A Non-overlapping Domain Decomposition Method for Scattering Problems

In this paper, a domain decomposition method for the exterior Helmholtz problem is investigated. The unboundary domain is divided into some non overlapping subdomains. The natural integral operator is used as the artificial boundary conditions on the exterior boundary of the computational domains. The convergence of the algorithm is given in the sense of energy norm. Finally, the discrete problem is discussed and some numerical examples are presented.     

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.     

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.     

Boundary Element Methods with Non-overlapping Doma in Decomposition for Elastodynamics

This paper presents the numerical implementation of boundary element method incorporated into a non overlapping domain decomposition method for solving the Navier equations of linear elastodynamics problems by Fourier transformation. Several examples are presented.     

三维轴对称功能梯度材料瞬态热传导问题的自然单元法

为了更有效地求解三维轴对称功能梯度材料瞬态热传导问题,对无网格自然单元法应用于此类问题进行了研究,并发展了相应的计算方法。基于几何形状和边界条件的轴对称性,三维的轴对称问题可降为二维平面问题。为了简化本质边界条件的施加,轴对称面上的温度场采用自然邻近插值进行离散。功能梯度材料特性的变化由高斯点的材料参数进行模拟。时间域上,采用传统的两点差分法进行离散求解,进而得到瞬态温度场的响应。数值算例结果表明,提出的方法是行之有效的,理论及方法不仅拓展了自然单元法的应用范围,而且对三维轴对称瞬态热传导分析具有普遍意义。     

Analytic Method for the Free Vibration of Plane Frame Structure

The analytic method for the free vibration of plane frame structure is presented. It is achieved by transforming the general eigenvalue problem of natural frequency and vibration mode of continuously distributed property system into typical boundary value problem of ordinary differential equation (ODE). A series of ODE, corresponding to the general eigenvalue problem, is formed, and then is solved by an general ordinary differential equation solver-COLSYS. Each component of a frame structure is regarded as an element in the analytic method. This makes it more efficient than that of finite element method which needs to increase elements to get the value of high-class natural frequency and vibration mode accurately. Some applications of the method show that it can solve the free bending vibration of plane-frame structure with various types of displacement constraint. The result indicates that this method is precise and efficient.     

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.     

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.     

VBEM Analysis for Signorini Problem

the virtual boundary element method is applied for the numerical solution of a Signorini problem, which is conduced by an electro-paint process. The present method base on a switching algorithm, which can be applied to the problem defined in a non-regular domain, and it can reach good accuracy with less elements.     

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.     

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.     

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.     

BOUNDARY ELEMENT METHODS AND ERROR ANALYSIS FOR THE SOLUTION OF ELLIPTIC NEUMANN PROBLEM IN R~2

This paper presents boundary element methods for the solution of elliptic partial differential equation with Neumann boundary problem in R~2. Total details of error analysis are given for general condition, and optimal estimates are obtained. For simplicity Laplace's equation is discussed in illustration.     

Study on Strengthening and Toughening Mechanism of Nb V Bearing High Strength Steels

It is an important subject to solve numerically the problems of anisotropic unsteady seepage flow.This paper presents a numerical method for computing the anisotropic_orthotropic unsteady seepage flow with BEM.The paper introduces the so_called boundary_only technique to solve time dependent problem.The Volume domain integral is voided by using the Green formula again.Several numerical examples are presented.     

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.     

A Treatment to the Isoline for Finite Element Solution to Ferrite-Martensite Microstructure

The numerical solution of sample point stress field in finite element network of ferrite-martensite microstructure under finite deformation has a high-gradient distributive characteristic whose stress gradient between ferrite and martensite crystalline element is very high. If the interpolation method is used in the whole region,the isogram will be seriously smooth so that the original high-gradient sample point stress field will become untrue. This paper presents a new method for drawing isogram of ferrite-martensite microstructiire stress field by interpolating sample points in an element one by one. This method can make the post-treatment of high-gradient discrete stress field in crystalline network approach to a truer condition and solve the problem of arbitrary curve boundary of the deformed two phase steel.     

The Taylor series multipole boundary element method (TSM BEM)and its applications in rolling engineering

Rainfall is the main input for probabilistic analysis and prediction of rainfall-triggered landslide. The joint probabilistic structure of daily rainfall (DR) and cumulative rainfall (CR), which are dominant parameters of rainfall related on landslide in Chongqing region, was analyzed. Following the traditional technology, daily rainfall was translated into discrete variable by rainfall grade and cumulative rainfall became continuous variable if records with very small cumulative rainfall were ignored. Then joint probabilistic model of discrete variable and continuous one was derived, and transiting solution of conditional density function was put forward, together with its approximation via a family of Dirac δ sequences. Naturally, the proposed method was used to analyze conditional density function of cumulative rainfall in Chongqing region, and the numerical results were verified by comparison. However, most of the conditional density functions were irregular and not modeled by simple probability density function, thus the finite mixture distribution was introduced, which is of uncomplicated format and relatively high precision. At last, the joint probabilistic model of daily rainfall and cumulative rainfall was built up by combining frequency function of grade of daily rainfall with conditional density model of cumulative rainfall.     

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.     

AN ENTHALPY METHOD OF SOLVING PHASE CHANGE PROBLEMS BY COUPLING THE ENTHALPY AND THE MOVING BOUNDARY

A numerical enthalpy method is proposed for solving the multi-dimensional phase change problem by coupling the enthalpy and the moving boundary. Its basic feature is to sepasate appearent and latent heat terms by transforming the standard enthalpy relation and to couple them into the latent heat term by introducing the idea of phase change rate between the axes and simultaneously to correct the heat flux of the inflowing/outflowing phase change control volume element , so that the accuracy of numerical solution is improved and the moving boundary can be accurately followed.