Recent Advances in Reliability Estimation of Time-Dependent Problems Using the Concept of Composite Limit State

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1 Automotive Research Center A U.S. Army Center of Excellence for Modeling and Simulation of Ground Vehicles Recent Advances in Reliability Estimation of Time-Dependent Problems Using the Concept of Composite Limit State Zissimos P. Mourelatos Monica Majcher Vijitashwa Pandey Mechanical Engineering Department Oakland University REC 2014; Chicago, IL 1

2 Overview Background information Definition of time-dependent P f Out-crossing rate approach Proposed approach to estimate time-dependent P f using Composite Limit State (CLS) Identification of CLS Calculation of time-dependent P f Implementation points Summary and future work REC 2014; Chicago, IL 2

3 Background Input System Output Uncertainty (Quantified) Propagation Design Uncertainty (Calculated) Random Variables (Time-Independent) Random Processes (Time-Dependent) Challenges: Quantification of a Random Processes Estimation of time-dependent reliability REC 2014; Chicago, IL 3

4 Time-Dependent Probability of Failure P f, T P t 0, T: gx, Yt 0, t 0 t time t 1 0 t2 t 3 t n T P f 0, T P g X, Y t, t 0 i i i 1 n Series System Reliability Problem REC 2014; Chicago, IL 4

5 Time-Dependent Probability of Failure P f, T 1 1 P 0 T i exp t 0 f 0 dt Out-crossing rate approach Failure rate t X, Y( t), t 0 gx, Y( t ), t P g 0 t lim 0 t Up-crossing rate Accurate ONLY if up-crossings are statistically independent REC 2014; Chicago, IL 5

6 Characterization of Input Random Process K-L expansion is used to represent random process Y(t) as : Y T t t Φ t Y p i1 i i Z i t Y,Φ t Z : mean of random process Y(t) : eigenvalue and eigenvector of covariance matrix of Y(t) : standard normal random variable Y(t) is assumed Gaussian REC 2014; Chicago, IL 6

7 Calculation of Time-Dependent Probability of Failure (P f ) P f, T P t 0, T: gx, Yt 0, t 0 Concept of Composite Limit State is used The Composite Limit State defines a convex domain representing the intersection of safe regions of all instantaneous limit states REC 2014; Chicago, IL 7

8 Identification of Composite Limit State z 2 * z i * z k z 1 Composite limit state ik cos Correlation Coefficient Two-step approach to identify composite limit state REC 2014; Chicago, IL 8

9 Identification of Composite Limit State z 2 * z i * z k z 1 Composite limit state ij 0.99 instantaneous limit states (almost parallel) Step 1: Delete highly correlated REC 2014; Chicago, IL 9

10 Identification of Composite Limit State z 2 * z i * z k z 1 Composite limit state Step 2: Delete instantaneous limit states that are not part of the composite If set Z : g Z, t 0, g Z, t j i i 1, i j 0 is null, the j th limit state is deleted Check by solving a series of LPs REC 2014; Chicago, IL 10

11 Calculation of Time-Dependent P f i z 2 * z i * z k z 1 Composite limit state P f ij t ij 0 t i t i, j t j ; zdz Bivariate standard normal vector REC 2014; Chicago, IL 11

12 Calculation of Time-Dependent P f Our approach calculates P f exactly as by eliminating ALL other terms using the convex polyhedron of the safe domain This is a substantial contribution in both Time-Dependent Reliability and System Reliability REC 2014; Chicago, IL 12

13 Calculation of Time-Dependent P f I g 4 g 6 H E F D z 2 g 5 C g 3 * z 2 If is a positive linear * * combination of z 1 and z 3, * * * ( jz j kz k zi, j, k 0 ) g 2 can be eliminated enlarging the safe domain (SD) so that : z * 1 z 2 * A G B z 1 z 3 * Composite limit state P f F 1 P 1 P P ABCDEFA GCDEFG ABG Original SD Enlarged SD f f f g 0, g 0, g P P P f PABG P REC 2014; Chicago, IL 13

14 Calculation of Time-Dependent P f I g 4 g 6 z 2 g 5 Finally: H D E F z * 1 z 2 * A G B C g 3 z 1 z 3 * Composite limit state This is an EXACT calculation of P f involving the convex polyhedron of the SD P f F 1 P ABCDEFA 1 PGCDEFG PABG 1 PGCHFG f f f f f P P 1 P P P P ABG DHE GCI ABG DHE f f PGCI P34 P41 P13 REC 2014; Chicago, IL 14 FHI f 1

15 Illustration of Composite Limit State Example : Hydrokinetic Turbine Blade under Time-dependent River Flow Loading * g t vt P F allow *Hu, Z. and Du, X., (2012), Time-dependent Reliability Analysis by a Sampling Approach to Extreme Values of Stochastic Processes, REC 2014; Proceedings Chicago, of IL the ASME 2012 IDETC/CIE 15 M flap EI t 2 t1 v Cm t allow 2EI : Gaussian random process with autocorrelation coefficient function v allow allow, t 1, I,C m, E t, t cos t t : Allowable strain : Random variables : Constants is calculated from 0 to 12 months 2 1 1

16 Illustration of Composite Limit State t = 8.2 months Safe Region Composite Limit State Instantaneous Limit State REC 2014; Chicago, IL 16

17 Reliability Index Illustration of Composite Limit State t 0.2 month discretization t = 8.2 months Time, months REC 2014; Chicago, IL 17

18 Illustration of Composite Limit State t = 5.2 months Safe Region Composite Limit State Instantaneous Limit State REC 2014; Chicago, IL 18

19 Reliability Index Illustration of Composite Limit State t 0.2 month discretization t = 5.2 months Time, months REC 2014; Chicago, IL 19

20 6 Illustration of Composite Limit State 0.1 month discretization 4 2 z z1 REC 2014; Chicago, IL 20

21 Illustration of Composite Limit State month discretization 4 2 z REC 2014; z1chicago, IL 21

22 6 Illustration of Composite Limit State 0.01 month discretization 4 2 z z1 REC 2014; Chicago, IL 22

23 Probability of Failure Calculation REC 2014; Chicago, IL 23

24 Implementation Points Observations: Proposed approach requires a timeindependent analysis (beta and MPP) at ALL time steps. Only low-beta limit states contribute to composite limit state. To avoid calculating beta and MPP at ALL time steps: Build a surrogate of beta curve using Kriging Build composite limit state (CLS) progressively starting with times where beta is low. Stop if CLS does not change further. REC 2014; Chicago, IL 24

25 Reliability Index Estimation using Kriging REC 2014; Chicago, IL 25

26 Reliability Index Estimation using Kriging REC 2014; Chicago, IL 26

27 Reliability Index Estimation using Kriging REC 2014; Chicago, IL 27

28 Reliability Index Estimation using Kriging REC 2014; Chicago, IL 28

29 Reliability Index Estimation using Kriging REC 2014; Chicago, IL 29

30 Reliability Index Estimation using Kriging REC 2014; Chicago, IL 30

31 Reliability Index Estimation using Kriging REC 2014; Chicago, IL 31

32 Reliability Index Estimation using Kriging REC 2014; Chicago, IL 32

33 Reliability Index Estimation using Kriging Only 14 Evaluations REC 2014; Chicago, IL 33

34 β(t) Progressive Estimation of Composite 5Δt 10Δt 15Δt 20Δt Time t REC 2014; Chicago, IL 34

35 Total Probability Theorem Proposed approach is based on FORM. FORM s accuracy deteriorates if Limit state is nonlinear Random variables are non-normal and / or correlated. The Total Probability Theorem can increase accuracy P f 0, T P F P F f wdw W W Mean value of conditional probability REC 2014; Chicago, IL 35

36 Example : Total Probability Theorem X, t X 2X X X t X t X Y t 3X Y t 0 g Y t, t t Y1, Y2 : Gaussian Processes X, X X 2 4, 5 : Non-normal R.V.s W X X X Y t P F If X , then is calculated using W , t x 2x x X t x t x y t x Y t g Realization of X 1 Normal R.V. Function of Normal R.V.s Linear function of Normal R.V.s REC 2014; Chicago, IL 36

37 Key Points of Composite Limit State Method Efficient estimation of reliability index Kriging metamodel using prediction variance estimation Identification of Composite Limit State using Convex domain formed by instantaneous limit states with smallest betas t t Calculation of P f using Composite Limit State Exact calculation using the convex polyhedron of the safe domain REC 2014; Chicago, IL 37

38 Summary and Future Work Developed a new time-dependent reliability estimation method using the concept of Composite Limit State (CLS). It Identifies the CLS automatically Calculates time-dependent probability of failure exactly for convex linear safe sets Future Work Improve computational efficiency (e.g. use Kriging to estimate MPP locus) Apply method to estimating remaining life due to fatigue failure REC 2014; Chicago, IL 38

39 Thank you for your attention REC 2014; Chicago, IL 39

40 Our Approach A novel MC-based method to calculate the time-dependent reliability (cumulative probability of failure) based on : short-duration data and an exponential extrapolation using MCS or Importance Sampling (Infant Mortality) Poisson s assumption (Useful Life) REC 2014; Chicago, IL 40

41 Efficient MC Simulation Approach 0 MCS 1 ˆ d b 0 dt t0 Exponential Extrapolation ˆ ( t) 0 e bt F t int c T ( t) t ˆ 0 1 e 1 (1 F t dt c T Poisson s Assumption ( t int )) e m ( tt int ), t [0, t, t [ t int int, t f ] ] REC 2014; Chicago, IL 41

42 Objectives and Scientific Contributions Objectives for Project 5.3 Develop a methodology to estimate reliability and remaining life of a vehicle system using time-dependent reliability / durability principles. Use the methodology to improve existing accelerated Life Testing (ALT) methods by Shortening testing time, and Using realistic testing conditions Implement all developments in TARDEC s Physical Simulation Lab Fundamental Scientific Contributions Developed advanced statistical methods to calibrate a math model using a limited number of tests. Developed a novel time-dependent reliability method using the concept of composite limit state. The method also advances state-ofthe-art in system reliability Developed a new paradigm for Accelerated Life Testing REC 2014; Chicago, IL 42

43 Accelerated Life Testing (ALT) Relates reliability measured under high stress conditions to expected reliability under normal conditions Advantage: Shortens testing time Disadvantage: Our Goal: Shorten testing time and use realistic testing conditions Uses unrealistic testing conditions REC 2014; Chicago, IL 43

44 Problem Description Random Variables Durability/Performance Measures in Time Road Height, in Longitudinal Distance, ft Terrain, Engine Load, etc. Vertical Acceleration in G, S(d,X,t) Time Vehicle speed : 20 mph; Mission distance : 100 miles Simulation can be practically performed for a short-duration time REC 2014; Chicago, IL 44

45 Random Variables Observations / Challenges Durability/Performance Measures in Time Road Height, in Longitudinal Distance, ft Terrain, Engine Load, etc. Vertical Acceleration in G, S(d,X,t) Time Major Challenges: Model input random processes (terrain, engine load) Develop detailed, and yet simple and accurate vehicle math models Run math models for long time REC 2014; Chicago, IL 45

46 Model input random processes (terrain, engine load) Time series and spectral decomposition methods Develop detailed, and yet simple and accurate vehicle math models Our Approach to Address Challenges Use available math models Calibrate math models using tests to improve their accuracy (Model V&V approach) Run math models for long time Run calibrated math models for a short duration REC 2014; Chicago, IL 46

47 Proposed Approach for ALT Ct Degraded vehicle parameter (e.g. or ) k s b s Response years Random process characterized by time series yearst years Available math model calibrated using tests REC 2014; Chicago, IL 47

48 Response Main Task in Proposed ALT Calculate reliability through time Reliability at time t is the probability that the system has not failed before time t. Must calculate time-dependent probability of failure months t years REC 2014; Chicago, IL 48

49 Definitions / Observations Reliability: Ability of a system to carry out a function in a time period [0, t L ] p c f c t F t P t L T L Prob. of Time to Failure F c T t Pt, t, such that gxt L 0, t L 0 Cumulative Prob. of Failure F i T t PgXt L L, t 0 L Instantaneous Prob. of Failure Time-Invariant Reliability t 0 L F i t T L time Time-Variant Reliability 0 tl time F c t T L 7/23/2014 REC 2014; Chicago, IL 49

50 Definitions / Observations Reliability: Ability of a system to carry out a function in a time period [0, t L ] p c f c t F t P t L T L Prob. of Time to Failure F c T t Pt, t, such that gxt L 0, t L 0 Cumulative Prob. of Failure F i T t PgXt L L, t 0 L Instantaneous Prob. of Failure Time-Invariant Reliability t 0 L F i t T L time Time-Variant Reliability 0 tl time F c t T L 7/23/2014 REC 2014; Chicago, IL 50

51 Calculation of Cumulative Probability of Failure State-of-the-art Approaches 7/23/2014 PHI2 method (Andrieu-Renaud, et al., RESS, 2004) Set-Based approach (Son and Savage, Quality & Rel. Engin., 2007) 0 t1 t t 2 t F tk 2 t K 1 t F time Kt State-of-the-art approaches are in general, inaccurate due to: Choice of t Not including contribution of all discrete times REC 2014; Chicago, IL 51

52 F c T What is Reliability? Cumulative Probability of Failure Reliability at time t is the probability that the system has not failed before time t. t Pt, t, such that gxt F L i T t PgXt L 0, t L L, t 0 L 0 Cumulative Prob. of Failure Instantaneous Prob. of Failure Calculation Methods for Maximum Response Method Niching GA & Lazy Learning Local Metamodeling MCS / Importance sampling F c T t Analytical F c T t 1 exp[ t 0 t dt] Simulation-based REC 2014; Chicago, IL 52

53 Control Arm Test Fixture at TARDEC HMMWV Control arm / Spring STM Table Actuation point for road input Output : Stress or Strain at different locations on control arm REC 2014; Chicago, IL 53

54 HMMWV Lower Control Arm Fixture MotionView Math Model Physical Model REC 2014; Chicago, IL 54

55 Key Points of Proposed Approach Time series and Spectral Decomposition Characterization of random processes Composite Limit State Approach Calculation of time-dependent probability of failure (or reliability) Companion project Model validation through calibration x, m x, c, x, x, y t y Prediction bias Zero mean random error REC 2014; Chicago, IL 55