稳定广义有限元法的研究与若干典型工程应用

项目名称： 稳定广义有限元法的研究与若干典型工程应用

项目编号： No.11471343

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 张庆辉

作者单位： 中山大学

项目金额： 60万元

中文摘要： 广义有限元法（GFEM）和拓展有限元法（XFEM）是最近二十多年新兴的数值模拟技术。两者可以用单元分解（PU）的概念统一起来，只是在整体或局部富集上有差别，文中统一用GFEM来代替二者。GFEM对规则网格上的有限元富集（附加）一些可以表征真解局部特征的基函数，来克服传统有限元网格划分的困难和不能有效模拟非光滑特征的缺点。多数情况下，富集函数与有限元PU函数具有较强的线性相关性，导致GFEM刚度矩阵的条件数很高、甚至是奇异的。本项目致力于稳定GFEM的研究，主要想法是保持传统GFEM网格与逼近性优势的同时，将其刚度矩阵条件数控制到与有限元同阶。我们将稳定GFEM应用到四类典型非光滑性问题，裂纹、界面、奇性和高阶问题。我们还研究数值积分、本质边界条件、blending元等GFEM执行过程中的难题。最后，我们将评价稳定GFEM与有限元等方法的优缺点，识别稳定GFEM优于有限元的工程问题类别。

中文关键词： 广义有限元法；稳定性分析；逼近阶；断裂力学问题；界面接触问题

英文摘要： Generalized finite element methods (GFEM) and extended finite element methods (XFEM), developed in last two decades, are novel technique of numerical simulation. Two methods can be unified under concept of partition of unity (PU), and differ in global or local enrichments only. We represent these two methods in terms of GFEM in this proposal. It is well-known that conventional finite element methods (FEM) suffer from mesh generation and can not simulate non-smooth features of real solution efficiently. GFEM enrich FEM on uniform mesh with functions characterizing the local feature of the real solutions to overcome these difficulties. In most situations, the enrichments and FEM PU functions have strong linear dependence, and cause bad conditioned and even singular stiffness matrices. This project is devoted to developing the stable GFEM, and the main idea consists in both retaining the same advantages as the conventional GFEM in uniform mesh and approximation properties and making their conditioning not worse than that of the standard FEM. We apply the stable GFEM in four classes of typical non-smooth problems: crack, interface, singular, and high-order problems. We also study difficulties of the stable GFEM in computation, such as numerical integration, essential boundary condition, blending element, and so on. Eventually, we will set up benchmark problems to evaluate performance of the stable GFEM and FEM, and identify the types of the engineering problems where the stable GFEM excel FEM.

英文关键词： generalized finite element method;stability analysis;approximation order;crack problem;interface problem