【作者】 周巍巍；

【导师】 高效伟；

【作者基本信息】 大连理工大学 ， 航空宇航科学与技术， 2015， 硕士

【摘要】 随着现今材料学科的大力发展和整体科技行业水平的提升,功能梯度材料作为一种材料性质随空间坐标呈梯度变化的复合材料,凭借其耐高温、耐腐蚀和高强度的特点得到众多学者的关注,并在工程中得到广泛应用。但在功能梯度材料的制备中,其内部结构不可避免地存在多重裂纹、孔隙和其他颗粒夹杂等缺陷,针对这一问题,人们提出使用多边形网格单元来模拟裂纹和孔隙等问断问题。相比传统的三角形和四边形单元,多边形网格单元的划分更加灵活,同等条件的精度更高,多边形单元不仅能够很好地模拟复杂的边界情况,并且能够很好地吻合以多边形为基本单胞的材料。但在有限元方法中,要想得到该类问题较为满意的结果,网格划分的工作量往往很巨大,且对多边形单元的积分计算有一定的难度。边界单元法在处理非均质和非线性问题时会有域积分出现在积分方程中,相比于将域积分转换成边界积分仅使用边界单元进行域积分计算的方法,用内部网格积分法计算域积分的方法具有计算速度快的优点。然而,目前还没有学者将多边形网格用于边界元法中域积分的计算中,其中的原因之一是多边形网格积分很难直接通过高斯积分来计算,而文献中常用的三角化积分法,需要先将多边形单元分解为多个三角形单元,这将带来额外的计算量与精度的损失。本文提出一种用多边形网格计算二维弹性力学问题边界元分析中域积分的新方法：首先推导变系数问题用规格化位移表述的边界-域积分方程,通过用有理函数插值,构造了由任意多边组成的多边形网格形函数,用于几何与物理量的插值；然后,用径向积分法将多边形域积分转换成沿多边形周边的线积分,从而解决了用统一方法计算各类多边形网格的积分问题；最后,基于所述理论和相关公式,编写了Fortran语言计算程序,通过三个具有代表性的数值算例验证了本文提出的方法的有效性。更多还原

【Abstract】 With the rapid development of modern materials science and improvement of the overall level of technology industry, Functionally Graded Materials (FGMs), as a new class of composite materials with the material properties varying gradually with respect to spatial coordinates, have attracted the attention of many scholars due to its high temperature resistance, corrosion resistance and high strength characteristics, and obtained extensive applications in the engineering. But in the industrial fabrication of functional gradient materials, inevitably there exist multiple cracks, voids and particle inclusions and other internal defects. In order to solve this problem, people proposed to use polygonal mesh elements to simulate the discontinuous problem in cracks and pores. Compared to the traditional triangular and quadrilateral elements, the discretization of polygonal mesh is more flexible and under the same conditions, the computational results are more accurate. Polygonal elements not only can well simulate the complex boundary geometry, but also can well fit the shapes of materials with polygonal elements as basic unit cells. But in the finite element method, to obtain a satisfactory result for these problems, the workload of generating elements is usually very huge and evaluation of related domain integrals over polygonal elements is sometimes difficult.When using the boundary element method (BEM) to solve non-homogeneous and nonlinear problems, there will be domain integrals appearing in the integral equations. Using the internal cell integration method to evaluate these domain integrals has the advantage of fast computation over the boundary element only based method established from the transformation of domain integral to boundary integral. However, there have been not yet people applying polygonal meshes in the domain integrals in BEM. One of reasons is that it is often difficult to evaluate integrals over polygonal meshes directly using the Gauss quadrature. As commonly discussed in references, the triangulation integration scheme induces additional computational costs due to the discretization of a polygonal mesh into a number of small triangles.In this paper, a new method is proposed to evaluate domain integrals in two dimensional elasticity BEM analysis using the polygonal mesh approach. Firstly, the boundary-do main integral equations with variable coefficients are derived, and shape functions for irregular polygonal meshes are constructed for interpolations of geometry and physical quantities based on the rational function interpolation. Then, the domain integrals over polygonal meshes are converted into line integrals along the polygon perimeter by employing the radial integration method, which results in a unified scheme to evaluate various polygonal mesh integrals. Finally, based on the proposed theory and derived formulation, a Fortran code is written and three representative numerical examples are analyzed to demonstrate the effectiveness of the proposed method in the paper.更多还原