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

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

余波,
孙文涧

合肥工业大学土木与水利工程学院, 合肥 230009

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
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- 被引次数: 0
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

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

YU Bo,
SUN Wenjian

College of Civil Engineering, Hefei University of Technology, Hefei 230009, P.R.China

摘要

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。
- 裂纹尖端识别 /
- 比例边界有限元法 /
- 灰狼优化算法 /
- 粒子群优化算法
Abstract: Based on the scaled boundary finite element method (SBFEM) and the grey wolf optimization algorithm (GWO), an identification method for crack tips was proposed. Firstly, the special advantages of the SBFEM were used to solve the fracture mechanics problem, the displacements of measurement points required in the inversion process were quickly and accurately calculated, and the correctness of the solution to the forward problem was verified in advance. Then, the objective function related to the crack tip position was established, and the identification of the crack tip position was converted to the optimization problem of solving the minimum value of the objective function. Finally, the GWO was used to optimize the objective function, that is, to search for the optimal position of the crack tip. The numerical example results show that, it is very effective to solve forward problems in the inversion process with the high precision and semi-analytical advantages of the SBFEM. The grey wolf optimization algorithm has good global convergence property, and can search for the crack tip position more quickly and accurately compared with the classical particle swarm optimization. The grey wolf optimization algorithm has good noise resistance.
- crack tip identification /
- scaled boundary finite element method /
- grey wolf optimization algorithm /
- particle swarm optimization algorithm

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图 1 比例边界有限单元

Figure 1. A scaled boundary finite element

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图 2 狼群社会等级

Figure 2. The social hierarchy of grey wolves

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图 3 GWO算法流程图

Figure 3. The GWO algorithm flowchart

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图 4 中心裂纹板

Figure 4. The center-cracked plate

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图 5 中心裂纹板单元划分

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

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图 6 狼群初始位置设置

Figure 6. Initial position setting of wolves

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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

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图 8 边缘裂纹板

Figure 8. The edge-cracked plate

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图 9 边缘裂纹板单元划分

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

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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

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图 11 斜裂纹板

Figure 11. The oblique crack plate

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图 12 迭代过程中不同的单元划分

Figure 12. Different element divisions during the iteration process

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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

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图 14 带裂纹扳手

Figure 14. The wrench with a crack

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图 15 带裂纹扳手的单元划分

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

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图 16 不带误差的目标函数

Figure 16. The objective function values without measurement errors

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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

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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

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表 1 中心裂纹应力强度因子

Table 1. The stress intensity factor of the center crack

SBFEM K_I / (MPa·m^1/2)	analytical solution K_I / (MPa·m^1/2)	relative error δ / %
1.4886	1.4819	0.452

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表 2 中心裂纹尖端识别结果

Table 2. Identification results of center crack tips

case	crack tip position (x, y) / m	objective 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

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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)

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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)

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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 ε	1%	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)

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