1. INTRODUCTION

Consider first a class of monohull ships moving steadily ahead on a flat calm infinitely-deep sea. Fix the displacement, draft, speed, and hull shape. The length is then essentially the only variable allowed. At any given length, adjust the beam by uniform scaling of all offsets, so as to achieve the prescribed displacement; longer ships are thinner. Now vary the length until the total (viscous plus wave) drag R_t=R_v+R_w is minimised.

The above simplified ship optimisation problem, with length as the only variable, usually possesses a non-trivial solution, i.e. a finite optimum length, for the following reason. Viscous drag R_v is predominantly skin friction, which is proportional to surface area, and as a body of a given volume gets longer and thinner, its surface area increases. Hence viscous drag increases with length at fixed displacement.

On the other hand, for conventional ships at conventional speeds, wave resistance R_w generally decreases as the ship length increases. Since we are holding the speed U fixed, as we increase the length L, we are decreasing the length-based Froude number F=U/sqrt(gL). At fixed displacement, and at most relatively low Froude numbers, wave resistance is a (rapidly) increasing function of Froude number, and therefore decreases with increasing length. There are wobbles in the graph of wave resistance versus Froude number, so this is not an absolute conclusion, but it does hold in most cases, and of course this advantage of long ships is very much part of the naval architectural art.

Since there are opposite trends with length in the two constituents of the total drag, there must be a minimum for their sum, at some non-trivial intermediate length. That length is generally somewhat larger than for conventional ships.

In fact, sometimes there is more than one local minimum in the graph of total drag versus length, and this phenomenon is discussed in more detail in the following section. There is often a delicate interplay between local and global optima, which makes for an optimisation process that is quite difficult to analyse. In order to deal with this problem, we use here a powerful general purpose technique, described later, called "genetic algorithms".

In the present paper, we perform an exhaustive treatment of this optimisation problem for a family of monohull and multihull vessels, covering a very large range of speeds and displacements. We hold the hull shape, displacement and speed fixed, and allow the draft (and for multihulls, various other parameters) as well as the length to vary until the minimum total drag is achieved. We treat viscous drag as the sum of skin friction (estimated by the 1957 ITTC line) and a generally small but sometimes crucial form drag contribution which is estimated by an empirical formula. We use Michell's integral for the wave resistance, which is only accurate for thin ships. However, this is a more than usually good assumption for the class of extremely fine hulls that arise from this optimisation process.

When we treat multihull vessels, each separate hull is taken from the same shape family as the monohulls. For catamarans consisting of two identical side-by-side hulls, there is thus just one further parameter that participates in the optimisation, namely the lateral hull separation distance. There are some speed ranges where there is an optimal finite choice for this separation, and others where the best separation is infinite - that is, the optimum "catamaran" is actually two unconnected hulls. We find that, from the point of view of total drag (with no length restriction), a catamaran can never compete with a length-optimised monohull of the same total displacement. This is essentially because of the increased wetted surface area created by splitting the hull in two, which increases further the already dominant viscous drag component.

As the number of hulls increases, many more parameters are involved in the optimisation of multihull vessels, even within the constraint that all hulls have the same shape. For example, for laterally-symmetric trimarans there are a total of seven parameters; namely two drafts, two lengths, one ratio between side-hull and total displacement, one longitudinal and one lateral separation distance. The side-hull displacement ratio parameter is somewhat special, in that the trimaran reduces to a monohull when this ratio is near zero and to a catamaran when it is near unity. If that parameter is included in the optimisation process, a monohull results automatically whenever a length-optimised monohull is superior to a trimaran (which is always!), and similarly a catamaran would result if a catamaran was superior to a trimaran. Hence in order to confine attention to true trimarans, we use only a 6-parameter optimisation, then repeating this optimisation for a range of values of the displacement ratio. The results show that (strictly from the total-resistance point of view of the present paper) trimarans are also uncompetitive with length-optimised monohulls or catamarans.

When there are length (or other) restrictions, and hence (for shorter-than-optimal ships) a greater contribution of wave resistance to the total drag, multihulls can have less total drag than monohulls of the same length, because of the potential for favourable hull-hull cancellation of wave resistance. For example, in work to be reported elsewhere, we have examined a 3500 tonne vessel of length 160m designed to operate at 40 knots speed. In that case, the best catamaran has 10% less total drag than the best monohull of the same length, and there are indications that further improvements are possible with optimised trimarans.

The main purpose of the present study is to provide a benchmark, from which extended studies can follow. One class of such extensions obviously involves allowing the shape of the hull to vary. For the present study, we have used a very fine type of hull, appropriate for high-speed and sporting-type vessels, and there is a need to repeat the study with more commercial shapes of hull.

However, perhaps of greater importance is inclusion of further constraints. When the only quantities held fixed are speed and displacement, it is not surprising that the resulting ship proportions are somewhat (but not outrageously) unconventional. Further constraints, such as constraints on maximum length or minimum beam, arise inevitably from commercial, structural, safety, seakeeping, or sporting requirements. When these constraints are imposed, the ship proportions will return to the more conventional range, but at a price in terms of increased total drag. It is of value to know just how much of a price is being paid.

1.1 An illustrative example

Fig. 1.1: Comparison of total resistance for two one tonne monohulls.

Figure 1.1 shows two typical examples of graphs of total drag versus length (in metres) at a fixed speed, for such a vessel. For the present purpose, it is not essential how the drag is determined or scaled, but we should note that it does include an allowance for form drag, discussed later. The blue curve is at a fixed speed of 5.56 knots and the red curve at only a very slightly higher speed of 5.59 knots. In both cases, there are two prominent minima, i.e. two distinct (and remarkably different) lengths are locally favourable, and define "best" and "second-best" ships. At the lower speed, the longer ship (13.2 metres length) is better than the shorter ship (9.8m), whereas at the higher speed, the shorter ship (9.6m) is superior to the longer ship (12.3m). Thus, as we vary the speed and other parameters, there may occur an interchange between two local optima, so that the optimum length may appear to change discontinuously. These changes can occur over a remarkably narrow range of speeds.

Fig. 1.2: Optimum length for a one tonne monohull

This type of discontinuity in the optimum length is shown in Figure 1.2, again taken from the family of one-tonne monohulls. This figure gives the optimum length in metres as a function of the speed in knots. The discontinuities indicated above occur only at relatively low speeds, notably at about 5.6 knots (where the change between the two curves of Figure 1.1 occurs) and 4.3 knots, with smaller discontinuities at even lower speeds.

At speeds between the discontinuities, the Froude number based on shiplength remains essentially constant, and examination of the variation of wave resistance with Froude number indicates that this constant value corresponds to a local minimum of wave resistance. What is happening as we increase the speed is that, in attempting to design for minimum total drag, we simultaneously increase the shiplength, in order to stay at that local minimum. This continues as long as possible while we increase the speed, and when it is no longer possible, the optimum ship suddenly decreases its length, so that the Froude number suddenly jumps to the next higher local minimum, avoiding the local maximum in between. This process is intuitively like changing gears!

The length variation in the example of Figure 1.2 is continuous for all speeds above 5.6 knots. However, as is discussed later, if form drag is neglected, there can also be an apparent high-speed discontinuity. It is important to note that, as indicated by Figure 1.1, there is no discontinuity in the actual total drag at these speeds, merely an interchange in the roles of "best" and "second-best" ships. At the speeds where the optimum length changes discontinuously, the residual total drag tends to reach a local maximum, where its rate of change with respect to speed changes discontinuously.

Although these discontinuities are of interest in their own right, they are not necessarily the most important feature of Figure 1.2. They depend on the fact that the wave resistance possesses minima, and these minima are to a certain extent magnified by the theoretical procedure (here Michell's integral) used to compute wave resistance. If more empirical means are used to estimate wave resistance, with the effect of smoothing out the humps and hollows in the wave resistance variation, there will be a consequent reduction in the size of the discontinuities. However, so long as there are at least two minima in the wave resistance curve, a discontinuity is inevitable, no matter what method is used to estimate wave resistance.

Above 5.6 knots, the optimum length of a one-tonne vessel varies smoothly, and it is unlikely that the optimum length is sensitive to the procedure for wave resistance computation. In fact, the range of speeds above that where discontinuous length changes occur is the one of greatest interest in practice; for example, it is the competitive speed range for rowing shells.