四边形常数单元离散下的声学非奇异BIE

奇异积分是基于Burton-Miller方程的声学边界元法实现过程的难点之一。关于三角形单元离散的积分单元的已经比较成熟,研究四边形常数单元离散下的声学边界积分方程(BIE),通过构造围绕配点的极小半球面进行积分,求得积分中的发散项,推导四边形常数单元离散下边界积分方程及其法向求导的非奇异表达式,从而得到非奇异Burton-Miller方程。运用Gauss Legendre积分公式计算BIE的S(x)的数值解,对比解析解的计算结果,得出了数值解、解析解以及二者的绝对误差、相对误差随ka的变化规律。实际应用时,当给定精度和ka的值后,可以通过改变所需要的截断项数,使得误差满足给定的精度要求。     

耦合边界元法和等效源法的稳健CHIEF法

针对声学边界元法中解的非唯一性和奇异积分问题,基于组合亥姆霍兹积分方程公式(combined helmholtz integral equation formulation, CHIEF)法思想,将常规边界元方程和等效源方程进行联立,并利用两者方程系数矩阵间的耦合等价关系,间接替换计算常规边界元法中的奇异系数矩阵,进而提出一种具有全频域唯一解、高计算精度和高稳定性的耦合CHIEF法。该方法将等效源方程作为补充方程,不仅解决了传统CHIEF法内点补充方程失效的问题,而且矩阵的间接替换计算避免了直接计算奇异积分,显著提高了计算效率和精度。通过声辐射和声散射的典型算例对比了所提方法、常规边界元法、常规Burton-Miller法和等效源法的计算效果。结果表明,所提方法不仅在全波数域内均能获得唯一解,且其计算精度和效率均优于常规边界元法和常规Burton-Miller方法,其系数矩阵条件数远低于等效源法。     

Helmholtz声学边界积分方程中奇异积分的计算

提出了一种非等参单元的四边形坐标变换,它将积分的曲面单元映射为另一四边形单元,通过两次坐标变换引入的雅可比行列式可以消除Helmholtz声学边界积分方程中的弱奇异型O(1／r))积分．而且利用δr／δn以及坐标变换可以同时消除坐标变换无法消除的Cauchy型(O(1／r^2))奇异积分,并给出了消除奇异性的详细证明．该方法给Helmholtz声学边界积分方程中的弱奇异积分与Cauchy奇异积分的计算以及编程提供了极大便利。     

应力强度因子计算的边界元法

本文利用非连续元离散边界的积分方程，推导了奇异积分的具体表达式，将非连续边界元和多域缩聚法用于二维弹性断裂应力强度因子计算，得到了合理的计算结果。     

与权函数对应的积分方程及其联立解法

边界元法一般采用控制方程的基本解作为权函数,这往往能在控制方程为齐次时可避免域积分。但当问题复杂,基本解不能求得时,此法便产生了困难。虽有人也曾偶尔用非基本解函数作为权函数,但本文将系统地探讨函娄与边界积分方程的关系,所涉及的各类定解问题都用Laplace基本解和kelvin基本解作为权函数,并提出边界点公式和内点公式联立求解的方法。这不但避免了求基本解的困难,同时也为编制能求解多种问题的多功能电算程序提供了方便,使程序的长度缩短了,编程和调试的难度也降低了。     

求解夹杂问题的有限部积分──边界元法

利用Kelvin解及有限部积分的概念和方法,导出求解含夹杂二维有限弹性体的超奇异积分方程,继而使用有限部积分与边界元结合的方法,为其建立了数值求解方法,即有限部积分与边界元法.最后计算了若干典型数值例子夹杂端部的应力强度因子.      

正交各向异性厚板振动分析的边界积分方程法

本文采用正交各向异性厚板静力问题的基本解作为边界积分方程的核函数,利用加权残数法建立了正交各向异性厚板振动分析的边界积分方程。文中详细地讨论了边界积分方程的数值处理过程并给出了若干数值算例以论证本文方法的正确性。      

不连续位移超奇异积分方程法解三维多裂纹问题

本文采用Ｂｅｔｉ互等功定理，导出了三维不连续位移基本解的一般形式，然后以该基本解为核函数，建立了求解三维多裂纹问题的超奇异边界积分方程组，并采用三角形单元变换技术和有限部分积分的方法给出了超奇异积分的数值解法，引入非协调单元处理技术解决了法向不确定的角点问题。最后，由裂纹面间断位移可直接求得裂纹前沿任意点的应力强度因子     

论Helmholtz方程的一类边界积分方程的合理性

本文导出了Helmholtz 方程超定边值问题有解的一个充要条件,和用非解析开拓法证明了文[1]中的Helmholtz 方程在外域中的解的边界积分表示式的合理性,并将此类边界积分表示式推广用于带空洞的有限域。这样就比较严密而又浅近地证明了基于该表示式建立起来的间接变量和直接变量边界积分方程的合理性。     

横观各向同性结构边界元分析

本文利用边界单元法分析三维横观各向同性结构,引用三个位势函数并利用叠加原理导出了基本解,并且利用基本解的新型结构形式避免了在分界单元计算中所遇到的所谓“奇异积分”的问题。     

An immersed finite element method with integral equation correction

We propose a robust immersed finite element method in which an integral equation formulation is used to enforce essential boundary conditions. The solution of a boundary value problem is expressed as the superposition of a finite element solution and an integral equation solution. For computing the finite element solution, the physical domain is embedded into a slightly larger Cartesian (box‐shaped) domain and is discretized using a block‐structured mesh. The defect in the essential boundary conditions, which occurs along the physical domain boundaries, is subsequently corrected with an integral equation method. In order to facilitate the mapping between the finite element and integral equation solutions, the physical domain boundary is represented with a signed distance function on the block‐structured mesh. As a result, only a boundary mesh of the physical domain is necessary and no domain mesh needs to be generated, except for the non‐boundary‐conforming block‐structured mesh. The overall approach is first presented for the Poisson equation and then generalized to incompressible viscous flow equations. As an example of fluid–structure coupling, the settling of a heavy rigid particle in a closed tank is considered. Copyright © 2010 John Wiley & Sons, Ltd.     

An initial value method for dual integral equations

An initial value method is derived for a set of dual integral equations encountered in solving mixed boundary value problems in mathematical physics with a circular line of separation of boundary conditions. It is shown that the solution itself, not just a transform of the solution, of the dual integral equations satisfies a Fredholm integral equation. The initial value problem is derived from this Fredholm equation.     

A boundary integral technique for the numerical modelling of the air flow for the Aaberg exhaust system

A boundary integral technique has been developed for the numerical simulation of the air flow for the Aaberg exhaust system. For the steady, ideal, irrotational air flow induced by a jet, the air velocity is an analytical function. The solution of the problem is formulated in the form of a boundary integral equation by seeking the solution of a mixed boundary-value problem of an analytical function based on the Riemann–Hilbert technique. The boundary integral equation is numerically solved by converting it into a system of linear algebraic equations, which are solved by the process of the Gaussian elimination. The air velocity vector at any point in the solution domain is then computed from the air velocity on the boundary of the solution domain.     

美式期权价格的积分表示及其数值方法

本文利用Laplace变换方法得到带连续红利的美式看涨期权价格的积分表示,以及最优执行边界满足的一个非线性的第二类Volterra积分方程.然后用数值积分公式给出了积分方程的数值解,从而得到了带连续红利的美式看涨期权价格及其执行边界的数值解.     

A new boundary integral equation for notch problem of antiplane elasticity

In this paper the notch problem of antiplane elasticity is discussed and a new boundary integral equation is formulated. In the problem, the distributed dislocation density is taken to be the unknown function. Unlike the usual choice, the resultant force function is taken as the right hand term of the integral equation; therefore, a new boundary integral equation for the notch problem of antiplane elasticity with a weaker singular kernel (logarithmic) is obtained. After introducing a particular fundamental solution of antiplane elasticity, the notch problem for the half-plane is discussed and the relevant boundary integral equation is formulated. The integral equations derived are compact in form and convenient for computation. Numerical examples demonstrated that high accuracy can be achieved by using the new boundary equation.     

美式期权价格的积分表示及其数值方法(英文)

本文利用Lalplace变换方法得到带连续红利的美式石看涨期权价格的积分表示,以及最优执行边界满足的一个非线性的第二类Volterra积分方程。然后用数值积分公式给出了积分方程的数值觯,从而得到了带连续红利的美式看涨期权价格及其执行边界的数值解。In this paper, we apply Laplace transform to obtain an integral representation for the solution for American call options with continuous dividend, and get a nonlinear Volterra integral equation of the second kind for the optimal exercise boundary. Then we give the numerical solution to the integral equation using the quadrature formulae, and so get the numerical solution of the price of American call option with continuous dividend and the optimal exercise boundary.     

A complex variable boundary element method: Development

A generalized boundary integral equation method for the solution of the Laplace equation is developed based on the Cauchy integral theorem for analytical complex variable functions. Although the approach is complicated by the utilization of complex variable theory, the resulting model involves direct integration along straight-line boundary segments (elements) rather than using quadrature formulae that are required in current real variable boundary element formulations. Previously published boundary integral equation methods based on the Cauchy integral theorem are shown to be a subset of the generalized model theory developed in this paper.     

Boundary integral equation method for elastic beams in frictionless contact problems

A boundary integral equation algorithm for the contact analysis of elastic beams is presented in this paper. The analysis of this sort is complicated by the unknown and moving boundary points. The new algorithm incorporates the interface compatibility equation, derived from the principle of minimum potential energy, into the general boundary integral equation so that the locations of boundary points may be correctly identified. The incremental and iterative solution procedure is presented. Accuracy and efficiency of the new algorithm are demonstrated using examples whose classical solutions exists.     

A boundary integral approach to analyze the viscous scattering of a pressure wave by a rigid body

The paper provides boundary integral equations for solving the problem of viscous scattering of a pressure wave by a rigid body. By using this mathematical tool uniqueness and existence theorems are proved. Since the boundary conditions are written in terms of velocities, vector boundary integral equations are obtained for solving the problem. The paper introduces single-layer viscous potentials and also a stress tensor. Correspondingly, a viscous double-layer potential is defined. The properties of all these potentials are investigated.By representing the scattered field as a combination of a single-layer viscous potential and a double-layer viscous potential the problem is reduced to the solution of a singular vectorial integral equation of Fredholm type of the second kind.In the case where the stress vector on the boundary is the main quantity of interest the corresponding boundary singular integral equation is proved to have a unique solution.