An investigation of eigenfrequencies of boundary integral equations and the Burton-Miller formulation in two-dimensional elastodynamics

In this study, we investigate the distribution of eigenfrequencies of boundary integral equations (BIEs) of two-dimensional elastodynamics. The corresponding eigenvalue problem is classified as a nonlinear eigenvalue problem. We confirm that the Burton-Miller formulation can properly avoid fictitious eigenfrequencies. The boundary element method (BEM) is expected as a powerful numerical tool for designing sophisticated devices related to elastic waves such as acoustic metamaterials. However, the BEM is known that it loses its accuracy for certain frequencies, called as fictitious eigenfrequencies, for problems defined in the infinite domain. Recent researches It has also been revealed that not only the real-valued eigenfrequencies but also the complex-valued ones may affect the accuracy of the BEM results. We examine the distribution of complex eigenvalues obtained by BIEs for time-harmonic elastodynamic problems with the help of the Sakurai-Sugiura method which is applicable to nonlinear eigenvalue problems. We also examine its relation to the accuracy of the BEM numerical results. We also discuss an appropriate choice of the coupling parameter from a viewpoint of the distribution of fictitious eigenfrequencies.