Extrapolation reconstruction of wind pressure fields on the claddings of high-rise buildings

Abstract

Recent research about reconstruction methods mainly used the interpolation reconstruction of the fluctuating wind pressure field on the surface. However, to investigate wind pressure at the edge of the building, the work presented in this paper focuses on the extrapolation reconstruction of wind pressure fields. Here, we propose an improved proper orthogonal decomposition (POD) and Kriging method with a von Kármán correlation function to resolve this issue. The studies show that it works well for not only interpolation reconstruction but also extrapolation reconstruction. The proposed method does require determination of the Hurst exponent and other parameters analysed from the original data. Hence, the fluctuating wind fields have been characterized by the von Kármán correlation function, as an a priori function. Compared with the cubic spline method and different variogram, preliminary results suggest less time consumption and high efficiency in extrapolation reconstruction at the edge.

Keywords

extrapolation reconstruction / proper orthogonal decomposition / Kriging method / von Kármán function / Hurst exponent / rescaled range analysis

Introduction

With improvements in the understanding of the response mechanisms of structures to wind action, the phenomenon of the overall destruction of a building due to strong wind action is rare; in contrast, damage to local enclosures frequently occurs, e.g., the curtain wall of a high-rise building falling off in Fig. 1 or the eaves and ridges on a large-span roof experiencing serious damage. Hence, the related social impact and economic losses cannot be ignored. Since 1970, scholars have shown that the positive wind pressure areas on a high-rise building surface follow a Gaussian distribution, whereas the negative pressure region has a certain skewness. The most unfavourable wind pressure coefficients on the surfaces of typical high-rise buildings with rectangular cross-sections have been studied based on scale models of wind tunnel test [¹]. Thus, a series of non-Gaussian studies on building wind pressure were conducted. In the air separation area, the probability distribution of the wind pressure has obvious non-Gaussian properties, with the peak value of wind pressure predicted by the Gaussian distribution in the air separation zone being much lower than the actual one, especially at the edge, eaves and ridge. Thus, the wind pressure field around the building is one of the controlling factors in structural analysis and must be first determined. Wind pressure prediction on a building surface is mainly conducted by means of an atmospheric boundary layer wind tunnel test. To meet the requirement of the similarity ratio of the turbulence integral length scale, the scale ratio of 1:800‒1:300 is often chosen in aero-elastic models. As a result, the number and position of wind pressure taps at the edge, eaves and ridge of the test model are distinctly limited. The reconstruction of wind pressure fields on the surface is practical and necessary. Many methods of reconstruction have been proposed in recent decades, and their applications have been successful in structural wind resistance analysis [²–⁶]. However, few literature studies have considered the fluctuating extrapolation in the edge, eaves, or ridge region.

The research field of wind pressure prediction can be mainly divided into two categories: 1) artificial intelligence techniques can be used to reconstruct wind pressure distribution characteristics [⁷–⁹], but such models lack a clear physical mechanism and depend on a massive amount of sample data, and 2) proper orthogonal decomposition (POD), which was introduced by Armitt [¹⁰] and Lumley to address turbulence flows and wind-related problems [¹¹–¹³] and was later developed to obtain the fluctuating wind pressure on a building surface or structure, such as a square prism [¹⁴] and low-rise buildings [¹⁵,¹⁶]. Predictions of wind pressure fluctuations have been successfully employed for the performance evaluation and response prediction of high-rise buildings [¹⁷] and the wind-induced dynamic response in a single-layer latticed dome [¹⁸].

The fundamental implication of POD is space-time separation technology, which have been effectively implemented in random field problems. Hence, the covariance of the fluctuating wind pressure measurements can be decomposed into eigenvectors related to space and principal coordinates related to time. Consequently, the spatial distribution of the wind pressure field can be characterized by the spatial interpolation of the eigenvector corresponding to the measured taps. Combined with time-dependent principal coordinates, the fluctuating wind pressure field over the whole surface can be reconstructed [¹⁹].

Generally, there are deterministic and geostatistics methods among the spatial interpolation methods of eigenvectors. In deterministic methods, information similarity between different taps and the smoothness over the whole surface are considered key targets. The inverse distance weighted average interpolation (IDW), bicubic interpolation and spline function method are all deterministic methods. These deterministic methods have been applied in studies of Chinese buildings. IDW was used to obtain the fluctuating wind pressure data on a dome surface [²⁰] that was suitable for uniformly distributed wind pressure taps and sensitive to extreme values. Bicubic interpolation was applied to the evaluate wind pressure field of a double-slope roof structure [²¹,²²]; however, loss of high frequency occurred because of the properties of the low pass filter. The thin plate spline interpolation method for a cylindrical lattice shell structure [⁴] does not depend on the potential statistical model, but a regular interpolation region is required, and the computational demand is large. All of the above studies focused on interpolating the reconstruction of wind pressure. For fluctuating wind extrapolation, only the spline function method was found to function relatively well among the deterministic interpolation methods.

Of the geostatistical methods, Kriging interpolation is a spatial interpolation method that can yield optimal, linear, and unbiased estimates of distributed data. Hence, in the case of limited sample data with an irregular distribution, the prediction accuracy is better than that of other interpolation methods [²³], but it is subject to the correlation of a variogram with the original data. Additionally, Kriging interpolation as a surrogate model needs to be constructed followed by a sensitivity analysis [²⁴,²⁵]. Kriging interpolation has been successfully used to predict the wind pressures of low-rise buildings during a typhoon [²⁶], a heliostat surface [¹⁹] and large-span structures [²³]. Geostatistical methods have obvious advantages, but no details of extrapolation reconstruction have been given in practical terms.

Furthermore, few variograms for Kriging extrapolating reconstruction have been studied in the wind engineering field. When airflow runs over a blunt edge of the cladding, the gradient of the wind pressure varies sharply, and the distribution pattern of the fluctuating pressure data exhibits non-Gaussian characteristics. However, a variogram must be reflected by the characteristics of the fluctuating wind field because it plays an important role in guaranteeing the extrapolation accuracy of reconstruction. The fluctuating wind pressure uses a certain fractal scale of the stochastic process to describe the characteristics of the experimental sequence. The fractal scale can be represented by the Hurst exponent and be solved by the rescaled range (R/S) method. The von Kármán correlation function originated from the turbulent wind velocity field, and the Hurst exponent is a critical parameter for the von Kármán function. Consequently, the von Kármán correlation function can be used to resolve this issue of extrapolation at an edge or ridge.

In the present paper, the POD-Kriging algorithm with the von Kármán correlation function is studied and adopted to extrapolate the fluctuating wind pressure at the edge and corner regions on a scaled model from the Tokyo Polytechnic University (TPU) Aerodynamic Database. In this study, the computational procedure following the workflow is illustrated in Fig. 2. The measured data are used to extract the eigenvectors and the prior parameters of the von Kármán function. Next, the fluctuating wind pressure is extrapolated at the edge and corner regions of the building surface, and the performance of the extrapolation method is assessed through comparisons with the original data.

The fundamental theory

Proper orthogonal decomposition

The POD based on Karhunen-Loeve decomposition (KLD) [²⁷] is a powerful method of coping with a discrete random vibration signal [²⁸], including the fluctuating wind pressure measured from wind tunnel tests.

Suppose that the data of the matrix of the fluctuating wind pressure \mathit{P}(t) are recorded at measured taps of(x_1,y_{\mathrm{1}}),(x_2,y_2),\cdots ,(x_N,y_N), where {\mathit{p}}_i(t)=\mathit{p}(x_i,y_i,t)(i=\mathrm{1},\mathrm{2},\cdots ,N)is a vector of the fluctuating pressure on the i^th measured tap. Thus, \mathit{P}(t) can be expressed as follows.

(1)\mathit{P}(t)=\{{\mathit{p}}_{\mathrm{1}}(t),{\mathit{p}}_{\mathrm{2}}(t),\cdots ,{\mathit{p}}_N(t)\}.

Next, we denote {\mathit{R}}_p as a covariance matrix of the fluctuating wind pressure from Eq. (1). The eigenvalue matrix \Lambda and eigenvector matrix \mathit{\Phi } can be obtained from

(2){\mathit{R}}_p\mathit{\Phi }=\mathbf{\Lambda }\mathit{\Phi },

where \mathbf{\Lambda } is a diagonal matrix composed of n-order eigenvalues in descending order and\mathit{\Phi }=\{{\mathit{\varphi }}_{\mathrm{1}},{\mathit{\varphi }}_{\mathrm{2}},\cdots ,{\mathit{\varphi }}_N\}is the eigenvector matrix, where {\mathit{\varphi }}_ican span the N-dimensional orthogonal space.

\mathit{a}(t)={\{{\mathit{a}}_{\mathrm{1}}(t),{\mathit{a}}_{\mathrm{2}}(t),\cdots ,{\mathit{a}}_n(t)\}}^{\mathrm{T}}is the maximum projection of \mathit{P}(t) based on an orthogonal matrix\mathit{\Phi } and is taken as the principal coordinate vector:

(3)\mathit{a}(t)={\mathit{\Phi }}^{\mathrm{T}}\mathit{P}(t).

According to the orthogonality of the eigenvector matrix \mathit{\Phi }, Eq. (3) can be expressed in the following form.

(4)\mathit{P}(t)=\mathit{\Phi }a(t)={\displaystyle \sum _{n=\mathrm{1}}^N{\varphi }_n{\mathit{a}}_n(t)}.

Hence, the eigenvector matrix \mathit{\Phi } with principal coordinate matrix \mathit{a}(t)can be applied to reconstruct the fluctuating wind pressure field by POD. Traditionally, the energy sum corresponding to the first few orders of modes comprises the overwhelming majority of the overall energy; thus, the truncation of modes can be performed to reduce time consumption while achieving reasonable accuracy.

Let Np>N denote the number of all points, including measured taps and investigated points; Ns is the number of sampling data related to time series at the measured taps. Wind pressure matrix {\widehat{P}}_{\mathrm{0}}(t) at Npinvestigated points can be given by

(5){\widehat{\mathit{P}}}_{\mathrm{0}}(t)={\widehat{\mathit{\Phi }}}_{Np\times M}{\mathit{a}}_{M\times Ns}(t)={\displaystyle \sum _{n=\mathrm{1}}^M{\widehat{\mathit{\varphi }}}_n{\mathit{a}}_n(t)}.

The eigenvector matrix {\widehat{\mathit{\Phi }}}_{Np\times M}of all points can be obtained in terms of the interpolation method. The Kriging method is chosen as one of the interpolation methods.

Kriging interpolation

The Kriging method, which was originally presented in geoscience by D. G. Krige, provides uniquely unbiased predictions of statistical variables with minimum and known variance from sparse sample data. After its introduction, Matheron formalized theoretical and systematic concepts of Kriging interpolation [²⁹]. To meet the second-order or weak stationary demand, the fluctuation wind pressure in the wind tunnel test is restricted to be random and spatially dependent, its mean is restricted to be a constant and its variance is restricted to depend only on the lag distance and direction between different points.

Practically, Kriging interpolation is intrinsically a spatial local interpolation method [³⁰] and can be used to characterize the local effect of wind pressure for an irregular surface; however, the interpolation correlation region around the predicted point must be reasonably determined. Within this region, few taps data must be included to meet the validity of stationarity.

Let Nt<N denote the number of taps around the investigated points; the n^th order eigenvector {\widehat{\mathit{\varphi }}}_{n(x_{\mathrm{0}},y_{\mathrm{0}})} at predicted point(x_{\mathrm{0}},y_{\mathrm{0}}) is derived by Punctual Kriging interpolation [³⁰,³¹]as follows:

(6){\widehat{\mathit{\varphi }}}_{n(x_{\mathrm{0}},y_{\mathrm{0}})}={\displaystyle \sum _{i=\mathrm{1}}^{N_t}{\lambda }_i}{\varphi }_{ni(x_i,y_i)}\begin{array}{ll}(n=\mathrm{1},\mathrm{2},\cdots ,M),& \end{array}

where {\varphi }_{ni(x_i,y_i)} is the n^th order eigenvector of the i^th measured taps(x_i,y_i). To ensure that the estimator is unbiased, the sum of weights is set to{\displaystyle \sum _{i=\mathrm{1}}^{N_t}{\lambda }_i}=\mathrm{1}.

The proposed method is based on the theory of random spatial processes under the condition of stationarity. Hence, the general mean might not be constant, and we suppose the expected differences would be zero as follows:

(7)E[{\widehat{\mathit{\varphi }}}_{n(x_{\mathrm{0}},y_{\mathrm{0}})}-{\mathit{\varphi }}_{n(x_{\mathrm{0}},y_{\mathrm{0}})}]=\mathrm{0.}

In addition, the variance of prediction is expressed by

(8)\mathrm{var}\left[{\widehat{\mathit{\varphi }}}_{n(x_{\mathrm{0}},y_{\mathrm{0}})}\right]={\sigma }_{n(x_{\mathrm{0}},y_{\mathrm{0}})}^{\mathrm{2}}=E\left[{\{{\widehat{\mathit{\varphi }}}_{n(x_{\mathrm{0}},y_{\mathrm{0}})}-{\mathit{\varphi }}_{n(x_{\mathrm{0}},y_{\mathrm{0}})}\}}^{\mathrm{2}}\right]=\mathrm{2}{\displaystyle \sum _{i=\mathrm{1}}^{N_t}{\lambda }_i\gamma (h_i)}-{\displaystyle \sum _{i=\mathrm{1}}^{N_t}{\displaystyle \sum _{i=\mathrm{1}}^{N_t}{\lambda }_i{\lambda }_j\gamma (h_{ij})}}.

where \gamma (h_i) is the semivariance between the predicted point (x_{\mathrm{0}},y_{\mathrm{0}}) and the i^th measured tap(x_i,y_i)and\gamma (h_{ij}) is the semivariance between the i^th measured tap (x_i,y_i)and the j^th measured tap (x_j,y_j). Regarding the anisotropy of the wind pressure field correlation, lag distances h_i andh_{ij}can be revised by introducing an anisotropy coefficient k

(9)h_i=\sqrt{{(x_{\mathrm{0}}-x_i)}^{\mathrm{2}}+k^{\mathrm{2}}{(y_{\mathrm{0}}-y_i)}^{\mathrm{2}}},

where the anisotropy is k={\mathit{a}}_x/{\mathit{a}}_y, and is described by the correlation length in different directions. The anisotropic coefficient k is defined as the ratio of the correlation length in the along-wind direction {\mathit{a}}_x to that in the across-wind direction {\mathit{a}}_y.

The semivariances can be calculated from the variogram and depend on sufficient sampling points with suitable configurations. If a prediction point is set at the measured taps, then the estimation of the variance is equal to zero.

The weight factor{\lambda }_i is difficult to determine while strictly satisfying Eq. (8). The next-best approach is to minimize the difference between two sides of the equation. In this case, {\lambda }_i can be obtained by solving the minimum of the estimation variance {\sigma }_E^{\mathrm{2}}under the condition of {\displaystyle \sum _{i=\mathrm{1}}^{N_t}{\lambda }_i}=\mathrm{1}.

(10){\sigma }_{E(x_{\mathrm{0}},y_{\mathrm{0}})}^{\mathrm{2}}=\mathrm{2}{\displaystyle \sum _{i=\mathrm{1}}^{N_t}{\lambda }_i\gamma (h_i)}-{\displaystyle \sum _{i=\mathrm{1}}^{N_t}{\displaystyle \sum _{i=\mathrm{1}}^{N_t}{\lambda }_i{\lambda }_j\gamma (h_{ij})}}-{\sigma }_{n(x_{\mathrm{0}},y_{\mathrm{0}})}^{\mathrm{2}}.

The extreme problem with the above constraint condition can be converted to an unconditional extreme by introducing the Lagrangian multiplier. In this manner, the objective function Eq. (10) is-modified as

(11){\sigma }_{E(x_{\mathrm{0}},y_{\mathrm{0}})}^{\mathrm{2}}=\mathrm{2}{\displaystyle \sum _{i=\mathrm{1}}^{N_t}{\lambda }_i\gamma (h_i)}-{\displaystyle \sum _{i=\mathrm{1}}^{N_t}{\displaystyle \sum _{i=\mathrm{1}}^{N_t}{\lambda }_i{\lambda }_j\gamma (h_{ij})}}-{\sigma }_{n(x_{\mathrm{0}},y_{\mathrm{0}})}^{\mathrm{2}}+\mathrm{2}\mu \left({\displaystyle \sum _{i=\mathrm{1}}^{N_t}{\lambda }_i}-\mathrm{1}\right).

Here,\mathrm{2}\mu is the Lagrangian multiplier. The solutions of the extreme problem Eq. (11) are

(12)\begin{array}{l}\begin{array}{ll}{\displaystyle \sum _{j=\mathrm{1}}^{N_t}{\lambda }_i\gamma (h_{ij})}-\mu =\gamma (h_i)& \begin{array}{ll}for& i=\mathrm{1},\mathrm{2},\mathrm{...},N_t\end{array}\end{array}\\ {\displaystyle \sum _{j=\mathrm{1}}^{N_t}{\lambda }_i-\mathrm{1}=\mathrm{0}}\end{array}.

Next, the weight components {\lambda }_{\mathrm{1}},{\lambda }_{\mathrm{2}},\mathrm{...},{\lambda }_{N_t}and the Lagrangian multiplier \mu are obtained, and the estimation variance of the Kriging method can be computed by Eq. (12). As a consequence, a suitable semivariance model g must be introduced to describe the spatial covariance.

von Kármán correlation function

The relation between the semivariance and covariance can be written as follows:

(13)\gamma (h)=C(\mathrm{0})-C(h),

where C(\mathrm{0})={\sigma }^2is the covariance at lag distance zero and C(h) is the covariance function of lag distance h. However, it is difficult to define the covariance function because the mean of the new point is unknown. The semivariance enables a quantitative description of the lag h. Thus, the variogram is more widely used than the covariance function.

To reconstruct the fluctuating wind pressure fields on the roof, the variograms are critical. To obtain an approximation to an experimental variogram, the authorized functions can be fitted to satisfy a characteristic of the empirical model. Generally, the semivariance \gamma (h) is expected to pass through the origin point at h=0; however, in practice, it passes at a certain positive value, which is denoted as the nugget variance (c₀). The measurement error and variation mostly caused by lag distances less than the shortest sampling lag are contained in the nugget variance. Hence, a pure nugget (c) does not represent lag distances larger than the correlation length(a). Figure 3 shows an experimental variogram form.

The von Kármán covariance function originated from the seemingly chaotic, random velocity fields [³²] and has been successfully applied in conditional geostatistical simulations of the porosity distribution [³³,³⁴] and consistent earthquake rupture models [³⁵]. Nevertheless, few studies of wind pressure fields have used this function as a variogram for the Kriging method.

First, von Kármán covariance functions can be constructed

(14)C(h)={\sigma }_k^{\mathrm{2}}{\mathrm{2}}^{\mathrm{1}-\nu }{(h/a)}^{\nu }{K}_{\nu }(h/a)/\mathrm{\Gamma }(\nu ),

where {\sigma }_k^{\mathrm{2}}is the a priori variance of the autocovariance function and a the correlation length or range of spatial dependence. Taking into account the gamma function [³⁶]

(15)\mathrm{\Gamma }(\nu )={\displaystyle {\int }_{\mathrm{0}}^{\infty }{e}^{-t}{t}^{\nu -\mathrm{1}}}dt.

K_{\nu }(r/a) is the modified Bessel function of the second kind for the Hurst exponent 0<v<1, which can be written [³⁷]

(16)K_{\nu }(r/a)=\left({\displaystyle \frac{\pi }{\mathrm{2}}}\right){\displaystyle \frac{I_{-\nu }(r/a)-I_{\nu }(r/a)}{\sin (\nu \pi )}},

whereI_{\nu }(r/a)can be expressed as

(17)I_{\nu }(r/a)={(r/a)}^{\nu }{\displaystyle \sum _{k=\mathrm{0}}^{\infty }\frac{{(r/a)}^{\mathrm{2}k}}{k!\mathrm{\Gamma }(\nu +k+\mathrm{1})}}.

Second, transforming the von Kármán covariance functions into a variogram in terms of Eq. (13) and Eq. (14), the following expression is obtained:

(18)\gamma (h)=c_0+c(\mathrm{1}-{\mathrm{2}}^{\mathrm{1}-\nu }{(h/a)}^{\nu }{K}_{\nu }(h/a)/\mathrm{\Gamma }(\nu )),

where c₀ is the nugget variance, C(\mathrm{0})=c_{\mathrm{0}}+c is the sill variance, and C(\mathrm{0}) is the variance{\sigma }_{}^{\mathrm{2}}.

The von Kármán function is a family of variograms in which the Hurst exponent 0<v<1. The turbulence in the wind field exhibits self-affinity for Hurst exponents of and self-similarity for Hurst exponents of [³³,³⁸]. For special cases, white noise is a self-affine fractal with n=0, Brownian noise is a self-affine fractal with n=0.5, and n=1 indicates an exactly self-similar process [³⁹]. To investigate the Hurst exponent of the fluctuating wind pressure at measured taps, R/S analysis has been applied in Appendix I. Originally, the method was used to determine the reservoir capacity for irrigation along the Nile by Hurst [⁴⁰]. The Hurst exponent is commonly used for fractal scaling in stochastic processes and is calculated to characterize experimental data series [⁴¹–⁴³]. Hence, the application of the von Kármán function in the POD-Kriging algorithm is studied, and other parameters derived from the prior are determined by the least-square method, then discussed in an engineering example. Additionally, if statistical result of the data series satisfy the Gaussian distribution, a maximized target result of variogram can also be obtained by the maximum likelihood function method [⁹,⁴⁴]. Notably, the probability distribution of the fluctuating wind pressure at the edge has obvious non-Gaussian properties.

Extrapolation reconstruction of wind pressure fields

Example study

A scaled wind tunnel test model from the Tokyo Polytechnic University Aerodynamic Database is chosen to assess the performance of the extrapolation method. The scale model is 1:400 and has geometric dimensions of 0.3 m (B) × 0.1 m (D) × 0.5 m (H). The fluctuating wind pressure data of the windward surface are used as validation data in the present work. A total of 24 measured taps are arranged to extract characteristic parameters. There are 15 measured taps used as prediction points, where 6 predicted data sets are used to test the extrapolation accuracy and 3 predicted sets are applied to verify the interpolation accuracy. The measured taps and the layout of the prediction points are shown on the windward surface. The central region is usually in a positive pressure area, and the edge regions are subjected to high gradient characteristics. The wind pressure variation from negative to positive is due to airflow separation around the building. Thus, the investigated points representing airflow characteristics with different sites on the surface are highlighted in Fig. 4.

Analysis of POD results

The magnitudes of all the eigenvalues can be extracted from the POD and the associated accumulated energy distribution of the first j order eigenvalue, as shown in Fig. 5. Obviously, mode 1, which accounts for 98.9% of the proportion of all the accumulated energy and has a significantly higher value than the others, plays an important role in the energy contribution. Statistics show that modes 1‒9 account for 100% of the accumulated energy. However, the first modes 1‒9 are not sufficient for reconstructing the pressure field.

Figure 6 shows the original time series and the frequency domain reconstruction of tap161 on the corner from modes 1‒3 to modes 1‒18. The reconstruction results of imperfect modes will be overestimated. Successively, the disagreement between reconstruction results and original data is gradually reduced as the number of high-frequency mode calculations increases. The reconstruction results and original data agree well until complete modes are taken into account.

Hurst exponents

The Hurst exponent is one of the most important parameters for the von Kármán function, which can be used to indicate the fractal scaling of stochastic process characterization. Figure 7 shows the Hurst exponents following the R/S analysis.

The distribution contours of the mean wind pressure coefficients on the windward surface are symmetrical. The gradients of wind pressure vary sharply at the edge, in contrast with those in the central area. The Hurst exponents (n) are calculated based on fluctuating wind pressure data at the investigated taps. Many of the investigated data exhibit Hurst exponents greater than 0.5, varying from 0.75 and 0.83. This result indicates that the fluctuating wind pressure is a self-similar process.

Interpolation reconstruction results

In Fig. 8, considering tap 171 and tap 206 as the investigated points in the central area, the original data agree well with the reconstruction results using the spline and POD/Kriging methods.The interpolation accuracy of the POD/Kriging method and the spline model is consistent. Physically, the interpolation precision is much better than that of the extrapolation results. The interpolation data are almost identical to the original data at the investigated points. Additionally, the discrepancy in the time series and frequency domain analysis for tap 162 at the edge can be attributed to the sparse sample data and large pressure gradients.

Extrapolation reconstruction results (I)

The fluctuating wind pressure at the investigated points was measured in a wind tunnel test to validate the agreement between the original data and the extrapolation results. Deterministic and spatial extrapolation results are compared and analysed in Fig. 9. In the deterministic interpolation method, the spline model is used for interpolation of the experimental data because the interpolation accuracy is better than that of other deterministic interpolation methods with sparse sample points. Hence, the cubic spline method, as is the penalized spline regression model [⁴⁵], is applied to reconstruct wind pressure data compared with the POD-Kriging algorithm. Typically, the original data are chosen as investigated points at the edges to verify the extrapolation accuracy. In general, the results show that the extrapolation accuracy of the POD-Kriging method is much better than that of the spline method. Notably, for tap 1 at the corner, the extrapolation precision can be guaranteed by the POD-Kriging method with the von Kármán function.

Extrapolation reconstruction results (P)

In Fig. 10, the extrapolation results using POD-Kriging with different variograms are compared to the original data. The trends of the two extrapolation results are generally similar. However, the results show that the extrapolation accuracy with the von Kármán function is much better than that with the linear variogram because the von Kármán correlation function has several adjustable parameters obtained from the statistics of the original data to satisfy different wind field changes. For tap 401 at the edge, the extrapolation results in the time domain with the von Kármán function are higher than that with the linear variogram. Therefore, the energy spectrum density at a low frequency is also higher than that of the target values in the frequency domain.

Non-Gaussian analysis of the extrapolation results

The root mean square (rms) of pressure coefficients collected from different studies are identical in Table 1. However, the Kurtosis and skewness are considerably different among the original data and the extrapolation results. For the windward surface, the extrapolation results using POD-Kriging with the von Kármán function, as well as the original data, exhibit Gaussian characteristics. However, the extrapolation results using POD with the spline method exhibit non-Gaussian characteristics because the Kurtosis is larger than 1. Thus, these observations confirm the importance and necessity of the use of an appropriate method when reconstructing edge regions via extrapolation.

Discussion

Proper orthogonal decomposition is a popular technique in processing large amounts of high-dimensional data to obtain low-dimensional descriptions, which can be captured as a random process of interest and save computing resources under accuracy conditions. Here, eigenvectors related to space about original data can be decomposed by POD for extension at investigated locations.

Kriging is a method of interpolation governed by prior covariances and relies on the semi-variogram. The variance of all the original data can be graphed out with distance lag to meet the validity of stationarity. In Table 1, the Kurtosis is less than 1, and we found that wind pressure data have a normal distribution on the windward surface. Importantly, Kriging relies on the semi-variogram with incorporated uncertainty.

The semi-variogram quantifies its autocorrelation, and closer distances are more related while farther distances are less related. It is critical to determine a reasonable shape of the semi-variogram. The von Kármán correlation function is a family of different Hurst exponents, which have exponential behaviour but different decay rates. Notably, the Hurst exponent reveals the fractal scaling of stochastic process characterization.

Additionally, it should be noted that the Kriging method is not optimal for abrupt changes, such as a large change gradient of wind pressure at the edge. However, we have noted that the Hurst exponent can be deduced at any new spatial location, and then a closer agreement to the original data is obtained. This is one of the reasons why POD-Kriging with the von Kármán correlation function is better than the other methods of extrapolation reconstruction.

Conclusions

The wind pressure field around a building is an important factor in wind resistance design. However, the mechanisms of wind-induced failure at edges or corners are complicated, because large wind pressure gradients exist in those regions when performing a site measurement. The POD-Kriging algorithm with the von Kármán function is an efficient method for fluctuating wind pressure extension. The validity of this method was confirmed based on the following arguments:

1) The POD-Kriging algorithm with the von Kármán function for wind pressure extrapolation reconstruction was presented and studied. This method was being improved to provide satisfactory predictions in the extrapolation of fluctuating data. The POD-Kriging algorithm has a clear advantage over deterministic methods with sparse sample points, and the extrapolation accuracy with the von Kármán function is much better than that with the linear variogram. The issues of extrapolation reconstruction at corners may be resolved by the proposed algorithm. The result shows that the proposed algorithm is an efficient and accurate method of wind pressure field extension for limited sample data.

2) Regarding fluctuating wind pressure at the edge closures, two issues should be taken into account: the spatial interpolation method for wind pressure should be adequately adopted, and an appropriate variogram for the Kriging method, such as the von Kármán function, should be further studied for the wind field.

3) The Hurst exponent, which is used as a measure of the long-term memory of a time series can be obtained by the R/S analysis. Here, the Hurst exponent of [0.75-0.85] denotes long-term positive autocorrelation on a windward surface. This range indicates that the fluctuating wind pressure is a self-similar process in the determination of stable reconstruction results. Another important factor is the correlation length, which may be associated with the characteristic scales of the building height (H) or width (B).

4) Moreover, the POD-Kriging algorithm with the von Kármán function was found to function reasonably well on the windward surface under a Gaussian distribution condition. However, the extrapolation reconstruction of the negative wind pressure region among the lateral and leeward surfaces must be further studied in future work.

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