Page 69 - Practical Design Ships and Floating Structures

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The optimal outer-hull arrangement is determined in the second step for the optimal center and outer
hull forms obtained in the first step. The present hull form optimization uses a gradient-based
technique, which requires a field solution for each design variable. A very simple CFD tool based on
the zeroth-order slender ship approximation is ideally suited for such an optimization technique
because of its extreme simplicity and efficiency. It has been shown in Yang et. al. (2000) that this
simple zeroth-order slender-ship theory, first given by Nobless (I 983), is adequate for the purpose of
determining the optimal hull arrangements for a wave cancellation multihull ship. This simple CFD
tool has also been used with success for hull-form optimization in Letcher et. al. (1987) and Wyatt and
Chang (1 994).

The hull surface is represented by a triangulation. This triangulation can be used to evaluate the wave
drag using present CFD tool. In order to obtain smooth hulls in the optimization process, the (very fast)
pseudo-shell approach developed by Soto et. al. (2001) is employed. The surface of the hull is
represented as a shell. The shell equations are solved using a stabilized finite element formulation with
given boundary conditions to obtain the rotation and displacement fields. Almost every grid point on
the hull surface can be chosen as design parameter, which leads to a very rich design space with
minimum user input.

The optimal outer-hull arrangement for the optimal center and outer hulls is determined by searching
the entire parameter space for each given Froude number for the purpose of minimizing the wave drag,
that is evaluated very efficiently using the present simple CFD tool. Results indicate that the new
design can achieve a fairly large wave drag reduction in comparison to the original design
configuration.

2 OPTIMAL SHAPE DESIGN
Any CFD-based optimal shape design procedure consists of the following ingredients:
- A set of design variables that determine the shape to be optimized;
- A set of constraints for these variables in order to obtain sensible shapes;
- An objective function I to measure the quality of a design;
- A field solver to determine the parameters required by I (e.g. drag, lift, moment, etc.);
- An optimization algorithm to determine how to change the design parameters in a rational and
expedient way.

The present hu11 form optimization uses a gradient-based technique. The gradients are obtained via
finite differences. This implies that for each design parameter, a field solution has to be obtained,
making the use of extremely fast solvers imperative. One optimization step may be summarized as
follows:
- Evaluate the objective function I for the original geometry Zs .
- Evaluate the gradient of the objective function for each design variable k = I, Nd:
a) Perturb the coordinates of the k-rh design variable in its deformation direction by a
small F, the rest of design parameters are not moved;
b) Solving the pseudo-shell problem using given boundary condition; this yields a new
perturbed geometry Z, ;
c) Evaluate the objective function I' for the perturbed geometry E; ;
d) Obtain the gradient of the objective function with respect to the k-th design variable by
finite differences as (I - f)/& .
- Make a line search in the negative gradient direction to find a minimum.
The detailed discussion about this approach can be found in Soto et. al. (2001). In the sequel, we
describe the objective function, the surface representation and the CFD solver used.