Efficient Observation of Random Phenomena

Transcription

1 Lecture 9 Efficient Observation of Random Phenomena Tokyo Polytechnic University The 1st Century Center of Excellence Program Yukio Tamura POD Proper Orthogonal Decomposition Stochastic Representation of Factor Analysis Karhunen-Lo Loève Decomposition

2 POD Find a deterministic coordinate function φ (x,y) that best correlates with all the elements of a set of randomly fluctuating wind pressure fields p (x,y,t). φ (x,y) is derived to maximize the projection from the wind pressure field p (x,y,t) to the deterministic coordinate function φ (x,y). Maximization of Projection Realize from the probabilistic standpoint: p(x,y,t)φ(x,y) dxdy = max By normalizing: p(x,y,t)φ(x,y) dxdy = max { φ(x,y) dxdy } 1/

3 Maximization of Projection p(x,y,t) can take positive or negative values Maximization is made by a mean square method p(x,y,t)φ (x,y) dxdy p(x',y',t x',y',t)φ (x',y' x',y') dx'dy' max φ (x,y) dxdy Eigenvalue Problem Rp (x,y,x',y' x,y,x',y') φ (x',y' x',y') dx'dy' λ φ (x,y) Rp (x,y,x',y' x,y,x',y') Spatial Correlation of p(x,y,t) Uniformly Distributed dx dy dy Matrix Matrix Form [Rp]{φ} λ{φ} [Rp]] : Spatial Correlation Matrix M M Square Matrix {φ} : Eigenvector of Spatial Correlation Matrix [Rp][ λ : Eigenvalue of Spatial Correlation Matrix [Rp][

4 Fluctuating Pressure Field p(x,y,t) M p(x,y,t) = Σ a m (t)φ m (x,y) m = 1 p(x,y,t)φ m (x,y)dxdy a m (t) = φ m (x,y) dxdy : m-th Principal Coordinate φ m (x,y) : m-th Eigenvector (Eigen Mode) Correlation Between Principal Coordinates No correlation between different modes : a m (t) a n (t) = 0 m n Mean square of the m-th principal coordinate : a m (t) = λ m = Eigenvalue

5 Mean Square of Wind Pressure Mean-square of wind pressure at point(x,y) M p(x,y,t) = Σ λ m φ m (x,y x,y) m = 1 Field-total sum of mean-square wind pressures M p(x,y,t) dxdy dxdy = Σ λ m m = 1 M = Σ a m (t) m = 1 Reconstruction in Lower Modes N ^ p(x,y,t) Σ a m (t)φ m (x,y) m = 1 N M

6 Fluctuating Pressure Field (Fluctuating Component Only) p 1 (t) p (t) p 3 (t) t t t t 1 t t 3 t 4 t 5 State Locus of Fluctuating Pressure Field p 3 (t) t 5 t 3 t t 4 p 1 (t) t 1 p (t)

7 State Locus on Same Plane η ( t ) t 5 a 1 (t ) t 1 t t 4 t 3 ξ( t ) State Locus on Same Line t 5 a 1 (t) t 1 t 4 t t 3

8 Projection of State Locus onto Principal Coordinate a 15 a 1 a 11 t 5 a 5 a 1 (t) t 1 a 1 a 4 t t 4 a a 3 t 3 a 14 a 13 a (t) Maximization of Projection Maximization of variance M σ a1 = 1 Σa M 1j j =1 not maximization of mean square analyze fluctuating component only (zero mean)

9 Principal Coordinates a 1 (t) a (t) a 3 (t) Linear combination of original (physical) coordinates p 1 (t) p (t) p 3 (t) a 1 (t) = φ 11 p 1 (t) + φ 1 p (t) + φ 13 p 3 (t) a (t) = φ 1 p 1 (t) + φ p (t) + φ 3 p 3 (t) a 3 (t) = φ 31 p 1 (t) + φ 3 p (t) + φ 33 p 3 (t) {a} = [ [φ ]{p} Coordinate Transformation Matrix [φ ] [φ ] = [φ ij ] m-th row vector {φ m } = { {φ m1, φ m, φ m3 } T m-th Eigenvector

10 Maximization of Variance of the 1st Mode Principal Coordinate a 1 (t) Assumption - No correlation between principal coordinates a 1 (t) a (t) a 3 (t) - Unit norm of eigenvector φ m1 +φ m +φ m3 = 1 Variance of 1st Principal Coordinate a 1 (t) a1 = {a{ 1 (t)} σ a1 σ a1 = {φ 11 p 1 (t) + φ 1 p (t) + φ 13 p 3 (t)} = φ 11 σ 1 + φ 1 σ +φ 13 σ 3 + φ 11 φ 1 σ 1 + φ 11 φ 13 σ 13 + φ 1 φ 13 σ 3 a1 : Variance of the 1st principal coordinate a 1 (t) σ m : Variance of fluctuating pressure p m (t) σ mn : Covariance of p m (t) and p n (t)

11 Lagrange s s Method of Indeterminate Coefficients Maximizing variance σ a1 of a 1 (t) is equivalent to maximizing the following value L: L = σ a1 +λ (φ 11 + φ 1 + φ 13 1) λ : constant ( φ 11 + φ 1 + φ = 1 1 L Differential Coefficient of L by φ 1m L = 0 φ 11 L φ 1 L φ 13 φ 1m (σ 1 λ) φ 11 +σ 1 φ 1 +σ 13 φ 13 = 0 (a) σ 1 φ 11 + (σ λ) φ 1 +σ 3 φ 13 = 0 (b) σ 31 φ 11 +σ 3 φ 1 + (σ 3 λ) φ 13 = 0 (c)

12 Condition for Non-trivial Solution Determinant of Coefficients = 0 (σ 1 λ) σ 1 σ 13 σ 1 (σ λ) σ 3 = 0 σ 31 σ 3 (σ 3 λ) (d) Eigenvalue Problem of Matrix [R p ] Covariance Matrix [R p ] = σ 1 σ 1 σ13 σ 1 σ σ 3 σ 31 σ 3 σ 3 σ i : Variance σ ij = σ ji : Covariance

13 Equation (d) has three roots: λ = λ 1, λ, λ 3 (λ 1 λ λ 3 ) Eq.(a) φ 11 +Eq.(b) Eq.(b) φ 1 +Eq.(c) Eq.(c) φ 13 φ 11 σ 1 +φ 1 σ +φ 13 σ 3 +φ 11 φ 1 σ 1 +φ 11 φ 13 σ 13 +φ 1 φ 13 σ 3 λ (φ 11 + φ 1 + φ 13 ) = 0 σ a1 = a1 = λ Max. Variance of Principal Coordinate = Eigenvalue σ a1 Max. λ 1

14 Orthogonality of Eigenvectors of Matrix {φ m } T {φ n } = Σ φ mj φ nj = φ m1 φ n1 + φ m φ n + φ m3 φ n3 = δ mn δ mn mn : Kronecker s Delta = { 0, 1, m = n 0, m n Reconstruction of Pressure Field Reconstruction by principal coordinates a 1 (t),, a (t), a 3 (t) and eigenvectors p 1 (t) = φ 11 a 1 (t) + φ 1 a (t) + φ 31 a 3 (t) p (t) = φ 1 a 1 (t) + φ a (t) + φ 3 a 3 (t) p 3 (t) = φ 13 a 1 (t) + φ 3 a (t) + φ 33 a 3 (t) {p} = [ [φ ] T {a}

15 Proportion of m-th Mode Proportion of m-th Principal Coordinate Variance of m-th Principal Coordinate Variance of Original Pressure Field c m σ 1 + σ + σ 3 σ a1 σ am σ am a1 + σ a + σ a3 λ m a3 λ 1 + λ + λ 3 Proportion and Cumulative Proportion Proportion of m-th Principal Coordinate λ m c m MΣλ m = 1 m Cumulative Proportion up to N-th Mode C N N Σ c m = 1 m

16 Fluctuating Pressure Field P 1 (t) P (t) P 3 (t) t t t 1 t t 3 t 4 t 5 t Low-rise Building Model

17 Randomly Fluctuating Pressures Mean and Standard Deviation of Fluctuating Pressures Wind Pressures on Surfaces of a Low-rise Building Model Mean Pressure C pmean Standard Deviation C p

18 Eigenvectors of the Lowest Three Modes 1st Mode nd Mode 3rd Mode Eigenvalues,, Proportions and Cumulative Proportions Wind Pressures on Surfaces of a Low-rise Building Model Mode 1 st nd 3 rd 4 th 5 th 6 th 7 th 8 th 9 th 10 th Eigenvalue Proportion (%) Cumulative Proportion (%) st nd rd th th th th th th

19 (a) Mean pressure coefficient C p (b) Fluctuating pressure coefficient C p Pressure distributions on a low-rise building model (θ = 45, D : B : H = 4:4:1) Lowest Three Eigenvectors 1st Mode nd Mode 3rd Mode

20 High-rise Building Model Pressure Measurement Model and Analytical Lumped Mass Model Length Scale = 1/400 Mean Wind Speed Profile α = 1/6 500 Pressure Taps t = sec T = 4 sec (3,768 samples)

21 Mean and Standard Deviation of Fluctuating Pressures Wind Pressures on Surfaces of a High-rise Building Model Mean Pressure Standard Deviation Eigenvectors of Lowest Two Modes Fluctuating pressures acting on a high-rise building model 1st Mode nd Mode

22 Eigen Vectors of 3rd, 4th and 5th Modes Fluctuating pressures acting on a high-rise building model 3rd Mode 4th Mode 5th Mode 4th Mode Along-wind force coefficients at 0.H and 0.8H by 4th mode (Suburban flow, α=1/6)

23 5th Mode Along-wind force coefficients at 0.H and 0.8H by 5th mode (Suburban flow, α=1/6) Eigenvalues,, Proportions and Cumulative Proportions Wind Pressures on Surfaces of a High-rise Building Model Mode 1 st nd 3 rd 4 th 5 th Eigenvalue Proportion (%) Cumulative Proportion (%) st nd rd th th th th th th th

24 Power Spectral Densities of Wind Forces Generalized Wind Forces Lowest Five Principal Coordinates Fluctuating Wind Force Coefficients by Each Mode (α = 1/6) rms C Fx Along-wind Force rms C Fy Across-wind Force

25 Fluctuating Wind Force Coefficients by Each Mode (α = 1/6) rms C Mz Torsional Moment rms C Mz RMS Force Coefficients Reconstructed by Selected Dominant Modes ( (α = 1/6) 1,5,10,11,13,14,16,18, 1 and 31st Modes

26 Power Spectra of Generalized Wind Forces Reconstructed by Selected Dominant Modes ( (α = 1/6 = 1/6) Generalized Wind Forces Reconstructed from Selected Dominant Modes Along-wind Force Across-wind Force Torsional Moment

27 Response Analysis in Time Domain Analytical Condition Coupled oscillation of along- wind, across-wind and torsional components Newmark β method : β = 1/41 Time interval : t = 0.71s Calculation length : T = 600s Tip mean wind speed : (H=00m) V H = 55m/s (100y-recurrence in Tokyo) Analytical Model and Forces 5 Lumped masses 3DOF Fundamental natural periods : T 1X = T 1Y = 5s T 1θ = 1.3s Damping ratios : % to the critical Wind forces: - based on original pressures - based on reconstructed pressures by selected dominant modes Along-wind : nd, 3 rd & 4 th Across-wind : 1 st & 5 th Torsional : 1 st, 5 th...31 st (10 modes) Responses due to Wind Forces Reconstructed from Selected Dominant Modes Along-wind Displacement Across-wind Displacement Torsional Angle

28 Responses due to Wind Forces Reconstructed from Selected Dominant Modes Displacement Shear Force Responses due to Wind Forces Reconstructed from Selected Dominant Modes Wind Forces Response at Top (H=00m)( Along-wind Across-wind Angular Displacement Displacement Displacement (cm) (cm) (cm) Max S.D. Max S.D. Max S.D. Original Reconstructed from Selected Dominant Modes ( nd, 3 rd and 4 th ) (1 st and 5 th ) (10 selected*) Error (%) * 1 st, 5 th, 10 th, 11 th, 13 th, 14 th, 16 th, 18 th, 1 st and 31 st Modes

29 Coordinate Transformation Matrix [A][ Matrix [A][ ] as an operator for transforming the coordinate {b} = [A][ {a} Vector {a}{ is transformed to another Vector {b}{ of a different magnitude and a different direction by operation of Matrix [A][. Eigenvalue and Eigenvector [ ] ex Eigenvalues : λ = 4.41, 1.59 Eigenvectors: {1,.41} T, {1, 0.41} T

30 Coordinate Transformation Matrix and Eigenvalue / Eigenvector {b} = [ ] {a} 1 st Eigenvector st Eigenvector nd {b} {a} nd Eigenvector Eigenvector Coordinate Transformation Matrix and Eigenvalue / Eigenvector {b} {b} = [ ] {a} = λ {a} λ 1 1 {a} Magnification Factor = Eigenvalue λ 1 = 4.41 λ = 1.59 λ a : Unit Vector a = = 1

31 Coordinate Transformation Matrix and Eigenvalue / Eigenvector {b} = [ ] {a} {b} {a} 1 st Eigenvector st Eigenvector nd Eigenvector nd Eigenvector Coordinate Transformation Matrix and Eigenvalue / Eigenvector {b} = [ ] {a} λ {b} {a} 1 nd 1 st Eigenvector st Eigenvector nd Eigenvector Eigenvector

32 POD of Random Field with a Singular Condition [ ] ex. a 0 0 a = [R p ] Eigenvalues : λ = a (multiple root) Eigenvectors: Indeterminate! Uncorrelated with the same variances! Example of Eigenvectors Sample A 1st Mode nd Mode c 1 = 9.% c = 5.4%

33 Example of Eigenvectors Sample B 1st Mode nd Mode c 1 = 30.6% c = 4.7% State Locus by a 1 (t)) and a (t) nd Principal Coordinate nd Principal Coordinate 1st Principal Coordinate Sample A 1st Principal Coordinate Sample B

34 Eigenvalues and Proportions with/without Inclusion of Mean Value Mode 1 st nd Without Mean Value With Mean Value Eigenvalue Proportion (%) Eigenvalue Proportion (%) st nd p (t) 1 a (t) a 1m (t) a 1 (t) (p 1m, p m ) a m (t) p 1 (t) Merits of POD Observe phenomena by most efficient coordinates Extract hidden systematic structures from random information A significant reduction in amount of information that needs to be stored