A New Multiphysics Finite Element Method for a Biot Model with Secondary Consolidation

Abstract

In this paper, we propose a new multiphysics finite element method for a Biot model with secondary consolidation in soil dynamics. To better describe the multiphysical processes underlying in the original model and propose stable numerical methods to overcome "locking phenomenon" of pressure and displacement, we reformulate the swelling clay model with secondary consolidation by a new multiphysics approach, which transforms the fluid-solid coupling problem to a fluid coupled problem. Then, we give the energy law and prior error estimate of the weak solution. Also, we design a fully discrete time-stepping scheme to use multiphysics finite element method with P₂-P₁-P₁ element pairs for the space variables and backward Euler method for the time variable, and we derive the discrete energy laws and the optimal convergence order error estimates. Also, we show some numerical examples to verify the theoretical results and there is no "locking phenomenon". Finally, we draw conclusions to summarize the main results of this paper.

Keywords

Biot model / multiphysics finite element method / optimal convergence order / secondary consolidation