Ultimate strength prediction of I-core sandwich plate based on BP neural network
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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βp、腹板柔度系数βw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。 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. -
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
${{{\sigma _{{\rm{u}}}}} \mathord{\left/ {\vphantom {{{\sigma _{{\rm{u}}}}} {{\sigma _{{\rm{Y}}}}}}} \right. } {{\sigma _{{\rm{Y}}}}}}$ 图 15 输入变量βp,βw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 βp, βw, 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tp /mm 2,3,4 腹板厚度tw /mm 2,4,6,8 腹板高度hw /mm 40,60,80 腹板间距dw /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σY /MPa 235,315,390 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 Ux Uy Uz Rx Ry Rz 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-K20 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 Ux Uy Uz Rx Ry Rz 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 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 Ux Uy Uz Rx Ry Rz 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 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 σu /MPa 1/8hw 183.671 7 1/4hw 184.319 9 1/2hw 189.535 6 hw 207.345 3 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 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 表 8 有限元仿真结果
Table 8. The results of FE simulation
序号 λ βp βw σu /σY 序号 λ βp βw σu /σY 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 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 wij bj wjk bk 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 表 10 I型金属夹芯板参数
Table 10. Parameters of I-core sandwich panels
序号 λ βp βw σu /σY 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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