On the spurious eigensolutions for the real-part boundary element method

In this paper, the eigensolutions are derived by using seven determinants of the direct-searching approach. Seven determinants include the complex valued, real and imaginary parts of determinant using the complex-valued kernel as well as the determinant by using the real-part, and imaginary-part kernels and the multiple reciprocity method (MRM). It is found that spurious eigensolutions of the real-part BEM match well with those of the MRM for the one-dimensional case. To satisfy the time-domain causal constraint, it is found that the real-part and imaginary-part kernels are not fully independent but are governed by the Hilbert transform. The idea of the combined Helmholtz exterior integral equation formulation method (CHEEF) in conjunction with the singular value decomposition (SVD) technique can be applied to suppress the occurrence of spurious eigenvalues. Updating terms and updating documents are employed to extract out the true and spurious eigenvalues. Possible failure CHEEF points are also examined. Also, the SVD structure of four influence matrices are examined. It is found that true and spurious eigenvectors are imbedded in the right and left unitary vectors in the SVD. Two- and three-dimensional cases are straightforward to be extended. The Hilbert transform pair for real and imaginary kernels was also examined.