Variational approach to shape derivatives for elasto-acoustic coupled scattering fields and an application with random interfaces

We establish the theoretical results, governed by Helmholtz equation and Lamé system, of shape derivatives of solutions to the elasto-acoustic coupled scattering problem. The primary techniques use the variational approach and the admissible perturbation characterized by the velocity method. Unlike perturbations of the boundary in the normal direction, the velocity method is introduced to conduct sensitivity analysis for an arbitrary domain with the least smooth conditions on a geometric boundary. In view of different boundary regularities, shape derivatives are investigated only in suitable Sobolev spaces. As a further application of our results, we derive the first order shape derivatives of solutions to stochastic elasto-acoustic equations with random interfaces, which can be used to obtain the approximation expectation, variance, and high order moments through Taylor shape expansion.