This paper presents a strategy for computation of super-convergent solutions of multi-dimensional problems in the finite element method(FEM) by recursive application of the one-dimensional(1D) element energy projection(EEP) technique. The main idea is to conceptually treat multi-dimensional problems as generalized 1D problems,based on which the concepts of generalized 1D FEM and its consequent EEP formulae have been developed in a unified manner. Equipped with these concepts, multi-dimensional problems can be recursively discretized in one dimension at each step, until a fully discretized standard finite element(FE) model is reached. This conceptual dimension-bydimension(D-by-D) discretization procedure is entirely equivalent to a full FE discretization. As a reverseD-by-D recovery procedure, by using the unified EEP formulae together with proper extraction of the generalized nodal solutions, super-convergent displacements and first derivatives for two-dimensional(2D) and three-dimensional(3D) problems can be obtained over the domain. Numerical examples of 3D Poisson's equation and elasticity problem are given to verify the feasibility and effectiveness of the proposed strategy.
无穷域问题广泛存在于实际工程中,半解析、半离散的数值计算方法有限元线法(Finite Element Method of Lines,简称FEMOL)对其具有较好的适应性。在已有的映射型FEMOL无穷单元理论的基础上,基于单元能量投影(Element Energy Projection,简称EEP)法的自适应FEMOL被应用于二维无穷域问题的求解。用户只需输入稀疏的初始网格和误差限,算法即自动生成优化的FEMOL网格,该网格上常规单元和无穷单元的FEMOL解均按最大模度量满足给定误差限。文中首先介绍二维FEMOL的原理策略、无穷单元的构建,然后概述基于EEP法的自适应FEMOL算法,并讨论其对无穷域问题的适用性,之后对圆柱绕流的Poisson方程问题、带孔无穷大板单向拉伸的弹性力学平面问题、受圆形均布荷载半空间体的三维轴对称问题进行了自适应分析,最终不仅给出了满足误差限的函数(位移)解,也给出了具有优良性态的导数(应力)解,从而为无穷域问题的求解提供了一种高效可靠的新途径。
