Uncertainty Propagation and Global Sensitivity Analysis in Hybrid Simulation using Polynomial Chaos Expansion

Transcription

1 Uncertainty Propagation and Global Sensitivity Analysis in Hybrid Simulation using Polynomial Chaos Expansion EU-US-Asia workshop on hybrid testing Ispra, 5-6 October 2015 G. Abbiati, S. Marelli, O.S. Bursi, B. Sudret and B. Stojadinovic 1

2 Acknowledgements 1. The speaker gratefully acknowledges the Workshop Organizing Committees for the invitation 2. The authors gratefully acknowledges the financial supports from the European Union through the SERIES project (Grant number: ). 3. The authors gratefully acknowledge the financial supports of the University of Trento for Lab. activities

3 Hybrid Simulation of a piping system response 3D model of the piping+support Dimensions and specifications of the piping Pipe Size 8 and 6 Schedule 40 Material API 5L Gr. X52 fy= 418 Mpa; fu = 554 Mpa; Elongation = 35.77% Liquid/Internal Pressure Water/ 3.2 MPa Page 3

4 Critical structural elements Test setup for a single elbow Hysteretic response of elbows 4

5 The experimental setup Seismic loading Page 5

6 Test setup at the University of Trento (Italy) Page 6

7 Deterministic testing assumptions Clamped piping ends 0.5% viscous damping Bursi O.S., Abbiati G., Reza Md.S., A Novel Hybrid Testing Approach for Piping Systems of Industrial Plants Smart Structures and Systems In press

8 HOW TO HANDLE MODEL UNCERTAINTIES IN HYBRID SIMULATION? 8

9 The benchmark problem r Seismic load 1 Ag [m/s 2 ] Time [s] ( β (( ) ) ) 2 1 γ n ( 2 1) r & = A sgn u u r + r u u & & & & 9

10 Input stochastic parameters r k1 [N/m] x k3 [N/m] x ξ [/] [/] (Damping) 10

11 Output response quantities r Displacement peaks: Restoring force peak: Total dissipated energy:, =arg,, =arg, =! 11

12 Method development objectives Uncertainty propagation: estimation of the variance of output response quantities given the variance of input stochastic parameters. Global sensitivity analysis: decomposition of the variance of output response quantities into components related to a generic subset of input stochastic parameters. 12

13 The testing protocol r Test Sampled input parameters Output response quantities 1 () (),, () () (),, (), ()... i (3) (3),, (3) (3) (3),, (3), (3)... N (4), (4), (4) (4), (4), (4), (4) Number of tests =8,16,32,64 10 * (Monte Carlo) 13

14 The surrogate model of the system response =8 9 7=,,, 9=,, OUTPUT INPUT 14

15 The Polynomial Chaos Expansion (PCE) 8 :; 9 =< = > Ψ A B C,D B E,F ={A: A <J} is the truncated set of multi-indices Ψ > = Multivariate polynomial with multi-index vector A = > = Coefficient of the single multivariate polynomial Marelli, S. & Sudret, B. UQLab: A Framework for Uncertainty Quantification in Matlab 257 Vulnerability, ICVRAM2014, Liverpool, United Kingdom,

16 Definition of multivariate polynomial E Ψ A 9 MΨ NO 3 3P 5 3 E A = 3PR 3 Degree of the univariate polynomial i-th Marelli, S. & Sudret, B. UQLab: A Framework for Uncertainty Quantification in Matlab 257 Vulnerability, ICVRAM2014, Liverpool, United Kingdom,

17 Definition of multivariate polynomial Ψ S,Ψ T = Ψ S Ψ T U V! W X =Y ST Probability density function Uniform Gaussian Gamma Beta Orthogonal polynomials Legendre Hermite Laguerre Jacobi Marelli, S. & Sudret, B. UQLab: A Framework for Uncertainty Quantification in Matlab 257 Vulnerability, ICVRAM2014, Liverpool, United Kingdom,

18 Uncertainty Propagation E 8 :; 9 =E < =\ N Ψ N 9 N ] ==\ 2 x 10-6 Var 8 :; 9 =E 8 :; 9 =\ = <=\ A N ] N^ Var σ Polynomial Chaos estimate -- Reference 95% Gaussian CI Training set size N 18

19 Global sensitivity analysis: Sobol' indices _ 3 = `3 ` _ 3 = a 3 `a ` First order Sobol' index = Fraction of the total output variance explained by the input parameter i-th alone Total Sobol' index = Fraction of the total output variance explained by the i-th input parameter in combination with all other parameters `a:; =Var 8 a :; 9 a = < =\ A _ a :; =`a :; `:; A B a Sobol' index `:; Var 8 :; 9 = < `a:; a,,e a^ whereais a generic subset of all input parameters 19

20 Global sensitivity analysis: Sobol' indices The 40% of the Variance of is related to the variance of f 20

21 Surrogate model of the entire response history HYBRID SIMULATION 1 9 = (), (), () 7 =,, S,, g... HYBRID SIMULATION i 9 3 = (3), (3), (3) 7 3 = 3,, 3 S,, 3 g... HYBRID SIMULATION N 9 4 = (4), (4), (4) 7 4 = 4,, 4 S,, 4 g 21

22 Surrogate model of the entire response history HYBRID SIMULATION 1 9 = (), (), () 7 =,, S,, g... HYBRID SIMULATION i 9 3 = (3), (3), (3) 7 3 = 3,, 3 S,, 3 g... HYBRID SIMULATION N 9 4 = (4), (4), (4) 7 4 = 4,, 4 S,, 4 g INSTANTANEOUS PCE? NOT EFFECTIVE 22

23 Surrogate model of the entire response history HYBRID SIMULATION 1 9 = (), (), () 7 =,, S,, g... HYBRID SIMULATION r 9 h = i,i, 7 h = h,, h S,, h g... HYBRID SIMULATION N 9 4 = (4), (4), (4) 7 4 = 4,, 4 S,, 4 g 23

24 Time warping transform 1/3 3 h Reference signal l 3 =f 3 +n 3 Time warping transform k(3) l Time warped signal 24

25 Time warping transform 2/3 Test Sampled input parameters Output response quantities 1 () (),, () f,n k, l... i (3) (3),, (3) f 3,n 3 k 3, l... N (4), (4), (4) f 4,n 4, k 4 l PCE time warping coefficients: PCE of the time warped response: f 9,n 9 k l,9 25

26 Time warping transform 3/3 Generic set of input parameter and time,9 3 Linear time warping l =f 9 3 +n 9 3 k l,9 3 \,9 3 26

27 Time-warping PCE performance NRMSE,\ Instantaneous PCE Time-warping PCE Training set size N 27

28 Conclusions In the current practice, numerical substructure design relies on deterministic assumptions and the probabilistic character of the emulated system response is completely missed. Polynomial Chaos Expansion is a robust framework for accommodating uncertainty propagation and global sensitivity analysis in Hybrid Simulation. About 20 hybrid simulations guarantee good estimates of both statistical moments and Sobol' indices of the response quantities for typical tested structures. According to the most widely used seismic performance-based design code models, such number agrees with the size of the ground motion set required to perform a reliable nonlinear dynamic analysis. 28

29 QUESTIONS? THANK YOU! 29