高性能有限元结点位移精度修正技术的研究

Finite element method (FEM) of displacement type is one of the most popular and versatile numerical method in engineering computation. Nodal displacements are the direct results of finite element (FE) analysis and the accuracy of all other quantities, such as displacements or stresses at interior points of elements, super-convergent results and adaptive mesh refinements etc., are all dependent on the quality of nodal displacements. Thus, the highly efficient recovery technology for nodal displacements which can improve the overall performance, quality and accuracy of FEM, is a breakthrough direction of fundamentally importance for high-performance FEM, which is also the goal of this research project. The basic idea of this study is to derive the equivalent nodal force vector of residuals based on the superior features of Element Energy Projection (EEP) method, originally proposed by the applicant for super-convergence computation, without changing the FE mesh and the global stiffness matrix. Then only by a simple back-substitution step can the nodal displacements of higher-order accuracy be obtained. The main task of this study is to investigate super-convergence mechanisms and characteristics of the proposed methodology, establish efficient algorithms and exploit effective practical applications by means of both numerical experiments and theoretical proofs. The intended outcome is to gain successful applications in problems from 1D to 3D, from linear to non-linear and from conventional to adaptive FEM. The proposed methodology in this study is expected to be characterized by its novel idea, advanced theory, unique technique and efficient algorithm, and hence possesses enormous potential of being a new and original progress for high-performance FEM.

位移型有限元法是当今工程计算中应用最普遍最广泛的数值计算方法，其直接结果便是结点位移，其他诸如单元内部位移和应力、超收敛位移和应力以及自适应网格划分和求解等技术，都直接依赖于结点位移的精度。对有限元结点位移的精度进行高效修正以提升有限元整体性能、质量和精度，是高性能有限元分析的一个重要突破方向。本研究旨在尝试并实现这一突破，基本思路是利用申请人业已提出的超收敛计算的EEP（单元能量投影）法之优良特性，在不改变有限元网格及其整体刚度矩阵的情况下，导出残差的等效结点荷载向量，只经回代过程即可得到更高阶精度的结点位移向量；主要任务是通过数值试验和理论证明相结合，探讨其超收敛的机理和特性，研发高效算法，开拓有效应用；预定目标是在一维至三维、线性和非线性、常规和自适应有限元领域均取得有效应用。本研究构思巧妙、理论新颖、技术独特、算法高效，有望成为高性能有限元分析的一个原创性新进展。

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1D到3D线性和非线性问题中已获得了广泛的成功。但以上的工作也存在一些难点和空白区。这是由于以上工作都是建立在有限元的结点精度高于内点精度的基础上的（如对m次一维单元，结点为2m阶、内点为m+1阶收敛）。而对于低次元，譬如对于常用的线性元，由于结点精度低（结点与内点均为2阶收敛），无法获得超收敛解，以至于一直排除在超收敛分析和自适应求解算法之外。为克服这一难点，本项目对有限元结点位移精度修正技术开展攻关研究，如期成功地提出了一套基于EEP技术的结点精度修正的理论和算法，有效提升了有限元整体性能、质量和精度，是高性能有限元分析的一个重要成果。其基本思路正如申请项目之初所期，利用超收敛计算的EEP法之优良特性，在不改变有限元网格及其整体刚度矩阵的情况下，导出残差的等效结点荷载向量，只经回代过程即可得到更高阶精度的结点位移向量。这一修正技术，在一维有限元中获得全面成功，修正后的结点位移精度提升了2阶，由原来的2m阶提升到2m+2阶；在二维和三维有限元中也获得不同程度的改进，特别是最常用的线性元，修正后的结点位移精度亦有翻倍的提升，由原来的2阶提升到4阶。在此基础上，进一步提出了修正的EEP算法，经理论证明和数值试验，修正的EEP解，对所有次数的单元都具有比有限元至少高一阶的超收敛性，因此可以作为误差估计器广泛应用于各类问题的自适应求解。项目按照预定目标，在一维至三维、线性和非线性、边值和初值、常规和自适应有限元领域均获得创新性成果和有效应用。本研究构思巧妙、理论新颖、技术独特、算法高效，具有更为深远和宽广的发展应用前景。