形状自由六面体低阶单元理论与应用研究

The performance of traditional finite elements is deeply affected by the quality of meshes, and mesh distortion usually leads to various numerical problems. However, high-quality hexahedral meshes can not be guaranteed for the complex three-dimensional structures. Furthermore, mesh distortion is also unavoidable in the simulation of large deformation situations. This project focuses on the sensitivity problem to mesh distortion of the 3D finite elements, intends to establish a novel shape-free low-order hexahedral element (8-node, 24-DOF), whose properties are independent of the distortion of mesh shape, and aims at effectively dealing with the complex 3D problems. This research can be treated as a beneficial supplement for the shortcomings and defects of traditional 3D finite elements. The innovations are as follows: 1. A shape-free low-order hexahedral element can be constructed by employing the novel natural coordinates and the generalized conforming method, and the difficulty of hexahedral mesh generation is expected to be solved; 2. Making use of the high performance of shape-free element under seriously distorted meshes, the sensitivity problem to mesh distortion in analysis of nonlinear problems, especially large deformation situations, may be solved; 3. The accuracy and efficiency of damage identification, such as locating damage position and quantification of severity, can be effectively improved by applying shape-free element, and finally the objective of enhancing computational performance from the element level can be achieved for the complex 3D problems.

传统有限单元的性能严重依赖于网格质量的优劣，网格畸变常常引起各种数值问题，而复杂三维结构又难以保证高质量的六面体网格，大变形模拟中网格形状的扭曲畸变更是无法避免的。本项目针对三维有限元的网格畸变敏感难题，意图发展精度高、且性能不受网格形状畸变影响的形状自由三维六面体低阶（8结点24自由度）单元模型，补充传统三维有限元的不足和缺陷，有效解决各类复杂三维问题。创新之处在于：1. 综合发挥新型自然坐标与广义协调法的各自优势，发展形状自由六面体低阶单元，解决三维六面体网格划分难题；2. 利用形状自由单元在严重畸变网格下依然可以保持高精度的优异性能，解决非线性问题尤其是大变形分析中的畸变敏感难题；3. 扩展形状自由单元在结构损伤识别分析中的应用，有效提高损伤定位、损伤程度识别等的精度和效率，实现从单元层次提升三维复杂问题的计算效能。

传统有限单元的性能严重依赖于网格质量的优劣，网格畸变常常引起各种数值问题，而复杂三维结构又难以保证高质量的六面体网格，大变形模拟中网格形状的扭曲畸变更是无法避免的。本项目针对网格畸变敏感难题，基于最小势能原理和三维斜坐标系，建立了新的三维六面体低阶（8结点24自由度）单元模型，该单元在处理弯曲问题时不出现各类数值问题，具有很强的抗畸变敏感性能，即使在严重畸变的网格下依然能够保持高精度，基本实现了单元“形状自由”的目标。此外，对带转角自由度三维六面体8结点单元的构造进行了探索研究。单元格式基于虚功原理，单元构造中的试探函数采用位移假设，位移场取带转角自由度的等参坐标位移场；测试函数采用应力假设，结合拟协调条件和直角坐标表示的满足所有控制方程的69项应力解析解对单元域内的应力场进行表达，由于采用了两套不同的插值函数，最终得到的单元的刚度矩阵为非对称的。该单元能够通过常应力/应变检验，但在计算弯曲问题中暴露出致命的缺陷，即对网格形状十分敏感。最后，针对单元在结构损伤分析中的应用，基于室内试验和声发射技术，开展了玄武岩纤维混凝土单轴受压损伤特性研究和玄武岩纤维改性沥青的断裂损伤特性研究，声发射参数能有效反映试验过程中材料的损伤特性，并表征损伤机理。