A theoretical and numerical determination of optimal ship forms based on Michell’s wave resistance

Abstract

Abstract. We determine the parametric hull of a given volume which minimizes the total water resistance for a given speed of the ship. The total resistance is the sum of Michell's wave resistance and of the viscous resistance, approximated by assuming a constant viscous drag coefficient. We prove that the optimized hull exists, is unique, symmetric, smooth and that it depends continuously on the speed. Numerical simulations show the efficiency of the approach, and complete the theoretical results.

Wave drag on asymmetric bodies

“…Many existing studies use the inviscid theories of Michell (1898) and Havelock (1919, 1932) to investigate the optimum design of ship hulls (Zakerdoost, Ghassemi & Ghiasi 2013; Zhao, Zong & Zou 2015; Dambrine, Pierre & Rousseaux 2016; Boucher et al. 2018).…”

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

“…there is a unique regular function f U,V ω which minimizes the total resistance. In [14], we actually assumed that ω was a rectangle, i.e. the length of the ship and its draft were given, but our results apply essentially in the same way for any open set ω whose boundary is regular enough.…”

“…Summing up, for a given f : ω → R + , the total resistance is given by (2.5), (2.1) and (2.10), and we denote it R total (f ) in order to stress its dependence on f . In [14], we considered the following optimization problem: for a given speed U , a given volume V of the hull, and a fixed set ω, find a nonnegative function f U,V ω which minimizes R total (f ) among (regular) nonnegative functions f : ω → R such that f = 0 on Γ − and ω f (x, z)dxdz = V /2. We proved that this problem is well-posed in an appropriate functional setting, i.e.…”

“…We stress that the integrals I 1 and I 2 (see (2.2)-(2.3)) are computed exacly (up to computer accuracy), whereas the integral (2.1) involving the parameter λ is computed by truncating the interval [1, +∞] into [1, Λ] with Λ large, and by using a numerical integration [14]. In particular, the kernel which is used for the numerical computation belongs to L ∞ (D × D): in this case, we have optimal regularity of the optimal shape (Theorem 6.1 with q > 2).…”

“…In [14], the authors and Rousseaux studied a similar problem in a H 1 (Sobolev space) setting, and they obtained existence and uniqueness of an optimal form. Because the optimal form was seeked in a class of nonnegative functions, the problem was no longer linear, as in the previous case, but only convex.…”