联合有限元法及其收敛性与精度分析

Combined finite element method and analyses on its convergence and accuracy

根据修正的H-R变分原理和最小势能原理构建一种联合有限元单元。首先对离散的H-R变分原理进行变分,可得到平面外应力与位移关系的欧拉方程;然后通过势能原理得到位移与外载荷关系的方程,独立求解该方程得到位移解,再将其代入平面外应力与位移关系的欧拉方程中,可得到平面外应力的解。考虑到结构可能由不同的材料组成,采用线性方程组的形式求解平面内应力,回避了直接依据本构关系按单元求解应力的方法。实验表明:线性8结点的非协调联合元(NCCFE)的数值结果稳定收敛、均衡、可靠,且收敛速度快、精度高。

Combined finite element formulation is built according to modified H-R variation principle and minimum potential principle.Firstly,the Euler equation expressing the relationship between out-plane stress and displacement is derived from discretized H-R variation principle.Secondly,the Euler equation expressing the relationship between displacement and out-load is deduced from potential principle,the displacement solution can be independently obtained from Euler equation.Substituting the displacement solution into the first Euler equation,the out-plane stress solution can be obtained.Considering the different materials consisting of the structure,the in-plane stress solution can be obtained by a set of linear equations,avoiding the direct stress solution method basing on constitutive relationship at element level.Example shows that the numerical result of linear eight-noded NCCFE is stably convergent,balanced and reliable with fast convergence speed and high accuracy.