UC Davis

The Virtual Element Method (VEM) is a recently introduced extension of the Finite ElementMethod (FEM) to general polygonal and polyhedral (polytopal) meshes. By using a set of virtual canonical basis functions, the method provides flexibility for meshing complex geometries with convex and nonconvex polygonal elements, as well as providing a simple approach to handling non-matching meshes and fracturing. Polynomial projection operators are introduced to provide approximation accuracy and to preserve polynomial consistency. However, one limitation of the VEM, is the need to devise a problem dependent stabilization term to retain coercivity. The choice of stabilization adds complexity in formulating new problems and an incorrect choice can adversely affect the solution accuracy. In this dissertation, we develop virtual element methods that do not rely on a stabilization term for problems in planar linear elasticity. We first present strain-based approaches, which use higher order polynomials to enhance the strain polynomial approximation. In these methods, the polynomials are only chosen to preserve the stability of the system. We give theoretical arguments for the stability, well-posedness and prove convergence estimates for the first-order case. These approaches are numerically tested on benchmark elasticity problems, and the results show that the methods attain optimal convergence rates and provide a viable alternative to the standard VEM for compressible materials. However, for thin structures or nearly-incompressible materials, we find that the strain-based approaches and the standard VEM are overly stiff and suffer from locking phenomena. To alleviate locking, we appeal to stress-based approaches that rely on the Hellinger–Reissner variational formulation. These methods use selectively chosen higher order divergence-free polynomials that preserve the stability as well as alleviate locking. Starting with quadrilateral elements, we use a five-parameter expansion of the stress field to construct a method that is free of volumetric and shear locking. For six-noded triangular elements, we find that a fifteen-term divergence-free stress expansion resulted in a method that does not require stabilization and shows immunity to locking. An alternative to using divergence-free polynomials is also explored. This approach uses a penalty term to weakly enforce the equilibrium equations. Numerical results reveal that the stress-based approaches provide optimal convergence, and robustness for compressible and nearly-incompressible problems.