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4.2
Multiplicative Response Correction
The surrogate model can be constructed either from sampled high-fidelity model data
using an appropriate approximation technique [28], or by utilizing a physically-based
low-fidelity model [19]. Here, we exploit the latter approach as we have a reliable
low-fidelity model at our disposal (see Sec. 3.3). Also, good physically-based surro-
gates can be constructed using a fraction of high-fidelity model data necessary to
build accurate approximation models [29].
There are several methods of constructing the surrogate from a physically-based
low-fidelity model. They include, among others, space mapping (SM) [19], various
response correction techniques [30], manifold mapping [31], and shape-preserving
response prediction [32]. In this paper, the surrogate model is created using a simple
multiplicative response correction, which turns out to be sufficient for our purposes.
An advantage of such an approach is that the surrogate is constructed using a single
high-fidelity model evaluation, and it is very easy to implement.
Recall that
C
p.f
(
x
) and
C
f.f
(
x
) denote the pressure and skin friction distributions of the
high-fidelity model. The respective distributions of the low-fidelity model are denoted as
C
p.c
(
x
) and
C
f.c
(
x
). We will use the notation
C
p.f
(
x
) = [
C
p.f.
1
(
x
)
C
p.f.
2
(
x
) ...
C
p.f.m
(
x
)]
T
, where
C
p.f.j
(
x
) is the
j
th component of
C
p.f
(
x
), with the components corresponding to different
coordinates along the
x
/
L
axis.
At iteration
i
, the surrogate model
C
p.s
(
i
)
of the pressure distribution
C
p.f
is con-
structed using the multiplicative response correction of the form:
()
i
()
i
()
i
()
i
T
C
() [
x
=
C
()
x
C
() .
x
C
()]
x
(8)
ps
.
ps
. .1
ps
. .2
psm
. .
C
()
i
()
x
=⋅
A
()
i
C
()
x
(9)
ps j
..
p j
.
pc j
..
where
j
= 1, 2, ...,
m
, and
C
()
i
(
x
x
()
i
)
pf j
..
A
()
.
i
pj
=
(10)
()
i
()
i
C
(
)
pc j
..
Similar definition holds for the skin friction distribution model
C
f.s
(
i
)
. Note that the
formulation (8)-(10) ensures zero-order consistency [33] between the surrogate and
the high-fidelity model, i.e.,
C
p.f
(
x
(
i
)
) =
C
p.s
(
i
)
(
x
(
i
)
). Rigorously speaking, this is not suffi-
cient to ensure the convergence of the surrogate-based scheme (7) to the optimal solution
of (5). However, because of being constructed from the physically-based low-fidelity
model, the surrogate (8)-(10) exhibits quite good generalization capability. As demonstrat-
ed in Sec. 5, this is sufficient for good performance of the surrogate-based design process.
One of the issues of model (8)-(10) is that (10) is not defined whenever
C
p.c.j
(
x
(
i
)
)
equals zero, and that the values of
A
p.j
(
i
)
are very large when
C
p.c.j
(
x
(
i
)
) is close to zero. This
may be a source of substantial distortion of the surrogate model response as illustrated in
Fig. 8. In order to alleviate this problem, the original surrogate model response is “smoo-
thened” in the vicinity of the regions where
A
p.j
(
i
)
is large (which indicates the problems
mentioned above). Let
j
max
be such that |
A
p.j
max
(
i
)
| >> 1 assumes (locally) the largest value.
Let
m
). The original values of
A
p.j
(
i
)
Δ
j
be the user-defined index range (typically,
Δ
j
= 0.01
⋅
are replaced, for
j
=
j
max
-
Δ
j
, ...,
j
max
-1,
j
max
,
j
max
+1, ...,
j
max
+
Δ
j
, by the interpolated values:
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