【摘要】：插值型無單元伽遼金比例邊界法(Interpolating Element-Free Galerkin Scaled Boundary Method,簡稱IEFG-SBM)是一種基于改進的插值型移動最小二乘法(Improved Interpolating Moving Least-Squares,簡稱IIMLS),結(jié)合了無單元伽遼金法(Element Free Galerkin Method,簡稱EFGM)與比例邊界法優(yōu)點的邊界型無網(wǎng)格法。該方法僅需在計算域的邊界上進行節(jié)點離散,減少了一維空間,且不需要邊界元法所需要的基本解;徑向上利用解析的方法進行求解,從而可獲得較高的精度。將這種方法應(yīng)用于彈性材料和壓電材料斷裂問題可以直接根據(jù)定義計算出裂紋尖端的強度因子。在改進的插值型移動最小二乘法中,不僅近似得到的試函數(shù)滿足插值性質(zhì),本質(zhì)邊界條件可以直接施加,而且權(quán)函數(shù)非奇異,避免了Lancaster提出的插值型移動最小二乘法因權(quán)函數(shù)奇異導(dǎo)致的計算不便。此外,相對于傳統(tǒng)的移動最小二乘法,改進的插值型移動最小二乘法計算形函數(shù)時待定系數(shù)減少了一個,從而可以提高計算效率,減小產(chǎn)生奇異矩陣的幾率。本文工作包括以下內(nèi)容:(1)建立彈性材料斷裂問題的基于改進的插值型移動最小二乘法的插值型無單元伽遼金比例邊界法,并推導(dǎo)了相應(yīng)的計算公式。通過編制Matlab程序計算了幾個典型的算例驗證了本文方法高精度和高效率的優(yōu)點。(2)在彈性材料斷裂問題的基礎(chǔ)上,將插值型無單元伽遼金比例邊界法應(yīng)用于壓電材料斷裂問題的計算中,并推導(dǎo)相應(yīng)的計算公式以及編制相應(yīng)的Matlab程序,從而更進一步驗證了該方法的有效性。(3)為進一步提高解的精度和網(wǎng)格劃分的靈活性,將結(jié)構(gòu)劃分為包含裂紋和不包含裂紋區(qū)域,建立了裂紋分析的插值型無單元伽遼金比例邊界法與有限元法(Finite Element Method,簡稱FEM)的耦合模型,并將其應(yīng)用于彈性和壓電材料斷裂問題,通過具體的算例驗證了該耦合方法的精確性與有效性。
[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.
【學(xué)位授予單位】：華東交通大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O241.8;O346.1

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