Hybrid-Trefftz finite elements for elastostatic and elastodynamic problems in porous media

Abstract

The displacement and stress models of the hybrid-Trefftz finite element formulation are applied to the elastostatic and elastodynamic analysis of two-dimensional saturated and unsaturated porous media problems. The formulation develops from the classical separation of variables in time and space, but it leads to two time integration strategies. The first is applied to periodic problems, which are discretized in time using Fourier analysis. A mixed finite element approach is used in the second strategy for discretization in time of non-periodic/transient problems. These strategies lead to a series of uncoupled problems in the space dimension, which is subsequently discretized using either the displacement or the stress model of the hybrid-Trefftz finite element formulation. The main distinction between the two models is in the way that the interelement continuity is enforced. The displacement model enforces the interelement compatibility, while the stress model enforces the interelement equilibrium. As is typical of Trefftz methods, for both models, the approximation bases are constrained to satisfy locally the homogeneous form of the domain (Navier) equations. The free-field solutions of these equations are derived in cylindrical coordinates and used to construct the domain approximations of the hybrid-Trefftz displacement and stress elements. If the original equations are non-homogeneous, the influence of the source terms is modelled using Trefftz-compliant solutions of the corresponding static problem. For saturated porous media, the finite element models are based on the Biot's theory. It assumes an elastic solid phase fully permeated by a compressible liquid phase obeying the Darcy's law. For the modelling of unsaturated porous media, the finite elements are formulated using the theory of mixtures with interfaces. The model is thermodynamically consistent and considers the full coupling between the solid, fluid and gas phases, including the effects of relative (seepage) accelerations. Small displacements and linear-elastic material behaviour are assumed for all models.