基于解析试函数的广义协调元

    本文研究基于解析试函数的广义协调元，探讨采用问题的解析基本解作为试探函数构造广义协调单元的方法。数值算例表明，基于解析试函数的广义协调元具有良好的性能，计算精度高，并较好地解决了有限元发展中的几个疑难问题。本文工作主要包括以下几个方面：
   1、根据分区混合能量原理构造了基于解析试函数的含裂纹应力元ATF-MS。分析了ATF-MS单元中应力项数与单元尺寸对计算结果的影响。根据数值实验经验给出了ATF-MS中应力项数和单元尺寸的合理取值。指出了该类单元存在多余零能模式，并且从理论上证明了在总体分析中即能自动消弭。这一结论从理论上保证了合理选择应力项数的灵活性，完善了分区混合元理论。
   2、采用解析试函数法构造了含切口单元ATF-VN。用此单元分析切口问题也非常有效。此外还分析了ATF-VN单元中应力项数与单元尺寸对计算结果的影响。给出了ATF-VN单元中应力项数和单元尺寸的合理取值。在Müller法的基础上提出了分区加速Müller法，利用分区加速Müller法计算了 V型切口平面与反平面特征根。其中，收边法是研究中首次发现的针对非线性特征根迭代算法的一个有效的预处理方法。
   3、在平面问题的讨论中，用平面问题的解析基本解构造出一组四结点四边形广义协调膜元：ATF-Q4、ATF-GCQ4、ATF-NCQ4，ATF- Q4θ。ATF-Q4、ATF-NCQ4和ATF-Q4θ都能破解MacNeal梁考题，消除梯形闭锁现象，对网格畸变极不敏感。这点是许多文献力图达到而未能达到的。这些成果的取得再一次显示出解析试函数广义协调元的优势。
   4、构造了基于解析试函数的广义协调厚板元ATF-PQ4、ATF-GCPQ4和薄板元ATF-BQ4、ATF-GCBQ4。这些单元精度高，且随着网格的加密能快速收敛到精确解。当问题退化到薄板结构时，厚板元ATF-PQ4、ATF-GCPQ4不出现剪切闭锁现象，也没有多余零能模式，无须进行缩减积分等处理，是一种完全的厚薄通用板单元。单元ATF-GCPQ4、ATF-GCBQ4对网格畸变极不敏感，具有良好的数值稳定性。
   5、针对构造基于解析试函数的广义协调元时的一些算例，本文对有限元的收敛性问题进行了探讨，提出了强式分片试验和弱式分片试验。


 Abstract
   The generalized conforming element based on analytical trial functions (the ATF element) is studied in this dissertation. It is concentrated on how to use the basic analytical solutions as the trial functions to formulate the generalized conforming element. Numerical examples show that the ATF elements exhibit excellent performance and some traditional difficult problems appeared in the development of the FEM are solved with them. The main contributions in this dissertation are as follows:
   1. Based on the sub-region mixed energy principle, a stress element with crack named ATF-MS is formulated. The factors that influence the precision of the solutions, such as the size of the element and the stress items'number in the element, are studied. The numerical test shows there exist a rational size and a rational stress item's number for this element. It is proved that spurious zero energy modes are present in the element but not present in the global analysis of the whole structure. So the stress items'number in this element can be more flexible. This study improves the theory of the sub-region mixed element.
   2. The ATF method is also applied to develop singular element in the V-notch analysis and an element with notch named ATF-VN is formulated. For the ATF-VN element, the size and the stress items'number also influence the precision of the solutions, so these two factors are studied and their optimal values are obtained. The sub-region accelerated Müller method is proposed from the Müller method, and used to analyze the eigenvalues of the plane and anti-plane problems of the V-notch. In the meantime, the boundary-withdraw method is also proposed, which is a new pretreatment method to the iterative solution of the nonlinear eigenvalue problems.
   3. In the plane problem, a group of generalized conforming elements is formulated by using the basic analytical solutions of the problem. They are named: ATF-Q4, ATF-GCQ4, ATF-NCQ4, ATF-Q4θ. ATF-Q4, ATF-NCQ4, and ATF-Q4θ can pass the T-shape locking of the MacNeal beam, and are not sensitive to the mesh distortion. It is what tried to realize but failed in many papers. These achievements show the advantage of the ATF element again.
   4. The thick plate elements ATF-PQ4, ATF-GCPQ4 and the thin plate elements ATF-BQ4, ATF-GCBQ4 are formulated with the ATF method. These elements are high-precision and converge to the exact solutions very fast with the mesh refinement. The thick plate elements ATF-PQ4 and ATF-GCPQ4 are free from shear locking. They also have not spurious zero energy, so technique of reduced integration is not necessary to these two elements. Thus, these two elements are the completely thick/thin plate elements. The ATF-GCPQ4 and the ATF-GCBQ4 are free from mesh sensitivity. Their solutions are high-precision and stable.
   5. Through some numerical examples, the convergence problem of the FEM, and two concepts: the strong patch test and the weak patch test are discussed.