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Article(id=1281203833183388106, tenantId=1146029695717560320, journalId=1240685776644648972, issueId=1281203336514867310, articleNumber=null, orderNo=null, doi=10.3969/j.issn.1007-7294.2026.04.008, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1761667200000, receivedDateStr=2025-10-29, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1783392397446, onlineDateStr=2026-07-07, pubDate=1776182400000, pubDateStr=2026-04-15, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1783392397446, onlineIssueDateStr=2026-07-07, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1783392397446, creator=13041195026, updateTime=1783392397446, updator=13041195026, issue=Issue{id=1281203336514867310, tenantId=1146029695717560320, journalId=1240685776644648972, year='2026', volume='30', issue='4', pageStart='507', pageEnd='658', issueExtLink='null', onlineDate='null', pubDate='1776182400000', pubDateStr='2026-04-15', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1783392279032, creator='13041195026', updateTime=1783395286077, updator='13701087609', preIssue=null, nextIssue=null, articleTotal=null, ext={EN=IssueExt(id=1281215949713945277, tenantId=1146029695717560320, journalId=1240685776644648972, issueId=1281203336514867310, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1281215949713945278, tenantId=1146029695717560320, journalId=1240685776644648972, issueId=1281203336514867310, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null, downloadFileDto=null}, startPage=591, endPage=600, ext={EN=ArticleExt(id=1281203836467528139, articleId=1281203833183388106, tenantId=1146029695717560320, journalId=1240685776644648972, language=EN, title=Data-driven weight-stability dual-objective optimization of slender ring-stiffened cylindrical shells, columnId=1242129251223274417, journalTitle=Journal of Ship Mechanics, columnName=Structural Mechanics, runingTitle=null, highlight=null, articleAbstract=

In the multi-objective optimization of slender ring-stiffened cylindrical shells, it is difficult to balance the structural lightweight and critical pressure for overall instability, which restricts the improvement of the performance of Autonomous Underwater Vehicle. In this paper, a data-driven multi-objective optimization strategy is proposed, which can improve the critical pressure for overall instability and reduce the weight of the structure. Firstly, based on the error convergence criterion and accuracy requirement, the appropriate training set size is found in this strategy through iteration, and a low-cost and high-precision surrogate model is constructed. Then, the second-generation non-dominated sorting genetic algorithm is used to obtain the Pareto solution set. The non-inferior solution is screened by the minimum distance method based on maximum and minimum normalization, and the optimal design scheme with a well-balanced performance is obtained by combining the local accuracy enhancement strategy of the surrogate model. Using the optimization strategy proposed in this paper, the structural weight of the slender ring-stiffened cylindrical shell is reduced by 9.1%, and the critical pressure for the overall stability is increased by 13.4%. An effective design to improve the stability of the ring-stiffened cylindrical shell without increasing its weight is to increase the thickness of the cylindrical shell, increase the height of the stiffeners, and reduce their number.

, authors=Yu-fan LIN, Jing-xia YUE, Jia-rui LIU, Hai-sen HE, Peng ZHANG, authorsList=Yu-fan LIN, Jing-xia YUE, Jia-rui LIU, Hai-sen HE, Peng ZHANG, authorCompany=null, correspAuthors=Jing-xia YUE, authorNote=null, correspAuthorsNote=null, copyrightStatement=Copyright ©2026 Journal of Ship Mechanics. All rights reserved., copyrightOwner=null, extLink=null, articleAbsUrl=null, sourceXml=null, magXml=null, pdfUrl=null, pdf=null, pdfFileSize=null, pdfExtLink=null, richHtmlUrl=null, mobilePdfUrl=null, reviewReport=null, pdfFirstPage=null, abstractGraph=null, abstractGraphContent=null, abstractVideo=null, citation=null, cebUrl=null, magXmlContent=null, mapNumber=null, fund=null), CN=ArticleExt(id=1281203888137159170, articleId=1281203833183388106, tenantId=1146029695717560320, journalId=1240685776644648972, language=CN, title=数据驱动的细长环肋圆柱壳重量-稳定性双目标优化设计方法, columnId=1241023038926410098, journalTitle=船舶力学, columnName=结构力学, runingTitle=null, highlight=null, articleAbstract=

在细长环肋圆柱壳的多目标优化问题中，结构轻量化和整体失稳临界压力优化的目标难以兼顾，制约了自主水下航行器性能的提升。本文提出了一种数据驱动的多目标优化策略，能在提高结构整体失稳临界压力的同时降低结构重量。该策略首先基于误差收敛准则和精度要求，通过迭代寻找合适的训练集大小，构建了低成本的高精度代理模型；然后采用二代非支配排序遗传算法得到Pareto解集，通过采用基于最大最小归一化的最小距离法进行非劣解筛选，并结合代理模型局部精度增强策略得到高均衡性的最优设计方案。采用本文的优化策略，细长环肋圆柱壳的结构重量减轻了9.1%，整体失稳临界压力增加了13.4%。不增加环肋圆柱壳重量提高其稳定性的有效设计是：增大圆柱壳厚度、增加加强筋高度，同时减少加强筋数量。

, authors=林宇帆, 乐京霞, 刘家睿, 何海森, 张鹏, authorsList=林宇帆, 乐京霞, 刘家睿, 何海森, 张鹏, authorCompany=null, correspAuthors=乐京霞, authorNote=

林宇帆（2001–），男，硕士研究生

, correspAuthorsNote=
乐京霞（1977–），女，博士，教授，通讯作者，E-mail：
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林宇帆（2001–），男，硕士研究生

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林宇帆（2001–），男，硕士研究生

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Properties of AL-7178-T62 aluminum alloy

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名称材料密度$ \rho /（{{\mathrm{g}}}\cdot {\text{cm}}^{{-3}}） $ 屈服强度σs/MPa弹性模量E/GPa泊松比μ
AL-7178-T623.1560800.32
), ArticleFig(id=1281203933804741186, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281203833183388106, language=CN, label=表1, caption=

AL-7178-T62铝合金性质

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名称材料密度$ \rho /（{{\mathrm{g}}}\cdot {\text{cm}}^{{-3}}） $ 屈服强度σs/MPa弹性模量E/GPa泊松比μ
AL-7178-T623.1560800.32
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Main parameters of slender ring-stiffened cylindrical shell

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参数名称/单位数值参数名称/单位数值
圆柱壳半径 R/mm320加强筋厚度 t0/mm42
圆柱壳整体长度 L0/mm5960舱段长度 L/mm1920
圆柱壳厚度 t/mm13强框架厚度 t1/mm50
加强筋数量 n15强框架长度 l1/mm50
加强筋高度 h0/mm33
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细长环肋圆柱壳主要参数

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参数名称/单位数值参数名称/单位数值
圆柱壳半径 R/mm320加强筋厚度 t0/mm42
圆柱壳整体长度 L0/mm5960舱段长度 L/mm1920
圆柱壳厚度 t/mm13强框架厚度 t1/mm50
加强筋数量 n15强框架长度 l1/mm50
加强筋高度 h0/mm33
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Table of optimization variable constraint

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优化变量nt1/mmh0/mmt0/mmt/mm
上限1870404515
下限83020105
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优化变量取值范围表

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优化变量nt1/mmh0/mmt0/mmt/mm
上限1870404515
下限83020105
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RMSD of historical iteration advantages of different surrogate models

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RMSDKriging迭代稳定性OPM迭代稳定性RSM迭代稳定性RBF迭代稳定性
σ17.2%稳定3.0%稳定2.9%稳定1.7%稳定
σ27.5%波动1.7%稳定3.4%稳定1.4%稳定
σf6.6%稳定3.3%稳定3.8%稳定1.5%稳定
Pcr8.2%波动0.9%稳定2.4%稳定0.7%波动
G8.7%稳定0.5%稳定0.7%稳定0.5%稳定
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不同代理模型历史迭代优点处的RMSD

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RMSDKriging迭代稳定性OPM迭代稳定性RSM迭代稳定性RBF迭代稳定性
σ17.2%稳定3.0%稳定2.9%稳定1.7%稳定
σ27.5%波动1.7%稳定3.4%稳定1.4%稳定
σf6.6%稳定3.3%稳定3.8%稳定1.5%稳定
Pcr8.2%波动0.9%稳定2.4%稳定0.7%波动
G8.7%稳定0.5%稳定0.7%稳定0.5%稳定
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Relative error check of weight and critical pressure for overall stability

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符号/单位第1次迭代第4次迭代约束条件
代理模型值有限元值相对误差代理模型值有限元值相对误差
G/kg444.4442.3−0.49%434.0433.2−0.19%−
Pcr/MPa33.833.50.90%31.531.3−0.73%≥27.6
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重量及整体失稳临界压力相对误差校核

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符号/单位第1次迭代第4次迭代约束条件
代理模型值有限元值相对误差代理模型值有限元值相对误差
G/kg444.4442.3−0.49%434.0433.2−0.19%−
Pcr/MPa33.833.50.90%31.531.3−0.73%≥27.6
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RMSD iteration of each value in optimization stage 2

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RMSD迭代次数σ1σ2σfPcrG
11.70%1.42%1.53%0.73%0.54%
21.68%1.45%1.52%0.62%0.42%
31.72%1.45%1.54%0.53%0.37%
41.67%1.49%1.56%0.50%0.30%
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优化阶段2各值RMSD迭代情况

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RMSD迭代次数σ1σ2σfPcrG
11.70%1.42%1.53%0.73%0.54%
21.68%1.45%1.52%0.62%0.42%
31.72%1.45%1.54%0.53%0.37%
41.67%1.49%1.56%0.50%0.30%
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Comparison of target value between the initial design scheme and the optimal design scheme

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结构重量/kg整体失稳临界压力/MPa
初始设计方案476.727.6
最优设计方案433.231.3
变化率−9.1%13.4%
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初始设计方案与最优设计方案目标值对比

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结构重量/kg整体失稳临界压力/MPa
初始设计方案476.727.6
最优设计方案433.231.3
变化率−9.1%13.4%
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Comparison of four selection methods

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方法整体失稳临界压力/MPa变化率δp结构重量/kg变化率δG评价因子β
初始设计方案27.6−476.7−−
基于最大最小归一化的
最小距离法		31.313.4%433.2−9.1%4.65
极大极小化法33.822.5%453.2−4.9%1.56
权重法（1:1）36.130.8%473.39−0.7%1.05
权重法（1:50）31.514.1%434.03−9.0%4.50
权重法（1:100）28.9−11.2%423.264.7%2.46
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四种选择方法的对比

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方法整体失稳临界压力/MPa变化率δp结构重量/kg变化率δG评价因子β
初始设计方案27.6−476.7−−
基于最大最小归一化的
最小距离法		31.313.4%433.2−9.1%4.65
极大极小化法33.822.5%453.2−4.9%1.56
权重法（1:1）36.130.8%473.39−0.7%1.05
权重法（1:50）31.514.1%434.03−9.0%4.50
权重法（1:100）28.9−11.2%423.264.7%2.46
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数据驱动的细长环肋圆柱壳重量-稳定性双目标优化设计方法
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林宇帆 , 乐京霞 , 刘家睿 , 何海森 , 张鹏
船舶力学 | 结构力学 2026,30(4): 591-600
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船舶力学 |结构力学 2026 , 30 (4) : 591 -600
数据驱动的细长环肋圆柱壳重量-稳定性双目标优化设计方法
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林宇帆（2001–），男，硕士研究生

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林宇帆（2001–），男，硕士研究生

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林宇帆, 乐京霞 , 刘家睿, 何海森, 张鹏
作者信息
  • 1.武汉理工大学 船海与能源动力工程学院，武汉 430063
通讯作者:
乐京霞（1977–），女，博士，教授，通讯作者，E-mail：
作者简介:

林宇帆（2001–），男，硕士研究生

Data-driven weight-stability dual-objective optimization of slender ring-stiffened cylindrical shells
Yu-fan LIN, Jing-xia YUE , Jia-rui LIU, Hai-sen HE, Peng ZHANG
Affiliations
  • 1.School of Naval Architecture, Ocean and Energy Power Engineering, Wuhan University of Technology, Wuhan 430063, China
出版时间: 2026-04-15 doi: 10.3969/j.issn.1007-7294.2026.04.008
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摘要
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在细长环肋圆柱壳的多目标优化问题中，结构轻量化和整体失稳临界压力优化的目标难以兼顾，制约了自主水下航行器性能的提升。本文提出了一种数据驱动的多目标优化策略，能在提高结构整体失稳临界压力的同时降低结构重量。该策略首先基于误差收敛准则和精度要求，通过迭代寻找合适的训练集大小，构建了低成本的高精度代理模型；然后采用二代非支配排序遗传算法得到Pareto解集，通过采用基于最大最小归一化的最小距离法进行非劣解筛选，并结合代理模型局部精度增强策略得到高均衡性的最优设计方案。采用本文的优化策略，细长环肋圆柱壳的结构重量减轻了9.1%，整体失稳临界压力增加了13.4%。不增加环肋圆柱壳重量提高其稳定性的有效设计是：增大圆柱壳厚度、增加加强筋高度，同时减少加强筋数量。

细长环肋圆柱壳  /  优化设计  /  整体失稳临界压力  /  最小距离法

In the multi-objective optimization of slender ring-stiffened cylindrical shells, it is difficult to balance the structural lightweight and critical pressure for overall instability, which restricts the improvement of the performance of Autonomous Underwater Vehicle. In this paper, a data-driven multi-objective optimization strategy is proposed, which can improve the critical pressure for overall instability and reduce the weight of the structure. Firstly, based on the error convergence criterion and accuracy requirement, the appropriate training set size is found in this strategy through iteration, and a low-cost and high-precision surrogate model is constructed. Then, the second-generation non-dominated sorting genetic algorithm is used to obtain the Pareto solution set. The non-inferior solution is screened by the minimum distance method based on maximum and minimum normalization, and the optimal design scheme with a well-balanced performance is obtained by combining the local accuracy enhancement strategy of the surrogate model. Using the optimization strategy proposed in this paper, the structural weight of the slender ring-stiffened cylindrical shell is reduced by 9.1%, and the critical pressure for the overall stability is increased by 13.4%. An effective design to improve the stability of the ring-stiffened cylindrical shell without increasing its weight is to increase the thickness of the cylindrical shell, increase the height of the stiffeners, and reduce their number.

slender ring-stiffened cylindrical shell  /  optimization design  /  the critical pressure for overall instability  /  minimum altitude distance method
林宇帆, 乐京霞, 刘家睿, 何海森, 张鹏. 数据驱动的细长环肋圆柱壳重量-稳定性双目标优化设计方法. 船舶力学, 2026 , 30 (4) : 591 -600 . DOI: 10.3969/j.issn.1007-7294.2026.04.008
Yu-fan LIN, Jing-xia YUE, Jia-rui LIU, Hai-sen HE, Peng ZHANG. Data-driven weight-stability dual-objective optimization of slender ring-stiffened cylindrical shells[J]. Journal of Ship Mechanics, 2026 , 30 (4) : 591 -600 . DOI: 10.3969/j.issn.1007-7294.2026.04.008
正文
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0　引　言
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细长环肋圆柱壳作为自主水下航行器AUV（Autonomous Underwater Vehicle）的核心承压部件，其轻量化与整体失稳临界压力优化是突破大潜深作业瓶颈的关键。细长环肋圆柱壳因其大长径比，在外部静水压力作用下易发生整体屈曲失效[1]。为提升稳定性，传统设计采用均匀加厚壳体或密集环肋布局，但这往往导致重量激增，直接影响AUV的续航与搭载能力。因此，针对细长环肋圆柱壳的结构重量和整体失稳临界压力进行双目标优化，对其结构安全与轻量化具有重要意义。
目前，国内外针对加筋圆柱耐压壳的优化研究主要集中于以重量或整体失稳临界压力为单一目标的设计改进。黄加强[2]以结构重量最小化为唯一目标，针对超长舱段加筋圆柱壳结构，采用遗传算法进行优化。结果表明，采用高强度材料并结合变刚度设计理论，可有效地实现减重。Foryś[3]将整体失稳临界压力作为单一目标，结合粒子群优化算法与有限元方法研究了环向加筋圆柱壳在外部压力下的优化设计，并通过调节内环几何参数与分布实现了优化目标的显著提升。Yang等[4]以结构重量最小化为目标，通过数据驱动方法优化了新型AUV耐压壳体的结构重量。Meng等[5]则结合等效刚度法和克里格模型（Kriging Model），提出了一种目标追踪的单目标优化方法，并用该方法对加筋圆柱壳结构进行了轻量化设计。由此可见，由于重量和稳定性的优化目标相互冲突，现有研究大多未对加筋圆柱耐压壳进行结构重量与整体失稳临界压力的双目标优化。
对于工程结构的优化问题，许多学者采用代理模型法进行多目标优化设计。丘文桢等[6]在三立柱半潜平台的优化设计中将最大垂荡运动响应和结构重量作为优化目标，利用径向基代理模型RBF（Radial Basis Function）进行多目标优化设计，得到了结果较好的Pareto解集。王威等[7]以复合装甲结构为研究对象，将Kriging模型应用到了弹道极限速度和面密度的多目标优化设计中。Kim等[8]通过融合深度学习的代理模型和多目标优化算法，有效实现了汽车刹车系统活塞表观行程与阻力扭矩的多目标优化。Lee等[9]提出了基于多保真度的神经网络代理模型与二代非支配排序遗传算法NSGA-II（Non-dominated Sorting Genetic Algorithm II）结合的方法，并针对车辆悬挂系统的行程、加速度及应力进行了多目标优化设计。但是，在高维度、强非线性的细长环肋圆柱壳多目标优化中，传统代理模型方法存在两个主要问题：一是局部近似精度不足；二是由于缺少先验知识，无法预先确定合适的训练集大小，从而难以一次性构建满足精度需求的代理模型[10]。
针对上述问题，本文提出一种数据驱动的多目标优化策略。该策略能够自主选择合适的训练集大小，提高代理模型的局部精度，并实现对细长环肋圆柱壳重量和整体失稳临界压力的双目标优化。在结构主尺度不变的条件下，本文将整体失稳临界压力最大化和结构重量最小化作为优化目标，以强度、稳定性为约束条件，并运用所提出的优化策略，得到了细长环肋圆柱壳的最优设计方案。
1　数据驱动的多目标优化策略
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1.1　数据驱动的二阶段优化策略
本文改良了基于目标追踪的优化策略[5]，提出一种数据驱动的多目标优化策略。优化阶段1通过误差收敛准则和精度要求，以低训练成本训练出高精度的代理模型；优化阶段2基于相对误差分析，通过迭代提高了代理模型的局部精度，从而保证了基于代理模型筛选出的最优设计方案的可靠性。其具体流程如图1所示。
在优化阶段1，首先通过拉丁超立方抽样方法LHS（Latin Hypercube Sampling）抽取样本点并通过仿真软件Abaqus进行有限元分析，得到有限元数据集。然后，将该数据分成测试集和训练集，并用训练集对代理模型进行训练。随后，通过测试集对代理模型进行精度校核，若模型的均方根误差小于最大允许均方根误差，则输出历史精度最高的代理模型至优化阶段2；若大于，则需要进行误差收敛性校核。当连续两次迭代的均方根误差满足式（1）时，判断为误差收敛，也进入优化阶段2。式（1）如下所示
$ |{\varepsilon }_{k}-{\varepsilon }_{k-\text{1}}| \lt \theta $
式中：εk为第k次迭代的均方根误差值，εk−1为第k−1次迭代的均方根误差值，θ为终止阈值，本文取θ = 1×10−3。
若误差收敛性不满足要求，则通过LHS方法抽取新的样本点添加至训练集重新训练模型，直到模型满足精度要求或收敛性要求。本文设定的最大允许均方根误差为1%。
在优化阶段2，基于优化阶段1的历史精度最高的代理模型，首先通过NSGA-II算法得到Pareto解集。然后，通过基于最大最小归一化的最小距离法选出最优设计方案，并对最优设计方案进行有限元验证，得到有限元计算结果和代理模型的预测值的相对误差。若相对误差不满足要求，则添加该有限元结果进入训练集，重新训练代理模型后重复上述流程；若满足相对误差要求，则输出该方案作为最优设计方案。本文设定的最大允许相对误差为1%。
1.2　优化阶段2最小距离法的实现
在优化阶段2得到的Pareto解集包含了大量非劣解，需从中选取一个最终设计方案用于工程实践。为此，常采用理想函数法进行处理，该方法主要包括权重法、极大极小化方法和最小距离法等[11]。权重法计算简单，但结果受权值影响显著[12]；极大极小化方法均衡性较强，但计算复杂且可能陷入局部最优；最小距离法全面性好，但其效果高度依赖于距离的定义方式。传统的最小距离法[13]在定义计算距离时，没有考虑多目标优化问题中各优化目标量纲不同的问题。本文采用最大最小归一化处理各优化目标，并设计了一种基于该方法的最小距离法。其均衡性较极大极小化方法更好，且消除了量纲的影响。改进后的最小距离法可用式（2）表示
$ \min D={\left\{{\left[{P}_{\tau }-\max ({P}_{\tau })\right]}^{2}+{\left[{G}_{\tau }-\min ({G}_{\tau })\right]}^{2}\right\}}^{1/2} $
式中：D为计算距离，Pτ、Gτ分别为第τ个Pareto解最大最小归一化后的整体失稳临界压力和结构重量；max（Pτ）、min（Gτ）分别为Pareto解集最大最小归一化后整体失稳临界压力的最大值和结构重量的最小值。最小的D所对应的优化方案，即为最优设计方案。
最大最小归一化的方式如式（3）所示
$ {{{X}^{\prime}_{\tau }}}=\frac{{X}_{i}-{X}_{\min }}{{X}_{\max }-{X}_{\min }} $
式中：$ {{{X}^{\prime}_{\tau }}} $为第τ个Pareto解集的最大最小归一化值，Xmax、Xmin分别为Pareto解集中的最大、最小值。
2　考虑重量和稳定性的细长环肋圆柱壳双目标优化
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在采用第1章的优化策略对细长环肋圆柱壳进行优化前，需建立有限元模型并确定优化模型设置，其具体设置将在本章中详细说明。
2.1　细长环肋圆柱壳的有限元模型
本文以1500 m潜深的细长环肋圆柱壳为优化对象，其整体及舱段结构如图2所示。整个耐压壳体由3个舱段组成，各舱段间通过强框架连接，每个舱段内部均采用等间距布置的加强筋进行加强。
铝合金具有比重小、强度高、抗腐蚀性强等良好性能，适用于超大潜深的细长环肋圆柱壳结构。本文采用AL-7178-T62铝合金作为细长环肋圆柱壳的材料，其材料性质如表1所示。
本文参考中国船级社《潜水系统与潜水器入级规范》[14]（简称《规范》），计算压力如式（4）所示
$ {P}_{{\mathrm{j}}}=n\rho g{h}_{{\mathrm{e}}} $
取计算深度he = 1500 m，海水密度ρ = 1020 kg/m3，重力加速度g = 9.81 m/s2，安全系数n = 1.5，计算得Pj = 22.5 MPa。
初始设计方案的结构尺寸如表2所示。
有限元模型的边界条件采用两端$ {u}_{\text{1}}={u}_{\text{2}}=0 $，在细长环肋圆柱壳表面施加压强形式的载荷，有限元模型如图3所示。通过收敛性分析，将7 mm作为网格划分尺寸，网格类型采用S4R。
2.2　优化模型
2.2.1　优化变量与目标函数
本文在优化过程中，保持细长环肋圆柱壳的半径R、长度L等主尺度保持不变，将加强筋数量n、强框架厚度t1、加强筋高度h0及厚度t0和壳体厚度t作为优化变量。优化变量的取值范围如表3所示，所有变量为整型变量。
目标函数如式（5）所示
$ \min f\left(x\right)=\left\{{f}_{1}\left(x\right),\frac{1}{{f}_{2}\left(x\right)}\right\} $
式中:$ {f}_{1}\left(x\right) $为结构重量，$ {f}_{2}\left(x\right) $为细长环肋圆柱壳的整体失稳临界压力。
2.2.2　约束条件
根据《规范》[14]要求，设定细长环肋圆柱壳的强度及稳定性约束条件如式（6）所示
$ \begin{cases} {\sigma }_{\text{1}}\leq \text{0.85}{\sigma }_{\text{s}}\\{\sigma }_{2}\leq 1.15{\sigma }_{\text{s}}\\{\sigma }_{\text{f}}\leq \text{0.6}{\sigma }_{\text{s}}\\{P}_{{\mathrm{cr}}}\geq 1.2{P}_{{\mathrm{j}}}\end{cases} $
式中：$ {\sigma }_{1} $为肋骨跨端壳板内表面周向应力，$ {\sigma }_{2} $为肋骨跨端壳板内表面轴向应力，$ {\sigma }_{\text{f}} $为肋骨应力，Pcr为细长环肋圆柱壳整体失稳的临界压力。
3　优化过程及结果分析
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3.1　优化阶段1迭代过程及结果
代理模型的训练误差和训练集大小的关系具有复杂性，在训练中可能出现强波动和过拟合问题，难以确定合适的训练集大小。基于1.1节的优化策略流程，图4展示了不同代理模型的整体失稳临界压力均方根误差RMSD（Root-Mean-Square Deviation）随样本点数量增加的变化趋势。由图可知，虽然RBF代理模型的RMSD总体随样本量增加呈下降趋势，但表现出强波动性。这是由于其鲁棒性较差，易受噪声影响，且整体失稳临界压力的非线性特征显著增加了拟合难度。Kriging代理模型的RMSD值随样本量增加呈上升趋势，这是由于训练集样本点增加时出现了过拟合现象。
为解决上述过拟合与波动性问题，我们采用了以下优化策略。通过优化阶段1，能够在给定误差范围内自动寻找适合的训练集大小，并在误差收敛或满足精度要求时停止迭代，进而避免增加训练成本。如图4所示，存在过拟合问题的Kriging代理模型在220个样本点时依据收敛性准则终止迭代，最终选定160个样本点作为最终训练集样本点数目。而对于具有强波动性的RBF代理模型，则根据精度要求在200个样本点及时停止迭代，并选择200作为最终训练集样本点数目。此外，响应面RSM（Response Surface Methodology）和正交多项式OPM（Orthogonal Polynomial Method）代理模型同样能适时终止迭代，选取合适的训练集大小。
表4统计了各模型在优化阶段1历史迭代优点处的RMSD。从表中可以看出，RBF代理模型只在Pcr的训练中具有一定的波动性，且其RMSD值最低、精度最高，故选用RBF代理模型进行后续优化。
3.2　优化阶段2分析
3.2.1　迭代过程与优化结果
高维度非线性的多目标优化问题中，其优化变量和结构响应之间的映射关系在局部可能非常复杂，这导致代理模型难以精确拟合，从而出现局部精度不足的问题。图5给出了优化阶段2不同迭代次数时的强度相对误差校核结果。结果显示，第1次迭代时周向应力σ1和轴向应力σ2的相对误差较大，且轴向应力σ2的有限元计算结果未满足强度约束。这表明较大的相对误差会降低最优设计方案选择的合理性，需对模型进行进一步修正。
优化阶段2能够提高代理模型的局部精度，保证最优设计方案选择的合理性。从图中可以看出随着迭代次数的增加，周向应力σ1、轴向应力σ2及肋骨应力σf的相对误差均下降，最终在第4次时满足精度要求。表5给出了结构重量及整体失稳临界压力的相对误差校核结果，从表中可以看出，第4次迭代时亦满足精度要求，故优化阶段2在第4次迭代时停止。
表6给出了各项关键响应（如各应力及临界压力） 在迭代过程中的RMSD变化情况。可以发现，优化阶段2各值RMSD的变化十分微小，基本不改变代理模型的整体精度。这是由于相对于第一阶段的训练集数量，第二阶段每次迭代添加的数据样本量是极小的，故对整体模型的精度影响有限。
图6对比了初始设计方案和最优设计方案的优化变量。可以发现，最优设计方案通过降低加强筋数量n和强框架厚度t1，适当增加加强筋高度h0和圆柱壳厚度t，实现了结构重量的降低与整体失稳临界压力的提升。
表7对比了初始设计方案和最优设计方案的结构重量和整体失稳临界压力。与初始设计方案相比，最优设计方案的结构重量下降了9.1%，整体失稳临界压力增加了13.4%，证明了该优化策略能够有效实现细长环肋圆柱壳结构重量和整体失稳临界压力的双目标优化。
鉴于初始设计方案和最优设计方案的失效变形都位于壳体的中部，图7对比了初始设计方案和最优设计方案的中部舱段位移云图。可以发现，最优设计方案的中部舱段的最大变形范围有所缩小，使整个结构更加趋于同步破坏，提高了结构的稳定性。
3.2.2　基于最大最小归一化的最小距离法有效性分析
为了验证基于最大最小归一化的最小距离法的有效性，本文将其与权重法（设置权重比分别为1∶1、1∶50和1∶100）及极大极小化法进行对比分析。本文采用评价因子β评价各选择方法。评价因子考虑了各目标的综合优化程度和优化的均衡性，其定义式如式（7）所示
$ \beta =\frac{\left| \left| {\delta }_{{\mathrm{p}}}\right| \text+\left| {\delta }_{G}\right| \right| }{\left| \left| {\delta }_{{\mathrm{p}}}\right| -\left| {\delta }_{G}\right| \right| } $
其中，δp为整体失稳临界压力的变化率，δG为结构重量的变化率，β为评价因子。评价因子越大，选择方法越优越。
对阶段2的第4次迭代得到的Pareto解集分别使用以上三种方法，得到最优设计方案的整体失稳临界压力、结构重量，以及其相对于初始设计方案的变化率和评价因子如表8所示。
从表8可以看出，当权重法取值为1∶50时其评价因子β最高，但其最优权重的范围在1∶1到1∶100之间，难以确定；采用极大极小化法虽能大幅提升整体失稳临界压力，但其对结构重量的优化却比较有限，故其评价因子β较小；而基于最大最小归一化的最小距离法在降低结构重量9.1%的同时，使整体失稳临界压力提升了13.4%，其评价因子β相较于其他选择方法最高，验证该方法的有效性和均衡性。
3.3　优化变量敏感性分析
本文基于优化阶段1的有限元分析数据，分析得到了各优化变量的全局影响因子。该全局影响因子通过回归分析得到，反映了优化变量对结构重量、整体失稳临界压力及周向、轴向、肋骨应力的影响程度。图8~12给出了全局影响因子的具体值，其中红色代表正相关，蓝色代表负相关。
由图8可知，加强筋高度h0对整体失稳临界压力的影响最大，故应该通过增大加强筋高度h0来提高细长环肋圆柱壳的稳定性。
由图9可知，对结构重量的影响最大的因素为圆柱壳厚度t和加强筋数量n，其影响因子分别为0.61和0.25。从数值上看，应该通过减小圆柱壳厚度t和减少加强筋数量n来降低结构重量。但从图10~12可以看出，圆柱壳厚度t对周向、轴向、肋骨应力影响较大，降低圆柱壳厚度t易导致设计方案不满足强度约束条件。因此，应该保持圆柱壳厚度t不变或增加，以满足强度约束条件；通过减少加强筋数量n，以降低结构重量。
4　结　论
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本文提出了一种数据驱动的多目标优化策略，并通过该优化策略对细长圆柱耐压壳的结构重量和整体失稳临界压力进行了双目标优化。在此基础上，得到如下结论：
（1）该数据驱动的多目标优化策略应用于细长环肋圆柱壳，使结构重量降低了9.1%，整体失稳临界压力提升了13.4%，验证了其有效性。
（2）优化阶段1能够在给定误差范围内自动确定合适的训练集规模，从而以较低的计算成本构建出满足精度要求的代理模型。优化阶段2能有效提升代理模型的局部拟合精度，将关键响应（如应力）的代理模型预测值与有限元计算值之间的相对误差控制在较低水平（如1%以内），从而保证了基于代理模型所选最优设计方案的可靠性。
（3）本文改良了最小距离法，提出了一种基于最大最小归一化的最小距离选择方法，并验证了该方法能够有效地实现多目标优化，且具备良好的均衡性。
（4）优化变量的全局影响因子分析表明，为同时实现减重和增强稳定性，应优先考虑增加加强筋高度以显著提升整体失稳临界压力，并减少加强筋数量以有效降低结构重量；而圆柱壳厚度对强度至关重要，需在满足约束条件下审慎确定，其增减对重量和临界压力的影响需综合权衡。

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2026年第30卷第4期
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doi: 10.3969/j.issn.1007-7294.2026.04.008
  • 接收时间：2025-10-29
  • 首发时间：2026-07-07
  • 出版时间：2026-04-15
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  • 收稿日期：2025-10-29
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    1.武汉理工大学 船海与能源动力工程学院，武汉 430063

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乐京霞（1977–），女，博士，教授，通讯作者，E-mail：
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