Topology optimization with implicit functions and regularization

Topology optimization is formulated in terms of the nodal variables that control an implicit function description of the shape. The implicit function is constrained by upper and lower bounds, so that only a band of nodal variables needs to be considered in each step of the optimization. The weak form of the equilibrium equation is expressed as a Heaviside function of the implicit function; the Heaviside function is regularized to permit the evaluation of sensitivities. We show that the method is a dual of the BendsHe–Kikuchi method. The method is applied both to problems of optimizing single material and multi-material con gurations; the latter is made possible by enrichment functions based on the extended nite element method that enable discontinuous derivatives to be accurately treated within an element. The method is remarkably robust and we found no instances of checkerboarding. The method handles topological merging and separation without any apparent di culties