Advances in Mechanical Engineering
by,theSobol’indicesarealsodynamicallychanged.usthe
time-dependent Sobol’ indices are dened by
𝑖
1
,...,𝑖
𝑠
(
)
=
𝑖
1
,...,𝑖
𝑠
(
)
(
)
,
1≤
1
≤⋅⋅⋅≤
𝑠
≤, =1,...,.
()
By denition, according to (), the following expression
is obtained:
𝑘
𝑖=1
𝑖
(
)
+
1≤𝑖<𝑗≤𝑛
𝑖𝑗
(
)
+⋅⋅⋅+
1,2,...,𝑛
(
)
=1.
()
Each
𝑖
1
,...,𝑖
𝑠
() is a quantitative index to answer how
uncertainty in the output of the model can be apportioned to
dierent sources of uncertainty in the model input parame-
ters {
1
,...,
𝑠
}at each time instance. In other words, the rst-
order indices
𝑖
()represent the inuence of each parameter
takenaloneandmeanthemaineectoftheparameter
𝑖
on (,𝜉). e high order indices
𝑖
1
,...,𝑖
𝑠
(), > 1, explain
possible interactive inuence of various parameters and mean
the joint eect of the input parameters {
1
,...,
𝑠
}on (,𝜉).
e total sensitivity function of parameter
𝑖
is dened by
𝑇𝑖
(
)
=
𝑖
(
)
+
𝑘
𝑗=1
𝑗 =𝑖
𝑖𝑗
(
)
+
𝑘
𝑟=𝑗+1
𝑟 =𝑖
𝑗 =𝑖
𝑖𝑗𝑟
(
)
+⋅⋅⋅+
𝑖𝑗𝑟,...,𝑘
(
)
,
()
which accounts for the total contribution to the output
variation due to the parameter
𝑖
at each time instant.
However, the Sobol’ indices are practically computed
using Monte Carlo simulation, which makes them hardly
applicable for computational simulation models [], espe-
ciallyastheyarerequiredtoperformateachtimeinstant.To
solve this problem, a surrogate model called time-dependent
PCE used to represent the response of the long-term degen-
eracy model is invested. e next section will introduce the
time-dependent PCE in detail.
4. Time-Dependent PCE for the Model Output
Polynomial chaos expansion is a promising surrogate model
that uses a set of orthogonal polynomial basis to approximate
the random space of the system response []. ere are two
types of PCE methods: intrusion approach and nonintrusive
approach according to the method to compute the coecients
of PCE. Intrusion approach is dicult, expensive, and time
consuming for many complex computational problems [].
us, this paper employs nonintrusive method to compute
the coecients of PCE.
e polynomial chaos for a dynamic system response can
be described as follows [–]:
(
,𝜉
)
=
0
(
)
+
∞
𝑖
1
=1
𝑖
1
(
)
Γ
1
𝑖
1
+
∞
𝑖
1
=1
𝑖
1
𝑖
2
=1
𝑖
1
𝑖
2
(
)
Γ
2
𝑖
1
,
𝑖
2
+
∞
𝑖
1
=1
𝑖
1
𝑖
2
=1
𝑖
2
𝑖
3
=1
𝑖
1
𝑖
2
𝑖
3
(
)
Γ
3
𝑖
1
,
𝑖
2
,
𝑖
3
,
()
where (,𝜉) is the random system response,
𝑖
() is the
coecient of PCE, Γ
𝑝
(
𝑖
1
,...,
𝑖
𝑝
) is the polynomial of the
selected basis, and is the polynomial degree. According
to the Wiener-Askey scheme, there are several families of
polynomials (e.g., Hermite, Legendre, Jacobi, etc.) []that
can be employed depending on the random distributions.
For example, when
𝑖
represents a standard normal random
variable, Hermite’s polynomials are selected as basis. e
multidimensional Hermite polynomials in ()aregivenby
Γ
𝑝
𝑖
1
,...,
𝑖
𝑝
=
(
−1
)
𝑝
(1/2)𝜉
𝑇
𝜉
𝑝
𝑖
1
⋅⋅⋅
𝑖
𝑝
(1/2)𝜉
𝑇
𝜉
,
()
where 𝜉 isthevectorofnormalrandomvariables{
𝑖
𝑘
}
𝑝
𝑘=1
.In
practical engineering, PCE contains limited input uncertain-
ties.us,() can be simplied as follows:
(
,𝜉
)
=
𝑁
𝑐
−1
𝑗=0
𝑗
(
)
𝑗
(
𝜉
)
,
()
where
𝑗
(𝜉)=
∏
𝑝
𝑖=1
𝑗
𝑚
𝑖
(
𝑖
)=Γ
𝑝
(
𝑖
1
,...,
𝑖
𝑝
)and
𝑐
is the
total number of PCE coecients, which can be calculated as
𝑐
=1+
!
(
−1
)
!
+
(
+1
)
!
(
−1
)
!2!
+⋅⋅⋅+
−1+!
(
−1
)
!!
=
+!
!!
,
()
where is the number of random variables in the system.
e multidimensional Hermite polynomials form an
orthogonal basis for the space of square-integrable PDFs, and
thePCEisconvergentinthemeansquaresense[]. In
general, the approximation accuracy rises with the order of
thePCE.ExceptforHermitepolynomials,onecanalsouse
other orthogonal polynomials from the generalized Askey
scheme for some standard non-Gaussian input uncertainty
distributions, for example, Laguerre polynomials for gamma
distributions, Jacobi polynomials for beta distribution, and
Charlier polynomials for Poisson distribution, and so forth
[]. For any arbitrary input distribution, a Gram-Schmidt
orthogonalization can be employed to generate the orthog-
onal family of polynomials [].
If the distribution types of each parameter are not unied,
we have to represent all uncertain parameters in terms of
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