Graphical Abstract Figure
A schematic representation of the new strategy for sensitivity estimation
Graphical Abstract Figure
A schematic representation of the new strategy for sensitivity estimation
Abstract
Time-variant reliability sensitivity (TRS) analysis can measure the effect of input factors on the structure/mechanism failure. The traditional method for TRS analysis employs a nested sampling procedure, with computational cost depending on the number of input factors. To address the above weaknesses, a single-loop method is developed for TRS analysis. Based on Bayes’ theorem, the sensitivity measure is derived and expressed by the difference between the probability density function (PDF) and the failure-conditional PDF. This derivation allows for TRS analysis to be performed with just one set of samples, where the computational complexity does not depend on the number of inputs. Then, the procedures for Monte Carlo simulation (MCS) are listed based on the innovative estimation of the sensitivity index. Three examples involving numerical and engineering problems are employed to validate the proposed strategy, with the direct MCS introduced for comparison. The results reveal that the proposed strategy provides satisfactory TRS analysis while significantly saving computational resources.
References
1.
Du
, X.
, 2014
, “Time-Dependent Mechanism Reliability Analysis With Envelope Functions and First-Order Approximation
,” ASME J. Mech. Des.
, 136
(8
), p. 081010
. 2.
Behtash
, M.
, and Alexander-Ramos
, M. J.
, 2022
, “A Reliability-Based Formulation for Simulation-Based Control Co-Design Using Generalized Polynomial Chaos Expansion
,” ASME J. Mech. Des.
, 144
(5
), p. 051705
. 3.
Zhao
, Q.
, Guo
, J.
, Zhao
, D.
, Yu
, D.
, and Hong
, J.
, 2021
, “Time-Dependent System Kinematic Reliability Analysis for Robotic Manipulators
,” ASME J. Mech. Des.
, 143
(4
), p. 041704
. 4.
MiarNaeimi
, F.
, Azizyan
, G.
, and Rashki
, M.
, 2019
, “Reliability Sensitivity Analysis Method Based on Subset Simulation Hybrid Techniques
,” Appl. Math. Model.
, 75
, pp. 607
–626
. 5.
Moustapha
, M.
, Parisi
, P.
, Marelli
, S.
, and Sudret
, B.
, 2024
, “Reliability Analysis of Arbitrary Systems Based on Active Learning and Global Sensitivity Analysis
,” Reliab. Eng. Syst. Saf.
, 248
, p. 110150
. 6.
Cadini
, F.
, Lombardo
, S. S.
, and Giglio
, M.
, 2020
, “Global Reliability Sensitivity Analysis by Sobol-Based Dynamic Adaptive Kriging Importance Sampling
,” Struct. Saf.
, 87
, p. 101998
. 7.
Wang
, X.
, Geng
, X.
, Wang
, L.
, Wang
, R.
, Chen
, X.
, and Fan
, W.
, 2018
, “Time-Dependent Reliability-Based Dimensional Synthesis for Planar Linkages With Unknown But Bounded Joint Clearances
,” ASME J. Mech. Des.
, 140
(6
), p. 061402
. 8.
Wang
, Z.
, Wang
, Z.
, Yu
, S.
, and Zhang
, K.
, 2019
, “Time-Dependent Mechanism Reliability Analysis Based on Envelope Function and Vine-Copula Function
,” Mech. Mach. Theory
, 134
, pp. 667
–684
. 9.
Zhang
, D.
, Shen
, S.
, Wu
, J.
, Wang
, F.
, and Han
, X.
, 2023
, “Kinematic Trajectory Accuracy Reliability Analysis for Industrial Robots Considering Intercorrelations Among Multi-Point Positioning Errors
,” Reliab. Eng. Syst. Saf.
, 229
, p. 108808
. 10.
Wang
, W.
, Shen
, G.
, Zhang
, Y.
, Zhu
, Z.
, Li
, C.
, and Lu
, H.
, 2021
, “Dynamic Reliability Analysis of Mechanical System With Wear and Vibration Failure Modes
,” Mech. Mach. Theory
, 163
, p. 104385
. 11.
Chen
, J.
, Chen
, L.
, Qian
, L.
, Chen
, G.
, and Zhou
, S.
, 2022
, “Time-Dependent Kinematic Reliability Analysis of Gear Mechanism Based on Sequential Decoupling Strategy and Saddle-Point Approximation
,” Reliab. Eng. Syst. Saf.
, 220
, p. 108292
. 12.
Rice
, S. O.
, 1944
, “Mathematical Analysis of Random Noise
,” Bell Syst. Tech. J.
, 23
(3
), pp. 282
–332
. 13.
Andrieu-Renaud
, C.
, Sudret
, B.
, and Lemaire
, M.
, 2004
, “The Phi2 Method: A Way to Compute Time-Variant Reliability
,” Reliab. Eng. Syst. Saf.
, 84
(1
), pp. 75
–86
. 14.
Zhang
, X.
, Lu
, Z.
, Zhao
, Y.
, and Li
, C.
, 2022
, “The Glo Method: An Efficient Algorithm for Time-Dependent Reliability Analysis Based on Outcrossing Rate
,” Struct. Saf.
, 97
, p. 102204
. 15.
Lu
, Z.
, Qiao
, H.
, Zhang
, X.
, and Zhao
, Y.
, 2023
, “An Innovative Method for Space-Time-Dependent Reliability Analysis
,” Struct. Saf.
, 102
, p. 102326
. 16.
Zhuang
, X.
, Li
, X.
, Liu
, C.
, Yu
, T.
, and Song
, B.
, 2024
, “Time-Dependent Kinematic Reliability of Motion Mechanisms With Dynamic Factors
,” Probabilist. Eng. Mech.
, 76
, p. 103598
. 17.
Savage
, G. J.
, and Son
, Y. K.
, 2016
, “An Extreme-Value Event Approach for Frequency-Domain Performance Reliability
,” IEEE Trans. Reliab.
, 65
(2
), pp. 1058
–1068
. 18.
Wang
, L.
, Zhang
, X.
, and Zhou
, Y.
, 2018
, “An Effective Approach for Kinematic Reliability Analysis of Steering Mechanisms
,” Reliab. Eng. Syst. Saf.
, 180
, pp. 62
–76
. 19.
Bae
, S.
, Park
, C.
, and Kim
, N. H.
, 2020
, “Estimating Effect of Additional Sample on Uncertainty Reduction in Reliability Analysis Using Gaussian Process
,” ASME J. Mech. Des.
, 142
(11
), p. 111706
. 20.
Zha
, C.
, Sun
, Z.
, Wang
, J.
, Pan
, C.
, Liu
, Z.
, and Dong
, P.
, 2022
, “A General Active-Learning Method for Surrogate-Based Structural Reliability Analysis
,” Struct. Eng. Mech.
, 83
(2
), pp. 167
–178
. 21.
Zhang
, C.
, and Bai
, G.
, 2012
, “Extremum Response Surface Method of Reliability Analysis on Two-Link Flexible Robot Manipulator
,” J. Cent. South Univ.
, 19
(1
), pp. 101
–107
. 22.
Ehre
, M.
, Papaioannou
, I.
, and Straub
, D.
, 2024
, “Variance-Based Reliability Sensitivity With Dependent Inputs Using Failure Samples
,” Struct. Saf.
, 106
, p. 102396
. 23.
Zha
, C.
, Pan
, C.
, Sun
, Z.
, and Liu
, Q.
, 2024
, “An Efficient Single-Loop Strategy for Time-Variant Reliability Sensitivity Analysis Based on Bayes’ Theorem
,” Structures
, 69
, p. 107464
. 24.
Chauhan
, M. S.
, OjedaTuz
, M.
, Catarelli
, R. A.
, Gurley
, K. R.
, Tsapetis
, D.
, and Shields
, M. D.
, 2024
, “On Active Learning for Gaussian Process-Based Global Sensitivity Analysis
,” Reliab. Eng. Syst. Saf.
, 245
, p. 109945
. 25.
Chiron
, M.
, Morio
, J.
, and Dubreuil
, S.
, 2023
, “Local Sensitivity of Failure Probability Through Polynomial Regression and Importance Sampling
,” Mathematics
, 11
(20
), p. 4357
. 26.
Papaioannou
, I.
, Breitung
, K.
, and Straub
, D.
, 2018
, “Reliability Sensitivity Estimation With Sequential Importance Sampling
,” Struct. Saf.
, 75
, pp. 24
–34
. 27.
Proppe
, C.
, 2021
, “Local Reliability Based Sensitivity Analysis With the Moving Particles Method
,” Reliab. Eng. Syst. Saf.
, 207
, p. 107269
. 28.
Vuillod
, B.
, Montemurro
, M.
, Panettieri
, E.
, and Hallo
, L.
, 2023
, “A Comparison Between Sobol’s Indices and Shapley’s Effect for Global Sensitivity Analysis of Systems With Independent Input Variables
,” Reliab. Eng. Syst. Saf.
, 234
, p. 109177
. 29.
Merritt
, M.
, Alexanderian
, A.
, and Gremaud
, P.
, 2023
, “Global Sensitivity Analysis of Rare Event Probabilities Using Subset Simulation and Polynomial Chaos Expansions
,” Int. J. Uncertain. Quan.
, 13
(1
), pp. 53
–67
. 30.
Sudret
, B.
, 2008
, “Global Sensitivity Analysis Using Polynomial Chaos Expansions
,” Reliab. Eng. Syst. Saf.
, 93
(7
), pp. 964
–979
. 31.
Zha
, C.
, Pan
, C.
, Sun
, Z.
, and Liu
, Q.
, 2024
, “Reliability Sensitivity Analysis for Water Hammer-Induced Stress Failure of Fluid-Conveying Pipe
,” Appl. Math. Model.
, 130
, pp. 51
–65
. 32.
Sobol
, I.
, 2001
, “Global Sensitivity Indices for Nonlinear Mathematical Models and Their Monte Carlo Estimates
,” Math. Comput. Simulat.
, 55
(1–3
), pp. 271
–280
. 33.
Homma
, T.
, and Saltelli
, A.
, 1996
, “Importance Measures in Global Sensitivity Analysis of Nonlinear Models
,” Reliab. Eng. Syst. Saf.
, 52
(1
), pp. 1
–17
. 34.
Borgonovo
, E.
, 2007
, “A New Uncertainty Importance Measure
,” Reliab. Eng. Syst. Saf.
, 92
(6
), pp. 771
–784
. 35.
Li
, L.
, Lu
, Z.
, Feng
, J.
, and Wang
, B.
, 2012
, “Moment-Independent Importance Measure of Basic Variable and Its State Dependent Parameter Solution
,” Struct. Saf.
, 38
, pp. 40
–47
. 36.
Cheng
, K.
, Lu
, Z.
, Ling
, C.
, and Zhou
, S.
, 2020
, “Surrogate-Assisted Global Sensitivity Analysis: An Overview
,” Struct. Multidiscip. Optim.
, 61
(3
), pp. 1187
–1213
. 37.
Wei
, P.
, Song
, J.
, Lu
, Z.
, and Yue
, Z.
, 2016
, “Time-dependent Reliability Sensitivity Analysis of Motion Mechanisms
,” Reliab. Eng. Syst. Saf.
, 149
, pp. 107
–120
. 38.
Wei
, P.
, Wang
, Y.
, and Tang
, C.
, 2017
, “Time-Variant Global Reliability Sensitivity Analysis of Structures With Both Input Random Variables and Stochastic Processes
,” Struct. Multidiscip. Optim.
, 55
(5
), pp. 1883
–1898
. 39.
Shang
, X.
, Zhang
, Z.
, Fang
, H.
, Jiang
, L.
, and Wang
, L.
, 2024
, “Global Sensitivity Analysis Using Polynomial Chaos Expansion Enhanced Gaussian Process Regression Method
,” Eng. Comput.
, 40
(2
), pp. 1231
–1246
. 40.
Jia
, D.
, and Wu
, Z.
, 2024
, “An Improved Adaptive Kriging Model for Importance Sampling Reliability and Reliability Global Sensitivity Analysis
,” Struct. Saf.
, 107
, p. 102427
. 41.
Zha
, C.
, Pan
, C.
, Sun
, Z.
, and Liu
, Q.
, 2024
, “Reliability Sensitivity Analysis for Set Pressure Tolerance of the Direct-Operated Relief Valve in a Vibration Environment
,” J. Mech. Sci. Technol.
, 38
(9
), pp. 4835
–4845
. 42.
Hu
, Z.
, and Mahadevan
, S.
, 2016
, “Global Sensitivity Analysis-Enhanced Surrogate (gsas) Modeling for Reliability Analysis
,” Struct. Multidiscip. Optim.
, 53
(3
), pp. 501
–521
. 43.
Liu
, Y.
, Li
, L.
, Chang
, Z.
, and Wang
, P.
, 2022
, “Kriging-Based Analytical Technique for Global Sensitivity Analysis of Systems With Multivariate Output
,” Struct. Multidiscip. Optim.
, 65
(10
), p. 288
. 44.
Idrissi
, M.
, Chabridon
, V.
, and Iooss
, B.
, 2021
, “Developments and Applications of Shapley Effects to Reliability-Oriented Sensitivity Analysis With Correlated Inputs
,” Environ. Model. Softw.
, 143
, p. 105115
. 45.
Zhang
, F.
, Xu
, X.
, Cheng
, L.
, Tan
, S.
, Wang
, W.
, and Wu
, M.
, 2020
, “Mechanism Reliability and Sensitivity Analysis Method Using Truncated and Correlated Normal Variables
,” Saf. Sci.
, 125
, p. 104615
. 46.
Xiao
, N.
, Huang
, H.
, Wang
, Z.
, Pang
, Y.
, and He
, L.
, 2011
, “Reliability Sensitivity Analysis for Structural Systems in Interval Probability Form
,” Struct. Multidiscip. Optim.
, 44
(5
), pp. 691
–705
. 47.
Wang
, W.
, Gao
, H.
, Wei
, P.
, and Zhou
, C.
, 2017
, “Extending First-Passage Method to Reliability Sensitivity Analysis of Motion Mechanisms
,” Proc. Inst. Mech. Eng., Part O: J. Risk Reliabil.
, 231
(5
), pp. 573
–586
. 48.
Zha
, C.
, Pan
, C.
, Sun
, Z.
, and Liu
, Q.
, 2024
, “A Single-Loop Reliability Sensitivity Analysis Strategy for Time-dependent Rare Events With Both Random Variables and Stochastic Processes
,” Reliab. Eng. Syst. Saf.
, 251
, p. 110373
. 49.
Shi
, Y.
, Lu
, Z.
, Cheng
, K.
, and Zhou
, Y.
, 2017
, “Temporal and Spatial Multi-Parameter Dynamic Reliability and Global Reliability Sensitivity Analysis Based on the Extreme Value Moments
,” Struct. Multidiscip. Optim.
, 56
(1
), pp. 117
–129
. 50.
Ling
, C.
, Lu
, Z.
, Zhang
, X.
, and Zhou
, S.
, 2020
, “An Efficient Method for Estimating Time-Dependent Global Reliability Sensitivity
,” Struct. Multidiscip. Optim.
, 62
(2
), pp. 851
–871
. 51.
Guo
, Q.
, Zhai
, H.
, Suo
, B.
, Zhao
, W.
, and Liu
, Y.
, 2023
, “Time-Variant Reliability Global Sensitivity Analysis With Single-Loop Kriging Model Combined With Importance Sampling
,” Probabilist. Eng. Mech.
, 72
, p. 103441
. 52.
Cadini
, F.
, Santos
, F.
, and Zio
, E.
, 2014
, “An Improved Adaptive Kriging-Based Importance Technique for Sampling Multiple Failure Regions of Low Probability
,” Reliab. Eng. Syst. Saf.
, 131
, pp. 109
–117
. 53.
Wang
, Y.
, Xiao
, S.
, and Lu
, Z.
, 2019
, “An Efficient Method Based on Bayes’ Theorem to Estimate the Failure-Probability-Based Sensitivity Measure
,” Mech. Syst. Signal Process.
, 115
, pp. 607
–620
. 54.
Zhao
, Z.
, Lu
, Z.
, Zhang
, X.
, and Zhao
, Y.
, 2022
, “A Nested Single-Loop Kriging Model Coupled With Subset Simulation for Time-Dependent System Reliability Analysis
,” Reliab. Eng. Syst. Saf.
, 228
, p. 108819
. 55.
Cui
, L.
, Lu
, Z.
, and Zhao
, X.
, 2010
, “Moment-Independent Importance Measure of Basic Random Variable and Its Probability Density Evolution Solution
,” Sci. China: Technol. Sci.
, 53
(4
), pp. 1138
–1145
. 56.
Botev
, Z. I.
, Grotowski
, J. F.
, and Kroese
, D. P.
, 2010
, “Kernel Density Estimation via Diffusion
,” Ann. Statist.
, 38
(5
), pp. 2916
–2957
. 57.
Zhou
, C.
, Shi
, Z.
, Kucherenko
, S.
, and Zhao
, H.
, 2022
, “A Unified Approach for Global Sensitivity Analysis Based on Active Subspace and Kriging
,” Reliab. Eng. Syst. Saf.
, 217
, p. 108080
. 
