Shape Taylor expansion for wave scattering problems

Abstract: The Taylor expansion of wave fields with respect to shape parameters has a wide range of applications in wave scattering problems, including inverse scattering, optimal design, and uncertainty quantification. However, deriving the high order shape derivatives required for this expansion poses significant challenges with conventional methods. This paper addresses these difficulties by introducing elegant recurrence formulas for computing high order shape derivatives. The derivation employs tools from exterior differential forms, Lie derivatives, and material derivatives. The work establishes a unified framework for computing the high order shape perturbations in scattering problems. In particular, the recurrence formulas are applicable to both acoustic and electromagnetic scattering models under a variety of boundary conditions, including Dirichlet, Neumann, impedance, and transmission types.