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>

Robust optimal ship hulls based on Michell's wave resistance

“…Moreover, u ⋆ is slightly longer than ū since it takes into account higher Froude numbers. We refer here to Dambrine and Pierre [11,Section 7] where it is seen that the length of the optimal domain increases with the Froude number. Despite these geometric differences, we can see in Figure 19 that the total resistance is very similar for both hulls on the whole interval [0.55, 0.95].…”

“…We fix an area a > 0. In earlier studies [10,11], Dambrine and Pierre studied the following shape optimization problem: For a given constant speed U, (2.15)…”

“…It could be interesting to consider C F depending continuously on 𝛼 as well, or as an independent random variable, but we assume for simplicity that C F is constant. Following Dambrine and Pierre [11], we introduce the normalized viscous resistance functional…”

“…We fix an area $$$ a>0 $$$. In earlier studies [10, 11], Dambrine and Pierre studied the following shape optimization problem: For a given constant speed $$$ U $$$, $$$$ \left\{\begin{array}{l}\mathrm{Find}\ \mathrm{a}\ \mathrm{set}\kern0.90em {D}_U\kern0.90em \mathrm{which}\ \mathrm{minimizes}\kern0.90em {R}_{total}\left(D,{f}_D,U\right)\\ {}\mathrm{a}\mathrm{mong}\ \mathrm{a}\mathrm{ll}\ \mathrm{bounded}\ \mathrm{regular}\ \mathrm{open}\ \mathrm{subsets}\kern0.90em D\kern0.90em \mathrm{of}\ \mathrm{the}\ \mathrm{lower}\ \mathrm{half}\hbox{-} \mathrm{plane}\\ {}\mathrm{such}\ \mathrm{that}\mid D\mid =a.\end{array}\right. $$$$ …”

“…In previous research [10,11], the more modern question of optimizing also the domain of definition of the hull function, for a given area, was considered. An optimal support was proved to exist, assuming for compactness that all the admissible supports belonged to a bounded hold-all domain.…”