Table Of ContentStochastic Finite Elements:
A Spectral Approach
Stochastic Finite Elements:
A Spectral Approach
Roger G. Ghanem
School of Engineering
State University of New York
Buffalo, New York
Pol D. Spanos
L.B. Ryon Chair in Engineering
Rice University
Houston, Texas
Springer-Verlag
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Roger G. Ghanem Pol D. Spanos
School of Engineering L.B. Ryon Chair in Engineering
and Applied Sciences Rice University
State University of New York Houston, TX 77251
Buffalo, NY 14260 USA
USA
Cover art: Description can be found in Figure 5.74 on page 176.
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9 8 7 6 5 432 1
ISBN-13: 978-1-4612-7795-8 e-ISBN-13: 978-1-4612-3094-6
DOl: 10.107/978-1-4612-3094-6
To my Mother and my Father.
R.C.C.
To my parents Demetri and Aicaterine, my first mentors in
quantitative thinking; to my wife Olympia, my permanent
catalyst in substantive living; and to my children Demetri and
Evie, a perpetual source of delightful randomness.
P.D.S.
" ... The principal means for ascertaining truth - induction and analogy -
are based on probabilities; so that the entire system of human knowledge
is connected with the theory (of probability) ... "
Pierre Simon de Laplace,
A Philosophical Essay on Probability, 1816.
" Nature permits us to calculate only probabilities, yet science has not
collapsed."
Richard P. Feynman,
QED: The Strange Theory of Light and Matter, 1985.
Preface
This monograph considers engineering systems with random parame
ters. Its context, format, and timing are correlated with the intention
of accelerating the evolution of the challenging field of Stochastic
Finite Elements. The random system parameters are modeled as
second order stochastic processes defined by their mean and covari
ance functions. Relying on the spectral properties of the covariance
function, the Karhunen-Loeve expansion is used' to represent these
processes in terms of a countable set of un correlated random vari
ables. Thus, the problem is cast in a finite dimensional setting.
Then, various spectral approximations for the stochastic response of
the system are obtained based on different criteria. Implementing
the concept of Generalized Inverse as defined by the Neumann Ex
pansion, leads to an explicit expression for the response process as a
multivariate polynomial functional of a set of un correlated random
variables. Alternatively, the solution process is treated as an element
in the Hilbert space of random functions, in which a spectral repre
sentation in terms of the Polynomial Chaoses is identified. In this
context, the solution process is approximated by its projection onto
a finite subspace spanned by these polynomials.
The concepts presented in this monograph can be construed as ex
tensions of the spectral formulation of the deterministic finite element
method to the space of random functions. These concepts are further
elucidated by applying them to problems from the field of structural
mechanics. The corresponding results are found in agreement with
those obtained by a Monte-Carlo simulation solution of the problems.
The authors wish to thank Rice University, the Houston Ad
vanced Research Center (HARC), and the National Center for Su
percomputing Applications (NCSA) for the extensive use of their
computational facilities during the course of the studies which have
viii PREFACE
led to the conception, development, and integration of the material of
this monograph. The financial support, over a period of years, from
the National Science Foundation, the National Center for Earthquake
Engineering Research at the State University of New York at Buffalo,
the Air Force Office of Scientific Research, and Rice University is
gratefully acknowledged. Further, the stimulating discussions with a
plethora of students and colleagues are greatly appreciated.
!t.G. Ghanem
P.D. Spanos
October 1990.
Contents
PREFACE
Vll
1 INTRODUCTION 1
1.1 Motivation ... 1
1.2 Review of Available Techniques. 3
1.3 The Mathematical Model 9
1.4 Outline ............. . 12
2 REPRESENTATION OF STOCHASTIC
PROCESSES 15
2.1 Preliminary Remarks ... . 15
2.2 Review of the Theory .. . 16
2.3 Karhunen-Loeve Expansion 20
2.3.1 Derivation ..... . 20
2.3.2 Properties ..... . 24
2.3.3 Solution of the Integral Equation 27
2.4 Homogeneous Chaos ...... . 45
2.4.1 Preliminary Remarks ...... . 45
2.4.2 Definitions and Properties . . . . 47
2.4.3 Construction of the Polynomial Chaos 50
3 STOCHASTIC FINITE ELEMENT METHOD:
Response Representation 67
3.1 Preliminary Remarks ..... . 67
3.2 Deterministic Finite Elements . 69
3.2.1 Problem Definition .. 69
3.2.2 Variational Approach . 70
3.2.3 Galerkin Approach . . . 72
3.2.4 p-Adaptive Methods, Spectral Methods and Hi-
erarchical Finite Element Bases . . . . . . . .. 73
x CONTENTS
3.3 Stochastic Finite Elements. . . . . . . . 74
3.3.1 Preliminary Remarks ...... . 74
3.3.2 Monte Carlo Simulation (MCS) . 75
3.3.3 Perturbation Method. . . . . . 77
3.3.4 Neumann Expansion Method . . 79
3.3.5 Improved Neumann Expansion . 81
3.3.6 Projection on the Homogeneous Chaos. 85
3.3.7 Geometrical and Variational Extensions 97
4 STOCHASTIC FINITE ELEMENTS:
Response Statistics 101
4.1 Reliability Theory Background . . . . . 101
4.2 Statistical Moments .......... . 104
4.2.1 Moments and Cummulants Equations . 104
4.2.2 Second Order Statistics . . . . . . . . . 113
4.3 Approximation to the Probability Distribution 115
4.4 Reliability Index and Response Surface Simulation . 118
5 NUMERICAL EXAMPLES 121
5.1 Preliminary Remarks. . . . . 121
5.2 One Dimensional Static Problem . 122
5.2.1 Formulation........ . 122
5.2.2 Results .......... . 127
5.3 Two Dimensional Static Problem . 150
5.3.1 Formulation........ . 150
5.3.2 Results .......... . 160
5.4 One Dimensional Dynamic Problem · 177
5.4.1 Description of the Problem · 177
5.4.2 Implementation. · 178
5.4.3 Results .......... . · 181
6 SUMMARY AND CONCLUDING REMARKS 185
BIBLIOGRAPHY 193
INDEX 211
Chapter 1
INTRODUCTION
1.1 Motivation
Randomness can be defined as a lack of pattern or regularity. This
feature can be observed in physical realizations of most objects that
are defined in a space-time context. Two sources of randomness are
generally recognized (Matheron, 1989). The first one is an inherent
irregularity in the phenomenon being observed, and the impossibility
of an exhaustive deterministic description. Such is the case, for in
stance, with the Uncertainty Principle of quantum mechanics and the
kinetic theory of gas. The other source of randomness can be related
to a generalized lack of knowledge about the processes involved. The
level of uncertainty associated with this class of problems can usually
be reduced by recording more observations of the process at hand
and by improving the measuring devices through which the process
is being observed. In this category fall, for instance, the econometric
series whereby the behavior of the financial market is modeled as a
stochastic process. This process, however, can be entirely specified,
in the deterministic sense, from a complete knowledge of the flow
of goods at the smallest levels. Another example that is of more
direct concern in the present study pertains to the properties of a
soil medium. These properties are uniquely defined at a given spatial
location within the medium. It is quite impractical, however, to
measure them at all points, or even at a relatively large number of
points. From a finite number of observations, these properties may be
modeled as random variables or, with a higher level of sophistication,
