二维有限元线法超收敛解答计算的EEP法

AN ELEMENT-ENERGY-PROJECTION METHOD FOR SUPER-CONVERGENT SOLUTIONS IN TWO-DIMENSIONAL FINITE ELEMENT METHOD OF LINES

有限元线法(FEMOL)是一种优良的半解析、半离散方法,但其解答存在解析方向和离散方向的精度不相称的弱点。本文提出将二维有限元线法比拟为广义一维问题的概念,遂可将新近提出的一维有限元超收敛计算的单元能量投影(EEP)法推广到二维有限元线法分析中。经有限元线法后处理中EEP超收敛计算而获得的解答,继承和保留了一维有限元中的出色表现,不但使任意一点的位移和应力的解答在两个方向具有相当的精度,而且都具有超收敛性质。文中以二维Poisson方程问题为例,具体给出了有限元线法EEP超收敛的公式,并给出了数值算例,用以表明本法的可行性和有效性。

While the finite element method of lines （FEMOL） is a general and powerful semi-analytical and semi-discretised method for two-dimensional BVP, its solution in the discrete direction and the analytical direction does not behave equally well. Using the analogy between the two-dimensional FEMOL and one-dimensional FEM, the present paper proposed the concept of the generalized one-dimensional problem for the two-dimensional FEMOL and then generalized the newly-developed EEP （Element Energy Projection） method for super-convergence computation in one-dimensional FEM to the two-dimensional FEMOL analysis. By applying the proposed EEP superconvergence computation in the post-processing of FEMOL analysis, both super convergent displacements and stresses at any point can be obtained with remarkable properties which not only balances the accuracy of the two directions but also makes the accuracy highly super-convergent. Detailed formulation and numerical examples for two-dimensional Poisson＇s equation are given in the paper to demonstrate the feasibility and effectiveness of the method.