球对称热应力问题的有限元误差分析

ERROR ANALYSIS OF FINITE ELEMENT CALCULATIONON THERMAL STRESS OF SPHERE

本文设计了一个求解球对称热应力的模型问题。从结点位移的解析解出发,利用有限元的位移线性插值函数,按照结点平均、单元平均或在单元重心取值等方法,计算应力和应变,然后与对应的解析解比较。结果表明,除按结点平均取值法算得的周向应变在理论上无误差外,其余量在内点具有二阶精度,在边界上为一阶精度;并表明了它们与温度梯度指数n的关系,应变的相对误差正比于n^2/31,应力的相对误差正比于n^3/31。因此,即使结点位移无误差,此时仍应以n^3Δr^2/31的误差阶作为单元划分的参考准则。

A model problem about thermal stress in a symmetric sphere has been designed for analyzing the error of thermal stress parameters which are calculated with finite element method, e. g. taking exact displacements directly on grids and using finite element linear interpolation function.The strains and stresses are obtained according to method of grid average and of element average or calculated at weight center in element . The comparison between these average values and exact solutions is shown that these parameters are of the second order precision on the interior grids and the first order precision on the border, and also shown that the error is proportionate to nz/31 for strains and n3/31 for stresses when exponent of temperature gradient n is to grow great. Therefore, the error order of n3△r2/3l still should be taken as reference principle to divide element, even displacements on the grids are exact.