zbMATH — the first resource for mathematics

Local integral equation method for potential problems in functionally graded anisotropic materials. (English) Zbl 1182.74237
Summary: Efficient computational techniques are developed for 2D potential problems in anisotropic media with continuously variable material coefficients. The method is based on integral relationships considered on local sub-domains and domain-type approximations of the field variable. Three different kinds of integral equations are combined with either a domain element interpolation or a meshless point interpolation. The physical background of the formulation is discussed briefly. The accuracy and the convergence of the proposed techniques are tested by several examples and compared with benchmark analytical solutions.

74S15 Boundary element methods applied to problems in solid mechanics
74E05 Inhomogeneity in solid mechanics
74E10 Anisotropy in solid mechanics
BEAN; Mfree2D
Full Text: DOI
[1] Brebbia, C.A.; Telles, J.C.F.; Wrobel, L.C., Boundary element techniques: theory and applications in engineering, (1984), Springer Berlin · Zbl 0556.73086
[2] Balas, J.; Sladek, J.; Sladek, V., Stress analysis by boundary element methods, (1989), Elsevier Amsterdam · Zbl 0681.73001
[3] Paris, F.; Canas, J., Boundary element method, (1997), Oxford University Press Inc. New York
[4] Wrobel, L.C., The boundary element method, vol. 1: applications in thermo-fluids and acoustics, (2002), John Wiley and Sons Ltd Chichester
[5] Clements, D.L., A boundary integral equation method for the numerical solution of a second order elliptic equation with variable coefficients, Jaustral math soc ser B, 22, 218-228, (1980) · Zbl 0452.65070
[6] Cheng, A.H-D., Darcy’s flow with variable permeability—a boundary integral solution, Water resources res, 20, 980-984, (1984)
[7] Cheng, A.H-D., Heterogeneities in flows through porous media by boundary element method, Topics in boundary element research: applications to geomechanics, vol. 4, (1987), p. 1291-344
[8] Shaw, R.P.; Makris, N., Green’s functions for Helmholtz and Laplace equations in heterogeneous media, Engng anal boundary elem, 10, 179-183, (1992)
[9] Shaw, R.P., Green’s functions for heterogeneous media potential problems, Engng anal boundary elem, 13, 219-221, (1994)
[10] Ang, W.T.; Kusuma, J.; Clements, D.L., A boundary element method for a second order elliptic partial differential equation with variable coefficients, Engng anal boundary elem, 18, 311-316, (1986)
[11] Gray, L.J.; Kaplan, T.; Richardson, J.D.; Paulino, G.H., Green’s functions and boundary integral analysis for exponentially graded materials: heat conduction, J appl mech trans ASME, 70, 543-549, (2003) · Zbl 1110.74461
[12] Martin, P.A.; Richardson, J.D.; Gray, L.J.; Berger, J.R., On Green’s function for a three-dimensional exponentially graded elastic solid, Proc R soc London A, 458, 1931-1947, (2002) · Zbl 1056.74017
[13] Sutradhar, A.; Paulino, G.H.; Gray, L.J., Transient heat conduction in homogeneous and non-homogeneous materials by the Laplace transform Galerkin boundary element method, Engng anal boundary elem, 16, 119-132, (2002) · Zbl 0995.80010
[14] Pan, E., Static Green’s functions in multilayered half spaces, Appl math model, 21, 509-521, (1997) · Zbl 0896.73002
[15] Partridge, P.W.; Brebbia, C.A.; Wrobel, L.C., The dual reciprocity boundary element method, (1992), Computational Mechanics Publications Southampton · Zbl 0758.65071
[16] Sladek, V.; Sladek, J., A new formulation for solution of boundary value problems using domain-type approximations and local integral equations, Elect J boundary elem, 1, 132-153, (2003)
[17] Sladek, V.; Sladek, J., Singular integrals in boundary element methods, (1998), Computational Mechanics Publications Southampton · Zbl 0961.74072
[18] Sutradhar, A.; Paulino, G.H., A simple boundary element method for problems of potential in non-homogeneous media, Int J num methods engng, 60, 2203-2230, (2004) · Zbl 1070.80005
[19] Sutradhar, A.; Paulino, G.H., The simple boundary element method for transient heat conduction in functionally graded materials, Comp methods appl mech engng, 193, 4511-4539, (2004) · Zbl 1073.80005
[20] Sutradhar A, Paulino GH, Gray LJ. On hypersingular surface integrals in the symmetric Galerkin boundary element method: application to heat conduction in exponentially graded materials. Int J Num Methods Engng 2005;62:122-57. · Zbl 1087.80007
[21] Sladek, V.; Sladek, J.; Zhang, Ch., Local integro-differential equations with domain elements for numerical solution of PDE with variable coefficients, J engng math, 51, 261-282, (2005) · Zbl 1073.65138
[22] Mikhailov, S.E., Localized boundary-domain integral formulations for problems with variable coefficients, Engng anal boundary elem, 26, 681-690, (2002) · Zbl 1016.65097
[23] Mikhailov SE, Nakhova IS. Numerical solution of a Neumann problem with variable coefficients by the localized boundary-domain integral equation method. In: Sia Amini, editor. Fourth UK Conference on Boundary Integral Methods. Salford University, UK, ISBN 0-902896-40-7, 2003. p. 175-84.
[24] Atluri, S.N.; Zhu, T., A new meshless local petrov – galerkin (MLPG) approach to nonlinear problems in computational modeling and simulation, Comput model simul engng, 3, 187-196, (1998)
[25] Atluri, S.N.; Zhu, T., A new meshless local petrov – galerkin (MLPG) approach in computational mechanics, Comput mech, 22, 117-127, (1998) · Zbl 0932.76067
[26] Atluri, S.N.; Zhu, T., The meshless local petrov – galerkin (MLPG) approach for solving problems in elasto-statics, Comput mech, 25, 169-179, (1999) · Zbl 0976.74078
[27] Atluri, S.N.; Shen, S., The meshless local petrov – galerkin (MLPG) method: a simple and less-costly alternative to the finite element and boundary element method, Comput model engng sci, 3, 11-52, (2002) · Zbl 0996.65116
[28] Atluri, S.N.; Shen, S., The meshless local petrov – galerkin (MLPG) method, (2002), Tech Science Press Encino · Zbl 1012.65116
[29] Atluri, S.N., The meshless method (MLPG) for domain and BIE discretizations, (2004), Tech Science Press Forsyth, GA · Zbl 1105.65107
[30] Sladek, J.; Sladek, V., Local boundary integral equation method for heat conduction problem in an anisotropic medium, () · Zbl 1129.74346
[31] Sladek, J.; Sladek, V.; Zhang, Ch., Heat conduction analysis in nonhomogeneous anisotropic solids, (), 609-614 · Zbl 1182.74258
[32] Sladek, J.; Sladek, V.; Atluri, S.N., Meshless local petrov – galerkin method for heat conduction problem in an anisotropic medium, Comput model engng sci, 6, 309-318, (2004) · Zbl 1084.80002
[33] Nowak, A.J.; Neves, A.C., The multiple reciprocity boundary element method, (1994), Computational Mechanics Publications Southampton · Zbl 0868.73006
[34] Hughes, T.J.R., The finite element method, Linear static and dynamic finite element analysis, (1987), Prentice-Hall Englewood Cliffs
[35] Liu, G.R., Mesh free methods, Moving beyond the finite element method, (2003), CRC Press Boca Raton
[36] Sladek, V.; Sladek, J.; Tanaka, M., Local integral equations and two meshless polynomial interpolations with application to potential problems in non-homogeneous media, Comput model engng sci, 7, 69-83, (2005) · Zbl 1099.74071
[37] Hardy, R.L., Theory and applications of the multiquadrics-biharmonic method (20 years of discovery 1968-1988), Comput math appl, 19, 163-208, (1990) · Zbl 0692.65003
[38] Golberg, M.A.; Chen, C.S.; Bowman, H., Some recent results and proposals for the use of radial basis functions in the BEM, Engng anal bound elem, 23, 285-296, (1999) · Zbl 0948.65132
[39] Chang, Y.P.; Kang, C.S.; Chen, D.J., The use of fundamental Green’s functions for the solution of problems of heat conduction in anisotropic media, J heat mass transfer, 16, 1905-1918, (1973) · Zbl 0263.35041
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.