Dual boundary integral equations at a corner using contour approach around singularity

A dual integral formulation for the Laplace equation problem with a corner is derived by using the contour approach surrounding the singularity. It is found that using the contour approach the jump term comes half and half from the free terms in the L and M kernel integrations, which is different from the limiting process from an interior point to a boundary point where the jump term comes from the L kernel only. Thus, the definition of the Hadamard principal value for hypersingular integration at the collocation point of a corner is extended to a generalized sense for both the tangent and normal derivative of double layer potentials in comparison with the conventional definition. Two regularized versions of dual boundary integral equations with corners are proposed to avoid the boundary effect and are tested by an example. The numerical implementation is incorporated in the BEPO2D program. Also, a numerical example with a Dirichlet boundary condition on the corner is verified to determine the validity of the dual integral formulation.     

On efficient computation of highly oscillatory retarded potential integral equations

This paper presents an efficient numerical method for retarded potential integral equations with highly oscillatory spatially time-harmonic incident waves, which is based on inverse Fourier transforms and efficient algorithms for the highly oscillatory Volterra integral equation of the first kind. From the integral equation, it leads to an efficient approximation by applying the Clenshaw–Curtis-type method which costs the same operations independent of large values of frequencies. Applying inverse Fourier transforms yields numerical results on solving the retarded potential integral equations. Preliminary numerical results show the efficiency and accuracy of the approximations.     

Finite element computation of absorbing boundary conditions for time-harmonic wave problems

This paper proposes a new method, in the frequency domain, to define absorbing boundary conditions for general two-dimensional problems. The main feature of the method is that it can obtain boundary conditions from the discretized equations without much knowledge of the analytical behavior of the solutions and is thus very general. It is based on the computation of waves in periodic structures and needs the dynamic stiffness matrix of only one period in the medium which can be obtained by standard finite element software. Boundary conditions at various orders of accuracy can be obtained in a simple way. This is then applied to study some examples for which analytical or numerical results are available. Good agreements between the present results and analytical solutions allow to check the efficiency and the accuracy of the proposed method.     

A complex variable boundary element method for elliptic partial differential equations in a multiple-connected region

A simple boundary element method based on the Cauchy integral formulae is proposed for the numerical solution of a class of boundary value problems involving a system of elliptic partial differential equations in a multiple-connected region of infinite extent. It can be easily and efficiently implemented on the computer.     

Numerical results for linear Fredholm integral equations of the first kind over surfaces in three dimensions

Linear Fredholm integral equations of the first kind over surfaces are less familiar than those of the second kind, although they arise in many applications like computer tomography, heat conduction and inverse scattering. This article emphasizes their numerical treatment, since discretization usually leads to ill-conditioned linear systems. Strictly speaking, the matrix is nearly singular and ordinary numerical methods fail. However, there exists a numerical regularization method – the Tikhonov method – to deal with this ill-conditioning and to obtain accurate numerical results.     

Characteristic-based operator-splitting finite element method for Navier-Stokes equations

A new finite element method, which is the characteristic-based operator-splitting (CBOS) algorithm, is developed to solve Navier-Stokes (N-S) equations. In each time step, the equations are split into the diffusive part and the convective part by adopting the operator-splitting algorithm. For the diffusive part, the temporal discretization is performed by the backward difference method which yields an implicit scheme and the spatial discretization is performed by the standard Galerkin method. The convective...     

Numerical solutions of the one-phase classical Stefan problem using an enthalpy green element formulation

The enthalpy method is exploited in tackling a heat transfer problem involving a change of state. The resulting governing equation is then solved with a hybrid finite element - boundary element technique known as the Green element method (GEM). Two methods of approximation are employed to handle the time derivative contained in the discrete element equation. The first involves a finite difference method, while the second utilizes a Galerkin finite element approach. The performance of both methods are assessed with a known closed form solution. The finite element based time discretization, despite its greater challenge, yields less reliable numerical results. In addition a numerical stability test of both methods based on a Fourier series analysis explain the dispersive characters of both techniques, and confirms that replication of correct results is largely attributed to their ability to handle the harmonics of small wavelengths which are usually dominant in the vicinity of a front.     

基于结构元的H模糊自适应边界元

针对实际工程系统初始设计阶段经常出现材料的物理特性、结构的几何尺寸以及承受的外来作用等不确定性的问题,研究具有模糊不确定性的边界条件,提出基于模糊结构元（Fuzzy Structure Element,FSE）理论的模糊边界元法.该方法能简便、高效地处理边界条件具有模糊不确定性的系统.数值方法本身具有误差且系统自身具有模糊不确定性,故进一步研究模糊边界元法的H自适应算法,给出实用的误差估计公式;对角点处进行H自适应分析,得到较高精度的解.数值算例验证该方法的有效性和优越性.     

基于格林函数的图像增强算法

由于传统的图像增强算法得到的增强图像存在细节的缺失,主观效果较差等缺陷,提出了一种图像增强算法。通过Retinex模型保证了增强图像具有较突出的细节特性;通过求解泊松方程满足了增强图像与原始图像在梯度域的一致性;采用自适应亮度映射得到适于显示的边界条件;对区域的边界进行采样降低算法的复杂度。实验对比了几种图像增强算法得到的增强图像以及相关的评价系数,验证了该算法能够有效地提高图像的对比度,增强图像的主观视觉效果较高。     

Worst case bounds on the point-wise discretization error in boundary element method for the elasticity problem

In this work, point-wise discretization error is bounded via interval approach for the elasticity problem using interval boundary element formulation. The formulation allows for computation of the worst case bounds on the boundary values for the elasticity problem. From these bounds the worst case bounds on the true solution at any point in the domain of the system can be computed. Examples are presented to demonstrate the effectiveness of the treatment of local discretization error in elasticity problem via interval methods.     

Three-dimensional eigenvalue analysis of the Helmholtz equation by multiple reciprocity boundary element method

The eigenvalue of the three-dimensional Helmholtz equation is determined efficiently by extending the previously developed method for the two-dimensional problem. Boundary integral equation is formulated in the realm of the multiple reciprocity method, using higher order fundamental solutions for the Laplace equation; yielding polynomial coefficient matrices in terms of unknown wavenumber (eigenvalue). The Newton iteration method with the help of LU decomposition is employed to search eigenvalue, which can reduce the computational task significantly.     

Numerical study for two-point boundary value problems using Green''s functions

In this paper, we study the Green's function to find numerical solutions of second-order ordinary differential equations for two-point boundary value problems. We derive some properties of Green's function which can be applied to a Green's function integral formula. And we discuss and analyze numerical solutions which are obtained by the Green's function method and a shooting method.     

On the computation of steady axi-symmetric withdrawal from a two-layer fluid

Simultaneous withdrawal of both fluids in a two-layer system is considered. The flow is assumed to be steady and axi-symmetric, and withdrawal takes place through a spherical sink of finite radius. The problem is solved numerically, using two coupled integral equations and the condition of continuous pressure across the interface of the two fluids. This non-linear axi-symmetric problem is extremely ill-conditioned, but its conditioning can be improved substantially by careful construction of an appropriate Green function for the integral equations. Numerical results are presented, and it is suggested that steady solutions are not possible for the limiting case when the spherical sink collapses to a point. In this respect, the solution is evidently very different to the corresponding two-dimensional case.     

基于进化策略的广义积分计算方法研究

提出了一种基于进化策略算法的广义积分计算新方法,该方法根据被积函数的变量区间任意选取分割点,作为进化策略的初始的群体,通过进化策略算法来优化这些分割点,最终可得到一些最优的分割点,然后再求和,再根据和函数定义适应度函数,在给定的终止条件下,可获的精度较高的积分值.最后,以广义积分(无穷积分),二重广义积分(瑕积分)为例,仿真结果表明,该算法相比传统的一些方法,具有计算精度高,自适应性强等特点.     

An efficient method for solving elliptic boundary element problems with application to the tokamak vacuum problem

A method for regularizing ill-posed Neumann Poisson-type problems based on applying operator transformations is presented. This method can be implemented in the context of the finite element method to compute the solution to inhomogeneous Neumann boundary conditions; it allows to overcome cases where the Neumann problem otherwise admits an infinite number of solutions. As a test application, we solve the Grad–Shafranov boundary problem in a toroidally symmetric geometry. Solving the regularized Neumann response problem is found to be several orders of magnitudes more efficient than solving the Dirichlet problem, which makes the approach competitive with the boundary element method without the need to derive a Green function. In the context of the boundary element method, the operator transformation technique can also be applied to obtain the response of the Grad–Shafranov equation from the toroidal Laplace n=1 response matrix using a simple matrix transformation.     

A finite element penalty method for the linearized viscoelastic Oldroyd fluid motion equations

In this paper, a fully discrete finite element penalty method is considered for the two-dimensional linearized viscoelastic fluid motion equations, arising from the Oldroyd model for the non-Newton fluid flows. With the finite element method for the spatial discretization and the backward Euler scheme for the temporal discretization, the velocity and pressure are decoupled in this method, which leads to a large reduction of the computational scale. Under some realistic assumptions, the unconditional stability of the fully discrete scheme is proved. Moreover, the optimal error estimates are obtained, which are better than the existing results. Finally, some numerical results are given to verify the theoretical analysis. The difference between the motion of the Newton and non-Newton fluid is also observed.     

On numerical stabilization in the solution of Saint-Venant equations using the finite element method

Solving the Saint-Venant equations by using numerical schemes like finite difference and finite element methods leads to some unwanted oscillations in the water surface elevation. The reason for these oscillations lies in the method used for the approximation of the nonlinear terms. One of the ways of smoothing these oscillations is by adding artificial viscosity into the scheme. In this paper, by using a suitable discretization, we first solve the one-dimensional Saint-Venant equations by a finite element method and eliminate the unwanted oscillations without using an artificial viscosity. Second, our main discussion is concentrated on numerical stabilization of the solution in detail. In fact, we first convert the systems resulting from the discretization to systems relating to just water surface elevation. Then, by using M-matrix properties, the stability of the solution is shown. Finally, two numerical examples of critical and subcritical flows are given to support our results.     

一类模糊限定线性方程及其结构元求解方法

提出了一种模糊限定线性方程,在区间分析的基础上研究了其解的形式,并基于模糊结构元方法给出了方程的解及其隶属函数的解析表示。同时给出了一个数值实例。     

基于Matlab的径向基函数数值计算软件包

为推动基于无网格方法的计算软件的发展,介绍基于Matlab自主开发的径向基函数（Radial Basis Function,RBF）数值计算软件包,阐述软件的理论基础、设计思路以及该软件包的功能和特点,并结合边界节点法（Boundary Knot Method,BKM）的数值实例给出软件的使用过程．该软件包可以根据不同的数学物理模型选择合适的数值算法来求解多种实际物理问题,也可对不同数值算法得到的结果进行比较．最后,总结应用Matlab进行数值计算软件开发的优缺点．     

The exterior problem for the Helmholtz equation with mixed boundary conditions in three dimensions

In this article, a new integral equation is derived to solve the exterior problem for the Helmholtz equation with mixed boundary conditions in three dimensions, and existence and uniqueness is proven for all wave numbers. We apply the boundary element collocation method to solve the system of Fredholm integral equations of the second kind, where we use constant interpolation. We observe superconvergence at the collocation nodes and illustrate it with numerical results for several smooth surfaces.