Environmental Engineering Reference
In-Depth Information
ingredient in this approach is the so-called
polynomial chaos expansions
(PCEs), which
allow one to represent a random output (e.g., the nodal displacement vector resulting from
a finite-element analysis) as a polynomial series in the input variables. Early applications of
such an SFEM to geotechnics can be found in Ghanem and Brzkala (1996); Sudret and Der
Kiureghian (2000); Ghiocel and Ghanem (2002); Clouteau and Lafargue (2003); Sudret
et al. (2004, 2006); and Berveiller et al. (2006).
During the past 10 years, PCEs have become a cross-field key approach to uncertainty
quantification in engineering problems ranging from computational fluid dynamics and
heat-transfer problems to electromagnetism. The associated computational methods have
also been somewhat simplified due to the emergence of
nonintrusive
spectral approaches,
as shown later.
The goal of this chapter is to give an overview on stochastic finite-element analysis using
PCEs, focusing more specifically on nonintrusive computation schemes. The chapter is orga-
nized as follows. Section 6.2 presents a versatile uncertainty quantification framework that
is now widely used by both researchers and practitioners (Sudret, 2007; De Rocquigny,
2012). Section 6.3 presents the machinery of PCEs in a step-by-step approach: how to con-
struct the polynomial chaos basis, how to compute the coefficients, how to estimate the
quality of the obtained series, and how to address large dimensional problems using
sparse
expansions. Section 6.4 shows how to postprocess a PC expansion for different applica-
tions, that is, compute statistical moments of the response quantities, estimate the model
output distribution, or carry out sensitivity analysis. Finally, Section 6.5 presents different
application examples in the field of geotechnics.
6.2 unCertaIntY ProPagatIon FraMeWork
6.2.1 Introduction
Let us consider a physical system (e.g., a foundation on a soft soil layer, a retaining wall,
etc.) whose mechanical behavior is represented by a computational model M:
M
x
∈⊂ =
D
y
M
()
x
∈
.
(6.1)
X
In this equation,
x
= {
x
1
, …,
x
M
}
T
gathers the
M
input parameters of the model while
y
is
the
quantity of interest
(QoI) in the analysis, for example, a load-carrying capacity, a limit
state equation for stability, and so on. In the sequel, only models having a single (scalar)
QoI are presented, although the derivations hold component-wise in case of vector-valued
models
y
q
As shown in
Figure 6.1
,
once a computational model M is chosen to evaluate the per-
formance of the system of interest (Step A), the sources of uncertainty are to be quantified
(Step B): in this step, the available information (expert judgment on the problem, databases
and literature, and existing measurements) is used to build a proper probabilistic model of
the input parameters, which is eventually cast as a random vector
X
described by a joint
probability density function (PDF)
f
X
. When the parameters are assumed statistically inde-
pendent, this joint distribution is equivalently defined by the set of marginal distribution of
all input parameters, say {,
=
M()
x
∈
.
X
i
= 1
…
If dependence exists, the copula formalism may
be used, see Nelsen (1999); Caniou (2012). As a consequence, the QoI becomes a random
variable
f
i
,
,
M
}.
Y
= M( ,
X
(6.2)
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