Response statistics of linear stochastic systems: A simultaneous diagonalisation approach C. F. Li & S Adhikari School of Engineering, University of Wales Swansea, Swansea, U.K. Email: C.F.Li@swansea.ac.uk URL: http://www.wimcs.ac.uk/members/stochastic/li c.html Email: S.Adhikari@swansea.ac.uk URL: http://engweb.swan.ac.uk/ ∼ adhikaris Palm Springs, CA, 5 May 2009 Simultaneous diagonalisation of stochastic systems – p.1/38
Outline Background and Motivation Discretisation of stochastic material parameters Current methods for response-statistics calculation Joint diagonalisation approach Numerical examples Conclusions Palm Springs, CA, 5 May 2009 Simultaneous diagonalisation of stochastic systems – p.2/38
Background In many stochastic mechanics problems we need to solve a system of linear stochastic equations: Ku = f . (1) K ∈ R n × n is a n × n real non-negative definite random matrix, f ∈ R n is a n -dimensional real deterministic input vector and u ∈ R n is a n -dimensional real uncertain output vector which we want to determine. This typically arise due to the discretisation of stochastic partial differential equations (eg. in the stochastic finite element method) Palm Springs, CA, 5 May 2009 Simultaneous diagonalisation of stochastic systems – p.3/38
Background In the context of linear structural mechanics, K is known as the stiffness matrix, f is the forcing vector and u is the vector of structural displacements. Often, the objective is to determine the probability density function (pdf) and consequently the cumulative distribution function (cdf) of u . This will allow one to calculate the reliability of the system. It is generally difficult to obtain the probably density function (pdf) of the response. As a consequence, engineers often intend to obtain only the fist few moments (typically the fist two) of the response quantity. Palm Springs, CA, 5 May 2009 Simultaneous diagonalisation of stochastic systems – p.4/38
Objectives We propose a joint diagonalisation method for the solution of stochastic linear systems. The method is based on the recently developed joint diagonalisation solution strategy and the Neumann expansion of inverse matrix. The joint diagonalisation method is applicable to stochastic linear systems with arbitrary number of random variables and has no registration on the type of probability distribution. Palm Springs, CA, 5 May 2009 Simultaneous diagonalisation of stochastic systems – p.5/38
Example of heterogeneous materials: 1D 4 baseline value 3.5 perturbed values 3 2.5 EI (Nm 2 ) 2 1.5 1 0.5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Length along the beam (m) An example of heterogeneous property in one dimension Palm Springs, CA, 5 May 2009 Simultaneous diagonalisation of stochastic systems – p.6/38
Example of heterogeneous materials: 2D An example of heterogeneous property in two dimensions Palm Springs, CA, 5 May 2009 Simultaneous diagonalisation of stochastic systems – p.7/38
Discretisation of stochastic material parameters middle point method local averaging method shape function method least-squares discretization method trigonometric series approximation K-L expansion method F-K-L expansion method Palm Springs, CA, 5 May 2009 Simultaneous diagonalisation of stochastic systems – p.8/38
Karhunen-Loève expansion A second-order stochastic field can be represented as: + ∞ � � b ( x , ω ) = E ( b ( x , ω )) + λ i ξ i ( ω ) ψ i ( x ) (2) i =1 The required eigen-structure is obtained through solving a generalized eigen-value problem � Cov ( b ( x 1 , ω ) , b ( x 2 , ω )) ψ ( x 1 ) d x 1 = λψ ( x 2 ) ⇒ Av = λ Bv D (3) Palm Springs, CA, 5 May 2009 Simultaneous diagonalisation of stochastic systems – p.9/38
Karhunen-Loève expansion After sorting from high to low the eigen-values of stochastic field, the K-L expansion is optimal in terms of approximation of the total variance of the random material parameter + ∞ � � λ i = Cov ( b ( x , ω ) , b ( x , ω )) d x (4) D i =1 Palm Springs, CA, 5 May 2009 Simultaneous diagonalisation of stochastic systems – p.10/38
Fourier-Karhunen-Loève expansion For stationary stochastic fields in regular domains, the Fourier-Karhunen-Loève expansion is developed by Li et al. 13,14 and by combining the representation theory of stationary stochastic fields, the F-K-L result is much more accurate and is obtained explicitly without solving any equation. � √− 1 xy dZ ( y , ω ) a ( x , ω ) = a o ( x ) + (5) R n e 1 � n R ( τ ) e −√− 1 τ y dτ f ( y ) = (6) (2 π ) n Palm Springs, CA, 5 May 2009 Simultaneous diagonalisation of stochastic systems – p.11/38
Current Approaches The random matrix can be represented as K = K 0 + ∆K (7) K 0 ∈ R n × n is the deterministic part and the random part: m m j � � � ξ j K I ξ j ξ l K II ∆K = j + jl + · · · (8) j =1 j =1 l =1 m is the number of random variables, K I j , K II jl ∈ R n × n , ∀ j, l are deterministic matrices and ξ j , ∀ j are real random variables. Palm Springs, CA, 5 May 2009 Simultaneous diagonalisation of stochastic systems – p.12/38
Perturbation based approach Represent the response as j m u = u 0 + ξ j u I � � ξ j ξ l u II j + jl + · · · . (9) j =1 l =1 where u 0 = K 0 − 1 f (10) j = − K 0 − 1 K I j u 0 , u I ∀ j (11) jl u 0 + K I jl = − K 0 − 1 [ K II u II j u I l + K I l u I and j ] , ∀ j, l. (12) Palm Springs, CA, 5 May 2009 Simultaneous diagonalisation of stochastic systems – p.13/38
Neumann expansion � � � K 0 − 1 ∆K Provided F < 1 , � � � � − 1 K − 1 = � K 0 ( I n + K 0 − 1 ∆K ) = K 0 − 1 − K 0 − 1 ∆KK 0 − 1 + K 0 − 1 ∆KK 0 − 1 ∆KK 0 − 1 + · · · . Therefore, u = K − 1 f = u 0 − Tu 0 + T 2 u 0 + · · · (13) where T = K 0 − 1 ∆K ∈ R n × n is a random matrix. Palm Springs, CA, 5 May 2009 Simultaneous diagonalisation of stochastic systems – p.14/38
Projection methods Here one ‘projects’ the solution vector onto a complete stochastic basis. Depending on how the basis is selected, several methods are proposed. Using the classical Polynomial Chaos (PC) projection scheme P − 1 � u = u j Ψ j ( ξ ) (14) j =0 where u j ∈ R n , ∀ j are unknown vectors and Ψ j ( ξ ) are multidimensional Hermite polynomials in ξ r . Palm Springs, CA, 5 May 2009 Simultaneous diagonalisation of stochastic systems – p.15/38
A partial summary Methods Sub-methods First and second order perturbation 11,22 , 1. Perturbation Neumann expansion 1,26 , based methods improved perturbation method 3 . Polynomial chaos expansion 7 , 2. Projection methods random eigenfunction expansion 1 , stochastic reduced basis method 16,18,19 , Wiener − Askey chaos expansion 23–25 , domain decomposition method 20,21 . Simulation methods 8,17 , 3. Monte carlo simulation Analytical method in references 5,6,9,10,15 , and other methods Exact solutions for beams 2,4 . Palm Springs, CA, 5 May 2009 Simultaneous diagonalisation of stochastic systems – p.16/38
Joint diagonalisation approach We consider a stochastic linear system: ( K 0 + ξ 1 ( ω ) K 1 + ξ 2 ( ω ) K 2 + · · · + ξ m ( ω ) K m ) u = f , (15) where K j , ∀ j are real symmetric deterministic matrices and ξ j ( ω ) , ∀ j are real random variables. The above stochastic linear system is commonly obtained in a SFEM formulation after discretising the random material parameters with the K-L (or F-K-L) expansion method. Palm Springs, CA, 5 May 2009 Simultaneous diagonalisation of stochastic systems – p.17/38
Joint diagonalisation approach By using a sequence of Givens transformations, stiffness matrices K j can be simultaneously diagonalised such that Q − 1 K j Q = Λ j + ∆ j ≈ Λ j , j = 1 , 2 , · · · , m (16) where Λ j , ∀ j are diagonal matrices and ∆ j , ∀ j are matrices with zero diagonal entries and small magnitude off-diagonal entries. The transform matrix Q is explicitly obtained as the product of the Givens rotation matrices Q = G T 1 G T 2 · · · G T (17) k where k is the total number of Givens transformations and G j , ∀ j are Givens rotation matrices Palm Springs, CA, 5 May 2009 Simultaneous diagonalisation of stochastic systems – p.18/38
Givens rotation matrix p q 1 ... 1 cos θ sin θ p 1 ... G = G ( p, q, θ ) ∆ = (18) . 1 − sin θ cos θ q 1 ... 1 Palm Springs, CA, 5 May 2009 Simultaneous diagonalisation of stochastic systems – p.19/38
Optimal Givens rotation angle For n × n real symmetric matrices K l let n n ∆ � � ( K l ) 2 off( K l ) = ( l = 1 , · · · , m ) (19) ij i =1 j =1 j � = i denote the quadratic sums of off-diagonal elements in K l . The m � aim here is to gradually reduce off( K l ) through a sequence of l =1 orthogonal similarity transformations that have no effect on � K l � F , the Frobenius norm of K l . Palm Springs, CA, 5 May 2009 Simultaneous diagonalisation of stochastic systems – p.20/38
Recommend
More recommend
