49 th AIAA SDM Conference, Schaumburg, IL, April 2008 PROBABILISTIC ANALYSIS OF DYNAMIC SYSTEMS WITH COMPLEX-VALUED EIGENSOLUTIONS EIGENSOLUTIONS Sharif Rahman The Uni ersit of Iowa The University of Iowa Iowa City, IA 52245 Work supported by U.S. National Science Foundation (CMMI-0653279)
OUTLINE Introduction Introduction Dimensional Decomposition Method Examples p Conclusions & Future Work
INTRODUCTION A General Random Eigenvalue Problem ( X N ) N N X : ( , ) ( , ) random eigenvector L or L random eigenvector L or L f Φ 0 ( X ); A X ( ), , A ( X ) ( X ) 1 K random matrices L L random eigenvalue or Example Random Eigenvalue Problem Problem/Application 1 1 Linear ; undamped or Linear ; undamped or Φ 0 proportionally damped systems ( X M X ) ( ) K X ( ) ( X ) 2 Quadratic ; non-proportionally Φ 0 2 ( damped systems; singularity X M X ) ( ) ( X C X ) ( ) K X ( ) ( X ) p problems 3 Palindromic ; acoustic emissions Φ 0 T ( X M X ) ( ) M ( X ) M ( X ) ( X ) ( X ) in high speed trains ( M 0 = M 1 T ) 1 0 1 4 Polynomial ; optimal control k Φ 0 problems ( X ) A ( X ) ( X ) k k 5 Rational ; plate vibration (m = 1) & m ( X C ) ( X ) Φ 0 fluid-solid structures (m = 2); k ( X M X ) ( ) K X ( ) ( X ) a ( X ) vibration of viscoelastic materials k k
INTRODUCTION Random Matrix Theory Approximate Methods Pioneering works by Wishart Dominated by perturbation (1928) Wigner Mehta and Dyson (1928), Wigner, Mehta, and Dyson methods methods Analytical solutions for classical Other methods ensembles (GOE, GUE, GSE) and Iteration method (Boyce) others Crossing theory (Grigoriu) C oss g t eo y (G go u) Asymptotic result yields statistical Reduced basis (Nair & Keane) Asymptotic method (Adhikari) solutions dependent only on the Polynomial chaos (Ghanem) global symmetry property of random matrices d t i Dim. Decomposition (Rahman) Impossible or highly non-trivial to Mostly used for real eigensolutions. apply for non-asymptotic or Complex-valued eigensolutions not general random matrices general random matrices studied extensively studied extensively Obj Objective: Develop dimensional decomposition method ti D l di i l d iti th d for solving complex-valued random eigenvalue problems
DIMENSIONAL DECOMPOSITION Decomposition for Quadratic Eigenvalue Problem D iti f Q d ti Ei l P bl ( ) x m 0 0 2 ( ) ( ) x M x x M x ( ) ( ) ( ) ( ) ( ) ( ) x C x x C x K x K x ( ) ( ) ( ) ( ) x x ( ) x ( ) x 1 ( ) x NONLINEAR N R I Input x Output SYSTEM ( ) ( ) x ( ) ( ) x 1 ( ) ( ) x R R I I N N N ( ) x ( x ) ( x , x ) ( x , , x ) m m ,0 m i , i m i i , i i m i , i i i 1 2 1 2 1 S 1 s Univariate Univariate i 1 i i , 1 i , , i 1 1 2 1 S i i i i i i i i 1 2 1 S (individual ˆ ( ) x , 1 m effects) Bivariate (2D cooperative effects) ˆ ( ) x m ,2 ˆ ( ) x m , S S variate (SD cooperative effects) S-variate (SD cooperative effects) Conjecture: Component functions arising in proposed decomposition will exhibit insignificant S -variate effects cooperatively when S N .
DIMENSIONAL DECOMPOSITION Lower-Variate Approximations L V i t A i ti Univariate Approximation reference point N N ˆ ˆ ( ) x ( , x , x ) ( , c , c , x c , , , c ) ( N 1) ( ) c m ,1 m ,1 1 N m 1 i 1 i i 1 N m i 1 ( x ) m i , i m ,0 Bivariate Approximation Bivariate Approximation ( x , x ) m i i ,1 2 i i 1 2 N ˆ ˆ ( ) x ( , x , x ) ( , c , c , x c , , , c , x , c , , c ) m ,2 m ,2 1 N m 1 i 1 i i 1 i 1 i i 1 N 1 1 1 2 2 2 i i , 1 1 2 i i i i 1 2 N ( 1)( 2) N N ( N 2) ( , c , c , x c , , , c ) ( ) c m 1 i 1 i i 1 N m 2 i 1 ( ) x m i , i m m ,0 ,0 n Lagrange ( ) j ( ) x ( ) x ( , c , c , x , c , , c ) m i , i j i m 1 i 1 i i 1 N shape h j j 1 1 n n functions ( j ) ( j ) ( x x , ) ( x ) ( x ) ( , c , c , x , c , , c , x , c , , c ) 1 2 m i i , i i j i j i m 1 i 1 i i 1 i 1 i i 1 N 1 2 1 2 1 1 2 2 1 1 1 2 2 2 j 1 j 1 2 1
DIMENSIONAL DECOMPOSITION Explicit Forms E li it F Univariate Approximation N N n ˆ ( j ) ( X ) ( X ) ( , c , c , x , c , , c ) ( N 1) ( ) c m ,1 j i m 1 i 1 i i 1 N m i 1 j 1 Bivariate Approximation Bivariate Approximation N n n ˆ ( j ) ( j ) ( X ) ( X ) ( X ) ( , c , c , x , c , , c , x , c , , c ) 1 2 m ,2 j i j i m 1 i 1 i i 1 i 1 i i 1 N 1 1 2 2 1 1 1 2 2 2 i i , 1 j 1 j 1 1 2 2 1 i i 1 2 N n ( N 1)( N 2) ( ) j ( N 2) ( X ) ( , c , c , x , c , , c ) ( c ) 1 1 1 j i m i i i N m 2 i 1 j 1 S-variate Approximation S variate Approximation S N S i 1 N n n ˆ i ( X ) ( 1) ( X ) ( X ) m S , j k j k i i 1 1 S i S i i i 0 0 k k , , k k 1 1 j j 1 1 j j 1 1 1 S i S i 1 k k 1 S i ( j ) ( j ) ( , c , c , x , c , , c , x , c , , c ) 1 S i m 1 k 1 k k 1 k 1 k k 1 N 1 1 1 S i S i S i
DIMENSIONAL DECOMPOSITION Computational Effort (Calculating Coefficients) C t ti l Eff t (C l l ti C ffi i t ) 2 ( 0 X M X ) ( ) ( X C X ) ( ) K X ( ) ( X ) 1 2 ( ) det c M c ( ) ( ) c C ( ) c K c ( ) 0 Char. 2 ( ) j ( ) j det ( , c , c , x , c , , c ) M ( , c , c , x , c , , c ) Eq. q 1 i 1 i i 1 N 1 i 1 i i 1 N nN (FEA) ( ) ( ) j j ( , c , c , x , c , , c ) C ( , c , c , x , c , , c ) 1 i 1 i i 1 N 1 i 1 i i 1 N ( ) j K ( , c , c , x , c , , c ) 0; i 1, , N j ; 1, , n 1 i 1 i i 1 N 2 2 ( ( j j ) ) ( ( j j ) ) d det ( , ( c , c , x , c , , c , x , c , , c ) ) 1 2 1 i 1 i i 1 i 1 i i 1 N 1 1 1 2 2 2 ( j ) ( j ) M ( , c , c , x , c , , c , x , c , , c ) 1 2 1 i 1 i i 1 i 1 i i 1 N 1 1 1 2 2 2 N ( N -1) n 2 /2 ( j ) ( j ) ( , c , c , x , c , , c , x , c , , c ) 1 2 1 i 1 i i 1 i 1 i i 1 N 1 1 1 2 2 2 ( j ) ( j ) C ( , c , c , x , c , , c , x , c , , c ) 1 2 1 i 1 i i 1 i 1 i i 1 N 1 1 1 2 2 2 ( j ) ( j ) K ( , c , c , x , c , , c , x , c , , c ) 0; , i i 1, , N j ; , j 1, , n 1 2 1 i 1 i i 1 i 1 i i 1 N 1 2 1 2 1 1 1 2 2 2 Univariate: nN + 1 (linear) Bivariate: N ( N -1) n 2 /2 + nN + 1 (quadratic)
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