A conditional simulation of non-normal velocity/pressure fields

Transcription

1 A conditional simulation of non-normal velocity/ fields K. R. urley, A. Kareem Department of Civil Engineering University of Florida, ainesville, FL, 36 Department of Civil Engineering and eological Sciences University of Notre Dame, Notre Dame, IN, Abstract he simulation of random velocity and signals at un-instrumented locations of a structure conditioned on measured records is often needed in wind engineering. Malfunctioning equipment may leave a hole in a wind data set or information may be lacking due to a limited number of instruments or difficulty in monitoring at certain locations. his paper presents a non-aussian conditional simulation technique in the context of non- aussian fluctuations in separated flow regions, or velocity fluctuations in atmospheric flows. First, a uni-variate non-aussian simulation technique is developed. his is extended to multi-variate, and subsequently utilized as a mapping technique in a conditional simulation algorithm. his technique is applied to both the extension of existing records beyond recorded lengths, and the simulation of missing or damaged records based on measured data at other locations.. Introduction Simulation methods are becoming very attractive for the analysis and prediction of nonlinear system response as computational expense decreases with increasing computer hardware speed. Implementation of time domain methods require simulated load time histories with specific statistical and spectral characteristics. Many of the studies encompassing analysis and modeling of wind effects on structures have tacitly assumed that the involved random processes are aussian. his assumption is valid for integrated load effects of the random field over large areas. However, regions under separated flows experience strong non-aussian effects in the distribution characterized by high skewness and kurtosis, and may be highly correlated over the entire separation region. he non-aussian effects lead to non-aussian local loads, and result in increased expected damage in glass panels and higher fatigue effects on other components of cladding. When the assumption of aussian wind loading is inappropriate, techniques for simulating non-aussian loading must be sought. his work introduces multi-variate non-aussian conditional simulation methods capable of producing realizations with a wide range of spectral and probabilistic characteristics conditional on measured records at some locations. he correlation between multiple locations is accurately simulated, while maintaining the appropriate spectral and probabilistic content of the processes at each location.. Background: Modified Hermite ransformation An earlier study [] presents the development of the so-called forward and backward Modified Hermite ransformations. Forward transformation produces a non-aussian process, x, through static transformation of a aussian process, u, using [] x u+ d 3 ( u ) + d 4 ( u 3 3u) ()

2 he forward Modified Hermite ransform identifies the optimum values for the polynomial coefficients and d 4 such that x matches the desired values of skewness and kurtosis. he backward Modified Hermite ransform identifies the appropriate coefficients in the inverse of Eq. necessary to produce a aussian process, u, from a non-aussian process, x. he forward and backward Modified Hermite ransformations are applied in a non- aussian simulation algorithm in the next section. 3. Spectral Correction: A non-aussian simulation technique Spectral Correction is a robust new non-aussian simulation method which utilizes user specified non- aussian characteristics and frequency content in the form of target moments through fourth order and a target power spectrum. Alternatively, the user may define non-aussian characteristics through a marginal PDF model. A schematic of the method is shown in Fig.. An iterative application of corrective transformations provides convergence to the desired spectral and probabilistic characteristics. Following from the top left in the schematic, the target spectrum is applied to produce a aussian process u using a standard aussian simulation routine [e.g. 3, 4]. u is sent through a moment correction transformation, consisting of a forward Modified Hermite ransformation to yield a process x which matches the desired skewness and kurtosis ( u x ). his non-aussian process x has a power spectrum x which no longer matches the target spectrum due to distortions in the transformation. he process x is sent through a spectral correction to produce x c which fits the target spectrum, and maintains the phase in x ( x x c ). he distorted skewness and kurtosis of are compared with the target values using err ( γ 3 ) γ 3 γ 3 x c + ( γ 4 ) γ 4 γ 4, () where, are the target, and γ 3, γ 4 are the measured skewness and kurtosis from x c. If the error, err, is unacceptably large, the second iteration begins by sending the process with the correct spectrum and distorted skewness and kurtosis, x c, back to the top of the loop to the moment correction. Here the skewness and kurtosis are again corrected using the backward and then forward Modified Hermite ransform to produce the second iteration of x ( x c x ). he iterations continue, checking the error, err, after each spectral correction transformation, until the distorted moments of x c converge to the target moments within a set tolerance. he spectrum of the resulting simulated process matches the target, and the skewness and kurtosis will be within user specified tolerance. ypically two or three iterations are required for convergence [5]. γ 3 γ 4 he Modified Hermite ransform may be replaced with a CDF-type mapping when the marginal PDF model associated with the Modified Hermite ransform is inappropriate [6]. his provides the option of describing the non-aussian characteristics of the desired process with either the first four moments, or a marginal PDF model. 3.. Example: simulated non-aussian suction on a rooftop his example uses full scale data measured on the roof of an instrumented building at exas ech University. he Spectral Correction algorithm is used to simulate measured samples of data in a separated flow region. he left side of Fig. shows a comparison of the measured data sample and a single simulation realization. he realization is generated using a target spectrum estimated from the data and the first four measured moments. Note the similarities between measured and simulated samples including strong negative skewness, dominant low frequency content, frequently clustered extreme peaks, and a zero rate of crossing into positive. Similarities are also observed through comparisons of the power spectral density and probability density function estimated from the measured data and simulated data in the right side of Fig.. he Spectral Correction simulation method closely emulates these important characteristics in the measured non-aussian data. d 3

3 aussian Simulation u target skewness and kurtosis γ 3, γ 4 Output YES x c Back ransformation Forward ransformation x FF x X Set X IFF X c X X c x c x c Moments target? moment correction spectral correction x c NO Fig.. Schematic of the Spectral Correction non-aussian simulation method measured data simulated data PSD power spectral density measured data spectrum simulated data spectrum frequency (Hz) probability density function.5 PDF measured data PDF simulated data PDF time (sec) 7 5 Fig.. Left: Measured and simulated data. Right: Power spectral density and probability density function of measured and simulated data 4. Multi-variate Spectral Correction Spectral Correction is extended to simulate multiple correlated non-aussian time histories. Multi-variate simulation becomes necessary for cases where non-aussian processes at spatially distributed locations are desired. For example, simulation of along the roof edge or along the corner region of a building, where under separated flow can be well correlated across the entire length. he uni-variate algorithm in the previous section begins with a single aussian simulation, and iteratively corrects the power spectrum and PDF of the simulated realization. he multi-variate Spectral Correction simulation algorithm starts with a multi-variate aussian simulation, applies iterative corrections to the power spectrum and PDF of each process, and additionally updates the cross-spectrum used to simulate the initial aus-

4 sian processes. he iterative corrections to the cross-spectrum result in a convergence to the desired correlation between the resulting simulated non-aussian processes. he correlation updating scheme operates on the inverse Fourier transform of the cross-spectrum, the cross correlation function, R D ij ( τ ). Pairs of simulated processes whose cross-correlation matches the target, x R N ij ( τ ) R ij ( τ ), will reflect the desired linear correlation between the i th and j th processes. A schematic representation of the multi-variate simulation algorithm is given in Fig. 3. Beginning at the top left, the target auto-spectrum, ii, and target skewness and kurtosis, γ 3, γ 4, of each process is input to the algorithm, along with the target cross-correlation function between each pair of processes, R ij. A standard multivariate aussian simulation algorithm [e.g. 7, 8] produces a set of aussian processes with the design cross-correlation function, R ij, and target auto-spectrum, ii. his design cross-correlation is initially set to the target D D cross-correlation function for the first iteration ( R ij R ij ). Each aussian process is then transformed to noni i aussian using Spectral Correction ( x x N ). he cross-correlation between these non-aussian processes x are measured, R N ij, and compared with the target values, R ij. If the error is too large, a second iteration is i begun, otherwise the set of processes x N are accepted as the output. wo or three iterations provides convergence to the target cross-correlation function. At the end of an iteration, the design cross-correlation function is updated for the next iteration using R D ij ( it + ) ρ D ρ D x ij ( it) ij ( it) [ ρ ij ρ N ij ( it) ] ( max ( Rij ) ), over all τ, (3) max( ρ D ij ( it) ) using the following, with the function of iteration notation ( it) replaced by the function of time lag ( τ ) notation ρ ij D D ( τ ) ( R ij ( τ ) ) ( max( R ij ( τ ) ) ), ρ ij ( τ ) ( R ij ( τ ) ) ( max( R ij ( τ ) ) ) ρ ij x N x N ( τ ) ( R ij ( τ ) ) ( max( R ij ( τ ) ) ). (4) it ii R D ij ( ) R ij, R ij aussian Multivariate Simulation ii, R D ij ( it) target skewness and kurtosis γ 3, γ 4 i...,, num j...,, num num # of variables or locations Iterative Loop it it + x i Spectral Correction i i x x N i...,, num i x N, x R N ij x R N ij? R ij No update using R ij D D ( it + ) R ij see Eq. 3 R ij, x N ( it), R ij i x N Yes: exit Fig. 3. Schematic of the Multivariate Spectral Correction Simulation method

5 4.. Example: rooftop at three locations his example again uses the full scale data, measured simultaneously at several locations on the roof of an instrumented building. We consider three locations on the roof in a separation zone where the is highly non-aussian. he goal is to generate realizations of three records which maintain the measured cross-spectral density, and properly reflect the higher order statistics and spectral contents at each location. Figure 4 is a diagram of the approximate locations of the taps. he offset leads to a non-zero phase relationship between locations. Figure 4 also compares the target and simulated cross-spectral density function and cross-correlation between locations and. he match between target and simulated cross-spectra is excellent. Figure 5 shows the three measured records, and their simulations using non-aussian multi-variate Spectral Correction. he low frequency correlation can be seen in both figures, along with the distinct non-aussian characteristics of the measured and simulated records. 5. Non-aussian conditional simulation he formulation for aussian conditional simulation is first presented. Extensions are then made to non- aussian conditional simulation using the Spectral Correction algorithm. 5.. aussian conditional simulation Consider a pair of correlated aussian random vectors V k and V u. Let the two variate normal distribution of these variables be denoted pv ( ) p V k N µ k C kk, V u µ u C uk C uu, (5) where µ i is the mean value of the variable i, and C ij is the cross-covariance between the variables i and j. If a sample of V k is measured and denoted as v k, then it is the conditional simulation of V u based on the measured record v k that is desired. his problem has been addressed in several forms [e.g. 9,,, ], each of which leads to an equivalent final form for the conditional PDF of V u given the informatio on V k, wind direction 3.5 ft ft freq. 4 cross correlation R phase() freq. rooftop.3.. data simulation. 4 6 Fig. 4. Left figure: Location of taps. Right figure: Comparisons of measured and simulated correlation statistics. op left: Absolute value of the cross-spectrum between locations and, op right: phase of the cross-spectrum between locations and Bottom: cross-correlation function

6 location measured location simulated location measured location simulated location 3 measured location 3 simulated time (sec) time (sec) time (sec) Fig. 5. Left: Measured rooftop at the three locations in Fig. 4. Right: Simulated rooftop at the three locations in Fig. 4. C kk C kk pv ( u V k v k ) Nµ ( u + ( v k µ k ), C uu ) A conditional simulation is then provided by [e.g. 9,, ]. (6) ( V u V k v k ) C kk ( v k V k ) + V u. (7) where V k, V u are unconditionally simulated variates, and v k is the known measured variate. Derivations of the covariance matrices and provide the information needed for aussian conditional simulation. C kk 5.. Frequency domain aussian conditional simulation Frequency domain conditional simulation is applicable for generating realizations of time series at unmeasured locations n...n based on measured records at other spatial locations...n. It is assumed that the spectral density matrix between the n known and the N n + unknown locations is available. C kk he covariance matrices and are represented in the frequency domain as... n, n, n, +... N, C... n, kk, n, n, +... N, (8) : : : : : : : : n, n,... n, n n, n n n +,... n he sub-matrices in Eq. 8 are composed of elements of the cross-spectral matrix between locations i and j., N

7 5.3. ime domain aussian conditional simulation Conditional simulation in the time domain is applied to extend the length of a measured process. he measured realization v k () t consists of discrete time components t...n, and the unknown realization v u () t is a continuation of the known realization over time t n...n. he simulated realizations V k, V u in Eq. 7 now consist of a simulation v...n ( ) based on the measured realization v k (...n ) that extends to time t N. his single simulated realization is partitioned into the know and unknown portions v...n ( ) [ v k (...n ) ( V u ( n...n)v k v k ) ]. (9) he covariance matrix in Eq. 7 consists of elements from the covariance vector c c c... c n c n c n + c n +... c N C c c c... c n kk, c n c n c n +... c N () : : : : : : : : : : c n c n c n 3... c c c c 3... c N n + where c i are elements of the cross-covariance vector from either the Fourier transform of the measured power spectrum or a temporal calculation Extension to non-aussian conditional simulation Conditional simulation using Eq. 7 requires that the variates V k, V u, v k be aussian. he conditionally simulated variate V u will then also be aussian. In order to conditionally simulate non-aussian records, eq. 7 is applied to an initial set of aussian processes, and the result is mapped into the desired non-aussian domain, using Spectral Correction. he correlation between the simulation at the unknown location and the measured records at other locations is used to iteratively update the cross-correlation matrix used to generate the initial set of aussian processes. A previous non-aussian conditional simulation algorithm [3] uses correlation distortion based transformations for the mapping, and is thus limited to simulation of wide banded processes. he use of Spectral Correction here is appropriate for wide banded processes, and additionally permits the conditional simulation of more narrow banded processes. Figure 6 is a schematic of the non-aussian frequency domain conditional simulation algorithm. he algorithm begins with the known non-aussian processes v k, the measured covariance matrix C kk, the target spectra of the unknown processes uu, the target skewness and kurtosis γ 3, γ 4, and the target cross-covariance matrix between the known and unknown processes. he design cross-covariance is set to the target crosscovariance for the first iteration. aussian simulations of V k, V u, denoted V D k, V u are generated using the input C kk C D, ku, uu, and a standard multivariate aussian simulation algorithm. A Modified Hermite ransform is applied to v k to produce a aussian variate v k. Equation 7 is then applied to produce a aussian condi- tional simulation at the unknown locations, denoted v u. he aussian process, v u, is sent through the Spectral Correction transformation to produce the non-aussian simulation v u with target skewness and kurtosis, and autospectra uu. he cross-covariance,, between v k and v u are measured and compared with the target cross-covariance. If the error between the measured and target covariance is not acceptable, the design crosscovariance,, used to generate V D k, V u is updated for use in the next iteration.

8 Inputs: v k, C kk,, uu,, C D ku it γ 3 γ 4 Iterative Loop multi-variate aussian simulation of V k, using C kk C D ku,, V u uu transform measured v k to aussian v k v k using modified Hermite it it + aussian conditional simulation of v u using V k V, u, C kk, trans. to non-aussian v u Eq.7 v u? No update D using Spectral Correction Yes output v u Fig. 6. Schematic of the conditional simulation algorithm using Spectral Correction Example: simulation of on building rooftop at a fourth locations using three measured locations he statistics of measured non-aussian roof suction records are used in conjunction with a standard wind velocity coherence model to generate four spatially separated non-aussian records. he locations are assumed to be 4 meters apart perpendicular to the wind direction, and no separation parallel to the wind direction. he skewness and kurtosis of each process is -.5, 5.5, respectively. he fourth location is assumed to be damaged, and the conditional simulation algorithm presented above is used to replace it based on knowledge of the other three records. he left side of Figure 7 is a view of the three known records and the simulated fourth record. Note the strong low frequency correlation among all four records. he right side of Figure 7 shows the target and measured coherence between locations and 4, and 4, and 3 and 4. he match demonstrates that the simulation is a suitable replacement for the missing record. he far right top plot in Fig. 7 is a comparison of the PDF of the missing recorded and its replacement. he use of Spectral Correction in the conditional simulation algorithm ensures excellent agreement between target and simulated marginal distribution and auto-spectrum Example: extension of a single non-aussian record An example of the time domain non-aussian conditional simulation method is shown in Fig. 8, where measured rooftop suction is conditionally simulated. he measured data consists of τ 5 points, and the total desired length is 4 points. A close up of the joining region is shown to demonstrate smooth transition from measured to simulated data. he upper right plot shows good comparison between the measured and extended marginal PDF, and the lower right plot compares the PSD of the measured and extended records. his example demonstrates the applicability of the time domain non-aussian conditional simulation algorithm to extend existing records while maintaining spectral and probabilistic characteristics.

9 measured location measured location measured location conditionally simulated location time (sec) coherence coherence Fig. 7. Left figure: hree known records and a conditionally simulated record at location 4. Right figure: Comparison of the target and simulated coherence functions between the simulated and measured locations. op right: PDF of the record at location 4 that is assumed missing (denoted measured), and histogram of its replacement using non-aussian conditional simulation. coh(,4) coh(,4) PDF coherence coh(,3) PDF of missing record at location 4 PDF of simulated record at location 4 coh(3,4) target coh simulation coh Non aussian measured data simulation PDF close up of joined region measured record conditionally simulated record 5 data simulated PSD Fig. 8. Left: Measured rooftop data and a non-aussian conditional simulation. Right: PDF and PSD of measured and full extended record.

10 6. Conclusions he non-aussian multi-variate and conditional spectral correction simulation algorithms are shown to be effective for generating realizations of time histories, at one or more spatial locations, which match target characteristics given in the form of an auto-spectrum, a marginal probability density function, and cross-correlation functions between multiple processes. Spectral Correction is extended to the conditional simulation of non- aussian processes to replace missing or damaged records. hese algorithms provide additional tools for generating input to Monte Carlo-based simulation methods of structural reliability analysis, particularly for cases where environmental loading differs significantly from aussian. 7. Acknowledgments he support for this work was provided in part by NSF rants CMS9496, and CMS , ONR rant N , and a grant from Lockheed-Martin. he first author was partially supported by a Department of Education AANNP Fellowship during this study. Experimental data was kindly provided by exas ech University. 8. References [] K. urley, A. Kareem and M. ognarelli, Simulation of a class of non-normal random processes, Int J. Non-Linear Mech., 3(5) (996) [] S.R. Winterstein, Nonlinear vibration models for extremes and fatigue, J. Eng. Mech., ASCE, 4() (988) [3] M. rigoriu, On the spectral representation method in simulation, Prob. Eng. Mech., 8, (993) [4] M. Shinozuka and C.M. Jan, Digital simulation of random processes and its applications, J. Snd. Vib., 5(), (97) -8. [5] K. urley and A. Kareem, Analysis, interpretation, modelling and simulation of unsteady wind and data, to appear in the J. Wind Eng. Ind. Aerodyn. (997). [6] M. rigoriu, Crossing of non-aussian translation processes, J. Eng. Mech., ASCE, (EM4) (984) 6-6. [7] M. Shinozuka, Simulation of multivariate and multidimensional random processes, J. of Acoust. Soc. Am., 49 (97) [8] L.E. Wittig, and A.K. Sinha, Simulation of multicorrelated random processes using the FF algorithm, J. Acoust. Soc. Am., 58(3) (975) [9] L.E. Borgman, Irregular Ocean Waves: Kinematics and Forces. he Sea, Prentice Hall (99). [] M. Hoshiya, Kriging and Conditional Simulation of a aussian Field, J. Eng. Mech, ASCE, (), (995) [] H. Kameda and H. Morikawa, An Interpolating Stochastic Process for Simulation of Conditional Random Fields, Prob. Eng. Mech. 7(4) (99) [] D.. Krige, wo-dimensional Weighted Moving Average rend Surfaces for Ore Evaluation, J. South African Inst. of Mining and Metallurgy, Proc. Symposium on Mathematical Statistics and Computer Applications for Ore Evaluation, Johannesburg, South Africa, (966) [3] I. Elishakoff, Y.J. Ren, and M. Shinozuka, M., Conditional Simulation of Non-aussian Random Fields, Eng. Struct., 6(7) (994)