Towards closed-walled designs in topology optimization using selective penalization

3D topology optimization using a density approach with penalization usually produces truss-like, open-walled structures. The coarser the mesh, the smaller the volume fraction, and the faster the penalization is increased, the more pronounced this effect tends to be. However, closed-walled designs are often more efficient and have other beneficial properties. For instance, closed walls can contribute to achieving self-supporting designs for additive manufacturing that potentially require fewer sacrificial support structures than truss-like designs. This paper presents a two-step optimization procedure for generating closed-walled designs using coarse meshes. The first step takes the usual Eulerian approach of performing a SIMP-based topology optimization on a fixed mesh. To keep thin geometrical features, like walls with a thickness below element size, penalization is switched off deliberately where the formation of such features is detected. Adopting a Lagrangian description, intermediate densities still present in the optimized design are subsequently eliminated in a second step by shrinking each element according to its density. By analogy with volumetric thermal contraction, this is accomplished by solving a fictitious thermo-elastic problem where the temperature has been replaced by a density expression. The outcome is a morphed mesh with a somewhat smoothed surface and a volume close to the specified material volume limit. This body-fitted representation of the design considerably simplifies the final conversion into a manufacturable CAD-type geometry. The two-step optimization procedure is applied to a cantilever, a torsion rod, and a disk reinforcement benchmark problem. Optimized designs are closed-walled and show very good agreement to those found for much finer meshes. Problem-specific stiffness improvements over truss-like designs between 6% and almost 30% were achieved and confirmed the findings previously reported by other authors.