A topology-motivated mixed finite element method for dynamic response of porous media (1506.06785v1)

Abstract: In this paper, we propose a numerical method for computing solutions to Biot's fully dynamic model of incompressible saturated porous media [Biot;1956]. Our spatial discretization scheme is based on the three-field formulation (u-w-p) and the coupling of a lowest order Raviart-Thomas mixed element [Raviart,Thomas;1977] for fluid variable fields (w, p ) and a nodal Galerkin finite element for skeleton variable field (u). These mixed spaces are constructed based on the natural topology of the variables; hence, are physically compatible and able to exactly model the kind of continuity which is expected. The method automatically satisfies the well known LBB (inf-sup) stability condition and avoids locking that usually occurs in the numerical computations in the incompressible limit and very low hydraulic conductivity. In contrast to the majority of approaches, our three-field formulation can fully capture dynamic behavior of porous media even in high frequency loading phenomena with considerable fluid acceleration such as liquefaction and biomechanics of porous tissues under rapid external loading. Moreover, we address the importance of consistent initial conditions for poroelasticity equations with the incompressibility constraint, which represent a system of differential algebraic equations. The energy balance equation is derived for the full porous medium and used to assess the stability and accuracy of our time integration. To highlight the capabilities of our method, a variety of numerical studies are provided including verification with analytical and boundary element solutions, wave propagation analyses, hydraulic conductivity effects on damping and frequency content, energy balance analyses, mass lumping considerations, effects of mesh pattern and size, and stability analyses. We also explain some discrepancies commonly found in dynamic poroelasticity results in the literature.