基于BP神经网络的I型金属夹芯板极限强度预测

卫钰汶; 仲强; 王德禹

doi:10.19693/j.issn.1673-3185.02335

基于BP神经网络的I型金属夹芯板极限强度预测

doi: 10.19693/j.issn.1673-3185.02335

上海交通大学海洋工程国家重点实验室，上海 200240

基金项目: 国家自然科学基金资助项目（51979163）

详细信息

作者简介:
卫钰汶，女，1998年生，硕士生。研究方向：船体结构极限强度。E-mail：yuwenwei@sjtu.edu.cn

仲强，男，1990年生，博士生。研究方向：船体结构极限强度。E-mail：zhongqiang@sjtu.edu.cn

王德禹，男，1963年生，博士，教授，博士生导师。研究方向：船舶与海洋工程结构力学，结构优化设计与可靠性分析，结构极限强度与试验技术研究。E-mail：dywang@sjtu.edu.cn

通信作者:
王德禹

中图分类号: U661.43
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出版历程
- 收稿日期: 2021-03-30
- 修回日期: 2021-05-25
- 网络出版日期: 2022-04-06
- 刊出日期: 2022-04-20

Ultimate strength prediction of I-core sandwich plate based on BP neural network

State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

基于BP神经网络的I型金属夹芯板极限强度预测由卫钰汶，等创作，采用知识共享署名4.0国际许可协议进行许可。

摘要

摘要: 目的针对过去对I型金属夹芯板的极限强度评估不完善的问题，提出一种采用 BP人工神经网络的方法来定量确定各相关参数对I型金属夹芯板极限强度的影响。方法首先，采用非线性有限元法研究I型金属夹芯板在面内轴向压缩载荷条件下的极限强度；然后，构造BP神经网络以对不同面板柔度系数β_p、腹板柔度系数β_w和梁柱柔度系数λ下I型金属夹芯板的极限强度进行预测；最后，提出采用人工神经网络权值和偏置法预测I型金属夹芯板极限强度的公式。结果针对所计算的算例尺寸，显示采用BP神经网络方法的极限强度预测的均方差MSE和相关系数R分别为0.001 2和0.981 8，所构建的神经网络模型具有较好的预测精度，最大误差不超过10%。结论所得结论可为I型金属夹芯板在船体结构中的应用提供参考。
- I型金属夹芯板 /
- BP人工神经网络 /
- 极限强度 /
- 非线性有限元法 /
- 预测
Abstract: Objectives In view of the incomplete evaluation of the ultimate strength of I-core sandwich panels in the past, a BP artificial neural network method is proposed to quantitatively determine the influence of relevant parameters on the ultimate strength of I-core sandwich panels. Methods First, the ultimate strength of I-core sandwich panels under axial compression are investigated using the nonlinear finite element method. Second, a BP neural network is constructed to predict the ultimate strength of I-core sandwich panels with different plate slenderness ratios between longitudinal webs, plate slenderness ratios of webs and column slenderness ratio of one longitudinal web. Finally, a formula for predicting the ultimate strength of I-core sandwich panels using the artificial neural network weight and bias method is proposed. Results The mean square error MSE and correlation coefficient R of ultimate strength prediction using the BP neural network method are 0.001 2 and 0.981 8 respectively. The proposed neural network model has good prediction accuracy, and the maximum error is less than 10%. Conclusions This study can provide references for the application of I-core sandwich panels in hull structures.
- I-core sandwich panels /
- BP neural network /
- ultimate strength /
- nonlinear finite element method /
- prediction

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图 1 I型金属夹芯板结构示意图

Figure 1. Schematic diagram of I-core sandwich panels

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图 3 纵向1/2+1+1/2跨（边界条件代替强横梁）模型示意图

Figure 3. Schematic diagram of longitudinal 1/2+1+1/2 spans (boundary conditions instead of strong beams) model

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图 4 纵向单跨（边界条件代替强构件）模型示意图

Figure 4. Schematic diagram of longitudinal 1 span (boundary conditions instead of strong members) model

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图 2 纵向三跨（有实际强构件）模型示意图

Figure 2. Schematic diagram of longitudinal 3 spans model with actual strong members

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图 5 不同模型范围下I型金属夹芯板轴压下载荷−端缩曲线对比

Figure 5. Comparison of load-deformation curves for I-core sandwich panels under axial compression with different range of models

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图 6 I型金属夹芯板轴压下载荷−端缩曲线对比

Figure 6. Comparison of load-deformation curves for I-core sandwich panel under axial compression

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图 7 模型试验与有限元仿真失效模式对比

Figure 7. Comparison of the failure mode by the FE analysis and model test

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图 8 预测I型金属夹芯板极限强度与屈服强度比的BP神经网络结构

Figure 8. Architecture of BP neural network for prediction of ultimate strength to yield strength ratio of I-core sandwich panels

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图 9 训练集中${{{\sigma _{{\rm{u}}}}} \mathord{\left/ {\vphantom {{{\sigma _{{\rm{u}}}}} {{\sigma _{{\rm{Y}}}}}}} \right. } {{\sigma _{{\rm{Y}}}}}}$的期待值与预测值的相关图

Figure 9. Correlation between the expected data and predicted outputs for ${{{\sigma _{{\rm{u}}}}} \mathord{\left/ {\vphantom {{{\sigma _{{\rm{u}}}}} {{\sigma _{{\rm{Y}}}}}}} \right. } {{\sigma _{{\rm{Y}}}}}}$ in training set

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图 10 验证集中${{{\sigma _{{\rm{u}}}}} \mathord{\left/ {\vphantom {{{\sigma _{{\rm{u}}}}} {{\sigma _{{\rm{Y}}}}}}} \right. } {{\sigma _{{\rm{Y}}}}}}$的期待值与预测值的相关图

Figure 10. Correlation between the expected data and predicted outputs for ${{{\sigma _{{\rm{u}}}}} \mathord{\left/ {\vphantom {{{\sigma _{{\rm{u}}}}} {{\sigma _{{\rm{Y}}}}}}} \right. } {{\sigma _{{\rm{Y}}}}}}$ in validation set

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图 11 测试集中${{{\sigma _{{\rm{u}}}}} \mathord{\left/ {\vphantom {{{\sigma _{{\rm{u}}}}} {{\sigma _{{\rm{Y}}}}}}} \right. } {{\sigma _{{\rm{Y}}}}}}$的期待值与预测值的相关图

Figure 11. Correlation between the expected data and predicted outputs for ${{{\sigma _{{\rm{u}}}}} \mathord{\left/ {\vphantom {{{\sigma _{{\rm{u}}}}} {{\sigma _{{\rm{Y}}}}}}} \right. } {{\sigma _{{\rm{Y}}}}}}$ in testing set

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图 12 全部集中${{{\sigma _{{\rm{u}}}}} \mathord{\left/ {\vphantom {{{\sigma _{{\rm{u}}}}} {{\sigma _{{\rm{Y}}}}}}} \right. } {{\sigma _{{\rm{Y}}}}}}$的期待值与预测值的相关图

Figure 12. Correlation between the expected data and predicted outputs for ${{{\sigma _{{\rm{u}}}}} \mathord{\left/ {\vphantom {{{\sigma _{{\rm{u}}}}} {{\sigma _{{\rm{Y}}}}}}} \right. } {{\sigma _{{\rm{Y}}}}}}$ in whole set

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图 13 测试集中${{{\sigma _{{\rm{u}}}}} \mathord{\left/ {\vphantom {{{\sigma _{{\rm{u}}}}} {{\sigma _{{\rm{Y}}}}}}} \right. } {{\sigma _{{\rm{Y}}}}}}$的期待值与预测值间误差

Figure 13. Error between the expected data and predicted outputs for ${{{\sigma _{{\rm{u}}}}} \mathord{\left/ {\vphantom {{{\sigma _{{\rm{u}}}}} {{\sigma _{{\rm{Y}}}}}}} \right. } {{\sigma _{{\rm{Y}}}}}}$ in testing set

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图 14 ${{{\sigma _{{\rm{u}}}}} \mathord{\left/ {\vphantom {{{\sigma _{{\rm{u}}}}} {{\sigma _{{\rm{Y}}}}}}} \right. } {{\sigma _{{\rm{Y}}}}}}$中期待值与预测值间误差

Figure 14. Error between the expected data and predicted outputs for ${{{\sigma _{{\rm{u}}}}} \mathord{\left/ {\vphantom {{{\sigma _{{\rm{u}}}}} {{\sigma _{{\rm{Y}}}}}}} \right. } {{\sigma _{{\rm{Y}}}}}}$

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图 15 输入变量β_p，β_w和λ对I型金属夹芯板结构${{{\sigma _{{\rm{u}}}}} \mathord{\left/ {\vphantom {{{\sigma _{{\rm{u}}}}} {{\sigma _{{\rm{Y}}}}}}} \right. } {{\sigma _{{\rm{Y}}}}}}$的影响程度

Figure 15. Relative importance of the input variables β_p, β_w, λ on the response variable of ${{{\sigma _{{\rm{u}}}}} \mathord{\left/ {\vphantom {{{\sigma _{{\rm{u}}}}} {{\sigma _{{\rm{Y}}}}}}} \right. } {{\sigma _{{\rm{Y}}}}}}$ of I-core sandwich panels

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表 1 I型金属夹芯板几何尺寸和材料参数

Table 1. Geometrical size and material parameters of I-core sandwich panels

参数	数值
上下面板厚度t_p /mm	2，3，4
腹板厚度t_w /mm	2，4，6，8
腹板高度h_w /mm	40，60，80
腹板间距d_w /mm	80，120，160
横梁间距a/mm	2 500，3 000，3 500
屈曲半波数e	15
屈服强度σ_Y /MPa	235，315，390

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表 2 纵向三跨（有实际强构件）模型的边界条件

Table 2. Boundary condition of longitudinal 3 spans model with actual strong members

施加范围	U_x	U_y	U_z	R_x	R_y	R_z
A1-B1，A1'-B1'， A2-B2，A2'-B2'			0		0	0
A1-A2，A1'-A2'	位移载荷	0	0	0		0
B1-B2，B1'-B2'	0	0	0	0		0
C1-D1，C2-D2，E1-F1，E2-F2， G1-H1，G2-H2，J1-K1，J2-K2			0

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表 3 纵向1/2+1+1/2跨（边界条件代替强横梁）模型的边界条件

Table 3. Boundary condition of longitudinal 1/2+1+1/2 spans (boundary conditions instead of strong beams) model

施加范围	U_x	U_y	U_z	R_x	R_y	R_z
A1-B1，A1'-B1'， A2-B2，A2'-B2'			0		0	0
A1-A2，A1'-A2'	位移载荷				0	0
B1-B2，B1'-B2'	0				0	0
K1-K2，L1-L2			0		0

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表 4 纵向单跨（边界条件代替强构件）模型的边界条件

Table 4. Boundary condition of longitudinal 1 span (boundary conditions instead of strong members) model

施加范围	U_x	U_y	U_z	R_x	R_y	R_z
A1-B1，A1'-B1'， A2-B2，A2'-B2'			0		0	0
A1-A2，A1'-A2'	位移载荷	0	0	0		0
B1-B2，B1'-B2'	0	0	0	0		0

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表 5 不同网格密度下的I型金属夹芯板数值计算结果

Table 5. Numerical results of I-core sandwich panels with different mesh densities

网格大小/mm	σ_u /MPa
1/8h_w	183.671 7
1/4h_w	184.319 9
1/2h_w	189.535 6
h_w	207.345 3

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表 6 不同训练函数的训练效果对比

Table 6. Performance comparison of different training function

函数	算法	迭代次数	迭代精度
Trainlm	Levenberg-Marquardt法	25	0.002 0
Traingd	梯度递减法	1 000	0.012 1
Traingdm	带动量因子的梯度递减法	1 000	0.034 7
Traingda	带自适应学习率的梯度递减法	1 000	0.005 9
Traingdx	带自适应学习率和动量因子的梯度递减法	1 000	0.010 1

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表 7 不同隐藏层层数及神经元个数的训练效果对比

Table 7. Performance comparison of different number of neurons

隐藏层1的神经元个数	隐藏层2的神经元个数	迭代次数	迭代精度
3	0	1 000	0.002 2
4	0	1 000	0.003 4
5	0	1 000	0.002 5
6	0	1 000	0.002 7
7	0	235	0.002 0
8	0	360	0.002 0
9	0	25	0.002 0
10	0	72	0.001 9
11	0	133	0.002 0
12	0	71	0.001 9
3	3	71	0.002 0
3	4	150	0.002 0
3	5	153	0.001 9
3	6	192	0.001 9
3	7	52	0.002 0
3	8	289	0.002 0
3	9	85	0.002 0

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表 8 有限元仿真结果

Table 8. The results of FE simulation

序号	λ	β_p	β_w	σ_u /σ_Y	序号	λ	β_p	β_w	σ_u /σ_Y
1	6.370 6	1.351 0	0.675 5	0.880 7	33	3.185 3	0.675 5	1.351 0	0.954 9
2	6.715 4	1.351 0	0.337 8	0.892 1	34	3.357 7	0.675 5	0.675 5	0.992 9
3	7.003 7	1.351 0	0.225 2	0.850 4	35	3.501 9	0.675 5	0.450 3	0.996 8
4	7.249 1	1.351 0	0.168 9	0.795 4	36	3.624 5	0.675 5	0.337 8	0.996 7
5	6.099 1	0.900 7	0.675 5	0.944 7	37	6.240 0	2.026 5	0.675 5	0.581 4
6	6.352 8	0.900 7	0.337 8	0.982 0	38	6.492 9	2.026 5	0.337 8	0.604 5
7	6.576 6	0.900 7	0.225 2	0.981 9	39	6.715 4	2.026 5	0.225 2	0.585 9
8	6.775 9	0.900 7	0.168 9	0.976 4	40	6.913 0	2.026 5	0.168 9	0.565 0
9	5.894 5	0.675 5	0.675 5	0.953 1	41	6.006 7	1.351 0	0.675 5	0.710 8
10	6.094 4	0.675 5	0.337 8	0.996 0	42	6.187 3	1.351 0	0.337 8	0.788 9
11	6.276 1	0.675 5	0.225 2	0.997 4	43	6.352 8	1.351 0	0.225 2	0.776 7
12	6.442 2	0.675 5	0.168 9	0.996 9	44	6.505 0	1.351 0	0.168 9	0.760 1
13	4.431 3	1.351 0	1.013 3	0.861 6	45	5.823 3	1.013 3	0.675 5	0.751 6
14	4.731 1	1.351 0	0.506 6	0.872 9	46	5.963 4	1.013 3	0.337 8	0.882 1
15	4.965 2	1.351 0	0.337 8	0.856 0	47	6.094 4	1.013 3	0.225 2	0.891 3
16	5.153 8	1.351 0	0.253 3	0.830 8	48	6.217 4	1.013 3	0.168 9	0.883 9
17	4.247 1	0.900 7	1.013 3	0.928 1	49	4.311 5	2.026 5	1.013 3	0.669 0
18	4.476 9	0.900 7	0.506 6	0.974 1	50	4.540 3	2.026 5	0.506 6	0.714 0
19	4.669 2	0.900 7	0.337 8	0.974 6	51	4.731 1	2.026 5	0.337 8	0.703 6
20	4.832 7	0.900 7	0.253 3	0.970 2	52	4.893 0	2.026 5	0.253 3	0.685 6
21	4.119 3	0.675 5	1.013 3	0.942 9	53	4.160 0	1.351 0	1.013 3	0.801 5
22	4.304 9	0.675 5	0.506 6	0.992 0	54	4.328 6	1.351 0	0.506 6	0.852 6
23	4.466 4	0.675 5	0.337 8	0.995 1	55	4.476 9	1.351 0	0.337 8	0.851 7
24	4.608 4	0.675 5	0.253 3	0.992 0	56	4.608 6	1.351 0	0.253 3	0.837 3
25	3.429 3	1.351 0	1.351 0	0.900 3	57	4.051 0	1.013 3	1.013 3	0.847 2
26	3.692 6	1.351 0	0.675 5	0.918 6	58	4.184 2	1.013 3	0.506 6	0.907 1
27	3.886 9	1.351 0	0.450 3	0.902 4	59	4.304 9	1.013 3	0.337 8	0.913 6
28	4.036 6	1.351 0	0.337 8	0.883 4	60	4.414 9	1.013 3	0.253 3	0.909 0
29	3.282 4	0.900 7	1.351 0	0.943 7	61	3.319 0	2.026 5	1.351 0	0.714 7
30	3.491 5	0.900 7	0.675 5	0.980 0	62	3.527 0	2.026 5	0.675 5	0.775 6
31	3.658 4	0.900 7	0.450 3	0.979 4	63	3.692 6	2.026 5	0.450 3	0.783 6
32	3.795 1	0.900 7	0.337 8	0.969 3	64	3.828 0	2.026 5	0.337 8	0.777 2

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表 9 I型金属夹芯板极限强度预测方程参数值

Table 9. Parameter values of prediction equation for ultimate strength of I-core sandwich panels

k	j	i	w_ij	b_j	w_jk	b_k
1	1	1	15.787 83	−8.060 5	−1.134 29	3.508 448
		2	−1.317 5
		3	1.242 68
	2	1	−3.184 67	14.235 29	0.202 705
		2	−12.944 9
		3	−1.725 95
	3	1	−5.953 61	1.084 991	−0.376 16
		2	11.236 42
		3	0.961 726
	4	1	29.143 18	−14.747 2	0.815 201
		2	−3.361 12
		3	4.365 897
	5	1	0.309 728	−3.401 43	−0.135 02
		2	15.531 94
		3	−12.604 4
	6	1	−1.301 16	−1.886 73	−2.492 59
		2	−6.731 54
		3	8.029 399
	7	1	6.911 944	−9.691 39	−0.172 38
		2	12.653 35
		3	6.526 226
	8	1	2.203 471	1.637 875	−2.538 26
		2	6.820 759
		3	−7.541 44
	9	1	−6.343 71	5.825 758	0.036 682
		2	−13.337 1
		3	−3.522 82

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表 10 I型金属夹芯板参数

Table 10. Parameters of I-core sandwich panels

序号	λ	β_p	β_w	σ_u /σ_Y
1	4.962 2	0.965 0	0.562 9	0.921 4
2	5.131 2	0.965 0	0.337 8	0.949 6
3	5.283 2	0.965 0	0.241 3	0.945 0
4	5.829 0	2.346 2	0.651 7	0.502 8
5	6.013 8	2.346 2	0.391 0	0.512 9
6	6.180 7	2.346 2	0.279 3	0.502 2
7	5.638 6	1.675 9	0.651 7	0.618 2
8	5.779 2	1.675 9	0.391 0	0.648 9
9	5.909 6	1.675 9	0.279 3	0.647 4
10	4.927 6	1.740 4	1.015 3	0.675 8
11	5.174 9	1.740 4	0.609 2	0.679 8
12	5.380 0	1.740 4	0.435 1	0.666 9
13	4.943 5	1.243 2	0.609 2	0.852 1
14	5.116 1	1.243 2	0.435 1	0.851 7
15	3.713 2	2.026 5	0.788 1	0.681 2
16	3.859 8	2.026 5	0.472 9	0.705 2
17	3.987 5	2.026 5	0.337 8	0.709 6
18	3.596 2	1.447 5	0.788 1	0.801 5
19	3.710 2	1.447 5	0.472 9	0.829 8
20	3.813 2	1.447 5	0.337 8	0.834 1

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