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

Abstract

<p style='text-indent:20px;'>We consider a ship hull design problem based on Michell's wave resistance. The half hull is represented by a nonnegative function and we seek the function whose domain of definition has a given area and which minimizes the total resistance for a given speed and a given volume. We show that the optimal hull depends only on two parameters without dimension, the viscous drag coefficient and the Froude number of the area of the support. We prove that, up to uniqueness, the optimal hull depends continuously on these two parameters. Moreover, the contribution of Michell's wave resistance vanishes as either the Froude number or the drag coefficient goes to infinity. Numerical simulations confirm the theoretical results for large Froude numbers. For Froude numbers typically smaller than <inline-formula><tex-math id="M1">\begin{document}$ 1 $\end{document}</tex-math></inline-formula>, the famous bulbous bow is numerically recovered. For intermediate Froude numbers, a "sinking" phenomenon occurs. It can be related to the nonexistence of a minimizer.</p>