Page 70 - Practical Design Ships and Floating Structures

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3 OBJECTIVE FUNCTION

From an engineering perspective, it is important to reduce the wave drag while still being able to
achieve a given displacement. For this reason, the objective function used for hull shape optimization
is given by:

where C, and Ci are the wave drag coefficient and its initial value, V and V' the hull displacement
and its initial value, and 0<0,,~$1 are relative weights. It was found to be very important to cast the
optimization function in this non-dimensional form. Otherwise the weights WIJ have to be adjusted for
different geometries.

4 SURFACE REPRESENTATION

There are many ways to represent surfaces. Analytical expressions given by B-Splines, NURBS or
Coon's patches are common. Another possibility is to take a surface triangulation and then allow every
point on the surface to move. This discrete surface representation can always be obtained from
analytical surface descriptions, and, for sufficiently fine surface triangulations, provides a very rich
design space with minimal user input. For this reason, this discrete surface description is used in the
present work.
During optimization, the individual points on the surface may move in such a way that a non-smooth
hull is produced. In order to obtain smooth hulls, 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 movement of
points is recast as a forcing term for the movement of the shell. The shell equations are solved using a
stabilized finite element formulation with given boundary conditions to obtain the rotation and
displacement fields. The boundary conditions in a shape optimization problem are dictated by the
design parameter displacement and the geometrical constrains. In the optimal design process, the user
only needs to generate the original surface mesh and a few design variables. The rest of the design
parameters and their respective deformation modes can be generated automatically by the method.

5 CFD SOLVEK FOR WAVE DRAG REPRESENTATION

Consider a ship advancing along a straight path, with constant speed U, in calm water of effectively
infinite depth and lateral extent. The x axis is taken along the path of the ship and points toward the
ship bow, the z axis is vertical and points upward, and the mean free surface is the plane z=U. Non-
dimensional coordinates (x, y, z) and velocities (u, v, w) are defined in terms of a characteristic length
L (taken as the length of the center hull for a wave cancellation multihull ship) and the ship speed U.
The wave drag CW is evaluated using the Havelock formula

for the energy radiated by the far-field waves. DW is the wave drag and vis defined as
U
o=- with F =- (1b)
2F2 G
Furthermore the wavenumber k in Eqn. la is defined in terms of the Fourier variable pas

k(p )=v+,/m (IC)