An electromagnetic shape optimisation for perfectly electric conductors by the time-domain boundary integral equations

This study proposes a shape optimisation framework for unsteady electromagnetic scattering problems on the basis of the time-domain boundary integral equation method, focusing on the perfectly electric conductors (PECs). The boundary-only formulation is ideal for treating a shape optimisation problem in an exterior domain. However, the electromagnetic shape optimisation in concern has been unrealised with the boundary integral approach regardless of the fact that the boundary-type shape derivative has been known in the literature. The first contribution of the present study is to derive a novel expression of the shape derivative in terms of the surface current densities of the primary and adjoint problems, by considering that the surface current density is handled by usual integral equations methods. The second contribution is to clarify the integral representations and equations of the adjoint electromagnetic fields in terms of the reversal time. These theoretical achievements possess a high affinity with the standard spatial discretising approach (i.e. RWG basis) whenever the temporal basis is sufficiently smooth. The numerical experiments confirmed the reliability of the proposed shape optimisation methodology and indicated the capability to deal with scientific and engineering applications.