【作者】 李茂军；

【导师】 祝家麟；

【作者基本信息】 重庆大学 ， 计算数学， 2009， 硕士

【摘要】 边界元法(BEM)是一种应用广泛的求解偏微分方程的方法,它具有精度高,降维等特点。无网格局部Petrov–Galerkin (MLPG)法是一种很受关注的数值方法,适合于求解非齐次,非线性,各向异性等问题。本文首先将无网格局部Petrov-Galerkin法和改进的移动最小二乘近似相结合,形成了改进的无网格局部Petrov-Galerkin法,并求解了二维类Helmholtz方程。在无网格局部Petrov-Galerkin法中,移动最小二乘近似被用来构造近似函数,在移动最小二乘近似中的代数方程组有时是病态的。因此改进的移动最小二乘近似被提出,改进的移动最小二乘近似采用加权正交函数系作为基函数,与传统的移动最小二乘近似相比,改进的移动最小二乘近似中的系数矩阵变成了非奇异的对角矩阵,因而无需计算系数矩阵的逆。数值算例的研究结果均表明改进的无网格局部Petrov-Galerkin法精度高,收敛速度快。无网格局部Petrov-Galerkin法是一个真正的无网格方法,它不需要单元或网格,但是它的计算量比有限元和边界元都大。因此本文基于边界元法和无网格局部Petrov-Galerkin法提出了一种直接耦合法,该方法将问题区域分解为不相重叠的边界元子域和无网格子域,连续性条件在两子域的公共边界上得到满足。然后将边界元方程、无网格方程以及连续性条件耦合成最终的方程组。在不同的子区域划分模式下讨论了该方法,一些数值算例被给出,证明了该方法的有效性。耦合法需要将边界元方程和无网格方程联立在一起,形成一个统一的大型方程组,因此本文又研究了基于边界元法和无网格局部Petrov-Galerkin法的区域分解法,该方法也将问题区域分解为不相重叠的边界元子域和无网格子域,连续性条件要在两子域的公共边界上得到满足,必须通过迭代程序。为了加速收敛,引进了固定松弛因子和动态松弛因子。然后通过丰富的数值算例详细讨论了两种松弛因子对迭代次数的影响以及公共边界上的初始值对迭代次数的影响。更多还原

【Abstract】 The boundary element method (BEM) is a widely used method because of its accuracy and dimensionality reduction. The meshless local Petrov–Galerkin (MLPG) method is a promising meshless method for solving partial differential equations. It is well-suited to problems involving non-homogeneous, anisotropic and non-linear problems.Firstly, the MLPG method and improved MLS (IMLS) approximation are combined, and an improved MLPG (IMLPG) method for two-dimensional analogical Helmholtz equation is discussed in this paper. In the MLPG method, the MLS approximation is used to obtain the approximation function. The algebra equation system in MLS approximation is sometimes ill conditioned. So a new method to establish the approximation function, IMLS approximation, was presented. In IMLS approximation, an orthogonal function system with a weight function is used as the basis function. In comparison with MLS approximation, the algebra equation system in IMLS approximation is not ill conditioned, and can be solved without having to obtain the inverse matrix. In our numerical tests, the numerical convergence of IMLPG method is studied and accurate results are obtained.Secondly, The MLPG method is a genuine meshless method which does not need‘‘elements’’or‘‘mesh,’’but uses a distributed set of nodes for both field interpolation and background integration. Although the MLPG method has many advantages, the method is more computationally expensive than the finite element method (FEM) and the boundary element method (BEM). So a coupled BEM and MLPG method for analyzing two-dimensional potential problems is presented in this paper. In this method, the analysis domain is divided into two non-overlapping regions (BEM region and MLPG region). The proposed coupling method directly couples the two methods without transition region, the continuity conditions are satisfied on the interface of the two sub-regions. The final system of algebra equation is composed of the BEM equation, the MLPG equation and the equation formed by the continuity conditions. This coupling method is numerically treated under several subdomain models. Some numerical examples for the potential problems governed by the Laplace and Poisson equations are presented to evaluate the accuracy and efficiency of the proposed technique. Thirdly, the coupling method must employs an entire unified equation for the whole domain, by combining the discretized equations for the BEM and MLPG sub-domains, so a non-overlapping domain decomposition algorithm based on the BEM and MLPG method is presented in this paper. This algorithm is iterative in nature. It essentially involves subdivision of the problem domain into subregions being respectively modeled by the two methods, as well as restoration of the original problem with continuity and equilibrium being satisfied along the interface. To speed up the rate at which the algorithm converges, static and dynamic relaxation parameters are employed. The validity of the algorithm is verified by solving some potential problems.更多还原