43 12 211 12 JOURNAL OF HARBIN INSIUE OF ECHNOLOGY Vol 43 No 12 Dec 211 POD - 1 2 1 1 1 159 sunnyhit@ hit edu cn 2 183 WAWS POD WAWS /POD WAWS /POD 2 U393 3 A 367-6234 211 12-13 - 5 WAWS / POD simulation of fluctuating wind field SUN Ying 1 LIN Bin 2 WU Yue 1 SUN Xiao-ying 1 1 School of Civil Engineering Harbin Institute of echnology 159 Harbin China sunnyhit@ hit edu cn 2 Beijing Construction Procuring & Contracting rade Center 183 Beijing China Abstract WAWS is the most common used method and it can produce random data time series with good accuracy but the low efficiency and memory exceed problem is inevitable at the case of large amout of variates and long time series An improved method combining WAWS and proper orthogonal decomposition POD technique is used in this paper to improve the computational efficiency which is obviously efficient in both time and memory consumption Furthermore an error standard is defined to estimate the simulation precision which can be used for choosing the order of POD method and error prediction prior to the simulation process Finally an example for numerical simulation of 2 points in random wind field is given to demonstrate the accuracy and effectiveness of this method Key words fluctuating wind velocity numerical simulation proper orthogonal decomposition weighted amplitude wave superposition AR ARMA DW 1 - WAWS 3 4 21-9 - 6 981521 59868 HI NSRIF 2998 LC2111 1976 1972 5-6 7
14 43 Cholesky θ jm ω H jm ω N 1 7 9 1 2 Cholesky n n 2 /2 8-11 Yang 8 FF 1 FF 1 1 9 Cholesky V j pδt = Re j h jm qδt exp i mδω { [ ( ) pδt m = 1 n ] } Cholesky 2 p = 1 2 M n - 1 M = 2N q p /M 11 q = 1 2 M - 1 Cholesky h jm qδt = M-1 B jm lδω exp i lδω qδt l = 3 B jm lδω B jm lδω = { 槡 2 Δω H jm ( lδω + mδω ) exp i ml l < N n N l < 2N POD 4 2 POD 2 WAWS /POD 2 1 WAWS /POD 2 POD 12-13 1 Shinozuka 7 n V t n V j t = 2 槡 Δω j N V t = [ V 1 t V n t ] t = 1 2 H jm ω ml m = 1 l = 1 N POD cos ω ml t - θ jm ω ml + ml j = 1 2 n V t = 1 a k t k = Φ a t k = 1 j N 5 Δω = ω up /N a t = a 1 t a 2 t a n t ω up ω > ω up V t S ω = ml a t = Φ V t 6 2π H jm ω ml Φ = 1 n C v H ω H ω S ω
12 POD - 15 C v k = λ k k 7 λ k λ n λ 1 - ωup S ω dω -ω up 8 7 8 珟 V t = 11 ωup S 21 a λ t 14 ω dω k λ k k 9 -ω up 14 C v Φ POD a t S a ω a n t λ n S a ω = Φ S ω Φ 1 9 1 V t 珘 S ω a t 珘 S ω = POD a n t 11 λ n [ 21 ] S a λ ω [ 11 21 ] 15 2 2 WAWS /POD 2 3 POD FF 珘 S ω S ω k 珘 S ω ij - S ω ij dω E ij = 1 k λ i S ω ij dω i = 1 V t S ω k λ E eigen k λ = 1 - S a λ ω k λ k λ 1 17 λ i S i = 1 a ω 17 k λ S a ω = S a11 ω k λ k λ S a12 ω k λ n-k λ [ ] S a21 ω n-k λ k λ S a22 ω n-k λ n-k λ 12 S a λ ω = 11 S 11 ω 11 + S 12 ω 21 + C v 21 S 21 ω 11 + S 22 ω 21 13 C v = R v = S ω dω k λ FF a λ t [ ] Π k = 1 11 16 λ i ij i j i = 1 E min E mean E max 9% Π k λ 9 11 k λ k λ k λ λ i i = 1 POD λ n λ 1 k λ n S ω λ n λ 1 Φ 1
16 43 2 k λ k λ 4 a i t k λ = 2 S a i ω 5% 15% 2 3 3 k λ CPU P84&2 26 GHz 2G PC 2 Matlab 2 x 6 m5 59 s z 3 m4 11 56 s B 1 m POD 95% 2 m /s Karman 15 fs v f = σ 2 v 4γ 1 + 7 8γ 2 5 /6 ( ) γ = fl v 珔 L V v = 1 z z 3 5 σv = I 珔 V z { 23 z 5 -α- 5 I = 1 z ( 35) 5 < z < 35 1 z 35 Coh i j f = { } exp - f C2 x x i - x j 2 + C 2 z z i - z j 2 1 /2 1 2 v珋 i + v珋 j C x = 16 C z = 1 1 s 49 6 s 1 4 8 12 16 2 选取的本征模态阶数 3 1 4 5 POD 2 a n t λ n 1-1 1 1-2 1-1 1 12 1 8 1-3 1-4 6 4 1-5 2 5 1 15 2 1-6 本征模态阶数 f/hz (a)k 姿 =2 时的 S 11(f) 时间主坐标均方值 / 本征值 35% 功率谱密度 误差 /% 2 4 3 2 1 12 1 8 6 4 2 1-2 误差评价标准最小误差平均误差最大误差 4 8 12 16 2 选取的本征模态阶数 结合 POD 的谐波合成法谐波合成法
12 POD - 17 1-1 1-2 功率谱密度 互相关函数 互相关函数 1-3 1-4 1-5 1-6 1-7 1-8 1-2 1-1 1 4 5 4 3 5 3 2 5 2 1 5 1 5-5 4 f/hz (b)k 姿 =2 时的 S 16(f) -1-5 5 1 (a)k 姿 =2 时的 R 11(t) 2 5 2 模拟值目标值 8 YANG J N On the normality and accuracy of simulated random processes J J Sound Vib 1973 26 3 417-428 1 5 1 9 GEORGE D Simulation of ergodic multivariate stochastic processed J J Eng Mech 1996 122 8 778-787 5 1 J 1998 31 3 72-79 -5-1 -5 11 5 1 J 27 35 7 15-19 (b)k 姿 =2 时的 R 16(t) 5 12 AMURA Y SUGANUMA S KIKUCHI H et al Proper orthogonal decomposition of random wind pressure field J Journal of Fluids and Structures 4 1 DEODAIS G SHINOZUKA M Autoregressive model for non-stationary stochastic processes J J Eng Mech 1988 114 11 1995-212 2 GERSCH W YONEMOO J Synthesis of multi-variate random vibration systems J J Sound Vib 1977 52 4 553-565 3 KOZIN F Auto-regressive moving-average models of earthquake records J Probab Eng Mech 1988 3 2 59-63 4 ROSSI R LAZZARI M Wind field simulation for structural engineering purposes J Int J Numer Meth Eng 24 61 738-763 5 KIAGAWA NOMURA A Wavelet-based method to generate artificial wind fluctuation data J J Wind Eng Ind Aerodyn 22 9 943-964 6 YAMADA M OHKIANI K Ortho-normal wavelet a- nalysis of turbulence J Fluid Dyn Res 1991 8 11-115 7 SHINOZUKA M JAN C M Digital simulation of random processes and its applications J J Sound Vib 1972 25 1 111-128 1999 13 169-195 13 CHEN X KAREEM A POD-based modeling analysis and simulation of dynamic wind load effects on structures J J Eng Mech 25 131 4 325-339 POD 14 HOLMES P LUMLEY J BEKOOZ G urbulence coherent structures dynamical systems and symmetry M Cambridge Cambridge University Press 1996 Cholesky 15 SOLARI G PICCARDO G Probabilistic 3 - D turbulence modeling for gust buffeting of structures Probabilistic Engineering Mechanics 21 16 73 - Cholesky 86