The Trefftz method for solving eigenvalue problems

For Laplace's eigenvalue problems, this paper presents new algorithms of the Trefftz method (i.e. the boundary approximation method), which solve the Helmholtz equation and then use an iteration process to yield approximate eigenvalues and eigenfunctions. The new iterative method has superlinear convergence rates and gives a better performance in numerical testing, compared with the other popular methods of rootfinding. Moreover, piecewise particular solutions are used for a basic model of eigenvalue problems on the unit square with the Dirichlet condition. Numerical experiments are also conducted for the eigenvalue problems with singularities. Our new algorithms using piecewise particular solutions are well suited to seek very accurate solutions of eigenvalue problems, in particular those with multiple singularities, interfaces and those on unbounded domains. Using piecewise particular solutions has also the advantage to solve complicated problems because uniform particular solutions may not always exist for the entire solution domain.     

A stochastic finite element method for real eigenvalue problems

A stochastic finite element method (SFEM) based on local averages of a random vector field is developed for both distinct and repeated eigenvalues. Formulae for the variances and covariances of the eigenvalues and eigenvectors are derived. It is shown in a numerical example that, as the number of elements increases, solutions obtained from the present SFEM formulation converge much faster than those obtained from the SFEM formulation based on mid-point discretization.     

A simultaneous iteration algorithm for symmetric eigenvalue problems

A FORTRAN IV algorithm is presented for determining sets of dominant eigenvalues and corresponding eigenvectors of symmetric matrices. It is also extended to the solution of the equations of natural vibration of a structure for which symmetric stiffness and mass matrices are available. The matrices are stored and processed in variable bandwidth form, thus enabling advantage to be gained from sparseness in the equations. Some of the procedures may also be used to solve symmetric positive definite equations such as those arising from the static analysis of structures loaded within the elastic range.     

Algebraic multigrid methods for solving generalized eigenvalue problems

Three algebraic multigrid (AMG) methods for solving generalized eigenvalue problems are presented. The first method combines modern AMG techniques with a non‐linear multigrid approach and nested iteration strategy. The second method is a preconditioned inverse iteration with linear AMG preconditioner. The third method is an enhancement of the previous one, namely the locally optimal block preconditioned conjugate gradient. Efficiency and accuracy of solutions computed by these AMG eigensolvers are validated on standard benchmarks where part of the spectrum is known. In particular, the problem of isospectral drums is addressed. Copyright © 2005 John Wiley & Sons, Ltd.     

A semi-analytic method for solving rib-type waveguide problems

A new semi-analytic method for solving optical rib-type waveguide problems is presented. In the method, the cross-section of a rib-type waveguide is divided into several regions. In each region, the refractive index profile and field distribution are expanded into Fourier cosine series, and then are substituted in the wave equation. A second-order differential matrix equation is then derived for each region, with a closed-form solution obtainable. With the boundary conditions used, an eigenmode equation for the rib waveguide can be derived and solved numerically to give the modal indices. Here, the presented method is used to deal with two rib waveguides in different geometric dimensions and/or compositions, respectively. Computational results show that the presented method is quite efficient, in terms of CPU time, in finding the modal indices accurately. The relative error in computing the modal index with the method is about 10^??5–10^??6.     

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.     

A new LBIE method for solving elastodynamic problems

A new local boundary integral equation (LBIE) method for the solution of elastodynamic problems in both frequency and time domain is proposed. Non-uniformly distributed points covering the analyzed domain are used for the interpolation of the involved fields. The key-point of the proposed methodology is that the support domain of each point is divided into parts with the aid of cells formed by connecting the point of interest with the nearby points. Then an efficient radial basis functions (RBF) interpolation scheme is exploited for the representation of displacements in each cell, while on the intersections between the local domains and the global boundary, tractions are treated as independent variables via conventional boundary elements. For each point the corresponding LBIE is written in terms of displacements only, since on the boundary of support domains tractions are eliminated with the aid of the elastostatic companion solution. The integration in support domains is performed easily and with high accuracy, while due to cells the extension of the method to three dimensions is straightforward. Transient solutions are obtained after inversion of frequency domain results with the inverse fast Fourier transform (FFT). Two representative numerical examples that demonstrate the accuracy of the proposed methodology are provided.     

A new high-order accurate continuous Galerkin method for linear elastodynamics problems

A new high-order accurate time-continuous Galerkin (TCG) method for elastodynamics is suggested. The accuracy of the new implicit TCG method is increased by a factor of two in comparison to that of the standard TCG method and is one order higher than the accuracy of the standard time-discontinuous Galerkin (TDG) method at the same number of degrees of freedom. The new method is unconditionally stable and has controllable numerical dissipation at high frequencies. An iterative predictor/multi-corrector solver that includes the factorization of the effective mass matrix of the same dimension as that of the mass matrix for the second-order methods is developed for the new TCG method. A new strategy combining numerical methods with small and large numerical dissipation is developed for elastodynamics. Simple numerical tests show a significant reduction in the computation time (by 5–25 times) for the new TCG method in comparison to that for second-order methods, and the suppression of spurious high-frequency oscillations.     

Least squares element method for boundary eigenvalue problems

Linear and non-linear boundary eigenvalue problems are discretized by a new finite element like method. The reason for the new construction principle is the non-linear dependence of the dynamic stiffness element matrix on an eigenparameter. The dynamic stiffness element matrix is evaluated for a fixed number of parameters and is then elementwise replaced by a polynomial in the eigenparameter by solving least squares problems. A fast solver is introduced for the resulting non-linear matrix eigenvalue problem. It consists of a combination of bisection method and inverse iteration. The superiority of the newconstructionprinciple in comparison with the finite or dynamic element method is demonstrated finally for some numerical examples.     

广义特征值问题的快速傅里叶变换法

针对广义特征值问题提出离散傅里叶变换法。该方法把结构的动力响应看作是一种信号,利用快速傅里叶变换进行分析,从而得到结构的振动频率。该方法避免对刚度矩阵求逆,可同时计算出所有的特征值,是一种直接方法。数值算例验证了该方法的正确性。     

广义特征值问题求解的改进Ｒｉｔｚ向量法

从提高算法的稳定性和计算效率入手,采取迭代及防止漏根、多根的措施,对传统的Ｒｉｔｚ向量法进行改进,提出改进的Ｒｉｔｚ向量法。此算法仅需生成ｒ维的Ｋｒｙｌｏｖ空间,大大降低投影矩阵阶数,减少投影矩阵特征值计算时间。引入重正交方案和模态比较法,并给出Ｒｉｔｚ向量块宽ｑ与生成步数ｒ的建议取值。最后通过四参数的谱变换法,不     

Subspace iteration accelrated by using Chebyshev polynomials for eigenvalue problems with symmetric matrices

Bathe's algorithm of subspace iteration for the solution of the eigenvalue problem with symmetric matrices is improved by incorporating an acceleration technique using Chebyshev polynomials. This method of acceleration is particularly effective for this kind of iteration. The rate of convergence of the iteration scheme presented is considerably improved when compared with the original one, and satisfactory rates of convergence can be obtained for a wider range of eigenvalues.     

A combined integral-equation and photoelastic method for solving contact problems

Summary The relation for the difference of principal stresses expressed in terms of Muskhelishvili's complex potential was transformed to an integral equation for the case of a half-plane in contact. Two methods for the numerical solution of the integral equation were developed and a study for the appropriate selection of the collocation points was made. The existence of a solution of the equation and the possibility of applying an iterative process was shown. As an example, the method was succesfully applied for solving the problem of a half-plane loaded by either a uniform, or a parabolic, distribution of forces.With 5 Figures     

A boundary element method for eigenvalue problems of polygonal membranes and plates

Summary An advanced Boundary Element formulation for eigenvalue problems of membranes and plates is developed. Polygonal membranes are considered, and are embedded into a proper basic domain in order to satisfy boundary conditions exactly as far as possible. Hence, boundary integrals have to be applied at the not coinciding boundaries only. Eigenvalues of the underlying Dirichlet's Helmholtz problem are calculated from frequency response functions evaluated by that method with Green's functions of finite domains, and natural frequencies of corresponding membranes and simply supported plates are determined by analogy. A numerical investigation is performed for parallelogram Mindlin plates. Natural frequencies and critical buckling eigenvalues are graphically represented in a nondimensional form, where the influence of skew angle and plate thickness is studied.With 4 Figures     

Radial integration boundary element method for acoustic eigenvalue problems

In this paper, the radial integration boundary element method is developed to solve acoustic eigenvalue problems for the sake of eliminating the frequency dependency of the coefficient matrices in traditional boundary element method. The radial integration method is presented to transform domain integrals to boundary integrals. In this case, the unknown acoustic variable contained in domain integrals is approximated with the use of compactly supported radial basis functions and the combination of radial basis functions and global functions. As a domain integrals transformation method, the radial integration method is based on pure mathematical treatments and eliminates the dependence on particular solutions of the dual reciprocity method and the particular integral method. Eventually, the acoustic eigenvalue analysis procedure based on the radial integration method resorts to a generalized eigenvalue problem rather than an enhanced determinant search method or a standard eigenvalue analysis with matrices of large size, just like the multiple reciprocity method. Several numerical examples are presented to demonstrate the validity and accuracy of the proposed approach.     

Efficient method for lasing eigenvalue problems of periodic structures

Lasing eigenvalue problems (LEPs) are non-conventional eigenvalue problems involving the frequency and gain threshold at the onset of lasing directly. Efficient numerical methods are needed to solve LEPs for the analysis, design and optimization of microcavity lasers. Existing computational methods for two-dimensional LEPs include the multipole method and the boundary integral equation method. In particular, the multipole method has been applied to LEPs of periodic structures, but it requires sophisticated mathematical techniques for evaluating slowly converging infinite sums that appear due to the periodicity. In this paper, a new method is developed for periodic LEPs based on the so-called Dirichlet-to-Neumann maps. The method is efficient since it avoids the slowly converging sums and can easily handle periodic structures with many arrays.     

A new space marching method for solving inverse heat conduction problems

A new space marching method is presented for solving the one-dimensional nonlinear inverse heat conduction problems. The temperature-dependent thermal properties and boundary condition on an accessible part of the boundary of the body are known. Additional temperature measurements in time are taken with a sensor located in an arbitrary position within the solid, and the objective is to determine the surface temperature and heat flux on the remaining part of the unspecified boundary. The temperature distribution throughout the solid, obtained from the inverse analysis, is then used for the computation of thermal stresses in the entire domain, including the boundary surfaces. The proposed method is appropriate for on-line monitoring of thermal stresses in pressure components. The three presented example show that the method is stable and accurate.     

A complex variable boundary element method for solving interface crack problems

The problem of finite bimaterial plates with an edge crack along the interface is studied. A complex variable boundary element method is presented and applied to determine the stress intensity factor for finite bimaterial plates. Using the pseudo-orthogonal characteristic of the eigenfunction expansion forms and the well-known Bueckner work conjugate integral and taking the different complex potentials as auxiliary fields, the interfacial stress intensity factors associated with the physical stress-displacement fields are evaluated. The effects of material properties and crack geometry on stress intensity factors are investigated. The numerical examples for three typical specimens with six different combinations of the bimaterial are given. This revised version was published online in July 2006 with corrections to the Cover Date.     

A fast multipole boundary element method for solving two-dimensional thermoelasticity problems

A fast multipole boundary element method (BEM) for solving general uncoupled steady-state thermoelasticity problems in two dimensions is presented in this paper. The fast multipole BEM is developed to handle the thermal term in the thermoelasticity boundary integral equation involving temperature and heat flux distributions on the boundary of the problem domain. Fast multipole expansions, local expansions and related translations for the thermal term are derived using complex variables. Several numerical examples are presented to show the accuracy and effectiveness of the developed fast multipole BEM in calculating the displacement and stress fields for 2-D elastic bodies under various thermal loads, including thin structure domains that are difficult to mesh using the finite element method (FEM). The BEM results using constant elements are found to be accurate compared with the analytical solutions, and the accuracy of the BEM results is found to be comparable to that of the FEM with linear elements. In addition, the BEM offers the ease of use in generating the mesh for a thin structure domain or a domain with complicated geometry, such as a perforated plate with randomly distributed holes for which the FEM fails to provide an adequate mesh. These results clearly demonstrate the potential of the developed fast multipole BEM for solving 2-D thermoelasticity problems.