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

Estimation of the failure probability of a floating wind turbine under environmental load Murangira, Achille; Zuniga, Miguel Munoz; Perdrizet, Timothée Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Estimation of the Failure Probability of a Floating Wind TurbineUnder Environmental LoadAchille MurangiraPost-doctoral Researcher, Dept. of Applied Mechanics, IFP Energies Nouvelles,Rueil-Malmaison, FranceMiguel Munoz ZunigaResearch Engineer, Dept. of Applied Mathematics, IFP Energies Nouvelles,Rueil-Malmaison, FranceTimothée PerdrizetResearch Engineer, Dept. of Applied Mechanics, Rueil-Malmaison, FranceABSTRACT: This present paper deals with the choice of stochastic expansion models for load processesin structural reliability of floating wind turbines. Two widely used models, the spectral representationmethod and the Karhunen-Loeve (KL) expansion are compared as far as their use in reliability assessmentbased on real world wind speed data. The use of expansion models makes it possible to compute theoutcrossing rate using time invariant reliability methods. We assume stationary conditions and defineand compare several outcrossing rate estimators based on a design point computation. We first adapt thePHI2 method to functionals, and specify a closed form of Koo et al. (2005)’s estimator for the KL model.An importance sampling scheme based on the design point with and without an additional Monte CarloMarkov Chain sampler is then suggested and its effectiveness is assessed for the KL model.Offshore wind farms are garnering considerableinterest because the generally superior wind speedmakes them more efficient than their onshoreequivalents. Due to the influence of wind and waveloads, the mechanical structure is prone to failurewhich raises the issue of the reliability of floatingwind turbine designs. Among structural reliabilityalgorithms, the first and second order reliabilitymethods (FORM/SORM)(Ditlevsen and Madsen(1996)) are popular because they usually requirea reasonable amount of response simulations.Nevertheless, the principal underlying hypothesis,namely that the response is approximately lineararound a unique design point, suggests the useof other algorithms. An obvious alternative toFORM/SORM is the traditional Monte CarloSimulation (MCS). Since the sought failure proba-bility is usually small, MCS may not be a suitableoption as it requires a significant amount of systemresponse simulations to achieve an acceptable levelof accuracy. Importance sampling (IS) can be usedin cases where a reasonable approximation of thefailure region is available. IS proceeds by samplingfrom a distribution whose main contribution takesplace around the failure domain. However, de-signing an adequate importance sampling densitythat yields reasonable estimation variance is nottrivial. An additional challenge of this specific timevarying reliability analysis is that the response at agiven time depends on functional inputs, the windspeed and wave elevation processes. It is thereforenecessary to use an expansion of the stochasticprocesses on a finite functional basis with randomcoefficients. This study focuses on the wind speedprocess modelling only, although the approach ismore general. Since the performance of virtuallyall the aforementioned reliability methods is highlydependent on the dimension of the input space, ex-pansion methods that lead to reasonable dimensionsare desirable. In this paper, we study the spectral112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015representation method (Shinozuka and Deodatis(1991)) and the Karhunen-Loève expansion (KL)(Ghanem and Spanos (1991)). The spectral rep-resentation method models the time dependentload as a stationary Gaussian process and iswidely used in the reliability of offshore structurescommunity (Jensen (2009)), while KL targetsmore general processes and has some attractiveoptimality properties. Based on available windspeed measurements, both the KL and spectralrepresentation models are fitted to the data. To thisend, the joint distribution of the KL coefficients isestimated via kernel density estimation (KDE) as inPoirion and Zentner (2014) and the spectrum in thespectral representation model is estimated directlyfrom the data assuming input stationarity. Thisstationarity assumption, motivated by fixed longterm conditions, enables the use of time-invariantreliability methods. The failure probability isassessed by estimating the wind turbine responseoutcrossing rate. Several estimates are providedbased on design points computations, which werecarried out after careful selection of an optimiza-tion algorithm. This work is a first step to thereliability analysis of wind turbines and will serveas point of comparison to other estimators whichdon’t rely on the design point (subset simulation,importance sampling, etc.). Two FORM approx-imations are used: the first one is a derivative ofthe PHI2 method (Andrieu-Renaud et al. (2004)),adapted to the case where the output depends onfunctional inputs, while the second one is basedon Koo et al. (2005)’s work. The outcrossing rateis then evaluated by importance sampling basedon the preliminary design point computation. Itis expressed as a product of a conditional proba-bility, evaluated by a Monte Carlo Markov Chain(MCMC) sampler, and an unconditional failureprobability, which is estimated through IS. Thesemethods are applied to the reliability analysis ofa recent wind turbine prototype for both inputload models in order to assess the impact on thecomputing cost and the final result.1. STOCHASTIC EXPANSION METHODS FORLOAD PROCESSES1.1. The spectral representation methodIn reliability analysis of offshore structures, it isfrequent to model both the wave elevation and windspeed as stationary Gaussian processes with spe-cific spectrum. The resulting expansion is a discreteformulation of the spectral representation theorem,which reads for stationary Gaussian processes:X(t) =n∑i=1(uiσi cos(ωit)− u¯iσi sin(ωit)) (1)ui, u¯i are standard uncorrelated normal variables,ωi are the frequencies with increment dωi =wi+1 − wi and σ 2i = S(ω)dωi where S(ω) is thepower spectrum density (p.s.d.) of {X(t), t ≥ 0}.The previous expansion is sometimes called non-deterministic spectrum amplitude (NSA) and sim-ilar to the original spectral representation modelwhich uses a deterministic amplitude with randomuniform phases (Shinozuka and Deodatis (1991)).As pointed out in Grigoriu (1993), there’s usuallyno clear reason to favour either method. It is tobe noted that a spectral representation with equallyspaced frequencies produces periodic sample pathswith period 2piω1 , ω1 being the minimum frequency(Shinozuka and Deodatis (1991)). Therefore, sim-ulation of sufficiently long paths requires large nin (1) which can be prohibitive from a reliabilityanalysis standpoint. To obtain longer trajectories, itis sometimes advocated to use non-equal frequencyspacing via iso-energy spectrum discretization forinstance: this consists in constructing the frequencygrid such that the energy between two successivefrequencies∫ ωi+1ωiS(ω)dω is constant. For a fixed n,this usually gives longer periods than the fixed fre-quency increment method (Jensen (2009)). A sim-ilar approach consists in expressing the spectrumin the period domain and using equally spaced dis-crete periods to discretize this spectrum.1.2. Karhunen-Loève expansion1.2.1. BackgroundLet {X(t), t ∈ [0,T ]} be a mean-square (m.s.)continuous stochastic process. Then there exists abasis of L2([0,T ]) eigenfunctions {φi, i ≥ 1}, such212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015that ∀t ∈ [0,T ]X(t) m.s.= E(X(t))+∞∑i=1√λiξiφi(t) (2)The zero-mean and uncorrelated random variablesξi, i ≥ 1 are the Karhunen-Loève coefficients, and{λi} is a sequence of decreasing eigenvalues. Thelatter are solutions to the integral equation∫ T0RX(s, t)φi(t)dt = λiφi(s)where RX is the process autocovariance function.The KL expansion is clearly more general than thespectral representation model since X need not bestationary nor Gaussian. In addition, it is optimal inthe sense that when truncated after a finite numbern of terms, it minimizes the mean-square error. Aformal criterion to identify the truncation order isthe percentage of explained variability defined asvarexpn =∫ T0E[(Xn(t)− ¯X(t))2]dt∫ T0E[(X(t)− ¯X(t))2]dt(3)where Xn is the expansion truncated after n termsand ¯X(t) = E(X(t)). n can be chosen so as to ac-count for a given percentage (e.g. 95 %) of the vari-ability of the process.In order to simulate sample paths according toa truncated KL expansion, the joint distribution ofthe KL coefficients ξ1, . . . ,ξn needs to be estimated.Note that for Gaussian processes, this distributionis also Gaussian but nothing is known in the gen-eral case. A useful tool is kernel density estimation(KDE) where the joint pdf is expressed as a mixtureof multivariate kernels centered around the sampleKL coefficients ξ (l), l = 1, . . . ,N.pˆξ (x) =1NhnN∑l=1K(x−ξ (l)h)(4)where ξ (l) = [ξ (l)1 , . . . ,ξ (l)n ]T is the vector ofKarhunen-Loève coefficients estimated from thedata viaξ (l)i = 1√λi∫ T0[Xl(t)− ¯X(t)]φi(t)dt (5)K is usually not influential on KDE accuracyso we assume a Gaussian kernel from now on.The bandwidth parameter h controls the degreeof smoothing and impacts significantly KDEperformance. Poirion and Zentner (2014) use KDEto estimate the pdf of ξ and show that, whenthe kernel is Gaussian, a suitable choice of thebandwidth is h = 1.06N− 15 .In practice, the KL expansion is con-structed from a database of N measurements{t 7→ Xi(t), i = 1, . . . ,N}. In this case, underregularity conditions, the KL expansion derivedfrom the empirical autocovariance function isstrongly consistent (Poirion and Zentner (2014))as N → ∞. Another aspect is that the measuredrealizations Xi are sampled at discrete time stepstk, k = 1, . . . ,M. Performing KL expansion can bedone by considering the interpolating processes˜Xi(t), s.t. ˜Xi(tk) = Xi(tk) and performing KL expan-sion on ˜X (see Ramsay and Silverman (2005)). Itcan then be shown that KL expansion of ˜X reducesto a matrix eigenvalue problem. In the sequel, thetilde will refer to KL expansion performed on theinterpolating process as defined by equations (2),(4) and (5).2. STRUCTURAL RELIABILITY METHODS2.1. Outcrossing approach and iso-probabilistictransformationLet X(t,ξ ) be the stochastic load process, whereξ is a random vector of expansion coefficients ina finite functional basis. We will assume thatresponse process at time t, Y (t,ξ ) only dependson the input load up to time t, that is Y (t,ξ ) =F(X[0,t](·,ξ )) where other deterministic structuralcharacteristics have been omitted. The limit statefunction G(t,ξ ) = s−Y (t,ξ ), where s is a scalarthreshold on the response, is strictly positive inthe safe domain and negative in the failure do-main. In time variant structural reliability, weare concerned with the evaluation of Pf (Th) =P(supt∈[0,Th]G(t,ξ ) ≤ 0) for a given time horizonTh. A quantity of interest is the outcrossing rateν+(t) defined asν+(t)= lim∆t→0P({G(t,ξ )> 0}∩{G(t +∆t,ξ )≤ 0})∆t(6)312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Indeed, ν+ is linked to the failure probabilitythrough the inequalityPf (Th)≤ P(G(0,ξ )≤ 0)+E(N+(0,Th)) (7)where E(N+(0,Th)) =∫ Th0 ν+(t)dt is the meannumber of outcrossings.In order to compute (6), it is convenient to ex-press the limit state function in the standard Gaus-sian space U. This is done thanks to an iso-probabilistic transform T, such as the Nataf trans-form , that maps the physical space to U. Lettingg(t,U) = G(t,T−1(U)) the limit state function inthe standard space, the outcrossing rate now readsν+(t)= lim∆t→0P({g(t,U)> 0}∩{g(t +∆t,U)≤ 0})∆t(8)We hereafter use the Nataf transform de-fined as T(ξ ) = T2 ◦ T1(ξ ), T1(ξ ) = Z =[Φ−1(F1(ξ1)) · · ·Φ−1(Fn(ξn))]T , T2(Z) = L−10 Zwhere L0LT0 = R0 and R0 is the correlation matrixof Z, Φ−1 is the inverse of the standard normal cu-mulative distribution function (cdf) and F1, . . . ,Fnare the marginal cdfs of the input KL coefficientvector. When using the spectral representationmodel, the Nataf transform is simply the identityfunction.If ξ is the vector of Karhunen-Loève coefficients,since Cov(ξ ) = In (see properties in 1.2.1), thecorrelation matrix of ξ is Cξ = In. Besides, ifCξi, j = 0, then R0,i j = 0 (Liu and Der Kiureghian(1986)). This leads to R0 = In. Using the kerneldensity estimate (4) for the distribution of ξ , thecorresponding marginal cdfs areFi = Fξi(x) =1NN∑l=1Φ(x−ξ (l)i˜h)From a numerical point of view, the inverse cdfcan be computed from the previous expression bypre-computing and storing y j(i) = Fi(x j(i)) wherexmin = x1(i)< x2(i)< · · ·< xr(i) = xmax(i) is a reg-ular grid. F−1i (y) can then be computed for all y byspline interpolation.Computation of (8) normally requires the evalua-tion of a joint exceedance probability for all in-volved time steps. However, in the case of station-ary input processes, where the response process attime t is only a function of X up until time t, the re-sponse is also stationary and ν+ is independent oft. Its computation thus only requires calculating theprobability (8) with a small ∆t at a fixed instant t.2.2. FORM/SORM approximations of the out-crossing rateFORM/SORM can be used to estimate ν+(t) viaone or two FORM analysis. PHI2 method makesuse of two FORM computations which yieldsν+PHI2 =Φ2(β (t),−β (t+∆t)),ρG(t, t+∆t))∆t(9)β (t) is the reliability index at time t, ∆t a small timeincrement and Φ2 the bivariate multinormal cdf.The reliability index is obtained as β (t) = ‖u∗(t)‖,where u∗(t) is the design point i.e. the most proba-ble failure point, a solution of constrained optimiza-tion problemu∗(t) = argming(t,u)=0‖u‖2 (10)The correlation factor ρG is defined as ρG(t, t +∆t) =−α(t)T α(t +∆t), where α(t) = u∗(t)β (t) .A second more economical method in terms oflimit state function evaluations, only resorts toone FORM computation. The main idea, due toKoo et al. (2005), is summed up by the followingobservation: let X∗1 be the critical wind episode cor-responding to design point u∗1 = u∗(t), and leads tofailure at fixed time t. Define X∗2 (τ) = X∗1 (τ −∆t)for ∆t ≤ τ and X∗2 (τ) = 0, for 0 ≤ τ ≤ ∆t. Then,it is clear that X∗2 leads to failure at time t + ∆tand may be used to obtain the corresponding de-sign point u∗2 = u∗(t +∆t). The following approxi-mation (Koo et al. (2005)) may then be used to es-timate ν+:ν+ ≈ ν+Koo =12piexp(−β22)[pi2+ arcsin(ρG)](11)where β = ‖u∗1‖ and ρG ≈−u∗1u∗2β 2 . When the spectralrepresentation is used, u∗2 can be computed analyti-cally (see Jensen and Capul (2005)). In the case ofKL expansion, we propose to compute a first orderapproximation of the design point at time t +∆t in412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015the physical space ξ ∗2 , and obtain u∗2 via the Nataftransform u∗2 = T(ξ ∗2 ). As justified in 1.2.1, ξ ∗2 isreplaced by ˜ξ ∗2 , the KL coefficient vector of the in-terpolating process ˜X∗2 . Using (5), ∀ j = 1, . . . ,n˜ξ ∗2, j = 1√˜λ j∫ T0˜X∗2,c(t) ˜φ j(t)dt (12)where ˜X∗2,c(t) = ˜X∗2 (t)− ¯˜X∗2 (t) is the centred in-terpolating critical process. Let ek, k = 1, . . . ,Mbe the following linear interpolating functions on]tk−1, tk+1]ek(t) =t − tk−1δτ 1]tk−1,tk](t)+tk+1 − tδτ 1]tk,tk+1](t)where the sampling interval δτ = tk+1 − tk issupposed constant. Then, X∗2 (t) ≈ ˜X∗2 (t) =∑Mk=1 X∗2 (tk)ek(t) and φ j(t)≈ ˜φ j(t) = ∑Mk=1 c jlel(t),c j,l = φ j(tl), j = 1, . . . ,n, l = 1, . . . ,M. Then, (12)becomes˜ξ ∗2, j = 1√˜λ j∫ T∆t˜X∗1,c(t −∆t)φ j(t)dt (13)= 1√˜λ j∑1≤k,l≤MX∗1,c(tk)c j,l∫ T∆tek(t −∆t)el(t)dt(14)After some basic integration, we have for ∆t → 0,˜ξ ∗2, j = ˜ξ ∗1, j + 1√˜λ jX∗1,c(t1)(c j,2 − c j,1)∆t2+ ∆t√˜λ jM−1∑k=2(c j,k −c j,k−12− c j,k+16)X∗1,c(tk)− 1√˜λ jX∗1,c(tM)(c j,M−1 + c j,M)∆t2+o(∆t) (15)Note that approximations (9) and (11) assumethere is a unique design point whose distance to theorigin is significantly less than other local minimas:this may not always be the case.2.3. Estimating the outcrossing rate via impor-tance samplingImportance sampling (IS) is a practical standardMonte Carlo variance reduction method. It is usefulto compute the expectation of a function of randomparameters, when the underlying density is eitherdifficult to sample from or yields high variance es-timate. Here, we are interested in approximating(8) via a finite element type approximation definedby the following expectationν+(t)≈E(1g(t,U)>0,g(t+∆t,U)≤0)∆t(16)Since we are in the standard space,U ∼ ϕn whereϕn is the standard n-dimensional multinormalpdf. The MC estimate of (16) is ν+MC(t) =1NMC∆t ∑NMCi=1 1g(t,ui)>0,g(t+∆t,ui)≤0 where ui ∼i.i.d. ϕn,i = 1, . . . ,NMC. Importance sampling proceeds bydrawing samples ui, i = 1, . . . ,NIS from a proposaldensity q and reweighting them appropriately pro-viding the IS estimate of (16)ν+IS(t)≈1NIS∆tNIS∑i=1ϕn(ui)q(ui) 1g(t,ui)>0,g(t+∆t,ui)≤0(17)A suitable choice of q usually leads to a lower vari-ance of the estimator. In this case, a frequent choiceis to choose a proposal density, for instance a Gaus-sian pdf, centred on the design point.The IS estima-tion variance readsσˆ 2IS =1NISNIS∑i=1(1g(t,ui)>0,g(t+∆t,ui)≤0ϕ2n (ui)∆t2q2(ui) −ν+IS)2(18)Since the numerator in (8) might be a few mag-nitude orders smaller than the failure probabil-ity P(g(t + ∆,U) ≤ 0), we decomposed it intoP(g(t,U) > 0 | g(t + ∆t,U) ≤ 0)P(g(t + ∆t,U) ≤0). p1 = P(g(t +∆t,U)≤ 0) can then be computedby an importance sampling procedure with a pro-posal centred on the design point at time t +∆t asoutlined previously. As for the conditional prob-ability p2 = P(g(t,U) > 0 | g(t + ∆t,U) ≤ 0), anMCMC sampler targeting P(· | g(t + ∆t,U) ≤ 0)may be used to enable the estimation. Indeed, whenestimatingP(g(t+∆t,U)≤ 0) by IS, the ns samplesthat fall in the failure domain {g(t + ∆t,U) ≤ 0}during the IS procedure can be used as seeds ofindependent Markov chains. If a population ofsize N is needed, then each chain can be run with512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015c⌈Nns⌉+ b iterations where b is a burnin parame-ter and c a thinning parameter used (meaning oneout of c MCMC samples is kept). In this study,the modified Metropolis-Hastings (MH) describedin (Au and Beck (2001)) was used. Letting pˆ1 bethe estimator of P(g(t +∆t,U)≤ 0) and pˆ2 that ofP(g(t,U) > 0 | g(t + ∆t,U) ≤ 0), the outcrossingrate estimate is ν+IS+MH =pˆ1 pˆ2∆t and the correspond-ing variance σ 2IS+MH can be estimated by notic-ing that the estimate pˆ1 is independent of pˆ2 sincethe dependency of the samples used to estimate pˆ2w.r.t. to those used for pˆ1, vanishes after sufficientburn-in in the Markov chain.σ 2IS+MH =Var( pˆ1)Var( pˆ2)∆t2 +Var( pˆ1)E2( pˆ2)∆t2 +Var( pˆ2)E2( pˆ1)∆t2Var(pˆ2) can be estimated as in Au and Beck (2001)while Var(pˆ1) is a standard IS variance estimate.Moreover, E(pˆ1) = p1 since IS provides unbiasedestimates. We can therefore estimate it by pˆ1. Asfor E(pˆ2), since pˆ2 is an MCMC estimate, it is bi-ased for a finite sample size. Nevertheless, we as-sume asymptotic conditions in which case E(pˆ2)≈p2.3. ASSESSMENT OF A WIND TURBINE SHORT-TERM FAILURE PROBABILITY SUBJECT TOWIND LOAD3.1. Test caseWe consider the problem of estimating the short-term failure probability of a wind turbine, that is thefailure probability assuming stationary conditions.In reliability analysis of offshore structures, theloads are classically described in terms of stochasticprocesses that are stationary within a given time in-terval. This window duration is 10 minutes for thewind speed process and 3 hours for the wave ele-vation. A bound on the short-term failure probabil-ity may be obtained by computing the outcrossingrate ν+(t) for fixed long-term parameters. We con-sider the 5 MW OC4 wind turbine, with a 90 m hubheight and 126 m rotor diameter whose responseis modelled by the NREL FAST code. To modelshort-term stationary conditions based on real windspeed data, we have extracted 10 minute record-ings of the wind speed at the Hornsrev site (Den-mark), available at Those selected65 measurements have a mean wind speed equalto the rated speed, i.e. U10 = 11.5 m/s and turbu-lence intensity Iu = 6%, i.e. σU = 0.7. Assumingstationarity during a 10 minute interval, the time-invariant outcrossing rate may be computed at anytime t in the interval. In practice, it is necessaryto discard the initial transient part of the responsewhich means that the stationarity is actually validafter the system memory effects have disappeared,say one minute after the beginning of the simulationif wind loads are only present and up to 3 or 4 min-utes if wave loading is considered. ν+ is computedat a fixed time t after the transient part. We there-fore only kept the first T = 5 minutes of each windspeed measurement. Without accounting for hydro-dynamic effects, it is sufficient to simulate the first60 seconds of the response. The limit state functionwe consider is therefore G(t0,ξ ) = 0.5−Y (t0,ξ )where Y is the wind turbine tower-top displacementin meters, where t0 = 1 minute.3.2. Stochastic model fittingWe have then fitted the spectral representationmodel as well as the Karhunen-Loeve model to thedata. For the spectral representation model, it isnecessary to specify a spectral density. This hasbeen done by computing averaged smoothed pe-riodograms of each wind speed measurement (seefig. 1). This p.s.d. estimate was then discretizedby selecting nharm frequencies ω1, . . . ,ωnharm via aniso-energy discretization scheme, where ω1 = 2piTand ωnharm = piδτ , δτ = 0.1s being the input sam-pling step. The comparison of both models wascarried out by setting n = 60 harmonics for the KLmodel and nharm = 50: this corresponds to a relativeL2 error of the autocovariance function less than0.1% for KL expansion and of 6% for the spectralrepresentation model. Be advised that the dimen-sion of the expansion is d = 2nharm = 100 in thespectral representation expansion.3.3. Simulation experiments3.3.1. Design point computationGiven both input stochastic models, several con-strained optimizations solvers were compared in or-der to identify the most probable failure point u∗612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20150 5 10 15 20 25 305e−045e−035e−025e−01ω(rad s)S(ω)(m²srad)Figure 1: One-sided power spectral density of windspeed processdefined by (10). The sequential quadratic program-ming (SQP) algorithm was the most efficient interms of computing cost and the gradient-free se-quential quadratic approximation (SQA) stood outin terms of number of calls to the limit state func-tion. For practical reasons SQP was used with aninitialization at the origin of the U space. Theoptimization is thus performed in dimension d =2nharm = 100 for the spectral representation modeland d = 60 for the KL expansion.3.3.2. Outcrossing rate estimation via a FORMapproximationAs stated in 2.2, we compute two approximationsfor the outcrossing rate: the extended PHI2 approx-imation ν+PHI2 and ν+Koo. The first one involvestwo FORM computations at time t0 and t0 + ∆twhile ν+Koo only requires one FORM analysis attime t0, the design point u∗2 at time t0 + ∆t beinga function of u∗1 and ∆t. For both approximations,∆t = 0.01 s. The results are presented in table 1,where the number of calls to the limit state func-tion is in brackets. Both estimators ν+PHI2 and ν+Kooseem to agree regardless of the input model. How-ever, the type of stochastic expansion plays a sig-nificant role even when the model parameters foreach expansion are estimated directly from the data.The outcrossing rate with a spectral representationof the input load yields is roughly 100 times lessthan that obtained with a KL expansion. This in-dicates that the assumption of a Gaussian processis a rather strong one for the wind velocity. As ex-pected the number of calls to the limit state functionfor the spectral representation is more important,most notably because of the use of finite differencesin the gradients computation. For validation pur-poses only, the multiple design point algorithm byDer Kiureghian and Dakessian (1998) was appliedand suggested no other significant failure mode.Table 1: Outcrossing rate estimation (FORM approxi-mation)spectral KLβ (t) 8.57 (4051) 7.96 (2295)ν+PHI2 2.68×10−17 (10039) 2.39×10−15 (7267)ν+Koo 2.90×10−17 (4051) 2.57×10−15 (2295)3.4. Outcrossing rate estimation via ImportancesamplingBased on the design point, we have used N =2000 samples to estimate ν+ via ν+IS or ν+IS+MH asdefined in 2.3 for the KL model. The estimatesare reported in table 2 along with the coefficientof variation and the number of g calls includingthe design point computation. The importance den-sity used for both estimators is the normal densitycentred at the design point and with identity co-variance matrix. Using "straight-up" importanceTable 2: Importance sampling estimators of the out-crossing rateIS IS+MHν+ 2.88×10−15 3.44×10−16c.o.v. 0.99 0.82# g calls 4295 10831sampling seems to yield high estimation varianceas evidenced by the c.o.v. The estimator ν+IS+MHwhich splits the probability into two terms, one be-ing estimated by IS and the other through MCMCby Metropolis-Hastings seems to lessen the estima-tion variance even if it remains non-negligible.4. CONCLUSIONFinite expansion models of stationary or non-stationary input loads are of practical importance in712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015reliability analysis of mechanical structures. Prac-titioners seek to account for maximum signal vari-ability while keeping a sufficiently compact repre-sentation in terms of the dimension of the randomparameters. This dimension impacts directly thecomputational cost of reliability algorithms sincethe number of calls to the simulator necessary toestimate failure probabilities grows rapidly with thedimension. We have compared two popular expan-sion schemes, the spectral representation methodand the Karhunen-Loève expansion by fitting bothmodels on real wind speed measurements in an ap-proximate stationary setting. Prior to any relia-bility analysis, it is confirmed that the Karhunen-Loève expansion is significantly more economicalin terms of number of random parameters. The esti-mation of the short-term failure probability is takenfrom the point of view of an outcrossing rate analy-sis. The identification of the design point is usedas a starting block for the outcrossing rate com-putations via FORM/SORM type approximationsor Importance sampling estimators with or with-out an MCMC sampler. The use of a KL modelmakes it possible to extend the PHI2 method tofunctional inputs, by rewriting the limit state func-tion in terms of the expansion coefficients. Thisleads to an estimator of the outcrossing rate thatrequires two FORM analyses. In the case of KLexpansion we derived an explicit approximation ofthe correlation factor in Koo’s outcrossing rate for-mula (Koo et al. (2005)) which only requires onedesign point computation. Additionally a MonteCarlo type estimator is studied. The estimation ofoutcrossing rate is split into an instantaneous fail-ure probability computation, estimated via impor-tance sampling based on a previously determineddesign point, and a conditional probability calcula-tion achieved through MCMC sampling. In futuredevelopments, design point(s) computation will bebased on the SQA optimization algorithm as it ismore economical in terms of calls to the limit statefunction which will mean a lesser computing costcompared to other solvers, one the wave loads havebeen taken into account. We also plan to adress theestimation of the outcrossing rate without a prelim-inary design point computation.5. REFERENCESAndrieu-Renaud, C., Sudret, B., and Lemaire, M.(2004). “The phi2 method: a way to compute time-variant reliability.” Reliability Engineering and Sys-tem Safety, 84, 75–86.Au, S.-K. and Beck, J. (2001). “Estimation of smallprobabilities of failure in high dimensions by sub-set simulation.” Probabilistic Engineering Mechan-ics, 16, 263–277.Der Kiureghian, A. and Dakessian, T. (1998). “Multi-ple design points in first and second order reliability.”Structural Safety, 20, 37–49.Ditlevsen, O. and Madsen, H. (1996). Structural Relia-bility Methods. John Wiley & Sons Inc.Ghanem, R. and Spanos, P. (1991). Stochastic finite ele-ment: A spectral approach. Springer-Verlag.Grigoriu, M. (1993). “On the spectral representationmethod in simulation.” Probabilistic Engineering Me-chanics, 8, 75–90.Jensen, J. (2009). “Extreme load predictions for float-ing offshore wind turbines.” 28th International Con-ference on Offshore Mechanics and Arctic Engineer-ing.Jensen, J. and Capul, J. (2005). “Extreme response pre-dictions for jack-up units in second order stochasticwaves by form.” Probabilistic Engineering Mechan-ics, 21, 330–337.Koo, H., Der Kiureghian, A., and Fujimura, K. (2005).“Design-point excitation for non-linear random vi-brations.” Probabilistic Engineering Mechanics, 20,136–147.Liu, P.-L. and Der Kiureghian, A. (1986). “Multivariatedistribution models with prescribed marginals and co-variances.” Probabilistic Engineering Mechanics, 1,105–112.Poirion, F. and Zentner, I. (2014). “Stochastic modelconstruction of observed random phenomena.” Prob-abilistic Engineering Mechanics, 36, 63–71.Ramsay, J. and Silverman, B. (2005). Functional DataAnalysis. Springer.Shinozuka, M. and Deodatis, G. (1991). “Simulation ofstochastic processes by spectral representation.” Ap-plied Mechanics Review, 44, 191–204.8


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