ibitfluiaiiotfi
o/
sioi
representation
jplf
0
{
b?k)
r ~
'
i3f
^p^QifHl
Masanoby Shinozyka
Department
of
Civil Engineering
and
Operations Research, Princeton University, Princeton
NJ
08544
George Deodatis
Department
of
Civil Engineering
and
Operations Research, Princeton University, Princeton
NJ
08544
The subject of this paper is the simulation of one-dimensional, uni-variate, stationary, Gaussian
stochastic processes using the spectral representation method. Following this methodology,
sample functions of the stochastic process can be generated with great computational efficiency
using a cosine series formula. These sample functions accurately reflect the prescribed
probabilistic characteristics of the stochastic process when the number N of the terms in
the cosine series is large. The ensemble-averaged power spectral density or autocorrelation
function approaches the corresponding target function as the sample size increases. In
addition, the generated sample functions possess ergodic characteristics in the sense that
the temporally-averaged mean value and the autocorrelation function are identical with the
corresponding targets, when the averaging takes place over the fundamental period of the
cosine series. The most important property of the simulated stochastic process is that it
is asymptotically Gaussian as A
r
—> oo. Another attractive feature of the method is that
the cosine series formula can be numerically computed efficiently using the Fast Fourier
Transform technique. The main area of application of this method is the Monte Carlo solution
of stochastic problems in engineering mechanics and structural engineering. Specifically, the
method has been applied to problems involving random loading (random vibration theory) and
random material and geometric properties (response variability due to system stochasticity).
CONTENTS
1 Introduction
2 Spectral Representation
of
Stationary Stochastic Processes
3 Simulation
of
Stochastic Processes
3.1 Simulation Formula
3.2 Rate
of
Convergence
of
Ensemble Autocorrelation Function
to
Target Autocorrelation Function
3.3 Gaussianness
of
Simulated Stochastic Process
3.4 Rate
of
Convergence
of
Simulated Stochastic Process
to
Gaussianness
4 Ergodicity
of
Simulated Stochastic Processes
4.1 Ergodicity
4.2 Rate
of
Convergence
of
Temporal Autocorrelation Function
to
Target Autocorrelation Function
4.3 Non-Ergodic Characteristics
of
Series Expression
in Eq (21)
5
Use of
Fast Fourier Transform (FFT) Technique
6 Numerical Examples
6.1 Stochastic Process Description
6.2 Simulation
by
Summation
of
Cosines
and
Zooming-in
6.3 Simulation
by FFT and
Zooming-in
6.4 Comparison Between Simulation
by
Summation
of
Cosines
and
Simulation
by FFT
Acknowledgment
References
1.
INTRODUCTION
In
the
last three decades
or
so, considerable progress
has
been made
in applying stochastic process theory
to the
general area
of
engineer-
ing mechanics
and
structural engineering
for the
purpose
of
assuring
the over-all structural safety
at a
higher level
of
reliability.
The
the-
ory
has
been initially applied
to
problems involving random loading
(random vibration theory)
and
during
the
last decade
to
problems
involving random material
and
geometric properties (response vari-
ability
due to
system stochasticity). Typical problems
in
random
vibration theory include among others
the
analysis
of
ship motions
Transmitted
by
Associate Editor Hayin Benaroya
caused
by
ocean waves,
the
analysis
of
aircraft response
to
gust
and
maneuver loads,
the
response analysis
of
offshore structures
to
wave
and wind forces,
the
study
of
vehicle vibrations caused
by
random
roadway roughness
and the
response analysis
of
structures subjected
to earthquake ground motion
or
atmospheric turbulence.
The
eval-
uation
of
response variability
due to
system stochasticity consists
of performing
the
response analysis
of
structural systems with ran-
domness
in
their material properties
(e.g.
elastic modulus)
or ge-
ometry
(e.g.
dimensions
of
structural members),
or
both.
The
most
widely-used method
to
solve random vibration problems
is the
fre-
quency domain analysis, while perturbation techniques
are the
most
frequently-used method
to
solve system stochasticity problems.
ASME Book
No
AMR094, $16.00
Appl Mech Rev,
vol
44,
no 4,
April
1991
131
© 1991 American Society
of
Mechanical Engineers
Downloaded 22 Mar 2010 to 128.42.164.80. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
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