基于EEP法的二维非线性有限元自适应求解研究

The self-adaptive analysis of Finite Element Method (FEM) for nonlinear problems is one of the central challenges over the research fields of computational mechanics and structural engineering. At present, the self-adaptive approach based on the Element Energy Projection (EEP) method has achieved full successes in the FE solution for various linear problems and successively in one-dimensional nonlinear problems as well, which has paved the way for its further application in various two-dimensional nonlinear problems. The present study is aimed at proposing a general and unified self-adaptive strategy for the FE solution of two-dimensional nonlinear problems. The basic idea is to introduce the present approaches directly into the nonlinear solution procedure by linearizing the original nonlinear problems into a series of linear ones, without the need for setting up a new set of formulas of super-convergent calculation and specific self-adaptive strategy for nonlinear problems. The main task of this study is to explore unified iterative schemes, establish complete algorithms, develop efficient computer codes, exploit effective practical applications, and establish new computational ideology. The expected goal is to be systematically successful in geometrically nonlinear problems, vibration and buckling problems and to make substantial breakthrough in material nonlinear problems and contacting problems. The proposed methodology in this study is expected to be characterized by its novel idea, advanced theory, efficient algorithm and reliable application, and hence possesses great potential of being a superior and competitive adaptive approach to nonlinear problems.

非线性问题的有限元自适应求解是当今计算力学和结构工程研究的前沿课题。申请人提出的基于超收敛EEP（单元能量投影）法的自适应分析方法在各类线性问题的有限元法中已获得广泛成功，继而对一维非线性问题也获得了成功的突破，这使得对二维非线性问题的挑战成为可能。本研究旨在基于EEP法构建一套一般的、统一的二维非线性有限元自适应求解算法：基本思路是通过对非线性问题线性化将现有算法直接引入非线性求解，而无需单独建立非线性问题的超收敛计算公式和自适应算法；主要任务是探讨统一迭代格式，构建成熟算法理论，研发高效代码程序，开拓有效实际应用，创立新型计算理念；预定目标是几何非线性和动力、稳定问题要系统成熟，材料非线性和接触非线性问题有重点突破。本研究所建立的方法可望具有思想新颖、理论先进、算法高效、可靠实用等特色，是一种颇具优势和竞争力的新式非线性问题求解方法。

由本课题组提出的有限元后处理超收敛计算的EEP（单元能量投影）法以及基于该法的自适应分析方法在二维线性有限元(FEM)和有限元线法(FEMOL)中已获得了广泛的成功，且对一维非线性问题也获得了成功的突破。在此基础上，本项目对二维非线性问题攻关研究，如期成功地提出了一套统一通用的基于EEP法的非线性自适应有限元求解方法。其基本思想是将非线性问题通过弱形式的牛顿法转化为一系列线性问题，直接引入现有的线性问题自适应求解方法，而无需单独建立非线性问题的超收敛计算公式和自适应算法；核心策略统一为非线性有限元解——超收敛解——网格细分这一“三步走”的策略。该法高效、稳定、通用、可靠，已成功应用于二维几何非线性问题、边界非线性问题、特征值问题等，颇具优势和竞争力。同时本研究对EEP超收敛算法进行了深入研究和探索，超前对三维非规则区域问题成功构建了有限元EEP超收敛计算方法，对二维问题局部加密网格下的超收敛和自适应分析也获得了初步成功，这些都为本研究提供了更为深远和宽广的发展应用前景。