01 Bucher Random Fields - Efficient Analysis and Simulation

8/10/2019 01 Bucher Random Fields - Efficient Analysis and Simulation

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Random FieldsEfficient Analysis and Simulation

Christian Bucher & Sebastian Wolff

Vienna University of Technology

& DYNARDO Austria GmbH, Vienna


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Overview

Introduction

Elementary properties

Conditional random fields

Computational aspects

Example

Concluding remarks

2/25 c Christian Bucher 2010-2014


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Random field

Real valued functionH(x) defined in ann-dimensional space

H

R; x= [x1, x2, . . . x n]

T

D Rn

Ensemble of all possible realisations

Describe statistics in terms of mean and variance

Need to consider the correlation structure between values of

Hat different locationsx and y

H(x, )

x, y

L

x

y



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Second order statistics of random field

Mean value function

H(x) =E[H(x)]

Autocovariance function

CHH(x, y) =E[

{H(x)

H(x)

}{H(y)

H(y)

}]

A random fieldH(x) is called weakly homogeneous if

H(x) = const. x D; CHH(x, x+) =CHH() x D

A homogeneous random field H(x) is called isotropic if

CHH() =CHH(||||)

Correlation distance (characteristic lengthLc)



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Example: Random field in a square plate

Simulated random samples of isotropic field



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Conditional Random Fields 1

Assume that the values of the random fieldH(x) are known

at the locationsxk, k = 1 . . . m

Stochastic interpolation for the conditional random field:

H(xi) =a(x) +

mk=1

bk(x)H(xk)

a(x) and bk(x) are random interpolating functions.

Make the mean value of the difference between the random

field and the conditional field zero

E[H(x) H(x)] = 0

Minimize the variance of the difference

E[(H(x)

H(x))2]

Min.



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Conditional Random Fields 2Mean value of the conditional random field.

H(x) =

CHH(x, x1) . . . CHH(x, xm)

C1HH

H(x1)

...

H(xm)

CHH denotes the covariance matrix of the random fieldH(x)

at the locations of the measurements.Covariance matrix of the conditional random field

C(x, y) =C(x, y)

CHH(x, x1) . . . CHH(x, xm)C1HH

CHH(y, x1)...

CHH(y, xm)

Zero at the measurement points.



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Spectral decomposition

Perform a Fourier-type series expansion using deterministic

basis functionskand random coefficients ck

H(x) =

k=1

ckk(x), ck R, k R, x D

Optimal choice of the basis functions is given by aneigenvalue (spectral) decomposition of the

auto-covariance function (Karhunen-Loeve expansion)

CHH =

k=1

kk(x)k(y),DCHH(x, y)x(x)dx=kk(y)

The basis functionskare orthogonal and the coefficientsckare uncorrelated.



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Discrete version

Discrete random field

Hi =H(xi), i = 1 . . . N (1)

Spectral decomposition is given by

Hi

E(Hi) =

N

k=1

k(xi)ck =

N

k=1

ikck (1)

In matrix-vector notation

H= c+H

Computation of basis vectors by solving for the eigenvalues

kof the covariance matrixCHH

CHHk =kk; k 0; k= 1 . . . N 9/25 c Christian Bucher 2010-2014


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Example: Random field in a square plate

Basis vectors



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Modeling random fields

Most important: Correlation structure, most significantparameter is the correlation length Lc.

Estimate for the correlation length can be obtained by

applying statistical methods to observed data

Type of probability distribution of the material/geometricalparameters. Statistical methods can be applied to infer

distribution information from observed measurements.

Helpful to identify the exact type of correlation (or

covariance) function, and to check for homogeneity. This will

be feasible only if a fairly large set of experimental data isavailable.



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Computational aspects

Need to set up the covariance matrix from covariance

function of field

Storage requirements ofO(M2)Covariance matrix is full

Karhunen-Loeve expansion is realised using numerical

methods from linear algebra (eigenvalue analysis)

Numerical complexity ofO(M3)


S


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Simulation for small correlation length

Assemble sparse covariance matrix (e.g. based on piecewise

polynomial covariance functions)

Cl ,p(d) = (1 d/l)p+, p >1

Perform a decomposition of the covariance matrix, possible

C=LLT, eg. by a sparse Cholesky factorization.

Simulate Nfield vectorsukof statistically independent

standard-normal random variables, one number for each

node.

Apply the correlation in standard normal space for each

samplek: zk =Luk.Transform the correlated field samples into the space of the

desired random field: xk,i =F(1) (N(zk,i)).

Does not reduce the number of variables


Si l i f l l i l h 1


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Simulation for large correlation length 1

Typical covariance function

Cl(d) = exp

d

2

2l2

A spectral decomposition is used to factorize the covariance

matrix byCov=

diag(i)

T

with eigenvaluesi andorthogonal eigenvectors = [i].

This decomposition is used to reduce the number of random

variables. Given a moderately large correlation length, only a

few (eg. 3-5) eigenvectors are required to represent more

than 90% of the total variability.

Perform a decomposition of the covariance matrix

CHH = diag(i i)T and choose m basis vectorsi being

associated with the largest eigenvalues.


Si l i f l l i l h 2


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Simulate Nvectors ukof statistically independentstandard-normal random variables, each vector is of

dimensionm.

Apply the (decomposed) covariance in standard normal space

for each samplek

zk =

mi

iiuk,i

Transform the correlated field samples into the space of thedesired random field: xk,j =F(1)

N(zk,j)

.


Si l ti f l l ti l th 3


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A global error measure may be based on the total variability

being explained by the selected eigenvalues, i.e.

= 1

i=1cii=1ni

= 1 1n

i=1

ci

wherein n is the number of discrete points.

This procedure allows the generation of random field samples

with relatively large correlation length parameters

It is based on a model order reduction, i.e. only a portion of

the desired variability can be retained.

Covariance matrix is stored as a dense matrix. Hence, the

size of the FEM mesh is effectively limited to 30.000nodes (covariance matrix has 9x108 entries, i.e. > 7GB).


Effi i t i l ti t t


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Efficient simulation strategy

Randomly select M support points from the finite element

mesh.

Assemble the covariance matrix for the selected sub-space.Perform a decomposition of the covariance matrix

C= diag(i)T and choosem basis vectorsi.

Create basis vectorsi by interpolating the values ofi on

the FEM mesh.

Simulate Nvectors ukof statistically independentstandard-normal random variables, each vector is of

dimensionm.

Apply the (decomposed) covariance in standard normal space

for each samplek

zk =mi

iuk,i

Transform the correlated field samples into the space of the

desired random field: xk,j =F(1) N(zk,j).


E pansion Optimal Linear Estimator 1


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Expansion Optimal Linear Estimator 1

Expansion Optimal Linear Estimation (EOLE) is an

extension of Kriging

Kriging interpolates a random field based on samples being

measured at a sub-set of mesh points.Assume that the sub-space is described by the field values

yk = {zk,1, . . . , z k,M} = {m

i=1

ii ,1uk,i, . . . ,m

i=1

ii ,Muk,i}




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Minimization of the variance between the target random fieldand its approximation under the constraint of equal mean

values of both results in:

i =CTzyC

1yy

i

iwith Cyydenoting the correlation matrix between the

sub-space points andCzydenoting the (rectangular)

covariance matrix between the sub-space points and the

nodes in full space.


Example


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Example

Sheet metal forming applicationModelled by 4-node shell elements using 8786 finite element

nodes

Homogeneoues field, exponential correlation function

Maximum dimension is 540 mm, correlation lengthparameter is chosen to be 100 mm.

Truncated Gaussian distribution with mean value 5, standard

deviation 15, lower bound20 and upper bound 30

Sub-space dimension is chosen to be small (between 50 and1000 points)


Example


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Example

M = 50 M = 200

M = 500


Example Basis vectors (M = 50)


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Example - Basis vectors (M= 50)


Example - Basis vectors (full)


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Example - Basis vectors (full)


Errors


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Errors

MAC values of various shapes (reference of comparison: full

model) for different numbers of support pointsn.

n MAC 1 MAC2 MAC5 MAC1050 0.999 0.999 0.949 0.393

100 0.999 0.999 0.999 0.986200 0.999 0.999 0.999 0.999

400 0.999 0.999 0.999 0.999

800 1 1 0.999 0.999

8786 1 1 1 1


Concluding Remarks


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Concluding Remarks

Karhunen-Loeve expansion is very useful for reduction of

number of variables

Solution of eigenvalue problem may run into computationalproblems (storage, time)

Suitable reduction methods reduce storage and time

requirements drastically

Software Statistics on Structures - SoSby


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01 Bucher Random Fields - Efficient Analysis and Simulation

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