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基于比例边界有限元法和灰狼优化算法的裂纹尖端位置识别

余波 孙文涧

余波,孙文涧. 基于比例边界有限元法和灰狼优化算法的裂纹尖端位置识别 [J]. 应用数学和力学,2021,42(11):1177-1189 doi: 10.21656/1000-0887.410381
引用本文: 余波,孙文涧. 基于比例边界有限元法和灰狼优化算法的裂纹尖端位置识别 [J]. 应用数学和力学,2021,42(11):1177-1189 doi: 10.21656/1000-0887.410381
YU Bo, SUN Wenjian. Identification of Crack Tip Positions Based on the Scaled Boundary Finite Element Method and the Grey Wolf Optimization Algorithm[J]. Applied Mathematics and Mechanics, 2021, 42(11): 1177-1189. doi: 10.21656/1000-0887.410381
Citation: YU Bo, SUN Wenjian. Identification of Crack Tip Positions Based on the Scaled Boundary Finite Element Method and the Grey Wolf Optimization Algorithm[J]. Applied Mathematics and Mechanics, 2021, 42(11): 1177-1189. doi: 10.21656/1000-0887.410381

基于比例边界有限元法和灰狼优化算法的裂纹尖端位置识别

doi: 10.21656/1000-0887.410381
基金项目: 国家自然科学基金(面上项目)(11872166)
详细信息
    作者简介:

    余波(1984—),男,副教授(通讯作者. E-mail:yubochina@hfut.edu.cn

  • 中图分类号: O346.1

Identification of Crack Tip Positions Based on the Scaled Boundary Finite Element Method and the Grey Wolf Optimization Algorithm

  • 摘要: 基于比例边界有限元法(SBFEM)和灰狼优化(GWO)算法,提出了一种裂纹尖端识别方法。首先,借助SBFEM解决断裂力学问题特有的优势,快速准确地计算出反演所需的测点位移,并验证了正问题求解的正确性。其次,建立与裂纹尖端位置有关的目标函数,将求解裂纹尖端位置转换为求解目标函数最小值的优化问题。最后,采用GWO算法对目标函数进行了优化,进而搜索裂纹尖端的最佳位置。数值算例结果表明:利用SBFEM的高精度、半解析的优点,在反演过程中采用其求解正问题是非常有效的;GWO算法具有良好的全局收敛性,且相比经典的粒子群算法,能够更快速准确地搜索出裂纹尖端的位置;GWO算法具有较好的抗噪性。
  • 图  1  比例边界有限单元

    Figure  1.  A scaled boundary finite element

    图  2  狼群社会等级

    Figure  2.  The social hierarchy of grey wolves

    图  3  GWO算法流程图

    Figure  3.  The GWO algorithm flowchart

    图  4  中心裂纹板

    Figure  4.  The center-cracked plate

    图  5  中心裂纹板单元划分

    Figure  5.  The element division of the center-cracked plate

    图  6  狼群初始位置设置

    Figure  6.  Initial position setting of wolves

    图  7  不同初始位置GWO算法搜索裂纹尖端的目标函数值

    Figure  7.  Objective function values for crack tips searched for in different initial positions with the GWO algorithm

    图  8  边缘裂纹板

    Figure  8.  The edge-cracked plate

    图  9  边缘裂纹板单元划分

    Figure  9.  The element division of the edge-cracked plate

    图  10  识别边缘裂纹尖端位置时的目标函数值

    Figure  10.  Objective function values for identifying the position of the edge crack tip

    图  11  斜裂纹板

    Figure  11.  The oblique crack plate

    图  12  迭代过程中不同的单元划分

    Figure  12.  Different element divisions during the iteration process

    图  13  识别斜裂纹尖端位置时的目标函数值

    Figure  13.  Objective function values for identifying the position of the oblique crack tip

    图  14  带裂纹扳手

    Figure  14.  The wrench with a crack

    图  15  带裂纹扳手的单元划分

    Figure  15.  The element division of the wrench with a crack

    图  16  不带误差的目标函数

    Figure  16.  The objective function values without measurement errors

    图  17  考虑正则化时不同测量误差的目标函数值

    Figure  17.  Objective function values with different measurement errors in view of the regularization scheme

    图  18  不同测量误差下目标函数值对比

    Figure  18.  Comparison of objective function values under different measurement errors

    表  1  中心裂纹应力强度因子

    Table  1.   The stress intensity factor of the center crack

    SBFEM
    KI / (MPa·m1/2)
    analytical solution KI / (MPa·m1/2)relative error
    δ / %
    1.48861.48190.452
    下载: 导出CSV

    表  2  中心裂纹尖端识别结果

    Table  2.   Identification results of center crack tips

    casecrack tip position (x, y) / mobjective function value
    1(−0.5001, 0.0096),(0.4999, 0.0086)2.49×10−13
    2(−0.5000, 0.0119),(0.5007, 0.0273)2.87×10−12
    3(−0.5013, 0.0419),(0.5008, 0.0131)6.03×10−12
    4(−0.4997, 0.0245),(0.5003, 0.0473)1.02×10−11
    下载: 导出CSV

    表  3  GWO和PSO算法识别边缘裂纹尖端位置结果

    Table  3.   The position of the edge crack tip identified with GWO and PSO algorithm

    real position (x, y) / m(x, y)GWO / m(x, y)PSO / m
    (0, 0)(−1.953×10−12, 3.621×10−12)(−2.877×10−12, 2.886×10−12)
    下载: 导出CSV

    表  4  GWO和PSO算法识别斜裂纹尖端位置

    Table  4.   The position of the oblique crack tips identified with GWO algorithm and PSO algorithm

    real position (x, y) / m(x, y)GWO / m(x, y)PSO / m
    (−0.5, 0)(−0.499 8, −9.2407×10−3)(−0.499 8, −1.0472×10−3)
    (0, 0.5)(2.2318×10−4, 0.503 0)(5.1873×10−4, 0.504 2)
    下载: 导出CSV

    表  5  考虑正则化时裂纹尖端识别结果

    Table  5.   Identification results of the crack tip in view of the regularization scheme

    error ε1%3%5%
    objective function value of GWO result 4.537×10−13 3.153×10−12 4.109×10−11
    objective function value of PSO result
    6.501×10−10 2.088×10−9 1.030×10−8
    crack tip position of GWO result (x,y)GWO /cm (1.904×10−4, −4.229×10−5) (8.294×10−4, 8.427×10−4) (2.501×10−3, 5.543×10−4)
    crack tip position of PSO result (x,y)PSO /cm (−2.713×10−2, 5.981×10−3) (−6.183×10−2, 1.237×10−2) (8.414×10−2, −1.315×10−2)
    下载: 导出CSV
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出版历程
  • 收稿日期:  2020-12-14
  • 修回日期:  2021-05-06
  • 网络出版日期:  2021-12-07
  • 刊出日期:  2021-11-30

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