Polyhedral finite elements for nonlinear solid mechanics using tetrahedral subdivisions and dual-cell aggregation

General polyhedral discretizations offer several advantages over classical approaches consisting of standard tetrahedra and hexahedra. These include increased flexibility and robustness in the meshing of geometrically complex domains and higher-quality solutions for both finite element and finite volume schemes. Currently, the use of general polyhedra is hampered by the lack of general-purpose polyhedral meshing algorithms and software. One approach for generating polyhedral meshes is the use of tetrahedral subdivisions and dual-cell aggregation. In this approach, each tetrahedron of an existing tetrahedral mesh is subdivided using one of several subdivision schemes. Polyhedral-dual cells may then be formed and formulated as finite elements with shape functions obtained through the use of generalized barycentric coordinates. We explore the use of dual-cell discretizations for applications in nonlinear solid mechanics using a displacement-based finite element formulation. Verification examples are presented that yield optimal rates of convergence. Accuracy of the methodology is demonstrated via several nonlinear examples that include large deformation and plasticity.