The Boundary-Element Approach to Modeling the Dynamics of Poroelastic Bodies

  • Leonid IgumnovEmail author
  • Svetlana Litvinchuk
  • Aleksandr Ipatov
  • Tatiana Iuzhina
Conference paper
Part of the Structural Integrity book series (STIN, volume 8)


The present paper is dedicated to dynamic behavior of poroelastic solids. Biot’s model of poroelastic media with four base functions is employed in order to describe wave propagation process, base functions are skeleton displacements and pore pressure of the fluid filler. In order to study the boundary-value problem boundary integral equations (BIE) method is applied, and to find their solutions boundary element method (BEM) for obtaining numerical solutions. The solution of the original problem is constructed in Laplace transforms, with the subsequent application of the algorithm for numerical inversion. The numerical scheme is based on the Green-Betty-Somigliana formula. To introduce BE-discretization, we consider the regularized boundary-integral equation. The collocation method is applied. As a result, systems of linear algebraic equations will be formed and can be solved with the parallel calculations usage. Modified Durbin’s algorithm of numerical inversion of Laplace transform is applied to perform solution in time domain. A problem of the three-dimensional poroelastic prismatic solid clamped at one end, and subjected to uniaxial and uniform impact loading and a problem of poroelastic cube with cavity subjected to a normal internal pressure are considered.


Poroelasticity Boundary element method (BEM) Boundary integral equation (BIE) Laplace transform inversion Durbin’s algorithm 



This work was supported by a grant from the Government of the Russian Federation (contract No. 14.Y26.31.0031).


  1. 1.
    Frenkel, J.: On the theory of seismic and seismoelectric phenomena in a moist soil. J. Phys. 8, 230–241 (1944)MathSciNetGoogle Scholar
  2. 2.
    Biot, M.A.: Theory of propagation of elastic waves in a fluid-suturated porous solid. J. Acoust. Soc. Am. 28(2), 168–191 (1956)CrossRefGoogle Scholar
  3. 3.
    Schanz, M.: Poroelastodynamics: linear models, analytical solutions, and numerical methods. Appl. Mech. Rev. 62(3), 030803-1–030803-15 (2009)CrossRefGoogle Scholar
  4. 4.
    Schanz, M.: Wave Propagation in elastic and Poroelastic Continua. Springer, Berlin (2001)CrossRefGoogle Scholar
  5. 5.
    Cruze, T.A., Rizzo, F.J.: A direct formulation and numerical solution of the general transient elastodynamic problem I. J. Math. Anal. Appl. 22(1), 244–259 (1968)CrossRefGoogle Scholar
  6. 6.
    Ugodchikov, A.G., Hutoryanskii, N.M.: Boundary element method in deformable solid mechanics. Kazan State University, Kazan (1986)Google Scholar
  7. 7.
    Durbin, F.: Numerical inversion of Laplace transforms: an efficient improvement to Dubner and Abate’s method. Comput. J. 17(4), 371–376 (1974)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Zhao, X.: An efficient approach for the numerical inversion of Laplace transform and its application in dynamic fracture analysis of a piezoelectric laminate. Int. J. Solids Struct. 41, 3653–3674 (2004)CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Leonid Igumnov
    • 1
    Email author
  • Svetlana Litvinchuk
    • 1
  • Aleksandr Ipatov
    • 1
  • Tatiana Iuzhina
    • 1
  1. 1.Research Institute for MechanicsLobachevsky State University of Nizhni NovgorodNizhny NovgorodRussia

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