Probabilistic Structural Dynamics: Parametric vs. Nonparametric Approach S Adhikari School of Engineering, Swansea University, Swansea, UK Email: S.Adhikari@swansea.ac.uk URL: http://engweb.swan.ac.uk/ ∼ adhikaris WIMCS Annual Meeting, 14 December 2009 Probabilistic Structural Dynamics – p.1/39
Research Areas Uncertainty Quantification (UQ) in Computational Mechanics Bio & Nanomechanics (nanotubes, graphene, cell mechanics) Dynamics of Complex Engineering Systems Inverse Problems for Linear and Non-linear Structural Dynamics Renewable Energy (wind energy, vibration energy harvesting) WIMCS Annual Meeting, 14 December 2009 Probabilistic Structural Dynamics – p.2/39
Outline of the presentation Uncertainty structural mechanics Parametric uncertainty quantification Stochastic finite element method Non-parametric uncertainty quantification Wishart random matrix approach Hybrid parametric and non-parametric uncertainty Both type uncertainties cover the entire domain Each type uncertainty is confined within non-overlapping subdomains Conclusions & collaboration opportunities WIMCS Annual Meeting, 14 December 2009 Probabilistic Structural Dynamics – p.3/39
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. WIMCS Annual Meeting, 14 December 2009 Probabilistic Structural Dynamics – p.4/39
Uncertainty propagation: key challenges The main difficulties are: the computational time can be prohibitively high compared to a deterministic analysis for real problems, the volume of input data can be unrealistic to obtain for a credible probabilistic analysis, the predictive accuracy can be poor if considerable resources are not spend on the previous two items, and WIMCS Annual Meeting, 14 December 2009 Probabilistic Structural Dynamics – p.5/39
Random continuous dynamical systems The equation of motion: ρ ( r , θ ) ∂ 2 U ( r , t ) ∂U ( r , t ) + L 1 + L 2 U ( r , t ) = p ( r , t ); r ∈ D , t ∈ [0 , T ] ∂t 2 ∂t (1) U ( r , t ) is the displacement variable, r is the spatial position vector and t is time. ρ ( r , θ ) is the random mass distribution of the system, p ( r , t ) is the distributed time-varying forcing function, L 1 is the random spatial self-adjoint damping operator, L 2 is the random spatial self-adjoint stiffness operator. Eq (1) is a Stochastic Partial Differential Equation (SPDE) [ie, the coefficients are random processes]. WIMCS Annual Meeting, 14 December 2009 Probabilistic Structural Dynamics – p.6/39
Stochastic Finite Element Method Problems of structural dynamics in which the uncertainty in specifying mass and stiffness of the structure is modeled within the framework of random fields can be treated using the Stochastic Finite Element Method (SFEM). The application of SFEM in linear structural dynamics typically consists of the following key steps: 1. Selection of appropriate probabilistic models for parameter uncertainties and boundary conditions 2. Replacement of the element property random fields by an equivalent set of a finite number of random variables. This step, known as the ‘discretisation of random fields’ is a major step in the analysis. 3. Formulation of the equation of motion of the form D ( ω ) u = f where D ( ω ) is the random dynamic stiffness matrix, u is the vector of random nodal displacement and f is the applied forces. In general D ( ω ) is a random symmetric complex matrix. 4. Calculation of the response statistics by either (a) solving the random eigenvalue problem, or (b) solving the set of complex random algebraic equations. WIMCS Annual Meeting, 14 December 2009 Probabilistic Structural Dynamics – p.7/39
Spectral Decomposition of random fields-2 Suppose H ( r , θ ) is a random field with a covariance function C H ( r 1 , r 2 ) defined in a space Ω . Since the covariance function is finite, symmetric and positive definite it can be represented by a spectral decomposition. Using this spectral decomposition, the random process H ( r , θ ) can be expressed in a generalized fourier type of series as ∞ � � H ( r , θ ) = H 0 ( r ) + λ i ξ i ( θ ) ϕ i ( r ) (2) i =1 where ξ i ( θ ) are uncorrelated random variables, λ i and ϕ i ( r ) are eigenvalues and eigenfunctions satisfying the integral equation � C H ( r 1 , r 2 ) ϕ i ( r 1 )d r 1 = λ i ϕ i ( r 2 ) , ∀ i = 1 , 2 , · · · (3) Ω The spectral decomposition in equation (2) is known as the Karhunen-Loève (KL) expansion. The series in (2) can be ordered in a decreasing series so that it can be truncated after a finite number of terms with a desired accuracy. WIMCS Annual Meeting, 14 December 2009 Probabilistic Structural Dynamics – p.8/39
Exponential autocorrelation function The autocorrelation function: C ( x 1 , x 2 ) = e −| x 1 − x 2 | /b (4) The underlying random process H ( x, θ ) can be expanded using the Karhunen-Loève expansion in the interval − a ≤ x ≤ a as ∞ � � H ( x, θ ) = ξ j ( θ ) λ j ϕ j ( x ) (5) 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 tan( ω j a ) = c λ j = j + c 2 , ϕ j ( x ) = , where , (6) ω 2 � ω j a + sin(2 ω j a ) 2 ω j and for even j are given by 2 c sin( ω j x ) tan( ω j a ) = ω j λ j = ω j 2 + c 2 , ϕ j ( x ) = , where − c. (7) � a − sin(2 ω j a ) 2 ω j WIMCS Annual Meeting, 14 December 2009 Probabilistic Structural Dynamics – p.9/39
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 ) + ρA ( x, θ ) ∂ 2 Y ( x, t ) � � = p ( x, t ) . (8) ∂x 2 ∂x 2 ∂t 2 Y ( x, t ) : transverse flexural displacement, EI ( x ) : flexural rigidity, ρA ( x ) : mass per unit length, and p ( x, t ) : applied forcing. Consider EI ( x, θ ) = EI 0 (1 + ǫ 1 F 1 ( x, θ )) (9) and ρA ( x, θ ) = ρA 0 (1 + ǫ 2 F 2 ( x, θ )) (10) 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 ) . Since, EI ( x, θ ) and ρA ( x, θ ) are strictly positive, F i ( x, θ ) ( i =1,2) are required to satisfy the conditions P [1 + ǫ i F i ( x, θ ) ≤ 0] = 0 . WIMCS Annual Meeting, 14 December 2009 Probabilistic Structural Dynamics – p.10/39
Example: A beam with random properties We express the shape functions for the finite element analysis of Euler-Bernoulli beams as N ( x ) = Γ s ( x ) (11) 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 . � Γ = and s ( x ) = (12) − 2 3 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 ( x ) dx = ′′ T ( x ) dx. (13) ′′ ( x ) EI ( x, θ ) N ′′ ( x ) N K e ( θ ) = EI 0 (1 + ǫ 1 F 1 ( x, θ )) N N 0 0 WIMCS Annual Meeting, 14 December 2009 Probabilistic Structural Dynamics – p.11/39
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 ) (14) where the deterministic and random parts are � ℓ e N K ′′ T ( x ) dx � ′′ ( x ) N � K e 0 = EI 0 and ∆K e ( θ ) = ǫ 1 ξ K j ( θ ) λ K j K ej . (15) N 0 j =1 The constant N K is the number of terms retained in the Karhunen-Loève 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 (16) 0 WIMCS Annual Meeting, 14 December 2009 Probabilistic Structural Dynamics – p.12/39
Example: A beam with random properties The mass matrix can be obtained as M e ( θ ) = M e 0 + ∆M e ( θ ) (17) The deterministic and random parts is given by � ℓ e N M N ( x ) N T ( x ) dx � � M e 0 = ρA 0 and ∆M e ( θ ) = ǫ 2 ξ M j ( θ ) λ M j M ej . (18) 0 j =1 The constant N M is the number of terms retained in Karhunen-Loève 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 = ρA 0 (19) 0 WIMCS Annual Meeting, 14 December 2009 Probabilistic Structural Dynamics – p.13/39
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