Topology Optimization with Wachspress and Voronoi Finite Elements

1. Abstract Traditionally, standard Lagrangian-type finite elements, such as linear quads and triangles, have been the elements of choice in the field of topology optimization. In general, finite element meshes with these elements exhibit the well-known checkerboard pathology in the iterative solution of topology optimization problems. Voronoi and Wachspress-type finite elements are less susceptible to such anomalies. Moreover, these elements provide more flexibility in mesh generation and are suitable for applications involving significant changes in the topology of the material domain. In particular, hexagonal Wachspress meshes include two-node connections (i.e. two elements are either not connected or connected by two nodes), and three edge-based symmetry lines per element. In contrast, quads can display one-node connections, which favor checkerboard configurations; and only have two edge-based symmetry lines. Thus checkerboard-free solutions are obtained without any further restrictions on the local variation of material density or filtering techniques (e.g. filter of sensitivities). We explore general Voronoi-type elements and present their implementation using a couple of approaches for topology optimization: e.g. element-based, and minimum length-scale control through projection functions. Examples are presented that demonstrate the advantages of the proposed elements in achieving checkerboard-free solutions and avoiding spurious fine-scale patterns from the design optimization process. Potential extensions and impact of this work will also be discussed. 2.