基于边界积分方程的Galerkin无网格方法
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摘要
无网格方法在近年来得到广泛关注,其基本特点是场函数建立在独立的节点上,节点之间无需网格联接。边界积分方程能使所考虑问题的维数降低一维,是求解线性问题和外部问题的一种有效工具。基于边界积分方程的无网格方法是无网格方法的一个重要分支。
本文首先回顾了无网格方法的发展历史和研究现状,综述了无网格方法数学理论的研究进展,介绍了无网格方法的基本原理,总结了无网格方法的特点、优越性以及目前无网格方法的难点和存在的问题。然后在大量前人工作的基础上,提出了一种新的基于边界积分方程的Galerkin无网格方法——Galerkin边界点法,并成功地将其应用于位势问题、弹性力学问题和流体力学问题的求解。
在Galerkin边界点法中,首先将边值问题归结为边界积分方程的弱形式或变分公式,然后利用移动最小二乘近似构造变分公式中的试探函数和检验函数,进而得到近似的有限维逼近空间。Galerkin边界点法利用了移动最小二乘近似的无网格思想和边界积分方程的降维特性,因此它的输入数据只是求解域边界上的离散分布的点。由于引入了变分公式,Galerkin边界点法能保持变分问题的对称性和正定性,该性质使得Galerkin边界点法是耦合有限元方法或者其它已经建立的区域型无网格方法(如无单元Galerkin法)的理想方法,这种耦合技术非常适合求解无限域问题。另外,虽然用移动最小二乘近似构造的形函数不具有插值特性,但是通过把边界函数与检验函数相乘并在边界上积分,Galerkin边界点法中的边界条件仍能容易地精确满足。
本论文针对Galerkin边界点法进行了理论分析和数值应用,具体研究工作如下:
首先研究了任意边界上的移动最小二乘近似,给出了移动最小二乘近似的性质。当节点和权函数满足一定条件时,证明了移动最小二乘近似的近似函数在Sobolev空间中的最优误差估计。误差结果表明,移动最小二乘近似的逼近误差与节点间距密切相关。
其次给出了Galerkin边界点法求解作为拟微分算子方程的边界积分方程的算法。边界积分方程首先被转化为相应的变分形式,然后用移动最小二乘近似的形函数构造解空间。为了计算积分,我们在边界上构造了背景网格。基于移动最小二乘近似的误差公式和拟微分算子理论,推导了用Galerkin边界点法求解边界积分方程的解的误差公式。从误差分析的过程中可以看出,Galerkin边界点法的误差主要来自两个方面:一是用背景网格上的积分去近似边界积分;二是用移动最小二乘近似去逼近边界变量。我们还考虑了边界积分方程的未知量需要满足一定约束条件的情形,此时我们采用Lagrange乘子放松这个约束,并给出了相应的数值实施过程和误差估计。
第三,用Galerkin边界点法求解了Laplace问题、双调和问题、线弹性问题和Stokes问题。把边值问题归化为等价的第一类边界积分方程,再用Galerkin边界点法进行求解。我们给出了数值求解过程,并就一般情形详细地进行了误差分析,得到了最佳渐进误差估计。当积分用的背景网格和边界重合时,我们进一步得到了近似解的能量模估计。数值算例表明了这种方法的有效性,并且数值结果和理论分析吻合。
The meshless (or meshfree) methods have drawn considerable attention in recent years. The main feature of this type of method is the absence of an explicit mesh, and the approximate solutions are constructed entirely based on a cluster of scattered nodes. Boundary integral equations (BIEs) are attractive computational techniques for linear and exterior problems as they can reduce the dimensionality of the original problem by one. The BIEs-based meshless method is an important branch of meshless methods.
This dissertation first reviews the recent developments of the meshless methods by means of their discretization scheme, together with some comments on the advantages and also some existing problems to be further studied. The developments of mathematical theory of the meshless methods are introduced in detail, and also the basic theory is expatiated thoroughly. Then, based on the large amount of work by the pioneers, a new BIEs-based meshless Galerkin method----a Galerkin Boundary Node Method (GBNM) is proposed and applied successfully to solve problems in potential theory, linear elasticity and fluid mechanics.
In the GBNM, an equivalent variational form of a BIE is used for representing the solution of boundary value problems, and the moving least-squares (MLS) approximation is employed to construct the trial and test functions of the variational form by a cluster of scattered boundary nodes instead of boundary elements. With the help of the MLS scheme, the GBNM is a boundary-type meshless method which requires only a nodal data structure on the bounding surface of the domain to be solved. A key advantage with the variational formulation is the GBNM can keep the symmetry and positive definiteness of the variational problems, a property that makes the method an ideal choice for coupling the finite element method or other established domain-type meshless methods such as the element-free Galerkin method for the problems with an unbounded domain. Besides, via multiplying the boundary function by a test function and integrating over the boundary, the boundary conditions of the original problem in the GBNM can be implemented exactly despite the MLS shape function lacking the delta function property.
The current dissertation concentrates on the theoretical analysis and numerical applications of the GBNM. The main contents of this thesis are as follows.
The first is the MLS approximation of a function on a generic boundary. The properties of the MLS approximation are studied comprehensively. Error estimates for the MLS approximation are deduced in Sobolev spaces when nodes and weight functions satisfy certain conditions. From the error analysis of the MLS scheme, we show that the error bound is directly related to the nodal spacing.
The second is the GBNM for a general BIE which can be characterized as pseudo-differential operator equation. The BIE is firstly converted into a variational formulation, and then the MLS shape functions are used to generate the approximate space. A background cell structure is constructed for purposes of numerical integration. Based on the error estimates of the MLS approximation and the pseudo-differential operator theory, error estimates of the solution of the integral equation are established in Sobolev spaces. From the error analysis, it is shown that the error results mainly from the approximation of the boundary by the cell structure, on which the numerical integration to be carried out, and approximation of the boundary variable by the MLS scheme. When the BIE has some constraints, a Lagrangian multiplier can be introduced in the process of numerical approximation. The corresponding numerical implementation and error analysis are also provided.
The third is the applications of the GBNM for the numerical solution of Laplace problems, biharmonic problems, linear elastic problems and Stokes problems. These boundary value problems are firstly reformulated as BIEs of the first kind, then the GBNM is used for obtaining the approximate solutions. The numerical implementations for these problems are described in depth. Total details of error analysis are given, and optimal asymptotic error estimates are obtained. When the integration cell structure is identical with the boundary, the estimates of the error in energy norms are also established. Numerical examples are presented to show the efficiency of the GBNM, and the numerical results are in consistency with the theoretical analysis.
本文首先回顾了无网格方法的发展历史和研究现状,综述了无网格方法数学理论的研究进展,介绍了无网格方法的基本原理,总结了无网格方法的特点、优越性以及目前无网格方法的难点和存在的问题。然后在大量前人工作的基础上,提出了一种新的基于边界积分方程的Galerkin无网格方法——Galerkin边界点法,并成功地将其应用于位势问题、弹性力学问题和流体力学问题的求解。
在Galerkin边界点法中,首先将边值问题归结为边界积分方程的弱形式或变分公式,然后利用移动最小二乘近似构造变分公式中的试探函数和检验函数,进而得到近似的有限维逼近空间。Galerkin边界点法利用了移动最小二乘近似的无网格思想和边界积分方程的降维特性,因此它的输入数据只是求解域边界上的离散分布的点。由于引入了变分公式,Galerkin边界点法能保持变分问题的对称性和正定性,该性质使得Galerkin边界点法是耦合有限元方法或者其它已经建立的区域型无网格方法(如无单元Galerkin法)的理想方法,这种耦合技术非常适合求解无限域问题。另外,虽然用移动最小二乘近似构造的形函数不具有插值特性,但是通过把边界函数与检验函数相乘并在边界上积分,Galerkin边界点法中的边界条件仍能容易地精确满足。
本论文针对Galerkin边界点法进行了理论分析和数值应用,具体研究工作如下:
首先研究了任意边界上的移动最小二乘近似,给出了移动最小二乘近似的性质。当节点和权函数满足一定条件时,证明了移动最小二乘近似的近似函数在Sobolev空间中的最优误差估计。误差结果表明,移动最小二乘近似的逼近误差与节点间距密切相关。
其次给出了Galerkin边界点法求解作为拟微分算子方程的边界积分方程的算法。边界积分方程首先被转化为相应的变分形式,然后用移动最小二乘近似的形函数构造解空间。为了计算积分,我们在边界上构造了背景网格。基于移动最小二乘近似的误差公式和拟微分算子理论,推导了用Galerkin边界点法求解边界积分方程的解的误差公式。从误差分析的过程中可以看出,Galerkin边界点法的误差主要来自两个方面:一是用背景网格上的积分去近似边界积分;二是用移动最小二乘近似去逼近边界变量。我们还考虑了边界积分方程的未知量需要满足一定约束条件的情形,此时我们采用Lagrange乘子放松这个约束,并给出了相应的数值实施过程和误差估计。
第三,用Galerkin边界点法求解了Laplace问题、双调和问题、线弹性问题和Stokes问题。把边值问题归化为等价的第一类边界积分方程,再用Galerkin边界点法进行求解。我们给出了数值求解过程,并就一般情形详细地进行了误差分析,得到了最佳渐进误差估计。当积分用的背景网格和边界重合时,我们进一步得到了近似解的能量模估计。数值算例表明了这种方法的有效性,并且数值结果和理论分析吻合。
The meshless (or meshfree) methods have drawn considerable attention in recent years. The main feature of this type of method is the absence of an explicit mesh, and the approximate solutions are constructed entirely based on a cluster of scattered nodes. Boundary integral equations (BIEs) are attractive computational techniques for linear and exterior problems as they can reduce the dimensionality of the original problem by one. The BIEs-based meshless method is an important branch of meshless methods.
This dissertation first reviews the recent developments of the meshless methods by means of their discretization scheme, together with some comments on the advantages and also some existing problems to be further studied. The developments of mathematical theory of the meshless methods are introduced in detail, and also the basic theory is expatiated thoroughly. Then, based on the large amount of work by the pioneers, a new BIEs-based meshless Galerkin method----a Galerkin Boundary Node Method (GBNM) is proposed and applied successfully to solve problems in potential theory, linear elasticity and fluid mechanics.
In the GBNM, an equivalent variational form of a BIE is used for representing the solution of boundary value problems, and the moving least-squares (MLS) approximation is employed to construct the trial and test functions of the variational form by a cluster of scattered boundary nodes instead of boundary elements. With the help of the MLS scheme, the GBNM is a boundary-type meshless method which requires only a nodal data structure on the bounding surface of the domain to be solved. A key advantage with the variational formulation is the GBNM can keep the symmetry and positive definiteness of the variational problems, a property that makes the method an ideal choice for coupling the finite element method or other established domain-type meshless methods such as the element-free Galerkin method for the problems with an unbounded domain. Besides, via multiplying the boundary function by a test function and integrating over the boundary, the boundary conditions of the original problem in the GBNM can be implemented exactly despite the MLS shape function lacking the delta function property.
The current dissertation concentrates on the theoretical analysis and numerical applications of the GBNM. The main contents of this thesis are as follows.
The first is the MLS approximation of a function on a generic boundary. The properties of the MLS approximation are studied comprehensively. Error estimates for the MLS approximation are deduced in Sobolev spaces when nodes and weight functions satisfy certain conditions. From the error analysis of the MLS scheme, we show that the error bound is directly related to the nodal spacing.
The second is the GBNM for a general BIE which can be characterized as pseudo-differential operator equation. The BIE is firstly converted into a variational formulation, and then the MLS shape functions are used to generate the approximate space. A background cell structure is constructed for purposes of numerical integration. Based on the error estimates of the MLS approximation and the pseudo-differential operator theory, error estimates of the solution of the integral equation are established in Sobolev spaces. From the error analysis, it is shown that the error results mainly from the approximation of the boundary by the cell structure, on which the numerical integration to be carried out, and approximation of the boundary variable by the MLS scheme. When the BIE has some constraints, a Lagrangian multiplier can be introduced in the process of numerical approximation. The corresponding numerical implementation and error analysis are also provided.
The third is the applications of the GBNM for the numerical solution of Laplace problems, biharmonic problems, linear elastic problems and Stokes problems. These boundary value problems are firstly reformulated as BIEs of the first kind, then the GBNM is used for obtaining the approximate solutions. The numerical implementations for these problems are described in depth. Total details of error analysis are given, and optimal asymptotic error estimates are obtained. When the integration cell structure is identical with the boundary, the estimates of the error in energy norms are also established. Numerical examples are presented to show the efficiency of the GBNM, and the numerical results are in consistency with the theoretical analysis.
引文
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