Mathematical analysis and numerical study to free vibrations of annular plates using BIEM and BEM

In this paper, the spurious eigenequations for annular plate eigenproblems by using BIEM and BEM are studied in the continuous and discrete systems. Since any two boundary integral equations in the plate formulation (4 equations) can be chosen, 6 (C) options can be considered instead of only two approaches (single‐layer and double‐layer methods, or singular and hypersingular equations) which are adopted for the eigenproblems of the membrane and acoustic problems. The occurring mechanism of the spurious eigenequation for annular plates in the complex‐valued formulations is studied analytically. For the continuous system, degenerate kernels for the fundamental solution and the Fourier series expansion for the circular boundary density are employed to derive the true and spurious eigenequations analytically. For the discrete system, the degenerate kernels for the fundamental solution and circulants resulting from the circular boundary are employed to determine the true and spurious eigenequations. True eigenequation depends on the specified boundary condition while spurious eigenequation is embedded in each formulation. It is found that the spurious eigenvalue for the annular plate is the true eigenvalue of the associated interior problem with an inner radius of the annular domain. Also, we provide three methods (SVD updating technique, Burton and Miller method and CHIEF method) to suppress the occurrence of the spurious eigenvalues. Several examples were demonstrated to check the validity of the formulations. Copyright © 2005 John Wiley & Sons, Ltd.     

The Method of Fundamental Solutions for Eigenfrequencies of Plate Vibrations

This paper describes the method of fundamental solutions (MFS) to solve eigenfrequencies of plate vibrations by utilizing the direct determinant search method. The complex-valued kernels are used in the MFS in order to avoid the spurious eigenvalues. The benchmark problems of a circular plate with clamped, simply supported and free boundary conditions are studied analytically as well as numerically using the discrete and continuous versions of the MFS schemes to demonstrate the major results of the present paper. Namely only true eigenvalues are contained and no spurious eigenvalues are included in the range of direct determinant search method. Consequently analytical derivation is carried out by using the degenerate kernels and Fourier series to obtain the exact eigenvalues which are used to validate the numerical methods. The MFS is free from meshes, singularities, and numerical integrations. As a result, the proposed numerical method can be easily used to solve plate vibrations free from spurious eigenvalues in simply connected domains.     

On the null and nonzero fields for true and spurious eigenvalues of annular and confocal elliptical membranes

In this paper, the dual boundary element method (BEM) and the null-field boundary integral equation method (BIEM) are both employed to solve two-dimensional eigenproblems. The positions of true and spurious eigenvalues for circular, elliptical, annular and confocal elliptical membranes are analytically examined in the continuous system and numerically studied in the discrete system. To analytically study eigenproblems, the polar and elliptical coordinates in conjunction with the Bessel functions, the Mathieu functions, the Fourier series and eigenfunction expansions are adopted. The fundamental solution is expanded into the degenerate kernel while the boundary densities of circular and elliptical boundaries are expanded by using the Fourier series and eigenfunction expansion, respectively. Dirichlet and Neumann eigenproblems are both considered as well as simply and doubly-connected domains are both addressed. By employing the singular value decomposition (SVD) technique in the discrete system, the common right unitary vectors corresponding to the true eigenvalues for the singular and hypersingular formulations are found while the common left unitary vectors corresponding to the spurious eigenvalues are obtained for the singular formulation or hypersingular formulation. True eigenvalues depend on the boundary condition while spurious eigenvalues depend on the approach, the singular formulation or hypersingular formulation of BEM/BIEM. Nonzero field in the domain are analytically derived and are numerically verified in case of the true eigenvalue while the interior null field and nonzero field for the complementary domain are obtained in case of the spurious eigenvalue. Four examples, circular, elliptical, annular and confocal elliptical membranes, are considered to demonstrate the finding of the present paper. After comparing with the analytical and numerical results, good agreements are made. The dual BEM displays the dual structure in the unitary vector and the null field.     

Mathematical analysis and numerical study of true and spurious eigenequations for free vibration of plates using real-part BEM

In this paper, a real-part BEM for solving the eigenfrequencies of plates is proposed for saving half effort in computation instead of using the complex-valued BEM. By employing the real-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. For the continuous system, mathematical analysis for the spurious eigenequation was done by using the degenerate kernel and Fourier series. Good agreement of the analytical solutions (continuous and discrete systems) is made. 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.     

Comparisons of fundamental solutions and particular solutions for Trefftz methods

In the Trefftz method (TM), the admissible functions satisfying the governing equation are chosen, then only the boundary conditions are dealt with. Both fundamental solutions (FS) and particular solutions (PS) satisfy the equation. The TM using FS leads to the method of fundamental solutions (MFS), and the TM using PS to the method of particular solutions (MPS). Since the MFS is one of TM, we may follow our recent book [20], [21] to provide the algorithms and analysis. Since the MFS and the MPS are meshless, they have attracted a great attention of researchers. In this paper numerical experiments are provided to support the error analysis of MFS in Li [15] for Laplace's equation in annular shaped domains. More importantly, comparisons are made in analysis and computation for MFS and MPS. From accuracy and stability, the MPS is superior to the MFS, the same conclusion as given in Schaback [24]. The uniform FS is simpler and the algorithms of MFS are easier to carry out, so that the computational efforts using MFS are much saved. Since today, the manpower saving is the most important criterion for choosing numerical methods, the MFS is also beneficial to engineering applications. Hence, both MFS and MPS may serve as modern numerical methods for PDE.     

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.     

Application of symmetric indirect Trefftz method to free vibration problems in 2D

A symmetric indirect Trefftz method is developed to solve the free vibration problem of a 2D membrane. It is proved that in this approach the spurious eigensolution exists, and an auxiliary matrix is constructed to help extraction of the spurious solution using the generalized singular‐value decomposition. In addition to the spurious eigensolution, this regular formulation suffers from its ill‐posed nature, i.e. the numerical instability. In order to deal with the numerical instability, the Tikhonov's regularization method, in conjunction with the generalized singular‐value decomposition, is suggested. The proposed approach has some merits when compared with other regular boundary element formulations reported so far; namely the capacity of representing eigenmodes and the ability to deal with a multiply connected domain of genus 1. Several numerical examples are demonstrated to show the validity of the current approach. Copyright © 2003 John Wiley & Sons, Ltd.     

A Matrix Decomposition MFS Algorithm for Biharmonic Problems in Annular Domains

The Method of Fundamental Solutions (MFS) is a boundary-type method for the solution of certain elliptic boundary value problems. In this work, we develop an efficient matrix decomposition MFS algorithm for the solution of biharmonic problems in annular domains. The circulant structure of the matrices involved in the MFS discretization is exploited by using Fast Fourier Transforms. The algorithm is tested numerically on several examples.     

A new boundary meshfree method with distributed sources

A new boundary meshfree method, to be called the boundary distributed source (BDS) method, is presented in this paper that is truly meshfree and easy to implement. The method is based on the same concept in the well-known method of fundamental solutions (MFS). However, in the BDS method the source points and collocation points coincide and both are placed on the boundary of the problem domain directly, unlike the traditional MFS that requires a fictitious boundary for placing the source points. To remove the singularities of the fundamental solutions, the concentrated point sources can be replaced by distributed sources over areas (for 2D problems) or volumes (for 3D problems) covering the source points. For Dirichlet boundary conditions, all the coefficients (either diagonal or off-diagonal) in the systems of equations can be determined analytically, leading to very simple implementation for this method. Methods to determine the diagonal coefficients for Neumann boundary conditions are discussed. Examples for 2D potential problems are presented to demonstrate the feasibility and accuracy of this new meshfree boundary-node method.     

Wave propagation in cracked elastic slabs and half-space domains—TBEM and MFS approaches

In this paper, the traction boundary element method (TBEM) and the method of fundamental solutions (MFS), formulated in the frequency domain, are used to evaluate the 3D scattered wave field generated by 2D empty cracks embedded in an elastic slab and a half-space. Both models overcome the thin-body difficulty posed when the classical BEM is applied.The crack exhibits arbitrary cross section geometry and null thickness. In neither model are the horizontal formation surfaces discretized, since appropriate fundamental solutions are used to take them into consideration.The TBEM models the crack as a single line. The singular and hypersingular integrals that arise during the TBEM model's implementation are computed analytically, which overcomes one of the drawbacks of this formulation. The results provided by the proposed TBEM model are verified against responses provided by the classical BEM models derived for the case of an empty cylindrical circular cavity.The MFS solution is approximated in terms of a linear combination of fundamental solutions, generated by a set of virtual sources simulating the scattered field produced by the crack, using a domain decomposition technique. To avoid singularities, these fictitious sources are not placed close to the crack, and the use of an enriched function to model the displacement jumps across the crack is unnecessary.The performances of the proposed models are compared and their limitations are shown by solving the case of a C-shaped crack embedded in an elastic slab and a half-space domain.The applicability of these formulations is illustrated by presenting snapshots from computer animations in the time domain for an elastic slab containing an S-shaped crack, after applying an inverse Fourier transformation to the frequency domain computations.     

An improved boundary distributed source method for two-dimensional Laplace equations

In this work, the boundary distributed source (BDS) method [EABE 34(11): 914-919] based on the method of fundamental solutions (MFS) is considered for the solution of two-dimensional Laplace equations. The BDS is a truly mesh-free method and quite easy to implement since the source points and field points are collocated on the domain boundary while the conventional MFS requires a fictitious boundary where the source points locate. The main idea of the BDS is that to avoid the singularities of the fundamental solutions the concentrated point sources in the conventional MFS are replaced by distributed sources over circles centered at the source points. In the original BDS, all elements of the system matrix can be derived analytically in a very simple form for the Dirichlet boundary conditions and off-diagonal elements for the Neumann boundary conditions, while the diagonal elements for the Neumann boundary conditions can be obtained indirectly from the constant potential field. This work suggests a simple way to determine the diagonal elements for the Neumann boundary conditions by invoking that the boundary integration of the normal gradient of the potential should vanish. Several numerical examples are addressed to show the feasibility and the accuracy of the proposed method.     

A boundary-only meshless method for numerical solution of the Eikonal equation

The radial basis function (RBF) collocation methods for the numerical solution of partial differential equation have been popular in recent years because of their advantage. For instance, they are inherently meshless, integration free and highly accurate. In this article we study the RBF solution of Eikonal equation using boundary knot method and analog equation method. The boundary knot method (BKM) is a meshless boundary-type radial basis function collocation technique. In contrast with the method of fundamental solution (MFS), the BKM uses the non-singular general solution instead of the singular fundamental solution to obtain the homogeneous solution. Similar to MFS, the RBF is employed to approximate the particular solution via the dual reciprocity principle. In the current paper, we applied the idea of analog equation method (AEM). According to AEM, the nonlinear governing operator is replaced by an equivalent nonhomogeneous linear one with known fundamental solution and under the same boundary conditions. Finally numerical results and discussions are presented to show the validity and efficiency of the proposed method.     

A fast multipole accelerated method of fundamental solutions for potential problems

The fast multipole method (FMM) is a very effective way to accelerate the numerical solutions of the methods based on Green's functions or fundamental solutions. Combined with the FMM, the boundary element method (BEM) can now solve large-scale problems with several million unknowns on a desktop computer. The method of fundamental solutions (MFS), also called superposition or source method and based on the fundamental solutions but without using integrals, has been studied for several decades along with the BEM. The MFS is a boundary meshless method in nature and offers more flexibility in modeling of a problem. It also avoids the singularity of the kernel by placing the source at some auxiliary points off the problem domain. However, like the traditional BEM, the conventional MFS also requires O(N²) operations to compute the system of equations and another O(N³) operations to solve the system using direct solvers, with N being the number of unknowns. Combining the FMM and MFS can potentially reduce the operations in formation and solution of the MFS system, as well as the memory requirement, all to O(N). This paper is an attempt in this direction. The FMM formulations for the MFS is presented for 2D potential problem. Issues in implementation of the FMM for the MFS are discussed. Numerical examples with up to 200,000 DOF's are solved successfully on a Pentium IV PC using the developed FMM MFS code. These results clearly demonstrate the efficiency, accuracy and potentials of the fast multipole accelerated MFS.     

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.     

Homotopy method of fundamental solutions for solving certain nonlinear partial differential equations

In this study, the homotopy analysis method (HAM) is combined with the method of fundamental solutions (MFS) and the augmented polyharmonic spline (APS) to solve certain nonlinear partial differential equations (PDE). The method of fundamental solutions with high-order augmented polyharmonic spline (MFS–APS) is a very accurate meshless numerical method which is capable of solving inhomogeneous PDEs if the fundamental solution and the analytical particular solutions of the APS associated with the considered operator are known. In the solution procedure, the HAM is applied to convert the considered nonlinear PDEs into a hierarchy of linear inhomogeneous PDEs, which can be sequentially solved by the MFS–APS. In order to solve strongly nonlinear problems, two auxiliary parameters are introduced to ensure the convergence of the HAM. Therefore, the homotopy method of fundamental solutions can be applied to solve problems of strongly nonlinear PDEs, including even those whose governing equation and boundary conditions do not contain any linear terms. Therefore, it can greatly enlarge the application areas of the MFS. Several numerical experiments were carried out to validate the proposed method.     

A method of fundamental solutions for transient heat conduction in layered materials

In this paper we investigate an application of the method of fundamental solutions (MFS) to transient heat conduction in layered materials, where the thermal diffusivity is piecewise constant. Recently, in Johansson and Lesnic [A method of fundamental solutions for transient heat conduction. Eng Anal Boundary Elem 2008;32:697–703], a MFS was proposed with the sources placed outside the space domain of interest, and we extend that technique to numerically approximate the heat flow in layered materials. Theoretical properties of the method, as well as numerical investigations are included.     

The method of fundamental solutions for a biharmonic inverse boundary determination problem

In this paper, a nonlinear inverse boundary value problem associated to the biharmonic equation is investigated. This problem consists of determining an unknown boundary portion of a solution domain by using additional data on the remaining known part of the boundary. The method of fundamental solutions (MFS), in combination with the Tikhonov zeroth order regularization technique, are employed. It is shown that the MFS regularization numerical technique produces a stable and accurate numerical solution for an optimal choice of the regularization parameter. A. Zeb on study leave visiting the University of Leeds.     

The quasi-linear method of fundamental solution applied to non-linear wave equations

This paper presents a new meshless method developed by combining the quasi-linear method of fundamental solution (QMFS) and the finite difference method to analyze wave equations. The method of fundamental solution (MFS) is an efficient numerical method for solution Laplace equation for both two- and three-dimensional problems. The method has also been applied for the solution of Poisson equations and transient Poisson-type equations by finding the particular solution to the non-homogeneous terms. In general, approximate particular solutions are constructed using the interpolation of the non-homogeneous terms by the radial basis functions (RBFs). The interpolation in terms of RBFs often leads to a badly conditioned problem which demands special cares. The current work suggests a linearization scheme for the non-homogeneous term in terms of the dependent variable and finite differencing in time resulting in Helmholtz-type equations whose fundamental solutions are available. Consequently, the particular solution is no longer needed and the MFS can be directly applied to the new linearized equation. The numerical examples illustrate the effectiveness of the presented method.     

Analytical study and numerical experiments for degenerate scale problems in boundary element method using degenerate kernels and circulants

For a potential problem, the boundary integral equation approach has been shown to yield a nonunique solution when the geometry is equal to a degenerate scale. In this paper, the degenerate scale problem in boundary element method (BEM) is analytically studied using the degenerate kernels and circulants. For the circular domain problem, the singular problem of the degenerate scale with radius one can be overcome by using the hypersingular formulation instead of the singular formulation. A simple example is shown to demonstrate the failure using the singular integral equations. To deal with the problem with a degenerate scale, a constant term is added to the fundamental solution to obtain the unique solution and another numerical example with an annular region is also considered.     

A meshless method for solving nonhomogeneous Cauchy problems

In this paper the method of fundamental solutions (MFS) and the method of particular solution (MPS) are combined as a one-stage approach to solve the Cauchy problem for Poisson's equation. The main idea is to approximate the solution of Poisson's equation using a linear combination of fundamental solutions and radial basis functions. As a result, we provide a direct and effective meshless method for solving inverse problems with inhomogeneous terms. Numerical results in 2D and 3D show that our proposed method is effective for Cauchy problems.