ASME 2015 IDETC/CIE Paper number: DETC2015-46168 Reliability Analysis for Multidisciplinary Systems Involving Stationary Stochastic Processes Zhifu Zhu, Zhen Hu, Xiaoping Du Missouri University of Science and Technology
Outline Background MDRA with stationary SP Examples Conclusions Acknowledgement 2
Multidisciplinary Systems Vehicles Aircrafts Wind Turbines Offshore Structures All pictures are taken from Wikimedia Commons, https://commons.wikimedia.org 3
Multidisciplinary Systems X: Random variables; Y: Stochastic processes 4
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
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
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
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
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
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
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
Example 1 For subsystem 1 L () t = X + X + Y () t 0.2 L () t 2 12 1 2 1 21 Z () t = X + Y () t + L () t + L () t For subsystem 2 2 2 1 2 1 12 21 L () t = L () t + X + Y () t 21 12 1 1 Z () t = X + Y () t + L () t + e 2 2 1 1 21 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
Example 1 0.045 0.04 0.035 Upcrossing FORM-MCS SORM-MCS MCS Probability of failure 0.03 0.025 0.02 0.015 0.01 0.005 0 0 2 4 6 8 10 Time 13
Example 2 System structure of the compound cylinders 14
0.1 0.09 0.08 Example 2 Upcrossing FORM-MCS SORM-MCS MCS Probability of failure 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 2 4 6 8 10 Time 15
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
Acknowledgement National Science Foundation through grant CMMI 1234855 The Intelligent Systems Center (ISC) at the Missouri University of Science and Technology 17