A Non-parametric Approach for Uncertainty Quantification in...
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A Non-parametric Approach forUncertainty Quantification in
ElastodynamicsS Adhikari
Department of Aerospace Engineering, University of Bristol, Bristol, U.K.
Email: [email protected]
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Stochastic structural dynamics
The equation of motion:
Mx(t) + Cx(t) + Kx(t) = p(t)
Due to the presence of uncertainty M, C and Kbecome random matrices.
The main objectives are:to quantify uncertainties in the systemmatricesto predict the variability in the responsevector x
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Current Methods
Two different approaches are currently available
Low frequency : Stochastic Finite ElementMethod (SFEM) - considers parametricuncertainties in details
High frequency : Statistical Energy Analysis(SEA) - do not consider parametricuncertainties in details
Work needs to be done : Hybrid method - some
kind of ‘combination’ of the above two
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Random Matrix Method (RMM)
The objective : To have an unified methodwhich will work across the frequency range.
The methodology :
Derive the matrix variate probability densityfunctions of M, C and K
Propagate the uncertainty (using MonteCarlo simulation or analytical methods) toobtain the response statistics (or pdf)
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Outline of the presentation
In what follows next, I will discuss:
Introduction to Matrix variate distributions
Maximum entropy distribution
Optimal Wishart distribution
Numerical examples
Open problems & discussions
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Matrix variate distributions
The probability density function of a randommatrix can be defined in a manner similar tothat of a random variable.
If A is an n × m real random matrix, the matrixvariate probability density function of A ∈ Rn,m,denoted as pA(A), is a mapping from thespace of n × m real matrices to the real line,i.e., pA(A) : Rn,m → R.
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Gaussian random matrix
The random matrix X ∈ Rn,p is said to have a matrix variateGaussian distribution with mean matrix M ∈ Rn,p andcovariance matrix Σ⊗Ψ, where Σ ∈ R
+n and Ψ ∈ R
+p provided
the pdf of X is given by
pX (X) = (2π)−np/2 |Σ|−p/2 |Ψ|−n/2
etr
{
−1
2Σ−1(X − M)Ψ−1(X − M)T
}
(1)
This distribution is usually denoted as X ∼ Nn,p (M,Σ ⊗ Ψ).
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Gaussian orthogonal ensembles
A random matrix H ∈ Rn,n belongs to the Gaussianorthogonal ensemble (GOE) provided its pdf of isgiven by
pH(H) = exp(
−θ2Trace(
H2)
+ θ1Trace (H) + θ0
)
where θ2 is real and positive and θ1 and θ0 are real.
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Wishart matrix
An n× n random symmetric positive definite matrix S is said tohave a Wishart distribution with parameters p ≥ n andΣ ∈ R
+n , if its pdf is given by
pS (S) =
{
21
2np Γn
(
1
2p
)
|Σ|1
2p
}
−1
|S|1
2(p−n−1)etr
{
−1
2Σ−1S
}
(2)
This distribution is usually denoted as S ∼ Wn(p,Σ).
Note: If p = n + 1, then the matrix is non-negative definite.
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Matrix variate Gammadistribution
An n × n random symmetric positive definite matrix W is saidto have a matrix variate gamma distribution with parameters a
and Ψ ∈ R+n , if its pdf is given by
pW (W) ={
Γn (a) |Ψ|−a}−1
|W|a−1
2(n+1) etr {−ΨW} ; ℜ(a) > (n − 1)/2 (3)
This distribution is usually denoted as W ∼ Gn(a,Ψ). Herethe multivariate gamma function:
Γn (a) = π1
4n(n−1)
n∏
k=1
Γ
[
a −1
2(k − 1)
]
; forℜ(a) > (n−1)/2 (4)
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Distribution of the systemmatrices
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 ofthe dynamic stiffness matrixD(ω) = −ω2M + iωC + K should exist ∀ω
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Distribution of the systemmatrices
The exact application of the last constraintrequires the derivation of the joint probabilitydensity function of M, C and K, which is quitedifficult to obtain.
We consider a simpler problem where it isrequired that the inverse moments of each ofthe system matrices M, C and K must exist.
Provided the system is damped, this willguarantee the existence of the moments of thefrequency response function matrix.
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Maximum Entropy Distribution
Suppose that the mean values of M, C and K aregiven by M, C and K respectively. Using thenotation G (which stands for any one the systemmatrices) the matrix variate density function ofG ∈ R
+n is given by pG (G) : R
+n → R. We have the
following constrains to obtain pG (G):∫
G>0
pG (G) dG = 1 (normalization) (5)
and∫
G>0
G pG (G) dG = G (the mean matrix)
(6)
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Further constraints
Suppose the inverse moments (say up to orderν) of the system matrix exist. This implies thatE
[∥
∥G−1∥
∥
F
ν]should be finite. Here the
Frobenius norm of matrix A is given by
‖A‖F =(
Trace(
AAT))1/2
.
Taking the logarithm for convenience, thecondition for the existence of the inversemoments can be expresses by
E[
ln |G|−ν] < ∞
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MEnt Distribution - 1
The Lagrangian becomes:
L(
pG)
= −
∫
G>0
pG (G) ln{
pG (G)}
dG+
(λ0 − 1)
(∫
G>0
pG (G) dG − 1
)
−ν
∫
G>0
ln |G| pG dG
+ Trace
(
Λ1
[∫
G>0
G pG (G) dG − G
])
(7)
Note: ν cannot be obtained uniquely!
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MEnt Distribution - 2
Using the calculus of variation
∂L(
pG)
∂pG= 0
or − ln{
pG (G)}
= λ0 + Trace (Λ1G) − ln |G|ν
or pG (G) = exp {−λ0} |G|ν etr {−Λ1G}
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MEnt Distribution - 3
Using the matrix variate Laplace transform(T ∈ Rn,n,S ∈ Cn,n, a > (n + 1)/2)
∫
T>0
etr {−ST} |T|a−(n+1)/2 dT = Γn(a) |S|−a
and substituting pG (G) into the constraintequations it can be shown that
pG (G) = r−nr {Γn(r)}−1
∣
∣G∣
∣
−r|G|ν etr
{
−rG−1
G}
(8)
where r = ν + (n + 1)/2.
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MEnt Distribution - 4
Comparing it with the Wishart distribution we have:Theorem 1. If ν-th order inverse-moment of asystem matrix G ≡ {M,C,K} exists and only the
mean of G is available, say G, then themaximum-entropy pdf of G follows the Wishartdistribution with parameters p = (2ν + n + 1) and
Σ = G/(2ν + n + 1), that is
G ∼ Wn
(
2ν + n + 1,G/(2ν + n + 1))
.
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Properties of the Distribution
Covariance tensor of G:
cov (Gij, Gkl) =1
2ν + n + 1
(
GikGjl + GilGjk
)
Normalized standard deviation matrix
δ2
G =E
[
‖G − E [G] ‖2F
]
‖E [G] ‖2F
=1
2ν + n + 1
1 +{Trace
(
G)
}2
Trace(
G2)
δ2
G≤
1 + n
2ν + n + 1and ν ↑ ⇒ δ2
G↓.
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Distribution of the inverse - 1
If G is Wn(p,Σ) then V = G−1 has the invertedWishart distribution:
PV(V) =2m−n−1n/2 |Ψ|m−n−1 /2
Γn[(m − n − 1)/2] |V|m/2etr
{
−1
2V−1Ψ
}
where m = n + p + 1 and Ψ = Σ−1 (recall thatp = 2ν + n + 1 and Σ = G/p)
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Distribution of the inverse - 2
Mean: E[
G−1]
=pG
−1
p − n − 1
cov(
G−1ij , G−1
kl
)
=(
2ν + n + 1)(ν−1G−1ij G
−1kl + G
−1ik G
−1jl + G
−1ilG
−1kj
)
2ν(2ν + 1)(2ν − 2)
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Distribution of the inverse - 3
Suppose n = 101 & ν = 2. So p = 2ν + n + 1 = 106 andp − n − 1 = 4. Therefore, E [G] = G and
E[
G−1]
=106
4G
−1= 26.5G
−1!!!!!!!!!!
From a practical point of view we do not expect them tobe so far apart!
One way to reduce the gap is to increase p. But thisimplies the reduction of variance.
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Optimal Wishart Distribution - 1
My argument: The distribution of G must be
such that E [G] and E[
G−1]
should be closest
to G and G−1
respectively.
Suppose G ∼ Wn
(
n + 1 + θ,G/α)
. We need tofind α such that the above condition is satisfied.
Therefore, define (and subsequently minimize)‘normalized errors’:ε1 =
∥
∥G − E [G]∥
∥
F/∥
∥G∥
∥
F
ε2 =∥
∥
∥G
−1− E
[
G−1]
∥
∥
∥
F/∥
∥
∥G
−1∥
∥
∥
F
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Optimal Wishart Distribution - 2
Because G ∼ Wn
(
n + 1 + θ,G/α)
we have
E [G] =n + 1 + θ
αG
and E[
G−1]
=α
θG
−1
We define the objective function to be minimized as
χ2 = ε12 + ε2
2 =(
1 − n+1+θα
)2+
(
1 − αθ
)2
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Optimal Wishart Distribution - 3
The optimal value of α can be obtained as bysetting ∂χ2
∂α = 0 orα4 − α3θ − θ4 + (−2 n + α − 2) θ3 +(
(n + 1) α − n2 − 2 n − 1)
θ2 = 0.
The only feasible value of α is
α =√
θ(n + 1 + θ)
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Optimal Wishart Distribution - 4
From this discussion we have the following:Theorem 2. If ν-th order inverse-moment of asystem matrix G ≡ {M,C,K} exists and only the
mean of G is available, say G, then the unbiaseddistribution of G follows the Wishart distributionwith parameters p = (2ν + n + 1) and
Σ = G/√
2ν(2ν + n + 1), that is
G ∼ Wn
(
2ν + n + 1,G/√
2ν(2ν + n + 1))
.
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Simulation Algorithm
Obtain θ =1
δ2
G
1 +{Trace
(
G)
}2
Trace(
G2)
− (n + 1)
If θ < 4, then select θ = 4.
Calculate α =√
θ(n + 1 + θ)
Generate samples of G ∼ Wn
(
n + 1 + θ,G/α)
(Matlabr command wishrnd can be used to generatethe samples)
Repeat the above steps for all system matrices and solvefor every samples
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Example: A cantilever Plate
00.5
11.5
22.5
0
0.5
1
1.5−0.5
0
0.5
1
X direction (length)
Output
Input
Y direction (width)
Fixed edge
ACantilever plate with a slot: µ = 0.3, ρ = 8000 kg/m3, t = 5mm,
Lx = 2.27m, Ly = 1.47mUncertainty Quantification – p.28/43
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Plate Mode 4
00.5
11.5
22.5
0
0.5
1
1.5−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
X direction (length)
Mode 4, freq. = 9.2119 Hz
Y direction (width)
Fourth Mode shape
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Plate Mode 5
00.5
11.5
22.5
0
0.5
1
1.5−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
X direction (length)
Mode 5, freq. = 11.6696 Hz
Y direction (width)
Fifth Mode shape
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Deterministic FRF
0 200 400 600 800 1000 1200−200
−180
−160
−140
−120
−100
−80
−60
−40
Frequency ω (Hz)
Log
ampli
tude
(dB)
log |H(448,79)
(ω)|
log |H(79,79)
(ω)|
FRF of the deterministic plate
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Random FRF - 1
Direct finite-element MCS of the amplitude of the cross-FRF of the plate with randomly placed
masses; 30 masses, each weighting 0.5% of the total mass of the plate are simulated.
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Random FRF - 2
Direct finite-element MCS of the amplitude of the driving-point FRF of the plate with randomly
placed masses; 30 masses, each weighting 0.5% of the total mass of the plate are simulated.
Uncertainty Quantification – p.33/43
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Wishart FRF - 1
MCS of the amplitude of the cross-FRF of the plate using optimal Wishart mass matrix,
n = 429, δM = 2.0449.
Uncertainty Quantification – p.34/43
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Newport, RI, May 1, 2006
Wishart FRF - 2
MCS of the amplitude of the driving-point-FRF of the plate using optimal Wishart mass
matrix, n = 429, δM = 2.0449.
Uncertainty Quantification – p.35/43
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Newport, RI, May 1, 2006
Comparison of Mean - 1
0 200 400 600 800 1000 1200−200
−180
−160
−140
−120
−100
−80
−60
−40
Frequency ω (Hz)
Log
ampli
tude
(dB)
of H (4
48,7
9) (ω)
DeterministicEnsamble average: SimulationEnsamble average: RMT
Comparison of the mean values of the amplitude of the cross-FRF.
Uncertainty Quantification – p.36/43
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Newport, RI, May 1, 2006
Comparison of Mean - 2
0 200 400 600 800 1000 1200−160
−140
−120
−100
−80
−60
−40
Frequency ω (Hz)
Log
ampli
tude
(dB)
of H (7
9,79
) (ω)
DeterministicEnsamble average: SimulationEnsamble average: RMT
Comparison of the mean values of the amplitude of the driving-point-FRF.
Uncertainty Quantification – p.37/43
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Newport, RI, May 1, 2006
Comparison of Variation - 1
0 200 400 600 800 1000 1200−180
−160
−140
−120
−100
−80
−60
−40
−20
Frequency ω (Hz)
Log
ampli
tude
(dB)
of H (4
48,7
9) (ω)
5% points: Simulation5% points: RMT95% points: Simulation95% points: RMT
Comparison of the 5% and 95% probability points of the amplitude of the cross-FRF.
Uncertainty Quantification – p.38/43
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Newport, RI, May 1, 2006
Comparison of Variation - 2
0 200 400 600 800 1000 1200−160
−140
−120
−100
−80
−60
−40
−20
Frequency ω (Hz)
Log
ampli
tude
(dB)
of H (7
9,79
) (ω)
5% points: Simulation5% points: RMT95% points: Simulation95% points: RMT
Comparison of the 5% and 95% probability points of the amplitude of the driving-point-FRF.
Uncertainty Quantification – p.39/43
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Newport, RI, May 1, 2006
Summary & conclusions
Wishart matrices may be used as the modelfor the system matrices in structural dynamics.
The parameters of the distribution wereobtained in closed-form by solving anoptimisation problem.
Only the mean matrix and and normalizedstandard deviation is required to model thesystem.
Numerical results show that uncertainty in theresponse is not very sensitive to the details ofthe correlation structure of the system matrices.
Uncertainty Quantification – p.40/43
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Next steps
Eigenvalue and eigenvector statistics
Steady-state and transient dynamic responsestatistics
Distribution of the dynamic stiffness matrix(complex Wishart matrix?) and its inverse (FRFmatrix)
Cumulative distribution function of the response(reliability problem)
Uncertainty Quantification – p.41/43
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Open issues & discussions
G is just one ‘observation’ - not an ensemblemean.
Are we taking account of model uncertainties(‘unknown unknowns’)?
How to incorporate a given covariance tensor ofG (e.g., obtained using the Stochastic Finiteelement Method)?
What is the consequence of the zeros in G arenot being preserved?
Uncertainty Quantification – p.42/43
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Structure of the Matrices
0 20 40 60
0
10
20
30
40
50
60
Column indices
Mass matrix
Ro
w in
dic
es
0 20 40 60
0
10
20
30
40
50
60
Column indices
Stiffess matrix
Ro
w in
dic
es
Nonzero elements of the system matrices
Uncertainty Quantification – p.43/43