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norm of the difference between the target and design distributions. The main difficul-
ty in this approach is to define the target distribution.
In this paper, we apply both the direct and the inverse approaches to the hydrody-
namic shape optimization of axisymmetric bodies. The direct approach involves the
minimization the drag coefficient (
C
D
) and we set
f
(
x
) =
C
D
(
x
). The drag coefficient
and other characteristic variables are defined in Sec. 3.4. In the inverse approach, a
target pressure distribution (
C
p,t
) is prescribed a priori and the function
f
(
x
) = ||
C
p
(
x
) -
C
p.t
||
2
, where
C
p
(
x
) is the hull surface pressure distribution, is minimized. No con-
straints are considered in this work, aside from the design variable bounds.
2.2
Solution Approach
The design problem (1) can be solved in a straightforward way using any available
algorithm to directly optimize the high-fidelity CFD model (Fig. 1(a)). In many cases,
this is impractical due to high computational cost of an accurate CFD simulation and
the fact that conventional optimization methods (e.g., gradient-based) normally re-
quire large number of objective function evaluations to yield an optimized design
[21]. This problem can be partially alleviated using cheap adjoint sensitivity [8], how-
ever, they are not always available and the number of required CFD simulations may
still be prohibitively large.
Here, the design process is accelerated using surrogate-based optimization (SBO)
[14], [15] exploiting physics-based low-fidelity models. The low-fidelity model inhe-
rits the knowledge about the system under consideration and it can be used to create a
fast and yet accurate representation of the high-fidelity model, a surrogate. The surro-
gate model can be then used as a prediction tool that yields an approximate high-
fidelity model optimum at a low computational cost [22]. The flow diagram of the
SBO process is shown in Fig. 1(b).
The surrogate model considered in this paper is constructed using an underlying
low-fidelity CFD model and a response correction technique. The low-fidelity CFD
model is constructed using the high-fidelity CFD model with a coarser mesh-
resolution and relaxed convergence criteria; called variable-resolution modeling [17].
A description of the variable-resolution modeling is given in Section 3. There are
various response correction techniques available, and the type appropriate in each
case depends on the nature of the response. A multiplicative response correction is
used in this work and is described in Section 4.
3
Computational Models
In this section, we describe the major components of the computational model of the
axisymmetric hull shape considered in this paper. In particular, we discuss the shape
parameterization, the high- and low-fidelity CFD models, as well as calculation of the
hull drag force.
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