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An Industrial Viewpoint onUncertainty Quantification inSimulation: Stakes,Methods, Tools, ExamplesAlberto PasanisiProject ManagerEDF R&D. Industrial Risk Management Dept.Chatou, Francealberto.pasanisi@edf.fr

SummaryCommon framework for uncertainty managementExamples of applied studies in different domainsrelevant for EDF :Nuclear Power GenerationHydraulicsMechanics2 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011

Common framework foruncertaintymanagement3 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011

Which uncertainty sources?The modeling process of a phenomenon contains many sourcesof uncertainty:model uncertainty: the translation of the phenomenon into a set of equations. Theunderstanding of the physicist is always incomplete and simplified,numerical uncertainty: the resolution of this set of equations often requires someadditional numerical simplifications,parametric uncertainty: the user feeds in the model with a set of deterministic values .According to his/her knowledgeDifferent kinds of uncertainties taint engineering studies; we focushere on parametric uncertainties (as it is common in practice)4 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011

Which (parametric) uncertainty sources?Epistemic uncertaintyIt is related to the lack of knowledge or precision of any given parameter which isdeterministic in itself (or which could be considered as deterministic under someaccepted hypotheses). E.g. a characteristic of a material.Stochastic (or aleatory) uncertaintyIt is related to the real variability of a parameter, which cannot be reduced (e.g. thedischarge of a river in a flood risk evaluation). The parameter is stochastic in itself.Reducible vs non-reducible uncertaintiesEpistemic uncertainties are (at least theoretically) reducibleInstead, stochastic uncertainties are (in general) irreducible (the discharge of a river willnever be predicted with certainty)A counter-example: stochastic uncertainty tainting the geometry of a mechanical piece Æ Can bereduced by improving the manufacturing line The reducible aspect is quite relative since itdepends on whether the cost of the reduction actions is affordable in practice5 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011

A (very) simplified exampleFlood water level calculationUncertaintyQZcZmKsZvStrickler’s FormulaZc : Flood level (variable of interest)Zm et Zv : level of the riverbed, upstream and downstream(random)Q : river discharge (random)Ks : Strickler’s roughness coefficient (random)B, L : Width and length of the river cross section (deterministic)General frameworkInputVariablesModelUncertain : XFixed : dOutput variablesof interestG(X,d)Z G(X, d)6 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011

Which output variable of interest?Formally, we can link the output variable of interest Z to a number ofcontinuous or discrete uncertain inputs X through the function G:d denotes the “fixed” variables of the study, representing, for instance a given scenario. In thefollowing we will simply note:The dimension of the output variable of interest can be 1 or 1Function G can be presented as:an analytical formula or a complex finite element code,with high / low computational costs (measured by its CPU time),The uncertain inputs are modeled thanks to a random vector X,composed of n univariate random variables (X1, X2, , Xn) linked by adependence structure.7 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011

Which goal?Four categories of industrial objectives:Industrial practice shows that the goals of any quantitative uncertainty assessmentusually fall into the following four categories:Understanding: to understand the influence or rank importance of uncertainties, thereby guidingany additional measurement, modeling or R&D efforts.Accrediting: to give credit to a model or a method of measurement, i.e. to reach an acceptablequality level for its use.Selecting: to compare relative performance and optimize the choice of a maintenance policy, anoperation or design of the system.Complying: to demonstrate the system’s compliance with an explicit criteria or regulatorythreshold (e.g. nuclear or environmental licensing, aeronautical certification, .)There may be several goals in any given study or along the time: for instance,importance ranking may serve as a first study in a more complex and long study leadingto the final design and/or the compliance demonstration8 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011

Which criteria?Different quantities of interestThese different objectives are embodied by different criteria upon the output variable ofinterest.These criteria can focus on the outputs’:rangecentral dispersion“central” value: mean, medianprobability of exceeding a threshold : usually, the threshold is extreme. For example, inthe certification stage of a product.Formally, the quantity of interest is a particular feature of the pdfof the variable of interest Z9 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011

Why are these questions so important?The proper identification of:the uncertain input parameters and the nature of their uncertainty sources,the output variable of interest and the goals of a given uncertainty assessment,is the key step in the uncertainty study, as it guides the choice ofthe most relevant mathematical methods to be appliedWhat is really relevant in the uncertainty study?σPfµMean, median, variance,(moments) of Zthreshold(Extreme) quantiles, probability ofexceeding a given threshold10 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011

A particular quantity of interest: the “probabilityof failure”G models a system (or a part of it) in operative conditionsVariable of interest Z Æ a given state-variable of the system (e.g. a temperature, a deformation, awater level etc.)Following an “operator’s” point of viewThe system is in safe operating condition if Z is above (or below) a given “safety” thresholdSystem “failure” event:Classical formulation (no loss of generality) in which the threshold is 0 and the system fails when Z isnegativeStructural Reliability Analysis (SRA) “vision”: Failure if C-L 0 (Capacity – Load)Failure domain:Problem: estimating the mean of the randomvariable “failure indicator”:XjDfXi11 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011

Need of a generic and shared methodologyThere has been a considerable rise in interest in many industries in therecent decadeFacing the questioning of their control authorities in an increasingnumber of different domains or businesses, large industrial companieshave felt that domain-specific approaches are no more appropriate.In spite of the diversity of terminologies, most of these methods share infact many common algorithms.That is why many industrial companies and public establishments haveset up a common methodological framework which is generic to allindustrial branches. This methodology has been drafted from industrialpractice, which enhances its adoption by industries.12 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011

Shared global methodologyThe global “uncertainty” framework isshared between EDF, CEA andseveral French and Europeanpartners (EADS, Dassault-Aviation,CEA, JRC, TU Delft )Uncertainty handbook(ESReDA framework, 2005-2008)13 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011

Uncertainty management - the globalmethodologyStep C : Propagationof uncertainty sourcesStep B:Quantificationof uncertaintysourcesModeled by probabilitydistributionsStep A : Specification of the esVariablesofof interestinterestZZ G(x,d)G(x,d)QuantityQuantity ofofinterestintereste.g.:e.g.: variance,variance,quantilequantile.Step C’ : Sensitivity analysis,RankingComing back(feedback)14 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011Decision criterione.g.: probability 10-b

Some comments (Step B). Available informationDifferent context depending on the available informationScarce data (or not at all) Æ Formalizing the expert judgmentA popular method: the maximum entropy principle Æ Between all pdf complying with expertinformation, choosing the one that maximizes the statistical entropy :Measure of the “vagueness” ofthe information on Xprovided by f(x)InformationMaximum Entropy pdfUniformExponentialNormalAnother popular choice: Triangular distribution (range mode)Feedback data available Æ Statistical fitting (parametric, non-parametric) in a frequentistor Bayesian framework15 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011

Some comments (Step B). DependencyTaking into account the dependency between inputs is a crucial issue inuncertainty analysisUsing copulas structure Æ CDF of the vector Xas a function of the marginal CDF of X1 Xn:Example: All bivariate densities here have the samemarginal pdf’s (standard Normal) and the sameSpearman rank coeff. (0.5)Using conditional distributionsoften based on “causality” considerationsDirected Acyclic Graphs (Bayesian Networks) are helpful for representing the dependency structureparentSet of the “parents” of xi16 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011descendant

Some comments (Step C and C’). CPU timeMain issue in the industrial practice: the computational burden!In most problems, the “cost” depends on the number of runs of the deterministic “function” GIf the code G is CPU time consumingBe careful with Monte-Carlo simulations!Rule of thumb: for estimating a rare probability of 10-r, you need 10r 2 runs of G !Appropriate methods (advanced Monte Carlo, meta-modeling)Appropriate software tools for:Effectively linking the deterministic model G(X) and the probabilistic model F(X)Perform distributing computations (High Performance Computing)Avoid DIY solutions !17 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011www.openturns.org

Examples.Nuclear PowerGeneration18 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011

Nuclear production at EDF58 operating nuclear units in France, located in 19 powerstationsPWR (Pressurized water reactor) technology3 power levelsInstalled power: 63.1 GWThanks to standard technologies and exploiting conditions, afeedback of more than 1000 operating years19 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011

PWR Power unit principlesTwo separate loops:Primary (pressurized water)Secondary (steam production)Three safety barriers (fuel beams, vessel, containment structure)Highly important stakesin terms of safetyIn terms of availability: 1 day off about 1 M 20 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011

The nuclear reactor pressure vessel (NRPV)A key componentHeight: 13 m, Internal diameter: 4 m,thickness: 0,2 m, weight: 270 tContains the fuel barsWhere the thermal exchange between fuelbars and primary fluid takes placeIt is the second “safety barrier”It cannot be replaced !Nuclear Unit Lifetime Vessel LifetimeExtremely harsh operating conditionsPressure: 155 barTemperature: 300 CIrradiation effects: the steel of the vessel becomes progressively brittle, increasing therisk of failure during an accidental situation21 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011

NRPV Safety assessment: a particular UQproblemThe problem formulation is typical in most nuclear safetyproblems:Given some hard (and indeed very rare) accidental conditions, what is the “failureprobability” of the component?It is the case of “structural reliability analysis” (SRA)The physical phenomenon is described by a computer codeThe system is safe if Z is lower (or greater) than a fixedvalue (equal to zero, without loss of generality)State variableof the systemFailure condition: Z 0Failure probabilityXjRandom Input vectorDfXiDomain of failure22 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011

NRPV Safety assessment example[Munoz-Zuniga et al., 2009] (1/3) Step AAccidental conditions scenario: cooling water (about 20 C) is injected into thevessel, to prevent over-warmingÆ Thermal cold shock Æ Risk of fast fracture around a manufacturing flawThermo-mechanical fast fracture model:thermo-hydraulic representation of the accidental event (cooling water injection, primary fluid temperature,pressure, heat transfer coefficient)thermo-mechanical model of the vessel cladding thickness, incorporating the vessel material propertiesdepending on the temperature ta fracture mechanics model around a manufacturing flawOutputs: Stress Intensity KCP(t) in the most stressed pointSteel toughness, KIC(t) in the most stressed pointGoal: Evaluate the probability that for at least one t, the function G KIC - KCP is negativeVessel thicknessthermo--hydraulicthermotransitorywater23 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011flawhdcladsteel

NRPV Safety assessment example[Munoz-Zuniga et al., 2009] (2/3) Step BA huge number of physical variables In this example, three are considered as random. Penalized values are given tothe remaining variables1) Toughness low limit, playing in the steeltoughness law KIC(t)Normal dispersion around a reference valueKICRCC2) Dimension of the flaw h,3) Distance between the flaw and theinterface steel-clad d,A more complex example with 7 randomized inputs is given in [Munoz-Zuniga etal., 2010]24 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011

NRPV Safety assessment example[Munoz-Zuniga et al., 2009] (3/3) Step CA numerical challenge:High CPU time consuming modelStandard Monte Carlo Methods are inappropriate to give an accurate estimate of PfAn innovative Monte Carlo sampling strategy has been developed: “ADS-2” (Adaptive DirectionalStratification)f1n f1A numerical challenge:n Learning step:Standard transformationDirectional samplingAdaptive strategy to sample more“useful” directionsP̂1ρ1 w1w3 ρP̂2ρ2w4Wˆ1srWˆ3srWithout Recycling3P̂3P̂4w2 Estimation step:directional simulations according to the estimatedallocation and estimation of the failure probabilitystratification into quadrantsand directional simulationswith prior allocationρ4RecyclingExample of results. NB Pf is here conditional tothe occurrence of very rare accidentalconditions25 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011Wˆ 2srWˆ 4srWˆ1rWˆ3rPˆnrADS 2PˆrADS 2Wˆ 2rWˆ 4r

Examples.Hydraulics26 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011

Hydraulic simulation: a key issueHydraulic simulation is a key issue for EDFBecause EDF is a major hydro-power operatormean annual production: 40 TWh220 dams, 447 hydro-power stationsBecause (sea or river) water plays a key role in nuclearproduction27 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011

Example of UQ in hydraulic simulation: effects ofthe embankment’s failure hydrograph on floodedareas assessment [Arnaud et al., 2010] (1/5)Context: French regulations for large damsLarge dams are considered as potential sources of major risks (Law 22/07/1987)Emergency Response Plans (PPI) must be prepared by the local authority ("Préfet")after consultationRisk assessment study :Risk assessment in case of dam failure: Evaluation of the Maximum water level (Zmax) andwave front arrival time (Tfront)Seismic analysisEvaluation of the possibility and effect of landslide in the reservoirHydrology studyHypotheses for the dam failure:Concrete dams : the dam collapses instantaneouslyEarth dams : the dam failure is assumed to be progressive by the formation of abreach due to internal erosion or an overflowEmbankment failurehydrograph28 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011

Effects of the embankment’s failure hydrographon flooded areas assessment [Arnaud et al.,2010] (2/5)The complex physics at play during the progressive erosion isnot well knownthe emptying hydrograph H is not well known:The maximum discharge QmaxThe time of occurrence of the maximum discharge TmaxWe assume that the reservoir volume (V) is knownWe assume a triangular hydrographStep AQQmaxTmTimeMax water level in the mostdangerous points of the valley:Zmax(x)Time of occurrence of Zmax(x)(arrival of the flood front): Tfron(x)29 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011

Effects of the embankment’s failure hydrographon flooded areas assessment [Arnaud et al.,2010] (3/5)Known variables:Features of the damDam height 123 m, Reservoir volume: V 1200 Mm3Valley featuresLength : 200 km, no tributaries, no dams downstreamVery irregular geometry with huge width variation Æ Hydraulic jumpsStep B Æ Uncertainty assessmentQmax and Tmax (Hydrograph form)too small amount and imprecise data: the pdfcould not be assessed by a statisticalprocedureAccording to the expert advice the followingpdf’s for Qmax and Tm have been proposed:Prob. distr. funct1) Normal :MeanStandard dev.2) Uniform :Lower boundUpper boundQmax (m3/s)Tm (s)100 00025 0005 0002 00050 000150 0001 0007 200Friction coefficient KsNot “measurable” variableExpert advice, based on valley morphologyknowledgeProb. distr. funct1) Truncated Normal:MeanStandard dev.Bounds2) Uniform:Lower boundUpper bound30 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011Ks305[17.5, 47.5]2535

Effects of the embankment’s failure hydrographon flooded areas assessment [Arnaud et al.,2010] (4/5)Step C Æ Uncertainty propagationHydraulics software: “Mascaret” Code (EDF R&D-CETMEF)1D shallow water modeling based on the De St Venant equationsFinite volume scheme with CFL limitation on the time stepHydraulic modelingUn-stationary flow conditions, Space discretization: 100 mThe time step ( 1-2 s) is controlled by the CFL condition. Duration of the simulation : 13 000 timesteps3 values of Qmax : 50 000 m3/s, 105 000 m3/s and150 000 m3/sMean value of KsTwo points (Point 1 and Point 2) are particularlydangerous with respect to the flooding risk. Theyare both located downstream from a sectionnarrowing Æ hydraulic jumpsWe will mainly focus on these two points700Absolute altitude Z (m AMSL)First set of 3 runs of the model to lookfor the more dangerous points––– Zmax for Q 150·103 m3/sCote max 105 000––– 150000Zmax for Q 105·103 m3/s50000––– CoteZmaxfor Q 50·103 m3/sdu fond––– bottom of the valley650Point 1600550500450Point 240035030001000031 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011200003000040000500006000070000Distance from the dam (km)8000090000100000

Effects of the embankment’s failure hydrographon flooded areas assessment [Arnaud et al.,2010] (5/5)Propagation method: Surface response Monte CarloZ(m ASML)Point 1Some resultsPdf 1Pdf 2maxExtreme Quantiles of Zmaxin points 1 and 2 (flood riskassessment)Quantile 99.9%Quantile 99%Quantile 95%676.66675.57673.67676.64675.52674.25Point 2Pdf 1Pdf 2517.14 515.04516.49 515.57513.71 514.14Sensitivity analysis Æ evaluation of the Spearman ranks’ correlation coefficients forall values of the abscissa 00500006000070000800009000010000032 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 100000

A hydraulic benchmark: the Garonne case-studyLa RéoleNordStream directionBourdellesHydraulic modeling of a 50 kmlong section of the Garonne riverÆ “Mascaret” CodeSte BazeilleMeilhan eins2 kmFlood plainTwo examples:Le Mas d’AgenaisLagruèreEmbanked mainchannelInverse modeling to assess the pdf ofStrickler’s roughness coefficient KsEvaluating an extreme quantile of the floodwater level at a given abscissaOr evaluating the probability for the floodwater level in a given abscissa to begreater than a threshold valueSt. Perdoux du BreuilFourques s/GaronneCase study shared between thepartners of the OPUS projectKs is never directly observedOne should estimate the pdf of Ks, given aset of observed coupled data(discharge,water level)Marmandebanklow flowchannelTwo different Ks foreach section:low-flow Ks andmain-channel Ks.“OPen source platform for Uncertainty treatment in Simulation”10 partners, Tot. budget: 2.2 M , Leader: EDF33 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011

The Garonne case-study: Inverse modeling of Ks[Couplet, Le Brusquet et al., 2010] (1/2)La RéolePhysical hypothesisNordStream directionBourdellesSte Bazeille3 parts each one with given values of the 2 KsM eilhan s/GaronnePart T3M armandeCouthures/GaronneSt. Perdoux du BreuilFourques s/GaronneStatistical problem: assessing the pdf of KsTaillebourgSénestisTonneinsPart T22 kmIn this example, we will assess the pdf of the Ks of the T3 part (terminalpart between Marmande and La Réole)Data: couples (discharges Qi, water levels Zi) at Mas d’Agenais and MarmandeLe M as d’AgenaisLagruèrePart T1Hypotheses:The vector Ks and observation errors are normalThe standard measurement error is σεMean values of KsTricky likelihood expressionCovariance matrix of KsDensity of Ks34 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011Density of zi, given Qi and Ks

The Garonne case-study: Inverse modeling of Ks[Couplet, Le Brusquet et al., 2010] (2/2)Some resultsTwo solutionsLikelihood maximization (variants of the EM algorithm: ECME, SAEM)Bayesian solution: MCMC sampling from the posterior pdf of β:NB: Uniformprior used35 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011

The Garonne case-study: Flood risk assessment[Arnaud, Vazquez, Bect et al., 2010]Goal: Evaluating the quantile of probability α 0.99 of the water level in agiven sectionOriginal meta-modeling technique developed within the OPUS project [Vazquez et al, 2010]Empirical estimation of the quantile:Building an approximationofbased on the n m evaluations:The n pointsare chosen sequentially in order to minimize a statistical “cost” (e.g. aquadratic loss) betweenand the empirical estimator built according to the surrogatemodelWith a dozen runs of the model, itis possible to build a “specialized”kriging meta-model for the quantileestimation (here m 2000)36 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011

ExamplesMechanics37 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011

A longstanding experience at EDF R&DSeveral studies in the field of probabilistic mechanics:Reliability analysisSensitivity analysisInverse problems Æ Bayesian updating of the behavior law of the material (e.g. concretein civil works studies)Several research works on polynomial chaos expansionA useful tool to perform high CPU time-consuming calculations aboveNumerous applicationsCooling towers, containment structures, thermal fatigue problems, lift-off assessment offuel rod .We will focus on an application concerning reliability and sensitivity analysis of globevalves38 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011

Globe valve reliability and sensitivity analysis[Berveiller et al., 2010] (1/5)Industrial globe valves are used for isolating a pipingpart inside a circuitryHarsh operating conditions: water temperature,pressure, corrosion problems .Reliability assessment: the tightness of the valve hasto be assured even with a maximum pressure of thewaterSeveral uncertain variablesMaterial propertiesFunctional clearancesLoadTo ensure the reliability of the mechanism, thecontact pressures and the max displacement ofthe rod must be lower than given values39 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011

Globe valve reliability and sensitivity analysis[Berveiller et al., 2010] (2/5)The modeling problem is very complex. We will work here on a simplifiedmechanical modelingLimit condition: embedded beamRodPackingGlandLoadContact Rod/PackingCase-study of the OPUS projectContact Rod/Gland40 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011

Globe valve reliability and sensitivity analysis[Berveiller et al., 2010] (3/5)Step AVariables of interest:Contact pressuresMax displacement of the rod6 Uncertain input variables:Packing Young’s modulusGland Young’s modulusBeam Young’s modulusSteel (Rod) Young’s modulusLoadClearanceGoal of the study:assessing the sensitivity of the variable of interest withrespect to the uncertain inputsQuantities of interest: Sensitivity indicesReminder: Sobol’ variance decomposition**Xi’s independentSobol’ indices:Deterministic model G(·):FEM Numerical model of the simplifiedscheme using Code Aster software(www.code-aster.org)First orderSecond order“Total” indexThey measure the “part” of the global varianceexplained by a single input (or a set of inputs)Monte Carlo calculation is CPU expensive, as manymodel runs are needed Æ Meta-modeling approach41 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011

Globe valve reliability and sensitivity analysis[Berveiller et al., 2010] (4/5)Step BUncertainty modeling of input variables:VariableProb. densityMeanCoefficientof VariationPacking Young’s modulus (MPa)LogNormal100 00020%Gland Young’s modulus (MPa)LogNormal207 00010%Beam Young’s modulus (MPa)LogNormal6 00010%Steel (Rod) Young’s modulus (MPa)LogNormal200 00010%Load (N)Normal10 00010%Clearance (mm)Beta[0,0.1]0.0550%Steps C,C’Non intrusive polynomial chaos approximationIsoprobabilistic transformation of the input vector:Polynomial chaos (PC) approximation:Number of terms of the sum:PC approx. of order m and degree qcoefficients42 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011Set of the m-dimensionalHermite polynomials of degree q

Globe valve reliability and sensitivity analysis[Berveiller et al., 2010] (5/5)Benefits of PC approximationOnce coefficients are evaluated, PC expansion allows performing quick Monte Carlo simulations, by runningthe meta-model instead of the expensive numerical code G(·)Moreover, due to the orthogonality of the polynomials, the evaluation of Sobol’ indices is straightforward[Sudret, 2008]:The calculation burden (i.e. running several times the code G) is focused on the estimationof the coefficientsSet of polynomialscontaining only ξiSeveral techniques: projection, regression, simulation, sparse PC expansion (LARS) [Blatman & Sudret, 2010]Example of resultsSobol’ indices for rod displacementPC approximation built by two different methods & tools:LARS, NISP (CEA)ClearanceLoadESteelMost influent variables : clearance, load, Steel Young’smodulus43 - Working Conference on Uncertainty Quantification in Scientific Computing - Boulder, Co. Aug. 2011

Some referencesMunoz-Zuniga, M. Garnier, J. Remy, de Rocquigny E. (2009). Adaptive DirectionalStratification, An adaptive directional sampling method on a stratified space. ICOSSAR 09,Sept. 2009, Osaka.Munoz-Zuniga, M. Garnier, J. Remy, E. de Rocquigny E (2010). Adaptive DirectionalStratification for controlled estimation of the probability of a rare event. Submitted.Arnaud, A. Goutal N., de Rocquigny, E. (2010). Influence des incertitudes sur leshydrogrammes de vidange de retenue en cas de rupture progressive d’un barrage enenrochements sur les zones inondées en aval. SimHydro 2010. June 2010, SophiaAntipolis.Couplet, M. Le Brusquet, L. Pasanisi, A. (2010). Caractérisation des coefficients deStrickler d'un fleuve par inversion probabiliste. 42èmes Journées de Statistique. Mai 2010,Marseille.Arnaud, A. Bect, J. Couplet, C. Pasanisi, A. Vazquez E. (2010) Evaluation d’un risqued’inondation fluviale par planification séquentielle d’expériences. 42èmes Journées deStatistique . Mai 2010, Marseille.Bect, J. Ginsbourger, D. Ling, L. Picheny, V. Vazquez, E. (2010) Sequential design ofcomputer experiments for the estimation of a probability of failure . Statistics andComputing (in press)de Rocquigny, E. Devictor, N. Tarantola S. (eds.) (2008). Uncertainty in industrialpractice. A guide to quantitative uncertainty management. Chichester: J. Wiley & Sons.Berveiller, M. Blatman, G. Martinez J.M. (2010). Analyse de sensibilité d'un robinet àsoupape à l'aide de développements sur chaos polynomial. 42èmes Journées deStatistique . Mai 2010, Marseille.Sudret, B. (2008). Global sensitivity analysis using polynomial chaos expansions.Reliability Engineering & System Safety 93 964–979.Blatman, G. Sudret, B. (2010). Efficient computation of global sensitivity indices usingsparse polynomial chaos

The modeling process of a phenomenon contains many sources of uncertainty: model uncertainty: the translation of the phenomenon into a set of equations. The understanding of the physicist is always incomplete and simplified, numerical uncertainty: the resolution of this set of equations often requires some additional numerical simplifications,

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