In this paper, a theoretical formulation based on the collocation method is presented for the eigenanalysis of arbitrarily shaped acoustic cavities. This article can be seen as the extension of non-dimensional influence function (NDIF) method proposed by Kang et al. (1999Kang et al. ( , 2000a extending from two-dimensional to three-dimensional case. Unlike the conventional collocation techniques in the literature, approximate functions used in this paper are two-point functions of which the argument is only the distance between the two points. Based on this radial basis expansion, the acoustic field can be represented more exactly. The field solution is obtained through the linear superposition of radial basis function, and boundary conditions can be applied at the discrete points. The influence matrix is symmetric regardless of the boundary shape of the cavity, and the calculated eigenvalues rapidly converge to the exact values by using only a few boundary nodes. Moreover, the method results in true and spurious boundary modes, which can be obtained from the right and left unitary vectors of singular value decomposition, respectively. By employing the updating term and document of singular value decomposition (SVD), the true and spurious eigensolutions can be sorted out, respectively. The validity of the proposed method are illustrated through several numerical examples.
The complex-valued dual BEM has been employed by to solve the acoustic modes of a cavity with or without a thin partition. A novel method using only the real part of the complex-valued dual BEM was found by Chen (1998) to be equivalent to the dual MRM. However, spurious eigenvalues occur. In this paper, we propose the singular value decomposition technique to ®lter out spurious eigenvalues and to determine the multiplicity of true eigenvalues by combining the real-part dual equations. Also, the role of the real-part dual BEM for problems with a degenerate boundary is examined. Four examples, including a square cavity with multiple eigenvalues, a rectangular cavity, a rectangular cavity with a zero thickness partition and a rectangular cavity with a partition with ®nite thickness, are presented to demonstrate the validity of the proposed method. Also, the analytical solution if available, the FEM results obtained by Petyt et al. and obtained using ABAQUS and the experimental measurements are compared with those of the proposed method, and it is found that agreement between them is very good.
In this paper, an imaginary-part BEM for solving the eigenfrequencies of plates is proposed for avoiding singularity and saving half effort in computation instead of using the complex-valued BEM. By employing the imaginary-part fundamental solution, the spurious eigenequations in conjunction with the true eigenequation are obtained for free vibration of plate. To verify this finding, the circulant is adopted to analytically derive the true and spurious eigenequations in the discrete system of a circular plate. In order to obtain the eigenvalues and boundary modes at the same time, the singular value decomposition (SVD) technique is utilized. The analytical solutions are derived in the discrete system. Three cases, clamped, simply-supported and free circular plates, are demonstrated analytically and numerically to see the validity of the present method. SVD updating technique is adopted to suppress the ocurrence of the spurious eigenvalues, and a clamped plate is demonstrated analytically for the discrete system in this paper.
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