Optimization and Engineering, 4, 337–364, 2003 c  2003 Kluwer Academic Publishers. Manufactured in The Netherlands Sensitivity Analysis Methods to Design Optimal Ship Hulls MAURO VALORANI University of Roma “La Sapienza”, Department of Mechanical and Aeronautical Engineering, Via Eudossiana 18, 00184 Roma, Italy email: valorani@dma.ing.uniroma1.it DANIELE PERI, EMILIO F. CAMPANA INSEAN, The Italian Ship Model Basin, Via di Vallerano 139, 00128 Roma, Italy email: d.peri@insean.it email: e.campana@insean.it Received August 7, 2001; Revised October 21, 2002; Accepted February 12, 2003 Abstract. Automatic procedures for the design of ship hull geometries yielding minimal wave resistance and wave breaking are an attractive opportunity from both the economical and practical standpoints. Estimating the cost function gradient according to the Sensitivity Equation and Adjoint Methods (SEM, AM) instead of using the standard finite difference approximations has the potential of reducing the computational cost of the overall optimization procedure. Aim of this paper is to assess the actual extent of the cost reduction. Speed-up factors of up to 3.3 have been obtained in the evaluation of the cost function gradient and of about 1.6 in the overall optimization procedure applied to an optimal shape design problem of an existing tanker ship. The SEM and AM methods perform better than finite differences mainly because of (i) the smaller number of flow solutions needed to compute the cost function gradient and (ii) the opportunity of using the same LU factored matrix for both the flow solver and the SEM or AM equations, a circumstance arising as a consequence of having chosen a linearized potential flow model of the 3D free-surface problem. Keywords: sensitivity equation method, adjoint method, optimal shape design, numerical ship hydrodynamics 1. Introduction The increased availability of low-cost computing power, coupled with the maturing of numerical techniques able to accurately predict the effects of body-flow interactions, have made today the design of ship hull shapes with CFD tools affordable. The preliminary definition of the optimal hull shape by these methods is an attractive option from both the economical and practical standpoints, since it allows to reduce the total number of hull models required to be tested in towing tanks in order to complete the overall hydrodynamic design process of a ship, and to devise innovative hull shapes. Several criteria can be adopted to design optimal ship shapes, such as finding the shape yielding minimal wave or total ship resistance, or a shape not generating breaking waves. For many ships, wave resistance may typically amount, depending on the ship’s speed, from 10% up to 60% of the total resistance. A lower value of the wave resistance is associated