An Efficient Computational Solution Scheme of the Random Eigenvalue Problems

Transcription

1 50th AIAA SDM Conference, 4-7 May 2009 An Efficient Computational Solution Scheme of the Random Eigenvalue Problems Rajib Chowdhury & Sondipon Adhikari School of Engineering Swansea University Swansea, UK

2 Outline Introduction Random Eigenvalue Problem High Dimensional Model Representation (HDMR) Examples Conclusions 2

3 Sources of uncertainty (a) parametric uncertainty - e.g., uncertainty in geometric parameters, friction coefficient, strength of the materials involved; (b) model inadequacy - arising from the lack of scientific knowledge about the model which is a-priori unknown; (c) experimental error - uncertain and unknown error percolate into the model when they are calibrated against experimental results; (d) computational uncertainty - e.g, machine precession, error tolerance and the so called h and p refinements in finite element analysis, 3

4 Random Eigenvalue Problem ( t) + ( t) + ( t) = ( t) Mx Cx Kx p Due to the presence of uncertainties, mass, damping and stiffness matrices are random matrices. The primary objectives are To quantify the uncertainties in system matrices. To estimate the variability of system responses. 4

5 Random Eigenvalue Problem Random eigenvalue of linear structural system K( X) Φ( X ) = Λ( X) M( X) Φ( X ) Main issues To find probabilistic characteristics of eigenpair. To find the joint statistics (moments, correlation). Several approaches are available on random eigenvalue problem, which are based on Perturbation method (Boyce, 1968; Zhang & Ellingwood, 1995 ) Iteration method (Boyce, 1968) Ritz method (Mehlhose, 1999) Crossing theory (Grigorie, 1992) Stochastic reduced basis (Nair & Keane, 2003) Asymptotic method (Adhikari, 2006) 5

6 Perturbation Method In the mean-centered approach α is the mean of x Alternatively, α can be obtained such that the any moment of each eigenvalue is calculated most accurately 6

7 Multidimensional Integrals 7

8 Multidimensional Integrals 8

9 Multidimensional Integrals 9

10 Moments of Eigenvalues 10

11 Moments of Eigenvalues 11

12 Moments of Eigenvalues 12

13 Multivariate Gaussian Case 13

14 Maximum Entropy pdf 14

15 Maximum Entropy pdf 15

16 Maximum Entropy pdf With three moments 16

17 HDMR Input N x R Output y( x) SYSTEM R N N y( x) = y0 + yi( xi) + yii ( x (,, ) 12 i, x ) 1 i + + y 2 i1 i x 1 12 ( 1,, ) S i xi + + y s N x xn i= 1 i, i = 1 i,, i = 1 Firstorder S i1 < i2 i1 < < is = yˆ 1( x) = yˆ 2 ( x) = yˆ ( x) S N S-order (SD cooperative effects) Second-order (2D cooperative effects) Conjecture: Component functions arising in proposed decomposition will exhibit insignificant S-order effects cooperatively when S N. (Rabitz & Alis, 1999; Alis & Rabitz, 2001) 17

18 HDMR Lower-order Approximations First-order Approximation reference point N I I yˆ ( x) yˆ ( x,, x ) y( c,, c, x, c,, c ) ( N 1) y( c) 1 N 1 i 1 i i+ 1 N i= 1 = y ( x ) Second-order Approximation N 1 2 i1< i2 i i = y ( x, x ) = y ii 12 i1 i2 II II yˆ ( x) yˆ ( x,, x ) y( c,, c, x, c,, c, x, c,, c ) 1 N 1 i1 1 i1 i1+ 1 i2 1 i2 i2+ 1 N i, i = 1 N ( N 1)( N 2) + ( N 2) y( c,, c, x, c,, c ) + y( c) 1 i 1 i i+ 1 N i= 1 2 = yi( xi) = y 0 0 n ( j1) ( j2) ii ( 1 2 i, 1 i ) φ 2 j1j ( 2 i, 1 i ) ( 2 1,, i1 1, i, 1 i1+ 1,, i2 1, i, 2 i2+ 1,, N) j = 1 j = n ( j) i i φj i 1 i 1 i i+ 1 N j= 1 y ( x ) ( x ) y( c,, c, x, c,, c ) n y x x x x y c c x c c x c c Interpolation function 18

19 Two-dimensional Taylor Series Expansion (, ) (, ) Convergence Issue 2 ( 1, 2) ( ) ( ) ( ) 1, 2 ( ) 1, 2 ( ) y c c y c c y c c y x x = y c c + x c + x c + x c x1 x2 x1 ( 1, 2) 2 y( c ) ( ) 1, c2 x c ( x c )( x c ) 2 2 y c c x x1 x2 Taylor expansion at x 1 = c 1 and x 2 = c 2 2 One-dimensional Taylor Series Expansion (, ) (, ) 2 ( 1, 2) y( c ) ( ) 1, c2 ( ) y c c 2 y x c = y c c + x c + x c x1 x1 (, ) (, ) Taylor expansion at x 1 = c 1 2 ( 1, 2) y( c ) ( ) 1, c2 ( ) y c c 2 y c x = g c c + x c + x c x2 x2 Taylor expansion at x 2 = c 2 (Li et.al., 2001) 19

20 Two-dimensional Taylor Series Expansion (, ) (, ) Convergence Issue 2 ( 1, 2) ( ) ( ) ( ) 1, 2 ( ) 1, 2 ( ) y c c y c c y c c y x x = y c c + x c + x c + x c x1 x2 x1 ( 1, 2) 2 y( c ) ( ) 1, c2 x c ( x c )( x c ) 2 2 y c c x x1 x2 2 2D cooperative effect Sum of Two One-dimensional Taylor Series (, ) (, ) (, ) (, ) ( 1, 2) y( c ) ( ) 1, c2 ( ) y c c y x c + y c x y c c = y c c + x c + x c x1 x2 ( 1, 2) y( c ) ( ) 1, c2 x c ( x c ) 2 2 y c c x x2 (Li et.al., 2001) 20

21 Residual Error j1+ j2 1 y y( x) yˆ ( x) = c x c x c j j j! j! x x Exact Errors in HDMR Approximation Approximate < j j i i i i j ( )( i i ) ( i i ) ŷ(x) represents reduced dimensional approximation, because only N number of 1-dimensional model approximation are required, as opposed to one N-dimensional approximation in y(x). If higher partial derivatives are negligibly small, ŷ(x) provides a convenient approximation of y(x) First-order HDMR expansion is the sum of all Taylor series terms, which contains only variable x i. Similarly, secondorder HDMR expansion is the sum of all Taylor series terms, which contains only variable x i and x j. Therefore any truncated HDMR expansion provides better approximation of y(x) than any truncated Taylor series (e.g., FORM/SORM). j (Li et.al., 2001) 21

22 HDMR (Continued) First-order Approximation Reference point yˆ ( x) φ ( x ) yc (,, c, x, c,, c ) ( N 1) y( c) 1 N n i= 1 j= 1 j i ( j) 1 i 1 i i+ 1 N coefficients Interpolation function coefficients x 2 c One Variable x 1 c x 1 Two Variable 22

23 HDMR (Continued) Second-order Approximation yˆ ( x) φ ( x, ) yc c x c c x c c 2 N n n i, i = 1 j = 1 j = i1< i2 i= 1 j= ( j1) ( j2) ( 1,, i1 1, i, 1 i1 1,, i2 1, i, 2 i2 1,, N) jj i xi + + coefficients N n ( j) ( N 1)( N 2) ( N 2) φ j( xi) yc ( 1,, ci 1, xi, ci+ 1,, cn) + y( c) 2 x 2 c x 1 Two Variables 23

24 HDMR (Continued) Computational Effort (Calculating Coefficients) No. of FEA for a linear/nonlinear problem, y( c) 1 FEA yc (,, c, x, c,, c ( j) 1 i 1 i i+ 1 N ( i = 1,, N; j = 1,, n) ) nn FEA yc c x c c x c,, c ) ( j1) ( j2) (,,,,,,,, i 1 i i + 1 i 1 i i + 1 N ( i, i = 1,, N; j, j = 1,, n) N(N-1)n 2 /2 FEA First-order: (n-1)n + 1 (linear) Second-order: N(N-1)(n-1) 2 /2 + (n-1)n + 1 (quadratic) (Chowdhury & Rao, 2009) 24

25 Example 1: 2-DOF system k 1 m 1 m 2 k 3 k 2 m m 0 1 = 0 m 2 ( ) k x ( x) k + k k = k3 k2( x) + k3 m = 10kg. 1 m = 15kg. 2 ( x) = ( + ) ( x) = ( + ) ( x) = 100 N/m k x N/m 1 1 k x N/m k { x,x} T x = 1 2 ; μ x =0, x = I 25

26 Example 1: 2-DOF system Exact eigenvalues for the 2-DOF system 26

27 Example 1: 2-DOF system MCS FO-HDMR SO-HDMR Probability densities for the 2-DOF system 27

28 Example 1: 2-DOF system ρ 12 = ρ 12 = ρ 12 = Scattered plot of λ 1 & λ 2 28

29 Example 2: 3-DOF system k 6 k 1 k 3 m 1 k 4 k 5 m 3 m 2 k 2 29

30 ( ) m1 x 0 0 M( x ) = 0 m2 ( x) m3 x ( ) ( x) + ( x) + ( x) ( x) ( x) 4( ) 4( ) 5( ) 2( ) 5( ) k ( x) k ( x) k ( x) + k ( x) + k ( x) k1 k4 k6 k4 k6 k( x) = k x k x + k x + k x k x ( x) ( x) i i+ 3 i+ 3 i+ 3 Example 2: 3-DOF system m = μ x; i = 1, 2, 3 with μ = 1. 0kg; i = 1, 2, 3 i i i i k =μ x ; i= 1,,6 with μ = 10N/m;. i= 1,,5 ( ) ( ) μ = 3. 0 N/m Case1 & Case 3 ; μ = N/m Case2 9 9 T { x1,, x9} ;,. ( Case 1 & Case 2) ( Case 3 ); 2 x = μ x =0 x =ν I ν= 015 ; ν= 30

31 Case 1: Well separated engenvalues Probability densities for the 3-DOF system 31

32 Case 1: Well separated engenvalues ρ 12 = ρ 12 = ρ 12 = Scattered plot of λ 1 & λ 2 32

33 Case 1: Well separated engenvalues ρ 13 = ρ 13 = ρ 13 = Scattered plot of λ 1 & λ 3 33

34 Case 1: Well separated engenvalues ρ 23 = ρ 23 = ρ 23 = Scattered plot of λ 2 & λ 3 34

35 Case 2: Closely spaced engenvalues Probability densities for the 3-DOF system 35

36 Case 3: Large statistical variation of input Probability densities for the 3-DOF system 36

37 Conclusions The statistics of the eigenvalues of linear stochastic dynamic systems has been Considered HDMR approximation method has been developed for efficient scheme for random eigenvalue problems. Pdf of the eigenvalues are obtained using using the maximum entropy method Yields accurate and convergent solutions Future works will look into joint moments and pdf of the eigenvalues and eigenvectors 37

38 References H. Rabitz and Ö. Alis, General Foundations of High Dimensional Model Representations, Journal of Mathematical Chemistry, 25, (1999). Ö. Alis and H. Rabitz, Efficient Implementation of High Dimensional Model Representations, Journal of Mathematical Chemistry, 29, (2001). G. Li, C. Rosenthal, and H. Rabitz, High Dimensional Model Representations, Journal of Physical Chemistry A, 105, (2001). R. Chowdhury and B.N. Rao, Assessment of High Dimensional Model Representation Techniques for Reliability Analysis, Probabilistic Engineering Mechanics, 24(1), , (2009). W.E. Boyce, Probabilistic Methods in Applied Mathematics I, Academic Press, New York, P.B. Nair and A.J. Keane, An Approximate Solution Scheme for the Algebraic Random Eigenvalue Problem, Journal of Sound and Vibration, 260(1), 45 65, (2003). S. Adhikari and M.I. Friswell, Random Matrix Eigenvalue Problems in Structural Dynamics, International Journal for Numerical Methods in Engineering, 69(3), , (2007). S. Mehlhose, J.V. Scheidt and R. Wunderlich, Random Eigenvalue Problems for Bending Vibrations of Beams, Zeitschrift für Angewandte Mathematik und Mechanik, 79, , (1999). M.A. Grigoriu, A Solution of the Random Eigenvalue Problem by Crossing Theory, Journal of Sound and Vibration, 158(1), 69 80, (1992). 38

39 39