**1. Introduction**

Computational fluid dynamics (CFD) is one of the general tools for estimating the resistance of a ship in calm water. Naval architects iterate the hull form variation, grid generation, and CFD calculation to minimize the resistance. Even though the hull form variation is extremely localized and low, the grid must be regenerated. Many shipbuilding and design companies are exerting efforts to automate the procedure to reduce the time and cost. Moreover, many studies on CFD-based design optimization have been conducted. An optimization algorithm to minimize the iterations is the most important and the hull form variation method according to the design parameter is very essential.

Kim and Yang [1] applied two surface modification methods for hull form optimization. One was based on a radial basis function (RBF), while the other was based on a sectional area curve. Kim and Yang's [1] RBF method uses only 6 control points as design variables to minimize the resistance of Korea Research Institute for Ships and Ocean Engineering (KRISO) Container Ship (KCS) in three speeds. The resistance of the modified hull was evaluated using a method based on the Neumann–Michell

theory, which uses only a surface mesh. Kim et al. [2] applied the same method used by Kim and Yang [1] to improve the resistance and seakeeping performance of the US Navy surface combatant David Taylor Model Basin (DTMB) 5415. Mahmood and Huang [3] optimized a bulbous bow to minimize the total resistance using a genetic algorithm. They used ANSYS FLUENT and GAMBIT (Ansys, In., Canonsburg, PA, USA) for resistance calculation and mesh generation, respectively. A GAMBIT journal file was created to automate the hull form variation and volume mesh generation in accordance with the design parameters. Zhang et al. [4] proposed an improved particle swarm optimization algorithm, where Siemens STAR-CCM+ (Siemens Industry Software Ltd., Plano, TX, USA) was used for volume mesh generation and resistance calculation. The hull form was varied using an arbitrary shape deformation (ASD) technique proposed by Sun et al. [5]. The ASD technique is based on a B-spline and requires that the volume is set up outside the body with many control points and connections.

The volume mesh deforming method has been developed to simplify the optimization process and reduce the turnaround time as shown in Figure 1. Mesh deformation is much faster than grid generation, and the simulation with a deformed mesh uses the results of the original mesh as the initial condition. Successive calculation also reduces the calculation time. Morris et al. [6] developed a mesh deformation method based on the RBF method. The control points of the RBF method were used as design parameters. The method was independent of both the flow solver and grid generator. Morris et al. [6] applied a method for optimizing airfoils with feasible sequential quadratic programming. They concluded that the method was extremely fast and efficient, and the deformed mesh quality was very high. Sieger et al. [7] compared the classical free-form deformation (FFD), direct manipulation FFD, and RBF methods with each other. They concluded that the RBF method was much faster and more precise than the other two methods. Luke et al. [8] proposed a mesh deformation method based on inverse distance weighted (IDW) interpolation. Their method interpolated the translational displacement and rotational displacement using the IDW method. The parallelization of the algorithm was also described. They showed that the non-orthogonality of the boundary layer of the deformed mesh using the IDW method is better than that by the RBF method if the rotation of the body surface is high. He et al. [9] applied the IDW method to optimize an airfoil that starts with a circle. To show the robustness of the IDW method, a two-dimensional (2D) mesh for the circle was deformed to the mesh of NACA 0012. They concluded that the IDW method is better than the RBF method in terms of non-orthogonality of the boundary layer. TransFinite Interpolation (TFI) method is also a popular and efficient method for structured grid. However, the TFI method is difficult to apply to polyhedral mesh because of the irregular distribution of mesh points [10].

In this paper, Section 2 introduces the RBF, IDW and improved RBF methods. To compare the quality and time for deformation, a polyhedral mesh for circular cylinder are deformed to the mesh for NACA 0012. The results show that the RBF method have problem with non-orthogonality in boundary layer cell. The IDW method takes much longer time than that of the RBF method. The non-orthogonality of the improved RBF method is as good as IDW method and the turnaround time is shorter than any other methods. To check the applicability of the improved RBF method, the polyhedral mesh for Japan bulk carrier (JBC) resistance calculation is deformed and the mesh is calculated in Section 3. Because of the mesh topology is identical with the original mesh, the result of the original mesh is used as the initial condition of deformed mesh. Therefore, the time for solution converging is reduced by two-thirds.

**Figure 1.** Procedure of computational fluid dynamics (CFD)-based optimization. (**a**) Re-meshing method. (**b**) Mesh deforming method.
