Wishart Random Matrices for Uncertainty Quantification of Complex Dynamical Systems S Adhikari School of Engineering, Swansea University, Swansea, UK Email: S.Adhikari@swansea.ac.uk URL: http://engweb.swan.ac.uk/ ∼ adhikaris ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.1/44
Outline of the presentation Uncertainty Quantification (UQ) in structural dynamics Review of current approaches Wishart random matrices Parameter selection Computational method Analytical method Experimental results Conclusions & future directions ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.2/44
Complex aerospace system Complex aerospace system can have millions of degrees of freedom and signifi- cant uncertainty in its numerical (Finite Element) model ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.3/44
� � � The role of uncertainty in computational science Real System� Measured output� Input� (�eg� , velocity,� (�eg�, earthquake,� acceleration� ,� turbulence� )� stress)� uncertain� experimental� error� system identification� System Uncertainty� Input Uncertainty� calibration/updating� parametric uncertainty� uncertainty in time� model inadequacy� history� model uncertainty� uncertainty in� location� calibration uncertainty� model validation� Physics based model� Simulated Input� L� (u) =� f� (time or frequency� (� eg� , ODE/�PDE�/�SDE�/� SPDE�)� domain)� Computational� Uncertainty� verification� machine precession,� Total Uncertainty =� error tolerance� input + system +� h� ’ and ‘� p� ‘� ’ refinements� computational� uncertainty� Model output� Computation� (�eg� , velocity,� (�eg�,�FEM�/� BEM� /Finite� acceleration� ,� difference/� SFEM� /� MCS� )� stress)� ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.4/44
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. ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.5/44
Problem-types in computational sciences Input System Output Problem name Main techniques Known (deter- Known (deter- Unknown Analysis (forward FEM/BEM/Finite ministic) ministic) problem) difference Known (deter- Incorrect (deter- Known (deter- Updating/calibration Modal updating ministic) ministic) ministic) Known (deter- Unknown Known (deter- System identifica- Kalman filter ministic) ministic) tion Assumed (de- Unknown (de- Prescribed Design Design optimisa- terministic) terministic) tion Unknown Partially Known Known Structural Health SHM methods Monitoring (SHM) Known (deter- Known (deter- Prescribed Control Modal control ministic) ministic) Known (ran- Known (deter- Unknown Random vibration Random vibration dom) ministic) methods ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.6/44
Problem-types in computational sciences Input System Output Problem name Main techniques Known (deter- Known (ran- Unknown Stochastic analysis SFEM/SEA/RMT ministic) dom) (forward problem) Known (ran- Incorrect (ran- Known (ran- Probabilistic updat- Bayesian calibra- dom) dom) dom) ing/calibration tion Assumed (ran- Unknown (ran- Prescribed (ran- Probabilistic de- RBOD dom/deterministic) dom) dom) sign Known (ran- Partially known Partially known Joint state and pa- Particle Kalman dom/deterministic) (random) (random) rameter estimation Filter/Ensemble Kalman Filter Known (ran- Known (ran- Known from Model validation Validation methods dom/deterministic) dom) experiment and model (random) Known (ran- Known (ran- Known from dif- Model verification verification meth- dom/deterministic) dom) ferent computa- ods tions (random) ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.7/44
Structural dynamics The equation of motion: M ¨ q ( t ) + C ˙ q ( t ) + Kq ( t ) = f ( t ) (1) Due to the presence of (parametric/nonparametric or both) uncertainty M , C and K become random matrices. The main objectives in the ‘forward problem’ are: to quantify uncertainties in the system matrices to predict the variability in the response vector q Probabilistic solution of this problem is expected to have more credibility compared to a deterministic solution ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.8/44
UQ approaches: challenges The main difficulties are due to: 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 as the state-of-the art methodology stands now (such as the Stochastic Finite Element Method), only very few highly trained professionals (such as those with PhDs) can even attempt to apply the complex concepts (e.g., random fields) and methodologies to real-life problems. ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.9/44
Main objectives Our work is aimed at developing methodologies [the 10-10-10 challenge] with the ambition that they should: not take more than 10 times the computational time required for the corresponding deterministic approach; result a predictive accuracy within 10% of direct Monte Carlo Simulation (MCS); use no more than 10 times of input data needed for the corresponding deterministic approach; and enable ‘normal’ engineering graduates to perform probabilistic structural dynamic analyses with a reasonable amount of training. ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.10/44
Current UQ approaches - 1 Two different approaches are currently available Parametric approaches : Such as the Stochastic Finite Element Method (SFEM): aim to characterize parametric uncertainty (type ‘a’) assumes that stochastic fields describing parametric uncertainties are known in details suitable for low-frequency dynamic applications ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.11/44
Current UQ approaches - 2 Nonparametric approaches : Such as the Statistical Energy Analysis (SEA) and Wishart random matrix theory: aim to characterize nonparametric uncertainty (types ‘b’ - ‘e’) does not consider parametric uncertainties in details suitable for high/mid-frequency dynamic applications extensive works over the past decade → general purpose commercial software is now available ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.12/44
Random Matrix Method (RMM) The objective : To have an unified method which will work across the frequency range. The methodology : Derive the matrix variate probability density functions of M , C and K Propagate the uncertainty (using Monte Carlo simulation or analytical methods) to obtain the response statistics (or pdf) ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.13/44
Matrix variate distributions The probability density function of a random matrix can be defined in a manner similar to that of a random variable. If A is an n × m real random matrix, the matrix variate probability density function of A ∈ R n,m , denoted as p A ( A ) , is a mapping from the space of n × m real matrices to the real line, i.e., p A ( A ) : R n,m → R . ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.14/44
Gaussian random matrix The random matrix X ∈ R n,p is said to have a matrix variate Gaussian distribution with mean matrix M ∈ R n,p and covariance matrix Σ ⊗ Ψ , where Σ ∈ R + n and Ψ ∈ R + p provided the pdf of X is given by p X ( X ) = (2 π ) − np/ 2 det { Σ } − p/ 2 det { Ψ } − n/ 2 � � − 1 2 Σ − 1 ( X − M ) Ψ − 1 ( X − M ) T etr (2) This distribution is usually denoted as X ∼ N n,p ( M , Σ ⊗ Ψ ) . ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.15/44
Wishart matrix A n × n symmetric positive definite random matrix S is said to have a Wishart distribution with parameters p ≥ n and Σ ∈ R + n , if its pdf is given by � � 1 � � − 1 � � − 1 2 np Γ n 1 1 1 2 p 2 Σ − 1 S 2 ( p − n − 1) etr p S ( S ) = 2 2 p det { Σ } | S | (3) This distribution is usually denoted as S ∼ W n ( p, Σ ) . Note: If p = n + 1 , then the matrix is non-negative definite. ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.16/44
Recommend
More recommend
