摘要
无网格方法与有限元法、边界元法等传统的数值分析方法相比具有许多突出的优点。近年来,国内外学者在无网格方法研究方面已经取得了许多具有开创性的重要成果。
无网格局部Petrov-Galerkin方法(简称MLPG法)是近几年发展起来的一种新的数值方法,由于它不需要任何有限元或边界元网格,不管这种网格是用于能量积分还是进行插值的目的,所以分析问题更显灵活和方便,被誉为是有发展前景的“真正的无网格方法”。近年来,Atluri等和龙述尧等在MLPG法及应用研究上取得可喜进展。在龙述尧等的工作基础上,本文提出Kirchhoff板问题的MLPG方法,进一步研究和发展了MLPG方法。
本文首先综述了无网格方法的发展历史和国内外的研究现状,对目前各种主要无网格方法进行了回顾和评价,总结了无网格法的特点、优越性以及目前无网格法的难点和存在的问题。综述中,特别地概述了目前板壳问题的无网格方法研究情况。然后介绍了基于Kirchhoff假设的板方程、解变量的移动最小二乘近似方法和数据点及函数拟合算例。
文中对Kirchhoff板静力问题提出了MLPG方法,通过对各向同性和各向异性板、弹性地基板等分别采用加权残值法在局部子域建立Kirchhoff板控制微分方程的等效积分对称弱形式,并对挠度变量采用移动最小二乘近似函数进行插值,使所有的积分都在规则形状的子域及其边界上进行。因为用移动最小二乘法来近似挠度变量,不容易直接施加本质边界条件,所以采用罚因子法施加本质边界条件。数值实施中对非对称线性系统求解采用了广义最小余量迭代算法。通过各种形状、不同支承及荷载的平板静力弯曲算例,探索了MLPG最优权函数支持域的形状和大小,检验了Kirchhoff板静力分析MLPG法的有效性和可行性。本文还提出了薄板稳定性或屈曲的局部Petrov-Galerkin方法,计算了各种不同形状和边界支撑的各向同性和各向异性板的稳定性系数。
对Kirchhoff板动力问题的分析,是在空间域上采用局部Petrov-Galerkin方法来离散和提出用虚节点值与实际节点值变换法处理本质边界条件的施加问题,时间域上的离散则采用Newmark-β方法,并采用子空间迭代法来分析板的固有振动。通过各向同性板和各向异性板的算例,研究了板的自振特性、强迫振动下的变形及内力响应、计算效率等问题。
通过本文的研究表明,无网格局部Petrov-Galerkin法不但能够求解二阶微分方程的边值问题,而且求解四阶偏微分方程的边值问题也很有效,也具有收敛快、稳定性好、对挠度和内力都具有精度高的特点;MLPG法具有无网格Galerkin法
Meshless methods possess many outstanding advantages over traditional numerical methods such as finite element method, boundary element method et al. A lot of the important pioneering effort has been done on meshless methods by scholars in a recent decade, and further investigations into many key problems of meshless methods should be made.The meshless local Petrov-Galerkin(MLPG) method is a new numerical technique presented in the recent years, for it doesn't need any finite element or boundary element meshes, no matter meshes for the use of energy integral or for the purpose of interpolation, it can analyze the problem flexibly and conveniently and is named as "a truly meshless method" with the great applied prospect. In the recent years, Atluri and Long SY et al have made a lot of investigations on the MLPG approach and its applications. On the basis of their work, applications of the MLPG method to the Kirchhoff plate are presented in this dissertation, which enriches theory of the MLPG and extends applications of the MLPG.At the beginning of the dissertation, recent developments of meshless methods are briefly reviewed. All kinds of meshless methods are reviewed and commented. Characteristics, advantages and disadvantages for a variety of meshless methods are pointed out. Among which, the research and development on meshless methods of plate and shell problems are reviewed in detail. Then, the basic equations of the plate based on the Kirchhoff hypotheses are formulated, and the moving least square (MLS) approximation for the solution variable of the Kirchhoff plate are presented.In the dissertation, the MLPG method to the static problem of the Kirchhoff plate is proposed. The symmetric weak forms of the equivalent integration equations to the governing differential equations for the isotropic, anisotropic plates and the plate on the elastic foundation in local subdomains are formulated by the weighted residual method (WRM). The deflection is interpolated by using the moving least squares (MLS) approximation. All integrals are carried out in subdomains and their boundaries with a regular shape. For the moving least square approximation does not possess properties of the Kronecker-delta function, the essential boundary condition can not be enforced directly, the penalty method is used to imposed the essential boundary condition. In the numerical implementation, the unsymmetrical linear system is solved by the iterative algorithm of the generalized minimal residual
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