弹性和压电材料断裂问题的插值型无单元伽辽金比例边界法

发布时间：2019-04-01 22:20

【摘要】：插值型无单元伽辽金比例边界法(Interpolating Element-Free Galerkin Scaled Boundary Method,简称IEFG-SBM)是一种基于改进的插值型移动最小二乘法(Improved Interpolating Moving Least-Squares,简称IIMLS),结合了无单元伽辽金法(Element Free Galerkin Method,简称EFGM)与比例边界法优点的边界型无网格法。该方法仅需在计算域的边界上进行节点离散,减少了一维空间,且不需要边界元法所需要的基本解;径向上利用解析的方法进行求解,从而可获得较高的精度。将这种方法应用于弹性材料和压电材料断裂问题可以直接根据定义计算出裂纹尖端的强度因子。在改进的插值型移动最小二乘法中,不仅近似得到的试函数满足插值性质,本质边界条件可以直接施加,而且权函数非奇异,避免了Lancaster提出的插值型移动最小二乘法因权函数奇异导致的计算不便。此外,相对于传统的移动最小二乘法,改进的插值型移动最小二乘法计算形函数时待定系数减少了一个,从而可以提高计算效率,减小产生奇异矩阵的几率。本文工作包括以下内容:(1)建立弹性材料断裂问题的基于改进的插值型移动最小二乘法的插值型无单元伽辽金比例边界法,并推导了相应的计算公式。通过编制Matlab程序计算了几个典型的算例验证了本文方法高精度和高效率的优点。(2)在弹性材料断裂问题的基础上,将插值型无单元伽辽金比例边界法应用于压电材料断裂问题的计算中,并推导相应的计算公式以及编制相应的Matlab程序,从而更进一步验证了该方法的有效性。(3)为进一步提高解的精度和网格划分的灵活性,将结构划分为包含裂纹和不包含裂纹区域,建立了裂纹分析的插值型无单元伽辽金比例边界法与有限元法(Finite Element Method,简称FEM)的耦合模型,并将其应用于弹性和压电材料断裂问题,通过具体的算例验证了该耦合方法的精确性与有效性。
[Abstract]:Interpolation element-free Galerkin proportional Boundary method (IEFG-SBM) is a modified interpolation-based moving least Squares (Improved Interpolating Moving Least-Squares, (IIMLS),) combined with element-free Galerkin method (Element Free Galerkin Method,. Abbreviated as EFGM) and boundary meshless method with advantages of proportional boundary method. The method only needs to discretize the nodes on the boundary of the computational domain, which reduces the one-dimensional space and does not need the basic solution required by the boundary element method, and the analytical method is used to solve the problem in radial direction, so that the accuracy can be obtained. The strength factor of crack tip can be calculated directly according to the definition by applying this method to the fracture problems of elastic materials and piezoelectric materials. In the improved interpolation-type moving least squares method, not only the approximate trial function satisfies the interpolation property, the essential boundary conditions can be directly applied, and the weight function is not singular. The interpolation moving least squares proposed by Lancaster can avoid the computational inconvenience caused by the singularity of the weight function. In addition, compared with the traditional moving least squares method, the modified interpolation moving least square method can reduce the undetermined coefficients when calculating the shape function by one, which can improve the computational efficiency and reduce the probability of generating singular matrix. The main contents of this paper are as follows: (1) the interpolation element-free Galerkin proportional boundary method based on the improved interpolation moving least square method for elastic material fracture problems is established and the corresponding calculation formulas are derived. By programming Matlab program, several typical examples are given to verify the advantages of this method with high precision and high efficiency. (2) based on the fracture problem of elastic materials, the results show that the proposed method has the advantages of high accuracy and high efficiency. In this paper, the interpolation element-free Galerkin proportional boundary method is applied to the calculation of piezoelectric material fracture problem, and the corresponding calculation formula and the corresponding Matlab program are deduced. The effectiveness of the proposed method is further verified. (3) in order to further improve the accuracy of the solution and the flexibility of mesh generation, the structure is divided into two regions: one with and without cracks. The coupled model of interpolation element-free Galerkin proportional boundary method and finite element (Finite Element Method, (FEM) for crack analysis is established and applied to elastic and piezoelectric material fracture problems. The accuracy and effectiveness of the coupling method are verified by an example.
【学位授予单位】：华东交通大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8;O346.1

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