Hydrodynamic Shape Optimization of Axisymmetric Bodies Using Multi-fidelity Modeling - Simulation and Modeling Methodologies, Technologies and Application - page 208

4 Surrogate Modeling and Optimization

In this section, a generic surrogate-based optimization scheme exploited here, as well
as the methodology for creating a surrogate model, is described. The specific method
utilized to align the responses of the low- and high-fidelity model, both the hull pres-
sure and skin friction distributions, is a multiplicative response correction.

4.1 Surrogate-Based Optimization Scheme

Our design task is formulated as a nonlinear minimization problem of the form

*

(6)

x = arg min f ( x )

lxu ≤≤

where f ( x ) is the objective function, x is the design variable vector, whereas l and u
are the lower and upper bounds, respectively. The high-fidelity model evaluation is
CPU-intensive so that solving the problem (6) directly, by plugging in the high-
fidelity model into the optimization loop, may be impractical. Instead, we exploit
surrogate-based optimization (SBO) [14,15] that shifts the optimization burden into
the computationally cheap surrogate, and, thus, allows us to solve (6) at a low compu-
tational cost.
The generic SBO optimization scheme is the following

(1) i + () i

x x = arg min s ( )

(7)

x

where x ( i ) , i = 0, 1, ..., is a sequence of approximate solutions to (6), whereas s ( i ) is the
surrogate model at iteration i . If the surrogate model is sufficiently good representa-
tion of the high-fidelity model f , the number of iterations required to find a satisfacto-
ry design is small [27].

Fig. 7. Edge of an element on the hull surface at radius r . The element length is Δ x and it makes
an angle θ to the x -axis. Pressure p acts normal to the hull surface. Shear stress τ acts parallel to
the surface.