形状自由的高性能有限元方法研究

项目名称： 形状自由的高性能有限元方法研究

项目编号： No.11272181

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 岑松

作者单位： 清华大学

项目金额： 78万元

中文摘要： 由于理论基础存在局限性，常规有限元方法的计算效果依赖于网格质量。而复杂情况下高质量的网格难以保证，形状畸变的单元常常导致各类数值问题发生。近期，本研究组分别基于最小余能原理和虚功原理，引入弹性力学平面问题应力函数的基本解析解作为试函数来发展新型有限元模型，既从理论源头消除了与网格形状相关的因素，又吸收了基本解析解的优势，成功构造了几种对网格严重畸变免疫的高阶杂交应力函数元和非对称元，称为"形状自由"有限元。本项目意图继续深化发展这一成果，探索并初步建立一种高精度、且精度不受网格形状影响的"形状自由"高性能有限元方法体系，包括完善和发展"形状自由"的高性能平面、轴对称、三维、板壳单元系列及其线性、非线性算法等。此外，此类方法对应力奇点、板壳边缘效应等复杂应力问题也有独到优势，可对裂纹扩展、板壳边缘效应等应力剧烈变化问题建立特殊的有限元模型和算法。该研究是对现行有限元方法体系的一种有益的补充

中文关键词： 有限元；形状自由；网格畸变；基本解析解；非对称

英文摘要： Due to the theoretical limitation, the performance of the conventional finite element method deeply depends on the quality of the mesh. However, mesh quality may be very low under complicated geometry or large deformation situation, which will lead to various numerical problems. Recently, based on the principle of the minimum complementary energy and the virtual work principle, respectively, two kinds of high-order "shape-free" plane element models, named hybrid stress-function element and improved unsymmetric element, were successfully developed by introducing the fundamental analytical solutions of the stress function as the trial functions. These shape-free models are immune to severe mesh distortions because they have eliminated the factors that cause the sensitivity problem to mesh distortion from the outset, and possess the advantages from both analytical and discrete methods. In this project, further studies on the shape-free finite element method will be performed. New kinds of shape-free high-performance finite element method whose performances are independent to element shapes, including linear and nonlinear element models of plane, axial symmetry, 3D, plate and shell, etc., are expected to be systematically developed. Furthermore, it is found that above methods also possess advantages when dealing wit

英文关键词： finite element；shape-free；mesh distortion；fundamental analytical solution；unsymmetric