Three‐dimensional hybrid‐Trefftz displacement elements for poroelastodynamic problems in saturated media

Abstract

The hybrid‐Trefftz displacement element formulation for poroelastodyamics in three‐dimensional saturated media is presented in this article. The governing equations are integrated in time using a wavelet decomposition which expands the time‐dependent problem into several spectral problems. Displacements are discretized in each finite element and tractions are approximated on each essential element boundary (displacement model). The displacement approximation functions satisfy the governing equations of the media (Trefftz constraint) and are not linked to the nodes. Trefftz bases are derived solving the governing equations and are composed of four types of modes: two compression waves and two shear waves. The weak enforcement of the Navier equation and boundary conditions yields a solving system where the unknown are the weights of the approximation bases. Numerical tests have been carried out to assess the convergence of the models. The results of the simulations are satisfactory: the numerical tests converge to known analytical solutions and the simulation of a propagation of a vertical shock wave is consistent to the one obtained with a 2D plane simulation under the same conditions. The results of a bender element test simulation in three dimensions are also presented.