Density-based shape optimization of 3D structures with mean curvature constraints

This paper presents a density-based shape optimization using an interface motion scheme as in level set methods. The aim is to generate optimal material distributions with high-quality interfaces within a uniform geometric representation for topology and shape optimization. This reduces the effort for post-processing and facilitates an automated conversion to CAD models. By using a density function, the proposed method can seamlessly adopt density-based topology optimization results as the initial design. Finite element analyses are performed using the same mesh and density field as for optimization. The interface motion is based on an advection equation and shape derivatives without a penalization of intermediate densities. This prevents the formation of large grey transition regions while avoiding mesh-dependent spatial oscillations of the interface. Thus, boundary-fitted meshes with smooth surfaces and sufficiently retained stiffness and volume can be extracted. In addition, an optional constraint for the mean curvature of the surface as well as the necessary shape derivative is introduced. Thus, the surface curvature can be limited to a technically justifiable value, which improves, for instance, the manufacturability and fatigue strength. The evaluation is carried out using a two-dimensional example starting from a topology-optimized design as well as four three-dimensional examples starting from a trivial design. The considered optimization objective is to minimize the volume with respect to compliance constraints and, partially, additional mean curvature constraints.