Compatible-Strain Mixed Finite Element Methods for 2D Compressible Nonlinear Elasticity

In this paper, using the Hilbert complexes of nonlinear elasticity, the approximation theory for Hilbert complexes, and the finite element exterior calculus, we introduce a new class of mixed finite element methods for 2D nonlinear elasticity -- compatible-strain mixed finite element methods (CSFEM). We consider a Hu-Washizu-type mixed formulation and choose the displacement, the displacement gradient, and the first Piola-Kirchhoff stress tensor as independent unknowns. We use the underlying spaces of the Hilbert complexes as the solution and test spaces. We discretize the Hilbert complexes and introduce a new class of mixed finite element methods for nonlinear elasticity by using the underlying finite element spaces of the discrete Hilbert complexes. This automatically enforces the trial spaces of the displacement gradient to satisfy the classical Hadamard jump condition, which is a necessary condition for the compatibility of non-smooth displacement gradients. The underlying finite element spaces of CSFEMs are the tensorial analogues of the standard Nedelec and Raviart-Thomas elements of vector fields. These spaces respect the global topologies of the domains in the sense that they can reproduce certain topological properties of the bodies regardless of the refinement level of meshes. By solving several numerical examples, we demostrate that CSFEMs have a good performance for bending problems, in the near incompressible regime, and for bodies with complex geometries. CSFEMs are capable of accurately approximating stresses and perform well in problems that standard enhanced strain methods suffer from the hourglass instability. Moreover, CSFEMs provide a convenient framework for modeling inhomogeneities.