A multiharmonic finite element method for scattering problems with small-amplitude boundary deformations

A finite element method in the frequency domain is proposed for solving scattering problems with moving or, more generally, deforming boundaries. First, the original problem is rewritten as an equivalent weak formulation set in a fixed domain. Next, this formulation is approximated as a simpler weak form based on asymptotic expansions when the amplitude of the movements or the deformations is small. Fourier series expansions of some geometrical quantities and of the solution, under the assumption that the movement is periodic, are next introduced to obtain a coupled multi harmonic frequency domain formulation. Standard finite element methods can then be applied to solve the resulting problem, and a block diagonal preconditioner is proposed to accelerate the Krylov subspace solution of the linear system for high-frequency problems. The efficiency of the resulting method is demonstrated on a radar sensing application for the automotive industry.