Topology optimization using polytopes

Meshing complex engineering domains is a challenging task. Arbitrary polyhedral meshes can provide the much needed flexibility in automated discretization of such domains. The geometric property of polyhedral meshes such as its unstructured nature and the connectivity of faces between elements makes them specially attractive for topology optimization applications. Numerical anomalies in designs such as the single node connections and checkerboard pattern can be naturally circumvented with polyhedrons. In the current work, we solve the governing three-dimensional elasticity state equation using the Virtual Element Method (VEM) approach. The main characteristic difference between VEM and standard finite element methods (FEM) is that in VEM the canonical basis functions are not constructed explicitly. Rather the stiffness matrix is computed directly utilizing a projection map which extracts the linear component of the deformation. Such a construction guarantees the satisfaction of the patch test (used by engineers as an indicator of optimal convergence of numerical solutions under mesh refinement). Finally, the computations reduce to the evaluation of matrices which contain purely geometric surface facet quantities. The present work focuses on the first-order VEM in which the degrees of freedom are associated with the vertices. The features of the current optimization approach are demonstrated using numerical examples for compliance minimization and compliant mechanism problems.