Thermal-Hydraulics Model with Time-Dependent Output Damar Wicaksono*, Omar Zerkak, and Andreas Pautz NUTHOS-11 Conference, Gyeongju, Korea, 11.10.2016 *[email protected]
and Backward Given data from a Separate Effect Test facility… Inverted Annular Film Boiling (IAFB) Dispersed Flow (DFFB) Transitionary (Inverted Slug) TRACE Post-CHF Closure Laws Tclad evolution at different axial levels 4.114 [m] FEBA (KIT, Germany) 1 28 ⋮ TRACE Model …derive the uncertainties of TRACE reflood-related model parameters Forward Model : , → Backward Model Uncertain inputs (controllable & model parameters) Uncertainty Propagation Uncertain output Experimental data ? Statistical Analysis & Decision Making Inverse Quantification
Parameters Uncertainty FLECHT-SEASET Westinghouse, USA ACHILLES Winfrith, UK FEBA KIT, Germany Excerpt from the TRACE Code Theory Manual: 1. “…the approximate value of the coefficient in Eq. (4-119) was determined from data comparisons with FLECHT- SEASET high flooding rate reflood data…” (pp. 164) 2. “In TRACE, the above interfacial drag coefficient has been reduced by a factor of ¾ to better match FLECHT-SEASET high flooding rate reflood data, so…” (pp. 166) 3. “…in simulations of FLECHT-SEASET high flooding rate tests, this value appeared to be too large and thus was reduced to…” (pp. 223) RBHT PSU, USA What if the calibration was done on other, similar, separate effect test facilities?
Predicted Values: a Formulation for Model Calibration exp ; , = true ; , + ϵ true ; , = model , �; , + ; , Model Discrepancy, unknown Experimental Data is a distorted information of the true value Model prediction is a distorted information of the true value Controlled/Design Variables, known (e.g., pressure, heat flux, etc.) spatio-temporal process, high-dimensional, correlated Measurement error, random Model Parameters; True or Best-Fitted, but unknown Computer Model, expensive to run Given ; , and other unknowns, what is � and its uncertainties?
Bayesian Framework exp ; , = model , �; , + ; , + observed (data) Code runs; high-dimensional & expensive random ⇒ uncertain statistical meta-modeling ⇒ uncertain unknown ⇒ uncertain unknown ⇒ uncertain (unknown code input-output relationship for limited budget of runs) would like to know more about Each unknown is uncertain, treated probabilistically by assigning prior distribution to be updated with data
nonlinearities/ interaction Many initial parameters, but only handfuls are important (in some sense) 26 Parameters to specify the TRACE model: • 4 boundary conditions (Pressure, inlet, etc.) • 9 material properties (, , etc.) • 2 spacer grid model (HTCenh, Δ) • 10 Post-CHF closure relations (HTC, , etc.) • 1 quench temperature DFFB Interf. Drag Grid HT Enhc. IAFB Interf. Drag DFFB Wall HTC DFFB Vapor-Interf. HTC Power Inlet Velocity Quench Temp. Outlet Pressure IAFB Wall HTC Sensitivity analysis on average temperature by screening method (Morris) With flat independent prior uncertainties, either in linear or log scales Total effect sensitivity Indication of nonlinearities/ interaction Focus on the most important ones Tclad at mid-assembly
prior distribution for functions , ∗ ≡ Cov , ∗ = � exp − | − ∗ | () =1 f ~𝐺𝐺 (), 2 , ∗ Probability Distribution over function (of ) values Mean function, the usual: zero, constant, linear, etc. Common variance correlation function power-exponential correlation correlation length power parameter multi-parameters construction Gaussian Process Gaussian process is Gaussian with infinite variates: ∈ ℝ; ∈ ℝK Prior Data Posterior Learn about () and Used in Bayesian Non-linear Regression
Principal Component Analysis (PCA) Full temperature transient output of FEBA TRACE Simulation is an “image”: × 𝟐𝟐 = 𝟑𝟑𝟑𝟑′𝟔𝟔𝟔𝟔 dimensions Dimension reduction by Principal Component Analysis (PCA) ′ = ⋅ ⋅ = ⋅ PC1 = 75% PC 2 = 9% PC 3 = 5% Mean SVD Concatenate and centered outputs Eigenvectors (principal components) PC scores (depends on parameter values) K
based Gaussian Process Meta-modelling model ; , = �model , + � , ⋅ + =1 PCA is Proper Orthogonal Decomposition (or KL-transform, Empirical OF, etc.) Only the scores are function of truncation error GP Meta-model construction on : 1. Run TRACE train times following a design 2. Carry Out PCA 3. Regress PC Scores on 4. Validate on valid data (500 points) Fitting 1is easy But higher is harder PC 1 PC 8 Benefit of using more PC in reconstruction increasingly marginal time to evaluate TRACE ; , output: 6-10 [] → ~ 𝟎𝟎−𝟑𝟑[s] PC Reconstruction Error DOE Coefficient of determination, 2 (1 is good!)
Discrepancy is the difference between model with known, perfect parameters (but with missing physics, error, etc.) and perfect data. Error tends to be correlated. ; , = true ; , − model , �; , unknown Why? Avoid overfitting, otherwise parameters will be used to compensate model bias in a given dataset would like to know Difficulty: Too many unknowns ⋅ and �, identifiability issue Solution: Modular Bayesian Approach* - estimate bias using GP with data and prediction with best- estimate parameters values, and allow for uncertainties Bottom Assembly Top Assembly *Bias defined up until quenching Model with the true/best/perfect parameters value ; , ~𝐺𝐺 , Σ
Flat Prior: Posterior Formulation Bayes’ Theorem � exp = exp} � × � ∫ exp} � × � � � Posterior Normalization constant, multi-D integral, not necessary with MCMC Likelihood Prior The Metropolis-Hastings (MH) Algorithm to sample from an un-normalized density � exp , in a nutshell* (use emcee package): 1.Start at any random value 2.Generate sample close to the current value 3.Accept or reject based on (new)/ old exp} � ~ � + , Σ + > > > + Σ + (exp 2 +other 2 ) GP meta-model Principal Comp. truncation model discrepancy experiment and other Multiple sources of variance, avoid overfitting Likelihood exp , = model �; , + , + Gaussian Gaussian Gaussian given unknown Consider only data from 1 FEBA Test. No in formulation
are constrained by data – model formulation Spacer Grid HT Enhancement DFFB Wall HTC DFFB Interfacial Drag IAFB Interfacial Drag Quenching Temperature -- are prior ranges 50’000 MCMC Samples Some ranges simply implausible Indication of bias at particular instance Most parameters are constrained and not strongly correlated given data and statistical model
Future Work ? ? ? ? 1. Quantification was done on one FEBA Test (No. 216). The effect of experimental control variables (pressure and inlet velocity) was not taken into account (under investigation) 2. Some parameters were found to be collinear in initial 7-parameter model calibration: the effect of one compensates the effect of the other (non-identifiable). Solution: Subset the parameters or, perhaps, introduce additional code response – data pair (under investigation) 70’000 MCMC Samples
data from the FEBA facility and assumed statistical calibration model …various sources of uncertainties, some are correlated, were taken into account to hedge against overfitting …Propagation of posterior uncertainty on test case seems to be consistent (narrower but still enveloping) uncertainty bands But, …the effect of experimental condition was not yet taken into account …parameter sensitivity is not enough to guarantee identifiability, additional output-data pairs needed to break collinearities Wir schaffen Wissen – heute für morgen Thanks a lot for your attention! Acknowledgments: • Swiss Federal Nuclear Safety Inspectorate (ENSI) • Swiss Federal Office of Energy (BFM)
Nuclear Engineering • Swiss joint MSc program between ETHZ and EPFL • Established in 2008, more than 100 students • Two-year program, 120 ECTS credits • Students attend two top-ranked universities: • ETH Zurich #9 • EPF Lausanne #14 (based on QS) • Scientific support and research at the Paul Scherrer Institute • Grants and scholarships available
𝐸𝐸 ≡ Δ Δ = 1 , 2 , … , i + Δ , … , − 1 , 2 , … , , … , Δ For many trajectories (𝒓𝒓 ), the basic statistics of the elementary effects can be calculated and interpreted as global sensitivity measures: «Standarized Elementary Effect» or EE Interpretation on Input Parameter Importance or ∗ Low High Low Non- influential Influential Linear, Additive High Non- influential Influential Non-linearities, Interactions
Meta-Model 2 , � = 1 − ∑ − � 𝑣𝑣 =1 2 ∑ − 2 𝑣𝑣 =1 Coefficient of determination Training Design (35 Optimized Latin Hypercube Design) Validation Design (100 Select points for Hammersley Sequence) Residual Sum of Squares Total Sum of Squares
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