单元性能对网格形状畸变的敏感问题一直是有限元研究中备受关注的重要课题之一，如何解决这个难题仍然具有很大的挑战性，尤其对于低阶单元而言。本文致力于发展“形状自由”的有限单元法，破解MacNeal局限定理的限制，在线弹性范围内构造性能不受网格形状优劣影响的高性能低阶单元，为最终解决有限元网格畸变敏感难题探索道路。主要工作如下： 一、推导了第二类四边形面积坐标QACM-II (S, T)表示的各向异性问题的纯弯曲状态解析解，以及三维斜坐标(R, S, T)表示的各向同性和各向异性问题的线性应力、应变和二次位移解析解。这些局部自然坐标表示的解析解与先进的有限元技术相结合，可以有效地避免单元对坐标方向的依赖性问题，并提高单元性能。 二、首次组合三种先进的单元技术，即非对称单元法、第二类四边形面积坐标法和解析试函数法，将QACM-II表示的纯弯曲状态解析解引入到试探位移中，构造了平面四边形4结点8自由度非对称单元US-ATFQ4。该单元能够严格地满足一阶位移问题（C0分片测试）和二阶位移问题（纯弯曲测试），对高阶问题能得到高精度的计算结果，克服梯形闭锁和体积闭锁，且对坐标旋转没有依赖性。单元受网格形状的影响较小，甚至在极端网格畸变下仍保持高精度，是一类“形状自由”的单元。该单元破解了MacNeal局限定理对平面低阶单元的限制，使发展抗畸变敏感的有限元模型成为可能。 三、将三维斜坐标、解析试函数法与非对称有限元法相结合，取斜坐标表示的位移解析解为试探位移，构造了六面体8结点非对称单元US-ATFH8。新单元不包含任何可调参数，可以看做平面单元US-ATFQ4的拓展，它们有着类似的性能，都对网格质量的优劣依赖性小。该单元同样破解了MacNeal局限定理对三维低阶单元的限制，推动了抗畸变敏感的高性能单元的发展。 四、基于最小余能原理和直角坐标下应力解析解，构造了二次多边形杂交应力函数单元HSF-AP-19。该单元结构简单，不需要复杂的域内位移插值，能够直接退化为三角形和四边形单元。无论网格形状为凸还是凹多边形，单元都能得到高精度的位移和应力结果，形状非常自由，解决了目前多边形单元构造困难问题。

The sensitivity problem to mesh distortion has been one of the most important research topics in the finite element methods for a long time, and how to solve this problem is still a challenge that remains outstanding, especially for the low-order element models. The purpose of this dissertation is to develop so-called “shape-free” finite element methods which can break through the limitation defined by MacNeal’s theorem, and formulate high-performance elements whose properties are not affected by the quality of meshes. These works will open a new way for solving the sensitivity problem to mesh distortion from outset. The main contributions are as follows: I. For plane anisotropic problems, two sets of analytical solutions for strains and displacements in terms of the second form of quadrilateral area coordinate QACM-II (S, T) for pure bending states are derived; for three-dimensional isotropic and anisotropic problems, nine sets of analytical solutions for linear stresses, linear strains and quadratic displacements in terms of 3D local oblique coordinates (R, S, T) are derived. Together with some advanced finite element techniques, these analytical solutions in terms of local natural coordinates can effectively solve the dependence on the coordinate rotation, as well as improve element performance. II. By combination of three advanced element techniques, including the unsymmetric element method, the second form of quadrilateral area coordinate method (QACM-II) and the analytical trial function method, a 4-node, 8-DOF unsymmetric plane quadrilateral element US-ATFQ4 is successfully constructed by taking the analytical solutions in terms of QACM-II for pure bending states as the trial functions of displacement fields. The new element can strictly satisfy both the classical first-order displacement test (C0 patch test) and the second-order test for pure bending, provide excellent results for high-order problems, overcome the trapezoidal and volumetric locking, and hold the invariance for the coordinate rotation. The performance of this element does not depend on the shape of meshes, and it can still keep high precision even when various severely distorted meshes are used. So, it is a kind of shape-free element. Especially, this element perfectly breaks through the limitation of MacNeal’s theorem for the plane low-order elements, which means that it is possible to make further efforts for developing distortion immune elements. III. A 3D 8-node hexahedral element US-ATFH8 is constructed by employing the unsymmetric element method, the analytical trial function method and the oblique coordinate method. In the formulations of this new 3D low-order element, the analytical solutions for displacements in terms of 3D local oblique coordinate are used as trail functions for displacement fields, and there is no any adjustable factor. It can be treated as an extension from the plane element US-ATFQ4, so that they have the similar performance and are both insensitive to various severe mesh distortions. This new 3D hexahedral element also breaks through the limitation of MacNeal’s theorem, and promotes the development of high-performance elements with high distortion resistance. IV. Based on the principle of minimum complementary energy and the analytical solutions in terms of Cartesian coordinate system, a quadratic polygonal element model HSF-AP-19? is developed by using the hybrid stress-function element method. The construction procedure of this new element is quite simple. It can be changed into triangular, quadrilateral and other polygonal elements directly without complex displacement interpolation functions. No matter the shapes of meshes are convex or concave polygons, the element can still obtain high-precision results for displacements and stresses. So, the shape of the element is quite free. It successfully solves the difficulty for the construction of polygonal elements.