Stabilization of topology optimization problems using Voronoi tessellations

Abstract Topology optimization, which is widely used in mechanical and aerospace engineering, has found its way into civil engineering. Here, progress made in additive manufacturing as well as new high‐performance materials allow for the construction of novel and complex geometric designs. Polygonal elements facilitate more flexible mesh generation and proper representation of these complex‐shaped design domains as well as their optimized versions. Additionally, local h ‐refinement can be employed efficiently. In this contribution Voronoi tessellations are used, which represent a highly efficient discretization technique that enables the polygonization of any structure. However, special finite element formulations have to be introduced due to the polygonal element geometries. The focus in this contribution lies on the determination of a structure's optimal material distribution, while maximizing its stiffness, that is, minimizing its compliance. The topology optimization problem is analogous to a mixed variational problem and shows the same numerical instability, known as checkerboarding. In this case, in certain regions, void and solid elements are alternating. It is shown that the use of polygonal finite elements with Voronoi tessellations avoids this phenomenon and leads to stable solutions. Several numerical examples are considered and the results are compared to the standard finite element formulation using quadrilateral elements, which are polluted by spurious checkerboard modes.