形状自由的高性能有限元方法研究

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 with problems with complicated stress distributions, so, new simple models and strategies are planned to be established for the crack growth problem and the edge effect problem of plate. These contributions can be treated as beneficial supplements for the usual finite element method.

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

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。本研究组在前期成果的基础上，意图发展出不受严重网格畸变影响且保持高精度的形状自由有限元法，主要的创新性成果有：1）创立了基于位移解析试函数的形状自由非对称有限元方法，采用新型自然坐标求得了力学问题的基本解析解，并将其作为有限元的试函数，成功发展出低阶的平面四边形4结点8自由度实体单元和三维六面体8结点24自由度实体单元，可以同时精确通过任意网格下的常应变分片检验和畸变网格下的纯弯测试，破解存在近30年的MacNeal局限定理定义的限制，为发展抗畸变单元开辟了可能；2）利用应力函数的解析解和最小余能原理发展了形状自由的平面杂交应力函数元，包括实体元和断裂单元，对网格畸变极为不敏感，同时具有较高精度，突破了有限元计算精度对网格质量的依赖以及应力奇点计算难题，极大简化了模拟裂纹扩展的计算过程；3）提出基于位移函数解析试函数法和最小余能原理的形状自由中厚板问题杂交位移函数元方法，所发展的杂交位移函数板单元形状自由，可以在网格畸变到其他模型无法工作的情况下继续保持高精度结果，特别是所发展的边界效应特殊单元，用极少的单元就可以反应出剧烈变化的内力分布，破解中厚板边缘效应有限元计算的历史难题。项目相关的研究基础获得2013年国家自然科学奖二等奖；发表论文31篇，其中期刊论文24篇，SCI论文22篇，包括国际计算力学顶级期刊IJNME和CMAME发表8篇；组织国际学术交流和会议3次，在国内外学术会议上作邀请报告6次；主编SCI有限元专刊3辑。培养博士生8名，硕士生2名。项目负责人入选国际SCI知名期刊《Engineering Computations》编委，当选中国力学学会《工程力学》学报副主编和计算力学专业委员会委员。

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