Advertisement

Computational Mechanics

, Volume 47, Issue 6, pp 669–680 | Cite as

A regularized collocation boundary element method for linear poroelasticity

  • Michael Messner
  • Martin SchanzEmail author
Original Paper

Abstract

This article presents a collocation boundary element method for linear poroelasticity, based on the first boundary integral equation with only weakly singular kernels. This is possible due to a regularization of the strongly singular double layer operator, based on integration by parts, which has been applied to poroelastodynamics for the first time. For the time discretization the convolution quadrature method (CQM) is used, which only requires the Laplace transform of the fundamental solution. Furthermore, since linear poroelasticity couples a linear elastic with an acoustic material, the spatial regularization procedure applied here is adopted from linear elasticity and is performed in Laplace domain due to the before mentioned CQM. Finally, the spatial discretization is done via a collocation scheme. At the end, some numerical results are shown to validate the presented method with respect to different temporal and spatial discretizations.

Keywords

Linear poroelasticity Collocation BEM Regularized double layer operator CQM 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Banjai L, Sauter S (2008) Rapid solution of the wave equation in unbounded domains. J Numer Anal 47(1):227–249, 109–128Google Scholar
  2. 2.
    Becache E, Nedelec JC, Nishimura N (1993) Regularization in 3D for anisotropic elastodynamic crack and obstacle problems. J Elast 31: 25–46MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Biot MA (1956) Theory of propagation of elastic waves in a fluid-saturated porous solid. I. J Acoust Soc Am 28(2): 168–178MathSciNetCrossRefGoogle Scholar
  4. 4.
    Biot MA (1956) Theory of propagation of elastic waves in a fluid-saturated porous solid. II. J Acoust Soc Am 28(2): 179–191MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bonnet G (1987) Basic singular solutions for a poroelastic medium in the dynamic range. J Acoust Soc Am 82(5): 1758–1763CrossRefGoogle Scholar
  6. 6.
    Bonnet G, Auriault JL (1985) Dynamics of saturated and deformable porous media: homogenization theory and determination of the solid-liquid coupling coefficients. Physics of finely divided matter. Springer Verlag, Berlin, pp 306–316Google Scholar
  7. 7.
    Chen J, Dargush GF (1995) Boundary element method for dynamic poroelastic and thermoelastic analyses. Int J Solids Struct 32(15): 2257–2278zbMATHCrossRefGoogle Scholar
  8. 8.
    Guiggiani M, Gigante A (1990) A general algorithm for multidimensional cauchy principal value integrals in the boundary element method. J Appl Mech 57: 906–915MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Hackbusch W (1989) Integralgleichungen. Teubner, StuttgartzbMATHGoogle Scholar
  10. 10.
    Han H (1994) The boundary integro-differential equations of three-dimensional Neumann problem in linear elasticity. Numer Math 68: 268–281CrossRefGoogle Scholar
  11. 11.
    Kielhorn L (2009) A time-domain symmetric Galerkin BEM for viscoelastodynamics. In: Computation in engineering and science, vol 5. Verlag der Technischen Universtiät GrazGoogle Scholar
  12. 12.
    Kielhorn L, Schanz M (2008) Convolution quadrature method-based symmetric Galerkin boundary element method for 3-D elastodynamics. Int J Numer Methods Eng 76: 1724–1746MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Kim YK, Kingsbury HB (1979) Dynamic characterization of poroelastic materials. Exp Mech 19: 252–258CrossRefGoogle Scholar
  14. 14.
    Kupradze VD (1979) Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity. North-HollandGoogle Scholar
  15. 15.
    Lubich C (1988) Convolution quadrature and discretized operational calculus I. Numer Math 52: 129–145MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Lubich C (1988) Convolution quadrature and discretized operational calculus II. Numer Math 52: 413–425MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Manolis GD, Beskos DE (1989) Integral formulation and fundamental solutions of dynamic poroelasticity and thermoelasticity. Acta Mech 76: 89–104zbMATHCrossRefGoogle Scholar
  18. 18.
    Mantic V (1993) A new formula for the c-matrix in the somigliana identity. J Elast 33: 191–201MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Martin PA, Rizzo FJ (1989) On boundary integral equations for crack problems. Proc Roy Soc A 421(1861): 341–355MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Maue AW (1949) Zur Formulierung eines allgemeinen Beugungsproblems durch eine Integralgleichung. Zeitschrift für Physik 126(7–9): 609–618MathSciNetGoogle Scholar
  21. 21.
    Messner M, Messner M, Rammerstorfer F, Urthaler P (2010) Hyperbolic and elliptic numerical analysis BEM library. http://www.mech.tugraz.at/HyENA
  22. 22.
    Nedelec J (1982) Integral equations with nonintegrable kernels. Integral Equ Oper Theory 5: 563–672MathSciNetCrossRefGoogle Scholar
  23. 23.
    Nishimura N, Kobayashi S (1989) A regularized boundary intgral equation method for elastodynamic crack problems. Comput Mech 4: 319–328zbMATHCrossRefGoogle Scholar
  24. 24.
    Schanz M (2001) Wave propagation in viscoelastic and poroelastic continua. Lecture notes in applied mechanics, vol 2. Springer, New YorkGoogle Scholar
  25. 25.
    Schanz M (2009) Poroelastodynamics: linear models, analytical solutions, and numerical methods. Appl Mech Rev 62(3):030,803-1–030,803-15Google Scholar
  26. 26.
    Schanz M (2010) On a reformulated convolution quadrature based boundary element method. Comput Model Eng Sci 58(2): 227–249MathSciNetGoogle Scholar
  27. 27.
    Schanz M, Cheng AHD (2000) Transient wave propagation in a one-dimensional poroelastic column. Acta Mech 145: 1–18zbMATHCrossRefGoogle Scholar
  28. 28.
    Steinbach O (2008) Numerical approximation methods for elliptic boundary value problems. Springer, New YorkzbMATHCrossRefGoogle Scholar
  29. 29.
    Wiebe T, Antes H (1991) A time domain integral formulation of dynamic poroelasticity. Acta Mech 90: 125–137MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Institute of Applied MechanicsGraz University of TechnologyGrazAustria

Personalised recommendations