Stabilized Unfitted Finite Element Method for Poroelasticity With Weak Discontinuity

Abstract

Poromechanics problems in geotechnical and geological contexts often involve complex formations with numerous boundaries and material interfaces, which significantly complicate numerical analysis and simulation. The traditional finite element method (FEM) encounters substantial challenges in these scenarios because it requires the mesh to conform precisely to each boundary and interface. This requirement complicates preprocessing and necessitates meticulous manual control to achieve a high-quality mesh. In contrast, unfitted FEMs are well-suited for these problems as they do not require the mesh to align with the model geometry. We propose a stabilized unfitted FEM that incorporates Nitsche's method and ghost penalty stabilization techniques to address complex poroelasticity problems. This approach treats material interfaces as weak discontinuities and ensures that compatibility conditions are satisfied. The proposed method allows the mesh to be independent of both boundaries and material interfaces. Nitsche's method is used to weakly enforce both Dirichlet boundary conditions and interface compatibility conditions, resulting in a symmetric weak form. Additionally, three types of ghost penalty terms are introduced for elements intersected by boundaries or interfaces, effectively eliminating cut-induced ill-conditioning. The proposed methodology has been validated through benchmark and practical problems, demonstrating optimal convergence and exceptional stability. This approach significantly enhances the stability and efficiency of hydro-mechanical analyses for complex geotechnical and geological problems.