In modern computer aided ship design system, ship hull forms are mostly represented by the Non-Uniform Rational B-Spline surfaces and saved in IGES file format for data exchange between different CAD/CAM systems. Ship hull forms represented by NURBS surfaces have many good properties including visually fair and perfectly smooth compared with hull surfaces represented by discrete meshes. The local and global deformation of NURBS surfaces is simple by relocating the NURBS control points, changing the weight of control points, which possesses strong geometric explanation and is suited for the hydrodynamic optimization of ship hull forms. Therefore, a NURBS- based hull surface modification method is developed and integrated into an in-house solver OPTShip-SJTU for the hydrodynamic optimization of ship hull forms. The developed method is applied to the hydrodynamic optimization of Series 60 model. The optimization results demonstrate that the developed method is efficient and flexible for the deformation of hull surfaces and is suited for the optimization of real life ship.
With the development of computer technology and computational fluid dynamics, the CFD-based hydrodynamic optimization of ship hull forms has become a very powerful tool for the design of hull lines. It can be applied to the optimization of ship hull forms in terms of total resistance and sea-keeping performance in the preliminary stage. With the ship industry paying more and more attention to energy conservation and emission reduction such as energy efficiency design index proposed by IMO, the CFD-based hydrodynamic optimization of ship hull forms will play a more important role in the future. In general, the CFD-based hydrodynamic optimization of ship hull forms consists of four modules (Yang, 2016) including a hull surface representation and modification module, a hydrodynamic evaluation module, a surrogate module, and an optimization module. Among them, the hull surface modification module is used to modify the prototype hull forms to produce new hull forms in terms of hydrodynamic performance and design constraints. It is obvious that the ultimate optimization results are highly determined by the efficiency and quality of the hull surface modification module. The representation of hull model can be classified into two categories (Harries-Abt and Hochkirch, 2004): conventional modeling and parametric modeling. Parametric modeling defines hull model in a high level using form parameters. It can use few design variables to achieve the global deformation of ship hull forms and every form parameter has a specific meaning. But it is hard to achieve the local deformation of hull surfaces such as in the bow and stern. On the other hand, the typical example of conventional hull geometry modeling is ship hull forms represented by NURBS surfaces, which uses points to define curves and uses curves to define surfaces forming hierarchical structure. The NURBS surfaces have a simple and uniform mathematical expression to represent any complex surface as it is a mapping from two dimensional parameter space to surface in threedimensional space defined by B-spline basis function, knot vectors of parameters, NURBS control points, and weights of control points (Piegl and Tiller, 2012). It uses piecewise rational polynomials to avoid using high degree formula to fit complex surfaces and meet geometry constraints, which allows for the local deformation of hull surfaces by moving the control points and changing the weights. There are two hull surface modification method based on NURBS. One is directly moving the NURBS control points of hull surfaces. Kim (2009) selected 31 NURBS control points of Wigley hull as design variables to optimize the total drag coefficient at three given speed. Park and Choi (2013) used the movement of NURBS control points along x and z directions in the ship bow as design variables to optimize Series 60 model in terms of resistance. Wang (2015) used the direct NURBS-based hull surface modification method to generate a bulbous bow and a stern in Wigley hull. Then the hull with initial bulbous bow was optimized using radial basis function interpolation method (RBF). The direct NURBS-based hull surface modification method can achieve large and local deformation of hull surfaces. But it needs to define many design variables, which increases the computation cost. The other one is moving the NURBS control points of hull surfaces combined with other deforming methods including the RBF method, the Free Form Deformation Method and the shifting method. Noble and Clapworthy (1999) used the Free Form Deformation method (FFD) to move the control points of NURBS surfaces, which avoids the limitations of ordinary FFDs. Kim and Yang (2013) applied the shifting method and the RBF method to move the NURBS control points of Model 5279 hull to achieve the global and local deformation of ship hull forms. The optimal hull with a bulbous bow and a stern end bulb was obtained. Yang (2016) applied the NURBS-based modification technique to the optimization of a series of Joint High Speed Sealift with different bow configuration. Cheng et al (2018) employed the radial basis function interpolation method to the optimization of Series 60 model and reached a conclusion that the value of support radius of Wendland ψ3,1 basis function has a great impact on the deformation of hull surfaces. In order to avoid generating saddle-shaped surfaces during the modification of hull surfaces, the support radius should be twice the maximum length of the Delaunay triangulation edges. Pérez et al (2007) used the cubic B-spline curves to construct the body plan of bulbous bow subject to certain form parameters and the B-spline surfaces that fit these curves were constructed. Then the initial hull with bulbous bow generated above is optimized via CFD-based optimization method. Compared with the direct NURBS-based hull surface modification method, these methods have its merits. Only a small number of control points are required as design variables to achieve the flexible deformation of hull surfaces. The large deformation of hull surfaces can be achieved where various geometry constraints can be easily satisfied. The modified region can have a fairing connection with the original hull.