基于辛弹性力学解析本征函数的有限元应力磨平方法

A stress recovery method based on the analytical eigenfunctions of symplectic elasticity

在实际工程结构的结构强度与优化等力学数值分析中,应力计算结果的精度是非常重要的。有限元法是得到最广泛应用的一类数值方法,并形成了众多通用的有限元程序系统。这些程序系统采用的几乎都是基于最小总势能的位移法,虽然其分析给出的有限元位移场具有较高的精度,但所得到的有限元应力场的精度较位移场大大降低。基于极坐标辛对偶体系所提供的平面弹性力学的解析辛本征展开解,并借用有限元程序系统所得到的节点位移,本文提出了一个应力分析的改进方法。数值结果表明,本方法给出的应力分析精度得到大幅提高,并具有良好的数值稳定性,可用于有限元程序系统的后处理,以提高应力尤其是关键区域应力的分析精度。

The accuracy of stress is important in the engineering application analysis, such as structural strength,structural optimization,etc. The Finite Element Method （FEM） is one of the most widely ap- plied numerical methods based on which many general program systems have been built. The displace- ment method based on the minimum total potential energy principle is commonly used for these FEM program systems. So the displacement field of high accuracy can be obtained. However,it would lead to a stress field of much lower accuracy comparing with the displacement field obtained. In this paper,a stress recovery method is presented to improve the result of stress analysis,which make use of the symplectic eigenfunctions of plane problems in the polar coordinate system and node displacements provided by FEM. Numerical results show that, the new technique improve evidently accuracy of stress analysis and the numerical stability is also very well. Hence,it could be applied for the postprocessing of the general program system of FEM to improve the accuracy of stress analysis, especially the accuracy of stress on the key area.