On the stable finite element procedures for dynamic problems of saturated porous media

The solution of problems in which coupling occurs between the displacement of the soil skeleton and the pore fluid pressure is fundamental in soil dynamics. The formulation requires that the interpolation functions for the displacement and pressure in the finite element discretization must satisfy the so-called Babuska-Brezzi stability criteria or the patch test in the limit of nearly incompressible pore fluid and small permeability. The criteria are not fulfiled by elements with the same order of interpolation for both variables unless stabilization techniques are introduced. This paper summarizes the stabilization techniques that have been proposed in the literature to overcome volumetric locking for the incompressible or nearly incompressible soil dynamic behaviours. In particular, the staggered implicit-implicit algorithm (i.e. the fractional step method in an implicit form) and the direct α-method proposed by the first author and Zienkiewicz et al. are briefly reviewed. Attentions will be paid to the steady-state formulations resulted from both approaches. Based on the steady state formulations, the paper will then discuss the determination of the local stabilization parameters, with which a significant improvement for the obtained solutions of pore-pressure can be achieved. Further discussion on the limitations of the methods is also given. Finally, several numerical examples are presented to illustrate the effectiveness of the proposed techniques.