求解非线性偏微分方程是科学研究和工程计算中的重点与难点，通常采用数值方法，比如有限元法。而一个优化的网格往往能使得有限元法求解变得更加高效可靠，甚至对于某些问题而言这是一个基本的要求。自适应有限元法（Adaptive Finite Element Method）在传统有限元法的基础上，给出解的误差估计，结合网格划分技术最终提供一个优化的有限元网格和满足用户精度要求的解，其难点在于如何给出有效的误差估计和高效的网格划分技术。 单元能量投影（Element Energy Projection，简称EEP）法是一种有限元后处理超收敛计算方法，利用其优良特性， 可为自适应有限元分析提供可靠的误差估计。基于EEP法的自适应分析，已成功地应用于各类一维线性及非线性有限元问题，并被推广至二维、三维线性有限元分析中去。基于此，本文首次提出了一套基于EEP法的二维几何非线性有限元自适应分析求解方案，并成功应用于求解薄膜大挠度问题、Plateau问题、参数型极小曲面问题和中厚板弯曲大挠度问题等几何非线性问题。全文的主要工作总结如下：1. 提出了基于EEP法的二维几何非线性有限元自适应分析的基本策略。通过有机地结合非线性迭代技术、EEP超收敛计算方法和高效的网格细分技术，形成一套整体性的高性能求解算法。该算法以最大模为度量控制位移误差，给出满足用户精度要求的解答。 2. 建立了二维问题上基于弱形式的Newton迭代格式。基于弱形式，便于从微分方程的角度对问题性质进行分析以及后续EEP公式的推导，便于将精度较差的有限元二阶导项进行分部积分处理，避免迭代收敛较慢或发散。3. 建立了形式更为一般化的二维EEP超收敛计算公式。以往的二维EEP超收敛公式多以Poisson方程为例，非线性问题线性化后得到的微分方程形式一般较为复杂，需要对不同问题重新推导超收敛计算公式，并针对复杂的非齐次边界条件作相应的分部积分处理以提高超收敛解的精度。4. 提出了一种修正的Levenberg-Marquardt法。基于问题的弱形式，论述了Newton法在求解参数型极小曲面问题时失效的原因，并针对相应的难点，给出了一种修正的Levenberg-Marquardt法，可应用于求解像参数型极小曲面问题这样一类Jacobian矩阵奇异的非线性问题。
Solving nonlinear partial differential equations (PDEs) is the key and difficult point in scientific research and engineering computations. The analytic solution is generally hard to obtain and thus numerical approximation becomes an inevitable approach. The finite element method (FEM) is a powerful numerical method that has been widely used. An optimal mesh can make nonlinear finite element (FE) analysis more efficient and reliable, and it is very fundamental for some nonlinear problems involving large strains or localization. Adaptive finite element methods (AFEM) are now increasingly used in engineering computations, aiming to optimize the computation of certain physical quantities of interest by properly adapting the FE meshes. The two key points to AFEM are reliable error estimation and efficient mesh refinement technique.The element energy projection (EEP) method is an effective and reliable point-wise super-convergent displacement recovery strategy for linear FE analysis. Based on EEP method, a series of adaptive FE analyses have been developed and have successfully applied to various one-dimensional (1D) FE, two-dimensional (2D) and three-dimensional (3D) linear FE adaptive analyses. This dissertation proposes a nonlinear adaptive FE strategy for 2D geometric nonlinear problems based on EEP method, which is successfully applied to solving some problems of large deflection of membrane problem, Plateau’s problem, parametric minimal surface problem and large deflection problem of thick plate. The main research works in this dissertation are as follows:1. The basic adaptive strategy of 2D geometric nonlinear FEM based on EEP method is proposed. By organically combining nonlinear iteration methods, EEP technique and efficient mesh refinement technique, a high-performance algorithm is formed. The employment of the maximum norm makes point-wisely error control on the solutions possible and makes the proposed method distinguished from other adaptive FE analyses.2. The formulae of Newton’s method based on weak form for 2D problems are derived. At the level of weak form, it is convenient to judge the nature of the problem and to derive the subsequent EEP formulae. Also, it is convenient to take a treatment of integration by parts for the second-order derivative term when needed.3. A more general EEP formulae for 2D problems are derived, since the linearized PDEs for nonlinear problems in this research are more complicated. Besides, a treatment of integration by parts is taken to improve the accuracy of the super-convergent solutions for the problems under the non-homogeneous boundary conditions. 4. A modified Levenberg-Marquardt method is proposed. Based on the weak form of the problem, the difficulties inherent in parametric minimal surface problems and the reason for the failure of Newton's method are discussed. A modified Levenberg-Marquardt method is given, which can be expected to apply to such a class of nonlinear problems with singular Jacobian matrix.
