Virtual element method (VEM)-based topology optimization: an integrated framework

We present a virtual element method (VEM)-based topology optimization framework using polyhedral elements, which allows for easy handling of non-Cartesian design domains in three dimensions. We take full advantage of the VEM properties by creating a unified approach in which the VEM is employed in both the structural and the optimization phases of the framework. In the structural problem, the VEM is adopted to solve the three-dimensional elasticity equation. Compared to the finite element method (FEM), the VEM does not require numerical integration and is less sensitive to degenerated elements (e.g., ones with skinny faces or small edges). In the optimization problem, we introduce a continuous approximation of material densities using VEM basis functions. As compared to the standard element-wise constant one, the continuous approximation enriches geometrical representations of structural topologies. Through two numerical examples with exact solutions, we verify the convergence and accuracy of both the VEM approximations of the displacement and material density fields. We also present several design examples involving non-Cartesian domains, demonstrating the main features of the proposed VEM-based topology optimization framework. The source code for a MATLAB implementation of the proposed work, named PolyTop3D, will be made available in the (electronic) Supplementary Material accompanying this publication.