摘要
工程中的数值方法，如有限元法和边界元法等目前已取得了很大成功。但是，这些方法网格的形成和存在对其应用也造成了一定的困难。目前正在发展的无网格方法可以彻底或部分地消除网格，是当前科学和工程计算方法研究的热点，也是科学和工程计算发展的趋势。
本文针对移动最小二乘法存在的病态性、精度和效率问题，提出了复变量移动最小二乘法，其优点是其试函数中所含的待定常数减少了，可有效地提高求解效率；讨论了复变量移动最小二乘法的基函数，得到了基于正交基函数的复变量移动最小二乘法的公式，其优点是在构造形函数时不会形成病态方程组，求解精度更高；在此基础上提出了复变量无网格方法，具有配点少、精度高、计算速度快的优点；对复变量无网格方法，讨论了权函数、基函数、节点影响域大小和节点分布密度对其求解精度的影响，得到了新的结论；针对裂纹问题，本文提出了扩展的复变量无网格方法，利用裂纹尖端位移场的解析解来扩展复变量无网格方法的基函数，可以较好地解决含裂纹的问题，提高了复变量无网格方法求解裂纹问题的精度；针对无网格方法难以处理边界条件的问题，本文提出了复变量无网格方法与有限元法的耦合法，提出了新的耦合逼近函数，解决了以前相关的耦合方法存在的问题，可提高求解精度；本文用所提出的复变量无网格方法对混凝土构件进行了允许开裂的非线性分析，并提出了适用于在无网格方法中裂缝处理的点弥散域方法，计算结果表明用复变量无网格方法解决非线性混凝土问题是可行的。
西安理工大学博士学位论文
本文对复变量移动最小二乘法和复变量无网格方法的一系列创新性的研究工作，
将促进无网格方法的研究和发展，也为无网格方法的工程应用提供了更为有效的方
法。
关键词:复变量移动最小二乘法，复变量无网格方法，扩展的复变量无网格方法，复
变量无网格方法与有限元法祸合，非线性混凝土问题
才一、
The numerical methods in engineering such as FEM (finite element method) and BEM (boundary element method) have been successfully used. But the existence and formation of mesh in these methods also cause some difficulties when they are used. The meshless method being presently developed will wholly or partially eliminate the mesh. This method is the hot point and recent trend in the study of engineering calculation.With the problems like ill-conditioning, precision and efficiency involved in moving least-square approximation, the moving least-square approximation with complex variables is presented in this paper. The advantage of the moving least-square approximation with complex variables shows that the number of the undetermined constants in the trial function is decreased, therefore the solving efficiency is increased. Meanwhile, The basis function of the moving least-square approximation with complex variables is discussed in this paper, and the formula based on the orthogonal basis function for the moving least-square approximation with complex variables is obtained. And the advantage of the method reveals that the ill-conditioning equation system won't be formed when the shape functions are obtained. As a result, the solving precision gets much better. With this basis, the meshless method with complex variables is given in this paper, and this method has the following advantages such as less distribution nodes, higher precision as well as faster calculation. To the meshless method with complex variables, the influences of the weight function, basic function, the influence domain of the node as well as the distribution density of nodes to the
solving precision are studied in this paper, and some new conclusions are obtained. To the problem of crack, the enriched meshless method with complex variables is presented in this paper. By using the analysis solution of displacements at the tip of a crack, the basis function of meshless method with complex variables will be enriched, and the precision that the meshless method with complex variables used to solve the crack problems will be increased. Since the meshless method can't deal with the boundary conditions well, so the coupling of meshless method with complex variables and finite element method is presented in this paper. And the new coupling approximation function is discussed. The problems existed in the old coupling methods are solved, and the solving precision also will be increased. Moreover, The nonlinear analysis to the concrete member allowed to crack is carried with the meshless method with complex variables proposed in this paper. The method of nodal diffusion domain which is suitable for the treatment of cracks in the meshless method is shown. And the calculations prove that it is very well to deal with the nonlinear concrete problems by using meshless method with complex variables.A series of creative studies for the moving least-square approximation with complex variables as well as the meshless method with complex variables are carried out in this paper. These studies will promote the development of meshless method, and at the same time the study on the meshless method is available for the application in engineering.
引文
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