Maths and Ship Design

I'd like to take this opportunity to report some optimum sailboat midsection shapes based on maximizing the RM for a given girth. I wasn't able to find anything published on this subject at all. There are no closed form solutions to the basic isoperimetric problem description, and when you enforce symmetry and include an arbitrary VCG, the problem is quite complex.

The basic problem description is as follows: Given a girth measurement (assumed sheer to sheer), an allowable heel angle that immerses the sheer, and a VCG, find the section shape and displacement that maximizes RM. Additional constraints can be added, such as displacement limits, box rules, curvature restrictions, and tangency restrictions at the keel and sheer. Wetted girth can be factored in to the measure of merit, and some 3D fudge factors that correct the wetted girth to wetted area of the middle part of the hull. Instead of RM, the area under the RM curve can be used as a measure of merit. This is actually a bit easier to do in practice.

In the simplest case with VCG at the waterline, the heeled beam to windward should be about 1/2 the heeled beam to leeward. The curvature of the section should be proportional to the heeled beam (measured from the heeled CG) . This is directional, and will go negative if not constrained by hull symmetry. It turns out that the heeled immersed shape is the same as a freezer bag filled with water (turned 90 degrees) , or a 2D hot air balloon (turned 90 degrees the other way).

But the mathematical landscape of optimal RM shapes is overall pretty flat but lumpy, with multiple local maxima, and the midsection shapes can change dramatically with small changes to the constraints. For instance, most modern sailboats will want a midship section area that is much less than that which is needed to develop maximum RM. And STIX ratings that restrict the area under the RM curve also have dramatic affects on STIX constrained optimal midsection shapes. Manufacturing and styling considerations that restrict curvature and tumblehome dramatically alter the optimum shapes. But the actual reductions in the RM values due to these restriction are typically rather small.

I used Excel and the free Xlam solver extension to run hundreds of traces with different objectives (max RM, max area of RM curve, RM divided by some function of wetted girth), various VCG conditions, various immersed area constraints, and various heel angles. Getting Xlam to solve problems with 40 or more degrees of freedom takes some research into how it works. I generally had to use a two step process. First, use the Evolutionarily solver to narrow the domain to a convex patch, then refine with the GRG non-linear solver. Start with the heel a degree more than you want, then change to the heel you want. Repeat starting with the heel a degree less than you want, then change to the one you want. Check to see the results converge. It takes five to ten minutes to run a trial, and I can chart five of them on a trace.

My hull half-section model was a chain of 20 links, each with a fixed length, that had a total length of 20. The angle of each link was a solver variable. All calculations were done on the heeled section, so the properties of the level condition had to be computed afterwards. This makes specifying VCG, beam, and draft constraints a bit awkward. They exist only as solver constraints, with Heeled VCG subject to solver modification. For any given trace, the level VCG is the same, but the heeled VCG associated with it differ among the trials.

Below is a trace of section shapes with constant girth, section area, and VCG. They all maximize RM at 25 degrees of heel, subject to an increase in the required area under the RM curve from 0 to 25 degrees. The blue curve has no area requirement. The red curve is the maximum area under the RM curve, and the rest lie in between.



This is the heeled trace

 

Phil, an interesting study.

What is the significance of maximizing RM for a fixed sheer-to-sheer girth?

Is submerged area fixed for an individual "Case"?

Is zero heel freeboard fixed or variable for an individual "Case"

Is the optimum section shape dependent on VCG? If so, why?

Any doubts or concerns raised by the optimum solutions being "lumpy"?

 

Without thinking too much about it, it looks to me that (Blue = Catamaran) & (Red = Trimaran).

 

Both types of ship mean independent hulls (2 or 3 but independent) joined by a structure that completes the whole. Therefore, in my opinion, it is always a monohull.

 

The main flaw is that the assumption is that the water surface is flat. In fact the wave making characteristics of any hull need to be taken into account. There is a rise on the bow, and then a dip and another rise.

 

Gonzo makes a really good point. A classic hull shape ends up being supported more on the centerline by the bow and stern waves with less flotation in the water at the widest points to contribute to the righting moment. For a uniform girth, this won't mean as much, since displacement is constant.

-Will

 

I think that the shape and size of the submerged volume, which changes with speed, is what needs to be considered. Sailboats can have a relative large beam above the static waterline because when heeled over the upper middle of the hull often does not touch the water.

 

1. This is basically about power. The more RM you can generate, the more power from the sails. But an RM value at a single angle of heel and single displacement is not terribly interesting. Getting a feel for how much you can permute the ideal shape and still have, say, 95% of the maximum RM is very helpful when designing. I chose girth as the reference value because it reflects shell weight, and it is computationally convenient. Plus I wanted a metric that was invariant under rotations. Girth is also a traditional measurement for sailboat ratings and class compliance. So with 20 links per side, the scale of everything is 1/40 girth in the model. Areas, VCG location, everything is computed in this scale. All computations can be scaled to real world dimensions by comparing the model beam to the real world beam. This is why I didn't post tabulated values - they are all in model scale.

2. It can be. Immersed area is a constrained variable in the solver. The easiest way to handle this with XLAM is to find the (unconstrained) optimal immersed area for what you are doing, and then walk the solver away from this in the direction you want using a one-sided bounding constraint. A lot of what went into this model is based on the proper care and feeding of the solver. Think of it as Excel's high maintenance girlfriend.

3. Strictly variable. The heeled freeboard is zero. The model computes a section shape in the heeled condition using all of the girth, then stands it up and finds the necessary unheeled parameters such as waterline, VCB, beam, draft, etc. Awkwardly, the model starts with the VCG input in the heeled condition, and computes the level VCG at the very end. If it isn't the VCG you wanted, the solver changes the heeled VCG and tries again. This works better than it might sound because this is one of the better behaved constraints. It is monotonic with respect to nearly every other variable the solver plays with. And I can guess starting values pretty well now. Running traces, the solution for one case is the starting condition for the next one, so that has a lot to do with organizing the constraints and scaling everything so that the solver works all of the variables as best it can. I did a fair bit of error analysis the hard way since XLAM doesn't exactly tell you how it determines convergence. I had it build a lot of circles and squares according to rules that optimize for circles and squares.

4. Yes, it is. The curvature of the hull is proportional to the lateral distance from the CG in the heeled condition. If the CG is above the keel line, there will be a hollow garboard. If the VCG is below the keel there will be a convex garboard. And for fixed immersed areas, the higher CG case will have a greater beam. I'll post a VCG trace later.

5. They are lumpy in an understandable way. If the centerline of the section remains immersed when heeled, there is a symmetry constraint placed on the portion of the section that is immersed on both sides. This contravenes the fact that the curvature should continue to change linearly from one side to the other. The result is that the section has a smooth first derivative from sheer to sheer, but the second derivative is discontinuous at two points corresponding to the heeled waterline. This means that the radius of curvature is also discontinuous. And this can produce some numerical artifacts. When increasing the heel angle from one trial to the next, the solver resists pushing a point that was just below the waterline to just above the waterline (on the high side). Instead, it adapts the shape to keep the point just below the waterline. Then it will "pop" and run the next point up to the waterline. So I know to inspect the points near the waterline. The actual difference in RM is tiny, like 1/100 %, but it can be visible as a kink in the curve. The underling math wrt the GRG algorithm is really messy here. I actually run a crude fuzzy logic calculation to defeat this little problem. It smears the waterline effect, giving the solver a gradient instead of a jump. This is all caused by the fact that near the optimal solution, all the differentials go to zero except where there is a discontinuity. I'm running 10^(-8) on the convergence criteria, and I have non-dimentionalized the important constraints to ensures uniform sensitivity. All of the calculated values for the section properties are based on polygons. I believe the accuracy is at least five places for all reported values for any normal sectional shape.

 

If VCG varies with the hull shape and displacement remaining fixed then righting moment varies linearly with VCG for a given heel angle. The relationship of righting moment and VCG (neglecting longitudinal trim effects) is

RM2 - RM1 = Displacement * (VCG2 - VCG1) * sine (heel angle)

Note that shape of the immersed shape does not have any effect on variation of righting moment with VCG at a fixed heel angle.

This means that for a fixed heel angle the shape which results in maximum righting moment is independent of the VCG.

 

VCG trace. If you hold the immersed area constant, and the heel angle constant, then you get a very similar shape over a wide range of VCG. This trace has VCGs of 4.5, 3, 0, -3,and -6.

There are slight variations. But if you allow displacement to vary, then there are much greater variations. The higher VCG will have a lesser best displacement and be beamier.

Don't confuse a formula for adjusting RM of a particular hull with the problem of optimizing the shape of the hull for a particular VCG. Those are different things. There are slight differences in the curvatures due to the point on the hull being a different lateral distance from the VCG in the heeled state.

 

Looks the solutions for range of VCG are slowly converging on a single solution, limited perhaps by finite precision arithmetic effects.

Not confusing them. I would expect the optimum solution to vary for a 2D optimization with the "displacement". The optimum shape should depend only on the non-dimensional ratio square root of immersed area / girth, and the heel angle.

Proof that the optimum shape is independent of VCG.

Consider the optimum shapes for two distinct VCG; VCG1 and VCG2

Assume that the shape for optimum righting moment varies with VCG when heel angle and area are fixed.
Assume that there is only one optimum shape for a given VCG when heel angle and area are fixed.
VCG1 results in shape A with righting moment RM1A
VCG2 results in shape B with righting moment RM2B

Righting moment of shape A and VCG2: RMA2 = RMA1 + (VCG1 - VCG2) * Area * Sine (heel angle) [1]
Righting moment of shape B and VCG1: RMB1 = RMB2 + (VCG2 - VCG1) * Area * Sine (heel angle) [2]

If Shape A is the optimum shape for VCG1 then RMA1 > RMB1 [3]
If Shape B is the optimum shape for VCG2 then RMB2 > RMA2 [4]

From [3] and [2]
RMA1 > RMB2 + (VCG2 - VCG1) * Area * Sine (heel angle) [5]

From [4] and [1]
RMB2 > RMA1 + (VCG1 - VCG2) * Area * Sine (heel angle) [6]

Rearranging [5]
RMA1 - RMB2 > (VCG2 - VCG1) * Area * Sine (heel angle) [7]

Rearranging [6]
RMB2 - RMA1 > (VCG1 - VCG2) * Area * Sine (heel angle)
RMA1 - RMB2 < (VCG2 - VCG1) * Area * Sine (heel angle) [8]

Both [7] and [8] can not both be true as VCG1 does not equal to VCG2.

The assumption that the optimum shape varies with VCG is incorrect.