disc embedded in a multilayered transversely isotropic medium. The solutions were validated with previous results from the literature and new results for the behavior of the rigid disc on layered soil models were presented. These solutions contribute to the study of the dynamic response of foundations and anchors embedded in the soil.


281

References [1] R. K. N. D. Rajapakse and Y. Wang. Y. “Green’s Function for Transversely Isotropic Elastic Half Space”. Journal of Engineering Mechanics, 119, n 9 (1993). [2] R. Y. S. Pak and A. T. Gobert; “Forced Vertical Vibration of Rigid Discs with Arbitrary Embedment”. J. Eng. Mech. 117, issue 11, 2527 (1991). [3] Y. Wang. Fundamental Solutions for Multi-Layered Transversely Isotropic Elastic Media and Boundary Element Applications. PhD Thesis (1992). [4]

A. Sommerfeld. Partial Differential Equations, Academic Press, New York (1949).

Appendix In Eqs. (3) to (8), the index m corresponds to the Fourier expansion with respect to the școordinate. In the case of axisymmetric vertical loading, m=0 [Rajapakse and Wang, 1993; Wang, 1992]. The other parameters depend on the material properties and are given by (for i=1,2):

ςϑi δζ ª( m − 1) J m −1 ( δζr ) − ( m + 1) J m +1 ( δζr ) º¼ − ªδ2ζ 2βϑi − ( κ − 1) δ2ξi2 º J m ( δζr ) ¬ ¼ 2r ¬ 2 2 2 2 º ª b 2i = αδ ξi − ( κ − 1) δ ζ ϑi J m ( δζr ) ¬ ¼ a7 a8 = = J m ( δζ r ) δξ1 δξ 2

(20)

a b a b 4i δζ a3 = 4 = 5 = = 53 = ª J m −1 ( δζr ) + J m +1 ( δζr ) º¼ ϑ1 ϑ2 (1 + ϑi ) δξi δξ3 2 ¬

(22)

b5i b δζ a a = 43 = ª J m −1 ( δζr ) − J m +1 ( δζr ) º¼ a6 = 1 = 2 = ϑ1 ϑ2 (1 + ϑi ) δξi δξ3 2 ¬

(23)

b6i = −

(19)

(21)

In which,

ϑ1,2 =

2 αξ1,2 − ζ2 + 1

κζ 2 1 ξ1,2 ( ζ ) = γζ 2 − 1 − α ± Φ 2α

(

(

Φ ( ζ ) = γζ 2 − 1 − α

)

2

(

(24)

)

1 2

(25)

)

− 4α βζ 4 − βζ 2 − ζ 2 + 1

(26)

ξ3 = ± ςζ 2 − 1

(27)

c c +c ρa 2 2 c c −c ω and γ = 1 + αβ − κ 2 α = 33 , β = 11 , κ = 13 44 , ς = 11 12 , δ = c44 c 44 2c 44 c 44 c 44

(28)

In Eqs. (19) to (23), Jm(δζr) denote Bessel functions of the first kind and order m.

282


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295

On a strategy to implement a fast and expedite boundary element method Ney Augusto Dumont and Carlos Andrés Aguilar Marón Department of Civil Engineering Pontifical Catholic University of Rio de Janeiro, PUC-Rio 22451-900, Brazil. e-mail: [email protected] Keywords: Boundary elements, variational methods, meshless methods, fast-multipole approach, largescale problems

Abstract. The present developments result from the combination of the variationally-based, hybrid boundary element method with a consistent formulation of the conventional, collocation boundary element method. The procedure is simple to implement and turns out to be computationally faster than other similar numerical methods – and almost as accurate – for the analysis of large-scale, two-dimensional and three-dimensional problems of potential and elasticity of general shape and topology, also applicable to time-dependent problems. Both the double-layer and the single-layer potential matrices of the collocation boundary element method are obtained in an expedite way that circumvents almost any numerical integration - except for a few regular integrals. Since the resultant matrices do not differ in nature from the ones of the collocation boundary element method, the developments are suited for an iterative solution in terms of a GMRES algorithm, for generally mixed boundary conditions. The proposed strategy - and core of the present contribution - aims at the solution of very large problems by using a semi-hierarchical procedure, for successively refined discretization meshes that are intertwined in the iterative solution process. Although it was not the initial intention, it comes out that the fast-multipole approach plays a decisive role in the complete formulation in order to speed up the numerical solution of a given problem no matter how convoluted its geometry might be. The numerical implementation is in its final stage and some numerical examples are being prepared to validate the formulation as well as to investigate its application possibilities. These numerical examples shall be presented in the extended version of the paper. 1.

Introduction

The collocation boundary element method (CBEM), whenever applicable, is a simple, powerful numerical analysis tool [1]. The present contribution is an attempt to show that the CBEM can be still more efficient and powerful – and still easier to implement computationally especially when dealing with very large problems. (A not lesser contribution is the demonstration that simplicity can be achieved without resorting to exotic concepts such as node displacements from corner points or regularizations.) Some precursory works have already been published on the subject [8, 11, 15]. The present contribution is a natural sequel of Refs. [12] and [15]. Besides the presentation of some slight conceptual improvements, it focuses on the possibilities of application of the method to very large problems. The next Section introduces the basic elasticity problem. It actually unavoidably in part repeats and in part condenses Sections 2, 3 and 4 of Ref. [12]. After a brief outline of the resultant numerical and linear-algebra problem, the basic ideas affected by the expedite formulation are presented, for a solution in terms of a hierarchical approach in connection with the fast-multipole method. 2.

Problem Formulation

An elastic body is submitted to body forces bi in the domain Ω and traction forces ti on part Γσ of the boundary. Displacements ui are known on the complementary part Γu of Γ. The task is to find an adequate approximation of the stress field that satisfies equilibrium in the domain, σ ji, j + bi = 0 in Ω

(1)

also satisfying the boundary equilibrium and compatibility equations, σ ji n j = ti

along Γσ ,

u i = ui

on

Γu

(2)

296


where n j is the outward unit normal to Γ. Indices i, j, (also k, l) may assume values 1, 2 or 3, as they refer to the coordinate directions x, y or z, respectively, for a general 3D analysis. Summation is indicated by repeated indices. Particularization to 2D analysis as well as to potential problems is straightforward. Stress and displacement assumptions. Three independent fields are used in the following developments. The displacement field is explicitly approximated along the boundary by udi , where ( )d means displacement assumption, in terms of polynomial functions uim with compact support and nodal displacement parameters d d = [dm ] ∈ Rn , for nd displacement degrees of freedom of the discretized model. An independent stress field p s s σi j , where ( ) stands for stress assumption, is given in the domain in terms of some particular solution σi j plus ∗ ∗ ∗ ∗ a series of fundamental solutions σi j m with global support, multiplied by force parameters p = [pm ] ∈ Rn applied at the same boundary nodal points m to which the nodal displacements dm are attached (n∗ = nd ). Displacements uis are obtained from σisj . Then, udi = uim dm

σisj

=

σ∗i jm

p∗m

+

⇒

on

p σi j

Γ such that

such that

uis = u∗im p∗m +

udi = u¯i ∗ σ jim, j = 0 and p ui + uris C sm p∗m

u∗im

on Γu p

and

σ ji, j = bi

(3)

in Ω

(4)

in Ω

(5)

σ∗i jm .

are displacement fundamental solutions corresponding to Rigid body motion is included in where r ∗ terms of functions uris multiplied by in principle arbitrary constants C sm ∈ Rn ×n , where nr is the number of rigid body displacements (r.b.d.) of the discretized problem [6, 9]. The fundamental solutions σ∗i jm are used as weight functions in the CBEM. In the variational BEMs and in the EBEM, in particular, they represent domain interpolation functions. The third independent field is used to approximate traction forces along the boundary by tit , where ( )t means traction assumption, as required in the conventional boundary element method, given as [7] tit =

|J|(at ) ui t ≡ ti t |J|

(6) t

where ui are polynomial interpolation functions with compact support and t = [t ] ∈ Rn are traction-force parameters. The index i refers to the coordinate directions whereas the index refers to any of the nt tractionforce degrees of freedom of the problem (thus denoting both location and orientation), for nodes adequately distributed along boundary segments of Γ. The interpolation functions ui have the same properties of uin , as presented in eq (3). Equation (6) holds as ti = ti t along Γσ , in particular, according to eq (2). In the above equation, |J|(at ) is the value of the Jacobian of the global (x, y, z) to natural (ξ, η) coordinate transformation at the nodal point and the term |J|(at ) /|J| features a term in the denominator that cancels the Jacobian term of the infinitesimal boundary segment dΓ = |J|dξdη in the numerator of two integral expressions introduced in eqs (9) and (13). This not only improves the capacity of tit to represent the traction forces along curved boundary segments but also simplifies the numerical integration of the related terms. The numbers of degrees of freedom for traction forces nt and displacements nd are not necessarily the same, since more than one traction-force parameter are needed to represent tractions that are not single valued at the boundary surface, generally at nodes where adjacent boundary segments present different outward normals [7]. Then, it results that nt ≥ nd , as t in eq (6) are traction-force attributes on boundary segments, whereas uin in eq (3) are displacement attributes at nodal points. The fact that nt ≥ nd leads to some rectangular matrices – the same eqs (13) and (9) of the CBEM, which have been just referred to, plus a third one, introduced in eq (11). Boundary approximation of the particular solution. Although neither conceptually nor formally necessary, the following approximation may render all subsequent equations simpler and more elegant [6]. Given a p p sufficiently refined boundary mesh, the displacements ui and the traction forces ti related to an arbitrary particular solution of the non-homogeneous governing eq (1), whenever available, can be approximated accurately d t p p enough by nodal displacement parameters d p = [dn ] ∈ Rn and traction force parameters t p = [t ] ∈ Rn , respectively, in terms of the interpolation functions of eqs (3) and (6): p

p

ui ≈ uin dn ,

p

p

ti ≈ ti t

on Γ

(7)


297

It is assumed with the above equations that a particular solution for the domain forces bi in eq (1) is known in p p terms of displacements ui and stresses σi j . The means to obtain such particular solutions other than in close form are not discussed herein (see, for instance, Partridge et al [20]). The conventional, collocation boundary element method. The matrix equation of the CBEM [1] may be expressed as [7] (8) H d − dp = G t − tp where H = [Hmn ] ∈ Rn ×n is a kinematic transformation matrix [4, 6, 9] and G = [Gm ] ∈ Rn ×n is a flexibility-like matrix (that is in general rectangular, as proposed). The formal definition of these matrices is Hmn = σ∗jim n j uin dΓ , Gm = ti u∗im dΓ (9) d

d

d

Γ

t

Γ

where n j are the projections of the unit outward normal to the boundary. The double-layer and single-layer potential matrices Hmn and Gm comprise in their definition singular and improper integrals, respectively, when source (m) and field (either n or ) indexes refer to the same nodal points. The singular integrals can be always evaluated mathematically in correspondence to simple mechanical meanings [1]. A conceptual assessment of eq (9) is given in Reference [7]. Approximation of displacements and traction forces on the boundary. In the present formulation, stress and displacement results at internal points can be given directly by eqs (4) and (5) in terms of force parameters p∗m that would be evaluated after the solution of eq (8) (a simpler and faster approach is introduced in Section 3.1.). This kind of evaluation of results, which circumvents the computational intensive use of the Somigliana’s identity of the CBEM, is borrowed from the hybrid boundary element method – HBEM [5, 6, 7, 8, 9]. According to that, eqs (5) and (4) may be applied to the boundary nodes [6, 9]: U∗ p∗ = d(p∗ )

u.p.f. and r.b.d excluded

T∗ p∗ = t(p∗ )

u.p.f. excluded

(10) (11)

where u.p.f. and r.b.d. mean unbalanced point forces and rigid-body displacements, which cannot take part in the outlined linear transformations. The definition of r.b.d. is intuitive and straightforward. The definition of u.p.f. is not intuitive, but also straightforward in terms of linear algebra [11, 10]. The terms in brackets in the above equations, (p∗ ), indicate the the nodal displacement and traction-force attributes are functions of the point-force paramenters of the fundamental solution. The above equations are very simple statements, except that there are embedded amounts of rigid-body displacements and of unbalanced forces that cannot be transformed. Moreover, the coefficients of the displacement ∗ ] ∈ Rnd ×nd as well as of the traction-force matrix T∗ = [T ∗ ] ∈ Rnt ×n∗ are undefined (and not matrix U∗ = [Umn m infinite!) when their indices refer to the same nodal point [10]. Moreover, it may be convenient to express the boundary traction approximations of eq (6) in terms of equivalent nodal forces, obtained from a virtual work statement [12]: p(t) = LT t

(12)

where the interpolation functions of eqs (3) and (6) were used, thus defining t d ti uim dΓ L = [Lm ] ∈ Rn ×n = Γ

(13)

According to eq (6), the Jacobian of dΓ = |J|dξdη, as for a 3D problem, cancels with the denominator of ti , such that the coefficients of Lm become pre-defined numbers that are independent from the problem’s geometry.

298

3.


The expedite boundary element method (EBEM)

∗ , T ∗ and L Using the definitions of the matrices Unm m in eqs (10), (11) and (13), eq (8) can be approximated m by its expedite boundary element version [12, 15],

˜ d − d p = U∗T LT t − t p H

or

˜ d − d p = U∗T p − p p H

(14)

˜ substitutes for T∗T L as an approximation of H. The second representation where, for simplicity of notation, H above is an alternative in terms of equivalent nodal forces, according to eq. (12), and a formulation more akin to the finite element method. However, these equations are useful only if the still undefined coefficients of U∗ and T∗ can be obtained. ˜ The traction-force matrix T∗ is rectangular. However, the Evaluation of the undefined coefficients of H. ˜ ≡ T∗T L above are the actual subjects of interest. The undefined coefficients of the square matrix given as H matrix L, as defined in eq (13), has the same numbers of rows and columns as T∗ , but is banded, with non-zero coefficient Lm only if the nodal displacement δdm and the traction-force attribute t refer to the same boundary ˜ refers to node points that are segment (element). As detailed in Refs. [12, 15], when a coefficient H˜ i j of H adjacent (lie on the same boundary element), it cannot and should not be approximated, that is, H˜ i j = Hi j . This includes the need of evaluating the coefficients about the main diagonal, which can always be carried out ˜ can by applying some spectral properties (such as orthogonality to rigid body displacements). Besides that, H always be directly evaluated as the product T∗T L. Evaluation of the undefined coefficients of U∗ . Once the undefined coefficients of the product T∗T L are evaluated, the best (and only) way of obtaining the undefined coefficients of U∗ , is to apply either eq (14) to a set of simple, analytical solutions (Da , Ta ) or (Da , Pa ) and then write the least-squares statement ∗T a T LD − U∗T LT Ta = min or T∗T LDa − U∗T Pa = min (15) For potential problems, there are either two or three constant fluxes, for 2D or 3D problems, respectively, as the simplest, applicable non-trivial solutions, and just one unknown per node. For general elasticity problems, there are either three or six constant stress states, for 2D or 3D problems, and either 2 × 2 or 3 × 3 unknowns (if symmetry is not enforced), which justifies the proposed least-squares solution. This solution scheme is similar to the one adopted in the HBEM for the evaluation of the undefined coefficients of the flexibility matrix F [2, 3, 4, 9, 10, 16, 19]. 3.1.

Solution of the problem’s equation and evaluation of results at internal points

Given a general mixed-boundary problem, either one of eq (14) can be solved for the problem’s unknowns and, afterwards, results at internal points are obtained by using the Somigliana’s identity. However, in the present framework the results can be directly obtained as functions of the point-force parameters p∗ , according to eqs (4) and (5), which circumvents any further integrations. An efficient way of handling the problem proposed in the above paragraph is by finding a means of evaluating all nodal unknowns – nodal displacements and forces, besides the point-force parameters p∗ – by solving just one matrix system. Equation (10) can be combined with one of the equations obtained in the hybrid boundary ˜ element method [2, 3, 4, 9, 10, 16, 19], here adapted for the approximate matrix H, ˜ T p∗ = p − p p H

(16)

to formulate a solution scheme for a general mixed boundary problem. As presented in Ref. [15], let subvectors of d and p have subscripts D and N assigned to characterize whether the boundary conditions are of Dirichlet or Neumann type. Then, eqs (10) and (16) can be represented as ⎧ ⎧ ⎡ ∗⎤ ⎡ ˜ T⎤ p⎫ p⎫ ⎢⎢⎢UN ⎥⎥⎥ ∗ ⎪ ⎢⎢⎢HN ⎥⎥⎥ ∗ ⎪ ⎪ ⎪ ⎪ ⎪p N − p N ⎪ ⎨dN − dN ⎪ ⎬ ⎬ ⎢ ⎥⎦⎥ p = ⎨ , (17) ⎢⎣⎢ ∗ ⎥⎦⎥ p = ⎪ ⎢ ⎪ ⎪ ⎣ ⎪ ⎪ ⎪ p⎪ p⎪ T ⎩ ⎩ ⎭ ˜ HD UD dD − dD pD − pD ⎭


299

This equation system is firstly solved for p∗ , provided that the problem is well posed, with the subsequent evaluation of the remaining boundary force and displacement parameters, as indicated below: ⎧ ⎧ ⎫ ⎧ p⎫ ⎡ ∗ ⎤ ⎡ ˜ T⎤ p⎫ ⎪ ⎢⎢⎢HN ⎥⎥⎥ ∗ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨pN − pN ⎪ ⎬ ⎬ ⎪ ⎬ ⎢⎢⎢⎢UN ⎥⎥⎥⎥ ∗ ⎨dN ⎪ ⎨dN ⎪ (18) , =⎪ ⎢⎢⎣ ∗ ⎥⎥⎦ p = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + ⎢⎣ ˜ T ⎥⎦ p ⎪ ⎪ p⎪ p⎪ ⎩ ⎭ ⎭ ⎩ ⎩ U dD − d pD p ⎭ H D

D

D

D

Finally, results at internal points are obtained by directly using eqs (4) and (5). A conceptual remark on the present developments. References [12, 15] use the very simple triangular 2D domain on the left in Figure 1 – discretized with three quadratic boundary elements – to illustrate the ˜ ≡ T∗T L. Although referring to formulation of the double-layer potential matrix in terms of the product H the same geometric point, the traction forces obtained as T∗ p∗ , eq (11), depend on the outward normal to the boundary, which is the reason for the assignment of double (for 2D problems) or more (as in a general 3D problem) values of t at points with tangent discontinuities1 . However, pre-defining the location of n boundary points (three points along each boundary segment in the illustration on the left in Figure 1) means that the ∗ t , on which the proposed expedite formulation relies [12, 15], is a reasonable approximation σ∗jim n j ≈ T m i one for a numerical integration that is exact only for a polynomial of degree n − 1. On the other hand, if these points are located as the roots of a Legendre polynomial, as illustrated on the right in Figure 1, the in eq (9) underlying Gauss-Legendre integration is exact for a polynomial of degree 2n − 1, which is a remarkable numerical improvement. The expedite implementations carried out so long have used the approach illustrated on the left in Figure 1, with reasonably accurate results for 2D problems [12, 13, 15], but with only slowly converging results in the case of 3D problems [14, 15]. Then, the approach with points represented as on the ∗ t – and, right in Figure 1 shall always be used. It is worth emphasizing that the approximation σ∗jim n j ≈ T m i ˜ ≡ T∗T L – only applies when and m refer to points that are not on the same consequently, the product H ˜ is otherwise evaluated as traditionally: H ˜ ≡ H. boundary element and that H 5

7

6 1

6

7

8

6

9

9

1

1 5

4

2

*

6

*

8

*

* 5

*4

1

2

2

3

5

*

4

3

*2 3

*

*4 3

Figure 1: Illustration of a triangular domain with six nodes and three quadratic boundary elements. The representation on the right is an improved numerical discretization version.

4.

Hierarchical boundary element discretization and representation

Codes for 2D and 3D potential and elasticity problems, with some numerical results already available in Refs. [11, 12, 14, 13, 15], are just being developed in the frame of a hierarchical mesh discretization that can be used also in the analysis stage of the problem. As implemented, linear, quadratic or cubic elements can be dealt with in a single code for 2D problems, while a code for 3D problems may indistinctively deal with linear, quadratic or cubic, triangular or quadrilateral elements. Figure 3 illustrates the local numbering pattern of four child 1 This feature is well known in the boundary element community and has led to several formulation variations (such as the discontinuous BEM) that are in general not exempt from conceptual mistakes. It is worth remarking, however, that in the CBEM this "problem" is only perceived in relation to the evaluation of the single-layer potential matrix.

300


elements generated from quadratic triangular and quadrilateral elements, for a general 3D problem with curved boundaries. The nodal numbers refer to nodal displacements. The nodes for the traction force attributes are not shown. They would correspond to the abscissas of an adequate numerical integration scheme over triangles, for the element on the left, or just the 3 by 3 points of a Gauss-Legendre quadrature for the quadrilateral element. The use of a reduced-integration scheme for far-field source points is being numerically assessed at present. Concerns on this subject are also brought up in the next Section. Successively refined meshes

η

10

3

6

1

ξ

9 4 13 2

12

1 7

15

14

4

5 3 14

11

8

15

2

8

η

16

4

c) Quadratic triangle T6 (te =6, oe =3)

3

13

7 4 20

1

1

12 19 21 17

3

ξ

18

6

2

11

9

2 5 10 d) Quadratic quadrilateral Q8 (te =8, oe =4)

Figure 2: Natural coordinates and numbering pattern used in the generation of quadratic triangular and quadrilateral elements.

numbered k = 2 . . . nmesh are generated from a first, coarse mesh. For each newly obtained mesh, the Cartesian coordinates of the generated nodes are sequentially added to vectors X, Y, Z, as for a 3D problem. The boundary elements keep the relative numbering of their parent elements (that is, for the elements shown in Figure 3, the four partitioned elements obtained from element ie are globally numbered iep = [4ie − 3, 4ie − 2, 4ie − 1, 4ie ]. A given mesh k has nel [k] elements (for the elements of Figure 3, nel [k + 1] = 4nel [k]) and ngl [k] nodes (which must be counted for each level k). After complete generation of the data for all mesh levels, the dimension of the vectors of coordinates X, Y, Z is ngl [nmesh ]. The local to global nodal mapping is given by the array of nodal incidences inc[k][ie , in ], where k refers to the generated mesh level, ie = 1 . . . nel [k] is the global element number and in = 1 . . . te is the local (element-referred, as indicated in Figure 3) node number. The proposed hierarchical boundary element discretization is very convenient when one is dealing with a largescale problem and a lower-level graphical representation of the problem’s geometry as well as of results is needed. During the partitioning process of a given element for the generation of its child elements, the Cartesian coordinates of the arising new nodes can be either interpolated from the parent element’s geometry (as nodes 7 . . . 15 on the left, or nodes 9 . . . 21 on the right in Figure 3) or given separately in order to characterize reentrances or rugosities that could not be resolved with a coarse mesh. Moreover, the concomitant availability of data for several mesh discretizations can be used in the solution of a problem from the bottom up. 5.

Implementation in the frame of a fast-multipole approach

Figure 3 represents part of a 2D problem, showing a boundary element, numbered as ie [1] = 1[1], whose adjacent elements are n[1] and 2[1] for the first mesh level k = 1. This element is successively partitioned into two elements, for mesh levels k = 2, 3, 4, and each time the adjacent elements of a generated element is obtained, thus creating a structure of element adjacencies elad j [k], which is actually a set whose nad j [k] entries are the elements of level k that are adjacent to the k-th child or grandchild element of ie [1]. This structure of element adjacencies is extremely simple, as conceived, since it exists at a given time only for one element on one level k, and can be generated very fast, as it only requires the knowledge of the mesh topology built in the table of nodal incidences inc[k][ie , in ]. The generation of the adjacency structure can be carried out regardless of topology (for either 2D or 3D problems that may also be multiply connected) in the frame of a recursive algorithm inside the nested loops that go through all elements and all mesh levels. (For a 2D problem, the number nad j [k] is obviously always equal to 3, if an element is considered as adjacent to itself, and the set nad j [k] can be obtained in a straight forward way, if one only takes care of the possibility of a domain being multiply connected.)

n[1 ] 2n

[2]

4n[ 3]

8n[4 ] 1[ 4]

2[4]

1[3]

1[2]

3[4]

4[4]

5[4]

2[3]

6[4] 3[3]

7[4]

8[4]

301

2[ 1]


9[4]

5[3]

] 3[2

4[3] 2[2]

1[1]

Figure 3: Hierarchical mesh discretization of a 2D problem with representation of the elements that are adjacent to a given element (number 1) on successive mesh levels.

The implemented algorithm is very fast, since it works only on the topology of the generated meshes in terms of Boolean operations with integer numbers. However, when boundary elements are farther apart than just the distance comprised by a few elements, and particularly if one is dealing with a structure that is very convoluted, the concept of topological distance must be replaced with the concept of geometrical distance. A fast-multipole procedure is being implemented to work together with the described structure of element adjacencies, since the use of cells and of the geometrical distance between them is ultimately unavoidable for very large problems of general shape and intricacy. The fast-multipole structure and algorithms are being constructed from scratch, although always relying on Ref. [17] and not forgetting the background knowledge of Refs. [21, 18]. Besides the combination of cells (geometry) with element adjacencies (topology), the method under development is different as the problem in focus is different, too. In fact, it deals with the solution of the first of eqs (18). The ˜ ≡ T∗T L appears as the transpose of the corresponding matrix in the CBEM double-layer potential matrix H and is proposed as an approximation when the source point is far from the field point. The matrix L can be built from just one small block matrix that is pre-evaluated as representative of all elements of a given type and independently from the geometry of the curved boundary [14, 15]. Then, the only requirement is the evaluation, in terms of fast multipoles, of the terms related to T∗ at points along the field element, as discussed by means of Figure 1. Moreover, the nodal displacements comprised by the matrix U∗ , as needed in part of eq (18), are evaluated in the frame of the same fast multipole algorithm. It comes out that no boundary integrations need be carried out (in the traditional sense) when the source and the field elements are non-adjacent. Conclusions This paper is a continuation of Ref. [14], for an expedite formulation of the boundary element method that requires no integrations in the traditional sense, except for a few regular ones for a narrow band of coefficients above and below the main diagonal of the matrix that approximates the double-layer potential matrix H. For mixed boundary conditions, the matrix system on the left in eq (18) is structured in such a way that an efficient iterative solver (GMRES) of very large equation systems can be used for the evaluation of the problem’s unknowns, at the same time that the parameters for the straightforward representation of results at internal points are obtained. As proposed, the EBEM combines a topological hierarchy of elements with the concept of fast multipoles, which renders a code that is simple to implement and applicable to families of generally curved boundary elements. At present, a first code for the solution of the Laplace equation is being tested. An extended version of the present manuscript is being prepared, in which convergence of results and computational efficiency are assessed for several numerical examples. Acknowledgments This project was supported by the Brazilian agencies CAPES, CNPq and FAPERJ.

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References [1] C. A. Brebbia, J. F. C. Telles, and L. C. Wrobel. Boundary Element Techniques. Springer-Verlag, Berlin, 1984. [2] R. A. P. Chaves. The Simplified Hybrid Boundary Element Method Applied to Time-Dependend Problems (in Portuguese). PhD thesis, Pontifical Catholic University of Rio de Janeiro, 2003. [3] M. F. F. de Oliveira and N. A. Dumont. Conceptual completion of the simplified hybrid boundary element method. In E. J. Sapountzakis and M. H. Aliabadi, editors, BETeq 2009 - International Conference on Boundary Element Techniques, pages 49–54, Athens, Greece, July 2009. [4] N. A. Dumont. The hybrid boundary element method: an alliance between mechanical consistency and simplicity. Applied Mechanics Reviews, 42(11):S54–S63, 1989. [5] N. A. Dumont. An assessment of the spectral properties of the matrix G used in the boundary element methods. Computational Mechanics, 22:32–41, 1998. [6] N. A. Dumont. Variationally-based, hybrid boundary element methods. Computer Assisted Mechanics and Engineering Sciences (CAMES), 10:407–430, 2003. [7] N. A. Dumont. The boundary element method revisited. In C. A. Brebbia, editor, Boundary Elements and Other Mesh Reduction Methods XXXII, pages 227–238, Southampton, U.K., 2010. WITPress. [8] N. A. Dumont. From the collocation boundary element method to a meshless formulation. In M. A. Storti E. N. Dvorkin, M. B. Goldschmit, editor, Mecánica Computacional, MECOM 2010 – IX Argentinean congress on Computational Mechanics and II South American congress on Computational Mechanics, XXXI CILAMCE – XXXI Iberian Latin-American Congress on Computational Methods in Engineering, pages 4635–4659 (on CD), Buenos Aires, Argentina, 2010. [9] N. A. Dumont. The hybrid boundary element method – fundamentals (to be submitted). Engineering Analysis with Boundary Elements, 2011. [10] N. A. Dumont. Toward a meshless formulation of the simplified hybrid boundary element method (to be submitted). 2011. [11] N. A. Dumont and C. A. Aguilar. The expedite boundary element method. In C. A. Brebbia and V. Popov, editors, Boundary Elements and Other Mesh Reduction Methods XXXII, pages 179–190. WITPress, Southampton, 2011. [12] N. A. Dumont and C. A. Aguilar. Expedite implementation of the boundary element method. In E. L. Albuquerque and M. H. Aliabadi, editors, Advances in Boundary element Techniques XII, pages 162–179. EC, Ltd., UK, 2011. [13] N. A. Dumont and C. A. Aguilar. Expedite implementation of the boundary element method for elasticity problems. In Proceedings CILAMCE - XXXII Iberian Latin-American Congress on Computational Methods in Engineering, page 16 pp (on CD), Ouro Preto, Brazil, 2011. [14] N. A. Dumont and C. A. Aguilar. Three-dimensional implementation of the expedite boundary element method. In Extended Abstracts of the IABEM2011, Symposium of the International Association for Boundary Element Methods, pages 113–118, Brescia, Italy, September 2011. [15] N. A. Dumont and C. A. Aguilar. The best of two worlds: The expedite boundary element method (in press). Engineering Structures, 2012. [16] N. A. Dumont and R. A. P. Chaves. General time-dependent analysis with the frequency-domain hybrid boundary element method. Computer Assisted Mechanics and Engineering Sciences, (10):431–452, 2003. [17] Y. Liu. Fast Multipole Boundary Element Method, Theory and Applications in Engineering. Cambridge, 2009. [18] N. Nishimura. Fast multipole accelerated boundary integral equation methods. Applied Mechanics Review, 55:299–324, 2002. [19] M. F. F. Oliveira. Conventional, hybrid and simplified boundary element methods (in Portuguese). Master’s thesis, Pontifical Catholic University of Rio de Janeiro, 2004. [20] P. W. Partridge, C. A. Brebbia, and L. C. Wrobel. The Dual Reciprocity Boundary Element Method. Computational Mechanics Publications, Southampton, 1992. [21] V. Rokhlin. Rapid solution of integral equations of classical potential theory. Journal of Computational Physics, 60:187–207, 1985.


Analysis of Anisotropic Symmetric Plates by the Adaptive Cross Approximation R. Q. Rodr´ıguez1 , P. Sollero1 and E. L. Albuquerque2 1

Faculty of Mechanical Engineering, University of Campinas, Brazil, {reneqr87,sollero}@fem.unicamp.br 2

Faculty of Mechanical Engineering, University of Brasilia, Brazil, [email protected]

Keywords: boundary element method, plates, hierarchical matrices, Adaptive Cross Approximation.

Abstract. This work analyses the application of hierarchical matrices and low rank approximations in anisotropic symmetric plates. Several applications for isotropic materials are available in literature. However, few works are found for anisotropic applications. First, a revision of the anisotropic fundamental solution is carried out. Then, the use of hierarchical matrices and low rank approximations is depicted. Finally, a numerical example is performed to demonstrate the feasibility of using ACA in anisotropic symmetric plates. Introduction. Over recent years, the boundary element method (BEM) has demonstrated to be a powerful numerical tool for the analysis and solution of many physical and engineering problems [1]. However, one of the big disadvantages of BEM when compared with the finite element method (FEM) is the fully populated and not symmetric system matrices, implying in higher memory requirements and solution times. Many research studies have focused in improving the BEM, such as block-based solvers [2, 3], the lumping technique [4] or iterative solvers [5]. A complete review of these methods is available in [8]. Many investigations have been carried out to speed up the solution process. One of the most famous methods is the Fast Multipole Method [6]. However, the main disadvantage of this method is the necessity of the harmonic expansion of the kernels. From the algebraic point of view however, the integration of a degenerate kernel over a cluster of suitably selected boundary elements corresponds to the approximation of the related matrix block by a low rank block [14]. As a consequence, it is possible to use purely algebraic algorithms to generate the approximation of suitable blocks of the collocation matrix, using only few entries of the original blocks [15]. This technique is referred to as the Adaptive Cross Approximation (ACA). Analogously to FMMs, the use of the hierarchical format is aimed at reducing the storage requirement and the computational complexity arising in the Boundary Element Method. After representing the coefficient matrix in hierarchical format, the solution of the system can be obtained either directly, by inverting the matrix in hierarchical format, or indirectly, by using iterative schemes with or without preconditioners [8, 9, 16]. In this paper the application of hierarchical matrices and ACA in symmetric anisotropic plates is illustrated. First, the anisotropic fundamental solution is briefly explained. Then, the use of hierarchical matrices and ACA is discussed. A simple numerical example and results are shown in the next section. Finally conclusions are pointed out.

303

304


Anisotropic Fundamental Solution. The displacement fundamental solution for elastostatics is given by [7]: u•ji zk , z k = 2Re qi1 A j1 ln z1 − z 1 + qi2 A j2 ln z2 − z 2

(1)

where z k is the source point z k = x 1 + µk x 2 , k = 1, 2

(2)

and zk is the field point given by equation zk = x1 + µk x2 , k = 1, 2

(3)

µk are the complex roots of characteristic polynomial, x 1 and x 2 are the source point coordinates, x1 and x2 are the field point coordinates and qik is given by qik =

a11 µk2 + a12 − a16 µk a11 µk + a22 µk − a26

(4)

where ai j are the anisotropic elastic constants. Due the requirement of unit load at zk and displacement continuity of fundamental solution, the complex coeeficients Aik are obtained by the solution of the linear system ⎡ ⎢ ⎢ ⎢ ⎣

1 −1 1 −1 µ1 −µ¯ 1 µ2 −µ¯ 2 q11 −q¯11 q12 −q¯12 q21 −q¯21 q22 −q¯22

⎤⎧ ⎪ ⎪ ⎥⎪ ⎨ ⎥ ⎥ ⎦⎪ ⎪ ⎪ ⎩

⎧ ⎪ δ j2 (2πi) ⎪ ⎪ ⎨ −δ (2πi) j1 = ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ 0

⎫ ⎪ A j1 ⎪ ⎪ A¯ j1 ⎬

⎫ ⎪ ⎪ ⎪ ⎬

A j2 A¯ j2

⎪ ⎪ ⎪ ⎭

(5)

where δi j is the Kronecker delta. In the case of isotropic material the roots of characteristic equation is the pure imaginary i and −i. This values become singular the linear system (5). Because of this, it is not possible to use isotropic materials to compare this formulation with the isotropic formulation. To do this comparison, quasi isotropic materials will be used (Young moduli E1 and E2 are made almost equal). The traction fundamental solution for elastostatics is given by pi j zk , z k = 2Re

1 1 g j1 (µ1 n1 − n2 ) Ai1 + g j2 (µ2 n1 − n2 ) Ai2 (z1 − z 1 ) (z2 − z 2 )

(6)

where

and nk are the normal vector components.

g jk =

µ1 µ2 −1 −1

(7)


305

Hierarchical Matrices. The main reference in this section was [8]. In this book chapter, a complete study of the use of hierarchical matrices and low rank approximations applied to large-scale 3D elastic problems was assessed. The specific application in 3D anisotropic crack problems was also accounted by Benedetti et al. [11]. Moreover, a formal definition and description of hierarchical matrices can be also found in [10, 12]. The objective of hierarchical matrices is to reduce the storage requirements as well as to speed up the time required to complete all matrix operations. In this method the matrix is represented as a collection of blocks, some of which admit a particular approximated representation that can be obtained by computing only few entries from the original blocks. These special blocks are called low rank blocks. The blocks that cannot be represented in this way must be computed and stored entirely and are called full rank blocks. Low rank blocks constitute an approximation of suitably selected blocks of the discrete integral operator based, from the analytical point of view, on a suitable expansion of the kernel of the continuous integral operator [14, 15]. This expansion, and consequently the existence of low rank approximants, is based on the asymptotic smoothness of the kernel functions, i.e. on the fact that the kernels Ui∗j and Ti∗j are singular only when x = y [14, 15, 16]. This represents a sufficient condition for the existence of low rank approximants and it does not exclude strongly or hyper-singular kernels like those appearing in the traction boundary integral equation. A low rank block M of size mxn has the following representation k

Mk = ∑ ai · bi T = A · BT

(8)

i=1

where A is a matrix of size mxk and B is a matrix of size nxk. For admissible blocks, k is low and the representation showed in Eq.(8) requires the storage of (m + n)k real numbers instead of the of the mxn original block. It is moreover apparent how it speeds up the matrix-vector product of the corresponding block. For a detailed analysis of the memory savings as well as the speed-up allowed see [17, 18]. A hiererchical approximation of large dense matrices arising from some generating function having diagonal singularity consist of three steps [19]: • Construction of clusters • Finding of possible admissible blocks • Low rank approximation of admissible blocks The construction of clusters was implemented based on the algorithm showed in [19]. First, the mass and centre of each cluster are stored, then, the covariance matrix of the cluster is obtained by Eq.(9) C=

n

∑ gk (xk − X) (xk − X)T

(9)

k=1

where n is the number of elements of the cluster, gk is the element length and X is the centre of the cluster. Then, the largest eigenvalue of C shows in the direction of the longest extension of the cluster. The separation line (plane in 3D) x ∈ ℜ2 : (x − X, v1 ) = 0 goes through the centre X of the cluster and is orthogonal to the eigenvector v1 . This algorithm will be applied recursively to the sons until they contain less than or equal to some prescribed number nmin of elements. Next, cluster pairs which are geometrically well separated are identified. They will be regarded as admissible cluster pairs. An appropiate admissibility criterion is the following simple geometrical condition. A pair of clusters (Clx ,Cly ) with nx > nmin and my > nmin elements is admissible if

306


min(diam(Clx ), diam(Cly )) ≤ ηdist(Clx ,Cly ),

(10)

where η is called the admissibility parameter. This parameter influences the number of admissible blocks on one hand and the convergence speed of the adaptive approximation of low rank blocks on the other hand [17]. A full study of this parameter was assessed by Benedetti et al. [8]. They showed that the choice of η directly affects the quality of the ACA-generated matrix and a good choice of this parameter results in a matrix closer to the optimal matrix produced by the coarsening procedure, this fact was also justified by the reduction in the number of blocks. In the present work, the actual diameters and the distance between two clusters were calculated and no approximation, as suggested by [8, 16, 19, 20], was made. Approximations are easily computable, however, result in a more rough and restrictive criterion. Once the clusters were defined and all admissible blocks were detected, we use the Adaptive Cross Approximation (ACA) to approximate by low rank these blocks. The original ACA algorithm was proposed by Bebendorf [14]. Three years later was further developed by Bebendorf and Rjasanow [15]. Several ACA algorithms and variations, as the so-called ACA+, are available in the literature [16, 19, 21]. The algorithm used in this work for the low rank approximations was the same as showed by Kurz et al. [19]. Results obtained after the low rank approximation of the admissible blocks by ACA, can be further recompressed, taking advantage of the reduced singular value decomposition (SVD) [16], increasing the gain in memory storage. This fact was first accounted by [16], then was corroborated by [8]. Moreover, Benedetti et al. [8] recommend a coarsening procedure, based on [13], to reduce the memory storage and finally a SVD new recompression (with less accuracy) to obtain the pre-conditioner matrix. In this work, a first recompression of the low rank blocks (with a high accuracy) is done in order to reduce the memory storage, then a final SVD recompression (with a low accuracy) is done in order to generate the preconditioner matrix. Finally, the system is solved by the generalized minimum residual method (GMRES). Numerical results To test the efficiency of the presented method in terms of memory storage and solution time, a simple configuration is analysed. A four-layer symmetric anisotropic plate is modelled with 2D constant elements. Layers are oriented at 45/-45/-45/45 degrees. All layers have the same dimensions: 10x10x1. The plate was clamped in one side and loaded in the opposite side with a 1Kgf/mm constant load, as can be seen in Figure 1. Material properties of the anisotropic plate are shown in Table 1.

Figure 1: Boundary conditions of the four-layer symmetric anisotropic plate.


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Table 1: Material properties of the plate.

Carbon/Epoxi

E1 (GPa) 142.17

E2 (GPa) 9.255

G12 (GPa) 4.795

ν12 0.3340

Six different meshes (120, 400, 800, 1200, 1600 and 4000 elements) are tested to show the variation of the memory savings and the speed-up ratios with respect to the conventional BEM standard time. The ACA accuracy is set to εc = 10−5 , the recompression SVD accuracy is set to εr = 10−5 and the pre-conditioner matrix accuracy is set to ε p = 10−1 .

Table 2: Memory storages and speed-up ratios of the five meshes. Elements

Stor. A

Stor. B

ACA (%)

120 400 800 1200 1600 4000

61.02% 31.43% 27.56% 23.36% 23.14% 22.90%

61.01% 28.00% 17.71% 14.19% 13.30% 11.53%

74.10% 85.09% 79.86% 82.13% 79.69% 78.57%

Standard time (s) 18.36 208.57 902.78 2143.30 4020.56 35034.52

GMRES iterations 1(10) 2(2) 2(3) 2(3) 2(4) 2(5)

Speed-up ratio 2.49 1.43 1.09 0.99 0.89 0.59

x−x ˜ L2 xL2

4.80x10−6 5.07x10−6 5.09x10−6 2.35x10−6 2.80x10−6 4.83x10−6

Table 2 reports: memory storages before (Storage A) and after the recompression (Storage B), the quantity in percentage of elements generated by ACA, the standard time, the number of GMRES iterations, the Speed-up ratio (Hierarchical time/Standard time) and finally the approximate solution accuracy in terms of the L2 norm. The L2 norm (best known as frobenius norm) used in Table 2 does not give insight into the quality of the approximation for engineering purposes. A node by node check of the solution confirmed, that, for a selected accuracy εc , the average errors are to the order of 0.1±1.0%. Bigger percentage errors can occur for degrees of freedom whose standard solution values are smaller than the requested accuracy. This consideration suggests that it is advisable to set the accuracy at the same order of magnitude as that of the smaller quantities of interest in the analysis [8]. It is important to notice that the speed-up ratio is less than one beyond a certain number of elements in the mesh (1200 elements). That means, that, the method works better after this value. Both, memory requirements and time comparison for differente meshes are shown in Figure 2.

308


4

70

4

x 10

Memory requirement 3.5 60

BEM ACA

Time [s]

Storage [%]

3 50

40

2.5

2

1.5

30 1 20 0.5

10

0

500

1000

1500

2000

2500

3000

Number of elements

3500

4000

0

0

500

1000

1500

2000

2500

3000

3500

4000

Number of elements

Figure 2: Memory requirements (storage) and time comparison for differente meshes. Finally, Figure 3 shows the block-wise structure of the collocation matrix for the finest mesh before and after the recompression, as well as, the preconditioner matrix generated by a coarse recompression. Setting the minimum number of elements (nmin ) per cluster as 100 and an admissibility parameter (η) of √ 2, 947 blocks were created, from which, 166 are admissible pairs. Every block is coloured showing the ratio between the memory required for low rank representatio and the memory in full rank format. The red color (ratio = 1), means that the block was generated as full rank.

Figure 3: Block-wise structure of the collocation matrix, (a)before recompression, (b)after recompression and (c)preconditioner matrix. Conclusions. In this work, the use of hierarchical matrices and low-rank approximations applied to symmetric anisotropic plates has been presented. Low rank approximations were accomplished by the use of ACA. This method is suitable for memory and time savings, especially in the case of large-scale problems. The code was implemented in Matlab, so better results are expected if more adequate programming languages are used, such as Fortran, C or C++. Future implementations will be done in Fortran with the intention to couple ACA, BEM and multiscale problems. Acknowledgment. The authors would like to thank the National Council for Scientific and Technological Development (CNPq) for the financial support of this work.

Advances in Boundary Element and Meshless Techniques References [1] M. H. Aliabadi. The Boundary Element Method. vol2: Applications in Solids and Structures. John Wiley & Sons, Ltd Chichester, 2002. [2] J. M. Crotty. A block equation solver for large unsymmetric matrices arising in the boundary integral equation method. International Journal for Numerical Methods in Engineering. 18:997–1017, 1982. [3] R. H. Rigby and M. H. Aliabadi. Out-of-core solver for large, multi-zone boundary element matrices, International Journal for Numerical Methods in Engineering. 38:1507–1533, 1995. [4] J. H. Kane and B. L. Kashava Kumar and S. Saigal. An arbitrary condensing, noncondensing solution strategy for large scale, multi-zone boundary element analysis, Computer Methods in Applied Mechanics and Engineering. 79:219–244, 1990. [5] W. J. Mansur and F. C. Araujo and E. B.Malaghini. Solution of BEM sustems of equations via iterative techniques, International Journal for Numerical Methods in Engineering. 33:1823–1841, 1992. [6] H. Rokhlin. Rapid solution of integral equation of classical potential theory Journal of Computational Physics, 60:187–207, 1985. [7] P. Sollero, and M. H. Aliabadi. Fracture mechanic analysis of anisotropic plates by the boundary element method. International Journal of Fracture, 64:269–284, 1993. [8] I. Benedetti, A. Milazzo, and M. H. Aliabadi. Fast hierarchical boundary element method for largescale 3-D elastic problems. In: M. H. Aliabadi and P. H. Wen. Boundary element methods in Engineering and Sciences, vol 4. Imperial College Press, 2011. [9] M. Bebendorf. Hierarchical LU decomposition-based-precoditioners for BEM. Computing, 74:225– 247, 2005. [10] W. Hackbusch. A sparse matrix arithmetic based on H–matrices. Part I. Computing, 62:89–108, 1999 2002. [11] I. Benedetti, A. Milazzo and M. H. Aliabadi. A fast dual boundary element method for 3D anisotropic crack problems. International Journal for Numerical Methods in Engineering, 80:1356–1378, 2009. [12] W. Hackbusch and Z. P. Nowak. A sparse H–matrix arithmetic. Part II. Application to multidimensional problems. Computing, 64:21–47, 2000. [13] W. Hackbusch, B. N. Khoromskij and R. Kriemann. Hierarchical matrices based on weak admissibility criterion. Computing, 73:207–243, 2004. [14] M. Bebendorf. Approximation of boundary element matrices. Numerische Mathematik, 86:565–589, 2000. [15] M. Bebendorf and S. Rjasanow. Adaptive low-rank approximation of colloacation matrices. Computing, 70:1–24, 2003. [16] L. Grasedyck. Adaptive recompression of H–matrices for BEM. Computing, 74:205–223, 2005.

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[17] S. Borm, L. Grasedick and W. Hackbusch. Introduction to hierarchical matrices with applications. Engineering Analysis with Boundary Elements, 27:405–422, 2003. [18] L. Grasedyck and W. Hackbusch. Construction and arithmetics of H–matrices. Computing, 70:295– 334, 2003. [19] S. Kurz, O. Rain and S. Rjasanow. Fast boundary element methods in computational Electromagnetism. In: M. Schanz and O. Steinbach. Boundary Element Analysis, Mathematical Aspects and Applications. Springer, 2007. [20] K. Gieberman. Multilevel approximation of boundary integral operators. Computing, 67:183–207, 2001. [21] M. Bebendorf and R. Grzhibovskis. Accelerating Galerkin BEM for linear elasticity using adaptive cross approximation. Mathematical Methods in the Applied Sciences, 29:1721–1747, 2006.


The Method of Fundamental Solutions Applied to Crack Problems using Tikhonov's Regularization and the Numerical Green's Function Procedure E.F. Fontes Jr.1,a, J.A.F. Santiago1,b. J.C.F. Telles1,c 1

Department of Civil Engineering, COPPE/UFRJ, CaixaPostal 68506, CEP21941-972, Rio de Janeiro, RJ, Brazil, [email protected], [email protected], [email protected]

Keywords: MFS, Green’s function, Tikhonov

Abstract The method of fundamental solutions (MFS) is applied to solve linear elastic fracture mechanics (LEFM) problems. The approximate solution is obtained by means of a linear combination of fundamental solutions containing the same crack geometry as the actual problem. In this way, the fundamental solution is the very same one applied in the numerical Green's function (NGF) BEM approach, in which the singular behavior of embedded crack problems is incorporated. Due to severe ill-conditioning present in the MFS matrices generated with the numerical Green's function, a regularization procedure (Tikhonov's) was needed to improve accuracy, stabilization of the solution and to reduce sensibility with respect to source point locations. Finally, accurate stress intensity factors could be obtained by a superposition of the generalized fundamental crack openings. This mesh-free technique presents good results when compared with the boundary element method and analytical solutions for the stress intensity factor calculations. 1 Introduction Modeling of linear elastic fracture mechanics (LEFM) problems to obtain accurate stress intensity factors (SIF) is naturally a difficult problem. A geometrical problem arises when a crack is present, which gives singular stress fields in the vicinity of the crack tips. To simulate this singular behavior we use the numerical Green’s function (NGF) procedure [1], successfully used to solve LEFM problems with the boundary element method [2], its first implementation was presented by Telles et al. [3], where the numerical procedure to calculate the fundamental solution shows accurate results for SIF computations. The method of fundamental solutions (MFS), has been selected to discretize the Navier equation, that models LEFM problems. The MFS, firstly developed by Kupradze and Aleksidze [4], is a meshless boundary-type method; an alternative to mesh-type methods like the finite element method (FEM) and BEM. In order to construct the solution, the MFS uses only a superposition of fundamental solutions associated to the problem. This is done without using any integrals, greatly simplifying its implementation. The MFS is also an efficient numerical technique and has successfully been applied for solving several kinds of partial differential equations. For example, potential problems [5], elastostatics problems [6], acoustic problems [7]. Details of MFS alternative formulations can be found in the review paper of Fairweather and Karageorghis [8]. Among the few studies found in literature to deal with the use of MFS for LEFM problems, it is worth noting the recently work of Guimaraes and Telles [9], where the MFS has been successfully applied to deal with fracture mechanics problems in Reissner’s plate using the NGF procedure. Other works have been developed to perform SIF computations using the domain decomposition technique or considering symmetry (whenever possible) to model the crack surface [10,11,12]. Here, the MFS formulation is developed with the implementation of the numerical Green’s function counterpart acting as the fundamental solution. Thus the discretization of the crack surface is elegantly avoided. This is

311

312


interesting for mesh-reduction methods like the MFS. A detailed description of how the NGF is built can be seen in references [1,3]. The usual ill-conditioning generated by the MFS matrices and the proper location of source points are always difficulties found in MFS implementations, hence the singular value decomposition (SVD) idea has been often used to reduce at least the conditioning effects [13]. Another possibility is to use a regularization technique like Tikhonov’s regularization. Recently, the paper of Lin et al. [14] discussed the use of regularization techniques with the MFS to solve direct problems, of the Laplace and the Helmholtz equations. The purpose of the present paper is to use the fundamental numerical Green’s function for embedding a precise crack representation with the MFS. Furthermore, in order to mitigate the severe ill-conditioning present in the MFS matrices generated with the NGF like the fundamental solution and the question of proper location of the source points, a Tikhonov regularization has been used to obtain a stabilized solution for the system of linear equations, improving the accuracy. 2 Governing Equations For a two-dimensional linear elastic body ȍ, bounded by the boundary ī, the well-known Navier equation in terms of displacements ‫ݑ‬௜ (generalized i directions displacements) can be written in the form: ‫ܩ‬ (1) ‫ݑ‬ ൅ ܾ௝ ൌ ͲǤ ͳ െ ʹߥ ௞ǡ௞௝ Where ‫ ܩ‬is the shear modulus, ߥ is the Poisson's ratio and ܾ௝ is the body force components. The displacement ‫ݑ‬௜ is solved from the Eq. (1) satisfying the boundary conditions: ‫ݑܩ‬௝ǡ௞௞ ൅

‫ݑ‬௜ ൌ ‫ݑ‬ത௜ ǡ‫݊݋‬ī௨ (2) ‫݌‬௜ ൌ ߪ௝௜ ݊௝ ൌ ‫݌‬௜ ǡ‫݊݋‬ī௣ In above equation, ‫ݑ‬ത௜ and ‫݌‬௜ are the prescribed displacements and tractions on the boundary ī௨ and ī௣ , respectively. The external boundary of the body is ī ൌ ī௨ ‫ ׫‬ī௣ . Note that the LEFN problems are formulated based on the linear elasticity theory above. But due to ҧ the presence of cracks in the elastic medium, there will be surfaces sharing the same geometric position, ҧ this produces some problems in the implementation of numerical methods. Difficulties like singularity of the system matrix or degeneration of the boundary integral equation [3] are expected to occur. So the MFS formulation needs special devices like the NGF procedure discussed in the next section to accommodate this. 3 Fundamental numerical Green’s function The fundamental solution used in this work is the numerical Green’s function [3]. The NGF is written in terms of a superposition of the Kelvin fundamental solution and a complementary part, which ensures that the final result is equivalent to an embedded crack unloaded within the infinite elastic medium subject to a unit applied load, given by ‫כ‬ሺ ௞ሺ ௖ ሺ ‫ݑ‬௜௝ ߦǡ ߯ሻ ൌ ‫ݑ‬௜௝ ߦǡ ߯ሻ ൅ ‫ݑ‬௜௝ ߦǡ ߯ሻ

(3)

‫כ‬ሺ ߦǡ ߯ሻ ‫݌‬௜௝

(4)

ൌ

௞ሺ ‫݌‬௜௝ ߦǡ ߯ሻ

௖ሺ ൅ ‫݌‬௜௝ ߦǡ ߯ሻ

‫כ‬ሺ ‫כ‬ሺ where ‫ݑ‬௜௝ ߦǡ ߯ሻ and ‫݌‬௜௝ ߦǡ ߯ሻ are the fundamental displacements and tractions in j direction at the field ௞ሺ point ߯ due to unit point loads applied at the source point ߦ in i direction, respectively and ‫ݑ‬௜௝ ߦǡ ߯ሻ and ௞ሺ ‫݌‬௜௝ ߦǡ ߯ሻrepresent the known Kelvin’s fundamental solutions for the uncracked body defined in [2]. Here ௖ ሺ ௖ሺ ‫ݑ‬௜௝ ߦǡ ߯ሻ and ‫݌‬௜௝ ߦǡ ߯ሻ stand for complementary components of the problem defined as an infinite space containing crack(s) of arbitrary geometry under applied distributed loads required to cancel the Kelvin's tractions as required in the original fundamental problem. The NGF procedure presents a suitable feature


313

for mesh-free methods, it eliminates the necessity of discretization of the crack surfaces, reducing the point generations of the problem. This is achieved due to the traction-free condition introduced by the ௖ሺ ௖ሺ ߦǡ ߯ሻ and ‫݌‬௜௝ ߦǡ ߯ሻ. complementary solutions ‫ݑ‬௜௝ Analytical expressions for Eqs. (3-4) are limited to general 2-D geometries and usually employ complex variable theory which is expensive in terms of computational implementation. A general alternative to obtain the complementary solutions in a real variable numerical approach can be found in the detailed paper of Telles et al. [3]. Consider ߯ ‫ ב‬ī୤ and using Somigliana’s identity, the complementary solutions can be defined in terms of the following boundary integral equations ௖ ሺ ௞ ሺ ‫ݑ‬௜௝ ߦǡ ߯ሻ ൌ න ‫݌‬௝௞ ߯ǡ ߞ ሻܿ௜௞ ሺߦǡ ߞ ሻ݀īሺߞሻ

(5)

௖ሺ ௞ሺ ‫݌‬௜௝ ߦǡ ߯ሻ ൌ න ܲ௝௞ ߯ǡ ߞ ሻܿ௜௞ ሺߦǡ ߞ ሻ݀īሺߞሻ

(6)

īష

īష

௖ ሺ ௖ ሺߦǡ ߞ ି ሻ is the crack opening displacements of the Green’s function in ߦǡ ߞ ା ሻ െ ‫ݑ‬௜௞ where ܿ௜௞ ሺߦǡ ߞ ሻ ൌ ‫ݑ‬௜௞ ା ି which ī and ī stand for superior and inferior surfaces of the crack ī୤ ൌ īା ‫ ׫‬īି with ߞ ‫ א‬īି and ௞ሺ ܲ௝௞ ߯ǡ ߞ ሻ is originated from the hyper-singular formulation given [1]: ௞ሺ The components ܲ௜௝௞ ሺ߯ǡ ߞ ሻ and ‫݌‬௜௝ ߯ǡ ߞ ሻ are known, hence only the crack opening displacements ܿ௜௞ need to be computed to produce the complementary components of displacement and traction (Eqs. (5-6)) and generate the fundamental numerical Green’s function defined in Eqs. (3-4). Equations (5-6) can be ௖ሺ ௞ሺ ߦǡ ߞ ሻ ൌ െ‫݌‬௜௝ ߦǡ ߞ ሻ and evaluating the solved numerically using a Gaussian quadrature [3]. Making ‫݌‬௜௝ ି limit of Eq. (6) as ߯ հ ī , a hyper-singular boundary integral equation for unknowns ܿ௜௞ can be written ௞ሺ ߦǡ ߞ ሻ ܲ௜௝௞ ሺߞ ǡ ߞ ሻܿ௜௞ ሺߦǡ ߞ ሻ݀īሺߞሻ ൌ െ‫݌‬௜௝

(7) ି

where, the symbol "=" on integral above indicates Hadamard’s finite part integral and ߞ ‫ א‬ī . The same point collocation technique adopted in the reference [3] to solve Eq. (7) is used in this work; hence the ҧ ҧ following square system of equations in matrix notation is generated as ҧ (8) ‫܋܁‬௜ ሺߦ ሻ ൌ െ‫ܘ‬௞௜ ሺߦ ሻ where S is a square matrix with dimension ܰ‫ ݅݌‬ൈ ܰ‫ ݅݌‬dependent only on the crack geometry (one matrix for each crack) and vectors ࢉ௜ ሺߦ ሻ and ࢖௞௜ ሺߦሻ are the unknowns vectors (crack openings) and the independent vector (traction values), respectively, in normal and transversal directions, due to the unit point load at source point ߦ in i direction. Finally, the numerical counterpart of Eqs. (3-4)-i.e., the NGF for LEFM problems is written as ே௣௜ ‫כ‬ሺ ௞ሺ ௞ ሺ ߦǡ ߯ሻ ൌ ‫ݑ‬௜௝ ߦǡ ߯ሻ ൅ ෍ ‫݌‬௝௞ ߯ǡ ߞ௡ ሻ ܿ௜௞ ሺߦǡ ߞ௡ ሻȁ‫ ܬ‬ȁ௡ ܹ௡ ‫ݑ‬௜௝

(9)

௡ୀଵ ே௣௜ ‫כ‬ሺ ௞ሺ ௞ሺ ‫݌‬௜௝ ߦǡ ߯ሻ ൌ ‫݌‬௜௝ ߦǡ ߯ሻ ൅ ෍ ܲ௝௞ ߯ǡ ߞ௡ ሻ ܿ௜௞ ሺߦǡ ߞ௡ ሻȁ‫ ܬ‬ȁ௡ ܹ௡

(10)

௡ୀଵ

4 The method of fundamental solutions Let the boundary value problem in an elastic solid of domain ષ enclosed by a boundary ડ governed by the Navier Equation (1), subjected to mixed boundary conditions as given in Eq. (2) in the absence of body forces. The method of fundamental solutions (MFS) [4] establish that the approximate solution can

314


be constructed by a summation of similar problems solution given by the following superposition ே ‫כ‬ሺ ߦǡ ߯ሻ݀௜ ሺߦ௡ ሻ ‫ݑ‬෤௝ ሺ‫ݔ‬ሻ ൌ ෍ ‫ݑ‬௜௝

(11)

௡ୀଵ ே ‫כ‬ሺ ߦǡ ߯ሻ݀௜ ሺߦ௡ ሻ ‫݌‬෤௝ ሺ‫ݔ‬ሻ ൌ ෍ ‫݌‬௜௝

(12)

௡ୀଵ

where ‫ݑ‬෤௝ ሺ‫ݔ‬ሻ and ‫݌‬෤௝ ሺ‫ݔ‬ሻ are approximations for the displacements and the tractions corresponding to a ‫כ‬ ‫כ‬ and ‫݌‬௜௝ have already been defined (Eqs. (9-10)), and usually ߦ௡ ‫ ב‬ȍ ‫׫‬ point ‫ א ݔ‬ȍ ‫ ׫‬ī, respectively, ‫ݑ‬௜௝ ī is the source point (virtual sources), of unknown intensity ݀௜ , associated with direction i. The indirect problem must be solved to compute the intensity factors ݀௜ . Therefore, the boundary conditions over ī are assumed to be satisfied at M boundary field points ‫ݔ‬௠ , N source points ߦ௡ are arbitrarily chosen as distributed in a similar boundary manner as forming a fictitious boundary surrounding ī (see Fig. 1) and satisfaction of the boundary conditions (2) are discretely imposed at the M points in Eqs. (11-12). This produces a linear system of equations: ே ‫ כ‬ሺ ෍ ‫ݑ‬௜௝ ߦ௡ ǡ ‫ݔ‬௠ ሻ݀௜ ሺߦ௡ ሻ ൌ ‫ݑ‬ത௝ ሺ‫ݔ‬௠ ሻǡ‫ݔ‬௠ ‫ א‬ī௨ ǡ݉ ൌ ͳǡ ǥ ǡ ‫ܯ‬

(13)

௡ୀଵ ே ‫ כ‬ሺ ෍ ‫݌‬௜௝ ߦ௡ ǡ ‫ݔ‬௠ ሻ݀௜ ሺߦ௡ ሻ ൌ ‫݌‬௝ ሺ‫ݔ‬௠ ሻǡ‫ݔ‬௠ ‫ א‬ī௣ ǡ݉ ൌ ͳǡ ǥ ǡ ‫ܯ‬

(14)

௡ୀଵ

ҧ

Figure 1: MFS discretization of a square domain ȍ with a center crack. Applying either Eqs. (11) or (12) for the M discrete field points ‫ݔ‬௠ , The linear system of equations above can be written in matrix notation (15) ‫ ܌ۯ‬ൌ ‫܊‬ where ࡭ is the coefficient matrix, ࢊ the unknown intensity factor vector, and ࢈ the right-hand side vector. Once all the values of ݀௜ are determined, the displacements and the tractions at any point on the boundary can be evaluated using Eq. (11) and Eq. (12), respectively. In addition, the displacement at any point inside the domain can be evaluated using Eq. (11). Note that the matrix ࡭ is not necessarily a square matrix. 4.1 The stress intensity factors Following the idea of the MFS, the stress intensity factors (SIF) can be calculated by the superposition of the fundamental generalized openings of the crack ܿ௜௝ obtained as the solution of Eq. (8) and the intensity factors ݀௜ , written as


315

ே

(16)

ܿ௝ ሺߞ ሻ ൌ ෍ ܿ௜௝ ሺߦ௡ ǡ ߞ ሻ݀௜ ሺߦ௡ ሻ

ǁ

௡ୀଵ

where ܿଵ and ܿଶ are the generalized openings of the crack calculated over the crack line. In this work ǁ ܿ௝ ሺߞሻ is calculated directly at a Gauss station close to the crack tip, using the same Gauss stations that are used to solve Eq. (7) and the SIF are calculated by the expression ǁ ǁ ߤξʹߨ ܿଶ (17) ‫ܭ‬ூ ൌ Ͷሺͳ െ ߥሻ ξ‫ݎ‬௧௜௣ ߤξʹߨǁ ܿଵ (18) Ͷሺͳ െ ߥሻ ξ‫ݎ‬௧௜௣ is the distance from point ߞ, where ܿ௝ is calculated, to the crack tip and ߤ is the known Lamé’s ǁ ‫ܭ‬ூூ ൌ

where ‫ ݎ‬௧௜௣ parameter.

ǁ 5 Tikhonov regularization The MFS frequently gives an ill-conditioned coefficient matrix [13]. Furthermore, in this paper, it was found that the complementary solutions (Eqs. (5-6) ) produce noise like disturbances, causing a severe illconditioning in the coefficient matrix A. Here, in order to overcome this difficulty, the so called Tikhonov’s regularization (TR) has been used to improve the accuracy and stabilize the solution. In the singular value decomposition (SVD) [15] any M by N matrix ࡭ can be factored into ‫ ۯ‬ൌ ‫ ܄܁܃‬୘

(19)

where ‫ ܃‬ൌ ሾ‫ܝ‬ଵ ǡ ‫ܝ‬ଶ ǡ ‫ ڮ‬ǡ ‫ܝ‬ெ ሿ is an M by M orthogonal matrix, ‫ ܄‬ൌ ሾ‫ܞ‬ଵ ǡ ‫ܞ‬ଶ ǡ ‫ ڮ‬ǡ ‫ܞ‬ே ሿ is an N by N orthogonal matrix and ‫ ܁‬is an M by N diagonal matrix with nonnegative diagonal elements ߪ௜ , called singular values, arranged in decreasing value order, ߪଵ ൒ ߪଶ ൒ ‫ ڮ‬൒ ߪ୫୧୬ሺெǡேሻ . In this work, the number o virtual sources can be smaller than the number of field points, so ߪ୫୧୬ሺெǡேሻ ൌ ߪே . Given a SVD of matrix ‫ ۯ‬in Eq. (19), the solution of the least square problem (15) can be obtained without difficulty, inverting matrix ‫ ۯ‬using the following ே

‫ ܌‬ൌ ‫ିۯ‬ଵ ‫ ܊‬ൌ ‫ି ܁܄‬૚ ‫܃‬୘ ‫ ܊‬ൌ ෍

൫‫ܝ‬୘ ൯௜ ‫܊‬

௜ୀଵ

ߪ௜

‫ܞ‬௜

(20)

The terms in the Eq. (20) with small values of ߪ௜ will be large producing an unstable solution; hence, one can use the regularization technique to filter the errors related to these small singular values. Consider the least squares problem obtained by augmenting the least squares problem of Eq. (15) in the following manner [16]: ݉݅݊ሼԡ‫܌ۯ‬ఈ െ‫܊‬ԡଶଶ ൅ ԡ۷ߙ‫܌‬ఈ ԡଶଶ ሽ

(21)

where ԡήԡଶ denotes the Euclidean norm, ۷ denotes the identity matrix and ߙ is the so called regularization parameter. Based on SVD, the TR solution can be written as ே

‫܌‬ఈ ൌ ෍ ݂௜ ௜ୀଵ

൫‫ܝ‬୘ ൯௜ ‫܊‬ ߪ௜

‫ܞ‬௜

(22)

where ݂௜ is the so called filter factors, defined as ݂௜ ൌ

ߪ௜ଶ ǡ݅ ൌ ͳǡʹǡ ‫ ڮ‬ǡ ܰ ߪ௜ଶ ൅ ߙ ଶ

(23)

316


Note that if ߙ ଶ is small, then the fit will be good, but the solution will be dominated by the rounding off errors naturally generated by the MFS-NGF. In addition, another kind of regularization has been studied, the damped singular value decomposition (DSVD), which instead of using the filter factor in Eq. (23), ݂௜ ൌ ߪ௜ Ȁሺߪ௜ ൅ ߙሻ [16] was used. Since unsatisfactory results have been obtained, only the TR method has been finally adopted here. The abbreviation MFS-NGF-TR will be used to denote the method of fundamental solutions described in section 4, using the numerical Green’s functions defined in Eqs. (9-10) as the fundamental solution and the Tikhonov regularization presented above. 6 Numerical results For all the examples, the virtual sources were distributed in similar fashion to the actual boundary, at a fixed distance d of the boundary points, as shown in Fig. 1. No improvement was observed for different shapes of the fictitious boundary. The adopted distance d depends on the crack length and the dimensions of the problem, it has been defined as ݀ ൌ ʹܽ െ ‫ݓ‬Ȁʹ‫ܮ‬. The number of field points ‫ ݔ‬is always the same as the number of virtual sources ߦ, producing a square system of equations (15). It is worth noticing that the MFS-NGF formulation can give poor results if the system of equations (15) is solved with standard least square or Gauss elimination without use of regularization for the values of d. There are some values of distances d and some values of crack length a that the traditional least square or the Gauss elimination may still give good results, but to discover such geometries can a complex task [8]. The major feature of the numerical procedure presented here is the possibility of establishing a fixed distance d of the virtual sources, for each problem, not too far from the boundary and the accuracy of the solution is found not too sensitive with respect to the virtual source locations as discussed in [13], leaving only the regularization parameter ߙ to be selected. The NGF procedure described in ref. [3] for the BEM was strictly followed. The crack line subdivision has been fixed in 6 with 12 Gauss points in each segment, for numerical integration purposes. The BEM code used to compare results with the MFS-NGF was the one developed by Telles et al. [3]. The boundary discretization used for the BEM-NGF examples uses linear elements, and the functional nodes are at the same points used for the MFS-NGF results. Finally, the stress intensity factors, SIF, are normalized accordingly, therefore units are omitted.

Figure 2: Plate with a center crack.

Figure 3: Plate with an edge crack.

6.1 Example 1 In this example the plate with a center crack problem of Fig. 2 is analyzed using the MFS-NGF-TR procedure. The boundary was discretized with 108 points, concentrating points over boundary sectors


presenting high solution gradients and 108 virtual source points have been used. The Young modulus is ‫ ܧ‬ൌ ʹʹͲͲǤͲ, Poisson ratio ߥ ൌ ͲǤͳ, traction ‫ ݌‬ൌ ͳǤͲ and the dimensions W=10 and L=10. 2a/W 0.1 0.2 0.3 0.4 0.5 0.6 0.7

‫ܭ‬ூ BEM-NGF 1.79052 1.82144 1.88057 1.97274 2.10965 2.30596 2.64450

‫ܭ‬ூ MFS-NGF 1.79231 1.81180 1.85984 1.94332 2.07980 2.32333 2.57857

ߙ 0.3725 0.1905 0.1830 0.1685 0.1435 0.1000 0.0525

Table 1: Comparative results for the center crack for various values of a. As can be seen in Table 1, the SIF computations for the MFS-NGF-TR procedure and the BEM-NGF show only small differences for all crack lengths a. 6.2 Example 2 In this example the edge crack problem of Fig. 3 is analyzed using the BEM-NGF procedure described in Ref. [3] and the MFS-NGF-TR. As before, for the MFS-NGF-TR solution, the boundary was described with 119 points (no symmetry), concentrating points over boundary sectors presenting high solution gradients and 119 virtual source points have been used. The Young’s modulus used is ‫ ܧ‬ൌ ͷǤͲ, Poisson ratio ߥ ൌ ͲǤ͵, traction ‫ ݌‬ൌ ͳǤͲ and the dimensions W=10 and L=50. a/W 0.1 0.2 0.3 0.4 0.5

‫ܭ‬ூ BEM-NGF 1.1870 1.3487 1.6307 1.9495 2.4112

‫ܭ‬ூ MFS-NGF 1.1816 1.3893 1.6859 2.1416 2.8950

ߙ 0.002185 0.002845 0.002432 0.001112 0.000101

Table 2: Comparative results for the edge crack for various values of a. The SIF approximations for the numerical procedures, in Table 2, show quite near results for most the practical crack lengths, in spite of the fact that the computed SIF values for the MFS-NGF-TR procedure have been directly calculated from the displacement fields whereas the SIF for the BEM-NGF have been obtained from stresses [3]. 7 Concluding remarks The application of the MFS-NGF-TR procedure to solve LEFM problems is discussed in this work. Here, the main objective was achieved using regularization to reduce the customary sensitivity of the results due to the location of the sources. The examples presented show satisfactory results and the MFS-NGF with TR has been seen to profit from not requiring any mesh or integrations over the main problem boundary. This has been possible, however, mainly due to the Tikhonov regularization employed. Acknowledgements: CNPq and PEC/COPPE/UFRJ. References [1] Guimaraes S, Telles JCF. General Application of Numerical Green’s Functions for SIF Computations With Boundary Elements. Computer Modelling in Engineering and Sciencess 2000; 1:131-139.

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[2] Brebbia, CA, Telles, JCF, Wrobel, LC. Boundary Elements Techniques: Theory and Applications in Engineering 1984. Springer-Verlag, Berlin Heidelberg New York [3] Telles JCF, Castor GS, Guimaraes S. A numerical Green’s function approach for boundary elements applied to fracture mechanics. Int. Journal for Numerical Methods in Engineering 1995; 38:3259-74. [4] Kupradze VD, Aleksidze MA. The method of functional equations for the approximate solution of certain boundary value problems. U.S.S.R. Computational Mathematics and Mathematical Phisics 1964;4(4):82-126. [5] Marin L. Stable MFS Solution to Singular Direct and Inverse Problems Associated with the Laplace Equation Subjected to Noisy Data. Computer Modelling in Engineering and Sciences 2008; 37:203242. [6] Poullikkas A, Karageorghis A, Georgiou G. The method of fundamental solutions for threedimensional elastostatics problems. Computers and Structures 2002; 80:365-370. [7] Antonio J, Tadeu, A, Godinho L. Sound wave propagation modeling in a 3D absorbing acoustic dome using the Method of Fundamental Solutions. ICCES 2007; 3:157-162. [8] Fairweather G, Karageorghis A. The method of fundamental solutions for elliptic boundary value problems. Advances in Computational Mathematics 1998; 9:69-95. [9] Guimaraes S, Telles JCF. The method of fundamental solutions for fracture mechanics-Reissner’s plate application. Engineering Analysis with Boundary Elements 2009; 33:1152-1160. [10] Alves CSJ, Leitao VMA. Crack analysis using an enriched MFS domain decomposition technique. Engineering Analysis with Boundary Elements 2006; 30:160-6. [11] Karageorghis A, Poullikkas A, Berger JR. Stress intensity factor computation using the method of fundamental solutions. Comput Mech 2006; 37:445-454. [12] Berger JR, Karageorghis A, Martin PA. Stress intensity factor computation using the method of fundamental solutions: mixed-mode problems. IJNME 2007; 69:469-483. [13] Ramachandran PA. Method of fundamental solutions: singular value decomposition analysis. Communications in Numerical Methods in Engineering 2002; 18:789-891. [14] Lin J, Chen W, Wang F. A new investigation into regularization techniques for the method of fundamental solutions. Mathematics and Computers Simulation 2011; 81:1144-1152. [15] Golub GH, Van Loan CF. Matrix Computations. The Johns Hopkins University Press, Baltimore, 1996; 3 rd ed. [16] Aster RC, Borchers B, Thurber CH. Parameter estimation and inverse problems. Elsevier Academic press 2005; USA.


An SGBEM implementation of an energetic approach for mixed mode delamination Roman Vodiˇcka1 , Vladislav Mantiˇc2 1

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

[email protected] 2

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

[email protected] Keywords: symmetric Galerkin BEM, weak interface, crack, delamination, debonding, energetic approach, damage, plastic slip.

Abstract. Recent increase of applications of layered structures requires developing suitable models for the onset and propagation of damage in this kind of structures. The interface between layers, usually represented by a relatively thin adhesive layer, can be partially or fully damaged. This situation is usually referred to as delamination. In the present contribution, a numerical approach for solving delamination problems is developed and tested numerically to show the behavior of the model and to assess its suitability in a particular situation. The energetically based approach employed includes damage of the interface in the form of interface cracks and distinguishes between fracture modes by introducing an interface plastic slip variable. A model of interface failure Let us consider a body defined by a planar domain Ω⊂R2 with a bounded Lipschitz boundary ∂ Ω=Γ. Let n denote the unit outward normal vector defined at ΓS – the smooth part of Γ. A split into two non-overlapping subdomains ΩA and ΩB whose respective boundaries are denoted as ΓA =∂ ΩA and ΓB =∂ ΩB , will be considered hereinafter for the sake of simplicity, Figure 1.

wA

nA ΓtA

ΓAu

Γc

ΩB

ΓtB nB

ΓBu

x2

O

ΩA

x1

wB

Figure 1: Problem configuration including an adhesive layer at the interface between two subdomains The common part of ΓA and ΓB called interface is denoted as Γc =ΓA ∩ ΓB . The boundary conditions prescribed on a part of the outer boundary represent a hard-device loading, i.e. prescribed displacements u=w over ΓAu or ΓBu . The remaining parts of the outer boundaries, denoted as ΓtA or ΓtB , are traction free, i.e. prescribed tractions t=0. The interface Γc is considered as a very thin adhesive layer represented by a continuous spring distribution with normal and tangential elastic stiffnesses kn and kt , respectively. Following [6, 7, 8] it is assumed that the subdomains can debond along the interface Γc , this debonding process being quasistatic and rate-independent. During this process the material of the adhesive layer is damaged. This is modelled by a scalar damage variable ζ which varies at each interface point between one and zero: values one and zero, respectively, corresponding to undamaged and fully damaged adhesive at a particular point. In addition to this variable, a plastic tangential slip variable π is considered at the interface, which allows us to distinguish between fracture modes I and II in

319

320


view of the experimental observations of interface crack growth. These observations indicate that the energy dissipated in mode II is usually significantly greater than that dissipated in mode I and also correspondingly the associated plastic zones in the adjacent bulk are larger in mode II than in mode I. Thus some additional dissipated energy is considered for interface fracture in mode II as a useful and practical approximation of the plastic phenomena appearing in relatively narrow plastic zones in the bulk located in the interface vicinity. Let the energy stored in the structure [6, 8] obeying the aforementioned type of interface damage and a kinematic-hardening-plasticity model [7], with the plastic slope kH , be expressed as ⎧ 1 η η ⎪ ⎪ u · t dΓ+ ⎪ ∑ ⎪ Γη 2 uη = wη (τ) on Γηu , ⎪ ⎨ η=A,B , if E (τ,u,ζ ,π) = (1) [u]n ≥ 0, 0 ≤ ζ ≤ on Γc 1 1 1 ⎪ ζ kn [u]2n + kt ([u]t −π)2 + kH π 2 dΓ ⎪ ⎪ 2 2 2 Γc ⎪ ⎪ ⎩ +∞, elsewhere. The first integral, representing the elastic strain energy in the bulk Ωη , is expressed in its boundary form (taking into account that tη =tη (uη )), which is advantageous when the numerical technique for solving elastic problems in Ωη is also boundary based. The first condition on Γc is the Signorini condition of unilateral contact, where the relative normal displacement [u]n = uB −uA ·nA is introduced. Similarly, the relative tangential displacement [u]t is defined. The dissipation potential for a rate independent process can be represented by a degree 1 homogeneous functional [4]. Considering both processes of the interface damage and plastic slip, the interface dissipation potential is given as follows: ⎧

⎨ ˙ dΓ, if ζ˙ ≤ 0 on Γc Gd |ζ˙ | + σyield |π| ˙ = Γc (2) R(ζ˙ , π) ⎩+∞, elsewhere. The parameter Gd is the (minimum) interface fracture energy required to fully debond the interface following the linear elastic-brittle part of the interface constitutive law. In particular, Gd represents the interface fracture energy in mode I. σyield is the interface yield shear stress for initiating the plastic slip along the interface. The rate-independent evolution is governed by the system of doubly nonlinear variational inclusions ∂u E (t, u, ζ , π) 0, ˙ ˙ + ∂ζ E (t, u, ζ , π) 0, ∂ζ˙ R(ζ , π)

(3)

˙ + ∂π E (t, u, ζ , π) 0, ∂π˙ R(ζ˙ , π) where the symbol ∂ refers to partial subdifferential relying on convexity of the pertinent functionals with respect to each particular variable, see [6]. The solution of nonlinear inclusions in engineering practice can be replaced by a generalization of the weak solution concept called energetic solution which was developed, for example, in [5]. An approach based on this energetic solution leads to a relatively simple computational implementation and also provides, from a mathematical point of view, a generalization of convex optimization to the case of non-convex energies which unfortunately occurs in the present model. The energetic solution to the rate-independent problem (3) is a process (u(τ), ζ (τ), π(τ)), τ∈[0; T ], for which the following three conditions are satisfied: Energy equality E (T, u(T ), ζ (t), π(T )) +

T 0

˙ R(ζ˙ (τ), π(τ))dτ =

T 0

∂τ E (τ, u(τ), ζ (τ), π(τ))dτ + E (0, u0 , ζ0 , π0 ), (4a)

˜ it holds ˜ ζ˜ , π) Stability inequality For any suitable (u, ˜ + R(ζ˜ − ζ (τ), π˜ − π(τ)). ˜ ζ˜ , π) E (τ, u(τ), ζ (τ), π(τ)) ≤ E (τ, u,

(4b)

Initial conditions u(0) = u0 ,

ζ (0) = ζ0 ,

π(0) = π0 .

(4c)


321

Numerical implementation of the model A numerical approach to obtain the above defined energetic solution usually considers time and spatial discretizations separately. As long as the problem can be formulated in terms of the boundary data only, see (1) and (2), the spatial discretization naturally leads to boundary elements. The standard approximation of the boundary distributions for u, ζ , π uses linear boundary elements, though, due to its nature, ζ can be approximated also by constant boundary elements. The time discretization provides the solution at time-steps defined by an increment δ such that τ k =kδ for k=1, 2, . . . Tδ . The time-stepping procedure starts by the solution at k=1 calculated from the initial conditions (4c). The stability condition provides the minimization problem for the solution at the successive step k, once the solution for the time step k−1 is known, c.f. [6], minimize H k (u, ζ , π) = E (kτ, u, ζ , π) + R(ζ − ζ k−1 , π − π k−1 ).

(5)

Unfortunately, the functional H is not convex. Its non-convexity requires applying a special numerical treatment in a minimization algorithm. The alternative minimization algorithm (AMA) proposed in [1] has been used to split the minimization to alternation between minimization with respect to (u, π) and with respect to ζ , each of these being a minimization of a convex functional. Such alternation, however, does not necessarily lead to a global minimizer which represents the energetic solution. Therefore a back-tracking algorithm (BTA) to control the process has been utilized, providing that the energy equality, in discrete form converted to a twosided inequality is satisfied, see [8]. The two-sided inequality can be written in the following form to compare the energies in two subsequent time steps: k

kδ (k−1)δ

∂τ E (τ, uk , ζ k , π k )dτ ≤ E (kδ , uk , ζ k , π k ) − E ((k−1)δ , uk−1 , ζ k−1 , π k−1 ) + R(ζ k − ζ k−1 , π k − π k−1 ) ≤ kδ

(k−1)δ

∂τ E (τ, uk−1 , ζ k−1 , π k−1 )dτ. (6)

Although there is no guarantee that the process converges to the global minimum, the practical experience with BTA, however, shows that it provides a solution with lower energy than that obtained by mere AMA. The role of the Symmetric Galerkin BEM (SGBEM) in the present computational procedure is to provide the elastic solution for given boundary data in each subdomain separately in order to calculate the elastic strain energy in these subdomains. Thus, in each time step and in each iteration of the minimization algorithm, the SGBEM code calculates unknown tractions along Γc ∪ Γu , assuming the displacements at Γc to be known from the used minimization procedure, in the same way as proposed and tested using a collocational BEM code in [7]. As long as the whole model is based on energies, the chosen SGBEM can also be deduced from energetical principles [11, 12]. This fact guarantees the positive definite character of the strain energy computed by SGBEM, in difference to the classical collocational BEM. As follows from the previous explanations, the SGBEM code is used merely to elastic strain energy computation in the bulk and does not include the solution of the whole interface problem, which is left to the suitable minimization algorithm, see also [7]. A conjugate gradient based algorithm with constraints, see [3], can be used in the minimization procedure with respect to u as the pertinent energies are quadratic. Numerical example The present formulation has been tested numerically by a computer code, which uses the SGBEM for finding elastic solution in each subdomain and a constrained minimization conjugate gradient based method for finding the interface unknowns: damage ζ k , plastic slip π k and displacements uk at each time step. The geometry and load configuration of the example shown in Figure 2 are motivated by the pull-push shear test [2] known in several engineering applications. In this case the prescribed displacements in Figure 2 are w= (w1 , 0). For a comparison, the case motivated by the double cantilever beam test [9] is also analysed with

322


x2

x1

Γt

Γc

Γt

Γu

h

L w

Lc Figure 2: Geometry for the model considered in the numerical solution the prescribed displacements given by w= (0, w2 ), see Figure 2. In either case, debonding occurs between the elastic domain and the rigid foundation. The plane strain state is considered. The dimensions of the layer are L=250mm and h=12.5mm. The layer is bonded to the rigid foundation along a part of its bottom face in the extent of Lc =225mm. The layer is made of aluminum with elastic properties E=7×104 MPa and ν=0.35. The adhesive material is epoxy resin, with elastic properties Ee =2.4×103 MPa and νe =0.33. Considering the adhesive layer thickness he =0.2mm, the corresponding stiffness parameters are Ee (1−νe ) e) =1.8×104 MPa mm−1 and kknt = 2(1−ν computed following [10] as kn = he (1+ν 1−2νe =4. e )(1−2νe ) The parameters that govern the crack growth in√the adhesive layer are: the fracture energy in mode I Gd =1×10−2 mJ mm−2 , plastic slip stress σyield =0.56 2kt Gd . The hardening slope for plastic slip is kH = k9t . The loading is applied on the right-hand side of the aluminum layer. The prescribed displacements of the hard-device are increasing during the loading process. The incrementally prescribed loading is given by the relation wλi =2λ ×10−4 wi , where w1 =1mm or w2 =1mm, the factor λ changing from an initial value λ =1 until the total breakage of the interface occurs. The relation between the resultant reaction-force along the adhesive zone and the imposed displacement is shown in Figure 3. Let us look at the horizontal loading. The slope of the curves changes for λ =21 −3 (w21 1 =4.2×10 mm) due to first plastic deformations in the interface. The first sudden decrease of the reaction −2 forces, both horizontal F1 and vertical F2 , corresponds to the crack initiation for λ =54 (w54 1 =1.08×10 mm). Subsequent crack propagation can be seen at load-steps λ =81 and λ =93, where a decrease of total reaction force appears. At the end of the process for λ =94, the rest of the elements along Γc abruptly breaks in one time-step. In fact, such a behaviour may be expected in view of the related results of the pull-push shear test, see [2, 7]. The case of vertical loading behaves differently. First, no plastic deformation in the interface occurs, it is directly damaged as the applied load is normal with respect to the interface. Nevertheless, the crack initiates at −2 the same time step λ =54 (w54 2 =1.08×10 mm). Subsequent crack propagation proceeds in many steps with =0.2444mm), the an apparent decrease of total reaction force. At the end of the process for λ =1222 (w1222 1 rest of the elements along Γc abruptly breaks in one time-step, however, unlike the previous load case, at this moment the crack extends to almost three quarters of the initially bonded interface. For the horizontal loading, Figure 4(a) shows the distributions of displacements and tractions along Γc before the crack appears, λ =53, and right after its initiation. The normal u2 and tangential u1 displacements are plotted scaled by a factor kn or kt , respectively, in order to show the relation ku=−t in an undamaged adhesive. It is clearly seen that in front of the crack tip a contact zone appears before and also after crack initiation. It is the part where the normal displacements are zero and the tractions exhibit a concentration. The tangential components do not obey this relation in the zone of the plastic deformation in the adhesive which developed before the crack has appeared. It is also clear that after debonding the traction along a part of Γc vanishes, where the surface separates from the foundation. Where Γc remains in contact, normal displacement is zero and contact traction is positive (negative in the graphs because −t is plotted). Similar pictures are also plotted for the vertical loading, see Figure 4(b). The contact zones in the front of the crack tip are also observed. As no plasticity appears, the relation ku=−t is confirmed in Γc , with the exception of the contact zone and of the debonded end of the interface. Conclusions An energy based model for interface debonding with a mode sensitive crack growth under rate-independent conditions has been considered. The sensitivity of the model to opening Mode I and shearing Mode II cracks has been achieved by considering two internal variables along the interface: damage parameter ζ and plastic

250 200 150 100 50 0

323

25 20 15 10 5 0

F1 F2

0.004

0.008

0.012

F2 [Nmm−1 ]

F1 [Nmm−1 ]


0.016

w1 [mm]

(a) 40 F1 F2

30

30

20

20

10

10

0 0

0.05

0.1

0.15

0.2

F2 [Nmm−1 ]

F1 [Nmm−1 ]

40

0 0.25

w2 [mm]

(b) Figure 3: Resultant force as a function of imposed displacement: (a) horizontal load, (b) vertical load slip π. The numerical implementation of spatial discretization via SGBEM has permitted the whole problem to be defined only by boundary and interface data. A simple 2D example has validated the model. The two load cases result in totally different delamination mechanisms which are nicely illustrated by the plots of resultant forces as functions of imposed displacements. Acknowledgement The authors are grateful to Prof. Tomáš Roubíˇcek (Charles University of Prague) and Dr. Christos Panagiotopoulos (University of Seville) for fruitful discussions. The part of the work has been accomplished during the stage of R.V. at the University of Seville, supported by the Spanish Ministry of Education (Ref. SAB2010-0082). R. V. acknowledges partial support from the grant VEGA 1/0201/11. V.M. acknowledges support from the Junta de Andalucía and European Social Fund through the Project of Excellence P08-TEP-04051. References [1] B. Bourdin, A. Francfort, and J.J. Marigo. Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids, 48, 797–826, (2000). [2] P. Cornetti and A. Carpinteri. Modelling of FRP-concrete delamination by means of an exponentially softening law. Eng. Struct., 33, 1988–2001, (2011). [3] Z. Dostál. Optimal Quadratic Programming Algorithms, Vol23 Springer Optimization and Its Applications. Springer, (2009). [4] A. Mielke, T. Roubíˇcek, and J. Zeman. Complete damage in elastic and viscoelastic media and its energetics. Comput. Methods Appl. Mech. Engrg., 199, 1242–1253, (2010). [5] A. Mielke and F. Theil. Mathematical model for rate-independent phase transformations with hysteresis. Nonl. Diff. Eqns. Appl., 11, 151–189, (2004). [6] T. Roubíˇcek, M. Kružík, and J. Zeman. Delamination and adhesive contact models and their mathematical analysis and numerical treatment. V. Mantiˇc (Ed) Mathematical Methods and Models in Composites, Imperial College Press, (2012). [7] T. Roubíˇcek, V. Mantiˇc, and C. Panagiotopoulos. Quasistatic mixed-mode delamination model. Discrete and Cont. Dynam. Syst., accepted, (Preprint No.2011-020, Neˇcas center, Prague). [8] T. Roubíˇcek, T. Scardia, and C. Zanini. Quasistatic delamination problem. Continuum Mech. Thermodyn., 21, 223–235, (2009).

25 20 15 10 5 0 -5

kt u1 ,λ =53 −t1 ,λ =53

25 20 15 10 5 0 -5

50

100 150 x1 [mm]

50

100 150 x1 [mm]

kn u2 ,λ =53 −t2 ,λ =53

0

kt u1 ,λ =54 −t1 ,λ =54

0

4 3 2 1 0 -1 -2

200 kn u2 , (−t2 ) [MPa]

0 kt u1 , (−t1 ) [MPa]

kn u2 , (−t2 ) [MPa]


kt u1 , (−t1 ) [MPa]

324

4 3 2 1 0 -1 -2

200

50

100 150 x1 [mm]

200

kn u2 ,λ =54 −t2 ,λ =54

0

50

100 150 x1 [mm]

200

0 kt u1 , (−t1 ) [MPa]

kn u2 , (−t2 ) [MPa]

kt u1 ,λ =53 −t1 ,λ =53

8 6 4 2 0 -2 -4

50

100 150 x1 [mm]

kt u1 ,λ =54 −t1 ,λ =54

0

50

100 150 x1 [mm]

5 4 3 2 1 0 -1

200

200

kn u2 ,λ =53 −t2 ,λ =53

0 kn u2 , (−t2 ) [MPa]

kt u1 , (−t1 ) [MPa]

(a) 8 6 4 2 0 -2 -4

5 4 3 2 1 0 -1

50

100 150 x1 [mm]

200

kn u2 ,λ =54 −t2 ,λ =54

0

50

100 150 x1 [mm]

200

(b) Figure 4: Distribution of tractions and displacements along the bonded and debonded interface before and right after the crack initiation, (a) horizontal loading, (b) vertical loading. [9] L. Távara, V. Mantiˇc, E. Graciani, J. Cañas and F. París. Analysis of a crack in a thin adhesive layer between orthotropic materials: an application to composite interlaminar fracture toughness test. CMES – Comp. Model. Eng. Sci., 58, 247–270, (2010). [10] L. Távara, V. Mantiˇc, E. Graciani, and F. París. BEM analysis of crack onset and propagation along fibermatrix interface under transverse tension using a linear elastic-brittle interface model. Eng. Anal. Bound. Elem., 35, 207–222, (2011). [11] R. Vodiˇcka, V. Mantiˇc, 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. Sci., 17, 173–203, (2007). [12] R. Vodiˇcka, V. Mantiˇc, and F. París. Two variational formulations for elastic domain decomposition problems solved by SGBEM enforcing coupling conditions in a weak form. Eng. Anal. Bound. Elem., 35, 148–155, (2011).


MULTIDOMAIN SBEM ANALYSIS OF TWO DIMENSIONAL ELASTOPLASTIC-CONTACT PROBLEMS Panzeca T.1,a, Terravecchia S. 1,b and Zito L.1,c 1

Dip.to di Ingegneria Civile, Ambientale e Aerospaziale, University of Palermo, Italy. a [email protected], [email protected], [email protected].

Keywords: symmetric BEM; elastoplasticity; contact-detachment; substructuring. Abstract: The Symmetric Boundary Element Method based on the Galerkin hypotheses has found application in the nonlinear analysis of plasticity and contact-detachment problems, but dealt with separately. In this paper we wants to treat these complex phenomena together. This method works in structures by introducing a subdivision into sub-structures, distinguished into macroelements, where elastic behaviour is assumed, and bem-elements, where it is possible for plastic strains to occur. In all the sub-structures, elasticity equations are written and regularity conditions in weighted (weak) form and/or in nodal (strong) form between boundaries have to be introduced, to attain the solving equation system. Introduction The present paper shows a strategy to perform the elastoplastic-contact-detachment analysis by using the Symmetric Boundary Element Method based on the Galerkin (SGBEM) hypotheses in a simultaneous analysis obtained by solving the non-linear problems of elastoplasticity and contact-detachment using Linear Complementarity Problem (LCP) in an incremental approach. - Plasticity problems. In the plastic analysis carried out using the symmetric BEM, it is necessary to distinguish a computing phase for the elastic response to all the actions, including the volumetric (body forces and plastic strains) ones, and a subsequent one for plastic strain evaluation, stored during the loading process [1]. In a first phase the Somigliana Identities (SIs) of the displacements and the tractions, both evaluated on the boundary, are employed through a weighting process. In a subsequent phase the stresses have to be evaluated in each bem-element and a predictor-corrector process has to be performed in order to evaluate the plastic strains stored in the bem-elements where the stress violates the elastic yield domain. In both phases, strong singular integrals are involved in the domain integrals when stresses and tractions caused by volumetric actions have to be evaluated [1,2]. - Contact-detachment problems. On the basis of the boundary integral method, in its symmetric formulation, the frictionless unilateral contact between two elastic bodies was studied according to the Signorini formulation [3]. A boundary discretization by boundary elements of the two bodies in contact leads to an algebraic formulation in the form of linear complementarity problem. - Elastoplasticity and Contact-detachment. The analysis of two bodies in contact having elastoplastic behaviour can be performed simultaneously, using an LCP analysis by alternating the contact-detachment phenomenon with the plasticity. This proves to be advantageous when this analysis is carried out through the symmetric BEM, mainly for two reasons: The contact-detachment process proves to have immediate execution because it is carried out through comparison between generalized quantities evaluated along the boundary elements and the reference values in weighted form. At every step characterizing the previous phase, an elastoplastic analysis is made in accordance with the predictor-corrector strategy. 1. The equation system governing the elastoplastic-contact problems In this section the integral equations governing the elastoplastic-contact problems are shown.

325

326


Let us consider a bi-dimensional body having domain : and boundary * , subjected to actions acting in its plane: - forces f2 at the portion * 2 of free boundary, - displacements u1 imposed at the portion *1 of constrained boundary, - body forces b and plastic strains İ p in : . The external actions f2 , u1 , b may increase separately or simultaneously through the multiplier E . In the hypothesis that the physical and geometrical characteristics of the body are zone-wise variables, an appropriate subdivision of the domain into bem-elements is introduced. This subdivision involves the introduction of an interface boundary * 0 between contiguous bem-elements and, as a consequence, two new unknown quantities arising in the analysis problem, i.e. the displacements u 0 and the tractions t 0 vectors, both referring to interface boundaries. Let us start by imposing for each bem-e the classical SIs:

³ G f d * ³ G ( u ) d * ³ G V İ d : ³ G b d : ³ G f d * ³ G (u) d * ³ G V İ d : ³ G b d : ³ GV f d * ³ GV (u) d * ³ GVV İ d : ³ GV b d : p

u

uu

*

ut

*

u

:

uu

:

p

t

*

ı

tu

tt

*

t

:

tu

:

(1a-c)

p

u

*

*

t

:

:

u

These provide the displacements, tractions and stresses in the unbounded domain caused by layered mechanical jumps f and double-layered kinematical ones ( u ) as well as by volumetric actions İ p (volumetric distortions) and b (body forces) both in the : domain. The operators G pq are the Fundamental Solution matrices, whose symbolism was introduced by Maier and Polizzotto [4]; the sub-indices p=u,t,ı and q=u,t,ı indicate the effect and the dual quantity in an energetic sense associated with the cause, respectively. For each substructure we can write the following integral eqautions 0 AX A 0 X 0 AV p E Lˆ X A X A p E Lˆ Z A 0

0

00

0

0V

0

(2a-c)

ı aV X aV 0 X 0 aVV p E ˆlV

These are obtained by weighting all the coefficients of eqs.(1a,b) computed on the boundaries according to the Galerkin approach [3-5]. The vector X collects the sub-vectors F1 , U 2 , whereas the vector X0 collects the sub-vectors F0 , U0 along the interfaces. The vector Z 0 collects the generalized (or weighted) diplacement and traction vectors defined in the interface boundary elements, obtained as a weighted response to all the known actions, amplified by E , and unknown actions, regarding boundary and domain quantities. The vector ı represents the stress, evaluated at the Gauss points, due to all the known actions, amplified by E , and unknown actions. The vector p represents the plastic strains defining the plastic distribution İ p Ȍ p p inside each bem-e. By performing a variable condensation through the replacement of the X vector extracted from eq.(2a) into eqs.(2b,c), one obtains:

X A 1[ A 0 X0 AV p E Lˆ ] Z D X D p E Zˆ 0

00

0

0V

0

(3a-c)

ı dV 0 X0 dVV p E ıˆ The latter are the characteristic equations of each bem-e. Because the body is subdivided into m bem-elements, for each of these the eqs.(3a-c) can be written. Thus we obtain three global relations connecting all the generalized quantities and the stresses related to the bemelements considered, formally equal to eqs.(3a,c), but regarding the characteristic equations of the system. Let us introduce the strong and weak coupling conditions between adjacent bem-elements:


X 0 Eȟ 0 strong coupling condtions Z =0 E weak coupling condtions 0

327

(4a,b)

ȟ 0 being the nodal interface vector which collects the mechanical F0 and kinematical U 0 unknowns of the assembled system. Using eqs.(4a,b), eqs.(3a-c) become: X A 1[ A 0 Eȟ 0 AV p E Lˆ ] K ȟ K p E fˆ 0 0V

00 0

(5a-c)

0

ı k V 0 ȟ 0 k VV p E ıˆ Let us perform a new variable condensation through the replacement of ȟ 0 vector extracted from eq.(5b) into eqs.(5a,c), thus obtaining:

X A 1[ A 0 Eȟ 0 AV p E Lˆ ] ȟ K 1[K p E fˆ ] 0

0V

00

(6a-c)

0

ı Kp E ıˆ e Eqs.(6a,b) provide the elastic solution in terms of nodal forces F1 , F0 and in terms of nodal displacements U 2 , U 0 . In detail, the quantities F0 on * 0 and U 2 on * 2 govern the contact-detachment phenomenum. Eq.(6c) provides the stress at the strain points of each bem-e as a function of the volumetric plastic strain p and of the external actions ıˆ e , the latter amplified by ȕ . The matrix K , defined as the self-stress influence matrix of the assembled system, is a square matrix having 3mx3m dimensions, with m bem-elements. It is fully-populated, non-symmetric and semi-definite negative. The evaluation of this matrix only involves knowledge of the material elastic characteristics and of the structure geometry. This equation is used to evaluate the trial stress in the predictor phase, whereas the first term is utilized to perform the corrector phase, in order to obtain the collapse load factor. The reader can refer to Zito et al. [5] for a more detailed discussion of the characteristics of this equation introduced for a multidomain SGBEM problem. 2. The incremental contact-detachment algorithm Inside the topic of the SGBEM, in order to reach the analytical solution to this frictionless contactdetachment problem, an iterative LCP procedure can be employed once the incremental elastic analysis has been performed using eqs.(6a,b). For this purpose we remember that the unknown vectors F0 n+1 , U 2A n+1 and U 2B n+1 , obtained by eqs.(6a,b) refer to the nodes of the in-contact boundary and to the nodes of the detached one at the generic load increment n 1 . Indeed, the vector F0 F0A F0B represents the nodal forces of the body A, computed in the contact boundary * 0A and the vectors U 2A and U 2B represent the nodal displacements of the free boundaries of * 2A and * 2B respectively. With reference to the system of the two in-contact bodies, whose boundaries are discretized into boundary elements, the contact-detachment phenomenon can be computed by rewriting in discrete form the classical Signorini equations via SGBEM [3] A N 2

U

A 2 n+1

U 2B n+1 H n+1 d 0

AF N Cd 0 0 0 n+1 ªN A «¬ 2

U

A 2 n+1

AF º U 2B n+1 H n+1 º» ª¬ N 2 2 n+1 ¼ ¼

A UA AF ªN U 0B n+1 º ª¬ N C º¼ ¬ 0 0 n+1 ¼ 0 0 n+1

0

0

gap condition

(7a)

contact condition

(7b)

complementarity condition on * 2

(7c)

complementarity condition on * 0

(7d)

328


where N 0A diag (" n 0A ") and N 2A diag (" n 2A ") are global matrices collecting the normal vector associated with the boundary * 0A , * 2A of the body A. The vector H n+1 collects all the nodal gaps between the corresponding nodes of the boundaries * 2A and * 2B at the load increment n 1 , in the zone of potential contact, whereas the vector C collects the cohesion between the nodes which are in contact, in the zone of potential detachment * 0 . In detail, through the eq.(7a), all the nodes of the free boundary * 2 , where the condition A UA N U 2B n+1 H n+1 0 occurs, change into the contact boundary * 0 , thus defining a new contact 2 2 n+1 boundary. Vice versa, through eq.(7b), all the nodes of the contact boundary * 0 , where the condition AF N C 0 occurs, change into the free boundary * 2 , thus defining a new detachment boundary. 0 0 n+1

3. The incremental elastoplastic analysis for active macro-zones

A brief description of the strategy utilized for incremental elastoplastic analysis via Multidomain SGBEM, called elastoplastic active macro-zone analysis, is provided in this section. The complete version can be found in [5]. For each loading step and at each bem-e this analysis uses eq.(6c) both to evaluate the trial stresses in the predictor phase and to compute the plastic strains in the corrector phase. Let us start by evaluating the trial stresses, i.e. the purely elastic response at the load increment n 1 in each m bem-element of the discretized body. For this purpose eq.(6c) provides all the predictors ı* n 1 as a function of the plastic strain vector p n , stored up at step n , of all the increments 'p inside step n 1 , and of the external load ıˆ e , amplified by ȕ n 1 : ı* n 1

K p n 1 ȕ n 1 ıˆ e

with p n 1

p n 'p

(8)

where ȕ n 1 ȕ n 'ȕ is the load factor and K matrix is fully-populated and regards all the bem-elements, the place of nonlinear phenomenon. A check on the plastic consistency condition of the stresses, computed at appropriately chosen points inside each bem-e, is performed using the yield condition expressed in this context through the von Mises law, i.e.: F [ı*i n 1 ]

1 2

ıTi n 1 M ı i n 1 ı 2y d 0

with i 1...m

(9)

where M is a matrix of constants and ı y the uni-axial yield stress. In the a bem-elements (with a d m ) where this inequality is violated, a return mapping phase is required in order to evaluate the plastic strain increments. This phase, called the corrector phase, uses the first term of eq.(6c) to obtain the elastoplastic solution at every bem-e where the plastic consistency condition is violated. In this phase the vector ı , representing the end step stress, as well as the increment of the volumetric plastic strain vector 'p , are unknown quantities. The latter are the plastic strains to impose on every plastically active bem-element in order to have the stress on the yield boundary of the elastic domain, through which the direction of the plastic flow can be defined. Obviously, inside each loading step the corrector phase has to be repeated until all the predictors satisfy the plastic consistency conditions. In detail, the elastoplastic algorithm allows one to write, for all the active h bem-elements ( h = 1,...,a ), a nonlocal system at the n 1 load step simultaneously in all the plastically active macro-zones identified in the previous predictor phase, i.e.:

ı a ( n 1)

ı*a ( n 1) K aa 'p a ( n 1)

F [ı a ( n 1) ] d 0 , 'ȁ a ( n 1) t 0 , 'ȁ a ( n 1) F [ı a ( n 1) ] = 0

(10) (10a-c)

where eqs.(10a-c) are the plastic admissibility conditions for the a bem-elements. The index a characterizes the vectors and matrices connecting the mechanical and kinematical quantities relating to all the active a bem-elements. The K aa matrix coefficients are derived from the K matrix present in eq.(6c), by extracting the blocks relating to the a plastically active bem-elements. In the following equations the subscript n 1 has been omitted for simplicity.


329

In the hypothesis that, for each h-th bem-e, the shape function defined in İ hp Ȍ p p h is the same as the shape function relating to the plastic multiplier, i.e. 'Oh ȥ p 'ȁ h with ȥ p t 0 , the plastic strain increment for the a-th active bem-elements is expressed as:

'p a

'ȁ a w ıa F [ı a ] 'ȁ a M a ı a

(11)

The solving nonlinear system for all the active a bem-elements is the following: * °ı a ı a 'ȁ a K aa M a ı a ® °¯ F [ı a ] 0

0

(12)

or, in explicit form, using the von Mises yield law and the plastic flow rule given by eq.(11) ı 1 ı*1 '/ 1 K 11 M ı 1 ! '/ a K 1a M ı a 0 ° °# °ı ı* '/ K M ı ! '/ K M ı 0 a 1 a1 1 a aa a ° a ®1 T 2 ı ı M ı 0 1 1 y °2 °# ° °¯ 12 ıTa M ı a ı 2y 0

(13)

where ı a is the stress solution located on the yield surface of the elastic domain of all the active bemelements, ı*a the elastic predictor, and 'ȁ a K aa M a ı a the corrective components (containing local and nonlocal contributions). The approximate solution of this nonlinear problem involving all the plastically active bem-elements, in terms of ı a and 'ȁ a , can be obtained by applying the Newton-Raphson procedure. 4. The elastoplastic-contact-detachment procedure In this section the sequence of steps concerning the proposed procedure for obtaining the numerical results is described. Step 0: Imput data.

ȕ n 'ȕ . 1 ˆ ȟ ° 0 n 1 K 00 K 0V p n ȕ n 1 f0 o X0 n 1 Eȟ 0 n 1 o F0 n 1 . Step 2: Nodal unknown update: ® o U 2 n 1 ° X n 1 A 1 A 0 Eȟ 0 n 1 AV p n ȕ n 1 Lˆ ¯ A UA AF Step 3: Check on contact-detachment N U 2B n 1 H n 1 d 0 , N Cd 0. 2 0 2 n 1 0 n 1 Step 1: Load update ȕ n 1

If true then go to Step 4 If false then modify topologically some boundaries * 0 o * 2 or/and * 2 o * 0 and go to Step 2 Step 4: Computing the elastic predictor ı* n 1

K p n 1 ȕ n 1 ıˆ s .

Step 5: Check for yielding If F [ı*i n 1 ] d Tol with i 1...m then go to Step 1, If F [ı*i n 1 ] ! Tol with i 1...m then go to Step 6. Step 6: Identification of the active bem-elements (a being the active macro-zone) in the corrector phase, °ı a ı*a 'ȁ a K aa M a ı a 0 ® °¯ F [ı a ] 0 Step 7: Plastic strain vector update: p a n 1

p a n 'ȁ a n 1 M a ı a n 1 , go to Step 1

330


5. Numerical results In order to show the efficiency of the proposed method, a numerical test was performed in the hypothesis of elastic-contact-detachment only. Let us consider the detachment problem regarding a beam A supported by two elastic blocks B, without friction or sliding, symmetrically loaded. The analysis is performed on half the structure, as shown in Fig.1a. q

A

10

B

20

a)

10

10

20

b)

Fig.1. Beam supported on elastic blocks: a) geometric description, b) strained shape obtained by iterative LCP analysis. The geometrical and mechanical characteristics of the structure are the same as those utilized by Vodicka [6]. Thus the beam A, having unitary thickness, is characterized by the Young modulus E a 30.6x104 Mpa and Poisson ratio X 0.3 and is subjected to vertical force distribution q 1020 daN / m . The body B is characterized by the Young modulus E b 30.6x106 Mpa and by same thickness and Poisson coefficient. In order to discretize the free and constrained boundaries of the solids A and B, a step p 2 cm and p 0.1 cm were introduced, respectively on * 2 and *1 boundaries. Fig.1b shows the final strained shape obtained by iterative LCP. In Table 1 the detachment length is shown in comparison with the solution obtained by Vodicka [6]. In all the cases shown in Table 1 the detachment length proves to be very similar.

Method

Detachment length [cm]

SGBEM Iterative LCP

8.4

BEM (R. Vodicka [6])

8.3

Table 1: Comparison of detachment lengths.

References. [1] Panzeca T., Terravecchia S., Zito L., (2010). “Computational aspects in 2D SBEM analysis with domain inelastic actions”. Int. J. Numer. Meth.Engng., 82, 184-2004. [2] Gao X.W., (2002). “The radial integration method for evaluation of domain integrals with boundary-only discretization”. Engng. Anal. Bound. Elem., 26, 905-916. [3] Panzeca T., Salerno M., Terravecchia S., Zito L., (2008). “The symmetric boundary element method for unilateral contact problems”. Comput. Methods Appl. Mech. Engrg., 197, 2667–2679. [4] Maier G., Polizzotto C., (1987). “A Galerkin approach to boundary element elastoplastic analysis”. Comput. Methods Appl. Mech. Engrg., 60, 175-194. [5] Zito L., Cucco F., Parlavecchio E., Panzeca T., (2012). “Incremental elastoplastic analysis for active macro-zones”. Int. J. Numer. Meth.Engng., In press. [6] Vodicka R., (2000). “The first and the second-kind boundary integral equation systems for solution of frictionless contact problems”. Eng. Anal. Bound. Elem., 24, 407–426.


HOW TO USE THE SBEM IN THE PRACTICAL ENGINEERING? T. Panzeca1, F. Cucco2, M. Salerno3 1

Dip.to di Ingegneria civile, ambientale, aerospaziale, dei materiali. University of Palermo, Italy. 1 [email protected] 2 University Kore of Enna, Italy. [email protected] 3 Dip.to di Costruzioni e metodi matematici in Architettura. University Federico II of Naples, Italy. [email protected] Key words: symmetric BEM, substructuring, displacement approach, closed form coefficients. Abstract: The aim of the present paper is to show the main difficulties implied in dealing with the Symmetric Boundary Element Method and to suggest the necessary strategies for overcoming them. Specifically, some basic characteristics will be shown for making the method valid and computationally effective, and that is to say: strategies for computing all the coefficients in closed form, sub-structuring process of the domain, characteristic matrix calculation in order to determine the elasticity equation and the load vector related to the boundary known actions, assembling process to determine the solving equation system, and transfer of the volume quantities (like body forces and plastic strains) to the boundary through the RIM technique. Introduction Recent developments in the Symmetric Boundary Element Method (SBEM) have highlighted its possible employment in engineering practice, thanks to its peculiarities, like for example fundamental solutions used as transfer functions and easy management of the discretization introduced on the boundary only. In structural engineering, especially for nonlinear analysis problems, like plasticity, fracture, contactdetachment and damage, this method appears more and more competitive. Many researchers have suggested strategies and provided solutions relating to several topics and some of these appear in the relative references shown in [1-3]. But, despite the peculiarities of the method and the development of the research, the right employment of this methodology in practical applications has not been found. Indeed, in their activities researchers have mainly promoted results obtained by means of the Finite Element Method because of the ease of finding of several calculus codes. In the work which we would like to present, a strategy to be pursued will be shown in order to give the right computational consistency to the SBEM. The structural systems considered in the present paper are loaded by in-plane actions, but the suggestions are valid for systems loaded by out-of-plane actions, and also for systems made up of mutually connected plates subjected to in-plane and out-of-plane loads. In the present paper we want to indicate the necessary steps for achieving all the strategies suited to making this method computationally effective. 1. Strategies for computing all the coefficients in closed form Let us consider a solid of domain : and boundary * , having assigned elastic characteristics (Young modulus E, Poisson coefficient Ȟ ) and thickness s, embedded in the infinite domain having the same physical characteristics and the same thickness. Let the boundary of the body be discretized into boundary elements. In each node, obtained by means of discretization, a force F and a displacement jump V = (-U) , both known and unknown, has to be introduced in the boundary nodes, having as a consequence the presence of a force distribution and a displacement jump along the boundary elements contiguous to the nodes. The response along the boundary, in terms of tractions and displacements, can be obtained as a generalized (weighted) effect according to the Galerkin strategy on two boundary element contiguous to the chosen node and the generalized quantity obtained may be thought of as connected to the node being examined. This strategy involves considering each coefficient as made up of a double integral, the inner one of which regards the distribution of the cause, the second one the weighting effect. Obviously, in the kernel of the double integral an appropriate foundamental solution is included, depending on the coordinates of the cause and effect points.

331

332


y'

1

Ys'

a

f 6

b

5

d

yg

4

Ye'

2

e

Og

y', y

n'

O'

c

3

O' O s

dac

X

e

Y

x', x

Ys

d

s

Ye

b e

n

xg

a

x'

z

b)

a)

Fig.1: a) Topology of the boundary, local and general reference system, b) Two boundary elements with the related reference systems. Two types of coefficients can be found and regard the cases in which: a) The cause distribution and effect weighting are related to the same boundary elements, involving the computation of the singular or hypersingular integrals. Their computation in closed form has been developed in several researches, including papers [4-7]. b) The cause distribution and effect weighting are located distant from one another, and no singularities are possible. In this case it is easy to find the closed form response througth the following technique: with reference to Fig.1, at first we utilize the projection technique at the local axis ( 0, x ', y ' , associated with

the element where the cause is distributed), regarding the quantities defined on the element where the effect is applied. Subsequently, the transformation of the functions into natural coordinates is introduced, and lastly the block coefficient is transformed from the local reference system ( 0, x ', y ' ) to the general one ( 0 g , xg , y g ) through appropriate transformation matrices. To clarify the strategy to compute these coefficients in closed form, we want to follow the sequence utilized by Terravecchia in [6] and shown in Fig.1: The cause is modelled on the boundary element “a”, characterized by a reference system ( 0, x ', y ' ), whereas the effect is weighted on the boundary element “b.” Let us consider the coordinates X and Y of the start node of the element “b” with respect to the start node of the element “a”, and the angle G between the two boundary elements. Using appropriate S.I. denoted by G (x; x ') , it is possible to compute the displacement and the traction in x :f , caused by a distribution : ( x ') of forces or displacement discontinuities acting in x ' of the boundary element “a” , through the following integral: x' a

³

G (x; x ',0) : '( x ') dx '

(1)

x' 0

Now the latter integral must be computed when the point x lies on the element “b.” For this purpose the axis z is introduced along “b”, having its origin at the start node. The geometrical relations expressing the projection of a point on “b” on the axes (O, x, y ) { (O ', x ', y ') are x

X z cos G

y Y z sin G

,

(2)

The integral, given in eq. (1), weighted by the function : ( z ) , along the element “b” gives rise to the following coefficient: z b

x' a

z 0

x' 0

³ : ( z) ³

G (x; x ',0) : ' ( x ') dx ' dz

(3)

where it is necesary to remember that z is related to the (O, x, y ) reference system through eqs.(2). Let us introduce the following linear shape functions, expressed in terms of natural coordinates ( [, [ ' ), both associated with the node ends of the boundary elements “a” and “b”, that is :e' ( x ') [ ' ,

:e ( z )

z b

[

1 2

As a consequence the integral in eq.( 3) can be modified as follows

(4)


1

1

0

0

³ [ ³ G ( X b [ cos G , Y b [ sin G ; a [ ',0) [ ' a d[ ' b d[

333

(5)

where “a” and “b” are the Jacobian terms

J'

dx ' d[ '

a ; J

dz d[

b

(6)

All cases regarding the coincident element or touching at the extreme points of boundaries can be obtained in a similar way to that shown before, but introducing a particular strategy able to bypass the difficulties connected with the singularity or hypersingularity present in the kernels of the double integrals. 2. Characteristic matrix

A second step is to write the characteristic matrix of the body being examined, whose coefficients relate generalized (tractions and displacements) quantities to nodal values (forces and displacement jumps), through computation in closed form of the coefficients as described in the previous section. Again in the hypothesis of the body being embedded in an unlimited domain, this matrix is written making no distinction regarding the type of boundary, that is to say whether it is constrained, free or a mixed type. Working in this way, the implementation of the coefficients of this pseudo-stiffness matrix is particularly simple, because it can be built for columns or rows, in a repetitive way, through the strategy introduced in the previous section to compute the coefficients in closed form. The characteristic matrix of the body, thus obtained, is symmetric and non-definite, and it appears as follows: BX=0

(7)

where X is the vector collecting the nodal boundary quantities (forces and displacement jumps). During the test phase of the program, it is necessary to perform two checks: the first regarding the block columns connected to the nodal displacement jumps, through the so-called rigid motion technique, the second regarding the flexibility of the sub-matrix having as its cause the external force vector and as its effect the generalized displacements. 3. Sub-structuring process of the domain

When the physical and geometrical characteristics of the structure are zone-wise variable, a domain discretization can be performed into sub-structures, distinguishing them into macro-elements (sub-structures having big dimensions) and into bem-elements (sub-structures having small or very small dimensions). The geometry of each sub-structure is not pre-defined, as instead happens in the Finite Element Method. The distinction between macro-elements and bem-elements is necessary when nonlinear problems are dealt with, because the presence of big linear macro-elements, where the elastic response is expected, allows one to perform a large reduction in variables, whereas the bem-elements are the places where inelastic behaviour occurs. For each sub-structure, an equation of a constitutive type can be obtained by means of eq.(7), performing a subdivision between known and unknown quantities, a reordering of matrix B , an appropriate variable identification of the vector X and a condensation process of the boundary variables (see Panzeca et al. [3]), thus obtaining: P = DU + P*

(8)

where the generalized tractions P evaluated along the interface sides and connected to the related nodes are expressed as functions of the displacements of the nodes of the same interface and of the known terms. This equation is formally analogous to the force-displacement equation written in the Finite Element Method. 4. Assembling process to determine the solving equation system

The assembling process is disciplined by means of the regularity conditions between the interface contiguous elements. Indeed, in order to reach this goal we do the following: - we utilize equation (8) again, but written for all the sub-structures obtained through the sub-structuring process, where the super-vectors P and P* collect the unknown and known generalized tractions of all the sub-structures, while the super-vector U collects the displacement vectors of the interface nodes and D is a block diagonal matrix;

334


- we perform a condensation process of the displacements, thus reducing the variables to the quantities which represent the solution of the analysis problem in accordance with the displacement method, as also occurs in the Finite Element Method, i.e. (9)

U=Zu

where Z is the topological matrix; - we utilize the action and reaction principle between the generalized tractions P of the boundary

elements facing each other, written in compact form, that is to say ZT P = 0 .

(10)

The latter equations are the regularity conditions of the assembled system, regarding the nodal compatibility of the displacements and the equilibrium of the generalized boundary tractions. The use of Eq.(8) written for all the substructures and of eqs. (9,10) allows one to obtain the equation system assembled in terms of interface nodal displacements [3] Ku = f *

(11) *

where the matrix K and the known vector f take on an analogous role to the one taken on in the Finite Element Method. An equation system is thus obtained through which it is possible to have the response of the structural system, in terms of nodal displacements, to the known boundary actions (forces on the free boundary and imposed displacements on the constrained one). Further use of eq.(9) gives the displacement vector u of each sub-structure and in succession (see Panzeca et al. [3]) it is possible to obtain all the boundary quantities of each sub-structure by means of eq.(7) and therefore, through the Somigliana Identities, the elastic response in each sub-domain. Moreover, from the computational point of view, the assemblage which connects sub-structures often having considerably different dimensions, i.e. macro-elements and bem-elements, does not involve problems of numerical instability because of the computation in closed form of all the coefficients of matrix B , as was pointed out regarding aplication of the Karnak program [8] to several cases of practical engineering. 5. Transfer of the volume quantities (like body forces and plastic strains) to the boundary through RIM technique

In order to make this method greatly competitive, it is necessary for the volume actions (like body forces p and plastic strains - ) to be computable through knowledge of the boundary geometry only. The procedure has to consider two types of problems: - in the analysis phase, evaluation of the generalized (weighted) displacements and tractions on the boundary elements to be introduced in the equation system as the symmetric BEM requires; - in the return phase, the computation of the displacements, strains, stresses and tractions in the domain of each sub-structure, as an effect of the boundary quantities and body actions through the use of the Somigliana Identities. Some singularities in the integrals (double integrals in the analysis phase and single integrals in the return phase) are present, involving different levels of difficulties. In our opinion, the presence of these singular coefficients represents the main difficulty for making the method valid and computationally effective in comparison with other approaches, like the Finite Element Method or the Collocation Boundary Element Method. There are several strategies to solve the singularity problem regarding the evaluation of the body actions, mainly based on the RIM technique [9]. This latter works on the Fundamental Solutions of the displacements and of the tractions performing a transformation from Cartesian coordinates into polar ones and subsequently, after appropriate mathematical simplification, making an inverse transformation from polar coordinates into Cartesian ones, thus obtaining a modified Fundamental Solution, which it is easy to integrate.


dS

335

d n

R x'

x

Fig.2: Cause point x ' variable between the effect point x and the boundary. When there is a need to find the traction inside the domain : of each sub-structure, the fundamental solutions obtained through the RIM technique include in their expression the Bui free term [10] in implicit form. Further, in the analysis phase, in order to find the load vector, the generalized traction caused by plastic strains has to be evaluated on every boundary element. The weighting of the traction on the

boundary elements can be made by using a particular strategy linked to the presence of the Bui free term in implicit form. A simpler alternative way of finding the generalized displacements and tractions is through a limit operation of the displacement and the traction, the latter having the same slope as the boundary element examined [11,12]. In the hypothesis of a constant value of the plastic strain inside the sub-structure, the displacement on the boundary is obtained as the sum of the contribution of the external force and displacement jump distributions along the boundary and of the RIM technique regarding the plastic strain action, that is to say:

³G

u

uu

*

1 d* f d * v³ G ut (u) d * u ³ G uV 2 * *

(12)

whereas the traction on the boundary can be obtained, through a limit operation from the inside, as the sum of the contribution of the external force and displacement jump distributions, of a regular boundary integral regarding the transfer of the plastic strains by means of the RIM technique, which takes on the role of a Cauchy Principal Value, of the Bui free term J written in explicit form and of the trigonometric expression J b associated with the slope of the boundary considered, that is: t

v³ G *

tu

1 d * - NT E ( J J ) - NT E I f d * f ³ G tt (u) d * v³ G tV b 2 *

(13)

with

J

5 4Q 2 1 1 4Q 4(1 Q ) 2

1 4Q 2 5 4Q 2

0

Jb

0

0 0

(14)

3 4Q

1 1 1 ª º Q Cos[G ] Cos[4G ] (1 2Q Cos[2G ]) Sin[2G ]» « (1 Q )Cos[2G ] 4 Cos[4G ] 4 2 « » 1 « 1 1 1 Q Cos[G ] Cos[4G ] (1 Q )Cos[2G ] Cos[4G ] (1 2Q Cos[2G ]) Sin[2G ]» « » 4(1 Q ) 4 4 2 « » 1 3 3 « » 4Cos [G ]Sin[G ] 4Cos[G ]Sin [G ] Cos[4G ]) 2 ¬« ¼»

(15)

and where one has set

G uV

(nT r ) G uV ,

G tV

NT E N ' G uV

(16)

with (nT r ) the scalar product between the distance x (effect point) and x ' (cause point), introduced as a

336


consequence of the RIM technique; N ' the (3x2) direction cosine matrix obtained as a consequence of the transfer process of the plastic strains on the boundary, E the (3x3) elasticity operator, NT the (2x3) trasposed direction cosine matrix connected to the boundary on which the traction has to be computed. The previous expressions are particularly useful in the analysis phase, in order to find the load vector regarding the generalized displacements and the tractions caused by plastic strain - to be evaluated for every sub-structure. References [1] Bonnet M., Maier G., Polizzotto C., (1998). “Symmetric Galerkin boundary element method”. Appl.

Mech. Rev., 51, 669-703. [2] Pérez-Gavilán J.J., Aliabadi M.H., (2001). “A Symmetric Galerkin Bem for multi-connected bodies: a

new approach”. Eng. Anal. Bound. Elem., 25, 633-638. [3] Panzeca T., Cucco F., Terravecchia S., “Symmetric boundary element method versus Finite element

method”. Comput. Meth. Appl. Mech. Engng., 191, 3347-3367. [4] Holzer S., (1993), “How to deal with hypersingular integrals in the symmetric BEM”. Comm. Num.

Meth. Engng., 9, 219-232. [5] Gray L.J., (1998), “Evaluation of singular and hypersingular Galerkin boundary integrals: direct limits

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

and symbolic computation. In: Sladek, J., Sladek, V. (Eds.), Singular Integrals in Boundary Element Methods. Computational Mechanics Publications, Southampton. Terravecchia S., (2006), “Closed form coefficients in the symmetric boundary element approach”. Eng. Anal. Bound. Elem., 30, 479–488. Panzeca T., Fujita Yashima H., Salerno M., (2000), “Direct stiffness matrices of BEs in the Galerkin BEM formulation”, Eur. J. Mech. A, Solids, 20, 277–298. Panzeca T., Cucco F., and Terravecchia, S., (2002), The program Karnak.sGbem, Release 2.1, Dep.nt DICAM, Palermo University, Italy. Gao X. W., (2002), “The radial integration method for evaluation of domain integrals with boundaryonly discretization”. Engng. Anal. Bound. Elem., 26, 905–916.

[10] Bui H. D., (1978), “Some remarks about the formulation of three-dimensional thermo-

elastoplastic problems by integral equation”. Int. J. Solids Struct., 14, 935–939. [11] Panzeca T., Terravecchia S., Zito L., (2010), “Computational aspects in 2D SBEM analysis with domain

inelastic actions”, Int. J. Num. Meth. Engng., 82, 184-204. [12] Zito L., Parlavecchio E., Panzeca T., (2011), “On the computational aspects of a symmetric multidomain

BEM approach for elastoplastic analysis”, J. Strain Analysis for Engng Des., 46, 103-120.


337

Vibration Analysis of Vehicle cab by BEM with Block SS Method

H.F. Gao1,a , T. Matsumoto1,b , T. Takahashi1,c , T. Yamada1,d 1 Department

of Mechanical Science and Engineering, Nagoya University Furo-cho, Chikusa-ku, Nagoya City, 464-8603, Japan

a [email protected], b [email protected], c [email protected],d [email protected]

Keywords: Acoustics analysis, Eigen frequency, Nonlinear eigenvalue problem, Block SS method

Abstract Boundary element method (BEM) is applied to eigenfrequency analyses of a simplified 2D model of vehicle cab structure. For calculating eigenfrequencies by BEM, block SS-method that is based on evaluating a path integral is used. This method results in solving a general eigenvalue problem whose matrix components are calculated by evaluating path integrals of a regular function defined by the coefficient matrices of BEM. The size of the related general eigenvalue problem is rather small and computation time spent for solving general eigenvalue problems is negligible compared with the total CPU time. The main part of the computation cost is spent for evaluating the path integrals. Also, present method enables calculating the eigenvalues in a given range, multiple eigenvalues, and also eigenvectors. The effectiveness of the method is demonstrated through calculations of eigenfrequencies of a simplified 2D model of vehicle cab structure. Introduction Natural frequency calculation is always playing a important role in the engineering field. Boundary element methods, which are considered as one of the significant numerical methods, and several BEM-based methods have been applied to eigenvalue analyses [1, 3, 2, 4, 5] [6, 7]. However the nonlinear eigenvalue problems resulting from BEM-based formulation make the solving process difficult and time consuming. Recently, a novel method based on contour integral to solve nonlinear eigenvalue problems contour integral was proposed by Sakurai and Sugiura [8], and its block version [9, 10] (block SS method) was developed for obtaining eigenvalues whose multiplicities are greater than one. This method has been applied to core-excited-state calculation [11] for its capability of selecting solved range. In this paper, we apply BEM combined with block SS method to calculate the natural frequencies of a vehicle cab structure and show the applicability of BEM to natural frequency calculation. The numerical results are compared with those obtained by using a commercial FEM software. Basic formulations We consider a time-harmonic acoustic wave propagation in a homogeneous and isotropic medium. The governing equation for this state is the following Helmholtz equation: r 2 p.x/ C k 2 p.x/ D 0 in D; where p.x/ is the sound pressure at a point x in the domain D, number. The boundary conditions are assumed as

r2

p.x/ D p.x/ N on Sp ; @p q.x/ D .x/ D i! v.x/ N D q.x/ N @n p.x/ D zv.x/ on Sv ;

(1)

is aplace’s operator, and k is the wave

(2) on Sq ;

(3) (4)

where n is the outward normal direction to the boundary, i the unit imaginary number, the density, ! the circular frequency, v the particle velocity in n direction, and z the acoustic impedance. A quantity with a bar .N/ denotes that it is a given quantity.

338


Eq (1) can be transformed into an integral representation by using the fundamental solution of eq (1). In this paper, we consider only two-dimensional problems, and the corresponding fundamental solution is given as i .1/ p .x; y/ D H0 .kr/; 4

(5) .1/

where x and y are two different points in D, r is the distance between x and y, and H0 denotes the zerothorder Hankel’s function of the first kind. Then, an integral representation relating the sound pressure at an arbitrary point in D with the sound pressure and its normal derivative on the boundary is obtained as follows: Z Z @p .x; y/ @p p.y/ D p .x; y/ .x/dS.x/ p.x/dS.x/: (6) @n @n.x/ S S The boundary integral equation is obtained by taking the limit y 2 D ! y 2 S, as follows: Z Z @p @p .x; y/ p .x; y/ .x/dS.x/; p.x/dS.x/ D cy p.y/ C @n.x/ @n S S

(7)

where cy D 12 if y is located at a smooth part of the boundary S. Eq (7) can be converted to a regularized form, which is used in the numerical computations in this paper, as follows: Z @p .x; y/ Œp.x/ p.y/ dS.x/ S @n.x/ Z @p .x; y/ @P .x; y/ dS.x/ p.y/ C @n.x/ @n.x/ Z S @p D p .x; y/ .x/dS.x/; (8) @n S where P .x; y/ is the fundamental solution of Laplace’s equation. Substituting the homogenous boundary condition and using quadratic isoparametric element for discretizing eq (8), we obtain a system of linear algebraic equations, as follows: A.k/x D 0;

(9) p

@p =@n,

where A.k/ is a matrix whose components are obtained by evaluating the integrals either of or and x is a column vector comprising unknown values at the nodes on the boundary. Because k is included implicitly in Hankel’s function, the eigenvalue problem corresponding to eq (9) becomes a nonlinear one. The block SS method [9, 10] can treat such a nonlinear eigenvalue problem. It extracts the eigenvalue for eq (9) lying inside a Jordan curve on the complex plane. The block SS method starts from the following moment matrices. Z 1 UH A.z/1 Vz m dz; (10) Mm D 2 i where V and U are arbitrary nonzero matrices. Both of the numbers of columns V and U are assumed to be an integer number l 1. With these matrices, we can treat the multiplicity of eigenvalues less that l. In this paper, we take U D V. The contour integral can be evaluated by using N -point trapezoidal rule. < Using the moment matrices defined by eq (10), following two Hankel matrices HKl and HKl are constructed. HKl D ŒMiCj 2 K i;j D1

(11)

< D ŒMiCj 1 K HKl i;j D1 ;

(12)

where MiCj 2 and Mi Cj 1 are l l sub-matrices with i C j 2 and i C j 1 order moment, respectively, where i; j D 1; 2; ; K. Therefore, the dimension of the Hankel matrices becomes Kl.


339

Fig. 1

Vehicle cab structure.

We can obtain the original eigenvalues k1 ; k2 ; ; kKl contained in the closed curve by solving eigenvalues of the matrix pencil: < kHKl : (13) HKl After the Hankel matrices are constructed, the singular value decomposition of HKl is carried out to make a rank test, as (14) HKl D C † EH ; where C and E are complex unitary matrices, † is a diagonal matrix with nonnegative real numbers at its diagonal elements. The standard linear eigenvalue problem of following matrix < B D CH HKl E †1

(15)

is solved instead of the original nonlinear one. Numerical simulation To examine the application of the proposed method, 2D area of a simplified vehicle cab structure is considered as shown in Fig.1. All the boundary condition set as Neumann condition which means the boundary is assumed to be rigid wall. The density of air and the sound speed are assumed as D 1:25Œkg=m3 and c D 333Œm=s, respectively. We divide the boundary into 119 constant elements as shown in Fig.2, where the solid circular symbols represent the edges of constant elements. For the Block SS method, a circular contour integration path D C Rei is used. In this numerical example, we used D .397:887; 0/ and R D 159:155. Singular values in † can be dropped to zeros if they are smaller than a certain threshold value. By some preliminary computations [12], we found that the threshold for the singular values is not constant but depends on the location of eigenvalues in the selected domain. Therefore, we propose a better way to detect the gap in the singular values. We check the differences of the singular values, i : i D

log10 .i / log10 .i 1 / h

i D 2; 3; :::; Kl

(16)

From the peak of the curve of i , we can identify the appropriate gap of the singular values. In Fig.3, we show the present i with h D 1. In the present example, we give a threshold for the effective peak as i < ı D 1, then we have 35 non-zero singular values with the gap larger than 104 . We obtain 35 eigenvalues by solving eq (15), among which 10 complex eigenvalues are include. These complex eigenvalues with large imaginary parts are fictitious ones and can be removed from the results.

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Fig. 2

Boundary element model

We also use FEM to compute the natural frequencies of the same cab structure to compare the numerical results obtained based on BEM. The domain is discretized with 61184 triangular linear elements for FEM analysis using a commercial FEM software: COMSOL. Both results by BEM and FEM are shown in Table 1, from which it is found that the 119 elements model yields a highly accurate results. To check the validity of the results of the natural frequencies, we consider an additional example to calculate the frequency responses with a harmonic excitation added on a part of the boundary. On the part of excitation, particle velocities of 1 [m/s] are given, while on other parts of the boundary given is the particle velocity 0 [m/s] as shown in Fig.4. The observation point at which sound pressure is calculated is set at (0.8,1.0). From the response curve shown in Fig.5, we find that there are peaks of the sound pressure in the vicinity of of the natural frequencies obtained by eigenvalue analysis. Conclusions An eigenvalue problem for a simplified 2D vehicle cab model is solved by BEM combined with Block SS method. The reduced eigenspace makes the matrix smaller than original size. Comparing the obtained results with those by FEM, it is found that BEM with small number of elements can give highly accurate eigenvalues. The only drawbacks in the present BEM-based approach is that the block SS method requires solving BEM repeatedly. However, these BEM calculations can be done independently each other and can be parallelized easily using a multi-core CPU. Acknowledgements The authors acknowledge the support from China Scholarship Council (CSC) (File No. 2009612004). References [1] J.O.Adeyeye, M.J.M.Bernal and K.E.Pitman International Journal for Numerical Methods in Engineering, 21, 779-787 (1985). [2] G.Bezine Mechanics Research Communications, 7, 141-150 (1980). [3] D.Nardini and C.A.Brebbia Applied Mathematical Modelling, 7, 157-162 (1983). [4] A.J.Nowak and C.A.Brebbia Engineering Analysis with Boundary Elements, 6, 164-167 (1989). [5] S.M.Kirkup and S.Amini Method. International Journal for Numerical Methods in Engineering, 36, 321330 (1993). [6] N.Kamiya, E.Andoh and K.Nogae Engineering Analysis with Boundary Elements, 12, 151-162 (1993).


Fig. 3

Differences of singular values.

observation point

harmonic excitation

Fig. 4

An excitation model.

[7] A.Ali, C.Rajakumar and S.M.Yunus Computers & Structures, 56, 837-847 (1995). [8] T.Sakurai and H.Sugiura Journal of Computational and Applied Mathematics, 159, 119-128 (2003). [9] T.Ikegami, T.Sakurai and U.Nagashima Journal of Computational and Applied Mathematics, 233, 19271936 (2010). [10] J.Asakura, T.Sakurai and H.Tadano, T.Ikegami and K.Kimura JSIAM Letters, 1, 52-55 (2009). [11] T.Tsuchimochi, M.Kobayashi, A.Nakata, Y.Imamura and H.Nakai Journal of Computational Chemistry, 29, 2311-2316 (2008). [12] H.F.Gao, T.Matsumoto, T.Takahashi and T.Yamamda Transactions of JASCOME, 11, 107-110 (2011).

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Fig. 5

Table 1

Frequency response of the structure.

Numerical results by BEM and FEM.

BEM (119 elements) 243.85229 261.84336 296.66396 305.64692 321.42724 325.60239 342.85348 359.90599 364.37458 369.31028 393.80344 406.87844 429.34371 437.55548 440.48749 459.30350 461.59526 475.82795 494.74607 507.56284 511.12976 511.96479 535.55759 547.26810 561.38522

FEM (61184 elements) 244.41101 263.92295 297.34892 307.11039 324.07146 326.13527 343.10725 359.90549 364.64592 370.91059 394.77935 407.50231 431.14975 438.28669 440.94774 459.62643 461.90236 475.73870 496.05505 508.10822 514.13825 515.05543 535.67821 548.59432 561.06965


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Curvature Singularities on Surfaces of Water Waves Gregory Baker Department of Mathematics, The Ohio State University, 231 W. 18th Ave, Columbus, OH43210. [email protected]

Jeong-Sook Im Department of Mathematics, The Ohio State University, 231 W. 18th Ave, Columbus, OH43210. [email protected]

Abstract: Boundary integral methods have proved very useful in the simulation of free surface motion, in part, because only information at the surface is necessary to track its motion. However, the velocity of the surface must be calculated quite accurately, and the error must be reasonably smooth, otherwise the surface buckles as numerical inaccuracies grow, leading to a failure in the simulation. For two-dimensional motion, the surface is just a curve and the boundary integrals are simple poles that may be removed, allowing spectrally accurate numerical integration. For three-dimensional motion, the singularity in the integrand, although weak, presents a greater challenge to the design of spectrally accurate quadrature. One way forward is to take advantage of a polar coordinate representation around the singularity point. Of course the typical grids used in boundary integral methods don’t lend themselves to transformation to local polar coordinates, but the range of integration can be split into two regions, one near the singularity where polar coordinates can be used with suitable interpolation and an outer region where standard methods apply. We provide details and results of some tests that confirm spectral accuracy in the method. Key–Words: Free Surface Flows, water waves, curvature singularities.

1

Introduction

The use of boundary integrals for tracking free surface flows in incompressible, inviscid fluids is now well established. They arise primarily as solutions to Laplace’s equation. Alternatively, the boundary integrals can be viewed as dipole sheets or vortex sheets, see for example [1]. The addition of a solid boundary can be treated with a source distribution for the velocity potential, or as a dipole distribution for the streamfunction [2]. These boundary integral formulations have several natural advantages: they reduce the spatial dimension by one; they can represent the surface very accurately; surface markers are adaptive as they follow the fluid motion. What is often overlooked is the important mathematical information carried by boundary integrals. In particular, they are very useful in studies of possible singularity formation in the curvature. Important progress has emerged from the viewpoint that the location of the surface can be analytically continued into the complex plane of the surface parameter and that the boundary integrals can also be analytically continued to produce simpler evolution equations that reveal the analytic structure in the solutions. This approach has led to explanations for the formation of curvature singularities on vortex sheets [3] and during the two-fluid Rayleigh-Taylor instability [4]. Here we report on the possible formation of curvature singularities for water waves. To fully appreciate the context of our work, it is useful to review what is known theoretically. The challenge in rigorous mathematical analysis is the possibility of waves breaking or forming singularities in slope or curvature in finite time. Two striking examples confirm the possibility is real. [5] proves that the limiting Stokes wave has a corner of angle 120 degrees that retains this form as the wave propagates. [6] suggests that the limiting standing waves reaches a crest with a corner of 90 degrees momentarily before subsiding with the disappearance of the corner until the next event. Consequently, long time existence in Sobolev spaces is difficult to establish without restrictions on the initial data that exclude these possibilities. Nevertheless, some progress has been made. Existence and uniqueness of solutions have been proven locally in time if the initial conditions are analytic by [7]. Recently, new results by [8], [9] prove existence and uniqueness in certain Sobolev spaces for both two- and threedimensional motion for finite time, provided the initial wave surface doesn’t self-intersect or that the initial motion satisfies the Taylor condition, that is, the surface does not accelerate faster than gravity in the inward direction normal to the water surface. Provided the initial data is small enough, long time existence and uniqueness has been established for two-dimensional motion by [10] and global existence and uniqueness in time has been established for three-dimensional motion by [11]. All these results hold under the assumption that the surface returns to a flat

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surface in the far field, but it seems plausible that similar results will hold for periodic domains, a case that is more practical in ocean studies. While general rigorous theory establishes the mathematical context in which water waves appear, it is often specific, generic solutions that shed insight. Progress has been made here too, based on numerical simulations and the extension of the ideas behind the analytic nature of the solutions in the complex spatial plane [12]. The work assumes two-dimension fluid flow and deep water. A review of the results is provided in 2 before the extension of the work is described in 3 for water waves of finite depth.

2

Deep water waves

Assume the fluid is two-dimensional with x a coordinate in the horizontal direction and y vertically upwards. Express the location of the water surface in parametric form xF (p, t), yF (p, t) with a dipole distribution µF (p, t) along the surface. The surface parameter p will be a Lagrangian variable marking the location of a particle that moves with the water. Assume that the motion is 2π periodic: xF (p, t) − p, YF (p, t) and µF (p, t) are 2π-periodic in p. Since water may be regarded as incompressible and the flow inviscid, the motion can be described by means of a velocity potential φ and a streamfunction ψ, both of which must satisfy Laplace’s equation. By introducing the complex notation z = x + iy, the complex potential Φ = φ + iψ may be expressed as the boundary integral, 2π 1 Φ(z) = µF (q) zF,q (q) cot 12 z − zF (q) dq . (1) 4πi 0 The dependency on t has been suppressed for convenience and the subscript q on zF indicates differentiation. The complex velocity w = u + iv is given by dΦ w∗ (z) = (z) , (2) dz where ∗ indicates complex conjugation. When evaluated on the water surface, the Lagrangian motion of a surface marker is given by ∂zF∗ Φp (p) (p) = w∗ (p) = . (3) ∂t zF,p (p) A partial derivative in t emphasizes that it is taken keeping p fixed. Bernoulli’s equation provides an evolution equation for the velocity potential, ∂φ 1 (p) = w∗ (p)w(p) − gyF (p) , ∂t 2

(4)

where g is the gravitational constant. Unfortunately, there is no evolution equation for the stream function ψ. Instead, the limit of (1) as z approaches the surface from below provides a Fredholm integral equation of the second kind for µF , µF (p) Φ(p) = Φ zF (p) = IF F (p; µF ) + , (5) 2 where 2π 1 µF (q) zF,q (q) cot 12 zF (p) − zF (q) dq (6) P IF F (p; µF ) = 4πi 0 is an integral in the principal-valued sense. Now by taking the real part of (5) gives the integral equation µF (p) + 2 IF F (p; µF ) = 2φ(p) .

(7)

It is also possible, as done in [12], to substitute (7) into (4) to obtain an evolution equation, still a Fredholm integral equation of second kind, for the dipole strength µ directly. Given that the motion is assumed periodic in space, it is natural to use spectral methods to evolve the surface numerically. Starting with known profiles in x(p), y(p), φ(p), the integral equation (7) must be solved first to obtain µ(p). The integral may be evaluated by the trapezoidal rule. The pole singularity is easily removed to ensure spectral accuracy [12]. The integral equation can be solved very efficiently by iteration. Once µ(p) is known, then Φ(p) can be calculated by (5). By differentiating the Fourier series representations for z(p) and Φ(p)


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Figure 1: The plunging breaker for ε = 0.5. The profiles are shown in sequence at t = 3.1 3.2 . . . , 4.0. analytically, the complex velocity can be determined from (3). Care must be taken to apply an appropriate spectral filter, examples are given in [12], otherwise round-off errors in amplitudes of the Fourier series will be amplified when differentiation is performed. The result is that the rate of change of ZF (p) and φ can be evaluated by (3) and (4) respectively and the results passed to any ODE solver. We use the standard fourth-order Runge-Kutta solver. We choose initial conditions associated with the propagation of a linear wave, x(p) = p − ε sin(p),

y(p) = ε cos(p) ,

φ(p) = ε sin(p) .

(8)

The length scale is set with 2π and the time scale with g = 1. Note that this initial profile is analytic in p. We pick ε = 0.5. Obviously, the amplitude is large enough that the propagating wave will not remain linear, but soon adjusts to nonlinear effects. The result is a breaking wave as shown in Fig. 1. Also evident is the formation of a very sharp tip to the breaking wave and the question naturally arises whether this tip forms a curvature singularity in finite time. The curvature is given by xp ypp − yp xpp κ(p) = 3/2 . x2p + yp2

(9)

and its profile at t = 4.0 is shown on the left in Fig. 2. The curvature has an extremely narrow spike that corresponds to the very tip of the plunging breaker. On the right of Fig. 2 , we show the time evolution of the minimum curvature and it certainly seems plausible that the grow of the magnitude of the curvature could become infinite in finite time. In general, it can be quite difficult to assess whether a singularity forms in finite time. One way forward that is both intellectually pleasing and practical is to search for isolated singularities in the complex p plane. If found, they can be tracked to see whether they will reach the real axis in finite time. This approach has been used successfully in several studies of curvature singularity formation in free surface flows, for example, [13], [14], [4], [3]. Basically, the method uses the asymptotic behaviour of the Fourier coefficients for large wavenumbers to detect the nearest isolated singularity in the complex p-plane. Despite the analyticity of x(p) and y(p) initially, the curvature has inverse 3/2-branch point singularities since κ(p) =

ε2 − ε cos(p) 1 + ε2 − 2ε cos(p)

3/2 ,

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Figure 2: Profile of the curvature at t = 4.0 and the time evolution of the minimum of the curvature from t = 3.56 to t = 4.0 in increments of 0.2. and the denominator can be written as

1 − ε eip

3/2

1 − ε e−ip

3/2

.

The singularities occur at ps = ±i ln ε. Any concerns that these singularities are an artifact of the Lagrangian parametrization can be removed expressing the curvature in terms of the arclength parameter. At ps , the curvature has a pole singularity in the complex arclength plane at sc [12]. What happens to the initial pole singularities in the curvature when the wave begins to move? If we assume that the pole singularity simply moves about the complex plane, sc (t) = γ(t) + iδ(t). The way to confirm this behaviour is to note the connection between a branch point singularity f (s) ∼ A(s − sc )µ and the tail of the Fourier spectrum of f (s), fˆk ∼ C|k|−µ−1 exp(−kδ + ikγ). A special form-fit applied to the Fourier spectrum of the curvature at any time produces the confirmation that µ = −1. In addition, the distance δ of the singularity from the real axis of the complex arclength plane can be tracked in time to assess whether the pole singularity is likely to reach the real axis in finite time. The distance δ(t) is shown in Fig. 3 for several choices of the initial amplitude ε of the wave as given in (8). If the amplitude is small enough, the singularity remains bounded away from the real axis, but for larger amplitudes, the singularity approaches the real axis very closely. Despite appearance, a closer view of the close approach establishes a dramatic slow down in approach and the evidence indicates that the singularity does not reach the real axis in finite time [?]. The close approach also coincides with the formation of plunging breaker. While the behavior of δ shows a clear distinction between plunging breakers and propagating waves, a better distinction can be made by regarding the wave slope yx = yp /xp . Form-fits of the spectrum of y(x) establish the presence of square root singularities y(x) ∼ A x − xb (t) in the complex x-plane which do reach the real axis in finite time, at which moment the wave slope is vertical as the wave begins to break [12].


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=0.2 =0.25 =0.3 =0.35 =0.4 =0.5

1

0.8

0.6

0.4

0.2

0 0

5

10

15 t

20

25

30

Figure 3: The distance δ from the real axis in the complex arclength plane for different ε.

3

Water waves of finite depth

Naturally, the questions arises whether the nature of these singularities changes if the water lies above a solid boundary. We’ll provide numerical evidence that the curvature singularities are still poles in the complex arclength plane and can approach the real axis very closely. There is also evidence of the possible formation of a corner wave. The presence of a solid boundary below the water surface requires a simple extension of the boundary integral technique. A dipole distribution µB (p) along the solid boundary, parametrized as zB (p), in the streamfunction can be added to (1), 2π 2π 1 1 1 Φ(z) = µF (q) zF,q (q) cot 12 z − zF (q) dq + 4π (10) 0 µB (q) zB,q (q) cot 2 z − zB (q) dq . 4πi 0 Consequently, (5) is replaced by µF (p) , Φ(p) = Φ zF (p) = IF F (p; µF ) + IF B (p; µB ) + 2 with IF b (p; µB ) =

1 4π

2π 0

µB (q) zB,q (q) cot 12 zF (p) − zB (q) dq .

(11)

(12)

The solid boundary must be a constant streamline (no normal velocity); in other words, Φ zB (p) is a constant set to zero for convenience. Thus, (13) µB (p) − 2 IBF (p; µF ) + IBB (p; µB ) = 0 , where 2π 1 µB (q) zB,q (q) cot 12 zB (p) − zB (q) dq , P 4π 0 2π 1 µF (q) zF,q (q) cot 12 zB (p) − zF (q) dq . IBF (p; µF ) = 4πi 0

IBB (p; µB ) =

Acknowledgements: The research was supported by NSF (grant OCE-0620885).

(14) (15)

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References: [1] G.R. Baker, Boundary Element Methods in Engineering and Sciences, Chapter 8, Imperial College Press, 2010. [2] G.R. Baker, D.I. Meiron and S.A. Orszag, Generalized vortex methods for free-surface flow problems, J. Fluid Mech., 123, 1982, pp. 477–501. [3] S.J. Cowley, G.R. Baker and S.A. Tanveer, On the formation of Moore curvature singularities in vortex sheets, J. Fluid Mech., 378, 1999, pp. 233–267. [4] G.R. Baker, R.E. Caflisch and M. Siegel, Singularity formation during Rayleigh-Taylor instability, J. Fluid Mech., 252, 1993, pp. 51–78. [5] J. Toland, Stokes waves, Topol. Meth. in Nonlinear Anal., 7, 1996, pp. 1–48. [6] M. Okamura, On the enclosed crest angle of the limiting profile of standing waves, Wave Motion, 28, 1998, pp. 79–87. [7] M. Shinbrot, The initial value problem for surface waves under gravity, Indiana U. Math J., 25, 1976, pp. 281–300. [8] S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in 2-D, Invent. Math., 130, 1997, pp.39–72. [9] S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in 3-D, J. Amer. Math. Soc., 12, 1999, pp. 445–495. [10] S. Wu, Almost global well-posedness of the 2-D full water wave problem, Invent. Math., 177, 2009, pp. 45– 135. [11] P. Germain, N. Masmoudi and J. Shatah, Global solutions for the gravity water waves equation in dimension 3, C. R. Acad. Sci. Paris, Ser.1346, 2009, pp.897–902. [12] G.R. Baker and C. Xie, Singularities in the complex physical plane for deep water waves, J. Fluid Mech., [13] R. Krasny, A study of singularity formation in a vortex sheet by the point-vortex approximation, J. Fluid Mech., 167, 1986, pp.65–93. [14] M. Shelley, A study of singularity formation in vortex-sheet motion by a spectrally accurate vortex method, J. Fluid Mech. 244, 1992, pp.493–526.


An Isogeometric Boundary Element Method for Interior Acoustic Analysis using T-splines *Robert N. Simpson1 , Michael A. Scott2 ,Haojie Lian1 1 Institute

2

of Mechanics and Advanced Materials, School of Engineering, Cardiff University, Queen’s Buildings, The Parade, Cardiff CF24 3AA, Wales, UK [email protected]

Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, Texas 78712, USA

Keywords: isogeometric analysis, boundary element method, T-splines, acoustics Abstract: The present paper outlines an isogeometric boundary element method for interior acoustic analysis using T-splines to discretise both the geometry of the problem and unknown fields. The motivation for such a method arises from the difficulties often encountered in creating a suitable ‘mesh’ for analysis which can often dominate the engineering design/analysis cycle. Instead, the present formulation uses the boundary representation provided completely by computer aided design (CAD) in the form of T-splines that allows extremely complicated geometries to be modelled as a single patch. The method is verified against an analytical solution imposed on a three-dimensional cube for a variety of wavenumbers. In addition, to demonstrate the ability of the method to analyse geometries of arbitrary complexity the method is applied to a propeller geometry.

Introduction The task of producing suitable discretisations or meshes before analysis can be performed is known to be arduous and time-consuming, particularly in the case of complicated 3D geometries. Any method with either reduces or eliminates the need to mesh is therefore attractive due to the significant cost and time-savings that can be realised. Several such methods have arisen in recent years including meshless methods [1], immersed boundary methods [2], enriched methods [3] and several others. More recently, attention has been focussed on the direct coupling of numerical methods with CAD under the general title of ‘isogeometric analysis’ (IGA) [4]. The idea is to use the same discretisation used by CAD also for approximating the unknown fields during analysis and has been shown to exhibit advantages for shells [5][6][7], phase-field models [8][9], fluid-structure analysis [10][11][12] and many other applications. A particularly attractive application of the concept is within the framework of the boundary element method (BEM) which relies on a discretisation of the boundary of the problem only. NonUniform Rational B-Spline (NURBS) based CAD packages provide a boundary representation that can be used directly for boundary element analysis and this forms the basis of the isogeometric boundary element method. The concept has been illustrated in several recent developments [13][14][15]. Much of the previous work on isogeometric BEM has focused on potential and linear elastic problems, but an area which shows great promise is the topic of acoustic analysis. We introduce an isogeometric BEM for interior acoustic analysis while emphasising the benefits of using a CAD discretisation of the geometry for analysis. The paper briefly outlines the acoustic boundary integral equation along with the use of T-splines to discretise both the geometry and unknown fields. Two examples are shown to verify the accuracy of the method.

Isogeometric Boundary Element Method The inclusion of the isogeometric concept within the boundary element method is conceptually quite simple. The traditional Lagrangian basis functions that are used to approximate both the geometry and unknown fields (e.g. displacement and traction for linear elasticity) are replaced by the basis functions defined by CAD. However, this change has significant advantages over the traditional approach where the task of generating a boundary mesh has been either removed or drastically reduced and the geometry is represented exactly at all stages of analysis. Most of the IGA literature up to present has focused on the use of NURBS as the basis function choice since they are predominant in CAD. But as is well-known in the computational geometry community, they exhibit deficiencies that often require repairing. In addition, the global nature of the basis functions presents challenges

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for refinement during analysis. As a solution to this, T-splines have been proposed [16] which create ‘watertight’ geometries and allow for local-refinement. The IGA community has embraced this technology with several important developments made [17][18]. More recently, the technology has been employed within a boundary element framework [19] for linear elasticity. The work is now carried forward for interior acoustic analysis. The boundary integral equation for acoustics can be written as ∂G(s, x) ∂φ(x) φ(x) dΓ = dΓ G(s, x) C(s)φ(s) + − ∂n ∂n Γ Γ

(1)

where s is the source point, x is the field point, φ denotes the acoustic potential, ∂φ/∂n denotes the acoustic velocity, C(s) is a jump term, G(s, x) is the Green’s function that satisfies the Helmholtz equation, Γ is the boundary of the domain and − denotes the integral is taken in the Cauchy Principal Value limiting sense. Differentiating equation (1) with respect to the direction of the normal at the source point, a hypersingular boundary integral equation is produced as ∂ 2 G(s, x) ∂G(s, x) ∂φ(x) ∂φ(s) += φ(x) dΓ = dΓ (2) C(s) ∂n(s) Γ ∂n(x)∂n(s) Γ ∂n(s) ∂n(x) with = denoting a Hadamard finite part integral. For exterior acoustic analysis a common method is to use a linear combination of (1) and (2) to avoid spurious results at eigenfrequencies of the interior problem [20], but for interior analysis either (1) or (2) can be used. In the present work we choose to implement the latter in a regularised form [21] where the integrals are at most weakly singular and can be evaluated using polar integration. To discretise (2), the following expressions are used φ(x) =

ncp

φA RA (x)

(3)

A=1

ncp ∂φ(x) = v(x) = vA RA (x) ∂n

(4)

A=1

where ncp is the number of control points (defined by a CAD discretisation), RA (x) is the set of global T-spline basis functions and φA , vA are acoustic potential and velocity coefficients. These expressions can be substituted into (2) to arrive at the desired discrete form. On edges and corners, a semi-discontinuous form of (4) is used [19]. The system of equations is formed through collocation defining the position of each point in a suitable manner [19] (since the usual definition of nodal collocation would result in points lying outside the boundary). The collocation points are defined in such a way that no point is ever located on an edge or corner and is at least C 1 continuous. The use of equations (3) and (4) represents a simple change to a boundary element implementation but has farreaching consequences. Given a T-spline geometry, the use of these equations allows the unknown fields to be discretised directly circumventing the need to generate a boundary mesh. Admittedly, what may be a suitable geometry discretisation may not be as suitable for analysis, but there exists a variety of refinement algorithms [4] that can be used to provide a richer approximation space. To verify the accuracy of the method and to demonstrate the ability to model complex geometries, two examples are now given.

Examples Cube The first problem considered is that of a cube of unit dimension in each direction with the centre of the cube located at the origin. For three dimensions, the following expression can be shown to be a solution to the Hemholtz equation √ √ √ (5) φ(x) = sin(kx/ 3) sin(ky/ 3) sin(kz/ 3). Given a particular wavenumber k, equation (5) can be applied over the entire boundary of the problem while solving for the acoustic velocity ∂φ/∂n. To assess the ability of the formulation for varying wavenumbers the mesh shown in Figure 1 is considered. The mesh is analysed with k = 15, 30, 60, 90. The exact solution for k = 15 along with the numerical solution is illustrated in Figure 2. To compare each of the numerical solutions against the exact solution, a subset of results are taken along a line located on the boundary with x, z = constant and y varying (shown in Figure 2). The results are shown in Figure


Figure 1: Mesh and collocation point locations for cube problem.

Figure 2: Cube problem: exact and numerical solutions for k = 30.

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352


Figure 3: Comparison of exact and numerical solutions along boundary line shown in Figure 2 for increasing wavenumber.


Figure 4: Bézier mesh used for propeller analysis 3. The breakdown of the results for k = 90 is evident. It is therefore imperative that consideration is given to the quality of the CAD discretisation before acoustic analysis is performed. This may take the form of a fine discretisation or the use of special basis functions that can capture the oscillatory nature of the solution (e.g. [22]). Propeller Finally, to demonstrate that the method can be applied to arbitrarily complicated geometries, the method is applied to a propeller using the same analytical solution as used in the previous section. An wavenumber of k = 1.0 is used. The numerical and exact solutions are shown in Figure 5 where good agreement can be seen. What should be particularly stressed in this example is that the geometry used has been taken directly from CAD with no meshing step required. In addition, the exact geometry is used throughout the analysis.

Summary This paper briefly outlined the application of the boundary element method for interior acoustic analysis using the recently developed computational geometry tool known as T-splines. The method falls within the framework of ‘isogeometric analysis’ whereby the exact geometry is used for all stages of analysis. The isogeometric boundary element formulation was outlined and the method of constructing a system of equations through collocation briefly described. The method was verified against an analytical solution prescribed over a cube demonstrating the behaviour for a variety of wavenumbers. And finally, to illustrate the ability of the method to handle complicated geometries, the method was applied to the problem of a propeller.

References [1] T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, P. Krysl, Meshless methods: An overview and recent developments, Computer Methods in Applied Mechanics and Engineering 139 (1-4) (1996) 3–47. [2] R. Mittal, G. Iaccarino, Immersed boundary methods, Annual Review of Fluid Mechanics 37 (2005) 239– 261. [3] N. Moës, J. Dolbow, T. Belytschko, A finite element method for crack growth without remeshing, International Journal for Numerical Methods in Engineering 46 (1) (1999) 131–150. [4] T. J. R. Hughes, J. A. Cottrell, Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement, Computer Methods in Applied Mechanics and Engineering 194 (2005) 4135– 4195. [5] J. Kiendl, K.-U. Bletzinger, J. Linhard, R. Wüchner, Isogeometric shell analysis with Kirchhoff-Love elements, Computer Methods in Applied Mechanics and Engineering 198 (49-52) (2009) 3902–3914. doi:DOI: 10.1016/j.cma.2009.08.013.

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Figure 5: Propeller geometry: exact and numerical acoustic velocity solution for problem defined by equation (5) [6] D. J. Benson, Y. Bazilevs, M. C. Hsu, T. J. R. Hughes, Isogeometric shell analysis: The Reissner-Mindlin shell, Computer Methods in Applied Mechanics and Engineering 199 (5-8) (2010) 276–289. [7] D. J. Benson, Y. Bazilevs, M. C. Hsu, T. J. R. Hughes, A large deformation, rotation-free, isogeometric shell, International Journal for Numerical Methods in Engineering, 200 (2011) 1367 – 1378. [8] H. Gomez, V. M. Calo, Y. Bazilevs, T. J. R. Hughes, Isogeometric analysis of the Cahn-Hilliard phase-field model, Computer Methods in Applied Mechanics and Engineering 197 (2008) 4333–4352. [9] M. J. Borden, M. A. Scott, C. V. Verhoosel, C. M. Landis, T. J. R. Hughes, A phase-field description of dynamic brittle fracture, Computer Methods in Applied Mechanics and Engineering 217 (2012) 77 – 95. [10] Y. Bazilevs, M. C. Hsu, M. A. Scott, Isogeometric fluid-structure interaction analysis with emphasis on nonmatching discretizations, and with application to wind turbines, Computer Methods in Applied Mechanics and Engineering submitted for publication. [11] Y. Bazilevs, J. R. Gohean, T. J. R. Hughes, R. D. Moser, Y. Zhang, Patient-specific isogeometric fluidstructure interaction analysis of thoracic aortic blood flow due to implantation of the Jarvik 2000 left ventricular assist device, Computer Methods in Applied Mechanics and Engineering 198 (45-46) (2009) 3534–3550. [12] Y. Bazilevs, V. M. Calo, T. J. R. Hughes, Y. Zhang, Isogeometric fluid-structure interaction: Theory, algorithms, and computations, Computational Mechanics 43 (2008) 3–37. [13] R. Simpson, S. Bordas, J. Trevelyan, T. Rabczuk, A two-dimensional isogeometric boundary element method for elastostatic analysis, Computer Methods in Applied Mechanics and Engineering 209-212 (2012) 87–100. [14] K. Li, X. Qian, Isogeometric analysis and shape optimization via boundary integral, Computer Aided Design 43 (11) (2011) 1427–1437. [15] C. Politis, A. I. Ginnis, P. D. Kaklis, K. Belibassakis, C. Feurer, An isogeometric BEM for exterior potentialflow problems in the plane, in: 2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling, SPM ’09, 2009, pp. 349–354. [16] T. W. Sederberg, J. Zheng, A. Bakenov, A. Nasri, T-splines and T-NURCCs, ACM Trans. Graph. 22 (2003) 477–484. [17] Y. Bazilevs, V. M. Calo, J. A. Cottrell, J. A. Evans, T. J. R. Hughes, S. Lipton, M. A. Scott, T. W. Sederberg, Isogeometric analysis using T-splines, Computer Methods in Applied Mechanics and Engineering 199 (5-8) (2010) 229–263.


[18] M. A. Scott, M. J. Borden, C. V. Verhoosel, T. W. Sederberg, T. J. R. Hughes, Isogeometric Finite Element Data Structures based on Bézier Extraction of T-splines, International Journal for Numerical Methods in Engineering, 88 (2011) 126 – 156. [19] M. Scott, R. Simpson, J. Evans, S. Lipton, S. Bordas, T. Hughes, T. Sederberg, Isogeometric boundary element analysis using unstructured T-splines, Submitted to Computer Methods in Applied Mechanics and Engineering (June 2012). [20] A. Burton, G. Miller, Application of integral equation methods to numerical solution of some exterior boundary-value problems, Proceedings Of The Royal Society Of London Series A-Mathematical And Physical Sciences 323 (1553) (1971) 201–210. [21] Y. Liu, S. Chen, A new form of the hypersingular boundary integral equation for 3-d acoustics and its implementation with C0 boundary elements, Computer Methods in Applied Mechanics and Engineering 173 (1999) 375–386. [22] O. Laghrouche, P. Bettess, E. Perrey-Debain, J. Trevelyan, Plane wave basis finite-elements for wave scattering in three dimensions, Communications in Numerical Methods in Engineering 19 (9) (2003) 715–723.

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A Novel Local Active Noise Control Strategy Using a Control Volume Formulated by a Fast Boundary Element Approach A. Brancati1,2, M.H. Aliabadi3,4 1

Facoltà di Ingegneria, Architettura e delle Scienze Motorie, Università di Enna “Kore”, Cittadella Universitaria, 94100 Enna, IT 2 3

[email protected]

Department of Aeronautics, Imperial College London, South Kensington, SW7 2AZ, UK, 4

[email protected]

Keywords: Active Noise Control, Control Volume, Adaptive Cross Approximation/Hierarchical matrix format.

Abstract. In this paper a novel local active noise control (ANC) strategy using a Control Volume (CV) is presented. A 3D Boundary Element Method (BEM) in conjunction with the Adaptive Cross Approximation (ACA), the Hierarchical Matrix (H-matrix) format and the GMRES solver has been utilised to speed up the computational cost and to simulate large scale problems. The primary disturbance is attenuated in the CV by minimising the square modulus of two acoustic quantities, the pressure and one component of the particle velocity. Formulations for one and two control sources are described. A large scale engineering problem demonstrate the accuracy of the proposed strategy. 1.

Introduction

The origins of Active Noise Control (ANC) can be traced back to the pioneering work of Paul Lueg [1] in 1933 and Conover [2] in 1956. In the last decades the problem of Active Noise Control (ANC) has been widely explored. In the free field, Nelson et al. [3] claimed that a significant global noise reduction (at least 10 dB power attenuation) can be achieved only if the separation distance between the primary monopole source and the control source is less than one-tenth of the wavelength of the disturbance. In the case where the control source is placed at half wavelength from the primary source, no reduction can be accomplished. In an enclosed space Nelson et al. [4] investigated and developed a computer simulation of ANC and verified their models experimentally for harmonic enclosed sound fields. They established that a disturbance can be globally reduced for resonance frequencies and the control source does not require to be separated by less than one half wave-length from the primary noise source as for the free field case, even for considerable number of sources. Early works on the development of local active noise control approach are due to the theoretical and experimental study of Joseph et. al [5] and the numerical work of David and Elliott [6]. It was reported [6] that a 10 dB reduction zone can be obtained for frequencies above the Schroeder frequency and for uniform and diffuse primary noise, and the reduction can be larger, up to one tenth of the wavelength, if the cancellation point is further from the secondary source. Moreover, the sound pressure level (SPL) away from the cancellation point is almost unaffected. The local ANC approach has been further developed by Garcia-Bonito and Elliott [7]. In the subsequent work Garcia-Bonito and Elliott [8] demonstrated that the reduction zone can be enlarged by canceling the pressure and the secondary particle velocity at two different points. This paper describe a strategy to reduce noise in a 3D free field by a local ANC approach. The BEM is utilised for simulations at monotone frequencies. The solution is accelerated by the ACA in conjunction with the H -matrix format and the GMRES. A novel and general local ANC strategy that can be directly applied to a conventional digital signal processing (DSP) system without significant changes is presented. The basic idea is to minimise the square modulus of the pressure and the square modulus of the total particle velocity in one direction in a predefined volume rather than at a single points or discrete points. This technique aims to extend the noise reduction volume into a larger zone than for a standard point cancellation procedure. The problem is approached using a single and two control sources, respectively. A large scale engineering problem of noise reduction inside an aircraft cabin is presented.

Advances in Boundary Element and Meshless Techniques 2.

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Control Volume Strategy: Single Control Source

A novel approach for attenuating an unwanted noise in a prescribed volume is here presented. The approach consists of reducing the noise in an area of interest, here called control volume (CV) and denoted as D 1) with XD belonging to D and where P(XD) collects the pressure at the CV points. As evident, for a given primary noise distribution, the above cost function is minimised by the secondary source field, that is generated by a 3D object with hard boundary conditions (i.e., q=0 ) everywhere except for a vibrating portion of the surface (i.e., q≠0 ) The optimum vibration velocity of the secondary source surface q’s can be evaluated using a constant α that relates the optimum secondary field solution to a solution obtained by any velocity of secondary source vibrating surface qs as follows 2) q’s = α qs The pressure P and particle velocity P’ at any point of the domain (including the boundary points) is generated by both the primary and the secondary fields. Due to the linearity of the wave equation, each of these fields can be evaluated separately, and summed together as follows

3) where the subscripts p and s refer to the primary and secondary quantities, respectively, and the subscripts R and I refer to the real and imaginary parts, respectively. Substituting relations (3) into the cost function (1) yields the following expression 4) The complex constant α is determined by minimising the cost function (i.e., setting the cost function derivative with the respect to αR and αI equal to zero). Hence, a system with two equations is obtained where the solution α can be written as follows 5) where the constants values in the above expression are shown in Table 1.

Table 1. Values of the terms on eq (5).

In order to obtain the value of α, the conventional BEM system of equations (AX=B) is solved first for the primary field and again for the secondary source field with any prescribed values of the boundary conditions at the vibrating surface. The solution (5) represents the optimum noise attenuation that can be achieved since the second derivative of fc with respect to α is always positive. 3.

Control Volume Strategy: Two Control Sources

The addition of the second control source requires another cost function that should be minimised contemporaneously with eq (1). Such a cost function can be written as follows 6) where P’1 is the component along the x1-direction of the particle velocity. The second loudspeaker is also modelled as a hard 3D object with a vibrating surface. Its optimal vibration velocity q’s is evaluated by a second constant as for the previous case: q’s=β qs. The total pressure and the total component of the total particle velocity along one direction at a generic point are evaluated, respectively, as in expression (3) by adding an extra term that refers to the second control source as follows

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7) where the subscripts s1 and s2 refer to the first and the second control source, respectively. Expressions in (7) are substituted into the cost functions (1) and (6) that are minimised by α and β (i.e., the cost function derivatives with the respect to αR, αI, βR and βI are set equal to zero). A system with four equations and four unknowns (αR, αI, βR and βI) is obtained whose solution is displayed below. The solution of such system can be simplified, as shown below, by considering the primary field as constant at the CV points. Each term of the α and β requires solution of the BEM solving system (AX=B) and to evaluate the solution at internal points three times, first for the primary field, by considering both the secondary sources as rigid surfaces, and again twice for the two secondary sources, acting independently, for any initial value of the vibrating velocity and by considering the other secondary source as a rigid sphere. The solution for the standard procedure in terms of αR, αI, βR and βI can be written as follows

8)

9)

10)

11) where P’PI and P’PR are the real and imaginary parts of component along the x1 -direction of the primary particle velocity, respectively. All the other quantities in the above solutions are reported in Table 2. It should be noted that each term of this table is integrated in the control volume.

Table 2. Values of the terms on eq (8-11).

4.

The Adaptive Cross Approximation

Advances in Boundary Element and Meshless Techniques Due to the fact that the BEM system matrix is non-symmetric and fully populated, the solutions of the BEM solving systems (AX=B) are time consuming, since they need to be solved twice or three times in case of single and double secondary source, respectively. Moreover, each term of the cost functions (1) and (6) requires evaluation of the internal point quantities that can considerably increase the solution time in the case of large sized control volumes. In this paper, to accelerate the CPU time for all simulations the Adaptive Cross Approximation (ACA) in conjunction with the H-matrix format and the iterative solver GMRES is utilized. The ACA is a pure algebraic technique widely investigated [9] and applied in many engineering fields [10]. The solving matrix is divided into two groups of blocks, full-rank and low-rank blocks. The former blocks are calculated entirely, while the latter blocks are replaced by few entries. The idea behind this technique is based on the consideration that the integrals of contiguous elements due to a single collocation point are almost identical, especially for high density meshes. The same consideration is valid for the integrals of a single element due to a number of contiguous collocation points. The main advantage of the ACA is that the kernel is not substituted (as happens in the Fast Multiple Method) and only few entries are required, therefore most part of existing BEM codes can be used without meaningful changes. Further details on this subject can be found in [11]. 5.

Numerical Results

This section presents the results obtained by the proposed strategy when applied to a large scale engineering problem, i.e., the ANC in an aircraft cabin. The model of the cabin represents a portion of fuselage limited by a rear and a front panel and it is included in a cuboid of dimensions 2 x 1.9 x 2.3 m (see Fig. 1). Inside, the cabin is composed of two lines of three seats. Each line is 0.5 x 1.35 m with height 1.054 m. Due to the geometrical symmetry, only half cabin is utilised. The cabin mesh is composed of 3914 nodes and 7699 constant elements in order to deal with up to 250 Hz (i.e., 10 elements per wavelength are guaranteed). The headrest is 0.46 m long (along the x2- axis) and the CV is a 0.2 x 0.32 x 0.2 m cuboid (so that it covers most of the headrest extension) and it is located in the front line at the seat close to the edge of the fuselage (see Fig.2). The integration in the CV is performed by subdividing it into 160 cuboid linear elements. The boundary conditions (BCs) were set to represent a possible real circumstance. The front and rear panel are modelled with soft BCs (p=0 ), whereas the symmetry plane as a hard panel (q=0 ). In the model the seats, the floor and the ceiling have sound absorption properties with absorbing coefficients varying linearly at different frequencies. In particular at 70 Hz the absorbing coefficients of seats, floor and ceiling have absorbing coefficient equal to 0.14, 0.10 and 0.06, respectively, and at 200 Hz equal to 0.53, 0.30 and 0.20. The primary disturbance is generated by a monopole located at the point (1.87, 0.25, 1.90) close to the symmetry panel and it is evaluated in the centre of the CV and maintained constant. The secondary source field is created by a sphere of radius 0.04 m with a constant active segment of 120° and its centre is located at the point (0.760, 1.525, 1.460) close to the ceiling above the CV. Its vibration portion is headed in the negative x3 direction. Four frequencies are analysed, i.e., 82, 110, 142, 182 Hz, being close to the resonance frequencies of the cabin at a typical jet noise frequency range (50-200 Hz). Table 3 shows the cost function without and with the ANC, and the noise reduction level (in dB) obtained inside the CV for each analysed frequency.

Fig. 1. Aircraft cabin geometry..

Fig 2. Zoom on the aircraft cabin geometry..

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Fig. 3 and 4 compare the SPL (in dB) inside the cabin at two frequencies, i.e., 82 and 182 Hz, without a) and with b) the proposed ANC technique.

Fig. 3. SPL inside an aircraft cabin at 82 Hz: a) without ANC, and b) with ANC.

6.

Fig 4. SPL inside an aircraft cabin at 182 Hz: a) without ANC, and b) with ANC.

Consideration and Future work

The proposed strategy is appealing as its application to a conventional DSP can be achieved since only the related transfer function (to include the CV parameters) needs to be modified. This is due to the fact that the primary disturbance is required only at a single point, and the extension of the noise reduction area depends only upon the secondary source response. It was also demonstrated that the use of an additional secondary source can provide a greater noise reduction. Furthermore, the secondary particle velocity can be used instead of the total particle velocity to make the strategy practical and to accelerate the execution time of the DSP system without affecting the noise attenuation level. An optimised control source location can increase the performance of the proposed strategy in enclosed space. The effectiveness of the proposed CV approach would need to be tested experimentally in the future. References [1] P Lueg. Process of Silencing Sound Oscillations. German Patent DPR No. 655 508 (1933). [2] W.B. Conover. Fighting noise with noise. Noise Control 2, 78-82 (1956). [3] P.A. Nelson, A.R.D. Curtis, S.J. Elliott, A.J. Bullmore. The Minimum Power Output of Free Field Point Sources and The Active Control of Sound. Journal of Sound and Vibration 116, 397-414, (1987). [4] P.A. Nelson, A.R.D. Curtis, S.J. Elliott, A.J. Bullmore. The active minimization of harmonic enclosed sound fields: part i, ii, iii. Journal of Sound and Vibration 117, 1-58 (1987). [5] P. Joseph, S.J. Elliott, P.A. Nelson. Near field Zone of quite. Journal of Sound and Vibration 172, 605-627 (1994). [6] A. David, S.J. Elliott. Numerical Studies of Actively Generated Quite Zones. Applied Acoustics 41, 63-79 (1994). [7] J. Garcia-Bonito, S.J. Elliot. Local Active Control of Diffracted Diffuse Sound Fields. J. Acoust. Soc. Am. 98, 1017-1024 (1995). [8] J. Garcia-Bonito, S.J. Elliot. Active Cancellation of Acoustic Pressure and Particle velocity in the Near Field of a Source. Journal of Sound and Vibration 221, 85-116 (1999). [9] M.Bebendorf, S.Rjasanow. Adaptive low-rank approximation of collocation matrices. Computing, 70(1), 1-24 (2003). [10] I.Benedetti, M.H.Aliabadi, G.Daví. A fast 3D dual boundary element method based on hierarchical matrices. International Journal of Solids and Structures, 45(7-8), 2355-2376 (2007). [11] A.Brancati, M.H.Aliabadi, I.Benedetti. Hierarchical Adaptive Cross Approximation GMRES Technique for Solution of Acoustic Problems Using the Boundary Element Method. CMES: Computer Modeling in Engineering & Sciences, 43(2), 149-172 (2009).


A Fast 3D-BEM Approach to Local Active Noise Control: a Sensitivity Analysis A. Brancati1,2, M.H. Aliabadi3,4 and V. Mallardo5,6 1

Facoltà di Ingegneria, Architettura e delle Scienze Motorie, Università di Enna “Kore”, Cittadella Universitaria, 94100 Enna, IT 2 3

[email protected]

Department of Aeronautics, Imperial College London, South Kensington, SW7 2AZ, UK, 4 5

[email protected]

Dip. Architecture, University of Ferrara, Italy, 6

[email protected]

Keywords: Active Noise Control, Control Volume, Sensitivity analysis, Optimum secondary source location/orientation, Adaptive Cross Approximation/Hierarchical matrix format.

Abstract. In this paper a fast Boundary Element (BE) formulation for optimising a local Active Noise Control (ANC) strategy is presented. The Helmholtz differential equation is rapidly solved in a threedimensional field using an Adaptive Cross Approximation (ACA) approach in conjunction with the Hierarchical matrix format and the GMRES solver. The noise is reduce in a local sense by attenuating a primary disturbance within a predefined volume, called control volume, utilising a control source modelled as an object with a vibrating surface. A sensitivity analysis is utilised to evaluate the optimum position of the secondary source and the control volume and the optimum orientation of the secondary source. The proposed formulation is valid both scattering and internal wave propagation problems. The accuracy and efficiency of the proposed formulation is demonstrated through example simulating real circumstances. 1.

Introduction

The field of Active Noise Control (ANC) has been extensively investigated in the last three decades. The initial effort was put on globally reducing a primary offending noise in an unbounded space by placing a secondary speakers in the proximity of the primary source. Elliott et al. [1] demonstrated the achievability of a system able to reduce noise in local sense. Studies to maximise the level of noise attenuation through the optimization of the actuator locations have been explored only recently. The work of Baek and Elliott [2] is a pioneering study for the evaluation of the optimum location of the secondary source. Another important study was conducted by Martin and Roure [3] who determine the optimum location of the actuators. Many researchers proposed methods to solve the shape design sensitivity analysis for the acoustic problems using the BEM, mainly in terms of accuracy (see for instance [4-6]). It has been shown that a direct differentiation approach is more accurate and is able to overcome the inaccuracy of the finite difference method (FDM). The main drawback of first order optimisation methods is that they necessitate the first derivatives of the cost function. Hence, it is also required to evaluate the first derivatives of the BEM system solution with respect to the design variables. Consequently, the problem requires solving two groups of systems of equations, one for the direct problem and a second group for evaluating the sensitivities. The computational effort can be large and it can be one of the main drawbacks of the sensitivity analysis. In recent years, only a few strategies have been proposed to overcome this difficulty (see [7-9]), employing preconditioned iterative equation solvers, a coupled FE and BE approach and the fast multipole method. In this paper a novel BEM formulation for the optimization of a local ANC is presented. The noise attenuation strategy is based upon the formulation presented in [10] where the noise, simulated in a 3D field, is reduced within a fixed enclosure, called a control volume (CV), with a fixed secondary source. Here, the three main original contributions are: i) first order derivative sensitivity analysis applied to a local ANC approach solved with the BEM; ii) optimization of the noise attenuation area location, of the control source location and, overall, orientation for any secondary source shape;

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iii) reduction of the computational cost of the proposed analysis utilising the Adaptive Cross Approximation (ACA) technique coupled with the Hierarchical matrix (H-matrix) format and the GMRES (RABEM code [11]). BEM formulations are presented and applied to the three dimensional (3D) Helmholtz equation for monotone frequencies both to determine the solution of the direct problem and to evaluate the sensitivities by an implicit differentiation approach. A superparametric formulation with linear-constant elements has been used. The Adaptive Cross Approximation (ACA) is utilised to generate both the system matrix and the right hand side vector, the H-matrix format is used for the storage requirements and the GMRES is used to solve the linear systems of equations. The same strategy is also adopted to evaluate the potential at selected internal points, the differential of the system matrix and of the potential with respect to the design variables. Examples are presented to demonstrate the accuracy of the proposed procedure. 2.

Optimisation strategy

The ANC has been approached by attenuating an unwanted noise in the Control Volume (CV), a confined region [10]. The proposed approach is able to optimise both the location and the orientation of the secondary source for fixed geometry and CV, and to optimise the location of the CV for given geometry and control source position and orientation. Due to the nature of the Helmholtz equation, the unwanted primary noise Up and the control source fields Us are independently evaluated and linearly summed to generate the total field U , i.e. 1) where α is a constant utilized to drive the optimum control field, the subscripts p and s refer to the primary and the secondary results, respectively, and the subscripts R and I refer to the real and imaginary parts, respectively. The problem is solved by minimising a specific cost function, i.e. the square modulus of the potential U within the CV in space as D 2) where a and b are the position vectors of the CV and the secondary source, respectively (see Fig. 1).

Fig. 1. Geometry of the problem: secondary source, remaining boundary, control volume, position vectors of the control volume (a) and of the secondary source (b).

Fig. 2. Euler angles.

The approach evaluates the optimum design variables iteratively and the final solution is achieved if the difference between the cost functions at two consecutive design variables k and k+1 is below a prescribed positive tolerance ε [12] 3) The vector containing the design variables a (or b) at the k+1 iteration can be evaluated by adding to the k+1 k k value of the vector at the previous iteration a particular variation of the design variable, a = a +∆a , with ∆ak=αkdk, where dk is the desirable search direction to seek the optimum solution and αk is the step size, a positive scalar quantity, that indicates the step in that direction. The optimisation procedure adopted here is based upon the Broyden-Fletcher-Goldfarb-Shanno (BFGS) variant of the Davidon-Fletcher-Powell (DFP) method [12] that is a first order method. Hence, at each iteration the derivative of the cost function (2) with respect to each single component m of a design variable (either a or b) is required and it is evaluated using eq. (1) as follows 4) where the Leibniz Integral Rule has been used and ,m stands as the partial derivative with respect to the design variable. Therefore, the optimization process can be updated if the sensitivities, i.e., the partial

Advances in Boundary Element and Meshless Techniques derivative of the primary and secondary potentials with respect to the position vector a or b at every CV point, are evaluated at every iteration. 3.

The BIE Differentiation

The sensitivities involved in eq. (4) can be determined by differentiating the system matrix Hu=Gq+p obtained by collocating the discretised form of the classic boundary integral equation as follows H,mu+Hu,m= G,mq+ Gq,m+p,m 5) where u and q are the vectors containing the potential and flux at the boundary, H and G are the solving matrix, p is a vector collecting the contribution of the extra sources such monopole and planewaves and ,m has the usual meaning. The boundary conditions can be easily accounted into the system matrix to give AY=BJ+p=F+p 6) where Y is the vector containing the unknown boundary potentials and fluxes, J is a vector collecting the prescribed BCs, A and B are two coefficient matrices which are non-symmetric and densely populated, and they are composed by the columns of the matrices H and G that correspond to the unknowns and the prescribed BCs, respectively. Finally, F is obtained by multiplying the matrix B with the vector J. The differential form of eq. (6) gives AY,m=A,mY+B,mJ+p,m 7) The potential derivative U,m with respect to the design variable at the internal points are evaluated utilizing the conventional post-processing step as follows U,m=-H,mu-Hu,m+G,mq+Gq,m+P,m 8) where H and G are two coefficient matrices similar to H and G but evaluated at selected internal points. 4.

Sensitivity Analysis

The partial derivative of the fundamental solutions with respect to the design variable m can be evaluated as follows

9) with r the distance between the boundary element x and the internal (or boundary) collocation point X’ (or x’) and i the imaginary unit. In order to calculate the partial derivative of r and r,n with respect to the design variable, the chain rule was utilized and the final expressions depend upon the type of optimisation problem under analysis, i.e., optimum control volume location or optimum secondary source location or optimum secondary source orientation. Optimum Location: Control Source and Control Volume. The optimisation procedure to evaluate the control source location that maximises the noise reduction level requires the value of r,m that can be calculated as follows 10) where j = 1; 2; 3 refers to the coordinate system, δjm is Kronecker delta, rj= xj-xj‘ is the single component of r along the j-axes and rm is the component along the direction m. The sign of the expression (10) depends upon the element over which the integral is calculated. In particular, if the element belongs to the secondary source, m coincides with the direction of xj and the sign of (10) is positive, whereas if the collocation point is on the secondary source boundary, m coincides with the direction of xj‘ and (10) is negative. The second derivative of r with respect to n and m is given by the following relation 11) The rule utilised previously in the eq. (10) to determine the sign is here recalled, hence r,nm has positive/negative sign when r,m is positive/negative. The optimisation procedure to evaluate the control volume location requires solving eq.s (10) and (11) that has in this case always negative signs. Optimum Orientation: Control Source. The proposed strategy to evaluate the optimum secondary source orientation involves the evaluation of the derivative of the fundamental solution through the previous

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relations in eq. (9). Fig. 2 shows the Euler angles used in the proposed approach. Let Xj system, with j=1, 2, 3, be fixed and let the xj system rotate with the control source. N is the line of nodes. Both systems have origins in the centre of the secondary source. The first rotation is by an angle φ around the X3-axes, the second by an angle θ around x1-axes and the last by an angle ψ around x3-axes. The partial derivatives with respect to m (φ or θ or ψ) of the distance r can be evaluated as follows: 12) where rj,m with j=1, 2, 3 is the derivative of the component rj along the j -axes with respect to the design variable m. The derivative of r,n with respect to m can be written as follows: 13) where nj,m indicates the derivative of the outward normal nj along the j-axes with respect to the design variable m. In order to evaluate Eq.s (12) and (13), only the derivative with respect to the design variable of the node coordinates xj in the global coordinate system is required. It should be noted that rj,m coincides with xj,m when the element node belongs to the secondary source boundary, whereas it coincides with -xj,m when the point source is on the secondary surface. As evident in Eq. (24), the value nj,m vanishes when an element does not belong to the secondary surface. The derivative of the nodal coordinates can be obtained by superposition of the source centre coordinates Cj and the source local coordinates Xj, that is 14) Hence, only the derivatives with respect to each of the three rotations in the fixed coordinate system are required. Finally, one obtains

15) where R-1j,m is the derivative of the j-th inverse rotation matrix with respect to m and they are written as follows:

16) The derivative of the normal n with respect to the rotation design variables is obtained similarly and is omitted here. 5.

Numerical Aspect

At each iteration of the optimisation procedure the two solving systems of eq.s (6) and (7) are modified by the changes of the secondary source position, hence they both need to be assembled and solved twice, for the primary and for the secondary fields (as evident, in the case of optimum control source rotation the primary field is evaluated only once). However, it is easy to demonstrate that not all the coefficients of the matrices H, G, H,m and G,m require a new value at each step. The proposed procedure locates the element coefficients referred to the secondary source at the beginning of the matrices and those referred to the remaining boundary of the domain at the end. Consequentially the solving matrices are partitioned into four components on the basis of the position of the source points and of the integration elements: two parts along the diagonal (Mii and Mij) and the remain two in the off diagonal section (Mji and Mjj). Mii and Mij refer to the effect on the secondary source generated by itself and by the remaining part of the boundary, respectively, while the Mji and Mjj parts collect the integrals when the source point is placed anywhere except on the secondary source and the integration element moves both on the secondary source (Mji) and on the remaining boundary (Mjj), respectively. As evident, Mii and Mjj remain unchanged for a new secondary source location/orientation, whereas the matrices H,m and G,m have zero coefficients in such portions. Hence, at each iteration the H, G, H,m and G,m matrices are evaluated only at the Mji and Mij portions for both the primary and secondary fields. Rapid Solver. It is well known that the BEM generates non-symmetric and fully populated system matrices, that slow down the solution time, especially in problems where a few different solutions are calculated at every step of the optimisation procedure. In our case, each term of the cost function (2) and its

Advances in Boundary Element and Meshless Techniques derivative (4) requires evaluation of the internal point quantities that can considerably increase the solution time in the case of large sized control volumes. To accelerate the CPU time an Adaptive Cross Approximation (ACA) approach in conjunction with the H-matrix format and the iterative solver GMRES is utilized. The ACA is a pure algebraic technique widely investigated [13] and applied in many engineering fields [14]. The solving matrix is divided into two groups of blocks, full-rank and low-rank blocks. The former blocks are calculated entirely, while the latter blocks are replaced by few entries. The idea behind this technique is based on the consideration that the integrals of contiguous elements due to a single collocation point are almost identical, especially for high density meshes. The same consideration is valid for the integrals of a single element due to a number of contiguous collocation points. The ACA has been chosen since it is not required to substitute the kernel (as happens in the Fast Multiple Method) and only few entries are required, therefore most part of existing BEM codes can be used without meaningful changes. Further details on this subject can be found in [11]. 6.

Numerical Results

In this section two different examples are presented to test the proposed procedure. The secondary field is created by a hard (q=0) spherical speaker (with radius equal to 8 cm except in the last example where it has been reduced to 4 cm) with a 120° vibrating segment (with constant q) and meshed with 120 constant elements (see fig. 3). The optimum secondary source location is obtained in terms of the coordinates of the sphere's centre. Optimisation of the secondary source orientation. A box with dimensions 1x0.5x0.5 (in metres) contains a small cubic (0.3x0.3x0.3 m) CV, with the two opposite corners (0.1, 0.05, 0.05) and (0.4, 0.35, 0.35). The box is meshed by 1002 nodes and 2000 superparametric (linear geometry, constant unknown) elements. The primary noise has unitary amplitude, and it is generated by a vibrating portion of a wall of the box, with opposite corners at (0.75, 0.25, 0.5) and (0.85, 0.4, 0.5) (see Fig.3). The remaining walls are modelled as hard (q=0). The frequency chosen is 335 Hz, close to the first resonance frequency (340 Hz). The centre of the secondary source is located in front of the source of noise at the point (0.8, 0.3, 0.4) and its vibrating surface is turned towards the negative x2-axes (initial guess). At this position the cost function value is fc=5.163·10-5. The geometry of the problem as depicted in Fig. 5, where the CV, the secondary source and the vibrating surface of the wall can be easily distinguished.

Fig. 3. Geometry of the problem, CV, control source (initial guess) and primary noise locations.

Fig 4. Sound pressure level distribution (dB) of the total field generated by the active noise control strategy: (a) initial guess and (b) optimal value.

Fig. 4 shows the SPL contour plot in the whole box for two different secondary source orientations (i.e. (a) initial guess (φ=0, θ=0, ψ=0) and (b) optimal configuration (φ=0.305, θ=0.857, ψ=0.092)) is shown. The optimum cost function value is fc=1.576 10-5, and the obtained noise attenuation amounts to 5.15 dB inside the CV. This example shows that the orientation of the secondary source influences the level of noise; its optimum value may reduce considerably the SPL inside the CV. Optimization of the secondary source location in a large-scale engineering problem. This simulation concerns the evaluation of the optimum location/orientation of a secondary source inside an airplane cabin. The model of the analysed cabin consists of two lines of three seats surrounded by the aircraft fuselage. Because of the geometrical symmetry, only one-half of the cabin is considered. The cabin is included in a cuboid of dimensions 2 x 1.9 x 2.3 m, and each line of seats has dimensions 0.5 x 1.35 with height 1.054 (see Fig. 5). The headrest is 0.46-m long (along the x2-axes), and the CV is located in the middle front seat and has dimensions 0.2 x 0.32 x 0.2 m; therefore, it covers most of the headrest extension. The cabin is meshed by 3914 nodes and 7699 constant elements in order to deal with up to 250 Hz (i.e. 10 elements per wavelength are guaranteed). The integration in the CV is performed by

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subdividing it into 160 cuboid linear elements. The boundary conditions have been set in order to simulate a real case; hence, the rear and front panel are modelled as soft surfaces (u=0). The remaining surfaces, that is, seats, floor and ceiling, have all been modelled with absorbing coefficients being 0.85, 0.2 and 0.15. In order to evaluate the impedance value, the impedance phase is set to zero. The noise source is generated by a monopole source located at point (1.46, 0.52, 0.87) in front of the rear seat close to the symmetry panel, whereas the secondary field is modelled as a rigid sphere (centred in (0.76, 1.02, 1.46)) with an active segment turned towards the negative x3-axes. Initially, a series of simulations were performed to evaluate the resonance frequencies of the cabin. Two values have been evaluated: 104 and 182 Hz, hence, two simulations were performed. In the first simulation, the final location of the control source centre is at point (8.576 10-1, 5.182 10-1, 1.641), whereas the vibrating portion is slightly modified by the procedure (-9.279 10-4, -2.058 10-2, 6.113 10-3). In the second simulation the optimal location and orientation of the secondary source are (7.195 10-1, 1.085, 1.558) and (1.446 10-4, 5.133 10-2, -2.908 10-3), respectively. The optimisation procedure is able to reduce the noise level of 8.58 dB at 104 Hz (see Fig. 6) and of 1.33dB at 182 Hz. However, the presence of the secondary source in its optimised configuration reduces the noise levels of 22.49 and 11.75 dB at 104 and 182 Hz, respectively.

Fig. 5. Geometry of the cabin: CV, initial secondary source and primary noise locations.

Fig 6. Sound pressure level (SPL) at 104 Hz generated by the (a) initial guess and (b) final optimised secondary source location/orientation.

References [1] J.Garcia-Bonito, S.J.Elliott. Active Cancellation of Acoustic Pressure and Particle velocity in the Near Field of a Source. Journal of Sound and Vibration; 221(1), 85-116 (1999). [2] K.H.Baek, S.J.Elliott. Natural Algorithms For Choosing Source Locations In Active Control Systems. Journal of Sound and Vibration, 186(2), 245-267 (1995). [3] T.Martin, A.Roure. Active Noise Control Of Acoustic Sources Using Spherical Harmonics Expansion and a Genetic Algorithm: Simulation and Experiment. J. Sound and Vib., 212(3), 511-523 (1998). [4] V.Mallardo, M.H.Aliabadi. A BEM sensitivity and shape identification analysis for acoustic scattering in fluid-solid problems. Int. J. for Num. Methods in Eng., 41(8), 1527-1541 (1998). [5] J.H. Kane JH, Mao S, Everstine GC. A boundary element formulation for acoustic shape sensitivity analysis. Journal of the Acoustical Society, 90(1), 561-573 (1991). [6] R.D.Ciskowski, C.A.Brebbia. Boundary element methods in acoustics. Computational Mechanics Publications Elsevier Applied Science, 1991. [7] K.Guru Prasad, J.H.Kane. Shape reanalysis and sensitivities utilizing preconditioned iterative boundary solvers. Structural Optimization, 4,224-235 (1992). [8] D.Fritze, S.Marburg, H.J.Hardtke. FEM–BEM-coupling and structural–acoustic sensitivity analysis for shell geometries. Computers and structures, 83, 143-154 (2005). [9] N.Nemitz, M.Bonnet. Topological sensitivity and FMM-accelerated BEM applied to 3D acoustic inverse scattering. Engineering Analysis with Boundary Elements, 32(11), 957-970 (2008). [10] A.Brancati, M.H.Aliabadi. Under review. Boundary Element Simulations For Local Active Noise Control Using An Extended Volume. [11] A.Brancati, M.H.Aliabadi, I.Benedetti. Hierarchical Adaptive Cross Approximation GMRES Technique for Solution of Acoustic Problems Using the Boundary Element Method. CMES: Computer Modeling in Engineering & Sciences, 43(2), 149-172 (2009). [12] W.H.Press. Numerical recipes: the art of scientific computing. Cambridge University Press (2007). [13] M.Bebendorf, S.Rjasanow. Adaptive low-rank approximation of collocation matrices. Computing, 70(1), 1-24 (2003). [14] I.Benedetti, M.H.Aliabadi, G.Daví. A fast 3D dual boundary element method based on hierarchical matrices. International Journal of Solids and Structures, 45(7-8), 2355-2376 (2007).


Effect of aspect ratio on the stability analysis of uniaxially loaded plates with eccentric holes by the boundary element method P. C. M. Doval1 , E. L. Albuquerque2 and P. Sollero3 1

Department of Mechanical and Material, Federal Institute of Maranhão Av. Get´ ulio Vargas,04, São Lu´ıs, Maranh˜ ao, Brazil, [email protected] 2

Faculty of Technology, University of Bras´ılia, Campus Universitário Darcy Ribeiro Bras´ılia, DF, Brazil, CEP 70.910-900, [email protected] 3

Department of Computational Mechanics , State University of Campinas Rua Mendeleiev, 200, Campinas, S˜ ao Paulo, Brazil, CEP 13.083-970, [email protected]

Abstract. This paper presents a boundary element formulation to investigate the onset of instability of square composite laminate perforated plates with length a. Composite laminate plates with simply supported and clamped edges in the out-off plane direction and subjected to uniaxial end compression in their longitudinal direction are considered. Stresses caused by external loads are calculated by the formulation of plane elasticity boundary element method. These stresses are introduced as body forces in the classical formulation of plates. The domain integrals due to body forces are transformed into boundary integrals using the radial integration method. In this method, body forces are approximated by a sum of radial basis functions, called approximation functions, multiplied by coefficients to be determined. Some numerical examples are analyzed in which critical loads, buckling modes, and coefficients of buckling are calculated. The potential to increase the elastic critical load for composite laminated unperforated plates by the introduction of a small perforation is reported. Keywords: Stability of structures, buckling, composite laminate plates, radial integration method, boundary element method.

1

Introduction

An understanding of buckling of structural components under compressive load has become particularly important with the introduction of high-strength composite material in engineering structures, which resulted in more optimized components than those used in previous projects. Buckling analysis of compression panels also is particularly important in aerospace structures. Mainly, in aerospace structures, often need to make holes in plates to allow passage through this duct. For example, for servicing through bulkheads or similar. Such perforation are manufactured to meet the cabling and routing through them, and the preferred placement is seldom central. The holes are normally circular in shape to ensure that no stress concentration arising, but sometimes it is necessary to form holes rectangular with rounded corners to avoid excessive stress concentrations. Plates structural built with these materials and slender members may fail when subjected to compressive loads in your plan. In some cases these failures are not by direct compression, but for lateral buckling. The formulation of the boundary element method for anisotropic plane elasticity was developed by [1] to problems of fracture mechanics and elasto-static and extended to other problems in the work of [2]. The formulation of boundary elements for the classical theory of anisotropic plates without domain integrals was developed by [3, 4], and extended to other problems in the work of [5]. The formulation for stability analysis of thin perforated plates of laminated composite was developed by [6]. The last work has used the radial integration method for the transformation of domain integrals that remain in the formulation into boundary integrals. The computational cost of this formulation was high however, especially for anisotropic formulation. This paper presents a boundary element formulation to investigate the onset of instability of square composite laminate perforated plates, with simply supported and clamped edges in the out-off plane

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direction and subjected to uniaxial end compression in their longitudinal direction. Stresses caused by external loads are calculated by the formulation of plane elasticity boundary element method. These stresses are introduced as body forces in the classical formulation of plates. The domain integrals due to body forces are transformed into boundary integrals using the radial integration method. In this method, body forces are approximated by a sum of radial basis functions, called approximation functions, multiplied by coefficients to be determined. The presented formulation does not require neither domain discretization nor computation of particular solutions. The main contribution of this paper is to examine the effect of aspect ratio on the stability analysis of thin perforated plates of laminated composite subject to uniaxially loaded by the boundary element method. As quoted in the literature, in [7] and [8], the radial integration method demand few integration points to obtain a solution close to the analytical solution. In these two studies, good results are obtained with ten integration points. However, this was not carried out for stability analysis of thin perforated plates of laminated composite. The main focus of this paper is to assess the number of integration points that are needed to obtain results with good accuracy. Some numerical examples are analyzed in which critical loads, buckling modes, and coefficients of buckling are calculated. The potential to increase the elastic critical load for composite laminated unperforated plates by the introduction of an eccentric perforation is reported.

2

Governing equations

Basically, the classic problem of buckling is a geometrically nonlinear problem described by a set of three differential equations which can be uncoupled and linearized in the case of elastic critical loads. In the absence of body forces, equations that describe the buckling of plates are given by: Nij,j = 0,

(1)

D11 u3,1111 + 4D16 u3,1112 + 2(D12 + D66 )u3,1122 + 4D26 u3,1222 + D22 u3,2222 = Nij u3,ij ,

(2)

where i, j, k = 1, 2; uk is the displacement in directions x1 and x2 , u3 stands for the displacement in the normal direction of the plate surface; Nij are the in-plane stress components, D11 , D22 , D66 , D12 , D16 , and D26 are the anisotropic thin plate bending stiffness coefficients.

2.1

Boundary integral equations

The determination of in-plane stress resultants in the domain is the first step in the solution of plate buckling. The in-plane boundary integral equation for displacements, obtained by applying the reciprocity and Green theorems in equation (1), is given by [9]:

cij uj (Q) +

Γ

t∗ik (Q, P )uk (P )dΓ(P ) =

Γ

u∗ik (Q, P )tk (P )dΓ(P ),

(3)

where ti = Nij nj is the traction in the boundary of the plate in the plane x1 − x2 , and nj is the normal at the boundary point; P is the field point; Q is the source point; and asterisks denote fundamental solutions. The anisotropic plane elasticity fundamental solutions can be found, for example, in [10]. The constant cij is introduced in order to take into account the possibility that the point Q can be placed in the domain, on the boundary, or outside the domain. The in-plane stress resultants at a point Q ∈ Ω are written as:

cik Nkj (Q) +

Γ

∗ Sikj (Q, P )uk (P )dΓ(P ) =

Γ

∗ Dijk (Q, P )tk (P )dΓ(P ),

(4)


369

where Dikj and Sikj are linear combinations of the plane-elasticity fundamental solutions. Due to stress concentrations in the geometry, stress resultants are non-uniform over the domain. The plate buckling equations are derived from the plate bending equations. Critical load factors are introduced into the equations as multiplication factors of body forces or transverse loads. Critical buckling loads are loads at which plates suddenly undergo considerable deflections in the transverse direction due to loads applied in the plane of the plate. The relation between the applied load and critical loads are given by the critical load factor λ by the following equation: Nijc = λNij

(5)

Nijc

where are critical stress resultants that are obtained when critical loads are applied. The integral equation for the plate buckling formulation, obtained by applying reciprocity and Green theorems at equation (2), is given by:

Ku3 (Q) +

=

Γ

Nc i=1

Rci (P )u∗3ci (Q, P ) +

+λ

Vn∗ (Q, P )w(P ) − m∗n (Q, P )

Ω

u3 Nij u∗3,ij dΩ +

Γ

Γ

Nc ∂w(P ) dΓ(P ) + Rc∗i (Q, P )u3ci (P ) ∂n i=1

Vn (P )u∗3 (Q, P ) − mn (P )

∂u∗3 (Q, P ) dΓ(P ) ∂n

ti u∗3 u3,i − ti u3 u∗3,i dΓ ,

(6)

where ∂() ∂n is the derivative in the direction of the outward vector n that is normal to the boundary Γ; mn e Vn are, respectively, the normal bending moment and the Kirchhoff equivalent shear force on the boundary Γ; Rc is the thin-plate reaction of corners; u∗3ci is the transverse displacement of corners; λ is the critical load factor; the constant K is introduced in order to take into account the possibility that the point Q can be placed in the domain, on the boundary, or outside the domain. As in the previous equation, an asterisk denotes a fundamental solution. Fundamental solutions for anisotropic thin plates can be found, for example, in [4]. A second integral equation is necessary in order to obtain the thin plate buckling boundary element formulation. This equation is obtained by the derivative of equation (6) in respect to the normal direction at the source point Q. This equation is given by: 3 K ∂u ∂m (Q) +

=

Nc

∂Vn∗ Γ

∂m

(Q, P )w(P ) −

∂u∗3 ci i=1 Rci (P ) ∂m (Q, P )

+λ

+

∂u∗3,ij Ω u3 Nij ∂m

∂Mn∗ ∂u3 (P ) ∂m (Q, P ) ∂n

∂u∗3 (Q,P ) Γ Vn (P ) ∂m

dΩ +

Γ

ti u∗3

∂u3,i ∂m

dΓ(P ) + −

Nc ∂Rc∗i i=1 ∂m

(Q, P )u3ci (P )

∂ 2 u∗3 mn (P ) ∂n∂m (Q, P )

−

∂u∗3,i ti u3 ∂m

dΓ ,

dΓ(P ) (7)

∂() where ∂m is the derivative with respect to the direction of the outward vector m that is normal to the boundary Γ at the source point Q. As it can be seen in equations (6) and (7), domain integrals arise in the formulation owing to the contribution of in-plane stresses to the out of plane direction. In order to transform these integrals into boundary integrals, consider that a body force b is approximated over the domain as a sum of M products between approximation functions fm and unknown coefficients γm , that is:

b(P ) ∼ =

M

γ m fm .

(8)

m=1

Equation (8) can be written in a matrix form, considering all boundary and domain source points, as:

370

Eds: P Prochazka and M H Aliabadi b = Fγ

(9)

γ = F−1 b.

(10)

Thus, γ can be computed as:

Body forces of integral equations (6) and (7) depend on displacements. So, using equation (10) and following the procedure presented by [11], domain integrals that come from these body forces can be transformed into boundary integrals. As it can be seen in equations (6) and (7), the body force that generates domain integrals is given by:

b = Nij u3 .

(11)

So, it’s need to compute Nij in each integration points. However, we have only the values of Nij at nodes and internal points. Values of Nij in integration points is computed by: Ni (x1 , x2 ) = f (r)F −1 Nij .

(12)

The implementation of the buckling formulation is quite similar to shells. Details on the implementation of the radial integration method for shells are given by [12].

2.2

Approximation functions

Two approximation functions are used in this work. The first is a radial basis function that has been used extensively in the Dual Reciprocity Method and is given by: fm1 = 1 + R.

(13)

The second is the well known thin plate spline: fm2 = R2 log(R),

(14)

used with the augmentation function given by equations (13) and (14). It has been shown in some works from literature that this approximation function can give excellent results for many different formulations [see Partridge (2000), and Golberg , Chen, and Bowman (1999)].

2.3

Eigenvalue problem

After the discretization of equations (6) and (7) into boundary elements and collocation of the source points in all boundary nodes, a linear system is generated. It is worth notice that the only loads considered in the linear buckling equations are that related to the in-plane stress Nij and tractions ti that are multiplied by the critical load factor λ. Furthermore, all the known values of w, ∂w/∂n, Mn , Vn , wci , Rci (boundary conditions) are set to zero. Dividing the boundary into Γ1 e Γ2 (Figure 1), this linear system can be written as:

H11 H12 H21 H22

w1 w2

−

G11 G12 G21 G22

V1 V2

=λ

M11 M12 M21 M22

w1 w2

,

(15)

where Γ1 stands for the part of the boundary where displacements and rotations are zero and Γ2 stands for the part of the boundary where bending moment and tractions are zero. Indices 1 and 2 stand for boundaries Γ1 and Γ2 , respectively. Matrices H, G, and M are influence matrices of the boundary element method due to integral terms of equations (6) and (7). As w1 = 0 and V2 = 0, equation (15) can be written as:


371

3 Γ1: u3 = ∂u ∂n = 0

Ω

Γ2: Vn = Mn=0 Figure 1: Domain with constrained and free degrees of freedom.

H12 w2 − G11 V1 = λM12 w2 , H22 w2 − G21 V1 = λM22 w2

(16)

ˆ 2, ˆ 2 = λMw Hw

(17)

ˆ = H22 − G21 G−1 H12 , H 11 ˆ = M22 − G21 G−1 M12 . M 11

(18)

or,

ˆ e M, ˆ are given by: where, H

The matrix equation (17) can be rewritten as an eigen vector problem 1 w2 , λ

(19)

ˆ ˆ −1 M. A=H

(20)

Aw2 = where,

Provided that A is non-symmetric, eigenvalues and eigenvectors of equation (19) can be found using standard numerical procedures for non symmetric matrices.

3

Numerical results

In order to assess the accuracy of the proposed formulation, in this work it is considered a square thin plate of laminated composite with rectangular holes, under uniformly uniaxial compression and the critical load parameter is computed considering all edges simply-supported and all edge clamped, shown in figures 2 and 3. The geometry and material properties of the plate are: ratio between length a and thickness h of the square plate is a/h = 100; ratio between the edge length of the plate and the edge length of the hole is a/b = 5; elastic moduli E1 = 181 GP a and E2 = 10.3 GP a and Poisson ratio ν12 = 0.28, and shear modulus G12 = 7.17 GP a. The numerical results are presented in terms of the dimensionless critical load parameter Kcr which is given by: Ncr a2 D22 is the critical load and a is the edge length of the square plate. Kcr =

where, Ncr

(21)

372


The mesh used has 32 quadratic discontinuous boundary elements (20 elements of equal length at the external boundary and 12 elements of equal length at the hole) with uniformly distributed internal points. 1.2 1 0.8 0.6 0.4 0.2 0 Ŧ0.2 Ŧ0.2

Figure 2: Geometric configuration

0

0.2

0.4

0.6

0.8

1

1.2

Figure 3: Boundary element model

The results for a square composite laminate plate with square hole, with constant dimensions c/a = 0.25 and e/a = 0, are shown in Table 1 using the radial integration method in boundary element formulation.

3.1

Load axis perpendicular to major dimention

The results presented in table 1 shows that as the eccentricity of the perforation from the central axis increase, a decrease in buckling load factor is observed. The sections at the side of the plate may be considered as narrow portions of plate, but their dimensions do not change by moving the slot down the plate. As the perforation approaches the centre of the plate howevwr, the buckling load factor increases. The effect is more significant when the major dimension if the perforation is large. As the minor dimension of the perforation increases, the buckling load factor decreases. The ”side plate” (portion of the plate either side of the perforation) aspect ratio increases, which should theoretically decrease the buckling load, and is therefore consistent with the results. With a larger major dimension, this effect is more pronounced, and might be expected since aspect ratio will became larger. With a central perforation, an increase in the major dimension of the perforation causes the buckling coefficient to increase. This is because the ”side plate” increase in aspect ratio and become dominant. If, however, the perforation is eccentrically positioned, the buckling load factor decreases with an increase in major dimension.

3.2

Load axis parallel to major dimention

The results presented in table 2 shows that when the plate major perforation axis is parallel to the load axis, the behaviour is somewhat different to that discribed above. The load applied to the plate is predominantly carried by the large quantity of remaining material but the effect of boundary conditions may be greater. As the eccenticity of the perforation increases, a decrease in buckling load factoris observed (Tab. 2). At hight eccentricities of perforation, the plate can be considered as a wide strut and a narrow strut. The narrow side plate will not buckle as easily as the wide side plate but the compressive stress will tend to the carried by the wider side plate. This combination should cause failure of the overall plate at lower buckling load factors. As the perforation is moved further towards the centre, the load becomes more uniformily distributed, and the buckling load of the material either side increases.

Advances in Boundary Element and Meshless Techniques Table 1: Critical load parameter for a perforated square plate (c/a = 2.5), major dimension perpendicular to load axis Ratiod/a 0.25 0.25 0.25 0.25 0.30 0.30 0.30 0.30 0.40 0.40 0.40 0.40 0.50 0.50 0.50 0.50 0.60 0.60 0.60 0.60

4

Ratiof/a 0.00 0.10 0.20 0.30 0.00 0.10 0.20 0.30 0.00 0.10 0.20 0.30 0.00 0.10 0.20 0.30 0.00 0.10 0.20 0.30

Kcr 11.38 6.60 6.50 6.00 8 116.85 100.28 96.08 8 116.85 100.28 96.08 8 116.85 100.28 96.08 8 116.85 100.28 96.08

373

Table 2: Critical load parameter for a perforated square plate (d/a = 2.5), major dimension perpendicular to load axis Ratioc/a 0.25 0.25 0.25 0.25 0.30 0.30 0.30 0.30 0.40 0.40 0.40 0.40 0.50 0.50 0.50 0.50 0.60 0.60 0.60 0.60

Ratiof/a 0.00 0.10 0.20 0.30 0.00 0.10 0.20 0.30 0.00 0.10 0.20 0.30 0.00 0.10 0.20 0.30 0.00 0.10 0.20 0.30

Kcr 8 116.85 100.28 96.08 8 116.85 100.28 96.08 8 116.85 100.28 96.08 8 116.85 100.28 96.08 8 116.85 100.28 96.08

Conclusions

This paper presented an analysis of performance of the radial integration method, considering accuracy and computational cost, when it is used in a boundary element formulation for the instability analysis of composite laminate plates with non-uniform stress field. Domain integrals are transformed into boundary integrals by the radial integration method. This result makes this method very suitable for the treatment of these types of problems, since the computational cost is not high and the fact that we do not need to calculate particular solutions makes the radial integration method advantageous, because of the easy implementation, when compared with the dual reciprocity boundary element method. Acknowledgment The first author would like to thank the Coordination of Improvement of Higher Education Personnel(CAPES), Brazil and State Univesity of Campinas (UNICAMP), Brazil, for financial support for this work.

References [1] P. Sollero. Fracture mechanics analysis of anisotropic laminates by the boundary element method. PhD thesis, Wessex Institute of Technology, 1994. [2] E. L. Albuquerque. Numerical analysis of dynamic anisotropic problems using the boundary element method. PhD thesis, Unicamp, Dept. Mec. Comput., July 2001. In Portuguese. [3] W. P. Paiva. An´ alise de problemas est´ aticos e dinˆ amicos em placas anisotr´ opicas usando o método dos elementos de contorno. PhD thesis, Universidade Estadual de Campinas, Campinas, 2005. In Portuguese.

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[4] E. L. Albuquerque, P. Sollero, W. Venturini, and M. H. Aliabadi. Boundary element analysis of anisotropic kirchhoff plates. International Journal of Solids and Structures, 43:4029–4046, 2006. [5] A. Reis et al. Computation of moments and stresses in laminated composite plates by the boundary element method. Engineering Analysis with Boundary Elements, 35:105–113, 2011. [6] P. C. M. Doval, E. L. Albuquerque, and P. Sollero. Stability analysis of composite laminate plates under non-uniform stress filds by the boundary element method. In XI International Conference on Boundary Element and Meshless Techniques, july 2011. [7] X. Gao. The radial integration method for evaluation of domain integrals with boundary only discretization. Engn. Analysis with Boundary Elements, 26:905–916, 2002. [8] L. J. M. Jesus, E. L. Albuquerque, K. R. Sousa, and P. Sollero. Further developments in the radial integration method. In XXXI CILAMCE - Congresso Ibero Latino Americano de Mtodos Computacionais em Engenharia, Buenos Aires, Argentina, November 2010. [9] M. H. Aliabadi. Boundary element method, the application in solids and structures. John Wiley and Sons Ltd, New York, 2002. [10] P. Sollero and M. H. Aliabadi. Fracture mechanics analysis of anisotropic plates by the boundary element method. Int. J. of Fracture, 64:269–284, 1993. [11] E. L. Albuquerque, P. Sollero, and W. P. Paiva. The radial integration method applied to dynamic problems of anisotropic plates. Communications in Numerical Methods in Engineering, 23:805– 818, 2007. [12] E. L. Albuquerque and M. H. Aliabadi. A boundary element formulation for boundary only analysis of thin shallow shells. CMES - Computer Modeling in Engineering and Sciences, 29:63– 73, 2008.


An Isogeometric Boundary Element Method for Interior Acoustic Analysis using T-splines *Robert N. Simpson1 , Michael A. Scott2 ,Haojie Lian1 1 Institute

2

of Mechanics and Advanced Materials, School of Engineering, Cardiff University, Queen’s Buildings, The Parade, Cardiff CF24 3AA, Wales, UK [email protected]

Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, Texas 78712, USA

Keywords: isogeometric analysis, boundary element method, T-splines, acoustics Abstract: The present paper outlines an isogeometric boundary element method for interior acoustic analysis using T-splines to discretise both the geometry of the problem and unknown fields. The motivation for such a method arises from the difficulties often encountered in creating a suitable ‘mesh’ for analysis which can often dominate the engineering design/analysis cycle. Instead, the present formulation uses the boundary representation provided completely by computer aided design (CAD) in the form of T-splines that allows extremely complicated geometries to be modelled as a single patch. The method is verified against an analytical solution imposed on a three-dimensional cube for a variety of wavenumbers. In addition, to demonstrate the ability of the method to analyse geometries of arbitrary complexity the method is applied to a propeller geometry.

Introduction The task of producing suitable discretisations or meshes before analysis can be performed is known to be arduous and time-consuming, particularly in the case of complicated 3D geometries. Any method with either reduces or eliminates the need to mesh is therefore attractive due to the significant cost and time-savings that can be realised. Several such methods have arisen in recent years including meshless methods [1], immersed boundary methods [2], enriched methods [3] and several others. More recently, attention has been focussed on the direct coupling of numerical methods with CAD under the general title of ‘isogeometric analysis’ (IGA) [4]. The idea is to use the same discretisation used by CAD also for approximating the unknown fields during analysis and has been shown to exhibit advantages for shells [5][6][7], phase-field models [8][9], fluid-structure analysis [10][11][12] and many other applications. A particularly attractive application of the concept is within the framework of the boundary element method (BEM) which relies on a discretisation of the boundary of the problem only. NonUniform Rational B-Spline (NURBS) based CAD packages provide a boundary representation that can be used directly for boundary element analysis and this forms the basis of the isogeometric boundary element method. The concept has been illustrated in several recent developments [13][14][15]. Much of the previous work on isogeometric BEM has focused on potential and linear elastic problems, but an area which shows great promise is the topic of acoustic analysis. We introduce an isogeometric BEM for interior acoustic analysis while emphasising the benefits of using a CAD discretisation of the geometry for analysis. The paper briefly outlines the acoustic boundary integral equation along with the use of T-splines to discretise both the geometry and unknown fields. Two examples are shown to verify the accuracy of the method.

Isogeometric Boundary Element Method The inclusion of the isogeometric concept within the boundary element method is conceptually quite simple. The traditional Lagrangian basis functions that are used to approximate both the geometry and unknown fields (e.g. displacement and traction for linear elasticity) are replaced by the basis functions defined by CAD. However, this change has significant advantages over the traditional approach where the task of generating a boundary mesh has been either removed or drastically reduced and the geometry is represented exactly at all stages of analysis. Most of the IGA literature up to present has focused on the use of NURBS as the basis function choice since they are predominant in CAD. But as is well-known in the computational geometry community, they exhibit deficiencies that often require repairing. In addition, the global nature of the basis functions presents challenges

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for refinement during analysis. As a solution to this, T-splines have been proposed [16] which create ‘watertight’ geometries and allow for local-refinement. The IGA community has embraced this technology with several important developments made [17][18]. More recently, the technology has been employed within a boundary element framework [19] for linear elasticity. The work is now carried forward for interior acoustic analysis. The boundary integral equation for acoustics can be written as ∂G(s, x) ∂φ(x) φ(x) dΓ = dΓ G(s, x) C(s)φ(s) + − ∂n ∂n Γ Γ

(1)

where s is the source point, x is the field point, φ denotes the acoustic potential, ∂φ/∂n denotes the acoustic velocity, C(s) is a jump term, G(s, x) is the Green’s function that satisfies the Helmholtz equation, Γ is the boundary of the domain and − denotes the integral is taken in the Cauchy Principal Value limiting sense. Differentiating equation (1) with respect to the direction of the normal at the source point, a hypersingular boundary integral equation is produced as ∂ 2 G(s, x) ∂G(s, x) ∂φ(x) ∂φ(s) += φ(x) dΓ = dΓ (2) C(s) ∂n(s) Γ ∂n(x)∂n(s) Γ ∂n(s) ∂n(x) with = denoting a Hadamard finite part integral. For exterior acoustic analysis a common method is to use a linear combination of (1) and (2) to avoid spurious results at eigenfrequencies of the interior problem [20], but for interior analysis either (1) or (2) can be used. In the present work we choose to implement the latter in a regularised form [21] where the integrals are at most weakly singular and can be evaluated using polar integration. To discretise (2), the following expressions are used φ(x) =

ncp

φA RA (x)

(3)

A=1

ncp ∂φ(x) = v(x) = vA RA (x) ∂n

(4)

A=1

where ncp is the number of control points (defined by a CAD discretisation), RA (x) is the set of global T-spline basis functions and φA , vA are acoustic potential and velocity coefficients. These expressions can be substituted into (2) to arrive at the desired discrete form. On edges and corners, a semi-discontinuous form of (4) is used [19]. The system of equations is formed through collocation defining the position of each point in a suitable manner [19] (since the usual definition of nodal collocation would result in points lying outside the boundary). The collocation points are defined in such a way that no point is ever located on an edge or corner and is at least C 1 continuous. The use of equations (3) and (4) represents a simple change to a boundary element implementation but has farreaching consequences. Given a T-spline geometry, the use of these equations allows the unknown fields to be discretised directly circumventing the need to generate a boundary mesh. Admittedly, what may be a suitable geometry discretisation may not be as suitable for analysis, but there exists a variety of refinement algorithms [4] that can be used to provide a richer approximation space. To verify the accuracy of the method and to demonstrate the ability to model complex geometries, two examples are now given.

Examples Cube The first problem considered is that of a cube of unit dimension in each direction with the centre of the cube located at the origin. For three dimensions, the following expression can be shown to be a solution to the Hemholtz equation √ √ √ (5) φ(x) = sin(kx/ 3) sin(ky/ 3) sin(kz/ 3). Given a particular wavenumber k, equation (5) can be applied over the entire boundary of the problem while solving for the acoustic velocity ∂φ/∂n. To assess the ability of the formulation for varying wavenumbers the mesh shown in Figure 1 is considered. The mesh is analysed with k = 15, 30, 60, 90. The exact solution for k = 15 along with the numerical solution is illustrated in Figure 2. To compare each of the numerical solutions against the exact solution, a subset of results are taken along a line located on the boundary with x, z = constant and y varying (shown in Figure 2). The results are shown in Figure


Figure 1: Mesh and collocation point locations for cube problem.

Figure 2: Cube problem: exact and numerical solutions for k = 30.

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Figure 3: Comparison of exact and numerical solutions along boundary line shown in Figure 2 for increasing wavenumber.


Figure 4: Bézier mesh used for propeller analysis 3. The breakdown of the results for k = 90 is evident. It is therefore imperative that consideration is given to the quality of the CAD discretisation before acoustic analysis is performed. This may take the form of a fine discretisation or the use of special basis functions that can capture the oscillatory nature of the solution (e.g. [22]). Propeller Finally, to demonstrate that the method can be applied to arbitrarily complicated geometries, the method is applied to a propeller using the same analytical solution as used in the previous section. An wavenumber of k = 1.0 is used. The numerical and exact solutions are shown in Figure 5 where good agreement can be seen. What should be particularly stressed in this example is that the geometry used has been taken directly from CAD with no meshing step required. In addition, the exact geometry is used throughout the analysis.

Summary This paper briefly outlined the application of the boundary element method for interior acoustic analysis using the recently developed computational geometry tool known as T-splines. The method falls within the framework of ‘isogeometric analysis’ whereby the exact geometry is used for all stages of analysis. The isogeometric boundary element formulation was outlined and the method of constructing a system of equations through collocation briefly described. The method was verified against an analytical solution prescribed over a cube demonstrating the behaviour for a variety of wavenumbers. And finally, to illustrate the ability of the method to handle complicated geometries, the method was applied to the problem of a propeller.

References [1] T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, P. Krysl, Meshless methods: An overview and recent developments, Computer Methods in Applied Mechanics and Engineering 139 (1-4) (1996) 3–47. [2] R. Mittal, G. Iaccarino, Immersed boundary methods, Annual Review of Fluid Mechanics 37 (2005) 239– 261. [3] N. Moës, J. Dolbow, T. Belytschko, A finite element method for crack growth without remeshing, International Journal for Numerical Methods in Engineering 46 (1) (1999) 131–150. [4] T. J. R. Hughes, J. A. Cottrell, Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement, Computer Methods in Applied Mechanics and Engineering 194 (2005) 4135– 4195. [5] J. Kiendl, K.-U. Bletzinger, J. Linhard, R. Wüchner, Isogeometric shell analysis with Kirchhoff-Love elements, Computer Methods in Applied Mechanics and Engineering 198 (49-52) (2009) 3902–3914. doi:DOI: 10.1016/j.cma.2009.08.013.

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Figure 5: Propeller geometry: exact and numerical acoustic velocity solution for problem defined by equation (5) [6] D. J. Benson, Y. Bazilevs, M. C. Hsu, T. J. R. Hughes, Isogeometric shell analysis: The Reissner-Mindlin shell, Computer Methods in Applied Mechanics and Engineering 199 (5-8) (2010) 276–289. [7] D. J. Benson, Y. Bazilevs, M. C. Hsu, T. J. R. Hughes, A large deformation, rotation-free, isogeometric shell, International Journal for Numerical Methods in Engineering, 200 (2011) 1367 – 1378. [8] H. Gomez, V. M. Calo, Y. Bazilevs, T. J. R. Hughes, Isogeometric analysis of the Cahn-Hilliard phase-field model, Computer Methods in Applied Mechanics and Engineering 197 (2008) 4333–4352. [9] M. J. Borden, M. A. Scott, C. V. Verhoosel, C. M. Landis, T. J. R. Hughes, A phase-field description of dynamic brittle fracture, Computer Methods in Applied Mechanics and Engineering 217 (2012) 77 – 95. [10] Y. Bazilevs, M. C. Hsu, M. A. Scott, Isogeometric fluid-structure interaction analysis with emphasis on nonmatching discretizations, and with application to wind turbines, Computer Methods in Applied Mechanics and Engineering submitted for publication. [11] Y. Bazilevs, J. R. Gohean, T. J. R. Hughes, R. D. Moser, Y. Zhang, Patient-specific isogeometric fluidstructure interaction analysis of thoracic aortic blood flow due to implantation of the Jarvik 2000 left ventricular assist device, Computer Methods in Applied Mechanics and Engineering 198 (45-46) (2009) 3534–3550. [12] Y. Bazilevs, V. M. Calo, T. J. R. Hughes, Y. Zhang, Isogeometric fluid-structure interaction: Theory, algorithms, and computations, Computational Mechanics 43 (2008) 3–37. [13] R. Simpson, S. Bordas, J. Trevelyan, T. Rabczuk, A two-dimensional isogeometric boundary element method for elastostatic analysis, Computer Methods in Applied Mechanics and Engineering 209-212 (2012) 87–100. [14] K. Li, X. Qian, Isogeometric analysis and shape optimization via boundary integral, Computer Aided Design 43 (11) (2011) 1427–1437. [15] C. Politis, A. I. Ginnis, P. D. Kaklis, K. Belibassakis, C. Feurer, An isogeometric BEM for exterior potentialflow problems in the plane, in: 2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling, SPM ’09, 2009, pp. 349–354. [16] T. W. Sederberg, J. Zheng, A. Bakenov, A. Nasri, T-splines and T-NURCCs, ACM Trans. Graph. 22 (2003) 477–484. [17] Y. Bazilevs, V. M. Calo, J. A. Cottrell, J. A. Evans, T. J. R. Hughes, S. Lipton, M. A. Scott, T. W. Sederberg, Isogeometric analysis using T-splines, Computer Methods in Applied Mechanics and Engineering 199 (5-8) (2010) 229–263.

Advances in Boundary Element and Meshless Techniques [18] M. A. Scott, M. J. Borden, C. V. Verhoosel, T. W. Sederberg, T. J. R. Hughes, Isogeometric Finite Element Data Structures based on Bézier Extraction of T-splines, International Journal for Numerical Methods in Engineering, 88 (2011) 126 – 156. [19] M. Scott, R. Simpson, J. Evans, S. Lipton, S. Bordas, T. Hughes, T. Sederberg, Isogeometric boundary element analysis using unstructured T-splines, Submitted to Computer Methods in Applied Mechanics and Engineering (June 2012). [20] A. Burton, G. Miller, Application of integral equation methods to numerical solution of some exterior boundary-value problems, Proceedings Of The Royal Society Of London Series A-Mathematical And Physical Sciences 323 (1553) (1971) 201–210. [21] Y. Liu, S. Chen, A new form of the hypersingular boundary integral equation for 3-d acoustics and its implementation with C0 boundary elements, Computer Methods in Applied Mechanics and Engineering 173 (1999) 375–386. [22] O. Laghrouche, P. Bettess, E. Perrey-Debain, J. Trevelyan, Plane wave basis finite-elements for wave scattering in three dimensions, Communications in Numerical Methods in Engineering 19 (9) (2003) 715–723.

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Structural Health Monitoring of Sensorised Panels using DBEM Z. Sharif Khodaei1,a, S.P.L. Leme2,b and M. H. Aliabadi1,c 1

Department of Aeronautics, Imperial College London, South Kensington Campus, Roderic Hill Building, Exhibition Road, SW7 2AZ, London, UK., a

[email protected], [email protected],

2

Departamento de Engenharia Civil e Ambiental, Universidade de Brasilia – UnB

Keywords: Structural Health Monitoring, Elastodynamics, DBEM, Laplace Transform Method. Abstract. This paper presents the application of Dual Boundary Element Method (DBEM) in dynamics for determination of the static response of a sensorised plate under external load. Piezoelectric sensors bonded to the plate are modeled as beams. Based on the coupled piezoelectric effect, the analytical relation between the dynamic strain field in the sensor and the output voltage is presented. Based on the sensor signals, a methodology Structural Health Monitoring technique can be developed to evaluate to the state of the structure. The DBEM allows the implementation of cracks in the structure; therefore the response of the damaged structure can be evaluated and compared to the healthy state for damage detection and characterization.

Introduction The increase usage of composite materials in aerospace industry, has led to greater interests in Structural Health Monitoring (SHM) techniques as a potential method to reduce the maintenance activities and operational costs, as well as to improve the aircraft safety and reliability. Composite materials have high stiffness and strength; however their properties can degrade severely in presence of damage. Impact events which can occur during the service life of an aircraft may result in Barely Visible Damage (BVID) which is a cause for concern for composite structures. Therefore, early and reliable detection of damages and defects in the structure is vital for the safety of the structure. SHM techniques are based on real time monitoring of the structure without the need to disassemble structural parts. The main principle of SHM techniques is based on distributed sensor-actuator networks which can continuously monitor the status of the structure and compare it to its healthy status. The presence of damage in the structure can affect its behavior which is captured through the sensor system. This change, in comparison to the pristine baseline, can be translated into damage detection and identification. Surface waves, after generation, propagate through thin medium guided by its surface. The propagational properties of the wave depend on the mechanical and material properties of the medium it travels through. The presence of any damage or discontinuity will alter the propagation of the wave. The damage reflected waves can be captured by surface mounted or embedded sensors in the structure and used in fault detection. Piezoelectric (PZT) transducers are one of the established sensors systems adopted in SHM techniques. They have been vastly used in passive sensing [1] and active sensing methods [2, 3]. Piezoelectric materials have direct and inverse piezoelectric effect. It means that an applied mechanical stress will generate a voltage and an applied voltage will change the shape of the sensor. This property makes them suitable to be used as transducers. They can be used simultaneously as actuators for generating surface waves (such as Lamb waves) and sensors for receiving the generated waves (see Fig. 1). The PZT sensor readings can then be used for impact and damage detection and characterization.


383

(a) (b) Fig. 1 (a) Lamb wave generated in a PZT actuator, (b) Sensor reading of a generated wave due to an impact

Dual Boundary Element Method using Laplace Transform for Dynamic Analysis The solution of general crack problem cannot be achieved with the direct application of the BEM; therefore the DBEM was developed by Portela et. al [4] by incorporating two independent boundary integral equations: the displacement equation applied for collocation on one of the crack surfaces and the traction equation on the other. DBEM is capable of analyzing configuration with any number of edges and embedded cracks in any geometry. In this paper, the DBEM is used in Laplace domain for dynamic analysis of a sensorised plate. Assuming zero initial displacement and velocity, the general field equation for elastodynamics of a homogeneous, isotropic linear elastic body is given as [5]:

(c12 c22 )ui ,ij ( x, Z ) c22u j ,ii ( x, Z )

s 2u j ( x, Z ) U b j ( x, Z ); i, j 1, 2.

(1)

where c1=(Ȝ+2ȝ/ȡ)1/2 is the P-wave velocity; c2=(ȝ/ȡ)1/2 is the S-wave velocity; Ȝ and ȝ are the Lame constants, ȡ is the density; bj is the body force vector, ui(x,Z) is the displacement vector and Z is a frequency parameter (Laplace parameter). The displacement equation, for a source point x’ at the boundary ī of a finite plate can be determined from the transformed dynamic equivalent of Somigliana’s identity as [6]:

cij x ' u j x ', s ³ Tij x ', x, s u j x, s d * *

³ U x ', x, s t x, s d * *

ij

j

³³ U ij x ', X , s b j X , s d :

(2)

:

where U ij x ', x, s and Tij x ', x, s are the Laplace transform of fundamental solutions for elastodynamics, respectively. cij is a constant, which depends on the position x’. For a 2D problem the fundamental solutions are:

U ij ( xc, x, Z )

Tij ( xc, x, Z )

1 2S

1 ª\ G lk F r,l r,k º¼ 2SU c22 ¬

ª§ d\ 1 · § w r · 2 § dF wr · wr r,l r,k F ¸ ¨ G lk r,k n,l ¸ F ¨ n,k r,l 2r,l r,k ... «¨ ¸2 dr r n r n dr n w w w © ¹ © ¹ © ¹ ¬ º § c2 ·§ d\ dF 1 · ... ¨¨ 12 2 ¸¸¨¨ F ¸¸r,l n,k » d r dr r ¹ © c2 ¹© ¼»

(3)

(4)

where P is the shear modulus, r is the distance between the collocation and integration point, and ni is the component of the outward normal at the boundary point. Moreover,

384


\

k0 ( k2 r )

º c2 1 ª « k1 (k2 r ) k1 (k1r ) » ; F k2 r ¬ c1 ¼

k2 ( k2 r )

c22 k1 (k1r ). c12

(5)

with k0 and k1 being the modified Bessel functions of the second kind. Sensor modeling. In this paper, rectangular PZT sensors are modeled as beams in the DBEM analysis. The sensors are surface mounted on the plate by a layer of adhesive, as depicted in Fig 2. When this configuration is subjected to a set of boundary loads and displacement (voltage or strain), the plate and the sensors share interaction forces. By writing the displacement compatibility equation, the relationship between the relative displacement and forces in the plate and sensors are expressed.

Fig 2 PZT transducer model The compatibility condition is based on the assumption that the displacement uj of a point X’ X ' *Sn at the attachment region in the plate and ujS of a corresponding point at sensor, has to be compatible with the shear deformation of the adhesive layer connecting the sensor to the plate as:

'u j X ' 'u j S X '

hadh 'W j adh X ' Gadh

(6)

where hadh is the thickness, Gadh is the coefficient of shear deformation and Ĳjadh is the shear stress of the adhesive layer. Sensor Equation. Once the strain (displacement) in the sensor is known, the output voltage can be computed. The coupled linear electro-mechanical constitutive relation (describing direct and converse piezoelectric effect) of a piezoelectric material can be expressed as [7]:

Q1 ½ ° ° °Q2 ° °Q3 ° ° ° °H11 ° ° ° ®H 22 ¾ °H ° ° 33 ° °H 23 ° ° ° °H13 ° °¯H12 °¿

ª p1 «0 « «0 « «0 «0 « «0 «0 « « d15 «0 ¬

0

0

0

0

0

0

d15

p2 0 0 0 0 d15 0

0 p3 d31 d31 d33 0 0

0 d31 c11 c12 c13 0 0

0 d31 c12 c11 c13 0 0

0 d33 c13 c13 c33 0 0

d15 0 0 0 0 c55 0

0 0 0 0 0 0 c55

0

0

0

0

0

0

0

0 º K1 ½ »°K ° »° 2 ° » ° K3 ° »° ° » °V 11 ° » °®V 22 °¾ , »° ° » °V 33 ° » °V ° » ° 23 ° » °V 13 ° c66 ¼» °¯V 12 ¿° 0 0 0 0 0 0 0

(7)

Where the Qi components are the electric displacement (equals to charge over PZT area), and Ki are components of the electric fields, i.e. voltage over PZT thickness. Vij and Hij are stresses and strains of the PZT material (i,j =1,2,3). p, d and c denote the dielectric permittivity, piezoelectric strain constant and compliance constants, respectively. For a rectangular PZT sensor with length l, width b and thickness t, under plane stress condition and no external electric field, the direct piezoelectric effect reduces to [8]


Q3

d31 (V 11 V 12 )(H11 H 22 ) [2d312 (V 11 V 12 ) p3V ]K 3 .

385

(8)

Considering the electric boundary condition of the PZT patch and that the in-plain normal strain are assumed constant through its thickness we can obtain the relationship between the output voltage generated by the PZT patch as a function of in-plain strain [8]:

Vout

d31 E pzt t ³³ (H11 H 22 )dxdy lb[ p3 (1 X ) / 2d 312 E pzt ]

(9)

Discussion In this paper an accurate computation of dynamic response of surface mounted sensors are presented using Dual Boundary Element Method together with Laplace transform. Moreover, a sensor model has been developed for generating the received signals in a PZT sensor under dynamic load. The sensor signals can be used effectively in Structural Health Monitoring assessment of the structure. The dynamic response of the pristine structure varies from damaged case. This difference can be captured by the sensor signals and utilized for damage detection and characterization. To validate this methodology, numerical examples will be presented at the conference. References [1] [2] [3] [4]

[5] [6] [7] [8]

Ghajari, M., Z. Sharif Khodaei, and M. Aliabadi, Impact Detection Using Artificial Neural Networks. Key Engineering Materials, 2012. 488: p. 767-770. Sharif Khodaei, Z. and M.H. Aliabadi, Damage Identification Using Lamb Waves. Key Engineering Materials, 2011. 452: p. 29-32. Sharif Khodaei, Z., R. Rojas-Diaz, and M. Aliabadi, Lamb-Wave Based Technique for Impact Damage Detection in Composite Stiffened Panels. Key Engineering Materials, 2012. 488: p. 5-8. Portela, A., M. Aliabadi, and D. Rooke, The dual boundary element method- Effective implementation for crack problems. International Journal for Numerical Methods in Engineering, 1992. 33(6): p. 1269-1287. Dominguez, J., Boundary elements in dynamics1993: Computational Mechanics Publications. Leme, S. and M. Aliabadi, Dual Boundary Element Method for Dynamic Analysis of Stiffened Plates. Theoretical and Applied Fracture Mechanics, 2011. Su, Z. and L. Ye, Identification of Damage Using Lamb Waves: From Fundamentals to Applications2009: Springer Verlag. di Scalea, F.L., H. Matt, and I. Bartoli, The response of rectangular piezoelectric sensors to Rayleigh and Lamb ultrasonic waves. The Journal of the Acoustical Society of America, 2007. 121: p. 175.

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