ORIGINAL PAPER M. Schanz Æ T. Ru¨berg Æ V. Struckmeier Quasi-static poroelastic boundary element formulation based on the convolution quadrature method Received: 25 May 2004 / Revised: 15 February 2005 / Published online: 4 June 2005 Ó Springer-Verlag 2005 Abstract Convolution Quadrature Method (CQM)- based Boundary Element formulations are up to now used only in dynamic formulations. The main difference to usual time-stepping BE formulations is the way to solve the convolution integral appearing in most time- dependent integral equations. In the CQM formulation, this convolution integral is approximated by a quadra- ture rule whose weights are determined by the Laplace transformed fundamental solutions and a linear multi- step method. In principle, for quasi-static poroelasticity there is no need to apply the CQM because time-dependent fun- damental solutions are available. However, these fun- damental solutions are highly complicated yielding very sensitive algorithms. On the contrary, the CQM based BE formulation proposed here is very robust and yields comparable results to other methodologies. This for- mulation is tested in 2-d in comparison with a Finite Element Method and analytical results. 1 Introduction Convolution Quadrature Method (CQM)-based Boundary Element (BE) formulations are first published in 1997 [19, 20] with applications in elasto- or viscoel- astodynamics. The main difference to usual time-step- ping BE formulations is the way to solve the convolution integral appearing in most time-dependent integral equations. In the CQM formulation, this convolution integral is approximated by a quadrature rule whose weights are determined by the Laplace transformed fundamental solutions and a multi-step method [13, 14]. An overview of this BE formulation is given in [18]. There are mainly two reasons to use a CQM-based BEM instead of usual time-stepping procedures. One reason is to improve the stability of the time-stepping procedure [19, 1]. The other reason is to tackle problems where no time-dependent fundamental solutions are available, e.g., for inelastic material behavior in vis- coelastodynamics [16], in poroelastodynamics [17], or for functional graded materials [25]. Also, this method is used to avoid highly complicated fundamental solutions in time domain [2, 23, 24]. However, up to now, the CQM-based BEM is used only in dynamic formulations. Clearly, for quasi-static problems in poroelasticity there is no need to apply the CQM because time-dependent fundamental solutions are available [7]. However, these fundamental solutions are highly complicated yielding very sensitive algo- rithms. Therefore, it is promising to apply the CQM also to the quasi-static integral equations in poroelasticity. This approach is presented for quasi-static viscoelastic- ity and poroelasticity for 3-dimensional continua in [21]. Here, the 2-d case for poroelasticity is discussed. Here, at first, poroelastic constitutive equations are recalled based on Biot’s theory [3]. It should be mentioned that the proposed method can also be applied to mixture theory based theories as the Theory of Porous Media [8] because the mathematical operator of the governing equations is equal to that of Biot’s theory, as shown for the dynamic case by Schanz and Diebels [22]. The singular behavior of the 2-d fundamental solutions in Laplace domain is discussed. The explicit expressions of these quasi-static fundamental solutions in Laplace domain may be found in [5] or the respective time domain solu- tions in the survey article by Cheng and Detournay [6]. Subsequent to the formulation of the constitutive and governing equations, the respective integral equations Comput Mech (2005) 37: 70–77 DOI 10.1007/s00466-005-0699-9 M. Schanz (&) Institute of Applied Mechanics, Graz University of Technology, Technikerstr. 4, 8010 Graz, Austria E-mail: m.schanz@tugraz.at Tel.: +43-316-8737600 Fax: +43-316-8737641 T. Ru¨berg Æ V. Struckmeier Institute of Applied Mechanics, Technical University Braunschweig, P.O. Box 3329, D-38023 Braunschweig, Germany