Compatible-strain mixed finite element methods for nonlinear elasticity

Abstract

A new family of mixed finite element methods--compatible-strain mixed finite element methods (CSFEMs)--are introduced for compressible and incompressible nonlinear elasticity problems in dimensions two and three. A Hu-Washizu-type mixed formulation is considered and the displacement, the displacement gradient, and the first Piola-Kirchhoff stress are chosen as the independent unknowns. To impose incompressibility, a pressure-like field is introduced as the fourth independent unknown. Using the Hilbert complexes of nonlinear elasticity that describe the kinematics and the kinetics of motion, we identify the solution spaces that the independent unknown fields belong to. In particular, we define the displacement in H1, the displacement gradient in H(curl), the stress in H(div), and the pressure field in L2. The test spaces of the mixed formulations are chosen to be the same as their corresponding solution spaces. In a conforming setting, we approximate the solution and the test spaces with some piecewise polynomial subspaces of them. Among these approximation spaces are the tensorial analogues of the standard Nédélec and Raviart-Thomas finite element spaces of vector fields. This approach results in mixed finite element methods that, by construction, satisfy both the Hadamard jump conditions and the continuity of traction at the discrete level regardless of the refinement level of the mesh. This, in particular, makes CSFEMs quite efficient for modeling heterogeneous solids. We assess the performance of CSFEMs by solving several numerical examples in dimensions two and three and demonstrate their good performance for bending problems, for bodies with complex geometries, for different material models, and in the nearly incompressible regime. Using CSFEMs, one can model deformations with very large strains and accurately approximate stresses and the pressure field. Moreover, in our numerical examples, we do not observe any numerical artifacts such as checkerboarding of pressure, hourglass instability, or locking.