Static and harmonic BEM solutions of gradient elasticity problems with axisymmetry

An advanced boundary element method/fast Fourier transform (BEM/FFT) methodology for treating static and time harmonic axisymmetric problems in linear elastic structures exhibiting microstructure effects, is presented. These microstructure effects are taken into account with the aid of a simple strain gradient elastic theory proposed by Aifantis and co-workers [Aifantis (1992), Altan and Aifantis (1992), Ru and Aifantis (1993)]. Boundary integral representations of both static and dynamic gradient elastic problems are employed. Boundary quantities, classical and non-classical (due to gradient terms) boundary conditions are expanded in complex Fourier series in the circumferential direction and the problem is decomposed into a series of problems, which are solved by the BEM by discretizing only the surface generator of the axisymmetric body. The BEM integrations are performed by FFT in the circumferential directions simultaneously for all Fourier coefficients and by Gauss quadrature in the generator direction. All the strongly singular integrals are computed directly by employing highly accurate three-dimensional integration techniques. The Fourier transform solution is numerically inverted by the FFT to provide the final solution. The accuracy of the proposed boundary element methodology is demonstrated by means of representative numerical examples.The authors acknowledge with thanks for the support provided by I.K.Y. through the program IKYDA 2002 (scientific cooperation between the University of Patras, Greece and the Ruhr-University Bochum, Germany).     

The vectorization expressions of Taylor series multipole-BEM for 3D elasticity problems

The Taylor Series Multipole Boundary Element Method (TSMBEM) can improve the computational efficiency of Boundary Element Methods (BEM) efficiently, which only requires O(N) computational costs (operations and memory) for a problem with N unknowns. But the Taylor expansions of fundamental solutions are generally expressed using tensor form in the literatures about TSMBEM. Although these kinds of formulations are easy to program, many repetitious operations are executed and many equivalent terms are saved, it will result in the waste of memory. It is presented that the vectorization expressions of Taylor series multipole boundary element formula for elasticity problems, which take account of the symmetric properties of fundamental solutions and the characteristic of 3D components. The vectorization formulations reduce the computational operations and storage required, and improve the computational efficiency. The validity and efficiency of proposed scheme are demonstrated by the numerical experiments.     

Integrated FEM/BEM approach to the dynamic and acoustic analysis of plate structures

An integrated finite element/boundary element method approach to the prediction of the interior acoustic radiation of open ended box structures is presented. Dynamic response of the structure is predicted in terms of the nodal displacements under sinusoidal point force excitation using the finite element method. Theoretical results obtained in terms of frequency response functions are verified using the results from tests performed on a box structure. The interior acoustic field is then examined by the boundary element method using the boundary conditions obtained from the finite element analysis. Sound pressure levels produced inside the structure are calculated and the results are compared with the experimental measurements.     

各向异性外问题的非均匀网格的自然边界元法及其耦合法

本文对于无界区域上各向异性外问题提出了在椭圆边界非均匀网格上的自然边界元法及其与有限元法的耦合法,证明相应的收敛定理和误差估计式,并且在这两种方法中引入基于等分布原理的移动网格技巧.最后,通过数值结果表明了误差收敛理论的正确性以及所提方法和技巧的有效性.     

A symmetric Galerkin BEM implementation for 3D elastostatic problems with an extension to curved elements

 Like the finite element method (FEM), the symmetric Galerkin boundary element method (SGBEM) can produce symmetric system matrices. While widely developed for two dimensional problems, the 3D-applications of the SGBEM are very rare. This paper deals with the regularization of the singular integrals in the case of 3D elastostatic problems. It is shown that the integration formulas can be extended to curved elements. In contrast to other techniques, the Kelvin fundamental solutions are used with no need to introduce the new kernel functions. The accuracy of the developed integration formulas is verified on a problem with known analytical solution. Received 6 November 2000     

加纵肋平底圆柱壳振动和声辐射的FEM/BEM研究

建立了两端带平底板的加纵肋圆柱壳水中声辐射计算的FEM／BEM三维模型，探究了加肋的高度、宽度、数目对平底圆柱壳表面平均速度、辐射功率、辐射效率、声场指向性的影响规律。计算方法是在有限元软件ANSYS中做加肋平底圆柱壳建模、模态分析基础上，将有关数据（网格、模态）导入边界元软件SYSNOISE中计算流体-结构耦合状态下的辐射声场特性。结果表明：（1）纵肋的高度、宽度以及数目增大都可以引起平底圆柱壳的表面平均速度、辐射功率、辐射效率随频率变化曲线峰的移动，同时使声辐射效率增大，但使表面平均速度、辐射功率变化不明显。（2）纵肋的高度、宽度增大都使低频声辐射中两端平底板的贡献量增大，而使研究频域内的高频声辐射在激励力的反方向上增强。当径向激励力作用在纵肋上时适当调整均布纵肋的数目可以改变平底圆柱壳辐射声场的指向性。这对于水下结构辐射噪声预报以及噪声抑制具有重要意义。     

On DR/BEM for eigenvalue analysis of 2-D acoustics

This paper discusses the efficiency of several DR/BEM formulations and other boundary techniques for the eigenvalue extraction of two-dimensional acoustic cavities. First, the paper shows that the well-known conical radial basis functions lead to extremely ill conditioning results in cases that the height of the cone is not properly chosen. Moreover, the accuracy of other known high-degree basis functions is tested. Second, the use of Pascals triangle is proposed as a better approximation of the inertial forces at least for the case of rectangular domains. Using Gordons blending-function formula, a systematic procedure is proposed for the selection of the proper monomials. Third, it is shown that the aforementioned functional set can be also used to establish an alternative boundary-type method where both inertial and static terms are treated in a consistent manner. The solution quality of these formulations is investigated by calculating the eigenvalues of a rectangular and a circular acoustic cavity where analytical solutions are known.     

A numerical solution of the steady MHD flow through infinite strips with BEM

The magnetohydrodynamic (MHD) flow of an incompressible, viscous, electrically conducting fluid in infinite channels in the presence of a magnetic field is investigated. The fluid is driven either by a pressure gradient or by the currents produced by electrodes placed parallel in the middle of the walls. The applied magnetic field is perpendicular to the infinite walls which are combined from conducting and insulated parts. A boundary element method (BEM) solution has been obtained by using a fundamental solution which enables to threat the convection-diffusion type equations in coupled form with general wall conductivities. Constant elements are used for the discretization of the walls by keeping them as finite since the boundary integrals are restricted to these boundaries due to the regularity conditions as x,y→±∞. The solutions are presented in terms of equivelocity and induced magnetic field contours for several values of Hartmann number and conducting lengths. The effect of the parameters on the solution is visualized.     

On the numerical solution of linear exterior problems via the uncoupling method

The application of the uncoupling of boundary integral and finite element methods to solve exterior boundary value problems in R ² yields a weak formulation that contains only one boundary term. This is the so-called uncoupling term, which is determined by the boundary integral operator of the single-layer potential acting on a circle centered at the origin. The purpose of this paper is to provide a suitable formula, which combines analytical and numerical methods, to approximately integrate the uncoupling term to any exacteness. Our method provides sharper error estimates than the one that uses Truncated Infinite Fourier Series (TIFE). As a model we consider the exterior Dirichlet problem for the Laplacian, and use linear finite elements for the corresponding Galerkin scheme. Some numerical experiments are also presented. © 1998 John Wiley & Sons, Ltd.     

Revisit of degenerate scales in the BIEM/BEM for 2D elasticity problems

The boundary integral equation method in conjunction with the degenerate kernel, the direct searching technique (singular value decomposition), and the only two-trials technique (2 × 2 matrix eigenvalue problem) are analytically and numerically used to find the degenerate scales, respectively. In the continuous system of boundary integral equation, the degenerate kernel for the 2D Kelvin solution in the polar coordinates is reviewed and the degenerate kernel in the elliptical coordinates is derived. Using the degenerate kernel, an analytical solution of the degenerate scales for the elasticity problem of circular and elliptical cases is obtained and compared with the numerical result. Further, the triangular case and square case were also numerically demonstrated.     

On the BEM lumped mass formulations of wave equation problems

This paper continues earlier research on the performance of several consistent mass matrices for the boundary element method (BEM) dynamic analysis (wave equation problems), examining now the applicability of lumped mass matrices for several BEM formulations. In the beginning, the lumped masses are distributed along the boundary and three relative BEM formulations are proposed. Finally, the lumped matrices are introduced into the Partial Differential Equation (PDE) as Dirac-type domain inertial terms.     

BEM and FEM analysis of Signorini contact problems with friction

In this paper, we present a comparative study of the boundary element method (BEM) and the finite element method (FEM) for analysis of Signorini contact problems in elastostatics with Coulomb's friction law. Particularities of each method and comparison with the penalty method are discussed. Numerical examples are included to demonstrate the present formulations and to highlight its performance.     

Numerical solution of the variation boundary integral equation for inverse problems

In this paper a procedure to solve the identification inverse problems for two‐dimensional potential fields is presented. The procedure relies on a boundary integral equation (BIE) for the variations of the potential, flux, and geometry. This equation is a linearization of the regular BIE for small changes in the geometry. The aim in the identification inverse problems is to find an unknown part of the boundary of the domain, usually an internal flaw, using experimental measurements as additional information. In this paper this problem is solved without resorting to a minimization of a functional, but by an iterative algorithm which alternately solves the regular BIE and the variation BIE. The variation of the geometry of the flaw is modelled by a virtual strainfield, which allows for greater flexibility in the shape of the assumed flaw. Several numerical examples demonstrate the effectiveness and reliability of the proposed approach. Copyright © 2000 John Wiley & Sons, Ltd.     

A BEM sensitivity and shape identification analysis for acoustic scattering in fluid–solid problems

In this paper a boundary element formulation for the sensitivity analysis of structures immersed in an inviscide fluid and illuminated by harmonic incident plane waves is presented. Also presented is the sensitivity analysis coupled with an optimization procedure for analyses of flaw identification problems. The formulation developed utilizes the boundary integral equation of the Helmholtz equation for the external problem and the Cauchy–Navier equation for the internal elastic problem. The sensitivities are obtained by the implicit differentiation technique. Examples are presented to demonstrate the accuracy of the proposed formulations. © 1998 John Wiley & Sons, Ltd.     

A self-regularized approach for rank-deficient systems in the BEM of 2D Laplace problems

The Laplace problem subject to the Dirichlet or Neumann boundary condition in the direct and indirect boundary element methods (BEM) sometimes both may result in a singular or ill-conditioned system (some special situations) for the interior problem. In this paper, the direct and indirect BEMs are revisited to examine the uniqueness of the solution by introducing the Fichera’s idea and the self-regularized technique. In order to construct the complete range of the integral operator in the BEM lacking a constant term in the case of a degenerate scale, the Fichera’s method is provided by adding the constraint and a slack variable to circumvent the problem of degenerate scale. We also revisit the Fredholm alternative theorem by using the singular value decomposition (SVD) in the discrete system and explain why the direct BEM and the indirect BEM are not indeed equivalent in the solution space. According to the relation between the SVD structure and Fichera’s technique, a self-regularized method is proposed in the matrix level to deal with non-unique solutions of the Neumann and Dirichlet problems which contain rigid body mode and degenerate scale, respectively, at the same time. The singularity and proportional influence matrices of 3 by 3 are studied by using the property of the symmetric circulant matrix. Finally, several examples are demonstrated to illustrate the validity and the effectiveness of the self-regularized method.     

On the removal of the non‐uniqueness in the solution of elastostatic problems by symmetric Galerkin BEM

A study of the removal of the non‐uniqueness in the solution of elastostatic problems by means of the symmetric Galerkin boundary element method is presented. The paper focuses on elastic problems defined on domains with cavities, where cavity boundaries are subjected to traction boundary conditions. A simple method consisting in a direct application of support conditions and several methods based on the Fredholm theory of linear operators are introduced, implemented and analysed. Numerical examples demonstrate the performance of the proposed methods and accuracy of their results, a comparative evaluation of the methods developed being finally presented. Copyright © 2005 John Wiley & Sons, Ltd.     

A fast multipole boundary element method for 3D multi-domain acoustic scattering problems based on the Burton-Miller formulation

A fast multipole boundary element method (FMBEM) for 3D multi-domain acoustic scattering problems based on the Burton-Miller formulation is presented in this paper. A multi-tree structure is designed for the multi-domain FMBEM. It results in mismatch of leaves and well separate cells definition in different domains and complicates the implementation of the algorithm, especially for preconditioning. A preconditioner based on boundary blocks is devised for the multi-domain FMBEM and its efficiency in reducing the number of iterations in solving large-scale multi-domain scattering problems is demonstrated. In addition to the analytical moment, another method, based on the anti-symmetry of the moment kernel, is developed to reduce the moment computation further by a factor of two. Frequency sweep analysis of a penetrable sphere shows that the multi-domain FMBEM based on the Burton-Miller formulation can overcome the non-unique solution problem at the fictitious eigenfrequencies. Several other numerical examples are presented to demonstrate the accuracy and efficiency of the developed multi-domain FMBEM for acoustic problems. In spite of the high cost of memory and CPU time for the multi-tree structure in the multi-domain FMBEM, a large BEM model studied with a PC has 0.3 million elements corresponding to 0.6 million unknowns, which clearly shows the potential of the developed FMBEM in solving large-scale multi-domain acoustics problems.     

A new algorithm for numerical solution of 3D elastoplastic contact problems with orthotropic friction law

3D elastoplastic frictional contact problems with orthotropic friction law belong to the unspecified boundary problems with nonlinearities in both material and geometric forms. One of the difficulties in solving the problem lies in the determination of the tangential slip states at the contact points. A great amount of computational efforts is needed so as to obtain high accuracy numerical results. Based on a combination of the well known mathematical programming method and iterative method, a finite element model is put forward in this paper. The problems are finally reduced to linear complementarity problems. A specially designed smoothing algorithm based on NCP-function is then applied for solving the problems. Numerical results are given to demonstrate the validity of the model and the algorithm proposed.The project is jointly supported by the National Natural Science Foundation (10225212, 50178016, 10302007), the National Key Basic Research Special Foundation (G1999032805), the Special Funds for Major State Basic Research Projects and the Foundation for University Key Teacher by the Ministry of Education of China. The authors are also grateful to the referees for their careful reading and detailed remarks on an earlier version of the paper.     

An adaptive fast multipole boundary element method for three-dimensional acoustic wave problems based on the Burton–Miller formulation

The high solution costs and non-uniqueness difficulties in the boundary element method (BEM) based on the conventional boundary integral equation (CBIE) formulation are two main weaknesses in the BEM for solving exterior acoustic wave problems. To tackle these two weaknesses, an adaptive fast multipole boundary element method (FMBEM) based on the Burton–Miller formulation for 3-D acoustics is presented in this paper. In this adaptive FMBEM, the Burton–Miller formulation using a linear combination of the CBIE and hypersingular BIE (HBIE) is applied to overcome the non-uniqueness difficulties. The iterative solver generalized minimal residual (GMRES) and fast multipole method (FMM) are adopted to improve the overall computational efficiency. This adaptive FMBEM for acoustics is an extension of the adaptive FMBEM for 3-D potential problems developed by the authors recently. Several examples on large-scale acoustic radiation and scattering problems are presented in this paper which show that the developed adaptive FMBEM can be several times faster than the non-adaptive FMBEM while maintaining the accuracies of the BEM.