Information Technology Reference
In-Depth Information
AI
i
pj
()
.
=
({[
j
−
2
Δ
j
...
j
− Δ −
j
1]
∪
max
max
∪+ Δ−
[
j
j
1...
j
+
2
Δ
∪
j
]},
max
max
(11)
{[
A
()
i
...
A
()
i
]
pj
.
−Δ
2
j
pj
.
−Δ−
j
1
max
max
()
i
()
i
∪
[
A
...
A
]},
j
)
pj
.
−Δ
2
j
pj
.
−Δ−
j
1
max
max
where
I
(
X
,
Y
,
Z
) is a function that interpolates the function values
Y
defined over the
domain
X
onto the set
Z
. Here, we use cubic splines. In other words, the values of
A
p.j
(
i
)
in the neighbourhood of
j
max
are “restored” using the values of
A
p.j
(
i
)
from the
surrounding of
j
=
j
max
-
j
. Figure 9(a) shows the “smoothened” surrogate
model response corresponding to that of Fig. 8. Figure 9 shows the surrogate and the
high-fidelity model responses, both
C
p
and
C
f
, at
x
(
i
)
and at some other design
x
.
Δ
j
, ...,
j
max
+
Δ
1
0.5
0
-0.5
-1
0
0.2
0.4
0.6
0.8
1
x/L
Fig. 8.
Surrogate model
C
p.s
(
i
)
(7)-(9) at
x
(
i
)
(- - -), and at some other design
x
(▬). By defini-
tion,
C
p.s
(
i
)
(
x
(
i
)
) =
C
p.f
(
x
(
i
)
). Note that
C
p.s
(
i
)
(
x
) has large spikes around the points where
C
p.s
(
i
)
(
x
(
i
)
)
is close to zero.
Fig. 9.
(a) Smoothened surrogate model (7)-(10)
C
p.s
(
i
)
(
x
(
i
)
) =
C
p.f
(
x
(
i
)
) (—),
C
p.s
(
i
)
(
x
) (- - -),
C
p.c
(
x
) (⋅ ⋅ ⋅), and
C
p.s
(
x
) (▬); (b) Smoothened responses
C
f.s
(
i
)
(
x
(
i
)
) =
C
f.f
(
x
(
i
)
) (—),
C
f.s
(
i
)
(
x
)
(- - -),
C
f.c
(
x
) (⋅ ⋅ ⋅), and
C
f.s
(
x
) (▬)
Search WWH ::
Custom Search