Validating Expensive Simulations with Expensive Experiments: A Bayesian Approach

Transcription

1 Validating Expensive Simulations with Expensive Experiments: A Bayesian Approach Dr. Arun Subramaniyan GE Global Research Center Niskayuna, NY 2012 ASME V& V Symposium Team: GE GRC: Liping Wang, Natarajan Kumar GE Aviation: Don Beeson, Gene Wiggs, Vaira Saravanan 1

2 Outline Motivation Overview of Bayesian Techniques Example Turbo-machinery Problems Key Challenges Summary 2

3 Motivation For a typical turbo-machinery component simulation RANS for heat transfer Steady: 4+ days on 100 proc. Unsteady: 2 weeks on 100 proc. LES for jet acoustics 4-6 weeks on 1000 proc. Input Factors (Xs) FEA for microstructure plasticity/life 2-3 weeks on 20 proc. Typical Engine test $1-2M setup costs $30-50K per test (operating cost) Non-linear & Non-monotonic behavior 3

4 Types of Uncertainty in Engineering Experimental Measurement Setup Input Uncontrolled parameters Challenges: Curse of dimensionality Non-monotonic behavior Small tail probability Unknown unknowns Model Parameters F = Kx + Mx Form Geometry Initial & boundary conditions 4

5 Types of Models Design parameters Xs Simulation Models d(x) Test Data e(x) Calibration parameters s Case 1: Well-established Physics Model Example: Heat Transfer, CFD Aero Case 2: Empirical or approximate Physics Model Example: Combustion dynamics Operating Conditions Xs Model parameters (immeasurable) qs Model uncertainty d(x) d(x) e(x) Engine A Engine B New Engine (may not have test data) Epistemic uncertainty Confidence level? Prediction uncertainty? 5 5

6 Enhanced Bayesian Hybrid Modeling * Based on Kennedy & O Hagan s Model Calibration with Emulators δ x y(x) (x,q ) Test Data = y x η x, θ + ε(x) Simulation Data 1 Calibrated parameter (posterior distribution) Uncertain Input Parameters Hybrid Modeling Framework (Bayesian Model Calibration, Updating & Prediction) Discrepancy d Sensitivity of Y to input parameters 2 3 Calibrated Predictive Models for Y and d (Gaussian Process) 5 Prediction of Y (mean, confidence and uncertainty) 4 Inputs: - Objective & Simulation Data 6

7 Enhanced Bayesian Hybrid Modeling Ability to handle large Xs problems (100+ Xs) Improved speed of MCMC Flexibility of handling different scenarios: Forward UQ, model calibration, model discrepancy, prediction with or without test data Transient Extension to Model Validation (V&V) Data confidence & optimization Real-time Visualization (from prior to posterior distributions, calibrated GP emulators, etc) Validation with many complicated engineering applications Prior Posterior GP predictions with confidence intervals 7

8 System Level Non-linear Engine Structural Model Objective: Probabilistic data matching Calibrate model parameters to match test data Model validation Challenges: High Dimensions 15+ Calibration Parameters (contact stiffness, heat transfer coefficients, etc.) 4 Outputs (Deflections) Complex physics Highly nonlinear, partial unknown physics, etc Limited test data 1 test point 8

9 System Level Calibration Results Initial Mismatch Calibrated Results q 1 q 2 q 3 q 4 q 5 q 6 q 7 q 8 q 9 q 10 9

10 Validation using Model Discrepancy δ 1 P δ 1 > ~ P δ 2 > ~ 1 δ 2 δ 3 δ 4 P δ 3 > ~ 0 P δ 4 > ~ Model NOT valid over entire design space 10

11 Large Scale Applications of Bayesian Hybrid Modeling PT_ratio 1. Engine level thermal model (100Xs) ~2 hours with improved MCMC speed on PC (4 processors) 2. Combustion Dynamics Improved Model predictive capability 3. Combustor Forced Vibration Model prediction 4. Cure Cycle Optimization Manufacture composite with target properties 5. Carbon fiber wind blade failure model (UQ) 6. Engine level structural model 7. RANS based heat transfer models (UQ and V&V) 8. CMC manufacturing spec-limits (UQ) 9. Engine level performance models -Model prediction & discrepancy 10.Gas turbine blade temperature model 11.Engine service models - Discrepancy Calibrated Performance 1 D BRM with Maps model at correction Speed = 105% 105% speed-line 105% speed-line 12.Engine cycle deck performance data matching Model prediction Total Life Model for CBM - Model prediction Test d(x) Hybrid Modeling Calibrated Simulator only Corrected flow 11

12 BHM Technical Challenges Effect of prior knowledge on validation Non-identifiability issue Handling high-dimensional problems Parallel MCMC, Adaptive MCMC, Manifold learning Existence of test and simulation data Lack of data Multi-scale and multi-fidelity system level models 12

13 Simple Example Problem y test = 1.5 x 2 + x/2 y sim = η = θx 2 Generate 10 Experiments & 100 Simulations x: 1 1, a: Uniform θ 13

14 Calibration Results y test = 1.5 x 2 + x/2 y sim = η = a x 2 + = 14

15 Prior knowledge scenarios Uniform distribution: True solution 1.5 Case Min Max Comments Typical, large bounds Narrow, true solution INSIDE bounds Very large bounds Narrow, true solution OUTSIDE upper bound Narrow, true solution OUTSIDE lower bound 15

16 Effect of parameter bounds = 1.43, = = 1.57, = Narrow bounds Large bounds = 1.14, = 0.2 = 1.69, = Solution outside upper bound Solution outside lower bound

17 Effect of priors on delta = 0.56, = 0.05 = 0.47, = Narrow bounds Large bounds = 0.86, = 0.19 = 0.29, = Solution outside upper bound Solution outside lower bound 17

18 Summary Bayesian techniques perform well with limited test and simulation data Bayesian Hybrid Models (BHM) can be used in noisy and missing data scenarios Assessing model parameters along with model discrepancy is important for validation Prior information can be used intelligently to alleviate lack of data Several challenges remain to be addressed 18

19 Thank you. 19