Polyhedral finite-element approximants in three-dimensional solid mechanics

Two novel finite-element-type based approximation schemes for solid-mechanics applications are presented in a three dimensional setting. The first, the Variable Element Topology Finite Element Method (VETFEM) retains all the powerful characteristics of the conventional Finite Element Method (FEM), while providing a level of simplicity and flexibility in the spatial discretization of the domain that compares to that of particle methods. The VETFEM employs elements that can take the form of general polyhedra, which are subject only to mild geometric restrictions in contrast to the conventional FEM. As such, the VETFEM can be especially useful for problems that involve complex geometry, adaptive remeshing or crack propagation.;The second novel method presented here, the Discrete Data Polyhedral Finite Element Method (DDPFEM), unlike other finite-element-based methods, does not define the shape functions pointwise on the element domain. Instead, the DDPFEM formulation involves values of the shape functions at the element nodes, and values of their gradients at the element integration points only. The method extends the applicability of arbitrary polyhedral elements to spatial discretizations that admit a greater degree of concavity in the elements, compared to VETFEM elements.;Convergence of the VETFEM is discussed and analyzed through an experimental analysis, and overall behavior observed through computational examples, including patch tests.