Improved Third Medium Formulation for 3D Topology Optimization with Contact

This paper introduces a novel regularization term, denoted ”HuHu-LuLu”, for Third Medium Contact (TMC), and shows its usefulness in 2D and 3D topology optimization problems. The HuHu-LuLu regularization, where HuHu and LuLu espectively refer to the contraction of the Hessian and Laplacian of the the displacement field, is designed to reduce penalization on bending and quadratic compression deformations while maintaining penalization on excessive skew deformations. This approach allows for more accurate modeling of curved contacts, which is particularly beneficial when the domain discretization is coarse. Our results demonstrate that the HuHu-LuLu regularization can significantly improve the accuracy of TMC in topology optimization. The application of the method to a snap problem in 2D, made possible through a damped pseudo time-stepping scheme, represents the first such application of TMC to model snap-through. Furthermore, by adjusting the discretization of the design variable field in the topology optimization from linear finite elements, which are conventionally used in topology optimization, to second-order finite elements, which is made feasible
through a matching Gauss-Lobatto integration which circumvents evaluation of negative material densities, we achieve increased design freedom on coarse discretizations. These advancements significantly enhance the fidelity of state-of-the-art TMC topology optimization frameworks for a given discretization, enabling the creation of advanced designs in 3D.