Computer Simulation of Stochastic Wind

Hindawi Publishing CorporationAdvances in Civil EngineeringVolume 2010, Article ID 749578, 20 pagesdoi:10.1155/2010/749578

Research Article

Computer Simulation of Stochastic Wind Velocity Fields forStructural Response Analysis: Comparisons and Applications

Filippo Ubertini1 and Fabio Giuliano2

1Department of Civil and Environmental Engineering, University of Perugia, Via G. Duranti 93, 06125 Perugia, Italy2Department of Structural and Geotechnical Engineering, University of Rome La Sapienza, Via Eudossiana 18, 00184 Rome, Italy

Correspondence should be addressed to Filippo Ubertini, [email protected]

Received 14 October 2009; Revised 1 April 2010; Accepted 13 April 2010

Academic Editor: Chungxiang Li

Copyright 2010 F. Ubertini and F. Giuliano. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

The digital simulation of wind velocity fields, modeled as multivariate stationary Gaussian processes, is a widely adopted toolto generate the external input for response analysis of wind-sensitive nonlinear structures. The problem does not entail anytheoretical diculty, existing already a large number of well-established techniques, such as the accurate weighted amplitudewave superposition (WAWS) method. However, reducing the computational eort required by the WAWS method is sometimesnecessary, especially when dealing with complex structures and high-dimensional simulation domains. In these cases, approximateformulas must be adopted, which however require an appropriate tuning of some fundamental parameters in such a way to achievean acceptable level of accuracy if compared to that obtained using the WAWS method. Among the dierent techniques availablefor this purpose, autoregressive (AR) filters and algorithms exploiting the proper orthogonal decomposition (POD) of the spectralmatrix deserve a special attention. In this paper, a properly organized way for implementing stochastic wind simulation algorithmsis outlined at first. Then, taking the WAWS method as a reference from the viewpoint of the accuracy of the simulated samples,a comparative study between POD-based and AR techniques is proposed, with a particular attention to computational eort andmemory requirements.

1. Introduction

The simulation of wind velocity fields has been one of themain topics of wind engineering for the last decades. In thisframework, wind velocity is usually idealized as the sum of amean part, assumed as constant within a conventional timeinterval, and a fluctuating part representing the atmosphericturbulence. This last is usually modeled as a stationary zero-mean Gaussian random process [1, 2].

Several techniques were proposed in the literature inorder to simulate Gaussian wind velocity fields to beemployed in structural analysis [35]. Among those, theclassic WAWS method, based on the pioneering work byShinozuka and Jan [6] and then modified by Deodatis[1] in such a way to achieve ergodic realizations and tobe eciently implemented through fast Fourier transform(FFT) algorithms, has proved to guarantee the best qualityof the obtained results [7]. Nevertheless, such a proce-

dure requires the Cholesky factorization of the spectralmatrix, which unfortunately leads to high computationalexpenses, especially when dealing with complex structuresand high-dimensional simulation domains. These dicultiesare mainly related to memory allocation and time consumingoperations, thus requiring, on the one hand, the reduction ofthe problem size. On the other hand, an accurate wind simu-lation is essential for predicting the wind-induced responseof flexible structures, such as transmission power lines,tall buildings, suspension, and cable-stayed bridges. Lessdemanding, yet approximate, procedures may be obtained byexploiting the properties of the POD decomposition of thespectral matrix, proposed in the papers by Li and Kareem [8],Di Paola [9], and Solari and Carassale [10]. A POD-basedtechnique, in particular, was recently applied to simulate thewind velocity field on a domain representing a long-spansuspension bridge [11] with significantly low computationaleorts. A third well-established class of simulation formulas

2 Advances in Civil Engineering

Calculate and store deterministic data in memoryPhase 0

Phase 1

Stochastic generations of wind velocity fields

Sim 1 Sim 2 Sim n

Figure 1: Adopted computational scheme for wind simulation.

0

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x2(m)

9

5

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1000 0 1000x2 (m)

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Figure 2: Grids for wind simulation: example 1 (a); example 2 (b).

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Figure 3: Example 1: wind velocity field sample generated using the WAWS method: along-wind velocities (a); across-wind velocities (b).

Advances in Civil Engineering 3

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Figure 4: Example 1 using WAWS method for N = 214 and Nt = T0/t (i.e., a process duration equal to one period T0; black linesdenote target functions, red lines denote numeric approximations): autospectrum of along-wind velocity at point 5 (a); correspondingautocorrelation function (b); cross-spectrum between along-wind velocities at points 5 and 9 (c); corresponding cross-correlation function(d).

is represented by autoregressive (AR) and autoregressivemoving average (ARMA) filters, early introduced in thepaper by Samaras et al. [12] and applied to wind simulationby Mignolet and Spanos [13], by Li and Kareem [8] and,more recently, by Di Paola and Gullo [14].

This paper addresses, at first, the problem of theecient implementation of wind simulation algorithms byproposing a two-steps approach which allows to properlyhandle large numbers of simulations, which is usually thecase for performing structural response analysis. Then, theWAWS method is adopted as a reference for calibrating theparameters of POD-based and AR techniques. These lastmethods are finally compared in terms of computationaleort and memory requirements. To this end, two numericalexamples are presented. The former, represented by thetower of a suspension bridge, is considered for modelscalibration. The latter, represented by an entire suspensionbridge, is a more demanding case, which is here chosenas a benchmark in order to test the capability of AR andPOD-based techniques in handling complex simulationsand to compare their computational eciency. Finally, theconvergence rates of the accuracies of the two simplifiedmethods are also investigated, in order to provide indicationsfor tuning the parameters that aect their results.

The considered formulas for wind simulation are onlysome of those currently available in the literature and thiswork does not attempt in any way to give a comprehensiveliterature review on this large topic. In particular, it must beunderlined that non-Gaussian techniques were also studiedand applied in the literature with the aim of directlysimulating wind pressure fields on structures. As examples,this problem was analyzed by Grigoriu [15, 16], by Popescu etal. [17], by Giore` et al. [18] and by Borri and Facchini [19],among others. Non-Gaussian responses of wind-excitedstructures were also studied (see, e.g., Gusella and Materazzi[20, 21]). A quite exhaustive review of all the approachescurrently available for simulating wind velocity fields andstructural responses was recently proposed by Kareem [22]also considering the nonstationary and non-Gaussian cases.

2. Stochastic Wind Modeling

Let Ox1x2x3 be the global Cartesian reference system with theorigin O lying on the ground. In the following developments,without loss of generality, it is assumed that the x3 axis isparallel to the gravity direction and the mean wind velocityU is parallel to the x1 axis. Following these definitions,


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Figure 5: Example 1 using WAWS method (black lines denote target functions, gray lines denote numeric approximations): autospectrumof along-wind velocity at point 5 (a); corresponding autocorrelation function (b); cross-spectrum between along-wind velocities at points 5and 9 (c); corresponding cross-correlation function (d).

the wind velocity field V(x, t) is idealized as the sum ofa mean value U(x3), function of the elevation from theground x3, and a stationary zero-mean fluctuation u(x, t) =(u1,u2,u3), that depends on the position x = (x1, x2, x3)and varies in time t. The three components u1, u2, u3of the vector u(x, t) represent therefore the longitudinal,lateral and vertical components of turbulence, respectively.As customary in wind engineering, the classic logarithmiclaw [4] is assumed to represent the variability of the meanwind velocity U with x3.

The atmospheric turbulence u is usually modeled as azero-mean, Gaussian, stationary random field that dependson time. If P and P denote two points located at xand x, then, from a probabilistic point of view, thecomplete characterization of this field is ensured by theknowledge of the correlation function Ruiuj (x, x

, ) forevery pair of turbulence components ui,uj , being the timelag. Assuming that u is ergodic, Ruiuj (x, x

, ) is given bythe Fourier transform of the cross power spectral density(CPSD) function Suiuj (x, x

,) between ui and uj , beingthe circular frequency. Most of the theoretical modelsadopted in wind engineering express the CPSD in terms ofauto-spectra Suiui(x,n), Sujuj (x

,n) and coherence functionCohvivj (x, x

,n), for i, j = 1, 2, 3, where n = /2 denotesthe frequency [9]. Several models were developed in the

literature to give an analytical representation to Sujuj (x,n),for j = 1, 2, 3. Here, the model by Solari and Piccardo [23]is adopted for this purpose and Cohvivj (x, x

,n) is expressedby means of the classic exponential law [11]. The imaginarypart of the CPSD (called quadrature spectrum) introduces atime lag between the simulated velocities, which may becomesignificant for points placed in the along-wind direction x1.Without loss of generality, the quadrature spectrum is hereneglected, since the considered examples focus on simulationdomains located on the x2x3 plane. This, however, does notlimit the generality of the presented results, which apply alsoto the case in which the CPSD has a nonnegative imaginarypart.

For simulation purposes, the spatial domain is dis-cretized into N points which usually represent significantnodes of the case study structure. The position of the kth

point is identified by the vector x(k) = (x(k)1 , x(k)2 , x(k)3 ),with k = 1, 2, . . . ,N . This spatial discretization allows torepresent the wind field V(k)(t) = V(x(k), t) as the sumof the mean velocity vectors U(k) = U(x(k)) and theturbulence vectors u(k)(t) = u(x(k), t) = (u(k)1 ,u(k)2 ,u(k)3 ).Following this approach, the time-dependent random fieldu(x, t) is transformed into a 3N-variate stationary randomprocess u(t), where u(t) is a 3N-order vector containingthe components u(k)1 , u

(k)2 , u

(k)3 of the turbulence vectors


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Figure 6: Example 1 using WAWS method (black lines denote target functions, gray lines denote numeric approximations): autospectrumof across-wind velocity at point 5 (a); corresponding autocorrelation function (b); cross-spectrum between across-wind velocities at points5 and 9 (c); corresponding cross-correlation function (d).

u(k)(t) for k = 1, 2, . . . ,N . The complete characterizationof the process u(t) is given by its power spectral density(PSD) matrix Su u(n) containing PSD and CPSD functionsbetween turbulent velocities. Equivalently, the two-side PSDmatrix Gu u() can also be utilized [11]. Having neglected thequadrature spectrum, both matrices Su u(n) and Gu u() arereal and positive definite.

3. Digital Simulation Formulas

Realizations of the process u(t) are generated along asequence t j for j = 1, 2, . . . ,Nt, of time instants with constanttime step t. The circular frequency domain is limited withinthe interval [c,c] where c denotes the cut-o circularfrequency. Such an interval is discretized by means of anequally spaced sequence k for k = 1, 2, . . . ,N of circularfrequencies, with step amplitude , being thereby c =N/2.

The most accurate and well-established algorithm forsimulating samples of Gaussian processes is the spectralrepresentation method, early proposed by Shinozuka and Jan[6]. This last was significantly improved by Deodatis [1] whoproposed a simulation formula that ensures the generationof ergodic sample functions and which is suitable to be

implemented using ecient FFT algorithms. The Shinozuka-Deodatis formula is based on the following frequency-dependent decomposition of Gu u(k):

Gu u(k) = T(k)T (k)T , k = 1, 2, . . . ,N, (1)where denotes the complex conjugate operator. It is worthnoting that the decomposition (1) is not unique. To thisregard, in the Shinozuka-Deodatis method the Choleskyfactorization algorithm is adopted which provides T(k) inlower triangular form. Based on this approach, the followingdigital simulation formula was obtained [1]:

ui(t j) =

i

r=1

N

k=1Tir(rk) cos

(rkt j + rk

),

i = 1, 2, . . . , 3N , j = 1, 2, . . . ,Nt,(2)

where rk = k + r/3N , Tir(rk) are the elements ofmatrix T(rk) and rkrk are independentrandom phasesuniformly distributed in the interval [0, 2]. Without loss ofgenerality, (2) is written under the assumption of neglectingthe quadrature spectrum, therefore T(rk) being a realmatrix. This method is sometimes called weighted ampli-tude waves superposition (WAWS) method. As observed


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Figure 7: Example 1 using POD method (black lines denote target functions, gray lines denote numeric approximations): autospectrum ofalong-wind velocity at point 5 (a); corresponding autocorrelation function (b); cross-spectrum between along-wind velocities at points 5and 9 (c); corresponding cross-correlation function (d).

by Di Paola [9], the central limit theorem ensures that theprocess simulated by means of (2) is asymptotically Gaussianas N becomes large. Equation (2) can be eciently imple-mented through a fast Fourier transform (FFT) algorithm,as discussed in [1].

An alternative approach to the WAWS formula is repre-sented by the POD-based technique proposed by Carassaleand Solari [11]. With such an approach the matrices T(k),in (1), are calculated as

T(k) =3N

r=1(k)r

(k)r , k = 1, 2, . . . ,N, (3)

where (k)r , for r = 1, 2, . . . , 3N , are the eigenvalues ofGu u(k) and

(k)r are the corresponding eigenvectors. These

last assume the well-known physical meaning of windblowing mode shapes [9]. The use of (3) leads to thefollowing simulation formula:

u(t j) =

3N

r=1

N

k=1ekt j

(k)r

(k)r p

(k)r , j = 1, 2, . . . ,Nt,

(4)

where is the imaginary unit and p(k)r are complex-valuedGaussian random numbers with unit variance. Likewise (2),

(4) can be eciently implemented through a fast Fouriertransform (FFT) algorithm as discussed in [11], assumingagain N = Nt and = 2/Ntt.

Equation (4) generates u(t) as the superposition of3N independent fully coherent stochastic processes, whichrepresent the contributions of the dierent wind modes. Avery convenient way for reducing both computer time andallocated memory using (4) is to calculate eigenvectors andeigenvalues of Gu u(k) only for a small number N Nof circular frequencies distributed along a reduced sequencek, for k = 1, 2, . . . , N, and then use interpolation formulaselsewhere. This approach, proposed by Carassale and Solari[11], is here adopted and briefly recalled in Appendix A.

Autoregressive methods generate the wind velocity fieldby filtering a 3N-order vector a(t) of uncorrelated band-limited Gaussian white noises aj(t), for j = 1, 2, . . . , 3N , withunit variance. The turbulent wind process is thus expressedin the following form:

u(t j) =

p

i=1iu

(t j it

)

+q

i=0Bia(t j it

), j = 1, 2, . . . ,Nt,

(5)


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Figure 8: Example 1 using POD method (black lines denote target functions, gray lines denote numeric approximations): autospectrum ofacross-wind velocity at point 5 (a); corresponding autocorrelation function (b); cross-spectrum between across-wind velocities at points 5and 9 (c); corresponding cross-correlation function (d).

where i and Bi are convenient 3N 3N matrices thatcan be easily calculated by imposing that the simulatedprocess satisfies the covariance structure of the target one[12]. Equation (5) represents a general ARMA method, inwhich p denotes the order of autoregression and q is theorder of the moving average component. As it is well known[7] an ARMA(p, q) can be approximated by an AR(p1),with p1 p, where the AR(p1) filter is readily obtainedby assuming q = 0 in (5). Regarding this point, it mustbe mentioned that AR(20) and ARMA(5, 5) provided goodresults for turbulent wind simulations in many dierentenvironmental conditions [7]. The necessary derivations forcalculating the coecient matrices of an AR(p1) filter arereported in Appendix A.

4. Efficient Implementation ofWind Simulation Methods

The three simulation algorithms presented in Section 3 areimplemented in the MATLAB R2009a [24] environment togenerate samples of along-wind and across-wind turbulencevelocities u1 and u3, respectively. Without loss of generality,the turbulent wind field is thus reduced to a 2N-variatestationary Gaussian process.

Equation (2) is implemented as the simulation formulaof the WAWS method, while (4) is adopted in the POD-based technique. Both in the WAWS and the POD methods,FFT algorithms are employed, as discussed in [1, 11], toimprove their computational eciency. An AR (p1) filter,given by (5) with q = 0, is considered to representautoregressive methods assuming a suciently large p1. Atwo-step approach is adopted in the implemented codes,following the diagram reported in Figure 1. Accordingly, aninitial phase is established (indicated as phase 0), whichmust be run only once, after which a general number n ofsimulations (Sim) can be performed (phase 1). Typically, inthe phase 0, all the data which are needed for the successivesimulations (e.g., spectral matrix, factorizations, coecients,etc.) are calculated and collected. This allows minimizingthe number of operations when simulating several windrealizations, which is usually the case for performing MonteCarlo simulations of the structural response. The results ofthe phase 0 are deterministic and must be stored in thecomputer memory.

The two main aspects that should be considered whencomparing dierent wind simulation techniques, from thecomputational point of view, are the allocated memory inthe phase 0 and the time needed for each single simulation


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Figure 9: Example 1 using AR method (black lines denote target functions, gray lines denote numeric approximations): autospectrum ofalong-wind velocity at point 5 (a); corresponding autocorrelation function (b); cross-spectrum between along-wind velocities at points 5and 9 (c); corresponding cross-correlation function (d).

Tsim in the phase 1. The former aspect is crucial sincecollecting a lot of data in the computer memory may causeoverflows and thus it may preclude the use of a certainprocedure when the dimension of the simulation domainbecomes large. The latter aspect is even more important sinceit gives the measure of the computational eort requiredby the considered algorithm. On the other hand, when theaccuracy of the dierent methods is concerned, the mostrelevant points that should be considered are the agreementbetween target and simulated characteristics of the stochasticprocess, mainly spectra and correlation functions, and peakvalues.

The three methods described above are implementedas schematically outlined in the pseudocodes provided inAppendix B. Accordingly, the main computational steps aresummarized in Table 1. The comparison between the oper-ations required by the considered algorithms immediatelyoutlines that the WAWS technique is the heaviest one fromthe computational viewpoint. On the contrary, both PODand AR techniques significantly reduce the computationaleort with respect to the WAWS method. Table 1 alsosummarizes the memory that is allocated in the phase0 in each of the three implemented codes. It is worthunderlying that the ratio between the memory allocated

in the POD-based method and the one allocated in theWAWS method is almost equal to N/N. In the nextsection it will be shown that the POD-based algorithm givessuciently accurate results with N = 50, if comparedto the corresponding WAWS method with N = 214. ThePOD-based technique allows therefore drastically reducingthe memory required by the WAWS method. This practicallyeliminates the risk of memory overflows in many technicalconditions. Nonetheless, in the most demanding cases, anAR(p1) filter can be adopted as it further reduces memoryallocation even with respect to the POD-based method.Indeed, the ratio between the memory allocated by the ARmethod and the one allocated by the POD method is equalto (p1 + 1)/N and, usually, p1 + 1 N.

5. Numerical Tests and Discussion

5.1. The Case Studies. The above described wind simulationtechniques are commonly employed for the definition of theexternal inputs for time domain response analyses of slenderstructures, as described in [2528], among others. Timedomain response analysis are essential for evaluating thestructural safety of wind sensitive structures, whose dynamic


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Figure 10: Example 1 using AR method (black lines denote target functions, gray lines denote numeric approximations): autospectrum ofacross-wind velocity at point 5 (a); corresponding autocorrelation function (b); cross-spectrum between across-wind velocities at points 5and 9 (c); corresponding cross-correlation function (d).

behavior is strongly nonlinear, for instance, in the case ofstructural cables and long-span bridges [29, 30].

Two numerical examples are considered in order toevaluate the performances of the simplified models, whentheir parameters are chosen in such a way to obtain asimilar accuracy with respect to the WAWS method. In theformer example the simulation domain is composed by 9significant points disposed along the height of the tower of asuspension bridge (see Figure 2(a)). This case is here adoptedin order to calibrate the parameters of POD-based and ARtechniques. In the latter example the simulation domain isrepresented by an entire suspension bridge. In this case, thespatial domain is composed by 83 nodes located on themean plane of the bridge (see Figure 2(b)). In both cases,the direction of the mean wind velocity is orthogonal to theplane of the simulation domain and its modulus is assignedby means of the classic logarithmic profile. In particular, amean velocity of 48.0 m/s is assumed at the top of the towerin the former example (node number 9 in Figure 2(a)) and amean velocity of 40.1 m/s is assumed at the mid-span of thebridge in the latter example (node 70 in Figure 2(b)). Theexponential decay coecients that appear in the expressionof the coherence function are assumed to be equal to those

reported in [11], while the variances 2u(k)j

of u(k)j , for j =1, 2, 3, are calculated following [23]. A nil coherence isassumed between orthogonal turbulence components u1 andu3. The following parameters are adopted in the simulations:t = 0.05 s, Nt = N = 214, = 0.00767 rad/s, c =62.8 rad/s. The simulations have been performed with the aidof a Core2 Duo Intel processor.

5.2. Calibration of POD-Based and AR Methods. Beforediscussing the performances of the simplified formulas,the parameters of POD-based and AR techniques must beproperly calibrated in order to achieve similar accuracieswith respect to the WAWS method. This task is hereaccomplished in the case of the first example. Literaturestudies [7, 11] suggest to choose, in similar cases, a reducednumber of factorizations (for POD-based technique only)N = 50 and an order of autoregression (for AR filter only)p1 = 20. After performing several tests, it was found thatsimilar values could be also adopted here, as it will be clearerin the rest of the paper by comparing the quality of theresults obtained by means of the three methods. Therefore,the values N = 50 and p1 = 20 are chosen here, thus alsoallowing a direct comparability with the results presented


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100

200

300400500

600

700800900

1000

S u(1

)1

u(7

0)1

(m2/s

)

103 102 101 100 101

n (Hz)

(e)

0

5

10

15

20

25

30

Ru

(1)

1u

(70)

1(m

2/s

2)

0 2 4 6 8 10 (s)

(f)

Figure 11: Example 2 using POD method (black lines denote target functions, gray lines denote numeric approximations): auto-spectra ofalong-wind velocities at points 1 and 70 (a), (b); corresponding autocorrelation functions (c), (d); cross-spectrum of along-wind velocitiesin points 1 and 70 (e); corresponding cross-correlation function (f).

in other literature works. As an example, a realization ofthe turbulent wind field generated by means of the WAWSmethod is shown in Figure 3.

The quality of the generated samples is analyzed inFigures 410. To this end, the turbulence velocities simulatedin two points (number 5 and 9 in Figure 2(a)) of the spatialdomain are considered. First of all, by virtue of the ergodicityof the process simulated by means of (2), it is expected thatthe WAWS method provides a perfect accuracy when oneentire period T0 = 2N 2/ (for a 2N-variate process) of

the process is simulated. This result has been confirmed hereas shown, for instance, in the results presented in Figure 4,which have been obtained by assuming Nt = T0/t andN = 214. In this case, as expected, target and computedspectra and correlation functions are overlapped (somemeaningless residual dierences are related to numeric FFTcalculations). Figures 5 and 6 refer to along-wind and across-wind (vertical) velocities, respectively, simulated by means ofthe WAWS method assuming Nt = 214, that is, Nt T0/t.The corresponding results obtained by means of POD-based


103

102

101

100

nS u

(1)

3u

(1)

3/

2 u(1

)3

103 102 101 100 101

n (Hz)

(a)

103

102

101

100

nS u

(70)

3u

(70)

3/

2 u(7

0)3

103 102 101 100 101

n (Hz)

(b)

0

1

2

3

4

5

6

7

8

9

10

Ru

(1)

3u

(1)

3(m

2/s

2)

0 2 4 6 8 10

(s)

(c)

0

1

2

3

4

5

6

7

8

9

10

Ru

(70)

3u

(70)

3(m

2/s

2)

0 2 4 6 8 10

(s)

(d)

5

0

5

10

15

20

S u(1

)3

u(7

0)3

(m2/s

)

103 102 101 100 101

n (Hz)

(e)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Ru

(1)

3u

(70)

3(m

2/s

2)

0 2 4 6 8 10

(s)

(f)

Figure 12: Example 2 using POD method (black lines denote target functions, gray lines denote numeric approximations): auto-spectraof across-wind velocities at points 1 and 70 (a), (b); corresponding autocorrelation functions (c), (d); cross-spectrum between along-windvelocities at points 1 and 70 (e); corresponding cross-correlation function (f).

and AR techniques, for the chosen parameters values, areshown in Figures 7, 8, 9, and 10. From the presented resultsit points out that the WAWS method provides a very goodmatch between target and computed spectra and correlationfunctions, even for Nt T0/t. The chosen POD-basedand AR methods are obviously less accurate than the WAWSformula, which is especially apparent by looking at the errorson auto- and cross-correlation functions. In particular, theanalysis of the results, also extended to other points of the

spatial domain, confirms that POD-based and AR techniquesproduce errors in the correlation functions, although, ingeneral, the agreement with the target functions is improvedas the coherence is decreased (as, e.g., in the across-windcomponents). Clearly, the desired levels of accuracy couldbe obtained in both cases of POD-based and AR algorithmsby choosing larger values of either the reduced number offactorizations N, adopted in the former case, or the finiteorder p1 chosen in the latter one. For the purposes of this


103

102

101

100

nS u

(1)

1u

(1)

1/

2 u(1

)1

103 102 101 100 101

n (Hz)

(a)

103

102

101

100

nS u

(70)

1u

(70)

1/

2 u(7

0)1

103 102 101 100 101

n (Hz)

(b)

0

5

10

15

20

25

30

Ru

(1)

1u

(1)

1(m

2/s

2)

0 5 10

(s)

(c)

0

10

20

30

Ru

(70)

1u

(70)

1(m

2/s

2)

0 5 10

(s)

(d)

1000

100

200

300400500

600

700800900

1000

S u(1

)1

u(7

0)1

(m2/s

)

103 102 101 100 101

n (Hz)

(e)

0

5

10

15

20

25

30

Ru

(1)

1u

(70)

1(m

2/s

2)

0 2 4 6 8 10 (s)

(f)

Figure 13: Example 2 using AR method (black lines denote target functions, gray lines denote numeric approximations): auto-spectra ofalong-wind velocities at points 1 and 70 (a), (b); corresponding autocorrelation functions (c), (d); cross-spectrum between along-windvelocities at points 1 and 70 (e); corresponding cross-correlation function (f).

study, the parameters N = 50 and p1 = 20 are consideredas good compromises between accuracy and computationaleciency. Indeed, with such choices, the accuracies of thesamples generated using POD-based and AR techniques areessentially comparable to each other and the errors withrespect to the WAWS method can be regarded as acceptablein view of a structural response analysis [28]. In any case,the comments reported in the following developments of thework could be readily extended to other possible choices ofthe models parameters.

5.3. Discussion. The results presented so far have permittedto choose the parameters of POD-based and AR methodsthat guarantee ecient simulations with an accuracy that isessentially comparable to that of the WAWS method. Thesecond example is now worth considering to better comparePOD and AR methods from the computational viewpoint.

The accuracy of the simulated samples is analyzed inFigures 11, 12, 13, and 14. To this end, to the along-windand across-wind velocities at the points number 1 (top of onetower) and 70 (bridge mid-span), indicated in Figure 2(b),


103

102

101

100

nS u

(1)

3u

(1)

3/

2 u(1

)3

103 102 101 100 101

n (Hz)

(a)

103

102

101

100

nS u

(70)

3u

(70)

3/

2 u(7

0)3

103 102 101 100 101

n (Hz)

(b)

0

1

2

3

4

5

6

7

8

9

10

Ru

(1)

3u

(1)

3(m

2/s

2)

0 2 4 6 8 10

(s)

(c)

0

1

2

3

4

5

6

7

8

9

10

Ru

(70)

3u

(70)

3(m

2/s

2)

0 2 4 6 8 10

(s)

(d)

5

0

5

10

15

20

S u(1

)3

u(7

0)3

(m2/s

)

103 102 101 100 101

n (Hz)

(e)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Ru

(1)

3u

(70)

3(m

2/s

2)

0 2 4 6 8 10

(s)

(f)

Figure 14: Example 2 using AR method (black lines denote target functions, gray lines denote numeric approximations): auto-spectra ofacross-wind velocities at points 1 and 70 (a), (b); corresponding autocorrelation functions (c), (d); cross-spectrum between along-windvelocities at points 1 and 70 (e); corresponding cross-correlation function (f).

are considered. The presented results are analogous to thoseobtained in the first example, thus indicating that the qualityof the simulated process is, as expected, independent onthe dimension of the simulation domain. In particular, theresults confirm that POD-based and AR methods, for thechosen values of N and p1, essentially guarantee similaraccuracies. The samples generated by means of the threealgorithms are also in good agreement as it concerns the peak

values u(k)1 max and u(k)3 max of u

(k)1 and u

(k)3 . As examples, some

relevant results are summarized and compared in Table 2 forthe two numerical examples.

The computer times required by the considered methodsin the two numerical examples are summarized in Table 3.These results emphasize that, as expected, POD and ARmethods are significantly more computationally ecientthan the WAWS formula. Although it is implicit that theseresults strongly depend on the environment of simulation,they still allow to derive some conclusions. Particularly,


0

20

40n(

k) r

102 101 100 102101

(rad/s)

(a)

0

5

n(

k) r

102 101 100 101

(rad/s)

(b)

Figure 15: Example 2: first ten nondimensional eigenvalues (POD) of atmospheric turbulence as a function of the circular frequency (a);detailed view (b).

1

(a)

2

(b)

3

(c)

4

(d)

5

(e)

6

(f)

Figure 16: Example 2: first 6 blowing mode shapes evaluated for the circular frequency value = 0.07 rad/s.


1

4N

orm

aliz

eder

ror

5 10 100 200

Order of outoregression

(a)

0

2

4

Nor

mal

ized

erro

r

4 10 100 1000

Reduced number of factorizations

(b)

Figure 17: Normalized average errors of simulated samples using AR filter (a) and POD-based technique (b) by increasing the order ofautoregression (AR filter) and the reduced number of factorizations (POD-based method).

Table 1: Two-steps implementation of the considered simulationmethods and corresponding memory allocation for a 2N-variateprocess (the symbol denotes proportionality).

WAWS POD AR

Phase 0

calculate2N NCholesky

matrices T(rk)of dimension2N 2N andstore 2N N

column vectorsT:,r(rk) oflength 2N

calculate andstore N

matrices ofeigenvectors (k)r

of dimension2N 2N and

2N Neigenvalues (k)r

calculate andstore p1

matrices i ofdimension

2N 2N andone matrix B0 of

dimension2N 2N

Phase 1

sumN(2N + 1)N

scalar numbersand performN(2N + 1)

inverse FFTs

perform Ninterpolations of(k)r and

(k)r ,

sumN (2N + 1)

vectors of length2N and perform2N inverse FFTs

perform(p1 + 1)Nt

multiplicationsof matrices i

and B0 byvectors of length

2N and sum(p1 + 1)Nt

vectors of length2N

Allocatedmemoryin phase 0

(2N)2N (2N)2N (2N)2(p1 + 1)

Table 2: Comparison between peak values of samples generatedusing the three simulation methods.

18-variate process 166-variate process

(example 1) (example 2)

Methodu(9)1 max(m/s)

u(9)3 max(m/s)

u(70)1 max(m/s)

u(70)3 max(m/s)

WAWS 18.9 9.5

POD 19.8 10.2 22.0 9.2

AR 18.0 9.7 21.0 9.5

the POD-based method seems to be more computationallyecient than the AR filter, both in performing prelimi-nary calculations (phase 0), which could become rele-vant when computing only a few wind simulations, andwind simulations (phase 1). Moreover, the computationalimprovement in using the POD-based technique instead of

Table 3: Computational eorts required by the three dierentmethods for digital wind simulation (T0 and Tsim denote thecomputer times necessary for performing the phase 0 and onesingle simulation in the phase 1, resp.).

18-variate process 166-variate process

(example 1) (example 2)

MethodMemory(Phase 0)

T0 (s) Tsim (s)Memory(Phase 0)

T0 (s) Tsim (s)

WAWS 11.3 MB 347.2 134.0

POD 0.1 MB 0.6 1.1 10.0 MB 5.3 8.2

AR 0.3 MB 10.8 1.7 2.2 MB 650.2 97.3

the AR filter seems to become more relevant as long as thedimension of the simulation domain becomes larger. Onthe other hand, as expected, the AR filter is the methodthat mostly reduces memory allocation, at least in the caseof high-dimensional simulation domains, which might begreatly beneficial in practical cases.

It is also worth mentioning that the eigenvalue analysisof the spectral matrix, performed in the POD-based method,reveals interesting properties of the wind field in view ofa structural analysis. In particular, as it is well-known,eigenvectors and eigenvalues of the spectral matrix representrespectively the shapes and the associated powers of the3N stochastic processes summed in (4). The eigenvalueanalysis usually reveals that only a few of these spectralmodes have large associated powers and that the eigenvectorscorresponding to the largest eigenvalues are very similar tothe fundamental mode shapes of the structure. Thus, theresponse of a slender structure is expected to be dominatedby the contribution of these waves [9].

As an example, Figure 15(a) shows the first ten eigenval-

ues (k)r (r = 1, . . . , 10) multiplied by the frequency n andplotted versus the circular frequency , in the case of thesecond example. As inferable from this figure, the first feweigenvalues are, as expected, much greater than the others. Itis also worth mentioning that, as shown in Figure 15(b), theeigenvalues of the spectral matrix are distinct and thus the

lines n(k)r versus do not cross. The turbulence eigenvectorscorresponding to the first six eigenvalues, calculated fora fixed value of the circular frequency ( = 0.07 rad/s),are shown in Figure 16. The figure evidences that the firstturbulence eigenvector involves the whole bridge in the


out-of-plane (along-wind) direction and it is very similarto the typical first structural mode shape of a suspensionbridge. Similar observations can be made for the remainingeigenvectors shown in Figure 16. It is noteworthy that hybridhorizontal/vertical eigenvectors are not detected since anil coherence has been assumed between along-wind andacross-wind turbulence components.

The presented results have indicated that the POD-basedtechnique exhibits a significant computational eciency,while the AR filter is especially convenient for preserving theallocated memory. Another important aspect that needs tobe addressed in the viewpoint of practical applications is therate of convergence of the accuracies of the AR filter and thePOD-based technique towards the target, when increasingthe relevant control parameters, that is, the order of autore-gression, p1, in the case of the AR filter, and the reducednumber of factorizations, N, in the case of the POD-basedtechnique. This aspect is analyzed here, by considering theerror, , between correlation functions of generated samples,

Rgenerated

u(i)1 u( j)1

and Rgenerated

u(i)3 u( j)3

, and the corresponding targets, Rtarget

u(i)1 u( j)1

and Rtarget

u(i)3 u( j)3

. This error is here defined as:

=N

i=1

N

j=i

Rgenerated

u(i)1 u( j)1

Rtargetu(i)1 u

( j)1

+N

i=1

N

j=i

Rgenerated

u(i)3 u( j)3

Rtargetu(i)3 u

( j)3

(6)

and calculated in the time lag interval 010 sec. The errordefined in (6) is realization dependent and, so, an averagingof among a conveniently large number of generations Ngenis sought, where Ngen is here taken as 200.

Following the previous definitions, Figure 17 shows plotsof the average errors versus p1 and N, for the AR filterand the POD-based technique, in the case of the firstnumerical example. For clarity of presentation, the resultsin Figure 17 are normalized with respect to the residualerror of the WAWS method with Nt = N = 214.The results presented in Figure 17 confirm the expectedtrends that the errors decrease with increasing p1 and N.Moreover, they show that, in the case of the POD-basedtechnique, the rate of convergence appears to be very fastfor small values of N. Namely, already for N = 24 thesystem reaches an accuracy which is only slightly worse thanthat of the WAWS method with Nt = N = 214 (thenormalized average error is about equal to 1.3), while forlarger values of N the convergence rate becomes smallerwith some erratic fluctuations of less than 5%. This resultindicates that the interpolation expressions of eigenvectorsand eigenvalues along the unequally spaced sequence kof circular frequencies, proposed by Carassale and Solari[11] and recalled in Appendix A, is very accurate whenapplied in the case of turbulence characteristics similar tothose considered here. The results presented in Figure 17might also suggest to choose values of N that are smallerthan the one adopted by Carassale and Solari [11] in asimilar application and chosen in the numerical examplespresented above (N = 50), without substantially decreasingthe resulting accuracy. Nonetheless, calculations performed

aside have shown that the average error in predicting peakvalues, calculated with respect to the WAWS method andconsidering 200 wind generations, rapidly decreases for Nup to about 50, where it is almost equal to 0.5%, whileremaining essentially constant for larger values of N. Thisresult shows that, in the present case, choosing N = 50 isconvenient in terms of accuracy in predicting peak values.

In the case of the AR filter, the results presented inFigure 17 show that for large values of p1 the error is stillslightly larger than the error of the WAWS method withNt = N = 214 (the ratio between the two is about 2.5).On this respect, a value of p1 equal to 200 can be regardedas large if compared to the value p1 = 20 recommended inthe literature in similar conditions [7] and adopted in thenumerical examples presented above. Looking at the resultspresented in Figure 17, the ratio between the average error ofthe AR filter and the residual error of the WAWS method,for p1 = 20, is about equal to 3.2. Considering the highaccuracy of the WAWS method, this result can be regardedas acceptable in many practical applications. It is also worthmentioning that, for p1 = 20, the average error in predictingpeak values is also small and approximately equal to 2.5%.

It is important to note that the results presented inFigure 17 do not allow to derive conclusions on the relativeaccuracy of the two simplified methods one with respectto the other. Indeed, for a fair and complete comparisonbetween the two methods, dierent turbulence characteris-tics should be considered, which, however, goes beyond thepurposes of the present investigation.

6. Conclusions

Simulating the wind velocity field in the case of high-dimensional domains may become a very demandingcomputational task due to memory occupation and timeconsuming simulations. In the present paper, a properlyorganized way for implementing wind simulation methodsis presented at first. Then, the widely adopted method usingwaves superposition (WAWS method) is taken as a referencefor calibrating the parameters of two well-known simplifiedmethods (POD-based and AR formulas). These last are thencompared, by devoting a special care to algorithm structureand computational expense.

Two numerical examples, with increasing complexity, areconsidered in the comparative study. The results indicatethat POD-based and AR techniques significantly reduce bothcomputer time and memory allocation with respect to theWAWS method, at the expense of smaller accuracies. ThePOD-based method appears to be more computationallyecient than the AR filter when the relevant parametersaecting their results are tuned in such a way to achievesimilar accuracies. On the other hand, the AR filter is greatlypreferable in terms of reduction of allocated memory.

Concerning the accuracy of the simulated samples, theWAWS method is known to guarantee the ergodicity ofthe generations. As shown in the paper, this means thatif an entire period of the process is simulated, the matchbetween target and simulated samples is perfect. Even when


considering shorter simulations, the accuracy of the WAWSmethod is seen to be very high, namely the dierencesbetween target and computed correlation functions are, forthe considered parameters values, almost unnoticeable. Theresults presented in this paper also confirm that, althoughless accurate, both POD-based and AR techniques canprovide good results if the models parameters are properlychosen. It must be also mentioned that, regardless the chosensimulation formula, the POD decomposition of the spectralmatrix is always a worth task to be tackled, as it providesuseful information about the stochastic wind process in viewof a structural analysis. Finally, the rates of convergence of theaccuracies of the two simplified approaches with increasingtuning parameters have also been investigated in order togive preliminary indications for choosing these parametersin practical cases.

Appendices

A. Additional Details onthe Adopted Simulation Formulas

The rules proposed by Carassale and Solari [11] for reducingthe number of eigenvectors and eigenvalues of matrixGu u(k) allocated in the computer memory, are described, atfirst, in this section. Then, the derivations for calculating thecoecient matrices of an AR (p1) filter are briefly recalled.

Concerning the POD-based technique, the followingsequence of circular frequencies must be introduced:

h = 2Ntt

(Nt2

)h1/N1,(h = 1, . . . , N

). (A.1)

Eigenvectors and eigenvalues of matrix Gu u, calculated forthe circular frequency values reported in (A.1), are denoted

by (h)r and (h)r , respectively. Then, by defining the sequence

sk:

sk = ln(k 1)ln(Nt/2)(N 1

)+ 1

(k = 2, . . . , Nt

2+ 1)

(A.2)

the eigenvalues (k)r of matrix Gu u(k) are interpolated as:

(k)r = (sk round+(sk))(round(sk))r

+(1 sk + round+(sk)

)(round

+(sk))r

(k = 2, . . . , Nt

2+ 1)

,

(k)r = (Nt+2k)r(k = Nt

2+ 2, . . . ,Nt

),

(A.3)

where round+() and round() provide the upper and lowerinteger rounds of the argument respectively. Finally, theeigenvectors are approximated stepwise as:

(k)r =

(round(sk))

r

(k = 2, . . . , Nt

2+ 1)

,

(Nt+1k)r

(k = Nt

2+ 2, . . . ,Nt

),

(A.4)

where round() returns the closest integer to the argument.The simulation formula of an AR(p1) filter is obtained,

by assuming q = 0 in (5), as:

u(t j)=

p

i=1iu

(t j it

)+ B0a

(t j). (A.5)

In order to calculate the coecient matrices i and B0, thecorrelation matrix Ru u() must be defined as:

Ru u() = E[

u(t)uT(t + )]. (A.6)

Assuming the shorter notation Ru u(kt) = Ru u(k), the fol-lowing hyper-system of algebraic equations can be obtained[12]:

K[1 2 p1

]T = b,

K =

Ru u(0) Ru u(1) Ru u(p 1)

Ru u(1) Ru u(0) Ru u(p 2)

......

. . ....

Ru u(1 p) Ru u

(2 p) Ru u(0)

b =[Ru u(1) Ru u(2) Ru u

(p)]T

(A.7)

Once the correlation matrices are calculated by means ofinverse FFT algorithms applied to CPSD functions, thecoecient matrices i can be easily determined by solvingthe algebraic hyper-system (A.7). Then it is possible tocalculate matrix B0, by post-multiplying (A.1) by u(t j)

T andtaking the average, namely:

E[

u(t j)

uT(t j)]=

p

i=1iE

[u(t j it

)uT(t j)]

+ B0E[

a(t j)

uT(t j)]

.

(A.8)

By definition of correlation matrix, (A.8) may be rewritten inthe following form:

Ru u(0) =p

i=1iRu u(i) + B0Ra u(0). (A.9)

From (A.9), it follows that the covariance structure of theprocess is preserved if the following condition is satisfied:

B0Ra u(0) = Ru u(0)p

i=1iRu u(i). (A.10)

Thus, the choice of matrix B0 is not unique and a possiblestrategy is to assume B0 = Ra u(0)T and to obtain it throughthe Cholesky decomposition.

B. Implemented Algorithms

The wind simulation algorithms implemented in this studyare schematically outlined in this section, for a 2N-variateprocess. The simulated wind field is denoted by the matrixvariable field u. Variables definitions and adopted functionsare implicit from the presented algorithms.


B.1. WAWS Algorithm

Step 0.

initialize wind simulation parameters

for r = 1 : 2Nfor k = 1 : Ncalculate r(k) as a function of r and k

for i = 1 : 2Nfor j = i : 2N

G(i, j) = Guuij (r(k))end

end

matrG = G + upper triangular part(G)T

T = Cholesky decomposition (matrG)T

matrT(:, k, r) = T(:, r)end

end

save wind simulation parameters

save matrT .

Step 1.

load wind simulation parameters

load matrT

field u = zero matrix(2N ,Nt)time = [t, 2t, . . . ,Ntt]phi = 2 uniformly distributed random numbers

(2N ,N)

for i = 1 : 2Nfor r = 1 : i

for k = 1 : NB rk(k) = matrT(i, k, r) exp( phi(r, k))

end

add field u = N Inverse FFT(B rk)for j = 1 : Nt/Nfield u(i, ( j 1) N + 1 : j N)= field u(i, ( j 1) N + 1 : j N)+ real(add field u. exp( r/2N

time(( j 1) N + 1 : j N) ))end

end

end

save field u.

B.2. POD-Based Algorithm

Step 0.

initialize wind simulation parameters

for k = 1 : Nfor i = 1 : 2N

for j = i : 2NG(i, j) = Guuij (k)

end

end

matrG hat = G + upper triangular part(G)TmatrPSI hat(:, :, k)

= matrix eigenvectors(matrG hat)matrGAMMA hat(:, k)

= eigenvalues(matrG hat)end

save wind simulation parameters

save matrPSI hat and matrGAMMA hat

Step 1.


load matrPSI hat and matrGAMMA hat

field u = zero matrix(2N ,Nt)R = 1/2 normally distributed random numbers

(N, 2N)

I = 1/2 normally distributed random numbers(N, 2N)

time = [t, 2t, . . . ,Ntt]for k = 1 : N

if k Nt/2 + 1sk = log(k 1)/ log(Nt/2) (N 1) + 1PSI k = matrPSI hat(:, :, round(sk))GAMMA k = (skfloor(sk))matrGAMMA hat

(:, :, ceil(sk))

+(1 sk + floor(sk)) matrGAMMA hat(:, :,floor(sk))

end

if k > Nt/2 + 1

sk2 = log(N + 1 k)/ log(Nt/2) (N 1) + 1PSI k = matrPSI hat(:, :, round(sk2))GAMMA k = (sk2 floor(sk2))matrGAMMA hat(:, :, ceil(sk2))+(1 sk2 + floor(sk2)) matrGAMMA hat(:, :,floor(sk2))

end


add field u = zero matrix(2N , 1)for r = 1 : 2N

add field u = add field u+ GAMMA k(r, 1)

PSI k(:, r) (R(k, r) +1 I(k, r))end

field u(:, k) = field u(:, k) + add field uend

for r = 1 : 2Nfield u(r, :) = N Inverse FFT(field u(r, :))end

field u = real(field u)save field u

B.3. AR Algorithm

Step 0.

for i = 1 : 2Nfor j = i : 2N

for k = N/2 + 1 : NmatrS(i, j, k N/2) = Suuij (k)

end

matrR(i, j, :) = Inverse FFT(matrS(i, j, :))end

end

for j = 1 : p1 + 1matrR(:, :, j) = matrR(:, :, j)+upper triangular part(matrR(:, :, j))T

end

arrange matrK = K and matrb = b from matrRsolve hyper-system (A.7) matr phi = matrK \matrbB0 = matrR(:, :, 1)for i = 1 : p1

phi(:, :, i) = matr phi(2N(i 1) + 1 : 2Ni, :)TB0 = B0 phi(:, :, i) matrR(:, :, i + 1)

end

B0 = Cholesky decompostion(B0)Tsave phi and B0

Step 1.


load phi and B0

a = normally distributed random numbers(2N ,Nt + p1)

field u = zero matrix(2N ,Nt)

for j = p1 + 1 : Nt + p1for i = 1 : p1

field u(:, j) = field u(:, j)+phi(:, :, i) field u(:, j i)

end

field u(:, j) = field u(:, j) + B0 a(:, j)end

field u = field u(:, p1 + 1 : Nt + p1)save field u.

References

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Computer Simulation of Stochastic Wind

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