  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 the polyhedral meshes
such as the unstructured nature and the facial connectivity between elements
makes them specially attractive for topology optimization applications.
Numerical anomalies in designs such as the single node connections and
checkerboard pattern, which are difficult to manufacture physically, are
naturally alleviated with polyhedrons. Special interpolants such as Wachspress,
mean value coordinates, maximum entropy shape functions are available to handle
arbitrary shaped elements. But the finite elements approaches based on these
shape functions face some challenges such as accurate and efficient computation
of the shape functions and their derivatives for the numerical evaluation of
the weak form integrals. In the current work, we solve the governing
three-dimensional elasticity state equation using a 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
associated with the vertices. Utilizing polyhedral elements for topology
optimization, we show that the mesh bias in the member orientation is
alleviated.

Optimization and Control;Numerical Analysis