Arc-Length Continuation Algorithm for Nonlinear Finite Element Equations

doi:10.13209/j.0479-8023.2017.072

Abstract

Abstract:

Stability analysis of engineering structures requires tracing equilibrium path of the structure when member’s buckling or material softening occurs. In nonlinear finite element analysis, the traditional Newton method fails at limit point and bifurcation point. The arc-length continuation method can overcome these numerical difficulties. To develop a nonlinear finite element code for stability analysis, the standard iteration formulation of Newton method is presented for the arc-length continuation method. Two practical formulations of the arc-length continuation method and their relationships with the standard form are also discussed. The applicability of the practical formulation is examined by the finite element analysis of stability for a slope.

Key words: arc-length continuation algorithm, equilibrium path curve, nonlinear analysis, Newton iteration method, finite element method

摘要：

研究工程结构因构件屈曲和材料软化导致的稳定性问题, 就需要追踪结构的平衡路径。当采用非线性有限元进行分析时, 传统的牛顿迭代法会在极值点和分叉点处失效, 而弧长延拓方法能很好地解决这一数值计算难题。针对结构稳定性非线性有限元分析程序的编制, 给出弧长延拓算法牛顿迭代的标准格式和两种实用的迭代格式, 并讨论它们之间的关系。通过一个边坡稳定性的有限元分析, 验证了实用迭代格式的有效性。

关键词: 弧长延拓算法, 平衡路径曲线, 非线性分析, 牛顿迭代法, 有限元方法

Youquan YIN, Yuan DI, Zaixing YAO. Arc-Length Continuation Algorithm for Nonlinear Finite Element Equations[J]. Acta Scientiarum Naturalium Universitatis Pekinensis, 2017, 53(5): 793-800.

殷有泉, 邸元, 姚再兴. 非线性有限元方程组的弧长延拓算法[J]. 北京大学学报自然科学版, 2017, 53(5): 793-800.

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URL: https://xbna.pku.edu.cn/EN/10.13209/j.0479-8023.2017.072

https://xbna.pku.edu.cn/EN/Y2017/V53/I5/793

Figures/Tables 6

Fig. 1 Arc-length continuation procedure for the simplified Newton method

Fig. 2 Finite element mesh of the slope

Fig. 3 Deformed mesh of the slope

Fig. 4 Von Mises stress distribution of the slope

Fig. 5 Hydrostatic stress distribution of the slope

Fig. 6 Equilibrium path curves for different strength reduction factors ζ

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