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Matrix decomposition MFS algorithms for elasticity and thermo-elasticity problems in axisymmetric domains. (English) Zbl 1206.35067

Summary: We propose an efficient matrix decomposition algorithm for the Method of Fundamental Solutions when applied to three-dimensional boundary value problems governed by elliptic systems of partial differential equations. In particular, we consider problems arising in linear elasticity in axisymmetric domains. The proposed algorithm exploits the block circulant structure of the coefficient matrices and makes use of fast Fourier transforms. The algorithm is also applied to problems in thermo-elasticity. Several numerical experiments are carried out.

MSC:

35E05 Fundamental solutions to PDEs and systems of PDEs with constant coefficients
65F05 Direct numerical methods for linear systems and matrix inversion
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
35J40 Boundary value problems for higher-order elliptic equations
41A30 Approximation by other special function classes
65N38 Boundary element methods for boundary value problems involving PDEs
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[1] M. A. Aleksidze, Fundamentalnye funktsii v priblizhennykh resheniyakh granichnykh zadach [Fundamental functions in approximate solutions of boundary value problems], Spravochnaya Matematicheskaya Biblioteka [Mathematical Reference Library], “Nauka”, Moscow, 1991. (in Russian); (with an English summary.).; M. A. Aleksidze, Fundamentalnye funktsii v priblizhennykh resheniyakh granichnykh zadach [Fundamental functions in approximate solutions of boundary value problems], Spravochnaya Matematicheskaya Biblioteka [Mathematical Reference Library], “Nauka”, Moscow, 1991. (in Russian); (with an English summary.). · Zbl 0752.65079
[2] Berger, J. R.; Karageorghis, A., The method of fundamental solutions for layered elastic materials, Eng. Anal. Bound. Elem., 25, 877-886 (2001) · Zbl 1008.74081
[3] J.R. Berger, A. Karageorghis, P.A. Martin, Stress intensity factor computation using the method of fundamental solutions: mixed-mode problems, Internat. J. Numer. Methods Eng., to appear.; J.R. Berger, A. Karageorghis, P.A. Martin, Stress intensity factor computation using the method of fundamental solutions: mixed-mode problems, Internat. J. Numer. Methods Eng., to appear. · Zbl 1194.74086
[4] Bialecki, B.; Fairweather, G., Matrix decomposition algorithms for separable elliptic boundary value problems in two space dimensions, J. Comput. Appl. Math., 46, 369-386 (1993) · Zbl 0781.65031
[5] Bialecki, B.; Fairweather, G., Orthogonal spline collocation methods for partial differential equations, J. Comput. Appl. Math., 128, 55-82 (2001), Numerical Analysis 2000, vol. VII, Partial Differential Equations · Zbl 0971.65105
[6] Bogomolny, A., Fundamental solutions method for elliptic boundary value problems, SIAM J. Numer. Anal., 22, 644-669 (1985) · Zbl 0579.65121
[7] C.A. Brebbia, S. Walker, E. Alarcón, A. Martin, F. Paris, F. Hartmann, J.C.F. Telles, L.C. Wrobel, J. Dominguez, M. Margulies, W.L. Wendland, Progress in Boundary Element Methods, vol. 1, Wiley, New York, 1981, Edited and with a preface by A. Brebbia, Halsted Press Book, New York.; C.A. Brebbia, S. Walker, E. Alarcón, A. Martin, F. Paris, F. Hartmann, J.C.F. Telles, L.C. Wrobel, J. Dominguez, M. Margulies, W.L. Wendland, Progress in Boundary Element Methods, vol. 1, Wiley, New York, 1981, Edited and with a preface by A. Brebbia, Halsted Press Book, New York.
[8] Burgess, G.; Maharejin, E., A comparison of the boundary element and superposition methods, Comput. & Structures, 19, 697-705 (1984) · Zbl 0552.73075
[9] P.J. Davis, Circulant Matrices, Wiley, New York, Chichester, Brisbane, 1979, A Wiley-Interscience Publication, Pure and Applied Mathematics.; P.J. Davis, Circulant Matrices, Wiley, New York, Chichester, Brisbane, 1979, A Wiley-Interscience Publication, Pure and Applied Mathematics. · Zbl 0418.15017
[10] Fairweather, G.; Karageorghis, A., The method of fundamental solutions for elliptic boundary value problems. Numerical treatment of boundary integral equations, Adv. Comput. Math., 9, 69-95 (1998) · Zbl 0922.65074
[11] Fairweather, G.; Karageorghis, A.; Smyrlis, Y.-S., A matrix decomposition MFS algorithm for axisymmetric biharmonic problems, Adv. Comput. Math., 23, 55-71 (2005) · Zbl 1067.65135
[12] Fenner, R. T., A force supersition approach to plane elastic stress and strain analysis, J. Strain Anal., 36, 517-529 (2001)
[13] Fichera, G., Il teorema del massimo modulo per l’equazione dell’elastostatica tridimensionale, Arch. Rational Mech. Anal., 7, 373-387 (1961) · Zbl 0100.30801
[14] Golberg, M. A.; Chen, C. S., Discrete Projection Methods for Integral Equations (1997), Computational Mechanics Publications: Computational Mechanics Publications Southampton · Zbl 0903.76065
[15] M.A. Golberg, C.S. Chen, The method of fundamental solutions for potential, Helmholtz and diffusion problems, in: Boundary Integral Methods: Numerical and Mathematical Aspects, Computational Engineering, vol. 1, WIT Press/Computational Mechanics Publications, Boston, MA, 1999, pp. 103-176.; M.A. Golberg, C.S. Chen, The method of fundamental solutions for potential, Helmholtz and diffusion problems, in: Boundary Integral Methods: Numerical and Mathematical Aspects, Computational Engineering, vol. 1, WIT Press/Computational Mechanics Publications, Boston, MA, 1999, pp. 103-176. · Zbl 0945.65130
[16] A. Karageorghis, C.S. Chen, Y.-S. Smyrlis, A matrix decomposition RBF algorithm: approximation of functions and their derivatives, Appl. Numer. Math., to appear.; A. Karageorghis, C.S. Chen, Y.-S. Smyrlis, A matrix decomposition RBF algorithm: approximation of functions and their derivatives, Appl. Numer. Math., to appear. · Zbl 1107.65305
[17] Karageorghis, A.; Fairweather, G., The method of fundamental solutions for axisymmetric elasticity problems, Comput. Mech., 25, 524-532 (2000) · Zbl 1011.74005
[18] Karageorghis, A.; Poullikkas, A.; Berger, J. R., Stress intensity factor computation using the method of fundamental solutions, Comput. Mech., 37, 445-454 (2006) · Zbl 1138.74336
[19] A. Karageorghis, Y.-S. Smyrlis, Matrix decomposition MFS algorithms, in: Boundary elements and other mesh reduction methods, XXVIII (Skiathos, 2006), International Series on Advances in Boundary Elements, vol. 42, WIT Press, Southampton, 2006, pp. 61-68.; A. Karageorghis, Y.-S. Smyrlis, Matrix decomposition MFS algorithms, in: Boundary elements and other mesh reduction methods, XXVIII (Skiathos, 2006), International Series on Advances in Boundary Elements, vol. 42, WIT Press, Southampton, 2006, pp. 61-68.
[20] A. Karageorghis, Y.-S. Smyrlis, T. Tsangaris, A matrix decomposition MFS algorithm for certain linear elasticity problems, Numer. Algorithms, to appear.; A. Karageorghis, Y.-S. Smyrlis, T. Tsangaris, A matrix decomposition MFS algorithm for certain linear elasticity problems, Numer. Algorithms, to appear. · Zbl 1104.74060
[21] Kitagawa, T., On the numerical stability of the method of fundamental solution applied to the Dirichlet problem, Japan J. Appl. Math., 5, 123-133 (1988) · Zbl 0644.65060
[22] Kołodziej, J. A., Review of applications of the boundary collocation methods in mechanics of continuous media, Solid Mech. Arch., 12, 187-231 (1987) · Zbl 0625.73096
[23] J.A. Kołodziej, Zastosowanie metody kollokacji brzegowej w zagadnieniach mechaniki [Applications of the Boundary Collocation Method in Applied Mechanics], Wydawnictwo Politechniki Poznańskiej, Poznań, 2001 (in Polish).; J.A. Kołodziej, Zastosowanie metody kollokacji brzegowej w zagadnieniach mechaniki [Applications of the Boundary Collocation Method in Applied Mechanics], Wydawnictwo Politechniki Poznańskiej, Poznań, 2001 (in Polish).
[24] Kupradze, V. D., On a method of solving approximately the limiting problems of mathematical physics, Ž. Vyčisl. Mat. i Mat. Fiz., 4, 1118-1121 (1964)
[25] Kupradze, V. D.; Aleksidze, M. A., An approximate method of solving certain boundary-value problems, Soobšč. Akad. Nauk Gruzin. SSR, 30, 529-536 (1963), (in Russian)
[26] Kupradze, V. D.; Aleksidze, M. A., The method of functional equations for the approximate solution of certain boundary value problems, Comput. Methods Math. Phys., 4, 82-126 (1964) · Zbl 0154.17604
[27] V.D. Kupradze, T.G. Gegelia, M.O. Basheleshvili, T.V. Burchuladze, Trekhmernye zadachi matematicheskoi teorii uprugosti i termouprugosti [Three-dimensional problems in the mathematical theory of elasticity and thermoelasticity.], Izdat. “Nauka”, Moscow, 1976 (in Russian).; V.D. Kupradze, T.G. Gegelia, M.O. Basheleshvili, T.V. Burchuladze, Trekhmernye zadachi matematicheskoi teorii uprugosti i termouprugosti [Three-dimensional problems in the mathematical theory of elasticity and thermoelasticity.], Izdat. “Nauka”, Moscow, 1976 (in Russian).
[28] Kythe, P. K., Fundamental Solutions for Differential Operators and Applications (1996), Birkhäuser Boston Inc.: Birkhäuser Boston Inc. Boston, MA · Zbl 0854.35118
[29] Love, A. E.H., A Treatise on the Mathematical Theory of Elasticity (1944), Dover Publications: Dover Publications New York · Zbl 0063.03651
[30] Maharejin, E., An extension of the superposition method for plane anisotropic elastic bodies, Comput. & Structures, 21, 953-958 (1985) · Zbl 0587.73123
[31] Marin, L.; Lesnic, D., The method of fundamental solutions for the Cauchy problem in two-dimensional linear elasticity, International J. Solids Structure, 41, 3425-3438 (2004) · Zbl 1071.74055
[32] Meyer, A.; Rjasanow, S., An effective direct solution method for certain boundary element equations in 3D, Math. Methods Appl. Sci., 13, 43-53 (1990) · Zbl 0703.65071
[33] Patterson, C.; Sheikh, M. A., On the use of fundamental solutions in Trefftz method for potential and elasticity problems, (Brebbia, C. A., Boundary Element Methods in Engineering, Proceedings of the Fourth International Seminar on Boundary Element Methods (1982), Springer: Springer New York), 43-54 · Zbl 0513.73096
[34] Poullikkas, A.; Karageorghis, A.; Georgiou, G., The method of fundamental solutions for three-dimensional elastostatics problems, Comput. & Structures, 80, 365-370 (2002)
[35] Raamachandran, J.; Rajamohan, C., Analysis of composite using charge simulation method, Eng. Anal. Bound. Elem., 18, 131-135 (1996)
[36] Redekop, D., Fundamental solutions for the collocation method in planar elastostatics, Appl. Math. Modelling, 6, 390-393 (1982) · Zbl 0492.73095
[37] Redekop, D.; Cheung, R. S.W., Fundamental solutions for the collocation method in three-dimensional elastostatics, Comput. & Structures, 26, 703-707 (1987) · Zbl 0612.73091
[38] Redekop, D.; Thompson, J. C., Use of fundamental solutions in the collocation method in axisymmetric elastostatics, Comput. & Structures, 17, 485-490 (1983)
[39] Rjasanow, S., Effective algorithms with circulant-block matrices, Linear Algebra Appl., 202, 55-69 (1994) · Zbl 0804.65042
[40] Rjasanow, S., Optimal preconditioner for boundary element formulation of the Dirichlet problem in elasticity, Math. Methods Appl. Sci., 18, 603-613 (1995) · Zbl 0827.73072
[41] Rjasanow, S., The structure of the boundary element matrix for the three-dimensional Dirichlet problem in elasticity, Numer. Linear Algebra Appl., 5, 203-217 (1998) · Zbl 0935.74076
[42] Y.-S. Smyrlis, Applicability and applications of the method of fundamental solutions, Technical Report TR-03-2006, Department of Mathematics and Statistics, University of Cyprus, February 2006.; Y.-S. Smyrlis, Applicability and applications of the method of fundamental solutions, Technical Report TR-03-2006, Department of Mathematics and Statistics, University of Cyprus, February 2006.
[43] Smyrlis, Y.-S.; Karageorghis, A., Some aspects of the method of fundamental solutions for certain harmonic problems, J. Sci. Comput., 16, 341-371 (2001) · Zbl 0995.65116
[44] Smyrlis, Y.-S.; Karageorghis, A., Some aspects of the method of fundamental solutions for certain biharmonic problems, CMES Comput. Model. Eng. Sci., 4, 535-550 (2003) · Zbl 1051.65110
[45] Smyrlis, Y.-S.; Karageorghis, A., A matrix decomposition MFS algorithm for axisymmetric potential problems, Eng. Anal. Bound. Elem., 28, 463-474 (2004) · Zbl 1074.65134
[46] Smyrlis, Y.-S.; Karageorghis, A., Numerical analysis of the MFS for certain harmonic problems, M2AN Math. Model. Numer. Anal., 38, 495-517 (2004) · Zbl 1079.65108
[47] Tsangaris, T.; Smyrlis, Y.-S.; Karageorghis, A., A matrix decomposition MFS algorithm for problems in hollow axisymmetric domains, J. Sci. Comput., 28, 31-50 (2006) · Zbl 1098.65117
[48] D.-L. Young, C.L. Chiu, C.M. Fan, C.C. Tsai, Y.C. Lin, Method of fundamental solutions for multidimensional stokes equations by the dual-potential formulation, Eur. J. Mech. B Fluids, to appear.; D.-L. Young, C.L. Chiu, C.M. Fan, C.C. Tsai, Y.C. Lin, Method of fundamental solutions for multidimensional stokes equations by the dual-potential formulation, Eur. J. Mech. B Fluids, to appear. · Zbl 1106.76020
[49] Young, D.-L.; Jane, S. J.; Fan, C. M.; Murugesan, K.; Tsai, C. C., The method of fundamental solutions for 2d and 3d stokes problems, J. Comput. Phys., 211, 1-8 (2006), (Short note) · Zbl 1160.76332
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