11.4 Random Field Simulation

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The use of wavelets for random field synthesis can be examined within the more general framework of

scale-type methods. The latter have been developed to improve the computational performances of

Monte Carlo simulations. Classical methods such as the spectral approach [71] or the autoregressive

moving average (ARMA) [72] are not readily applicable for this purpose, especially when using

nonuniform meshes or when enhancement of local resolution is desirable. To address these

shortcomings, Fournier et al. [73] have proposed a “random midpoint method” to synthesize fractional

Brownian motion; that is, a scale-type method where values of the random field for points within a

coarser scale are generated first, and then the generated samples are used to determine values for a finer

scale. This approach has been extended by Lewis [74] into a “generalized stochastic subdivision method,”

suitable for a broad class of stationary processes, and by Fenton and Vanmarcke [75] into a “local average

subdivision method,” which includes a random field smoothing procedure producing averages of the

field for an increasingly finer scale.

An interpretation of scale-type approaches in the context of random field synthesis has been given by

Zeldin and Spanos [39] using compactly supported Daubechies wavelets. Specifically, a synthesis

algorithm has been developed that includes the previous methods proposed by Lewis [74] and Fenton

and Vanmarcke [75] as a particular case. To synthesize a sample of a given process, the closed-form

expressions

r j;i

k;l ¼ E

h

d j

k di

l

i

¼

ð1

21

ð1

21

Rf ðx1; x2Þcj;k ðx1Þci;lðx2Þdx1 dx2 ð11:67Þ

b j;i

k;l ¼ E

h

c j

k di

l

i

¼

ð1

21

ð1

21

Rf ðx1; x2Þfj;kðx1Þci;lðx2Þdx1 dx2 ð11:68Þ

a j;i

k;l ¼ E

h

c j

k ci

l

i

¼

ð1

21

ð1

21

Rf ðx1; x2Þfj;kðx1Þfi;lðx2Þdx1 dx2 ð11:69Þ

given in Refs. [21,39] are considered to relate the autocorrelation function Rf ðx1; x2Þ of the process to the

coefficients of its wavelet transform, which in this case are random variables. The synthesis algorithm is

based on the wavelet reconstruction algorithm developed by Mallat [34,35], which proceeds from coarse

to fine scales to determine the wavelet coefficients. Some relevant properties of wavelet ensure the

computational efficiency of the algorithm. Specifically, using the quasi-differential properties of

wavelets showed by Belkin [76], the coefficients d j

k ’s are derived directly from c j

k’s by the approximate

11-14 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

linear combination

d j

k ¼

X

l

a j

k;lc j

l þ b j

kuk ð11:70Þ

where uk’s are uncorrelated, zero mean, unit variance random variables, statistically independent of c j

k ’s.

For a wide class of stochastic processes, wavelet coefficients prove weakly correlated as the difference k 2 l

increases and, for this, the summation in Equation 11.70 is generally restricted to adjacent elements only.

The algorithm is completed by an error-assessment procedure which allows refining of the triggering

scale j in order to fit the sought target statistical properties of the synthesized field.

Further studies on the role of wavelet analysis in stochastic mechanics applications may be found in

Ref. [21], which has showed how wavelet bases can be used in approximate Karhunen – Loe`ve expansions.

Any stationary process can then be represented as

f ðtÞ ¼

X

j;k

d j

kcj;k ðtÞ ð11:71Þ

where d j

k’s are uncorrelated random variables and cj;kðtÞ are a nonorthogonal wavelet-like basis.

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