Environmental Engineering Reference
In-Depth Information
Table 6.2 Hermite polynomial chaos basis M = 2
standard normal variables, p = 3
j
Ψ α Ψ j
α
0
(0, 0)
Ψ 0 = 1
1
(1, 0)
Ψ 1 = U 1
2
(0, 1)
Ψ 2 = U 2
3
(2, 0)
1 2
Ψ 3
=−
(
U
12
)
/
4
(1, 1)
Ψ 4 = U 1 U 2
5
(0, 2)
(
U
2
12
)
Ψ 5
=−
2
6
(3, 0)
(
UU
1 3
3
)
6
Ψ 6
=−
1
7
(2, 1)
(
UU
1 2
1
2
Ψ 7
=−
2
8
(1, 2)
(
UU
2
1
2
Ψ 8
=−
2
1
9
(0, 3)
(
UU
3
3
)
6
Ψ 9
=−
2
2
As a conclusion, the random response of our computational model Y = M (T ( U 1 , U 2 )) will
be approximated by a 10-term polynomial series expansion in ( U 1 , U 2 ):
~
def
Y
=
PC M 12
(, )
U
Uy
=
+
yU
+
yU
+
yU
(
2
12
)
+
yUU
0
1
1
2
2
3
412
+
yU
(
2
12+
)
yU
(
3
3
U
)
6
+
y UU
(
2
1
)
2
52
61
1
7
1
2
(6.23)
+
yU
(
2
1
)
U
2
+
y
(
UU
3
3
)
6
.
82
1
9
2
2
6.3.4 Computation of the coefficients
6.3.4.1 Introduction
Once the truncated basis has been selected, the coefficients {}
α α∈ shall be computed.
Historically, the so-called intrusive computation schemes have been developed in the
context of stochastic finite-element analysis (Ghanem and Spanos, 1991). In this setup,
the constitutive equations of the physical problem (e.g., linear elasticity for estimating
the settlement of foundations) are discretized both in the physical space (using standard
finite-element techniques) and in the random space using the PCE. This results in coupled
systems of equations that require ad-hoc solvers, thus the term “intrusive.” The applica-
tion of such approaches to geotechnical problems may be found in Ghiocel and Ghanem
(2002); Berveiller et al. (2004b); Sudret et al. (2004, 2006); and Sudret and Berveiller
(2007, Chapter 7 ).
In the past decade, alternative approaches termed nonintrusive have been developed
for computing the expansion coefficients. The common point of these techniques is that
they rely upon the repeated run of the computational model for selected realizations
of random vector X , exactly as in MCS. Thus, the computational model may be used
without modification. The main techniques are now reviewed with an emphasis on least-
square minimization.
y
A M,p
 
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