三维Helmholtz方程边值问题的新型边界积分-微分方程及其边界元解法

用双层位势表示的二维Neumann边值问题的边界归化方法,将原始问题归化为新型边界积分-微分方程,由此导出一种新的既能保持原始问题的自伴性,又具有可积弱奇性积分核的边界变分方程.本文将此法推广到三维Helmholtz方程Neumann边值问题,并给出最优能量模误差估计和内部最大模超收敛估计.     

求解二维多区域声波散射问题的边界元方法

对于多散射区域的声波散射问题的外Neumann边值问题,用单层位势来逼近每个散射域上的散射波,再利用位势理论的跳跃关系将问题转换为第二类边界积分方程组的求解问题,然后用Nystrom方法进行了求解.对多个随机散射区域的声波散射问题,数值例子体现了该求解方法的可行性和准确性.     

三维重调和方程的双方程边界积分方程法

以守恒积分为工具,推导了三维重调和方程的新的边界积分方程,所得出的新方程与传统的边界积分方程相比较,降低了奇异性,避免了传统边界元方法中的强奇异积分的计算.对不同边界都采用第二类积分方程,得到了三维重调和方程的双方程方法.     

混合边界下开弧外区域声波散射问题的数值解法

关于时间调和声波在一个无限长圆柱形导体上的散射,可以转化为R2中一段光滑开弧上的散射问题.利用单双层位势来逼近散射波,通过单双层位势在开弧两侧的跳跃关系建立了混合边界的积分方程组,然后对此方程组进行参数化和离散化,最终得到离散化后的积分方程组.此边界积分方程组的解是存在唯一的.     

弹性力学问题解唯一的边界积分方程

从积分方程式出发,应用基本解的特性分析,说明在力边值问题中,位移边界积分方程和面力边界积分方程的位移解不唯一.提出了位移解唯一的条件,建立了唯一解的位移边界积分方程和面力边界积分方程.实例计算结果表明唯一解的边界积分方程是有效的.     

POISSON方程新的边界积分方程

POISSON方程边界值问题边界元法所应用的边界积分方程,其类型,关于未知位势导数是第一类积分方程,关于未知位势是第二类积分方程。本本文从格林公式出发,通过建立位势的单、双场守恒积分公式,推导出POISSON方程新的边界积分方程,其类型与经典方程相反,关于未知位势是第一类积分方程,关于未知位势导数是第二类积分方程。     

边界元方法的抽象误差估计及其应用

§1.引言 边界元方法是近二十年来发展的一种求解偏微分方程的数值方法,其基本思想是:先利用Green公式或位势将区域上的偏微分方程转化成边界上的积分方程,此时偏微分方程的解由边界积分方程的解表出;然后数值求解边界积分方程,进而求得偏微分方程的近     

用单层位势求解尖角区域上的Dirichlet外问题

采用Kress变换以及处理第一类奇异核的积分方法,运用Nystrom方法利用单层位势求解尖角区域上的Dirichlet外问题.给出具体的算法和数值例子,通过数值例子可以看出用单层位势求解尖角区域上的Dirichlet外问题与用单双层结合求解所得的结果基本上一致,说明这种方法是有效的和可行的.     

求解二维多区域声波传播问题的一种边界元方法

针对多区域中声波的传播问题,其中每个散射区域的介质是相同的,将散射区域内的声波用一种单双层混合位势的形式来表示,再应用Green定理表示出外部介质区域中的声波,并形成相应的边界积分方程.如果区域个数为M时,传统的边界元方法最终将形成2M个边界积分方程并对应2M个未知函数,而本文的边界元方法最终只形成M个边界积分方程以及对应M个未知函数,从而使得求解的方程和未知数的个数都减少了一倍.最后,通过对数值算例的求解,验证了该方法的可行性及精确性.     

弹性薄板弯曲问题的等价的边界积分方程

用非解析开拓数学方法建立平面弹性薄板弯曲问题理论中具有间接变量的等价边界积分方程,并采用变分法进行了严格的说明.以往出现的三种间接变量边界积分方程,它们都不是等价的,对此我们进行了深入的讨论.     

Convergence of the Neumann Series in BEM for the Neumann Problem of the Stokes System

A weak solution of the Neumann problem for the Stokes system in Sobolev space is studied in a bounded Lipschitz domain with connected boundary. A solution is looked for in the form of a hydrodynamical single layer potential. It leads to an integral equation on the boundary of the domain. Necessary and sufficient conditions for the solvability of the problem are given. Moreover, it is shown that we can obtain a solution of this integral equation using the successive approximation method. Then the consequences for the direct boundary integral equation method are treated. A solution of the Neumann problem for the Stokes system is the sum of the hydrodynamical single layer potential corresponding to the boundary condition and the hydrodynamical double layer potential corresponding to the trace of the velocity part of the solution. Using boundary behavior of potentials we get an integral equation on the boundary of the domain where the trace of the velocity part of the solution is unknown. It is shown that we can obtain a solution of this integral equation using the successive approximation method.     

A new integral equation approach to the Neumann problem in acoustic scattering

We suggest a new approach of reduction of the Neumann problem in acoustic scattering to a uniquely solvable Fredholm integral equation of the second kind with weakly singular kernel. To derive this equation we placed an additional boundary with an appropriate boundary condition inside the scatterer. The solution of the problem is obtained in the form of a single layer potential on the whole boundary. The density in the potential satisfies a uniquely solvable Fredholm integral equation of the second kind and can be computed by standard codes. Copyright © 2001 John Wiley & Sons, Ltd.     

Integral equation methods for scattering by infinite rough surfaces

In this paper, we consider the Dirichlet and impedance boundary value problems for the Helmholtz equation in a non‐locally perturbed half‐plane. These boundary value problems arise in a study of time‐harmonic acoustic scattering of an incident field by a sound‐soft, infinite rough surface where the total field vanishes (the Dirichlet problem) or by an infinite, impedance rough surface where the total field satisfies a homogeneous impedance condition (the impedance problem). We propose a new boundary integral equation formulation for the Dirichlet problem, utilizing a combined double‐ and single‐layer potential and a Dirichlet half‐plane Green's function. For the impedance problem we propose two boundary integral equation formulations, both using a half‐plane impedance Green's function, the first derived from Green's representation theorem, and the second arising from seeking the solution as a single‐layer potential. We show that all the integral equations proposed are uniquely solvable in the space of bounded and continuous functions for all wavenumbers. As an important corollary we prove that, for a variety of incident fields including an incident plane wave, the impedance boundary value problem for the scattered field has a unique solution under certain constraints on the boundary impedance. Copyright © 2003 John Wiley & Sons, Ltd.     

Boundary Integral Solution of the Time-Fractional Diffusion Equation

Here we consider initial boundary value problem for the time–fractional diffusion equation by using the single layer potential representation for the solution. We derive the equivalent boundary integral equation. We will show that the single layer potential admits the usual jump relations and discuss the mapping properties of the single layer operator in the anisotropic Sobolev spaces. Our main theorem is that the single layer operator is coercive in an anisotropic Sobolev space. Based on the coercivity and continuity of the single layer operator we finally show the bijectivity of the operator in a certain range of anisotropic Sobolev spaces.      

Edge asymptotics for the radiosity equation over polyhedral boundaries

In the present paper we consider the radiosity equation over the boundary of a polyhedral domain. Similarly to corresponding results on the double‐layer potential equation, the solution of the second kind integral equation with non‐compact integral operator is piecewise continuous. The partial derivatives, however, are not bounded. In the present paper we derive the first term in the asymptotic expansion of the solution in the vicinity of an edge. Note that, knowing this term, optimal mesh gradings can be designed for the numerical solution of this equation. Copyright © 1999 John Wiley & Sons, Ltd.     

阻尼边界条件散射问题的数值解法

该文研究了光滑区域上二维Helmholtz方程阻尼边界条件外问题的数值解法, 应用单双层位势组合来逼近散射场, 因此积分方程中含有超奇异算子. 给出了超奇异算子的离散化方法, 在Holder空间中给出了误差估计和解析边界的收敛性分析. 最后针对该方法给出数值实例, 以表明该方法的有效性.     

On the global existence and uniqueness of solutions to the nonstationary boundary layer system

In this paper, we study the problem of boundary layer for nonstationary flows of viscous incompressible fluids. There are some open problems in the field of boundary layer. The method used here is mainly based on a transformation which reduces the boundary layer system to an initial-boundary value problem for a single quasilinear parabolic equation. We prove the existence of weak solutions to the modified nonstationary boundary layer system. Moreover, the stability and uniqueness of weak solutions are discussed.     

Numerical method for the solution of integral equations in a problem with directional derivative for the Laplace equation outside open curves

By using a simple layer potential and an angular potential, one can reduce the problem with a directional derivative for the Laplace equation outside several open curves on the plane to a uniquely solvable system of integral equations that consists of an integral equation of the second kind and additional integral conditions. The kernel in the integral equation of the second kind contains singularities and can be represented as a Cauchy singular integral. We suggest a numerical method for solving a system of integral equations. Quadrature formulas for the logarithmic and angular potentials are represented. The quadrature formula for the logarithmic potential preserves the property of its continuity across the boundary (open curves).     

A symmetric boundary element method for the Stokes problem in multiple connected domains

We consider a symmetric Galerkin boundary element method for the Stokes problem with general boundary conditions including slip conditions. The boundary value problem is reformulated as Steklov–Poincaré boundary integral equation which is then solved by a standard approximation scheme. An essential tool in our approach is the invertibility of the single layer potential which requires the definition of appropriate factor spaces due to the topology of the domain. Here we describe a modified boundary element approach to solve Dirichlet boundary value problems in multiple connected domains. A suitable extension of the standard single layer potential leads to an operator which is elliptic on the original function space. Copyright © 2003 John Wiley & Sons, Ltd.     

Finite part representations of hyper singular integral equation of acoustic scattering and radiation by open smooth surfaces

The Green’s function solution of the Helmholtz's equation for acoustic scattering by hard surfaces and radiation by vibrating surfaces, lead in both the cases, to a hyper singular surface boundary integral equation. Considering a general open surface, a simple proof has been given to show that the integral is to be interpreted like the Hadmard finite part of a divergent integral in one variable. The equation is reformulated as a Cauchy principal value integral equation, but also containing the potential at the control point. It is amenable to numerical treatment by conventional methods. An alternative formulation in the better known form, containing the tangential derivative of the potential is also given. The two dimensional problem for an open arc is separately treated for its simpler feature.