Shape Taylor expansion for wave scattering problems
Gang Bao, Haoran Ma, Jun Lai, Jingzhi Li
TL;DR
This work addresses the challenge of high-order shape derivatives in wave scattering by introducing a recurrence framework based on exterior differential forms, Lie derivatives, and material derivatives to compute the shape Taylor expansion for acoustic and electromagnetic problems. It unifies Dirichlet, Neumann, impedance, and transmission boundary cases, deriving recursion formulas that propagate higher-order boundary data from lower-order quantities. A key innovation is the trace operator decomposition, which avoids complex surface differential operators and enables compact, recursive updates for all derivative orders. The results, complemented by vector proxies for second-order derivatives, open pathways for improved inverse scattering, design optimization, and uncertainty quantification in shape-perturbed 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.