A Reduced Orthogonal Projection Approach for Stochastic Finite - PowerPoint PPT Presentation

A Reduced Orthogonal Projection Approach for Stochastic Finite Element Analysis S Adhikari 1 1 Swansea University, UK The University of Liverpool Adhikari (Swansea) Reduced Projection Approach for SFEM 14 September 2010 1 / 58

Outline of the talk Introduction 1 Uncertainty in computational mechanics Stochastic elliptic PDEs Spectral decomposition in a vector space 2 Projection in a finite dimensional vector-space Properties of the spectral functions Error minimization in the Hilbert space 3 The Galerkin approach Computational method Numerical illustration 4 ZnO nanowires Results for larger correlation length Results for smaller correlation length Conclusions 5 Acknowledgements 6 Adhikari (Swansea) Reduced Projection Approach for SFEM 14 September 2010 2 / 58

Introduction Uncertainty in computational mechanics 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, and (e) model uncertainty - genuine randomness in the model such as uncertainty in the position and velocity in quantum mechanics, deterministic chaos. Adhikari (Swansea) Reduced Projection Approach for SFEM 14 September 2010 3 / 58

Introduction Stochastic elliptic PDEs Stochastic elliptic PDE We consider the stochastic elliptic partial differential equation (PDE) − ∇ [ a ( r , θ ) ∇ u ( r , θ )] = p ( r ); r in D (1) with the associated boundary condition u ( r , θ ) = 0 ; r on ∂ D (2) Here a : R d × Θ → R is a random field, which can be viewed as a set of random variables indexed by r ∈ R d . We assume the random field a ( r , θ ) to be stationary and square integrable. Based on the physical problem the random field a ( r , θ ) can be used to model different physical quantities. Adhikari (Swansea) Reduced Projection Approach for SFEM 14 September 2010 4 / 58

Introduction Stochastic elliptic PDEs Discretized Stochastic PDE The random process a ( r , θ ) can be expressed in a generalized fourier type of series known as the Karhunen-Lo` eve expansion ∞ � √ ν i ξ i ( θ ) ϕ i ( r ) a ( r , θ ) = a 0 ( r ) + (3) i = 1 Here a 0 ( r ) is the mean function, ξ i ( θ ) are uncorrelated standard Gaussian random variables, ν i and ϕ i ( r ) are eigenvalues and eigenfunctions satisfying the integral equation � C a ( r 1 , r 2 ) ϕ j ( r 1 ) d r 1 = ν j ϕ j ( r 2 ) , ∀ j = 1 , 2 , · · · (4) D Adhikari (Swansea) Reduced Projection Approach for SFEM 14 September 2010 5 / 58

Introduction Stochastic elliptic PDEs Exponential autocorrelation function The autocorrelation function: C ( x 1 , x 2 ) = e −| x 1 − x 2 | / b (5) The underlying random process H ( x , θ ) can be expanded using the Karhunen-Lo` eve expansion in the interval − a ≤ x ≤ a as � ∞ � H ( x , θ ) = ξ j ( θ ) λ j ϕ j ( x ) (6) j = 1 Using the notation c = 1 / b , the corresponding eigenvalues and eigenfunctions for odd j are given by cos ( θ j x ) 2 c c λ j = j + c 2 , ϕ j ( x ) = , tan ( θ j a ) = , (7) where θ 2 � θ j sin ( 2 θ j a ) � � a + � 2 θ j and for even j are given by 2 c sin ( θ j x ) θ j λ j = θ j 2 + c 2 , ϕ j ( x ) = , where tan ( θ j a ) = . (8) � − c sin ( 2 θ j a ) � � � a − 2 θ j Adhikari (Swansea) Reduced Projection Approach for SFEM 14 September 2010 6 / 58

Introduction Stochastic elliptic PDEs Example: A beam with random properties The equation of motion of an undamped Euler-Bernoulli beam of length L with random bending stiffness and mass distribution: � � ∂ 2 EI ( x , θ ) ∂ 2 Y ( x , t ) + ρ A ( x , θ ) ∂ 2 Y ( x , t ) = p ( x , t ) . (9) ∂ x 2 ∂ x 2 ∂ t 2 Y ( x , t ) : transverse flexural displacement, EI ( x ) : flexural rigidity, ρ A ( x ) : mass per unit length, and p ( x , t ) : applied forcing. Consider EI ( x , θ ) = EI 0 ( 1 + ǫ 1 F 1 ( x , θ )) (10) and ρ A ( x , θ ) = ρ A 0 ( 1 + ǫ 2 F 2 ( x , θ )) (11) The subscript 0 indicates the mean values, 0 < ǫ i << 1 ( i =1,2) are deterministic constants and the random fields F i ( x , θ ) are taken to have zero mean, unit standard deviation and covariance R ij ( ξ ) . Adhikari (Swansea) Reduced Projection Approach for SFEM 14 September 2010 7 / 58

Introduction Stochastic elliptic PDEs Example: A beam with random properties We can express the shape functions for the finite element analysis of Euler-Bernoulli beams as N ( x ) = R s ( x ) (12) where − 3 2   1 0 ℓ e 2 ℓ e 3       − 2 1   0 1    ℓ e 2 ℓ e 2  1 , x , x 2 , x 3 � T .   � R = s ( x ) = (13)   and   3 − 2   0 0   ℓ e 2 ℓ e 3         − 1 1   0 0 ℓ e 2 ℓ e 2 The element stiffness matrix: � ℓ e � ℓ e ′′ T ′′ T ′′ ′′ K e ( θ ) = N ( x ) EI ( x , θ ) N ( x ) dx = EI 0 ( 1 + ǫ 1 F 1 ( x , θ )) N ( x ) N ( x ) dx . (14) 0 0 Adhikari (Swansea) Reduced Projection Approach for SFEM 14 September 2010 8 / 58

Introduction Stochastic elliptic PDEs Example: A beam with random properties Expanding the random field F 1 ( x , θ ) in KL expansion K e ( θ ) = K e 0 + ∆ K e ( θ ) (15) where the deterministic and random parts are � ℓ e � N K � ′′ T ( x ) dx ′′ ( x ) N K e 0 = EI 0 N and ∆ K e ( θ ) = ǫ 1 ξ K j ( θ ) λ K j K ej . 0 j = 1 (16) The constant N K is the number of terms retained in the Karhunen-Lo` eve expansion and ξ K j ( θ ) are uncorrelated Gaussian random variables with zero mean and unit standard deviation. The constant matrices K ej can be expressed as � ℓ e ′′ T ( x ) dx ′′ ( x ) N K ej = EI 0 ϕ K j ( x e + x ) N (17) 0 Adhikari (Swansea) Reduced Projection Approach for SFEM 14 September 2010 9 / 58

Introduction Stochastic elliptic PDEs Example: A beam with random properties The mass matrix can be obtained as M e ( θ ) = M e 0 + ∆ M e ( θ ) (18) The deterministic and random parts is given by � ℓ e � N M � N ( x ) N T ( x ) dx M e 0 = ρ A 0 ∆ M e ( θ ) = ǫ 2 ξ M j ( θ ) λ M j M ej . and 0 j = 1 (19) The constant N M is the number of terms retained in Karhunen-Lo` eve expansion and the constant matrices M ej can be expressed as � ℓ e ϕ M j ( x e + x ) N ( x ) N T ( x ) dx . M ej = ρ A 0 (20) 0 Adhikari (Swansea) Reduced Projection Approach for SFEM 14 September 2010 10 / 58

Introduction Stochastic elliptic PDEs Example: A beam with random properties These element matrices can be assembled to form the global random stiffness and mass matrices of the form K ( θ ) = K 0 + ∆ K ( θ ) and M ( θ ) = M 0 + ∆ M ( θ ) . (21) Here the deterministic parts K 0 and M 0 are the usual global stiffness and mass matrices obtained form the conventional finite element method. The random parts can be expressed as N K � N M � � � ∆ K ( θ ) = ǫ 1 ξ K j ( θ ) λ K j K j and ∆ M ( θ ) = ǫ 2 ξ M j ( θ ) λ M j M j j = 1 j = 1 (22) The element matrices K ej and M ej can be assembled into the global matrices K j and M j . The total number of random variables depend on the number of terms used for the truncation of the infinite series. This in turn depends on the respective correlation lengths of the underlying random fields. Adhikari (Swansea) Reduced Projection Approach for SFEM 14 September 2010 11 / 58

Introduction Stochastic elliptic PDEs Discrete equation for stochastic mechanics Truncating the KL expansion upto the M -th term and discretising the displacement field, the equation for static deformation can be expresses as � � M � A 0 + ξ i ( θ ) A i u ( θ ) = f (23) i = 1 The aim is to efficiently solve for u ( θ ) . Adhikari (Swansea) Reduced Projection Approach for SFEM 14 September 2010 12 / 58

Introduction Stochastic elliptic PDEs Polynomial Chaos expansion Using the Polynomial Chaos expansion, the solution (a vector valued function) can be expressed as ∞ � u ( θ ) = u i 0 h 0 + u i 1 h 1 ( ξ i 1 ( θ )) i 1 = 1 ∞ i 1 � � + u i 1 , i 2 h 2 ( ξ i 1 ( θ ) , ξ i 2 ( θ )) i 1 = 1 i 2 = 1 ∞ i 1 i 2 � � � + u i 1 i 2 i 3 h 3 ( ξ i 1 ( θ ) , ξ i 2 ( θ ) , ξ i 3 ( θ )) i 1 = 1 i 2 = 1 i 3 = 1 i 1 i 2 i 3 ∞ � � � � + u i 1 i 2 i 3 i 4 h 4 ( ξ i 1 ( θ ) , ξ i 2 ( θ ) , ξ i 3 ( θ ) , ξ i 4 ( θ )) + . . . , i 1 = 1 i 2 = 1 i 3 = 1 i 4 = 1 Here u i 1 ,..., i p ∈ R n are deterministic vectors to be determined. Adhikari (Swansea) Reduced Projection Approach for SFEM 14 September 2010 13 / 58

Introduction Stochastic elliptic PDEs Polynomial Chaos expansion After the finite truncation, concisely, the polynomial chaos expansion can be written as P � ˆ u ( θ ) = H k ( ξ ( θ )) u k (24) k = 1 where H k ( ξ ( θ )) are the polynomial chaoses. The value of the number of terms P depends on the number of basic random variables M and the order of the PC expansion r as r � ( M + j − 1 )! P = (25) j !( M − 1 )! j = 0 Adhikari (Swansea) Reduced Projection Approach for SFEM 14 September 2010 14 / 58