Topology Optimization for Fluid Flow Problems using the Virtual Element Method

Abstract

This work presents some applications of topology optimization for fluid flow problems using the Virtual Element Method (VEM) (Veiga et al. 2013) in arbitrary two-dimensional domains. The idea is to design an optimal layout for the incompressible Newtonian fluid flow, governed by the Stokes equations, to minimize the viscous drag. The porosity approach proposed by (Borrvall and Petersson, 2003) is used in the topology optimization formulation. To solve the governing boundary value problem,
the recently proposed VEM is used. The key feature that distinguishes the VEM from the classical finite element method is that the interpolation functions in the interior of the elements are not required to be computed explicitly. The use of appropriate local projection maps allows for the extractions of the rigid body motion and the constant strain components of the deformation. Therefore, the computation of the local matrices is reduced to the evaluation of geometric quantities on the boundaries of the elements. Finally, several numerical examples are provided to demonstrate the efficiency and applicability of the VEM for the topology optimization of fluid flow problems.