欧拉梁弹性稳定分析的有限元p 型超收敛算法1)
A p -TYPE SUPERCONVERGENT RECOVERY METHOD FOR FE ELASTIC STABILITY ANALYSIS OF EULER BEAMS1)
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摘要: 本文将有限元p型超收敛算法应用于欧拉梁弹性稳定分析。该法基于有限元解答中失稳载荷和失稳模态结点位移的超收敛特性，建立了单元上失稳模态近似满足的线性常微分方程边值问题，在每个单元上，对该边值问题采用一个高次元进行求解，获得失稳模态的超收敛解，再将失稳模态的超收敛解代入瑞利商的解析表达式，最终获得失稳载荷的超收敛解。该法思路简明，通过少量计算即可显著提高失稳载荷和失稳模态的精度与收敛阶。数值算例表明，该法高效、可靠，值得进一步研究和推广到各类杆系结构。Abstract: This paper extends the p-type superconvergent recovery method to the finite element elastic stability analysis of Euler beams. Based on the superconvergence properties of the buckling loads and the nodal displacements in the buckling modes in regular FE solutions, a linear ordinary differential boundary value problem (BVP) is set up, which approximately governs the buckling mode in each element. This linear BVP within an element is solved with a higher order element, and a more accurate buckling mode is recovered. Then by substituting the recovered buckling mode into the Rayleigh quotient in analytic form, the buckling load is recovered. This method is simple and clear. It can improve the accuracy and the convergence rate of the buckling loads and the buckling modes significantly with a small amount of computation. Numerical examples demonstrate that this method is reliable and efficient and is worth further extending to other skeletal structures.
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