Optimizing Boat Hull Shape for Maximum Speed: A Calculus of Variations Approach

[Bafsemann]

Was wondering if it was possible to derive the best possible shape of a boat hull to achieve maximum speed? As it is the equation on how to calculate the speed of a sphere moving in water... or else I am just totally wrong and you can bluntly ignore this post

 

[HallsofIvy]

That's an extremely difficult problem. Since you are trying to find the shape- i.e. a function- that maximizes a property it is a "Calculus of Variations" problem.

 

[saltydog]

Was wondering if it was possible to derive the best possible shape of a boat hull to achieve maximum speed? As it is the equation on how to calculate the speed of a sphere moving in water... or else I am just totally wrong and you can bluntly ignore this post

Guys, I'm just interested in this Ok; know very little about the Calculus of variation. Could someone at least formulate the problem? Let me begin with a conceptual approach:

Assume we have the shape of the boat in the form of a surface above the x-y plane. Now it seems to me that this surface can be a very nicely behaved function. The speed of the boat, I'll just guess will be a function of frictional forces which in turn are dependent on the shape of the surface as well as the boundary which I'll call $\Omega$. Now, let me take a leap of faith:

We wish to minimize the following integral which sums up all the frictional forces on the hull:

$$I=\int\int_{\Omega} G[x,y,f(x,y),f_x,f_y]ds$$

Where G is some functional relationship of these forces to the shape of the hull (and I suppose it's slopes as well) and our objective is to find f(x,y) which minimizes the integral (constrained by realistic limits of course, such as manufacturing ones).

I know for the simple case of a function of a single variable f(x), wishing to minimize the integral:

$$I=\int_a^b F(x,y,y')dx$$

we can do some calculus and come up with Euler's equation which must be satisfied:

$$\frac{\partial F}{\partial y}-\frac{d}{dx} \frac{\partial F}{\partial y'}=0$$

I suppose there is an equivalent one for a double integral? How about for just any old integral of that form say for:

$$I=\int_0^1\int_0^1 (f+f_x+f_y)dxdy$$

How would I find the function f(x,y) which minimizes (or maximized) this integral)? Am I getting off-subject?