On the potential use of performance bounds based on semidefinite programs for topology optimization

Topology optimization has matured to become a powerful engineering design tool capable of designing exceptional structures and materials for targeted applications. Despite its undeniable success unanswered questions remain, one of which being: How far from the global performance optimum is a given topology optimized design? Typically this is a hard question to answer, as almost all interesting topology optimization problems are non-convex, i.e., local minima exist in the design space. In this work, we investigate performance bounds for structural optimization problems via a computational framework that utilizes Lagrange duality theory. The approach is applicable to the subset of optimization problem formulations that can be equivalently recast as Quadratically Constrained Quadratic Programs (QPQPs). This approach provides a viable measure of how "close" a given design is to the global optimum. To bound the primal QCQP, its dual problem is solved, which is shown to coincide with a convex Semidefinite Program (SDP) ensuring a single global optimum. The method's capabilities and limitations are explored via several numerical examples, considering the design of mode converters and resonating plates.