Advances in Boundary Element & Meshless ...

The Conferences on Boundary Element and Meshless Techniques are devoted to fostering the continued involvement of the research community in identifying new problem areas, mathematical procedures, innovative applications, and novel solution techniques. Previous conferences devoted to Boundary Element and Meshless Techniques were held in London, UK (1999), New Jersey, USA (2001), Beijing, China (2002), Granada, Spain (2003), Lisbon, Portugal (2004), Montreal, Canada (2005), Paris, France (2006), Naples, Italy (2007), Seville, Spain (2008), Athens, Greece (2009), Berlin, Germany (2010), Brasilia, Brazil (2011), Prague, Czech Republic (2012), Paris, France (2013) and Florence, Italy (2014)

ISBN 978‐0‐9576731‐2‐0

Advances in Boundary Element & Meshless Techniques XVI

EC ltd

Advances in Boundary Element & Meshless Techniques XVI

Edited by V Mantič A Sáez M H Aliabadi

Advances In Boundary Element and Meshless Techniques XVI

Advances In Boundary Element and Meshless Techniques XVI

Edited by V Mantič A Sáez M H Aliabadi

EC

ltd

Published by EC, Ltd, UK Copyright © 2015, Published by EC Ltd, P.O.Box 425, Eastleigh, SO53 4YQ, England Phone (+44) 2380 260334

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ISBN: 978-0-9576731-2-0

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 XVI 6-8 July 2015, Valencia, Spain Organising Committee: Prof. Vladislav Mantič Prof. Andrés Sáez Department of Continuum Mechanics and Theory of Structures School of Engineering University of Seville Seville 41092 Spain [email protected], [email protected] Prof. Ferri M H Aliabadi Department of Aeronautics Imperial College, South Kensington Campus London SW7 2AZ UK [email protected] International Scientific Advisory Committee Abe, K (Japan) Baker, G (USA) Benedetti, I (Italy) Beskos, D (Greece) Blázquez (Spain) Chen, Weiqiu (China) Chen, Wen (China) Cisilino, A (Argentina) Darrigrand, E (France) De Araujo, F C (Brazil) Denda, M (USA) Dong, C (China) Dumont, N (Brazil) Estorff, O.v (Germany) Gao, X.W. (China) García-Sánchez (Spain) Hartmann, F (Germany)

Hematiyan, M.R. (Iran) Hirose, S (Japan) Kinnas, S (USA) Liu, G-R (Singapore) Mallardo, V (Italy) Mansur, W. J (Brazil) Mantič, V (Spain) Marin, L (Romania) Matsumoto, T (Japan) Mesquita, E (Brazil) Millazo, A (Italy) Minutolo, V (Italy) Ochiai, Y (Japan) Panzeca, T (Italy) Pérez Gavilán, J J (Mexico) Pineda, E (Mexico) Procházka, P (Czech Republic) Qin, Q (Australia) Sáez, A (Spain) Sapountzakis, E.J. (Greece) Sellier, A (France) Semblat, J-F (France) Seok Soon Lee (Korea) Shiah, Y (Taiwan) Sládek, J (Slovakia) Sládek, 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) Wrobel, L C (UK) Yao, Z (China) Zhang, Ch (Germany)

PREFACE The Conferences on Boundary Element and Meshless Techniques are devoted to fostering the continued involvement of the research community in identifying new problem areas, mathematical procedures, innovative applications, and novel solution techniques in both boundary element methods (BEM) and boundary integral equation methods (BIEM). Previous successful conferences devoted to Boundary Element Techniques were held in London, UK (1999), New Jersey, USA (2001), Beijing, China (2002), Granada, Spain (2003), Lisbon, Portugal (2004), Montreal, Canada (2005), Paris, France (2006), Naples, Italy (2007), Seville, Spain (2008), Athens, Greece (2009), Berlin, Germany (2010), Brasilia, Brazil (2011), Prague, Czech Republic (2012), Paris, France (2013) and Florence, Italy (2014). The present volume is a collection of edited papers that were accepted for presentation at the Boundary Element Techniques Conference held at the Ayre Hotel Astoria Palace, Valencia, Spain during 6-8th July 2015. Research papers received from 18 countries formed the basis for the Technical Program. The themes considered for the technical program included solid mechanics, fluid mechanics, potential theory, composite materials, fracture mechanics, damage mechanics, contact and wear, optimization, heat transfer, dynamics and vibrations, acoustics and geomechanics. The conference organizers would also like to express their appreciation to the International Scientific Advisory Board for their assistance in supporting and promoting the objectives of the meeting and for their assistance in the form of reviews of the submitted papers. Editors

CONTENT Solution of 3D thermo-piezoelectric problem by the Local Point Interpolation – Boundary Element Method (LPI-BEM) G Pierson, A Rachid Korbeogo, B Kaka Bonzi, R Kouitat Njiwa Evaluation of stress intensity factors and T-stress by finite block method: statics P H Wen, C Shi and M H Aliabadi An extended finite point method for problems with singularities A Shojaei, F Mossaiby, M Zaccariotto and U Galvanetto Dynamic behaviour of a finite elastic solid matrix with nanoinclusions S.L. Parvanova, G.D. Manolis, P.S. Dineva Hybrid BEM-FEM algorithm in time-domain for numerical modelling of soil-tunnel interaction G. Vasilev, S. Parvanova Gravity-driven migration of bubbles and/or solid particles near a free surface J Sladek, VSladek and P Stana Mixed convection flow in a lid-driven cavity with hydromagnetic effect using DRBEM F.S. O˘glakkaya and C. Bozkaya Thermoelastic stress analysis of 3D generally anisotropic bodies by the boundary element method Y.C. Shiah, C.L. Tan, Y.H. Chen Electrostatic field analysis in anisotropic conductive media using a voxel-based static method of moments S Hamada A new BEM approach for the slow 2D MHD flow about interacting solid particles A. Sellier, S. H. Aydin and M. Tezer-Sezgin Migration of solid particles near a porous slab when subject to a Stokes ambient flow S. Khabthani, A. Sellier and F. Feuillebois

1

h-adaptive refinement applied to 2D elastic Boundary Element Method R. R. Reboredo, E. L. Albuquerque and A. Portela

79

8 16 22 30 38 44 50 59 67 73

Stokes flow in a lid-driven rectangular duct under the Influence of a point source magnetic field Pelin Senel and M. Tezer-Sezgin Application of the isogeometric approach to the boundary element method V. Mallardo, E. Ruocco Meshless thermal stress analysis of nonhomogeneous mater Y OchiaiI and K Toyoda A topology optimisation for cloaking devices with the boundary element method K. Nakamoto, H. Isakari, T. Takahashi, T. Matsumoto Boundary element analysis of two-dimensional thermo-elastic problems with curved line heat sources M. Mohammadi, M.R. Hematiyan Numerical investigation of an orthotropic microdilatation approach in living tissue modeling A Voignier, J-P Jehl and R Kouitat Njiwa Two-dimensional Stokes flow of an electrically conducting fluid in a channel under uniform magnetic field M Gürbüz and M. Tezer-Sezgin BEM analysis of fibre-matrix interface crack onset and propagation under biaxial transverse loads using Finite Fracture Mechanics on elastic interfaces M. Munoz-Reja, L. Tavara, V. Mantic, P. Cornetti Natural convection flow of a nanofluid in an enclosure under a uniform magnetic field M. Tezer-Sezgin, Canan Bozkaya and Önder Türk Boundary element analysis of piezoelectric films under spherical indentation L. Rodrıguez-Tembleque, F.C. Buroni and A. Saez A comparison of three evaluation methods for Green’s function and Its derivatives for 3D general anisotropic solids Longtao Xie, Ch Zhang, Ch Hwu, J Sladek and V Sladek MFS-fading regularization method for inverse boundary value problems in 2D linear elasticity L Marin, F Delvare, and A Cimetiere Cohesive-zone modelling of shear connection with Coulomb friction and interface damage: An SGBEM implementation and application J Kšinan, F Kšinan R Vodicka A background decomposition meshfree method for evaluating fracture parameters of 2D linear cracked solids A. Khosravifard, M.R. Hematiyan, T.Q. Bui

85 91 97 105 112 118 126 132

138 144 150 156 162 170

Topological derivatives for acoustics with various boundary conditions H Isakari, Mi Hanada and T Matsumoto Boundary elements for three-dimensional anisotropic elastic solids with fundamental solutions obtained by radon-stroh formalism Chung-Lei Hsu, Ch Hwu, and Y.C. Shiah A method for appropriate determination of the location of sources in the method of fundamental solutions M.R. Hematiyan, A. Haghighi and A. Khosravifard An implicit enrichment approach in the boundary element method framework for stress intensity factors calculation in anisotropic materials G Hattori, I A Alatawi and J Travelyan DRBEM solution of mixed convection flow of non-Newtonian nanofluids with a moving wall and sinusoidal temperature profile S Gumgum A computationally effective 3D Boundary Element Method for polycrystalline micromechanics V Gulizzi, I Benedetti Analysis of beams Including cross sectional warping and In-plane deformation by DEM I.C. Dikaros and E.J. Sapountzakis An embedded variationally-hybrid boundary element formulation for the solution of Navier Stokes equations over Irregular twodimensional domains N A. Dumont, E Y. Mamani, C A. Aguilar and H F. C. Peixoto A Vortex Approach for Unsteady Insect Flight Analysis in 2-D A. M. Denda Modelling steady flow past a two-dimensional bluff body by using Eulerlets E Chadwick Numerical investigation of unsteady natural convection from a heated cylinder in a square enclosure C Bozkaya Intergranular fracture in polycrystalline materials using the dual reciprocity boundary element method A. F. Galvis, R. Q. Rodrıguez and P. Sollero Hierarchical matrices for the scattering by rough surfaces P Daquin, R Perrussel, J-Rene Poirier Advanced beam element under longitudinal external loading by BEM E.J. Sapountzakis, L.N. Tsellos and I.C. Dikaros

178 184 189 197

203 209 215 223

231 239 248 254 259 267

Performance of compact radial basis functions in the direct interpolation boundary element method for solving potential problems C F Loeffler, L Zamprogno, A Bulcao, W.J. Mansur Application of KL expansion technique to stochastic SFBEM with random fields in elastostatic problems X Fan, Zhi Xu, Z Qin, C Su A 2.5D BEM-FEM formulation to model acoustics and elastic waves within a half-space formation A. Romero, A. Tadeu, P. Galvın and J. Antonio Analysis of the neutral layer offset of bimetal composite plate in the straightening process using boundary element subfield method H.L. Gui, Q.X. Huang, G.X. Shen, C.X. Yu and Q. Li Inverse Green element solutions of pollution sources and their contaminant plumes in groundwater transport E Onyari and ATaigbenu

273

279 287 294 300

Advances in Boundary Element and Meshless Techniques XVI

1

Solution of 3D thermo-piezoelectric problem by the Local Point Interpolation – Boundary Element Method (LPI-BEM) Gaël Pierson1, Aly Rachid Korbeogo2, Bernard Kaka Bonzi3, Richard Kouitat Njiwa4 1 Université de Lorraine, Institut Jean Lamour, Dpt N2EV, UMR 7198 CNRS. Parc de Saurupt, CS 14234, 54042 Nancy Cedex France. [email protected] 2 Université de Ouagadougou, Laboratoire de Matériaux et Environnement (LAME), Ecole doctorale ST 03, B.P. 7021 Ouaga 03 Burkina Faso. [email protected] 3 Dpt. Mathématique et Informatique, Ecole doctorale ST, Université de Ouagadougou 03, B.P. 7021 Ouaga 03 Burkina Faso. [email protected] 4 Université de Lorraine, Institut Jean Lamour, Dpt N2EV, UMR 7198 CNRS. Parc de Saurupt, CS 14234, 54042 Nancy Cedex France. [email protected] Keywords: thermos-piezoelectricity, BEM, Meshfree strong form, Radial basis function.

Abstract. Piezoelectric materials are widely used in the growing field of intelligent structures and control systems such sensors and actuators. Some of the piezoelectric materials also exhibit a pyroelectric effect. This means that an electric charge or a voltage is generated when the sample experiences temperature variation. It is then needed to account for the coupling of the thermal, electric and mechanical fields. The aim of the presented work is to demonstrate the effectiveness of the LPI-BEM in this case. Recall that this simple to implement method combines the advantage of conventional isotropic-BEM and the local point interpolation applied to the strong field equations and has already proved efficient in a number of situations. Introduction New and interesting material properties are highlighted in the coupling multi-physics phenomena. This is evidenced by the abundant literature on the solution of multi-physics problems which covers the range from couple isotropic thermo-elasticity to the nowadays considered thermo-magneto-electric-elasticity with anisotropic material parameters as well as growth of biological tissue with account of chemical exchanges. The coupled thermo-electro-elasticity results from the interaction between thermal, electrical and mechanical fields. A fine understanding of these coupling effects under various loadings is of paramount importance for the prediction of the lifetime of the designed smart structure. The associated coupled field equations can be solved analytically in some cases with simple geometries [13]. More generally and for more complex geometries, numerical methods are unavoidable. There are commercial tools, based on the finite element method (FEM), such as COMSOL Multiphysics® which are able to efficiently solve multi-physics problems. However, this powerful and versatile method has its own shortcomings such as the difficulty of a correct generation of the 3D mesh near singular regions in thin structures and an adequate treatment of these singularities. Thus, it is not unreasonable to look for alternatives to the powerful FEM. When the differential equation of the problem at hand is linear with well-established fundamental solution, the boundary element method (BEM) is a powerful alternative to the FEM especially in presence of singularities (see e.g. [4-6]). BEM has also been applied successfully for the solution of problems with anisotropic material parameters including piezoelectric solids (e.g. [7-13]). In these cases, the implementation of the method involves tedious calculations of the derivatives of the Green's functions. As another alternative to the FEM, an attractive and very easy to implement meshfree method is the point collocation method applied to the strong form field equations. The local point collocation method with radial basis function has been applied for the solution of some problems (e.g. [14,15]). However, its accuracy deteriorates when dealing with natural boundary conditions. Then Liu et al. [15] proposed the weak strong form method in order to improve the accuracy of the method. The approach presented in this paper combines the advantages of well-established conventional BEM with the ease of implementation of the strong form point collocation method. The method is referred to as the local point interpolation-boundary element method (LPI-BEM). It has proven efficient for the solution of

1

Eds: V Mantic, A Saez, M H Aliabadi

anisotropic elasticity [16] and static piezoelectricity [17]. It is here extended to the case of thermo-electromechanical problems. In the method, the primary field variables are partitioned into a complementary part and a particular one. The complementary fields satisfy a homogeneous isotropic linear partial differential equation (PDE) with well-established analytical fundamental solution (Navier equation, Laplace equation). The particular integrals are obtained by solving strong form differential equations using the local radial point interpolation. First of all, we present the governing equations of a thermo-piezoelectric medium. Then, the main steps of the LPI-BEM are described. Finally, the validity of this simple to implement approach is investigated using some simple examples. Governing Equations Throughout, indicial notation and the associated Einstein summation convention over repeated indices are adopted. The effects of body force, electric current and free charge are not taken into account. The heat generation by electric field is assumed negligible. Under quasi-electrostatic assumption, the governing equations of the small strain motion of an elastic thermo-electric solid occupying the domain ȳwith boundary Ȟ, are expressed in Cartesian coordinates as [18]: ߪ௜௝ǡ௝ ൌ Ͳ in ȳ ‫ܦ‬௝ǡ௝ ൌ Ͳ in ȳ ݂௜ǡ௜ ൌ Ͳ in ȳ with ݂௜ ൌ െ‫ܭ‬௜௝ ߠǡ௝ and where V ij is the stress tensor, Di the electric displacement, Kij tensor and T the temperature change.

the heat conduction

The constitutive relations are as follows ߪ௜௝ ൌ ‫ܥ‬௜௝௞௟ ߝ௞௟ ൅ ݁௞௜௝ ߮ǡ௞ െ ߣ௜௝ ߠ ‫ܦ‬௝ ൌ ݁௝௞௟ ߝ௞௟ െ ݄௝௞ ߮ǡ௞ ൅ ܲ௝ ߠ

(4) (5)

Note that ‫ܧ‬௞ ൌ െ߮ǡ௞ . In these equations, Cijkl is the elastic tensor, H ij the small strain deformation tensor,

ekij the piezoelectric tensor, Ei the electric field vector, M the electric potential, Oij the thermal elastic coupling tensor, hij the dielectric tensor and Pi the pyroelectric coefficients vector. With ݊௝ the outward normal vector on the boundary, the traction vector and electric and thermal fluxes are given respectively by: ‫ݐ‬௜ ൌ ߪ௝௜ ݊௝ on Ȟ (6) ‫ݍ‬ఝ ൌ ‫ܦ‬௝ ݊௝ on Ȟ (7) on Ȟ (8) ‫ݍ‬ఏ ൌ ݂௜ ݊௜

Solution Method Our purpose is to propose a simple BEM based solution procedure which conserves the advantage of the reduction of the problem dimension by one, without the tedious implementation of the anisotropic Green functions. First, assume that the kinematical primary variables are the sum of a complementary term and a particular term. That is ‫ ݑ‬ൌ ‫ ܥݑ‬൅ ‫ ܲݑ‬, ߮ ൌ ߮‫ ܥ‬൅ ߮ܲ and ߠ ൌ ߠ‫ ܥ‬൅ ߠܲ . Let C be a constant isotropic elastic tensor 0 0 0 0 with parameters G0 and Q0. Introduce the scalar tensors H ij h G ij and Kij k G ij . 0

The complementary fields satisfy the following homogeneous equations: ଴ ஼ ‫ܥ‬௜௝௞௟ ‫ݑ‬௞ǡ௟௝ ൌͲ

(9)

2


3

଴ ஼ ‫ܪ‬௝௞ ߮ǡ௞௝ ൌ Ͳ ଴ ஼ ‫ܭ‬௜௝ ߠǡ௝௜ ൌ Ͳ

(10) (11)

The boundary element formulations of equations (9-11) are well documented. They lead to systems of equations of the forms:

> H @^ u ` > G @^ t `

(12)

ª¬ H M º¼ ^ M c ` ª¬GM º¼ ^ qMc `

(13)

> H @^ T ` > G @^ q `

(14)

c

c

u

u

c

T

c

T

T

The particular fields solve: ௉ ଴ ‫ܥ‬௜௝௞௟ ‫ݑ‬௞ǡ௟௝ ൅ ߜ‫ܥ‬௜௝௞௟ ‫ݑ‬௞ǡ௟௝ ൅ ݁௞௜௝ ߮ǡ௞௝ െ ߣ௜௝ ߠǡ௝ ൌ Ͳ ଴ ௉ ‫ܪ‬௝௞ ߮ǡ௞௝ ൅ ߜ݄௝௞ ߮ǡ௞௝ െ ݁௝௞௟ ‫ݑ‬௞ǡ௟௝ െ ܲ௝ ߠǡ௝ ൌ Ͳ ௉ ‫ܭ‬௜௝଴ ߠǡ௝௜ ൅ ߜ‫ܭ‬௜௝ ߠǡ௝௜ ൌ Ͳ

(15) (16) (17)

where [G C ] [C ] [C ] , [G h ] [ h ] [ H ] and [G K ] [ K ] [ K ] . 0

0

0

The introduced partition affects Neumann type boundary conditions. Indeed, the traction vector and electric and thermal fluxes can be written as: ଴ ଴ ஼ ௉ ‫ݑ‬௞ǡ௟ ݊௝ ൅ ‫ܥ‬௜௝௞௟ ‫ݑ‬௞ǡ௟ ݊௝ ൅ ߜ‫ܥ‬௜௝௞௟ ‫ݑ‬௞ǡ௟ ݊௝ ൅ ݁௞௜௝ ߮ǡ௞ ݊௝ െ ߣ௜௝ ߠ݊௝ ‫ݐ‬௜ ൌ ‫ܥ‬௜௝௞௟ ଴ ஼ ଴ ௉ ‫ݍ‬ఝ ൌ ‫ܪ‬௝௞ ߮ǡ௞ ݊௝ ൅ ‫ܪ‬௝௞ ߮ǡ௞ ݊௝ ൅ ߜ݄௝௞ ߮ǡ௞ ݊௝ െ ݁௝௞௟ ‫ݑ‬௞ǡ௟ ݊௝ െ ܲ௝ ߠ݊௝ ‫ݍ‬ఏ ൌ െ‫ܭ‬௜௝଴ ߠǡ௝஼ ݊௜ െ ‫ܭ‬௜௝଴ ߠǡ௝௉ ݊௜ ൅ ߜ‫ܭ‬௜௝ ߠǡ௝ ݊௜

(18) (19) (20)

We can rewrite these equations in the compact forms: ‫ݐ‬௜ ൌ ‫ݐ‬௜ு ൅ ‫ݐ‬௜௉ ൅ ߜ‫ݐ‬௜ ൅ ‫ݐ‬ఝ ൅ ‫ݐ‬ఏ ‫ݍ‬ఝ ൌ െ‫ݍ‬ఝ஼ െ ‫ݍ‬ఝ௉ െ ߜ‫ݍ‬ఝ ൅ ߜ‫ݍ‬ఝ௨ ൅ ߜ‫ݍ‬ఝఏ ‫ݍ‬ఏ ൌ ‫ݍ‬ఏ஼ ൅ ‫ݍ‬ఏ௉ ൅ ߜ‫ݍ‬ఏ

(21) (22) (23)

In the local radial point interpolation, a field

Z x is approximated as: w x

N

M

¦ R r a ¦ P x b i

i

i 1

N

with the constraint condition:

¦ P x a j

i

i

j

j

.

j 1

0 j 1 m .

i 1

Ri r is the radial basis functions, N the number of nodes in the neighbourhood (support domain) of point x and M is the number of monomial terms in the polynomial basis ܲ௝ ሺ‫ݔ‬ሻ. Coefficients ai and b j can be determined by enforcing the approximation to be satisfied at the N nodes in the support domain. After some algebraic manipulation the interpolation is written in the compact form: (24) Z x >) x @^Z / L ` Using this relation, for each internal collocation point, equations (12-14) read:

> B @T ª¬C 0 º¼ > B @> )3 @û/pL ` > B @T >G C @> B @> ) 3 @û/ L ` > B @T > e@T >)1 @^M / L ` > B @

T

Ô`> )1 @^T / L `

0

(25)

3


h

0

k

0

^`T >)1 @^M /pL ` ^`T >G h @>)1 @^M / L ` ^`T > e@> B @> )3 @û/ L ` ^`T ^ P`> )1 @^T / L `

^` >)1 @^ T

T /pL

` ^`

T

In these equations, ^z/ L `

^z / L `

z

1

z2

z3

>G K @>)1 @^T / L `

z

1 1

z12

V 22

V 33 V 12

Ô ` O11

O22

O33

O12

given in terms of a vector z

T

z1N

T

O23 , ^ P` T

z1, z2 , z3 T

z2N

z3N

T

for a 3 dimensional vector field and

§ w © wx1

for a scalar field, ^` ¨¨

V 13 V 23 , ^H ` O13

(27)

0

. . . zN

. . .

^V ` V 11

z13

(26)

0

H11

P1

by B( z )

H 22 P2

ª z1 «0 « «¬ 0

H 33

2 H12

P3 , ^D` T

T

w · ¸ , wx 3 ¸¹

w wx 2

2 H 23 , T

2 H13

D1

0 z2

0 0

z2 z1

z3 0

0

z3

0

z1

D3 , and matrix B is T

D2 T

0º z3 » . >)1 @ and >) 3 @ are » z2 »¼

properly constructed matrices from interpolation (24). Now, consider all internal collocation points and if the particular integrals are selected such that u P

Mp

û ` ^M ` ^T ` p

p

p

0

0 and T p 0 at all boundary points. , Equations (25-27) can then be solved for the particular fields. (28) ª Bus º û` > Cu @^M ` > Du @^T ` ¬ ¼ (29) ª¬ BM º¼ ^M ` ª¬CM º¼ û` ª¬ DM º¼ ^T ` ª BTs º ^T `

¬

(30)

¼

Using equations (18-20) and consider relations (21-23), elimination of the complementary fields leads to final equations:

ª¬ H u º¼ û` ¬ª H uM ¼º ^M ` > H uT @^T ` >Gu @^t` ¬ª H M ¼º ^M ` ¬ª H M u ¼º û` ¬ª H MT ¼º ^T ` ª¬GM ¼º ^qM ` > HT @^T ` ª¬GT º¼ ^qT ` Remarkably, the final equations contain boundary primary variables and internal kinematical unknowns as in traditional BEM. Boundary conditions can be taken into account as usual and the resulting system of equations solved by a standard direct solver.

Numerical Examples In this work the generalized multi-quadrics radial basis functions are adopted: Ri ( r ) ( ri c ) where 2

ri

2

q

x xi , c and q are known as shape parameters. In our numerical experiment, the shape parameter c

is assumed to be proportional to a minimum distance d 0 defined as the maximum value among the minimum distances in the x, y and z directions between collocation centres. More specifically we set c E d 0 . The approach has already proved efficient for anisotropic and piezoelectric problems [16-17]. In these cases, method is accurate for a wide range of variation of the multi-quadrics shape parameters. Then, the ability of the approach to deal with static thermo-electro-elastic problems is investigated. 3.1. Anisotropic thermal problem

4


5

In this case, we are concerned only with eq. (3) Let us consider the problem of anisotropic heat conduction in a unit cube. Our numerical results will be compared with those obtained by Sladek et al [19] using the MLPG method. The boundary of the unit cube is subdivided into 24 nine noded elements. For the presented results the boundary nodes are supplemented with 27 internal collocation centres. At the top surface of the cube with normal ‫ݔ‬ଷ ( ‫)ݖ‬, the prescribed temperature change is 1 and at the bottom surface it is 0. The remaining faces of the cube are insulated. The non-zero heat conduction coefficients are collected in the table 1 below: Table 1 Thermal conductivity tensor components ‫ܭ‬ଵଵ 1

‫ܭ‬ଶଶ 1.5

‫ܭ‬ଷଷ 1

‫ܭ‬ଵଶ 0.5

‫ܭ‬ଵଷ 0.5

‫ܭ‬ଶଷ 0.5

The results from our simulation are presented in figure 1a and 1b below. These results agree very well with those presented by Sladek et al. (J Sladek et al., 2008)

(A)

1,2

x/a = 1 and y/a = 1 x/a = 1 and y/a = 0 x/a = 0 and y/a = 0 x/a = 0 and y/a = 1

1,0

Temperature (T)

0,8

0,6

0,4

(B)

0,2

0,0 0,0

0,2

0,4

0,6

0,8

1,0

z/a

Figure 1 Temperature variation in the unit cube: a/ iso-surfaces b/ variations with ࢞૜ coordinate.

3.2. Anisotropic thermo-piezoelectricity The material parameters considered in this case are collected in the table below. They are those of a PZT-5H and are taken in the paper by Shang and Kuna[20] Table 2 Non-zero coefficients of a thermo-piezoelectric solid Elastic coefficients ሺ‫ܽܲܩ‬ሻ ‫ܥ‬ଵଵ 126

‫ܥ‬ଵଶ 55

‫ܥ‬ଵଷ 53

Heat conduction coefficients ሺܹ‫ି ܭ‬ଵ ݉ିଵ ሻ ‫ܭ‬ଵଵ ‫ܭ‬ଷଷ 50 75

‫ܥ‬ଷଷ 117

Piezoelectric coefficients (‫ି݉ܥ‬ଶ ሻ ‫ܥ‬ସସ 35.3

݁ଷଵ -6.5

݁ଷଷ 23.3

Thermal modulus ሺͳͲ଺ ܰ‫ି ܭ‬ଵ ݉ିଶ ሻ ߣଵଵ 1.97

ߣଷଷ 1.42

݁ଵହ 17.0

Dielectric coefficients ሺͳͲି଼ ‫ି ܸܥ‬ଵ ݉ିଵ ሻ ݄ଵଵ ݄ଷଷ 1.51 1.30

Pyroelectric coefficient ሺͳͲି଺ ‫ି ܭܥ‬ଵ ݉ିଶ ሻ ܲଷ -5.48

We still consider the case of the unit cube. The top and bottom surfaces of the unit cube (normal ‫ݔ‬ଷ ) are prescribed with the temperature changes 100°C and 0°C respectively. The lateral surfaces of the cube are insulated. As for the mechanical boundary conditions, the top and lower surfaces of the cube are fully constrained against displacement and the remaining surfaces are free of traction. The electric potential is zero at the bottom surface, while the lateral faces are

5

Eds: V Mantic, A Saez, M H Aliabadi free of induction. The obtained results are presented in figures 2 below. As expected the von Mises stress distribution is symmetrical about ‫ݔ‬ଵ and ‫ݔ‬ଶ axes (Fig. 2a) in the thermo-elasticity case. This is also the case for the thermopiezoelectric case. Note however that, the higher stress values are obtained at the lower surface where the temperature increase is zero. The situation is reversed compared to the thermo-elastic case. Probably a different situation should be obtained with different material parameters.

Z

(A)

Y

(B)

X

Z

X

Y

sig_von_mises

sig_von_mises

0.4

0.2

0

z -0.4

0.4

0.2

0

z

-0.2

1.60E-02 1.38E-02 1.17E-02 9.49E-03 7.31E-03 5.14E-03 2.97E-03 8.00E-04

0.28 0.258469 0.236938 0.215408 0.193877 0.172346 0.150815 0.129285 0.107754 0.0862231 0.0646923 0.0431615 0.0216308 0.0001

-0.2

-0.4

-0.4

-0.2 -0.4

0

-0.4 -0.2

y

0.2

0 0.2 0.4

0.4

-0.2

x -0.4

0 -0.2

y

x

0.2

0 0.2 0.4

0.4

Figure 2: Distribution of von Mises stress. a/ thermo-elastic case. b/ thermo-piezoelectric case

Conclusion In this work a boundary element based (LPI-BEM) solution procedure to multi-physics problems is presented. The approach which uses a partition of the primary field variables is illustrated on the case of a piezoelectric solid with anisotropic material parameters. The method couples conventional isotropic BEM with local radial point interpolation of a strong form differential equation. The implementation of the method has been detailed and it is shown that only minor modification of conventional boundary element implementation is required. Compared to other approaches using radial basis functions, the influence of the shape parameters on the solution accuracy is substantially reduced. This alternative to other mesh reduction approach seems promising for a wide range of problems. Its applicability to transient multi-physics problems will be the subject of a future paper.

References [1] F. Shang, Z. Wang, Z. Li Engineering Fracture Mechanics. 55(5) 737-750 (1996) [2] XJ Zheng,YC Zhou, MZ Nin International Journal of Solids and Structures 39 3935-3957 (2002) [3] H.L. Dai, X. Wang International Journal of Solids and Structures 43 5628-5646 (2006) [4] . C.A. Brebbia, J. Dominguez, Boundary Elements. An introductory course. Computational Mechanics Publications,1992 [5] J. Balas, J. Sladek, V. Sladek, Stress analysis by boundary element methods. Elsevier,1989 [6] M. Bonnet, Boundary integral equation methods for solids and fluids. John Wiley and Sons, New York,1999 [7] D.M. Barnett Physica. Status Solidi (b) 741-748(1972) [8] L.J. Gray, A. Griffith, L. Johnson, P.A. Wawrzyneek Electronic Journal of Boundary Elements. 1 (2) 6894(2003) [9] S.M. Vogel, F.J. Rizzo Journal of Elasticity. 3 203-216 (1973) [10] T. Chen, F.Z. Lin Computational Mechanics. 15 485-496(1995) [11] V.-G. Lee Mechanic Research Communications 30 241-249 (2003) [12] H. Ding, J. Liang Computer & Structures. 71 445-455(1999) [13] M. Denda, C.Y.Wang Computer Methods in Applied Mechanics and Engineering. 198 2950-2963(2009) [14]G.R. Liu, Y.T. Gu. Journal of Sound and Vibration, 246,29–46 (2001). [15] G.R. Liu, Y.T. Gu Computational Mechanics, 33 2-14 (2003) [16] R. Kouitat Njiwa Engineering Analysis with Boundary Elements 35 611-615 (2011)

6


7

[17] N. Thurieau, R. Kouitat Njiwa, M. Taghite Engineering Analysis with Boundary Elements, 36 (11), 1513-1521(2012) [18] M. Aouidi International journal of Solids and structures, 43, 6347-6358 (2006) [19] V. Sladek, J. Sladek, C.L. Tan, S.N. Atluri Computer Modelling in Rngineering and Sciences, 32, 161174 (2008) [20] F. Shang, M. Kuna Computational Materials Science, 26, 197-201 (2003)

7


Evaluation of stress intensity factors and T-stress by finite block method: statics P.H. Wen1,a, C. Shi1,b and M.H. Aliabadi2,c 1

School of Engineering and Material Sciences, Queen Mary University of London, UK 2

Department of Aeronautics, Imperial College, London, UK

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

Keywords: Williams series of stress function, Lagrange series interpolation, finite block method, stress intensity factor, T-stress. Abstract. In this paper, the Finite Block Method (FBM) for computing the stress intensity factors (SIFs) and T-stress under static load is presented. Based on one-dimensional first order partial differential matrices derived from the Lagrange series interpolation, the higher order partial differential matrices can be determined directly. In order to capture the stress intensity factor and the T-stress, the Williams' series of stress function are introduced in the circular core centred at crack tip with consideration of traction and displacement continuities along the circumference of the circular core. Introduction It is known that there is a singularity of stress at the crack tip existing in a homogeneous medium for any shaped crack for both static and dynamic loading. The subject of dynamic fracture mechanics involves the analysis of structures with cracks when the load or the crack size changes rapidly. In these cases inertial effects must be taken into account. The significance of these effects depends on loading conditions, material properties and the geometrical configuration of the body. The determination of the stress intensity factors and T-stress is a fundamental task for elastoplastic fracture mechanics in practical engineering. It is well known that the stress intensity factors (SIFs) play a significant role in linear elastic fracture mechanics as they characterize the crack-tip stress and strain fields. However, the T-stress, which is the second constant in the asymptotic expression of the crack-tip stress fields, has to be concerned in elastoplastic fracture mechanics because it has been found that the T-stress has of certain effects on the crack growth direction, shape and size of the plastic zone, crack-tip constraint and fracture toughness etc (see Du and Hancock [1]; Larsson and Carlson [2]; O’Dowd et al. [3]). Cotterell and Rice [4] showed that the sign of T-stress determines the stability of a straight mode I path, i.e. it is stable if T < 0 and unstable if T > 0. It has been now established that the positive T-stress strengthens the level of crack tip stress triaxiality and leads to high crack tip constraint, while negative T-stress reduces the level of crack tip stress triaxiality and leads to the loss the crack tip constraint. An comprehensive overview of past research on T-stress was presented in [5] by Gupta et al. The different phenomena associated with T-stress such as crack path stability, isochromatic fringes pattern, plastic zone influence and constraint parameter were discussed by the authors. In this paper, the finite block method [6,7,8] combined with of stress and displacement fields of Williams’ series around crack tip is presented to evaluate stress intensity factors, T-stress and other coefficients for the rest regular terms in the stress fields. With the first order partial differential matrices, the partial differential matrices of any order can be obtained by mapping technique for each block. A set of linear algebraic equations from the governing equations in strong form, connection conditions along the interfaces between blocks and circular core and boundary conditions are presented. Lagrange interpolation for two-dimensions One dimensional uniformly distributed nodes is shown in Figure 1(a) with the nodes collocated at [i 1 2(i 1) /(M 1) , i 1,2,..., M , M is number of collocation points (seed number). Any one dimensional function u ([ ) cane be approximated using Lagrange series as

u ([ )

M

M

([ [ m ) ui i [m )

¦ ([ i 1 m 1 mzi

M

¦ F ([ , [ )u i

i 1

i

(1)


where

Kj

9

ui is the nodal value of point i. For two dimensional problem, in the same manner for axis K , 1 2( j 1) /( N 1) , j 1,2,..., N , where N is the number of nodes along the K axis. Therefore,

the function u ([ ,K ) can be approximated by M

ª

N

M

º ([ [ m ) N (K K n ) » u » l i [ m ) n 1 (K j K n ) nz j ¼»

M

¦¦ «« ([

u([ ,K )

i 1 j 1

m 1

¬« mzi

N

¦¦ F ([ , [ )G(K,K i

j

)ul

(2)

i 1 j 1

where M

([ [ m ) , G (K ,K j ) i [m )

N

([

F ([ , [ i )

m 1 m zi

(K K n ) j Kn )

(K n 1 nz j

(3)

( j 1) M i . The number of nodes in total is T

and the subscript in u l , l

M u N . Then the first order

partial differential is determined easily with respects to [

wu w[

M

wF ([ , [ i ) G (K ,K j )u l w[ 1

N

¦¦ i 1 j

(4)

where

wF ([ , [ i ) w[ wu wK and

M

w w[

M

m 1, m zi

N

¦¦ F ([ , [ ) i

i 1 j 1

wG(K,K j ) wK

([ [ m ) i [m )

([

wG(K ,K j ) wK

w N (K K n ) wK n 1, (K j K n ) nz j

N

M

M

¦ ([ [ l 1 k 1, k z i , k z l

M

k

)/

([

m 1, m z i

i

[m ) ,

(5)

ul

(6)

N

¦ (K K l 1 k 1, k zi , k zl

(a)

N

k

)/

(K

n 1, n z j

j

Kn )

(7)

(b)

Figure 1. Two-dimensional node distribution in mapping domain: (a) the local number system of node; (b) square domain with 8 seeds for the mapping geometry. For a block with arbitrary shape of boundary, the mapping technique is introduced combined with Lagrange series method. Same as the finite element method, a quadratic element for two dimensional problem in the Cartesian coordinate ( x , y ) can be mapped into a square : ' in the mapping domain ([ ,K ) [ d 1; K d 1 as shown in Figure 1(b) by using a set of quadratic shape functions with 8 seeds. The first order partial differentials of function u ( x, y ) in the Cartesian coordinate system defined


1§ wu wu · wu ¨ E 11 ¸, E 12 J ¨© w[ wK ¸¹ wy

wu wx where

1§ wu wu · ¨ E 21 ¸, E 22 J ¨© w[ wK ¸¹

(8)

wy wy wx wx , J E 22 E 11 E 21 E 12 , E 12 , E 21 , E 22 wK w[ wK w[ Then, the first order partial differentials can be obtained in terms of the nodal values

E 11

wu wx

wG (K ,K j ) º wF ([ , [ ) 1 M N ª ¦¦ «E11 w[ i G (K ,K j ) E12 F ([ , [ i ) wK »u l J i 1 j 1¬ ¼

D xl ([ ,K )u l ,

wu wy

wG (K ,K j ) º wF ([ , [ ) 1 M N ª ¦¦ «E 21 w[ i G (K ,K j ) E 22 F ([ , [ i ) wK »ul J i 1 j 1¬ ¼

D yl ([ ,K )u l ,

(9)

(10)

where again l ( j 1) u M i . We can also define the first order partial differentials in (10) at each node in a matrix form [17] as

ux where

Dy

Dxu , u y

^D

D yu ,

(11)

the vector of nodal value of displacement u

yl

û1 , u2 ,...,uT `

T

,T

M u N , Dx

^Dxl ([k ,Kk )` ,

([k ,Kk )` (k , l 1,2,...,T ) . In addition, for order L derivatives in two dimension with respect to both

coordinates [ and K ,

uxy( mn) ( x, y)

w mnu ( x, y) , mn wx mwy n

L.

(12)

The nodal values of the above high order partial differentials can be written in matrix form approximately, in terms of the first order partial differential matrices Dx and Dy , as

u (xymn)

D mx D ny u.

(13)

Strategy and formulations of the finite block method Consider the 2D elasticity with domain : enclosed by boundary * in isotropic media in polar coordinate system. The constitute equations for two dimensional plane-stress problems in polar coordinate system are

ªV r º «V » « T» «¬W rT »¼

ªc11 c12 «c « 21 c 22 «¬ 0 0

0 ºª H r º 0 »» «« H T »», c33 »¼ «¬J rT »¼

where

c11

c 22

E , c12 1 Q 2

c 21

(14)

QE , c 33 1 Q 2

E 2 (1 Q )

G,

(15)

in which, E , Q are Youngs modulus and Poison ratio, G is shear modulus. The strains can be expressed with displacements as

Hr

wu r ,HT wr

1 wu T u r , J rT r wT r

wu T 1 w u r u T . wr r wT r

(16)

For two-dimensional static problem in the polar coordinate, the equilibrium equations are

wV r 1 w W rT V r V T b r 0, r wT r wr wW rT 1 wV T 2W rT bT 0. wr r wT r

(17)

where br and bT are body forces. Introducing the first order differential matrices in Eq.(11) in stresses in Eq.(14) yields


11

ˆ D u u , c11 D r u r c12 R T T r ˆ D u u , c 21 D r u r c 22 R T T r ˆD u R û , c D u R

Sr ST S rT

33

r

T

-

T

r

(18)

ˆ diag [1/ r ] , the {V r1 ,V r 2 ,...,V rT } , ST {V T 1 , V T 2 ,...,V TT }T , S rT {W rT 1 ,W rT 2 ,...,W rTT }T , R k coordinate correspondence r o x and T o y and then: D r D x , D T D y . Applying Eq.(11) over T

where Sr

equilibrium equations in Eq.(17) for each collocation point P in the domain gives

ˆD S R ˆ S S b 0, Dr S r R T rT r T r ˆ D S 2R ˆS b D r S rT R 0 . T T rT T

P:

(19)

Substituting Eq.(18) into Eq.(19) gives a set of linear algebraic equations in terms of the nodal values of displacements as following

>D Rˆ c D c Rˆ c Rˆ D Rˆ D Rˆ c D c Rˆ @ u ˆ R ˆ c R ˆ D D R ˆ c R ˆ D @ u b >c D R 0, >c D 2Rˆ Rˆ D Rˆ D c D c Rˆ @ u ˆ D R ˆ c R ˆD R ˆD @ u b >c D 2R 0, r

11

r

12

r

12

33

T

r

33

r

33

T

33

r

T

21

2

T

r

T

12

T

22

r

T

r

22

T

(20a)

r

r

22

22

r

T

T

(20b)

T

For static elasticity, the boundary conditions of the displacement and traction are specified as

u r (P)

u r0 ( P ),

uT ( P )

u T0 ( P ),

V r ( P ) n r ( P ) W rT ( P ) nT ( P )

t r0 ( P ),

W rT ( P ) n r ( P ) V T ( P ) nT ( P )

tT0 ( P ),

0

0

0

P*u

(21)

P *V

(21)

0

in which ur ( P), uT ( P), t r ( P) and tT ( P) are the boundary values of displacements and tractions on and

*u

*V respectively.

Williams series of stress function In order to obtain dimensionless coefficients, it is of advantage to normalise the crack-tip distance on the characteristic length w . For a cracked body, the general solutions in a series representation for Airy stress function given by Williams [27,29] for symmetric type f 1 n 1/ 2 3 º ª n 1 / 2 ) V 0 w 2 ¦ r / w An «cos(n )T cos(n )T » 2 n 3/ 2 2 ¼ r d r , 0 dT dS , ¬ n 1 (23) 0 f

V 0 w 2 ¦ r / w Bn >cos(n 1)T cos(n 1)T @ n 1

n 1

where

V 0 is

characteristic stress such as a uniformly distributed load on the boundary,

r0 is radius of the

circular core centered at crack tip ( r0 / w 1) and An , Bn are dimensionless coefficients. The components of stress in the circular core are given by

V Tc

f

ª ¬

1 2

V 0 ¦ r / w n 3 / 2 (n 1 / 2)(n 1 / 2) An «cos(n )T n 1

f

V 0 ¦ r / w

n 1

n 1 / 2 3 º cos(n )T » n3/ 2 2 ¼

(24a)

(n 1)nBn >cos(n 1)T cos(n 1)T @

n 1

V rc

ª n 2 4n 7 / 4 º cos(n 3 / 2)T (n 1 / 2) cos(n 1 / 2)T » ¬ n 3/ 2 ¼

f

V 0 ¦ r / w n 3 / 2 (n 1 / 2) An « n 1

f

V 0 ¦ r / w n 1

n 1

nBn >(n 3) cos(n 1)T (n 1) cos(n 1)T @

r d r0

(24b)


W rcT

f

V 0 ¦ r / w n3 / 2 (n 1 / 2)(n 1 / 2) An >sin(n 1 / 2)T sin(n 3 / 2)T @ n 1

f

(24c)

V 0 ¦ r / w nBn >(n 1) sin(n 1)T (n 1) sin(n 1)T @ n 1

n 1

and the displacement fields by (1 Q )V 0 w f r / w n1 / 2 2n 1 An ª«(n 4Q 7 / 2) cos(n 3 )T (n 3 / 2) cos(n 1 )T º» u rc ¦ 2 ¼ E 2n 3 ¬ 2 n 1

(1 Q )V 0 w f r / w n Bn >(n 4Q 3) cos(n 1)T (n 1) cos(n 1)T @ U 0 cosT , ¦ E n 1

uTc

(1 Q )V 0 w f r / w n1 / 2 2n 1 An ª«(n 3 / 2) sin(n 1 )T (n 4Q 5 / 2) sin(n 3 )T º» ¦ E 2n 3 ¬ 2 2 ¼ n 1 (1 Q )V 0 w f n r / w Bn >(n 1) sin(n 1)T (n 4Q 3) sin(n 1)T @ U 0 sin T ¦ E n 1

where U0 is the horizontal displacement at crack tip ( r stress intensity factor and T-stress are obtained by

KI

V 0 18Sa A1 , T

(24d)

(24e)

0) . In terms of the coefficients An and Bn , the

4V 0 B1 .

(25)

For simple case, one block is sufficient in numerical procedure shown in Figure 2(a). In the Williams series of stress and displacement in Eq.(24), we consider the finite terms with truncation number n . On the interface between the block and circular core (r points and four connection equations

r0 , 0 d T d S ) , we suppose that there are N collocation

ur (r0 ,T ) urc (r0 ,T ), uT (r0 ,T ) uTc (r0 ,T ),

V r (r0 ,T ) V rc (r0 ,T ), W rT (r0 ,T ) W rcT (r0 ,T ) ,

T Ti , i 1,2,...N

Figure 2. Finite blocks and circular core centred at crack tip of radius

(26)

r0 in polar coordinate with


If the number of truncation terms is chosen as to determine displacements (uri , uTi ), i

n

13

N , we have 2T 2n linear algebraic equations in total

1,2,..., T , and coefficients ( An , Bn ) , n

1,2,..., n . However, in

Bn should be removed if the horizontal motion at the crack tip U0 is considered. Therefore, the unknowns in the Williams series are ( An , Bn ) , n 1,2,..., n 1 , An and U0 . In the case with more than one block as shown in Figure 2(b), there are interface with connection

the Williams series, the last term of truncation

conditions on ab and cd. Therefore, the number of connection equations at the collocation points on the interface between the blocks and circular core is N I N II N III 4 . Then the number of truncation is chosen as n N I N II N III 4 in order to determine the same number of coefficients in the Williams series. Numerical example

Firstly, consider an edge-cracked circular disk loaded by constant normal tractions V 0 along the circumference shown in Figure 3(a). Due to symmetry about horizontal axis, only half of disk is modeled with the boundary conditions

W rT

uT

ur

0,

V r V 0 ,W rT

W rT

VT

T 0, 0 r d R , T 0, r R ,

0,

(27)

0 d T d S,r R ,

0,

T S , 0 r d R.

0,

Suppose that the crack tip is located at the centre of the disk and crack length a R . Number of collocation point along the interface is N and then the number of truncation terms should be n N with characteristic length w( D

2 R ) . Again, the last term Bn should be removed in order to consider the

U0 at crack tip in Eq.(24). Due to the disk is centered at crack tip, one block with 8 seeds is enough to for mapping of geometry precisely shown in Figure 3(b) and the number of seed is selected as M N for

motion

convenience. Table 1 and Table 2 show the numerical results of the stress intensity factor and T-stress against the number of seed N when r0 / R 0.2 and the radius of Williams core zone r0 / R when N 15. 0 The results given by Fett [9] are considered as benchmark, which are K I

3.1716V 0 Sa and T 0

1.896V 0 respectively. It is evident that high accurate solutions can be obtained in the region of N t 9 and 0.1 d r0 / R d 0.3 . Table 1. Normalized stress intensity factor and T-stress versus the number of seed. N K I K I0 / K I0 (%) T T 0 / T 0 (%) T /V0 K I / V 0 Sa 9 3.1800 0.266 1.9259 11 3.1751 0.110 1.9042 13 3.1738 0.071 1.8993 15 3.1735 0.059 1.8981 Table 2. Normalized stress intensity factor and T-stress versus the ratio r0 / a .

1.578 0.433 0.171 0.109

r0 / a

K I / V 0 Sa

K I K I0 / K I0 (%)

T /V0

T T 0 / T 0 (%)

0.10 0.15 0.20 0.25 0.30

3.1794 3.1741 3.1735 3.1732 3.1681

0.247 0.079 0.059 0.051 0.078

1.9171 1.9000 1.8981 1.8982 1.9004

1.112 0.213 0.109 0.116 0.233


S

V0

r0 R

r

crack

r0

R

(b)

I (a)

Figure 3. Half disk with crack and distribution of collocation points: (a) cracked disk under tensile load; (b) mapping of geometry with 8 seeds ( ) and collocation points ( ). Secondly, consider a cracked circular disk bounded with a outer ring with different material loaded by

constant normal tractions V 0 along the circumference shown in Figure 4. Comparing with the first example above, one more block is added with one more interface between block I and block II. The number of collocation points in total is T M I N I M II N II and the number of truncation terms in Williams series is chosen as n N I and width of component in Williams series w 2R0 . Continuous of displacements and stresses along the interface of blocks I and II have to be satisfied. Same as the first case, the quadratic block with 8 seeds is employed. In the computation, the parameters are selected r0 / R0 0.2 , M I M II 17 ,

N I N II 9 and Q I Q II 0 . 3 . The numerical results of the stress intensity factor and T-stress against the ratio of Young’s module E II / E I and ratio R / R0 are shown in Figures 5(a)(b).

V0 II I

R0

R

I Figure 4. Half cracked disk (I) bounded with a ring (II) under tensile load.

Conclusion Computational procedure was demonstrated by four examples. Following observations can be summarised: (1) Finite block method can be easily applied to fracture problems combined with Williams’ stress function; (2) Stress intensity factors and T-stress in Williams’ series can be obtained with high accuracy; (3) This method can be extended to determine the stress intensity factor and T-stress under dynamic load.


15

Figure 5. Two disks with different materials: (a) Normalized stress intensity factor; (b) Normalized T-stress. References [1] Du ZZ, Hancock JW. The effect of nonsingular stresses on crack tip constraint. J Mech Phys Solids 1991;39:555-567. [2] Larsson SG, Carlson AJ. 1973. Influence of non-singular stress terms and specimen geometry on small scale yielding at crack tips in elastic–plastic materials. J Mech Phys Solids 1973;21:263-277. [3] O’Dowd NP, Shih C, Dodds R. The role of geometry and crack growth on constraint and implications for ductile/brittle fracture. Am Soc Test Mater. 1995;2:134-159. [4] Cotterell B, Rice JR. Slightly curved or kinked cracks. Int J Fract 1980;16:155-69. [5] Gupta M, Alderliesten RC, Benedictus R. A review of T-stress and its effects in fracture mechanics. Engineering Fracture Mechanics 2015;134:218-241. [6] Wen PH, Cao P, Korakianitis T. Finite block method in elasticity. Engineering Analysis with Boundary Element 2014;46:116-125. [7] Li M, Wen PH. Finite block method for transient heat conduction analysis in functionally graded media. Int J Numerical Methods in Engineering 2014;99(5):372-390. [8] Li M, A. Munjiza, P.H. Wen. Finite block method for contact analysis in functionally graded materials. International Journal for Numerical Methods in Engineering (accepted 2015). [9] Fett T. Stress intensity factors, T-stresses, weight functions. Karlsruhe, Germany: University of Karlsruhe; 2008.


An Extended Finite Point Method for Problems with Singularities Arman Shojaei1, Farshid Mossaiby2, Mirco Zaccariotto3 and Ugo Galvanetto4 1,3,4

2

Department of Industrial Engineering, University of Padua, Via Venezia 1, 35131, Padova, PD, Italy

Department of Civil Engineering, University of Isfahan, Isfahan 81744-73441, Isfahan, Iran,

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

and

[email protected]

Keywords: Meshless, Finite Point Method, Singular Problems, Singular basis functions, Laplace equation

Abstract This paper aims at applying the meshless finite point method (FPM) to problems with singularities. The study is a preliminary one to show the capability of FPM to be extended for a class of singular problems governed by Laplace and Poisson equations. In this way, some new residual free influence functions are developed and added to the series expansion of the functions used in the FPM approximation. This enrichment technique is only done for the nodes whose clouds (sub-domain) cover singular regions. This strategy contributes to increase considerably the accuracy of the results with high convergence rate for FPM in solution of such singular problems. Introduction Attempts for developing numerical methods requiring no mesh have led to emerging of a new generation of numerical method so called “Meshless” methods. In recent decades element free or meshless methods have gained significant popularity. Among them are the element free Galerkin method [1], the meshless local Petrov-Galerkin method [2], and the finite point method [3]. A major advantage of these approaches, over mesh-based methods, is that arbitrary distribution of nodes can be used; morover, adaptive schemes can be introduced more easily by distributing more nodes, instead of elements, in regions of high error [4]. The finite point method (FPM), proposed by Oñate et al. in 1996 is one of the most prominent meshless techniques in the realm of computational mechanics. The method is based on the use of a weighted least square interpolation scheme using typical complete monomial basis functions. Since no integration is involved in this approach, FPM is categorized as a truly meshless method, for it offers many advantages in terms of computational cost and simplicity over many other meshless methods. A fast growing attention by many researchers on this subject can be seen in [5–9]. Similar to most of numerical methods, the proper efficiency of FPM deteriorates in the solution of problems that contain singular boundary points. For instance, it is well known that the spatial derivatives of the potential field governed by the Laplace and Poisson equations can become infinite (undefined) at corners or edges. However, due to the smooth nature of the bases used in the approximation scheme, significant numerical errors arise near such singular points which affect even the accuracy of farther parts. In this paper this limitation is addressed by generating some new residual free basis functions that can form such spatial derivatives near singular points. These functions are incorporated into the series expansion of the bases used in sub-domains close to singular points. The content of the paper is structured as follows. In the next section the problem of interest is described. The basis of FPM formulation is presented next. Then the generated singular bases are introduced and in a subsequent section their efficiency is examined through the well-known Motz’s problem. Finally, the conclusion section is presented. Problem Description


17

We consider a 2D problem domain as : bounded by a boundary * * D * N where * D * N I (see Fig.1). * D and * N stand for boundaries with Dirichlet and Neumann conditions, respectively. The unknown function u (field variable) should be determined through the governing differential equation of the problem considering the prescribed boundary conditions. In this paper, we assume that the problem is governed by a general homogeneous Laplace problem which has many applications in a wide range of scientific and engineering problems

2u LDu

0

in :

fD

in * D ,

LN u

fN

in * N

(1)

in which 2 is the Laplacian differential operator, f D and f N are the prescribed boundary conditions defined on the Dirichlet and Neumann boundaries, respectively. Consequently, L D and L N are the boundary operators applied for the Dirichlet and Neumann boundaries. As it is demonstrated in Figure 1, *S represents some boundary points which are potential to be singular on the basis of the boundary conditions defined on.

Figure 1: A general solution domain with possible singular points

In the following section, we shall abstractedly explain the approximation scheme of FPM in the solution of partial differential equations; we particularly focus on the Laplace equation. The Finite Point Approximation Scheme

Let N i : , i 1, 2, , n be a collection of n nodes distributed within the problem domain and on its boundaries (Figure 1). Around each of these nodes, a sub-domain :i , so called cloud, along with a local coordinate system centered on the node is considered. Each cloud specifies the neighboring nodes of N i through which the unknown function is to be approximated locally with respect to the nodal values associated to the neighboring nodes. The approximation of u , say uˆ , in the local coordinate system of each cloud can be stated as

ˆ x) u (x) # u(

m

¦ p (x)D l

pT (x)

l

x

l 1

>x

y@ , T

(2)

where p(x) is a vector of base monomials and is a vector of coefficients. Ensuring that the bases are complete, for a 2D problem we can specify p(x) [1 x

y

x2

xy

y 2 ]T

for m

6 .

The local coordinates of the nodes inside a cloud can be assigned in a vector as x Rj ^x j , j one can sample u (x) at the n R nodes as follows

(3) 1, 2, n R ` ,


u

u1 °u ° 2 ® ° °u n ¯ R

½ uˆ1 ° ° uˆ ° ° 2 ¾#® ° ° ° °uˆn ¿ ¯ R

½ ° ° ¾ ° ° ¿

pT1 ° T ° p2 ® ° °pTn ¯ R

½ ° ° ¾ C . ° ° ¿

(4)

Assuming that n R is generally greater than m , satisfaction of eq (4) entails a weighted least square procedure that leads to minimization of a norm as nR

J

¦w

i

(x j )[u j uˆ j ]2

j 1

nR

¦w j 1

2

i

( x j ) ¬ªu j pTj ¼º ,

(5)

where w i (x) is a suitable weight function that we take as recommended in [8],

exp( r 2 c 2 ) exp( rm2 c 2 ) , 0 d rm d r ° w (r) ® , (6) 1 exp( rm2 c 2 ) °0 r ! rm ¯ in which r represent the distance between a node and the center node, c and rm are two factors that must be taken proportional to rmax , which is the distance between the center node and the most remote node in the cloud. In a same way as in [8], we take c 0.25rmax and rm 2rmax . Likewise, the norm in eq (5) should be minimized in the view of (4) as wJ 0. (7) w The above equation results in a linear system of equations as nR

A

Bu,

A

¦w

i

(x j )p j pT ,

j 1

B

ª¬w i ( x1 )p1 ,w i ( x 2 )p 2 ,,w i ( x nR )p n R º¼ .

(8)

By solving the above equation and inserting the obtained into (2), the approximation for ith cloud will be obtained as uˆ(x) pT (x) A 1Bu (x)u , (9) where (x) contains all the shape functions in the approximation scheme of the cloud. The discretized system of equations in FPM is found by introducing the approximated field variable in (9) to the equations given in eq (1); then collocating the differential equation at each node, based on its position in the domain, leads to the final system of linear equations as KU F , (10) where K , U and F are the coefficient matrix, the unknown nodal values and the vector of known values for the global system of linear equations, respectively. Influence basis functions

There are many practical engineering problems in which the spatial derivatives of potential field governed by Poisson or Laplace equations become infinite at corners or edges. This phenomenon can be abundantly found in many problems such as seepage flow net under a dam as well as elastic torsion of prismatic rods [10, 11]. However, performance of FPM deteriorates in solution of such problems due to the smooth nature of the monomials used in eq (3). In order that, inspired by a recent study of the authors in [12], we propose some new influence functions that can be applied locally for any desired singular region of the solution. The proposed influence functions are as follows ps (x) [ x iy e P (x iy ) x iy e P (x iy ) ( y ix )e P ( y ix ) ( y ix )e P ( y ix ) ]T , (11) where P is a constant coefficient. It should be pointed out that the proposed functions satisfy the Laplace differential equation, and they are somewhat similar to the residual free bases that are used in the meshless local exponential basis function methods (MLEBF) as in [13, 14]. However, the difference is that the spatial derivatives of the proposed functions are singular, for they can form the aforementioned desired


19

singularity on singular boundary nodes. In the current strategy, we apply the influence functions only for the nodes whose cloud contain the singular nodes (see the red nodes in Figure 1). It should be remarked here, for these clouds the origin of the local coordinate system should be put on the singular nodes as their gradients are infinite at x y 0 . One can find a comprehensive study in [14] regarding the optimal selection of P when exponential basis functions are used locally. In the same line, as suggested in [14], we consider S 2 d P d S 2 . Numerical Example In this example we employ the proposed method in the solution of Motz’s problem. The problem was first introduced in 1974 for relaxation method [15]. Since then, many researchers have standardized it as a prototype of problems with a singularity to verify the efficiency of other methods. The problem is governed by a Laplace equation with the following boundary conditions u 500, on * 4 u

on * 2

0,

(12)

n u 0, on *1 * 3 * 5 Over a rectangular domain (see Figure 2).

Figure 2: (a) The solution domain in Motz’s problem (b) a sample of distribution of nodes

The problem is known as a singular problem due to the sudden change in the boundary conditions at the origin of the axes. The exact solution of the problem is given as [16] f

¦d

u (r,T )

k

r k 1 2 cos(k 1 2)T ,

(13)

k 0

where d k stands for the coefficients to be evaluated by expansion of the boundary conditions, also (r,T ) are the polar coordinates with origin at (0,0) . For the propose of testing a numerical method, an admissible numerical function with finite terms for this problem can be considered as (see [16]) u N (r,T )

NK

¦D

k

r k 1 2 cos(k 1 2)T

(14)

k 0

In this regard, to elucidate the accuracy of the present method for this problem, we use eq (14) as a benchmark with the D k coefficients reported in [16] for N k 34 . We use a distribution of nodes as shown in Figure 2b to represent the solution domain. For this example we take m 15 in eq (3) by using 30 numbers of nodes inside each cloud. For the enriched clouds, whose center are shown by red nodes in Figure 2b, we use 12 influence functions by taking P [ S 2,0, S 2] and 16 number of nodes inside the cloud. Moreover, to evaluate the accuracy and convergence rate of the method, an error norm over the whole domain nodes is considered as n

eu

¦ (u l 1

l

u exact )2 l

n

¦ (u

exact l

)2

(15)

l 1

where u l denotes the lth nodal value for the field variable u . Convergence plot of the solution obtained by both the conventional FPM, without using the influence functions, and that of the current approach, referred to as XFPM, is given in Figure 3. It is should be noted that h is the average node spacing. As expected, because of the smooth nature of the bases used, the convergence rate of FPM considerably deteriorates due to the presence of the singular point. In the same


figure we have reported the result of XEBF pertaining to the use of singular bases near the singular point. The results are indicative of this fact that the introduction of the new bases significantly improves the accuracy and convergence of the results. 0

Log(eu)

1 2 1.27

3

2.86

4

FPM

1

XFPM

5 1.4

1.2

1

0.8

0.6

Log(h)

Figure 3: Convergence plot of the results

The contour plot of the exact solution and that of the obtained results for, for a grid of 946 nodes is shown in Figure 4 . In the same figure we have included the contour plot of the errors. As is expected, the maximum error is obtained at the location of the singular point. 1.0

1.0

1.0

0.8

0.8

0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0.0

0.0 1.0

0.5

0.0

0.5

1.0

0.0 1.0

A

0.5

0.0

0.5

1.0

1.0

b

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.5

0.0

0.5

1.0

c

0.2

0.0

0.0 1.0

0.5

0.0

0.5

1.0

1.0

0.5

d

0.0

0.5

1.0

e

Figure 4: Contour plot of (a) exact solution (b) the solution obtained by XFPM (c) the solution obtained by FPM (d)

uˆ u N for XFPM (e) uˆ u N for FPM Conclusion In this paper, an extended version of the meshless finite point method has been developed. In this way, some suitable residual free basis functions have been introduced to be added to the series expansion of the approximation bases for the cloud of nodes near the singular points. The robustness of the current approach has been investigated via the well-known Motz’s problem. A glimpse into the obtained results revealed that the performance of FPM deteriorates in the regions near the singularities; however, using the proposed strategy culminates in results with high accuracy and convergence rate. This study may pave the road for future investigations to apply FPM for a wider class of problems with singularities.


21

References 1. Belytschko T, Lu YY, Gu L (1994) Element-free Galerkin methods. Int J Numer Methods Eng 37:229–256. doi: 10.1002/nme.1620370205 2. Atluri SN, Zhu T-L (2000) The meshless local Petrov-Galerkin (MLPG) approach for solving problems in elastostatics. Comput Mech 25:169–179. doi: 10.1007/s004660050467 3. Oñate E, Idelsohn S, Zienkiewicz OC, Taylor RL (1996) A finite point method in computational mechanics. applications to convective transport and fluid flow. Int J Numer Methods Eng 39:3839–3866. doi: 10.1002/(SICI)1097-0207(19961130)39:223.0.CO;2-R 4. Dipasquale D, Zaccariotto M, Galvanetto U (2014) Crack propagation with adaptive grid refinement in 2D peridynamics. Int J Fract 190:1–22. doi: 10.1007/s10704-014-9970-4 5. Oñate E, Idelsohn S (1998) A mesh-free finite point method for advective-diffusive transport and fluid flow problems. Comput Mech 21:283–292. doi: 10.1007/s004660050304 6. Oñate E, Sacco C, Idelsohn S (2000) A finite point method for incompressible flow problems. Comput Vis Sci 3:67–75. doi: 10.1007/s007910050053 7. Ortega E, Oñate E, Idelsohn S (2007) An improved finite point method for tridimensional potential flows. Comput Mech 40:949–963. doi: 10.1007/s00466-006-0154-6 8. Boroomand B, Tabatabaei AA, Oñate E (2005) Simple modifications for stabilization of the finite point method. Int J Numer Methods Eng 63:351–379. doi: 10.1002/nme.1278 9. Boroomand B, Najjar M, Oñate E (2009) The generalized finite point method. Comput Mech 44:173–190. doi: 10.1007/s00466-009-0363-x 10. Young D-L, Fan C-M, Tsai C-C, Chen C-W (2006) The method of fundamental solutions and domain decomposition method for degenerate seepage flownet problems. J Chinese Inst Eng 29:63–73. 11. Gorzelaczyk P, Koodziej JA (2008) Some remarks concerning the shape of the source contour with application of the method of fundamental solutions to elastic torsion of prismatic rods. Eng Anal Bound Elem 32:64–75. doi: 10.1016/j.enganabound.2007.05.004 12. Mossaiby F, Bazrpach M, Shojaei A (2015) Extending the method of exponential basis functions to problems with singularities. Eng Comput. doi: 10.1108/EC-01-2014-0019 13. Soleimanifar E, Boroomand B, Mossaiby F (2014) A meshless method using local exponential basis functions with weak continuity up to a desired order. Comput Mech 53:1355–1374. doi: 10.1007/s00466-014-0979-3 14. Shojaei A, Boroomand B, Mossaiby F (2015) A simple meshless method for challenging engineering problems. Eng. Comput. In press: 15. Li ZC, Lu TT, Hu HY, Cheng AHD (2008) Trefftz And Collocation Methods. WIT Press 16. Lu TT, Hu HY, Li ZC (2004) Highly accurate solutions of Motz’s and the cracked beam problems. Eng Anal Bound Elem 28:1387–1403. doi: 10.1016/j.enganabound.2004.03.005


Dynamic behaviour of a finite elastic solid matrix with nano-inclusions S.L. Parvanova1, G.D. Manolis2, P.S. Dineva3 1

Assoc. Prof., Dept. of Struct. Engineering, UACEG, 1046 Sofia, Bulgaria, [email protected] 2

Prof., Dept. of Civil Engineering, AUTH, 54124 Thessaloniki, Greece, [email protected] 3

Prof., Institute of Mechanics, BAS, 1113 Sofia, Bulgaria, [email protected]

Keywords: In-plane elastodynamics; Finite-sized solid; concentration; Gurtin-Murdoch model; Boundary elements.

Nano-cavities;

Nano-inclusions;

Stress

Abstract. The 2D elastodynamic problem for a finite-sized solid matrix containing multiple nano-cavities and/or elastic nano-inclusions of arbitrary shape and configuration is solved herein. The presence of nanoheterogeneities within the elastic matrix gives rise to both wave scattering and stress concentration phenomena, the latter being responsible for fracture of the matrix material. These phenomena can only be studied by taking into consideration the surface/interface properties, the size of the inclusions and the interaction between the inclusions and solid matrix interfaces. The method of solution used is the boundary integral equation method (BIEM), which is first verified against benchmark examples and subsequently used for numerical simulations. The BIEM formulation combines classical elastodynamic theory for the bulk solid with non-classical boundary conditions and the localized constitutive law for the matrixheterogeneity interfaces within the framework of the Gurtin-Murdoch theory. Also, frequency-dependent fundamental solutions for the equations of motion in the bulk solid are used as kernels. Results drawn from the numerical simulations reveal the degree of dependence of the scattered wave field and of the dynamic stress concentration factor (DSCF) on the shape, size, number and geometrical configuration of multiple nano-cavities and nano-inclusions in the finite elastic matrix. These results have applications in material science, computational fracture mechanics and nondestructive testing evaluation of nano-composites. Introduction The diffraction of elastic waves by microstructure (voids, inclusions, cracks) in solids remains a fundamental issue in mechanics, both from a theoretical and an engineering applications viewpoint. Due to the increasing ratio of surface/interface area to volume in nano-composites, these surfaces and interfaces may have significant effects on the mechanical properties of the surrounding medium. It is well known that classical continuum theory is deficient in predicting material behaviour at the nanoscale, where size effects are no longer negligible. A mechanical model taking into consideration the presence of interface/surface stresses within the framework of the continuum mechanics approach was first proposed by Gurtin and Murdoch [1] and further developed in Ref. [2-5]. In this model, the surface domain is assumed to be very thin, with different material properties in reference to the surrounding bulk material. More specifically, its deformation is described by a linear stress-strain constitutive relation, and it adheres to the bulk solid without slipping. As a consequence, the equilibrium equations along the surface/interface yield to nonclassical boundary conditions. In the following short review of the results obtained so far in the field, we are interested on the dynamic behaviour of solids with nano-heterogeneities in the frame of the Gurtin and Murdoch [1] theory. The state-of-the-art review in the field shows that: (a) There are few, if any, results for transient external loads; (b) there is a lack of BIEM-generated results for solution of wave scattering by nano-heterogeneties. BIEM results currently available in the literature are for static problems, see Ref. [6], except for very recent ones on wave diffraction by nano-inclusions in an infinite matrix, see Ref. [7]; (c) results are for elliptical and circular inclusions; (d) it should be noted that the assumption of infinite solids has been introduced in most of these papers. We note that in the infinite boundary model, the sizedependent effect at the edges of nano-structured materials is ignored. However, the presence of edges in a finite elastic plate leads to high stress concentration effects in the nano-sized structure, which is quite different from those observed in an infinite domain model. To date, relatively little work on the size effect of finite nano-composites has been done. This was the motivation for proposing herein an efficient computational BIEM tool for solution of elastodynamic problems for finite-sized solid matrices containing multiple nano-cavities and/or elastic nano-inclusions under time harmonic loads.


23

Problem statement and boundary conditions Wave motion in the plane x3=0 in a Cartesian coordinate system O-x1x2x3 is considered, see Fig. 1 where a perforated finite, homogeneous elastic solid with boundary * is subjected to time-harmonic loads with frequency ω. The solid matrix contains multiple inclusions with boundaries * nI or holes * nH , n=1, 2,…N of arbitrary geometrical shape, number, size and geometrical configuration. We assume that the heterogeneities do not intersect each other and denote their total surface is * I

N

* IN and * H

n 1

N

N . *H

n 1

The total boundary is denoted as S * * H in the case of holes and as S * * I in the case of inclusions. The material properties are OM , PM , UM for the solid matrix and OI ,n , PI ,n , U I ,n for the n-th inclusion. The only nonzero quantities here are displacement vector components ui x1 , x2 , Z , stresses

V ij x1 , x2 , Z , plus the corresponding traction vector components ti x1 , x2 , Z V ij x1 , x2 , Z n j , where ni is

the outward unit normal vector and (i,j=1,2). In the bulk solid, the equation of motion in the absence of body forces is:

V ij ,i UZ 2u j where V ij

M M Cijkl uk ,l in the matrix ° ,C ® I ,n I ,n C u in the n th inclusion ijkl ° ijkl k ,l ¯

(1)

0

OG ijG kl P G ik G jl G il G jk , U

UM ® ¯ U I ,n

(2)

In here, comma subscripts denote partial differentiation with respect to the spatial coordinates and the summation convention over repeated indices is implied. Following Ref. [1], the n-th interface between each nano-inclusion and its surrounding matrix is regarded as a thin material with its own mechanical properties O S ,n , P S ,n and surface tension W 0,n . More specifically, W 0,n is the residual surface tension under unstrained condition that will induce additional static deformation, but in dynamic analysis this is often ignored. We assume that for all inclusion surfaces, we have the same material properties denoted by O S , P S ,W 0 . The /j constitutive equation along the j-th interface * M between solid matrix and j-th inclusion is I

V sur , j

W 0 2P s O s H sur , j

(3)

/j . In the above, V sur , j , H sur , j respectively are the stress and the corresponding strain along the interface * M I

σ0

σ0

Model A

Model B

Model C 1.5d 1.5d

d

10d

1.5d 1.5d φ

Model D

1.5d 1.5d

Figure 1: Elastic plate with various cavity and inclusion configurations: Geometry, dimensions and placement for Models A-D In order to complete the boundary-value problem (BVP), we have the following boundary conditions:


/j Boundary conditions along any inclusion/matrix interface * M : Taking into consideration I surface/interface effect, we have: (a) Continuity of displacements along the j-th interface coming from the side of the heterogeneity uij and from the side of the surrounding matrix uiM , uij uiM , i 1,2 ; (b) Interface /j equilibrium conditions along the arc length s j , i.e., on the undeformed interface s j { * M written in I

terms of the local normal and tangential (n, t ) coordinates defined on s j { *ij . These lead to the following relation between traction ti j along the j-th interface coming from the side of the heterogeneity t i j and from the side of the surrounding matrix tiM , see Ref. [7]: t1j t1M ½ ® j M¾ ¯t2 t2 ¿

^f ` ª¬T sur j

sur j

u j ½ º¼ ® 1j ¾ ¯u2 ¿

(4)

We note that matrix Tjsur and the vector f jsur are defined as follows:

^f ` sur

j

sur

ªT j ¬

º ¼

ª n1 «n ¬ 2

ª « « n2 º « n1 »¼ «§ § « D «¨ ¨ «¨¨ ¨© U j ¬©

W0 Uj

§ 2 · ¨ D W 0 w ¸ ¨ U 2 ws 2j ¸ j © ¹ w D ws j U 2 j

wU j · W 0 w ·¸ ¸ ws j ¸ U j ws j ¸¸ ¹ ¹

n1 ½ ® ¾ ¯n2 ¿

(5)

§ § 0 · ·º 0 ¨ D w ¨ W w W wU j ¸ ¸ » 2 ¨¨ U j ws j ¨ U j ws j U j ws j ¸ ¸¸ » n © ¹ ¹» ª 1 © » « n § · 2 0 »¬ 2 ¨D w W ¸ » 2 2 ¨ ws j U ¸ » j © ¹ ¼

n2 º (6) n1 »¼

Here D 2P S O S , n1 n1j n1M ; n2 n2j n2M are the components of the normal vector as viewed from either side of an interface, and U j is the curvature radius of the j-th interface s j . /j Boundary condition along any cavity/matrix interface * M H : Again, by taking into consideration

surface/interface effects, this is the same as condition as Eq. (4), keeping in mind that t1j

t2j

0.

Boundary conditions along the solid matrix outer boundaries: These depend on the applied loads and are specified for any particular problem. The finite solid contours are assumed free of surface effects. BIEM formulation and solution The aim is to determine the stress-strain field at any point of the finite solid and the SCF along the perimeter of any nano-inclusion and nano-cavity. To this end, the BVP defined above can be reformulated via a set of the boundary integral equations along the boundary S based on the frequency-dependent fundamental solutions of elastodynamics for in-plane wave motion as follows: cij u j x, Z

³U x, y,Z t y,Z dS y ³ P x, y,Z u y,Z dS y , x S * ij

S

* ij

j

j

(7)

S

In the above, cij are jump terms dependent on the local geometry at the collocation point x x1 , x2 , U ij* is the displacement fundamental solution of Eq. (1) and Pij*

* CijqlU qk ,l nk is the corresponding traction

fundamental solution. The displacements and stresses at any point inside the solid can be obtained from the well-known integral representation formulas using the solutions of Eq. (7). The sub-structuring version of the direct BIEM is used here. More specifically, a simple modification of this technique is used by eliminating the interface tractions in order to reduce the number of degrees-offreedom (DOF) at the interfaces. As a consequence, this modified version requires less approximation and produces a smaller system matrix size that leads to smaller solution times. The proposed numerical scheme was programmed using the commercial MATLAB software package.


25

Verification studies Verification of the proposed BIEM scheme is achieved through a number of comparisons with available results in the literature, for both static and time-harmonic loads. So far, known analytical and numerical solutions are for infinite domains, so the size of the finite region considered here is extended to infinity with respect to the heterogeneity size. In all test examples, a single nano-hole or nano-inclusion is considered in a finite square matrix. When the model is subjected to static loads, the solution for a finite solid starts with the size of the square matrix ballooning to 20, 50 and 100 times the radius of the basic inclusion, so as recover the corresponding solution for the infinite plane. In the case of time-harmonic loads at higher frequencies of vibration, best accuracy with the infinite plate solution is obtained when the matrix size balloons to 100 times the radius of the reference inclusion. Test example 1: Nano scale circular cavity in a square matrix under a static load Consider a square matrix of size 100 a0 containing a circular hole of radius a0 1.109 m . The vertical edges of the square matrix are subjected to uniform normal tractions t1 V 0 , whereas the horizontal edges have normal tractions t2 V 0Q / 1 Q , with a Poisson ratio Q

0.33098 . The excitation frequency is low

and equal to CP/1000, where CP (OM 2PM ) UM is the P-wave velocity, in order to simulate quasistatic conditions. The benchmark solution is for an infinite elastic plane with a single, nanoscale cylindrical hole under far-field loading. The surface stresses acting at the hole boundary are modelled as a thin film with surface elastic parameters O S and P S , keeping the properties of the surrounding medium fixed: OM 64.43GPa, PM 32.9GPa . Next, the far field loading is the static equivalent of a longitudinal Pwave,

namely

traction

t1 V 0 ; t2 V 0Q / 1 Q .

Surface

effects

are

defined

by

parameter

K S 2P S O S W 0 , and the special case of W 0 0 is used in the verification study. This test example was solved analytically in Grekov and Morozov [8], who reduced the BVP to a hypersingular integral equation with respect to the unknown surface stress by applying the Goursat-Kolosov complex potential theory and Muskhelishvili’s technique. Figure 2a is a comparison between our BIEM results and the analytical solution in Ref. [8] for the normalized hoop stress VMM / V 0 along the perimeter of the circular hole with fixed radius

a0 and for different values of parameter K S (in N/m). The BIEM mesh comprised 32 quadratic BE for the hole boundary and 64 quadratic BE for the square matrix, with 16 BE per side. We observe that the BIEM solution reproduces the analytical solution extremely well.

(a)

(b) 2.5

3

2.5 2

[8], Ks=20 N/m

[8], Ks=40 N/m

[8], Ks=80 N/m

[8]-classical

BEM, Ks=20 N/m

BEM, Ks=40 N/m

BEM, Ks=80 N/m

BEM-classical

BEM, s=0

BEM, s=0.1

2

BEM, s=0.5 BEM, s=2

1.5

|σφφ/σ0|

σϕϕ/σ0

1.5

1

finite domain

1

0.5

0.5

ϕ, [rad]

0

φ/π

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

0

0.2

0.4

0.6

0.8

1

Figure 2: Stress ratio VMM / V 0 along (a) the circular hole perimeter for different surface elastic KS values and (b) along the circular inclusion perimeter for different surface elastic s values Test example 2: Nano scale circular inclusion in a square matrix under a static load Consider the same problem as in the test example 1, but now a nano-inclusion with radius a0 is considered. The dimensionless parameter which captures interface effects at the nanoscale is defined as


s K S / 2PM a0 . The stiffness ratio of the inclusion to the matrix phases is PI PM 0.2 , and different values of the surface parameter are considered, namely s 0; 0.1; 0.5; 2.0 . Two BVPs are actually solved here: (1) An infinite plane with a circular nano-inclusion under far field static load that is equivalent to a longitudinal P-wave stress field, i.e., t1 V 0 ; t2 V 0Q / 1 Q , where Q 0.26 . The corresponding BIEM solution based on the elastostatic fundamental solution is reported by the present authors in Ref. [7]; (2) a finite square matrix of size 100 a0 containing a circular nano-inclusion under uniform normal tractions t1 along the vertical edges and t 2 along the horizontal edges of the finite domain. The second problem is solved by the BIEM based on the frequency-dependent elastodynamic fundamental solutions at the low excitation frequency CP/1000, i.e., for quasi-static conditions. All results are plotted concurrently in Fig. 2b, where again the good quality of the BIEM solution in terms of accuracy is clearly observable. Test example 3: Nano scale circular inclusion in a square matrix under dynamic load This benchmark example is for a circular inclusion in an infinite plane under an SV-wave propagating with frequency Z along the 0x1 axis, which was solved in Ref. [9] using the wave function expansion method and resolved by the present authors in Ref. [7]. The finite domain model is solved with respect to the scattered wave field, and the total tractions and displacements are obtained by superposition of scattered and incident quantities. In order to ensure zero scattered displacements along the finite boundaries of the solid matrix, a 5% material damping is introduced, while material damping for the inclusion is set at 0%. We compare in Fig. 3 the dynamic stress concentration factor defined as DSCFS

VMM / V 0 at points

along the interface perimeter and at a fixed non-dimensional frequency :M kS , M

kS ,M a , where

Z CS ,M is the corresponding wave number with respect to matrix domain. The DSCF is normalized

by the incident stresses V 0

PM u0 CS2,M . The densities of inclusion and solid matrix correspond to a

frequency ratio of :I :M 3.0 . The Fig. 3 plots are for :M 0.2 and S, where excellent agreement is observed between infinite and finite domain models. The dimensionless surface parameter s is defined in the same manner as in the test example 2. Finally, the BIEM mesh used 32 BE for the interface discretization at the low frequency value :M 0.2 and 64 BE at the higher value :M S . The square finite domain contour was discretized using 64 BE for both cases. (b)

(a) 4

BEM, s=0

3.5

BEM, s=0.1

Ω=0.2

3

|σφφ/ σ0|

finite domain

2 1.5

BEM, s=2 finite domain

2.5 2 1.5

1

1

0.5

0.5 0

0 0

.

Ω= π

BEM, s=0.1

BEM, s=0.5

BEM, s=2

2.5

BEM, s=0

3.5

BEM, s=0.5

3

|σφφ/ σ0|

4

0.2

0.4

φ/π

0.6

0.8

1

0

0.2

0.4

0.6

φ/π

0.8

1

Figure 3: DSCF along a circular nano-inclusion in infinite and finite domains for different surface elastic parameters s values and at fixed non-dimensional frequency values of the propagating SV wave: (a) :M=0.2; (b) :M=S Numerical simulation studies In what follows, we conduct a series of numerical simulations to reproduce the dynamic stress concentration field that develops in a finite solid matrix with nano-heterogeneities. We consider the basic


27

case where the solid matrix is a square plate under time-harmonic tensile forces of magnitude V 0 applied at the vertical sides. Figure 1 shows the four basic configurations (Models A, B, C, D) involving one, three and five nano-heterogeneities. These heterogeneities are either circular cavities or inclusions with radius a0 . The size of the square plate is 10d, with d the heterogeneity diameter. As before, the dimensionless parameter which captures interface effects at the nanoscale is defined as s K S / 2P M a0 , where P M is the shear modulus of the matrix material. When the heterogeneity is an inclusion, then the stiffness ratio of both material phases is PI PM 0.2 and the densities correspond to a frequency ratio of :I :M 3.0 . In all simulations, the material damping is set to 5% and Poisson’s ratio is 0.26 for both matrix and inclusion. In order to avoid rigid body motion, four edge nodes belonging to the axes of symmetry are restrained, as shown in Fig. 1. In Fig. 4, we plot normalized hoop stresses spectra for a representative point with polar angle M S / 2 at the heterogeneity interface versus normalized frequency for a single cavity and for a single inclusion (Model A). The DSCF is defined as DSCFS

VMM / V 0 , where V 0 is the magnitude of applied tensile

stresses. Four different values of the surface parameter are considered, namely s 0; 0.1; 0.5; 1.0 . The problem is solved for a frequency range going up to :M 1 , and divided into 50 equal steps. For clarity, the plots are presented up to :M 0.8 . In terms of boundary discretization, the BIEM meshes used to model the perimeter of any given heterogeneity comprise 32 quadratic BE resulting in 64 nodal points. The outer contour of the plate is modelled by 32, equal length BE, with 8 along each side.

(a)

35

35 s=0

30

|σφφ/ σ0|

|σφφ/ σ0|

s=0.5

20

Observer point

30

s=0.1

25

(b)

s=1

15

s=0.1

25

s=0.5

20

s=1

15

10

10

5

5

0

s=0

d

0 0

0.2

0.4

Ω=ωd/CP

0.6

0.8

0

0.2

0.4

Ω=ωd/CP

0.6

0.8

Figure 4: DSCF for Model A at observer point M S / 2 on: (a) The cavity interface; (b) the inclusion interface. As expected, the normalized hoop stresses in the case of circular inclusion are smaller compared to those of the single cavity,. The strongest amplification in the DSCF is observed at a dimensionless frequency of :M 0.24 for both models and for all surface parameter values. As the surface parameter s value increases, the DSCF values drop, which means that the strong molecular surface action reduces the classical mechanical hoop stresses observed at the macro-scale. This holds truth when the surface parameter s is positive. Next, Fig. 5 plots the DSCF versus frequency at a fixed observation point (shown in red colour) with radial coordinate M S / 2 along the central reference cavity perimeter. The solid matrix contains additional cavities arranged in the pattern depicted as Models A, B, C,D (see Fig. 1). All plots are for fixed surface parameter s=0.1. By increasing the number of cavities, we observe a shift of the first DSCF peak to the left. This is so because the reduced stiffness of the relevant model leads to a lower fundamental frequency value in the composite structure. Also, the three horizontal (or vertical) cavities arrangement reduces the DSCF in the central cavity. More specifically, this reduction is substantial in case of Model B, i.e., the horizontal cavity arrangement. Furthermore, the normalized DSCF distribution along the central cavity perimeter for a fixed frequency value is also shown in Fig. 5. The results for all four model layouts


are also depicted at the normalized frequency value that produces the maximum DSCF previously observed, and this plot is along the entire central cavity perimeter. The conclusion is that interaction effects with a sizeable increase in the DSCF are manifested only when the number of cavities increases to five. The presence of three inhomogeneities arranged vertically is of no consequence in the DSCF that develops in the central cavity. Their horizontal arrangement, however, leads to a reduction in the central cavity DSCF, which indicates that the two external cavity act protectively as a stress relief mechanism for the central one. (a)

40

Observer point

35

d

30

(b) s=0.1

40

model A

model A, Ω=0.24

model B, Ω=0.22

model C, Ω=0.24

model D, Ω=0.20

model B

25

|σφφ/ σ0|

|σφφ/ σ0|

50

s=0.1

model C

20

model D

15 10

30 20 10

5 0

0 0

0.2

0.4

0

0.6

0.2

0.4

Ω=ωd/CP

0.6

0.8

1

φ/π

Figure 5: (a) DSCF versus normalized frequency for Models A-D at the central cavity and for observer point M S / 2 ; (b) polar distribution of the DSCF for different models at a fixed frequency value corresponding to the maximum amplification (a)

25

Observer point

model A

d

20

(b) s=0.1 model A, ΩM=0.24 model C, ΩM=0.24

25

model B

20

model C

15

model B, ΩM=0.24 model D, ΩM=0.22

|σφφ/ σ0|

|σφφ/ σ0|

30

s=0.1

model D

15

10

10

5

5

0

0 0

0.2

0.4

ΩM=ωd/CP

0.6

0

0.2

0.4

0.6

0.8

1

φ/π

Figure 6: (a) DSCF versus normalized frequency for Models A-D at the central inclusion and at observer point M S / 2 ; (b) polar distribution of the DSCF for different models at fixed frequency values The same set of results, but for the case of inclusions instead of cavities, are plotted in Fig. 6, where the DSCF at the inclusion-matrix interface from the matrix side is presented. Generally speaking, the picture that emerges is similar to the cavity case, although some differences can be noticed. For instance, Model B with the three inclusions arranged horizontally leads to insignificant reduction in the DSCF, whereas the three vertical inclusion arrangement (Model C) cause stress amplification at the central inclusion interface, all as compared to the single inclusion case. Also, the overall influence of the inclusion arrangement is not as significant as in the case of cavities, which can be deduced by comparing Figs. 5b and 6b. Finally, we examine the influence of the cavity shape on the DSCF in Fig. 7. The square matrix domain now contains a single cavity, whose shape changes from a circle to a horizontally elongated ellipse and then to a vertically elongated ellipse. The normalized frequency value is fixed as : Zd / CP 0.24 and the surface parameter is s=0.1. The semi-axes are chosen so that a+b=d and their dimensions vary from a/b=1 to a/b=2 in Fig. 8a and then to a/b =0.5 in Fig. 8b. In all cases, DSCF amplification is observed along the perimeter at points where the shape of the ellipse becomes sharp. In the former case, amplification of about


29

25% at M 0 and the same de-amplification value at M S / 2 are observed. In the latter case, the amplification at M S / 2 is 38% , while the deamplification at M 0 is 15%.

1.000

30

1.077

a

1.160

25 20

10

φ/π

1.348

20

0.2

0.4

0.6

0.8

1

1.000 0.929

a

0.862 0.800 0.742 0.688

15

1.700

10

1.842

5

2.000

0 0

25

1.571

5

30

1.250 1.455

|σφφ/ σ0|

15

35

a/b:

d

|σφφ/ σ0|

d

a+b=d

35

(b)

40

a/b:

b

(a) b

40

0.636 0.588 0.543

φ/π

0.500

0 0

0.2

0.4

0.6

0.8

1

Figure 7: Polar distribution of the DSCF at frequency : 0.24 for a cavity contour whose shape varies from: (a) Circle to horizontally elongated ellipse; (b) circle to vertically elongated ellipse. Conclusions In this work, the in-plane wave field that develops in a finite-sized elastic solid containing multiple cylindrical nano-cavities or nano-inclusions of arbitrary shape, size, configuration and elastic properties is studied. The computational tool is the BIEM with sub-structuring capabilities for the heterogeneous finite solid, a formulation that is modified to include the Gurtin-Murdoch surface elasticity model. This formulation yields a highly effective numerical scheme, and the MATLAB-based software developed herein is first verified against benchmark examples and subsequently used in extensive parametric studies. The results from the numerical simulations demonstrate the potential of the BIEM to produce highly accurate results for the dynamic behaviour of finite solids strengthened (or weakened) by multiple nanoheterogeneities, which leads to further studies in computational nano-mechanics. Acknowledgments: The authors acknowledge support by the Bulgarian NSF under Grant no. DFNI-I02/12. References [1] M.E. Gurtin and A.I. Murdoch Archives Rational Mechanics Analysis, 57, 291–323 (1975) [2] J. W. Cahn and F. Larché Acta Metallurgica 30,51-56 (1982) [3] R.C. Cammarata Progress in Surface Science 46, 1-38 (1994) [4] P. Sharma, S. Ganti and N. Bhate Applied Physics Letters 82, 535- 537 (2003) [5] H.L. Duan, J. Wang, Z.P. Huang and B.L. Karihaloo Journal of the Mechanics and Physics of Solids 53, 1574-1596 (2005) [6] C.Y. Dong and E. Pan Engineering Analysis with Boundary Elements 35, 996-1002 (2011) [7] S. Parvanova, G.D. Manolis and P. Dineva Engineering Analysis with Boundary Elements 56, 57–69 (2015) [8] M.A. Grekov and N. Morozov Journal of Ningbo University 25(1), 60-63 (2012) [9] Y. Ru, G.F. Wang and T.J. Wang Journal of Vibration and Acoustics 13: 061011 1-7 (2009)


Hybrid BEM-FEM Algorithm in Time-Domain for Numerical Modelling of Soil-Tunnel Interaction G. Vasilev1, S. Parvanova2 1

2

Department of Structural Engineering, UACEG, 1046 Sofia, Bulgaria, e-mail: [email protected]

Department of Structural Engineering, UACEG, 1046 Sofia, Bulgaria, e-mail: [email protected]

Keywords: boundary element method, hybrid models, wave propagation, ANSYS

Abstract. The main aim of this paper is to develop and verify effective time-dependent hybrid algorithm for modelling of dynamic behaviour and seismic response of complex soil structure systems in plane-strain state. The hybrid technique is based on the boundary element method (BEM) and finite element method (FEM). The infinite geological half-plane is modelled by direct BEM based on the full space elastodynamic fundamental solutions in Laplace and Fourier domains and Lubich Operational Quadrature in order to obtain time-dependent stiffness matrix and load vector of the seismically active far-field zone. The structure and the surrounding near soil profile are treated by FEM and ANSYS software package. Both domains work together and the full interaction between incident and scattered waves is taken into account. The accuracy and verification study is discussed. The presented simulations reveal the complex character of the seismic field in an inhomogeneous and heterogeneous geological medium containing additionally an underground structure. Introduction The seismic response of soil-structure interaction (SSI) systems has attracted the attention of researchers for many years. The underground transportation structures and utility networks (e.g. rail-way and road tunnels, hydraulic tunnels, lifelines for transportation of water, oil, natural gas, etc.) are less prone to damages compared to over-ground buildings but underestimating the effect of earthquakes on them may lead to major damages or even fatal collapse. The seismic analysis of such systems requires an adequate numerical modelling of both the geological stratum and the structure which could be achieved by application of hybrid numerical schemes combining two numerical methods [1-6], or analytical with numerical approach. The hybrid techniques allow us to benefit from the advantages of different numerical methods and to avoid their respective drawbacks. It is well known that the finite element method (FEM) is the most useful approach in handling various classes of civil engineering problems, including nonlinearity, anisotropy and inhomogeneity, while the boundary element method (BEM) is the most appropriate method in modelling of infinite domains. That is why the BEM-FEM combination seems to be the most promising technique for modelling and further analysis of the SSI problem in hand. One of the first applications of the BEM-FEM combination was presented in [1,2], where the authors used the hybrid approach in modelling and analysis of 3D and 2D foundations on an elastic soil in time domain. More recently a combination between the method of fundamental solutions (MFS) and FEM was developed in [3] for modelling and dynamic analysis of 2D SSI systems, and hierarchical BEM-FEM coupling was presented in [4] for 3D SSI problems, all in frequency domain. In [6] authors proposed an efficient frequency dependent hybrid approach for modelling of underground structures in layered geological stratum embedded in an elastic infinite half-plane. The BEM formulations using elastodynamic fundamental solutions are based on either the direct timedomain approach or the transformed-domain approach, where time dependence is removed by taking a Fourier, Laplace or other integral transform with respect to the time variable. The direct integration in time is the most appropriate approach for solving SSI problems, with structural nonlinearities and/or nonlineraties in the near soil region. In the classical time-domain BEM, the spatial discretization and the time discretization are not independent of each other, which is one of the reasons why more BEM results are based on the frequency-dependent fundamental solutions. The main disadvantage of the frequency-


31

dependent BEM approach is that non-linear behaviour cannot be modelled. The operational quadrature methods, developed by Lubich [7] and applied by Shanz [8] where the Riemman convolution integral is numerically approximated by a quadrature formula whose weights are determined by the Laplace transformed fundamental solutions, lead directly to time-dependent solutions obtained by linear multistep method. This procedure allows using BEM based on the frequency-dependent fundamental solutions in complex domains (Laplace or Fourier) followed by derivation of the solutions directly in the real timedomain. In this work an efficient hybrid time-dependent BEM-FEM numerical scheme is developed for modelling and analysis of different soil-tunnel systems. The proposed computational scheme is realized as follows: 1) the discrete mechanical BEM formulation of the seismically active geological environment and the seismic excitation is generated in complex (Laplace or Fourier) domain; 2) by using Lubich’s Operational Quadrature approach [7, 8] the complex mechanical BEM formulation is converted in discrete transient FEM formulation to finally obtain the time-dependent stiffness matrix and load vector for the infinite far field domain; 3) the far filed infinite domain is coupled with the FEM model of the near field and the structure, and linear multi-step method is applied for determination of the transient response of the whole soil-structure system. Problem formulation Consider 2D elastodynamic problem of a layered soil-tunnel interaction system, embedded in an elastic isotropic half-plane containing transient seismic source with a prescribed magnitude f 0,i , (i=1, 2). Planestrain state and respectively in-plane wave motion in the plane x3 Ox1x2 x3 is assumed, see Fig. 1. x2

ΓFS ΓP

1

x02

Oi, μi, ρi, ξi Γ2t f02

Free surface ΓFS x

1

FED

3

λt, μt, ρt, ξt Γ1t

f0

ΓINT BED

f01 x

0 of a Cartesian coordinate system

2

x01

O0, μ0, ρ0, ξ0

Figure 1: Soil-Tunnel interaction scheme. The numerical model is divided into two domains: a) boundary element domain (BED) describing semiinfinite homogeneous far-field geological region with Lame constants, density and material damping λ0, μ0, ρ0, ξ0. This domain is delineated by the flat free surface boundary * FS and contact interface * INT and contains an embedded line seismic source at a point x 0 ; b) finite element domain (FED) comprising of N non-parallel layers with material parameters of the i-th layer λi, μi, ρi, ξi, i = 1, 2,…, N. This domain is outlined by the free surface boundary * FS * P , where * P is the boundary of the surface relief of arbitrary geometry, and contact interface * INT . The FED contains an embedded lined infinite cylindrical tunnel with material properties λt, μt, ρt, ξt, whose internal and external boundaries are denoted by *1t and * 2t respectively. The concentrated at point x0 seismic body force is Fi x, t

f0i g t G x, x0 , where its

time function is g(t) and G x, x0 is the Dirac delta function. The dynamic equilibrium equation is:

V ij , j x1 , x2 , t U

w 2ui x1 , x2 , t wt2

Fi x1 , x2 , t

(1)


where: V ij

Cijkl x2 uk ,l , Cijkl x2 O x2 GijG kl P x2 Gik G jl Gil G jk , G ij is Kronecker’s delta symbol

and comma subscripts denote partial differentiation with respect to the spatial coordinates, while the summation convention over repeated indices is implied. The boundary conditions are as follows: a) inside FED we have zero tractions along boundaries * FS * P *1t and displacement compatibility and traction equilibrium conditions hold along boundaries *i *2t2t for i = 1, 2,…, N; b) outside FED the tractions along the flat part * FS of the free surface belonging to the BED are zero, compatibility and equilibrium conditions hold along interface boundary * INT between FED and BED, and finally Sommerfeld radiation condition is satisfied at infinity. Hybrid BEM-FEM coupling and analysis BEM formulation in complex (Laplace or Fourier) domain for the infinite half-plane. The BEM formulation is based on the complex domain approach where time dependence is removed by taking a Fourier, Laplace or other integral transform with respect to the time variable. In this case the original hyperbolic partial differential equation of motion (1) is reduced to elliptic one that is easier for mathematical modelling. The boundary-value problem in the BED (far-field semi-infinite region with the seismic source) is reformulated via boundary integral equation (2) along * BED * FS * INT with respect to the complex variable z: cij u (jBED ) x, z

U ij*( BED ) x, ξ, z t (jBED ) ξ, z dS ξ

³

* BED

³

(2)

Pij*( BED ) x, ξ, z u (jBED ) ξ, z dS ξ f 0i g z U ij*( BED ) x, x0 , z .

* BED

Here: z is the circular frequency ω or the Laplace variable s, related with the well-known relation s iZ l Z is , where i is the imaginary unit; g(z) is the Fourier or Laplace transform of the time-history function describing the dynamic excitation; cij is the jump term depending on the local geometry at the collocation point x; x and ξ are the position vectors of the source-receiver couple; ti tractions and n j are the components of the outward normal vector; fundamental solution of the governing equation in complex domain;

Pij*( BED )

BED ) U*( ij

V ij n j are the

is the displacement

is the corresponding traction.

Conversion of the BEM model of the BED into one macro-finite element (FE). In order to combine infinite part of the system (or BED) with the finite one (or FED), the boundary element model will be converted into one macro-finite element. For that purpose, firstly, reduction of the degrees of freedom (DOF) of the boundary element model is necessary because the contact zone between both domains is presented by common interface * INT . For the aim of DOF condensation the BED is divided into 3 parts: 1) left flat free surface, part of the * FS belonging to the BED; 2) common interface * INT ; 3) right flat free surface, part of the * FS , which falls into BED, see Fig. 1. Taking into account the zero traction boundary conditions at contours 1 and 3 the equation (3) is obtained that relates traction t INT and displacement u INT vectors along * INT : t INT

Bu INT P.

(3) 1

ª¬H21 H23 º¼ Au H22 , where Here the matrix B of block size 1x1 is equal to G22 ª¬H21 H23 º¼ At Gij and Hij are the well-known influence submatrices along the boundary * BED obtained after

discretization of Eq. (2). Here matrices At

ª H11 H13 º «H » ¬ 31 H33 ¼

1

ª G12 º «G » and Au ¬ 32 ¼

and

At

H13 º ªH « 11 » H H 33 ¼ ¬ 31

1

Au

of block size 2x1 are equal to

ª H12 º « H » . Subscripts denote boundary (1 for the left ¬ 32 ¼


33

flat surface, 2 for the interface contour, 3 for the right flat free surface). Left subscript is associated with source or collocation point contour, the right one is for receiver or field point boundary. In that formulation 1 § t ½ · the vector where P G22 ¬ª H21 H23 º¼ At ¨ ª¬ H21 H23 º¼ Ψ Φ2 ª¬G21 G23 º¼ ® 1 ¾ ¸ , ¨ t3 ¿ ¸¹ ¯ ©

1

ª H11 H13 º § Φ1 ½ ª G11 G13 º t1 ½ · «H » ¨¨ ® ¾ « » ® ¾ ¸¸ is vector of block size 2x1 and the load vector Φ2 takes ¬ 31 H33 ¼ © ¯Φ3 ¿ ¬G31 G33 ¼ ¯t3 ¿ ¹ into consideration the last term in Eq. (2). Details about this condensation could be found in [6]. The second step in macro-FE generation is conversion of the BEM system into FEM like system, respectively replacement of the interface tractions t INT with equivalent nodal forces FINT . This relation is Ψ

1

obtained as FINT

³N

T

Mt INT , where the matrix M

NJJd d[ (J is the Jacobian, [ is intrinsic

e 1

coordinate), whose components are numerically derived as a product of the traction shape functions for the particular boundary element e, see [6]. Substituting eq (3) in this relation the following expression is obtained, which is a FEM like discrete system of equations, necessary for formulation of the BED as a single macro-finite element: FINT

K ( BED)u INT R.

(4)

( BED )

Here K MB is the stiffness matrix in complex domain and R MP is the generalized load vector of nodal forces dependent on the traction load vector P after the condensation procedure, see [6]; Derivation of time-dependent stiffness matrix. The stiffness matrix for the BED model is firstly obtained in Laplace domain, subsequently Lubich’s Convolution Quadrature (LCQ) rules [8] are applied and the stiffness matrix in time domain is derived. The k-th time dependent stiffness matrix, at time tk k 't ( 't is ˆ (s) ω ('t ) / 't . Here K ˆ is the stiffness matrix derived in Laplace the time step), is K (t ) L1 K k

k

k

ˆ s is expressed through the matrices ω being the domain, while its inverse Laplace transform L1 K k

Lubich’s convolution quadrature weights (see [7]). The nodal-force vector R is firstly obtained in frequency (Fourier) domain, then by inverse discrete fast Fourier transform the vector is derived in time domain. The k-th time dependent load vector, at time tk k 't , is R k tk F 1 R(Z ) . Implementation of the macro-finite element in ANSYS software via the ANSYS user programmable features (UPFs). The macro-FE is implemented in ANSYS by using for its formulation the time-dependent stiffness matrix and the load vector. The FED is modelled by 8-noded plane finite elements with 16 DOF for plane strain state (PLANE82). Both the FE and the BE time-dependent models are coupled by satisfaction of the nodal compatibility and equilibrium conditions required for all available DOF on the BED-FED contact surface. The final discrete system of equations of the coupled model in time domain, which is obtained after application of the convolution theorem, is as follows: j

¦ K k u j k 1

R j ,

j 1..N ,

k 1

where N is the total number of time steps. The real dynamic stiffness matrices are K k

(5)

K (kBED) K (kFED) ,

k 't ) at which the quantities are calculated. In this system the only unknowns are the components of the displacement vector u j at time t j , since all the where the subscript designates the discrete time value ( tk

displacement components in the previous time steps, included in the discrete sum, are previously calculated. This dependence on the previous time steps and not on the future ones is consequence of the causality of the system. The stiffness matrices K (kBED ) and load vectors R j are generated with the authors’ code developed in MATLAB, while K (kFED ) is generated by the ANSYS software;


Solution of the algebraic system Ku = F in time domain by the corresponding solvers in ANSYS. One of the major differences between BEM and FEM is that the later method leads to sparse symmetric positive definite matrices, while the former one, based on a collocation technique, leads to full non-symmetric matrices. When full method of solution in transient FEM analysis is chosen and ANSYS detects the presence of structure with unsymmetrical and fully populated matrices it automatically chooses a solver for dealing with the arisen unsymmetrical system of equations. Verification study To verify the accuracy of the hybrid algorithm in time domain and the solution of the corresponding transient analysis we compare the authors’ results obtained by the present hybrid tool against results derived via the conventional BEM authors’ software. We have the validated BEM software codes for modeling of anti-plane and in-plane elastodynamic problems in the half-plane with surface relief and sub-surface peculiarities in frequency domain [10,11]. x2

a)

b)

a

a

Fictitious FED contact μ ρ ν ξ boundary ΓINT 1 1 1 1

Vertical Displacement (u2)

0.4

x1 k.a

x0(x01,x02)

BED μ0 ρ0 ν0 ζ0

x2 x1

0.2

observation point 0 0

1

-0.4

fi

2

3

4

f0i G x, x0 exp(iZt )

5

6

Time, [s]

-0.2

BEM followed by Fourier transform ANSYS model

-0.6 -0.8

Figure 2: Verification example: a) Geometry; b) Vertical displacement time-history at the bottom of the canyon. The numerical example, considered here, is a semi-circular canyon of radius a = 1m located in an elastic, isotropic and homogeneous half-plane with material parameters: Poisson’s ratio Q 0 1/ 3 ; shear modulus

P0 1.106 KPa ; density U0

2000 t/m3 ; and hysteretic damping ratio [0

an embedded source of magnitude f0i

6

(0, 2.10 ) , located at a point x0

0% . The load is presented by (0, 2.5) which is in the FED.

The time-history variation of the source magnitude is defined by the function g t

f0eD t sin(Z0t ) , where

D 1 and Z0 10 > rad/s@ . This function has continuous Fourier transform in frequency domain f (Z )

f 0Z0 / (Z0 )2 (D iZ )2 .

Two numerical analyses are performed: (1) reference (benchmark) BEM solution performed by authors’ software based on conventional BEM and frequency dependent full space fundamental solutions followed by inverse fast Fourier transform of the results. The BE mesh comprises of 144 quadratic boundary elements, 48 of which located along the canyon contour. The length of the discretized flat free surface is 40 meters or 40a, measured left and right to the canyon center. The Nyquist frequency is 10 Hz and the frequency range is divided into 300 equal steps; (2) The hybrid BEM-FEM model comprises of two domains: infinite homogeneous half plane (BED) and semi-circular FED, of radius 5a, as shown in Fig. 2a. Both domains have identical material properties. Firstly the infinite half-plane is modelled by BEM in complex (Laplace) domain then the convolution quadrature rule is applied and time-dependent stiffness matrix for macro-FE formulation is obtained. The FED is modeled in ANSYS by using 8-noded FE with 16 DOF (PLANE 82), the macro-FE (MATRIX 50) consists of 97 nodes related by the stiffness matrix. Finally, transient analysis is performed to the FE model, where Lubich’s Convolution Quadrature algorithm is used along with Newmark’s integration scheme. The unknown displacement components of the whole BEM-FEM numerical model are derived directly in time domain.


35

Comparisons of the results obtained by both analyses are shown in Fig. 2 b, where the vertical displacement components for the bottom of the canyon ( x 0, 1 ) versus time variable are depicted. Obviously very good plotting accuracy is achieved. Simulation study The aim of this section is to demonstrate the application and efficiency of the proposed hybrid technique for modelling of relatively complex soil-structure systems. The computational model is an illustrative example of a practical tunnel construction and properties of the soil profile imitate real geotechnical data, although they are not a real case study. We consider a multi-story tunnel construction of circular cross section with inner radius a=7 m as shown in Fig. 3a, which is embedded in a multi-layered FED depicted in Fig. 3b. The tunnel contains two train lanes on the first level and several travel road lanes on the upper level. All dimensions of the tunnel liners and internal plates and walls are given in Fig. 3a. The depth of burial is h=3a=21 m measured from the flat free surface. The material properties of all components of the tunnel construction are as follows: modulus of elasticity Et 30000 MPa , Poisson’s ratio Q t 0.2 , material density Ut 2.5 t/m3 , and constant material damping ξt=0%.The near soil region, presented by FED, comprises of 5 geological layers of arbitrary outline, which in case of real practical problem could be reported from the geological profile. The material properties of the adopted soil strata are as follows: μ 1=620 MPa, ν1=1/3, ρ1=1.8 t/m3; μ2=500 MPa, ν2=0.4, ρ2=1.75 t/m3; μ3=200 MPa, ν3=0.45, ρ3=1.8 t/m3; μ4=100 MPa, ν4=0.46, ρ4=1.85 t/m3; μ5=65 MPa, ν5=0.48, ρ5=1.9 t/m3. The material damping of all layers is assigned as ξ1= ξ2= ξ3=ξ4=ξ5=0%. The outline of the FED is delineated by upper traction free surface containing hill with height h=14 m and semi-width b=30 m, and semi-circular lower boundary of radius 10a=70 m, which is the BED-FED common interface. The shape of the hill is described by the following function x2 ( x1 ) 0.5h 1 cos S x1 / b . The FED itself is embedded

in an elastic homogeneous half-plane (BED) with material parameters μ0=700 MPa, ν0=1/3, ρ0=2 t/m3 and material damping again ξ0=0%. The load is presented by in-plane waves radiating by an embedded line transient source of magnitude f0i (0, 1.106 kN) located at a point x0 (0, 150) below the flat surface

0.3

7.4

0.4

0.3

b)

x2

μ5 ρ5 ν5 ξ 5

h

0.40

a=7

4.9

f0i g t G x, x0 follows the time-history function g (t ) defined in the

b

μ4 ρ4 ν4 ξ 4

x1

a

3a

in BED. The excitation Fi x, t previous section. a)

μ3 ρ3 ν3 ξ 3 μ2 ρ2 ν2 ξ 2 μ1 ρ1 ν1 ξ 1

10a

Figure 3: Geometry of the illustrative example: a) tunnel construction; b) geological strata The stiffness matrix and load vector for the macro–FE, having all features of the infinite half-plane, are generated after BEM solution in Laplace and frequency domain respectively. The BE model comprises of 144 quadratic boundary elements, 48 of which located along the common interface * INT . The length of the discretized flat free surface is 2800 meters or 400a (40 times radius of the contact contour). The Nyquist frequency is 15 Hz which results in time step equal to 1/30 sec. The FEM mesh employed in the numerical disctretization for the FED comprises of 8325 quadratic finite elements (ANSYS code PLANE 82) and one macro-finite element containing 97 nodes (ANSYS code MATRIX 50). Two numerical models are considered in order to point out the influence of the stratified near soil profile: (1) tunnel construction embedded in homogeneous, elastic, and isotropic FED with material properties of the soil deposit identical with those of the half-plane μ0= μ1-5 =700 MPa, ν0= ν1-5=1/3, ρ0= ρ1-

Eds: V Mantic, A Saez, M H Aliabadi 3 5=2 t/m and zero material damping; (2) stratified FED with homogeneous layers and different material properties as reported at the beginning of this section. The first set of the results concerns displacement components at the free surface. The surface distribution of both horizontal and vertical displacements, at fixed time t=0.4s and t=0.7333s are presented in Fig. 4a,b. The chosen time values correspond to the first two extreme values in the vertical displacements at top of the hill versus time.

a)

0.3

0.1 0.05

u1, u2

0.1

u1, u2

b)

0.15 u1-layered FED u2-layered FED u1-homogeneous HP u2-homogeneous HP

0.2

0 -0.1

0

u1-layered FED u2-layered FED u1-homogeneous HP u2-homogeneous HP

-0.05

-0.2

-0.3

-0.1

-0.4

-0.15 -10

-5

0

5

10

-10

-5

x1/a

0

5

10

x1/a

Figure 4: Displacement components along the free surface: a) t = 0.4 sec; b) t = 0.7333 sec Fig. 4 illustrates the sensitivity of the synthetic wave field at the free surface to the heterogeneous character of the wave path. Of interest is the fact that the influence of the stratified near-soil geological profile is highly pronounced only inside the FED. The reason for this phenomenon could be explained as a consequence of the primary P-wave propagation path generated from the relatively deep source. The Pwaves reach the free surface at both ends of the contact interface propagating mainly in the infinite halfplane and barely passing through the first two layers. This is the reason why at x1/a=±10 we observe nearly the same vertical displacement components for both considered soil profiles, see Fig. 4. a)

σφφ. [MPa]

60

20 0

-20

1

2

3

t, [sec]

φ= π/2

20

40

0

b)

30

homogeneous HP layered FED

φ=0

4

σφφ. [MPa]

80

homogeneous HP layered FED

10 0 0 -10

1

2

3

t, [sec]

4

-20

-40

-30

-60

-40

Figure 5: Hoop stresses along the soil-tunnel interface for fixed representative points, from the tunnel side: a) polar angle φ=0; b) polar angle φ=π/2

Figure 6: Polar distribution of the stresses in [kN/m2] at t=0.4 sec: a) hoop stresses at soil-tunnel interface from the tunnel side; b) radial and hoop stresses for the homogeneous half-plane from the soil side; c) radial and hoop stresses for the layered FED from the soil side


37

Fig. 5 plots the hoop stresses at two observer points with polar angles φ=0 and φ=π/2 versus time for both types of soil deposit: homogeneous and layered. According to these plots layered FED causes amplification at polar angle φ=0, while the hoop stresses at polar angle φ= π/2 , are generally reduced for the case of layered soil profile compared to those in the case of homogeneous half-plane. The next set of results is for stress distribution along the soil-tunnel interface from the tunnel and the soil side. Fig. 6a depicts polar distribution of the hoop stresses at fixed time t=0.4 sec at soil-tunnel interface from the tunnel side for homogeneous half-plane and layered FED. Here the results obtained by both models are quite different as far as the tunnel is in the core of the stratified FED and all in-plane waves surely propagate through the soil strata in order to reach this interface independently of the source location. Figs. 6b and 6c depict radial and hoop stresses at the contact soil-tunnel interface from the soil side for homogeneous half-plane and layered FED respectively. The polar stress distribution in all plots in Fig. 6 shows singularities in hoop and radial stresses due to the contact of the tunnel wall with the slabs and other structural elements. Conclusion In this paper a hybrid BEM-FEM numerical algorithm in time domain has been developed for modelling of dynamic behaviour and seismic response of complex 2D plane-strain soil-structure interaction problems. The numerical model of the far field soil region is developed by BEM, based on the full space elastodynamic fundamental solutions in Laplace and Fourier domains and Lubich Operational Quadrature in order to obtain time-dependent stiffness matrix and load vector of the seismically active infinite halfplane. The structure and the near field region are treated by FEM and ANSYS software package. The main advantage of the coupled technique is the ability to handle in one model the time-dependent behaviour of the whole system defined by the seismic source with its specific geophysical properties, the inhomogeneous and heterogeneous wave path with complex geometry, the local geotechnical region with free- and subsurface relief and finally the underground structure with its specific structural properties. This hybrid approach could be successfully extended for treating of both geometrical (large displacements, uplifting phenomena) and physical (inelastic material, fluid saturated soils) type of nonlinearities. References [1] D. L. Karabalis, D. E. Beskos, Soil Dyn Earthquake Eng 4(2): 91-101 (1985). [2] C. Spyrakos, D. Beskos, Soil Dyn Earthquake Eng 5(2): 84-96 (1986). [3] L. Godinho, P. Amado-Mendes, A. Pereira, D. Jr. Soares, Computers and Structures 129: 74-85 (2013) [4] P. Coulier, S. François, G. Lombaert and G. Degrande, Int. J. Numer. Meth. Engng. 97: 505–530 (2014). [5] G. Manolis, S. Parvanova, K. Makra, P. Dineva, Bulletin of Earthquake Engineering, DOI: 10.1007/s10518-014-9698-6 (2015). [6] G. Vasilev, S. Parvanova, P. Dineva, F. Wuttke, Soil Dynamics and Earthquake Engineering 70: 104– 117 (2015). [7] C. Lubich, Numerische Mathematic 52(2): 129-145 (1988). [8] M. Schanz, Communications Numerical Methods Engineering 15: 799–809 (1999). [10] S. L. Parvanova, P. S. Dineva, G. D. Manolis, F. Wuttke, Bull Earthquake Eng 12: 981–1005 (2014). [11] S. L. Parvanova, P. S. Dineva, G. D. Manolis, Acta Mechanica 225(7): 1843-1865 (2014).


Scaled Boundary Finite Element Method for Thermoelasticity in Voided Materials Jan Sladek1, Vladimir Sladek1 and Peter Stanak1 1

Institute of Construction and Architecture, Slovak Academy of Sciences, 84503 Bratislava, Slovakia, email: [email protected]

Keywords: Uncoupled thermoelasticity, Representative volume element (RVE), circular voids, 2d problems, stationary boundary conditions.

Abstract. Thermoelastic problems in materials with voids are analyzed by the scaled boundary finite element method (SBFEM). The SBFEM combines the main advantages of the finite element method (FEM) and the boundary element method (BEM). In this method, only the boundary is discretized with elements leading to a reduction of spatial dimension by one. In contrast to the BEM, no fundamental solution is required, which permits to analyze general boundary value problems. The computational homogenization technique is applied in voided materials. The evolution of the mechanical and thermal fields at the macroscopic level is resolved through the incorporation of the microstructural response. The microstructural analyses are performed on the representative volume element (RVE), where essential physical geometrical information about the microstructural components is included.

Introduction The presence of voids and cracks affects material properties and functionality of these elements in structures. The influences of voids on the effective properties have been studied by many authors (see review paper [1]). The shape and distribution of voids can be arbitrary. In principle it would be possible to refer directly to the microscopic scale, but such microscopic models are often far too complex to handle for the analysis of a large structure. Therefore, the multiscale modelling is a convenient technique to consider voided microstructure properties and transfer them into the macroscopic models [2]. The microstructure can be accounted in macro-structural analyses via homogenization when the effective (overall) material coefficients are obtained from solutions of appropriate boundary value problems on micro-scale level in a representative volume element (RVE) [3]. Earlier homogenization techniques have been based on analytical approaches. Mainly self-consistent and Mori-Tanaka analytical approaches are utilized to get effective material coefficients [4,5]. A comprehensive validation of analytical homogenization models is given by Ghossein and Levesque [6]. Later, numerical approaches have been developed to determine effective material properties in composite materials [7,8]. Various multiscale methods are reviewed by Kanoute et al. [9] in the context of modelling mechanical and thermomechanical responses of composites. In the present paper a pure numerical approach has been developed to evaluate thermomechanical effective material properties in a voided material. Numerical analyses are performed on the RVE. The RVE contains sufficient microstructural information to be representative of any similar volume taken from any location in the voided solid. In uncoupled thermoelasticity, the temperature field is not influenced by displacements. Therefore, the heat conduction equation is solved first to obtain the temperature distribution. The equation of motion is subsequently solved for mechanical quantities. In homogenization techniques the RVE under specific boundary conditions is analyzed for determination of influence of voids on material properties. Therefore, we need to have a reliable computational method to solve boundary value problems on the RVE. In the present paper, the scaled boundary finite element method (SBFEM) is developed for 2D boundary value problem in a porous elastic solid under stationary thermoelastic boundary conditions. Up to now the SBFEM have been successfully applied to elastostatic, elastodynamic problems, thermopiezoelectricity and piezoelectric crack problem too [10-14]. The SBFEM is applied here on the micro-level (RVE) of the elastic solids with voids to compute effective material properties.


39

Scaled boundary finite element method in uncoupled thermoelasticity Governing equations in uncoupled stationary thermoelasticity are given by the balance of forces and stationary heat conduction equations [15] V ij , j (x) 0 , (1)

kijT,ij (x) 0 ,

(2)

where V ij , T and ui are the stresses, temperature difference and displacements, respectively. Symbol kij is used for the thermal conductivity tensor. Constitutive equations are given by the well known DuhamelNeumann constitutive law for the stress tensor V ij ( x) cijkl H kl ( x) J ijT (x) , (3) where cijkl are the materials elastic coefficients and J ij is the stress-temperature modulus. The stresstemperature modulus can be expressed through the elastic coefficients and the coefficients of linear thermal expansion D kl as J ij cijklD kl . (4) For 2-D plane problems, the constitutive equation (3) is frequently written in terms of the second-order tensor of elastic constants [16]. The constitutive equation for orthotropic materials and plane strain problems has the following form ªV 11 º ª c11 c12 0 º ª H11 º ª c11 c12 c13 º ªD11 º ª H11 º «V » «c » « H » «c » «D » T C « H » T , c 0 c c (5) 22 12 22 22 12 22 23 22 « » « »« » « »« » « 22 » 0 c66 ¼» ¬« 2H12 ¼» ¬« 0 0 0 ¼» ¬«D 33 ¼» ¬«V 12 ¼» ¬« 0 ¬« 2H12 ¼» ª c11 c12 c13 º ªD11 º ª J 11 º «c »« » « » « 12 c22 c23 » «D 22 » «J 22 » . «¬ 0 0 0 »¼ «¬D 33 »¼ «¬ 0 »¼ In scaled boundary finite element method (SBFEM) a scaling center O is selected at a point from which the whole boundary is directly visible [10]. The boundary S is scaled by the dimensionless radial coordinate[ pointing from the scaling center. Only that boundary is discretized with line elements S e . The whole with

analyzed domain V is decomposed into triangular sectors V e associated with the boundary line elements S e . The global Cartesian coordinates ( x, y ) on a line element S e (the superscript e denotes the element) on the boundary are parametrized as e e x1 S e x K S e : ª¬ N K º¼ ^ x` , x2 S e y K S e : ¬ª N K ¼º ^ y` ,

with K being the local coordinate K [1, 1] , ª¬ N K º¼ N1 K , N 2 K is the matrix of shape functions. The analyzed domain is described by scaling the boundary with the dimensionless radial coordinate [ pointing from the scaling center O ( [ 0 ) to a point on the boundary ( [ 1 ). The Cartesian coordinates

x1 x2 of a point inside the triangular sector V e are parametrized as x1 ([ K ) V [ x K S [ ª¬ N e K º¼ ^ x1` e

x2 ([ K ) V e

(6)

e

[ y K S

e

[ ª¬ N e K º¼ ^ x2 `

(7)

where [ and K are called the scaled boundary coordinates, [ [0,1] , K [1,1] . In the first step the heat conduction (2) is analyzed. Let T a ([ ) for a 1, 2, ..., n be the temperature field

parametrized along the radial line passing the scaling center and the node x1a x2a

on the boundary.

e

Furthermore, we assume the linear interpolation within the boundary element S , i.e. T (x) S e ª N e (K ) º ^T ([ 1)` . ¬ ¼

(8)


Now, we extend the assumed interpolation on the boundary element, i.e. the approximation of the dependence on the parameter K when [

T ( x) V e

1 , also to interior points in the sector V e as

ª N (K ) º ^T ([ )` , ¬ ¼ e

(9)

with new unknowns T a ([ ) being dependent on one parameter. The governing equation (2) and the prescribed boundary conditions result from the variational formulation (10) ³ GT,i (x)qi (x)dV ³ GT (x)q (x)dS 0 . V

Sq

Bearing in mind the approximations the transformation of Cartesian coordinates into scaled boundary coordinates the variational formulation yields the following governing equation T § · [ 2 ª E 0 º ^T ([ )` [ ¨ ª E 0 º ª E1 º ª E1 º ¸ ^T ([ )` ª E 2 º ^T ([ )` 0 , (11) ,[[ ,[ ¬ ¬ ¼ ¬ ¼¹ ¼ ©¬ ¼ ¬ ¼ where the coefficient matrices ª E 0 º , ª E1 º and ª E 2 º are given by Li et al. [13]. ¬ ¼ ¬ ¼ ¬ ¼ A matrix function solution technique is adopted to solve the scaled boundary finite element equation (11). Introducing T

^q ([ )` : [ ª¬ E 0 º¼ ^T ([ )`,[ ª¬ E1 º¼ ^T ([ )` ,

(12)

one can eliminate the 2nd order derivative ^T ([ ,W )` equations for 2n variables ^T ([ )` , ^q ([ )` °^T ([ )`°½ ¾ ¯°^q ([ )` ¿°,[

[®

,[[

in (11) and get the set of the 1st order differential

°^T ([ )`°½ >Z @® ¾ ¯°^q ([ )` ¿°

(13)

with the Hamiltonian coefficient matrix 1 T 1 ª º ª E 0 º ª E1 º , ªE0 º « » ¬ ¼ ¬ ¼ ¬ ¼ (14) Z > @ « » . 1 T 1 2 1 0 1 1 0 «ª E º ª E º ªE º ªE º , ª E º ª E º » «¬ ¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼ »¼ The eigenvalue method [17] has been applied to solve the system of ordinary differential equations (ODE) and the solution is written by T ([ ) > < 21 @ ª¬[ Oi º¼ ^c1` , (15)

where Oi are eigenvalues and < ij are eigen-functions. The temperature on the boundary ^T `

^c1` > L@û ` > L@[u1 u2 ]T

with the linear differential operator ª w º 0 » « w x « 1 » « w » > L@ « 0 » . wx2 » « « w w » « » «¬ wx2 wx1 »¼

(16)

(17)


The linear differential operator [ L ] in equation (17) is transformed to the coordinates [ K as w 1 2 w [ L] [b1 K ] ªb K º¼ w[ [ ¬ wK where ª y K 0 º K » 1 « 1 ª¬b K º¼ « 0 xK K » , | J K | « » y K K » x K ¬« K ¼ ª y K 0 º 1 « » ª¬b 2 K º¼ K » 0 x | J K | « «¬ x K y K »¼

,

41

(18)

(19)

(20)

where J (K ) is Jacobian of coordinate transformation. The displacements at any point ([ K ) inside the sector V e is obtained by interpolating û ([ )` with the shape functions as

ª¬ N e K > I @º¼ {u [ } , where > I @ is a 2 u 2 identity matrix. {u ([ K )} V e

(21)

The stresses ^V [ K ` can be obtained at any point ([ K ) inside the sector V e from equations (5) and (21) as

^V [ K `

>C @ ª¬ B1e K º¼ û [ `[ >C @ ª¬ B 2e K º¼ û [ ` ^T [ ,K ` , 1

Ve

[

(22)

where ª¬ B1e K º¼ [b1 (K )] ª¬ N e K > I @º¼ , ª¬ B 2e K º¼ [b 2 (K )] ª¬ N ,eK K > I @º¼ . On the principle of virtual work for considered problem it is possible to derive the scaled finite element equation for thermoelasticity

[ 2 [ E 0 M ]{u ([ )},[[ [ [ E 0 M ] [ E1M ]T [ E1M ] {u ([ )},[ [ E 2 M ]{u ([ )} ^F ([ )` 0 ,

(23)

where coefficient matrices ª E º , ª E º and ª E º are given by Song [11] and ¬ ¼ ¬ ¼ ¬ ¼ 1§ T T · ^F ([ )` [J ³ ¨ ª¬ B1 (K ) º¼ [ ^T ` ,[ ª¬ B 2 (K ) º¼ ^T ` ¸ J (K ) dK . ¹ 1© 0M

1M

2M

The scaled boundary finite element equation (23) is solved in the first step as homogeneous equation. Then, the non-homogeneous equation (23) is solved by the technique of variation of parameters [17].

Effective thermo-mechanical material properties Let us consider a rectangular RVE sample : {x ( x1 , x2 ); x1 [0, a ], x2 [0, b]} . Inside the rectangular RVE domain there are generally some microstructural elements with arbitrary geometry. Then, the average values of the conjugated fields within the analysed sample are given as


If the boundary conditions are selected as shown in Fig. 1, the average values of the secondary fields are given as T ,1 -1 const , T ,2 0 , Then, we can get the following effective heat conduction coefficients q q k11eff 1 , k21eff 2 ,

-1

-1

where

Fig. 1 Boundary conditions appropriate for evaluation of k11eff , k21eff

q1

1 § b ¨ a ³ q1 ab © 0 x1

q2

1 § a ¨ b ³ q2 ab © 0

a

a

dx2 ³ ª« q2 0¬

x2 b

b

x2 b

dx1 ³ ª« q1 x 1 0¬

a

q2

º

·

x dx x2 0 »¼ 1 1 ¸

q1 x

1

¹ º x dx · . 0 ¼» 2 2 ¸ ¹

Similarly one can select convenient boundary conditions on the RVE to evaluate mechanical and thermomechanical effective coefficients. In numerical example it is analyzed one circular void in square domain (a x a), where various values of void radii are considered. The material parameters corresponding to cadmium selenide ceramic material [Li at al. (2015)].The analyzed domain is divided into polygons and each polygon is treated as a scaled boundary finite element subdomain. To have a good visibility from the scale centre on boundaries of the analyzed domain, it is necessary to introduce subdomains. We have considered 12 polygons in numerical analyses with getting quasi uniform fictitious triangulation which is appropriate for approximation accuracy. There are totaly 216 degrees of freedom (DOFs) in this mesh. The volume fraction of voids is defined as f S r02 / a 2 , where r0 is the radius of the circular void. The numerical results for effective material parameters are presented in Fig. 2-5.

Fig. 2 Variation of effective elastic coefficients eff c11eff , c22 on porosity

Fig. 3 Variation of effective elastic coefficient eff c44 and c12eff on porosity


Fig. 4 Variation of effective thermal conductivities on porosity

43

Fig. 5 Variation of effective stress-temperature eff on porosity. moduli J 11eff and J 22

Acknowledgements This work is supported by the Slovak Science and Technology Assistance Agency registered under number APVV-0014-10 and the Slovak Grant Agency VEGA-2/0011/13, which are gratefully acknowledged.

References [1] I. Jasiuk, C. Chen, M.F. Thorp Applied Mechanics Review 47, 18-28 (1994). [2] B. Engquist, X. Li, W. Ren, E. Vanden-Eijnden Communication in Computational Physics 2, 367-450 (2007). [3] R. Hill Journal of the Mechanics and Physics of Solids 11, 357-372 (1963). [4] B. Budiansky Journal of the Mechanics and Physics of Solids 13, 223-227 (1965). [5] T. Mori, K. Tanaka Acta Metallica 21, 571-574 (1973). [6] E. Ghossein, M. Levesque Mechanics of Materials 75, 135-150 (2014). [7] H. Bohm, W. Han Modeling Simulation Material Science in Engineering 9, 47-65 (2001). [8] O. Pierard, C. Friebel, I. Doghri Composite Science Technology 64, 1587-1603 (2004). [9] P. Kanoute, D.P. Boso, J.L. Chaboche, B.A. Schrefler Archive Computational Methods in Engineering 16,31-75 (2009). [10] A.J. Deeks, J.P. Wolf Computational Mechanics 28, 489-504 (2002). [11] Ch. Song Computer Methods in Applied Mechanics and Engineering 193, 2325-2356 (2004). [12] I. Chiong, E. Tat Ooi, Ch. Song, F. Tin-Loi Engineering Fracture Mechanics 131, 210-231 (2014). [13] Ch. Li, E. Tat Ooi, Ch. Song, S. Natarajan International Journal of Solids Structures 52, 114-129 (2015). [14] Ch. Li, Ch. Song, H. Man, E. Tat Ooi, W. Gao International Journal of Solids Structures 51, 20962108 (2014). [15] W. Nowacki Thermoelasticity, Pergamon, Oxford (1986). [16] S.G. Lekhnitskii Theory of Elasticity of an Anisotropic Body, Holden Day, San Francisco

(1963). [17] Ch. Song, J.P. Wolf Computer Methods in Applied Mechanics and Engineering 180, 117-135 (1999).


Mixed convection flow in a lid-driven cavity with hydromagnetic effect using DRBEM 1 and C. Bozkaya2 ˘ F.S. Oglakkaya 1,2

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

Keywords: MHD, mixed convection, DRBEM.

Abstract. The present numerical study is conducted to investigate the two-dimensional, steady, laminar, incompressible hydromagnetic mixed convection flow and heat transfer characteristics in a vertical lid-driven cavity involving a heated source on one vertical wall with Joule heating effect. The mixed convection phenomena is a result of the forced convection from moving left wall, and the natural convection from buoyant effect due to thermal nonhomogeneity of the cavity walls. The cavity is filled with an electrically conducting fluid and the right wall is partially heated by two different types of sources such as a rectangular and a semicircular heater. The horizontal walls of the cavity are adiabatic whereas the left wall moving upward is kept cold. The governing equations given in stream function-vorticity-temperature form are discretized using dual reciprocity boundary element method (DRBEM) with constant elements. The DRBEM is a boundary only nature technique which approximates all the terms except the Laplacian by means of radial basis functions. Numerical simulations are carried out at a fixed Reynolds number of Re = 100 for the controlling parameters, such as the Hartmann number (Ha), Rayleigh number (Ra) and Joule heating parameter (J). The fluid flow and temperature distributions are analyzed through streamlines and isotherms to assess the effect of the various combination of aforementioned parameters.

Introduction The mixed convection flow in a lid-driven cavity is the combination of the shear and buoyancy effects. The shear and the buoyancy forces are the results of the movement of one wall and the temperature differences, respectively. The analysis of the mixed convection flow and temperature distribution are fundamentally important and computationally challenging in many engineering applications due to the existence of both shear and buoyancy effects. The natural and forced convection problems occur in a wide variety of industrial applications such as metallurgical and material processing, air conditioning and electronic cooling. On the other hand, magnetohydrodynamics studies the behavior of electrically conducting fluids in the presence of an electromagnetic field. The investigation of the MHD problem finds extensive applications such as crystal growth, cooling of nuclear reactor, liquid metal, turbulance control, heat and mass transfers control. Recent years, numerical studies on MHD mixed convection problems have attracted the attention of many researchers and there have been many investigations in the literature on mixed convection flow in lid-driven cavity. Rahman et al. [1] studied on MHD mixed convection flow with Jolue heating parameter in a liddriven cavity with a heated semi-circular source. They used the finite element method for the discretization and observed the results for the control parameters such as the Rayleigh number, Hartmann number and Joule heating parameter. On the other hand, Chatterjee [2] worked on MHD mixed convection in a lid-driven cavity with rectangular and semi-circular heat sources by using a finite volume approach based on the SIMPLEC algorithm. In the present study, we focuses on the numerical solution to hydromagnetic mixed convection flow and heat transfer in a vertical lid-driven square enclosure involving a heated source on the right vertical wall with Joule heating effect. Two different types of sources ( namely, rectangular and semicircular heaters) are considered. Simulations are performed by using the dual reciprocity boundary element method for the physical controlling parameters, such as the Rayleigh number (103 ≤ Ra ≤ 105 ), Hartmann number (0 ≤ Ha ≤ 50), and Joule heating parameter (0 ≤ J ≤ 5), keeping the Reynolds number fixed at Re = 100. The fluid flow and temperature


45

distributions are analyzed through streamlines and isotherms to assess the effect of various combination of aforementioned parameters.

Problem Definition and Governing Equations The schematic presentation of the system is given in Figure 1. This is a two-dimensional square cavity with side length . The left wall of cavity has a movement from bottom to top in its own plane at a constant velocity V0 and is kept at a constant cold temperature Tc . The right wall is the source for heat with a heated semirectangle or semicircle at constant hot temperature Th . The length of the vertical side of rectangular heater and diameter of semicircle is taken as 0.2L. Horizontal walls are adiabatic. The fluid is assumed to be electrically conducting, while walls of the cavity are considered to be insulated. The gravity acts in the vertical direction whereas the magnetic field, with a uniform strength B0 , is effective in the horizontal direction normal to the moving wall. The magnetic Reynolds number is assumed to be small enough, so that the induce magnetic field is neglected. In addition, the effect of Joule heating is considered, but the pressure work and the viscous dissipation are assumed to be negligible. Thus, the governing equations of the steady, laminar flow of a viscous and electrically

V0

V0

g

g

(a)

B0

(b)

B0

Figure 1: Geometry of the problem for (a) rectangular, (b) semicircular heaters conducting fluid subjected to a uniform horizontal magnetic field can be written in the non-dimensional stream function, vorticity, temperature form as ∇2 ψ = −w (1)

∂ψ ∂w ∂ψ ∂w Ra ∂ θ Ha2 ∂ 2 ψ 1 − = ∇2 w + 2 + ∂y ∂x ∂x ∂y Re Re Pr ∂ x Re ∂ x2

(2)

∂ψ ∂θ ∂ψ ∂θ ∂ψ 2 1 (3) − = ∇2 θ + J(− ) ∂y ∂x ∂x ∂y RePr ∂x in these equations, the non dimensional parameters are defined as: The Reynolds number; Re = V0 /ν , the 3 c ) Prandtl number; Pr = ν /α , the Hartmann number; Ha = B0 σ /μ , the Rayleigh number; Ra = gβ (Thv−T α σ B2 V

and J = ρ c p (T0h −T0 c ) , Joule heating parameter. Here ν , α , β , σ , μ , c p are the kinematic viscosity, the thermal diffusivity, the thermal expansion coefficient, electrical conductivity, the viscosity coefficients of the fluid and the specific heat at constant pressure, respectively. The non-dimensional boundary conditions corresponding to the considered problem are On the top and bottom walls: On the left vertical wall: On the right vertical wall except source: At the source:

ψ ψ ψ ψ

= ψx = ψy = 0, ∂∂ θn = 0 = ψy = 0, ψx = −1, θ = 0 = 0, ψx = 0, ψy = 0, ∂∂ θn = 0 = 0, ψx = ψy = 0, θ = 1 .

(4)

On the other hand, the unknown boundary values for the vorticity will be obtained from the stream function equation (1) by using a radial basis function approximation during the application of DRBEM.


Numerical Method The governing equations of MHD mixed convection problem, obtained from the Navier-Stokes equations of fluid dynamics and energy equations under the influence of magnetic field, are discretized by using DRBEM. The aim of DRBEM is to reduce the domain integral of the governing equations to the boundary integrals by 1 1 using the fundamental solution of Laplace equation u∗ = ln( ). Thus, by weighting the equations (1)-(3) 2 π r by u∗ and applying the Divergence theorem, we get the following equations. ci ψi +

Γ

(q∗ ψ − u∗

∂ψ )dΓ = − ∂n

Ω

(−w)u∗ dΩ

(5)

∂w ∂w ∂ψ ∂w ∂ψ ∂ 2ψ Ra ∂ θ ci wi + (q∗ w − u∗ )dΓ = − − )− − Ha2 2 u∗ dΩ Re( ∂n ∂x ∂y ∂y ∂x RePr ∂ x ∂x Γ Ω ∂ θ ∂ θ ∂ ψ ∂ θ ∂ ψ ∂ ψ ci θi + (q∗ θ − u∗ )dΓ = − PrRe − − J(− )2 u∗ dΩ ∂n ∂x ∂y ∂y ∂x ∂x Γ Ω

q∗

(6) (7)

∂ u∗ /∂ n,

where = Γ is the boundary of the domain Ω and the constant ci depends only on the boundary geometry at the point i under consideration. The domain integrals on the right hand side of equations are treated as nonhomogeneities and they are approximated by radial basis functions f j (x, y). These functions are linked to the particular solutions uˆ j through the equation ∇2 uˆ j = f j [3]. The approximations for each domain integrals are given by N+L

N+L

N+L

j=1

j=1

j=1

∑ α j f j (x, y), ∑ β j f j (x, y), ∑ γ j f j (x, y)

for ψ , w, T respectively. The coefficients α j , β j , γ j are undetermined constants, N and L denote the number of boundary and interior nodes respectively. Thus, the equations (5)-(7) result in ci ψi +

ci wi + ci θi +

Γ

(q∗ ψ − u∗

N+L ∂ψ )dΓ = ∑ α j ci uˆ ji + (q∗ uˆ j − u∗ qˆ j )dΓ ∂n Γ j=1

(8)

(q∗ w − u∗

N+L ∂w )dΓ = ∑ β j ci uˆ ji + (q∗ uˆ j − u∗ qˆ j )dΓ ∂n Γ j=1

(9)

Γ

Γ

(q∗ θ − u∗

N+L ∂θ )dΓ = ∑ γ j ci uˆ ji + (q∗ uˆ j − u∗ qˆ j )dΓ ∂n Γ j=1

(10)

which contain only the boundary integrals and qˆ = ∂ uˆ j /∂ n. After the discretization of the boundary by using constant elements, the matrix-vector for of the equations (8)-(10) is obtained as ∂ψ ˆ −1 {−w} = (HUˆ − GQ)F ∂n 2 ∂w ˆ −1 Re( ∂ w ∂ ψ − ∂ w ∂ ψ ) − Ra ∂ θ − Ha2 ∂ ψ ) = (HUˆ − GQ)F (Hw − G ∂n ∂x ∂y ∂y ∂x RePr ∂ x ∂ x2 ∂θ ˆ −1 PrRe[ ∂ θ ∂ ψ − ∂ θ ∂ ψ − J(− ∂ ψ )2 ] . (H θ − G ) = (HUˆ − GQ)F ∂n ∂x ∂y ∂y ∂x ∂x Hψ − G

(11)

(12) (13)

The matrices Uˆ and Qˆ are constructed by taking each of the vectors uˆ j and qˆ j as columns respectively. Each column of F consists of a vector f j containing the values of the function f j at the (N + L) collocation points. The components of matrices H and G are Hi j = ci δi j +

1 2π

Γj

∂ 1 ln( ) dΓ j , ∂n r

Hii = −

N

∑

, Hi j

j=1, j=i

Gi j =

1 2π

Γj

1 ln( ) dΓ j , r

Gii =

A (ln(2/A) + 1) 2π

where r is the distance from node i to element j, A is the length of the element and δi j is the Kronecker delta function.


47

The nonlinear coupled DRBEM equations (11)-(13) are solved iteratively. First, equation (11) is solved for the stream function with a given initial value of vorticity. Once the interior and boundary values of stream function are obtained, the unknown boundary conditions for w are calculated from the stream function equation ∂ F −1 ∂ ψ ∂ F −1 ∂ ψ by the use of radial basis functions as w = −( F + F ). Then, the determined boundary ∂x ∂x ∂y ∂y conditions of vorticity and initial estimate for the temperature are inserted in the vorticity equation (12) and the unknown vorticity values are obtained throughout the domain. The energy equation (13) is solved in a similar manner. This iterative process is terminated when a preassigned tolerance between two successive iterations (e.g. 10−5 ) is reached.

Numerical results The DRBEM analysis for the two-dimensional MHD mixed convection flow under consideration is performed at a fixed Reynolds number of Re = 100 to investigate the effects of the Hartmann number, Rayleigh number and Joule heating parameter on the flow and temperature fields. The boundaries of the cavity are discretized by using an adequate number of constant boundary elements. For example, maximum N = 320 constant boundary elements are used for the case when Ha = 50 and Ra = 105 . ψ ψ T T

Ha = 0

Ha = 20

Ha = 50

(a) (b) Figure 2: The effect of Hartmann number on streamlines and isotherms at Re = 100, Ra = 105 , J = 1 when Ha = 0, 20, 50 (from top to bottom): (a) rectangular heater, (b) semicircular heater. First, the numerical simulations are conducted for the values of Hartmann number Ha = 0, 20, 50 at fixed values of Ra = 105 and J = 1 to see the effect of Ha on the flow field and temperature distribution. Figure 2 and Figure 3 represent the streamlines and isotherms when the left wall moves vertically upward and is heated by (a) rectangular heater and (b) semicircular heater. It is observed that two unequal counter rotating vortices are formed inside the enclosure for streamlines in all cases. These two vortices are basically an outcome of the forced flow as a result of the combination of lid movement and the buoyancy-driven flow due to differential heating. The vortex adjacent to the left moving wall has a clockwise (negative sense of) rotation, since the lid moves vertically from bottom to top. The main vortex close to the right hot wall have a counterclockwise (positive sense of) rotation due to the thermal buoyancy effect. As the Hartmann number increases, the magnetic force increases and as a result the effect due to the thermal buoyancy on the flow and temperature is reduced. The positive main vortex of an almost elliptical shape formed at the center of the cavity, moves slightly upward


ψ

T

ψ

T

J=0

J=3

J=5

(a) (b) Figure 3: The effect of Joule heating parameter on streamlines and isotherms at Re = 100, Ra = 105 , Ha = 10 when J = 0, 3, 5 (from top to bottom): (a) rectangular heater, (b) semicircular heater. ψ ψ T T

Ra = 103

Ra = 104

Ra = 105

(a) (b) Figure 4: The effect of Rayleigh number on streamlines and isotherms at Re = 100, J = 1, Ha = 10 when Ra = 103 , 104 , 105 (from top to bottom): (a) rectangular heater, (b) semicircular heater. and towards to the heater, and becomes circular as Hartmann number increases. As the strength of the magnetic field increases, the strength of the main flow decreases. On the other hand, following the upward motion of the left wall, a negative weak vortex is formed close to the left wall. This secondary vortex expands horizontally with an increase in Ha. A thin thermal boundary layer in isotherms is observed along the vertical left wall


49

and around the heater for low Ha values. The isotherms become more parallel to the vertical walls for higher Ha, indicating that the heat transfer is dominated by conduction. Generally, no significant effect is observed due to the shape of heated sources. However, the heat transfer from the heated surface is more in the case of semicircular heat source when compared to its rectangular counterpart, since the length of the heated portion is larger when the source is semicircular. Secondly, the effect of Joule heating parameter on the fluid flow and temperature distribution is investigated by taking J = 0, 3, 5 when Re = 100, Ra = 105 and Ha = 10 for rectangular heater and semicircular heater (see Figure 3). It is observed that, as the Joule heating parameter increases the strength of the main vortex for streamlines decreases following the formation of a third negative vortex near the right top corner. On the other hand, the cavity heates more with an increase in J and the high temperature is captured near the top wall. Finally, Figure 4 displays the streamlines and isotherms for different values of Rayleigh number at Re = 100, J = 1 and Ha = 10. At a low value of Rayleigh number, Ra = 103 , in the profile of streamlines there is a main cell which is in clockwise direction and the core of this cell has a circular behavior. When the Rayleigh number increases, the main cell starts to extend horizontally taking a more elliptical shape and it shrinks towards the left wall following the formation of a positive secondary vortex infront of the heater. This secondary vortex extends towards the left wall and the flow is dominated by this circulation as Rayleigh number increases. On the other hand, the isotherms accumulated above the heater at Ra = 103 , disperse horizontally along the top wall with an increase in Rayleigh number to Ra = 104 . Further, the isotherms become horizontal and form a boundary layer along the moving left wall for the highest value of Ra = 105 .

Conclusion The mixed convection MHD flow and heat transfer problems in a vertical lid-driven cavity with a semicircular and rectangular heat sources are solved numerically by using the dual reciprocity boundary element method. It is well observed that two counter rotating circulations are formed in streamlines irrespective of the values of physical parameters and type of sources. The positive main vortex of an almost elliptical shape formed at the center of the cavity, moves slightly upward and towards to the heater, and becomes circular as Hartmann number increases. As the strength of the magnetic field increases, the strength of the main flow decreases. On the other hand, following the upward motion of the left wall, a negative weak vortex is formed close to left wall. This secondary vortex expands horizontally with an increase in Ha. A thin thermal boundary layer in isotherms is observed along the vertical left wall and around the heater for low Ha values. The isotherms becomes more parallel to the vertical walls for higher Ha, indicating that the heat transfer is dominated by conduction. On the other hand, the strength of the main vortex of streamlines decreases following the formation a third vortex close to heater at higher Joule heating parameter whereas no significant effect of J on the temperature distribution is observed. It is also observed that, an increase in values of Ra has an opposite effect on the isotherms when compared to the case with high Hartmann numbers. That is, the isotherms becomes horizontal as Ra increases indicating that the heat transfer is convected dominated. All the obtained results are in good agreement with the previous results given in the works [1, 2].

References [1]

Rahman, M.M., Oztop, H.F., Rahim, N.A., Saidur, R., Al-Salem, K.: MHD mixed convection with Joule heating effect in a lid-driven cavity with a heated semi-circular source using the finite element technique 60: 543–560 (2011)

[2]

Chatterjee, D.: MHD mixed convection in a lid-driven cavity including a heated source, Numeric. Heat Transfer, Part A, 64: 235–254 (2013)

[3]

Brebbia, C.A., Partridge, P.W., Wrobel, L.C.: The Dual Reciprocity Boundary Element Method. Computational Mechanics Publications, Southampton, Boston (1992)


Thermoelastic Stress Analysis of 3D Generally Anisotropic Bodies by the Boundary Element Method Y.C. Shiah1, C.L. Tan*2, Y.H. Chen3 1 Department of Aeronautics and Astronautics National Cheng Kung University, Tainan 701, Taiwan, R.O.C. 2

Department of Mechanical & Aerospace Engineering Carleton University, Ottawa, Canada K1S 5B6

3

Department of Computer Science and Information Management Providence University, Taichung, Taiwan, R.O.C. (* Corresponding author; Email: [email protected])

Keywords: Three-dimensional anisotropic thermoelasticity, boundary element method, analytical volume-to-surface integral transformation. Abstract. In the boundary element method (BEM) for stress analysis, it is well known that thermal loads give rise to an additional volume integral in the primary form of the boundary integral equation (BIE). This volume integral needs to be further transformed to surface ones in order to retain the characteristic of the BEM as a boundary solution technique. In this study of the BEM for 3D thermoelasticity in general anisotropy, the fundamental solutions are expressed as Fourier series of an explicit-form Green’s function [1]. In the exact volume-to-surface integral transformation associated with the term for the thermal effects in the BIE, a new kernel function is constructed. All formulations are implemented in an existing BEM code for 3D elastostatic analysis. Some numerical examples are presented to demonstrate the veracity of the formulations and the implementation, where the numerical results are compared with those obtained using the finite element method (FEM). Introduction In the direct formulation of the boundary integral equation (BIE) for the boundary element method (BEM), an additional volume integral arises when thermal loads are involved. If this integral is to be evaluated directly, it would require "cell-discretization" throughout the solution domain, destroying the notion of the BEM as a boundary solution technique. Over the years, several schemes have been proposed to overcome the need for domain discretization. They include the dual reciprocity method [2], the multiply reciprocity method [3], the particular integral approach [4], and the exact transformation method (ETM) [5]. Among these schemes, the ETMis most appealing, not least because it restores the BIE to a true boundary integral equation without requiring analytical and/or numerical approximations, unlike the other schemes. For steady state, isotropic thermoelasticity, this volume integral has been exactly transformed to boundary ones using the ETM for both 2D and 3D cases [5]. It has also been successfully achieved for 2D anisotropic thermoelasticity [6] via a domain mapping technique [7]. However, the extension to the same end in BEM analysis for 3D generally anisotropic thermoelasticity poses a great challenge due to the mathematical complexity of the associated fundamental solutions. The topic of numerical evaluation of the fundamental solution and its derivatives for 3D generally anisotropic elastic bodies has also remained a focus of numerous investigations for several decades (e.g. [8]-[14]). This is due to the fact that the Green’s function presented in these cited works are not in closed, algebraic form. The fully explicit forms of the Green’s function have only been developed in more recent years, see [1], [15]-[18]. They allow their implementation into a BEM code with relatively greater ease, although they are still mathematically very elaborate. Indeed, the present lead authors further improved on the implementation, as well as the computational efforts for the numerical evaluation of the fundamental solution and its derivatives in BEM, by representing these quantities as a double Fourier series [19, 20]. Following this success, they further established that the double Fourier series representation of the Green’s function and its derivatives can provide a very expedient means to facilitate the volume-to-surface integral transformation for the thermoelastic effects in the BIE in 3D general anisotropy [21]. This paper reports on the successful implement of the ETM described in [21] for BEM thermoelastic analysis in 3D


51

general anisotropy. A brief review of the analytical basis will first be presented below, followed by some numerical examples.

BIE for Anisotropic Thermoelasticity For a generally anisotropic elastic solid, the constitutive relationship between the stress ij and the strain ij with temperature change, , is governed by the well-known Duhamel-Neumann relation:

V ij

Cijkl H kl J ij 4, (i, j, k , l 1, 2,3),

(1)

where Cijkl and J ij denote the elastic constants (stiffness coefficients) and thermal modulii of the material, respectively. The stiffness coefficients C to be used for the analysis are arranged in the order according to

T V 11 , V 22 , V 33 , V 23 , V 13 , V 12 ,

T H11 , H 22 , H 33 , 2H 23 , 2H13 , 2H12 .

(2)

Also, the thermal modulii in Eq. (1) are given by

J ij Cijkl D kl ,

(3)

where D kl denotes the coefficients of thermal expansion. As in the usual manner to treat steady-state sequentially coupled thermoelasticity, the resulting elastic field is determined from the temperature distribution corresponding to the boundary conditions prescribed for heat conduction analysis. For this, the thermal field can be solved independently but must be first obtained before solving the elastostatic problem. Under steady-state condition with zero heat source, the anisotropic heat conduction is governed by K ij 4,ij

(4)

0, (i, j 1, 2,3),

in the Cartesian coordinate system, where Kij are the thermal conductivity coefficients. As has been shown in [22], Eq. (4) can be transformed to the canonical form of Laplace equation by a simple coordinate transformation, ˆx T

F xT ,

(5)

where ˆx and x represent the transformed and the original coordinates, respectively; F denotes the transformation matrix with its coefficients defined by the thermal conductivity coefficients [22]. Under the transformed coordinate system, the heat conduction is now governed by 4 ,ii

0, (i 1, 2,3) ,

(6)

where the underscore denotes the mapped coordinate system. More details of this domain-mapping treatment may be found in [7]. Once the temperature field in the body is determined via solving the BIE for the mapped domain, the solution for the corresponding elastic field of the solid body can then proceed. In the direct BEM formulation, the displacements ui and the tractions ti at the source point P and the field point Q on the surface S of an elastic body are related by the following integral equation, * C ij ( P ) u i ( P ) + ³s u i (Q ) Tij ( P, Q ) dS

= ³ t i (Q ) U ij* ( P, Q ) dS s

³J s

ik

n k ( Q ) 4 ( Q ) U ij* ( P, Q ) dS ³ J ik 4 ,k ( q ) U ij* ( P, q ) d : .

(7)

:

The last term in Eq. (7), namely, the volume integral, needs to be transformed analytically to surface integrals, as shown in [21], a review of this will be presented following that for the Fourier series representation of the Green’s function and its derivatives.


As derived in [1], the Green's function of displacements can be expressed in terms of the spherical coordinates (r, T, I) as 1 1 4 ˆ (n ) (8) U* (x)= ¦ qn , 4S r n= 0 where r represents the radial distance between the source and the field point; and the quantities qn , ˆ ( n ) , and are given by -1 ° ° 2 E1 E 2 E 3 qn = ® ° 1 ° 2E E E ¯ 1 2 3

*ˆ ij(n)

ª ° 3 º ptn °½ « Re ®¦ ¾ -G n 2 » p -p ¬« ¯° t=1 t t+1 pt -pt+2 ¿° ¼» ptn- 2 pt 1 pt 2 °½ ° 3 Re ®¦ ¾ p ¯° t=1 t -pt+1 pt -pt+2 ¿°

for n= 0, 1, 2,

(9a)

for n= 3, 4,

* (n) * (n) , (i, j 1,2, 3) , (i1)( j1)(i2)( j2) (i1)( j2)(i2)( j1)

(9b)

N ik =Cijks m j ms ,

(9c)

m=( sin T , cos T , 0).

(9d)

In Eq. (9a), the Stroh's eigenvalues, pi , appear as three pairs of complex conjugates:

pv =D v + i E v , E v >0 , (Q =1, 2, 3) ,

(10)

whose conjugates are denoted by pv . More details of the variables in Eq. (9a)-(9d) can be found in [15]. It is evident that taking spatial derivatives of Eq. (8) to obtain the fundamental solution for tractions, Tij* , and performing the

corresponding numerical computations can be quite intricate. To simplify this, the present lead authors rewrite the Green's function into a Fourier series form, as follows [19, 20],

U* =H(T , I ) / 4S r , H uv (T , I )

a

(11a)

a

¦ ¦O

( m, n ) uv

e

i mT n I

,

m a n a

u, v

1, 2, 3 ,

(11b)

where a is an integer large enough to ensure convergence of the series; Ouv( m,n ) are unknown coefficients determined, from the theory of Fourier series, by

Ouv( m ,n )

1 4S 2

S

S

³ S ³ S H T , I e uv

i mT nI

dT d I .

(12)

From numerical experiments, a=16 is found to be sufficiently large to ensure the series converge even for very high degree of anisotropy. Performing differentiations in the spherical coordinate system, the 1st-order derivatives of U*, denoted by U*', may be expressed as

U uv* ,l

a a ( m , n ) i mT n I ª cos T sin I i n cos I º ° ¦ ¦ Ouv e « » for l 1 °m a n a ¬ i m sin T / sin I ¼ ° 1 ° a a ( m , n ) i mT n I ª sin T sin I i n cos I º . ® ¦ ¦ Ouv e « » for l 2 4S r 2 ° m a n a ¬ i m cos T / sin I ¼ ° a a ° ¦ ¦ Ouv( m ,n ) e i m T n I ª¬ cos I i n sin I º¼ for l 3 °m a n a ¯

(13)


53

and its computation by Eq. (13) is fairly straightforward.

Analytical Transformation of the Volume Integral For brevity, the extra volume integral associated with thermal effects in the integral equation, Eq. (7), is denoted by Vj, i.e.

³ J ik 4 ,k U ij* d : , :

Vj

(14)

where the notations of P and q in the integrand that denote the source and the field points, respectively, have also been omitted for brevity. Since the details of the volume-to-surface integral transformation have been reported in [21], only the key steps are reviewed here. Similar to the transformation process for the corresponding 2D case [6], the volume integral is, first, redefined in the mapped domain by ˆ, ³ ˆ *ik 4,k U ij* d :

Vj

(15)

:

where * ik represents the transformed thermal modulii, defined by

K11

Z '3

FT ,

(16)

and

'

K11 K 22 K12 2 , Z

K11K13' K11K12 K132 K11K12 K13 K23 K232 K112 .

(17)

Following the same treatment as in [6], the volume integral can be exactly transformed to the mapped surface Sˆ , given by

Vj

³

Sˆ

(18)

U *ij ,k .

(19)

* ik ª« 4 Wijk* ,t Wijk* 4 ,t nˆ t 4 U ij* nˆ k º» dSˆ , ¬ ¼

where Wijk* is a new kernel function introduced to satisfy

Wijk* ,tt

Similarly, the U *ij ,k in Eq. (19) is given by Eq. (13) but now defined in the mapped coordinate system. In the mapped spherical coordinate system, the constraint condition specified in Eq. (19) is expressed as * 2 * * w 2Wijk* N ijk (Tˆ, Iˆ ) 2 wWijk 1 w Wijk cot Iˆ wWijk 1 , (20) + + + = wrˆ 2 rˆ wrˆ rˆ 2 wIˆ 2 rˆ 2 wIˆ rˆ 2 rˆ 2 sin 2 Iˆ wTˆ 2 where the explicit expression for N (Tˆ, Iˆ) representing the right-hand-side of Eq.(19), is given by Eq. (13). From

w 2Wijk*

+

ijk

the constraint condition, it is obvious that Wijk* must be independent of rˆ , and thus, Eq. (20) is simplified to

sin 2 Iˆ

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

In a similar manner, Wijk* is also expressed as a double-Fourier series:

N ijk (Tˆ, Iˆ) sin 2 Iˆ .

(21)


a

a

¦ ¦ C

Wijk*

( m , n ) i ( m Tˆ n Iˆ ) ijk

e

,

(22)

m a n a

( m,n ) where C ijk are unknown coefficients to be determined. By substituting Eq. (22) into Eq. (21), one obtains

1 a a ( m , n ) 2 ˆ ˆ ¦ ¦ Cijk n 2m 2 n 2 cos 2Iˆ i n sin 2Iˆ ei ( mT nI ) 2 m a n a

N ijk (Tˆ, Iˆ) sin 2 Iˆ .

(23)

For determining the unknown coefficients, Eq. (23 )is integrated as follows:

1 2S 2 1

S2

S S

a

a

³S ³S ¦ ¦ C

( m ,n ) ijk

m a n a

n

2m 2 n 2 cos 2Iˆ i n sin 2Iˆ ei( mT nI ) e i ( pT qI ) dTˆ dIˆ ˆ

ˆ

ˆ

ˆ

.

S S

³

2

(24)

2 i( pT qI ) ˆ dT dIˆ ³ N ijk (Tˆ, Iˆ) sin Iˆ e ˆ

ˆ

S S

Performing the indicated integrations for integers p and q ranging from -a to +a, and solving a banded system of ( m,n ) simultaneous equations [21], C ijk may be calculated without any difficulty. Direct differentiation of Eq. (22) under the mapped spherical coordinates, also yields

Wijk* ,t

a a ( m ,n ) i( mTˆ nIˆ ) i n cos Tˆ cos Iˆ i m sin Tˆ / sin Iˆ ° ¦ ¦ Cijk e °m a n a 1 ° a a ( m ,n ) i( mTˆ nIˆ ) i n sin Tˆ cos Iˆ i m cos Tˆ / sin Iˆ ® ¦ ¦ Cijk e rˆ ° m a n a ° a a ( m ,n ) i( mTˆ nIˆ ) i n sin Iˆ ° ¦ ¦ C ijk e ¯m a n a

(for t 1)

(for t

2) .

(for t

3)

*

With the explicit expressions for Wijk and

(25)

Wijk* ,t now determined, the volume integral can calculated in a

straightforward manner using Eq. (18). Numerical Examples All the formulations presented above have been implemented in a BEM code for elastostatic analysis. To demonstrate the veracity of the formulations and implementation, a few examples are presented here. The first example considers an isotropic cube with side length L of 2 units. This is shown in Fig. 1 with the boundary element mesh employed. For the purpose of verification, the analysis was first carried out through the generally anisotropic algorithm with the isotropic properties and the results are compared with the corresponding ones obtained from the conventional BEM isotropic analysis. The numerical values for the elastic and thermal properties of the material were, for simplicity, taken to be as follows: Young's modulus E=1000, Poisson’s ratio = 0.3, the thermal expansion coefficient D = 10-3, and the thermal conductivity K = 1. For the boundary conditions, the six surfaces are restrained from displacements in their respective normal directions while the domain is subjected to a uniform heat flow with a temperature difference '4 = 100o between the bottom and the top surfaces; all side surfaces are thermally insulated. Although advantage can be taken of symmetry for this isotropic problem, the full model of the physical problem was used, with a total of 96 quadratic isoparametric elements. Table 1 shows some sample results, comparing the numerical values of the resultant displacements ( uo

u12 u22 u32 ) and the non-zero principal

stresses at boundary nodes along AB obtained using the anisotropic and the isotropic algorithms, respectively. As can be seen from the table, the results are in excellent agreement.


55

Table1: Comparison of results for isotropic cube: anisotropic and isotropic BEM algorithms -Example1

00

D

x3

B

x3/L

V1

x1 C

-0.125

0

0.125

0.25

0.375

Aniso. -1.2500 -1.2500 -1.2500 -1.2500 -1.0714 -0.8928 -0.7143

ED '4

A L

1000

-0.25

Aniso. 0.1016 0.1741 0.2176 0.2321 0.2176 0.1741 0.1016 u0 LD '4 Iso. 0.1016 0.1741 0.2176 0.2321 0.2176 0.1741 0.1016

x2

L

-0.375

Aniso. -1.7857 -1.6072 -1.4286 -1.2500 -1.0714 -0.8928 -0.7143

V2 ED '4

L=2

Iso. -1.7857 -1.6072 -1.4286 -1.2500 -1.0714 -0.8928 -0.7143 Aniso. -1.7857 -1.6072 -1.4286 -1.2500 -1.2500 -1.2500 -1.2500

V3

Fig.1: Problem definition and the BEM mesh - Example 1.

Iso. -1.2500 -1.2500 -1.2500 -1.2500 -1.0714 -0.8928 -0.7143

ED '4

Iso. -1.7857 -1.6072 -1.4286 -1.2500 -1.2500 -1.2500 -1.2500

For further verification of the anisotropic formulation, the material properties of alumina Al2O3 crystal were chosen for analysis of the same cube. It has the following stiffness constants [22]: C11= 465 GPa, C33= 563 GPa, C44= 233 GPa, C12= 124 GPa, C13= 117 GPa, C14= 101 GPa.

(26)

Also, the following thermal properties were used in the analysis: * K11 18 (W / m 0 C),

D

* 11

* K 22

8.1u 10 (1 / C), D 6

0

* 22

10 (W / m 0 C),

* K 33

5.4 u 10 (1 / C), D 6

0

* 33

25 (W / m 0 C), 9.2 u 10 6 (1 / 0 C).

(27)

To treat the problem as a generally anisotropic case, the principal material axes were rotated with respect to the x1, x3, x1-axis by 60o, -30o, -45o counterclockwise, respectively, in succession. This results in the following thermal constants in the global Cartesian coordinate system:

K

§ 18.1250 3.1250 1.5309 · ¨ ¸, ¨ 3.1250 12.1250 3.9804 ¸ ¨ 1.5309 3.9804 22.7500 ¸ © ¹

§ 7.9844 8.9688 1.3013 · ¨ ¸ 6 , ¨ 8.9688 5.9594 9.5683 ¸ u 10 ¨ 1.3013 9.5683 8.7563 ¸ © ¹

(28)

Similarly, the stiffness coefficients with respect to the global Cartesian system becomes

§ 590.7459 43.5588 83.0704 25.2795 84.4116 20.2070 · ¨ ¸ ¨ 43.5588 590.7459 83.0704 -84.4116 -25.2795 20.2070 ¸ ¨ 83.0704 83.0704 608.1093 31.2692 -31.2692 -51.7889 ¸ . C ¨ ¸ ¨ 25.2795 -84.4116 31.2692 190.3829 -60.4764 -3.9995 ¸ ¨ 84.4116 -25.2795 -31.2692 -60.4764 190.3829 3.9995 ¸ ¨¨ ¸ 3.9995 107.4338 ¸¹ © 20.2070 20.2070 -51.7889 -3.9995

(29)

It can be seen that the respective material property matrices are now fully populated, denoting full, general anisotropy with respect to the global Cartesian coordinate system. Due to the high degree of anisotropy thereby generated for this test, a relatively refined mesh (384 elements) as shown in Fig. 2(a) was employed for the BEM


analysis. For verification of the results, the problem was also analyzed using ANSYS, the commercial FEM software whereby 8000 SOLIDS226 elements were used (Fig. 2(b)). Shown in Fig. 3 are the computed variations of the resultant displacement of the points along AB and CD . It can be seen that both analyses are in excellent agreement.

u0/LD11'4

The calculated normalized principal stresses along AB and CD are plotted in Fig.4. Again, the BEM results are in excellent agreement with those from the FEM analysis.

ANSYSBEM AB CD

(a)

(b)

x3/L

Fig.3: Variations of the normalized displacements along

and

.

V1/C11D11'4

V2/C11D11'4

Fig.2: Mesh employed for analysis by (a) BEM, (b) ANSYS.

x3/L

V3/C11D11'4

x3/L

ANSYS BEM AB CD

x3/L

Fig. 4: Distribution of the normalized principal stresses along AB and CD .


57

Figure 5(a) shows the next example treated. It is a hollow cylinder made of the same alumina crystal with material properties governed by Eqs. (28) and (29) with respect to the global Cartesian system. It has a height h = 8 and the inner and outer radii are ri =1 and ro = 2, respectively. The inner and outer circumferential surfaces are prescribed with temperatures 1000 and 00, respectively, while the top and bottom surfaces are insulated. For the elastic boundary conditions, the outer surface is fully constrained and all other surfaces are free to displace in any direction. For the BEM model, 280 quadratic elements were employed (Fig. 5(a)); the ANSYS analysis employed11400 SOLIDS226elements (Fig. 5(b)). Also, the displacements of a few internal points at r= (ri+ro)/2 on the x1-x2 plane were calculated. The computed resultant displacements for these internal points and the boundary nodes of the inner surface on the x1-x2 plane are shown plotted in Fig. 6. Figure 7 shows the calculated hoop stresses of the boundary nodes on the x1-x2 plane. Once again, excellent agreements of all the corresponding BEM and FEM results were obtained.

ro=2 ri=1

u0/LD11'4

x3

x2

h=8

x1

(a)

ANSYS BEM r=1 r=1.5

T(Deg.) T (Deg.)

(b)

Fig. 5: Mesh employed for analysis by (a) BEM, and (b) ANSYS.

Fig. 6: Distribution of the normalized resultant displacement of the surface nodes on the x1-x2 plane.

Conclusions

VTT/C11D11'4

The exact transformation method that transforms the additional volume integral associated with thermal effects in the BIE to surface integrals has been achieved and implemented for BEM in 3D general anisotropic thermoelasticity. This has never been successfully undertaken in the literature because of the mathematical complexity of the Green’s function. The approach followed the same key steps developed by the lead authors for the 2D case previously, together with the use of the double Fourier representation of the 3D Green’s function. The transformed surface integrals have been successfully implemented into an existing BEM code. This has been illustrated by some examples in which numerical results have been compared with those obtained using ANSYS FEM analysis using very refined meshes, and excellent agreement between them have been obtained.

ANSYS BEM r=1 r=2

T (Deg.)

Fig. 7: Distribution of the normalized hoop stress of the surface nodes on the x1-x2 plane.


References [1] Ting, T.C.T.; Lee. V.G., 1997, Q. J. Mech. Appl. Math. 50, 407-426. [2]

Nardini, D.; Brebbia, C. A., 1982, A new approach to free vibration analysis using boundary elements, Boundary Element Methods in Engineering, Computational Mechanics Publications, Southampton.

[3]

Nowak, A.J.; Brebbia, C.A., 1989, Engng. Analy. Boundary Elem.6(3): 164-167.

[4]

Deb, A.; Banerjee, P.K., 1990, Commun. Appl. Num. Meth. 6:111-119.

[5]

Rizzo, F.J.; Shippy D.J.,1977,Int. J. Numerical Methods Engng. 11:1753-1768.

[6]

Shiah, Y.C.; Tan, C.L., 1999, Comput. Mech. 23: 87-96.

[7]

Shiah, Y.C.; Tan, C.L.,1997, Engng. Analy. Boundary Elem. 20: 347-351.

[8]

Vogel, S.M.; Rizzo, F.J., 1973, J. Elasticity 3: 203-216.

[9]

Wilson, R.B.; Cruse, T.A., 1978, Int. J. Numer. Methods Engng. 12: 1383-1397.

[10] Sales, M.A.; Gray, L.J., 1998, Comp. & Struct., 69, 247-254. [11] Pan, E.; Yuan, F.G.,2000,Int. J. Numer. Methods Engng. 48, 211-237. [12] Tonon, F.; Pan, E.; Amadei, B., 2001,Comp. & Struct. 79, 469-482. [13] Phan, P.V.; Gray, L.J.; Kaplan, T., 2004,Methods Engng. 20, 335-341. [14] Wang, C.Y.; Denda, M., 2007. Int. J. Solids Struct. 44, 7073-7091. [15] Shiah, Y.C.; Tan, C.L.; Lee, V.G., 2008, CMES-Comp. Modeling Engng. Sc. 34, 205-226. [16] Tan, C.L.; Shiah, Y.C.; Lin, C.W., 2009, CMES-Comp. Modeling Engng. Sc.41, 195-214. [17] Lee,V.G.,2003,Mech. Res. Comm. 30, 241–249. [18] Lee,V.G., 2009, Int. J. Solids Struct. 46, 1471-1479. [19] Tan, C.L., Shiah, Y.C., Wang, C.Y., Int. J. Solids Struct. 50: 2701-2711, 2013. [20] Shiah, Y.C.; Tan, C.L.; Wang, C.Y., 2012,Engng. Analy. Boundary Elem. 36: 1746-1755. [21] Shiah, Y.C.; Tan, C.L., 2014,CMES-Comp. Modeling Engng. Sc.102(6), 425-447. [22] Shiah, Y.C.; Tan, C.L., 2004, Engng. Analy. Boundary Elem., 28, 43-52.


Electrostatic Field Analysis in Anisotropic Conductive Media Using a Voxel-based Static Method of Moments Shoji Hamada 1 1

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

Keywords: voxel-based analysis, static method of moments, indirect boundary element method, electrostatic field, biological tissue, anisotropic conductivity, diffusion tensor imaging

Abstract. A voxel-based static method of moments (MoM) is proposed in order to analyze electrostatic fields in biological tissues with anisotropic conductivities, such as nerve fiber. This method represents a cubic voxel of an anisotropic medium with a sextet of square surface charge elements that cover the voxel surface, and the sum of the six charges is constrained to be zero in order to properly address the anisotropy. Moreover, it represents a voxel cluster of an isotropic medium with elements that cover the cluster. Each element has a uniform surface charge density, which is governed by a boundary equation that is described using the quantities on one side of the boundary. Because this MoM does not use volume integrals, it is considered to be a kind of indirect boundary element method (IBEM) and can be concurrently used with the voxel-based IBEM. After confirming the validity of the MoM, we calculate the electrostatic field in a simplified human head model that includes anisotropic conductive tissues and is constructed using diffusion tensor imaging data. The voxel side length of the model is set at 2.5 mm and a homogeneous 50-Hz magnetic field is applied. It is demonstrated that the proposed voxel-based MoM is applicable to field analyses in voxel models that are composed of isotropic and anisotropic tissues. Introduction Numerical electromagnetic field analyses that are based on voxel models are widely conducted using multiple methods, such as the finite difference method and the finite element method (FEM) [1-3]. Advantages of the voxel-based analysis include the facile production of realistic models from threedimensional image data and a simple data structure that is suitable for storage, handling, and visualization. One of the typical applications is field analysis in an anatomical human model constructed using magnetic resonance imaging (MRI) data [3, 4]. Some biological tissues, such as nerve fiber, muscle, and bone, have anisotropic conductivities, and their isotropic approximations can produce field inaccuracies. By considering anisotropy, for example, FEM analyses were conducted using head models that were based on diffusion tensor imaging (DTI) data [2, 5]. DTI is a type of MRI and visualizes the apparent diffusion tensor Da of water molecules inside the tissues. Conductivity tensors are then approximately estimated using Da . On the other hand, a voxel-based indirect boundary element method (IBEM), which used the Laplace kernel fast multipole method (FMM) to handle large-scale problems, was developed in [6-8]. This IBEM analyzed the electrostatic fields in cubic-voxel models that described the conductivities of biological tissues. For example, it sufficiently managed a voxel model composed of 2.2 billion voxels and 61 million boundary elements using a personal computer. However, this method could not address anisotropic conductivities. In this study, in order to enhance the versatility of the voxel-based IBEM, the static method of moments (MoM) [9-11] is modified to a voxel-based static MoM, which is designed to address anisotropic conductivities and to be concurrently applicable with the voxel-based IBEM. This MoM does not require volume integrals; thus, it is considered to be a kind of IBEM [11]. After validating the MoM, we calculate the electrostatic field in a simplified human head model including anisotropic conductive tissues, which is produced using DTI data. It is demonstrated that the proposed MoM and the IBEM are concurrently applicable to field analyses in voxel models that are composed of isotropic and anisotropic tissues. Voxel-based Indirect BEM and Voxel-based Static Method of Moments

59


Magnetically Induced Electrostatic Field in Biological Samples. The basic equations of magnetically induced low-frequency faint currents in a biological sample were provided by, for example, Dawson [1]. Here, both the displacement current and secondary magnetic field are assumed to be negligibly small. When an external magnetic flux density B0 and a vector potential A0 , which satisfy B0 u A0 , are applied, the magnetically induced electric field E and the electric current density J satisfy the following equations: E jZA0 I , J VE , 2I 0 , (1) where j, Z , V , and I are the imaginary unit, angular frequency, conductivity, and scalar potential, respectively. The following boundary equation (BEQ) holds on a boundary with a unit normal vector n : V E x n V E x n . (2) Subscripts r indicate the positive and negative sides with respect to n . This BEQ is equivalent to the following pair of BEQs, each of which is described using quantities defined on one side of the boundary: s V 0 V r 1 E x n , s V 0 V r 1 E x n , (3)

where s, V r , and V 0 are surface charge density, specific conductivity, and an arbitrary scalar baseline value of conductivity, respectively. These quantities are analogous to those describing magnetization or dielectric polarization. Anisotropic conductivity is represented by a second-order tensor, V V 0V r . Voxel-based Indirect Boundary Element Method. The voxel-based IBEM regards a square boundary sandwiched by two voxels having different conductivities as a square boundary element [6]. The element has uniform surface charge density and numerically simulates I in eq (1). Here, n is +i, +j, or +k, which are parallel to the x, y, and z-axes, respectively. Eq (2) produces the following BEQ for the IBEM: V 0V r En V 0V r En , En r E r x ndS " 2 , (4)

³

S

where " is the voxel side length. Eq (3) provides the following BEQs, which are equivalent to eq (4):

s

V 0 V r 1 En ,

s

V 0 V r 1 En- ,

³ s dS

sr

S

"2 .

r

(5)

Let us approximate the sr by uniform charge densities r of surface charge elements as follows:

V 0 V r 1 En ,

V 0 V r 1 En- .

(6)

The and are governed by these BEQs, each of which is described using quantities defined on only one side of the elements, and we refer to these elements as one-sided elements (see Fig. 1(a)). The surface charge density of the element of the IBEM equals the sum of and when they are in the same position. Now, the elements of the IBEM can also be allocated using the following procedure (see Fig. 2): [i] The one-sided elements are allocated to cover each isotropic medium that is represented as a voxel cluster. The non-conductive open region outside the tissues is also classified as an isotropic voxel cluster. [ii] A pair of one-sided elements that are in the same position is replaced with an element of the IBEM. y z z

En

+

En+ 4

2

3

3

En2

En3

En3

En4

+x, +y, or +z

(a)

En1 En5 5

1

4

x

En4

En1 En6 6

(b)

1

5

x

En5

En2 En6 6

2

y

Fig. 1: (a) Pair of one-sided elements and (b) sextet of one-sided elements for an anisotropic voxel. Voxel-based Static Method of Moments. We propose a voxel-based static MoM, which represents a cubic voxel of an anisotropic medium using a sextet of one-sided elements that are defined on the inner surfaces of the voxel (see Figs. 1(b) and 2). It is assumed that the surface charge densities, 1 through 6 ,

in Fig. 1(b) are approximately evaluated by the following equations using En1 through En 6 , similar to the way r are evaluated by Enr in eq (6):


61

, § E n1 · ¨ ¸ § ACAT Y Z ACAT Y Z · ¨ E n 2 ¸ ACAT 11 x11 V rxx 1 ¨ ¸ 2 6 2 6 ¸ ¨ E n 3 ¸ , ACAT x22 V ryy 1 , V0¨ 22 T T ¨ ACA Y Z ACA Y Z ¸ ¨ E n 4 ¸ ¸ ¨ ¸ ACAT ¨ x V rzz 1 33 33 2 6 2 6 ¹ ¨ En5 ¸ © Ë ¸ © n6 ¹ 0 0 · 0· § u x v x wx · § V ru 1 § x11 0 § x11 x22 x33 · ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ 0 ¸ , Y ¨ 0 x22 0 ¸ , Z ¨ x11 x22 x33 ¸ , (7) A ¨ u y v y wy ¸ , C ¨ 0 V rv 1 ü v w ¸ ¨ 0 ¨0 ¨x ¸ 0 V rw 1¸¹ 0 x33 ¸¹ z z ¹ © z © © © 11 x22 x33 ¹ T where A denotes the transpose of A , and V rxx , V rxy , V rxz , V ryy , V ryz , and V rzz are components of tensor V r in the xyz-coordinate system. Vectors u (ux , uy , uz ) , v (vx , vy , vz ) , and w (wx , wy , wz ) are the unit eigenvectors of V r , and V ru , V rv , and V rw are scalar specific conductivities in the corresponding directions. The terms of ACAT and Y describe, for example, that 1 depends on En1 , ( En 2 + En 5 )/2, and ( En3 + En6 )/2. The terms of Z constrain

¦

6

i 1

i to be zero, and those of Y and Z vanish when " tends to zero. These

properties allow eq (7) to address anisotropy. When the following procedures are added to those listed in the previous section, concurrent use of the voxel-based MoM and IBEM is possible. This is illustrated in Fig. 2. [i] The sextet of elements is allocated to every anisotropic voxel. [ii] Sextet charges cannot be replaced with elements of the IBEM.

0

0 0 0

0 0

0

2 2 1 2 2 1

1 0

0 0

1 1 1

1 0

0

0 1 1

0 0

0

0 0 0

0 0

1 0

[i]

IDNO0:air(isotropicmedium) IDNO1:isotropicmedium IDNO2:anisotropicmedium

[ii]

:sextetofelements :onesidedelement

:elementofIBEM

Fig. 2: Two-dimensional schematic explanation of the arrangement of elements for a voxel model involving isotropic and anisotropic media. Governing Simultaneous Linear Equations. The elements of the IBEM, the one-sided elements, and the sextet charge elements are collectively rearranged in a new order and assigned a serial number between 1 and N. The charge density for the ith element is denoted xi . Considering Coulomb’s law and the

externally applied field, the Enr on the ith element, Enr [i ] , is represented by the following integral equation:

³

jZA0 x ndS

N

°

x j ri r j

½ "2 ° x , (8) dS ¾ x ndS r Si Si Sj 2 V 0 k {i } k j 1, j z{i } °¿ ¯ 4SV 0 ri r j where ri is the position vector on the ith element, {i} denotes i if no other element is in the location of the ith element, and {i} denotes the set containing i and the other element’s number if they are in the same position. The governing linear equations, Cx b , are derived using eqs (4, 6–8), where C , x , and b are an En r [ i ]

¦ ³ ®³ °

3

¦


N u N coefficient matrix, N u 1 unknown vector, and N u 1 constant vector, respectively. Because this formulation does not involve volume integrals, the MoM is considered to be a kind of IBEM. After solving for x , fields E and J are calculated for all voxel centers using an integral equation similar to eq (8). Simplified Human Head Model Constructed Using Diffusion Tensor Images

In order to analyze electrostatic fields in an anisotropic biological sample, a simplified model of a human head was generated using DTI data [2, 5]. DTI is a kind of MRI that produces a set of MR images of in vivo biological tissues, from which apparent diffusion tensor Da of water molecules inside the tissues can be evaluated for every voxel location. Rullmann presented one of the simplest methods for evaluating conductivity tensor V from Da [2]. This method assumes that V is proportional to Da and that the magnitude ratio of V to Da equals the ratio of the isotropic conductivity value found in literature to the geometric mean of the three eigenvalues of Da . We obtained Da data using DTI Studio and DTI sample data of a human head [12]. The dimensions of the original Da data were 256 u 256 u 58 and were resized to 88 u 88 u 60. The cubic voxel side length was set at 2.5 mm. On the basis of T2-weighted MR images, the volume was classified into five regions (see Fig. 3): air, outside the skull, skull, inside the skull, and eyeballs. The top and bottom slices of the voxel model were considered to be “air” slices. The number of voxels and the average value of the isotropic conductivities found in literature are summarized in Table 1. The average was calculated using the anatomical human voxel model found in [3, 4], in which the tissue conductivities ranged between VL = 0.02 Sm-1 and VU = 2.0 Sm-1, inclusive. The VL and VU were the respective values of the cortical bone and cerebrospinal fluid (CSF). The conductivity tensor was approximately evaluated using a method that is similar to Rullmann’s method [2], however, the gray matter, white matter, and CSF regions were not separated in the “inside the skull” region. We adjusted V 0V ru , V 0V rv , and V 0V rw to the upper limit value VU, when this limit was exceeded because the conductivity of CSF would not be exceeded. We also adjusted these values to a lower limit value of 0.002 Sm-1, which was one tenth of VL, when they were less than the limit or negative. The “inside the skull” region adopted the evaluated V tensor and the other regions adopted the isotropic average values of conductivities that are listed in Table1.

ID No. 0 1 2 3 4 Fig. 3: Simplified head model.

Region Air Outside the skull Skull Inside the skull Eyeballs

No. of voxels 287,967

Average conductivity / Sm-1 0.00

33,123

0.21

52,881

0.02

89,903

0.47

766

1.45

Table 1: The numbers of voxels and average conductivities.

Computing Environment and Settings for Calculation

A personal computer running 64-bit Microsoft Windows 7 was used for the calculations. It had an Intel Core i7-4960X CPU (6 CPU cores, 3.6 GHz) with 64 GiB of RAM. The source code was compiled using Intel Visual FORTRAN Compiler XE Ver. 14. The applied homogeneous magnetic field, B0 ( B0 x , Boy , Boz ) , was 50-Hz AC. The vector potential was defined to be A0

0.5( B0 y z B0 z y )i 0.5( B0 z x B0 x z ) j 0.5( B0 x y B0 y x)k , and V 0 was set to 1.0 Sm-1.

The Coulomb interaction between the surface charge elements was calculated using three-dimensional fast Fourier transformation (FFT) routines in the Intel Math Kernel Library (MKL) instead of the FMM. Because the number of voxels and unknowns used in this study, which did not exceed 176,673 and 563,822,


63

respectively, were not substantially large, the FMM was not required. Also, the iterative solver used was the GBi-CGSTAB(s, L) [13], where s = 3 and L = 2. Results Fields in a Rectangular Solid Conductor. In order to validate the proposed MoM, we calculated the electrostatic field in a homogeneous and anisotropic conductive rectangular solid. The dimensions of the conductor were -0.1 m d x d 0.1 m, -0.07 m d y d 0.07 m, and -0.25 m d z d 0.25 m. The voxel side length was set at 0.01 m. Therefore, this conductor is represented as a voxel model that does not include shape representation error, which is sometimes called staircasing error. Thus, we can confirm the validity of the MoM without incurring this type of error. First, because an analytical solution of the induced field was available, the following settings were adopted: u = i , V ru = V rx = 8, v = j , V rv = V ry = 4, w = k , V rw = V rz = 2, B0 = B0 k , and B0 z = 1.0 P T.

The analytical solution is provided by the following equations [14]: n § ( 2n 1)Sy V · § (2n 1)Sb V · · 8ZB0 z a V y f §¨ 1 § (2n 1)Sx · x ¸ x ¸¸ Ex cos¨ / cosh¨ , (9) ¸ sinh ¨ 2 ¨ ¸ ¨ S V x n 0 ¨ 2n 1 2 2a 2 a V 2a V y ¸¹ ¸ © ¹ y ¹ © © © ¹ n § (2n 1)Sx V y · § ·· 8ZB0 z b V x f §¨ 1 § (2n 1)Sy · ¸ / cosh ¨ (2n 1)Sa V y ¸ ¸ . (10) Ey cos¨ ¸ sinh ¨ 2 ¨ ¸ ¨ ¨ 2 V y n 0 2n 1 2b 2b Vx ¹ 2b V x ¸¹ ¸ © ¹ S © © © ¹ Here, 2a and 2b are the side lengths of the rectangular solid in the x and y directions, respectively. Furthermore, we also calculated the induced field using a direct BEM, which incorporated triangular patches. The fundamental solutions and the boundary integral equation are as follows [15]:

¦

¦

1

2 2 2 I * ( r , ri ) §¨ 4S V rvV rw ui u V rwV ru vi v V ruV rv wi w ·¸ ,

©

(11)

¹

q* ( r , ri ) V ru

wI (r , ri ) wI (r , ri ) wI (r , ri ) u x n V rv v x n V rw wxn, wu wv ww *

*

*

(12)

c(ri ) I (ri ) I * (r , ri ) q(r )dS q* (r , ri )I (r )dS , (13) S S 4S where r uu vv ww and ri ui u vi v wi w are position vectors on the boundary, n is the normal vector at r , and c(ri ) is a constant that depends on the boundary shape at ri . The number of conductive voxels and the number of unknowns were 14,000 and 87,960, respectively. The direct BEM utilized 21,674 unknowns. Figs. 4(a) and 4(b) display three kinds of electrostatic fields on a cross-section for z = -0.005 m. However, the differences between them are difficult to distinguish in Fig. 4(a). Although we see a slight decline in the accuracy of the MoM in the vicinity of the corners of the rectangular solid in Fig. 4(b), the calculated fields generally agreed.

³

Ex / 10 Vm

-1

³

-6 -1

0

-6

-0.05 -0.1

0

x/m

0.1

20

0

0

-20 0 20

Ey / 10 Vm

y/m

0.05

20

-0.065 m 0) and each particle experiences a rigid-body migration. More precisely, the particle (n) (n) Pn , with attached point On and closed contour Cn , translates at the velocity U(n) = U1 e1 + U2 e2 (n) (n) and rotates at the angular velocity Ω = Ω e3 . A MHD two-dimensional liquid flow with velocity u and pressure p takes place in the liquid domain D which, assuming vanishing Reynolds number and magnetic Reynolds number (see the introduction), satisfies the well-posed following boundary-value


69

Stokes problem μ∇2 u + σB 2 (u ∧ e1 ) ∧ e1 = ∇p and ∇.u = 0 in D, (u, p) → (0, 0) as r = |OM| → ∞,

u = ud on C = C1 ∪ C2

(2) (3)

where the provided velocity ud is here given by the relations ud = U(n) + Ω(n) e3 ∧ On M on Cn (n=1,2).

(4)

The flow (u, p), with stress tensor σ, exerts on the cluster contour C = C1 ∪ C2 a traction f = σ.n where n designates the unit normal on C pointing into the liquid. The challenging issue addressed in the present study is to propose a relevant approach to accurately obtain, once ud is precsribed, the solution (u, p) to the Stokes problem (2)-(3) and also the resulting traction f on the contour C. Fundamental flow and resulting collection of exact solutions As shown in [7], it is possible to determined in closed form the so-called fundamental flow (u, p) produced by a concentrated point force of strength s = s1 e1 + s2 e2 located at the point x0 and such that μ∇2 u + σB 2 (u ∧ e1 ) ∧ e1 − ∇p = −δ(x − x0 )s and ∇.u = 0 for x = x0

(5)

(u, p) → (0, 0) as |x − x0 | → ∞

(6)

where δ is the two-dimensional delta pseudo-function. Adopting henceforth the usual tensor summation convention over repeated indices (all belonging for the present two-dimensional problem to {1, 2}), this flow has a stress tensor σ(x) = σij (x, x0 )ei ⊗ ej and, by linearity, one has ui (x) =

1 1 1 Gij (x, x0 )sj , p(x) = Pj (x, x0 )sj , σik (x, x0 ) = Tijk (x, x0 )sj 4πμ 4π 4π

(7)

with above functions Pj (x, x0 )j , Gij (x, x0 ) and Tijk (x, x0 ) obtained (using a method analogous to the one employed in [8] for another three-dimensional problem) and expressed in [7] in terms of the usual modified Bessel functions K0 or K1 (see also [9]) of order zero and one, respectively. Clearly, one thus obtains a collection of exact flows (u, p) solutions to (2)-(3) by putting a unit force s at any point x0 located inside the body P1 or P2 and taking for ud = u on the cluster contour C. Proposed boundary approach This section presents a boundary formulation which is efficient to solve the challenging boundaryvalue problem (2)-(3) for a general 2D MHD flow (u, p). This procedure appeals to the previous fundamental tensors G and T (with above Cartesian components Gij and Tijk ) for the velocity field u and both to the previous vector P = Pj ej and another function Q for the pressure field p. Key velocity and pressure boundary integral representations It is possible to prove that for a flow (u, p) solution to (2)-(3) key integral representations exist for the velocity and pressure fields u and p. Setting vi = v.ei for a vector v and extending the material given in [10] for a usual 2D Stokes flow free from body force, one arrives at (curtailing the details) 1 1 Gji (x0 , y)fi (y)dl(y) + ui (y)Tijk (y, x0 )nk (y)dl(y) for x0 in D, (8) uj (x0 ) = − 4πμ C 4π C 1 Pi (x0 , y)fi (y)dl(y) p(x0 ) = − 4π C ∂Pi ∂Pk μ ui (y)nk (y) δik Q(x0 , y) − (y, x0 ) − (y, x0 ) dl(y) for x0 in D (9) + 4π C ∂yk ∂yi


with the following function (not derived in [7] but not established here for conciseness) 1 rˆ rˆ x x ˆ1 x ˆ1 ˆ1 , d = ( μ/σ)/B Q(x, x0 ) = 2 cosh( )K0 ( ) + sinh( )K1 ( ) d 2d 2d 2d 2d rˆ

(10)

ˆ .ei and also rˆ = |ˆ ˆ = x − x0 , x î = x x|. where x Associated suitable coupled boundary-integral equations Inspecting the previous integral representations (8)-(9) reveals that in getting the flow in the entire liquid domain D it is actually sufficient to know the velocity u and the traction f = σ.n on the cluster contour C = C1 ∪ C2 . For the problem (2)-(3) we know u = ud on C and thus are let with the determination of f there. This task is handled by letting in (8) the point x0 tend onto the contour C. Curtailing the regularization treatment, one then arrives at at the following coupled Fredholm boundary-integral equations of the first kind (here j = 1, 2) σB 2 1 Gji (x, y)fi (y)dl(y) = uj (x) + [(u(x) ∧ e1 ) ∧ e1 ].y(1) ni (y)Gji x, y)fi (y)dl(y) − 4πμ C 4πμ C1 1 1 ui (y)Tijk (y, x)nk (y)dl(y) − [ui (y) − ui (x)]Tijk (y, x)nk (y)dl(y) for x on C1 , (11) − 4π C2 4π C1 σB 2 1 Gji (x, y)fi (y)dl(y) = uj (x) + [(u(x) ∧ e1 ) ∧ e1 ].y(2) ni (y)Gji x, y)fi (y)dl(y) − 4πμ C 4πμ C2 1 1 ui (y)Tijk (y, x)nk (y)dl(y) − [ui (y) − ui (x)]Tijk (y, x)nk (y)dl(y) for x on C2 (12) − 4π C1 4π C2 where we set y(n) = y − OOn . In summary, we solve in practice the governing problem (2)-(3), for a 2D MHD Stokes flow about the cluster, from its prescribed value u = ud on C by first inverting the boundary-integral equations (11)-(12) for the unknown surface traction f = σ.n on C and then, if needed, subsequently compute the velocity u and the pressure p in the liquid domain by appealing to the integral representations (8)-(9). Numerical implementation and benchmark tests Even for one disk it has not been possible to derive a simple theoretical treatment (see [3, 6]). Therefore, a numerical treatment is needed to solve, using the advocated boundary formulation, the more challenging case of two interacting arbitrary-shaped bodies P1 and P2 experiencing prescribed rigid-body motions (U(1) , Ω(1) e3 ) and (U(2) , Ω(2) e3 ), respectively. This section first briefly describes the achieved numerical implementation and then reports a few numerical benchmark tests of the method against the previously-exhibited exact solutions. Implemented boundary element method Each encountered boundary-integral equation (11), (12) is numerically discretized using high-order boundary elements (with quadratic interpolation) on the cluster contour. More precisely, we put Nn nodal points on each contour Cn and enforce at each node (and for i = 1, 2) the boundary-integral equations (11)-(12). Setting N = N1 +N2 one then ends up for the 2N unknown Cartesian components fi = f .ei at those nodes with a linear system. This linear system has a (2N ) × (2N ) fully-populated square matrix and is here inverted by the iterative linear system subroutine solver dsgesv from open source Lapack Library. Of course, the number of nodes required on each particle to reach accurate results depends on both the cluster geometry (particles shapes and also locations) but also on the Hartmann number M = a/d. For instance, and not surprisingly, more nodal points are necessary when M increases because close each body the flow structure has typical length scale d = a/M. Comparisons for a general 2-disk cluster


71

One basic step in the present work is to test the implemented boundary method. This is here done by appealing to the exact solutions produced by putting a unit force with strength s at some pole x0 inside one body. At each point x on the cluster contour C one obtains the exact velocity u and traction f = σ.n from the knowledge of the Green tensors G(x, x0 ) and T(x, x0 ). The exact values of u are put in the right-hand sides of (11)-(12) and the numerical prediction of f is gained by inverting those boundary-integral equations. For a sake of conciseness, we restrict in the present proceedings our comparisons to the case of a 2-disk cluster similar to the one depicted in Fig. 1(b). This cluster is made of a small disk with radius a1 = a and center O1 = O and of a second big disk with radius a2 = 2a and center O2 . The center-tocenter distance is d = |O1 O2 | > 3a and the unit vector e = O1 O2 /O1 O2 has angle π/4 with respect to the e1 direction. The numerical comparisons for the velocity components u1 , u2 and the pressure p induced by a unit force with strength s put at O1 are made here at three points P1 , P2 and P3 located in the liquid domain and defined by OP1 = (1.1a)e, OP2 = (d − 1.1a)e and OP3 = (OP1 + OP2 )/2. Those tests have been performed for several values of the disk-disk separation d and Hartmann number M = a/d using different (coarse or refined) meshes on the cluster contour C.

P t1 M =1

P t2 P t3 P t1

M = 10

P t2 P t3

P t1 M =1

P t2 P t3 P t1

M = 10

P t2 P t3

d = 4.0

μu1 d = 5.0

d = 7.0

B E B E B E B E B E B E

0.105372 0.105371 0.071141 0.071141 0.085267 0.085267 0.007356 0.007356 0.001724 0.001724 0.003493 0.003493

0.105372 0.105371 0.048949 0.048949 0.068230 0.068230 0.007357 0.007356 0.000322 0.000322 0.001451 0.001451

0.105372 0.105371 0.027469 0.027469 0.047355 0.047355 0.007357 0.007356 0.000013 0.000013 0.000273 0.000273

B E B E B E B E B E B E

0.032845 0.032844 0.026703 0.026703 0.029684 0.029684 0.003196 0.003196 0.000735 0.000735 0.001500 0.001500

0.032845 0.032844 0.020380 0.020380 0.025996 0.025995 0.003196 0.003196 0.000136 0.000136 0.000618 0.000618

0.032845 0.032844 0.012119 0.012119 0.019839 0.019839 0.003196 0.003196 0.000006 0.000006 0.000115 0.000115

μu2 d = 4.0 d = 5.0 d = 7.0 Case I : s = (1, 0) 0.032845 0.032845 0.032845 0.032844 0.032844 0.032844 0.026703 0.020380 0.012119 0.026703 0.020380 0.012119 0.029684 0.025996 0.019839 0.029684 0.025995 0.019839 0.003197 0.003196 0.003196 0.003196 0.003196 0.003196 0.000735 0.000136 0.000006 0.000735 0.000136 0.000006 0.001501 0.000618 0.000115 0.001500 0.000618 0.000115 Case II : s = (0, 1) 0.039684 0.039684 0.039684 0.039683 0.039683 0.039683 0.017735 0.008189 0.003230 0.017735 0.008189 0.003231 0.025899 0.016240 0.007677 0.025899 0.016240 0.007677 0.000964 0.000964 0.000964 0.000964 0.000964 0.000964 0.000254 0.000050 0.000002 0.000254 0.000050 0.000002 0.000493 0.000215 0.000043 0.000493 0.000215 0.000043

d = 4.0

p d = 5.0

d = 7.0

0.115544 0.115533 0.071616 0.071605 0.088127 0.088117 0.073593 0.073552 0.017256 0.017239 0.034959 0.034931

0.115537 0.115533 0.048458 0.048454 0.068413 0.068410 0.073566 0.073552 0.003218 0.003217 0.014512 0.014507

0.115535 0.115533 0.027322 0.027321 0.046869 0.046868 0.073562 0.073552 0.000132 0.000132 0.002732 0.002731

0.088675 0.088667 0.045566 0.045558 0.061131 0.061124 0.031997 0.031986 0.007357 0.007351 0.015011 0.015003

0.088670 0.088667 0.026401 0.026399 0.042697 0.042695 0.031992 0.031986 0.001359 0.001358 0.006179 0.006178

0.088669 0.088667 0.012903 0.012902 0.025244 0.025243 0.031991 0.031986 0.000055 0.000055 0.001153 0.001153

Table 1: Computed (B) and exact (E) values of the quantities μu1 , μu2 and p at points P1 , P2 , P3 located in the liquid domain surrounding a 2-disk cluster similar to the one sketched in Fig. 1(b). The results obtained, spreading 768 nodes on the small disk and 1536 nodes on the big disk, for Hartmann numbers M = 1, 10 and separations d = 4a, d = 5a (the case shown in Fig. 1(b)) and d = 7a are given in Table 1. Clearly, there is very good agreement between the analytical and the numerical values both for a unit force s parallel with (s = e1 ) or normal to (s = e2 ) the imposed uniform ambient magnetic field Be1 . Conclusions A new boundary procedure has been proposed to efficiently obtain the two-dimensional steady viscous MHD flow (u, p) of a conducting liquid about a given arbitrary 2-particle cluster provided the velocity u is prescribed on the cluster boundary C. The method consists in inverting on C four coupled regularized boundary-integral equations, which govern the cartesian components of the traction taking


place there. One should note that such a boundary approach also permits one to subsequently gain in the liquid domain the flow (u, p). As shown by the numerical comparisons reported for specific exact flows about a 2-disk cluster, the proposed numerical implementation nicely agrees with theoretical predictions in a large range of Hartman number. Therefore, the advocated approach opens the way to the solution of the motivating problem of two interacting solid particles experiencing prescribed rigid-body motions. This nice feature will be illustrated, again for a 2-particle cluster, at the oral presentation. References [1] R. J. Moreau Magnetohydrodynamics, Fluid Mechanics and its applications. Kluwer Academic Publisher. (1990). [2] K. Gotoh Stokes flow of an electrically conducting fluid in a uniform magnetic field. Journal of the Physical Society of Japan, 15 (4), 696-705 (1960). [3] H. Yosinobu and T. Kakutani Two-dimensional Stokes flow of an electrically conducting fluid in a uniform magnetic field. Journal of the Physical Society of Japan, 14 (10), 1433-1444 (1959). [4] J. Hartmann Theory of the laminar flow of an electrically conductive liquid in a homogeneous magentic field. Det Kgl . Danske Videnskabernes Selskab . Mathematisk-fysiske Meddelelser, XV (6), 1-28 (1937). [5] H. Yosinobu A linearized theory of Magentohydrodynamic flow past a fixed body in a parallel magnetic field. Journal of the Physical Society of Japan, 15 (1), 175-188 (1959). [6] A. B. Tsinober and A. G. Shtern Stokes flow around a circular cylinder in a transverse magentic field. MagnetoHydrodynamics, 3 (4), 146-147 (1967). [7] A. Sellier, S. H. Aydin and M. Tezer-Sezgin Free-Space Fundamental Solution of a 2D Steady Slow Viscous MHD Flow. CMES, 102 (5), 393-406 (2014). [8] J. Priede Fundamental solutions of MHD Stokes flow arXiv: 1309.3886v1. Physics. fluid. Dynamics, (2013). [9] M. Abramowitz and I. A. Stegun Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York. (1965). [10] C. Pozrikidis Boundary integral and singularity methods for linearized viscous flow, Cambridge University Press, (1992).


73

Migration of solid particles near a porous slab when subject to a Stokes ambient flow S. Khabthani1 , A. Sellier2 and F. Feuillebois3 ´ Laboratoire Ingénierie Mathématique, Ecole Polytechnique de Tunisie, rue El Khawarezmi, BP 743, La Marsa, Tunisia 2 LadHyx. Ecole polytechnique, 91128 Palaiseau C´ edex, France 3 LIMSI - CNRS, BP 133, 91403 Orsay C´ edex, France [email protected], 2 [email protected], 3 [email protected] 1

1

Keywords: Porous slab, Stokes flow, Darcy flow, Beavers & Joseph conditions, Green tensor, Indirect Boundary-integral equation.

Abstract. This work examines the interacting slow viscous migration of two solid particles immersed above a porous slab in a Newtonian liquid subject to a prescribed ambient steady Stokes flow. The (undisturbed and disturbed) flows above and below the porous slab, which is bounded by two parallel planes, satisfy the Stokes equations while the incompressible flow inside the slab obeys the Darcy law. In addition those flows are coupled at the slab boundaries by the so-called Beavers & Joseph conditions. An efficient indirect boundary approach is then proposed and implemented to compute each particle incurred rigid-body motion. It rests on the use of a specific Green tensor determined elsewhere and reduces the task to the (numerical) solution of 13 Fredholm boundary-integral equations of the first kind on the cluster surface. In addition to some informations about the implemented collocation boundary element technique, the manuscript also presents preliminary results for a 2-sphere cluster made of equal spheres. Introduction In many applications one has to determine the flow of a suspension made of solid particles immersed in a Newtonian liquid and subject to a prescribed imposed ambiant far-field flow with velocity ua and pressure pa . In general, the suspension flow is unsteady and governed by the non-linear incompressible Navier-stokes equations. It also deeply depends upon the suspended solid bodies (shapes, locations and nature). Even for unbounded suspensions the problem is then of great complexity and it therefore requires a numerical treatment. Even more complicated is the case, widely-encountered in practice, of bounded suspensions for which both particle-particle and particle-boundary interactions take place. Those two kinds of interactions depend on the particles and boundaries shape and physical surface properties with no-slip, slipping, permeable or impermeable surfaces. As often done in the literature when the particles typical size a is small compared with the local average radius of curvature R of a close boundary (a R), we here assume the suspension to be bounded by one motionless plane wall Σ with equation x3 = 0 and to be located (above the wall) in the x3 > 0 domain. Adopting Cartesian coordinates (O, x1 , x2 , x3 ) attached to Σ and denoting by v and q the liquid flow and pressure Σ one then writes, depending upon the wall nature, different bondary conditions. Amongst the most employed conditions for a motionless plane wall one can distinguish the following ones: (i) The usual case of a solid impermeable and no-slip wall at which one requires the fluid velocity to vanish, i. e. v = 0. (ii) The case of a slip impermeable wall for which one usually adopts the famous Navier slip condition [1], which setting vi = v.ei , reads v1 = λ

∂v2 ∂v1 , v2 = λ and v3 = 0 at Σ(x3 = 0) ∂x3 ∂x3

(1)

where λ ≥ 0 is the wall slip length. The condition (1) has been experimentally confirmed [2,3] and the larger λ is the more slipping is the wall. Of course, the case (i) is retrieved for λ → 0 while the limit λ → ∞ gives the case of a free surface Σ.


(iii) The case of a slip permeable wall for which the velocity component v3 is non-zero. In proposing adequate boundary conditions on the permeable wall it is necessary to take into account of the medium (liquid or porous medium) located below the wall in the x3 < 0 domain where the liquid now has divergence-free velocity v and pressure q . For a porous medium in which the Darcy law v = −K∇q /μ is used, with K > 0 the so-called uniform permeability, the prescribed and so-called Beavers & Joseph boundary conditions on the x3 = 0 permeable wall read [4,5] ∂v1 σ ∂v2 σ = √ (v1 − v1 ), = √ (v2 − v2 ), v3 = v3Q , = Q at Σ(x3 = 0) (2) ∂x3 ∂x3 K K √ where the associated slip length K/σ is related to both the permeability K > 0 and another dimensionless coefficient σ√> 0. One should note that (2) encompasses (1) which is retrieved by taking √ σ = K/λ and letting K tend to zero so that v vanishes. For dilute suspensions it is possible to neglect particle-particle interactions and to consider the case of a single particle interacting with the wall Σ. This has been done in the literature for the previous different conditions (i)-(iii) by neglecting all inertial effects and assuming a quasi-steady Stokes flow (v, q) about the particle. The reader is referred to [6] for (i) and different particle shapes, to [7,8] for a sphere with conditions (ii) on the wall. Recently, [9,10] proposed a new indirect boundary approach to deal with the challenging case of a porous slab with parallel planes boundaries at which the extended Beavers & Joseph conditions are applied. More precisely, [9] obtains a specific Green tensor complying with those boundary conditions and proposes the indirect boundary approach which is then numerically worked out for different particle shapes while [10] derives an asymptotic treatment for a particle sufficiently distant from the porous slab. Unfortunately, there is (to the authors very best knowledge) no work dealing with the challenging case of several particles interacting in the vicinity of a porous slab. This work therefore extends the material given in [9] to the case of two solid particles freely suspended, above the porous slab, in a liquid subject to a given ambient steady Stokes flow (ua , pa ). Addressed problem and useful decomposition This section presents the addressed problem together with its decomposition, by linearity, into two different tasks. Governing assumptions and equations As depicted in Fig. 1(a), we consider two solid particles P1 and P2 immersed in a Newtonian liquid, with uniform viscosity μ and subject to a prescribed ambient steady Stokes flow (ua , pa ), above a porous slab S with uniform permeability K > 0, constant thickness e > 0 and parallel plane x3

u∞

g

μ D

S1 P1

D

K, σ

(b)

O2

O1

2a/λ

Σu (x3 = 0) x1

O S

S2 P2

e

S2 O2

Σl (x3 = −e)

O1

S1

a/δ Σ(x3 = 0)

x1

Figure 1: Two solid particles P1 and P2 suspended near a porous slab in a Newtonian liquid subject to the ambient Stokes flow (ua , pa ). (a) Arbitrary cluster. (b) Case of two equal in-line spheres.


75

upper and lower boundaries Σu (x3 = 0) and Σl (x3 = −e). The solid particle Pn has smooth surface Sn , attached point On and a rigid-body motion characterized by its translational velocity U(n) (the velocity of the point On ) and angular velocity Ω(n) . Henceforth, we use Cartesian coordinates (O, x1 , x2 , x3 ) attached to the motionless slab and the usual tensor summation notation with, for instance, v = vi ei . Let us now consider three flows: a flow with velocity v and pressure q in the liquid domain D above the slab, a flow with velocity v and pressure q in the porous slab domain S and also a flow with velocity v and pressure q in the liquid domain D below the slab. Those flows will be termed “coupled” if they satisfy the Stokes and Darcy equations μ∇2 v = ∇q and ∇.v = 0 in D, μ∇2 v = ∇q and ∇.v = 0 in D ,

(3)

v = −K∇q /μ and ∇.v = 0 in S

(4)

together with following Beavers & Joseph boundary conditions ∂vj σ = √ (v1 − vj )for j = 1, 2;v3 = v3 , q = q at Σu (x3 = 0), ∂x3 K ∂vj σ = − √ (vj − vj )for j = 1, 2;v3 = v3 , q = q at Σl (x3 = −e). ∂x3 K

(5) (6)

In absence of particles the liquid experiences the given ambient Stokes flow (ua , pa ) in the entire x3 > 0 half space and “coupled” ambient Stokes flow (ua , pa ) in the entire x3 < 0 half space and Darcy flow (ua , pa ) in the slab. Those ambient flows are disturbed by the particles so that in our problem the prevailing flows become (ua + u, pa + p) in D, (ua + u , pa + p ) in D and (ua + u , pa + p ) in S where (by linearity) the Stokes flows (u, p), (u , p ) and the Darcy flow (u , p ) are not only“coupled”, i. e. obey (3)-(6), but also satisfy (due to the no-slip velocity boundary condition on each particle) u = −ua + U(n) + Ω(n) ∧ On M on Sn for n = 1, 2.

(7)

Being a Stokes flow also inside each particle, (ua , pa ) exerts no force and no torque on each surface Sn . Denoting by σ the stress tensor associated to (u, p), the force F(n) and the torque C(n) (about the point On ) experienced by each particle Pn then read σ.ndS, C(n) = On M ∧ σ.ndS. (8) F(n) = Sn

Sn

Key problems One basic issue is to determine each particle rigid-body motion (U(n) , Ω(n) ). Neglecting inertia this is done by requiring each particle to be force-free and torque-free, i. e. by imposing the additional relations F(1) = F(2) = 0 and C(1) = C(2) = 0. One convenient method to proceed is two distinguish, by linearity, two different problems for three “coupled” flows (u, p), (u , p ) and (u , p ) : Problem 1: The so-called hydrodynamic problem for which there is no ambient flow and the particles experience prescribed rigid-body migrations. By linearity, it is clear that (n),(m)

F(n) = Ft

(n),(m)

.U(m) + Fr(n),(m) .Ω(m) , C(n) = Ct

.U(m) + Cr(n),(m) .Ω(m)

(n),(m) FL

(n),(m) CL

(9)

with Cartesian components of the occurring fourth-rank tensors and (with L = t or L = r) evaluated by determining the exerted forces F(n) and torques C(n) in 12 different cases: the ones in which one particle is at rest while the second one either translates or rotates with unit velocity ek (successively taking k = 1, 2, 3). Problem 2: The case of two particles held fixed in the ambient flow (ua , pa ). In that case one has to (n) (n) calculate the force Fa and torque Ca applied on the particules (again by employing the definitions (8)).


Once above problems have been solved by actually dealing with 13 different flows (u, p) the rigidbody motions (U(1) , Ω(1) ) and (U(1) , Ω(1) ) are gained for the freely-suspended (torque-free and forcefree) particles by solving the (well-posed) linear system (n),(m)

F(n) = Ft

(n),(m)

.U(m) + Fr(n),(m) .Ω(m) = −Fa(n) , C(n) = Ct

.U(m) + Cr(n),(m) .Ω(m) = −Ca(n) . (10)

Employed indirect boundary approach As shown in the previous section, the task reduces to the evaluation of the force and torque acting on each particle for the problems (i)-(ii). This is done by appealing to an indirect bounday approach in which each encountered velocity field u receives the following integral representation 1 u(x) = − G(x, y).d(y)dS for x in D (11) 8πμ S1 ∪S2 with d an unknown surface density having nothing to do with the (local) surface traction σ.n taking place on the entire cluster surface S = S1 ∪S2 and G(x, y) a specific second-rank Green tensor obtained in [9] and admitting a pretty-involved dependence upon the points x and y located above the slab. At that stage, three nice properties play a key role: (i) The Green tensor G is calculated in order that the resulting “coupled” Stokes flows (u, p), (u , p ) and Darcy flow (u , p ) indeed satisfy (3)-(6). (ii) Although d = σ.n on S one can establish the relations ddS, C(n) = On M ∧ ddS. (12) F(n) = Sn

Sn

F(n)

and torques C(n) for our 13 previously-mentioned which permit one to evaluate the required forces flows (encountered for Problem 1 and Problem 2). (iii) The Green tensor G(x, y) turns out to be regular as x → y and this enables one to also write (11) for x located on S = S1 ∪S2 therefore ending, invoking (7), up with the Freholm boundary-integral equation of the first kind 1 G(x, y).d(y)dS = −ua + U(n) + Ω(n) ∧ On M on Sn for n = 1, 2. (13) − 8πμ S1 ∪S2 Numerical implementation and preliminary results This section briefly gives informations regarding the implemented numerical strategy and aslo presents preliminary numerical results. Adopted collocation boundary element technique As explained earlier, one has to numerically invert the indirect boundary-integral equation (13) 13 times (by solely changing the right-hand side). To do so we implemented a standard collocation boundary element technique by putting Nn nodal points on Sn where quadratic 6-node curved triangular boundary elements are used. As a result, the discretized boundary-equation (13) becomes a linear system with, setting N = N1 + N2 , a fully populated (3N ) × (3N ) influence matrix. This linear system is solved by Gaussian elimination. Preliminary results for a 2-sphere cluster For a sake of conciseness, we solely report numerical results for a 2-sphere cluster made of two in-line equal spheres with radius a and 02 01 = de1 where d = 01 02 > 2a. As sketched in Fig. 1(b), we introduce two sphere-sphere separation parameter λ and wall-sphere separation parameter δ such that (14) 0 < λ = 2a/d < 1, 0 < δ = a/(001 .e3 ) < 1. The cluster is immersed in the ambient “coupled” flows √ √ ua = γ1 [x3 + K/σ]e1 , ua = γ1 [x3 + e − K/σ]e1 , ua = 0, pa = pa = pa = 0

(15)


77

1,02

(a) 1

0,98

0,96

0,94

0,92

0,9

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

λ

0,9

λ

0,9

0,4

(b) 0,3

0,2

0,1

0

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

Figure 2: Normalized velocities versus the sphere-sphere separation parameter λ for δ = 0.1(◦), δ = 0.3(•), δ = 0.5(∗), δ = 0.7() and δ = 0.9(). Here e = 1, σ = 1 and K = 0.1a2 . (a) velocity u. (b) angular velocity w. with arbitrary independent constant parameters γ1 and γ1 . For symmetry reasons it is found that √ √ K K (1) (2) (1) (2) (16) ]ue1 , aΩ = aΩ = γ1 [h + ]we2 U = U = γ1 [h + σ σ with normalized translational velocity u and angular velocity w now depending upon the porous slab properties (K, σ, e) and the cluster location (δ, λ). Those velocities, tending to unity when both δ and λ vanish, are plotted versus λ in Fig. 2 for several values of δ. Clearly, the wall-cluster interaction are seen to promote the spheres rotation while by contrast decreasing the spheres translational migration (increase δ for a given value of λ). Conclusions A new boundary procedure has been proposed and implemented to determine the rigid-body migration of two freely-suspended and interacting arbitrary-shaped solid particles immersed in a Newtonian liquid in the vicinity of a porous slab when the liquid is subject to a prescribed ambient steady Stokes


flow. The trick consists in exploiting a new boundary integral representation of the liquid velocity field about the particle. This representation appeals to a specific Green tensor determined earlier in a previous work and actually involves on the cluster boundary a surface density which has no physical meaning but permits one to calculate the total force and torque exerted on each particule. Preliminary results for two equal spheres immersed in an ambient shear show that particle-particle interactions affect the solution obtained in [9] for a single particle interacting with porous slab. Other clusters will be considered 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] N.V Churaev, V.D Sobolev and A. N. Somov Slippage of liquids over lyophobic solid aurfaces. J. Colloid Int. Sci.,97, 574-581 (1984). [3] J. Baudry, E. Charlaix, A. Tonck and D. Mazuyer Experimental evidence for a large slip effect at a nonwetting fluid-solid interface. Langmuir,17, 5232-5236 (2001). [4] G. Beavers and D. Joseph Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30, 197-207 (1967). [5] P.G. Saffman On the boundary condition at the surface of a porous medium. Stud. Appl. Maths. 50, 50-93 (1971). [6] J. Happel, H. Brenner Low Reynolds number hydrodynamics, Martinus Nijhoff, (1973). [7] 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). [8] 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). [9] S. Khabthani, A. Sellier, L. Elsami and F. Feuillebois Motion of a solid particle in a shear flow along a porous slab. J. Fluid Mech. 713, 271-306 (2012). [10] S. Khabthani, A. Sellier and F. Feuillebois Motion of a distant solid particle in a shear flow along a porous slab. Z. Angew. Math. Phys. 64, 1759-1777 (2013). [11] C. Pozrikidis Boundary integral and singularity methods for linearized viscous flow, Cambridge University Press, (1992). [12] M. Bonnet Boundary Integral Equation Methods for Solids and Fluids, John Wiley & Sons Ltd, (1999).


79

h-adaptive refinement applied to 2D elastic Boundary Element Method formulation R. R. Reboredo1, E. L. Albuquerque2and A. Portela3 1

University of Brasilia - UnB

Faculty of Mechanical Engineering Brasilia, DF, Brazil [email protected] 2


Faculty of Mechanical Engineering Brasilia, DF, Brazil [email protected] 3


Faculty of Civil Engineering Brasilia, DF, Brazil [email protected]

Keywords:error estimator, mesh refinement, adaptive boundary element method Abstract.This paper is concerned with the effective numerical implementation of the adaptive boundaryelement method (BEM) for two-dimensional elastic problems. The method uses a simple error estimator, based on the discontinuity of the solution across the boundaries between adjacent elements and implements the h version of the adaptive mesh refinement for 2D elastic BEM formulation. In order to validate the algorithm, the classical Timoshenko's slender cantilever, a quasi-singular problem, is solved using both conventional BEM and its adaptive form.

1

Introduction

Portela [10] states that a posteriori error estimation and adaptivity is now well established for the finite element method, as demonstrated in the Zienkiewiczand Zhu [14, 15] works. In order to guarantee the necessary accuracy of the numerical solution, a reliable estimation of the boundaryelement discretization errors is required. [10] introduces a new error estimator based on the use of non-conforming parametric boundaryelements, considering the 2D potential DBEM formulation. The distinctive advantage of this approach, as he states, which is intrinsic to any boundary-element analysis with non-conforming elements, is the simplicity of the error estimator. The objective of this paper is to present an error estimator for adaptive two dimensional elastic BEM analysis. In the final section, a quasi-singular problem, namely the Timoshenko's slender cantilever, is investigated in order to compare the performance of both BEM formulation: the classical one, using the Conventional Boundary Integral Equation (CBIE), and the h-adaptive refinement BEM algorithm as proposed.


2

Boundary Integral Equations

To deduce the Boundary Integral Equations (BIEs) for elastostatics, we first consider Somigliana’s Identity corresponding to governing equations for elastic fields, namely the equilibrium equations, the straindisplacement relation, the stress-strain relation and the boundary conditions for an elasticity problem. Following this, the boundary integral equation of the displacement field in domain Ω, considering any value of x in the domain Ω, will be: ܿ௜௝ ሺ࢞ሻ‫ݑ‬௝ ሺ࢞ሻ ൅ න ܶ௜௝ ሺ࢞ǡ ࢟ሻ‫ݑ‬௝ ሺ࢟ሻ ݀Γሺ࢟ሻ ൌ න ܷ௜௝ ሺ࢞ǡ ࢟ሻ‫ݐ‬௝ ሺ࢟ሻ ݀Γሺ࢟ሻǡሺͳሻ Γ

Γ

in which the coefficient ܿ௜௝ is expressed as ͳൗʹ ߜ௜௝ when the boundary is smooth. In other cases, ܿ௜௝ follows a specific expression that can be found in [1]. ܷ௜௝ and ܶ௜௝ are displacement and traction components of the fundamental solutions, respectively, for 2D plane-strain elasticity problems. These components are given by:

ܷ௜௝ ሺ࢞ǡ ࢟ሻ ൌ ܶ௜௝ ሺ࢞ǡ ࢟ሻ ൌ

3

ͳ ͳ ͳ ൤ሺ͵ െ Ͷߥሻߜ௜௝ ൬ ൰ ൅ ‫ݎ‬ǡ௜ ‫ݎ‬ǡ௝ െ ߜ௜௝ ൨ǡሺʹሻ ‫ݎ‬ ʹ ͺߨߤሺͳ െ ߥሻ

߲‫ݎ‬ ͳ ൜ ൣሺͳ െ ʹߥሻߜ௜௝ ൅ ʹ‫ݎ‬ǡ௜ ‫ݎ‬ǡ௝ ൧ െ ሺͳ െ ʹߥሻሺ‫ݎ‬ǡ௜ ݊௝ െ ‫ݎ‬ǡ௝ ݊௜ ሻൠǤሺ͵ሻ Ͷߨሺͳ െ ߥሻ‫߲݊ ݎ‬

Solving Strategy

This paper extends to the elastic BEM formulation the simple error estimator of the boundary-element discretization presented in [10], which is based on the discontinuities of the solution that possibly occur across the boundaries of adjacent elements. The solving strategy can be summarized as follows: x the whole boundary Γ is modeled with quadratic discontinuous boundary-elements; x collocation is always carried out at the element nodes; x the algorithm computes the error percentages per boundary-element, identifying which element require a refinement; x the h-adaptive refinement is applied. 4

Error Estimators and Percentage Errors

Considering the most important source of error in the boundary-element analysis is the discretization of the boundary [10],especially when the state variables are approximated, it is important to find a way in which the approximation error can be reduced. Following this idea, an adaptive mesh-refinement by reducing the boundary-element size, calledh-adaptive refinement, is proposed in order to reduce the approximation error aforementioned. The discretization errors of the state variables can be defined as: ݁௝௨ ൌ ‫ݑ‬௝଴ െ ‫ݑ‬௝ ൌ ߜ‫ݑ‬௝ ǡሺͶሻ for the displacement, where ݆ ൌ ͳǡ ʹ and ݁௝௧ ൌ ‫ݐ‬௝଴ െ ‫ݐ‬௝ ൌ ߜ‫ݐ‬௝ ǡሺͷሻ for the traction, in which ‫ݑ‬௜଴ , ‫ݐ‬௜଴ and ‫ݑ‬௜ , ‫ݐ‬௜ represent, respectively, the exact and the approximate solutions of the state variables for the 2D elastic formulation, along the boundary.


81

The norm of the errors of the state variables is computed, representing the correctness with which the state variables are modeled by the boundary-elements and are used to generate computable estimates of the remaining discretization errors. The global estimator error is defined, therefore, as the sum of the norms of the errors described in Eqs. (4) and (5). These norms are defined as error estimators. The error indicator per element show how much of the estimated global error is a contribution from each boundary-element. The approximation errors can be defined as: ݁௝௨ ൌ ‫ܬ‬௝௨ ሺ͸ሻ and ݁௝௧ ൌ ‫ܬ‬௝௧ ǡሺ͹ሻ where ‫ܬ‬௝௨ and ‫ܬ‬௝௧ are the solutions jumps of the BEM approximation at the inter-element boundary points of each boundary-element for both directions (݆ ൌ ͳǡ ʹሻ. The solutions jumps are computed for every element node, being linearly approximated between the respective values of the solution jumps of the two interelement boundary points by the shape functions. The element error indicators can be defined as: ଵȀଶ

ԡ݁ଵ௨ ԡ௜ ൌ ቈන ሺ‫ܬ‬ଵ௨ ሻ;݀Ȟ቉

ൌ

݄௜ ௨ǡଵ ൣ൫‫ ܬ‬൯; ൅ ሺ‫ܬ‬ଵ௨ǡଶ ሻ;൧ǡሺͺሻ ʹ ଵ

ൌ

݄௜ ௨ǡଵ ൣ൫‫ ܬ‬൯; ൅ ሺ‫ܬ‬ଶ௨ǡଶ ሻ;൧ǡሺͻሻ ʹ ଶ

୻೔

ଵȀଶ

ԡ݁ଶ௨ ԡ௜ ൌ ቈන ሺ‫ܬ‬ଶ௨ ሻ;݀Ȟ቉ ୻೔

for both directions, and, ଵȀଶ

ԡ݁ଵ௧ ԡ௜ ൌ ቈන ሺ‫ܬ‬ଵ௧ ሻ;݀Ȟ቉

ൌ

݄௜ ௧ǡଵ ൣ൫‫ ܬ‬൯; ൅ ሺ‫ܬ‬ଵ௧ǡଶ ሻ;൧ǡሺͳͲሻ ʹ ଵ

ൌ

݄௜ ௧ǡଵ ൣ൫‫ ܬ‬൯; ൅ ሺ‫ܬ‬ଶ௧ǡଶ ሻ;൧ǡሺͳͳሻ ʹ ଶ

୻೔

ଵȀଶ

ԡ݁ଶ௧ ԡ௜ ൌ ቈන ሺ‫ܬ‬ଶ௧ ሻ;݀Ȟ቉ ୻೔

again for both directions, where ‫ܬ‬௝௨ǡଵ , ‫ܬ‬௝௨ǡଶ, ‫ܬ‬௝௧ǡଵ , ‫ܬ‬௝௧ǡଶ are the solution jumps at the two inter-element boundary points for both directions (݆ ൌ ͳǡ ʹ), ݄௜ represents the length of the generic element Ȟ௜ , ฮ݁௝௨ ฮ , ฮ݁௝௧ ฮ are the ௜ ௜ local error estimators of the state variables for both directions and per boundary-element ݅. The percentage error indicators of the BEM analysis are defined as: ൫ߣ௝௨ ൯ ൌ ൥ ௜

ฮ݁௝௨ ฮ ;

௜ ൩ǡሺͳʹሻ ԡ‫ݑ‬ԡ; ൅ ฮ݁௝௨ ฮ;

for the displacement in both directions, ൫ߣ௝௧ ൯ ൌ ൥ ௜

ฮ݁௝௧ ฮ ;

௜ ൩ሺͳ͵ሻ ԡ‫ݐ‬ԡ; ൅ ฮ݁௝௧ ฮ;

for the traction in both directions, where ฮ݁௝௨ ฮ and ฮ݁௝௧ ฮ are the global errors, defined as a sum of the every boundary-element local errors.


5

Numerical Results

The problem solved in order to validate the h-adaptive refinement algorithm was the classical Timoshenko's slender cantilever. This problem has a simple analytical solution, which will work as a standard to be compared with the numerical solution by the BEM adaptive analysis. [12] shows that analytical vertical displacement solution for a cantilever beam with a shear tip load can be determined by eq. (14). ‫ݑ‬௬ ሺ‫ݔ‬ǡ ‫ݕ‬ሻ ൌ

ܲ‫;݄ ݕ‬ ቈ ሺͷߥ ൅ Ͷሻ‫ ݔ‬൅ ͵ߥሺ‫ ܮ‬െ ‫ݔ‬ሻ‫ ݕ‬ଶ ൅ ሺ͵‫ ܮ‬െ ‫ݔ‬ሻ‫;ݔ‬቉ǡሺͳͶሻ ͸‫ ܫܧ‬Ͷ

where E is Young’s modulus, ߥ is Poisson’s ratio and I is the second moment of area of the cross-section. Appling eq. (14) for ‫ ܧ‬ൌ ͳͲ଺, ߥ ൌ ͲǤ͵, ܲ ൌ ͳܰ and ‫ ݐ‬ൌ ͳ݉, we have: ‫ݑ‬௬ ൌ ͶǤͲͲͲ͵݉Ǥሺͳͷሻ To Boundary Element Analysis, thin shaped problems as Timoshenko's slender cantilever have proved themselves to be a not so simple issue to solve. Kane [5] says that kind of geometry presents "definite pathological problems for single zone analysis". [6] stated problems of this type were normally relegated to the more effective treatment of Finite Element Method (FEM). In the present scenario, however, special techniques are being developed in order to treat the uniqueness inherent to the nature of thin-shaped structures - which turns a simple problem, such as a cantilever beam with a shear tip load, into something that requires special treatment. According to [13], conventional boundary element method (CBEM) using Gaussian quadrature has difficulties to generate reliable results for thin-walled structures. The major reason for the failure is the kernels' integration presents near singularities, establishing a situation in which a node on one side of the thin-shaped body is too close to the node on the opposite side. Although nearly singular integrals are mathematically different from singular integrals, they both represent an obstacle that prevents the use of conventional numerical quadrature in the sense of numerical analysis, since the integrand oscillates profusely under the integration interval. In the case of singular integrals, many algorithms have been developed and applied successfully, from which we can mention [2], [4], [7], [8], [11] and many others. In dealing with nearly singular integrals, other several direct and indirect algorithms were developed, presented in the works of [3], [9]. On the present work, the main objective is to compare the performance of the CBEM algorithm and the hadaptive refinement techniquewithout using a special technique to treat near singularities, considering a thinshaped elastic problem in two dimensions. In this case, we are trying to validate the adaptive refinement algorithm performance, since the Timoshenko's slender cantilever has a known analytical solution. Using CBEM with 10 integration points, we refined the mesh into two different ways: firstly, subdividing the boundary-elements equally; and, finally, using the h-adaptive refinement, which implies in refine the mesh in specific locations, according with the maximum boundary-element percentage error indicator (eq. 12 and 13). The h-adaptive refinement algorithm indentifies the boundary-element with the maximum error, allowing a more efficient refinement process. The results obtained are shown below. Table 1. Results for vertical displacement using 10 integration points (CBIE)

Number of Collocation Points 84 164 244 404 606 808

Vertical Displacement (m)

Relative Error (%)

Elapsed Time (s)

3,4613 3,5845 3,707 3,8347 3,8915 3,9225

13,47 10,39 7,33 4,13 2,72 1,94

7,52 25,39 53,97 141,54 315,31 556,49


83

Table 2. Results for vertical displacement using h-adaptive refinement

Iteration 0 1 2 3 4

Number of Collocation Points 84 124 164 244 324

Vertical Displacement (m) Relative Error (%) 3,4613 3,7400 4,1157 4,0730 3,9975

Image 1. Vertical displacement for CBIE and adaptive refinement

6

13,47 6,51 2,88 1,82 0,0687

Elapsed Time (s) 7,52 15,06 26,10 53,94 93,39

Image 2.Elapsed time for CBIE and adaptive refinement algorithms

Conclusion

This paper assesses a h-adaptive refinement BEM technique applied to 2D elastic problems as an extension of the works of [10]. The technique consists on the computation of every boundary-element percentage error. The problem chosen in order to validate the algorithm was the classical Timoshenko's slender cantilever, considering a 1:100 aspect ratio. The results obtained with this technique show the algorithm efficiency. With only 4 iterations, we obtained a 0,0687% error result within 93 seconds of time processing, considering a 324 collocation points. The best result obtained with the CBIE was a 1,94% within 556 seconds of time processing, considering a 808 collocation points. 7

References

[1]Brebbia, C. A. & Dominguez, J.,1992. Boundary Elements: An Introductory Course. Wit Press Computational Mechanics Publications. [2]Brebbia, C. A., Tells, J. C. F., and Wrobel, L. C., 1984. Boundary Element Techniques, Springer-Verlag, Berlin, Germany. [3] Granados, J. J. and Gallego, G., 2001. Regularization of Nearly HypersingularIntegrals in the Boundary Element Method. Engineering Analysis withBoundary Elements, Vol. 25, No. 3, pp. 165-184. [4]Hong, H.-K. and Chen, J. T.,1988. Derivations of Integral Equations of Elasticity. Journal of Engineering Mechanics, ASCE, Vol. 114, No. 6, pp. 1028-1044.


[5] Kane, J. H., 1994. Boundary Elements Analysis in Engineering Continuum Mechanics. Prentice-Hall, Englewood Cliffs, New Jersey. [6] Krishnasamy, G., Rizzo, F. J., Liu, Y, 1994. Boundary Integral Equations for Thin Bodies. International Journal for Numerical Methods in Engineering, Vol. 37, pp. 107-121. [7] Liu, C. S.,2006. A Modified Collocation Trefftz Method for the Inverse Cauchy Problem of Laplace Equation.Engineering Analysis with BoundaryElements, Vol. 32, No. 9, pp. 778-785. [8] Liu, C. S., Chang, C. W., and Chiang, C. Y., 2008. A Regularized Integral Equation Method for the Inverse Geometry Heat Conduction Problem. International Journal of Heat and Mass Transfer, Vol. 51, No. 21, pp. 5380-5388. [9] Mukerjee, S., Chati, M. K., and Shi, X. L., 2000. Evaluation of Nearly Singular Integrals in Boundary Element Contour and Node Methods for Three Dimensional Linear Elasticity.International Journal of Solids and Structures,Vol. 37, No. 51, pp. 7633-7654. [10] Portela, A., 2011. Dual boundary-element method: Simple error estimator and adptivity. International Journal for Numerical Methods in Engineering. [11] Sladek, V. and Sladek, J., 1998. Singular Integrals in Boundary Element Methods, Computational Mechanics Publications, Southampton, UK andBoston. [12]Timoshenko, S., 1936. Theory of Elasticity 3E. Engineering Societies Monographs. McGraw-Hill. [13] Zhang, Y., Gu, Y., Chen, J., 2011. Boundary Element Analysis of Thin Structural Problems in 2D Elastostatics. Journal of Marine Science and Technology, Vol. 19, pp. 409-416. [14] Zienkiewicz O. C., Zhu J. Z.,1992. The Superconvergent patch recovery and a posteriori error estimates. Part 1: the recovery technique. International Journal for Numerical Methods in Engineering; 33:1331-1364. [15] Zienkiewicz O. C., Zhu J. Z., 1992. The Superconvergent patch recovery and a posteriori error estimates. Part 2: error estimates and adaptivity. International Journal for Numerical Methods in Engineering; 33:1365-1382.


85

Stokes Flow in a Lid-Driven Rectangular Duct Under the Influence of a Point Source Magnetic Field Pelin Senel1 and M. Tezer-Sezgin2 1

2

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

Keywords: DRBEM, Stokes flow, magnetic source.

Abstract. In this paper, we investigate the two-dimensional, laminar, viscous, fully developed slow flow of an incompressible, non-conducting fluid in a lid-driven rectangular duct under the influence of a point source magnetic field. The magnetization of the fluid is considered as a function of the magnetic field intensity, and affects the flow behavior in the duct. The continuity and momentum equations for the Stokes flow (low Reynolds number) including magnetic force components are solved in terms of the velocity and the pressure of the fluid. The Dual Reciprocity Boundary Element Method (DRBEM) is used by taking all the terms other than diffusion as inhomogeneity in the Poisson’s equations for velocity components and the pressure. This allows one to use the fundamental solution of Laplace equation in Boundary Element Method (BEM) formulation. The unknown pressure boundary conditions are obtained through momentum equations by using finite difference approximations of pressure gradients at the boundary and interior points. All the space derivatives are calculated by using DRBEM coordinate matrix. Constant elements are taken for the discretization of the boundary of the duct with sufficient number of interior points to depict the flow and the pressure behavior of the fluid. The use of DRBEM results in considerably small sized discretized system due to the boundary nature of the method. The obtained numerical results show that for the values of M = Mn/Re (Mn magnetic number and Re r ξ; x @Hdl ³ 2Sk lsource H

(7)

Euclidian distance r from source point ξ

r

x xs

2

( y ys ) 2

(8)

x and y can be written as:

x

( xs , y s ) to a point x ( x, y) on the quadratic heat source is:

N1 x1 N 2 x2 N 3 x3 ,

y

N1 y1 N 2 y2 N 3 y3

(9)

where the quadratic shape functions N1 , N 2 and N 3 are:

N1

1 K (K 1) 2

(K 1)(K 1)

N2

N3

1 K (K 1) 2

(10)

K is a local coordinate attached to the quadratic heat source that varies from -1 to 1. The infinitesimal element dl can be written as: JdK

dx 2 dy 2

dl

(11)

where J is the Jacobian of transformation and can be written as: 2

2

dN 3 · § dN1 dN 3 · dN 2 dN 2 § dN1 ¨¨ x1 ¸¸ ¨¨ y1 ¸ x2 x3 y2 y3 dK dK ¹ © dK dK dK ¸¹ © dK Considering a quadratic variation for intensity of the heat source, sx becomes: sx N1 g1 N 2 g 2 N 3 g 3 J

(12)

(13)

Substituting eq (11) and eq (13) into eq (7) leads to:

1 N1 g1 N 2 g 2 N 3 g 3 ln>r K @JdK 2Sk ³1 1

Ig

(14)

By using eq (4), eq (11) and eq (13), the heat source integral in the displacement integral equation, i.e. eq (3) becomes:

D (1 Q ) >N1 g1 N 2 g 2 N 3 g 3 @>ri K 2 ln>r K @ 1 @JdK 8S (1 Q )k ³1 1

I gi

where ri are

r1

x xs ,

r2

y ys

(15)

(16)

Similarly the heat source integral in the stress integral eq (5) can be expressed as:

ED 4S (1 Q )k

I gij

1

³ >N g 1

1

1

ª§ º 1 2Q · G ij N 2 g 2 N 3 g 3 @«¨ ln>r K @ r,i K r, j K » JdK ¸ 2 ¹ 1 2Q ¬© ¼

(17)

where r,i can be expressed as follows:

r,i

ri r

Integrals in eq (14), eq (15), and eq (17) can be evaluated using standard numerical integration methods.

(18)


113

Numerical Examples To show the accuracy of the presented method, two examples are analyzed. In each example the results obtained by the presented method are compared with those of the FEM with a very fine mesh. Example 1: A circular heat source with uniform intensity in a circular domain. In the first example, a circular domain with R 0.5m is considered. The structural and thermal boundary conditions of the problem are shown in Fig. 3. This problem is analyzed under plane strain condition with E 200 GPa ,

60 W m $ C . The problem boundary is modeled by 32 linear boundary elements. A curved heat source, which is distributed over a circle with radius r 0.25m is considered. The strength of the heat source is considered to be constant over the circle and equal to s 4000Wm 1 . Q

0.3 , D

11.7 u 10 6

1

$

C

, k

Figure 3: The BEM discretization and boundary conditions of the circular domain with a circular heat source In the BEM analysis, the circular heat source is modeled by only 4 quadratic heat sources. The presented BEM results are compared with the FEM results obtained using the ANSYS software. In the FEM analysis, the circular heat source is modelled by a distributed heat source in a ring with the small width w 0.01m . The FEM mesh (9461 second-order quadrilateral elements) is shown in Fig. 4.

Figure 4: The FEM mesh of the circular domain with the circular heat source


The obtained results for the temperature, vertical displacement, the stress in the x-direction and the stress in the y-direction along the y-axis are shown in Fig. 5. The results obtained by the proposed BEM show a good agreement with those of the FEM with the fine mesh. (a)

(b)

(c)


115

(d)

Figure 5. The FEM and BEM results along the y-axis for the circular domain with the circular heat source: (a) temperature, (b) vertical displacement, (c) stress in the x-direction, and (d) the stress in the y-direction. Example 2: A heat source with an elliptical shape and non-uniform intensity in a rectangular domain. In the second example a rectangular domain with a 0.15m and b 0.3m is considered. The structural and thermal boundary conditions of the problem are shown in Fig. 6. This problem is analyzed under plane strain condition with E 200 GPa , Q 0.3 , D 11.7 u 10 6 1 C , k 60 W m $ C . The problem boundary is $

modeled by 48 linear boundary elements. A curved heat source with an elliptical shape, centered at (0.085,0.065) is considered. The major and minor radius of the ellipse are r1 0.04m and r2 0.02m , respectively.

Figure 6: The boundary conditions of the rectangular domain with an elliptic heat source The strength of the heat source is considered to be a function of E [0 2S ] with the following form:

s 400001 cosE Wm 1

(19)


where E is the angle of the ellipse radius from the x-axis. In the BEM analysis, the elliptical heat source is modeled by only 8 quadratic heat sources. The presented BEM results are compared with the FEM results obtained using the ANSYS software. In the FEM analysis, the elliptic heat source is modelled by a distributed heat source in an elliptic ring with the small width w 0.0025m .The FEM mesh (4356 secondorder quadrilateral elements) is shown in Fig. 7.

Figure 7: The FEM mesh of the rectangular domain with the elliptic heat source The obtained results for the temperature, vertical displacement and the horizontal stress along line A-B are shown in Fig. 8. Again, the results obtained by the proposed BEM show a good agreement with those of the FEM with a fine mesh. (a)


117

(b)

(c)

Figure 8. The FEM and BEM results along line AB for the rectangular domain with the elliptic heat source: (a) temperature, (b) vertical displacement, and (c) stress in the x-direction. Conclusions A boundary element formulation for analysis of two-dimensional thermo-elastic problems with curved line heat sources was presented. The shape and the intensity function of the curved line heat source can be arbitrary and complicated. For accurate modeling of a curved line heat source in the FEM, a distributed heat source over a small part of the domain and a large number of elements should be considered. On the other hand, curved line heat sources can be efficiently modeled in the proposed BEM without considering additional degrees of freedom. Two examples were presented to show the efficiency and the effectiveness of the presented BEM. The obtained numerical results show that the proposed method gives accurate results even with a small number of boundary elements. References [1] Y. C. Shiah, T. L. Guao and C. L. Tan, Comput. Model. Engng Sci., 7(3), 321–338, (2005). [2] M. R. Hematiyan, M. Mohammadi and M. H. Aliabadi, J. Strain Anal. Engng Design, 46, 227-242, (2010). [3] L.C.Wrobel and M.H Aliabadi The Boundary Element Method, Vol1 :Applications in Thermo-Fluids and Acoustics, Vol2: Applications in Solids and Structures, Wiley (2002).

[4] M. Mohammadi, M. R. Hematiyan and M. H. Aliabadi, J. Strain Anal. Engng Design, 45(8), 605–627, (2010).


Numerical investigation of an orthotropic microdilatation approach in living tissue modeling Arnaud Voignier1, Jean-Philippe Jehl2 and Richard Kouitat Njiwa3 Université de Lorraine, Institut Jean Lamour - Dpt N2EV - UMR 7198 CNRS Parc de Saurupt, CS 14234, F-54042 Nancy Cedex [email protected], [email protected], [email protected]

Keywords: Orthotropic, microdilatation, Isotropic BEM, Meshfree strong form. Abstract. Extended continuum mechanical approaches are now becoming increasingly popular for modeling various types of microstructured materials, such as foams and porous solids. The potential advantages of the microcontinuum approach are currently being investigated in the field of biomechanical modeling. In this field, conducting a numerical investigation of the material response is evidently of paramount importance. This study sought to investigate the potential of the orthotropic microdilatation modeling method. The problem’s field equations have been solved by applying a numerical approach combining the conventional isotropic boundary elements method with local radial point interpolation. Our resulting numerical examples investigate the influence of transverse isotropy on the material response.

Introduction It is becoming increasingly clear that the microstructure of material must be taken into account when establishing its corresponding constitutive equation. Mechanical behavior modeling considers the microstructural information of a material either implicitly or explicitly. Living tissue, such as heart tissue, is non homogeneous with a complex micro-organization. In this context, it is not easy to apply multi-scale modeling or classical mathematical homogenization approaches; phenomenological approaches are still required. It is for this reason that we chose to continue the work of Rosenberg and Cimrman [1] by first considering an isotropic medium [2,3]. In this work, the previous study is extended in order to take into account the anisotropy of the material. The theory behind using micromorphic media, as described by Eringen and Suhubi [4], is that it enables us to capture the impact of the microstructure on the overall response of the material. For a full description, a material point of a micromorphic medium possesses twelve degrees of freedom: the three traditional components of displacement and nine components of a micro-deformation tensor. The material is typically specialized depending on the salient microscopic motions. When the material point of the medium can rotate and stretch, the medium is known as a microstretch material. In the microdilatation medium, the material point can only dilate or contract, a definition already applied to model foam and some porous media [5]. The literature on microdilation media is not extensive because, it is, in fact, difficult to understand and qualify the responses of this type of media using conventional mechanical tests. We believe that numerical experiments are extremely useful in this field. On account of this, we have opted to investigate the response of a 3D microdilatation medium to loading. Our study presented herein is essentially numerical and based on a specifically developed numerical tool. First of all, we present the governing equations of an orthotropic microdilatation medium. Then, we describe the adopted numerical method, called the “local point interpolation – boundary element method” (LPI-BEM). Finally, the numerical results are presented.


119

Governing Equations In the theory of microdilatation medium occupying the domain ȳ with boundary Ȟ, the material point ‫ ݔ‬is attached to a triad of directors that can stretch. The material point possesses four degrees of freedom: the three components of the traditional displacement vector (‫ݑ‬௜ ), and the scalar microdilatation (߮). In the absence of body loads, the problem consists of finding the solution of the system of partial differential equations obtained from the equilibrium conditions [5]: (1) ߪ௝௜ǡ௝ ሺ‫ݔ‬ሻ ൌ Ͳ ‫ݏ‬௞ǡ௞ ሺ‫ݔ‬ሻ െ ‫݌‬ሺ‫ݔ‬ሻ ൌ Ͳ (2) In these equations, ߪ௜௝ ሺ‫ݔ‬ሻ represents the stress tensor, ‫ݏ‬௞ ሺ‫ݔ‬ሻ the vector of internal hypertraction, and the scalar ‫݌‬ሺ‫ݔ‬ሻ is the generalized internal body load (microstress function). The latter can be viewed as an internal pressure. Next we considered the case of a quasi-homogeneous and orthotropic solid, producing the following corresponding constitutive relations: ߪ௜௝ ሺ‫ݔ‬ሻ ൌ ‫ܥ‬௜௝௥௦ ߝ௥௦ ൅ ‫ܣ‬௜௝ ߮ ‫ݏ‬௜ ൌ ‫ܦ‬௜௝ ߛ௝ ‫݌‬ሺ‫ݔ‬ሻ ൌ ‫ܣ‬௥௦ ߝ௥௦ ൅ ܾ߮

(3) (4) (5)

ଵ

ߝ௜௝ ൌ ൫‫ݑ‬௜ǡ௝ ൅ ‫ݑ‬௝ǡ௜ ൯ and ߛ௜ ൌ ߮ǡ௜ ଶ

With ݊௝ as the outward normal vector on the boundary, the tractions acting at a regular point of the boundary are given by: ‫ݐ‬௜ ൌ ߪ௝௜ ݊௝ and ‫ ݏ‬ൌ ‫ݏ‬௝ ݊௝

(6)

Solution Method In the case of linear problems with a well-established analytical fundamental solution, the boundary element method has already proven highly efficient. In our study, no fundamental solution of the field equations existed; the boundary element method (BEM) was found to lose its principal appeal, namely the reduction of the problem dimension by one, due to traditional volume cells being needed in the “field boundary element method”. In order to overcome this obstacle, a number of strategies have been proposed, such as the dual reciprocity method (DRM) or radial integration method (RIM), which enable the conversion of volume integrals into surface ones. In recent years, a large number of researchers have invested in the development of so-called meshless or meshfree methods. Among the various meshless approaches, the local point interpolation method is highly appealing on account of how simple it is to implement. This approach fails in accuracy in the presence of Neumann boundary conditions, which are almost an inevitability when solving solid mechanic problems. Liu et al. [6] have suggested a way to circumvent this difficulty by adopting the “weak-strong-form local point interpolation” method. In a recent publication, Kouitat [7] proposed a novel strategy that combines the best elements of both the conventional BEM and local point interpolation methods. This LPI-BEM approach has proved efficient in the context of anisotropic elasticity [7], piezoelectric solids [8], and nonlocal elasticity [9]. We adopted this method in our study, detailing below the principal steps followed in the context of a microstretch medium. It should also be mentioned that a solution procedure using the finite element method was also presented by Kirchner and Steinmann [10]. Our calculations were based on the assumption that the kinematical primary variables are the sum of a complementary part and a particular term. Namely: ‫ݑ‬௜ ൌ ‫ݑ‬௜஼ ൅ ‫ݑ‬௜௉ , ߮ ൌ ߮ ஼ ൅ ߮ ௉ . The complementary fields satisfied the following homogeneous equations:

ሾ‫ܤ‬ሺ‫׏‬ሻሿ் ሾ‫ ܥ‬଴ ሿሾ‫ܤ‬ሺ‫׏‬ሻሿሼ‫ݑ‬஼ ሽ ൌ Ͳ

(7)

݀ ଴ ሼ‫׏‬ሽ் ሼ‫߮׏‬஼ ሽ ൌ Ͳ

(8)


Accordingly, the particular integrals were determined by solving:

ሾ‫ܤ‬ሺ‫׏‬ሻሿ் ሾ‫ ܥ‬଴ ሿሾ‫ܤ‬ሺ‫׏‬ሻሿሼ‫ݑ‬௉ ሽ ൅ ሾ‫ܤ‬ሺ‫׏‬ሻሿ் ሺሾߜ‫ܥ‬ሿሼߝሽ ൅ ሼ‫ܣ‬ሽ߮ሻ ൌ Ͳ

(9)

‫ ܦ‬଴ ሼ‫׏‬ሽ் ሼ‫߮׏‬௉ ሽ ൅ ሼ‫׏‬ሽ் ሾߜ‫ܦ‬ሿሼ‫߮׏‬ሽ െ ሼ‫ܣ‬ሽ் ሼߝሽ െ ܾ߮ ൌ Ͳ

(10)

In equations (7-10),ሾߜ‫ܥ‬ሿ ൌ ሾ‫ܥ‬ሿ െ ሾ‫ ܥ‬଴ ሿ,ሾߜ‫ܦ‬ሿ ൌ ሾ‫ܦ‬ሿ െ ሾ‫ ܦ‬଴ ሿ, ሼ‫׏‬ሽ ൌ ሺ߲Ȁ߲‫߲ ݔ‬Ȁ߲‫߲ ݕ‬Ȁ߲‫ݖ‬ሻ் , ሼ‫ݖ‬ሽ ൌ ሺ‫ݖ‬ଵ ‫ݖ‬ଶ ‫ݖ‬ଷ ሻ் ,

‫ݖ‬ଵ ሾ‫ܤ‬ሺ‫ݖ‬ሻሿ ൌ ൥ Ͳ Ͳ

Ͳ ‫ݖ‬ଶ Ͳ

Ͳ ‫ݖ‬ଶ Ͳ ‫ݖ‬ଵ ‫ݖ‬ଷ Ͳ

‫ݖ‬ଷ Ͳ ‫ݖ‬ଵ

Ͳ ் ‫ݖ‬ଷ ൩ ‫ݖ‬ଶ

଴ ‫ ܥ‬଴ is a constant isotropic elastic tensor with shear modulus (‫ܩ‬௨ҧ ) and Poisson ratio (ߥ෤௨ ). The scalar tensor ‫ܦ‬௜௝ ൌ ‫ܦ‬଴ ߜ௜௝ .

Equations (7 and 8) were solved by the conventional boundary element method, thus producing the following systems of equations: ሾ‫ܪ‬௨ ሿሼ‫ݑ‬஼ ሽ ൌ ሾ‫ܩ‬௨ ሿሼ‫ ݐ‬஼ ሽ , and ൣ‫ܪ‬ఝ ൧ሼ߮ ஼ ሽ ൌ ൣ‫ܩ‬ఝ ൧ሼ‫ ݏ‬஼ ሽ

(11)

The particular fields solved: ‫ܩ‬௨ҧ ቂȟ‫ݑ‬௜௉ ൅

ଵ ‫ݑ‬௉ ቃ ෥ೠ ௝ǡ௝௜ ଵିଶఔ

൅ ߜ‫ܥ‬௜௝௞௟ ‫ݑ‬௞ǡ௟௝ ൅ ‫ܣ‬௜௝ ߮ǡ௝ ൌ Ͳ

௉ ‫ܦ‬଴ ߮ǡ௞௞ ൅ ߜ‫ܦ‬௜௝ ߮ǡ௜௝ െ ܾ߮ െ ‫ܣ‬௥௥ ‫ݑ‬௥ǡ௥ ൌ Ͳ

(12)

(13)

The tractions at a regular point on the boundary were written as: ஺ ‫ݐ‬௜ ൌ ‫ݐ‬௜஼ ൅ ‫ݐ‬௜௉ ൅ ߜ‫ݐ‬௜ with ‫ݐ‬௜஺ ൌ ൫ߣߝ௥௥ ߜ௜௝ ൅ ʹߤߝ௜௝஺ ൯݊௝ , ሺ‫ ܣ‬ൌ ‫ܲݎ݋ܥ‬ሻ

and ߜ‫ݐ‬௜ ൌ ൫ߜ‫ܥ‬௜௝௞௟ ߝ௞௟ ൅ ‫ܣ‬௜௝ ߮൯݊௝

(14)

‫ ݏ‬ൌ ‫ ݏ‬஼ ൅ ‫ ݏ‬௉ ൅ ߜܵ with ‫ ݏ‬஺ ൌ ‫ܦ‬଴ ߰ǡ௝஺ ݊௝ ሺ‫ ܣ‬ൌ ‫ܲݎ݋ܥ‬ሻ and ߜ‫ ݏ‬ൌ ߜ‫ܦ‬௜௝ ߮௜ ݊௝

(15)

Following this, we then considered the solution of equations (12-13), using a local radial point collocation method. In this method [6], a field ߱ሺ‫ݔ‬ሻ was approximated as: ெ ߱ሺ‫ݔ‬ሻ ൌ σே ௜ୀଵ ܴ௜ ሺ‫ݎ‬ሻܽ௜ ൅ σ௝ୀଵ ‫݌‬௝ ሺ‫ݔ‬ሻܾ௝

with the following constraints: σே ௜ୀଵ ‫݌‬௝ ሺ‫ݔ‬ሻܽ௜ ൌ Ͳ , ݆ ൌ ͳ െ ‫ ܯ‬and ݅ ൌ ͳ െ ܰ. Here ܴ௜ ሺ‫ݎ‬ሻ is the selected radial basis functions, ܰ the number of nodes in the neighborhood (support domain) of point ‫ݔ‬, and ‫ ܯ‬the number of monomial terms in the selected polynomial basis ܲ௝ ሺ‫ݔ‬ሻ. Coefficients ୧ and ୨ could be determined by enforcing the approximation to be satisfied at the ܰ centers. Following some algebraic manipulations, coefficients ୧ and ୨ could be expressed in terms of the field nodal values, and the interpolation could be written in the following compact form:

߱௛ ሺ‫ݔ‬ሻ ൌ ሾȰሺ‫ݔ‬ሻሿ൛߱Ȁ௅ ൟ When adopting interpolation (16) for all kinematical fields, at a given collocation center, equations of the following forms were obtained:

(16)


121

௉ ሾܽ௨௨ ሿ൛‫ݑ‬Ȁ௅ ൟ ൅ ሾܾ௨௨ ሿ൛‫ݑ‬Ȁ௅ ൟ ൅ ൣܾ௨ఝ ൧൛߮Ȁ௅ ൟ ൌ Ͳ ௉ ൣܽఝఝ ൧൛߮Ȁ௅ ൟ ൅ ൣܾఝఝ ൧൛߮Ȁ௅ ൟ െ ൣܾఝ௨ ൧൛‫ݑ‬Ȁ௅ ൟ ൌ Ͳ

(17)

On collection of the above equations for all the internal collocation centers, taking the assumption that the particular integrals are identically zero at all boundary points, the following solutions of systems of equations were obtained: ሼ‫ݑ‬௉ ሽ ൌ ሾ‫ܣ‬௨ ሿሼ‫ݑ‬ሽ ൅ ሾ‫ܤ‬௨ ሿሼ߮ሽ ሼ߮ ௉ ሽ ൌ ൣ‫ܣ‬ఝ ൧ሼ߮ሽ ൅ ൣ‫ܤ‬ఝ ൧ሼ‫ݑ‬ሽ

(18) (19)

Following a similar strategy, the tractions at the boundary points could be written in the following forms: ሼ‫ݐ‬ሽ ൌ ሼ‫ ݐ‬ு ሽ ൅ ሾ‫ܭܣ‬௨௨ ሿሼ‫ݑ‬ሽ ൅ ൣ‫ܭܣ‬௨ఝ ൧ሼ߮ሽand ሼ‫ݏ‬ሽ ൌ ሼ‫ ݏ‬ு ሽ ൅ ൣ‫ܭܣ‬ఝ௨ ൧ሼ‫ݑ‬ሽ ൅ ൣ‫ܭܣ‬ఝఝ ൧ሼ߮ሽ After conducting some algebraic manipulations, the final coupled systems of equations were of the following forms: ഥ௨ ሿሼ‫ݑ‬ሽ ൅ ൣ‫ܪ‬௨ఝ ൧ሼ߮ሽ ൌ ሾ‫ܩ‬௨ ሿሼ‫ݐ‬ሽ ሾ‫ܪ‬ ഥఝ ൧ሼ߮ሽ ൅ ൣ‫ܪ‬ఝ௨ ൧ሼ‫ݑ‬ሽ ൌ ൣ‫ܩ‬ఝ ൧ሼ‫ݏ‬ሽ ൣ‫ܪ‬

(19) (21)

Particularly worthy of mention, the final equations contained similar boundary primary variables and internal kinematic unknowns to those of a traditional BEM. Boundary conditions could be taken into account as in standard practice and the resulting system of equations was solved by a standard direct solver.

Numerical examples ௤

In our work, we applied the multi-quadrics radial basis functions as follows: ܴ௜ ሺ‫ݎ‬ሻ ൌ ൫‫ݎ‬௜ଶ ൅ ܿ ଶ ൯ , where ‫ݎ‬௜ ൌ ԡ‫ ݔ‬െ ‫ݔ‬௜ ԡ andܿ and ‫ ݍ‬were known as shape parameters. Shape parameter was taken proportionally to minimum distance଴ , defined as the maximum value among the minimum distances in the x, y, and z directions between collocation points. Firstly, we had to validate the proposed tool on a cubic specimen. The boundary of the cube was subdivided into 24 nine-node quadrilateral elements. The boundary nodes were supplemented by 27 internal collocation centers. The non-zero material parameters used in the simulation are collected in tables 1 and 2 below. ‫ܥ‬ଵଵ ͶͳͲͺǤͺ

‫ܥ‬ଶଶ ͶͳͲͺǤͺ

‫ܥ‬ଷଷ ʹʹ͸͹ǤͶͷ

‫ܥ‬ଵଶ ʹͶͲͺǤͺ

‫ܥ‬ଵଷ ͳ͹ͲͲǤ͸

‫ܥ‬ଶଷ ͳ͹ͲͲǤ͸

‫ܥ‬ସସ ͺͷͲ

‫ܥ‬ହହ Ͷ͹ͷǤͺ͸

‫଺଺ܥ‬ Ͷ͹ͷǤͺ͸

Table 1: Material parameters of the microstrech medium (‫)ܽܲܯ‬ ‫ܣ‬௜௝ Classic Auxetic

‫ܣ‬ଵଵ 3 E04 -1 E04

‫ܣ‬ଶଶ 3 E04 -1 E04

‫ܣ‬ଷଷ 2 E04 2 E04

Table 2: coupling macro/micro values for classic and auxetic cube case (‫ܽܲܯ‬ሻ Two sets of parameters ‫ܣ‬௜௝ (equation (3)) were adopted. The first set lead to a lateral contraction of the specimen under traction while the second set induces a lateral expansion. In the latter case the material is auxetic, that is it behaves as a material with a negative Poisson ratio. On the top surface of the sample a uniform traction load of ͳͲିଶ ܰ was applied. The bottom surface was simply supported and the lateral faces of the cube were free of traction. Deformed shapes of both samples are shown in figures 1a and b. Let us point out that for representation purpose the displacement were multiplied by a factor which worth ͷͳͲହ in the classical case and ͳͲ଻ in the auxetic case.


a)

b)

Fig. 1 – Cubic specimen a) Cube under tension – classic case b) Cube under tension – auxetic case The results obtained for each case are the expected ones as compared to analytical solution. Indeed, for the classical specimen, the area of each cross section of the cube is reduced. Conversely, this area is increased for the auxetic specimen. It is well known that the shape parameters of radial basis functions may affect the accuracy of the numerical calculation. In other to check this point we considered the results at a boundary point (‫ܣ‬ሺെͲǤʹͷǢ െͲǤʹͷǢ ͲǤͷሻ) and an internal point (‫ܤ‬ሺͲǤʹͷǢ ͲǤʹͷǢ െͲǤʹͷሻ) as these parameters are varied. The relative errors on the strain values as compared to analytical solutions are collected in table 3.

a)

b)

qc ͲǤͷ͵

ͲǤ͹ͷ

ͲǤͻ͹

ͳǤͲ͵

ͳǤʹͷ

ͳǤͶ͹

ͳͲିଵ

ͳͲିଶ

ͳͲିଷ

ͳͲିସ

ͳͲିଵ

qc

ͳͲିଶ

ͳͲିଷ

ͳͲିସ

ͷǤͷͻ͸ െ ͲͶ

ߜሺߝଵଵ ሻ

ͺǤ͵͵ͻ‫ ܧ‬െ ͲͶ

ͳǤʹ͸ʹ െ ͲͶ

ͳǤͲͻͷ െ Ͳͷ

ͳǤͲͻͷ െ Ͳͷ

ߜ ሺߝଵଵ ሻ

ͳǤͶͷͳ െ Ͳ͵

ͶǤʹʹͶ െ ͲͶ

ͷǤͷͻ͸ െ ͲͶ

ߜሺߝଶଶ ሻ

ͳǤͲͳͺ‫ ܧ‬െ Ͳ͵

ͳǤͶͺͳ െ ͲͶ

ͳǤͲͻͷ െ Ͳͷ

ͳǤͲͻͷ െ Ͳͷ

ߜ ሺߝଶଶ ሻ

ͶǤͲͲͷ െ ͲͶ

͸Ǥͻ͸ͺ െ ͲͶ

ͷǤͷͻ͸ െ ͲͶ

ͷǤͷͻ͸ െ ͲͶ

ߜሺߝଷଷ ሻ

ͺǤͳͷͷ‫ ܧ‬െ Ͳʹ

ͺǤͳͻ͸ െ Ͳʹ

ͺǤʹͲʹ െ Ͳʹ

ͺǤʹͲͷ െ Ͳʹ

ߜ ሺߝଷଷ ሻ

ͳǤ͸Ͳ͹ െ Ͳͳ

ͳǤ͸Ͳ͸ െ Ͳͳ

ͳǤ͸Ͳ͸ െ Ͳͳ

ͳǤ͸Ͳ͸ െ Ͳͳ

ߜሺߝଵଵ ሻ

ͳǤͶͷͳ െ Ͳ͵

ʹǤͳ͸͹ െ ͲͶ

͹Ǥͻͷ͵ െ Ͳͷ

ͶǤ͸ͻͳ െ ͲͶ

ߜሺߝଶଶ ሻ

ͳǤ͸͵ͷ െ Ͳ͵

͵Ǥ͵ʹͲ െ ͲͶ

ͳǤͻͶͺ െ ͲͶ

͵Ǥͷ͵ͻ െ ͲͶ

ͲǤͷ͵

ͲǤ͹ͷ

ߜ ሺߝଵଵ ሻ

ʹǤͲ͸ͺ െ Ͳ͵

ͺǤ͵͵ͻ െ ͲͶ

͸Ǥʹͺʹ െ ͲͶ

ͳǤͶͺͳ െ ͲͶ

ߜ ሺߝଶଶ ሻ

ͳǤͲͳͺ െ Ͳ͵

ʹǤͳ͸͹ െ ͲͶ

ͶǤʹʹͶ െ ͲͶ

ͻǤͲʹͷ െ ͲͶ

ߜ ሺߝଷଷ ሻ

ͳǤ͸Ͳ͹ െ Ͳͳ

ͳǤ͸Ͳͺ െ Ͳͳ

ͳǤ͸ͳͲ െ Ͳͳ

ͳǤ͸ͳͲ െ Ͳͳ

ߜ ሺߝଵଵ ሻ

ʹǤͶ͵͸ െ Ͳʹ

ͳǤ͹Ͳʹ െ Ͳʹ

ͶǤ͹ʹͳ െ Ͳ͵

ͳǤͲͷͲ െ Ͳʹ

ߜሺߝଷଷ ሻ

ͺǤͳͶͳ െ Ͳʹ

ͺǤʹͳͺ െ Ͳʹ

ͺǤʹͶ͸ െ Ͳʹ

ͺǤʹͷ͹ െ Ͳʹ

ߜሺߝଵଵ ሻ

ʹǤ͵͸͹‫ ܧ‬െ Ͳʹ

ͳǤ͸ͶͲ‫ ܧ‬െ Ͳʹ

ͷǤ͵͵ͺ‫ ܧ‬െ Ͳ͵

ͻǤͺͺ͹‫ ܧ‬െ Ͳ͵

ߜሺߝଶଶ ሻ

ʹǤ͵ͻʹ െ Ͳʹ

ͳǤ͸ͷͻ െ Ͳʹ

ͷǤͳͷͷ െ Ͳ͵

ͳǤͲͲ͹ െ Ͳʹ

ߜ ሺߝଶଶ ሻ

ʹǤ͵͵ͳ െ Ͳʹ

ͳǤͷͻ͹ െ Ͳʹ

ͷǤ͹͹ʹ െ Ͳ͵

ͻǤͶͷ͵ െ Ͳ͵

ߜሺߝଷଷ ሻ

ͺǤͳʹʹ െ Ͳʹ

ͺǤʹͳͲ െ Ͳʹ

ͺǤʹͶͲ െ Ͳʹ

ͺǤʹͷͻ െ Ͳʹ

ߜ ሺߝଷଷ ሻ

ͳǤ͸Ͳ͹ െ Ͳͳ

ͳǤ͸Ͳͻ െ Ͳͳ

ͳǤ͸ͳͳ െ Ͳͳ

ͳǤ͸ͳʹ െ Ͳͳ

ͲǤͻ͹

ͳǤͲ͵

ߜ ሺߝଵଵ ሻ

͹Ǥͻͷ͵ െ Ͳͷ

ͷǤͲͺ͸ െ Ͳ͵

͹Ǥ͵Ͷͻ െ Ͳ͵

͸ǤͲͻ͵ െ Ͳ͵

ߜ ሺߝଶଶ ሻ

ͻǤ͹ͳͳ െ ͲͶ

ͶǤͲ͵ͷ െ Ͳ͵

͸Ǥʹͻͻ െ Ͳ͵

͹ǤͳͶ͵ െ Ͳ͵

ߜሺߝଵଵ ሻ

͸ǤͲ͸͵ െ ͲͶ

ͶǤͶͲͲ െ Ͳ͵

͸Ǥ͹͵ʹ െ Ͳ͵

͸Ǥ͹ͳͲ െ Ͳ͵

ߜሺߝଶଶ ሻ

͵Ǥͷ͵ͻ െ ͲͶ

ͶǤ͸ͷ͵ െ Ͳ͵

͸Ǥͻͳ͸ െ Ͳ͵

͸Ǥͷʹ͸ െ Ͳ͵

ߜሺߝଷଷ ሻ

ͺǤͳͳͶ െ Ͳʹ

ͺǤʹͲʹ െ Ͳʹ

ͺǤʹ͵ʹ െ Ͳʹ

ͺǤʹͶ͸ െ Ͳʹ

ߜሺߝଵଵ ሻ

ͳǤͺͶͳ െ Ͳ͵

ͶǤ͹ͲͶ െ Ͳʹ

ͳǤ͹ͻͺ െ Ͳʹ

ͶǤͶͲͲ െ Ͳ͵

ߜሺߝଶଶ ሻ

ͳǤͷʹͲ െ Ͳ͵

ͶǤ͸͹ͺ െ Ͳʹ

ͳǤͺʹ͵ െ Ͳʹ

ͶǤ͹ʹͳ െ Ͳ͵

ߜ ሺߝଶଶ ሻ

ʹǤʹͲ͸ െ Ͳ͵

ͶǤ͹ͶͲ െ Ͳʹ

ͳǤ͹ͷͷ െ Ͳʹ

ͶǤͲ͵ͷ െ Ͳ͵

ߜሺߝଷଷ ሻ

ͺǤͲͺͻ െ Ͳʹ

ͺǤͳ͸͸ െ Ͳʹ

ͺǤͳͺ͵ െ Ͳʹ

ͺǤͳͺͷ െ Ͳʹ

ߜ ሺߝଷଷ ሻ

ͳǤ͸Ͳ͹ െ Ͳͳ

ͳǤ͸Ͳͻ െ Ͳͳ

ͳǤ͸ͳͲ െ Ͳͳ

ͳǤ͸ͳͲ െ Ͳͳ

ߜሺߝଵଵ ሻ

ͷǤͲͳ͹ െ Ͳ͵

ͳǤ͵ʹͷ െ Ͳʹ

ͳǤ͹ͻ͸ െ Ͳʹ

͵Ǥʹͳʹ െ Ͳ͵

ߜሺߝଶଶ ሻ

ͷǤ͵͵ͺ െ Ͳ͵

ͳǤ͵ͷ͹ െ Ͳʹ

ͳǤ͹͸Ͷ െ Ͳʹ

ʹǤͺͻͳ െ Ͳ͵

ߜሺߝଷଷ ሻ

ͺǤͲͷͻ െ Ͳʹ

ͺǤͳʹͷ െ Ͳʹ

ͺǤͳ͵ͳ െ Ͳʹ

ͺǤͳ͵ͳ െ Ͳʹ

ͳǤʹͷ

ͳǤͶ͹

ߜ ሺߝଷଷ ሻ

ͳǤ͸Ͳ͹ െ Ͳͳ

ͳǤ͸Ͳͻ െ Ͳͳ

ͳǤ͸ͳͳ െ Ͳͳ

ͳǤ͸ͳʹ െ Ͳͳ

ߜ ሺߝଵଵ ሻ

ͳǤͳͷͷ െ Ͳ͵

ͶǤ͸͵ͷ െ Ͳʹ

ͳǤͺ͸Ͳ െ Ͳʹ

ͷǤͲͺ͸ െ Ͳ͵

ߜ ሺߝଵଵ ሻ

ͷǤ͹Ͳ͵ െ Ͳ͵

ͳǤ͵ͻ͵ െ Ͳʹ

ͳǤ͹ʹ͹ െ Ͳʹ

ʹǤͷʹ͹ െ Ͳ͵

ߜ ሺߝଶଶ ሻ

ͶǤ͸ͷ͵ െ Ͳ͵

ͳǤʹͺͺ െ Ͳʹ

ͳǤͺ͵ʹ െ Ͳʹ

͵Ǥͷ͹͹ െ Ͳ͵

ߜ ሺߝଷଷ ሻ

ͳǤ͸Ͳ͸ െ Ͳͳ

ͳǤ͸Ͳͺ െ Ͳͳ

ͳǤ͸Ͳͺ െ Ͳͳ

ͳǤ͸Ͳͺ െ Ͳͳ

Table 3 – Relative error ߜ for the principals strains (%) Auxetic case – a) Point on the top of the cube – b) Point inside of the cube


123

The maximum error is 0.17 % when the multi-quadric shape parameters were varied in the ranges [0.5; 1.5] for q and [10-4; 10-1] for c. The proposed approach is clearly effective and accurate is this case of uniform loading. Similar accurate results are obtained when adopting a cylindrical specimen. Next, let us consider a tubular specimen which may be viewed as a simplified representation of the left ventricle or as a part of an artery. The boundary of this tube was subdivided into 144 nine-node quadrilateral elements. The boundary nodes were supplemented by 192 internal collocation centers. Moreover, the multiquadric shape parameters taken for each simulations with the tube are 1.03 for q and ͳͲିଷ for c. In a first example, the tube is submitted to an internal pressure (32 kPa). Its outer lateral surface is free of traction while the end surfaces are constrained against axial displacement. As expected, the displacement is symmetrical in x and y as shown in fig 2. The developed numerical tool was also able to treat the case of torsion loading of the specimen as shown in fig 3. Indeed the rotation of cross sections incresase gradually from zero at the clampbed lower surface to a maximum value at the upper surface.

a)

b)

c)

Fig. 2 – a) Pressure distribution b) Slice of the tube – X displacement c) Slice of the tube – Y displacement

Fig. 3 – Tubular shear problem a) Total displacement – Top view b) Total displacement – All tube


4 Conclusion The LPI-BEM method were applied to the solution of anisotropic microdilatation problem. The corresponding equations are those of anisotropic poroelasticity. The results obtained in the case of uniform loading of a cubic specimen are accurate and stable with respect to the multi-quadrics shape parameters. This alternative to other mesh reduction approach seems promising for a wide range of problems. The approach was also successful for different geometries and loading conditions. This numerical work illustrated the great potential of the “local point interpolation – boundary element” method for addressing problems relating to the microstretch modeling of living tissue. Following on from this study, another study is scheduled to continue with our investigations, taking into account the non homogeneity of the micro-macro coupling tensor in order to simulate necrotic areas in a living tissue. Extension to large deformation will also be proposed in a future work.

References [1] J. Rosenberg and R. Cimrman, « Microcontinuum approach in biomechanical modeling », Mathematics and Computers in Simulation 61 (2003) 249-260 [2] J-Ph. Jehl and R. Kouitat Njiwa, « A (Constrained) Microstretch Approach in Living Tissue Modeling: a Numerical Investigation Using the Local Point Interpolation – Boundary Element Method »Beteq 2014 [3] J-Ph. Jehl and R. Kouitat Njiwa, « A (Constrained) Microstretch Approach in Living Tissue Modeling: a Numerical Investigation Using the Local Point Interpolation – Boundary Element Method », Computer Modeling in Engineering & Sciences Vol.102, No.5, 2014 [4] A. C. Eringen and E. S. Suhubi, « Nonlinear theory of simple micro-elastic solids—I », International Journal of Engineering Science, vol. 2, no 2, p. 189-203, mai 1964. [5] D. Iesan, " Deformation of orthotropic porous elastic bars under lateral loading", Arch. Mech. vol 62 n° 1, p. 3-20, 2010 [6] G. R. Liu and Y. T. Gu, « A local radial point interpolation method (LRPIM) for free vibration analyses of 2-D solids », Journal of Sound and Vibration, vol. 246, no 1, p. 29-46, sept. 2001. [7] R. Kouitat Njiwa, « Isotropic-BEM coupled with a local point interpolation method for the solution of 3D-anisotropic elasticity problems », Engineering Analysis with Boundary Elements, vol. 35, no 4, p. 611-615, avr. 2011. [8] N. Thurieau, R. Kouitat Njiwa, and M. Taghite, « A simple solution procedure to 3D-piezoelectric problems: Isotropic BEM coupled with a point collocation method », Engineering Analysis with Boundary Elements, vol. 36, no 11, p. 1513 1521, nov. 2012. [9] M. Schwartz, N. T. Niane, and R. Kouitat Njiwa, « A simple solution method to 3D integral nonlocal elasticity: Isotropic-BEM coupled with strong form local radial point interpolation », Engineering Analysis with Boundary Elements, vol. 36, no 4, p. 606 612, avr. 2012. [10] N. Kirchner and P. Steinmann, « Mechanics of extended continua: modeling and simulation of elastic microstretch materials », Computational Mechanics, vol. 40, no 4, p. 651 666, sept. 2007. [11] Johnston P. R., 2003, “A cylindrical model for studying subendocardial ischaemia in the left ventricle,” Mathematical Biosciences, 186(1), pp. 43–61. [12] Nardinocchi P., Puddu P. E., Teresi L., and Varano V., 2012, “Advantages in the torsional performances of a simplified cylindrical geometry due to transmural differential contractile properties,” European Journal of Mechanics - A/Solids, 36(0), pp. 173–179. [13] Cohn J. N., Johnson G. R., Shabetai R., Loeb H., Tristani F., Rector T., Smith R., and Fletcher R., 1993, “Ejection fraction, peak exercise oxygen consumption, cardiothoracic ratio, ventricular arrhythmias, and plasma norepinephrine as determinants of prognosis in heart failure. The V-HeFT VA Cooperative Studies Group,” Circulation, 87(6 Suppl), pp. VI5–16. [14] Juillière Y., Barbier G., Feldmann L., Grentzinger A., Danchin N., and Cherrier F., 1997, “Additional predictive value of both left and right ventricular ejection fractions on long-term survival in idiopathic dilated cardiomyopathy,” Eur. Heart J., 18(2), pp. 276–280.


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[15] Hallstrom A., Pratt C. M., Greene H. L., Huther M., Gottlieb S., DeMaria A., and Young J. B., 1995, “Relations between heart failure, ejection fraction, arrhythmia suppression and mortality: analysis of the Cardiac Arrhythmia Suppression Trial,” J. Am. Coll. Cardiol., 25(6), pp. 1250–1257. [16] Curtis J. P., Sokol S. I., Wang Y., Rathore S. S., Ko D. T., Jadbabaie F., Portnay E. L., Marshalko S. J., Radford M. J., and Krumholz H. M., 2003, “The association of left ventricular ejection fraction, mortality, and cause of death in stable outpatients with heart failure,” Journal of the American College of Cardiology, 42(4), pp. 736–742. [17] Cho G.-Y., Marwick T. H., Kim H.-S., Kim M.-K., Hong K.-S., and Oh D.-J., 2009, “Global 2Dimensional Strain as a New Prognosticator in Patients With Heart Failure,” Journal of the American College of Cardiology, 54(7), pp. 618–624. [18] Reimer K. A., Lowe J. E., Rasmussen M. M., and Jennings R. B., 1977, “The wavefront phenomenon of ischemic cell death. 1. Myocardial infarct size vs duration of coronary occlusion in dogs,” Circulation, 56(5), pp. 786–794.


Two-dimensional Stokes Flow of an Electrically Conducting Fluid in a Channel Under Uniform Magnetic Field Merve Gürbüz1 and M. Tezer-Sezgin2 1

2

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

Keywords: Stokes MHD flow, radial basis functions.

Abstract. This paper deals with the two-dimensional flow of a viscous, incompressible and electrically conducting fluid in a lid-driven rectangular duct under the effect of a uniform horizontally applied magnetic field. The hydrodynamic and electromagnetic equations are solved simultaneously using Stokes approximation and neglecting convection terms. The governing partial differential equations in terms of velocity components, pressure, stream function and vorticity are solved iteratively by using radial basis function approximations to terms other than diffusion. The procedure includes the satisfaction of boundary conditions during the approximations of the right hand sides for each equation and obtaining not only the particular solution but at the same time the solution itself. The unknown boundary conditions for the pressure and the vorticity are obtained from the momentum equations and the vorticity definition, respectively. For obtaining vorticity boundary values, stream function equation is discretized using finite difference which includes also interior values. Pressure boundary conditions are derived by using coordinate matrix for space derivatives and finite difference for pressure gradients. The numerical results show that as Hartmann number increases flow is flattened and boundary layers are developed close to the moving lid and at the separation region of the main and secondary flows. This is an expected behavior in MHD flow. The increase in the intensity of the magnetic field increases pressure magnitudes and separates pressure into four vortices forming also boundary layers in the separation region. The solution is obtained in a considerably low computational cost through the use of radial basis functions in the approximation.

Introduction Magnetohydrodynamics (MHD) focuses on fluid flows under the effect of a magnetic field. Its popularity comes from the wide range of industrial applications such as MHD pumps, geothermal energy extraction, metal production, plasma physics and nuclear fusion. The MHD governing equations are the coupled system of Navier-Stokes equations and Maxwell’s equations of electromagnetics through Ohm’s law. In this study, the fluid is taken as incompressible, viscous and electrically conducting. The flow is assumed to be slow (Stokes approximation due to small values of Reynolds number). Induced magnetic field is also neglected due to the assumption of small magnetic Reynolds number. The present study applies the radial basis function (RBF) approximations to solve two-dimensional MHD Stokes equations in a lid-driven rectangular duct under a uniform magnetic field applied in the x−direction. Stokes flow has been solved by using method of fundamental solutions for 2D and 3D cases in lid-driven cavities by Young et al. [1]. RBF approximation has been applied by Kutanaei et al. [2] and Bustamante et al. [3] to solve Stokes equations in cavities and backward step flow regions. Magnetic field influence has been added to Stokes flow in Mramor et al. [4] in the RBF formulation. In this study, we combine the effective use of RBF approximation in solving MHD Stokes flow equations for several values of Hartmann number.


127

Mathematical formulation Consider the two-dimensional steady flow of a viscous, incompressible and electrically conducting fluid in a lid-driven rectangular duct under the effect of a uniform horizontally applied magnetic field. The governing coupled equations are the continuity equation and the Navier-Stokes equations of hydrodynamics, and the Maxwell’s equations of electromagnetics through Ohm’s law. These equations are given in non-dimensional form as [5] ∇·u = 0 (1) Re(u · ∇)u = −∇p + ∇2 u + M 2 (u × H) × H

(2)

Rm (u × H) = ∇ × H

(3)

where H is the magnetic field, M is the Hartmann number, and Re and Rm are the Reynolds and magnetic Reynolds numbers, respectively. u = (u, v) is the velocity field. Taking MHD Stokes approximation (Re 0). This criterion, proposed originally in [11],is: min

t(x)

0≤x≤Δa tc (ψ(x))

≥ 1,

where

t(x) =

σ 2 (x) + τ 2 (x)

and

tc (ψ(x)) =

σc2 (ψ(x)) + τc2 (ψ(x)).

(7) It should be noticed that the traction vector, as well as the previously defined fracture toughness, depends on the mixity at the analysed point x. The normal and shear critical tractions can be expressed in terms of the normal critical value for pure mode I (¯ σc ) and dimensionless functions, as was similarly done for the fracture toughness in (4). Then, following [2, 8]: |ψ| ≤ π2 , ˆ c (ψ(x)) · cos ψ(x), (8a) σc (ψ(x)) = σ ¯c G −| cot ψ(x)|, |ψ| ≥ π2 , π ˆ c (ψ(x)) · sin ψ(x), |ψ| ≤ π2 , (8b) τc (ψ(x)) = σ ¯c ξ G sign ψ(x), |ψ| ≥ 2 ,


Glass-Epoxy

Ef (GPa) 70.8

νf 0.22

Em (GPa) 2.79

νm 0.33

GIc (Jm−2 ) 2

σ ¯c (MPa) 90

kn (MPa/μm) 2025

kt /kn 0.25

Table 1: Material and interface properties (kn for μ = 1) A way to characterize the FFF+LEBIM coupled criterion used in the present work, is by means of ¯ Ic 2 /¯ = σmax σc2 , the dimensionless characteristic parameter defined in [7], for pure mode I is μ = 2knσ¯G 2 c ¯c , are the maximum and critical traction associated to the energy and stress based where σmax and σ criterion, respectively. Thus, when μ = 1 the present model becomes the original LEBIM. When μ value increases the interface becomes stiffer, then if μ → ∞ a perfect (rigid) interface may be obtained. As can be seen from the previous sections the fracture toughness, strength and stiffness of the interface are independent in the present FFM + LEBIM approach. Notice that in the original LEBIM, these variables were directly related by an equation. Cylindrical inclusion under biaxial transverse loads A plane strain problem of a fibre embedded in an “infinite” matrix is considered, an undamaged interface is considered initially. The fibre-matrix system is subjected to a biaxial remote loading. Fibre radius a and a 2H side square matrix with H/a = 200/3 are used, see figure 1. Both inclusion

σy

σy

y

y

σx

r=a

x

σx

2H

σx

r=a

B

σ ψσ τ θd θo

σy

σy

2H

2H

(a)

A x

σx

2H

(b)

Figure 1: Inclusion problem configuration under biaxial remote transverse loads (a) without and (b) with a partial debond. and matrix are considered to be isotropic linear elastic materials, whose characteristics are presented in Table 1. LEBIM models the interface as a continuum spring distribution, with kn and kt given in Table 1, which corresponds to μ = 1. For larger values of μ, kn increases proportionally. θd is defined as the debond angle which defines the crack size in each step. When a crack propagation occurs springs placed within the area of θd lose the capability of transmit the load between solids (stiffness becomes zero). Then, the springs located at the ends of the area determined by θd represent the crack tips. The applied remote loads, σx∞ and σy∞ with σx∞ ≥ σy∞ , are shown in Figure 9. The following general load-biaxiality parameter is used to represent the biaxility relation between the applied remote loads: σx∞ + σy∞ , −1 ≤ χ ≤ 1, χ= (9) 2max{|σx∞ | , σy∞ } In the present paper the numerical models solved fulfill σx∞ ≥ σy∞ . Convergency study of the BEM mesh in the problem under study The procedure proposed in the present work requires the evaluation of the energy in a critical point (i.e. crack tip) in each step. This fact may lead to numerical errors if the mesh used is not small


135

enough. It should be mentioned that, although interface tractions in LEBIM are bounded at the crack tip, local tractions in the zone close to the interface crack tip follow an asymptotic law [3, 4]. Then, errors are caused because tractions at a critical point converges to the Griffith (perfect interface) solution when the fibre-matrix interface becomes stiffer. The solved numerical examples of the single

3

G

2 GI

1 0

4

0.10 0.16 0.50 1.50

5

º º º º

G

3 2

GII

σx

GII

mesh mesh mesh mesh

GI

1

0

50

100

θd (º)

0

150

G E *, GI E *, GII E * a σx a σx a

5

º º º º

50

0

(a) μ = 1

4

0.10 0.16 0.50 1.50

º º º º

3 G

2 GII

1

100

θd (º)

mesh mesh mesh mesh

σx

4

0.10 0.16 0.50 1.50


mesh mesh mesh mesh

σx


5

0

150

0

50

(b) μ = 4

θd (º)

GI

100

150

(c) μ = 8

Figure 2: ERRs for different crack sizes (different θd values) with χ =0.5 (uniaxial case) for different mesh sizes and (a) μ =1, (b) μ =4 and (c) μ =8.

σ σx

2 1

τ

0 60

70

2 1

at crack tip

θ (º)

90

100

(a) μ = 1 and θd = 36◦

110

4 3 2

σ at

crack tip

1

0

80

σ with mesh 0.10º σ with mesh 0.16º σ with mesh 0.50º σ with mesh 1.50º τ with mesh 0.10º τ with mesh 0.16º τ with mesh 0.50º τ with mesh 1.50º τ at crack tip

5

τ σx

3

6

,

τ σx

4

,

3

,

τ σx

σ at crack tip

σ with mesh 0.10º σ with mesh 0.16º σ with mesh 0.50º σ at crack tip σ with mesh 1.50º τ with mesh 0.10º τ with mesh 0.16º τ with mesh 0.50º τ with mesh 1.50º τ at crack tip

5

σ σx

5 4

6

σ with mesh 0.10º σ with mesh 0.16º σ with mesh 0.50º σ with mesh 1.50º τ with mesh 0.10º τ with mesh 0.16º τ with mesh 0.50º τ with mesh 1.50º

σ σx

6

0

60

70

80

θ (º)

90

100

(b) μ = 4 and θd = 48◦

110

60

70

80

θ (º)

90

100

110

(c) μ = 8 and θd = 48◦

Figure 3: Normal and shear tractions along the interface for a specific crack size (θd value) with χ =0.5 (uniaxial case) for different mesh sizes and (a) μ =1 (b) μ =2, (c) μ =4 and (a) μ =8. fibre problem, see figure 1, model the interface with a distribution of springs between the nodes along the fibre and matrix. The elements used in the mesh are lineal and continuous. Thus, each element has two geometrical nodes (one at each extreme) which coincides with the integration points. Then, when a spring fails (creating a portion of an interface crack), the minimum Δθd size is restricted by the mesh size. Due to the facts mentioned above, before studying closely the considered problem, a convergency study of the mesh used is necessary. The single fibre problem for a glass-epoxy system with χ = 0.5 (uniaxial traction with σx∞ = 0), for different values of μ (1, 4 and 8) are taken into account. As the solution of this problem is symmetric, for this convergency study a symmetry plane coincident with x-axis is considered. Thus, a mesh refinement is feasible and at same time computational time is reduced. The single modes and total ERRs for different crack sizes (θd ) are depicted in figure 2. Four types of mesh are considered for the interface between fibre and matrix. Then, the element sizes considered are defined by the arc circumferences determined by the polar angles 0.10◦ , 0.16◦ , 0.50◦ and 1.50◦ .


It can be seen that when the interface becomes stiffer, increasing μ, the convergency of the results is more difficult. Then, in order to get an adequate value of the ERR for a stiff interface a fine mesh is required. This fact is a consequence of the difficulty in the approximation of tractions in a critical point (i.e. crack tip) by the use of linear elements in the BEM code. The distribution of normal and shear tractions for a specific crack size (θd ) is represented in figure 3. The crack size chosen for each value of μ, is the one with more differences in the ERR figures. In figure 3, different slopes produced are observed for different μ values. A large μ value (stiff interface) has an abrupt change of slope in the zone close to the crack tip. Then, in order to obtain an adequate approximation of tractions in that zone, a fine mesh is necessary. From the results obtained in this convergency study a slight mesh dependency is observed. Thus, in order to get reliable results in the model under study, a mesh size with a polar angle of 0.15◦ is used in the following. Numerical Results The interface crack behaviour is shown in figure 4. This plot represents the crack size (θd ) versus the necessary remote applied load in x-direction that causes the onset or propagation of the interface crack. Specifically, the plot represented in figure 4(a) is the solution for the single fibre problem in a glass-epoxy system, under a biaxial traction (σx∞ =σy∞ , i.e. χ = 0.75). The Hutchinson and Suo [10] interface failure criterion is used for these results shown in figure 4(a). It should be noticed from figure 4(a), that the solution for the coupled criterion introduced in this paper for μ = 1 is similar to the solution obtained with the original LEBIM formulation. It should also be noticed that a slight 1

120 FFM+LEBIM μ=4 FFM+LEBIM μ=2 FFM+LEBIM μ=1

110 100

0.8

LEBIM μ=1

90

0.7

σx

∞

σx σ c

70

∞

(MPa)

80

60

36.2

35.8

40

0.6 0.5

36

50

0.4

35.6 144

30 20

χ = 0.75 χ = 0.50 χ = 0.00 χ =-0.25

0.9

145

146

0

50

100

θ d (º) (a)

θa

0.3

θa 150

200

0.2

0

50

100

θ d (º)

150

200

(b)

Figure 4: (a)Normalized remote applied load in x-direction with respect to the semidebond angle θd , for different μ values. (b) Normalized remote applied load in x-direction with respect to the semidebond angle θd , for different biaxial load combinations using μ = 4. decrease of the critical loads σx∞ is obtained at the same time that the interface becomes stiffer. This effect may be caused by the influence of the increase of stiffness in the mixity angle. A detailed study showed that when μ increases, in this problem, the mixity angle changes in every undamaged interface point. In figure 4(b) the crack size, θd , versus the remote applied load in x-direction, σx∞ , for different configurations of the biaxial loading is presented, for μ = 4. Where, χ = 0.75 represents an biaxial traction, χ = 0.50 represent the uniaxial traction case, and χ = 0 and χ = −0.25 are two biaxial traction-compression cases, see (9). From the figure it can be seen that a compressive remote load, acting as a secondary load, makes easier crack onset (i.e. a lower critical load is necessary to produce crack onset). At the same time in this cases the unstable crack size, denoted by the arrest angle θa , is smaller than when a traction is acting as a secondary load. This fact may be produced due to a compressive state may lead to the close of the interface crack when it reaches a certain size.


137

Conclusions A novel computational procedure has been developed and implemented in a 2D BEM code, it combines the FFM and LEBIM formulation. This procedure opens new possibilities to study the onset and propagation of cracks along interfaces and adhesive layers. This is due to the use of realistic values of strength, fracture toughness and in particular interface stiffness, which can be significantly higher than in the original LEBIM, and are decoupled in the present procedure. It is noticeable that when μ = 1 the present procedure may become the original LEBIM. It is interesting to observe that for the present fibre-matrix system the predictions of the crack onset and propagation obtained by FFM and LEBIM differ only slightly from those obtained by the original LEBIM, which indicates only a moderate dependence of these predictions on the interface stiffness. Acknowledgements The work was supported by the Junta de Andaluc´ıa and European Social Fund (Projects of Excellence TEP-1207, TEP-2045, TEP-4051, P12-TEP-1050), the Spanish Ministry of Education and Science (Projects TRA2006-08077 and MAT2009-14022) and Spanish Ministry of Economy and Competitiveness (Projects MAT2012-37387 and DPI2012-37187). References [1] F. Par´ıs, E. Correa and V. Mantiˇc. Kinking of transverse interface cracks between fiber and matrix. Journal of Applied Mechanics, 74:703–716, 2007. [2] L. T´ avara, V. Mantiˇc, E. Graciani and F. Par´ıs. BEM analysis of crack onset and propagation along fiber-matrix interface under transverse tension using a linear elastic-brittle interface model. Engineering Analysis with Boundary Elements, 35:207–222, 2011. [3] L. T´ avara, V. Mantiˇc, E. Graciani, J. Ca˜ nas and F. Par´ıs. Analysis of a crack in a thin adhesive layer between orthotropic materials. An application to composite interlaminar fracture toughness test. CMES-Computer Modeling in Engineering and Sciences, 58(3):247–270, 2010. [4] S. Lenci. Analysis of a crack at a weak interface. International Journal of Fracture, 108:275–290, 2001. [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, 46:1287–1304, 2009. [6] V. Mantiˇc and I.G. Garc´ıa. Crack onset and growth at the fibre–matrix interface under a remote biaxial transverse load. Application of a coupled stress and energy criterion. International Journal of Solids and Structures, 49:2273–2290, 2012. [7] P. Cornetti, V. Mantiˇc and A. Carpinteri. Finite fracture mechanics at elastic interfaces. International Journal of Solids and Structures, 49:1022–1032, 2012. [8] V. Mantiˇc, L. T´ avara, A. Blázquez, E. Graciani and F. Par´ıs. Application of a linear elasticbrittle interface model to the crack initiation and propagation at fibre-matrix interface under biaxial transverse loads. ArXiv preprint. arXiv:1311.4596, 2013. [9] A. Carpinteri, P. Cornetti and N. Pugno. Edge debonding in FRP strengthened beams: Stress versus energy failure criteria. Engineering Structures, 31:2436–2447, 2009. [10] J.W. Hutchinson and Z. Suo. Mixed mode cracking in layered materials. Advances in Applied Mechanics. 29:63–191, 1992. [11] D. Leguillon. Strength or toughness? a criterion for crack onset at a notch. European Journal of Mechanics A/Solids, 21:61–72, 2002.


Natural convection flow of a nanofluid in an enclosure under a uniform magnetic field M. Tezer-Sezgin1 , Canan Bozkaya1 and Önder Türk2 1

Department of Mathematics, Middle East Technical University, 06800, Ankara, Turkey, e-mail: [email protected] 2 Department of Mathematics, Gebze Technical University, 41400, Kocaeli, Turkey.

Keywords: DRBEM, FEM, Natural convection, nanofluid, magnetic field.

Abstract. In this study, the natural convection in square enclosures filled with water based aluminum oxide (Al2 O3 ) under the influence of an externally applied horizontal magnetic field, is considered numerically. The flow is steady, two-dimensional and laminar, the nanoparticles and water are assumed to be in thermal equilibrium. The nanofluid is considered as Newtonian and incompressible. The induced magnetic field, dissipation and Joule heating are neglected. No-slip boundary conditions for the velocity are imposed on the walls of the enclosure. The vertical walls are isothermal, and the horizontal walls are adiabatic. The mathematical modeling of the problem results in a coupled nonlinear system of partial differential equations in terms of the velocity, pressure and the temperature of the nanofluid. The equations are solved in terms of stream function, vorticity and temperature using both dual reciprocity boundary element method (DRBEM) and the finite element method (FEM) for several values of characteristic flow parameters, namely, solid volume fraction (φ ), Rayleigh (Ra) and Hartmann (Ha) numbers. The nonlinear terms are treated as inhomogeneity in DRBEM approach, which enables the use of the fundamental solution of the Laplace equation. The discretization of only the boundary of the region is the main advantage of DRBEM giving small algebraic systems to be solved at a small expense. Finite element method on the other hand, is capable of giving more accurate results especially for very high characteristic flow parameters, but it results in large sized algebraic systems requiring high computational cost. Numerical results are obtained for 0 ≤ φ ≤ 0.2, and Rayleigh number and Hartmann number values up to 107 and 500, respectively. The buoyancy-driven circulating flows undergo inversion of direction as Ra and Ha increase, and magnitudes of streamlines and vorticity contours increase as Ra increases but decrease as Ha increases. The isotherms have a horizontal profile for high Ra values as a result of convective dominance over conduction. As Ha is increased, the effect of the convection on the flow is reduced, and the isotherms tend to have vertical profiles. An increase in the solid volume fraction decreases the strength of streamlines and the temperature increase is more pronounced in the lower half part of the enclosure.

Introduction The natural convection under the influence of a magnetic field have received considerable attention over the last decades due to their wide variety of engineering applications, such as crystal growth, nuclear reactor cooling, microelectronic devices, and solar technology. The nanofluid is a liquid-solid mixture in which metallic or nonmetallic nanoparticles are suspended. The suspended particles change transport properties and heat transfer performance of the nanofluid, which exhibits a great potential in enhancing heat transfer. The mechanism of heat transfer of nanofluids is investigated by many researchers in recent years. The dual reciprocity boundary element method (DRBEM) solution of the unsteady natural convective flow of nanofluids in enclosures with a heat source is presented in [1]. The natural convection in a square enclosure filled with a nanofluid under the effect of a magnetic field is investigated numerically using control volume formulation SIMPLE algorithm in [2]. The influence of an external magnetic field on the flow and heat transfer of a nanofluid in a semi annulus enclosure with sinusoidal hot wall is investigated in [3], using a control volume based finite element method. In the present study, the natural convection in square enclosures filled with a nanofluid (water-Al2 O3 ) is investigated numerically. The steady, two-dimensional and laminar flow is under the influence of an horizontally applied external magnetic field. The nanoparticles and water are assumed to be in thermal equilibrium. The


139

nanofluid is considered as Newtonian and incompressible, and the induced magnetic field, dissipation and Joule heating effects are neglected. The problem is described by a coupled, nonlinear system of PDEs with appropriate boundary conditions. The solution is obtained iteratively using two different numerical approaches, namely, DRBEM and FEM. The discretization of only the boundary of the region resulting in small algebraic systems, constitutes the main advantage of DRBEM, whereas, well developed and widely used reliable numerical approach FEM is used for achieving high accuracy.

Governing equations The steady, two-dimensional natural convection flow of a nanofluid (water-Al2 O3 ) is considered in a square enclosure. The governing equations are given in stream function ψ , vorticity w and temperature T as [2] ∇2 ψ = −w

μn f 2 ∇ w = ρn f α f

(1)

(ρβ )n f ∂ T ∂w ∂ψ ∂w ∂ψ − − RaPr − Ha2 Pr 2 ∂x ∂y ∂y ∂x ρn f β f ∂ x ∂x

(2)

αn f 2 ∇ T αf

(3)

∂ 2ψ

=

∂T ∂ψ ∂T ∂ψ − ∂x ∂y ∂y ∂x

gβ 3 (T −T )

σ

where Ra = f ν f αhf c , Pr = ν f /α f , and Ha = B0 ρn fnνf f are the nondimensional Rayleigh, Prandtl, and Hartmann numbers, respectively. Here, σ and ν are the electrical conductivity and kinematic viscosity, respectively. The thermo-physical properties of water, aluminum oxide and the nanofluid are determined as [2] ρn f = (1 − φ )ρ f + φ ρ p , σn f = (1 − φ )σ f + φ σ p , αn f = kn f /(ρ Cp )n f , μn f = μ f (1 − φ )−2.5 , (ρβ )n f = (1 − φ )(ρβ ) f + φ (ρβ ) p , (ρ Cp )n f = (1 − φ )(ρ Cp ) f + φ (ρ Cp ) p , kn f = k f [k p + 2k f − 2φ (k f − k p )]/[k p + 2k f + φ (k f − k p )]

(4)

where φ is the solid volume fraction, ρ is the density, α is the thermal diffusivity, Cp is the specific heat, β is the thermal expansion coefficient, μ is the effective dynamic viscosity, k is the thermal conductivity, is the characteristic length and the subscripts ‘n f ’, ‘ f ’ and ‘ p ’ refer to nanofluid, fluid and nanoparticle, respectively. y

ψ = 0, 1

∂T ∂y

=0

B0

ψ =0 T =1

O

ψ =0 T =0

ψ = 0,

∂T ∂y

=0

1

x

Figure 1: Domain and boundary conditions of the problem. No-slip boundary conditions for the velocity are imposed on the walls. The vertical walls are heated and cooled at x = 0 and x = 1, respectively, and the horizontal walls are adiabatic. The problem geometry and the boundary conditions are shown on Figure 1.

Numerical Methods DRBEM formulation The aim of the DRBEM is to transform the governing equations of the problem into boundary integral equations by treating all the terms except Laplacian operator as inhomogeneity, [4]. In this manner, the governing Equations (1)-(3) are weighted with the two-dimensional fundamental solution of Laplace equation,


u∗ = 1/2π ln(1/r). The application of the Green’s second identity to Equations (1)-(3) results in ci ψi + ci wi +

Γ

(q∗ w − u∗

∂w )dΓ = − ∂n ci Ti +

q∗

Γ

Γ

(q∗ ψ − u∗

ρn f α f Ω μn f

(q∗ T − u∗

∂ψ )dΓ = − ∂n

Ω

(−w)u∗ dΩ

(5)

(ρβ )n f ∂ T ∂w ∂ψ ∂w ∂ψ ∂ 2ψ − − RaPr − Ha2 Pr 2 ∂x ∂y ∂y ∂x ρn f β f ∂ x ∂x

∂T )dΓ = − ∂n

Ω

u∗ dΩ

αf ∂T ∂ψ ∂T ∂ψ ∗ − )u dΩ ( αn f ∂ x ∂ y ∂y ∂x

(6) (7)

∂ u∗ /∂ n,

where = Γ is the boundary of the domain Ω and the subscript i denotes the source point. The constant ci is given by ci = θi /2π with the internal angle θi at the source point. The integrands of the domain integrals on the right hand side of Equations (5)-(7) except u∗ are treated as inhomogeneity. Thus, they are approximated by a set of radial basis functions f j (x, y) linked with the particular solutions uˆ j to the equation ∇2 uˆ j = f j , [4]. The approximations for these integrands are given by N+L N+L ∑N+L j=1 α j f j (x, y), ∑ j=1 β j f j (x, y) and ∑ j=1 γ j f j (x, y), respectively, for Equations (5), (6) and (7). The coefficients α j , β j and γ j are undetermined constants. The numbers of the boundary and the internal nodes are denoted by N and L, respectively. Now, the right hand sides of Equations (5)-(7) also involve the multiplication of the Laplace operator with the fundamental solution u∗ , which can be treated in a similar manner by the use of DRBEM, [4], to obtain the following boundary only integrals. The use of constant elements for the discretization of the boundary results in the corresponding matrix-vector form of Equations (5)-(7) ∂ψ ˆ −1 {−w} , = (HUˆ − GQ)F ∂n 2 ∂w ˆ −1 ρn f α f ∂ w ∂ ψ − ∂ w ∂ ψ − RaPr (ρβ )n f ∂ T − Ha2 Pr ∂ ψ ) = (HUˆ − GQ)F (Hw − G ∂n μn f ∂x ∂y ∂y ∂x ρn f β f ∂ x ∂ x2 α ∂T ∂ T ∂ ψ ∂ T ∂ ψ f ˆ −1 ) = (HUˆ − GQ)F − (HT − G ∂n αn f ∂ x ∂ y ∂y ∂x Hψ − G

(8) (9) (10)

where the matrices Uˆ and Qˆ are constructed by taking each of the vectors uˆ j and qˆ j as columns, respectively. The coordinate matrix F of size (N + L) contains the radial basis functions f j as columns (e.g. f = 1 + r j ). The components of the matrices H and G are Hi j = ci δi j +

1 2π

Γj

1 ∂ ln( ) dΓ j , ∂n r

Hii = −

N

∑

j=1, j=i

Hi j ,

Gi j =

1 2π

Γj

1 ln( ) dΓ j , r

Gii =

A 2 (ln( ) + 1) 2π A

where r is the distance from node i to element j, A is the length of the element and δi j is the Kronecker delta function. The resulting nonlinear and coupled DRBEM equations are solved by an iterative process with initial estimates of vorticity and temperature. First, the stream function equation (8) is solved by giving an initial estimate for vorticity. Then, with the use of an initial estimate for the temperature the vorticity equation (9) is solved. Once the vorticity values are obtained at all points in the domain, a similar procedure is employed for the solution of the energy equation (10). In each iteration, the required space derivatives of the unknowns ψ , w and T , and also the unknown vorticity boundary conditions are obtained by using the coordinate matrix F as

∂ R ∂ F −1 = F R, ∂x ∂x

∂ R ∂ F −1 = F R, ∂y ∂y

w = −(

∂ 2 F −1 ∂ 2F F ψ + 2 F −1 ψ ) 2 ∂x ∂y

where R is one of the unknowns ψ , w or T . The iterative procedure will stop when a preassigned tolerance is reached between two successive iterations.

FEM formulation The FEM formulation of Equations (1)-(3) is introduced by first obtaining the weak form. The weak form is developed by multiplying Equations (1)-(3) with the weight functions ω1 , ω2 and ω3 which are assumed to be twice differentiable with respect to x and y, and taken as equal to the shape functions used for an element approximation in Galerkin approach [5]. Then, the application of the divergence theorem results in, −

Ω

∂ ω1 ∂ ψ ∂ ω1 ∂ ψ + ∂x ∂x ∂y ∂y

dΩ +

Ω

ω1 wdΩ +

∂Ω

ω1

∂ψ ds = 0 ∂n

(11)


−

μn f ρn f α f

Ω

141

∂ ω2 ∂ w ∂ ω2 ∂ w ∂ψ ∂w ∂w ∂ψ + dΩ − ω2 − dΩ ∂x ∂x ∂y ∂y ∂x ∂y ∂y ∂x Ω

(ρβ )n f ρn f β f

μn f ∂T ∂ 2ψ ∂w dΩ + Ha2 Pr ω2 2 dΩ + ds = 0 ω2 ∂x ∂x ρn f α f ∂ Ω ∂ n Ω Ω αn f αn f ∂ ω3 ∂ T ∂ ω3 ∂ T ∂T ∂ψ ∂T ∂ψ ∂T − + dΩ − w3 − dΩ + ds = 0 ω3 αf Ω ∂x ∂x ∂y ∂y ∂x ∂y ∂y ∂x α f ∂Ω ∂n Ω +RaPr

ω2

(12)

(13)

where the boundary integrals drop due to the property of shape functions to be vanished for Dirichlet boundary conditions, and zero normal derivative conditions. The region is discretized by using 6-nodal triangular elements and quadratic shape functions are used in the approximation of ψ , w and T over each element. Assembly procedure for all elements results in matrix-vector system of equations [K] {ψ } = [M] {w}

(14)

μn f (ρβ )n f {F1 } + Ha2 Pr {F2 } [K] {w} + [A] {w} = RaPr ρn f α f ρn f β f αn f [K] {T } + [A] {T } = {0} αf

(15) (16)

where the matrices [K] and [M] and are the stiffness and mass matrices, respectively. The matrix [A] contains the convective terms. The right hand side vectors {Fk }, k = 1, 2, are computed from previously obtained ψ and T values. In the computations of the element matrices and vectors, the integrations are carried out using an isoparametric interpolation and Gaussian quadrature. The equations are solved iteratively in terms of stream function, vorticity and temperature, and the missing vorticity boundary values are calculated using Taylor series expansion of the stream function with two inner values and using Equation (1).

Results and discussion The steady natural convection in an enclosure filled with a water-(Al2 O3 ) nanofluid under the influence of a magnetic field is analyzed by two powerful numerical techniques, DRBEM and FEM. The obtained numerical results are presented including a comparison of these two methods. Numerical tests are carried out for several values of Ra, Ha and φ . The FEM solutions are obtained by using 1152 quadratic triangular elements, whereas in DRBEM solutions 120 constant boundary elements are used. ψ

w

T

(a)

(b)

Figure 2: Streamlines, vorticity contours and isotherms for Ha = 60, Ra = 105 , φ = 0.03: (a) DRBEM, (b) FEM.


The comparison of the DRBEM and FEM solutions to the system (1)-(3) under the effect of an horizontally applied magnetic field is displayed in Figure 2 for the case when Ha = 60, Ra = 105 , φ = 0.03. These results show the agreement of the solutions obtained by both FEM and DRBEM methods which are also in well agreement with the solutions given in the work of Ghasemi [2]. On the other hand, the variation in stream function, vorticity and temperature along the vertical centerline x = 0.5, 0 ≤ y ≤ 1 at different solid volume fractions (φ = 0, 0.03, 0.08, 0.15, 0.2) is displayed in Figure 3 when Ha = 60, Ra = 105 . The results are obtained by using DRBEM. It is observed that the stream function values decrease in magnitude with an increase in solid volume fraction, which indicates a reduction in the fluid flow rate along the vertical centerline of the cavity. However, there is no significant effect of φ on the vorticity variation. On the other hand, as φ increases, the temperature increases in the lower part of the cavity (0 < y < 0.5) whereas it decreases in the upper part (0.5 < y < 1). ψ

w

T

Figure 3: The variation of stream function (left), vorticity (middle) and temperature (right) along the the vertical centerline x = 0.5, 0 ≤ y ≤ 1 at different solid volume fractions (φ = 0, 0.03, 0.08, 0.15, 0.2) when Ha = 60, Ra = 105 . In Figure 4, FEM solutions are illustrated in terms of stream function-vorticity-temperature contours, for Ha = 100 and (a) Ra = 104 and (b) Ra = 106 . When Rayleigh number is increased, the strength of the buoyancy driven circulation increases and undergo an inversion which is an indication of convection domination in the flow. As a consequence, the isotherms tend to have change profiles from vertical to horizontal. The results are in good agreement with the ones obtained in [2]. ψ

w

T

(a)

(b)

Figure 4: Streamlines, vorticity contours and isotherms by FEM for Ha = 100, φ = 0.03: (a) Ra = 104 , (b) Ra = 106 . The effect of the increase in Hartmann number is investigated for a high Rayleigh number, Ra = 107 using FEM and the results are illustrated in Figure 5. It is observed that, the horizontal profiles of streamlines and isotherms are slightly distorted into a diagonal form. The magnitudes of both the streamlines and the vorticity contours are decreased showing the flattening tendency of MHD flow. The convection domination effect on

Advances in Boundary Element and Meshless Techniques XVI ψ

w

143 T

(a)

(b)

Figure 5: Streamlines, vorticity contours and isotherms by FEM for Ra = 107 , φ = 0.03: (a) Ha = 150, (b) Ha = 300. the flow due to the high Rayleigh number is slightly overwhelmed by an increase in the values of Hartmann number from 150 to 300. It will be more pronounced for higher values of Ha.

Conclusion The natural convection flow in an enclosure filled with a nanofluid is solved under the effect of a magnetic field by using two numerical techniques, namely FEM and DRBEM. The effects of the physical parameters on the flow behaviour and the temperature distribution are investigated. It is observed that increases in Hartmann number and solid volume fraction result in a decrease in the magnitude of stream function and vorticity whereas they increase as Rayleigh number increases. Furthermore, the flow becomes convection dominated for high values of Ra which leads to a horizontal profile for isotherms. On the other hand, an increase in Ha has an opposite effect on the profiles of isotherms, that is the isotherms become vertical following the reduction in the effect of convection. The results obtained by using DRBEM and FEM are in good agreement with each others and also with the results given in the literature [2].

References [1] S. Gümgüm and M. Tezer-Sezgin. Drbem solution of natural convection flow of nanofluids with a heat source. Engineering Analysis with Boundary Elements, 34:727–737, 2010. [2] B. Ghasemi, SM. Aminossadati, and A. Raisi. Magnetic field effect on natural convection in a nanofluidfilled square enclosure. International Journal of Thermal Sciences, 50:1748–1756, 2011. [3] M. Sheikholeslami and D. D. Ganji. Ferrohydrodynamic and magnetohydrodynamic effects on ferrofluid flow and convective heat transfer. Energy, 75:400–410, 2014. [4] C. A. Brebbia, P. W. Partridge, and L. C. Wrobel. The Dual Reciprocity Boundary Element Method. Computational Mechanics Publications, Southampton, Boston, 1992. [5] J. N. Reddy. An introduction to the finite element method. The McGraw-Hill Companies, New York, 2006.


Boundary element analysis of piezoelectric films under spherical indentation L. Rodr´ıguez-Tembleque∗ , F.C. Buroni and A. Sáez Escuela Técnica Superior de Ingenier´ıa, Universidad de Sevilla (US) Camino de los Descubrimientos s/n, E41092 Sevilla, SPAIN. ∗ [email protected]

Keywords: Piezoelectric Films, Indentation, Electro-Elastic Materials, Contact Mechanics, Boundary Element Method.

Abstract. Piezoelectric materials are available in a variety of shapes and forms in smart systems. One of the most common shapes for sensor or actuator applications is the film form. To have a better understanding of the indentation behavior of these systems, this work analyzes piezoelectric films under different contact boundary conditions. The boundary element method (BEM) is used for modeling piezoelectric films under 3D contact, in the presence of electric fields. The BEM proves to be a very suitable numerical method for these 3D interface interaction problems where the number of degrees of freedom per node is increased due to fact that the electric field is taken into account. The BEM considers only the boundary degrees of freedom what makes it possible to reduce the number of unknowns and to obtain a very good accuracy with less number of elements than finite element formulations [1, 2, 3, 4]. The boundary element methodology is validated by comparison with analytical solutions presented in by Wang et al. [5] and then applied to study piezoelectric films under spherical indentation. Introduction Let us consider a 3D anisotropic piezoelectric (PE) body Ω ⊂ R3 with a piecewise smooth boundary ∂Ω, in a Cartesian coordinate system (xi ) (i = 1, 2, 3). Two partitions of the boundary ∂Ω are considered to define the mechanical and the electrical boundary conditions. The first one divides ∂Ω into three disjoint parts: ∂Ωt on which tractions are prescribed t˜i , ∂Ωu with imposed diplacements u ˜i and ∂Ωc that represents the potential contact surface, which have outward unit normal vector ν ci . The second partition is: ∂Ω = ∂Ωϕ ∪ ∂Ωq ∪ ∂Ωc , being the electrical potential ϕ˜ prescribed on ∂Ωϕ , and the electrical charges q˜ assumed on ∂Ωq . The mechanical equilibrium equations of this problem, in the absence of body forces, and the electric equilibrium equations under free electrical charge are σij,j = 0 Di,i = 0

in Ω, in Ω,

(1)

where σij are the components of Cauchy stress tensor and Di are the electric displacements. The infinitesimal strain tensor γij and the electric field Ei are defined as γij = (ui,j + uj,i )/2 Ei = −ϕ,i

in Ω, in Ω,

(2)

The elastic and electric fields are coupled through the linear constitutive law σij = cijkl γkl − elij El Di = eikl γkl + il El

in Ω, in Ω,

(3)

where cijkl and il denote the components of the elastic stiffness tensor and the dielectric permittivity tensor, respectively; and eijk are the PE coupling coefficients. These tensors satisfy the following


145

symmetries: cijkl = cjikl = cijlk = cklij , ekij = ekji , kl = lk , being the elastic constant and dielectric permittivity tensors are positive definite. Mechanical and electrical boundary conditions are prescried on ∂Ω. The Dirichlet boundary conditions are ˜i on ∂Ωu , ui = u (4) ϕ = ϕ˜ on ∂Ωϕ , and the Neumann boundary conditions are given by σij νj = t˜i Di νi = q˜

on ∂Ωt , on ∂Ωq ,

(5)

with νi being the outward unit normal to the boundary. For a well-posed problem either displacement or traction and electric potential or normal charge flux must be prescribed at each boundary point outside the contact zone ∂Ωc . Under small displacement assumption, a common unit normal vector νci can be considered in ∂Ωc . So the nonlinear boundary conditions are: σij νcj = pi Di νci = −κ(ϕ − ϕo )

on ∂Ωc , on ∂Ωc ,

(6)

where pi is the contact traction, pν = p · ν c is the normal contact pressure, κ is the conductivity coefficient and ϕo denotes the electric potential of the foundation or the indenter. Mechanical contact conditions The unilateral contact law involves Signorini’s contact conditions in ∂Ωc : gν ≥ 0,

pν ≤ 0,

gν pν = 0,

(7)

where gν = (go − uν ), being go the initial gap between the bodies and uν = u · ν c . The normal contact constraints presented in (7) can be formulated as: pν − PR− (p∗ν ) = 0,

(8)

where PR− (•) is the normal projection function (PR− (•) = min(0, •)) and p∗ν = pν + rν gν is the augmented normal traction. The parameters rν is the normal dimensional penalization parameter (rν ∈ R+ ). Electrical contact conditions The electrical conductivity coefficient in (6) can be defined as κ = κ(pν ) what allows to describe perfect electrical contact conditions similarly to the Signorini’s contact conditions, 0 if pν = 0, (9) κ(pν ) = κ∗ if pν < 0, being κ∗ the conductivity parameter defined in [6]. So, according to (9), the electrical contact condition (6) shows that when there is no contact (i.e. pν = 0) on ∂Ωc the normal component of the electric displacement field vanishes, and when there is contact, electrical charges appear.


Boundary element equations The well-known mechanical boundary integral equation for displacements at the source point x ∈ ∂Ω in the absence of body forces can be extended to the corresponding PE problem as [7] ˇJK (x | x )tJ (x)dS(x) U cJK (x )uJ (x ) + − TˇJK (x | x )uJ (x)dS(x) = (10) ∂Ω

∂Ω

where uJ is the extended displacement vector (see Barnett & Lothe representation [8]); t J is the extended tractions vector; cJK depends on the local geometry of the boundary ∂Ω at the collocation point x and is equal to 12 δJK for a smooth boundary at x (being δJK the extended Kronecker ˇJK and TˇJK are the extended displacement fundamental solution and the extended traction delta); U fundamental solution at a boundary point x due to a unit extended source applied at point x , respectively. In this work, a scheme for the evaluation of the extended fundamental solutions is implemented, which posses the remarkable characteristics that it is explicit and valid for mathematical degenerate and non-degenerate materials in the Stroh formalism context [9]. The strongly singular integral on the left-hand side is evaluated in the Cauchy principal value sense, whereas the weakly singular integral on the right-hand side is evaluated as an improper integral. The boundary ∂Ω can be discretized with elements and a collocation procedure on boundary integral equation (10), after the boundary conditions are considered, leads to a system of equations that, once solved, represents an approximate solution of the corresponding PE problem. The integral equation (10) can be written as follows: cJK (x )uJ (x ) +

Ne e=1

− TˇJK (x | x)uJ (x)dS(x)

=

∂Ωe

Ne e=1

ˇJK (x | x)tJ (x)dS(x) , U

(11)

∂Ωe

where the boundary is discretized into Ne elements of surface ∂Ωe and the integrals over the boundary ∂Ω are replaced by the sum of the integrals over the surface of each element. The functions uJ and tJ are approximated over each element ∂Ωe using linear shape functions, as a function of the nodal values: 4 Ni (ξ, η)uiJ , (12) uJ = i=1

tJ =

4

Ni (ξ, η)tiJ ,

(13)

i=1

where uiJ and tiJ are the nodal extended displacements and tractions, respectively. After discretizing the boundary, Eq. (11) can be written as Hu = Gt,

(14)

where vectors u and t contains the values of all nodal extended displacements and tractions, respectively. Contact discrete variables and restrictions The boundary element approximation for PE contact problems (14) can be rearranged according to the boundary conditions as: (15) Ax x + Ap p = F, where (x)T = [(xe )T (uc )T (ϕc )T ] collects the nodal external unknowns (xe ) and the contact displacements (uc ) and electric potential (ϕc ). Ax = [Axe Auc Aϕc ] is constructed with the columns of matrices H and G, and Ap = [Apc Aqc ] with the columns of G belonging to the nodal extended contact traction unknowns (i.e. (p)T = [(pc )T (qc )T ]).


147

The discrete contact gap for every contact node i can be expressed as (g)i = (go )i − (uc )i .

(16)

In the above expression, vector g contains the normal and tangential mechanical gap vectors (i.e. g ν and gτ ). So vector go collects initial mechanical gap vectors. The electrical contact condition (6) and (9) on every contact node i are: (qc )i = −κ((pν )i )((ϕc )i − (ϕo )i ),

(17)

and the contact restrictions (8) is: (pν )i − PR− ( (pν )i + rν (gν )i ) = 0,

(18)

where pν contains the normal contact tractions of every contact node i. Solution method The nonlinear equations set (15–18) make it possible to compute the variables on load step (k), (k) z(k) = (x(k) , pc , g(k) ), when the variables on previous instant z(k−1) are known. In this work, z(k) is computed using the iterative Uzawa scheme proposed in [6]. Numerical studies The problem illustrated in Fig. 1 (a) presents a spherical indentation of a piezoelectric block, whose dimensions are 2L1 × 2L1 × t, being L1 = 50 × 10−3 m. The block is discretized by 1024 linear quadrilateral boundary elements, using 16 × 16 elements on the Lo × Lo potential contact zone (Lo = 5 × 10−3 m), as Fig. 1 (b) shows. The ceramic BaTiO3 with the symmetry axis coinciding with x3 -direction is considered in this case, being its properties presented in Table 1. The rigid sphere of radius R = 100 × 10−3 m is subjected to a normal indentation of 7 × 10−5 m and the PE block is assumed to be ideally bonded at the base (x3 = −t). Frictionless and insulated spherical indentation conditions are considered. Different indetations are studied varying thicknesses t from a piezoelectric half space (HS) configuration, where the contact radius a is much smaller than the thickness of the film (i.e. a/t → 0), to a thin film (TF) configuration (i.e. a/t → ∞). Fig. 2 shows a comparison between the proposed boundary element solution and the analytical solution presented by Wang et al. [5] for PE HS and PE TF. Fig. 2 (a) shows normal contact pressure distributions on ∂Ωc and Fig. 2 (b) presents the electric potentials for t = 80 × 10−3 m and

Table 1: Material properties of the piezoelectric ceramic BaTiO 3 . Elastic coefficients (GP a) c1111 c1122 c1133 c3333 c2323 Piezoelectric coefficients (C/m2 ) e113 e333 e311 Dielectric constants (10−9 F/m) 11 33

150.00 66.00 66.00 146.00 44.00 11.40 17.50 −4.350 9.868 11.151


(a)

(b)

Figure 1: (a) The physical setting: rigid indenter over a piezoelectric film. (b) Boundary element mesh details.

t = 0.5 × 10−3 m (i.e. a/t = 0.031 and a/t = 7.5, respectively). An excellent agreement between analytical and numerical solutions can be observed in both cases for HS and TF. Indentation responses of a finitely thick piezoelectric film are shown in Fig. 3. It seen from Fig. 3, the transition occurs at around a/t 1: Half space solutions are good approximation of the indentation responses of a finitely thick piezoelectric film when a/t < 0.1, whilst solutions for an infinitely thin film are more appropriate when a/t 10.

(a)

(b)

Figure 2: Comparisson between the boundary element solution and the analytical solution [5] for: normal pressure (a) and electric potential (b) distribution. Conclusions A boundary element formulation has been applied to study indentation of PE films in the presence of electric fields. The proposed formulation has been applied to analyze finitely thick piezoelectric films. The results present an excellent agreement with the analytical solution presented in [5]. So this boundary element formulation proves to be a very interesting numerical methodology to analyse these smart systems with a good accuracy. Finally, it should be noted that the present formulation can be


(a)

149

(b)

Figure 3: Normalized indentation response as function of contact radius for thick piezoelectric film subjected to insulating punch: (a) indentation force and (b) electric potential. applied not only to studying PE films under indentation conditions, but also it may be extended to consider magneto-electro-elastic (MEE) films. Acknowledgments This work is supported by the Ministerio de Ciencia e Innovaci´ on (Spain) through the research project DPI2013-43267-P. References [1] Han, W., Sofonea, M., Kazmi, K., Analysis and numerical solution of a frictionless contact problem for electro-elastic-visco-plastic materials. Comput. Methods Appl. Mech. Engrg., 2007; 196: 3915– 3926. [2] Barboteu, M., Fern´ andez, J.R., Ouafik, Y., Numerical analysis of two frictionless elasticpiezoelectric contact problems. J. Math. Anal. Appl., 2008; 339: 905–917. [3] Liu, M., Fuqian, Y., Finite element simulation of the effect of electric boundary conditions on the spherical indentation of transversely isotropic piezoelectric films. Smart Mater. Struct., 2012; 21: 105020(10pp). [4] H¨ ueber, S., Matei, A., Wohlmuth, B., A contact problem for electro-elastic materials. ZAMM-Z. Angew. Math. Me., 2013; 93: 789–800. [5] Wang, J.H., Chen, C.Q., Lu, T.J., Indentation responses of piezoelectric films, J. Mech. Phys. Solids., 2008; 56: 3331–3351. [6] Rodr´ıguez-Tembleque, L., Buroni, F.C., S´ aez, A., 3D BEM for anisotropic frictional contact of piezoelectric bodies, Comput. Mech., 2015; Submitted. [7] Hill, L.R., Farris, T.N., Three-Dimensional Piezoelectric Boundary Element Method, AIAA Journal, 1998; 36: 102–108. [8] Barnett, D.M., Lothe, J., Dislocations and Line Charges in Anisotropic Piezoelectric Insulators, Phys. Stat. Sol. (b), 1975; 67: 105–111. [9] Buroni, F.C., S´ aez, A., Three-dimensional Green’s function and its derivative for materials with general anisotropic magneto-electro-elastic coupling, Proc. R. Soc. A, 2010; 466: 515.


A Comparison of Three Evaluation Methods for Green’s Function and Its Derivatives for 3D General Anisotropic Solids Longtao Xie1, Chuanzeng Zhang1, Chyanbin Hwu2, Jan Sladek3 and Vladimir Sladek3 1

Department of Civil Engineering, University of Siegen, D-57068 Siegen, Germany, [email protected]; [email protected] 2 Institute of Aeronautics and Astronautics, National Cheng Kung University, Taiwan, [email protected] 3 Institute of Construction and Architecture, Slovak Academy of Sciences, Slovakia, ŵĂŝůƚŽ͗ũĂŶ͘ƐůĂĚĞŬΛƐĂǀďĂ͘ƐŬ;[email protected] Keywords: Anisotropic Green’s function, Derivatives of Green’s function, Computational efficiency

Abstract. A comparison of three different methods for the numerical evaluation of three-dimensional (3D) anisotropic Green’s function and its 1st and 2nd derivatives is presented. The integral form expressions of the Green’s function and its derivatives proposed by Mura [5] are the basis of this investigation. The main task in the evaluation of the Green’s function and its derivatives is accordingly the evaluation of three one-dimensional (1D) infinite integrals. In the first method, numerical integration is applied directly to compute the three integrals. In the second method, standard residue calculus is used to calculate the three integrals, which results in explicit expressions of the Green’s function and its derivatives for non-degenerate cases with simple poles. In the third method, an improved residue calculus technique valid for both degenerate and non-degenerate cases is applied to the three integrals, which leads to unified explicit expressions for the Green’s function and its derivatives. The three methods are implemented numerically in FORTRAN to make a direct comparison. Introduction Green’s function and its derivatives play an important role in the boundary integral equation or boundary element method (BEM). In homogeneous, isotropic and linear elasticity, these functions have a simple analytical form. They can be evaluated directly in a BEM program. However, in general anisotropic elasticity, Green’s function and its derivatives are much more complex. Though Wilson and Cruse [1] proposed in 1978 a practical algorithm by employing a cubic interpolation from tabulated pre-calculated values for the evaluation of the Green’s function and its derivatives in BEM, the direct evaluation of the anisotropic Green’s function and its derivatives was preferred and hence investigated by many researchers. The 3D anisotropic Green’s function and its derivatives can be deduced to expressions in terms of contour integrals [2-5]. For example, in a 3D general anisotropic solid with the elastic stiffness tensor Cijkl , the Green’s function can be represented by

Gij ( x ) =

1 −1 S Nij ( ȟ ) D ( ȟ ) ds ( ȟ ), 8π 2 r v

(1)

where r = |x|; S is a unit circle in a plane whose normal vector is along x; ȟ is located on S; Nij( ȟ ) and

D( ȟ ) are cofactors and determinant of Kik ( ȟ ) = Cijkl ξ jξl . As proved in many references, Eq. (1) can be transformed into a 1D infinite integral as

Gij ( x ) =

1 4π

2

r

+∞

−∞

N ij ( p ) D −1 ( p ) dp,

(2)


151

where Nij(p) and D(p) are deduced from Nij(ȟ) and D(ȟ) with the substitution of ȟ = n + pm ; n and m are two mutually orthogonal unit vectors on the oblique plane perpendicular to x. In particular, D(p) is a 6th-order polynomial. As long as the elastic strain energy is positive, the roots of D(p) are three pairs of complex conjugates known as Stroh’ eigenvalues. By applying the Cauchy’s residue theorem with the assumption that pi (i=1,2,3) are three distinct roots of D(p) with a positive imaginary part, Eq. (2) becomes

Gij ( x ) =

i

3

N ij ( pv )

D′ ( p ) , 2π r v =1

(3)

v

where D′( p ) = dD( p ) / dp and i = −1 . Eq. (3) results from a simple residue calculus applied on Eq. (2) and is known as Fredholm’s formula. However, when D(p) has repeated roots, Eq. (3) is not valid. To deal with the so-called degenerate situations with repeated roots, Ting and Lee [6] proposed a new expression which keeps valid even in degenerate cases. Shiah et al. [7] and Tan et al. [15] expressed the Green’s function and its derivatives as Fourier series, and they demonstrated that their method is very efficient from the numerical point of view. The derivatives of Green’s function were investigated also by many researchers [8-13]. Although the numerical integration method for the evaluation of the Green’s function and its derivatives was suggested many years ago, many researchers are still interested in the explicit expressions of Green’s function and its derivatives, which should be advantageous in the BEM. Phan et al. [10] used the Cauchy’s residue theorem to derive explicit expressions for the Green’s function and its 1st derivative in terms of the Stroh’s eigenvalues. However, their expressions were derived separately for three different cases, namely, three distinct eigenvalues (non-degenerate case), two identical eigenvalues (partially degenerate case) and three identical eigenvalues (degenerate case). Lee [11] also derived the Green’s function and its 1st derivative separately for three different cases as mentioned before. She mentioned also the way how to obtain the 2nd derivative, but no final expressions and examples for the 2nd derivative of the Green’s function were given. Buroni and Saez [12] presented novel unified explicit expressions for the 1st and 2nd derivatives of the Green’s function by using spherical coordinate, and their expressions are valid for both non-degenerate and degenerate cases. Recently, the authors [13,14] have also derived unified explicit expressions for the Green’s function and its 1st and 2nd derivatives, which have the same expressions for non-degenerate and degenerate cases in contrast to references [10,11]. Different from the work of Buroni and Saez [12], partial derivatives of the Green’s function were performed in [13,14] with respect to Cartesian coordinates instead of spherical coordinates, which is easier, especially for the 2nd derivative of the Green’s function. It is expected that the novel unified expressions are much more convenient to the numerical implementation, because we needn’t to distinguish different non-degenerate and degenerate cases in the programming. To the authors’ best knowledge, a direct and detailed comparison between the different evaluation methods based on the numerical integration and the explicit expressions for the Green’s function and its derivatives has not yet been reported in literature. In this paper, based on the contour integral expressions of the Green’s function and its derivatives in reference [5], three different methods are presented and compared. In the first method, a direct numerical integration of the corresponding 1D infinite integral expressions of the Green’s function and its derivatives is applied, while in the second method the residue calculus is applied to the 1D infinite integrals which leads to explicit expressions for the Green’s function and its derivatives. In the third method, unified explicit expressions of the Green’s function and its derivatives derived recently by the authors [13,14] are also presented. Like the Green’s function presented by Ting and Lee [6], the unified explicit expressions are valid for both non-degenerate and degenerate cases. The three methods are implemented numerically in FORTRAN to make a direct comparison of their accuracy and efficiency.

1D infinite integral expressions of the Green’s function and its derivatives Due to the length limitation of the paper, the contour form expressions for the derivatives of the Green’s function and the procedure to transform the contour integrals into 1D infinite integrals are not


given here. Instead, the final 1D infinite integral form expressions of the Green’s function and its derivatives given by Mura [5] are listed here directly as

1 Aij ( x ) , 2π r 1 − Aij ( x ) xk + Pijk ( x ) , Gij ,k ( x ) = 2π r 2

1 Gij ,kl ( x ) = 3 Aij ( x ) xk xl − Pijk ( x ) xl + Pijl ( x ) xk + Qijkl ( x ) , πr where x is the unit vector of x ; Aij(p), Pijk(p) and Qijkl(p) are 1D infinite integrals given by Gij ( x ) =

{

(4)

}

1 +∞ N ij ( p ) D −1 ( p ) dp, 2π −∞ 1 +∞ ξk H ij ( p ) D −2 ( p ) dp, Pijk ( x ) = 2π −∞ 1 +∞ ξk ξl M ij ( p ) D −3 ( p ) dp, Qijkl ( x ) = 2π −∞ Aij ( x ) =

(5)

in which Hij(p) and Mij(p) are given as follows

H ij ( p ) = Fim ( p ) N jm ( p ) , M ij ( p ) = Lij ( p ) − Rij ( p ) D ( p ) , Ehm ( p ) = C phmq ( x pξ q + xqξ p ) ,

(6)

Fim ( p ) = N ih ( p ) Ehm ( p ) , Lij ( p ) = H ih ( p ) Fjh ( p ) ,

Rij ( p ) = x p xq C phmq N ih ( p ) N jm ( p ) . In principle, the Green’s function and its derivatives can be evaluated by applying a numerical integration method to the three integrals Aij(p), Pijk(p) and Qijkl(p) given in Eq. (5). It should be noted here that the interval of a 1D infinite integral

+∞

−∞

f ( p )dp can be transformed

2 to [-1,1] by the variable change p = t / (1 − t ) , which results in

+∞

−∞

2 +1 t 1+ t f ( p)dp = f dt , 2 2 2 −1 1 − t (1 − t )

(7)

and can then be computed by using the standard Gaussian quadrature.ġ Standard residue calculus for 1D infinite integrals In this section, with the assumption that the roots of D(p) are distinct, Cauchy’s residue theorem is applied to the three 1D infinite integrals (5), which results in explicit expressions. According to Cauchy’s residue theorem, in the case of n different poles pk, k=1,2,…,n with Im(pk)>0 among the poles of f(p) we have


153

n

+∞

f ( p )dp = 2π i Res( pk ).

−∞

(8)

k =1

If pk is a pole of mth-order, then

Res( pk ) =

1 d m−1 lim m−1 [( p − pk )m f ( p )]. ( m − 1)! p→ pk dp

(9)

There are three poles pk in Aij(p), Pijk(p) and Qijkl(p). The order of pk could be thus 1, 2 and 3, respectively. By virtue of Eqs. (8) and (9), the following explicit expressions can be obtained 3

Aij ( x ) = − Im n =1 3

Pijk ( x ) = − Im

N ij ( pn ) , D′ ( pn ) ′ ( pn ) − D′′ ( pn ) Hˆ ijk ( pn ) D′ ( pn ) Hˆ ijk D′ ( pn )

n =1

3

Qijkl ( x ) = − Im n =1

1 2 D′ ( pn )

5

3

{

, (10)

′ ( pn ) D′ ( pn ) Mˆ ijkl ( pn ) − 3D′ ( pn ) D′′ ( pn ) Mˆ ijkl 2

}

+ 3 D′′ ( pn ) − D′′′ ( pn ) D′ ( pn ) Mˆ ijkl ( pn ) ,

2

where

Hˆ ijk ( pn ) = ξk H ij ( p ) , Mˆ ijkl ( pn ) = ξk ξl M ij ( p ) . (11) Since D ( p ) , Hˆ ijk ( pn ) and Mˆ ijkl ( pn ) are polynomials of p, they and their derivatives can be determined in an accurate way by using algorithms for polynomials in a computer program. It should be remarked here that the assumption of three distinct eigenvalues or poles can be ensured by a small perturbation of the material parameters in the degenerate cases when two or three eigenvalues are identical. Improved residue calculus for 1D infinite integrals In this section, we present the recently derived unified explicit expressions for the Green’s function and its derivatives briefly, which is based on an improved residue calculus method of the 1D infinite integrals [13]. In the Green’s function, Nij(p) is a 4th-order polynomial in p, thus we have

Gij ( x ) =

4

1 4απ

2

N r

( n) ij n

I ,

(12)

n =0

where Į is the coefficient of p6 in D(p), N ij( n ) are the coefficients of the polynomial and dependent on the material parameters and the position but independent of p, and +∞

pn

−∞

∏ ( p − p )( p − p )

In =

3

v =1

v

dp.

v

Further, I n can be expressed by the following two special integralsġ

(13)


+∞

1

−∞

∏ v=1 ( p − pv )( p − pv )

Un =

n

dp,

+∞

p

−∞

∏ v=1 ( p − pv )( p − pv )

Vn =

n

dp.

(14)

That is

I 0˙U 3ˈ˙ I1 V3ˈ 2

I 2˙U 2 + 2 Re( p3 ) I1 − p3 I 0 , 2

I 3˙V2 + 2 Re( p3 ) I 2 − p3 I1 ,

(15)

I 4 = U1 + 2 Re( p2 + p3 ) I 3 − [| p2 |2 + | p3 |2 +4 Re( p2 ) Re( p3 )]I 2 + 2[Re( p2 ) | p3 |2 + Re( p3 ) | p2 |2 ]I1 − | p2 |2 | p3 |2 I 0 . Explicit expressions of Un and Vn can be obtained by using a standard residue calculus followed by a reformation of the expressions, which are valid for both non-degenerate and degenerate cases. So we designate this procedure as the improved residue calculus method. By using a similar procedure, unified explicit expressions can be also obtained for the derivatives of the Green’s function, which are also valid for repeated roots or degenerate cases. Numerical experiments and discussions The aforementioned three methods for the evaluation of the Green’s function and its derivatives are implemented in FORTRAN. The material Mg is chosen in the numerical test. The non-zero elastic constants of Mg are

C11 = C22 = 59.7GPa, C33 = 61.7GPa, C13 = C23 = 21.7GPa, C12 = 26.2GPa,

C44 = C55 = 16.4GPa,

C66 = 16.75GPa.

(16)

The number of the Gaussian points in the numerical integration method is properly chosen as 25 to ensure a similar accuracy with the other two methods, namely, the standard residue calculus method and the improved residue calculus method. The evaluation point is chosen as (1,2,3). Table 1 shows the numerical results obtained by the three different methods. It verifies that all the three methods can give accurate numerical results for the Green’s function and its derivatives. Numerical integration

Standard residue calculus

Improved residue calculus

G11(10-13m)

8.37829813

8.37829728

8.37829799

G11,2(10-13)

-1.318380374

-1.318379883

-1.318380356

8.975786

8.975805

8.975786

-14

G11,23(10 )

Table 1. Numerical results for the Green’s function and its derivatives for Mg at the point (1,2,3) by the three methods. Table 2 gives the computing time for the evaluation of the Green’s function and its derivatives required by the three different methods. The results demonstrate that generally the two residue calculus methods have a higher efficiency than the numerical integration method for the evaluation of the Green’s function and its 1st derivative, but a lower efficiency for the numerical evaluation of the 2nd derivative. Besides the improved residue calculus method seems to be the most efficient one among the three methods.


155

Time [s] Numerical integration Standard residue calculus Improved residue calculus (x10,000) Gij

0.477288723

1.540374756*10-2

0.1587375999

Gij,k

0.298385859

0.107014179

7.51008391*10-2

Gij,kl

0.370131731

1.23520386

0.501444101

Total

1.338908

1.359703

0.673786

Table 2. The computing time for the numerical evaluation of the Green’s function and its derivatives for Mg at the point (1,2,3) required by the three methods. Conclusions In this paper, three different methods for computing the Green’s function and its derivatives are presented and compared. It is found that to ensure the same accuracy, the standard residue calculus method for computing the Green’s function and its 1st derivative has a higher efficiency but a lower efficiency for the calculation of the 2nd derivative than the direct numerical integration method of the 1D infinite integral expressions. And the improved residue calculus method seems to be the most efficient one for computing the Green’s function, its 1st and its 2nd derivatives. Acknowledgment The first author would like to thank the financial support by the China Scholarship Council (CSC, Project No. 2011626148).

References

[1] R.B. Wilson and T.A. Cruse International Journal for Numerical Methods in Engineering, 12(9), 1383-1397 (1978). [2] I. Fredholm. Acta Mathematica, 23(1), 1-42 (1900). [3] I. M. Lifshitz and L.N. Rozentsveig. Zh. Eksp. Teor. Fiz ,17(9),783-791 (1947). [4] D.M. Barnett Phys. Stat. Sol. (b), 49, 741-748 (1972). [5] T. Mura Micromechanics of Defects in Solids, Martinus Nijhoff Publishers, Dordrecht (1987). [6] T.C. Ting and V.-G. Lee The Quarterly Journal of Mechanics and Applied Mathematics, 50(3), 407-426 (1997). [7] Y.C. Shiah, C.L. Tan and C.Y. Wang Engineering Analysis with Boundary Elements, 36, 17461755, 2012. [8] M.A. Sales and L.J. Gray Computers and Structures, 69, 247-254 (1998). [9] V.-G. Lee Mechanics Research Communications, 30, 241-249 (2003). [10] A.-V. Phan, L.J. Gray and T. Kaplan Engineering Analysis with Boundary Elements, 29, 570576(2005). [11] V.-G. Lee International Journal of Solids and Structures, 46, 3471-3479 (2009). [12] F.C. Buroni and A. Sáez Journal of Applied Mechanics, 80, 14 pages (2013). [13] L.T. Xie, Ch. Zhang, Y.P. Wan and Z. Zhong Advances in Boundary Element & Meshless Techniques XIV, ed. Seller, A. and Aliabadi, M.H., EC Ltd, UK, 286-291 (2013). [14] L.T. Xie and Ch. Zhang, in preparation. [15] C.L. Tan, Y.C. Shiah and C.Y. Wang International Journal of Solids and Structures, 50, 27012711 (2013).


MFS-fading regularization method for inverse boundary value problems in 2D linear elasticity Liviu Marin1,2 , Franck Delvare3,4,5 , and Alain Cimetière6 1 Department

of Mathematics, Faculty of Mathematics and Computer Science, University of Bucharest, 14 Academiei, 010014 Bucharest, Romania 2 Institute of Solid Mechanics, Romanian Academy, 15 Constantin Mille, 010141 Bucharest, Romania 3 Normandie Univ, 4 UNICAEN, LMNO, F-14032 Caen, France, 5 CNRS, UMR 6139, F-14032 Caen, France 6 Institute Pprime, CNRS-ENSMA-University of Poitiers, UPR 3346, Boulevard Marie et Pierre Curie, BP 30179 F-86962 Chasseneuil Futuroscope Cedex, France Keywords: Linear elasticity; Inverse boundary value problem; Cauchy problem; Regularization; Method of fundamental solutions (MFS); Iterative method.

Abstract. We study the numerical reconstruction of the missing displacements and tractions on an inaccessible part of the boundary in the case of 2D linear isotropic elastic materials from the knowledge of over-prescribed noisy measurements taken on the remaining accessible boundary part. This inverse problem is solved using the fading regularization method, originally proposed by [1, 2] for the Laplace equation, in conjunction with the MFS. The stabilisation of the numerical method proposed herein is achieved by stopping the iterative procedure according to Morozov’s discrepancy principle [5]. Mathematical Formulation Consider a bounded domain Ω ⊂ R2 occupied by a homogeneous isotropic linear elastic material characterised by Poisson’s ratio ν ∈ (0, 0.5) and the shear modulus G > 0. We also assume that the boundary ∂Ω of the solution domain Ω is either a smooth or a piecewise smooth curve. In the absence of body forces, the equilibrium equations of isotropic linear elasticity, in terms of the displacement vector, are given by: L u(x) ≡ −∇ · σ(x) = 0 ,

x ∈ Ω,

(1)

where L is the Lamé or Cauchy-Navier differential operator, σ(x) is the stress tensor associated with the displacement vector u(x), whilst on assuming small deformations, the corresponding strain tensor (x) is given by the kinematic relations (x) =

1 ∇u(x) + ∇u(x)T , 2

x ∈ Ω = Ω ∪ ∂Ω .

The stress and strain tensors are related by Hooke’s constitutive law, i.e. ν tr ((x)) I , x ∈ Ω , σ(x) = 2G (x) + 1 − 2ν

(2)

(3)

where I = [δij ]1≤i,j≤2 is the identity matrix in R2×2 , whilst ν = ν in the plane strain state and ν = ν (1 + ν) in the plane stress state, respectively. The Lamé-Navier differential operator L in (1) is obtained by applying the differential operator −∇ to the constitutive law (3) and then using (2) 2ν ∇ ∇ · u(x) . L u(x) ≡ −G ∇ · ∇u(x) + ∇u(x)T + (4) 1 − 2ν Further, we let n(x) be the outward unit normal vector to the boundary ∂Ω of the solution domain Ω and t(x) be the traction vector at x ∈ ∂Ω defined by t(x) = σ(x) n(x) ,

x ∈ ∂Ω .

(5)

In the direct/forward formulation, the displacement and traction vectors are prescribed on the boundaries Γu and Γt , respectively, where Γu ∪ Γt = ∂Ω and Γu ∩ Γt = ∅. In numerous practical situations, only a part of the boundary, say Γ1 ⊂ ∂Ω, is accessible for measurements, while the remaining boundary part, Γ2 = ∂Ω \ Γ1 , is inaccessible and hence no boundary data is available on it. 141


157

In such a situation, additional measurements are available on a portion of or on the entire Γ1 , hence compensating for the lack of boundary data on Γ2 , and this corresponds to an inverse boundary value problem. Let H1 (Ω) be the Sobolev space of real valued functions in Ω endowed with the usual Sobolev norm. The space of traces functions from H1 (Ω) to ∂Ω is denoted by H1/2 (∂Ω), while the restrictions of the functions belonging to the space H1/2 (∂Ω) to the subsets Γu ⊂ ∂Ω and Γt ⊂ ∂Ω define the spaces H1/2 (Γu ) and H1/2 (Γt ), respectively. Herein, we use the following notation H1 (Ω) := H1 (Ω)×H1 (Ω), as well as similar notations for the other function spaces employed, i.e. H1/2 (Γu ) := H1/2 (Γu ) × H1/2 (Γu ) and H1/2 (Γt ) := H1/2 (Γt ) × H1/2 (Γt ). Finally, we denote by H−1/2 (Γt ) the dual space of H1/2 (Γt ). In the sequel, we consider the following inverse boundary value problem: Find u ∈ H1 (Ω) such that the governing partial differential equation (1) is satisfied together with the boundary conditions:

and

(x) , u(x) = u

x ∈ Γu ,

(6a)

t(x) = t(x) ,

x ∈ Γt ,

(6b)

where the boundaries Γu and Γt satisfy the following relations: ∅ = Γu , Γt ∂Ω ;

Γu ∪ Γt ∂Ω ;

Γu ∩ Γt = ∅ ;

(7)

∈ H1/2 (Γu ) and t ∈ H−1/2 (Γt ). It should be noted that the general inverse boundary value whilst u problem in two-dimensional isotropic linear elasticity given by (1), (6a), (6b) and (7) actually describes the following cases: Problem (A): The Cauchy problem in isotropic linear elasticity when Γu = Γt . Problem (B): The inverse boundary value problem in isotropic linear elasticity when Γu = Γt . and In (6a) and (6b), u t are prescribed boundary displacements and tractions, respectively. Each of the aforementioned inverse problems is considerably more difficult to solve both analytically and numerically than direct problems since its solution does not satisfy the general conditions of wellposedness [4]. Therefore, direct methods, such as the least-squares method, will fail to produce stable and physically meaningful solutions to these problems and hence suitable regularization procedures should be employed. Fading Regularization Method To describe the fading regularization method, we consider the Cauchy problem (A) with Γ := Γu = Γt and define the space of solutions of the equilibrium equations of isotropic linear elasticity (1) by

H(Ω) = u ∈ H1 (Ω) L u(x) = 0 , x ∈ Ω . (8) As a direct consequence of definition (8), the traces of elements u ∈ H(Ω) on ∂Ω, u∂Ω , t(u)∂Ω , where t(u)∂Ω := σ(u) n ∂Ω is the traction vector on ∂Ω (i.e. the Neumann boundary condition) corresponding to the displacement vector u∂Ω , generate the space of compatible data on ∂Ω, namely

H(∂Ω) = U = u, t(u) ∈ H1/2 (∂Ω) × H−1/2 (∂Ω) ∃ v ∈ H(Ω) : v∂Ω = u , t(v)∂Ω = t(u) . (9) It can be easily shown that the space of compatible data H(∂Ω) is a closed subspace of H1/2 (∂Ω) × H−1/2 (∂Ω) with respect to the scalar product

1 u(x) · v(x) dΓ(x) + 2 t u(x) · t v(x) dΓ(x) ,

U, VH(∂Ω) = G ∂Ω (10) ∂Ω ∀ U = u, t(u) , V = v, t(v) ∈ H(∂Ω) .

142


Analogously, one can define the space of restrictions on Γ of the space of compatible data H(∂Ω), denoted by H(Γ), which is endowed with the scalar product on H(∂Ω) defined by (10) and its induced norm · H(∂Ω) . The Cauchy problem (A) can be re-written in an equivalent form as follows: . = u , (11) t ∈ H(Γ) , find U = u, t(u) ∈ H(∂Ω) : UΓ = U Given U It should be noted that problem (11) is in general not solvable unless the given Cauchy data are ∈ H(Γ). If the solution of problem (11) exists, then this is unique, but it does not compatible, i.e. U depend continuously on the data, i.e. problem (11) is ill-posed in the sense of [4]. Consequently, this inverse problem should be regularized in order to obtain a stable and physically meaningful solution of it and this is achieved by employing the fading regularization method. Further, we briefly describe the principle of the fading regularization method. The idea for solving problem (11) consists of seeking the best solution satisfying the boundary conditions (6a) and (6b) on the accessible boundary, provided that the equilibrium equations (1) are fulfilled. This leads to defining the solution of Cauchy problem (1), (6a) and (6b) in terms of an approximate solution which solves the following optimisation problem: Given U = u , t ∈ H(Γ) , find U = u, t(u) ∈ H(∂Ω) : (12a) J(U) ≤ J(V) , ∀ V = v, t(v) ∈ H(∂Ω) , where

J(·) : H(∂Ω) −→ [0, ∞) ,

2 J(V) = V − U H(Γ) .

(12b)

We note that problem (12) is ill-posed. Actually, it is always possible to find a solution U ∈ H(∂Ω) while its restriction to ∂Ω \ Γ such that its restriction to Γ is as close as possible to the given data U, is unstable. Therefore, it is necessary to introduce a control term in the functional J to overcome this instability of the solution. Hence the optimisation problem (12) is replaced by the following one: Given U = u , t ∈ H(Γ) , find U = u, t(u) ∈ H(∂Ω) : (13a) Jλ (U) ≤ Jλ (V) , ∀ V = v, t(v) ∈ H(∂Ω) , where

Jλ (·) : H(∂Ω) −→ [0, ∞) ,

2 2 Jλ (V) = V − U H(Γ) + λ V − Φ H(∂Ω) .

(13b)

Here λ > 0 is a parameter to be specified, Φ ∈ H(∂Ω), whilst the norms · H(∂Ω) and · H(Γ) are the norms derived from the scalar product on H(∂Ω) defined by (10) and the corresponding scalar product induced on H(Γ), respectively. The control term λ V − Φ 2∂Ω in (13b) is defined on the entire boundary ∂Ω and may be considered as a regularizing term. The introduction of this control term makes problem (13) well-posed in the sense of [4] and, in particular, its solution depends continuously as well as the coefficient λ and the choice of Φ. on the data U, One way to obtain a solution to the Cauchy problem (1), (6a) and (6b), which is also independent of λ or Φ, is to introduce this solution, or the solution of optimization problem (12), as the limit of a sequence of well-posed optimization problems (13). Consequently, an iterative regularizing strategy is introduced [1, 2] and this makes the solution of problem (11) the fix point of an operator defined on H(∂Ω) and also taking values in H(∂Ω) or, equivalently, each iterate is the solution of a well-posed optimization problem. More precisely, the corresponding iterative algorithm reads as follows: Step 1: Choose the initial guess U(0) = 0. Step 2: For each k ≥ 1, solve the following minimization problem: Find U(k) ∈ H(∂Ω) : J k−1 (U(k) ) ≤ J k−1 (V) , ∀ V ∈ H(∂Ω) , λ λ (14) k−1 k−1 (k−1) 2 2 where Jλ (·) : H(∂Ω) −→ [0, ∞) , Jλ (V) = V − U H(Γ) + λ V − U H(∂Ω) . 143


159

Clearly, the aforementioned iterative procedure can also be written in the following equivalent form: Step 1: Choose the initial guess U(0) = 0. Step 2: For each k ≥ 1, solve the following problem: Find U(k) ∈ H(∂Ω) : VH(Γ) + λ U(k) − U(k−1) , VH(∂Ω) = 0 ,

U(k) − U,

∀ V ∈ H(∂Ω) .

(15)

This iterative process accounts, in a very good and precise manner, for the equilibrium equations (1) since at each iteration the optimal element is sought in the Sobolev space H(∂Ω), see (13)–(15). The minimising functional in (13b) or (14) consists of two terms, each of these having its own role. More precisely, the first term of the functional given by (13b) or (14) is defined on Γ only and measures the gap between the sought optimal element in H(∂Ω) and the given over-specified boundary conditions on Γ. The second term of the functional from (13b) or (14) is a regularization term and is defined not only on the under-specified boundary ∂Ω \ Γ, where the boundary data are reconstructed, but on the entire boundary ∂Ω of the solution domain, with the mention that this term controls the distance between the new optimal element sought at the present iteration and that obtained at the previous iteration. It should be noted that the norm of the regularization term decreases and tends to zero as the number of iterations increases. Hence at each iteration, the optimal element satisfies the , equilibrium equations (1) and agrees as well as possible with the over-specified boundary data U Γ while at the same time remains close to the optimal element obtained at the previous iteration. Finally, we stress out that the proposed fading regularization algorithm allows for both the reconstruction of the missing boundary data on the under-specified boundary ∂Ω \ Γ and the denoising of the perturbed over-specified boundary data on Γ. Method of Fundamental Solutions For any collocation point x ∈ Ω and any singularity or source point ξ ∈ R2 \ Ω, the fundamental solution matrix U(x, ξ) = [Uij (x, ξ)]i,j=1,2 , for the displacement vector in the Cauchy-Navier system in two-dimensional isotropic linear elasticity is given by: 1 xi − ξi xj − ξj Uij (x, ξ) = −(3 − 4ν) log x − ξ δij + , i, j = 1, 2 . (16) 8πG(1 − ν)

x − ξ x − ξ By taking x ∈ ∂Ω and differentiating (16) with respect to xj , j = 1, 2, one obtains, in a straightforward manner, the derivatives of the fundamental solution for the displacement vector. Further, by combining (16) with the definition of the traction vector (5) and Hooke’s constitutive law (3) of isotropic linear elasticity, the fundamental solution matrix T(x, ξ) = [Tij (x, ξ)]i,j=1,2 , for the traction vector in the case of two-dimensional isotropic linear elasticity is then obtained. More precisely, one obtains: ∂U1j (x, ξ) ∂U2j (x, ξ) 2G T1j (x, ξ) = +ν n1 (x) (1 − ν) 1 − 2ν ∂x1 ∂x2 (17a) ∂U1j (x, ξ) ∂U2j (x, ξ) + n2 (x), j = 1, 2 , +G ∂x2 ∂x1

∂U1j (x, ξ) ∂U2j (x, ξ) + n1 (x) ∂x2 ∂x1 ∂U1j (x, ξ) ∂U2j (x, ξ) 2G ν + (1 − ν) n2 (x), + 1 − 2ν ∂x1 ∂x2

T2j (x, ξ) = G

(17b) j = 1, 2 .

In the MFS, the displacement vector is approximated by a linear combination of fundamental solution matrices with respect to N sources, ξ(n) n=1,N ⊂ R2 \ Ω, in the form [3] u(x) ≈ uN (c, ξ; x) =

N

U(x, ξ(n) ) c(n) ,

n=1

144

x ∈ Ω,

(18)


(1)

(1)

(2)

(2)

(N )

(N ) T

where the vector c = c1 , c2 , c1 , c2 , . . . c1 , c2 ∈ R2N contains the unknown MFS coefficients, 2N ξ ∈ R is a vector containing the coordinates of the sources and the components of the fundamental solution matrix U(x, ξ(n) ) are given by (16). Analogously to (18), one can also approximate the tractionvector on ∂Ω by a linear combination of fundamental solution matrices with respect to the sources ξ(n) n=1,N as [3] t(x) ≈ tN (c, ξ; x) =

N

T(x, ξ(n) ) c(n) ,

x ∈ ∂Ω ,

(19)

n=1

where the components of the fundamental solution matrix T(x, ξ(n) ) are given by (17a) and (17b). Further, according to the fading regularization method, at each step k ≥ 0 of the minimization problem (13) or, equivalently, eq. (14), one has to approximate both the known boundary data u(k) Γu and t(k) Γt and the unknown boundary data u(k) ∂Ω\Γu and t(k) ∂Ω\Γt , at the same time accounting for ε Γ and tε . To do this, we collocate the corresponding the given perturbed boundary conditions u u (m) Γt on Γu and x(m ) m =1,Mt on Γt and boundary conditions (6a) and (6b) at the points x m=1,Mu also express for the unknown displacements and tractions at (19) the MFS approximations (18) and the points x(m) m=Mu +1,M on ∂Ω\Γu and x(m ) m =Mt +1,M on ∂Ω\Γt , respectively. Consequently, at each step k ≥ 0, the minimization problem (13) or, equivalently, eq. (14), is reduced to a linear minimisation problem with respect to the corresponding unknown MFS constants. Numerical Results We linear elastic material which occupies the annular domain Ω = consider an isotropic

further x ∈ R2 Rint < x < Rout , where Rint = Rout /2 = 1.0, and is characterised by the material constants G = 4.80 × 1010 N/m2 and ν = 0.34. We also assume that the elastic field associated with the material occupying the domain Ω corresponds to constant inner and outer radial pressures, σint = σout /2 = 1010 N/m2 , respectively. Such a direct problem admits the following analytical solution in displacements which describe a plane strain state 1−ν x 1 , x ∈ Ω, (20a) u(an) (x) = V −W 1+ν

x 2 2G where V≡−

σout R2out − σint R2int R2out − R2int

and

W≡

(σout − σint ) R2out R2int . R2out − R2int

(20b)

In this study, we consider the Cauchy problem (A) with Γu = Γt = Γout and perturbed boundary displacements (pu = 1%, 3%, 5%). We have considered Mu = Mt = 40 uniformly distributed collocation points on Γout , where M = Mu + Mt , as well as N = 60 uniformly distributed singularities corresponding to the outer (40 singularities) and the inner boundaries (20 singularities), respectively, which are preassigned and kept fixed throughout the solution process. A good option for the required stopping criterion would be to cease the iterative procedure when the relative root mean square errors for the known displacement uΓu and traction vectors tΓt , respectively, attain their corresponding minimum. However, since the fading regularization method may be viewed as a generalization of the classical Tikhonov regularization method applied to the continuous inverse problem under investigation, we prefer to employ another stopping criterion, which is rigorous and, at the same time, has a mathematical rationale, namely the discrepancy principle of Morozov [5]. To do so, we introduce the convergence error E defined by ε 2 , E(k) := U(k) Γ − U (21a)

ε and U(k) are the measured noisy data and the computed data at step k of the iterative where U Γ procedure on the boundary Γ, respectively. Note that the convergence error introduced above actually measures the fit of the numerical solution at step k into the discrete MFS system and represents the 145


161

discretised version of the first term in the minimisation functional Jλk−1 given by (13). The discrepancy principle [5] states that the iterative procedure should be stopped at the first iteration, kopt , when the convergence error has about the same order as the noise in the data, namely (21b) kopt = min k ≥ 1 E(k) ≤ ε , ε ε Γ 2 + t Γ − t Γ 2 is a discrete measure of the noise in the prescribed data. where ε = u Γu − u u t t 0.3

1

Analytical pu =1% pu =3% pu =5%

0.2 0.5

t2 /1010

0.1

u1

0 −0.1

0

−0.5

−0.2 Analytical pu =1% pu =3% pu =5%

−0.3 −0.4 −1

−0.8

−0.6

−1

−0.4

−0.2

θ/(2π)

(a) Γu = Γt = Γout : u1 ∂Ω\Γ

0

−1.5 −1

−0.8

−0.6

−0.4

−0.2

θ/(2π)

0

(b) Γu = Γt = Γout : t2 ∂Ω\Γ

u

t

Figure 1: The analytical and numerical displacement (a) u1 ∂Ω\Γu and traction (b) t2 ∂Ω\Γt , obtained using the MFS-fading regularization method, Morozov’s discrepancy principle, and various levels of noise added into uΓu . Figs. 1(a) and (b) present the numerical results for the displacement u1 and the traction t2 , respectively, on the under-specified boundary Γint , obtained using the MFS-fading regularization method, Morozov’s discrepancy principle, and pu = 1%, 3%, 5%, and their corresponding analytical values. It can be seen that the numerical solutions for the displacements and tractions on Γint are stable approximations to their corresponding exact solutions, free of unbounded and rapid oscillations, and they converge to the exact solutions as the level of noise decreases. Conclusions In this paper, the solution of inverse boundary value problems in 2D linear isotropic elasticity was investigated using the MFS-fading regularization method. The proposed iterative procedure was stopped according to the discrepancy principle of Morozov [5]. The MFS-fading regularization algorithm provides us with very accurate, convergent and stable numerical results for boundary data reconstruction problems in 2D isotropic linear elasticity, at the same time being a very robust and versatile iterative algorithm for inverse boundary value problems in elasticity. Acknowledgements. L.M. acknowledges the financial support received from the Romanian National Authority for Scientific Research (CNCS–UEFISCDI), project no. PN–II–ID–PCE–2011–3–0521. A part of this research was carried out whilst L.M. visited the University of Caen-Lower Normandy.

References

[1] A. Cimetière, F. Delvare, and F. Pons, Comptes Rendus de l’Académie des Sciences - Série IIb Mécanique 328, 639–644 (2000). [2] A. Cimetière, F. Delvare, M. Jaoua, and F. Pons, Inverse Problems 17, 553–570 (2001). [3] G. Fairweather and A. Karageorghis, Advances in Computational Mathematics 9, 69–95 (1998). [4] J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Yale University Press (1923). [5] V. A. Morozov, V. A., Soviet Mathematics Doklady 7, 414–417 (1996). 146


Cohesive-zone modelling of shear connection with Coulomb friction and interface damage. An SGBEM implementation and application ˇ ˇ Jozef Kšinan, Filip Kšinan Roman Vodiˇcka Technical University of Košice, Civil Engineering Faculty, Vysokoškolská 4, 042 00 Košice, Slovakia

[email protected], [email protected], [email protected] Keywords: interface damage, Coulomb friction, shear connection, Cohesive-Zone Model, SGBEM, quasi-static delamination, energy formulation, steel-concrete composites.

Abstract. A new mathematical model for the cohesive zone modelling of shear connection combining the interface damage and friction contact has been developed and applied. The analysis of debonding process at plane model coupling the Cohesive Interface Model (CIM) and Coulomb friction is considered. The concept of solution is based on quasi-static rate-independent evolution of debonding process at the interface. The numerical model captures the influence of the friction at the interface during the interfacial damage under compression and pull-push loading. The presented approach has been applied for an interface problem of steelconcrete reinforcement. In particular, the coupling of non-linear phenomena of friction and interface damage mechanisms occurring in shear connections of steel-concrete composites has been analysed. The proposed mathematical approach is based on an energetic formulation using the solution concept of maximally dissipative local solution. The solution is approximated by a time stepping procedure and Symmetric Galerkin Boundary Element Method (SGBEM). Achieved results affirm the applicability of the proposed model for an analysis of friction in debonding process at steel-concrete interface and asses the model applicability in the area of steel-concrete composites. Introduction Recently, the various engineering applications and numerical analysis of interface damage in the steel-concrete composites concerning with the combination of damage process and of friction contact at partially or fully damaged interface. A mechanism coupling the interface damage and frictional contact belong to challenging current problematic issues in nowadays. There are several approaches for an analysis of the contact problems by Boundary Element Method (BEM). The presented work is focused on the improvement of the energetic model of interface debonding proposed in [5] in order to cover also the frictional contact between the debonded parts of steel-concrete reinforced structure. One of the most efficient and useful ways for the numerical modelling of interface damage, especially the crack initiation and crack propagation, is by applying of the cohesive-zone models in combination with frictional contact. The Cohesive Interface Model (CIM), is introduced, taking into account the possibility of frictional contact, and implemented in a Symmetric Galerkin BEM (SGBEM) code. In the paper, the frictional law is regularized to cope with the energetic character of the model see [7]. The regularization is proposed so that convex quadratic energy functional are obtained and quadratic programming algorithms can be efficiently applied. In the following sections the developed model is described, its numerical solution is outlined and an example is solved to assess its applicability in the area of steel-concrete composites. Frictional Contact Model The investigation of the interface damage mechanism is a crucial aspect in development and prediction of debonding process in engineering practice. The presented work is focused on the improvement of the energetic model of the interface debonding proposed in [5] in order to cover also the frictional contact between the debonded parts of steel-concrete reinforced structure. A new mathematical model to combine interface damage and friction in a cohesive-zone model is proposed. In this paper, a numerical model able to predict interface damage considering the Coulomb friction contact between debonded surfaces is developed and implemented.


163

The effect of gradual increase of friction represents the natural outcome of the gradual increase of the interface damage, from the initial state to the complete decohesion. The process of damage initiation and evolution has been defined by cohesive interface contact model CIM, which is defined for the damage evolution and for the friction, the simple Coulomb friction law is adopted. The solution of the contact problem is based on the evolution of energies during the loading process: the elastic energy stored in the bulks and the energy dissipated due to friction. From the physical point of view, the frictional dissipation in our model is given by the functional ˙ R(t; u)=

Γc

˙ s dΓ , ts [u]

(1)

˙ s , and t denotes the contact traction where the rate of the relative tangential displacement [u]s is denoted [u] vector, and where ts denotes its tangential shear component. The problem arises when the contact traction in the dissipation functional R is approximated, hence from the mathematical point of view it is more convenient to express the traction vector t in terms of displacements. Such, the tangential traction can be formulated with reference to the displacements in the following way: first let us consider the classical Coulomb friction law |ts | ≤ μ |tn | with a constant friction coefficient μ≥0 as a relation between normal and tangential tractions, tn and ts and then the standard Signorini condition of unilateral normal contact tn [u]n =0, tn ≤0, [u]n ≥0 in the contact zone Γc is replaced by the normal compliance penalization condition: tn =kg [u]− n such that a small overlapping of solids is allowed originating a pertinent contact compression, where [u]− n expresses the negative part of the relative normal displacement. This penalization can also be explained by a presence of a very thin layer of a very high normal stiffness kg 0, which is compressed in contact and stress-free out of contact. The solution of the contact problem is given by the energy evolution during the loading process: the elastic energy stored in 2 Γc corresponding to the normal compliance penalization condition is given by the integral: ΓC 12 kg ([u]− n ) dΓ , whereas the rate of energy dissipated due to the friction is given by the integral: ΓC

˙ s dΓ . f (ζ ) μ kg [u]− n · [u]

(2)

The interpretation of the Coulomb law in the formulation of energy dissipation potential is represented by damage dependent friction function f (ζ ). This function designates the activation of friction with respect to decreasing interface stiffness, whereas 0 ≤ f (ζ ) ≤ 1 is a dimensionless function characterizing the switch between the interface shear stresses due to adhesive and friction forces. The effect of friction function f (ζ ) grows progressively with decreasing interface damage ζ whereby it holds 0 ≤ ζ ≤ 1 and f (0)=1. The function f (ζ ) characterizes the process of friction activation and its consequently increasing influence on the interface stiffness degradation. It can be determined and fitted by the experimentally obtained results. An versatile expression of a damage dependent friction function depending on two parameters is assumed in the present work: f (ζ )= (1 − ζ ) p

where

0 < p < ∞.

(3)

Cohesive Interface Models (CIM) In engineering and computational mechanics practise another approach is usually preferred, with a non-linear but continuous response of the interface model, including the so-called softening period. This kind of model is referred to as Cohesive Interface Model (CIM). CIM provides a simple and useful model of thin cohesive layer between two surfaces which breaks in a gradual way. An efficient and versatile approach to achieve the continuous non-linear interface response is based on the energy formulation, employing the stored energy functional E , as proposed in [5]. The expression of the stored interface energy is modified by an additional term including ζ 2 and some stiffness parameters. This model yields initially (for an undamaged layer) a linear response, that decays with decreasing ζ is given by the following relations between the normal and shear stresses, tn and displacements jumps across the layer, [u]n and [u]s : tn = kn1 ζ + kn2 ζ 2 [u]n and ts , and normal and tangential ts =ζ ks1 ζ + ks2 ζ 2 ) u]s , where ζ is a damage parameter, 0 ≤ ζ ≤ 1, initially ζ =1 representing an undamaged layer, whereas ζ =0 corresponds to a cracked layer, not able to transmit tensile stress. Considering, e.g., pure opening Mode I, the failure of a layer point occurs when the driving force (energy release rate) G reaches the


Figure 1: CIM response for the driving force G, the damage parameter ζ and the mechanical stress σ . activation threshold, usually referred to as fracture energy, Gd , and correspondingly both, the mechanical stress tn and relative normal displacement [u]n , achieve their critical values, respectively, In the first, linear elastic, part of the stress-relative√displacement diagram, this stress is a linear function n2 2Gd , whereas in the second, softening, part of this of [u]n up to achieving its critical value tnc = √kkn1 +k n1 +2kn2 diagram it evolves non-linearly until it vanishes for the critical relative (opening) displacement unc = 2Gd /kn1 , see Figure 1. Consequently, at that instant, the damage parameter and mechanical stress jump down to zero consequently, leading to a continuous response of the model, see Figure 1. The schematic illustration presents the model response in pure Mode I under a displacement-controlled 1D experiment. Mathematical concept of the delamination process This section reviews the mathematical formulation of the energetic approach of interface failure mechanism for CIM interface model covering the interfacial friction contact model. The solution is acquired by variational formulation, which exploits developed numerical treatment of inelastic process. Energetic formulation of the interface damage To define the energetic conception of the interface damage mechanism, let us consider the energy stored [5, 7, 9] in the structure (given by Ω η and Γc ) obeying the aforementioned type of the interface damage. Let us assume in the stored energy formulation the general formulation of the so-called damage dependent stiffness function φ (ζ ), which is induced for the CIM. Consequently, we can express these formulations for the normal and tangential direction, respectively in the form: φn (ζ )=ζ (kn1 + ζ kn2 ),

φs (ζ )=ζ (ks1 + ζ ks2 ).

(4)

Then the stored energy functional [5] is defined as E (τ,u, ζ ) =

ΩA

1 A A A ε :C :ε dΩ + 2

ΩB

1 B B B ε :C :ε dΩ 2 +

Γc

1 2 φn (ζ ) [u]2n + φs (ζ ) [u]2s + kg ([u]− n ) dΓ , (5) 2

with the admissible displacements uη = wη (τ) on Γuη and the small strain tensor ε η =ε(uη ). The first two integrals, representing the elastic strain energy in the adjacent subdomains Ω η are expressed in their boundary form (taking into account that tη =tη (uη ), which is advantageous when the numerical treatment for solving elastic problems in Ω η is also based on boundary formulation. Let t and [u] denote the traction and relative displacement vectors at the contact zone Γc , with tn and ts being the normal and shear stresses and [u]n and [u]s the normal and tangential relative displacements, e.g. [u]n =(uB −uA )·nA is defined at the contact zone. Similarly, the relative tangential displacement [u]s can be also defined at the interface, as shown in Fig. ??. The potential energy of the external forces (acting only along the boundary): F (τ,u) = −

Γt A

fA ·uA dΓ −

Γt B

fB ·uB dΓ .

(6)

The dissipation potential R can be defined by the following functional [5] and which reflects the rate-dependence of the debonding process.

˙ ζ˙ ) = ˙ s | + Gd |ζ˙ | dΓ , R(u, ζ ; u, f (ζ ) μ kg [u]− (7) n |·[u] Γc


165

where ε˙ η =ε(u˙ η ) expresses the strain rate and τr denotes the bulks viscosity parameter and 0 ≤ f (ζ ) ≤ 1 is a dimensionless function characterizing the switch between the interface shear stresses due to cohesive and friction forces. f (ζ ) is increasing with decreasing ζ (0 ≤ ζ ≤ 1) and f (0) = 1. The main feature of the proposed mathematical model is that energy functional is separately quadratic both in the [u] and ζ variable. This fact enables to apply very efficient quadratic programming algorithms for solving the minimization problem, see [2]. Based on the above assumptions, a quasi-static visco-elastic evolution is governed by the following non-linear inclusions: ˙ ζ˙ ) δu F (τ,u), ∂u E (τ, u, ζ ) + ∂u˙ R(u, ζ ; u, ˙ ζ˙ ) 0, ∂ E (τ, u, ζ ) + ∂ ˙ R(u, ζ ; u, ζ

ζ

(8)

where δ expresses the Gâteaux differential of a (smooth) functional and ∂ refers to partial subdifferential of (nonsmooth) energy functional relying on convexity of the used functionals, see [5, 7, 9]. Numerical solution and computer implementation The numerical procedure devised to solve the above problem is based on the concept of Maximally-Dissipative Local Solution [9] and considers time and spatial discretizations separately, as usual [5]. The procedure is formulated in terms of the boundary data only using the Symmetric Galerkin BEM (SGBEM) for the spatial discretization. Spatial discretization and SGBEM The role of the SGBEM in the present computational procedure is to provide a complete boundary-value solution for given boundary data in each solid Ω η in order to calculate the elastic strain energy stored in these solids and at their interface Γc . For this, it is convenient to change the bulk integrals in (5) to boundary integrals Ωη

ε(uη ):Cη :ε(uη )dΩ =

Γη

tη (uη ) · uη dΓ .

(9)

In the present procedure, the SGBEM code calculates unknown tractions along Γc ∪ Γuη and unknown displacements along Γt η , assuming the displacements jump at Γc to be known from the used minimization procedure, in the same way as proposed and tested in [7, 8]. Time discretization The semi-implicit time-stepping scheme is defined by a fixed time step size t0 such that k k−1 ˙ u −u t k =kt0 for k=1, 2, . . . . The displacement rate is approximated by the finite difference u≈ , where uk t0 k denotes the solution at the discrete time t . Similarly the damage-parameter rate can be approximated by k k−1 ζ˙ ≈ ζ −ζ . The differentiation with respect to the displacement and damage-parameter rates can be replaced t0 by the differentiation with respect to u and ζ , respectively, as well, i.e. u − uk−1 ζ − ζ k−1 ˙ ζ˙ ) ≈ t0 ∂u R(uk−1 , ζ k−1 ; , ), ∂u˙ R(uk−1 , ζ k−1 ; u, t0 t0 k−1 k−1 u−u ζ −ζ ˙ ζ˙ ) ≈ t0 ∂ζ R(uk−1 , ζ k−1 ; , ). ∂ζ˙ R(uk−1 , ζ k−1 ; u, t0 t0

(10) (11)

It means that the inclusions (8) are approximated at discrete times t k by the first order optimality condition for the total energy functional H k H k (u, ζ ) = E (kt0 , u, ζ ) + t0 R(uk−1 , ζ k−1 ;

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

(12)

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

(13)

The optimality solution, is denoted by Substituting the results of the previous time-step k − 1 into the dissipation potential due to friction, makes the functional H k (u, ζ ) separately convex with respect to the


unknowns u and ζ . This leads to the following fractional-step-like strategy in the time step k: first H k (u, ζ k−1 ) is minimized with respect to u defining uk , and second H k (uk , ζ ) is minimized with respect to ζ defining ζ k . The above formulation has been implemented in an SGBEM code [7, 8] in M ATLAB by using a conjugate gradient based method for constrained minimization [2]. Numerical example In this section, the above numerical procedure to analyse a quasi-static evolution of cohesive interface model in presence of friction contact is tested in a plane strain problem of shear connection of steel-concrete reinforcement. The main objective of this study is to demonstrate the capabilities of the procedure, and to show the applicability of the proposed cohesive interface model and to capture the influence of the friction contact in the analysed contact problem of steel-concrete reinforcement. Model description In particular, the response of a pull-push shear test of steel-concrete reinforcement is presented. The numerical model consist of two domains: the steel strip connector and concrete layer mutually joined by steel spine connector and put on each other. The steel component is fixed along its bottom side to a rigid foundation. The applied loading is assumed on the top domain in two steps. First, a vertical compress loading is applied which leads after the interface rupture to a receding contact problem. Second, a loading equivalent to standard pull-push shear test well known from several engineering applications is applied afterwards. The loading process defines the prescribed displacements (hard-device loading) increasing during the both steps of the loading process with prescribed: • normal displacements wk2 = − ut k for k=1, 2, . . .50, with u=0.01mm and t k =k t0 , t0 =1s, • tangential displacements wk1 = − ut k for k=25, . . .400 with u=0.01mm and t k =k t0 , t0 =1s. In total was considered k=400 load steps. The numerical analysis captures the dependency of the stresses and deformations on the shape of steel spine shear connector at the steel strip. Therefore, it have been assumed four different models with various shapes of steel shear spine, see Figure 2. The loading and boundary conditions are the same for all models, the geometry of particular model ”D” is depicted at the Figure 3.

Figure 2: Investigated shapes of tested shear steel connectors. Parameter Statement Top bulk is made of concrete layer with Young’ s modulus Ec =38GPa and Poisson’s ratio νc =0.3. It is considered that prior to loading, this concrete layer is joined to the steel strip by steel spine in the extent of the interface, see Fig. 3 (red line). The dimensions of the steel strip connector are L=220 mm, L1 =200 mm, L2 =20 mm and H=80 mm, H1 =40 mm, H2 =15mm, r1 =mm, r2 =2mm with elastic properties Es =38 GPa and νs =0.3. According to the fact that, the kinematic behaviour of the interface depends only on the tangential component [u˙s ] of the displacement jump, the normal component [un ] is not involved.


167

Therefore, the value of the normal stiffness kn has no significant effect. Such the corresponding interface stiffness parameters were considered the same in the tangential and the normal direction as kn =ks =16 MPa, according to [4]. The interface stiffness parameters were derived from the aforementioned concept of CIM. Both the normal and tangential stiffnesses were split into two parts according to the relations: kn =kn1 +kn2 , ks =ks1 +ks2 , kn1 =0.01 × kn , kn2 =0.99 × kn , ks1 =0.01 × ks , ks2 =0.99 × ks . The Coulomb friction coefficient was considered μ=0.5. The contact model requires the stiffness parameter to be set kg =100 × kn . The parameters that govern the evolution of the crack growth are the decohesion (fracture) energy, which depends on the concrete age. According to Bažant et al. [1], its value , at early age, can be taken Gd =100 mJ/m−2 . The critical value of the tangential stress tc =1.3 MPa at the interface has been derived from the value of fracture energy.

Figure 3: Geometry of shear connection in steel-concrete reinforced structure. Numerical results The achieved numerical solution of the investigated contact model is presented in the following figures. On the Figure 4 is presented the dependency of normal tractions on the proposed shapes of shear connectors. The presented tractions for all models are depicted for the same time step k=100 and under the same intensity of loading. The symbol L expresses the length of the steel-concrete interface. In the proposed model "A" of shear connector has been achieved the maximum value of the normal tractions tnA =31 MPa . After the applying of the load in the tangential direction, the unstick of the concrete layer has been occurred. This effect is from the engineering point of view inconvenient, hence were developed another different shapes of the shear connectors. In order to eliminate the effect of unstick of the concrete layer, the model ”B” has been proposed with the perpendicular sides of the steel spine, which resulted in increased normal tractions tnB =66 MPa. The second proposed shape was still inconvenient. In the model "C" for that reason of eliminating of the unstick of the concrete layer has been proposed the shape with the diagonal sides. This shape is from the point of effect of unstick convenient, however as can be observed from the Figure 4, in the corners of steel spine becomes the concentration of stresses. The normal tractions acquire values tnC =107 MPa, which from the engineering point of view does not correspond with real response of the concrete. Modifying of the previous model, designed with curved corners, obtaining the new model "D". Applying of such modification eliminating the stress singularities and also the effect of unstick of the concrete layer. The maximum value of the normal stress is tnD =58 MPa, that corresponds with real values of stresses in respect to the strength of concrete . For that reason it has been assumed that model "D" is the most convenient model for investigation of the shear connection at the steel-concrete interface. When an interface part reaches the required amount of stored energy per unit length Gd the corresponding damage parameter ζ changes from 1 to 0 continuously, which is characteristic for the softening stage in the CIM, see Figure 5. For the model "D" the process of the interface damage has been initiated, when the system reaches the required amount of decohesion (fracture) energy, which have been occurred for the time step k=88. From the Figure 5 can be observed the evolution of the damage parameter ζ which captures that the crack initialization becomes on the right side of interface. The considerable impact on the evolution of tangential stresses has the proposed steel spine connector which transverse the significant part of the shear displacement, which document the bulk deformations in tangential direction, see Figure 6. Based on obtained results and model response, it can be assumed that the proposed model "D" of steel spine shear connector has been designed efficiently for defined loading conditions.


Figure 4: Response of the normal tractions at the interface according to the pertinent shape of shear connector.

Figure 5: Evolution of distributions of damage parameter ζ at the pertinent load steps.


169

Figure 6: Deformed shapes for various time instances, at the scale of structure multiplied by a factor of five. Conclusions An energy-based model for solving the contact problem of steel-concrete reinforcement with effect of friction has been analysed. Simple 2D examples of the steel-concrete shear connection have been analyzed in order to investigate the response of steel strip connector subjected to pressure and shear test. Achieved results confirm the significant influence of the friction during the damage process of the interface. The cohesive approach was obtained by mere adding new cohesive stiffness parameters providing the required non-linear continuous dependence of the analysed parameters. Presented numerical model confirms the expected behaviour in accordance with the applied theory and asses its applicability in the area of steel-concrete composite materials. Acknowledgement The authors also acknowledge the financial support from the Grant Agency of Slovak Republic. The project numbers is VEGA Grant No. 1/0788/12. References [1] Z.P. Bažant, L. Zhengzbi, and M. Thomas Identification of Stress-slip law for bar or fibre pullout by size effect test. Journal of Engineering Mechanics, 620-625, 1995. [2] Z. Dostál. Optimal Quadratic Programming Algorithms, volume 23 of Springer Optimization and Its Applications. Springer, Berlin, 2009. [3] C.G. Panagiotopoulos, V. Mantiˇc, I.G. García, E. Graciani. Quadratic programming for minimization of the total potential energy to solve contact problems using the collocation BEM In Advances in Boundary Element Techniques XIV, A. Sellier and M.H. Aliabadi (Eds.), pages 292–297, EC ltd, Eastleigh, 2013. [4] M. Raous, M. A. Karray. Model coupling friction and adhesion for steel-concrete interfaces. International Journal of Computer Applications in Technology, Inderscience, 34 : 42–51, 2009. [5] T. Roubíˇcek, M. Kružík, J. Zeman. Delamination and adhesive contact models and their mathematical analysis and numerical treatment (Chapter 9). In V. Mantiˇc (Ed.), Mathematical Methods and Models in Composites, Imperial College Press, London, 2014. [6] N. Valaroso, L. Champaney. A damage-mechanics-based approach for modelling decohesion in adhesively bonded assemblies. Eng Fract Mech, 73:2774–2801, 2006. [7] R. Vodiˇcka, V. Mantiˇc. An SGBEM implementation with quadratic programming for solving contact problems with Coulomb friction. In Advances in Boundary Element Techniques XIV, A. Sellier and M.H. Aliabadi (Eds.), pages 444–449, EC ltd, Eastleigh, 2013. [8] R. Vodiˇcka, V. Mantiˇc, F. París. Symmetric variational formulation of BIE for domain decomposition problems in elasticity – an SGBEM approach for nonconforming discretizations of curved interfaces. CMES – Comp. Model. Eng. Sci., 17:173–203, 2007. [9] R. Vodiˇcka, V. Mantiˇc, T. Roubíˇcek. Energetic versus maximally-dissipative local solution of a quasi-static rate-independent mixed-mode delamination model. Meccanica, 49:2993–2963, 2014.


A background decomposition meshfree method for evaluating fracture parameters of 2D linear cracked solids

A. Khosravifard1,*, M.R. Hematiyan2,*, T.Q. Bui3,** *

Department of Mechanical Engineering, Shiraz University, Shiraz 71936, Iran Department of Mechanical and Environmental Informatics, Tokyo Institute of Technology, 2-12-1-W8-22, Ookayama, Meguro-ku, Tokyo 152-8552, Japan

**

1

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

Keywords: Fracture mechanics; Meshfree method; Background decomposition method; Stress intensity factor

Abstract. In this paper, a newly developed integration technique, namely the background decomposition method (BDM), is applied to the meshfree analysis of linear fracture mechanics problems. The BDM is a meshfree integration technique, which is suitable for evaluation of integrals in the domains in which the density of the nodal points vary drastically. The BDM is also an efficient technique for evaluation of the integral of functions, which exhibit singular behavior. In the present work, two different approaches have been adopted for the analysis of fracture problems. In the first approach, a nodal refinement is carried out in the vicinity of the crack tip and the regular meshfree methods are used for the analysis of the problem. In the second approach, the enriched basis functions are used to account for the asymptotic fields, while no nodal refinement is performed. The BDM is used in the two approaches for evaluation of the integrals. Values of the stress intensity factor for a benchmark problem are calculated by the proposed method and a close agreement with analytical solutions is obtained. Introduction In the past two decades, meshfree methods have been introduced and constantly developed to overcome the difficulties of mesh-based techniques. These difficulties are mainly due to the existence of a predefined mesh of elements. When the elements become highly distorted or when the problem domain requires some sort of re-meshing, a loss of accuracy occurs. In such situations, use of meshfree methods can be helpful. Analysis of fracture mechanics is one of interesting application areas of meshfree methods. Stationary and propagating cracks can be modeled by the meshfree methods accurately. Classically, two different approaches have been adopted for accurately capturing the singular behavior of the stress field at the crack tip. In the first approach, a nodal refinement is carried out in a local sub-domain in the vicinity of the crack tip [1]. In the second approach, the basis functions of the meshfree method are properly enriched to include a set of appropriate singular terms of the displacement and stress fields [2]. Fleming et al. [2] presented an enriched element free Galerkin (EFG) method for the analysis of fracture problems. To obtain the enriched formulation, they proposed two different methods. In the first method, the asymptotic fields are added to the trial function, while the basis functions are augmented by the asymptotic fields in the second method. Ventura et al. [3] proposed a vector level set method for modeling propagating cracks in the EFG method. In their analysis, the Westergard’s solution as the enrichment near the crack tip is used. Rao and Rahman [4] also employed the EFG for analysis of linear elastic cracked structures. In their formulation of the problem, a new weight function as well as a technique for accurate implementation of the essential boundary conditions is introduced. Later, they extended their method for calculation of the stress intensity factors of a stationary crack in functionally graded materials of arbitrary geometry [5].


171

Nguyen et al. [6] developed an extended meshfree radial point interpolation method (X-RPIM) for modeling quasi-static crack growth in two-dimensional elastic solids. They used enriched basis functions for capturing the singularity at the crack tip. In numerical analysis of fracture mechanics problems the evaluation of domain integrals near the crack tip requires a special treatment, which is due to the sharp variations of the integrand near the crack tip [2]. The common approach is to use a background mesh for evaluation of the domain integrals. Finer integration cells are usually used in the vicinity of the crack tip, and a high order Gaussian quadrature method is utilized in that cells. In this work, a newly developed integration technique is employed for the fracture analysis of cracked domains. This integration technique, the background decomposition method (BDM), has been previously used for evaluating the domain integrals in meshfree methods for the analysis of 2D and 3D elasto-statics problems without any crack. Herein, two different approaches are used for the analysis of cracked domains. In the first approach, a nodal refinement is performed at the crack tip, while the ordinary basis functions of the meshfree method are employed. The BDM is then utilized for accurate and efficient evaluation of the displacement and stress fields near the crack tip. In the second approach, no nodal refinement is performed at the crack tip, but the enriched basis functions are employed. In this case, a modified version of the BDM is proposed and used for efficient evaluation of the displacement and stress fields near the crack tip.

The BDM for evaluation of domain integrals in fracture problems The BDM was first proposed by Hematiyan et al. for efficient evaluation of domain integrals in meshfree methods, which are based on weak formulation [7]. The main idea behind the BDM is to account for the local density of the nodes in the problem domain, for determination of the position of the integration points. The BDM has been found to be mostly useful for problems in which the density of the nodal points varies in different parts of the domain, or in cases where the integrand has a singular behavior at some points of the domain. Therefore, analysis of cracked domains would be an interesting application of the BDM. Evaluation of an integral by the BDM in a domain where the nodal density varies drastically follows a fourstep procedure. In the first step, the scattered nodes are divided into some groups, based on their local density. The domain is then decomposed into some portions in the second step. In the third step, integration points and weight of each partition are evaluated. Finally, global vectors for evaluation of the domain integral are calculated in the fourth step. In the following, a brief discussion of these four steps is given. Step 1: Categorizing the nodes based on their local density. For determining the local density of the nodal points in a domain, the average spacing of each node to its neighboring nodes is calculated first. A spacing value (s) corresponding to each node is computed in this step. The maximum and minimum spacing of the nodes in the domain are denoted by smax and smin, respectively. If these values are relatively close to each other, the nodes need not be divided into different groups. Otherwise, the interval [smin smax] is divided into some sub-intervals. Each sub-interval is associated with a grade of nodal spacing, i.e. grade 1: s [ smin s2 ] , s1 ( smin s2 ) 2 grade 2: s [ s2

s3 ] ,

s2

( s 2 s3 ) 2

grade R: s [ s R

( s R smax ) 2 The values of si are selected such that 2 d si 1 si d 3 . smax ] , s R

Step 2: Partitioning of the integration domain. Based on the grading of the nodal points, the integration domain is divided into some partitions such that either all of the nodes in a partition belong to a single grade, or there are at most four nodes in a partition. To this end, the well-known quadtree partitioning technique is adopted. In this technique, a square that covers the original domain is considered. The square is recursively subdivided into four child quadrants. Each child square is subdivided into four new squares unless all the nodes in each square are in a same grade or the number of nodes in the square is equal or less than four. In Fig. 1(a), a typical cracked domain is schematically depicted. In this domain, the nodal points are concentrated at the crack tip. Fig. 1(b) demonstrates the partition of the integration domain.


Fig. 1: A typical cracked domain along with (a) the meshfree nodal points, (b) the integration partitions. Step 3: Evaluation of the integration points and weights of the BDM. In this step, using a ray sweep method, the positions of integration points of each partition are determined. An integration weight is associated to each integration point, and the value of the integral is evaluated by the weighted summation of the value of the integrand at the integration points, i.e. Nk

Ik

¦W

K

i

f ( xi , yi )

(1)

j 1

where Ik refers to the value of the integral

³

f ( x, y ) d: on the k-th partition. Wi K are the integration

:k

weights, and ( xi , yi ) are the coordinates of the integration points. Nk is the number of integration points in the partition and is selected according to the grade of the nodal points in the partition. This means that, more integration points are located in partitions with a fine nodal arrangement. In this way, each part of the domain receives as much integration point as is required for accurate evaluation of the domain integral. If an integration partition is fully inside the original integration domain, the standard Gaussian quadrature method is used for determination of the integration weights and points of that partition. Otherwise, the ray sweep method is used. Fig. 2 shows an integration partition which is partially inside the original domain. The integral of a function f ( x, y ) over this partition can be written as:

Ik

³

³ §¨© ³

f ( x, y ) d:

:k

d

b

c

a

g ( x, y ) dx ·¸ dy ¹

³

d

c

l ( y ) dy

(2)

where

³

l ( y)

b

a

g ( x, y ) dx

(3)

f ( x, y ) ( x , y ) : k ® ( x, y ) : k ¯ 0

(4)

and

g ( x, y )

The line integral on the right hand side of Eq. (2), as well as the integral in Eq. (3) are evaluated by the composite Gaussian quadrature method, i.e. n

Ik Ji

¦ §¨© ³

n

1 l ( y ) dy ·¸ ¦ §¨ ³ l y (K ) J i dK ·¸ (5) 1 yi ¹ © ¹ i 1 i 1 dy dK yi 1 yi 2 is the Jacobian of the i-th integration interval. Each of the integrals on the right yi 1

hand side of Eq. (5) is evaluated by the m-point Gaussian method to give n

Ik

m

¦¦ J w l ( y ) i

i 1 j 1

j

j

(6)


173

Fig. 2: Illustration of an integration partition which is partially inside the original domain.

y (K j ) , and wj and j are the integration weight and integration point of the Gaussian quadrature

where y j

method. In order to evaluate Ik by Eq. (6), the value of l ( y j ) should be computed by Eq. (3). The integral in Eq. (3) is evaluated along the line y

y j . This line is referred to as an integration ray and is depicted in

Fig. 3. From this figure it can be inferred that the number of intersection points of each ray with the boundary is always even. Therefore, l ( y j ) can be rewritten as follows:

l( y j )

³

b

a

³

g ( x, y j ) dx

x2

x1

x4

x2 q

x3

x2 q 1

f ( x, y j ) dx ³ f ( x, y j ) dx ³

f ( x, y j ) dx

(7)

If the composite Gaussian quadrature method is used for evaluation of each integral on the right hand side of Eq. (7), the following formula is obtained:

³

nc

x2 i

x2 i 1

¦³

f ( x, y j ) dx

r 1

where F ([ )

1

1

F ([ ) J rc d[

nc

mc

¦¦ J c w F ([ ) r

s

s

(8)

r 1 s 1

f x([ ), y j and J rc

x2i x2i 1 . 2 nc

Fig. 3: Illustration of an integration ray and its intersection with a domain. Upon the substitution of Eqs (7) and (8) into Eq. (6), the final form of the formula for evaluation of the integral in the k-th partition is obtained (see Eq. (1)). Step 4: Computation of the global vectors. After obtaining the value of the domain integral in each partition, the total value of the domain integral can be obtained by adding the values obtained from each partition: P

I

P

Nk

¦ I ¦¦W k

k 1

k 1 j 1

N

k j

f (Q kj )

¦W

n

n 1

f (Q n )

W T Fint

(9)


where p is the total number of partitions, and Q i are the coordinates of the integration points. The vector

W collects the values of the integration weights and the vector Fint includes the values of the integrand at the integration points. Fig. 4 depicts the integration points of the BDM for the domain shown in Fig. 1. It is clear that the density of the integration points conform to that of the nodal points.

Fig. 4: The BDM integration points for the domain shown in Fig. 1. As mentioned earlier, the BDM is also a robust tool for evaluation of domain integrals with integrands that exhibit singular behavior. For instance, the BDM can be used for evaluation of domain integrals that appear in the formulation of the meshfree methods that use enriched basis functions. In such cases, the general procedure of the BDM is similar to the above mentioned steps, except for the first step. Instead of categorizing the nodes, based on their local density, the distance of a node to the crack tip is also considered. In this way, parts of the region that are close to the crack tip will receive a denser distribution of integration points.

The enriched EFG method for analysis of fracture mechanics The element free Galerkin method is a meshfree technique which uses a set of irregularly scattered nodes on the problem domain and its boundaries to obtain a system of discretized equations. In the EFG, nodes are not connected to each other with a mesh, and only a CAD-like description of the geometry is needed to formulate the final system of equations. The EFG uses the moving least squares (MLS) method to approximate the displacement field. In the MLS, the displacement filed is written as follows: m

u h ( x)

¦ a ( x) p ( x ) j

aT P

j

(10)

j 1

where aj are coefficients, pj are the basis functions, and m is the number of basis functions. The basis functions are usually the monomials, selected from the Pascal triangle. However, when the EFG is used for the analysis of fracture mechanics problems, the asymptotic fields are added to monomial terms, i.e.

PT

ª «¬1 x

y

r cos

T 2

r sin

T

T

r sin sin T 2

2

T º r cos sin T » 2 ¼

(11)

where r and are measured from the crack tip. Following the standard procedure of the MLS, the displacement filed can be written as: n

u h ( x)

¦P

T

(x) A 1 (x)B(x)

n

¦I (x)u i

i

(12)

i 1

i 1

where n

A ( x)

Ii (x)

¦W (x)p(x )p i

i 1 m

¦ p (x)(A j

j 1

i

1

T

(x i ), B(x)

(x)B(x)) ji

>W1 (x)p(x1 ),W2 (x)p(x 2 ),,Wn (x)p(x n )@

P T (x)( A 1B)i

(13) (14)


175

In Eqs (12) to (14), n is the number of nodes in the support domain of point x, and Wi is the weight function of the i-th node in the support domain. Upon substitution of the displacement field, Eq. (12), into the Galerkin weak formulation of the problem, the discretized system of equations can be obtained [8]:

ªK «G T ¬

G º U ½ ® ¾ 0 »¼ ¯ ¿

F ½ ® ¾ ¯Q ¿

(15)

In Eq. (15), U is the displacement vector, and is the vector of Lagrange multipliers. For definitions of other matrices and vectors in Eq. (15), refer to [8].

Example: Determination of the stress intensity factor of a cracked domain To verify the usefulness of the BDM in evaluating domain integrals in the meshfree analysis of fracture problems, a plate containing an edge crack, as shown in Fig. 5(a), is analyzed. The numerical analysis is performed both by the EFG method and the meshfree radial point interpolation method (RPIM). In the formulation of the EFG, enriched basis functions are used, and no nodal refinement at the crack tip is carried out. In contrast, in the analysis of the problem with the meshfree RPIM, no enrichment technique is utilized, but a few nodes (6 to 12 nodes) are scattered around the crack tip. Due to the symmetry of the problem domain and the loading, only half of the domain is modeled. In Fig. 5(b), a typical nodal distribution of the meshfree RPIM is shown for a crack of length a 3.2 . The nodal refinement around the crack tip is also shown in this figure. The nodal distribution of the EFG method for the same crack length is depicted in Fig. 5(c). A structured nodal distribution without any refinement at the crack tip is used for the EFG. An analytical solution for the stress intensity factor of this problem is given by [9]:

KI

ª

§a· ©b¹

§a· ©b¹

2

§a· ©b¹

3

4 §a· º © b ¹ ¼»

V Sa «1.12 0.23¨ ¸ 10.55¨ ¸ 21.72¨ ¸ 30.39¨ ¸ », ¬«

a 0.6 b

(16)

Fig. 5: A cracked plate, (a) the geometry and loading, (b) nodal distribution of the meshfree RPIM, (c) nodal distribution of the EFG. For the numerical analysis of the problem with the mentioned meshfree methods, three different nodal arrangements are used. In the EFG three structured 7×7, 11×11, and 16×16 nodal distributions are used, while in the RPIM, nodal points are scattered irregularly. The number of nodes in the three different arrangements is 42, 123, and 210. For easy reference, these nodal arrangements are referred to as arrangement 1, 2, and 3, respectively. The domain integrals in both the EFG method and the RPIM are evaluated by the BDM. Fig. 6 depicts the distribution of the BDM integration points, for the nodal arrangements shown in Fig. 5. This figure shows that in case of a refined nodal arrangement, the


distribution of the integration points conform to that of the nodal points. Also, it can be seen that in case of the EFG with enriched basis functions, the density of the integration points near the crack tip is more than other parts of the region.

Fig. 6: Distribution of the BDM integration points for the (a) RPIM, and (b) EFG. For different ratios of the crack length to the plate width (a/b), the stress intensity factors are obtained with the EFG and RPIM and with the three different nodal arrangements. For calculation of the stress intensity factor, the J-integral method is employed. In tables 1, and 2 the results obtained by the RPIM and EFG, are compared with the analytical solutions obtained by Eq. (16). These tables clearly show the excellent agreement of the results obtained by the proposed method. Although the results of both the EFG and the RPIM methods reveal an acceptable accuracy, these tables suggest that by using the RPIM with a nodal refinement, more stable and reliable results can be obtained. Table 1: Stress intensity factors obtained by the RPIM and BDM Arrangement 1 Arrangement 2 Analytical a/b KI Error (%) Error (%) KI KI 0.2 3.07 3.10 1.0 3.06 0.3 0.3 4.56 4.29 5.9 4.48 1.7 0.4 6.67 6.48 2.8 6.61 0.9 0.5 10.02 9.82 2.0 9.93 0.9

Arrangement 3 Error (%) KI 3.07 0.0 4.53 0.6 6.65 0.3 10.01 0.1

Table 2: Stress intensity factors obtained by the EFG and BDM 7×7 nodes 11×11 nodes Analytical a/b KI Error (%) Error (%) KI KI 0.2 3.07 2.60 15.3 3.04 1.0 0.3 4.56 4.20 7.9 4.63 1.5 0.4 6.67 6.00 10.0 6.54 1.9 0.5 10.02 7.80 22.2 9.92 1.0

16×16 nodes Error (%) KI 3.12 1.6 4.56 0.0 6.62 0.7 9.99 0.3

The stress field ahead of the crack tip is also computed using the RPIM and depicted in Fig. 7. This figure corresponds to the crack length of a 3.2 , and the nodal arrangement 3, i.e. the arrangement with 210 nodes. The results of the proposed technique are compared to those obtained by ANSYS with a very fine mesh. It is seen that by using the BDM, not only the stress intensity factor, but also the distribution of the stress field can be obtained with a high accuracy. Considering that a small number of nodal points is used to obtain the RPIM results, the proposed technique is found to be promising for the fast and efficient analysis of fracture mechanics problems.


177

Fig. 7: Comparison of the (a) radial, and (b) circumferential stresses ahead of the crack tip at =0° for a=3.2 obtained by the proposed RPIM with BDM and the FEM using ANSYS.

Conclusions In this paper, a meshfree Galerkin method for the analysis of linear fracture mechanics problems is proposed. The method makes use of a robust integration technique, i.e. the background decomposition method (BDM), for accurately capturing the singular behavior of the problem. Two different approaches were used in this work for modeling of cracked domains. In the first approach, a nodal refinement was carried out at the crack tip and ordinary basis functions were used. In the second approach, no nodal refinement was performed, but the asymptotic fields were added to the basis function. The stress intensity factor of a cracked plate was then calculated and compared with an analytical solution. It was observed that by adding a few nodes (between 6 to12) near the crack tip and using conventional basis functions, very accurate results can be obtained. The results obtained by enriching the basis functions were also acceptable, but not as stable as those of the first approach. Enriching the basis functions increases the size of the moment matrix of the EFG, and therefore the computational time increases. Consequently, it is concluded that by adding only a few nodes in the vicinity of the crack tip and using the BDM for evaluation of the domain integrals, the stress intensity factor can be found more accurately and efficiently.

References [1] T. Belytschko, Y.Y. Lu, and L. Gu, International Journal for Numerical Methods in Engineering, 37(2), 229 – 256 (1994). [2] M. Fleming, Y.A. Chu, B. Moran, and T. Belytschko, International Journal for Numerical Methods in Engineering, 40, 1483 – 1504 (1997). [3] G. Ventura, J.X. Xu, and T. Belytschko, International Journal for Numerical Methods in Engineering, 54, 923 – 944 (2002). [4] B.N. Rao, and S. Rahman, Computational Mechanics, 26, 398–408 (2000). [5] B.N. Rao, and S. Rahman, Engineering Fracture Mechanics, 70, 1–27 (2003). [6] N.T. Nguyen, T.Q. Bui, Ch. Zhang, and T.T. Truong, Engineering analysis with boundary elements, 44, 87–97 (2014). [7] M.R. Hematiyan, A. Khosravifard, and G.R. Liu, Computers and Structures, 142, 64 – 78 (2014). [8] G.R. Liu, and Y.T. Gu, An Introduction to Meshfree Methods and Their Programming, Springer, (2005). [9] E.E. Gdoutos, Fracture Mechanics - An Introduction, 2nd edition, Springer, (2005).


Topological derivatives for acoustics with various boundary conditions

Hiroshi Isakari1,a , Moemi Hanada1,a and Toshiro Matsumoto1,c 1 Department

of Mechanical Science and Engineering, Nagoya University Furo-cho, Chikusa-ku, Nagoya, 464-8603, Japan

a [email protected], b m

[email protected], c [email protected]

Keywords: Topological derivative, Optimum design, Boundary element method, Acoustic wave

Abstract The topological derivative is essential to the topology optimisation which is considered as the most promising CAE-based design method. For wave problems, the topological derivative characterises a sensitivity of an objective function when an infinitesimal scatterer appears. We have investigated representations of the topological derivatives for acoustic scatterers with either the Neumann boundary or impedance boundary. We have also investigated the influence of the definition domain of the objective function on the topological derivatives. Introduction The topology optimisation is considered as the most promising optimal design method because of its design capability. Modern topology optimisation methods [1, 2, 3, 4, 5] utilise the topological derivative which characterises the sensitivity of the objective function to an appearance of an infinitesimal object in the design domain. As Novotny et al.[6] have pointed out, the topological derivative has a different representation for a different boundary condition on the infinitesimal object. Novotny et al. derived a variety of representations of the topological derivatives for the Laplace problem in 2D. In this paper, we investigate the topological derivatives for 2D acoustics with various boundary conditions, such as the Neumann and impedance boundary conditions. We also investigate the topological derivatives for “design-dependent” objective function, which is also defined on the surface of the infinitesimal object. Topological derivatives We are interested in finding a geometry (including its topology) of Ω which minimises the following objective functional J(Ω): J(Ω) =

M

∑

m=1

obs f (u(xm )) +

Γ

g(u, q)dΓ,

(1)

obs is an observation point, and M is the total number of the observation points. where f and g are functionals, xm Also, u and q solve the following boundary value problem (BVP):

Δu(x) + k2 u(x) = 0

x ∈ R2 \ Ω,

u(x) = u(x) ˆ ∂ u(x) q(x) := = q(x) ˆ ∂n iρω u(x) q(x) = z

∂ (u(x) − uin (x)) ∂ |x|

− ik(u(x) − uin (x)) = o(|x|− 2 ) 1

(2)

x ∈ Γu ,

(3)

x ∈ Γq ,

(4)

x ∈ Γz ,

(5)

as |x| → ∞,

(6)

where ˆ· indicates a known boundary data, n is an inward normal vector of Ω. k = ω

ρ κ

is the wave number,

where κ and ω are the bulk modulus and the angular frequency, respectively. Also, ρ , z and uin are the


179

density, the impedance and the incident field, respectively. Γu , Γq and Γz are disjoint subsets of Γ which satisfy Γ = Γu ∪ Γq ∪ Γz . Let us assume that a circular infinitesimal scatterer Ωε of radius ε appears in R2 \ Ω, and u and q becomes u + δ u and q + δ q, respectively. By subtracting the BVP (2)–(6) from the BVP for u + δ u, one obtains the following BVP for δ u and δ q: Δδ u(x) + k2 δ u(x) = 0

x ∈ R2 \ Ω ∪ Ωε ,

(7)

δ u(x) = 0 x ∈ Γu , ∂ δ u(x) δ q(x) := = 0 x ∈ Γq , ∂n iρωδ u(x) x ∈ Γz , δ q(x) = z F(u(x) + δ u(x), q(x) + δ q(x)) = 0 x ∈ Γε := ∂ Ωε , 1 ∂ δ u(x) − ikδ u(x) = o(|x|− 2 ) as |x| → ∞, ∂ |x|

(8) (9) (10) (11) (12)

where F is a function which represents the boundary condition on Γε to be discussed later. The objective function (1) also suffers from a change, denoted as δ J = J(Ω ∪ Ωε ) − J(Ω), because of the appearance of Ωε as follows: M ∂f ∂g ∂g ∂g ∂g obs δJ = ℜ ∑ δ u(xm ) +ℜ δ u + δ q dΓ + δ u + δ q dΓ. (13) g(u, q) + ℜ ∂q ∂u ∂q Γ ∂u Γε m=1 ∂ u With δ J, the topological derivative T is defined as follows:

δ J = T a(ε ) + o(a(ε )),

(14)

where a(x) is a monotonically increasing function for x > 0. Since a direct evaluation of eq. (13) is impractical, which requires the evaluation of the perturbations δ u and δ q on all of the observation points xobs and Γ ∪ Γε , we introduce the adjoint variables λ and μ to eliminate δ u and δ q in eq. (13). δ u and δ q are governed by the following BVP: Δλ (x) + k2 λ (x) +

obs ) ∂ f (xm obs δ (x − xm )=0 ∂ u m=1 M

∑

x ∈ R2 \ Ω,

∂ g(x) x ∈ Γu , ∂q ∂ λ (x) ∂ g(x) μ (x) := = x ∈ Γq , ∂n ∂ u iρω ∂ g(x) ∂ g(x) iρω + μ (x) = λ (x) + x ∈ Γε , z z ∂q ∂u 1 ∂ λ (x) − ikλ (x) = o(|x|− 2 ) as |x| → ∞. ∂ |x| λ (x) = −

(15) (16) (17) (18) (19)

With the help of the reciprocal theorem for δ u and λ in R2 \ Ω ∪ Ωε , eq. (13) can be evaluated as follows: ∂g ∂g δJ = ℜ (λ δ q − δ uμ ) dΓ + δ u + δ q dΓ. (20) g(u, q) + ℜ ∂u ∂q Γε Γε Note that δ u and δ q only on Γε are required for the evaluation of eq. (20). The explicit representations for δ u and δ q on Γε can be obtained once the explicit boundary condition on Γε is given (see eq. (11)) , which are shown in the following subsections. The case that Γε is the impedance boundary Let us first assume that the boundary condition on Γε is given


as the impedance one, which is expressed as follows: F(u + δ u, q + δ q) = (q + δ q) −

iρω (u + δ u) = 0. z

(21)

Also, let us, for the time being, assume that g vanishes on Γε . In this case, δ J in eq. (20) is written with the help of the Gauss theorem and eq. (21) as follows:

iρωλ iρωλ u ∇λ · ∇u − k2 λ u dΩ . − μ δu+ δJ = ℜ (22) dΓ + ℜ z z Γε Ωε

δ u on Γε has the following asymptotic expansion as ε → 0: iz δ u|Γε = −u(x0 )F0 − ∇u(x0 ) · nF1 , ρω 1 ρωπ ρω 2γ log kε ε + O(ε 2 ), ε+ F0 (ε ) = 1+i 2z π z 2 ρω F1 (ε ) = −i ε + O(ε 2 ). z Thus, we obtain the asymptotic expansion of RHS in eq. (22) as follows: iρωλ u δ J = 2πε ℜ + o(ε ), z which gives us the representation of the topological derivative T as follows: iρωλ (x)u(x) T (x) = ℜ . z

(23) (24) (25)

(26)

(27)

In the case that g does not vanish on Γε , the asymptotic expansion of the second term in RHS of eq. (20) is also required, which can be carried out after the explicit representation of g is given. For example, when 1 the objective function is given as the energy flux on Γ, i.e., g = 2ωρ ℑ [uq], ¯ the topological derivative has the following representation: iρωλ (x)u(x) |u(x)|2 . (28) T (x) = ℜ + z 2z The case that Γε is the Neumann boundary Let us then consider the case that the boundary condition on Γε is given as the Neumann one, which is expressed as follows: F(u, q) = q − qˆ = 0,

(29)

where qˆ is a prescribed Neumann data on Γε . Let us, for the time being, assume that g vanishes on Γε and qˆ = 0. In this case, δ J in eq. (20) is written with the help of the Gauss theorem and eq. (29) as follows:

∇λ · ∇u − k2 λ u dΩ − ℜ δJ = ℜ μδ udΓ (30) Ωε

Γε

δ u on Γε has the following asymptotic expansion as ε → 0: δ u|Γε = −ε ∇u · n + o(ε ). Thus, we obtain the following asymptotic expansion of RHS in eq. (30) as

2∇λ · ∇u − k2 λ u dΩ + o(ε 2 ), δJ = ℜ Ωε

(31)

(32)


181

which gives us the following representation for the topological derivative: T (x) = ℜ 2∇λ (x) · ∇u(x) − k2 λ (x)u(x) .

(33)

In the case of qˆ = 0, (30) is modified as follows:

∇λ · ∇u − k2 λ u dΩ − ℜ δJ = ℜ μδ udΓ + ℜ λ qdΓ ˆ , Ωε

Γε

Γε

(34)

the last term of which has the asymptotic expansion as 2πε ℜ [λ q] ˆ + o(ε ). Thus, the topological derivative in this case is expressed as follows: T (x) = ℜ [λ (x)q] ˆ.

(35)

In the case of g = 0 on Γε , the topological derivative is obtained once the explicit representation of g is given. 2 For example, when the objective function is given as g = |u|2 , eq. (33) is modified as T (x) =

|u(x)|2 , 2

(36)

and eq. (35) is modified as T (x) = ℜ [λ (x)q] ˆ +

|u(x)|2 . 2

(37)

Numerical examples In this section, we present some numerical examples which verify the validity of the topological derivatives presented in the preceding section. To this end, we have computed the topological derivatives along with the topological differences in the domain shown in Fig. 1. We here consider a plane wave incidence: uin = exp(ikx1 ) with k = 0.21. A single scatterer Ω = {(x1 , x2 )| x12 + x22 ≤ 10} is set to be either a material with impedance boundary or with the Neumann boundary. The boundary condition and the objective function considered are listed in Table 1.

Fig. 1

Experimental setting. The topological derivatives and differences are computed on lines A, B and C.

Fig. 2–8 show the topological derivatives and differences computed for each case. The agreement between the proposed derivatives and differences is satisfactory, from which the validity of the proposed derivatives is confirmed. From Fig. 3 and Fig. 4, one may observe that Case 2. and Case 3. have a considerably different distribution of the topological derivative. This indicates that the domain of the objective function may have a significant influence for the topological derivative. Conclusion We have derived some topological derivatives for acoustics with either the Neumann or impedance boundary condition. We have confirmed that the proposed topological derivatives agree with references. We found that the distribution of the topological derivatives is sensitive not only to the boundary condition but also to the domain of the objective function. We plan to develop a topology optimisation software combined with the proposed topological derivatives. We are especially interested in investigating a topology optimisation method for designdependent optimisation problems where the objective function is defined also on the morphing boundary which appears in the process of the optimisation.


Table 1 Boundary condition Case 1. Case 2. Case 3.

Γz = Γ ∪ Γε , z = 5, Γz = Γ ∪ Γ ε , z = 5 Γz = Γ ∪ Γε , z = 5,

Case 4. Case 5. Case 6. Case 7.

Γq = Γ ∪ Γε , qˆ = 0. Γq = Γ ∪ Γε , qˆ = 0. Γq = Γ ∪ Γε , qˆ = 1. Γq = Γ ∪ Γε , qˆ = 1.

obs )|2

f = 0, g = |u|2 on Γ, g = 0 on Γε . 2 f = 0, g = |u|2 on Γ ∪ Γε . |u|2 f = 0, g = 2 on Γ, g = 0 on Γε . 2 f = 0, g = |u|2 on Γ ∪ Γε . 2

D on A

T on B

-20

-10

0 x1

10

20

T on C

Topological derivative

0 -0.02 -0.04 -0.06 -0.08 -0.1 -0.12 -0.14 -30

30

-20

D on A

-10

0 x1

T on B

10

20

30

0.08 0.06 0.04 0.02 0 -0.02 -0.04 -0.06 -0.08 -0.1 -30

-20

-20

-10

0 x1

10

20

-10

D on C

0 x2

T on C 0.003

0

0.002

-0.02 -0.04 -0.06 -0.08 -0.1 -0.12 -30

30

D on B

0.02 Topological derivative


0.005 0 -0.005 -0.01 -0.015 -0.02 -0.025 -0.03 -0.035 -0.04 -30

10

20

30

-20

-10

0 x1

10

20

0.001 0 -0.001 -0.002 -0.003 -0.004 -30

30

D on C

-20

-10

0 x2

10

20

30

Topological derivatives for Case 2. (left): on A, (center): on B, (right): on C T on A

D on A

T on B


0.1 0.09 0.08 0.07 0.06 0.05 0.04 -30

-20

-10

0 x1

10

20

30

D on B

T on C

0.03

0.14

0.02

0.12


Fig. 3


D on B

0.02

T on A

Fig. 4

obs = (35, −10 + 2m), (m = 1, .., 17)., g = 0 , M = 17, xm 1 f = 0, g = 2ωρ ℑ [uq] ¯ on Γ, g = 0 on Γε . 1 f = 0, g = 2ωρ ℑ [uq] ¯ on Γ ∪ Γε .

Topological derivatives for Case 1. (left): on A, (center): on B, (right): on C

Fig. 2


|u(xm f = ∑M m=1 2



T on A -0.01 -0.02 -0.03 -0.04 -0.05 -0.06 -0.07 -0.08 -0.09 -0.1 -0.11 -0.12 -30

Boundary conditions and objective functions. Objective function

0.01 0 -0.01 -0.02 -0.03 -30

-20

-10

0 x1

10

20

30

D on C

0.1 0.08 0.06 0.04 0.02 0 -30

-20

-10

0 x2

10

20

30


References

[1] Martin Burger, Benjamin Hackl, and Wolfgang Ring. Incorporating topological derivatives into level set methods. Journal of Computational Physics, 194(1):344–362, 2004. [2] Takayuki Yamada, Kazuhiro Izui, Shinji Nishiwaki, and Akihiro Takezawa. A topology optimization method based on the level set method incorporating a fictitious interface energy. Computer Methods in Applied Mechanics and Engineering, 199(45):2876–2891, 2010. [3] Samuel Amstutz and Heiko Andrä. A new algorithm for topology optimization using a level-set method. Journal of Computational Physics, 216(2):573–588, 2006. [4] Grégoire Allaire, François Jouve, and Anca-Maria Toader. Structural optimization using sensitivity analysis and a level-set method. Journal of computational physics, 194(1):363–393, 2004. [5] Hiroshi Isakari, Kohei Kuriyama, Shinya Harada, Takayuki Yamada, Toru Takahashi, and Toshiro Matsumoto. A topology optimisation for three-dimensional acoustics with the level set method and the fast multipole boundary element method. Mechanical Engineering Journal, 1(4):CM0039–CM0039, 2014. [6] Antonio André Novotny, Raúl A Feijóo, Edgardo Taroco, and Claudio Padra. Topological sensitivity analysis. Computer Methods in Applied Mechanics and Engineering, 192(7):803–829, 2003.


-20

-10

0 x1

10

20

-0.04 -0.06 -0.08 -0.1 -20

-10

0 x1

T on B


D on A

-20

-10

0 x1

10

20

10

20

0.001 0 -0.001 -0.002 -0.003 -0.004 -30

30

-20

D on B

-10

0 x2

T on C

0.02

0.003

0

0.002

-0.02 -0.04 -0.06 -0.08 -0.1 -0.12 -30

30

T on A

-20

D on A

-10

0 x1

T on B

10

20

10

20

30

D on C

0.001 0 -0.001 -0.002 -0.003 -0.004 -30

30

-20

-10

0 x2

10

20

30

T on C

1 0 -1 -2 -20

-10

0 x1

10

20

5 0 -5 -10 -15 -30

30

D on C


2

-3 -30

D on B

10 Topological derivative

3

-20

-10

0 x1

10

20

0.6 0.4 0.2 0 -0.2 -0.4 -30

30

-20

-10

0 x2

10

20

30


Fig. 7

T on A

D on A

T on B


2 1 0 -1 -2 -3 -30

D on B

T on C

15

3

-20

-10

0 x1

10

20

30

10 5 0 -5 -10 -30

D on C



-0.02

D on C


Fig. 6


0.002


T on A 0.005 0 -0.005 -0.01 -0.015 -0.02 -0.025 -0.03 -0.035 -0.04 -30

Fig. 8

T on C 0.003

0

-0.12 -30

30

D on B

0.02


Fig. 5


T on B



D on A


T on A 0.005 0 -0.005 -0.01 -0.015 -0.02 -0.025 -0.03 -0.035 -0.04 -30

183

-20

-10

0 x1

10

20

30

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 -30

-20

-10

0 x2

10

20

30



Boundary Elements for Three-Dimensional Anisotropic Elastic Solids with Fundamental Solutions obtained by Radon-Stroh Formalism Chung-Lei Hsu, Chyanbin Hwu, and Y.C. Shiah* Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan 701, Taiwan, R.O.C. (* Corresponding to: [email protected]) Keywords: Boundary element method, anisotropic elasticity, Green’s function, fundamental solution, Radon-Stroh formalism, Radon transform. Abstract. This article presents a boundary element analysis of three-dimensional anisotropic elasticity by using the fundamental solutions obtained from Radon-Stroh formalism. Since the transformation used in Radon-Stroh formalism transforms two variables and keeps one variable unchanged, the Radon domain is a plane parallel to one of planes spanned by the coordinated axes. With this special feature, their associated derivatives become much easier to be evaluated than those of the traditional solutions based on the oblique plane. For demonstrating the veracity of the formulations and showing our implementation, two simple examples are presented.

Introduction For boundary element analysis, evaluation of the fundamental solutions plays a key role in computing its associated boundary integrals. The fundamental solutions for both two-dimensional (2D) and three-dimensional (3D) isotropic elastic bodies have been well documented in the literature. For 2D anisotropic elastic solids, classical solutions can be found in Lekhnitskii [1], Ting [2] and Hwu [3]. On the other hands, due to the mathematical complexity of the existing solution forms, those for 3D anisotropic cases still need further improvement. Over the years, several solutions, including the fundamental solution and its derivatives, for 3D anisotropic elastic solids have been presented in the literature and can be broadly categorized into five forms: (1) an integral on the oblique plane with normal coincident with the position vector (e.g. [4]), (2) an integral on the vertical or horizontal plane (e.g. [5]), (3) an analytical expression expressed in terms of Stroh’s eigenvalues (e.g. [6]), (4) an analytical solution expressed in terms of Stroh’s eigenvectors (e.g. [7]), and (5) an approximated series expansion (e.g. [8]). Most fundamental solutions presented in the literature start from the line integral on the oblique plane with normal coincident with the position vector, and subsequently, serious complexities arise in taking their derivatives on the oblique plane. Alternatively, they can be formulated by using the 2D Radon transform (e.g. [5], [9]), where the integral is defined on the vertical plane instead. As a result, the derivatives can be analytically derived on that plane without hassles. Taking this advantage, further improvement on the integral obtained from the 2D Radon transform was made by using the identities developed in the Stroh's formalism for 2D anisotropic elasticity [10]. In this paper, the fundamental solutions obtained by 2D Radon transform are employed to evaluate the boundary integral equations for 3D anisotropic elasticity. Verification was done by two simple examples. One is a cantilever beam, and the other is a solid cube. Comparison was also made, respectively, by the analytical solution of beam theory and the solutions calculated by the commercial finite element software ANSYS.

Boundary element formulation for 3D anisotropic elastic solids In absence of body forces, the boundary integral equation for 3D anisotropic elastostatics is written as [10]

cij ( )u j ( ) ³ tij* (, x ) u j ( x )d *( x ) *

³

*

uij* (, x ) t j ( x )d *( x ), i, j 1,2,3,

(1)


185

where * denotes the boundary of the elastic solid; u j ( x ) and t j (x ) are the displacements and surface tractions

along the boundaries; uij* (, x ) and tij* (, x ) are the fundamental solutions of displacements and tractions; cij ( ) are the free term coefficients of the source point , which equal to G ij / 2 for a smooth boundary and cij

G ij for

internal points, where G ij is the Kronecker delta. In practical applications, cij ( ) can be computed by considering rigid body motion. In the literature, there are several different expressions of the fundamental solutions. Among these, some are expressed in integral form, and others are in terms of Stroh’s eigenvalues. The former takes a lot of computational time on numerical integration, whereas the latter spends their time on the calculation of derivatives. Although an alternative approach has been proposed to avoid the numerical integration and derivatives, their fundamental solutions are approximated instead of exact [8]. To employ the exact closed-form solution and to avoid the trouble of numerical integration and derivatives, in this paper we try to utilize the fundamental solutions obtained by Radon-Stroh formalism [5, 10, 11]. By taking a 2D Radon transform on the governing equations of 3D anisotropic elasticity, it can be shown that the fundamental solutions can be written as [12]

[uij* ]

sgn( 'x3 ) 4S 2

³

S

0

sgn( 'x3 ) 4S 2

1ˆ N 2 (\ )dT , [tij* ] r

³

S

0

1 ˆT M1 (\ )dT , r2

(2a)

where

ˆ (\ ) N 2

ˆ (\ ) sin\ N 2 (\ ), M 1

cos\ s M1 (\ ) sin\ s M1* (\ ),

M1 (\ )

[sin 2 \ N1 (\ )]c cos 2\ I,

(2b)

\ ) [sin 2 \ N1* (\ )]c cos 2\ N1 ,

M1* (

N1* (\ )

N1N1 (\ ) N 2 N 3 (\ ).

In eq.(2a), sgn( 'x3 ) 1 for 'x3 ! 0 , sgn( 'x3 )

1 for 'x3 0 , and 'xi

xi xî , i 1,2,3 ; ( x1 , x2 , x3 ) and

( xˆ1 , xˆ 2 , xˆ3 ) are, respectively, the locations of field point x and source point ; Ni and Ni (\ ), i 1,2,3 are the

Stroh’s fundamental elasticity matrices [3], r and \ are the distance and polar angle related to the coordinates of Radon domain by r

cos\ s

wU / ws and sin\ s

( 'x1 cos T 'x2 sin T )2 ( 'x3 ) 2 , \

tan 1

'x3 . 'x1 cos T 'x2 sin T

(3)

wx3 / ws where s is a parameter denoting the tangential direction of the boundary

surface, and U and x3 are the coordinates used in 2D Radon-domain. The superscript T represents the transpose of a matrix; the prime xc denotes the derivative with respect to \ , and the derivative of the generalized fundamental matrices Ni (\ ) has been proved to be [3]

Note that Ni

N1c (\ )

{I N12 (\ ) N 2 (\ )N 3 (\ )},

Nc2 (\ )

{N1 (\ ) N 2 (\ ) N 2 (\ )N1T (\ )},

Nc3 (\ )

{N 3 (\ )N1 (\ ) N1T (\ )N 3 (\ )}.

Ni (0), i 1,2,3, and hence Nci

0.

(4)


Despite the presence of integral forms in eq. (2a), it is very promising to obtain their closed-form solutions expressed in terms of Stroh’s eigenvectors in a manner as in [11]. At this stage, the present work is just to implement the formulations in boundary element analysis and verify the results.

Numerical examples To verify the correctness of the presented formulations, two simple examples are analyzed using eq. (1). One is a cantilever beam subjected to a lateral point load at one end (Fig 1), and the other is a solid cube subjected to a tensile force on one side and fixed on the other (Fig 2). To show the applicability of the formulations to the degenerate case and the general anisotropy as well, the material for the first example has the following isotropic properties: E (Young's Modulus) 200 GPa, Q (Poisson's Ratio)=0.29; (5a) the stiffness coefficients of the second example are arbitrarily assumed to be

ª¬Cij º¼

ª 477.1 60.4 199.1 35.3 33.1 43.5º « 60.4 531.1 61 18.7 7.5 86.7 » « » 61 509.6 57.9 31.5 39.4 » « 199.1 « » , (unit: GPa). 12 » « 35.3 18.7 57.9 176.6 36.5 « 33.1 7.5 31.5 36.5 294.6 3.8 » « » 3.8 122.7 ¼ 12 ¬ 43.5 86.7 39.4

(5b)

Figure 1. (a) Cantilever beam; (b) BEM mesh.

Figure 2. (a) Solid cube; (b) BEM mesh.

For the first example, a point load P = 25000 kN is applied at the right end of a cantilever beam with the length L (Fig. 1(a)). For comparing the BEM results with the beam theory, various lengths are taken (L=H, 3H, 5H, 7H, and 10H, where H = 5m). As shown in Fig. 1(b), the BEM modeling employed 44 quadratic quadrilateral elements with total 146 nodes. The numerical integrations applied the Gauss quadrature rule using 8, 16, and 32 Gaussian points in each independent analysis to give respective results. For the second example, uniform


187

tension=1 (N/m2) is applied on top of the cube with side length S=5 (m). A total of 24 quadratic quadrilateral elements and 74 nodes were employed for the simulation as shown in Fig 2(b). Due to the highly anisotropic properties, relatively larger numbers of Gaussian points 16, 32 and 64 points were taken for the numerical integration. Fig 3 shows the computed deflection u2 and the axial stress V 11 on top surface along x1 axis of the cantilever beam when L=10H. By comparison with the beam theory and the analysis by the finite element method (FEM) using ANSYS (SOLID186 and BEAM188), it is seen that obvious discrepancy occurs only for case of 8point Gauss integration; otherwise, the results are in quite satisfactory agreements with theoretical values and FEM analysis as well. To see the influence of beam length, three additional comparisons are shown in Fig 4. From this figure we see that our results agree well with the analysis of ANSYS-SOLID186; however, the results of beam theory together with ANSYS-BEAM188 cannot provide ideal analysis for the short beam. Fig 5 shows u12 u22 u32 and V 0

the results of u0

3sij sij / 2 at surface points along AB & CD and at internal points

V ij V kk G ij / 3 are deviatoric stresses. It is seen that most data are in satisfactory agreement.

along EF , where sij

0.015

0

ANSYS (SOLID186) Simple bending stress Presented BEM (16 pts) Presented BEM (32 pts)

0.012

-10 ANSYS (SOLID186) ANSYS (BEAM188) Euler beam Presented BEM (8 pts) Presented BEM (16 pts) Presented BEM (32 pts)

-15

0.009

V11 / PLH

u2 E / PL

-5

0.006 0.003 0.000 -0.003

-20 0.0

0.2

0.4

0.6

0.8

0.0

1.0

0.2

0.4

0.6

0.8

1.0

1.2

x1 / L

x1 / L

0

0

-2

-1 u2 E / PL

u2 E / PL

Figure 3. (a) Deflection of beam (L=10H), (b) Stress.

-4 -6

-3

ANSYS (SOLID186) ANSYS (BEAM188) Euler beam Presented BEM (16 pts) Presented BEM (32 pts)

-8

-2 ANSYS (SOLID186) ANSYS (BEAM188) Euler beam Presented BEM (16 pts) Presented BEM (32 pts)

-4

-10

-5 0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0

0.2

x1 / L

0.4

0.6

0.8

1.0

1.2

0.8

1.0

1.2

x1 / L

0.0 0.0

u2 E / PL

u2 E / PL

-0.5

-1.0 ANSYS (SOLID186) ANSYS (BEAM188) Euler beam Presented BEM (16 pts) Presented BEM (32 pts)

-1.5

0.2

0.4

0.6

x1 / L

0.8

ANSYS (SOLID186) ANSYS (BEAM188) Euler beam Presented BEM (16 pts) Presented BEM (32 pts)

-0.2

-2.0 0.0

-0.1

1.0

1.2

-0.3 0.0

0.2

0.4

0.6

x2 / L

Figure 4. Deflection of beam for (a) L = 7H, (b) L = 5H, (c) L = 3H, (d) L = H.


× 10

-6

4

6

CD

4

ANSYS BEM (16-pts) BEM (32-pts) BEM (64-pts)

EF

3

-0.6

ANSYS BEM (16-pts) BEM (32-pts) BEM (64-pts)

3

V0 (MPa)

u0 (m)

5

EF

AB

2

1 -0.4

-0.2

0.0

0.2

0.4

0.6

-0.6

x1 / L

-0.4

-0.2

0.0

0.2

0.4

0.6

x1 / L

Figure 5. (a) Deformation, (b) equivalent stress.

Conclusions Using the fundamental solution obtained by Radon-Stroh formalism, numerical integration is still necessary in the boundary element analysis for three-dimensional anisotropic elastostatics. Through the two simple examples shown in this paper, it is seen that good agreements with other available solutions are obtained by using only 16 Gaussian points for the numerical integration. This is a good sign for our further study about the solutions obtained by Radon-Stroh formalism.

Acknowledgements The authors would like to thank the National Science Councils, Taiwan, Republic of China, for the support through Grant NSC 102-2221-E-006-290-MY3 and NSC 101-2221-E-006-056-MY3. References [1] S.G. Lekhnitskii Anisotropic Plates, (1968). [2] T. C. T. Ting Anisotropic Elasticity. Theory and Applications, Oxford University Press (1996). [3] C. Hwu Anisotropic Elastic Plates, Springer NewYork (2010). [4] D. Barnett Physica status solidi, 49, 741-748 (1972). [5] K.C. Wu Journal of Elasticity, 51, 213-225 (1998). [6] T. Ting, V.-G. Lee The Quarterly Journal of Mechanics and Applied Mathematics, 50, 407-426 (1997). [7] G. Nakamura, K. Tanuma, The Quarterly Journal of Mechanics and Applied Mathematics, 50, 179-194 (1997). [8] Y.C. Shiah, C.L. Tan, C. Wang, Engineering Analysis with Boundary Elements. 36, 1746-1755 (2012) [9] F.C. Buroni, M. Denda Advances in Boundary Element Techniques, (2014) [10] C.A. Brebbia, J.C.F. Telles and L.C. Wrobel Boundary Element Techniques: Theory and Applications in Engineering, Springer-Verlag (1984).

[11] L. Xie, C. Hwu and Ch. Zhang Advanced Methods for Calculating Green’s Function and Its Derivatives for Three-Dimensional Anisotropic Elastic Solids, submitted for publication (2015) [12] C. Hwu, C.L. Hsu and Y.C. Shiah, C. L. Green’s Functions for Three-Dimensional Anisotropic Elastic Solids via Radon-Stroh Formalism, under preparation (2015).


189

A method for appropriate determination of the location of sources in the method of fundamental solutions M.R. Hematiyan1, A. Haghighi2 and A. Khosravifard3 1, 2, 3

Department of Mechanical Engineering, Shiraz University, Shiraz 71936, Iran 1 2 3

[email protected]

[email protected]

[email protected]

Keywords: Method of fundamental solutions (MFS), Laplace equation; Source location; Location parameter

Abstract. A method based on the local smoothness of the solution on the boundary is presented for locating the source points in the method of fundamental solutions. The method is not based on an adaptive algorithm and is carried out without any optimization procedure. In this approach, the distance from a source point to the boundary of the solution domain is determined by accounting for the location of the neighbouring source points. The configuration of the source points determined by the present method results in a wellconditioned system of equations. The method is formulated for 2D problems governed by the Laplace equation, with the mention that it can also be developed for other governing equations. Several numerical examples are presented to prove the efficiency of the method. It is shown that by increasing the number of source points, the MFS with the presented method yields numerical solutions converging monotonically to the exact solution. Moreover, it is found that boundary conditions with a sharp variation of the field variable can be efficiently handled by the proposed method.

Introduction The method of fundamental solutions (MFS) is a simple boundary meshfree method, which has attracted the attention of scientists and engineers, especially over the last two decades. Analogous to the boundary element method (BEM), the MFS is based on the knowledge of a fundamental solution of the problem. However, unlike the BEM, the MFS is an integration-free method and this is the advantage of the MFS over the BEM. Since the MFS uses fundamental solutions as interpolating basis functions, the governing equations are exactly satisfied in the solution domain and on its boundary. Therefore, the MFS has the potential to provide very good numerical solutions simply by satisfying the boundary conditions as accurate as possible. For an accurate satisfaction of the boundary conditions in the MFS, one needs to locate the source points around the main boundary of the problem appropriately. In recent years, some researchers have investigated the location of the source points in the MFS. Gorzelańczyk and Kołodziej [1] used the MFS to solve several torsion problems with different configurations of the source points. They concluded that the error of the MFS is lower if the source points are located on a boundary geometrically similar to the boundary of the solution domain than when the source points are located on a circle enclosing the domain of interest. Karageorghis [2] presented a method for determining the optimum distance between the pseudo- and original boundaries in the MFS. Based on this method, an optimization problem with a single design variable should be solved. Wong and Ling [3] used an optimization method based on the effective condition number of the MFS system of equations for determining the configuration of the source points. Liu [4] proposed a distribution of the source points with different radial distances to solve the 2D Laplace equation using the MFS. Li et. al. [5] investigated the accuracy of the MFS for 2D boundary value problems associated with the Laplace equation and subject to non-harmonic boundary conditions. They suggested several ways for determining the location of collocation points and sources on the original boundary and the pseudo-boundary, respectively. Perhaps, the most important advantage of the MFS is its simplicity. Adaptive or dynamic approaches for locating the source points in the MFS require the solution of nonlinear or optimization problems, which make the MFS unattractive. The aim of this paper is to present a method for appropriate determination of the location of the MFS sources such that the following requirements are fulfilled:


x x

The method determines a configuration for the location of sources without any optimization algorithm. The configuration of sources results in a well-conditioned system of equations, i.e. the system can be solved by standard solvers such as the Gaussian elimination method. x By increasing the number of sources, the MFS-based solution converges to the exact one. x Boundary conditions with a sharp variation of the field variable can be handled by this method. The main idea of the method proposed herein is to locate the sources such that they have a locally smoothing effect on the boundary and they can handle boundary conditions with sharp variations. Moreover, the configuration of source and collocation points produces a system of equations with larger values on its main diagonal.

The MFS for 2D Laplace equation Consider the following boundary value problem governed by the Laplace equation in the 2D domain : and a general boundary condition on * , respectively, (1) 2 u 0 in : wu (2) f1u f 2 f 3 on * wn 2 Here, f1 , f 2 and f 3 are given functions on the boundary, denotes the Laplace operator, n is the outward normal direction to the boundary * and u is the primary variable. In the MFS, the solution is approximated by a linear combination of fundamental solutions, see e.g. [6], N

u( x )

¦ a u ( x, ξ ) *

i

i

(3)

i 1

where ξ i and a i are the location and intensity of the ith source located on the pseudo-boundary *c , respectively, N is the number of sources, and x is a point in the domain or on the boundary of the solution domain. The constants a i are the unknowns of the problem and they have to be found. The fundamental solution of the 2D Laplace equation is given by [6]: 1 (4) u * ( x, ξ i ) ln r 2S where r x - ξ i is the distance between the field point x and the source/singularity point ξ i . The MFS approximation of the partial derivatives of u can be found just by differentiating Eq. (3). For example, the normal derivative of u is approximated by wu * (x, ξ i ) wu(x ) N (5) ai ¦ wn wn i 1 To find the unknown constants a i , we consider M t N collocation points B1 , B1 , ..., B M on * and collocate the corresponding boundary condition at these points. From Eqs. (2), (3) and (5), one obtains N ª wu * (B i , ξ j ) º a j « f 1 ( B i )u * ( B i , ξ j ) f 2 ( B i ) (6) » f 3 (B i ), i 1, 2, ..., M ¦ wn j 1 ¼ ¬ which represents a system of M linear equations with N unknowns. In general, one can write system (6) as (7) AX F where the components of the matrix A R M uN and the vectors X R N and F R M can be expressed by wu* (Bi , ξ j ) (8) Aij f1 (Bi )u* (Bi , ξ j ) f 2 (Bi ) wn (9) X i ai , Fi f 3 (Bi ) If M = N, then system (7) can be solved using a standard solver such as the Gaussian elimination method, whereas if M ! N , system (7) is over-determined and can be solved in the least-squares sense, see e.g. [7]. In this work, the number of source points and collocation points are considered to be equal and the generated system of equations (7) is solved using the standard Gaussian elimination method. The system of equations in the MFS may be highly ill-conditioned. The solution of an ill-conditioned system of equations can be extremely inaccurate. The condition number of the coefficients matrix of a


191

system gives the information regarding how well-conditioned the system is. An upper bound for the relative error of the solution of a linear system of equations can be found using the following expression [8, 9]: (10) Erelative d N (A)H machine where N (A) represents the condition number of the coefficients matrix A , and Erelative is the relative error of the solution. H machine is the machine epsilon, which is the difference between one and the smallest floating point number greater than one in the computational machine.

Determination of the location of sources in the 2D MFS Consider a problem discretised using the MFS with N sources located at ξ1 , ξ 2 , ..., ξ N . We also assume that the number of collocation points on the boundary is N. The closest boundary point to the source point ξ i is considered as a collocation point and is denoted by B i as shown in Fig. 1. The points ξ i and B i are referred to as conjugated points, i.e., the point B i is referred to as the conjugated collocation point of ξ i . The distance from the source point ξ i to the collocation point B j is denoted by d ij . We arrange source and collocation points in a way that the distance from each collocation point to its conjugated source point is smaller than its distance to any other source point. This constraint can be expressed as follows: (11) d ii d ij for i 1, 2, ..., N and j z i In problems with Dirichlet boundary conditions, the constraint (11) guarantees that the diagonal element of a row in the coefficient matrix A is greater than other elements in the same row. Presence of larger values on the main diagonal of a matrix makes the condition number of the matrix smaller. For convenience, this constraint is further referred to as the condition improvement constraint. Now, we define a parameter, which describes the location of a source point relative to its neighbour source and collocation points. The ratios of d ii to d i ,i 1 and d ii to d i ,i 1 are denoted by K ic and K icc , respectively: K ic

d ii , d i ,i 1

K icc

d ii , d i ,i 1

0 Kic, Kicc 1

(12)

The location parameter for the source point ξ i is the minimum of the two parameters K ic and K icc : (13) Ki Min( Kic, Kicc), 0 Ki 1

Figure 1: Schematic diagram of the domain : and its corresponding boundary * , and several sources on the pseudo-boundary *c . It should be noted that if the location parameter of a source has a very small value (i.e. close to zero), then the source is closer to the boundary in comparison to its distance to the neighbouring source, whereas if the location parameter of a source has a value close to 1, then the source is closer to its neighbouring source than to the boundary. According to the condition improvement constraint (11), each source has the highest effect at its conjugated collocation point. When the location parameter of sources are very small, the sources show a local effect on the boundary, i.e. a large variation of the field variable occurs between the collocation points and therefore, the solution will not be locally smooth on the boundary. On the other hand, when the location parameter of sources approach unity, the sources produce a very smooth variation of the field variable on the boundary,


without effectively capturing the variation of the variable on the boundary. In the cases, where the location parameter is close to 1, the elements of the coefficients matrix of the generated system of equation will be very close to each other and therefore the system of equations becomes highly ill-conditioned. The selection of an appropriate value for the location parameter should be made such that not only it produces a locally smooth variation on the boundary, but also it enforces the satisfaction of the boundary conditions through a well-conditioned system of equations. Now, we investigate the influence of various values of the location parameter K on the numerical solution of boundary value problems for the 2D Laplace equation. It can be observed from Eq. (4) that the fundamental solution of the 2D Laplace equation and its derivatives are expressed in terms of the basis functions ln(r ) , r, x r , and r, y r , where r, x and r, y are the derivatives of r with respect to x and y, respectively. We consider a 2D domain whose boundary contains a straight-line segment as illustrated in Fig. 2. Without any loss of the generality, we assume that the boundary is directed along the x-coordinate. Seven equally spaced sources with equal strength and unit spacing are considered on the corresponding pseudo-boundary *c . To investigate the effect of the sources on the field variable and its derivatives on the boundary * , the following functions are evaluated on * : 7

7

g1

¦ a ln(r ) , i

i 1

i

g2

¦ a (r i

i 1

,x

7

r) ξ , g3 i

¦ a (r i

i 1

,y

r) ξ

(14) i

where ai 1 . Functions g1 , g 2 , and g 3 correspond to the fundamental solution of the 2D Laplace equation, and its x- and y-derivatives, respectively.

Figure 2: Schematic diagram of the boundary with 7 sources and their conjugated collocation points. The variation of functions g1 , g 2 , and g 3 on * is presented, for various values of K, in Fig. 3. It can be observed from Fig. 3 that functions g1 and g 2 show an oscillatory behaviour for K = 0.3 and K = 0.5. However, function g 3 still exhibits low oscillations on * even when K = 0.7. It is observed that K 0.8 is the lowest value of the location parameter for which smooth results are obtained.

Figure 3: Variation of (a) function g1 , (b) function g 2 , and (c) function g 3 on the boundary, for various values of the location parameter K. If K 0.8 , then the distance from a source point to its conjugated collocation point is sufficiently distinct from the distance between that source point and the conjugated collocation points of neighbouring source


193

points and hence the resulting MFS system of equations does not become ill-conditioned. If the sources are positioned such that K 0.8 , then a locally smooth variation of the field variable and its derivative on the boundary is not guaranteed. Consequently, for problems governed by the 2D Laplace equation, it is recommended that (15) 0.8 d K 1 This constraint is referred to as the local smoothness constraint. From the performed numerical experiments (see examples), it was observed that larger values of K (e.g. K = 0.95) still lead to a well-conditioned system of equations. However, larger values of K are suitable for problems with smooth boundary conditions, while lower values of K are suitable for source points corresponding to a part of boundary on which a condition with a sharp variation is prescribed. In the above numerical and graphical study of the effects of the MFS sources, we considered a 2D domain whose boundary contained a straight segment (Fig. 2), as well as equal strengths for the sources, i.e. ai 1 in Eq. (14). It was concluded that K = 0.8 can be considered as the lower bound for the location parameter. Considering a curved boundary instead of the straight one or using unequal values for a i instead of ai 1 has no effect on the local smoothness of the solution on the boundary; therefore K = 0.8 can be considered as the lower bound for the location parameter in general cases. The following steps and comments on the location of the MFS sources via the proposed method should be made: 1) Locate the collocation points on the boundary of the solution domain under investigation. 2) Consider a higher number of collocation points around a point on the boundary with a sharp change of boundary data. 3) Consider a sufficient number of collocation points in order to properly represent the boundary of the solution domain. 4) Modelling sharp concave/re-entrant corners in the MFS requires a special treatment such as, e.g., the domain decomposition method [1]. For an accurate modelling of a concave/re-entrant corner by the proposed method, one can approximately represent the corner by an arc with a small radius. 5) For the MFS analysis of 2D problems governed by the Laplace equation, it is recommended to use the location parameter for the source points in the range 0.8 d K 1 . Low values of K, i.e. close to 0.8, are more suitable for boundary points where a sharp variation of the boundary data occurs, while higher values of K are more suitable for a part of the boundary corresponding to a smooth boundary condition. 6) Locate the collocation and source points in a way that the condition improvement constraint is satisfied. Examples Several examples are presented in this section to assess the accuracy and convergence of the MFS when the presented method is used for locating the source points. For all examples investigated herein, a standard matrix inversion method, without any special algorithm, is used for solving the resulting system of equations. In the present computations, the relative error of solution, ( u ) at a boundary point is defined as

E

uexact uMFS umax

(16)

where uexact and uMFS are the exact and the MFS solutions, respectively, and umax is the maximum value of u on the boundary. The error of function u is evaluated at a large number of boundary points different from the MFS collocation points. The Laplace equation in a circle with various boundary conditions. In this example, we consider the 2D Laplace equation in a unit circle centred at the origin of the coordinates system, together with two different Dirichlet boundary conditions. In the first case, the following harmonic Dirichlet boundary condition is assumed: (17) u 3x 2 y y 3 In the second case, the following non-harmonic boundary condition is considered. (18) u 3x 2 y 2 y 3 Both boundary conditions (17) and (18) have a smooth variation over the boundary. Different number of source points and different values of the location parameter are considered in the analyses. The variation of


the condition number of the coefficients matrix with respect to the location parameter for different numbers of source points is shown in Fig. 4. It should be mentioned that the condition number is the same for the problem with different Dirichlet boundary conditions. It can be observed from Fig. 4 that the condition number of the coefficients matrix monotonically increases with respect to the location parameter for all the cases with different number of source points. In addition, it is seen that the condition number is almost the same for cases with the same location parameter and different number of source points.

Figure 4: Variation of the condition number with respect to the location parameter for different numbers of source points. The maximum error of the solutions with respect to the location parameter, are shown in Fig. 5. The convergence rate is high and very accurate solutions are obtained by considering a relatively low number of source points. It is observed from Fig. 5 that the error decreases with increase of K; however, for very large values of K (greater than 0.97) the error increases due to ill-conditioning of the coefficients matrix.

Figure 5: Maximum relative error of the results with respect to the location parameter and with various numbers of source points, (left) the problem with the harmonic boundary condition, (right) the problem with the non-harmonic boundary condition. The Laplace equation in a rectangle with a sharp boundary condition. In this example, a 2×1 rectangle with two opposite corners at (-1,0) and (1,1) and the following boundary conditions is considered. on x 1, x 1, y 1 (19) u 1 u 1

0.001(1 x 2 ) (0.001 x 2 )

on y

0

(20)

It can be seen from Eq. (20) that the boundary condition is nearly singular at the point (0,0). There is a sharp variation of the boundary data around this point; however, the boundary condition is smooth elsewhere. The problem is solved using 140 sources. The configuration of the sources and their conjugated collocation points is displayed in Fig. 6. We have considered more number of collocation points around the nearly singular point. The location parameter for all sources except the corner sources is K=0.9. The location parameter for the corner sources is set to K=0.8 to satisfy the condition improvement constraint. The exact and numerical values retrieved for u on the lower edge of the rectangle are presented in Fig. 7. As it can be seen, there is an excellent agreement between numerical and exact solutions. The maximum and average relative errors of the solution are 6.5×10-3 and 1.8×10-4, respectively. The condition number of the generated coefficients matrix is 2.5×104.


195

Figure 6: Configuration of the source and the collocation points for the rectangle.

Figure 7: Exact and numerical values of u on the lower edge of the rectangle. The Laplace equation in a Cassinian boundary. In this example, we consider a domain with a Cassinian boundary, described by (21) r(T ) (cos 4T 4 / 3 sin 2 4T )1/ 4 0 d T d 2S where r and T are polar coordinates. This curve includes convex and concave parts. The Laplace equation is solved over this domain for three different cases with 80, 120, and 160 sources. The configurations of source points and collocation points for the three different cases are shown in Fig. 8. The distance from a collocation point at a concave part of the curve to its conjugated source should be less than the radius of curvature. In each case, d ii is the same for all source points; however, both constraints (11) and (15) are satisfied. The error of the obtained solutions for different number of source points and boundary conditions are presented in Table 1. As it can be seen from Table 1, the errors of solutions are relatively small in all cases. It is also observed that the error monotonically decreases with increase of the number of source points.

Figure 8: Different configurations of source and collocation points, (left) N 80 , d ii 0.16 , K min 0.80 , K max 0.94 , K avg 0.84 , (middle) N 120 , d ii 0.11 , K min 0.82 , K max 0.91 , K avg 0.85 , (right) N

160 , d ii

0.08 , K min

0.82 , K max

0.89 , K avg

0.85 .


Table 1: Errors of solutions over the Cassinian curve obtained using the MFS with different number of source points and boundary conditions Boundary Number of source Maximum relative Average relative Condition number condition points error error 80 1.04×10-4 1.16×10-5 1.12×104 -5 -6 u 1 120 3.46×10 4.33×10 1.37×104 160 2.57×10-5 3.71×10-6 1.19×104 -3 -4 80 7.40×10 4.60×10 1.12×104 u x sin(4Sy ) 120 5.29×10-4 3.14×10-5 1.37×104 160 1.76×10-4 1.00×10-5 1.19×104 -2 -4 80 1.42×10 6.66×10 1.12×104 u y 120 9.58×10-3 3.14×10-4 1.37×104 -3 -4 160 7.21×10 1.80×10 1.19×104

Conclusions In this study, a method based on the local smoothness of the solution, was presented for determination of the location of the MFS source points. The appropriate location of the MFS sources was obtained without any optimization algorithm. The distance between a source point on the pseudo-boundary and its conjugated collocation point on the physical boundary of the solution domain was expressed according to the value of the location parameter at that point. In the proposed method the location of sources are determined by satisfying the condition improvement and the local smoothness constraints. By satisfying these constraints, a solution without any local oscillation on the boundary is obtained. Moreover, the condition number of the generated coefficients matrix is maintained in an appropriate range. The accuracy of the proposed method was investigated by considering several examples. In all of the examples, a well-conditioned system of equations was obtained by the configuration of source and collocation points determined by the proposed method. It was also observed that an increase in the number of source points, resulted in a monotonic decrease of the error of the numerical results retrieved by the MFS. It was also found that boundary conditions with a sharp variation of the field variable could be handled by the proposed approach. Increase of the value of the location parameter results in smoother solutions; however, it produces a coefficients matrix with a larger condition number. A proper configuration of the MFS source and collocation points in the vicinity of a concave/re-entrant corner requires more investigation and this will be considered in future works. We finally mention that the presented method in this paper is devoted to the 2D Laplace equation, but it can be extended to other linear partial differential operators admitting a fundamental solution.

References [1] P. Gorzelańczyk, J.A. Kołodziej, Engineering Analysis with Boundary Elements, 32(1), 64-75 (2008). [2] A. Karageorghis, Advances in Applied Mathematics and Mechanics, 1(4), 510-528 (2009). [3] K.Y. Wong, L. Ling, Engineering Analysis with Boundary Elements, 35(1), 42-46 (2011). [4] C.S. Liu, Engineering Analysis with Boundary Elements, 36(8), 1235-1245 (2012). [5] M. Li, C.S. Chen, A. Karageorghis, Computers and Math. with Applications, 66(11), 2400-2424 (2013). [6] G. Fairweather, A. Karageorghis, Advances in Computational Mathematics, 9(1-2), 69-95 (1998). [7] Y.S. Smyrlis, A. Karageorghis, Journal of Comp. and Applied Math., 227(1), 83-92 (2009). [8] A. Kaw Introduction to Matrix Algebra, autarkaw.com, (2008). [9] L.N. Trefethen, D. Bau III, Numerical linear algebra, Vol. 50, SIAM, (1997).


197

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N a a" uaj 7

" M " K˜ I " K˜ II 8 88 *89" " 9 " :2;


199

! " #!$ % " ! % # $ ! ! & cij (ξ)uj (ξ) + p∗ij (x, ξ)uj (x)dΓ(x) = u∗ij (x, ξ)pj (x)dΓ(x) Γ Γ cij (ξ)pj (ξ) + Nk s∗kij (x, ξ)uj (x)dΓ(x) = Nk d∗kij (x, ξ)pj (x)dΓ(x) Γ

Γ

#'$ #($

Γ # $ Ω Nk

% u∗ij p∗ij d∗kij s∗kij % )* u∗ij p∗ij % + " #$ " #'$ #($

cij (ξ)uj (ξ) + cij (ξ)pj (ξ) + Nk

Γ

Γ

p∗ij (x, ξ)uj (x)dΓ(x) +

s∗kij (x, ξ)uj (x)dΓ(x) + Nk

Γc

Γc

˜ l ψlj (ξ)dΓ = p∗ij (x, ξ)K

Γ

˜ l ψlj (ξ)dΓ = Nk s∗kij (x, ξ)K

u∗ij (x, ξ)pj (x)dΓ(x)

#,$

Γ

d∗kij (x, ξ)pj (x)dΓ(x)

#-$ Γc = Γ+ ∪Γ− Γ+ Γ− . p∗ij d∗rij s∗rij ' K˜ I K˜ II " " " % " % % L a=1

N a uaj

upper

=

L a=1

N a uaj

#/$

lower

L 0 " #/$ x y 1 " % % %

2 3

√ &

! 0 KI = σ∞ πa σ∞ a 4 2 5 % 6 7 C11 = 137.97 89 C12 = 5.78 89 C16 = 20.54 89 C22 = 12.45 89 C26 = 2.30 89 C66 = 12.98 89 : & %

;4

( 0

#$ % #