Chinese Journal of Ship Research, Volume. 18, Issue 3, 13(2023)
Application of uncertainty design optimization based on polynomial chaos expansions and maximum entropy method in ship design
Figures & Tables(16)
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Table 1. Polynomial coefficients of function
G under different expansion ordersView tableView in ArticleTable 1. Polynomial coefficients of function
G under different expansion orders系数 2阶展开 3阶展开 4阶展开 准确值 IPCM法 PCM法 IPCM法 PCM法 IPCM法 PCM法 6个样本 9个样本 10个样本 25个样本 15个样本 25个样本 $ {b_0} $ 4 682.300 4 682.300 4 682.300 4 682.300 4 682.300 4 682.300 4 682.299 $ {b_1} $ 2 864.99 2 864.99 2 865.00 2 865.00 2865.00 2865.00 2865.00 $ {b_2} $ 2 345.15 2 345.15 2 470.15 2 470.15 2 470.15 2 470.15 2 470.15 $ {b_3} $ 997.50 997.50 997.50 997.50 997.50 997.50 997.50 $ {b_4} $ 742.50 742.50 742.50 742.50 742.50 742.50 742.50 $ {b_5} $ 716.497 526.359 500.010 500.010 500.010 500.010 500.000 $ {b_6} $ 125.00 125.00 125.00 125.00 125.00 $ {b_7} $ 125.02 125.00 125.00 125.00 125.00 $ {b_8} $ 125.00 125.00 125.00 125.00 125.00 $ {b_9} $ 0.01 0 3.51×10−3 5.74×10−4 0 $ {b_{10}} $ 6.82×10−4 1.38×10−4 $ {b_{11}} $ 3.66×10−4 7.19×10−5 $ {b_{12}} $ 1.08×10−3 6.72×10−4 $ {b_{13}} $ 5.17×10−4 1.58×10−5 $ {b_{14}} $ 1.85×10−3 4.25×10−4
Table 2. First 4 order moments solved by PCE and MC
View tableView in ArticleTable 2. First 4 order moments solved by PCE and MC
方法名称 均值 标准差 偏态系数 峰度系数 3阶PCE法 4 682.30 4 227.46 1.880 3 8.636 2 MC法 4 684.59 4 231.60 1.887 2 8.684 1
Table 3. Engineering optimization model for bulk carriers
View tableView in ArticleTable 3. Engineering optimization model for bulk carriers
设计变量 参数值 船长L/m $150 \leqslant L \leqslant 274.32$ 船宽B/m $20 \leqslant B \leqslant 32.31$ 型深D/m $ 13 \leqslant D \leqslant 25 $ 吃水T/m $10 \leqslant T \leqslant 11.71$ 方形系数CB $ 0.63 \leqslant {C_{\text{B}}} \leqslant 0.75 $ 设计航速Vk/kn $ 14 \leqslant {V_{\text{k}}} \leqslant 20 $
Table 4. The mean and the standard deviation of the objective and the constraint under the influence of uncertainty
View tableView in ArticleTable 4. The mean and the standard deviation of the objective and the constraint under the influence of uncertainty
目标或约束的待定系数 均值 误差/% 标准差 误差/% MC法 PCE法 MC法 PCE法 TC 10.06 9.92 1.39 4.21 4.29 1.90 Fn≤0.32 0.169 9 0.169 9 0 0.018 0.018 0 25 000≤ DW ≤500 000 3.408×104 3.479×104 1.95 8 266 8 223 0.52 T－0.45DW 0.31 ≤0 0.309 0.304 1.64 1.097 1.133 3.18
Table 5. The failure probability
p f of constraints using MEM and MC methodsView tableView in ArticleTable 5. The failure probability
p f of constraints using MEM and MC methods约束条件的待定系数 失效概率pf /% MC法 MEM法 Fn≤0.32 0 0 25 000≤ DW ≤500 000 11.2 11.3 T－0.45DW 0.31 ≤0 47.0 46.7
Table 6. Design variables, uncertain variables, constraints and objectives of UDO
View tableView in ArticleTable 6. Design variables, uncertain variables, constraints and objectives of UDO
参数 优化设计类别 RDO RBDO RBRDO 设计变量 参见 表3 不确定量 X：μX=1.7，σX=0.1μX Y：μY=0.7，σY=0.1μY Z：μZ=0，σZ =0.1Vk 约束条件 参见式（21） pf (Fn ≤0.32)≤1% pf (25 000 ≤DW≤500 000)≤1% pf (T－0.45DW 0.31≤0)≤1% 目标函数 min μ(TC) 和 min σ(TC) min TC min μ(TC) 和 min σ(TC)
Table 7. The comparison of optimization results
View tableView in ArticleTable 7. The comparison of optimization results
目标或约束 系数值 初始方案[ 10 ]DO RDO RBDO RBRDO PCE法 MC法 PCE法 MC法 PCE法 MC法 TC 确定值 9.93 8.45 8.49 8.50 8.58 8.58 8.60 8.64 均值 10.13 9.79 9.80 10.08 10.09 10.04 10.02 标准差 4.29 3.55 3.57 4.13 4.15 4.09 4.08 Fn ≤0.32 失效概率pf 0 0 0 0 0 0 0 0 标准差 0.018 0.018 0.016 0.016 0.018 0.018 0.017 0.017 25 000 ≤DW≤500 000 失效概率pf 13.3% 11.3% 9.2% 9.0% 0 0 0.4% 0.1% 标准差 8 886 8 223 8 167 8 171 8 355 8 371 8 097 8 092 T－0.45DW 0.31≤0 失效概率pf 14.2% 46.7% 32.5% 32.3% 0 0 0.3% 0.1% 标准差 1.203 1.133 1.059 1.092 1.143 1.160 1.004 1.003
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Xiao WEI, Heng LI, Chenran HUANG. Application of uncertainty design optimization based on polynomial chaos expansions and maximum entropy method in ship design[J]. Chinese Journal of Ship Research, 2023, 18(3): 13
Paper Information
Category: Ship Design and Performance
Received: Oct. 24, 2021
Accepted: --
Published Online: Mar. 20, 2025
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