Stochastic Finite Element Analysis of Uncertain Structural Systems S Adhikari 1 1 Swansea University, UK University of Edinburgh, Edinburgh Adhikari (SU) SFEM for Structural Systems 16 June 2010 1 / 62
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 (SU) SFEM for Structural Systems 16 June 2010 2 / 62
Introduction Uncertainty in computational mechanics A general overview Adhikari (SU) SFEM for Structural Systems 16 June 2010 3 / 62
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 (SU) SFEM for Structural Systems 16 June 2010 4 / 62
Introduction Uncertainty in computational mechanics Problem-types in structural mechanics Input System Output Problem name Main techniques Known (determinis- Known (determinis- Unknown Analysis (forward FEM/BEM/Finite tic) tic) problem) difference Known (determinis- Incorrect (determinis- Known (determinis- Updating/calibration Modal updating tic) tic) tic) Known (determinis- Unknown Known (determinis- System identification Kalman filter tic) tic) Assumed (determin- Unknown (determin- Prescribed Design Design optimisation istic) istic) Unknown Partially Known Known Structural Health Mon- SHM methods itoring (SHM) Known (determinis- Known (determinis- Prescribed Control Modal control tic) tic) Known (random) Known (determinis- Unknown Random vibration Random vibration tic) methods Adhikari (SU) SFEM for Structural Systems 16 June 2010 5 / 62
Introduction Uncertainty in computational mechanics Problem-types in structural mechanics Input System Output Problem name Main techniques Known (determinis- Known (random) Unknown Stochastic analysis SFEM/SEA/RMT tic) (forward problem) Known (random) Incorrect (random) Known (random) Bayesian calibration Probabilistic updat- ing/calibration Assumed (ran- Unknown (random) Prescribed (random) Probabilistic design RBOD dom/deterministic) Known (ran- Partially known (ran- Partially known (ran- Joint state and param- Particle Kalman Fil- dom/deterministic) dom) dom) eter estimation ter/Ensemble Kalman Filter Known (ran- Known (random) Known from experi- Model validation Validation methods dom/deterministic) ment and model (ran- dom) Known (ran- Known (random) Known from differ- Model verification verification methods dom/deterministic) ent computations (random) Adhikari (SU) SFEM for Structural Systems 16 June 2010 6 / 62
Introduction Stochastic elliptic PDEs Stochastic elliptic PDE We consider the stochastic elliptic partial differential equation (PDE) − ∇ [ a ( r , ω ) ∇ u ( r , ω )] = p ( r ); r in D (1) with the associated boundary condition u ( r , ω ) = 0 ; r on ∂ D (2) Here a : R d × Ω → 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 , ω ) to be stationary and square integrable. Based on the physical problem the random field a ( r , ω ) can be used to model different physical quantities. Adhikari (SU) SFEM for Structural Systems 16 June 2010 7 / 62
Introduction Stochastic elliptic PDEs Discretized Stochastic PDE The random process a ( r , ω ) can be expressed in a generalized fourier type of series known as the Karhunen-Lo` eve expansion ∞ � √ ν i ξ i ( ω ) ϕ i ( r ) a ( r , ω ) = a 0 ( r ) + (3) i = 1 Here a 0 ( r ) is the mean function, ξ i ( ω ) 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 (SU) SFEM for Structural Systems 16 June 2010 8 / 62
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 ( ω j x ) 2 c c λ j = j + c 2 , ϕ j ( x ) = , tan ( ω j a ) = , (7) where ω 2 � ω j sin ( 2 ω j a ) � � a + � 2 ω j and for even j are given by 2 c sin ( ω j x ) ω j λ j = ω j 2 + c 2 , ϕ j ( x ) = , where tan ( ω j a ) = . (8) � − c sin ( 2 ω j a ) � � � a − 2 ω j Adhikari (SU) SFEM for Structural Systems 16 June 2010 9 / 62
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 (SU) SFEM for Structural Systems 16 June 2010 10 / 62
Introduction Stochastic elliptic PDEs Example: A beam with random properties We 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 (SU) SFEM for Structural Systems 16 June 2010 11 / 62
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 (SU) SFEM for Structural Systems 16 June 2010 12 / 62
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 (SU) SFEM for Structural Systems 16 June 2010 13 / 62
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