Regularity of Optimal Ship Forms Based on Michell’s Wave Resistance

Abstract

Abstract. We introduce an optimal shaping problem based on Michell's wave resistance formula in order to find the form of a ship which has an immerged hull with minimal total resistance. The problem is to find a function u ∈ H 1 0 (D), even in the z-variable, and which minimizes the functionalwith an area constraint on the set {(x, z) ∈ D : u(x, z) = 0} and with the volume constraint D u(x, z)dxdz = V ; D is a bounded open subset of R 2 , symmetric about the x-axis, and k is Michell's kernel. We prove that u is locally α-Hölder continuous on D for all 0 < α < 2/5, and locally Lipschitz continuous onThe main assumption is the nonnegativity of u. We also prove that the area constraint is "saturated". The results are first derived for a general kernel k ∈ L q (D ×D) with q ∈ (1, +∞]. A numerical simulation illustrates the theoretical result.

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

“…Formulation of the shape optimization problem. In this section, we recall the functional setting from [17]. The main novelty is subsection 3.3, where we set the problem in the whole plane R 2 .…”

“…In the discussion above, the domain of definition of the admissible hull functions was fixed. In [17], the authors proposed to consider the domain of definition of the hull function, or more precisely its support, as the unknown of the problem. The area of the support was kept fixed in order to be consistent with the thin ship assumptions.…”

“…This geometric shape optimization approach [1,11,29] allows to minimize even more the total resistance. In [17], an optimal support was proved to exist, assuming for compactness that the admissible supports all belonged to a bounded hold-all domain. A bulbous bow was obtained numerically.…”

“…Our main purpose in this paper is to understand how the optimal support depends on the speed of the ship, within the framework of [17]. We first recall the model from [17] in Section 2.…”

“…Our main purpose in this paper is to understand how the optimal support depends on the speed of the ship, within the framework of [17]. We first recall the model from [17] in Section 2. In Section 3, we introduce a nondimensional version of the problem, which shows that the optimal support depends only on a area Froude number, once the viscous drag coefficient is set.…”