A Fast Solver for the Elastic Scattering of Multiple Particles
Provides an efficient numerical method for elastic scattering problems with multiple particles, relevant to acoustics and solid mechanics.
The authors developed a fast solver for elastic wave scattering by multiple rigid particles in 2D, using Helmholtz decomposition and boundary integral equations with FMM acceleration, achieving high accuracy and efficiency.
Consider the elastic scattering of a time-harmonic wave by multiple well separated rigid particles in two dimensions. To avoid using the complex Green's tensor of the elastic wave equation, we utilize the Helmholtz decomposition to convert the boundary value problem of the elastic wave equation into a coupled boundary value problem of Helmholtz equations. Based on single, double, and combined layer potentials with the simpler Green's function of the Helmholtz equation, we present three different boundary integral equations for the coupled boundary value problem. The well-posedness of the new integral equations are established. Computationally, a scattering matrix based method is proposed to evaluate the elastic wave for arbitrarily shaped particles. The method uses the local expansion for the incident wave and the multipole expansion for the scattered wave. The linear system of algebraic equations is solved by GMRES with fast multipole method (FMM) acceleration. Numerical results show that the method is fast and highly accurate for solving the elastic scattering problem with multiple particles.