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An Efficient Computational Solution Scheme

• f the Random Eigenvalue Problems

Rajib Chowdhury & Sondipon Adhikari School of Engineering Swansea University Swansea, UK

50th AIAA SDM Conference, 4-7 May 2009

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Outline

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

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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,

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• 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.

( ) ( ) ( ) ( )

t t t t + + =

• Mx

Cx Kx p

Random Eigenvalue Problem

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• 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

• n

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)

Random Eigenvalue Problem

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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

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Multidim ensional I ntegrals

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Multidim ensional I ntegrals

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Multidim ensional I ntegrals

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Mom ents of Eigenvalues

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Mom ents of Eigenvalues

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Mom ents of Eigenvalues

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Multivariate Gaussian Case

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Maxim um Entropy pdf

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Maxim um Entropy pdf

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Maxim um Entropy pdf

• With three moments

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HDMR

SYSTEM

Input

N

∈ x R Output ( ) y ∈ x R

Conjecture: Component functions arising in proposed decomposition will exhibit insignificant S-order effects cooperatively when S → N.

1 2 1 2 1 1 1 2 1 1 2 1 1 2

, , 1 ˆ ( 12 1 , 1 1 ˆ ( ( ) ) ) ˆ

( ) ( , , ) ( , , ) ( ( , ) )

S s S S S

N i i i i i N i i i y N i i i i i i i i i i y N i y N

y x y x y y y x x x y x x

= < = = < = = = <

= + + + + + +

∑ ∑ ∑

x x x

x

• Second-order (2D cooperative effects)

S-order (SD cooperative effects) First-

• rder

(Rabitz & Alis, 1999; Alis & Rabitz, 2001)

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1 2 1 2 1 2 1 2 1 2 1 1 1 2 2 2 2 1

( ) 1 1 1 1 ( ) ( ) 1 1 1 1 1 1 1

( ) ( ) ( , , , , , , ) ( , ) ( , ) ( , , , , , , , , , , )

n j i i j i i i i N j n n j j i i i i j j i i i i i i i i N j j

y x x y c c x c c y x x x x y c c x c c x c c

− + = − + − + = =

≅ φ ≅ φ

∑ ∑∑

• Lower-order Approximations

1 1 1 1 1 ( )

ˆ ˆ ( ) ( , , ) ( , , , , , , ) ( 1) ( )

i i

N I I N i i i N i y y x

y y x x y c c x c c N y

− + = = =

≡ ≡ − −

∑

• x

c

1 2 1 2 1 1 1 2 2 2 1 2 1 2

( , ) 1 1 1 1 1 1 , 1 1 1 1 1 ( )

ˆ ˆ ( ) ( , , ) ( , , , , , , , , , , ) ( 1)( 2) ( 2) ( , , , , , , ) ( ) 2

i i i i i i

y x x N II II N i i i i i i N i i i i N i i i N i y x y

y y x x y c c x c c x c c N N N y c c x c c y

= − + − + = < − + = = =

≡ ≡ − − + − − +

∑ ∑

• x

c

• First-order Approximation

Second-order Approximation

Interpolation function reference point

HDMR

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• Two-dimensional Taylor Series Expansion

( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )

2 2 1 2 1 2 1 2 1 2 1 2 1 1 2 2 1 1 2 1 2 1 2 2 2 1 2 1 2 2 2 1 1 2 2 2 2 1 2

, , , , , , , y c c y c c y c c y x x y c c x c x c x c x x x y c c y c c x c x c x c x x x ∂ ∂ ∂ = + − + − + − ∂ ∂ ∂ ∂ ∂ + − + − − + ∂ ∂ ∂

• (

) ( ) ( )( ) ( )( )

2 2 1 2 1 2 1 2 1 2 1 1 1 1 2 1 1

, , , , y c c y c c y x c y c c x c x c x x ∂ ∂ = + − + − + ∂ ∂

• One-dimensional Taylor Series Expansion

( ) ( ) ( )( ) ( )( )

2 2 1 2 1 2 1 2 1 2 2 2 2 2 2 2 2

, , , , y c c y c c y c x g c c x c x c x x ∂ ∂ = + − + − + ∂ ∂

• Taylor expansion at x1 = c1

Taylor expansion at x2 = c2 Taylor expansion at x1 = c1 and x2 = c2

Convergence Issue

(Li et.al., 2001)

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• Two-dimensional Taylor Series Expansion

( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )

2 2 1 2 1 2 1 2 1 2 1 2 1 1 2 2 1 1 2 1 2 1 2 2 2 1 2 1 2 2 2 1 1 2 2 2 2 1 2

, , , , , , , y c c y c c y c c y x x y c c x c x c x c x x x y c c y c c x c x c x c x x x ∂ ∂ ∂ = + − + − + − ∂ ∂ ∂ ∂ ∂ + − + − − + ∂ ∂ ∂

• Sum of Two One-dimensional Taylor Series

2D cooperative effect

( ) ( ) ( ) ( ) ( )( ) ( )( ) ( )( ) ( )( )

1 2 1 2 1 2 1 2 1 2 1 2 1 1 2 2 1 2 2 2 2 2 1 2 1 2 1 1 2 2 2 2 1 2

, , , , , , , , y c c y c c y x c y c x y c c y c c x c x c x x y c c y c c x c x c x x ∂ ∂ + − = + − + − ∂ ∂ ∂ ∂ + − + − + ∂ ∂

• Convergence Issue

(Li et.al., 2001)

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• Residual Error
• ŷ(x) represents reduced dimensional approximation, because
• nly 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 xi. Similarly, second-

• rder HDMR expansion is the sum of all Taylor series terms,

which contains only variable xi and xj. Therefore any truncated HDMR expansion provides better approximation

• f y(x) than any truncated Taylor series (e.g., FORM/SORM).

( )(

) ( )

1 2 1 2 1 1 2 2 1 2 2 1 1 2 1 2

1 2

1 ˆ ( ) ( ) ! !

j j j j i i i i j j j j i i i i

y y y x c x c j j x x

+ ∞ ∞ <

∂ − = − − ∂ ∂

∑∑ ∑

x x c

Approximate Exact

Errors in HDMR Approximation

(Li et.al., 2001)

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1

x

One Variable

2

x

1

x

Two Variable

1 ) 1 ( 1 1 1 1

( , , , , , ˆ ( ) ( ) ( 1) , ) ( )

N n j i i j j i i i N

y c c x x N c c y y

= = − +

≅ φ − −

∑ ∑

c x

• First-order Approximation

coefficients coefficients

HDMR (Continued)

Reference point Interpolation function

c c

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Second-order Approximation

HDMR (Continued)

1

x

2

x Two Variables

1 2 1 1 1 1 2 1 2 1 2 2 1 1 2 2 2 2

( ) ( ) 1 1 1 1 1 2 , 1 1 1 1 1 1 ( ) 1 1

( , , , , , , , , , , ) ( , , , ˆ ( ) ( , ) ( 1)( 2) ( 2) ( ) , , , ) 2 ) (

N n n j j i i i i j j j i i i i i i N j i j i i N i i N n j i i j

y c c x c c x c c y c c x c c y x N N N x y x

− + − + − = = = < = = +

≅ φ − − − − φ +

∑ ∑∑ ∑ ∑

• c

x coefficients c

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• Computational Effort (Calculating Coefficients)

( ) y c

( ) 1 1 1

( , , , , , , ( 1, , ; 1, , ) )

j i i i N

i y c N j n c x c c

− +

= =

• 1

2 1 1 1 2 2 2

( ) ( ) 1 1 1 1 1 1 2 1 2

( , , , , , , , , ( , 1, , ; , 1, , ) , , )

j j i i i i i i N

i i N y c c x c c j n c j x c

− + − +

= =

• No. of FEA for a linear/nonlinear problem,

1 FEA nN FEA N(N-1)n2/2 FEA First-order: (n-1)N + 1 (linear) Second-order: N(N-1)(n-1)2/2 + (n-1)N + 1 (quadratic)

HDMR (Continued)

(Chowdhury & Rao, 2009)

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Example 1: 2-DOF system

k1 m1 m2 k3 k2

1 2

m m ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ m

( ) ( ) ( )

1 3 3 3 2 3

k k k k k k ⎡ ⎤ + − = ⎢ ⎥ − + ⎣ ⎦ x k x x

1 2

1 0kg 1 5kg m . m . = =

( ) ( ) ( ) ( ) ( )

1 1 2 2 1

1000 1 0 25 N/m 1100 1 0 25 N/m 100N/m k . x k . x k = + = + = x x x

{ }

1 2

; ,

T

x ,x = ∑ =

x x

x μ = 0 I

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Exact eigenvalues for the 2-DOF system

Example 1: 2-DOF system

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Probability densities for the 2-DOF system

MCS SO-HDMR FO-HDMR

Example 1: 2-DOF system

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12

0 312 . ρ =

12

0 293 . ρ =

12

0 295 . ρ =

Scattered plot of λ1 & λ2

Example 1: 2-DOF system

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m1 m3 m2 k1 k2 k3 k4 k5 k6

Example 2: 3-DOF system

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( ) ( ) ( ) ( )

m m m ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦

1 2 3

x M x = x x

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 4 6 4 6 4 4 5 2 5 6 5 5 3 6

k k k k k k k k k k k k k k k k ⎡ ⎤ + + − − ⎢ ⎥ = − + + − ⎢ ⎥ ⎢ ⎥ − − + + ⎣ ⎦ x x x x x x x x x x x x x x x x

{ } ( ) ( )

2 1 9

Case 1 & Case 2 , , ; , 0 15 ; Case 30 3 ;

T

x x . . = ∑ = ν ν = ν =

• x

x

x μ = 0 I

( ) ( ) ( ) ( )

3 3 3 9 9

; 1 2 3 with 1 0kg; 1 2 3 ; 1, ,6 with 1 0N/m; 1, ,5 3 0N/m Case1 & Case 3 ; 1 275N/m Case2

i i i i i i i i

m x i , , . i , , k x i . i . .

+ + +

= μ = μ = = = μ = μ = = μ = μ =

• x

x

Example 2: 3-DOF system

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Probability densities for the 3-DOF system

Case 1: Well separated engenvalues

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12

0 558 . ρ =

12

0 553 . ρ =

12

0 551 . ρ =

Scattered plot of λ1 & λ2

Case 1: Well separated engenvalues

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13

0 434 . ρ =

13

0 438 . ρ =

13

0 433 . ρ =

Scattered plot of λ1 & λ3

Case 1: Well separated engenvalues

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23

0 234 . ρ =

23

0 221 . ρ =

23

0 232 . ρ =

Scattered plot of λ2 & λ3

Case 1: Well separated engenvalues

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Probability densities for the 3-DOF system

Case 2: Closely spaced engenvalues

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Case 3: Large statistical variation of input

Probability densities for the 3-DOF system

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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

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References

• H. Rabitz and Ö. Alis, “General Foundations of High Dimensional Model Representations,” Journal of

Mathematical Chemistry, 25, 197-233 (1999). Ö. Alis and H. Rabitz, “Efficient Implementation of High Dimensional Model Representations,” Journal

• f Mathematical Chemistry, 29, 127-142 (2001).
• G. Li, C. Rosenthal, and H. Rabitz, “High Dimensional Model Representations,” Journal of Physical

Chemistry A, 105, 7765-7777 (2001).

• R. Chowdhury and B.N. Rao, “Assessment of High Dimensional Model Representation Techniques for

Reliability Analysis,” Probabilistic Engineering Mechanics, 24(1), 100-115, (2009). W.E. Boyce, Probabilistic Methods in Applied Mathematics I, Academic Press, New York, 1968. 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),562−591, (2007).

• S. Mehlhose, J.V. Scheidt and R. Wunderlich, “Random Eigenvalue Problems for Bending Vibrations of

Beams“, Zeitschrift für Angewandte Mathematik und Mechanik, 79, 693–702, (1999). M.A. Grigoriu, “A Solution of the Random Eigenvalue Problem by Crossing Theory”, Journal of Sound and Vibration, 158(1), 69–80, (1992).

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