A mixed least‐squares finite element formulation within the framework of the theory of porous media

Abstract

In the present contribution a mixed least-squares finite element method (LSFEM) based on the Theory of Porous Media (TPM) is presented. In detail, we investigate an incompressible binary model consisting of the phases solid and liquid. The main idea is based on the modeling of saturated porous structures. The resulting finite element is a four-field formulation in terms of solid displacements, liquid pressure, mixture stresses and a new variable related to the pressure gradient. The conforming discretization of the unknowns in the spaces H(div) and H 1 is realized by vector-valued Raviart-Thomas and standard Lagrange functions. Finally, a numerical example for liquid saturated porous structures considering an incompressible, linear elastic material behavior at small deformations, demonstrates the applicability of the LSFEM approach to the TPM.

A mixed least‐squares finite element method for a geometrically nonlinear TPM formulation

“…The advantages of the geometrically nonlinear formulation, that is large strains and large displacements, are the goal of upcoming research in ionic electroactive polymers. This would not be possible with the linar theory of Schwarz et al [35]. This expansion would provide a deeper understanding of the behavior of porous media under more extreme loading conditions.…”

“…The basic equations can be taken from the basic literature [1–3]. A similar approach was given in [35] for linear elasticity and is now extended to geometrically nonlinear elasticity. In the following, a minimization problem is to be solved based on a least‐squares functional $${\cal F}({\cal I})$$ with respect to the unknowns $${\cal I} = \lbrace \bm{v}, \bm{\sigma }, \ldots \rbrace$$.…”