Topology Optimization with the Homogenization and the Level-Set Methods

Abstract

After a brief review of the homogenization and level-set methods for structural optimization we make some comparisons of their numerical results. The typical problem is to find the optimal shape of an elastic body which is both of minimum weight and maximal stiness under specified loadings. This problem is known to be “ill-posed”, namely there is generically no optimal shape and the solutions computed by classical numerical algorithms are highly sensitive to the initial guess and mesh-dependent. The homogenization method makes this problem well-posed by allowing microperforated composites as admissible designs. It induces new numerical algorithms which capture an optimal shape on a fixed mesh. The homogenization method is able to perform topology optimization since it places no explicit or implicit restriction on the topology of the optimal shape. The level-set method instead does not change the ill-posed nature of the problem. It is a combination of the level-set algorithm of Osher and Sethian with the classical shape gradient (or boundary sensitivity). Although this last method is not specifically designed for topology optimization, it can easily handle topology changes. Its cost is also moderate since the shape is captured on a fixed Eulerian mesh. We discuss their respective advantages and drawbacks.