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

Generation of synthetic accelerograms compatible with a set of design specifications Batou, Anas; Soize, Christian Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Generation of Synthetic Accelerograms Compatible with a Set ofDesign SpecificationsAnas BatouAssistant professor, Université Paris-Est, Laboratoire Modélisation et SimulationMulti-Echelle, Marne-la-Vallée, FranceChristian SoizeProfessor, Université Paris-Est, Laboratoire Modélisation et Simulation Multi-Echelle,Marne-la-Vallée, FranceABSTRACT: The research addressed here concerns the generation of seismic accelerograms compatiblewith a given response spectrum and with other design specifications. The time sampling of thestochastic accelerogram yields a time series represented by a random vector in high dimension. Theprobability density function (pdf) of this random vector is constructed using the Maximum Entropy(MaxEnt) principle under constraints defined by the available information. In this paper, a newalgorithm, adapted to the high stochastic dimension, is proposed to identify the Lagrange multipliersintroduced in the MaxEnt principle to take into account the constraints. This novel algorithm is based on(1) the minimization of an appropriate convex functional and (2) the construction of the probabilitydistribution defined as the invariant measure of an Itô Stochastic Differential Equation in order toestimate the integrals in high dimension of the problem.1. INTRODUCTIONThis paper is devoted to the generation of seismicaccelerograms that are compatible with some de-sign specifications such as the Velocity ResponseSpectrum, the Peak Ground Acceleration (PGA),etc. The Maximum Entropy (MaxEnt) principleintroduced by Jaynes (1957a,b) in the frameworkof Information Theory constructed by E.Shannon(1948) is a powerful method which allows usto construct a probability distribution of a ran-dom vector under some constraints defined by theavailable information. This method has recentlybeen applied in Soize (2010) for the generation ofspectrum-compatible accelerograms as trajectoriesof a non-Gaussian non-stationary centered randomprocess represented by a high-dimension randomvector for which the probability density function(pdf) is constructed using the MaxEnt principle un-der constraints relative to (1) the mean value, (2)the variance of the components and (3) the meanvalue of the Velocity Response Spectrum (VRS).The objective of this paper is to take into accountadditional constraints related to some design spec-ifications. To achieve this objective, the methodol-ogy proposed in Soize (2010) is extended to takeinto account constraints relative to statistics on (1)the end values for the velocity and the displace-ment, (2) the PGA, (3) the Peak Ground Velocity(PGV), (4) the envelop of the random VRS and(5) the Cumulative Absolute Velocity (CAV). TheMaxEnt pdf is constructed and a generator of inde-pendent realizations adapted to the high-stochasticdimension of an accelerogram is proposed. Fur-thermore an adapted method (see Batou and Soize(2014)) for the identification of the Lagrange mul-tipliers is developed. In Section 2 the MaxEnt prin-ciple is used to construct the pdf of the accelera-tion random vector under constraints defined by theavailable information. Finally, Section 3 is devoted112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015to an application of the methodology for which thetarget VRS is constructed following the Eurocode 8(see CEN (2003)).2. CONSTRUCTION OF THE PROBABIL-ITY DISTRIBUTIONThe MaxEnt principle is used to construct the prob-ability distribution of the random vector associatedwith a sampled stochastic process under some con-straints defined by the available information.The random acceleration of the soil is mod-eled by a second-order centered stochastic pro-cess {A(t), t ∈ [0,T ]}. A time sampling of thisstochastic process is introduced with a samplingtime step ∆t such that T = N∆t, yielding a time se-ries {A1, . . . ,AN} for which the RN-valued randomvector A = (A1, . . . ,AN) is associated with. Theprobability distribution of the random vector A hasto be constructed.2.1. Maximum entropy principleThe objective of this section is to construct the pdfa 7→ pA(a) of the random vector A using the Max-Ent principle under the constraints defined by theavailable information relative to random vector A.The support of the pdf is assumed to be RN . LetE{.} be the mathematical expectation. The avail-able information is assumed to be written asE{g(A)}= f , (1)in which a 7→ g(a) is a given function from RN intoRµ and where f is a given (or target) vector in Rµ .Equation (1) can be rewritten as∫RNg(a)pA(a)da = f . (2)An additional constraint related to the normaliza-tion of the pdf pA(a) is introduced such that∫RNpA(a)da = 1 . (3)The entropy of the pdf a 7→ pA(a) is defined byS(pA) =−∫RNpA(a) log(pA(a))da , (4)where log is the natural logarithm. Let C be the setof all the pdf defined on RN with values in R+, ver-ifying the constraints defined by Eqs. (2) and (3).Then the MaxEnt principle consists in construct-ing the probability density function a 7→ pA(a) asthe unique pdf in C which maximizes the entropyS(pA). Then by introducing a Lagrange multiplierλ associated with Eq. (2) and belonging to an ad-missible open subset Lµ of Rµ , it can be shownthat the MaxEnt solution, if it exists, is defined bypA(a) = c0(λ sol)exp(−〈λ sol,g(a)〉) , (5)in which λ sol is such that Eq. (2) is satisfied andwhere c0(λ ) is the normalization constant definedbyc0(λ ) ={∫RNexp(−〈λ ,g(a)〉)da}−1. (6)2.2. Calculation of the Lagrange multipliersIn this section, we propose a general methodologyfor the calculation of the Lagrange multipliers λ sol.2.2.1. Objective function and methododologyUsing Eqs. (2) and (5), vector λ sol is the solutionin λ of the following set of µ nonlinear algebraicequations∫RNg(a)c0(λ )exp(−〈λ ,g(a)〉) = f . (7)A more convenient way to calculate vector λ solconsists in solving the following optimization prob-lem (see Golan et al. (1996)),λ sol = arg minλ∈Lµ⊂RµΓ(λ ) , (8)in which the objective function Γ is written asΓ(λ ) = 〈λ , f〉− log(c0(λ )) . (9)Let {Aλ , λ ∈ Lµ} be a family of RN-valued ran-dom variables for which the pdf is defined, for all λin Lµ , bypAλ (a) = c0(λ )exp(−〈λ ,g(a)〉) . (10)Then the gradient vector ∇Γ(λ ) and the Hessianmatrix [H(λ )] of function λ 7→ Γ(λ ) are written as∇Γ(λ ) = f−E{g(Aλ )} . (11)212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015[H(λ )] = E{g(Aλ )g(Aλ )T}−E{g(Aλ )}E{g(Aλ )}T , (12)in which uT is the transpose of u. It is assumed thatthe constraints defined by Eq. (2) are algebraicallyindependent. Consequently, the Hessian matrix ispositive definite and therefore, function λ 7→ Γ(λ )is strictly convex and reaches its minimum for λ solwhich is such that ∇Γ(λ ) = 0 for λ = λ sol. It canthen be deduced that the minimum of function λ 7→Γ(λ ) corresponds to the solution of Eq. (7). Theoptimization problem defined by Eq. (8) is solvedusing the Newton iterative methodλ i+1 = λ i−α [H(λ i)]−1∇Γ(λ i) , (13)in which α belongs to ]0 ,1] is an under-relaxationparameter that ensures the convergence towards thesolution λ sol. In general, for the non-Gaussian case,the integrals in the right-hand side of Eqs. (11) and(12) cannot explicitly be calculated and cannot bediscretized in RN . In this paper, these integrals areestimated using the Monte Carlo simulation methodfor which independent realizations of the randomvector Aλ are generated using a specific algorithmpresented below.2.2.2. Generator of independent realizationsThe objective of this section is to provide a gen-erator of independent realizations of the randomvector Aλ for all λ fixed in Lµ . A generatorof independent realizations for MaxEnt distribu-tions has been proposed in Soize (2008, 2010) inthe class of the MCMC algorithms. This method-ology consists in constructing the pdf of randomvector Aλ as the density of the invariant measurepAλ (a)da, associated with the stationary solution ofa second-order nonlinear Itô Stochastic differentialequation (ISDE). The advantages of this generatorcompared to the other MCMC generators such asthe Metropolis-Hastings (see Hastings (1970)) al-gorithm are: (1) The mathematical results concern-ing the existence and the uniqueness of an invari-ant measure can be used, (2) a damping matrix canbe introduced in order to rapidly reach the invari-ant measure, and (3) there is no need to introduce aproposal distribution which can induce difficultiesin high dimension. Below, we directly introducethe generator of independent realizations using adiscretization of the ISDE. Details concerning theconstruction of this generator can be found in Soize(2008, 2010).The ISDE is discretized using a semi-implicit inte-gration scheme in order to avoid the resolution of analgebraic nonlinear equation at each step while al-lowing significantly increase of the time step com-pared to a purely explicit scheme. Concerning theinitial conditions of the ISDE, the more the prob-ability distribution of the initial conditions is closeto the invariant measure, the shorter is the transientresponse and then the more efficient is the identi-fication algorithm of the Lagrange multipliers (seeBatou and Soize (2014)).2.2.3. Estimation of the mathematical expecta-tionsThe generator described hereinbefore allowsfor constructing ns independent realizationsA1λ , . . . ,Ansλ of random vector Aλ representing theacceleration of the soil. The mean value E{g(Aλ )}and the correlation matrix E{g(Aλ )g(Aλ )T}are estimated using the Monte Carlo simulationmethod byE{g(Aλ )} ≃1nsns∑ℓ=1g(Aℓλ ) , (14)E{g(Aλ )g(Aλ )T} ≃1nsns∑ℓ=1g(Aℓλ )g(Aℓλ )T . (15)3. APPLICATIONSThe acceleration stochastic process is sampled suchthat the final time T = 20 s. The time step is∆t = 0.0125 s. We then have N = 1600 (we assumeA(0) = 0 ms−2 almost surely).3.1. Available informationThe available information related to random vectorA is defined by:(1) The random vector A is centered.(2) The standard deviation of each component ofrandom vector A is imposed. The target values areplotted in Fig. 1.312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20150 5 10 15 2000.511.52Time (s)Standard deviation (ms−2)Figure 1: Target for the standard deviations of the com-ponents of random vector A.(3) The variance of the end-velocity (resultingfrom a numerical integration of random vector A)is zero.(4) The variance of the end-displacement (result-ing from two successive numerical integrations ofrandom vector A) is zero.(5) The target for the mean VRS (seeClough and Penzien (1975)), which is denotedby s, is constructed following the Eurocode 8 fora A-type soil and a PGA equal to 5 ms−2. It isdefined for a damping ratio ξ = 0.05 and for 20frequencies that are (in rad/s) 1.04, 1.34, 1.73,2.23, 2.86, 3.69, 4.74, 6.11, 7.86, 10.11, 13.01,16.74, 21.53, 27.70, 35.64, 45.86, 59.00, 75.91,97.67 and 125.66. The target of the mean VRS isplotted in Fig. 2.(6) Let slow be the lower envelop defined byslow = 0.5× s and sup be the upper envelop definedby sup = 1.5×s. The probability for random vectorA of being inside the region delimited by the twoenvelops is 0.99.(7) The mean PGA is 5 ms−2.(8) The mean PGV is 0.45 ms−1.(9) The mean CAV is 13 ms−2.3.2. ResultsFor the ISDE, the number of integration steps isM = 600. At each iteration, ns = 900 Monte Carlosimulations are carried out. The methodology de-veloped in Section 2.2.1 is applied using 30 itera-tions. The under-relaxation parameter is α = 0.3.101 10200.20.40.60.8Frequency (rad/s)mean VRS (ms−1)Figure 2: Target of the mean VRS.Figure 3 shows two independent realizations of therandom vector Aλ sol , which is generated using aclassical generator for Gaussian random variableand which are representative of two independent re-alizations of the random accelerogram. The corre-sponding trajectories of the velocity times series Vand of the displacement times series D result fromtwo successive numerical integrations of each real-ization of the random accelerogram and are plottedin Figs. 4 and 5. As expected, it can be seen thatthe end velocity and the end displacements are bothequal to zero. Figure 6 displays a comparison ofthe estimated standard deviation of the componentswith the target values.0 5 10 15 2000.511.52Time (s)Standard Deviation (ms−2)Figure 6: Variance: Target (thick dashed line) and esti-mation (thin solid line).Figure 7 shows a comparison of the mean VRS412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20150 5 10 15 20−6−4−2024Time (s)acceleration (ms−2)0 5 10 15 20−4−20246Time (s)acceleration (ms−2)Figure 3: Two independent realizations of the randomaccelerogram.with its target.101 10200.20.40.60.8Frequency (rad/s)VRS (ms−1 )Figure 7: Mean VRS: Target (dashed line), estimation(mixed line).The Figure 8 shows 100 trajectories of the ran-0 5 10 15 20−0.4−0.200.20.4Time (s)velocity (ms−1)0 5 10 15 20−0.4−0.3−0.2−0.100.10.20.3Time (s)velocity (ms−1)Figure 4: Two independent realizations of the randomvelocity.dom VRS and the envelops slow and sup.101 10200.20.40.60.811.21.4Frequency (rad/s)VRS (ms−1 )Figure 8: Random VRS: 100 trajectories (thin lines),lower and upper envelop (thick lines).It can be seen in Figs. 6 to 8 a good matching512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20150 5 10 15 20−0.1−0.0500.050.10.150.20.25Time (s)displacement (m)0 5 10 15 20−0.3−0.2−0.100.10.2Time (s)displacement (m)Figure 5: Two independent realizations of the randomdisplacement.between the estimated values and the target values.Concerning the PGA, the PGV and the CAV, theresults are summarized in Table 1. It can be seena good matching of the estimated mean values forthe PGA, the PGV and the CAV with the target val-ues. The results presented in Section ?? show aConstraint Estimation TargetMean PGA (ms−2) 4.98 5Mean PGV (ms−1) 0.46 0.45Mean CAV (ms−1) 13.04 13Table 1: For the PGA, the PGV, the CAV: comparisonof the estimated mean value with the target value.very good agreement between the target values andthe values estimated using the generated accelero-grams. Nevertheless, in Figs. 3-5, it can be seenthat the generated trajectories are not perfectly nat-ural. For instance, spurious low-frequency contentappears at the beginning of the signal. Such low-frequency content should appear later in the signal.The trajectories could be improved by adding con-traints related to the nonstationarity of the velocity(or response) spectrum.4. CONCLUSIONSA new methodology has been presented for the gen-eration of accelerograms compatible with a givenVRS and other properties. If necessary, additionalconstraints could easily be taken into account in ad-dition to those developed in this paper. The applica-tion shows a good matching between the estimatedvalues and the target values. The generated trajecto-ries could be improved by adding contraints relatedto time dependence of the VRS.5. ACKNOWLEDGMENTThis research was supported by the "AgenceNa-tionale de la Recherche", Contract TYCHE, ANR-2010-BLAN-0904.6. REFERENCESBatou, A. and Soize, C. (2014). “Generation of accelero-grams compatible with design specifications using in-formation theory.” Bulletin of Earthquake Engineer-ing, 12(2), 769–794.CEN (2003). Eurocode 8: Design of Structures forEarthquake Resistance-Part1: General Rules, Seis-mic Actions and Rules for Buildings. EN 1998-1:2003 European Committee for Standardization. Brus-sels.Clough, R. W. and Penzien, J. (1975). Dynamics ofStructures. McGraw-Hill.E.Shannon, C. (1948). “A mathematical theory of com-munication.” Bell System Tech. J., 27, 379–423 and623–659.Golan, A., Judge, G., and Miller, D. (1996). Maximumentropy econometrics: robust estimation with limiteddata. Wiley, New York.Hastings, W. K. (1970). “Monte carlo sampling methodsusing markov chains and their applications.” Biomet-rica, 109, 57–97.612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Jaynes, E. T. (1957a). “Information theory and statisticalmechanics.” Physical Review, 106(4), 620–630.Jaynes, E. T. (1957b). “Information theory and statisticalmechanics.” Physical Review, 108(2), 171–190.Soize, C. (2008). “Construction of probability distribu-tions in high dimension using the maximum entropyprinciple. applications to stochastic processes, ran-dom fields and random matrices.” International Jour-nal for Numerical Methods in Engineering, 76(10),1583–1611.Soize, C. (2010). “Information theory for generation ofaccelerograms associated with shock response spec-tra.” Computer-Aided Civil and Infrastructure Engi-neering, 25, 334–347.7

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