Spurious Eigensolutions Research Articles

Modified 2.5D singular boundary methods to deal with spurious eigensolutions in exterior acoustic problems

This paper studies the problem of spurious eigensolutions in the context of the singular boundary method (SBM) formulated in the two-and-a-half-dimensional (2.5D) domain and proposes two numerical schemes to overcome this numerical difficulty. The SBM can be seen as a version of the method of fundamental solutions where the source points are located at the physical boundary. Similar to other boundary-type discretization schemes, the SBM also encounters the non-uniqueness problem at the vicinity of the eigensolutions of the corresponding interior problem. In the 2.5D domain framework, the so-called fictitious eigenfrequencies appearing in 3D problems arise in the form of spurious dispersion curves associated with propagation modes of the corresponding interior problem. The two enhanced 2.5D SBM approaches proposed in this work, based on the Burton–Miller method in one case and the dual surface method in the other, are designed to filter out the spurious eigenvalues from the simulation results and deliver accurate solutions along the wavenumber-frequency spectrum. Three benchmark examples including the radiation problems of an infinitely long cylinder under Dirichlet and Neumann boundary conditions and the radiation problem of a longitudinally infinite object with a constant star-like cross section subjected to a Dirichlet boundary condition are considered to study the proposed methods. The results demonstrate the capability of the proposed numerical schemes to successfully avoid the non-uniqueness problem when the 2.5D SBM is employed.

Eigenproblems of two-dimensional acoustic cavities with smoothly varying boundaries by using the generalized multipole method

This paper presents a semi-analytical approach to solve the eigenproblem of a two-dimensional acoustic cavity with smoothly varying boundaries. The multipole expansion for the acoustic pressure is formulated in terms of Bessel and Hankel functions to satisfy the Helmholtz equation in the polar coordinate system. Rather than using the addition theorem, the multipole method and directional derivative are both combined to propose a generalized multipole method in which the acoustic pressure and its normal derivative with respect to non-local polar coordinates can be calculated. The boundary conditions are satisfied by uniformly collocating points on the boundaries. By truncating the multipole expansion, a finite linear algebraic system is acquired. The direct searching approach is applied to identify the natural frequencies using the singular value decomposition technique. Several numerical examples are presented, including those of an annulus cavity, a confocal elliptical annulus cavity and an arbitrarily shaped cavity with an inner elliptical boundary. The accuracy and numerical convergence of the proposed method are validated by comparison with results of the available analytical method and the commercial finite-element code ABAQUS. No spurious eigensolutions are found in the proposed formulation. Due to its semi-analytical character, excellent accuracy and fast rate of convergence are the main features of the proposed method.

True and Spurious Eigensolutions of an Elliptical Membrane by Using the Nondimensional Dynamic Influence Function Method

In this paper, we employ the nondimensional dynamic influence function (NDIF) method to solve the free vibration problem of an elliptical membrane. It is found that the spurious eigensolutions appear in the Dirichlet problem by using the double-layer potential approach. Besides, the spurious eigensolutions also occur in the Neumann problem if the single-layer potential approach is utilized. Owing to the appearance of spurious eigensolutions accompanied with true eigensolutions, singular value decomposition (SVD) updating techniques are employed to extract out true and spurious eigenvalues. Since the circulant property in the discrete system is broken, the analytical prediction for the spurious solution is achieved by using the indirect boundary integral formulation. To analytically study the eigenproblems containing the elliptical boundaries, the fundamental solution is expanded into a degenerate kernel by using the elliptical coordinates and the unknown coefficients are expanded by using the eigenfunction expansion. True and spurious eigenvalues are simultaneously found to be the zeros of the modified Mathieu functions of the first kind for the Dirichlet problem when using the single-layer potential formulation, while both true and spurious eigenvalues appear to be the zeros of the derivative of modified Mathieu function for the Neumann problem by using the double-layer potential formulation. By choosing only the imaginary-part kernel in the indirect boundary integral equation method (BIEM) to solve the eigenproblem of an elliptical membrane, spurious eigensolutions also appear at the same position with those of NDIF since boundary distribution can be lumped. The NDIF method can be seen as a special case of the indirect BIEM by lumping the boundary distribution. Both the analytical study and the numerical experiments match well with the same true and spurious solutions.

Acoustic eigenproblems of elliptical cylindrical cavities with multiple elliptical cylinders by using the collocation multipole method

Abstract Acoustic eigenproblems of 3-D elliptical cylindrical cavities having multiple elliptical cylinders are solved by means of a 3-D semi-analytical formulation based on a multipole expansion, directional derivative, collocation technique and singular value decomposition (SVD). The multipole expansion for the acoustic pressure is formulated in terms of angular and radial Mathieu functions due to elliptical boundaries considered here. The boundary conditions are satisfied by uniformly collocating points along the boundaries. When considering sound-hard or Neumann boundary conditions, the normal derivative of acoustic pressure with respect to non-local elliptic coordinates is derived using the directional derivative for multiply-connected domain problems. By truncating the multipole expansion, a finite linear algebraic system is acquired. The direct searching approach is applied to identify the natural frequencies by using the SVD. Several numerical examples are examined, including those of an elliptical cylindrical cavity, a confocal elliptical annulus cylindrical cavity and an elliptical cylindrical cavity with two elliptical cylinders. Convergence and comparison studies are done to demonstrate the validation and accuracy of the present method. No spurious eigensolutions are found in the proposed formulation. Excellent accuracy and fast rate of convergence are the main features of the present method thanks to its semi-analytical character.

Analytical and numerical investigation for true and spurious eigensolutions of an elliptical membrane using the real-part dual BIEM/BEM

The paper performs analytical and numerical investigation of the true and spurious eigensolutions of an elliptical membrane using the real-part boundary integral equation method (BIEM) following the successful work on a circular case by using the dual boundary element method (BEM) (Kuo et al. in Int. J. Numer. Methods Eng. 48:1401–1422, 2000). We extend to the elliptical case in this paper. To analytically study the eigenproblems of an elliptical membrane, the elliptical coordinates and Mathieu functions are adopted. The fundamental solution is expanded into the degenerate kernel by using the elliptical coordinates and the boundary densities are expanded by using the eigenfunction expansion. The Jacobian terms may exist in the degenerate kernel, boundary density and boundary contour integration but they can cancel each other out. Therefore, the orthogonal relations are reserved in the boundary contour integral. It is interesting to find that the BIEM using the real or the imaginary-part kernel to deal with an elliptical membrane yields spurious eigensolutions. This finding agrees with those corresponding to the circular case. The spurious eigenvalues in the real-part BIEM are found to be the zeros of the mth-order (even or odd) modified Mathieu functions of the second kind or their derivatives. To verify this finding, the BEM is implemented. Furthermore, the commercial finite-element code ABAQUS is also utilized to provide eigensolutions for comparisons. It is found that good agreement is obtained.

Natural mode analysis of an acoustic cavity with multiple elliptical boundaries by using the collocation multipole method

This paper presents a semi-analytical approach to solve the eigenproblem of an acoustic cavity with multiple elliptical boundaries. To satisfy the Helmholtz equation in the elliptical coordinate system, the multipole expansion for the acoustic pressure is formulated in terms of angular and radial Mathieu functions. The boundary conditions are satisfied by uniformly collocating points on the boundaries. The acoustic pressure at each point is directly calculated in each elliptical coordinate system. In different coordinate systems, the normal derivative of the acoustic pressure is calculated by using the appropriate directional derivative, an alternative to the addition theorem. By truncating the multipole expansion, a finite linear algebraic system is derived. The direct searching approach is employed to determine the natural frequencies by using the singular value decomposition (SVD). Numerical results are widely discussed for several examples including an elliptical cavity, a confocal elliptical annulus cavity and an elliptical cavity with two elliptical cylinders. The accuracy and numerical convergence of the presented method is validated by comparison with available results from the analytical method and the commercial finite-element code ABAQUS. No spurious eigensolutions are found in the proposed formulation. Excellent accuracy and fast rate of convergence are the key features of the present method thanks to its semi-analytical feature.

Analytical investigation for true and spurious eigensolutions of multiply-connected membranes containing elliptical boundaries using the dual BIEM

It is well known that the boundary element method may induce spurious eigenvalues while solving eigenvalue problems. The finding that spurious eigenvalues depend on the geometry of inner boundary and the approach utilized has been revealed analytically and numerically in the literature. However, all the related efforts were focused on eigenproblems involving circular boundaries. On the other hand, the extension to elliptical boundaries seems not straightforward and lacks of attention. Accordingly, this paper performs an analytical investigation of spurious eigenvalues for a confocal elliptical membrane using boundary integral equation methods (BIEM) in conjunction with separable kernels and eigenfunction expansion. To analytically study this eigenproblem, the elliptic coordinates and Mathieu functions are adopted. The fundamental solution is expanded into the separable kernel by using the elliptic coordinates and the boundary densities are expanded by using the eigenfunction expansion. The Jacobian terms may exist in the separable kernel, boundary density and boundary contour integration and they can cancel each other out. Therefore, the orthogonal relations are reserved in the boundary contour integration. In this way, a similar finding about the mechanism of spurious eigenvalues is found and agrees with those corresponding to the annular case. To verify this finding, the boundary element method and the commercial finite-element code ABAQUS are also utilized to provide eigensolutions, respectively, for comparisons. Good agreement is observed from comparisons. Based on the adaptive observer system, the present approach can deal with eigenproblems containing circular and elliptical boundaries at the same time in a semi-analytical manner. By using the BIEM, it is found that spurious eigenvalues are the zeros of the modified Mathieu functions which depend on the inner elliptical boundary and the integral formulation. Finally, several methods including the CHIEF method, the SVD updating technique and the Burton & Miller method are employed to filter out the spurious eigenvalues, respectively. In addition, the efficiency of the CHIEF method is better than those of the SVD updating technique and the Burton & Miller approach, since not only hypersingularity is avoided but also computation effort is saved.

COMPUTER ASSISTED PROOF OF SPURIOUS EIGENSOLUTION FOR ANNULAR AND ECCENTRIC MEMBRANES

Spurious eigensolutions in the boundary integral equation (BIE) or boundary e1ement method (BEM) for doub1yconnected domain problems, the eccentric and annular membranes, are studied analytically and numerically in this paper. For the mathematica1 ana1ysis, we employ the null-field integral equation, the degenerate kernels and the Fourier series to prove the existence of spurious eigensolutions in the continuous system. Examples of eccentric case, annular membrane and general shape are solved by using the null-field equation approach and BEM, respectively. Based on the numerical experiment, computer-assisted proof for the existence of spurious eigenvalues in companion with the trivial outer boundary data is given. The spurious eigenvalue is found to be the true eigenvalue of a circular membrane with the inner radius. The SVD structure for the four influence matrices is examined Also, the trivial outer boundary densities are found in case of spurious eigenva1ues and are shown in the bar chart.

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.

Analytical study and numerical experiments of true and spurious eigensolutions of free vibration of circular plates using real-part BIEM

Following the success of the CHEEF method [Chen IL, Chen JT, Kuo SR, Liang MT. A new method for true and spurious eigensolutions of arbitrary cavities using the combined Helmholtz exterior integral equation formulation method. J Acoust Soc Am 2001; 109(3):982–98] and the real-part BEM [Kuo SR, Chen JT, Huang CX. Analytical study and numerical experiments for true and spurious eigensolutions of a circular cavity using the real-part dual BEM. Int J Numer Methods Eng 2000; 48:1401–22] for solving the membrane eigenproblem, we extend to the plate problem in this paper. The boundary integral equation method (BIEM) using only the real-part kernel instead of the complex-valued kernel is employed to solve the plate eigenproblem for saving half effort in computation. The spurious eigenvalue that resulted due to insufficient constraint is examined. To deal with this problem, a combined Helmholtz exterior integral equation formulation method (CHEEF) is employed to provide sufficient constraints to filter out spurious eigenvalues. The constraint equations of the transverse displacement, normal derivative and tangent derivative for the exterior collocating points are derived. If these constraint equations are properly chosen, one collocating point was sufficient to filter out all the spurious eigenvalues easily and efficiently, even for the repeated spurious eigenvalues. Finally, numerical experiments are performed to verify the analytical results.

Eigensolutions of multiply connected membranes using the method of fundamental solutions

In this paper, the method of fundamental solutions (MFS) of single and double-layer potential approaches for solving the eigenfrequencies of multiply connected membranes is proposed. By employing the fundamental solution, the coefficients of influence matrices are easily determined. The spurious eigensolution accompanied by the true eigensolution appears. It is found that the spurious eigensolution using the MFS depends on the location of the inner boundary where the sources are distributed. To verify this finding, the true and spurious eigenvalues in an annular domain are analytically studied using the degenerate kernels and circulants for an annular membrane. In order to obtain the true eigensolution, the singular value decomposition (SVD) updating techniques and the Burton and Miller method are utilized to filter out the spurious eigensolutions. Two examples are demonstrated analytically and numerically to see the validity of the present method.

True and spurious eigensolutions for the problems with the mixed-type boundary conditions using BEMs

Boundary integral equations and boundary element methods were employed semi-analytically and numerically to study the occurrence of spurious eigenvalues for the interior Helmholtz equations with the mixed-type boundary conditions. The degenerate kernels of fundamental function and Fourier series were utilized in the null-field integral equation to derive the true and spurious eigenfunctions semi-analytically. The complex-valued boundary element method (BEM), the real-part BEM, the imaginary-part BEM and the multiple reciprocity method were utilized to solve the eigenproblem. The results are compared with those of FEM. A simple case of one-dimensional eigenproblem was also addressed. Moreover, the SVD updating techniques in conjunction with the Fredholm's alternative theorem were adopted to sort out the spurious eigenvalues and extract the true ones. It is emphasized that the occurrence of spurious eigenvalues depends on the adopted method (singular or hypersingular formulation) no matter what the given type of boundary conditions for the problem is. The illustrative examples were verified successfully and the numerical results matched well with those of FEM.

Acoustic eigenanalysis for multiply‐connected problems using dual BEM

AbstractIn this paper, the eigenanalysis for the multiply‐connected domain problem is studied by using the dual boundary element method. The occurrence and treatment of the spurious eigenvalues for multiply‐connected domain problem are reviewed when the complex‐valued BEM in used. Three approaches, Burton and Miller method, CHIEF concept and SVD updating techniques, are adopted to suppress the occurrence of spurious eigensolutions. Instead of using the singular and hypersingular formulations, the singularity‐free methods, the null‐field equation approach and the fictitious BEM, are also utilized to deal with the eigenproblem. Both the eigenvalues and eigenmodes are compared with the analytical solutions and those of FEM for the illustrative examples. Good agreement is made. Copyright © 2004 John Wiley & Sons, Ltd.

True and spurious eigensolutions for the problems with the mixed-type boundary conditions using BEMs

Boundary integral equations and boundary element methods were employed semi-analytically and numerically to study the occurrence of spurious eigenvalues for the interior Helmholtz equations with the mixed-type boundary conditions. The degenerate kernels of fundamental function and Fourier series were utilized in the null-field integral equation to derive the true and spurious eigenfunctions semi-analytically. The complex-valued boundary element method (BEM), the real-part BEM, the imaginary-part BEM and the multiple reciprocity method were utilized to solve the eigenproblem. The results are compared with those of FEM. A simple case of one-dimensional eigenproblem was also addressed. Moreover, the SVD updating techniques in conjunction with the Fredholm's alternative theorem were adopted to sort out the spurious eigenvalues and extract the true ones. It is emphasized that the occurrence of spurious eigenvalues depends on the adopted method (singular or hypersingular formulation) no matter what the given type of boundary conditions for the problem is. The illustrative examples were verified successfully and the numerical results matched well with those of FEM.

An asymmetric indirect Trefftz method for solving free-vibration problems

In this paper, a new asymmetric indirect Trefftz method (AITM) has been developed to solve free-vibration problems. The proposed method is categorized into a regular type of boundary element methods (BEMs) such that no singular or hypersingular integration is necessary. However, like other regular BEMs the proposed approach encounters the numerical instability as the number of elements increases. To deal with such an ill-posed behavior, Tikhonov's regularization method in conjunction with the generalized singular-value decomposition (GSVD) is adopted. It is proved that the degeneracy of the proposed indirect Trefftz method has the same mathematical structure as the direct Trefftz method. Thus, no special effort should be paid in programming. Besides, such an equivalency indicates that the current method does not have spurious eigensolutions. Furthermore, the proposed approach can easily treat a multiply connected domain of genus 1 as shown in Fig. 1. Due to its indirect nature, the present approach can also represent the mode shapes within its own mathematical formulations. Several numerical examples are given to show the validity of the proposed approach.

Study on the true and spurious eigensolutions of two-dimensional cavities using the dual multiple reciprocity method

In this paper, true and spurious eigensolutions for a circular cavity using the dual multiple reciprocity method (MRM) are analytically derived and numerically verified by the developed program. The roots of spurious eigenequation are found analytically by using symbolic manipulation software. A more efficient method is proposed by choosing a fewer number of equations from the dual MRM instead of all of the equations in the dual MRM. Numerical experiments are performed by using dual MRM program for comparison purposes. A circular cavity of radius 1 m with Neumann boundary conditions is considered, and the results match very well between the theoretical prediction and the numerical experiments for the first four true eigenvalues and the first two spurious eigenvalues. Also, a noncircular case of square cavity is numerically implemented. The true eigensolutions can be easily solved by the dual MRM program in conjunction with the singular value decomposition technique. At the same time, the boundary modes and the multiplicities of the true eigenvalues can also be determined.

THE BOUNDARY COLLOCATION METHOD WITH MESHLESS CONCEPT FOR ACOUSTIC EIGENANALYSIS OF TWO-DIMENSIONAL CAVITIES USING RADIAL BASIS FUNCTION

In this paper, a meshless method for the acoustic eigenfrequencies using radial basis function (RBF) is proposed. The coefficients of influence matrices are easily determined by the two-point functions. In determining the diagonal elements of the influence matrices, two techniques, limiting approach and invariant method, are employed. Based on the RBF in the imaginary-part kernel, the method results in spurious eigenvalues which can be separated by using the singular value decomposition (SVD) technique in conjunction with the Fredholm alternative theorem. To understand why the spurious eigenvalues occur, analytical study in the discrete system by discretizing the circular boundary is conducted by using circulants. By using the SVD updating terms and documents, the true and spurious eigensolutions can be extracted out respectively. Several examples are demonstrated to see the validity of the present method.

Boundary collocation method for acoustic eigenanalysis of three-dimensional cavities using radial basis function

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. (1999, 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.

A new method for true and spurious eigensolutions of arbitrary cavities using the combined Helmholtz exterior integral equation formulation method.

Integral equation methods have been widely used to solve interior eigenproblems and exterior acoustic problems (radiation and scattering). It was recently found that the real-part boundary element method (BEM) for the interior problem results in spurious eigensolutions if the singular (UT) or the hypersingular (LM) equation is used alone. The real-part BEM results in spurious solutions for interior problems in a similar way that the singular integral equation (UT method) results in fictitious solutions for the exterior problem. To solve this problem, a Combined Helmholtz Exterior integral Equation Formulation method (CHEEF) is proposed. Based on the CHEEF method, the spurious solutions can be filtered out if additional constraints from the exterior points are chosen carefully. Finally, two examples for the eigensolutions of circular and rectangular cavities are considered. The optimum numbers and proper positions for selecting the points in the exterior domain are analytically studied. Also, numerical experiments were designed to verify the analytical results. It is worth pointing out that the nodal line of radiation mode of a circle can be rotated due to symmetry, while the nodal line of the rectangular is on a fixed position.

Analytical study and numerical experiments for true and spurious eigensolutions of a circular cavity using an efficient mixed-part dual BEM

In this paper, we develop an efficient mixed-part dual BEM to solve the eigensolutions of a circular cavity analytically and numerically. The method is proposed by choosing a fewer number of equations from the dual BEM instead of all of the equations in the dual BEM developed by Chen and his coworkers. To solve this problem analytically, the spurious solution can be filtered out by adding constraints from the dual boundary integral equations. The proposed method is superior to the complex-valued BEM not only for half effort in constructing the influence matrix, but also for its fewer size of dimension. Also, numerical experiments are performed to compare with the analytical results and the true eigensolutions can be easily extracted out in conjunction with the singular value decomposition technique (SVD). The optimum number of collocation point and appropriate collocating positions for the additional constraints are discussed.