Genetic algorithm techniques are used to find optimum dimensions for small (300 kg displacement) monohulls and multihulls at design speeds of 7.5 knots, 10.0 knots, 12.5 knots and 15.0 knots. A three parameter hullform family is used to investigate the effects of waterline, cross-section, and keel-line shape. Michell's integral is used for the wave resistance, the 1957 ITTC line for the skin friction, and a simple empirical formula for the form drag. A catamaran with a design speed of 10 knots seems to be a good compromise choice, even though monohulls are always slightly superior. Trimarans are not competitive. It is shown that allowing the lateral separation of the demihulls of a catamaran to vary during the race can improve off-design performance. However, a catamaran with the demihull separation fixed at the maximum possible value is only slightly inferior.

The inaugural Advanced Technology Boat Race was held on Lake Burley Griffin in Canberra, Australia in April 1996. In part, the present report stems from an informal request by one Australian team for assistance in evaluating the performance of some candidate multihull designs for the 2nd event. As the team are now unlikely to enter a boat in the 1997 race, we have posted the present study to the web to assist other potential competitors.

1.1 Wave Resistance

We use Michell's integral (Michell 1898; see also Tuck 1989) to estimate the wave resistance of the vessels. This requires evaluation of a triple integral, one integral in each of the length-wise and draft-wise co-ordinate directions, and one integral with respect to the angle theta of propagation of the ship-generated waves. The numerical method used here for evaluating these integrals is described fully in Tuck (1987). We use 33 stations, 33 waterlines, and 640 intervals for the integration with respect to theta.

There is really no actual Michell integral for multihulls. What we use here is the assumption that each separate hull can be represented by the same singularity distribution (namely sources distributed over its own centreplane) as if that hull were alone. This neglects one type of interaction between the hulls, namely the influence of one hull on another in creating a cross-flow which modifies this singularity distribution, in particular inducing vortices as well as sources. On the other hand, it does not prevent interference between the wave systems generated by the centreplane sources. Little is known of the relative importance of these two types of interactions, but the present assumption seems to yield quite good results for the wave resistance (Tuck 1987).

It is notable that the assumption that there are no induced vortices due to other hulls can be exactly satisfied by allowing the hull centrelines to possess a suitable small camber (Lin 1974). This camber has no effect on the wave resistance, and may be desirable in eliminating induced drag. It is also important (Tuck 1987) that the size of this induced-crossflow effect relative to that due to the hull's own thickness is proportional to the draft/length ratio, and hence negligible for conventional (and a forteriori the present optimal) slender ships of small draft, even if formally of the same order as self-thickness effects for the thin ships of finite draft for which Michell derived his integral.

1.2 Viscous Resistance

The viscous resistance R_v can be written as

R_v=1/2*rho*U²*S*C_v

where rho is the water density (herein 1000 kg/m3) and S the wetted surface area of the hull. When skin friction dominates, the drag coefficient C_v approximately equals C_f , where C_f is a skin friction coefficient which can be estimated using the ITTC 1957 ship correlation line (Proc. 8th ITTC).

C_f = 0.075/(log₁₀ (R)-2)²

where R = UL/nu is the Reynolds number; nu is the kinematic viscosity (herein, 1.054 x 10^-6m²s^-1).

1.3 Form Effects

Including a form factor specific to the hullform under consideration can often give better estimates of the viscous drag. This factor is difficult to estimate and may vary with speed because of (among other things) changes in trim and sinkage. In their examination of eight-oared rowing shells, Scragg and Nelson (1993) found a simple empirical formula for the form factor of these hulls. The viscous resistance coefficient is written as

C_v = (1+k) C_f

where

k = 0.0097(theta_entry + theta_exit)

Here, theta_entry and theta_exit are the half-angles (in degrees) of the bow and stern, respectively, at the waterplane.

We decided to use Scragg and Nelson's empirical formula in the present study for a number of reasons. Firstly, the popular alternatives, Holtrop and Mennen's (1978) statistical method, and Holtrop's (1984) reanalysis, will not always be applicable: their method is restricted to length-based Froude numbers less than 0.45, although Holtrop extended this to planing speeds in 1984. Secondly, we know of no other simple alternatives. Thirdly, since Scragg and Nelson based their formula on empirical data for single, double and quad shells, as well as eight-oared shells, we suspect (almost as much as hope) that their formula will give reasonable estimates for the small, slender, convex hulls considered in the present study.

1.4 Shape Effects

A three parameter hullform family was used to investigate the effects of shape.

Let X(x) = 4x(x-1), with 0<=1.

Then the non-dimensional offsets, are given by

0 if X=0 or X^a₃<z,
Y(x,z; a₁,a₂,a₃) = 1/2FG^a₂ otherwise.

Here F = X^a₁, G = 1-z²X^-2a₃ and 0 <= x,z,a₁,a₂,a₃ <=1.

At any given length and draft, adjust the beam by uniform scaling of the non-dimensional offsets given by the above formula, so as to achieve the desired displacement.

In the above formulation, the first parameter, a₁, controls the shape of the waterlines, the second parameter, a₂, determines the cross-section shape, and the third parameter, a₃, controls the shape of the keel-line. If, for example, a₁=0.0, then the waterline shape is rectangular, if a₁=0.5 the waterline is elliptical and if a₁=1.0, the hull will have parabolic waterlines. And similarly for cross-sections and keel-lines.

Some other examples are shown in Table 1. For example, the PEP hull (a₁,a₂,a₃)=(1.0,0.5,1.0) has parabolic waterlines, elliptical cross-sections and a parabolic keel-line. The PER (a₁,a₂,a₃)=(1.0,0.5,0) hull is similar, but has a rectangular sideview.

Scragg and Nelson's form factor penalises hulls with large entrance and exit angles at the load waterline, and consequently there will be a strong push towards parabolic waterlines, i.e. toward a₁ = 1.0 in the formulation given above.

The computer program GODZILLA (Lazauskas 1996) was used to find the optimum dimensions of monohulls and multihulls over a wide range of design speeds. The program contains a variety of hill-climbing routines to assist in the search process, however for the highly multimodal objective function we are considering in the present study, the program's nonlinear components are essential. These include genetic algorithm techniques, and a number of other heuristics that can broadly be described as Artificial Life methods.

Optimisations were performed at four design speeds, 7.5 knots, 10.0 knots, 12.5 knots and 15.0 knots. At each of these design speeds we searched for the optimum monohull, for the optimum symmetric catamaran, and for two optimum symmetric trimarans, one where the central hull weighs 270 kgs and the outriggers each weigh 15 kgs (sigma=0.1 in our notation), and another trimaran where the central hull weighs 120 kgs, and the outriggers each weigh 90 kgs (sigma=0.6). Here sigma is the proportion of total displacement that is attributable to the sum of both outriggers.

For monohulls we have a five parameter optimisation problem, one length, one draft and three shape parameters. For catamarans, the problem has six parameters, the same five as for monohulls, plus the lateral separation distance of the demihulls. For each trimaran, sigma=0.1 and sigma=0.6, we have a twelve parameter problem, two lengths, two drafts, three shape factors for the central hull, three shape factors for the outriggers, one lateral hull separation distance, and one longitudinal hull separation distance. Each of these four candidate vessels, monohull, catamaran, sigma=0.1 trimaran, and sigma=0.6 trimaran, were optimised at the four design speeds, making 16 design problems in all.

In all cases considered here, the total displacement was fixed at 300 kg. Hull lengths and overall length for trimarans were constrained to be less than or equal to 6.0 metres. Overall width was constrained to be less than or equal to 2.5 metres.

The optimum dimensions and other parameters for the hulls are given in Tables 2(a) and 2(b).

Fig. 1: Some canonical hull configurations referred to in this study

3.1 Comparisons by Design Speed

3.1.1 Design Speed = 7.5 knots

The waterline shape factor, a₁, for the optimum monohull is 0.839 which means that the waterlines are slightly fuller than parabolic. The cross-section shape factor, a₂, is 0.293 which indicates that the cross-sections are slightly more elliptical than rectangular. The keel-line shape factor, a₃, is zero, indicating a constant draft hull (i.e. rectangular sideview).

In our examination of low drag rowing shells, Tuck and Lazauskas [1996], we also looked at monohulls of 300 kilogram displacement over a range of design speeds similar to that in the present study. We used "PEP" hulls (for Parabolic waterlines, Elliptical cross-sections and Parabolic keel-line), and found that the optimum boat at a design speed of 7.5 knots had a length of approximately 12.1 metres. Thus, under the rules of the present boat race, we are restricted to about half the optimum unconstrained length. The prismatic coefficient of the optimal monohull is 0.699, compared to 0.533 for a PEP hull.

The optimal catamaran demihulls have parabolic waterlines; the cross-section shape is 0.353, which is fuller than elliptical, and the keel-line is almost constant draft (a₃ =0.136). In our paper on low drag rowing shells we also examined 150 kg monohulls equivalent to the present catamaran demihulls. The optimum length for the PEP hull at a design speed of 7.5 knots is approximately 10.1 metres. Thus, the catamaran demihulls will be shorter than the optimum unconstrained length. However, the constraint is not as severe as for the monohull, and so the shape factors do not deviate from those of a PEP by as much as those of the optimum monohull. The prismatic coefficient of the demihulls is 0.644.

The demihulls are separated as widely as the competition constraint allows, i.e. the overall width is 2.5 metres. A true optimum catamaran at this speed would have an even wider separation.

For the sigma=0.1 trimaran the shape factors for the (small) wing hulls are almost the same as those for a PEP hull, however the cross-sections are slightly fuller than elliptical. Since the outriggers are quite small, they are not affected by the 6 metre length constraint. The shape factors for the large central hull are similar to those of the optimum monohull. This is as expected; since the central hull contains 90% of the displacement, it is still quite severely affected by the length constraint. This optimum trimaran is similar to the type I trimaran shown in figure 1.

The optimal sigma=0.6 trimaran for a design speed of 7.5 knots has a 3.267 metre long central hull and side hulls that are 3.433 metres. The longitudinal spacing (2.650 metres) is such that the overall length is 6.0 metres, the maximum allowable. See Type III Trimaran in figure 1. This category of trimaran tends to have the 3 hulls all rather similar in size, dimensions and shape, and there may be construction advantages in the use of three identical hulls.

Fig. 2(a): Comparison of Rt for 7.5 knot designs

Figure 2(a) shows the total resistance (in Newtons) as a function of boatspeed (in knots) of these four candidate designs, optimised for minimum total resistance at a design speed of U=7.5 knots.

Clearly the monohull has the least resistance of the four vessels. The sigma=0.6 trimaran is the worst performer, and has a noticeable hump between 3.5 and 5.5 knots. Both the catamaran and sigma=0.1 trimaran have very similar resistance near the design speed, but the catamaran has better off-design characteristics and superior static stability.

Although the trimaran with small outriggers (sigma=0.1) is not quite as good as the catamaran, it seems likely that a trimaran with even smaller outriggers, say sigma=0.075, will be more competitive. For sigma=0.075, however, the outrigger hulls would each weigh only 11.25 kgs so that there may be problems in achieving adequate static stability.

3.1.2 Design Speed = 10.0 knots

Fig. 2(b): Comparison of Rt for 10 knot designs

Figure 2(b) shows the total resistance of four vessels optimised at a design speed of 10.0 knots. The monohull is clearly best, next best is the catamaran. At this speed, there is a genuine optimum catamaran demihull separation, such that the overall width of the catamaran is about 2 metres, less than the maximum allowable by the rules.

3.1.3 Design Speed = 12.5 knots

Fig. 2(c): Comparison of Rt for 12.5 knot designs

In Figure 2(c), design speed = 12.5 knots, there is a small hump in the resistance of the sigma=0.1 trimaran between 3.0 and 5.5 knots.

3.1.4 Design Speed = 15.0 knots

Fig. 2(d): Comparison of Rt for 15 knot designs

The most noticeable feature in figure 2(d), design speed = 15 knots, is the peculiar behaviour of the trimaran with large outriggers (sigma=0.6) at low speeds. We shouldn't be too surprised by this. After all, we have not specified anything at off-design speeds. In any case, the trimaran is clearly unsuitable, since its low speed performance is so bad.

The resistance of the trimaran with small outriggers is very similar to that of the catamaran for speeds greater than about 10 knots. However, this trimaran has quite short outriggers, L_w=1.117 metres, and they are tucked in close together near the rear of the central hull. In effect, this trimaran reduces its total resistance by trying to behave like a monohull! Obviously, the static stability of such an arrangement is small, and there is little reason to use it in comparison to the monohull which it mimics. Even the optimal catamaran is somewhat monohull-like at this speed, with an overall width of less than 1 metre.

3.2 Comparisons by Vessel Type

In figures 3(a) to 3(d) we show the total resistance for the optimum vessels in non-dimensional form. Also, we group the results by the type of vessel, rather than comparing performance of different vessels at the same design speed as we did in figures 2(a)-2(d). In the following figures, the total resistance coefficient is given by

C_t = R_t/(1/2*rho*U²D^2/3)

where D is the total displacement of the vessel, here 0.3 tonnes. Please note the vertical scale of the graphs.

3.2.1 Monohulls

Fig. 3(a): Comparison of Ct for Monohulls

Figure 3(a) shows C_t as a function of speed in knots for the four optimal monohulls. In each case, note that the lowest drag is achieved at the design speed by the vessel that is optimised for that speed, not surprisingly. However, the 10-knot design speed vessel performs better than or hardly worse than all other monohull vessels over a large range of speeds, say from 6 knots to 14 knots.

3.2.2 Catamarans

Fig. 3(b): Comparison of Ct for Catamarans

Figure 3(b) shows C_t for catamarans. Off-design performance is qualitatively similar to the monohulls in figure 3(a). Again, the 10-knot design vessel is superior or only slightly inferior to other catamarans over a large range of speeds, say from 4 knots to 13 knots.

3.2.3 sigma=0.1 Trimarans

Fig. 3(c): Comparison of Ct for sigma=0.1 Trimarans

Figure 3(c) shows C_t for the trimarans with small outriggers. There seem to be two distinct types of curves in this figure. The trimarans optimised at design speeds of 7.5 knots and 10.0 knots have configurations similar to the type I trimarans depicted in figure 1. At design speeds of 12.5 knots and 15.0 knots, the configurations are similar to the type II trimarans in figure 1.

3.2.4 sigma=0.6 Trimarans

Fig. 3(d): Comparison of Ct for sigma=0.6 Trimarans

Figure 3(d) shows C_t for the trimarans with large outriggers. As with the sigma=0.1 trimarans, there are two distinct types of curves in the figure. Here though, the vessels at the lowest and the highest design speeds have type III configurations. At the other two design speeds the vessels are of type IV.

In summary, at all four design speeds, a monohull is superior. However, the catamaran is only slightly worse, and may have better stability. Trimarans are in general not competitive. Perhaps the catamaran designed for 10 knots is a good compromise choice, with good performance over a large range of speeds.

3.3 A Variable Width Catamaran

We will use the catamaran optimised at a design speed of 10.0 knots as a baseline vessel from which to investigate the effect of varying the lateral spacing of the demihulls during the race. This vessel has a lateral spacing of 1.691 metres, and an overall width of 2.005 metres in its optimal form for operation at design speed. However, other lateral spacings may be better at off-design speeds. Of course, if it is windy on the day of the race, the widest spacing possible may be necessary to get the maximum static stability. Incidentally, the keel-line shape factor for the demihulls of the baseline catamaran is 0.359.

Fig. 4: Variation of Rw with Catamaran Demihull Separation Distance and Speed

Figure 4 is a contour plot of wave resistance (in Newtons) with speed (in knots) on the horizontal axis and W, the lateral hull spacing (in metres), on the vertical axis. The maximum value of W in this plot means that the overall width can exceed the competition limits, but we include these values here to see what trends exist beyond the competition limits. The regions of highest wave resistance occur at high speeds and when the lateral spacing is small (the dark blue/purple region in the bottom righthand corner). The lowest wave resistance occurs for speeds below about 5.0 knots.

What we really want is a path of least resistance starting at the extreme lefthand edge and extending to the extreme righthand edge of the contour plot. The point to head for at the right edge is quite clearly at W=0.75, where the lowest wave resistance occurs for the highest speed considered. It is not clear on the contour plot where we should start at the lefthand edge, although, because the resistance is so low anyway, the choice is probably not critically important. In any case, this is almost a trivial search problem for GODZILLA. We keep the length and shape parameters fixed and vary only the lateral hull spacing. Optima were sought at speeds of 1.0, 2.0, ... ,10.0 m/sec, (or 1.9425, 3.8850, ..., 19.4250 knots). The results of these optimisations are shown in figure 5. There are a few wobbles in the graph, the search was not pathologically exhaustive, but the estimates fit well with the contour plot and our (human) expectations. In any case, the lessons are obvious enough: at low and high speeds, we should keep the hulls close together, at intermediate speeds the hulls should be widely spaced.

Fig. 5: Optimum Lateral Demihull Spacing

To estimate the effect of varying the lateral hull spacing at different speeds, we used the hull spacing regime shown in Table 3 when calculating the total resistance.

The spacing regime in the above table does not exactly match the optimal values shown in Figure 5. In effect it emulates a mechanism whereby the hull spacing can only take on discrete values.

Fig. 6: Improvement in Total Resistance over Baseline Catamaran

Figure 6 shows the expected relative improvement (i.e. reduction in drag) in employing a variable width catamaran. Also shown is the expected relative improvement achieved by using a catamaran with W fixed at 2.186 metres, which gives the maximum allowable overall width of 2.5 metres. The improvements shown are relative to the total resistance of the catamaran with W fixed at the optimum of 1.691 metres. Thus our baseline catamaran is the positive x-axis in this plot.

The graph clearly shows that using maximally spaced catamaran demihulls at speeds below 9.5 knots will lead to an improvement over the baseline catamaran. However, at speeds above 9.5 knots, performance will be worse than the baseline catamaran, but only by about one third of one percent. On the other hand, using the lateral hull spacing regime described in Table 3 which involves reducing the spacing below the baseline value of 1.691 metres, can improve performance for speeds greater than 10 knots. Of course, when the demihulls are close together they will interfere with each other to a greater degree. Who knows? We could end up doing worse this way than by using fixed widely-spaced hulls.

We have found optimum dimensions for small monohulls, catamarans, and trimarans over a range of design speeds. Monohulls have the lowest total resistance, but also potentially the lowest static stability. It is unlikely that trimarans will be competitive with optimum catamarans under the constraints imposed by the race rules. We have shown that small gains in off-design performance might be achieved with a variable width catamaran, although no practical means for achieving this end has been proposed.

More work needs to be done on trimarans with small outriggers. Variable lateral and longitudinal hull spacing, in the manner of a swing-wing aircraft, could improve off-design performance more than is possible with the catamaran examined in the present report. However, trimarans with small outriggers have less static stability than catamarans.

There many other ways to reduce the total resistance. However some of the following suggestions might not be acceptable.

• The use of riblets and other hull surface tricks are not explicitly proscribed.

• Helium-filled surface-piercing craft do not seem to be prohibited.

• Using the solar panel as a lifting surface in the wind to reduce displacement may or may not be allowed under the rules. We view it as a drag reduction mechanism rather than as a propulsion device which is prohibited under the race rules.

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Lin W.-C. "The force and moment on a twin hull ship in steady potential flow", 10th O.N.R. Symp. Naval Hydro., MIT, pp.493-516, 1974.

Michell, J. H. "The Wave Resistance of a Ship", Phil. Mag., Vol. 45, 1898, pp. 106-123.

Proceedings of the 8th ITTC, Madrid, Spain 1957, published by Canal de Experiencias Hidrodinamicas, El Pardo, Madrid.

Scragg, Carl A. and Nelson, Bruce D. "The Design of an Eight-Oared Rowing Shell", Marine Technology, Vol. 30, No. 2, April 1993, pp. 84-99.

Tuck, E. O. and Lazauskas, L. "Low Drag Rowing Shells", Proc. 3rd Conf. on Mathematics and Computers in Sport, Bond Uni., Queensland, Australia, 30 Sept. - 2nd Oct. 1996, pp. 17-34.

Tuck, E. O. "Wave Resistance of Thin Ships and Catamarans", The University of Adelaide, Applied Mathematics Report T8701, (Jan. 1987).