Polygonal finite elements for topology optimization: A unifying paradigm

TLDR

Topology optimization typically parameterizes designs on uniform Lagrangian finite-element grids, yet this approach suffers from checkerboard patterns, one-node connections, and mesh-dependent orientation bias that can produce suboptimal results. The study aims to mitigate the influence of mesh geometry on topology optimization outcomes by investigating unstructured meshes. To this end, the authors employ polygonal meshes derived from Voronoi tessellations, which offer greater geometric isotropy, flexibility for complex domains, and avoid numerical instabilities. © 2009 John Wiley & Sons, Ltd.

Abstract

Abstract In topology optimization literature, the parameterization of design is commonly carried out on uniform grids consisting of Lagrangian‐type finite elements (e.g. linear quads). These formulations, however, suffer from numerical anomalies such as checkerboard patterns and one‐node connections, which has prompted extensive research on these topics. A problem less often noted is that the constrained geometry of these discretizations can cause bias in the orientation of members, leading to mesh‐dependent sub‐optimal designs. Thus, to address the geometric features of the spatial discretization, we examine the use of unstructured meshes in reducing the influence of mesh geometry on topology optimization solutions. More specifically, we consider polygonal meshes constructed from Voronoi tessellations, which in addition to possessing higher degree of geometric isotropy, allow for greater flexibility in discretizing complex domains without suffering from numerical instabilities. Copyright © 2009 John Wiley & Sons, Ltd.