International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Fusing simulation results from multifidelity aero-servo-elastic simulators : application to extreme loads.. Abdallah, Imad; Sudret, Bruno; Lataniotis, Christos; Sørensen, John Dalsgaard; Natarajan, Anand Jul 31, 2015

You don't seem to have a PDF reader installed, try download the pdf

Item Metadata


53032-Paper_223_Abdallah.pdf [ 472.59kB ]
JSON: 53032-1.0076137.json
JSON-LD: 53032-1.0076137-ld.json
RDF/XML (Pretty): 53032-1.0076137-rdf.xml
RDF/JSON: 53032-1.0076137-rdf.json
Turtle: 53032-1.0076137-turtle.txt
N-Triples: 53032-1.0076137-rdf-ntriples.txt
Original Record: 53032-1.0076137-source.json
Full Text

Full Text

12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Fusing Simulation Results From Multifidelity Aero-servo-elasticSimulators - Application To Extreme Loads On Wind TurbineImad AbdallahTechnical University of Denmark, Department of Wind Energy, Roskilde, DenmarkBruno SudretETH Zürich, Chair of Risk, Safety and Uncertainty Quantification, Zürich, SwitzerlandChristos LataniotisETH Zürich, Chair of Risk, Safety and Uncertainty Quantification, Zürich, SwitzerlandJohn D. SørensenAalborg University, Department of Civil Engineering, Aalborg, DenmarkAnand NatarajanTechnical University of Denmark, Department of Wind Energy, Roskilde, DenmarkABSTRACT: Fusing predictions from multiple simulators in the early stages of the conceptual design ofa wind turbine results in reduction in model uncertainty and risk mitigation. Aero-servo-elastic is a termthat refers to the coupling of wind inflow, aerodynamics, structural dynamics and controls. Fusing the re-sponse data from multiple aero-servo-elastic simulators could provide better predictive ability than usingany single simulator. The co-Kriging approach to fuse information from multifidelity aero-servo-elasticsimulators is presented. We illustrate the co-Kriging approach to fuse the extreme flapwise bending mo-ment at the blade root of a large wind turbine as a function of wind speed, turbulence and shear exponentin the presence of model uncertainty and non-stationary noise in the output. The extreme responses areobtained by two widely accepted numerical aero-servo-elastic simulators, FAST and BLADED. Withlimited high-fidelity response samples, the co-Kriging model produced notably accurate prediction ofvalidation data.1. INTRODUCTIONAnalysts and designers increasingly use multiplecommercial and research-based aero-servo-elasticsimulators to compare the prediction of wind tur-bines’ structural response. The aero-servo-elasticsimulators are of varying fidelity and have differ-ent underlying assumptions. As a result, the aero-servo-elastic response may vary amongst simula-tors even if the external inflow condition is thesame. The sub-models with the largest impact onthe aero-servo-elastic response variability are aero-dynamic, structural, control systems and wind in-flow. The aero-servo-elastic simulators are vali-dated using test measurements from prototype windturbines. The current practice is to cover thediscrepancy amongst the simulators by imposingsafety factors resulting in a safe design. It is rea-sonable to assume that model uncertainty is of theepistemic type and can be estimated at the designstage with (usually) decreasing uncertainty whenmore simulations from multiple sources are avail-able.The objective in this paper is to fuse the extremeresponse from multiple aero-servo-elastic simula-tors of various fidelity and complexity to predict"the most likely" extreme response of a wind tur-bine. Forrester et al. (2007) used the co-Krigingtechnique in the optimization of a generic aircraftwing using one "cheap" and one "expensive" flowsolver. The Co-Kriging approach was also used by112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Han and Görtz (2012) to predict the mean aerody-namic lift and drag coefficients on a two dimen-sional airfoil and a three dimensional aircraft usinga low-fidelity Euler flow solver and a high-fidelityNavier-Stokes solver.The novelty in this paper is the implementationof the co-Kriging technique to predict the extreme(not the mean) response in the presence of non-stationary noise in the output (i.e. the magnitude ofnoise varies as a function of the input variables) inthe case when the low and high-fidelity aero-servo-elastic simulators of the same wind turbine are im-plemented by two independent engineers (i.e. hu-man error and uncertainty in the modelling and in-put assumptions are implicitly included). In thispaper, we demonstrate the co-Kriging methodol-ogy to fuse the extreme blade root flapwise bend-ing moment of a large multi-megawatt wind tur-bine by using two aero-servo-elastic simulators,FAST (Jonkman and Buhl, 2005) and BLADED(Bossanyi (2003a), Bossanyi (2003b)).2. THE CASE FOR DATA FUSIONWind turbine aero-servo-elastic simulators of vary-ing fidelities exhibit similarities and dependencein terms of the input variables and the underly-ing physical models (aerodynamic, structural, con-trol systems and wind inflow). The dependenceamongst various simulators may not be quantifiedby a single scalar number; it may well be thatthe dependence varies as a function of the designand input space (Christensen, 2012). Thus, we askthe fundamental question: Does it make any senseto fuse information from multifidelity aero-servo-elastic simulators Mi?• To a great extent, simulators {Mi, i=1,...,n}share similar (often identical) inputs and de-scribe similar (often identical) underlyingmodelling and physics assumptions.• The various simulators may have been cali-brated using the same test measurements.• The higher fidelity simulators may simply bean expansion of the lower fidelity simulationmodel by inclusion of additional physics.• Let us assume that for a given set of inputsX = [x(1), ...,x(N)], simulators Mi predict re-sponsesYi = [M1(x(1)), ...,Mi(x(N))]T . Then,Yi generally share the same trend and do notdiffer significantly from each other. In addi-tion, the simulators Mi do not exhibit clearbias in the predicted response Y i.• The various aero-servo-elastic simulators mayhave been coded by the same or cooperatingengineers, scientists and research institutes,and the same experts may have given their in-puts/reviews/recommendations during the de-velopment and validation of the various simu-lators Mi resulting in similar assumptions andbiases being used.• The various simulators Mi are certified by ac-credited institutes for use in the industry to de-sign wind turbines. The certification processinvolves a lengthy validation and verificationagainst measurements. Hence, no particularsimulator Mi is deemed better than the other.The implication of the argumentation above is thatrather than treating the aero-servo-elastic numeri-cal simulators as parts of a hierarchy, they are con-sidered as individual (but correlated) informationsources. Furthermore, the simulators are assumedto be black boxes and we focus on the output quan-tity of interest (response) Yi.3. METHODOLOGY3.1. Co-KrigingIn this section we present a brief theoretical de-scription of Kriging and Co-Kriging based on workby Sacks et al. (1989), Kennedy and O’Hagan(2000), Jones (2001), Forrester et al. (2007),Dubourg (2011), Han et al. (2012) , Pichenyet al. (2012), Sudret (2012) and Schöbi and Sudret(2014). Kriging is a stochastic interpolation tech-nique which assumes that the "true" model outputis a realization of a Gaussian process:Y (x) = µ(x)+Z(x) (1)where µ(x) is the mean value of the Gaussian pro-cess (trend) and Z(x) is a zero-mean stationaryGaussian process with variance σ2Y and a Covari-ance of the form:C(x,x′) = σ2Y R(x− x′ | θ)(2)212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015where θ gathers the hyperparameters of the au-tocorrelation function R. From a design of ex-periments X , one can build the correlation ma-trix with terms Ri j = R(x(i),x( j) | θ)represent-ing the correlation between the sampled (observed)points. In the case of simple Kriging µ(x) is as-sumed to be a known constant. In the case of or-dinary Kriging µ(x) is assumed to be an unknownconstant. In the case of universal Kriging µ(x) iscast as ∑mj=0β j f j(x), i.e. a linear combination ofunknown (to be determined) linear regression coef-ficients β j, j = 1, ...,m and a set of preselected ba-sis functions f j(x), j = 1, ...,m (usually predefinedpolynomial functions). The autocorrelation func-tion R may be a generalized exponential kernel:R(x,x′) = exp(−∑Mi=1θi(xi− x′i)pi),θi ≥ 0, pi ∈ (0,2](3)where M is the number of dimensions of the in-put space and θi and pi are unknown parame-ters to be determined. Other choices for R is aGaussian kernel, or a Matérn kernel, etc. In or-der to establish a Kriging surrogate model, a de-sign of experiments is formed X = [x(1), ...,x(N)]and a corresponding set of computer simulationsare performed. The output is gathered in a vectorY = [M (x(1)), ...,M (x(N))]T . The Kriging esti-mator (predicted response given the design of ex-periments) at a new point x∗ ∈ DX is a Gaussianvariable Yˆ (x∗) with mean µYˆ and variance σ2Yˆ de-fined as (Best Linear Unbiased Estimator):µYˆ (x∗) = E[Yˆ (x∗) |M(x(i))]= f T βˆ + rT R−1(Y −F βˆ) (4)σ2Yˆ (x∗) = Var[Yˆ (x∗) |M(x(i))]= σˆ2Y[1− rT R−1r +uT (F T R−1F )−1u](5)where the optimal Kriging variance σˆ2Y and optimalKriging trend coefficients βˆ (θ ) are given by:σˆ2Y =(Y −F βˆ )T R−1(Y −F βˆ )N(6)βˆ =(F T R−1F)−1F T R−1Y (7)and u, r and F are given by:u = F T R−1r− f (8)r =R(x∗− x(1); θˆ )...R(x∗− x(N); θˆ ) (9)F =[f j(x(i))]=f0(x(1)) . . . fm(x(1))...f0(x(N)) . . . fm(x(N)) (10)Note that r is the correlation matrix between thesampled points and the point where a predictionis to be made. In the general case of a-priori un-known correlation parameters θˆ , the optimal valuescan be estimated through Bayesian inference, max-imum likelihood estimate or a leave-one-out cross-validation (Bachoc, 2013).In case the outputs of the computer experimentscontain "noise", the Kriging model should regressthe data in order to generate a smooth trend. TheKriging thus amounts to conditioning Yˆ (x∗) onnoisy observations M(x(i))+ εi. The Kriging es-timator mean µYˆ (x∗) and variance σ2Yˆ (x∗) are givenby Equations 4 and 5, respectively by replacingthe correlation matrix R with R + λ 2I , where λ 2is the estimated variance of the noise term εi. Wenow consider how to build a surrogate model ofa highly complex and expensive-to-run aero-servo-elastic response that is enhanced with data fromcheaper and approximate analyses. This approachis traditionally known as co-Kriging (Kennedy andO’Hagan, 2000). Co-Kriging has been proposedunder various names such as "hierarchical Krig-ing", "multifidelity surrogate modelling", "variablefidelity surrogate modelling", "data fusion", "multi-stage surrogate modelling", "recursive co-Kriging",etc. The formulation of co-Kriging presented hereis based on Han and Görtz (2012): we consider lsets of response data obtained by running l aero-servo-elastic numerical simulators of varying fi-delity and computational expense. We denote bylevel s the response data of the highest level of fi-delity. For any given level 1≤ l ≤ s, co-Kriging can312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015be written as:µ(l)Yˆ= βˆ µ(l−1)Yˆ+ rT R−1(Y −F βˆ ) (11)where βˆ is a scaling factor with a similar expressionas in Equation 7, indicating how much the low-and high-fidelity responses are correlated to eachother. µ(l−1)Yˆis the trend in the kriging of the dataat level l and the expression R−1(Y −F βˆ ) dependsonly on the sampled data at level l. An appealingfeature of the above formulation is that it entailsvery little modifications to an existing Kriging codeif the latter is sufficiently modular.U(Z)LiftDragTrailing Vorticity in the wakeMtYawControlPitchControlWavesSoilFoundationTurbulenceWind SpeedWindshear MbFigure 1: A wind turbine. Mb is the flapwise bendingmoment at the blade root. U(Z) is the mean wind speedat height Z. Verical wind shear (dotted grey line) andturbulence (thick black line).4. APPLICATION TO EXTREME LOADSON WIND TURBINEWe illustrate co-Kriging in fusing the extreme flap-wise bending moment at the blade root of a windturbine (Figure 1) by using two numerical aero-servo-elastic simulators, FAST and BLADED.4.1. Aero-servo-elastic simulationsFAST is a time-domain aero-servo-elastic simulatorthat employs a combined modal and multibody dy-namics formulation. FAST models the turbine us-ing 24 Degrees of Freedom (DOFs). These DOFsinclude two blade-flap modes and one blade-edgemode per blade. It has two fore-aft and two side-to-side tower bending modes in addition to nacelleyaw. The other DOFs represent the generator az-imuth angle and the compliance in the drive trainbetween the generator and hub/rotor. The aerody-namic model is based on the Blade Element Mo-mentum theory (Hansen, 2001). A design of exper-iments (Table 1) is produced in order to examine theeffects of wind speed, inflow turbulence and shearvariations on the predicted extreme loads. For eachcombination of wind speed, turbulence level andshear exponent we generate realizations of windtime series with 24 stochastic seeds. Some of thewind speed, turbulence and shear exponent combi-nations are excluded because they are unphysical,resulting in a total of 33,480 10-minute time seriessimulations. One 10-minute wind time series sim-ulation in FAST takes approximately three minutesin real time. The output used from the simulationsare the blade root flapwise bending moment. Theglobal maxima of the bending moment data are ex-tracted for each of the 33,480 10-minute time se-ries.Table 1: Design of experiments for the FAST simula-tions. The variables are wind speed [m/s], turbulence[m/s] and the wind shear exponent.Wind Speed[m/s]Turbulence[m/s]Shear expo-nent [-]4,5, · · ·,25 0.1, 1, 2, 3, 4,5, 6+/-1.0, +/-0.6, +/-0.2,+/-0.1, 0, 1.5BLADED is a time domain aero-servo-elasticsimulator that is used to conduct the high-fidelityaero-servo-elastic simulations of the same tur-bine geometry. The structural dynamics withinBLADED are based on a modal model. The blade ismodelled using up to 12 modes, six blade-flap andsix blade-edge per blade. It also has three fore-aftand three side-to-side tower bending modes. So-phisticated power train dynamics are included. Theaerodynamic model is based on the Blade ElementMomentum theory. A design of experiments is pro-duced as shown in Table 2. For each combina-412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015tion of wind speed, turbulence level and shear ex-ponent we generate realizations of wind time se-ries with 12 stochastic seeds. Some of the windspeed, turbulence and shear exponent combinationsare excluded because they are unphysical, result-ing in a total of 4344 10-minute simulations. One10-minute wind time series simulation in BLADEDtakes approximately 25 minutes in real time. Theoutput used from the simulations are the blade rootflapwise bending moment. The global maxima ofthe bending moment data are extracted for each ofthe 4344 10-minute time series. The simulations inBLADED and FAST consider a wind turbine thathas a 110 meters rotor diameter and 2 MW ratedpower. The wind turbine is erected on a 90 meterstower. Both the FAST and BLADED aero-servo-elastic simulations were performed with exactly thesame control systems in the form of an externalDLL. The FAST and BLADED simulation modelsdo not use exactly the same input parameters in thestructural and aerodynamic sub-models.Table 2: Design of experiments for the BLADED simu-lations. The variables are wind speed [m/s], turbulence[m/s] and the wind shear exponent.Wind Speed[m/s]Turbulence[m/s]Shear expo-nent [-]4, 8, 10, 12,15, 20, 250.1, 1, 2, 3, 4,5, 6+/-1.0,+/-0.6,+/-0.2,+/-0.1,0,1.55. RESULTS AND DISCUSSIONSWe start with a simple generic example to demon-strate Kriging and co-Kriging. In Figure 2, thenoisy response of the low-fidelity simulator is plot-ted as a function of wind speed. A Universal Krig-ing model is fitted to the noisy response using aGaussian correlation function R and a 3rd-orderpolynomial basis. The low-fidelity Kriging modelis then used as the trend to fit a co-Kriging modelto the noisy high-fidelity response.In Figure 3, we compare the co-Kriging modelto a universal Kriging model (Gaussian correla-tion function R and a 2nd-order polynomial basis).0 5 10 15 20 25200030004000500060007000800090001000011000Vf(V)  Training Points LFKriging approximationLF FunctionSim. LF SamplesFigure 2: Response of the low-fidelity simulator at 6wind speeds with 24 stochastic repetitions each (blackcrosses). The mean of the 24 samples is calculatedand represented by the black dots. The Kriging modelwith noisy observations is the dotted red line. The low-fidelity (LF) function is the response if a large numberof stochastic simulations are performed.0 5 10 15 20 2530003500400045005000550060006500700075008000Vf(V)  Training Points HFKriging approximationHF FunctionCoKriging approximationSim. HF SamplesFigure 3: Response of the high-fidelity simulator at 3wind speeds with 6 stochastic repetitions each (blackcrosses). The mean of the 6 samples is calculated andrepresented by the black dots. The Kriging model withnoisy observations is the dashed green line. The Co-Kriging model with noisy observations is the dotted redline. The high-fidelity (HF) function is the response if alarge number of stochastic simulations are performed.Note that the high-fidelity responses are placed atonly three wind speeds (4 m/s: turbine starts, 25m/s: turbine shuts-down and 12 m/s: peak ro-512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015tor aerodynamic thrust). The co-Kriging predic-tions of the noisy high-fidelity response are notablybetter than the Kriging prediction based only onthe high-fidelity samples. UQLab (Marelli and Su-dret, 2014) is used to compute the Kriging and co-Kriging meta-models.5101520250123456020004000600080001000012000 Wind Speed [m/s]Shear Exponent = 0.2Turublence [m/s] My [kNm]FASTBLADEDFigure 4: Scatter plot of the blade root extreme flapwisebending moment My as a function of wind speed andturbulence for shear exponent α = 0.2. Note the vari-ability (noise) of My for a given turbulence and windspeed.A common practice during the design and op-timization of a wind turbine is to generate a sig-nificant number of stochastic simulations, typicallyusing two or more aero-servo-elastic simulators.Next, we show an example where the entirety ofthe loads simulations (as described in Section 4)are used to demonstrate a "real world" engineeringapplication of data fusion using co-Kriging in highdimensions. The FAST and BLADED simulatorswere prepared by two independent engineers (oneof whom is the first author of this paper). The sim-ulations output are shown in Figure 4; even thoughthe magnitude of the extreme flapwise bending mo-ment at the blade root for low and high-fidelity sim-ulators are not the same, they yield a similar trend.In Figure 4, for the same pair of turbulence andwind speed the output of the simulations is noisydue to the stochastic nature of the simulated windspeed time series. In addition, the magnitude ofscatter (noise) increases with increasing turbulencelevel. Note that the low and high-fidelity simulatorsare not sampled at exactly the same input variables.24003000300036003600420042004200480048004800480054005400540054005400600060006000600060006600660066006600660072007200720072007200 78007800 78007800Wind Speed [m/s]Turublence [m/s]Shear Exponent = 0.2  415553516163661974013516475060326426697774392901388052366324696075482490313940905162657671314 6 8 10 12 14 16 18 20 22 240.511.522.533.544.555.56 Kriging HFValidation setFigure 5: Projection of the Kriging model of the noisyhigh-fidelity (Bladed) extreme flapwise bending momentMy compared to a validation set at wind speeds V =8,12,15,20m/s and shear α = 0.2.2400 2400300030003000360036003600420042004200420048004800480048004800540054005400540054006000600060006000600066006600 66006600660072007200720072007800Wind Speed [m/s]Turublence [m/s]Shear Exponent = 0.2  415553516163661974013516475060326426697774392901388052366324696075482490313940905162657671314 6 8 10 12 14 16 18 20 22 240.511.522.533.544.555.56 Co−Kriging HFValidation setFigure 6: Projection of the Co-Kriging model of theof the noisy high-fidelity (Bladed) extreme flapwisebending moment My compared to a validation set atwind speeds V = 8,12,15,20m/s and shear α = 0.2.A Universal Kriging model is first fitted to allthe noisy load response of the low-fidelity simulator(FAST) using a Gaussian correlation function R and3rd-order polynomial basis. The low-fidelity Krig-ing model is then used as a model trend to fit a co-Kriging model to the noisy high-fidelity (Bladed)612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015load response. A subset of the high-fidelity datais used to build the co-Kriging model while the re-maining data is used as validation points. This sub-set corresponds to the load response at wind speedsV = [4,10,25]m/s as depicted in Figure 4. A uni-versal Kriging model is also fitted to the same sub-set of the noisy high-fidelity load response using aGaussian correlation function R and 2nd-order poly-nomial basis. A projection of the Kriging and co-Kriging models of the noisy high-fidelity load re-sponse together with validation points are shown inFigures 5 and 6, respectively. To allow visualiza-tion of the meta-models predictions we set the shearexponent to α = 0.2. Qualitatively, one can see thatthe co-Kriging model predictions are close to thevalidation points, while the Kriging model gener-ally over-predicts the extreme load response. Usingthe low-fidelity Kriging model as a trend improvesthe predictive accuracy of the co-Kriging model ofthe high-fidelity load response, even in the presenceof noise and by using very few high-fidelity samplepoints.This is shown more clearly in Figures 7–9 wherethe accuracy of the Kriging and co-Kriging mod-els of the high-fidelity extreme load response arecompared. In those figures the validation points areshown with the corresponding scatter. The Krigingmodel from the high-fidelity response points givesa poor approximation of the validation points, whilethe co-Kriging model performs notably better inhigh dimensions. Hence, despite the difference be-tween the output of the low and high-fidelity simu-lators, we were able to fuse both data sets so that theprediction error of the high-fidelity load response isreduced. The 95% confidence interval of the co-Kriging predictions is also shown in Figures 7, 8and 9. The confidence interval of the co-Krigingpredictions reflects a mix of epistemic (statistical)uncertainty due to the number of sampled pointsand due to the noise in the simulations output.6. CONCLUSIONSWe have shown a co-Kriging based methodology tofuse the "noisy" extreme flapwise bending momentat the blade root of a large wind turbine from a low-fidelity and a high-fidelity aero-servo-elastic simu-lator. With limited high-fidelity response samples,Figure 7: Comparison of the Kriging and co-Krigingmodels of the high-fidelity (HF) extreme flapwise bend-ing moment My as a function of turbulence for α = 0.2and V = 8m/s.Figure 8: Comparison of the Kriging and co-Krigingmodels of the high-fidelity (HF) extreme flapwise bend-ing moment My as a function of turbulence for α = 0.2and V = 12m/s.Figure 9: Comparison of the Kriging and co-Krigingmodels of the high-fidelity (HF) extreme flapwise bend-ing moment My as a function of turbulence for α = 0.2and V = 20m/s.the co-Kriging predictions compared well with val-712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015idation data. The notably accurate prediction per-formance is due to using the low-fidelity Krigingmodel as a model trend for the co-Kriging model. Itis straight forward to extend this method to multiplefidelity levels. The confidence interval on the pre-dictions of the co-Kriging model reflects a mix ofepistemic (statistical) uncertainty due to the num-ber of sampled points and due to the noise in thesimulations output. A future study could attemptto quantify these two sources of uncertainties sep-arately. In this paper, the main assumption is thatthe high and low-fidelity aero-servo-elastic simula-tions follow similar trends, which makes the fusionof results feasible. If the trend were not present thenfusing data using co-Kriging would become hard toperform and less reliable. Finally, extreme loadsresponse of a wind turbine are known not to followa Gaussian process; a future study could attemptto modify the co-Kriging methodology to includenon-Gaussian processes.7. REFERENCESBachoc, F. (2013). “Cross validation and maximumlikelihood estimations of hyper-parameters of Gaus-sian processes with model misspecifications.” Com-put. Stat. Data Anal., 66, 55–69.Bossanyi, E. (2003a). “GH Bladed theory manual.” Re-port No. 282-BR-009, DNV GL.Bossanyi, E. (2003b). “GH Bladed user manual.” ReportNo. 282-BR-010, DNV GL.Christensen, D. E. (2012). “Multifidelity methods formultidisciplinary design under uncertainty.” M.S. the-sis, Massachusetts Institute of Technology, Departe-ment of aeronautics and astronautics.Dubourg, V. (2011). “Adaptive surrogate models for reli-ability analysis and reliability-based design optimiza-tion.” Ph.D. thesis, Université Blaise Pascal - Cler-mont II, LaMI.Forrester, A. I., Sóbester, A., and Keane, A. J. (2007).“Multi-fidelity optimization via surrogate modelling.”Proceedings of the Royal Society A: Mathemati-cal, Physical and Engineering Sciences, 463(2088),3251–3269.Han, Z., Zimmerman, R., and Görtz, S. (2012). “Alterna-tive cokriging method for variable-fidelity surrogatemodeling.” AIAA Journal, 50(5), 1205–1210.Han, Z.-H. and Görtz, S. (2012). “Hierarchical Krigingmodel for variable-fidelity surrogate modeling.” AIAAJournal, 50(9), 1885–1896.Hansen, M. (2001). Aerodynamics of wind turbines :rotors, loads and structure. James & James (SciencePublishers) Ltd. pp. 152. ISBN 1902916069.Jones, D. R. (2001). “A Taxonomy of global optimiza-tion methods Based on response surfaces.” Journal ofGlobal Optimization, 345–383.Jonkman, J. and Buhl, M. (2005). “FAST user’s guide.”Report No. NREL/EL-500-38230, National Renew-able Energy Laboratory.Kennedy, M. and O’Hagan, A. (2000). “Predicting theoutput from a complex computer code when fast ap-proximations are available.” Biometrika, 87(1), 1–13.Marelli, S. and Sudret, B. (2014). “UQLab: a frame-work for uncertainty quantification in MATLAB.”Proc. 2nd Int. Conf. on Vulnerability, Risk Analysisand Management (ICVRAM2014), Liverpool, UnitedKingdom.Picheny, V., Wagner, T., and Ginsbourger, D. (2012). “Abenchmark of kriging-based infill criteria for noisyoptimization.Sacks, J., Welch, W., Mitchell, T., and Wynn, H. (1989).“Design and analysis of computer experiments.” Stat.Sci., 4, 409–435.Schöbi, R. and Sudret, B. (2014). “Polynomial-chaos-based kriging.” Int. J. Uncertainty Quantification,submitted.Sudret, B. (2012). “Meta-models for structural reliabil-ity and uncertainty quantification.” Fifth Asian-PacificSymposium on Structural Reliability and its Applica-tions, <>.8


Citation Scheme:


Citations by CSL (citeproc-js)

Usage Statistics



Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            async >
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:


Related Items