传统概率方法已被广泛用于处理实际结构中的不确定性问题，然而概率模型需要大量不确定信息用以构建概率密度函数来描述参数的不确定性，这对于复杂工程实际问题而言是十分困难的。反之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分析方法。研究内容主要如下：

       (1)一种结构不确定性量化的区间椭球交集模型。提出了一种新的不规则凸集模型-区间椭球交集模型，该模型通过对区间模型、椭球模型进行取交运算而构建，用于度量结构不确定域；讨论了区间椭球交集模型的性质，并给出求解模型不确定域体积的蒙特卡洛模拟法；通过切比雪夫不等式对区间椭球交集模型进行膨胀处理，可赋予模型对未来样本的预测能力。采用三个算例验证模型和方法的有效性和可行性。

       (2)基于区间椭球交集模型的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，提出一种半解析法对结构响应区间求解。其基本思想为通过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；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；通过三个算例验证所提方法有效性。

       (3)基于区间椭球交集模型的结构非概率可靠性分析。提出一种区间椭球交集模型结构非概率可靠性指标，将结构功能函数均值与离差的比值作为结构可靠性度量；揭示所提非概率可靠性指标与区间、椭球模型非概率可靠性指标之间的内在关联；通过一个工程案例揭示了所提非概率可靠性指标中的不一致性问题。通过三个算例验证了所提非概率可靠性指标的的有效性和可行性。

    The traditional probabilistic methods have been widely applied to deal with the uncertain problems. However, the complete information is required to define a probabilistic distribution by using the probability method, which is generally difficult for complex engineering problems. As the alternative of the probability method, the non-probabilistic convex model only requires the boundaries of structural uncertain parameters and is suitable for dealing with engineering problems with limited samples. However, the available convex models focus mainly on regular mathematical models, and thus may provide the excessive expansion of the uncertainty domain. In view of this, in this paper a new type of convex model, namely an interval and ellipsoidal intersection model, is proposed to bound the uncertainty domain, whereby the corresponding structural uncertainty analysis method is investigated. The main research contents are as follows:

     (1) An interval and ellipsoid intersection model for structural uncertainty quantification. In this paper a new type of convex model, namely an interval and ellipsoidal intersection model, is proposed to bound the uncertainty domain. Firstly, the interval and ellipsoidal intersection model is proposed to describe the uncertain domain, which can be constructed by taking the intersection of the interval model and the ellipsoidal model. Secondly, the corresponding characteristics of that model are investigated, and Monte Carlo Simulation (MCS) method is given for computing the volume of the uncertainty domain. Thirdly, Chebyshev inequality is used to inflate the interval and ellipsoidal intersection model to accommodate future possible data points. Finally, three numerical examples are used to verify the effectiveness and feasibility of the proposed model and methods.

     (2) A structural uncertainty propagation based on the interval and ellipsoid intersection model. The proposed model is applied to structural uncertainty propagation analysis, and two cases of the nonlinear response function are considered. For the weakly nonlinear response function, its linear approximation can be obtained by using the first-order Taylor series expansion, and then a semi-analytical method is developed to predict its structural response interval. The main idea of semi-analytical method is to determine the extremum of the interval ellipsoid intersection model by computing a series of extrema on the ellipsoid surfaces and verifying whether these extremum points lie within the uncertainty domain defined by the interval model. For the strong nonlinear response function, its nonlinear approximation can be obtained by using the second-order Taylor series expansion, and then the Sequential Quadratic Programming (SQP) method is adopted to predict its structural response interval. Three numerical examples are provided to demonstrate the effectiveness of the proposed methods.

    (3) A structural non-probabilistic reliability index based on the interval and ellipsoid intersection model. Firstly, based on the interval and ellipsoid intersection model, a novel non-probabilistic reliability index is proposed to measure structural reliability, which is defined as the ratio of the mean of the structural performance function to its radius. Secondly, the inherent connections between the proposed reliability index and those based on the interval and ellipsoid models are revealed. Thirdly, the invariance problem existing the proposed reliability index is discussed through an engineering case. Finally, Three numerical examples are provided to demonstrate the effectiveness of the proposed methods.

[1]周济. 智能制造“中国制造2025”的主攻方向[J]. 中国机械工程, 2015, 26(17): 2273-2284.

[2]张义民. 机械可靠性设计的内涵与递进[J]. 机械工程学报, 2010, 46(14): 167-188.

[3]国家自然科学基金委员会工程与材料科学部. 机械工程学科发展战略报告(2021-2035)[M]. 北京: 科学出版社, 2022.

[4]龙乐豪. 序[J]. 火箭推进, 2024, 50 (05): 3-4.

[5]赵国藩, 曹居易, 张宽权. 工程结构可靠度[M]. 北京:水利电力出版社, 1984.

[6]李杰著. 随机结构系统[M]. 北京: 科学出版社, 1996.

[7]毕仁贵. 考虑相关性的不确定凸集模型与非概率可靠性分析方法[D]. 长沙: 湖南大学, 2015.

[8]熊芬芬, 李泽贤, 刘宇等. 基于数值模拟的工程设计中参数不确定性表征方法研究综述[J]. 航空学报, 2023, 44(12): 028611.

[9]贾大卫, 吴子燕, 何乡. 基于BP神经网络和Laplace渐进积分法的结构可靠性计算[J]. 固体力学学报, 2021, 42(04): 477-489.

[10]Rackwitz R. Reliability analysis-a review and some perspectives[J]. Structural Safety, 2001, 23: 365-395.

[11]Keshtegar B, Meng Z. A hybrid relaxed first-order reliability method for efficient structural reliability analysis[J]. Structural Safety, 2017, 66: 84-93.

[12]Yang M D, Zhang D Q, Han X. New efficient and robust method for structural reliability analysis and its application in reliability-based design optimizations[J]. Computer Methods in Applied Mechanics and Engineering, 2020, 366: 113018.

[13]Jia B, Lu Z, Lei J. Fuzzy first-order and second moment method for failure credibility analysis in the presence of fuzzy uncertainty[J]. IEEE Transaction on Fuzzy Systems. 30, 3166.

[14]You L F, Zhang J G, Zhou S, et al. A novel mixed uncertainty support vector machine method for structural reliability analysis[J]. Acta Mechanica Sinica. 232, 1497.

[15]Bagheri M, Zhu S P, Ben Seghier M E A, et al. Hybrid intelligent method for fuzzy reliability analysis of corroded X100 steel pipelines[J]. Engineer Computations. 37, 2559.

[16]Sun W, Yang Z, Chen G. Structural eigenvalue analysis under the constraint of a fuzzy convex set model[J], Acta Mechanica Sinica. 34, 653.

[17]Ben-haim Y. A non-probabilistic concept of reliability[J]. Structures Safety, 1994, 14(4): 227-245.

[18]Elishakoff I, Ielisseeff P, Glegg S A L. Nonprobabilistic convex-theoretic modeling of scatter in material properties[J]. American Institute of Aeronautics and Astronautics Inc, 2012, 32: 843–849.

[19]Wang X J, Wang L, Qiu Z P. Response analysis based on smallest interval-set of parameters for structures with uncertainty[J]. Applied Mathematics and Mechanics, 2012, 33(9): 1153-1166.

[20]Jiang C, Ni B Y, Liu N Y, et al. Interval process model and non-random vibration analysis[J]. Journal of Sound and Vibration, 2016, 373: 104-131.

[21]Moens D, Munck, M D, Desmet W, et al. Numerical dynamic analysis of uncertain mechanical structures based on interval fields[J]. Springer Netherlands, 2011, 27: 71-83.

[22]Zhu L P, Elishakoff I, Starnes J H. Derivation of multi-dimensional ellipsoidal convex model for experimental date[J]. Mathematical and Computer Modelling, 1996, 24(2): 103-114.

[23]Jiang C, Han X, Lu G Y, et al. Correlation analysis of non-probabilistic convex model and corresponding structural reliability technique[J]. Computer Methods in Applied Mechanics and Engineering, 2011, 200(33-36): 2528-2546.

[24]Kang Z, Zhang W. Construction and application of an ellipsoidal convex model using a semi-definite programming formulation from measured data[J]. Computer Methods in Applied Mechanics and Engineering, 2016, 300: 461-489.

[25]Jiang C, Ni B Y, Han X, et al. Non-probabilistic convex model process: A new method of time-variant uncertainty analysis and its application to structural dynamic reliability problems[J]. Computer Methods in Applied Mechanics and Engineering, 2014, 268: 656-676.

[26]Jiang C, Zhang Q F, Han X, et al. Multidimensional parallelepiped model-a new type of non-probabilistic convex model for structural uncertainty analysis[J]. International Journal for Numerical Methods in Engineering, 2015, 103(1): 31-59.

[27]Ni B Y, Jiang C, Han X. An improved multidimensional parallelepiped non-probabilistic model for structural uncertainty analysis[J]. Applied Mathematical Modelling, 2016, 40(7-8): 4727-4745.

[28]Wang C, Matthies H G. A modified parallelepiped model for non-probabilistic uncertainty quantification and propagation analysis[J]. Computer Methods in Applied Mechanics and Engineering, 2020, 369: 113209.

[29]王攀, 臧朝平. 改进的平行六面体凸模型识别动力学不确定参数区间的方法[J]. 振动工程学报, 2019, 32(1): 97-106.

[30]乔心州, 张锦瑞, 方秀荣等. 一种结构不确定分析的改进多维平行六面体模型[J]. 兵工学报, 2024, 45 (06): 1877-1888.

[31]Ni B Y, Elishakoff I, Jiang C. Generalization of the super ellipsoid concept and its application in mechanics[J]. Applied Mathematical Modelling, 2016, 40: 9427-9444.

[32]Qiu Z P, Tang H J, Zhu B. A non-probabilistic convex polyhedron model for reliability analysis of structures with multiple failure modes and correlated uncertainties based on limited data [J]. Acta Mechanica Sinica, 2022, 39 (2): 42602-42615.

[33]Cao L X, Liu J, Xie L, et al. Non-probabilistic polygonal convex set model for structural uncertainty quantification[J]. Applied Mathematical Modelling, 2021, 89: 504-518.

[34]Qiu Z P, Wang X J, Chen J Y. Exact bounds for the static response set of structures with uncertain-but-bounded parameters[J]. International Journal of Solids and Structures, 2006, 43(21): 6574-6593.

[35]Wu J, Luo Z, Zhang Y, et al. Interval uncertain method for multi-body mechanical systems using Chebyshev inclusion functions[J]. International Journal for Numerical Methods in Engineering, 2013, 95: 608-630.

[36]Qiu Z P, Wang X J. Comparison of dynamic response of structures with uncertain non-probabilistic interval analysis method and probabilistic approach[J]. International Journal of Solids Structures, 2003, 40(20): 5423-5439.

[37]Wang X J, Wang L. Uncertainty quantification and propagation analysis of structures based on measurement data[J]. Mathematical and Computer Modelling, 2011, 54(11-12): 2725-2735.

[38]Sevillano E, Sun R, Perera R. Damage evaluation of structures with uncertain parameters via interval analysis and FE model updating methods. Structural Control & Health Monitoring, 2016, 4: 1-22.

[39]Zhou Y T, Jiang C, Han X. Interval and subinterval analysis methods of the structural analysis and their error estimations[J]. International Journal of Computational Methods, 2006, 03(2): 229-244.

[40]Qiu Z P, Elishakoff I. Antioptimization of structures with large uncertain-but-non-random parameters via interval analysis[J]. Computer Methods in Applied Mechanics and Engineering. 152 (1998) 361-372.

[41]Long X Y, Jiang C, Han X, Tang C, Guan F J. An enhanced subinterval analysis method for uncertain structural problems[J]. Applied Mathematical Modelling, 2018: 580-593.

[42]Ouyang H, Liu J, Han X, et al. Correlation propagation for uncertainty analysis of structures based on a non-probabilistic ellipsoidal model[J]. Applied Mathematical Modelling, 2020, 88: 190-207.

[43]Lv H, Li Z C, Huang X T, et al. Effective correlation analysis algorithms for uncertain structures based on multidimensional parallelepiped model[J]. Applied Mathematical Modelling, 2023, 120: 667-685.

[44]Xia B, Yu D. Modified sub-interval perturbation finite element method for 2D acoustic field prediction with large uncertain-but-bounded parameters[J]. Journal of Sound and Vibration, 2012, 331(16): 3774–3790.

[45]李贵杰, 吕震宙, 葛山增, 等. 用非概率理论分析混合不确定性结构的可靠性[J]. 国防科技大学学报, 2014, 36 (05): 21-25.

[46]刘浩, 杨开云, 张德权. 基于聚类椭球模型的不确定性传播分析[J]. 计算力学学报, 2018, 35 (04): 431-436.

[47] Elishakoff I. Discussion on: a non-probabilistic concept of reliability[J]. Structural Safety, 1995, 17(3): 195-199.

[48]郭书祥, 吕震宙, 冯元生. 基于区间分析的结构非概率可靠性模型[J]. 计算力学学报, 2002, 19(3): 332-335.

[49]Xiao N C, Huang H Z, Li Y F, et al. Non-probabilistic reliability sensitivity analysis of the model of structural systems with interval variables whose state of dependence is determined by constraints[J]. Proceedings of the institution of Mechanical Engineers, Part O: Journal of Risk and Reliability.2013, 227(5): 491-498.

[50]Xin T D, Zhao J G, Cui C Y, et al. A non-probabilistic time-variant method for structural reliability analysis[J]. Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability, 2020, 234(5): 664-75.

[51]Wang L, Wang X J, Chen X, et al. Time-variant reliability model and its measure index of structures based on a non-probabilistic interval process[J]. Acta Mechanica, 2015, 226(10): 3221-3241.

[52]曹鸿钧, 段宝岩. 基于凸集合模型的非概率可靠性研究[J]. 计算力学学报，2005, 22(5): 546-549.

[53]Jiang C, Zhang Q F., Han X, et al. A non-probabilistic structural reliability analysis method based on a multidimensional parallelepiped convex model[J]. Acta Mechanica, 2014, 225: 383-395.

[54]Hong L, Li H, Fu J, et al. Hybrid active learning method for non-probabilistic reliability analysis with multi-super-ellipsoidal model[J]. Reliability Engineering & System Safety, 2022, 222: 108414.

[55]Meng Z, Hu H, Zhou H. Super parametric convex model and its application for non-probabilistic reliability-based design optimization[J]. Applied Mathematical Modelling, 2018, 55: 354-370.

[56]Zhan J ,Luo Y ,Zhang X , et al. A general assessment index for non-probabilistic reliability of structures with bounded field and parametric uncertainties[J]. Computer Methods in Applied Mechanics and Engineering, 2020, 366.

[57]Jiang T, Chen J J. A semi-analytic method for calculating non-probabilistic reliability index[J]. Applied Mathematical Modelling, 2007, 31(7): 1362-1370.

[58]Chen X Y, Tang C Y, Tsui C P, et al. Modified scheme based on semi-analytic approach for computing non-probabilistic reliability index[J]. Acta Mechanica Solida Sinica, 2010, 23(2): 115-123.

[59]Jiang C, Lu G Y, Han X, et al. Some important issues on first-order reliability analysis with nonprobabilistic convex models[J]. Journal of Mechanical Design, 2014, 136(3): 031501-1-034501-5.

[60]Qiao X Z, Song L F, Liu P, et al. Invariance problem in structural non-probabilistic reliability index[J]. Journal of Mechanical Science and Technology, 2021, 35(11): 4953-4961.

[61]Liu H ,Xiao C N . An efficient method for calculating system non-probabilistic reliability index[J]. Eksploatacaja I Niezawodnosc-Maintenance and Reliability, 2021, 23 (3): 498-504.

[62]Qiao X , Liu Z , Fang X , et al. Linear Programming-Based Non-Probabilistic Reliability Bounds Method for Series Systems[J]. Applied Sciences, 2024, 14 (14): 6215-6215.

[63]王晓军, 邱志平, 武哲. 结构非概率集合可靠性模型[J]. 力学学报, 2007, 39(5): 641-646.

[64]Wang X J, Qiu Z P, Elishakoff I. Non-probabilistic set-theoretic model for structural safety measure[J]. Acta Mechanica, 2008, 198(1-2): 51-64.

[65]孙海龙, 姚卫星. 结构区间可靠性分析的可能度法[J]. 中国机械工程,2008,19(12): 1483-1488.

[66]乔心州, 仇原鹰, 孔宪光. 一种基于椭球凸集的结构非概率可靠性模型[J]. 工程力学, 2009, 26(11): 203-208.

[67]Hong L, Li H, Fu J, et al. Hybrid active learning method for non-probabilistic reliability analysis with multi-super-ellipsoidal model[J]. Reliability Engineering & System Safety, 2022, 222: 108414.

[68]Wang R X, Wang X J, Wang L, et al. Efficient computational method for the non-probabilistic reliability of linear structural systems[J]. Acta Mechanica Solida Sinica, 2016, 29(3): 284-299.

[69]Jiang C, Bi R G, Lu G Y, et al. Structural reliability analysis using non-probabilistic convex model[J]. Computer Methods in Applied Mechanics and Engineering, 2013, 254: 83-89.

[70]Qiao X, Zhao Y, Fang X, et al. An Improved Nonprobabilistic Monte Carlo Simulation Method for Structural Reliability Analysis Based on Convex Models [J]. International Journal of Computational Methods, 2024, 22 (01).

[71]姜峰, 李华聪, 符江锋, 等. 基于RBF和主动学习的非概率可靠度求解方法[J]. 航空学报, 2023, 44(02): 213-227.

[72]高冉, 周建方. 非概率可靠指标和可靠度定义的比较研究[J]. 应用力学学报, 2020, 37 (02): 873-881.

[73]Elishakoff I, Sarlin N. Uncertainty quantification based on pillars of experiment, theory, and computation. Part I: Data analysis [J]. Mechanical Systems and Signal Processing, 2016, 74 29-53.

[74]Elishakoff I, Sarlin N. Uncertainty quantification based on pillars of experiment, theory, and computation. Part II: Theory and computation [J]. Mechanical Systems and Signal Processing, 2016, 74 54-72.

[75]Ayyasamy S, Ramu P, Elishakoff I. Chebyshev inequality–based inflated convex hull for uncertainty quantification and optimization with scarce samples[J]. Structural and Multidisciplinary Optimization, 2021, 64: 2267-2285.

[76]邱志平. 非概率集合理论凸方法及其应用[M]. 北京: 国防工业出版社, 2005.

[77]Yan Y H, Wang X J, Li Y L. Structural reliability with credibility based on the non-probabilistic set-theoretic analysis[J]. Aerospace Science and Technology, 2022, 127: 107730.

[78]Meng Z, Zhang Z H, Zhou H L. A novel experimental data-driven exponential convex model for reliability assessment with uncertain-but-bounded parameters[J]. Applied Mathematical Modelling, 2020, 77: 773-787.

[79]Jiang C, Bi R G, Lu G Y, et al. Structural reliability analysis using non-probabilistic convex model[J]. Computer Methods in Applied Mechanics and Engineering, 2013, 254: 83-89.

[80]Jiang C, Lu G Y, Han X, et al. Some important issues on first-order reliability analysis with non-probabilistic convex models[J]. ASME Journal of Mechanical Design, 2014, 136(3): 1-5