On the topological derivative for the frequency derivative of the response in exterior wave problems

Abstract

Usually, the topology optimisation related to wave devices is formulated to minimise a shape function of a structural response to wave with a single frequency. The optimal structure obtained in such a way may not be robust with respect to the frequency change, i.e., the performance of the optimal structure may considerably deteriorate when the frequency of the incident wave is slightly different from the expected one. To remedy the robustness, it may be efficient to consider the shape function involving higher-order frequency derivatives of the structural response. In this paper, we discuss a computation for the topological derivative of higher-order frequency derivatives of acoustic response in a two-dimensional exterior domain. We show that we can “interchange” the topological and frequency derivatives. We also demonstrate that automatic differentiation is useful for computing higher-order frequency derivatives of the topological derivative.