Due to its unique and intriguing properties, polygonal and polyhedral discretization is an emerging field in computational mechanics. This project advances finite element and virtual element formulations for computational mechanics problems on general polygonal and polyhedral discretizations.

The construction of finite element approximations on polygonal and polyhedral meshes relies on generalized barycentric coordinates, which are non-polynomial (e.g. rational) 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, we contributed a general gradient correction scheme, which restores the polynomial consistency by adding a minimal perturbation to the gradient of the displacement field, and applied 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 (e.g., arbitrary non-convex shapes). The VEM is inspired by the mimetic methods, which mimics fundamental properties of mathematical and physical systems (e.g., exact mathematical identities of tensor calculus). 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. We have contributed novel VEM formulations for several classes of computational mechanics problems. First, to study soft materials, we presented 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 the 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 so-called impossible constructions). Second, to fully realize the potential of VEM in mesh adaptation (i.e., refinement, coarsening and local re-meshing), we created 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 a design viewpoint, we devised 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.

D. Heng Chi, who was supported by this project during his PhD, won the 28^th Robert J. MELOSH Medal in computational Mechanics in 2017. Moreover, he was part of the team that won the SIEMENS hackathon at Georgia Tech. The team combined machine learning and topology optimization to make computational design and digital manufacturing more efficient and effective.

Last Modified: 07/20/2021
Modified by: Glaucio Paulino