基于无网格伽辽金法（EFG）在线弹性断裂分析中的求解与应用

发布时间：2018-08-01 11:04

【摘要】：无网格(meshless method)法作为近十年来迅速兴起的一种数值分析方法,它以其新颖的数值思想和先进的数值技术,得到了学术界的初步认可和广泛关注。不同于有限元法,无网格法在构造形函数时,是建立在一系列离散的点上,不需要借助网格,消除了节点之间的网格束缚,减少了因网格畸变而带来的困难,从而摆脱了有限元繁杂的单元网格生成过程。在分析裂纹等不连续问题时,无网格法具有独特的优势,其中无网格伽辽金法(EFG)法,以其精度高、稳定性好和收敛速度快等特点成为最具发展前景的一种,本文采用这种方法对线性问题和断裂问题进行了研究。(1)以线弹性力断裂学为分析手段,总结并概括了连续介质和非连续介质的断裂数值方法,阐述了国内外无网格法的研究进展。(2)基于加权残量法基本理论,论述了无网格法的几种常见的插值方法并分析了各自的优缺点,对插值方法中移动最小二乘法(MLS)进行了重点的阐述。在此基础上,对形函数里面的矩阵存在奇异性情况,给出了解决方案,对不同种类的权函数进行数值计算和分析。(3)根据线弹性断裂力学,从应力强度因子和能量释放率两个方面对裂纹体的力学行为进行描述,基于无网格伽辽金法(EFG)并对其进行改进,分析了线弹性断裂力学中应力强度因子的数值求法并在程序中成功实现。(4)对EFG法中处理不连续问题的四大类数值方法进行了系统而全面的总结和分析,这些数值方法为不连续问题的选取准则提供了参考依据。本文的创新在于对扩充形函数进行分析,改进了EFG法并建立了基于单位分解思想的无网格伽辽金法的近似位移函数,详细阐述程序设计中的一些关键问题,制定程序流程图,并给出相应的主程序。(5)基于MATLAB软件,编写了改进无网格伽辽金法计算程序,计算了MLS法近似函数,有效的分析了线弹性问题即一维杆件的计算和二维悬臂深梁的求解,验证了形函数的性质及参数的变化规律;并通过程序计算了断裂问题中的应力集中现象及应力强度因子的计算,与扩展有限元法作对比得到了较高的近似精度,讨论了权函数、节点分布密度和影响域扩大系数对求解精度的影响及变化规律,通过算例验证本文编写计算程序的准确性。
[Abstract]:Meshless method (meshless) method, as a numerical analysis method which has been rising rapidly in the past ten years, has gained the initial recognition and widespread concern in the academic circle with its new numerical ideas and advanced numerical techniques. Grid, which eliminates the constraint of grid between nodes, reduces the difficulty caused by grid distortion, and gets rid of the complex element mesh generation process of finite element. In the analysis of the discontinuity problems such as cracks, the meshless method has unique advantages, in which the EFG method is high in precision, good in stability and fast in convergence. This method is the most promising one. In this paper, this method is used to study linear problems and fracture problems. (1) the fracture numerical method of linear elastic force is used to sum up and summarize the fracture numerical methods of continuous medium and discontinuous medium, and the research progress of the meshless method at home and abroad is expounded. (2) based on the weighted residual method In this theory, several common interpolation methods of meshless method are discussed and their advantages and disadvantages are analyzed, and the moving least square method (MLS) in the interpolation method is emphatically expounded. On this basis, the singular situation of the matrix in the shape function is given, and the solution is given, and the numerical calculation and classification of different kinds of weight functions are given. (3) according to the linear elastic fracture mechanics, the mechanical behavior of the cracked body is described from two aspects of the stress intensity factor and the energy release rate. Based on the meshless Galerkin method (EFG) and improving it, the numerical method for the stress intensity factor in the linear elastic fracture mechanics is analyzed and successfully realized in the program. (4) the treatment of the EFG method is not good. Four kinds of numerical methods for continuous problems are systematically and comprehensively summarized and analyzed. These numerical methods provide reference for the selection criteria of discontinuous problems. The innovation of this paper is to analyze the extended shape function, improve the EFG method and establish the approximate displacement function of the element free Galerkin method based on the unit decomposition idea. Several key problems in the program design are elaborated in detail, the program flow chart is formulated and the corresponding main program is given. (5) based on the MATLAB software, an improved grid free Galerkin method is compiled and the approximate function of the MLS method is calculated. The linear elastic problem, that is, the calculation of the linear elastic problem, and the solution of the two-dimensional cantilever deep beam, is effectively analyzed. The properties of the shape function and the change law of the parameters, and the calculation of the stress concentration and the stress intensity factor in the fracture problem are calculated by the program. The higher approximate accuracy is obtained by comparing with the extended finite element method. The influence of the weight function, the distribution density of the node and the enlargement coefficient of the influence domain on the solution accuracy are discussed. An example is given to verify the accuracy of the program written in this paper.
【学位授予单位】：河南理工大学
【学位级别】：硕士
【学位授予年份】：2014
【分类号】：TU312.3

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