Indirect boundary element method combining extra fundamental solutions for solving exterior acoustic problems with fictitious frequencies

Abstract

The authors propose an alternative approach to solve the problem of fictitious frequencies. It is different from the mixed potential approach in the indirect method as well as the Burton and Miller approach in the direct boundary element method (BEM). The authors add some fundamental solutions with unknown strength in the solution representation to complete the solution space. From the viewpoint of the adding source, the present idea is similar to the combined Helmholtz interior integral equation formulation (CHIEF) method. The difference between the added source point and null-field point of CHIEF is their role. The former supplies the deficient basis due to the fictitious frequency while the latter provides the extra constraint equation. It can be alternatively found by adding the right unitary vectors of zero singular value. The bordered matrix is invertible for the fictitious frequency if the extra source points do not locate at the failure position. This is the reason why the property is analogous to the CHIEF method in the direct BEM. Therefore, it can fill in the blank area of why there is no CHIEF method in the indirect method. The authors also analytically derive the locations of possible failure source points by using the degenerate kernel.