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Experimentaltests carried out by adopting both the original and the optimized bulbs have confirmed thesuccess of the optimization process in reducing the total resistance.These early attempts were not able to exploit the full potential of the optimization pro-cedure because only a small number of design parameters could be used to describe thehull geometry. Such limitation was related to the need of cutting down the overall costof the optimization procedure, which is formed by the product of the cost of a singleflow solution times the total number of flow solutions needed to reach the optimum de-sign condition. Furthermore, the quality of the optimal shape strongly depends on thequality of the flow model used. Thus the cost of a single flow solution is bound to growin the future—no matter how much the flow solvers could improve their computational efficiency. Variable-fidelity modeling promises to alleviate somewhat the severity of thislimitation.Therefore, reducing the total number of flow solutions appears to be the only way to cutdown the overall cost of the design procedure. Zero-order methods, such as the genetic al-gorithm approaches, have a very slow convergence rate to the optimum. Faster convergencerates are offered by first- and second-order methods, which require the calculation of thegradient and/or Hessian (or approximate Hessian) of the cost function. These vector andmatrix quantities are generally obtained by approximating their elements by finite differ-ences. Therefore, one cost function gradient evaluation could require as many as 2Ndp flowsolutions,where Ndp is the number of design parameters used to generate the newhull shape.Finding a more efficient way to compute the gradient, while retaining the high convergencerate, could therefore offer a significant reduction in the overall cost of the optimization.In summary, to improve the computational efficiency of the automatic design procedure,we followed these two guidelines:1. to apply the variable-fidelity strategy to calculate the cost function gradient accuratelyonce approaching the optimum and only approximately when far from it;2. to compute the cost function gradient by using Sensitivity Analysis techniques as de-scribed in Lions (1971), Jameson and Reuther (1988), Pironneau (1974), and Narducciet al. (1995) instead of using finite differences.The first option has already been investigated in Campana et al. 源]自{751^`论\文}网·www.751com.cn/ (1999) and Peri et al.(2001). There, the flow model is based on a linearized potential model of the 3D free-surface problem, and the cost function is expressed directly in terms of the potential and itsderivatives. The justification to choose this low-fidelity and cost-effective flow model stemsfrom the satisfactory agreement found between the model prediction and the experimentaldata, which is well within the typical accuracy bounds of an engineering approach. Thegradient information required to drive the optimization procedure are obtained by a FDapproach. Aim of the present work is to make the optimization procedure more efficient byobtaining gradients information via Sensitivity Analysis methods, instead of FD.In the FD method, the design parameters are perturbed one at a time, according to acentered difference scheme; the cost function is then computed by feeding the perturbedvalues of the design parameters to the flow solver, treating the flow solver as a blackbox. In contrast, the Sensitivity Equations Methods exploit the existence of a set of partialdifferential equations describing the flow dynamics to predict the sensitivity of the flowfield to perturbations of the design parameters. To achieve this goal only requires oneflow solution evaluated at the unperturbed design parameter values. This allows a drasticreduction in the number of flow solutions: from 2Ndp to just 1, plus the solution of somelinear systems.
These approaches becomemore andmore profitable as the number of designparameters increases.As a test case to validate the alternative approaches, the same ship hull investigated inCampana et al. (1999) and Peri et al. (2001) has been selected. The validity of the techniquewill be assessed by comparing the accuracy and the efficiency obtained by using the FD andSensitivity Analysis methods to compute the cost function gradient. As a further control, it will be demonstrated that the same optimal hull shape can be recovered by adopting boththe FD and Sensitivity Analysis methods in an optimization search.2. Shape parameterizationFinding an optimal hull shape requires the ability to modify the ship hull itself by alteringa finite number of design parameters ξk , with k = 1, Ndp.In this work, the selected designparameters are the control points of the Bezier perturbation surfaces described in Appendix,which uniquely identify a tentative hull shape as a perturbation of the baseline ship hullgeometry (figure 2). These perturbations do not need to be small.Thus, the number of degrees of freedomavailable tomodify the ship hull geometry equalsthe number Ndp of design parameters. However, the greater freedom gained in shapingthe hull is balanced by a higher computational cost of finding an optimum in a higherdimensional design parameter space.The polynomial functions used to generate the hull shape must be chosen so as to conferspecial geometrical properties on the optimal ship hull, such as the specific convexityproperties or bounds on the shape waviness. Otherwise, to achieve the same goals wouldrequire to explicitly add extra constraints to the cost function at the expense of a muchhigher computational cost.3. Flow solverThe design problem considered consists in optimizing the bulb of a ship moving forward ata constant speed V in calmwater. The bulb should be shaped so as tominimize the hull drag. The flow field generated by the hull shape is predicted under the simplifying assumptionsof inviscid and irrotational flow. These assumptions allow to determine the flow field aboutthe ship hull and the free-surface perturbations by means of a potential flow model. Thepotential flow equations allow to define the state of the flow at each point of the semi-infinite3D space bounded by the ship hull surface and the water free-surface as a function of adistribution of sources located at both the ship hull surface and the water free-surface.