Kernel-based collocation methods with TBC for the elastic wave scattering by obstacles

Elastic wave scattering plays a crucial role in medical imaging, seismic exploration, and non-destructive testing. In this paper, the kernel-based collocation methods are constructed for elastic wave scattering problems by multiple obstacles. Using Helmholtz decomposition, the original Navier equations for elastic waves in unbounded domain are reformulated into a Helmholtz equation system with two potential functions, coupled on the obstacle boundaries. To handle the unbounded domain, the transparent boundary conditions (TBC) are built based on the Dirichlet-to-Neumann (DtN) operator. The proposed method employs a kernel-based collocation method that combines radial basis functions (RBFs) with a weighted least-squares (WLS) method. The WLS formulations are proposed by setting more collocation points than trial centers and adding weights with respect to fill distance of collocation sets at obstacle and TBC boundary collocation terms. Using Whittle-Matérn-Sobolev kernels with kernel smoothness m, numerical experiments demonstrate that the proposed method can obtain solutions with the expected $m-2$ convergence rate for elastic scattering involving both single and multiple irregular obstacles. Furthermore, compared with the Kansa method and other mesh-dependent methods, the proposed method offers higher accuracy and more stable solutions for relatively large angular frequencies.