基于EEP技术的一维有限元结点位移误差计算

CALCULATION OF ERRORS OF NODAL DISPLACEMENTS IN ONEDIMENSIONAL FINITE ELEMENT METHODS USING ELEMENT ENERGY PROJECTION TECHNIQUE

利用单元能量投影(Element Energy Projection,简称EEP)法所计算的EEP超收敛解,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具有更高阶精度的结点位移的误差估计,使结点位移精度得到极大提高。该文以一般的二阶常微分方程边值和初值问题为例,给出算法和相应的数值算例。从中可以看出,本法十分简单而高效:对于m≥1次单元,采用EEP简约格式和凝聚格式修正后的结点位移,分别具有O(h^(2m+2))和O(h^(3m+mod(m,2)))的超常规的超收敛阶。该文给出了典型算例,并对该法的进一步拓展和应用作了讨论。

Using super-convergent solutions calculated by the Element Energy Projection(EEP)method,equivalent nodal load vectors from the residual load term were derived in this paper without changing the finite element(FE)meshes and the global stiffness matrices.The subsequent back-substitutions can generate highly accurate estimates for the errors of nodal displacements and hence greatly improve the nodal accuracy.Taking a general second-order ordinary differential equation as the model problem,the algorithm of the proposed method and associated numerical examples were given to show that the proposed method is simple and effective,and that using elements of degree m≥1,the improved nodal displacements can gain the super-super-convergence orders h2m+2 and h3m+mod(m,2)for simplified and condensed EEP forms,respectively.A variety of significant further extensions and applications were also discussed.