Nyström method for elastic wave scattering by three-dimensional obstacles

Abstract

Nyström method is developed to solve for boundary integral equations (BIE’s) for elastic wave scattering by three-dimensional obstacles. To generate the matrix equation from a BIE, Nyström method applies a quadrature rule to the integrations of smooth integrands over a discretized element directly and chooses the values of the unknown function at quadrature points as the system’s unknowns to be solved. This leads to a simple procedure to form the off-diagonal entries of matrix by simply evaluating the integrands without numerical integrations. For the diagonal or near diagonal entries corresponding to the integrals over a singular or near-singular element where the kernels are singular or near singular, we develop a systematic singularity treatment technique, known as the local correction scheme, based on the linear approximation of elements. The scheme differs from the singularity regularization or subtraction technique used in the boundary element method (BEM). It applies the series expansion of scalar Green’s function to the kernels and derives analytical solutions for the strongly singular integrals under the Cauchy principal value like (CPV-like) sense. Since the approach avoids the need for reformulating the BIE for singularity removal in BEM and solves for the Somigliana’s equation directly, it is easy to implement and efficient in calculation. Numerical examples are used to demonstrate its robustness.