ON THE FALSE DEGENERACY OF THE HELMHOLTZ BOUNDARY INTEGRAL EQUATIONS

• Published in: Journal of sound and vibration Vol. 258; no. 1; pp. 95 - 122
• Main Authors: YEIH, W., KUO, S.R., LIU, R.F., CHANG, J.R.
• Format: Journal Article
• Language: English
• Published: London Elsevier Ltd 2002; Elsevier
• Subjects: Boundary-integral methods; Computational techniques; Exact sciences and technology; Fundamental areas of phenomenology (including applications); Mathematical methods in physics; Physics; Solid mechanics; Structural and continuum mechanics; Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...); Vibrations and mechanical waves; Hole; Perforated plate; Free vibration; Degenerate system; Helmholtz equation; Boundary integral method; Integral equation; Numerical method; Eigenvalue problem; Structural analysis; Complex variable method
• ISSN: 0022-460X; 1095-8568
• DOI: 10.1006/jsvi.2002.5073

In this paper, the false degeneracy of the Helmholtz boundary integral equations is examined. A new theory to explain the false degeneracy of the Helmholtz boundary integral equations is given. In this proposed theory, a false degeneracy of the boundary integral equation is explained as finding a non-trivial source distribution such that it results in trivial field quantities inside the domain interested and non-trivial field quantities for its counter part, i.e., outside the domain interested. It is clearly explained that such a false degeneracy is independent of prescribed boundary conditions but dependent on the integral equation one selects. Moreover, the false degeneracy of the integral equation for the interested domain relates to the eigenproblem for its counter part. Under such a unified theory, the fictitious eigenvalue, spurious eigenvalue and pseudo-fictitious eigenvalue can be explained in a simple mathematical frame. It is concluded from our theoretical analysis that a multiply connected domain results in the pseudo-fictitious eigenvalue even the complex-valued formulations are used. In order to eliminate various kinds of false degeneracy, two methods are employed according to the previous research. A unified view of these two methods is provided such that they can be thought to be equivalent from mathematical point of view. Several numerical examples are given to show the validity of current approach.