Advertisement

Acta Mechanica

, Volume 223, Issue 8, pp 1751–1768 | Cite as

A symmetric Galerkin boundary element method for 3d linear poroelasticity

  • Michael Messner
  • Martin SchanzEmail author
Article

Abstract

In this paper, a symmetric Galerkin boundary element formulation for 3D linear poroelasticity is presented. By means of the convolution quadrature method, the time domain problem is decoupled into a set of Laplace domain problems. Regularizing their kernel functions via integration by parts, it is possible to compute all operators for rather general discretizations, only requiring the evaluation of weakly singular integrals. At the end, some numerical results are presented and compared with a collocation BEM. Throughout these studies, the symmetric Galerkin BEM performs better than the collocation method, especially for not optimal discretizations parameters, i.e. a bad relation of mesh to time-step size. The most obvious advantages can be observed in the fluid flux results. However, these advantages are obtained at a higher numerical cost.

Keywords

Boundary Integral Equation Laplace Domain Berea Sandstone Poroelastic Material Solid Displacement 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Banjai, L., Sauter, S.: Rapid solution of the wave equation in unbounded domains. J. Numer. Anal. 47, 227–249109–128–249109 (2008)Google Scholar
  2. 2.
    Banjai L., Schanz M.: Wave propagation problems treated with convolution quadrature and BEM. In: Langer, U., Schanz, M., Steinbach, O., Wendland, W.L. (eds.) Fast Boundary Element Methods in Engineering and Industrial Applications, chapter 5, pp. 147–187. Springer, Berlin (2012)Google Scholar
  3. 3.
    Biot M.A.: Theory of propagation of elastic waves in a fluid-saturated porous solid I. J. Acoust. Soc. Am. 28, 168–178 (1956)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Biot M.A.: Theory of propagation of elastic waves in a fluid-saturated porous solid II. J. Acoust. Soc. Am. 28, 179–191 (1956)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bonnet G.: Basic singular solutions for a poroelastic medium in the dynamic range. J. Acoust. Soc. Am. 82, 1758–1763 (1987)CrossRefGoogle Scholar
  6. 6.
    Bonnet M., Maier G., Polizzotto C.: Symmetric Galerkin boundary element method. Appl. Mech. Rev. 51, 669–704 (1998)CrossRefGoogle Scholar
  7. 7.
    Cheng, A.H.-D.: On free space Green’s function for higher order Helmholtz equations. In: Boundary Element Methods: Fundamentals and Applications, Proceedings of the IABEM Symposium, Kyoto, (1991)Google Scholar
  8. 8.
    Costabel M., Stephan E.P.: Coupling of finite and boundary element methods for an elastoplastic interface problem. SIAM J. Numer. Anal. 27, 1212–1226 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Guiggiani M., Krishnasamy G., Rudolphi T.J., Rizzo F.J.: A general algorithm for the numerical solution of hypersingular boundary integral equations. J. Appl. Mech. 59, 604–614 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Han H.: The boundary integro-differential equations of three-dimensional Neumann problem in linear elasticity. Num. Math. 68, 268–281 (1994)CrossRefGoogle Scholar
  11. 11.
    Kielhorn L., Schanz M.: Convolution quadrature method-based symmetric Galerkin boundary element method for 3-d elastodynamics. Int. J. Numer. Methods Eng. 76, 1724–1746 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Kim Y.K., Kingsbury H.B.: Dynamic characterization of poroelastic materials. Exp. Mech. 19, 252–258 (1979)CrossRefGoogle Scholar
  13. 13.
    Kupradze V.D.: Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity. North-Holland, Amsterdam (1979)zbMATHGoogle Scholar
  14. 14.
    Lubich C.: Convolution quadrature and discretized operational calculus I. Numer. Math. 52, 129–145 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Lubich C.: Convolution quadrature and discretized operational calculus II. Numer. Math. 52, 413–425 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Manolis G.D., Beskos D.E.: Integral formulation and fundamental solutions of dynamic poroelasticity and thermoelasticity. Acta Mech. 76, 89–104 (1989)zbMATHCrossRefGoogle Scholar
  17. 17.
    Maue A.W.: Zur Formulierung eines allgemeinen Beugungsproblems durch eine Integralgleichung. Zeitschrift für Physik 126, 609–618 (1949)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Messner M., Schanz M.: A regularized collocation boundary element method for linear poroelasticity. Comput. Mech. 47, 669–680 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Messner, M., Messner, M., Rammerstorfer, F., Urthaler, P.: Hyperbolic and Elliptic Numerical Analysis BEM library. http://www.mech.tugraz.at/HyENA (2010)
  20. 20.
    Sauter S., Schwab C.: Boundary Element Methods. Springer, Berlin (2010)Google Scholar
  21. 21.
    Schanz M.: Wave Propagation in Viscoelastic and Poroelastic Continua, Lecture Notes in Applied Mechanics, vol. 2. Springer, Berlin (2001)Google Scholar
  22. 22.
    Schanz M.: Poroelastodynamics: linear models, analytical solutions, and numerical methods. Appl. Mech. Rev. 62, 030803-1–030803-15 (2009)CrossRefGoogle Scholar
  23. 23.
    Schanz M., Cheng A.H.D.: Transient wave propagation in a one-dimensional poroelastic column. Acta Mech. 145, 1–18 (2000)zbMATHCrossRefGoogle Scholar
  24. 24.
    Schanz M., Steinbach O., Urthaler P.: A boundary integral formulation for poroelastic materials. PAMM 9, 595–596 (2009)CrossRefGoogle Scholar
  25. 25.
    Sirtori S., Maier G., Novati G., Miccoli S.: A Galerkin symmetric boundary-element method in elasticity: formulation and implementation. Int. J. Numer. Methods Eng. 35, 255–282 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Steinbach O.: Numerical Approximation Methods for Elliptic Boundary Value Problems. Springer, Berlin (2008)zbMATHCrossRefGoogle Scholar
  27. 27.
    Wiebe T., Antes H.: A time domain integral formulation fo dynamic poroelasticity. Acta Mech. 90, 125–137 (1991)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Institute of Applied MechanicsGraz University of TechnologyGrazAustria

Personalised recommendations