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Combined Finite Element Method Based on Two Variational Principles and Its Convergence and Accuracy

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 作者 单位 E-mail 卿光辉 中国民航大学 qingluke@126.com 毛俊慧 中国民航大学 刘艳红 中国民航大学

根据修正的H-R变分原理和最小势能原理构建了联合有限元列式，以联合有限元列式为基础的联合有限元方法归属于位移法。首先，对离散的H-R变分原理的平面外应力进行变分，可得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表示位移与外载荷关系（欧拉方程），独立求解该方程得到问题的位移解，将位移解代入平面外应力与位移关系欧拉方程中可得到平面外应力的解。考虑不同的结构材料，基于线性方程组的形式求解平面内应力是本文的另一特点，即回避了直接依据本构关系按单元求解应力的方法。数值实例表明线性8结点非协调联合元(NCCFE)的力学量稳定收敛、均衡和可靠，且收敛速度快，结果精度高。

Combined finite element formulation(CFEF) is established by the modified H- R varialtional principle and minimum potential principle. The combined finite element method(CFEFM), based on the combined finite element formulation(CFEF), belongs to the displacement method. Firstly, the Euler equation, which expresses the relation of the out-plane stresses and displacements, is derived from the discretized H-R variational principle by the variation of the stress variable. Secondly, the Euler equation, which expresses the relationship of displacements and forces is reduced from the potential principle, the displacement solution can be obtained from the Euler equation independently. Instituting the displacement solution into the relationship of the out-plane stresses and displacements, one can obtain the solution of the out-plane stress. It is another character in this paper that, considering different materials of a structure, the solution of in-plane stresses can be solved by the linear equation set. The method obtaining the stress solution by the constitutive relationship at the element level is avoided. The numerical examples show that the convergence rates of displacement and stress variables of the 8-noded no-compatible combined element(NCCFE) are balanced, stable fast and with a fine precision.
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