摘要
介绍弧长算法的完整的理论基础。首先讨论用普通牛顿选代法解1个自由度非线性问题时若在加载路径上出现奇点将面临的困难,然后引进弧长参数并论讨用它来发现和解决极限点或分叉点问题的可能性,讨论在2个自由度有限元问题中弧长法的理论并将它推广到 n 个自由度的情况,对 n+1维空间建立具有普遍性的弧长方程,首次提出一个实用的加载增量公式。所建议的算法很容易被任何现存的这类有限元软件采用。
A complete theoretical backgriound of the arc length algorithm is introduced. First discuss the difficulties of using ordinary Newton-Raphson method to solve an one DOF nonlinear problem while a singular point occurred in the load path.The arc length parameter is then introduced to study the possibility for finding and solving the limit point or bifurca- tion point problem.After,discussing the theory of two DOF finite element problem and ex- panding to n DOF problem ,the general arc length formulation in the n+1 dimension vector space is then established.A useful formula for the load increment is given in the first time. The proposed algorithm is easy to fit any of the large existing FEM software.Several useful results to determine the load increment,the are length increment and the criteria of enlarging or shortening the arc length are presented.Several interesting numerical results of tracing the complete load path of a nonlinear structural analysis are given,and some of them are first shown in the literatures.
关键词
有限元
弧长算法
全程追踪
finite element
arc length algorithm
tracing of complete load path
load path
load increment