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A matrix decomposition MFS algorithm for certain linear elasticity problems

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Abstract

We propose an efficient matrix decomposition Method of Fundamental Solutions algorithm for the solution of certain two-dimensional linear elasticity problems. In particular, we consider the solution of the Cauchy–Navier equations in circular domains subject to Dirichlet boundary conditions, that is when the displacements are prescribed on the boundary. The proposed algorithm is extended to the case of annular domains. Numerical experiments for both types of problems are presented.

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References

  1. Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II. Commun. Pure Appl. Math. 17, 35–92 (1964)

    MathSciNet  MATH  Google Scholar 

  2. Berger, J.R., Karageorghis, A.: The method of fundamental solutions for layered elastic materials. Eng. Anal. Bound. Elem. 25, 877–886 (2001)

    Article  MATH  Google Scholar 

  3. Bialecki, B., Fairweather, G.: Matrix decomposition algorithms for separable elliptic boundary value problems in two space dimensions. J. Comput. Appl. Math. 46(3), 369–386 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bogomolny, A.: Fundamental solutions method for elliptic boundary value problems. SIAM J. Numer. Anal. 22(4), 644–669 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  5. Burgess, G., Maharejin, E.: A comparison of the boundary element and superposition methods. Comput. Struct. 19(5–6), 697–705 (1984)

    Article  MATH  Google Scholar 

  6. Chen, C.W., Young, D.L., Tsai, C.C., Murugesan, K.: The method of fundamental solutions for inverse 2D Stokes problems. Comput. Mech. 37(1), 2–14 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cho, H.A., Golberg, M.A., Muleshkov, A.S., Li, X.: Trefftz methods for time dependent partial differential equations. Comput. Mat. Cont. 1(1), 1–37 (2004)

    Google Scholar 

  8. Davis, P.J.: Circulant Matrices. Wiley, New York-Chichester-Brisbane (A Wiley-Interscience Publication, Pure and Applied Mathematics) (1979)

    MATH  Google Scholar 

  9. Fairweather, G., Karageorghis, A.: The method of fundamental solutions for elliptic boundary value problems. Numerical treatment of boundary integral equations. Adv. Comput. Math. 9(1-2), 69–95 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fairweather, G., Karageorghis, A., Martin, P.A.: The method of fundamental solutions for scattering and radiation problems. Eng. Anal. Bound. Elem. 27, 759–769 (2003)

    Article  MATH  Google Scholar 

  11. Fenner, R.T.: A force supersition approach to plane elastic stress and strain analysis. J. Strain Anal. 36(5), 517–529 (2001)

    Article  Google Scholar 

  12. Golberg, M.A., Chen, C.S.: The method of fundamental solutions for potential, Helmholtz and diffusion problems. In: Boundary Integral Methods: Numerical and Mathematical Aspects. Comput. Eng., vol. 1, pp. 103–176. WIT/Comput. Mech. Publ., Boston, Massachusetts (1999)

  13. Kane, J.H.: Boundary Element Analysis in Engineering Continuum Mechanics. Prentice Hall, Engelwood Cliffs, New Jersey (1994)

    Google Scholar 

  14. Karageorghis, A., Fairweather, G.: The method of fundamental solutions for axisymmetric elasticity problems. Comput. Mech. 25(6), 524–532 (2000)

    Article  MATH  Google Scholar 

  15. Kitagawa, T.: On the numerical stability of the method of fundamental solution applied to the Dirichlet problem. Jpn. J. Appl. Math. 5(1), 123–133 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kołodziej, J.A.: Review of applications of the boundary collocation methods in mechanics of continuous media. Solid Mech. Arch. 12, 187–231 (1987)

    MATH  Google Scholar 

  17. Kupradze, V.D.: On a method of solving approximately the limiting problems of mathematical physics. Ž. Vyčisl. Mat. Mat. Fiz. 4, 1118–1121 (1964)

    MathSciNet  Google Scholar 

  18. 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)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kythe, P.K.: Fundamental Solutions for Differential Operators and Applications. Birkhäuser, Boston, Massachusetts (1996)

    MATH  Google Scholar 

  20. Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity, 4th edn. Dover, New York (1944)

    MATH  Google Scholar 

  21. Maharejin, E.: An extension of the superposition method for plane anisotropic elastic bodies. Comput. Struct. 21(5), 953–958 (1985)

    Article  Google Scholar 

  22. Marin, L., Lesnic, D.: The method of fundamental solutions for the Cauchy problem in two-dimensional linear elasticity. Int. J. Solids Struct. 41(13), 3425–3438 (2004)

    Article  MATH  Google Scholar 

  23. Mathon, R., Johnston, R.L.: The approximate solution of elliptic boundary-value problems by fundamental solutions. SIAM J. Numer. Anal. 14(4), 638–650 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  24. Patterson, C., Sheikh, M.A.: On the use of fundamental solutions in Trefftz method for potential and elasticity problems. In: Brebbia, C.A. (ed) Boundary Element Methods in Engineering. Proceedings of the Fourth International Seminar on Boundary Element Methods, pp. 43–54. Springer, Berlin Heidelberg New York (1982)

    Google Scholar 

  25. Poullikkas, A., Karageorghis, A., Georgiou, G.: The method of fundamental solutions for three-dimensional elastostatics problems. Comput. Struct. 80(3-4), 365–370 (2002)

    Article  MathSciNet  Google Scholar 

  26. Pozrikidis, C.: Boundary Integral and Singularity Methods for Linearized Viscous Flow. In: Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, UK (1992)

    Google Scholar 

  27. Raamachandran, J., Rajamohan, C.: Analysis of composite plates using charge simulation method. Eng. Anal. Bound. Elem. 18(2), 131–135 (1996)

    Article  Google Scholar 

  28. Redekop, D.: Fundamental solutions for the collocation method in planar elastostatics. Appl. Math. Model. 6(5), 390–393 (1982)

    Article  MATH  Google Scholar 

  29. Redekop, D., Cheung, R.S.W.: Fundamental solutions for the collocation method in three-dimensional elastostatics. Comput. Struct. 26(4), 703–707 (1987)

    Article  MATH  Google Scholar 

  30. Redekop, D., Thompson, J.C.: Use of fundamental solutions in the collocation method in axisymmetric elastostatics. Comput. Struct. 17(4), 485–490 (1983)

    Article  Google Scholar 

  31. Rjasanow, S.: Optimal preconditioner for boundary element formulation of the Dirichlet problem in elasticity. Math. Methods Appl. Sci. 18(8), 603–613 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  32. Smyrlis, Y.-S.: Applicability and applications of the method of fundamental solutions. Tech. Report TR–03–2006, Department of Mathematics and Statistics, University of Cyprus, Nicosia, Cyprus (2006)

    Google Scholar 

  33. Smyrlis, Y.-S., Karageorghis, A.: A matrix decomposition MFS algorithm for axisymmetric potential problems. Eng. Anal. Bound. Elem. 28, 463–474 (2004)

    Article  MATH  Google Scholar 

  34. Smyrlis, Y.-S., Karageorghis, A.: Numerical analysis of the MFS for certain harmonic problems. M2AN Math. Model. Numer. Anal. 38(3), 495–517 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  35. Smyrlis, Y.-S., Karageorghis, A.: The method of fundamental solutions for stationary heat conduction problems in rotationally symmetric domains. SIAM J. Sci. Comput. 26(4), 1493–1512 (2006)

    Article  MathSciNet  Google Scholar 

  36. Tsangaris, Th., Smyrlis, Y.-S., Karageorghis, A.: Numerical analysis of the MFS for harmonic problems in annular domains. Numer. Methods Partial Differ. Equ. 22, 507–539 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  37. 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), 1–8 (2006) (short note)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, 1678, Nicosia, Cyprus

    A. Karageorghis, Y. -S. Smyrlis & T. Tsangaris

Authors
  1. A. Karageorghis
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  2. Y. -S. Smyrlis
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  3. T. Tsangaris
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Correspondence to A. Karageorghis.

Additional information

Parts of this work were undertaken whilst the corresponding author was a Visiting Professor in the Department of Mathematical and Computer Sciences, Colorado School of Mines, Golden, Colorado 80401, USA

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Karageorghis, A., Smyrlis, Y.S. & Tsangaris, T. A matrix decomposition MFS algorithm for certain linear elasticity problems. Numer Algor 43, 123–149 (2006). https://doi.org/10.1007/s11075-006-9045-3

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