二维自适应技术新进展：从有限元线法到有限元法  

NEW PROGRESS IN SELF-ADAPTIVE ANALYSIS OF 2D PROBLEMS:FROM FEMOL TO FEM

作　　者：袁驷[1] 徐俊杰[1] 叶康生[1] 邢沁妍[1] 

机构地区：[1]清华大学土木工程系,土木工程安全与耐久教育部重点实验室,北京100084

出　　处：《工程力学》2011年第A02期1-10,共10页Engineering Mechanics

基　　金：国家自然科学基金项目(51078199,50678093,50278046,51078198);长江学者和创新团队发展计划项目(IRT00736)

摘　　要：二维有限元线法(FEMOL)的自适应分析已经取得成功,而且表现出色。然而,为了进一步推广应用领域,提高效率和效能,将其先进的自适应技术在最常用的有限元法(FEM)当中实现,便成为必然追求。经过近年的研究,已经基本实现了二维自适应分析技术从FEMOL到FEM的跨越,该文意在对这方面的进展作一简要综述与报道。从FEMOL出发,继承单元能量投影(EEP)法这一超收敛计算的核心技术,该文提出"逐维离散,逐维恢复"的基本求解策略。超收敛计算方案和基于单元边线解答的均差法,巧妙化解了整套算法由FEMOL到FEM转化中的一系列难点,形成一套新型的二维FEM自适应分析技术。整套方法继承了FEMOL的优点,可以对任意几何区域上的问题,按最大模度量给出逐点满足给定误差限的位移解答,同时克服了FEMOL解析方向精度冗余的弱点,增强了灵活性,显著提高了求解效率。该文给出充足的数值算例用以展示整套算法的可靠性和高效性。Remarkable success has been made in the self-adaptive analysis of the 2D Finite Element Method of Lines （FEMOL）. However, in order to further expand the application areas and improve the computing power and efficiency, it has been needed to extend the advanced self-adaptive technology of FEMOL to the realm of the most-commonly-used Finite Element Method （FEM）. With recent intensive studying, the technology transfer from FEMOL to FEM in the self-adaptive analysis of 2D problems has initially and successfully been achieved. The present paper gives a brief overview and report about this advancement. Based on the concept of FEMOL and the Element Energy Projection （EEP） method for super-convergence computation, a ＂discretization and recovery by dimension＂ scheme was proposed. By using the proposed super-convergence computation scheme and an error-averaging method for mesh generation based on the solution on element edges, a series of difficulties in the ＂technology transfer＂ from FEMOL to FEM were overcome smartly, and as a result, a new type of adaptive analysis strategy for 2D FEM was proposed. Like the excellent performance in FEMOL, the algorithm could adaptively produce FEM results on arbitrary geometric domains with the displacement accuracy point-wisely satisfying the user specified error tolerance in max-norm. In addition, the algorithm overcomes the disadvantage of precision redundancy in the analytical direction of FEMOL, and significantly improves computational efficiency and enhances application flexibility. Numerous representative numerical examples were given to demonstrate the reliability and efficiency of the proposed algorithm.

关 键 词：有限元法 有限元线法 单元能量投影 超收敛 自适应 二维边值问题 

分 类 号：O241.82[理学—计算数学] TU311.4[理学—数学]