Aspects of finite element formulations for the coupled problem of poroelasticity based on a canonical minimization principle

This work presents a new finite element treatment of the coupled problem of Darcy–Biot-type fluid transport in porous media undergoing large deformations, that is free from any stabilization techniques. The formulation bases on an incremental two-field minimization principle that is constrained by the equation of continuity for the fluid mass content and determines at a given state the deformation and the fluid mass flux vector. The big advantage of the minimization formulation over classical saddle point principles of poroelasticity is the omission of the inf-sup condition—a condition that makes the construction of stable and computationally efficient finite element formulations difficult. Due to the \(H(\hbox {Div}, {\mathcal B}_0)\) variational structure of the minimization principle on the fluid side, lowest order Raviart–Thomas elements are used for the conforming approximation of the fluid mass flux. Furthermore, a standard nodal-based element using bilinear interpolation for both fields combined with a reduced numerical integration of the (volumetric) coupling term is analyzed and used for the solution of the minimization principle. Representative numerical examples demonstrate the performance of the proposed finite element designs of the minimization principle and clearly underline advantages over finite element formulations of the classical two-field saddle point principle formulated in deformation and fluid potential.