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Article(id=1241838550346436703, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1241838547544641612, articleNumber=null, orderNo=null, doi=10.19636/j.cnki.cjsm42-1250/o3.2024.058, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1730908800000, receivedDateStr=2024-11-07, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1774006982646, onlineDateStr=2026-03-20, pubDate=1745337600000, pubDateStr=2025-04-23, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1774006982646, onlineIssueDateStr=2026-03-20, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1774006982646, creator=13701087609, updateTime=1774006982646, updator=13701087609, issue=Issue{id=1241838547544641612, tenantId=1146029695717560320, journalId=1241755870837649424, year='2025', volume='46', issue='2', pageStart='149', pageEnd='296', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1774006981978, creator=13701087609, updateTime=1774007061443, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1241838880882758461, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1241838547544641612, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1241838880882758462, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1241838547544641612, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=257, endPage=274, ext={EN=ArticleExt(id=1241838552116433003, articleId=1241838550346436703, tenantId=1146029695717560320, journalId=1241755870837649424, language=EN, title=A New Type of Non-probabilistic Convex Model for Structural Uncertainty Analysis, columnId=1241831201674171363, journalTitle=Chinese Journal of Solid Mechanics, columnName=Research Papers, runingTitle=null, highlight=null, articleAbstract=

The non-probabilistic convex model requires only the boundaries of structurally uncertain parameters, making it suitable for dealing with engineering problems with limited samples. However, existing convex models primarily focus on regular mathematical models, potentially leading to an excessive expansion of the uncertainty domain. This paper introduces a new type of convex model, namely the interval and ellipsoidal intersection model, to more accurately constrain the uncertainty domain, and examines its application in structural uncertainty propagation analysis. Firstly, the interval and ellipsoidal intersection model is proposed to describe the uncertainty domain, which is constructed by taking the intersection of the interval model and the ellipsoidal model. Subsequently, the proposed model is applied to structural uncertainty propagation analysis with two cases of nonlinear response functions. For the weakly nonlinear response function, a linear approximation is derived using the first-order Taylor series expansion, and then a semi-analytical method is developed to predict its structural response interval. For the strongly nonlinear response function, a nonlinear approximation is achieved using the second-order Taylor series expansion, and the sequential quadratic programming (SQP) method is adopted to predict its structural response interval. Finally, results from four numerical examples indicate that the proposed model generally offers a smaller uncertainty domain and narrower structural response interval compared to the traditional interval and ellipsoidal models. Additionally, the semi-analytical method is more efficient than the SQP method and the Monte Carlo simulation (MCS) method.

, correspAuthors=Xinzhou Qiao, authorNote=null, correspAuthorsNote=null, copyrightStatement=null, 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, authorCompany=null, fund=null, authors=null, authorsList=Xinzhou Qiao, Jiahui Li, Xiurong Fang, Peng Liu), CN=ArticleExt(id=1241838574706954951, articleId=1241838550346436703, tenantId=1146029695717560320, journalId=1241755870837649424, language=CN, title=一种结构不确定性分析的新型非概率凸集模型, columnId=1241831201896469478, journalTitle=固体力学学报, columnName=研究论文, runingTitle=null, highlight=null, articleAbstract=

非概率凸集模型仅需获知不确定性参数的范围或界限，适用于处理工程结构中常见的小样本问题. 本文提出一种新型非概率凸集模型-区间椭球交集模型来描述不确定域，并研究相应的结构不确定性传播分析方法. 首先提出区间椭球交集模型来描述不确定域，该模型可通过对区间模型和椭球模型进行取交运算构建. 其次，将区间椭球交集模型应用于结构不确定性传播分析. 针对弱非线性响应函数，对其进行泰勒一阶展开近似，通过半解析法对结构响应区间求解；针对强非线性响应函数，对其进行泰勒二阶展开近似，采用序列二次规划法对结构响应区间求解. 最后，四个算例分析验证了文中所提模型和方法的有效性和可行性.

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language=EN, label=Fig.9, caption=The roof truss structure, figureFileSmall=slWrXBGbRO/JWywR65ngsQ==, figureFileBig=IQdUbL/AE2vprn/7MPAGkg==, tableContent=null), ArticleFig(id=1241838579731731316, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838550346436703, language=CN, label=图9, caption=屋架结构, figureFileSmall=slWrXBGbRO/JWywR65ngsQ==, figureFileBig=IQdUbL/AE2vprn/7MPAGkg==, tableContent=null), ArticleFig(id=1241838579807228788, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838550346436703, language=EN, label=Table 1, caption=

The characteristics of different models

, figureFileSmall=null, figureFileBig=null, tableContent=
模型样本适应性能否处理相关性问题建模效率模型精度
区间弱不能高低
椭球弱能高低
平行六面体中能中中
多边凸集中能中中
凸包强能低高
), ArticleFig(id=1241838579874337653, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838550346436703, language=CN, label=表1, caption=

不同模型的特点

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模型样本适应性能否处理相关性问题建模效率模型精度
区间弱不能高低
椭球弱能高低
平行六面体中能中中
多边凸集中能中中
凸包强能低高
), ArticleFig(id=1241838580012749686, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838550346436703, language=EN, label=Table 2, caption=

The ratio of the volume of four models to that of convex hull

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模型体积比（模型体积/凸包体积）
第一组第二组第三组
区间模型3.7791.0503.625
椭球模型1.6051.4021.598
平行六面体模型1.0551.0501.047
区间椭球交集模型1.3181.0261.317
), ArticleFig(id=1241838580113412985, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838550346436703, language=CN, label=表2, caption=

四种模型体积与凸包体积的比值

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模型体积比（模型体积/凸包体积）
第一组第二组第三组
区间模型3.7791.0503.625
椭球模型1.6051.4021.598
平行六面体模型1.0551.0501.047
区间椭球交集模型1.3181.0261.317
), ArticleFig(id=1241838580184716152, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838550346436703, language=EN, label=Table 3, caption=

The first sample data

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No. F1（kN） F2（kN）No. F1（kN） F2（kN）
11.8085.57671.4565.376
21.3765.54081.6765.424
31.4285.64891.2165.092
41.2045.268101.9125.492
51.8725.468111.7125.836
61.5645.704121.0165.436
), ArticleFig(id=1241838580272796538, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838550346436703, language=CN, label=表3, caption=

第一组样本数据

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No. F1（kN） F2（kN）No. F1（kN） F2（kN）
11.8085.57671.4565.376
21.3765.54081.6765.424
31.4285.64891.2165.092
41.2045.268101.9125.492
51.8725.468111.7125.836
61.5645.704121.0165.436
), ArticleFig(id=1241838580381848445, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838550346436703, language=EN, label=Table 4, caption=

The second sample data

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No. F1（kN） F2（kN）No. F1（kN） F2（kN）
11.3525.09271.4225.436
21.0925.24381.2235.345
31.6035.73491.4035.365
41.3415.836101.3565.548
51.4135.624111.3015.213
61.2135.435121.3235.714
), ArticleFig(id=1241838580469928832, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838550346436703, language=CN, label=表4, caption=

第二组样本数据

, figureFileSmall=null, figureFileBig=null, tableContent=
No. F1（kN） F2（kN）No. F1（kN） F2（kN）
11.3525.09271.4225.436
21.0925.24381.2235.345
31.6035.73491.4035.365
41.3415.836101.3565.548
51.4135.624111.3015.213
61.2135.435121.3235.714
), ArticleFig(id=1241838580583175043, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838550346436703, language=EN, label=Table 5, caption=

The structural response interval of the cantilever beam

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模型第一组样本响应区间（kN·m）第二组样本响应区间（kN·m）
区间模型[17.3080，21.3320][17.4600，20.7140]
椭球模型[17.5435，21.0965][17.4227，20.7603]
区间椭球交集模型（半解析法）[17.5435，21.0965][17.5058，20.6716]
区间椭球交集模型（序列二次规划法）[17.5435，21.0965][17.5058，20.6716]
), ArticleFig(id=1241838580688032644, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838550346436703, language=CN, label=表5, caption=

悬臂梁的结构响应区间

, figureFileSmall=null, figureFileBig=null, tableContent=
模型第一组样本响应区间（kN·m）第二组样本响应区间（kN·m）
区间模型[17.3080，21.3320][17.4600，20.7140]
椭球模型[17.5435，21.0965][17.4227，20.7603]
区间椭球交集模型（半解析法）[17.5435，21.0965][17.5058，20.6716]
区间椭球交集模型（序列二次规划法）[17.5435，21.0965][17.5058，20.6716]
), ArticleFig(id=1241838580776113029, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838550346436703, language=EN, label=Table 6, caption=

The evaluation times of response function for the SQP method and the semi-analytical method

, figureFileSmall=null, figureFileBig=null, tableContent=
方法响应函数评价次数
第一组第二组
序列二次规划法5430
半解析法68
), ArticleFig(id=1241838580872582024, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838550346436703, language=CN, label=表6, caption=

序列二次规划法和半解析法响应函数评价次数

, figureFileSmall=null, figureFileBig=null, tableContent=
方法响应函数评价次数
第一组第二组
序列二次规划法5430
半解析法68
), ArticleFig(id=1241838580981633931, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838550346436703, language=EN, label=Table 7, caption=

Uncertainty sample data of the truss arm structure of a crane

, figureFileSmall=null, figureFileBig=null, tableContent=
序号 R1（mm） R2（mm） R3（mm） R4（mm）序号 R1（mm） R2（mm） R3（mm） R4（mm）
185.0579.9345.0037.993184.9480.0745.0137.99
284.9479.9844.9338.053284.9980.0245.0037.96
385.0680.0244.9637.993384.9979.9644.9038.07
485.0380.1844.9538.043485.0880.0745.1537.99
584.8579.8544.8937.903584.9979.9144.9137.97
684.9979.9344.9737.983684.8579.9044.8937.99
784.9980.1545.0437.963785.0879.9944.9637.96
884.9880.0444.9937.923885.0680.0245.0137.94
985.0080.0045.0438.033984.9980.0244.9738.13
1085.0080.0444.9638.014084.9279.9645.0238.08
1185.0479.9944.9338.004184.9880.0044.9738.02
1285.0880.1845.1537.934284.9979.9945.0237.94
1385.0280.0745.0238.064385.0180.0345.0437.96
1484.9980.0744.9938.024484.9980.0545.0937.99
1585.0380.0744.9937.994585.0280.0644.9938.04
1685.0180.0345.0738.004685.0279.9644.8937.93
1784.9579.9445.0137.994784.9480.0044.9637.88
1885.0580.0445.0137.914884.8579.9445.0737.93
1985.0280.0845.0837.994984.9679.9444.9538.02
2085.0180.0244.9637.965084.9780.0045.0538.02
2185.0880.1845.0337.955184.9880.0845.0138.02
2285.1080.0445.0437.945285.0879.9645.1537.99
2384.9579.9844.9937.975384.8579.8544.8938.01
2484.9980.0145.0137.905484.8579.9244.9937.98
2584.9979.9644.9438.055585.0680.0644.9438.04
2684.9780.0444.9438.035684.9579.9545.1537.93
2785.0879.9445.0138.005785.0580.0045.0438.02
2884.9679.9545.0438.005885.0280.0345.0737.96
2984.9879.9645.1337.965985.0080.0644.9537.98
3084.9679.8544.9738.056085.0180.0844.9838.03
), ArticleFig(id=1241838581073908620, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838550346436703, language=CN, label=表7, caption=

起重机桁架臂结构不确定样本数据

, figureFileSmall=null, figureFileBig=null, tableContent=
序号 R1（mm） R2（mm） R3（mm） R4（mm）序号 R1（mm） R2（mm） R3（mm） R4（mm）
185.0579.9345.0037.993184.9480.0745.0137.99
284.9479.9844.9338.053284.9980.0245.0037.96
385.0680.0244.9637.993384.9979.9644.9038.07
485.0380.1844.9538.043485.0880.0745.1537.99
584.8579.8544.8937.903584.9979.9144.9137.97
684.9979.9344.9737.983684.8579.9044.8937.99
784.9980.1545.0437.963785.0879.9944.9637.96
884.9880.0444.9937.923885.0680.0245.0137.94
985.0080.0045.0438.033984.9980.0244.9738.13
1085.0080.0444.9638.014084.9279.9645.0238.08
1185.0479.9944.9338.004184.9880.0044.9738.02
1285.0880.1845.1537.934284.9979.9945.0237.94
1385.0280.0745.0238.064385.0180.0345.0437.96
1484.9980.0744.9938.024484.9980.0545.0937.99
1585.0380.0744.9937.994585.0280.0644.9938.04
1685.0180.0345.0738.004685.0279.9644.8937.93
1784.9579.9445.0137.994784.9480.0044.9637.88
1885.0580.0445.0137.914884.8579.9445.0737.93
1985.0280.0845.0837.994984.9679.9444.9538.02
2085.0180.0244.9637.965084.9780.0045.0538.02
2185.0880.1845.0337.955184.9880.0845.0138.02
2285.1080.0445.0437.945285.0879.9645.1537.99
2384.9579.9844.9937.975384.8579.8544.8938.01
2484.9980.0145.0137.905484.8579.9244.9937.98
2584.9979.9644.9438.055585.0680.0644.9438.04
2684.9780.0444.9438.035684.9579.9545.1537.93
2785.0879.9445.0138.005785.0580.0045.0438.02
2884.9679.9545.0438.005885.0280.0345.0737.96
2984.9879.9645.1337.965985.0080.0644.9537.98
3084.9679.8544.9738.056085.0180.0844.9838.03
), ArticleFig(id=1241838581170377613, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838550346436703, language=EN, label=Table 8, caption=

Uncertainty propagation analysis results for the truss arm structure of a crane

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模型响应区间（mm）
区间模型[57.166，57.663]
椭球模型[57.154，57.678]
区间椭球交集模型（MCS法）[57.170，57.662]
区间椭球交集模型（序列二次规划法）[57.184，57.662]
区间椭球交集模型（半解析法）[57.183，57.661]
), ArticleFig(id=1241838581266846607, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838550346436703, language=CN, label=表8, caption=

起重机桁臂架结构不确定性传播分析结果

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模型响应区间（mm）
区间模型[57.166，57.663]
椭球模型[57.154，57.678]
区间椭球交集模型（MCS法）[57.170，57.662]
区间椭球交集模型（序列二次规划法）[57.184，57.662]
区间椭球交集模型（半解析法）[57.183，57.661]
), ArticleFig(id=1241838581338149777, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838550346436703, language=EN, label=Table 9, caption=

The evaluation times of response function for the MCS method, the SQP method and the semi-analytical method

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方法响应函数评价次数
MCS法108
序列二次规划法73
半解析法14
), ArticleFig(id=1241838581396870034, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838550346436703, language=CN, label=表9, caption=

MCS法、序列二次规划和半解析法响应函数评价次数

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方法响应函数评价次数
MCS法108
序列二次规划法73
半解析法14
), ArticleFig(id=1241838581459784597, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838550346436703, language=EN, label=Table 10, caption=

Uncertainty propagation analysis results for the roof truss structure

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方法结构响应区间上界结构响应区间下界
数值相对误差数值相对误差
MCS0.05164—0.01645—
一阶近似法0.0299942.05%0.0130121.21%
二阶近似法0.051320.619%0.016520.425%
), ArticleFig(id=1241838581543670681, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838550346436703, language=CN, label=表10, caption=

屋架结构不确定性传播分析结果

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方法结构响应区间上界结构响应区间下界
数值相对误差数值相对误差
MCS0.05164—0.01645—
一阶近似法0.0299942.05%0.0130121.21%
二阶近似法0.051320.619%0.016520.425%
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一种结构不确定性分析的新型非概率凸集模型
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乔心州 ** , 李佳辉 , 方秀荣 , 刘鹏
固体力学学报 | 研究论文 2025,46(2): 257-274
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固体力学学报 | 研究论文 2025, 46(2): 257-274
一种结构不确定性分析的新型非概率凸集模型
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乔心州** , 李佳辉, 方秀荣, 刘鹏
作者信息
  • 西安科技大学机械工程学院，西安，710054

通讯作者:

A New Type of Non-probabilistic Convex Model for Structural Uncertainty Analysis
Xinzhou Qiao** , Jiahui Li, Xiurong Fang, Peng Liu
Affiliations
  • College of Mechanical Engineering, Xi'an University of Science and Technology, Xi'an, 710054
出版时间: 2025-04-23 doi: 10.19636/j.cnki.cjsm42-1250/o3.2024.058
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摘要
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非概率凸集模型仅需获知不确定性参数的范围或界限，适用于处理工程结构中常见的小样本问题. 本文提出一种新型非概率凸集模型-区间椭球交集模型来描述不确定域，并研究相应的结构不确定性传播分析方法. 首先提出区间椭球交集模型来描述不确定域，该模型可通过对区间模型和椭球模型进行取交运算构建. 其次，将区间椭球交集模型应用于结构不确定性传播分析. 针对弱非线性响应函数，对其进行泰勒一阶展开近似，通过半解析法对结构响应区间求解；针对强非线性响应函数，对其进行泰勒二阶展开近似，采用序列二次规划法对结构响应区间求解. 最后，四个算例分析验证了文中所提模型和方法的有效性和可行性.

非概率凸集模型  /  区间椭球交集模型  /  不确定性传播分析  /  半解析法

The non-probabilistic convex model requires only the boundaries of structurally uncertain parameters, making it suitable for dealing with engineering problems with limited samples. However, existing convex models primarily focus on regular mathematical models, potentially leading to an excessive expansion of the uncertainty domain. This paper introduces a new type of convex model, namely the interval and ellipsoidal intersection model, to more accurately constrain the uncertainty domain, and examines its application in structural uncertainty propagation analysis. Firstly, the interval and ellipsoidal intersection model is proposed to describe the uncertainty domain, which is constructed by taking the intersection of the interval model and the ellipsoidal model. Subsequently, the proposed model is applied to structural uncertainty propagation analysis with two cases of nonlinear response functions. For the weakly nonlinear response function, a linear approximation is derived using the first-order Taylor series expansion, and then a semi-analytical method is developed to predict its structural response interval. For the strongly nonlinear response function, a nonlinear approximation is achieved using the second-order Taylor series expansion, and the sequential quadratic programming (SQP) method is adopted to predict its structural response interval. Finally, results from four numerical examples indicate that the proposed model generally offers a smaller uncertainty domain and narrower structural response interval compared to the traditional interval and ellipsoidal models. Additionally, the semi-analytical method is more efficient than the SQP method and the Monte Carlo simulation (MCS) method.

non-probabilistic convex model  /  interval and ellipsoidal intersection model  /  uncertainty propagation analysis  /  semi-analytical method
乔心州, 李佳辉, 方秀荣, 刘鹏. 一种结构不确定性分析的新型非概率凸集模型. 固体力学学报, 2025 , 46 (2) : 257 -274 . DOI: 10.19636/j.cnki.cjsm42-1250/o3.2024.058
Xinzhou Qiao, Jiahui Li, Xiurong Fang, Peng Liu. A New Type of Non-probabilistic Convex Model for Structural Uncertainty Analysis[J]. Chinese Journal of Solid Mechanics, 2025 , 46 (2) : 257 -274 . DOI: 10.19636/j.cnki.cjsm42-1250/o3.2024.058
正文
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0　引言
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工程实际结构广泛存在着与载荷、材料特性和边界条件等相关的诸多不确定性[1-4]. 概率模型是最为常见处理上述不确定性的数学模型，通常需要大量不确定样本信息构建变量的概率密度函数[5-7]. 然而，由于制造成本或测试困难等限制，在许多实际工程结构中仅能获得有限不确定样本信息. 针对上述情况，Ben-Haim和Elishakoff[8,9]提出了非概率凸模型理论来处理实验样本不足时的不确定性问题. 非概率凸集模型采用能包含所有样本数据且体积最小的凸集来描述样本的边界信息，因而代表了一系列凸集模型而非单一凸集模型，近年来受到学者的广泛关注[10-21].
根据模型的几何形状不同，现有的非概率凸集模型可分为规则和不规则凸集模型两类. 规则凸集模型主要包括区间模型、椭球模型和平行六面体模型，通常具备简洁的数学表达且易于解决工程中常见的高维问题. 区间模型采用多维长方体来描述参数的不确定域. Ben-Haim和Elishakoff[8,9]采用区间模型描述变量的不确定域. Wang等[10]提出采用旋转矩阵法构造能包含所有样本数据的最小体积区间. 针对区间模型不确定性传播分析问题，Qiu等[11]、Wang等[12]、Qiu等[13]和Wu等[14]分别提出顶点法、一阶泰勒展开法、区间摄动法、切比雪夫区间法和改进子区间法求解结构响应区间. 区间模型的不确定变量均是独立变化的，因此理论上仅能处理独立变量问题. 椭球模型则采用高维椭球描述变量不确定域，椭球的特征矩阵可在一定程度上表征不确定变量的相关性，适用于处理相关变量问题. Zhu等[15]利用旋转矩阵法在所有坐标系中寻找能够包络实验样本点的最小体积椭球. Jiang等[16]利用协方差矩阵构建最小体积椭球. Kang和Zhang[17]通过半定规划法来确定最小体积椭球. Ouyang等[18]提出了一种相关矩阵分析法，可同时获得椭球模型结构响应区间及响应之间的相关性. 相对于仅能分别处理独立变量和相关变量的区间模型和椭球模型，平行六面体模型采用多维平行六面体量化不确定性参数，将相关变量和独立变量置于统一框架下，可有效处理工程实际中常见的独立变量和相关变量共存问题. Jiang等[19]通过定义相关系数和边缘区间提出一种多维平行六面体模型，但未能提供该模型的数学表达式. Ni等[20]通过重新定义相关系数提出一种改进多维平行六面体模型，并给出该模型的显式数学表达式. 针对改进多维平行六面体模型对特殊角度和顶点的限定，乔[1]等提出一种新的多维平行六面体模型，该模型通常比改进多维平行六面体模型有更紧凑的不确定域及更好的样本适应性. LV等[21]提出一阶摄动法求解多维平行六面体模型结构响应区间.
相较于规则凸集模型，不规则凸集模型的数学表达形式相对复杂但通常具备更好的样本适应性，因而可以提供更紧凑的不确定域. Cao等[22]提出了多边凸集模型，该模型是传统区间模型和主成分分析区间模型的交集，通常比两者具备更紧凑的不确定域，且能处理工程中常见的变量相关和高维问题. Ayyasamy等[23]通过凸包描述不确定域，凸包在某种程度上被看作是有限个半空间和超平面的交集，理论上能够给出包络所有样本点的最小体积凸集，然而该模型难以处理复杂的变量高维问题，限制了其工程适用性. 表1列出了上述非概率凸集模型的特点.
本文提出一种新的非概率凸集模型-区间椭球交集模型来描述结构不确定域，并研究其相应的结构不确定性传播分析方法. 该模型的提出基于以下两点考虑：一是考虑到区间模型和椭球模型是两种最为常见的非概率凸集模型，因而区间椭球交集模型具备相对简洁的数学表达式；二是取交运算是一种常见的保凸运算，可保证新模型仍具备凸性和更紧凑的不确定域. 本文剩余部分安排如下：第一节通过取交运算构建区间椭球交集模型并研究模型性质. 第二节针对弱非线响应函数提出半解析法求解结构响应区间. 第三节针对强非线响应函数采用序列二次规划法求解结构响应区间. 第四节通过四个算例验证所提模型和方法的有效性和可行性. 第五节对全文进行总结.
1　区间椭球交集模型
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区间椭球交集模型可通过对区间模型和椭球模型进行取交运算获得，其数学表达式为
其中，Ω表示区间椭球交集模型的不确定域，是区间中点，是区间半径，是椭球中点，M是确定椭球的大小和方向的特征矩阵，数学上可表示为
如何根据给定样本点构建区间和椭球模型方法详见附录A.
（1）区间模型仅适用于处理独立变量问题，而椭球模型仅适用于处理相关变量问题，因而两种模型在表征不确定性时不可避免可能存在过估计，进而影响不确定性分析的精度. 而区间椭球交集模型所对应的集合是区间模型和椭球模型对应集合的子集，保证了模型的凸性且能够更精确的表征不确定性. 同时，考虑到区间模型和椭球模型均易于处理工程结构中的高维问题，因此区间椭球交集模型同样易于处理高维问题.
（2）根据位置关系不同，本文所提区间椭球交集模型中区间和椭球模型的相交情况可分成两种：相交不包含关系（图1）和相交且包含关系. 当椭球模型和区间模型为相交但包含关系时，若椭球模型的不确定域完全包含区间模型的不确定域时（图2（a）），区间椭球交集模型退化为区间模型. 若区间模型的不确定域完全包含椭球模型的不确定域时，区间椭球交集模型退化为椭球模型（图2（b））. 因此，区间模型和椭球模型可视为区间椭球交集模型的两个特例.
（3）区间椭球交集模型是区间模型和椭球模型通过取交运算获得，因而对于一个线性或近似线性结构响应函数，区间椭球交集模型有较大可能提供比椭球模型或区间模型更窄的结构响应区间. 当区间椭球交集模型出现图2所示情况时，可相应地按照区间模型或椭球模型的不确定传播分析方法求解结构响应区间；当区间椭球交集模型出现图1所示的情况时，按照下文半解析法求解模型对应结构响应函数的结构响应区间，此时，区间模型和椭球模型的不同位置关系将直接影响结构响应区间的求解难度，详细的示意图可参见2.3节图4.
2　结构不确定性传播分析的一阶近似方法
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结构不确定性传播指的是不确定性参数的波动引起结构响应波动. 当结构不确定参数采用非概率凸模型描述时，其结构响应通常为一区间. 因此，基于区间椭球交集模型的结构不确定性传播分析的主要任务是合理预测结构响应区间.
假设结构响应函数为
当工程结构上参数的不确定性水平较小时，为了减少计算量，将结构响应函数在椭球中点进行一阶泰勒展开.
这里采用椭球中点作为泰勒展开点，其原因在于本文通过求解一系列降维椭球表面的极值以获取区间椭球交集模型结构响应区间，显然椭球中点是最合理的近似展开点.
结构响应区间可通过如下优化问题进行求解.
对于式（5）的优化问题，其目标函数为线性函数，可行域为凸集，则根据凸优化理论，区间椭球交集模型的极值点其最优解必定出现在约束条件的边界上.
2.1　半解析法
如下提出以下半解析法求解式（5）的优化问题进而确定结构响应区间. 以极大值为例，首先求解结构响应函数在椭球模型不确定域上的极大值.
根据拉格朗日乘子法[24]，式（6）的极大值点为
其次，检验式（7）得到的极值点是否位于区间模型不确定域，即
若满足区间模型不确定域，则将式（7）带入式（6）得到结构响应函数的上界为
若极大值点Xmax不满足区间模型不确定域，则进一步确定结构响应函数在降维椭球模型不确定域的极大值.
不妨设结构响应上界所对应的极值点存在m（1≤m≤n）个参数X1，X2，…，Xm不满足区间模型不确定域即或. 此时，X1，X2，…，Xm将依据下式被赋予特定值.
将式（10）代入式（6）可得到如下优化问题来确定结构响应上界.
式中，X*=[Xm+1，Xm+2，…，Xn]T、特征矩阵M*和椭球中点分别为
式中A、c和λ分别为
同样根据拉格朗日乘子法[24]，式（11）的极值点为
如果X*max满足区间模型不确定域，则将式（14）带入式（11）得到结构响应函数的上界为
如果X*max仍然不满足区间模型不确定域，则重复式（10）-（14）过程，直至获得结构响应区间上界.
同理，结构响应下界也可按照相同的方法进行求解，只需要进行如下调整：
求解结构响应函数在椭球模型不确定域上的极小值点的优化问题为
根据拉格朗日乘子法[24]，式（16）的极小值点Xmin为
根据式（8）判断极小值点是否位于区间模型不确定域内，若满足则对应结构响应函数的极小值为
若存在p（1≤p≤n）个参数X1，X2，…，Xp不满足区间模型不确定域，则结构响应函数的下界需求解如下优化问题.
式中，依然由式（10）方法确定. X′=[Xp+1，Xp+2，…，Xn]T、椭球中点和特征矩阵M′可通过类似式（13）方法获取. 对应的极小值点为
如果X′min满足区间模型不确定域，则将式（17）带入式（16），得到结构响应函数的下界为
如果X′min不满足区间不确定域，重复上述步骤直到获得结构响应区间下界.
2.2　半解析法的步骤和实施流程
采用半解析法求解区间椭球交集模型响应区间上界的步骤归纳如下：
（1）依据式（4）得到结构响应函数g（X）的线性近似函数.
（2）以结构响应函数的线性近似作为目标函数，以椭球模型不确定域为约束条件，构造式（6）的优化问题.
（3）采用拉格朗日乘子法求解式（6）所示优化问题的极大值点Xmax.
（4）根据式（8）判断极大值点Xmax是否位于区间模型不确定域. 若满足，则根据式（9）输出结构响应上界；若不满足，则转步骤5.
（5）根据式（10）对不满足区间模型不确定域的不确定参数赋值，构建式（11）所示的优化问题确定结构响应函数上界. 其中，降维椭球由式（12）和（13）确定.
（6）根据式（14）给出式（11）所示优化问题的极大值点X*max.
（7）根据式（8）判断极大值点X*max是否位于区间模型不确定域，若满足，则根据式（15）输出结构响应上界；若不满足，若不满足则转步骤5直到获得结构响应区间上界.
图3给出了半解析法求解结构响应区间上界的实施流程图. 确定结构响应区间下界的步骤及实施流程与上界的步骤及实施流程类似. 需要指出的是，本文半解析法适用于处理结构响应函数为弱非线性或小范围不确定性情况下结构响应问题.
本文所提半解析法旨在通过拉格朗日乘子法求解一系列降维椭球上的极值，并通过验证椭球极值点是否在区间模型的不确定域内来确定结构响应函数的响应区间. 降维椭球的极值通过解析方法求解，半解析法体现在解析法确定极值和判断准则相结合以确定结构响应区间. 而序列二次规划法通常通过一系列的迭代步骤逐步改进解的质量，直到达到满足一定精度要求的近似最优解. 半解析法的最差情况下时间复杂度为O（n），序列二次规划法的时间复杂度为O（n3），其次，正如图4所示，受到极值点位置的不同分布的影响，半解析法通常仅需要较少的执行次数即可确定结构响应区间，因此在实际执行中半解析法的效率通常高于序列二次规划法，这一结论也可通过下一节的算例二和三验证.
2.3　半解析法的图解说明
本文所提半解析法的基本原理是求解一系列椭球表面的极值，并通过检验极值点是否位于区间模型不确定域内，来确定区间椭球交集模型的极值. 以下以图4所示的三维情形说明半解析法的求解过程. 若根据式（7）求得的极值点位于黄色区域时，显然极值点位于区间模型不确定域内，则极值点对应的结构响应函数值即为响应函数的极值；若根据式（7）求得的极值点位于绿色区域时，极值点可通过求解响应函数在黑色虚线表示的二维椭圆上的极值确定. 若根据式（7）求得的极值点位于粉色区域时，极值点可通过求解响应函数在红色线段表示的一维椭圆上的极值确定. 若根据式（7）求得的极值点位于蓝色区域时，此时意味着极值点的所有坐标均位于区间模型的不确定域外，则极值点可直接确定为黑点（区间模型的顶点）.
3　结构不确定性传播分析的二阶近似方法
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通过上述分析可知，当结构响应函数的非线性程度不高时，泰勒一阶近似可以提供足够的精度. 针对强非线性结构响应函数，采用一阶近似会造成较大的分析结果误差. 为提高分析精度，可采用泰勒二阶展开对结构响应函数进行近似.
对式（3）的结构响应函数在椭球中点进行二阶泰勒展开：
其中H为结构响应函数g（X）的海森矩阵，其表示为
结构响应区间通过对如下优化问题求解获得.
从几何角度而言，泰勒一阶近似和二阶近似分别采用超平面和抛物面对结构响应函数对应的超曲面进行近似，显然二阶近似能够考虑超曲面的曲率特性，因此，泰勒二阶近似通常具备更好的精度，更适用于强非线性结构响应函数. 然而，相比于泰勒一阶近似，泰勒二阶近似需要计算响应函数的海森阵，因而其效率通常低于一阶近似. 总而言之，二阶近似方法通常比一阶近似方法有较高的精度和较低的效率.
对于式（24）所示的优化问题，上一节的半解析法的二阶版本为该问题的求解提供了一个可行的思路. 然而，通过对算法精度（二阶版本算法与序列二次规划法时间复杂度均为O（n3））、效率和鲁棒性的综合考虑，这里采用成熟的序列二次规划法对该模型进行求解.
4　算例
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本节通过四个算例验证所提模型和方法. 算例一为数值算例，针对三组不同分布情况样本，分别构建区间模型、椭球模型、平行六面体模型[19]和区间椭球交集模型并进行对比分析；算例二、三分别涉及线性和弱非线性结构响应函数，用于对比分析基于三种模型的不确定性传播分析结果，以及不同方法求解结构响应区间的精度与效率. 算例四为强非线性结构响应函数，用于对比分析一阶近似和二阶近似方法的精度. 其中，算例二半解析法所得传播分析结果以序列二次规化法结果为精确解，原因在于其所涉及优化问题为凸优化问题，序列二次规划法即可给出全局最优解. 算例三半解析法所得传播分析结果以MCS法结果为参考解. 算例四所得传播分析结果以MCS法结果为参考解. MCS法确定的结构响应区间的步骤见附录B. 所有序列二次规划法均以椭球中点为初始点，迭代收敛准则为|gk-gk+1|≤10-6，其中gk为序列二次规划法第k次的函数值.
4.1　数值算例
考虑三组不确定样本数据，样本数据信息详见文献[25]的12-14页. 这三组样本数据分别涉及两个不确定变量之间正相关、独立和负相关情况.
根据三组样本数据确定的区间椭球交集模型分别为：
对于三组样本数据，采用凸包算法可求得其对应最优凸包的体积分别为20.90、76.22和21.33. 凸包体积可通过MATLAB中的convexHull函数确定. 区间、椭球、平行六面体及区间椭球交集模型的精度可采用模型体积与凸包体积的比值来衡量如表2所示，显然体积比越小，模型精度越高.
为便于对比分析，图5绘制了包含相同样本点的区间模型、椭球模型、平行六面体模型和区间椭球交集模型. 由表2和图5可以看出：
（1）由表2可以看出，当两变量相关时（图5（a）和5（c）），区间椭球交集模型、平行六面体模型和椭球模型的不确定域明显小于区间模型的不确定域，原因在于区间模型无法考虑变量之间的相关性因而扩大了变量的不确定域；区间椭球交集模型、平行六面体模型和椭球模型都适于量化相关变量；当两变量独立时（图5（b）），区间椭球交集模型、平行六面体模型和区间模型的不确定域明显小于椭球模型的不确定域，原因在于椭球模型无法处理变量独立问题因而扩大了变量的不确定域.
（2）与区间模型和椭球模型相比，区间椭球交集模型一方面可以获得更紧凑的不确定域，另一方面既适用于变量相关情形又适用于变量独立情形. 当变量相关时区间椭球交集模型的精度低于平行六面体模型，而当变量独立时区间椭球交集模型的精度优于平行六面体模型. 此外，需要特别指出的是，本算例仅针对三种二维情况进行讨论. 当涉及高维变量时，区间椭球交集模型仍是一种能够完全包络所有样本点的模型，而现有的平行六面体模型通常不能保证完全包络所有样本点[19,20].
（3）区间椭球交集模型需要分别构建区间模型和椭球模型，并对两者进行取交运算获得. 而区间模型、椭球模型和平行六面体模型均可依据样本直接构建. 因此就效率而言，区间椭球交集模型低于上述三种模型.
4.2　悬臂梁结构
以图6所示的悬臂梁结构为例[26]. 其分别在固定端b1=2 m，b2=3 m受到两个集中力F1和F2的作用. 结构的极限力矩为
作用力F1和F2为不确定参数. 其两组不确定样本数据分别见表3和4. 其中，第一组样本数据来自文献[26]，第二组样本数据为本文假设给出.
根据两组样本数据确定的区间椭球交集模型分别为
以下分别给出三种模型结构响应区间.
（1）区间模型响应区间
由于结构响应函数为不确定变量的单调线性函数，区间模型结构响应区间可由顶点法[11]给出.
其中，表示区间中点，表示区间半径.
（2）椭球模型响应区间
椭球模型结构响应区间可采用拉格朗日乘子法[24]求解.
其中，表示椭球中点，M表示椭球的特征矩阵.
（3）区间椭球交集模型结构响应区间
针对第一组样本数据，为确定区间椭球交集模型结构响应区间上界，可首先求解如下优化问题.
通过拉格朗日乘子法[24]可得到式（30）的极大值点为
可以看出，极大值点位于区间模型不确定域内，即
则区间椭球交集模型结构响应区间为
类似的，结构响应区间下界为
针对第二组样本数据，为得到其结构响应区间上界，同样首先求解如下优化问题.
利用拉格朗日乘子法[24]可得式（35）的极大值点为
式（36）可以看出极大值点中|5.8961-5.464|=0.4321>0.372. 则根据式（10），将不确定变量F2用区间上界5.836代替，进而得到如下优化问题.
其中降维后的不确定参数F*、椭球中点以及特征矩阵M*为
根据拉格朗日乘子法[24]，式（37）的极大值点为
式（39）的极大值点位于区间模型不确定域内即，则结构响应区间上界为
同理，结构响应区间下界为
表5列出了三种模型不确定性传播分析结果. 为验证半解析法分析结果，表5同样给出了序列二次规划法得出的区间椭球交集模型结构响应区间.
从表5可以看出：
（1）文中半解析法得到区间椭球交集模型结构响应区间与序列二次规划法得到的响应区间一致，验证了所提方法的有效性.
（2）对于区间模型和椭球模型而言，第一组数据中区间模型的结构响应区间完全包含椭球模型的结构响应区间；而第二组数据椭球模型的结构响应区间完全包含区间模型的结构响应区间. 表明当区间模型和椭球模型存在相交但不包含关系时，随着结构响应极值点的位置变化，两者的结构响应区间存在不同的关系.
（3）区间椭球交集模型的不确定域既是区间模型不确定域的子集也是椭球模型不确定域的子集，因此，对于两组数据而言，区间椭球交集模型结构响应区间完全包含于椭球模型和区间模型的结构响应区间；表明区间椭球交集模型不仅能够具有更紧凑的不确定域且能提供更合理的结构响应区间.
表6列出了序列二次规划法和半解析法的函数评价次数. 可以看出，与序列二次规划法相比，半解析法具备更少的响应函数调用次数，因此具有更高的效率.
4.3　起重机桁架臂结构
考虑图7所示起重机桁架臂结构. 结构的弹性模量E=2×105 MPa，泊松比μ=0.3. 结构起重端受到作用为F1=100 kN的竖直载荷. 由于制造与测量误差，上弦杆半径R1、下弦杆半径R2、垂向腹杆半径R3及斜向腹杆半径R4均为不确定变量. 表7列出了某型号起重机桁架臂的不确定变量的60个样本. 这里选取起重机桁架臂的最大位移为结构响应函数. 结构响应函数利用30个样本点通过二次响应面法构建，30个样本点由图8所示包含144个节点和304个单元的结构有限元分析模型获取. 结构响应函数为：
依据表7样本数据构建的区间椭球交集模型为
将式（42）所示结构响应函数在椭球中点进行一阶泰勒展开：
求解区间椭球交集模型的结构响应区间转换为求解如下优化问题：
表8给出了区间模型、椭球模型和区间椭球交集模型的结构响应区间，同时给出了MCS法（本例采用108个样本模拟）确定的区间椭球交集模型结构响应区间作为参考解.
通过表8可以看出：
（1）对于该非线性结构响应函数，基于半解析法的区间椭球交集模型结构响应区间依旧包含于椭球模型和区间模型的结构响应区间.
（2）基于半解析法的区间椭球交集模型结构响应区间与序列二次规划法和MCS方法给出的结构响应区间基本一致，其上界和下界最大相对误差分别为0.022%和0.0017%. 结果表明对于该弱非线性结构响应函数，半解析法给出的结构响应区间具有足够的精度.
表9列出了MCS、序列二次规划法和半解析法的函数评价次数. 可以看出，与序列二次规划法和MCS法相比，半解析法具备更少的响应函数评价次数，因此具有更高的效率. 对于本算例而言，采用序列二次规划法存在得不到全局最优解的可能.
4.4　屋架结构
考虑图9所示受均布载荷的屋架结构[27]，桁架的弦和压杆采用钢筋混凝土制成，底部的弦和拉杆采用钢材制作. 其结构响应函数为
其中，q（N/m）表示均布载荷，l（m）表示桁架杆杆长，Ac（m2）和As（m2）分别表示两种桁架杆的截面积，均为不确定变量. 不确定变量的不确定域为：
表10为屋架结构不确定性传播分析结果. 其中MCS法（108个样本）作为参考解. 由表10可知，对于该强非线性结构响应函数，一阶近似得到的结构响应区间的上界和下界的相对误差为42.05%和21.21%，而二阶近似法的结构响应区间的上界和下界的分别为0.619%和0.425%. 可见，二阶近似法的求解精度较高，而一阶近似法的求解精度较低，其结果工程上无法接受. 因此，对于该屋架结构涉及的强非线性结构响应函数，采用二阶近似求解结构响应区间更合理.
5　结论
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本文提出了一种结构不确定性分析的新型非概率凸集模型即区间椭球交集模型. 该模型具有比区间模型和椭球模型更紧凑的不确定域，也适用于处理实际工程结构中的高维问题，在精度和效率之间保持良好的平衡. 进一步基于区间椭球交集模型提出半解析法求解弱非线性结构的结构响应区间，其原理是通过求解一系列椭球表面的极值，进一步确定区间椭球交集模型的结构响应区间. 针对弱非线响应函数，对其进行泰勒一阶近似，并提出半解析法求解结构响应区间；针对强非线响应函数，对其进行泰勒二阶近似，并采用序列二次规划法求解结构响应区间. 四个算例表明，对比区间模型和椭球模型，区间椭球交集模型不仅拥有更小的不确定域，还具备更紧凑的响应区间. 与序列二次规划化法和MCS相比，半解析法处理弱非线性结构问题具备相似的精度但更高的效率. 对于强非线性结构响应函数，采用二阶近似法的精度明显高于一阶近似法. 作为一种新型非概率凸集模型，研究其其他应用如可靠性分析和可靠性优化设计当是进一步的研究工作.
文献
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2025年第46卷第2期
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doi: 10.19636/j.cnki.cjsm42-1250/o3.2024.058
  • 接收时间：2024-11-07
  • 首发时间：2026-03-20
  • 出版时间：2025-04-23
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    西安科技大学机械工程学院，西安，710054

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2种不同金属材料的力学参数
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Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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