Research Progresses on Reliability Analysis and Optimal Design of Automobile Structures

doi:10.3969/j.issn.1004-132X.2024.11.006

Abstract

Abstract: In order to study the reliability of automobile structures, the uncertainty of mechanical structure analysis and design was summarized, and the uncertainty of design variables and parameters of automobile structures was analyzed from three aspects: structural parameter uncertainty, material performance parameter uncertainty and load uncertainty. The research progresses of probabilistic reliability analysis and non-probabilistic reliability analysis methods were reviewed and summarized. The applications of reliability analysis method in automobile structures were listed. The mathematical model and algorithm of reliability were sorted out, and the applications of reliability optimization design in lightweight and crashworthiness of automobiles were studied, and the development trends of reliability analysis and optimization design of automobile structures were presented.

Key words: automobile structure, reliability analysis, reliability optimization design, uncertainty

摘要： 为研究汽车结构的可靠性，对机械结构分析和设计中的不确定性进行总结，从结构参数不确定性、材料性能参数不确定性和载荷不确定性3方面分析了汽车结构设计变量及其参数的不确定性；对概率可靠性分析和非概率可靠性分析方法的研究进展进行了梳理和综述；列举了可靠性分析方法在汽车结构中的应用；对可靠性的数学模型与算法进行梳理，重点研究了可靠性优化设计在汽车轻量化及耐撞性等方面的应用，并指出了汽车结构可靠性分析与优化设计中的发展趋势。

关键词: 汽车结构, 可靠性分析, 可靠性优化设计, 不确定性

CLC Number:

TP182

ZHANG Lei1, 2, 3, SUN Xuetao1, 2, CHEN Jie1, 2, SUN Yuanbo3, GUO Jiajia1, 2, ZHENG Jie1, 2. Research Progresses on Reliability Analysis and Optimal Design of Automobile Structures[J]. China Mechanical Engineering, 2024, 35(11): 1948-1962，1970.

张磊1, 2, 3, 孙学涛1, 2, 陈洁1, 2, 孙远波3, 郭佳佳1, 2, 郑杰1, 2. 汽车结构可靠性分析与优化设计研究进展[J]. 中国机械工程, 2024, 35(11): 1948-1962，1970.

References

［1］庞欢，石东阳，王道成，等. 基于质保数据的汽车可靠性评估及车企运营决策系统［J］. 汽车安全与节能学报，2022，13(2)：250-258.
PANG Huan，WANG Dongyang，WANG Daocheng，et al. Automotive Reliability Evaluation and Operational Decision-making System for Automotive Companies Based on Quality Assurance Data［J］. Journal of Automotive Safety and Energy，2022，13(2)：250-258.
［2］刘晓昂，焦湘和，武一民. 汽车独立悬架后桥悬置系统设计方法研究［J］. 中国机械工程，2020，31(5)：535-542.
LIU Xiaoang，JIAO Xianghe，WU Yimin. Research on the Design Method of Rear Axle Suspension System for Automobile Independent Suspension［J］. China Mechanical Engineering，2020，31(5)：535-542.
［3］DONG Z，LI B W，LI J Y，et. al. Online Reliability Assessment of Energy Systems Based on a High-order Extended-state-observer with Application to Nuclear Reactors［J］. Renewable and Sustainable Energy Reviews，2022，158：1-13.
［4］KESHTEGAR B，BAGHERI M，FEI C W，et al. Multi-extremum-modified Response Basis Model for Nonlinear Response Prediction of Dynamic Turbine Blisk［J］. Engineering with Computers，2022，38(2)： 1243-1254.
［5］WANG W，SHEN G，ZHANG Y M，et al. Dynamic Reliability Analysis of Mechanical System with Wear and Vibration Failure Modes［J］. Mechanism and Machine Theory，2021，163：1-16.
［6］高鹏，谢里阳. 基于改进发生函数方法的多状态系统可靠性分析［J］. 航空学报，2010，31(5)：934-939.
GAO Peng，XIE Liyang. Reliability Analysis of Multi-state Systems Based on Improved Occurrence Function Method［J］. Acta Aeronautica et Astronautica Sinica，2021，31(5)：934-939.
［7］SU S，CHANIOTAKIS E，NARAYANAN S，et al. Clustered Tabu Search Optimization for Reservation-based Shared Autonomous Vehicles［J］. Transportation Letters—the International Journal of Transportation Research，2022，14(2)：124-128.
［8］YAN L，ZHANG T，GAO Y，et al. Reliability Ana-lysis of Station Autonomous Computer System Based on Fuzzy Dynamic Fault Tree and Markov Model［J］. Engineering Reports，2021,3(8)：1-14.
［9］AI H，FAN Y H，ZHANG J L，et al. Topology Optimization of Computer Communication Network Based on Improved Genetic Algorithm［J］. Journal of Intelligent Systems，2022，31(1)：651-659.
［10］GUO Y Y，SUN Y C，LI L B，et al. Reliability Assessment for Multi-source Data of Mechanical Parts of Civil Aircraft Based on the Model［J］. Journal of Mechanical Science and Technology，2019，33(7)：3205-3211.
［11］LU C，FENG Y W，LIEM R P，et al. Improved Kriging with Extremum Response Surface Method for Structural Dynamic Reliability and Sensitivity Analyses［J］. Aerospace Science and Technology，2018，76：164-175.
［12］吴含冰. 汽车电子元器件可靠性标准研究［J］.大众标准化，2020(8)：39-40.
WU Hanbing. Research on Reliability Standards for Automotive Electronic Components［J］. Popular Standardization，2020(8)：39-40.
［13］武佳丽，庄恒国，张诗建，等. 国内外汽车可靠性评价研究现状［J］.科技风，2020(7)：268.
WU Jiali，ZHUANG Hengguo，ZHANG Shijian， et al. Current Status of Research on Automotive Reli-ability Evaluation at Home and Abroad［J］. Technology Style，2020(7)：268.
［14］康锐. 确信可靠性理论与方法［M］. 北京：国防工业出版社，2020.
KANG Rui. Believing in Reliability Theory and Methods［M］. Beijing：National Defence Industry Press，2020.
［15］TB/T 35482—2019机车车辆强度设计及试验鉴定规范总则［S］.北京：中国铁道出版社，2019.
TB/T 35482—2019 General Rules for Strength Design and Testing Appraisal of Locomotives and Vehicles［S］. Beijing:China Railway Publishing House,2019.
［16］ZHANG H，HUANG G，YU D. Numerical Modeling for the Frame Structure of Light Van-type Electric Truck［J］. Science Progress，2020，103(2)：1-27.
［17］ZHANG J，DENG Y，ZHENG B，et al. Lightweight Design of Low-load Electric Vehicle Frame［J］. Journal of Computational Methods in Sciences and Engineering，2021，21(2)：423-433.
［18］MI C，GU Z，ZHANG Y，et al. Frame Weight and Anti-fatigue Co-optimization of a Mining Dump Truck Based on Kriging Approximation Model［J］. Engineering Failure Analysis，2016，66：99-109.
［19］WANG S，WANG D. Crashworthiness-based Multi-objective Integrated Optimization of Electric Vehicle Chassis Frame［J］. Archives of Civil and Mechanical Engineering，2021，21(3)：103.
［20］WANG T，WANG L，WANG C，et al. Crashworthiness Analysis and Multi-objective Optimization of a Commercial Vehicle Frame：a Mixed Meta-Modeling-based Method［J］. Advances in Mechanical Engineering，2018，10(5)：1-24.
［21］WANG J F，CHEN J Q，ZHOU Y J，et al. Vector-angle Geometric Mapping-based Directional Importance Sampling Method for Reliability Analysis［J］. Struct Multidisciplinary Optimization，2022，65(5):154.
［22］HARIRI-ARDEBILI M A，POURKAMALI-ANARAKI F. Structural Uncertainty Quantification with Partial Information［J］. Expert Systems with Applications，2022，198：1-18.
［23］RILEY M E，GRANDHI R V. Quantification of Model-form and Predictive Uncertainty for Multi-physics Simulation［J］. Computers & Structures，2011，89(11)：1206-1213.
［24］WANG X，MENG Z，YANG B，et al. Reliability-based Design Optimization of Material Orientation and Structural Topology of Fiber-reinforced Composite Structures under Load Uncertainty［J］. Composite Structure，2022，291：1-14.
［25］ACAR E，BAYRAK G，JUNG Y，et al. Modeling，Analysis，and Optimization under Uncertainties：a Review［J］. Structural and Multidisciplinary Optimization，2021，64：2909-2945.
［26］DU X P，XU H Y，ZHU F. A Data Mining Method for Structure Design with Uncertainty in Design Variables［J］. Computers & Structures，2021，244：1-13.
［27］梁小冰，卢耀安，王成勇. 双转台五轴机床旋转轴位置无关几何误差的辨识［J］. 中国机械工程，2023，34(21)：2585-2591.
LIANG Xiaobing，LU Yaoan，WANG Chengyong. Identification of Position-independent Geometric Errors of Five-axis Machine Tools with Double Rotary Tables［J］. China Mechanical Engineering，2023，34(21)：2585-2591.
［28］邵健一，樊轶. 基于FMEA的汽车装配工艺可靠性分析与优化［J］. 质量与标准化，2023(5)：38-41.
SHAO Jianyi，FAN Yi. Reliability Analysis and Optimization of Automotive Assembly Process Based on FMEA［J］. Quality and Standardization，2023(5)：38-41.
［29］周文军，张能武. 焊接工艺实用手册［M］.北京：化学工业出版社，2020.
ZHOU Wenjun，ZHANG Nengwu. Practical Manual for Welding Processes［M］. Beijing：Chemical Industry Press，2020.
［30］马莲琴，孙红辉. 汽车车架骨骼精度实现的技术策略［J］. 焊接技术，2023，52(9)：106-110.
MA Lianqin，SUN Honghui. Technical Strategy for Realizing Skeleton Accuracy of Automobile Frames［J］. Welding Technique，2023，52(9)：106-110.
［31］沈晔超，宋守许，王玉琳，等. 基于汽车驱动桥壳再制造的堆焊可靠性研究［J］. 中国机械工程，2013，24(5)：676-680.
SHEN Yechao，SONG Shouxu，WANG Yulin，et al. Research on Reliability of Surfacing Welding Based on Remanufacturing of Automotive Drive Axle Housing［J］. China Mechanical Engineering，2013，24(5)：676-680.
［32］刘雨婷，胡代钧，刘献栋，等. 关于汽车结构设计中存在轻量化极限的分析［J］. 汽车工程，2019，41(8)：892-895.
LIU Yuting，HU Daijun，LIU Xiandong，et al. Analysis of Lightweight Limits in Automotive Structural Design［J］. Automotive Engineering，2019，41(8)：892-895.
［33］李志远，原崇新. 可靠性分析在航空复合材料结构上的应用及展望［J］. 民用飞机设计与研究，2023(2)：145-152.
LI Zhiyuan，YUAN Chongxin. Application and Outlook of Reliability Analysis in Aerospace Composite Structure［J］. Civil Aircraft Design & Research，2023(2)：145-152.
［34］SEYYED A L R，AMIN K，ZHANG J. Robust Topology Optimization under Material and Loading Uncertainties Using an Evolutionary Structural Extended Finite Element Method［J］. Engineering Analysis with Boundary Elements，2021，133：61-70.
［35］郑宇宁，王一凡，王淑玉，等. 考虑非概率不确定性的复合材料壁板颤振可靠性优化方法［J］. 振动工程学报，2024，37(2)：267-275.
ZHENG Yuning，WANG Yifan，WANG Shuyu，et al. A Flutter Reliability Optimization Method for Composite Panels Considering Non-probabilistic Uncertainties［J］. Journal of Vibration Engineering，2019，37(2)：267-275.
［36］WANG W X，GAO H S，ZHOU C C，et al. Reli-ability Analysis of Motion Mechanism under Three Types of Hybrid Uncertainties［J］. Mechanism and Machine Theory，2018，121：769-784.
［37］吴洁，张建国，游令非，等. 基于改进加权响应面的结构可靠度计算方法［J］.北京航空航天大学学报，2021，47(8)：1638-1645.
WU Jie，ZHANG Jianguo，YOU Lingfei，et al. Structural Reliability Calculation Method Based on Improved Weighted Response Surface［J］. Journal of Beijing University of Aeronautics and Astronautics，2021，47(8)：1638-1645.
［38］李永华，陈鹏，田宗睿，等. 基于改进BP神经网络的结构可靠度计算方法［J］.机械强度，2021，43(6)：1359-1365.
LI Yonghua，CHEN Peng，TIAN Zongrui，et al. Structure Reliability Calculation Method Based on Improved Neural Network［J］. Journal of Mechanical Strength，2021，43(6)：1359-1365.
［39］周策，白斌，叶楠. 自适应粒子群优化支持向量回归的工程系统可靠性预测［J］. 机械工程学报，2023，59(14)：328-338.
ZHOU Ce，BAI Bin，YE Nan. Reliability Prediction of Engineering System Based on Adaptive Particle Swarm Optimization Support Vector Regression［J］. Journal of Mechanical Engineering，2023，59(14)：328-338.
［40］THORVE S，HU Z，LAKKARAJU K，et al. A Framework for the Comparison of Agent-based Models［J］. Autonomous Agents and Multi-Agent Systems，2022，36(2)：32.
［41］DU X P，XU H Y，ZHU F. A Data Mining Method for Structure Design with Uncertainty in Design Variables［J］. Computers & Structures，2021，244：1-13.
［42］CHOI S K，CANFIELD R A，GRANDHI R. Reliability-based Structural Optimization［M］. London：Springer London，2007.
［43］陈振中，黄冬宇，田娇，等. 基于二阶抛物线近似的结构可靠性分析方法［J］. 工程设计学报， 2024，31(1)：50-58.
CHEN Zhenzhong，HUANG Dongyu，TIAN Jiao，et al. Structural Reliability Analysis Method Based on Second Order Parabolic Approximation［J］. Chinese Jourmal of Engineering Design, 2024,31(1):50-58.
［44］MENG Z，LI G，YANG D，et al. A New Directional Stability Transformation Method of Chaos Control for First Order Reliability Analysis［J］. Structural and Multidisciplinary Optimization，2017，55:601-612.
［45］HAO P，WANG Y T，LIU C，et al. A Novel Non-probabilistic Reliability-based Design Optimization Algorithm Using Enhanced Chaos Control Method［J］. Computer Methods in Applied Mechanics and Engineering，2017，318:572-593.
［46］ZHAO W，CHEN Y Y，LIU J. An Effective First Order Reliability Method Based on Barzilai-Borwein Step［J］. Applied Mathematical Modelling，2020，77：1545-1563.
［47］KESHTEGAR B，CHAKRABORTY S. A Hybrid Self-adaptive Conjugate First Order Reliability Method for Robust Structural Reliability Analysis［J］. Applied Mathematical Modelling，2018，53：319-332.
［48］ARAB G H，RASHKI M，ROSTAMIAN M，et al. Refined First-order Reliability Method Using Cross-entropy Optimization Method［J］. Engineering with Computers，2019，35：1507-1519.
［49］ZHU S P，KESHTEGAR B，BEN S M E A，et al. Hybrid and Enhanced PSO：Novel First-order Reliability Method-based Hybrid Intelligent Approaches［J］. Computer Methods in Applied Mechanics and Engineering，2022，393：1-19.
［50］YASEEN Z M，ALDLEMY M S，OUKATI S M. Non-gradient Probabilistic Gaussian Global-best Harmony Search Optimization for First-order Reliability Method［J］. Engineering with Computers，2020，36:1189-1200.
［51］HUANG X Z，LI Y X，ZHANG Y M，et al. A New Direct Second-order Reliability Analysis Method［J］. Applied Mathematical Modelling，2018，55:68-80.
［52］GASPAR B，NAESS A，LEIRA B J. System Reliability Analysis by Monte Carlo Based Method and Finite Element Structural Models［J］. Journal of Offshore Mechanics and Arctic Engineering，2014，136(3)：1-9.
［53］DUBOURG V，SUDRET B，DEHEEGER F. Metamodel-based Importance Sampling for Structural Reliability Analysis［J］. Probabilistic Engineering Mechanics，2013，33：47-57.
［54］BOURINET J M，DEHEEGER F，LEMAIRE M. Assessing Small Failure Probabilities by Combined Subset Simulation and Support Vector Machines［J］. Structural Safety，2011，33(6)：343-353.
［55］SONG J，VALDEBENITO M，WEI P，et al. Non-intrusive Imprecise Stochastic Simulation by Line Sampling［J］. Structural Safety，2020，84：1-17.
［56］VALENTINI F，SILVA O M，TORII A J，et al. Local Averaged Stratified Sampling Method［J］. Journal of the Brazilian Society of Mechanical Sciences and Engineering，2022，44(7)：1-23.
［57］HONG L，LI H，PENG K，et al. A Novel Kriging Based Active Learning Method for Structural Reliability Analysis［J］. Journal of Mechanical Science & Technology，2020，34(4)：1545-1556.
［58］VAPNIK V. An Overview of Statistical Learning Theory［J］. IEEE Transactions on Neural Networks，1999，10(5)：988-999.
［59］高进，崔海冰，樊涛，等. 一种基于自适应Kriging集成模型的结构可靠性分析方法［J］. 中国机械工程，2024，35(1)：83-92.
GAO Jin，CUI Haibing，FAN Tao，et al. A Structural Reliability Analysis Method Based on Adaptive Kriging Integrated Model［J］. China Mechanical Engineering，2024，35(1)：83-92.
［60］张明，王恩志，刘耀儒，等. 利用多项式混沌展开的结构可靠性分析［J］. 清华大学学报：自然科学版，2022，62(8)：1314-1320.
ZHANG Ming，WANG Enzhi，LIU Yaoru，et al. Structural Reliability Analysis Using Polynomial Chaos Expansion［J］. Journal of Tsinghua University：Natural Science Edition，2022，62(8)：1314-1320.
［61］许敏，胡丽丹. 径向基函数神经网络快速算法及其应用［J］. 统计与决策，2021，37(16)：52-56.
XU Min，HU Lidan. Fast Algorithm of Radial Basis Function Neural Network and Its Application［J］. Statistics & Decision，2021，37(16)：52-56.
［62］JIN R，CHEN W. Comparative Studies of Metamodelling Techniques under Multiple Modelling Criteria［J］. Structural & Multidisciplinary Optimization，2001，23(1)：1-13.
［63］杨旭锋，程鑫，刘泽清. 一种融合交叉熵自适应抽样与ALK模型的可靠性分析方法［J］. 机械工程学报，2023，59(12)：56-66.
YANG Xufeng，CHENG Xin，LIU Zeqing. Reli-ability Analysis Method Combining Cross-entropy Adaptive Sampling and ALK Model［J］. Journal of Mechanical Engineering，2023，59(12)：56-66.
［64］YANG S Y，HE Z Y，CHAI J B，et al. A Novel Hybrid Adaptive Framework for Support Vector Machine-based Reliability Analysis：a Comparative Study［J］. Structures，2023，58：1-21.
［65］WAN H P，GAN J R，ZHU Y K，et al. SS-MASVM：an Advanced Technique for Assessing Failure Probability of High-dimensional Complex Systems Using the Multi-class Adaptive Support Vector Machine［J］. Computer Methods in Applied Mechanics and Engineering，2024，418：1-16.
［66］张文鑫，吕震宙. 一种新的自适应Kriging法停止准则及其在涡轮盘疲劳寿命可靠性中的应用［J］. 机械工程学报，2022，58(6)：263-273.
ZHANG Wenxin，LYU Zhenzhou. New Stopping Criterion of Adaptive Kriging Method and Its Application in Fatigue Life Reliability for Turbine Disk［J］. Journal of Mechanical Engineering，2022，58(6)：263-273.
［67］XU J M，WANG H M，ZHANG L B，et al. Robust Twin Depth Support Vector Machine Based on Average Depth［J］. Knowledge-Based Systems，2023，274：1-14.
［68］LI G S，MA S C，ZHANG D Q，et al. An Efficient Sequential Anisotropic RBF Reliability Analysis Method with Fast Cross-validation and Parallelizability［J］. Reliability Engineering & System Safety，2024，241：1-14.
［69］IGUSA T，BUONOPANE S G，ELLINGWOOD B R. Bayesian Analysis of Uncertainty for Structural Engineering Applications［J］. Structural Safety，2002，24(2)：165-186.
［70］BACCOU J，CHOJNACKI E，MERCAT C，et al. Extending Monte Carlo Simulations to Represent and Propagate Uncertainties in Presence of Incomplete Knowledge:Application to the Transfer of a Radionuclide in the Environment［J］. Journal of Environmental Engineering，2008，134(5)：362-378.
［71］YAO H，ZHAO C B，CHEN P Y，et al. A Non-probabilistic Reliability Analysis Method with the Fuzzy Failure Criterion［J］. Structures，2023，58：1-9.
［72］DU X. Reliability-based Design Optimization with Dependent Interval Variables［J］. International Journal for Numerical Methods in Engineering，2012，91(2)：218-228.
［73］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.
［74］BAE H R，GRANDHI R V，CANFIELD R A. An Approximation Approach for Uncertainty Quantification Using Evidence Theory［J］. Reliability Engineering & System Safety，2004，86(3)：215-225.
［75］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，2023，45(5)：1-12.
QIAO Xinzhou，ZHANG Jinrui，FANG Xiurong，et al. An Improved Multidimensional Parallelepiped Model for Structural Uncertainty Analysis［J］. Acta Ordnance Engineering，2023，45(5)：1-12.
［76］BAI X，TU B Y. Ellipsoid Non-probabilistic Reli-ability Analysis of the Crack Growth Fatigue of a New Titanium Alloy Used in Deep-sea Manned Cabin［J］. Theoretical and Applied Fracture Mechanics，2021，115：1-11.
［77］檀中强，潘骏，胡明，等. 基于椭球模型的机构非精确概率可靠性分析方法［J］. 机械工程学报，2019，55(2)：168-176.
TAN Zhongqiang，PAN Jun，HU Ming，et al. Imprecise Probabilistic Reliability Analysis Method for Mechanism Based on Ellipsoid Model［J］. Journal of Mechanical Engineering，2019，55(2)：168-176.
［78］LI K, LIU H. Structural Reliability Analysis by Using Non-probabilistic Multi-cluster Ellipsoidal Model［J］. Entropy：Basel，2022，24(9):1-24.
［79］HONG L X，LI H C，FU J F，et al. Hybrid Active Learning Method for Non-probabilistic Reliability Analysis with Multi-super-ellipsoidal Model［J］. Reliability Engineering & System Safety，2022，222：1-14.
［80］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.
［81］CHEN L，ZHANG Z，YANG G，et al. Evidence-theory-based Reliability Analysis from the Perspective of Focal Element Classification Using Deep Learning Approach［J］. Journal of Mechanical Design，2023，145(7)：1-12.
［82］WANG S N，TANG Y C. An Improved Approach for Generation of a Basic Probability Assignment in the Evidence Theory Based on Gaussian Distribution［J］. Arabian Journal for Science and Engineering，2022，47(2)：1595-1607.
［83］ZHANG Z, RUAN XX，DUAN M F，et al. An Efficient Epistemic Uncertainty Analysis Method Using Evidence Theory［J］. Computer Methods in Applied Mechanics and Engineering，2018，339：443-466.
［84］贾大卫，吴子燕，何乡. 一种基于凸集-概率混合模型的结构可靠性分析法［J］. 固体力学学报，2020，41(5)：470-484.
JIA Dawei，WU Ziyan，HE Xiang. A Method for Structural Reliability Analysis Based on the Convex Set-probability Hybrid Model［J］. Chinese Journal of Solid Mechanics，2020，41(5)：470-484.
［85］梁云飞，张德权，彭周源. 一种高效的概率-证据混合工业机器人定位精度可靠性分析方法［J］. 燕山大学学报，2022，46(4)：309-318.
LIANG Yunfei，ZHANG Dequan，PENG Zhou-yuan. An Efficient Reliability Analysis Method for Positioning Accuracy of Industrial Robot through Probability and Evidence Theory［J］. Journal of Yanshan University，2022，46(4)：309-318.
［86］曹亮，陈国强，董丽君，等. 基于概率论和证据理论的小概率问题可靠性分析［J］. 湖南工程学院学报：自然科学版，2019，29(2)：30-35.
CAO Liang，CHEN Guoqiang，DONG Lijun，et al. Reliability Analysis of Small Probability Problem Based on Probality and Evidence Theory［J］. Journal of Hunan Institute of Engineering：Natural Science Edition，2019，29(2)：30-35.
［87］WANG C，MATTHIES H G. Epistemic Uncertainty-based Reliability Analysis for Engineering System with Hybrid Evidence and Fuzzy Variables［J］. Computer Methods in Applied Mechanics and Engineering，2019，355：438-455.
［88］胡启国，袁帅辉. 考虑失效相关的汽车盘式制动器的可靠性分析［J］. 汽车安全与节能学报，2020，11(2)：198-204.
HU Qiguo，YUAN Shuaihui. Reliability Analysis of Automotive Disc Brakes Considering Failure Correlation［J］. Journal of Automotive Safety and Energy Conservation，2020，11(2)：198-204.
［89］王俊峰，陈吉清，兰凤崇，等. 矢量-角度几何映射的方向重要抽样分布函数近似化方法及多失效模式可靠性分析［J］. 汽车工程，2023，45(6)：1081-1089.
WANG Junfeng，CHEN Jiqing，LAN Fengchong，et al. Approximation Method of Distribution Function Based on Directional Importance Sampling of Vector-angle Geometric Mapping and Reliability Analysis of Multiple Failure Modes［J］. Automotive Engineering，2023，45(6)：1081-1089.
［90］CHEN C H，LONG J Q，TIAN H L，et al. Reli-ability Analysis of the Light Truck Frame Based on Statics Structure and Response Surface Methodo-logy［C］∥2019 4th International Conference on System Reliability and Safety(ICSRS). Rome：IEEE， 2019：407-413.
［91］刘玉敏，谢启盛，承姿辛，等. 基于AK-MCS法的新能源电动汽车转向驱动桥空心半轴可靠度分析［J］. 机械设计，2023，40(5)：27-35.
LIU Yumin，XIE Qisheng，CHENG Zixin，et al. Reliability Analysis on Steering Drive Axles Hollow Half Shaft of New-energy Electric Vehicles Based on AK-MCS Method［J］. Journal of Machine Design，2023，40(5)：27-35.
［92］陈嘉慧，米洁，张胜伦，等. 考虑参数不确定性的疲劳断轴可靠性分析方法［J］. 机床与液压，2023，51(15)：216-220.
CHEN Jiahui，MI Jie，ZHANG Shenglun，et al. Reliability Analysis Method of Fatigue Broken Spindle Considering Parameter Uncertainty［J］. Machine Tool & Hydraulics，2023，51(15)：216-220.
［93］李光霁，刘新玲. 引擎盖的多目标优化和可靠性分析［J］. 现代制造工程，2020(12)：19-24.
LI Guangji，LIU Xinling. Multi-objective Optimization and Reliability Analysis of Engine Hood［J］. Modern Manufacturing Engineering，2020(12)：19-24.
［94］俞陆新，崔强，柳砚，等. Copula函数模型的汽车偏置碰前纵梁结构可靠性分析［J］. 佳木斯大学学报：自然科学版，2022，40(4)：59-61.
YU Luxin，CUI Qiang，LIU Yan，et al. Structural Reliability Analysis of Front Stringer of Automobile Offset Collision Based on Copula Function Model［J］. Journal of Jiamusi University：Natural Science Edition，2022，40(4)：59-61.
［95］LYU H，YAN K，HUANG X T，et al. Uncertainty and Correlation Propagation Analysis of Powertrain Mounting Systems Based on Multi-ellipsoid Convex Model［J］. Mechanical Systems and Signal Processing，2022，173：1-25.
［96］吕辉，钟顺江，李振聪，等. 基于多维子平行六面体模型的可靠性分析方法及应用［J］. 汽车工程，2023，45(6)：1090-1098.
LYU Hui，ZHONG Shunjiang，LI Zhencong，et al. Reliability Analysis Method Based on Multidimensional Sub-parallelepiped Model and Its Application［J］. Automotive Engineering，2023，45(6)：1090-1098.
［97］LYU H，FENG Q，CAI Z，et al. An Optimization Method for Brake Instability Reduction with Fuzzy-boundary Interval Variables［J］. Proceedings of the Institution of Mechanical Engineers，Part D：Journal of Automobile Engineering，2019，233(12)：3209-3221.
［98］MI C J，LI W T，WU W G. Fuzzy Fatigue Reliability Analysis and Optimization of A-type Frame of Electric Wheel Dump Truck Based on Response Surface Method［J］. Mechanics，2019，25(1)：44-51.
［99］LI F，ZHOU Y C，WEI T H，et al. An lnterval-probability Hybrid Reliability Method for Truck Frame［J］. Journal of Failure Analysis and Prevention，2022，22(3)：1260-1267.
［100］邱继伟，罗海胜. 基于混合不确定性的螺旋锥齿轮结构可靠性分析［J］. 吉林大学学报：工学版，2022，52(2)：466-473.
QIU Jiwei，LUO Haisheng. Structure Reliability Analysis of Spiral Bevel Gear Based on Hybrid Uncertainties［J］. Journal of Jilin University：Engineering and Technology Edition，2022，52(2)：466-473.
［101］李伟平，孔剑，陶祺臻，等. 驾驶室ROPS性能非概率可靠性分析［J］. 机械科学与技术，2016，35(5)：768-773.
LI Weiping，KONG Jian，TAO Qizhen，et al. Non-probabilistic Reliability Analysis of Cab ROPS Performance［J］. Mechanical Science and Technology，2016，35(5)：768-773.
［102］赵军，陶友瑞. 一种基于概率-证据理论的滑动轴承可靠性优化设计［J］. 润滑与密封，2021，46(9)：128-133.
ZHAO Jun，TAO Yourui. Reliability Optimization Design of Plain Bearing Based on Probability-evidence Theory［J］. Lubrication & Seals，2019，46(9)：128-133.
［103］顾方秋，苏小平，缪小冬. 某半挂车车架性能分析及结构优化设计［J］. 重庆理工大学学报：自然科学版，2022，36(7)：94-101.
GU Fangqiu，SU Xiaoping，MIAO Xiaodong. Performance Analysis and Structural Optimization Design of a Semi-trailer Frame［J］. Journal of Chongqing University of Technology：Natural Science，2022，36(7)：94-101.
［104］GUAN B，WAN M，WU X D. Reliability-based Design Optimization of Structures with Disturbed Design Variables and Dependent Uncertainties［J］. Journal of Physics：Conference Series，2022 (1)：1-9.
［105］王倩蓉，姜潮，方腾. 一种考虑参数相关性的可靠性优化设计方法［J］. 中国机械工程，2018，29(19)：2312-2319.
WANG Qianrong，JIANG Chao，FANG Teng. Reliability-based Design Optimization Considering Correlated Random Variables［J］. China Mechanical Engineering，2018，29(19)：2312-2319.
［106］XU X，HUANG C，LI C，et al. Uncertain Design Optimization of Automobile Structures：a Survey［J］. Electronic Research Archive，2023，31(3)：1212-1239.
［107］QIU Z P，HUANG R，WANG X J，et al. Structural Reliability Analysis and Reliability-based Design Optimization:Recent Advances［J］. Science China—Physics Mechanics & Astronomy，2013，56(9)：1611-1628.
［108］ZHAO H B，LI S J，CHEN B R. The Reliability-based Design Optimization of Considering Rock-support Interaction for Rock Tunnels［J］. Advances in Civil Engineering，2021，2021：1-13.
［109］MENG Z，YANG D，ZHOU H，et al. Convergence Control of Single Loop Approach for Reliability-based Design Optimization［J］. Structural and Multidisciplinary Optimization，2018，57(3)：1079-1091.
［110］WANG Z H，LI H，CHEN Z M，et al. Sequential Optimization and Moment-based Method for Efficient Probabilistic Design［J］. Structural and Multidisciplinary Optimization，2020，62(1)：387-404.
［111］陈龙，朱颖，徐力，等. 基于权重灵敏度分析的结构形状等几何优化方法［J］. 中国机械工程，2020，31(11)：1306-1314.
CHEN Long，ZHU Ying，XU Li，et al. Geometric Optimization Method for Structural Shapes Based on Analysis of Weight Sensitivity［J］. China Mechanical Engineering，2020，31(11)：1306-1314.
［112］张磊，许帅康，陈洁，等. 列车车体轻量化设计研究进展［J］. 机械工程学报，2023，59(24)：177-196.
ZHANG Lei，XU Shuaikang，CHEN Jie，et al. Research Progress on Lightweight Design of Train Body［J］. Journal of Mechanical Engineering，2023，59(24)：177-196.
［113］陈建良，赵清海，李信卿，等. 内燃机活塞点阵结构可靠性拓扑优化设计［J］. 机械强度，2023，45(1)：91-97.
CHEN Jianliang，ZHAO Qinghai，LI Xinqing，et al. Reliability Topology Optimization Design of Piston Lattice Structure for Internal Combustion Engine［J］. Journal of Mechanical Strength，2023，45(1)：91-97.
［114］承姿辛，刘玉敏，邱光琦. 基于可靠性优化的空心半轴轻量化设计［J］.机械设计，2023，40(1)：34-39.
CHENG Zixin，LIU Yumi，QIU Guangqi. Lightweight Design of Hollow Half Shaft Based on Reliability Optimization［J］. Journal of Machine Design，2023，40(1)：34-39.
［115］XU X，CHEN X B，LIU Z，et al. Reliability-based Design for Lightweight Vehicle Structures with Uncertain Manufacturing Accuracy［J］. Applied Mathematical Modelling，2021，95：22-37.
［116］魏彤辉，左文杰，郑宏伟，等. 基于降维算法的车身可靠性优化［J］. 汽车工程，2020，42(7)：941-948.
WEI Tonghui，ZUO Wenjie，ZHENG Hongwei，et al. Reliability-based Design Optimization of Car Body Based on Dimension Reduction Method［J］. Automotive Engineering，2020，42(7)：941-948.
［117］LI Q Y，CHEN Y Q，LYU W H. Reliability Optimization Design of Transmission Structure of Automotive Mechanical Transmission［J］. IOP Conference Series：Materials Science and Engineering，2018，452(2)：022036.
［118］LIU Z，ZHANG L，ZHU P，et al. Reliability-based Design Optimization of 3D Orthogonal Woven Composite Automobile Shock Tower［J］. Journal of Composite Materials，2021，55(30)：4617-4631.
［119］李方义，荣见华，胡林，等.基于概率凸集混合模型的汽车正面碰撞结构可靠性优化设计［J］.振动与冲击，2016，35(3)：215-220.
LI Fangyi，RONG Jianhua，HU Lin，et al. Reliability-based Optimization Design of Structure Subjected to Vehicle Frontal Impact Based on Probability-convex Hybrid Model［J］. Journal of Vibration and Shock，2016，35(3)：215-220.
［120］王琼，黄志亮. 含概率-区间混合不确定性的汽车正面碰撞可靠性优化设计［J］. 汽车工程，2019，41(5)：545-549.
WANG Qiong，HUANG Zhiliang. Optimal Design of Vehicle Frontal Crash Reliability with Probability-interval Mixed Uncertainty［J］. Automotive Engineering，2019，41(5)：545-549.
［121］刘鑫，张远洋. 基于混合模型汽车结构耐撞性的可靠性优化设计［J］. 长沙理工大学学报：自然科学版，2021，18(1)：95-101.
LIU Xin, ZHANG Yuanyang. Reliability-based Design Optimization for Vehicle Structural Crashworthiness Based on Hybrid Model［J］. Journal of Changsha University of Science & Technology(Natural Science),2021,18（1）：95-101.
［122］HUANG Z L，JIANG C，ZHOU Y S，et al. Reliability-based Design Optimization for Problems with Interval Distribution Parameters［J］. Structural and Multidisciplinary Optimization，2017，55(2)：513-528.
［123］姜平，鲍娣，夏梁，等. 基于组合近似模型的汽车结构耐撞性优化设计［J］. 合肥工业大学学报：自然科学版，2020，43(4)：443-449.
JIANG Ping，BAO Di，XIA Liang，et al. Structural Crashworthiness Optimization of a Vehicle Based on Ensemble of Metamodels［J］. Journal of Hefei University of Technology：Natural Science，2020，43(4)：443-449.
［124］张勇，林福泳. 铝泡沫填充薄壁结构耐撞可靠性优化设计［J］. 机械工程学报，2011，47(22)：93-99.
ZHANG Yong，LIN Fuyong. Crashworthiness Reliability Design Optimization of Aluminum Foam Filled Thin-wall Structures［J］. Journal of Mechanical Engineering，2011，47(22)：93-99.
［125］汪婷，李翼良，张代胜，等. 汽车前纵梁结构耐撞性分析及多目标优化设计［J］. 汽车实用技术，2023，48(8)：85-90.
WANG Ting，LI Yiliang，ZHANG Daisheng，et al. Crashworthiness Analysis and Multi-objective Optimization Design of Automobile Front Longitudinal Beam Structure［J］. Automobile Applied Technology，2023，48(8)：85-90.
［126］高剑武，李旗，李亚军，等. 基于PSO-SVR近似模型的汽车正碰可靠性优化［J］.合肥工业大学学报：自然科学版，2020，43(6)：725-730.
GAO Jianwu，LI Qi，LI Yajun，et al. Reliability Optimization of Vehicle Frontal Impact Based on PSO-SVR Surrogate Model［J］. Journal of Hefei University of Technology：Natural Science，2020，43(6)：725-730.
［127］WU H Q，KUANG S J，HOU H B. Research on Application of Electric Vehicle Collision Based on Reliability Optimization Design Method［J］. International Journal of Computational Methods，2019，16(7)：1-16
［128］陶威，刘钊，许灿，等. 三维正交机织复合材料翼子板多尺度可靠性优化设计［J］. 上海交通大学学报，2021，55(5)：615-623.
TAO Wei，LIU Zhao，XU Can，et al. Multiscale Reliability Optimization Design of Three-dimensional Orthogonal Woven Composite Fender［J］. Journal of Shanghai Jiao Tong University，2021，55(5)：615-623.
［129］吴自强，陈振中，夏雨. 平纹机织复合材料纯电动汽车电池箱多尺度可靠性优化设计［J］. 东华大学学报：自然科学版，2023，49(4)：66-75.
WU Ziqiang，CHEN Zhenzhong，XIA Yu. Multi-scale Reliability Optimization Design of Plain Woven Composite Battery Box for Pure Electric Vehicle［J］. Journal of Donghua University:Natural Science Edition，2019，49(4)：66-75.
［130］孙远波，张磊，唐嘉薇，等. 人机系统可靠性设计研究进展［J］. 包装工程，2023，44(18)：11-18.
SUN Yuanbo，ZHANG Lei，TANG Jiawei，et al. Research Progress of Man-machine System Reli-ability Design［J］. Packaging Engineering，2023，44(18)：11-18.

[1]	CUI Shuo, LIU Xiuli, LI Xiangjie, WU Guoxin. Prediction of RUL and Uncertainty Quantification Evaluation Methods for High-end Rotating Machinery [J]. China Mechanical Engineering, 2026, 37(1): 209-222.
[2]	Pengpeng ZHI, Xueqin LIU, Yi GUAN, Jiang WU, Zhonglai WANG. AMESim-PID-Kriging-based Reliability Analysis Method for Air-fuel Ratio Control of General Aviation Engines [J]. China Mechanical Engineering, 2025, 36(10): 2433-2443.
[3]	TIAN Hailong1, 2, SUN Yuzhi1, 2, YANG Zhaojun1, 2, LIU Zhifeng1, 2, CHEN Chuanhai1, 2, HE Jialong1, 2. Research Status and Development Trends of Failure Modes, Effects, and Criticality Analysis for CNC Machine Tool Reliability [J]. China Mechanical Engineering, 2025, 36(07): 1430-1441.
[4]	CHEN Zhiyong, LI Pan, YE Mingxu, LIN Xinyou. RBF Neural Network Adaptive Control for Autonomous Electric Vehicles Based on Parameter Prediction [J]. China Mechanical Engineering, 2024, 35(06): 982-992.
[5]	SONG Zhouzhou1, 2, ZHANG Hanyu1, 2, LIU Zhao3, ZHU Ping1, 2. A High-dimensional Uncertainty Propagation Method Based on Supervised Dimension Reduction and Adaptive Kriging Modeling [J]. China Mechanical Engineering, 2024, 35(05): 762-769，810.
[6]	LIU Yanhao, NI Bingyu, TIAN Wanyi, JIANG Chao. A Sequential Simulation Method for Dynamic Uncertainty Analysis of Rigid-flexible Coupling Systems under Interval Process Excitations [J]. China Mechanical Engineering, 2024, 35(05): 770-783.
[7]	LIU Jingfei1, JIANG Chao2, NI Bingyu2, WANG Zongtai3. High Dimensional Multioutput Uncertainty Propagation Method via Active Learning and Bayesian Deep Neural Network [J]. China Mechanical Engineering, 2024, 35(05): 792-801.
[8]	ZHANG Wei1, 2, XIE Jun1, DU Yanbin1, RAN Yan3, ZHANG Genbao3, 4. A Reliability Allocation Method for Meta-action of Mechanical Products Considering Double Uncertainty [J]. China Mechanical Engineering, 2024, 35(05): 802-810.
[9]	QU Xiaozhang, ZHANG Jiabei, ZHAI Fangzhi. Reliability-based Analysis for Two-phase Flow of Locomotive Side Wall Filtration Systems with Non-random Load Uncertainty [J]. China Mechanical Engineering, 2023, 34(15): 1881-1889.
[10]	WANG Qiao, DU Xuesong, SONG Chaosheng, ZHU Caichao, SUN Jianquan, LIAO Delin. Research on Accelerated Life Test Method of Harmonic Reducers [J]. China Mechanical Engineering, 2022, 33(19): 2317-2324.
[11]	CHEN Dongning, HU Yanlong, YAO Chengyu, WANG Kuantong, MA Lei, . Multi-dimensional Dynamic Bayesian Network and Its Importance Measure Analysis Method [J]. China Mechanical Engineering, 2022, 33(19): 2340-2346.
[12]	YI Qian, LI Congbo, PAN Jian, ZHANG You. Reliability Analysis of Machining Accuracy and Processing Parameter Optimization for Thin-plate Parts [J]. China Mechanical Engineering, 2022, 33(11): 1269-1277.
[13]	ZHANG Jiahua, LI Aiping, LIU Xuemei. Routing Planning for Assembly Line Material Distributions under Interval Uncertain Travel Time [J]. China Mechanical Engineering, 2021, 32(18): 2239-2246.
[14]	HAO Yaodong, GU Cansong, ZHOU Huanyang, DONG Junhong, LI Hongliang, DENG Jianghua. Stochastic Statistical Energy Method for High-frequency Analysis of Uncertain Acoustic-Structural Coupling Model#br# [J]. China Mechanical Engineering, 2021, 32(08): 921-929.
[15]	WANG Kang, QI Jinping, ZHOU Yahui, LI Shaoxiong, ZHAO Ruihu, GUO Hao. Reliability Analysis of Train Control Center Based on Discrete-time Bayesian Network [J]. China Mechanical Engineering, 2021, 32(04): 390-398.