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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

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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

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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

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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

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The role of uncertainty in computational science

Real System Input

(eg, earthquake,

turbulence ) Measured output (eg , velocity, acceleration , stress)

  • Physics based model

L

(u) = f ( eg , ODE/PDE/SDE/ SPDE) System Uncertainty parametric uncertainty model inadequacy model uncertainty calibration uncertainty Simulated Input (time or frequency domain) Input Uncertainty uncertainty in time history uncertainty in

location

Computation

(eg,FEM/ BEM /Finite difference/ SFEM / MCS )

calibration/updating uncertain experimental error Computational Uncertainty machine precession, error tolerance ‘ h ’ and ‘p ’ refinements Model output (eg , velocity, acceleration , stress) verification system identification Total Uncertainty = input + system + computational uncertainty model validation ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.4/44

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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

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Problem-types in computational sciences

Input System Output Problem name Main techniques Known (deter- ministic) Known (deter- ministic) Unknown Analysis (forward problem) FEM/BEM/Finite difference Known (deter- ministic) Incorrect (deter- ministic) Known (deter- ministic) Updating/calibration Modal updating Known (deter- ministic) Unknown Known (deter- ministic) System identifica- tion Kalman filter Assumed (de- terministic) Unknown (de- terministic) Prescribed Design Design

  • ptimisa-

tion Unknown Partially Known Known Structural Health Monitoring (SHM) SHM methods Known (deter- ministic) Known (deter- ministic) Prescribed Control Modal control Known (ran- dom) Known (deter- ministic) Unknown Random vibration Random vibration methods

ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.6/44

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Problem-types in computational sciences

Input System Output Problem name Main techniques Known (deter- ministic) Known (ran- dom) Unknown Stochastic analysis (forward problem) SFEM/SEA/RMT Known (ran- dom) Incorrect (ran- dom) Known (ran- dom) Probabilistic updat- ing/calibration Bayesian calibra- tion Assumed (ran- dom/deterministic) Unknown (ran- dom) Prescribed (ran- dom) Probabilistic de- sign RBOD Known (ran- dom/deterministic) Partially known (random) Partially known (random) Joint state and pa- rameter estimation Particle Kalman Filter/Ensemble Kalman Filter Known (ran- dom/deterministic) Known (ran- dom) Known from experiment and model (random) Model validation Validation methods Known (ran- dom/deterministic) Known (ran- dom) Known from dif- ferent computa- tions (random) Model verification verification meth-

  • ds

ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.7/44

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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

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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.

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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.

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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

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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

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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

  • r analytical methods) to obtain the response statistics

(or pdf)

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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 ∈ Rn,m, denoted as pA(A), is a mapping from the space of n × m real matrices to the real line, i.e., pA(A) : Rn,m → R.

ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.14/44

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Gaussian random matrix

The random matrix X ∈ Rn,p is said to have a matrix variate Gaussian distribution with mean matrix M ∈ Rn,p and covariance matrix Σ ⊗ Ψ, where Σ ∈ R+

n and Ψ ∈ R+ p provided the pdf of X

is given by pX (X) = (2π)−np/2det {Σ}−p/2 det {Ψ}−n/2 etr

  • −1

2Σ−1(X − M)Ψ−1(X − M)T

  • (2)

This distribution is usually denoted as X ∼ Nn,p (M, Σ ⊗ Ψ).

ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.15/44

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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 pS (S) =

  • 2

1 2 np Γn

1 2p

  • det {Σ}

1 2 p

−1 |S|

1 2 (p−n−1)etr

  • −1

2Σ−1S

  • (3)

This distribution is usually denoted as S ∼ Wn(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

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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

pW (W) =

  • Γn (a) det {Ψ}−a−1 det {W}a− 1

2 (n+1) etr {−ΨW} ;

ℜ(a) (4) This distribution is usually denoted as W ∼ Gn(a, Ψ). Here the multivariate gamma function: Γn (a) = π

1 4n(n−1)

n

  • k=1

Γ

  • a − 1

2(k − 1)

  • ; for ℜ(a) > (n − 1)/2

(5)

ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.17/44

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Wishart random matrix approach

The probability density function of the mass (M), damping (C) and stiffness (K) matrices should be such that they are symmetric and non-negative matrices. Wishart random matrix (a non-Gaussian matrix) is the simplest mathematical model which can satisfy these two criteria: [M, C, K] ≡ G ∼ Wn(p, Σ). Suppose we ‘know’ (e.g, by measurement or stochastic modeling) the mean (G0) and the (normalized) standard deviation (σG) of the system matrices:

σ2

G =

E

  • G − E [G] 2

F

  • E [G] 2

F

. (6)

ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.18/44

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Wishart parameter selection - 1

The parameters p and Σ can be obtained based on what criteria we select. We investigate four possible choices.

  • 1. Criteria 1: E [G] = G0 and σG =

σG which results p = n + 1 + θ and Σ = G0/p (7) where θ = (1 + β)/ σ2

G − (n + 1) and

β = {Trace (G0)}2 /Trace

  • G0

2

.

  • 2. Criteria 2: G0 − E [G]F and
  • G0

−1 − E

  • G−1
  • F are

minimum and σG = σG. This results: p = n + 1 + θ and Σ = G0/α (8) where α =

  • θ(n + 1 + θ).

ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.19/44

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Wishart parameter selection - 2

  • 1. Criteria 3: E
  • G−1

= G0

−1 and σG =

σG. This results: p = n + 1 + θ and Σ = G0/θ (9)

  • 2. Criteria 4: The mean of the eigenvalues of the distribution is

same as the ‘measured’ eigenvalues of the mean matrix and the (normalized) standard deviation is same as the measured standard deviation: E

  • M−1

= M0

−1, E [K] = K0, σM =

σM and σK = σK. (10)

ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.20/44

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A cantilever plate: front view

The test rig for the cantilever plate; front view.

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A cantilever plate: side view

The test rig for the cantilever plate; side view.

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Physical properties

Plate Properties Numerical values Length (Lx) 998 mm Width (Ly) 530 mm Thickness (th) 3.0 mm Mass density (ρ) 7860 kg/m3 Young’s modulus (E) 2.0 × 105 MPa Poisson’s ratio (µ) 0.3 Total weight 12.47 kg

Material and geometric properties

  • f

the cantilever plate considered for the experiment. The data presented here are available from http://engweb.swan.ac.uk/∼adhikaris/uq/.

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Mean of cross-FRF

500 1000 1500 2000 2500 3000 3500 4000 −180 −160 −140 −120 −100 −80 −60 Frequency (Hz) Mean of amplitude (dB) of FRF at point 1 RMT−1 RMT−2 RMT−3 RMT−4 SFEM

Mean of the amplitude of the response of the cross-FRF of the plate, n = 1200, σM = 0.1326 and σK = 0.3335.

ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.24/44

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Error in the mean of cross-FRF

500 1000 1500 2000 2500 3000 3500 4000 2 4 6 8 10 12 14 16 18 20 Frequency (Hz) Error in mean of amplitude of FRF at point 1 RMT−1 RMT−2 RMT−3 RMT−4

Error in the mean of the amplitude of the response of the cross-FRF of the plate, n = 1200, σM = 0.1326 and σK = 0.3335.

ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.25/44

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Mean of Driving-point-FRF

500 1000 1500 2000 2500 3000 3500 4000 −180 −160 −140 −120 −100 −80 −60 Frequency (Hz) Mean of amplitude (dB) of FRF at point 2 RMT−1 RMT−2 RMT−3 RMT−4 SFEM

Mean of the amplitude of the response of the driving-point-FRF of the plate, n = 1200, σM = 0.1326 and σK = 0.3335.

ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.26/44

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Error in the Mean of Driving-point-FRF

500 1000 1500 2000 2500 3000 3500 4000 2 4 6 8 10 12 14 16 18 20 Frequency (Hz) Error in mean of amplitude of FRF at point 2 RMT−1 RMT−2 RMT−3 RMT−4

Error in the mean of the amplitude of the response of the driving-point-FRF of the plate, n = 1200, σM = 0.1326 and σK = 0.3335.

ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.27/44

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Standard Deviation of Cross-FRF

500 1000 1500 2000 2500 3000 3500 4000 −180 −160 −140 −120 −100 −80 −60 Frequency (Hz) Standard deviation of amplitude of FRF at point 1 RMT−1 RMT−2 RMT−3 RMT−4 SFEM

Standard deviation of the amplitude of the response of the cross-FRF of the plate, n = 1200, σM = 0.1326 and σK = 0.3335.

ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.28/44

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Error in the Standard Deviation of Cross-FRF

500 1000 1500 2000 2500 3000 3500 4000 2 4 6 8 10 12 14 16 18 20 Frequency (Hz) Error in standard deviation of amplitude of FRF at point 1 RMT−1 RMT−2 RMT−3 RMT−4

Error in the standard deviation of the amplitude of the response of the cross-FRF

  • f the plate, n = 1200, σM = 0.1326 and σK = 0.3335.

ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.29/44

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Standard deviation of driving-point-FRF

500 1000 1500 2000 2500 3000 3500 4000 −180 −160 −140 −120 −100 −80 −60 Frequency (Hz) Standard deviation of amplitude of FRF at point 2 RMT−1 RMT−2 RMT−3 RMT−4 SFEM

Standard deviation of the amplitude of the response of the driving-point-FRF of the plate, n = 1200, σM = 0.1326 and σK = 0.3335.

ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.30/44

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Error in the standard deviation of driving-point-FRF

500 1000 1500 2000 2500 3000 3500 4000 2 4 6 8 10 12 14 16 18 20 Frequency (Hz) Error in standard deviation of amplitude of FRF at point 2 RMT−1 RMT−2 RMT−3 RMT−4

Error in the standard deviation of the amplitude of the response of the driving- point-FRF of the plate, n = 1200, σM = 0.1326 and σK = 0.3335.

ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.31/44

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Main observations

Error in the low frequency region is higher than that in the higher frequencies In the high frequency region all methods are similar Overall, parameter selection 3 performs best; especially in the low frequency region.

ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.32/44

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Standard deviation: low frequency

100 200 300 400 500 600 700 800 −180 −160 −140 −120 −100 −80 −60 Frequency (Hz) Standard deviation of amplitude of FRF at point 2 RMT−1 RMT−2 RMT−3 RMT−4 SFEM

Standard deviation of the amplitude of the response of the driving-point-FRF of the plate in the low frequency region, n = 1200, σM = 0.1326 and σK = 0.3335.

ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.33/44

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Error in the standard deviation: low frequency

100 200 300 400 500 600 700 800 2 4 6 8 10 12 14 16 18 20 Frequency (Hz) Error in standard deviation of amplitude of FRF at point 2 RMT−1 RMT−2 RMT−3 RMT−4

Error in the standard deviation of the amplitude of the response of the driving- point-FRF of the plate in the low frequency region, n = 1200, σM = 0.1326 and σK = 0.3335.

ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.34/44

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Finite element & Wishart matrix model

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 −0.5 0.5 1

6 4

X direction (length)

5

Outputs

2 3

Input

1

Y direction (width) F i x e d e d g e

Baseline Model: 25 × 15 elements, 416 nodes, 1200 degrees-of-freedom. Input node number: 481, Output node numbers: 481, 877, 268, 1135, 211 and 844, 0.7% modal damping is assumed for all modes.

ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.35/44

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Comparison of driving-point-FRF

500 1000 1500 2000 2500 3000 3500 4000 −40 −30 −20 −10 10 20 30 40 50 60 Frequency (Hz) Relative std of H(1,1) (ω)

Comparison of the mean and standard deviation of the amplitude of the driving- point-FRF , n = 1200, δM = 0.1166 and δK = 0.2711.

ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.36/44

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Comparison of driving-point-FRF: Low Freq

100 200 300 400 500 600 700 800 900 1000 −40 −30 −20 −10 10 20 30 40 50 60 Frequency (Hz) Relative std of H(1,1) (ω)

Comparison of the mean and standard deviation of the amplitude of the driving- point-FRF , n = 1200, δM = 0.1166 and δK = 0.2711.

ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.37/44

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Comparison of driving-point-FRF: Mid Freq

1000 1500 2000 2500 −40 −30 −20 −10 10 20 30 40 50 60 Frequency (Hz) Relative std of H(1,1) (ω)

Comparison of the mean and standard deviation of the amplitude of the driving- point-FRF , n = 1200, δM = 0.1166 and δK = 0.2711.

ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.38/44

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Comparison of driving-point-FRF: High Freq

2500 3000 3500 4000 −40 −30 −20 −10 10 20 30 40 50 60 Frequency (Hz) Relative std of H(1,1) (ω)

Comparison of the mean and standard deviation of the amplitude of the driving- point-FRF , n = 1200, δM = 0.1166 and δK = 0.2711.

ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.39/44

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Comparison of cross-FRF

500 1000 1500 2000 2500 3000 3500 4000 −40 −30 −20 −10 10 20 30 40 50 60 Frequency (Hz) Relative std of H(1,2) (ω)

Comparison of the mean and standard deviation of the amplitude of the cross- FRF , n = 1200, δM = 0.1166 and δK = 0.2711.

ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.40/44

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Comparison of cross-FRF: Low Freq

100 200 300 400 500 600 700 800 900 1000 −40 −30 −20 −10 10 20 30 40 50 60 Frequency (Hz) Relative std of H(1,2) (ω)

Comparison of the mean and standard deviation of the amplitude of the cross- FRF , n = 1200, δM = 0.1166 and δK = 0.2711.

ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.41/44

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Comparison of cross-FRF: Mid Freq

1000 1500 2000 2500 −40 −30 −20 −10 10 20 30 40 50 60 Frequency (Hz) Relative std of H(1,2) (ω)

Comparison of the mean and standard deviation of the amplitude of the cross- FRF , n = 1200, δM = 0.1166 and δK = 0.2711.

ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.42/44

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Comparison of cross-FRF: High Freq

2500 3000 3500 4000 −40 −30 −20 −10 10 20 30 40 50 60 Frequency (Hz) Relative std of H(1,2) (ω)

Comparison of the mean and standard deviation of the amplitude of the cross- FRF , n = 1200, δM = 0.1166 and δK = 0.2711.

ICNPAA, Genoa, 26 June 2008 UQ of complex systems – p.43/44

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Conclusions

When uncertainties in the system parameters (parametric uncertainty) and modelling (nonparametric uncertainty) are considered, the discretized equation of motion of linear dynamical systems is characterized by random mass, stiffness and damping matrices. Wishart matrices may be used as the model for the random system matrices in structural dynamics. Only the mean matrix and normalized standard deviation is required to model the system. Our results show that experimental results and Wishart matrix based results match well in the mid and high frequency region. ICNPAA, Genoa, 26 June 2008

UQ of complex systems – p.44/44