Uncertainty Quantification and Propagation in Aerospace Structural Dynamical Models Sondipon Adhikari Department of Aerospace Engineering, University of Bristol, Bristol, U.K. Email: S.Adhikari@bristol.ac.uk URL: http://www.aer.bris.ac.uk/contact/academic/adhikari/home.html Uncertainty in Aero Models – p.1/62 Strathclyde, 30 January 2007
Outline of the presentation Probabilistic structural dynamics Random matrix model for aerospace systems Wishart random matrices Uncertainty propagation Random eigenvalue problems Experimental validation Open problems & discussions Uncertainty in Aero Models – p.2/62 Strathclyde, 30 January 2007
Overview of Predictive Methods in Engineering There are five key steps: Physics (mechanics) model building Uncertainty Quantification (UQ) Uncertainty Propagation (UP) Model Verification & Validation (V & V) Prediction Tools are available for each of these steps. Currently, our focus is mainly on UQ and UP in linear dynamical systems. Uncertainty in Aero Models – p.3/62 Strathclyde, 30 January 2007
Complex aerospace model Su� bs� yst� em� 4� Subsystem 3� Subsystem 2� Subsystem 1� Possible subsystem models for an aircraft Uncertainty in Aero Models – p.4/62 Strathclyde, 30 January 2007
Why uncertainty? Different sources of uncertainties in the modeling and simulation of dynamic systems may be attributed, but not limited, to the following factors: Mathematical models: equations (linear, non-linear), geometry, damping model (viscous, non-viscous, fractional derivative), boundary conditions/initial conditions, input forces; Model parameters: Young’s modulus, mass density, Poisson’s ratio, damping model parameters (damping coefficient, relaxation modulus, fractional derivative order) Uncertainty in Aero Models – p.5/62 Strathclyde, 30 January 2007
Why uncertainty? Numerical algorithms: weak formulations, discretisation of displacement fields (in finite element method), discretisation of stochastic fields (in stochastic finite element method), approximate solution algorithms, truncation and roundoff errors, tolerances in the optimization and iterative methods, artificial intelligent (AI) method (choice of neural networks) Measurements: noise, resolution (number of sensors and actuators), experimental hardware, excitation method (nature of shakers and hammers), excitation and measurement point, data processing (amplification, number of data points, FFT), calibration Uncertainty in Aero Models – p.6/62 Strathclyde, 30 January 2007
Structural dynamics The equation of motion: M ¨ x ( t ) + C ˙ x ( t ) + Kx ( t ) = p ( t ) Due to the presence of 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 x Uncertainty in Aero Models – p.7/62 Strathclyde, 30 January 2007
Current Methods Two different approaches are currently available Low frequency : Stochastic Finite Element Method (SFEM) - assumes that stochastic fields describing parametric uncertainties are known in details High frequency : Statistical Energy Analysis (SEA) - do not consider parametric uncertainties in details Uncertainty in Aero Models – p.8/62 Strathclyde, 30 January 2007
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) Uncertainty in Aero Models – p.9/62 Strathclyde, 30 January 2007
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 . Uncertainty in Aero Models – p.10/62 Strathclyde, 30 January 2007
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 | Σ | − p/ 2 | Ψ | − n/ 2 � � − 1 2 Σ − 1 ( X − M ) Ψ − 1 ( X − M ) T etr (1) This distribution is usually denoted as X ∼ N n,p ( M , Σ ⊗ Ψ ) . Uncertainty in Aero Models – p.11/62 Strathclyde, 30 January 2007
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 ( p − n − 1) etr 2 Σ − 1 S p S ( S ) = 2 2 p | Σ | | S | (2) This distribution is usually denoted as S ∼ W n ( p, Σ ) . Note: If p = n + 1 , then the matrix is non-negative definite. Uncertainty in Aero Models – p.12/62 Strathclyde, 30 January 2007
Matrix variate Gamma distribution A n × n symmetric positive definite matrix random W is said to have a matrix variate gamma distribution with parameters a and Ψ ∈ R + n , if its pdf is given by ℜ ( a ) > 1 Γ n ( a ) | Ψ | − a � − 1 | W | a − 1 2 ( n +1) etr {− ΨW } ; � p W ( W ) = 2( n − (3) This distribution is usually denoted as W ∼ G n ( a, Ψ ) . Here the multivariate gamma function: n � � a − 1 1 � 4 n ( n − 1) Γ n ( a ) = π Γ 2( k − 1) ; for ℜ ( a ) > ( n − 1) / 2 (4) k =1 Uncertainty in Aero Models – p.13/62 Strathclyde, 30 January 2007
Distribution of the system matrices The distribution of the random system matrices M , C and K should be such that they are symmetric positive-definite, and the moments (at least first two) of the inverse of the dynamic stiffness matrix D ( ω ) = − ω 2 M + iω C + K should exist ∀ ω Uncertainty in Aero Models – p.14/62 Strathclyde, 30 January 2007
Distribution of the system matrices The exact application of the last constraint requires the derivation of the joint probability density function of M , C and K , which is quite difficult to obtain. We consider a simpler problem where it is required that the inverse moments of each of the system matrices M , C and K must exist. Provided the system is damped, this will guarantee the existence of the moments of the frequency response function matrix. Uncertainty in Aero Models – p.15/62 Strathclyde, 30 January 2007
Maximum Entropy Distribution Soize (2000,2006) used this approach and obtained the matrix variate Gamma distribution. Since Gamma and Wishart distribution are similar we have: Theorem 1. If ν -th order inverse-moment of a system matrix G ≡ { M , C , K } exists and only the mean of G is available, say G , then the maximum-entropy pdf of G follows the Wishart distribution with parameters p = (2 ν + n + 1) and Σ = G / (2 ν + n + 1) , that is � � G ∼ W n 2 ν + n + 1 , G / (2 ν + n + 1) . Uncertainty in Aero Models – p.16/62 Strathclyde, 30 January 2007
Matrix Factorization Approach (MFA) Because G is a symmetric and positive-definite random matrix, it can be always factorized as G = XX T (5) where X ∈ R n × p , p ≥ n is in general a rectangular matrix. The simplest case is when the mean of X is O ∈ R n × p , p ≥ n and the covariance tensor of X is given by Σ ⊗ I p ∈ R np × np where Σ ∈ R + n . X is a Gaussian random matrix with mean O ∈ R n × p , p ≥ n and covariance Σ ⊗ I p ∈ R np × np . Uncertainty in Aero Models – p.17/62 Strathclyde, 30 January 2007
Wishart Pdf After some algebra it can be shown that G is a W n ( p, Σ ) Wishart random matrix, whose pdf is given given by � − 1 � � 1 � � � − 1 2 np Γ n 1 1 1 2 p 2 ( p − n − 1) etr 2 Σ − 1 G p G ( G ) = 2 2 p | Σ | | G | (6) Uncertainty in Aero Models – p.18/62 Strathclyde, 30 January 2007
Parameter Estimation of Wishart Distribution The distribution of G must be such that E [ G ] and − 1 respectively. G − 1 � � E should be closest to G and G Since G ∼ W n ( p, Σ ) , there are two unknown parameters in this distribution, namely, p and Σ . This implies that there are in total 1 + n ( n + 1) / 2 number of unknowns. We define and subsequently minimize ‘normalized errors’: � � � � ε 1 = � G − E [ G ] F / � G � � F − 1 − E � G − 1 �� � − 1 � � ε 2 = F / � G � G � � � � � � F Uncertainty in Aero Models – p.19/62 Strathclyde, 30 January 2007
MFA Distribution Solving the optimization problem we have: Theorem 2. If ν -th order inverse-moment of a system matrix G ≡ { M , C , K } exists and only the mean of G is available, say G , then the distribution of G follows the Wishart distribution with parameters � p = (2 ν + n + 1) and Σ = G / 2 ν (2 ν + n + 1) , that is � � � G ∼ W n 2 ν + n + 1 , G / 2 ν (2 ν + n + 1) . Uncertainty in Aero Models – p.20/62 Strathclyde, 30 January 2007
Response statistics - 1 The equation of motion is Dx = p , D is in general n × n complex random matrix. The response is given by x = D − 1 p Consider static problems so that all matrices/vectors are real. Uncertainty in Aero Models – p.21/62 Strathclyde, 30 January 2007
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