Due to its unique and intriguing properties, polygonal and polyhedral discretization is an emerging field in computational mechanics. This thesis deals with developing finite element and virtual element formulations for computational mechanics problems on general polygonal and polyhedral discretizations.
The construction of finite element approximation on polygonal/polyhedral meshes relies on generalized barycentric coordinates, which are non-polynomial functions. Thus, the existing numerical integration schemes, typically designed to integrate polynomial functions, will lead to persistent consistency errors that do not vanish with mesh refinement. To overcome the limitation, this thesis presents a general gradient correction scheme, which restores the polynomial consistency by adding a minimal perturbation to the gradient of the displacement field, and applies it to formulate both lower- and higher-order polygonal finite elements for finite elasticity problems. With the gradient correction scheme, the optimal convergence is recovered in finite elasticity problems.
The Virtual Element Method (VEM) was recently proposed as an attractive framework to handle unstructured polygonal/polyhedral discretizations and beyond. The VEM is inspired by the mimetic methods, which mimics fundamental properties of mathematical and physical systems. Unlike the Finite Element Method (FEM), there are no explicit shape functions in the VEM, which is a unique feature that leads to flexible definitions of the local VEM spaces. This thesis also develops novel VEM formulations for several classes of computational mechanics problems. First, to study soft materials, we present a general VEM framework for finite elasticity. The framework features a nonlinear stabilization scheme, which evolves with deformation; and a local mathematical displacement space, which can effectively handle any element shape, including concave elements or ones with non-planar faces. We verify convergence and accuracy of the proposed virtual elements by means of examples using unique element shapes inspired by Escher (the Dutch artist famous for his impossible constructions). Second, to fully realize the potential of VEM in mesh adaptation, we develop a gradient recovery scheme and a posteriori error estimator for VEM of arbitrary order for linear elasticity problems. The a posteriori error estimator is simple to implement yet has been shown to be effective through theoretical and numerical analyses. Finally, from design viewpoint, we present an efficient topology optimization framework on general polyhedral discretizations by synergistically incorporating the VEM and its mathematical/numerical features in the underlining formulation. As a result, the tailored VEM space naturally leads to a continuous material density field interpolated from nodal design variables. This approach yields a mixed virtual element with an enhanced density field. We present examples that explore the aforementioned features of our VEM-based topology optimization framework and contrast our results with the traditional FEM-based approaches that dominate the technical literature.
Dr. Glaucio H. Paulino
Dr. Arash Yavari, Dr. Phanish Suryanarayana, Dr. David L. McDowell (ME), Dr. Oscar Lopez-Pamies (UIUC), Dr. Lourenco Beirao da Vega (U. Milan), Dr. Cameron Talischi (Industry), and Dr. Kyoungsoo Park (Yonsei U.)