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

A random field representation based on stochastic harmonic functions Chen, Jianbing; He, Jingran; Li, Jie Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  1A Random Field Representation Based on Stochastic Harmonic Functions Jianbing Chen  Professor, School of Civil Engineering & State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, Shanghai, China Jingran He  Ph.D. Student, School of Civil Engineering, Tongji University, Shanghai, China Jie Li  Professor, School of Civil Engineering & State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, Shanghai, China ABSTRACT: Random field models are widely adopted in engineering practice, for instance, to describe the mechanical properties of an engineering site or a civil structure. The most widely employed approach to represent a random field is the spectral representation. In the present paper, a new stochastic harmonic function representation by partitioning the wave number domain is proposed. The properties of the new representation are studied. A numerical example is studied to illustrate the proposed method. 1. INTRODUCTION Random field models are widely employed in engineering practice (Vanmarcke, 2010). For instance, the live load on the floor of a structure is actually a random field rather than a concentrated force or distributed force with constant intensity, simultaneously the mechanical property, e.g., the modulus of elasticity, of a concrete floor plate is also a random field. Usually the second order statistics of a random field, i.e., the mean and correlation function or power spectral density in the wave number domain (PSD for short) is available through physical reasoning or observed data. In engineering practice the representation of a random field, i.e., to express the random field by a function of space variables and a set of random variables is necessary. A variety of approaches, including the local average method (Vanmarcke, 2010), the spectral representation method (Shinozuka & Deodatis, 1991), the ARMA method (Spanos & Zeldin, 1998), the Karhunen-Loeve decomposition (Loeve, 1977; Huang et al., 2001; Spanos et al., 2007) and the double-orthogonal decomposition (Liu et al., 2011), ect., were studied by various researchers. All of these approaches could recover the PSD as the retained terms tends to infinity. However, in practice, truncation must be conducted. Generally, hundreds or even more than one thousand terms should be retained, and thus the same number of random variables is involved. This will induce difficulty in various aspects and thus to reduce the number of random variables is very important (Spanos et al, 2007; Chen et al, 2013).  The concept of stochastic harmonic function (SHF) was studied in Chen and Li (2010) to represent stochastic processes. It was extended to the representation of two-dimensional random fields by Liang et al. (2012) where the wave number domain was partitioned in a tensor product manner.  In the present paper, a new scheme of stochastic harmonic function (SHF) representation of a two-dimensional random field is proposed and verified. The expression of this kind of SHF representation is first introduced. 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  2Then the reproduced PSD and the target PSD are compared. An example of the deflection of a plate with the modulus of elasticity described by a random field is studied.  2. STOCHASTIC HARMONIC FUNCTION REPRESENTATIONS 2.1. The concept of Stochastic Harmonic Function (SHF) Let the one-sided PSD of a random field Y  be  ( )G k , where 1( , , )rk k=k   is the wave number vector, r  is the spatial dimension. The classical spectral representation of the random field is  SP1( ) ( )sin( )Ni i iiY A   x k k x    (1) where 1( , , )rx x=x   is the space variable vector, 1( , , )i i irk k=k   is the i-th discretized wave number vector, 1ri ij jj k x=⋅ = åk x  is the inner product of ik  and x , i ’s are independent random variables uniformly distributed over 0 2[ , ]p , the amplitude is given by       22 ii rA G d  k k k   (2) in which iW  is the rectangular-shaped partitioned subdomain to which ik  belongs. Note that in Eq. (1) the discretized wave numbers are deterministic, and thus the PSD of SP ( )Y x  could only tends to the target PSD ( )G k  as N ¥ . Based on the stochastic harmonic function (SHF) representation of stochastic processes (Chen et al. 2013), Liang et al. (2012) proposed the following representation for the random field Y   SHF1( ) ( )sin( )Ni i iiY A   x K K x    (3) where iK  is now the random wave number vector uniformly distributed over the  rectangular-shaped partitioned iW , i ’s are still independent random variables uniformly distributed over 0 2[ , ]p , and the amplitude is   2( ) ( )(2 )i i irA G S K K  (4) where ( )iS   is the volume of the hyper-rectangular i . To be clear, the PDF of iK  is  1  for   0         otherwise( ),( ) ,ii iSp W Î Wìïï= íïïîKkk  (5) where  lower upper lower upper lower upper1 1 2 2, , , , , ,[ , ] [ , ] [ , ]i i i i i r i r ik k k k k kW = ´ ´  (6) in which lower upper lower upper1 1 1 1 1, ,[ , ] [ , ]ri i ik k k k= =  and lower upper lower upper1 1 1 1, , , ,[ , ) [ , )i i j jk k k k = Æ  for any i j¹ , lower upper1 1,k k  are the cut-off lower and upper wave number in 1k . It is proved that the SHF representation in Eq.(3) could reproduce the target PSD exactly when the number of the retained terms N  is any arbitrary integer, even when it is one. However, because of the tensor product of the partitioning in Eq.(6), the number  1 2 1rr jjN N N N N== =  (7) where jN  is the partitioned number over lower upper[ , ]j jk k , i.e., the partitioned number in the direction of jk . Therefore, the total number of random variables is 12 r jj N= . This is still a large number for practical applications. In the present paper, the expressions in the form of Eqs.(3), (4) and (5) are still adopted, but the partitioned subdomain in Eq.(6) will be replaced by the Voronoi cell partition rather than the tensor product of rectangular-shaped partition, as illustrated in Figure 1. 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  3 Figure 1: Voronoi Cells on wave number domain  For clarity, we now replace iW  by iV , which are the Voronoi cells partitioning the wave number domain, Eqs.(4) and (5) now becomes,    2( ) ( )(2 )i i irA G S VK K  (8) and  1  for    0         otherwise( ),( ) ,ii iS V Vp Îìïï= íïïîKkk  (9) respectively. In this case, we could scatter some points arbitrarily, in principle, and then obtained the corresponding Voronoi cells (Li & Chen, 2009), which could be taken as the above subdomains iV . Therefore, the number N  is not related to the dimension as the one in Eq.(7), and thus great flexibility could be achieved. 3. AN EXAMPLE OF SHF REPRESENTATION OF A RANDOM FIELD 3.1. The samples of random fields An example is illustrated here to make things clear. Take a random field with the following PSD (Shinozuka & Deodatis, 1996)    2 22 1 2 1 1 2 21 2,1 1 ,1 ,2 2 ,2, exp ,4 2 2,  u u u uc c c k c kS k kK K K K K K                      (10) where the parameters 1c  and 2c  are related to the correlation distance, both assumed to be 1 here;   is also assumed to be 1. The cut-off wave number is upper upper1 2 5k k  . For the purpose of comparison, the spectral representation method is firstly employed to generate a sample of the random field, as shown in Figure 2. In this case, the two-dimensional cut-off wave number domain is partitioned into 100 100  subdomains which are identical squares, thus the number of random variables, i.e., the random phase angular, is 10000. Figure 2(a) is the 3-D view of the random field sample while Figure 2(b) is the contour lines of this sample.  (a)  (b) Figure 2 The random field sample generated by the spectral representation method 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.10.20.30.40.50.60.70.80.91K1K 2Contour lines of the random field sample based on The Spectral Representation Methodx1(m)x 2(m)0 1 2 3 4 5 6 7 8 9 1001234567891012th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  4 Then the SHF method adopting Voronoi cells is employed to generate a sample of the random field, as shown in Figure 2. Figure 2 (a) is the 3-D view of the random field sample while Figure 2 (b) is the contour lines of this sample. In this case the number of random variables = 25 pairs of random wave number coordinates + 25 random phase = 75.   (a)  (b) Figure 3 The random field sample generated by Stochastic Harmonic Function Representation  From the figures, it is observed that the appearance of these two random fields seems to be alike. In order to further verify the SHF representation in more details, the reproduced correlation function and PSD are studied. 3.2. The recovered PSD It is easy to verify that the correlation function of the SHF is     2 ,1 ,2 ,1 ,21,1 , cos2Nj j j jjRE A K K K K           (11) Based on this equation the Monte-Carlo method is employed to calculating the correlation function of the random field:     MC 21, 2, 1, 2,1 1MC,1 , cos2N Nij ij ij iji jRA K K K KN             (12) where MCN  is the number of the random field samples. The results are shown in Figures 4 and 5. Here we take MC 100000N  .  (a)  (b) Figure 4 The Correlation Function  Contour lines of the random field sample based on The Stochastic Harmonic Functionson Voronoi Cellsx1(m)x 2(m)0 1 2 3 4 5 6 7 8 9 10024681012th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  5 Figure 5 The cross-sectional view of the correlation function  Then a FFT method is employed to compute the PSD of the random fields. The results are shown in Figures 6 and 7.   (a)  (b) Figure 6 The PSD   Figure 7 The cross-sectional view of the PSD  Figures 4 through 7 show that the SHF’s second order statistics converge to the target ones with high accuracy, although only very few, compared to the classical spectral representation method, random variables are involved. Some errors occur on the top of the PSD, which means that some energy is lost in the vicinity of the peak, the reason is that the random wave number coordinates is sparse. This problem will vanish when the number of Voronoi cells increases. 4. EXAMPLE 4.1. An elastic plate with random modulus of elasticity The deflection of an elastic plate with random modulus of elasticity is analyzed.  The geometry of the plate is shown in Table 1.  Table 1: Geometry of the plate Length Width Thickness10m 10m 0.15m  The plate is clamped by its four corners, all six freedoms are fixed. The loads applied on the plate uniformly include the dead load 2.0kN/m2 and the live load 2.0kN/m2. The dead load and the live load are combined by the following equation (Chinese Code GB50068-2001, 2001): -5 -4 -3 -2 -1 0 1 2 3 4 500.20.40.60.81The cross-sectional view of the correlation functionx lag (m)The correlation function  Target correlation functionThe recovered correlation function-5 -4 -3 -2 -1 0 1 2 3 4 500.010.020.030.040.050.060.070.080.090.1The cross-sectional view of the PSDsWave number K1PSD  The recovered Power Spectral DensityThe target Power Spectral Density12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  6 G G Q c QS S S      (13) where G  and Q  represent the partial factors for the dead load and the live load, respectively. c  is the combination factor for the live load. Three cases are considered: Case 1: The correlation distance is large: L=10m. Case 2: The correlation distance is small: L=1m. Case 3: The correlation distance L=5m. The PSD of the random field of modulus of elasticity is the same as Eq.(10). But in the three cases, 1c  and 2c  are different, and could be obtained by (Shinozuka and Deodatis, 1996)  21 2 1 2,  2c c L c c    (14) Local average is applied. The relationship between the local average of the random field and the random field expression is related to the correlation distance (Vanmarcke, 2010).  The material of the plate is assumed to be concrete with the statistics listed in Table 2. The coefficient of variation of elastic modulus takes 0.1 according to Wu et al. (2010).  Table 2: The statistics of elastic modulus Mean Coefficient of Variation 3.25×1010N/m2 0.1  The FEM software Opensees is employed. The element type is ‘ShellMITC4’ which is a kind of plate fiber element.  The plate is meshed into 19×19 meshes, the number of the nodes are 20×20.  The deflection of the plate is considered as the main factor of failure. The resulted samples of the three cases are shown in Figures 8 through 10.  (a)  (b) Figure 8 The deflection of plate with L=10m   (a) Coordinate X (m)Coordinate Y (m)The contour view of the deflection  0 2 4 6 8 100246810-0.014-0.012-0.01-0.008-0.006-0.004-0.002012th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  7 (b) Figure 9 The deflection of plate with L=5m   (a)  (b) Figure 10 The deflection of plate with L=1m  The above figures show that the deflection is slightly different. The statistics of the deflection are calculated by the Monte Carlo method with 100000 samples and are listed in Table 3.   Table 3: The statistics of deflection Correlation distance L (m)Mean value of deflection (m) Standard deviation of deflection (m)10 -1.6129×10-2 2.0543×10-3 5 -1.6141×10-2 1.7092×10-31 -1.6013×10-2 0.4906×10-3  4.2. An elastic plate with deterministic modulus of elasticity In this case the deterministic elastic modulus takes the mean value of the above random elastic modulus. The geometric properties are the same as in the preceding subsection.  The deflection is now shown in Figure 11.  (a)  (b) Figure 11 The deflection of the plate with deterministic elastic modulus  The deflection of the center is -1.5645×10-2,  which is slightly different between the plate with random elastic modulus. The result also shows Coordinate X (m)Coordinate Y (m)The contour view of the deflection  0 2 4 6 8 100246810-0.014-0.012-0.01-0.008-0.006-0.004-0.0020Coordinate X (m)Coordinate Y (m)The contour view of the deflection  0 2 4 6 8 100246810-0.014-0.012-0.01-0.008-0.006-0.004-0.0020Coordinate X (m)Coordinate Y (m)The contour view of the deflection  0 2 4 6 8 100246810-0.014-0.012-0.01-0.008-0.006-0.004-0.002012th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  8that the variation of the deflection increases with the correlation distance. But the mean values are almost invariant against the correlation distance. 5. CONCLUDING REMARKS A new scheme of SHF representation of random fields is proposed. The wave number domain is partitioned by Voronoi cells over which random wave number of different harmonic components are uniformly distributed. By this the number of random variables could be considerably reduced whereas the PSD could be exactly reproduced. The deflection of a plate with random modulus of elasticity is studied. The results when the correlation length is different are analyzed, showing that the variation of deflection increases with the correlation distance, whereas the mean values changes little. 6. AKNOWLEDGEMENTS Financial supports from the National Natural Science Foundation of China (NSFC Grant Nos.11172210 and 51261120374), the National Key Technology R & D Program (Grant No. 2011BAJ09B03-02) and the State Key Laboratory of Disaster Reduction in Civil Engineering (Grant Nos. SLDRCE14-A-06 and SLDRCE14-B-17) are greatly appreciated. 7. REFERENCES Chen, J. B., Li, J., (2010), “Stochastic Harmonic Function and Spectral Representations,” Chinese Journal of Theoretical and Applied Mechanics, 39(10), 1413 Chen, J. B., Sun, W. 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