A stabilized assumed deformation gradient finite element formulation for strongly coupled poromechanical simulations at finite strain

Abstract

An adaptively stabilized finite element scheme is proposed for a strongly coupled hydro-mechanical problem in fluid-infiltrating porous solids at finite strain. We first present the derivation of the poromechanics model via mixture theory in large deformation. By exploiting assumed deformation gradient techniques, we develop a numerical procedure capable of simultaneously curing the multiple-locking phenomena related to shear failure, incompressibility imposed by pore fluid, and/or incompressible solid skeleton and produce solutions that satisfy the inf-sup condition. The template-based generic programming and automatic differentiation (AD) techniques used to implement the stabilized model are also highlighted. Finally, numerical examples are given to show the versatility and efficiency of this model. Darcy's velocity [7], by using inf-sup stable finite elements (e.g., Talyor-Hood, Raviart-Thomas finite elements) [9,[11][12][13][14], or by applying stabilization procedures to cure the otherwise unstable finite elements [7,[15][16][17][18]. The displacement-Darcy-velocity coupling scheme is relatively easy to implement. However, computation time will be significantly increased because of the extra degrees of freedom added for the pore fluid velocity. Implementing inf-sup stable displacement-pressure finite element requires only one extra degree of freedom for the pore pressure. However, inf-sup stable finite elements require special meshing, data structure, and preprocessing and postprocessing tools to accommodate the need to have different basis functions for the displacement and pore pressure solutions. This difficulty is accompanied with a significant increase in computation cost, as higher-order mixed finite elements lead to larger systems of equations and require more integration points to perform numerical integration, as pointed out in [14].An equal-order displacement-pore pressure mixed finite element formulation is computationally more efficient and does not require significant modification to data structures (except adding an additional degree of freedom to all nodes), and it is therefore more feasible for both code development and maintenance. The price for using the equal-order mixed finite element method, however, is that the mixed finite element formulation must be stabilized by either adding additional terms or introducing enhanced shape functions to eliminate the spurious mode encountered as a result of a failure to satisfy the inf-sup condition. In recent years, effort has been invested to develop stabilization procedures for poromechanics problem under the small strain assumption, for example, [7,[16][17][18]. However, there are numerous occasions in which porous solids experience significant deformation such that a finite strain formulation becomes essential. Examples include water-saturated soil near critical state [7] and hydrated biological tissue during normal physiological activities [19]. In those situations, a stabilized formulation for large deformation poromechanics inheriting the same ...

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