A NEW BOUNDARY INTEGRAL EQUATION METHOD FOR SOLVING EXTERIOR BOUNDARY VALUE PROBLEMS OF THREE-DIMENSIONAL HELMHOLTZ EQUATION

This paper pressnts a new boundary integral equation method for solving exteri-or boundary value problems of three-dimensional Heimheltz equation by using the multiple reciproc-ity method.Firstly,integral representations of the solution in an exterior domain as well as on itsboundary,which have the peculiarity that integral kernels are infin ite seriesea developed from thenormal fundamental solution of Laplace equation and independent of the wavenumber,are given andproved under the Dirichlet condition.Then,based on the representation of the solution on the bound-ary,boundary integral equations for solving the Dirichlet and the Neumann boundary value prob-lems are obtained,and remarks for some problems concerned with solving these integral equationsnumerically are made.Finally, the advantages of the proposed method,as compared with the conven-tional boundary element methods,are summarized.     

A BOUNDARY ELEMENT METHOD FOR THE NEUMANN PROBLEM OF HELMHOLTZ EQUATION BY USE OF A DOUBLE LAYER POTENTIAL IN R~3

This paper presents a new numerical solution for Neumann problem of Helmholtz equation in R~3. The expression of the solution for this problem is obtained by use of a double layer potential and it leads to a Fredholm boundary integral equation of the first kind. Then, the existence and unicity of the integral equation which is equivalent to the boundary value problem are obtained in a suitable Sobolev space. Finally, a variational form which is equivalent to the integral equation is applied to the construction of a finite element method and the error estimate is given.     

A BOUNDARY ELEMENT METHOD FOR SOLVING DIRICHLET PROBLEM OF THE HELMHOLTZ EQUATION IN R~2

This paper presents a boundary element method for solving Dirichlet bou-ndary value problem of the Helmholtz equation in R~2.First,the existence andthe uniqueness of an extended solution for the problem are obtained.Then,thesolution is expressed in terms of simple layer potentials,and this expression,which is suitable to the interior as well as the exterior problem,leads to aboundary integral equation of the first kind.Finally,a finite element approachis applied to solve a variational form which is equivalent to the boundaryintegral equation.     

Galerkin Boundary Element Method for Laplace Equation

Galerkin method based on the variation principle is used to solve differential and integral equations. The boundary problem of Laplace equation is changed into the variational equation which is equivalent to the boundary integral equation. Using linear element, it is solved by Galerkin boundary element method. In computation of stiffness matrix, the exactly integral formula is used in the first order integral expression, The numerical integral formula is used in the second order integral expression. Thus the problem of calculation of double singular integral is carried out. The numerical experiments also prove this method is reliable. The error of Galerkin boundary element is tested with numerical experimentation.     

Garlerkin Boundary Element Method with Hyper Singular integral kernel

A Galerkin Boundary Elements was applied to solve the first kind of integral equation with hyper-singularity, which can be deduced from the direct boundary integral formula for the Neumann problem of Laplace equation. The concept of integration by parts in the sense of distributions was used. When boundary rotation is introduced, the two order derivatives of singular kernel are shifted to the boundary rotation of unknown function in the Galerkin variational formulation. While linear boundary elements are used for 2-dimensional problems, the boundary rotation on each element can be discretized into a constant vector, so that the integration can be performed in a simple way and the difficulty of numerical calculation for hyper-singularity is overcome. The results of numerical examples demonstrate that the scheme presented is practical and effective.     

Galerkin Boundary Element Method for Direct Boundary Integral Eqution for Dirichlet Problem of 2-D Laplace Eqation

The direct boundary integral equation of two-dimensional Laplace equation for Dirichlet problem is(con-sidered).It is deduced by Green's formula and the fundamental solution.The most-used numerical method for solving(direct) boundary integral equation is collocation method,and seldom have been used the Galerkin scheme in this case.The direct boundary integral eqution is changed into the variational eqution.Using linear element,it is solved by Galerkin boundary method.In the variational eqution double integrations shall be carried out.The paper presents the analytical formula to calculate the inner integration and the Gaussian quadrature is used for the outer integration. The numerical experimentation proved thefaesibility and the efficiency.     

BEM of Two-dimension Multiharmonic Equation and Its Applications

The paper discusses BEM of 2_dimension non_homogeneous multiharmonic equation.Under supposing non_homogeneous term is m_degree homonic,the integral in domain is transformed boundary integrals,and boundary integral equations are obtained correspondly.     

Elastic Analysis of Beam-Plate Foundation with Winkler Model

The beam-plate foundation is divided into two parts.The boundary element methods areapplied to analyze the plates with Winklermodel.The finite element methods are used to evaluate thebeams The coupling equation is obtained by the coodination of both force and displacement.     

Study on the Nonlinear Vibration System with Slow-changing Parameters

In this paper a posteriori error estimates for Galerkin approximation of general operator equations is firstly presented in the framework of Sobolev spaces. Then a practical posteriori error estimates formula for the adaptive boundary element method solving the acoustic scattering problem with a finite plane screen is obtained by triangulations. The mathematical model of this problem is the three dimensional Neumann boundary value problem of Helmholtz equation with finite plane boundary.     

Boundary Element Methods with Overlapping Domain Decomposition for Parabolic Problem

Boundary element method is a numerical method for solving partial differential equations. There are several formulations of boundary element method (BEM) applied to solve a parabolic differential equation.The approach,which employs time- dependent fundamental solution,allows longer time steps in time integration than other approaches,and this can cut down on time for computer implementation with high precision.Domain decomposition method,which decompose the domain that a given problem is to be solved into subdomains,has the advantages of reducing the large problem into smaller ones and reducing the complex problem into simpler ones,and allows parallel computing.An overlapping domain decomposition method is applied combining a boundary element formulation with time-dependent fundamental solution to solve a diffusion equation. Firstly, by domain decomposition, the problem divided into two problems on subdomains, and then the initial-Boundary problems are solved by boundry element method on each subdomain.Some numerical examples are presented to illustrate feasibility and efficiency of the method. The numerical experiments show that the convergence rate of the method is dependent with the overlapping degree of the subdomains.     

BOUNDARY ELEMENT METHOD FOR MULTI-BODY CONTACT PROBLEM AND ITS APPLICATION IN HUGE EXCAVATOR

A boundary integrate equation and its discrete technique for multi-body contact problem is carried out on basic equations of boundary element method and its fortran program is compiled. Using the method ,the press distribution on the roller plate of the huge excavator with capacity 16 m3 was calculated and a distroy stress analysis of caterpillar of the excavator was carried out.     

SINGULARITIES IN BOUNDARY ELEMENT METHODS

This paper presents an all round review of recent deve-lopments in treating the singularities in boundary element methods bothfor numerical computing and for mathematical analysing.Approachesfor numerical treatment of singular and hyper-singular integrations arelisted.Singular behaviour of solution on non-smooth boundary are discus-sed and the mathematical tools for describing it,such as the Sobolevspaces defined on a part of boundary,the pseudo-differential operatorsare presented.In order to incorporate the singular behaviour into theboundary element approximation,the technique of introducing singularboundary element is suggested.     

Galerkin Boundary Element Method for Two-dimension Laplace Equation of Neumman Condition

The authors apply the Galenkin variational equation to solve the integral equation with hyper singularity, which can be deduced from the double layer solution for Neumann problem of Laplace equation. The scheme of partial integration in the sense of distributions is introduced to reduce the hyper singularity integral into a weak one with the boundary rotation of unknown function. The numerical implementation with linear boundary elements is presented. The numerical examples illustrate the feasibility and efficiency of the method.     

Remarks for Various Boundary IntegralEquations Reduced from the ExteriorNeumann Problem of Helmholtz Equation

This paper presents a discussion on various boundary integral equations reduced from the exterior Neumann problem of Helmholtz equation.The author analyses how the famous difficulty that some equations have no unique solution when the wave number k is an eigenvalue of an interior problem is arised in the course of reducing these equations from Helmholtz representations,and proposes a method of overcoming the difficulty,that is,introducing a direct boundary integral equation which has unique solution for all wave numbers k and is equivalent to the original boundary value problem.Besides,advantages and shortcomings for these integral equations are estimated respectively.     

Similar Boundary Element Method in Elastodynamics

For the boundary element method of elastodynamics, some properties of matrices are discussed in case of similar boundary elements and the similar boundary element method is presented. In a series of similar boundary elements, when the corresponding matrices of a boundary element are obtained, the ones of other boundary elements in the series can be obtained by proportion. Then the coefficient matrix of the last system of linear algebraic equations can be obtained by the method of superposition. Compared with the general boundary element method, the computing speed can be raised by the similar boundary element method given in this paper.     

Research on Shoelast CAD/CAM Technology

Some new developments of boundary contour method have been presented in this paper. The developments include the boundary contour method based on equivalent boundary integral equations, the traction boundary contour method as well as the application of the boundary contour method to crack problems and elastic thin plate bending problems.     

Exact Integration of Constant Element of Elastomer in Boundary Element Method

The boundary integral in Boundary Element Method affects the precision and the speed of the method. If the boundary integral with constant element, the nonsingular integrals are popularly calculated by the Gauss numerical integral, and the singular integrals are popularly calculated by the analytical integral. This paper presents an alternative way with Gauss formula to transform the double integral in elastic problem on 3-d into the linear integrals on the boundary of each subdomains, so that all the singular integrals and nonsingular integrals are calculated by analytical method. The example indicates that this method makes the precision and the speed of BEM improve.     

Boundary Element Method for Time-dependent Chloride Diffusion and the Chloride Distribution in Concrete

The process of chloride diffusion in concrete is time-dependent.The boundary element method (BEM) with a time-dependent diffusion coefficient is presented for chloride diffusion in concrete based on the suitable transformation of variables.The fundamental solution of the partial differential equation for time-dependent chloride diffusion in concrete is developed,and the compensation length of the diffusion field is defined as well as the compensation coefficient.The scheme of BEM with a time-dependent diffusion coefficient is developed.Two examples are given to demonstrate the accuracy and efficiency of the presented method and the rationality and the importance of the compensation length for the method.     

Dynamic Imaging for 3 - D Brain Hematoma

Based on disturbance method of electrical current field, numerical calculation about dynamic imaging for brain hematomas have been studied on a 3 - D sphere model. With current injected, we have found the law of the brain boundary potentials'change while the volume of brain hematoma changed. Some studies about the effect of brain skull with poor electrical conductivity to the brain boundary potentials. The results show that we can know the change of brain hematomas in terms of the measurement for the change of boundary potentials. The injected current can penetrate through the skull. These results are valuable to help doctor to accurately diagnose the brain hematoma and realize the noninvasive monitoring the brain hematoma.     

Boundary Element Method of Material Parameter Identification in 3-D Infinite Problem

A 3 D boundary element method and dual boundary control technique of material parameter identification were proposed in this paper. The infinite boundary element was used for infinite problem in geomechanics.The computational result shows the validity of the proposed method.