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5
Numerical Examples
The methodology of Section 4 is applied to the hydrodynamic shape optimization
of torpedo-type hulls, involving both the direct and inverse design approaches. De-
signs are obtained with the surrogate model optimized using the pattern-search algo-
rithm [34]. For comparison purposes, designs obtained through direct optimization of
the straightforward high-fidelity model using the pattern-search algorithm
[34] are
also presented.
For both the direct and the inverse design approaches, the design variable vector is
x
= [
a
x
n
y
n
x
t
y
t
]
T
, where
a
is the nose length, (
x
n
,
y
n
) and (
x
t
,
y
t
) are the coordinates of
the free control points on the nose and tail Bézier curves, respectively, i.e., points 3
and 8 in Fig. 2. See Section 3.1 for a description of the shape parameterization. The
lower and upper bounds of design variables are
l
= [0 0 0 80 0]
T
cm
and
u
= [30 30 10
100 10]
T
cm
, respectively. Other geometrical shape parameters are, for both cases,
L
=
100 cm
, d
= 20
cm,
and
b
= 50
cm
. The flow speed is 2
m
/
s
and the Reynolds number
is 2 million.
5.1
Direct Design
Numerical results for a direct design case are presented in Table 1. The hull drag
coefficient is minimized by finding the appropriate shape and length of the nose and
tail sections for a given hull length, diameter, and cylindrical section length. In this
case, the drag coefficient is reduced by 6.3%. This drag reduction comes from a re-
duction in skin friction and a lower pressure peak where the nose and tail connect
with the midsection (Figs. 10(a) and 10(b)). These changes are due to a more stream-
lined nose (longer by 6 cm) and a fuller tail, when compared to the initial design (Fig.
10(c)).
Our approach requires 3 high-fidelity and 300 low-fidelity model evaluations. The
ratio of the high-fidelity model evaluation time to the corrected low-fidelity model
evaluation time varies between 11 to 45, depending on whether the flow solver con-
verges to the residual limit of 10
-6
, or the maximum iteration limit of 1000. We ex-
press the total optimization cost of the presented method in the equivalent number of
high-fidelity model evaluations. For the sake of simplicity, we use a fixed value of 30
as the high- to low-fidelity model evaluation time ratio. The results show that the total
optimization cost of the presented approach is around 13 equivalent high-fidelity
model evaluations. The direct optimization method, using the pattern-search algo-
rithm [34] yields very similar design, but at the substantially higher computational
cost of 282 high-fidelity model evaluations.
5.2
Inverse Design
Inverse design of the hull shape was performed by prescribing a target pressure distri-
bution. The objective is to minimize the norm of the difference between the pressure
distribution of the hull design and the target pressure distribution. The numerical re-
sults are of the inverse design are presented in Table 1.
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