Optimal Ship Forms for Minimum Total Resistance

Abstract

A numerical scheme is developed by using the "tent" function to compute the hull surface area and then the ship frictional resistance in a quadratic form in terms of ship offsets. Combining with the wave resistance, a total ship resistance formula is derived in a standard quadratic form. With a set of linear-inequality constraints, the optimal solution of ship offsets for minimum total resistance can be obtained by applying a quadratic programming method to the problem. Computations have been carried out for three conditions which have exactly the same constraints as those required to obtain the optimal forms for minimum wave resistance shown in an earlier work [1].3 The optimal forms found for minimum total resistance also have either bow or midship bulbs. The new optimal bulbs have a relatively small size, higher vertical slope to the baseline, and less curvature at the tip of the bulb.

Continuity with respect to the speed for optimal ship forms based on Michell's formula

“…Now we assume by contradiction that J(u ) = 16π for some u ∈ C + (R × R ). Then by (31) we have J 0 (u ) = 16π and J wave (u ) = 0. In particular, u is a minimizer of J 0 in C + (R × R ).…”

“…This "sinking" process can be related to the non-existence of a minimizer in the case where no bounding box is added to the formulation of the problem. It gives a new interpretation to the midship bulbs obtained by some authors in the case of a fixed domain [24,30,31].…”

“…By solving a linear integro-differential equation, they showed that a unique solution exists in the class of continuous functions. Closely related problems involving Michell's formula were also studied numerically, theoretically and experimentally by several authors (see [18,23,31,35,36,38] and [49, p. 209]). It is also natural that the hull be represented by a nonnegative function, in order to avoid self-crossing [42].…”

“…We note that the H 1 Sobolev space is associated to the Dirichlet energy in a natural way. This minimization problem can be discretized for numerical simulations and solved in the framework of quadratic programming [18,23,31,35].…”

“…The numerical simulations in [18,31,35] showed that the famous bulbous bow [20,26] can be very efficient in minimizing the total resistance, for some values of the parameters. In some cases, a midship bulb was also found to be interesting [15,23,30,31].…”