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张小民, 毛建中, 雷从一, 段书用. 基于M-K模型计算FLC无条件收敛新算法. 力学学报, 2024, 56(4): 1-15. DOI: 10.6052/0459-1879-23-488
引用本文: 张小民, 毛建中, 雷从一, 段书用. 基于M-K模型计算FLC无条件收敛新算法. 力学学报, 2024, 56(4): 1-15. DOI: 10.6052/0459-1879-23-488
Zhang Xiaomin, Mao Jianzhong, Lei Congyi, Duan Shuyong. A new unconditional convergence algorithm for calculating flc based on m-k model. Chinese Journal of Theoretical and Applied Mechanics, 2024, 56(4): 1-15. DOI: 10.6052/0459-1879-23-488
Citation: Zhang Xiaomin, Mao Jianzhong, Lei Congyi, Duan Shuyong. A new unconditional convergence algorithm for calculating flc based on m-k model. Chinese Journal of Theoretical and Applied Mechanics, 2024, 56(4): 1-15. DOI: 10.6052/0459-1879-23-488

基于M-K模型计算FLC无条件收敛新算法

A NEW UNCONDITIONAL CONVERGENCE ALGORITHM FOR CALCULATING FLC BASED ON M-K MODEL

  • 摘要: M-K失稳模型的无条件收敛与快速求解对成形极限曲线(forming limit curve, FLC)的获取具有重要的理论意义与研究价值, 也是理论FLC计算的难点之一. 然而, 在M-K失稳模型框架内基于复杂屈服准则计算成形极限曲线时, 采用传统牛顿迭代算法无法保证收敛, 而固定增量法则计算耗时过长, 严重制约了M-K失稳模型的应用场景. 因此, 文章以锆合金为实验材料, 提出了一种自适应增量算法, 该算法通过迭代误差信息与震荡情况自适应调整迭代步长, 不仅能够保证计算精度, 满足无条件收敛, 还能大幅提升计算速度. 此外, 本研究还验证了采用自适应增量法计算锆合金FLC的准确性, 并分析了算法的迭代误差Err与初始应变增量对FLC预测结果的影响. 结果表明: 自适应增量法具有无条件保证收敛、计算速度快以及计算结果不受初始迭代步长影响的优势, 其计算耗时仅为固定增量法的0.44%; 此外, 本研究认为初始参数中迭代误差小于 10^-6 、初始应变增量小于0.001是得到稳定的FLC结果的前提条件; 根据对锆合金FLC预测结果可知, 在M-K模型框架内基于Yld2000-2d屈服准则理论采用自适应增量法计算可以很好地描述锆合金的塑性行为. 由于自适应增量迭代法简单易用且计算高效, 有望应用于有限元软件中子程序的开发, 加速其计算速度并提高锆合金数值模拟精度.

     

    Abstract: It has important theoretical significance and research value to meet the requirements of unconditional convergence and rapid solution of the M-K instability model for obtaining the forming limit curve (FLC) in sheet metal forming. And the above requirements are also considered as the difficulties of the theoretical calculation of FLC. However, when calculating the FLC based on advanced yield criteria within the framework of the M-K instability model, convergence results cannot be guaranteed by employing the traditional Newton iterative algorithm. Besides, the fixed increment rule takes too long to calculate, seriously restricting the application scenarios of the M-K instability model. Therefore, in this study, an adaptive incremental algorithm is proposed and zirconium alloys are consumed as the experiment material. This algorithm adaptively adjusts the iteration step size through iterative error information and oscillation conditions, which not only ensures calculation accuracy and meets unconditional convergence requirements, but also significantly improves calculation speed. Moreover, the calculation accuracy of the FLC of zirconium alloys by using the adaptive incremental method has been verified in this study while the impacts of the algorithm's iteration error as well as the initial strain increment on the FLC prediction results are also analyzed. The results show that the adaptive incremental method has the advantages of unconditionally ensuring convergence and fast calculation speed, and the calculation results are not affected by the initial iteration step size. While its calculation time is only 0.43% of that of the fixed incremental method. In addition, this investigation suggests that an iteration error of less than 10^-6 and an initial strain increment of less than 0.001 in the initial parameters are prerequisites for obtaining stable FLC results. According to the prediction results of the FLC, it can be concluded that the plastic behavior of zirconium alloys can be well described by employing the adaptive incremental method based on the Yld2000-2d yield criterion theory within the M-K model framework. Due to the advantages of simplicity, user friendly as well as the high computational efficiency, the adaptive incremental iteration method should be considered as a promising algorithm in the development of subroutines in finite element software, accelerating its calculation speed and improving the accuracy of numerical simulation of zirconium alloys.

     

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