开边界外调和方程的迦辽金边界元解法
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摘要
最典型也最简单的椭圆型偏微分方程是调和方程，又称Laplace方程。力学和物理学研究中的许多问题都归结为Laplace方程的边值问题。例如：弹性膜的平衡问题，稳定状态的热传导问题，不可压缩势流问题，静电场问题以及静磁场问题。这些问题虽然有完全不同的物理背景，却往往导致完全相同的数学表达式。本文利用Galerkin边界元方法对该问题加以研究，特别是针对边界是直线段或开弧段情况下无限域上的调和方程进行了研究。该问题属奇异边界问题，对应于裂纹、屏障等问题的数值模拟在实际应用的背景。因为把二维调和方程的边值问题转化为等价的边界积分方程时带有约束条件，用Galerkin边界元方法求解带约束条件的变分方程，在数值离散时，我们采用Lagrange乘子法处理约束条件。
Galerkin方法是基于变分原理基础上的一种把微分方程或积分方程转化为等价的变分方程，通过离散变分方程求原方程数值解的方法。本文把Laplace方程的边值问题转化为边界积分方程后，通过与边界积分方程等价的变分形式，采用线性单元，利用Galerkin边界元方法求解。在计算单元刚度矩阵时，对二重积分的第一重使用精确积分，第二重使用数值积分，从而有效的克服了奇异积分的计算。本文还把此方法推广到求解区域边界是直线段或者开弧段的情况，采用引入奇异边界单元的方法来模拟解在开边界端点附近的奇异性。最后利用Fortran Power Station4.0程序语言分别编写了用Galerkin方法求解区域边界为闭曲线及区域边界为直线段或开弧段Laplace方程的计算机程序，通过几个算例证明了该方法是有效的、是可行。而且通过数值实验也验证了Galerkin方法误差的理论结果。
The harmonic equation，or Laplace equation is a typical and simple elliptic partial differential equation. The mathematical model of many problems of mechanics and physics are reduced to boundary problems of Laplace equation. Such as: the balance of elastic membrane, the heat conduction of stationary state, the incompressible potential flow, the problem of electrostatic field and magnetostatics field. These problems have different physical nature, but can be reduced to the same mathematical expression. In this paper, we study the numerical solutions of these problems using Galerkin boundary element, especially for the problem with open boundary such as the problems exterior to an open segment or an open curve in the plane. Since the equivalent boundary integral equation for two dimensional Laplace equation has constraint condition. Lagrange multiplicator method is introduced in the numerical computation to release the constraint.
Galerkin method based on the variation principle is used to solve differential and integral equations. In this paper, 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 calculation of double singular integration is needed. The exactly integral formula is used in the inner integral expression, and the numerical integral formula is used in the outer integral expression. We extend this method to the problem of a domain which boundary is an open arc or an open segment. In this paper, we simulate the singularity of solution in extreme point of open arc or open segment by singular element. With Fortran Power Station 4.0, we make Galerkin boundary element method program for solving Laplace equation on region which boundary is a closed curve or an open arc, and the numerical experiments also prove this method is reliable. Last we test the error of Galerkin boundary element by numerical experimentation.
Galerkin方法是基于变分原理基础上的一种把微分方程或积分方程转化为等价的变分方程，通过离散变分方程求原方程数值解的方法。本文把Laplace方程的边值问题转化为边界积分方程后，通过与边界积分方程等价的变分形式，采用线性单元，利用Galerkin边界元方法求解。在计算单元刚度矩阵时，对二重积分的第一重使用精确积分，第二重使用数值积分，从而有效的克服了奇异积分的计算。本文还把此方法推广到求解区域边界是直线段或者开弧段的情况，采用引入奇异边界单元的方法来模拟解在开边界端点附近的奇异性。最后利用Fortran Power Station4.0程序语言分别编写了用Galerkin方法求解区域边界为闭曲线及区域边界为直线段或开弧段Laplace方程的计算机程序，通过几个算例证明了该方法是有效的、是可行。而且通过数值实验也验证了Galerkin方法误差的理论结果。
The harmonic equation，or Laplace equation is a typical and simple elliptic partial differential equation. The mathematical model of many problems of mechanics and physics are reduced to boundary problems of Laplace equation. Such as: the balance of elastic membrane, the heat conduction of stationary state, the incompressible potential flow, the problem of electrostatic field and magnetostatics field. These problems have different physical nature, but can be reduced to the same mathematical expression. In this paper, we study the numerical solutions of these problems using Galerkin boundary element, especially for the problem with open boundary such as the problems exterior to an open segment or an open curve in the plane. Since the equivalent boundary integral equation for two dimensional Laplace equation has constraint condition. Lagrange multiplicator method is introduced in the numerical computation to release the constraint.
Galerkin method based on the variation principle is used to solve differential and integral equations. In this paper, 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 calculation of double singular integration is needed. The exactly integral formula is used in the inner integral expression, and the numerical integral formula is used in the outer integral expression. We extend this method to the problem of a domain which boundary is an open arc or an open segment. In this paper, we simulate the singularity of solution in extreme point of open arc or open segment by singular element. With Fortran Power Station 4.0, we make Galerkin boundary element method program for solving Laplace equation on region which boundary is a closed curve or an open arc, and the numerical experiments also prove this method is reliable. Last we test the error of Galerkin boundary element by numerical experimentation.
引文
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