Irregular Frequencies in Ultrasonic Boundary Integral Equations for Spherical Scatterer

The boundary element method (BEM) is a well-established full-wave technique for simulating the scattering of ultrasonic waves in elastic materials. The ultrasonic wave scattering problem is usually formulated in terms of the so-called conventional and hypersingular boundary integral equations (CBIE and HBIE, respectively) to apply the BEM. Since both CBIE and HBIE admit multiple solutions at some wave frequencies, they render the BEM ineffective in obtaining a numerical solution. We analyze this problem in the case of a spherical scatterer, a common defect shape used in both benchmarking and practice. Specifically, we compute scattered fields using the CBIE and HBIE formulations and show that in some special cases, the scattered far-fields can be obtained accurately despite the BEM being ill-conditioned at those frequencies.