Reliability Analysis for Multidisciplinary Systems Involving Stationary Stochastic Processes

Transcription

1 ASME 2015 IDETC/CIE Paper number: DETC Reliability Analysis for Multidisciplinary Systems Involving Stationary Stochastic Processes Zhifu Zhu, Zhen Hu, Xiaoping Du Missouri University of Science and Technology

2 Outline Background MDRA with stationary SP Examples Conclusions Acknowledgement 2

3 Multidisciplinary Systems Vehicles Aircrafts Wind Turbines Offshore Structures All pictures are taken from Wikimedia Commons, 3

4 Multidisciplinary Systems X: Random variables; Y: Stochastic processes 4

5 Inputs: X: Random variables Response of subsystem i Problem Statement Y: Stationary stochastic processes Zi () t = gzi ( Xs, Xi, Ys (), t Yi (), t L i ()) t L: Linking variables Time-dependent reliability over [ t0, t s ] R( t, t ) = Pr{ Z ( t) = g ( X, X, Y ( t), Y( t), L ( t)) < e, t [ t, t ]} 0 s i Zi s i s i i 0 s Since the involvement of stochastic processes, the responses are time-dependent random variables, calculating the reliability is difficult. 5

6 Time-Dependent Reliability Methods Upcrossing rate methods -Asymptotic upcrossing rate of a Gaussian stochastic process (i.e. Lindgren 1984, Breitung 1984, 1988) - Vector out-crossing rate using parallel approach (i.e. Hagen, 1992) - The Rice s formula based method (i.e. Rice, 1944, Sudret, Lemaire, 2004, Hu and Du, 2012) - The joint-upcrossing rate method (i.e. Hu and Du, 2013) Surrogate model methods - Composite limit-state function method (i.e. Mourelatos, 2011) - Nested extreme value response method (i.e. Wang and Wang, 2014) - Mixed Efficient Global Optimization method (i.e. Hu and Du, 2015) Sampling methods - Importance sampling approach (i.e. Singh and Mourelatos, 2011) - Markov Chain Monte Carlo method (i.e. Wang and Mourelatos, 2013) - Sampling of extreme value distribution (i.e. Hu and Du, 2013) These methods are for components and may not be applicable for multidisciplinary systems. 6

7 Proposed Method Approximate a response w.r.t. X by FORM and SORM at MPP Then the response is a linear stationary Gaussian process (FORM), or a quadratic stationary Gaussian process (SORM) Use MCS 7

8 Step 1 MPP Search Optimization [1] min u ( ul, ) st.. Failure constraint gˆ Z ( u, ) 0 i i L i > L j ( t) = gl ( u, ), 1,2,, j j L j j = n Coupling between subsystems [1] Du, X., Guo, J., and Beeram, H., 2008, "Sequential optimization and reliability assessment for multidisciplinary systems design," Structural and Multidisciplinary Optimization. 8

9 Step 2 Approximation FORM gˆ gˆ gˆ * * * T Z ( U) Z ( ui) + Z ( ui)( Ui ui) i i i { Z t > e} = { Ht > β} Pr ( ) Pr ( ) i SORM gˆ gˆ gˆ * * * T Z ( U) Z ( ui) + Z ( ui)( Ui ui) i i i 1 + ˆ i 2 * 2 * * T ( Ui ui) gz ( ui)( Ui ui) 9

10 Step 3 Simulation The Expansion Optimal Linear Estimation (EOLE) method [2]. Ut p i T ( ) ϕ ρ ( tt, ) = i= 1 V η i i U i T V i are independent standard normal random variables; i and are the eigenvalues and eigenvectors of the matrix, respectively. ( ) ( t1, t1) ( t1, t2) ( t1, t ) ( t, t ) ( t, t ) ( t, t ) ρ ρ ρ ρ ρ ρ ρ ρ ρ U U U m U 2 1 U 2 2 U 2 m = ( t, t ) ( t, t ) ( t, t ) ρ t 1, t U 2 is the autocorrelation function of Ut (). N f pf ( t0, ts) = N [2] Li, C. C., Kiureghian, A. D., 1993, "Optimal discretization of random fields," Journal of Engineering Mechanics. η U m 1 U m 2 U m m m m ϕ i 10

11 Examples Two examples are solved using: 1. Proposed method based on FORM (FORM-MCS) 2. Proposed method based on SORM (SORM-MCS) 3. Upcrossing rate method (Upcrossing) 4. Direct MCS with the original limit-state function (MCS) 11

12 Example 1 For subsystem 1 L () t = X + X + Y () t 0.2 L () t Z () t = X + Y () t + L () t + L () t For subsystem L () t = L () t + X + Y () t Z () t = X + Y () t + L () t + e L 12 () t The autocorrelation coefficient function of Y () t 1 1 { 2 λ } ( ) = ( ) ρy t1, t2 exp t2 t1 / λ = 0.9 is a correlation length. Limit state e = 22 For MCS, Time interval [0,10] is divided into 200 time instants and 10 6 samples are generated at each time instant. 12

13 Example Upcrossing FORM-MCS SORM-MCS MCS Probability of failure Time 13

14 Example 2 System structure of the compound cylinders 14

15 Example 2 Upcrossing FORM-MCS SORM-MCS MCS Probability of failure Time 15

16 Conclusions A reliability analysis method for time-dependent multidisciplinary system with stationary stochastic processes is developed. The results of the examples showed the efficiency and accuracy of the proposed method. Explicit functions of time Non-stationary processes Higher efficiency Future Work 16

17 Acknowledgement National Science Foundation through grant CMMI The Intelligent Systems Center (ISC) at the Missouri University of Science and Technology 17