*2.3. Improved RBF Method*

The drawback of the RBF method is the non-orthogonality of the boundary layer. To reduce the non-orthogonality, the centroids of the first boundary cells are added as control points. The displacements of the new control points are calculated using the IDW method. To calculate the translational and rotational displacement, the RBF calculation is repeated twice. First, the boundary face is deformed with only the initial control points. The displacements of the centroids of the first boundary cells are calculated with the displacements of the deformed boundary face by the IDW method. Second, the displacements of the volume mesh grid points are calculated with the grid points of the deformed boundary and the centroids of the first boundary cells. To reduce the calculation time, every 4 points of the grid points and centroids is used as control points.

**Figure 5.** Deformed mesh using the inverse distance weighted (IDW) method. (**a**) whole domain (side view), (**b**) near body (side view), (**c**) around leading edge (side view), (**d**) around leading trailing (side view).

It is difficult to apply both the RBF and IDW methods to problems involving the free surface because the grid around the free surface has to be aligned to the free surface to minimize the numerical diffusion of the volume of fluid function (VOF). Therefore, the deformable region must be limited. The cells around the deformed boundary are designated as deformable cells, while those outside of the regions are set as frozen cells. The faces sheared by the deformable and frozen cells are designated as fixed faces. The points on the fixed faces are added to the fixed control points.

Figure 6 displays the deformed mesh. The non-orthogonality of the boundary cell is better than those by the RBF and IDW methods. The thickness of the boundary cells around the leading and trailing edges is well preserved. The cells in yellow circle are deformable cells.

**Figure 6.** Deformed mesh using the RBF method with additional control points. (**a**) whole domain (side view), (**b**) near body (side view), (**c**) around leading edge (side view), (**d**) around trailing edge (side view).

The quality of the deformed mesh is compared with the initial mesh in Table 3. The deformed mesh quality using the proposed method is as good as that by the IDW method. Moreover, the proposed method is much faster than the others owing to the limited deformation region.


**Table 3.** Quality of the deformed mesh using the proposed method.

#### **3. Mesh Deformation for Hull Form Variation**

The proposed mesh deformation method was applied to the mesh for ship resistance calculation to examine its applicability to CFD-based optimization. The ship used in the calculations is the JBC model. The scale ratio and the draft are 1:40 and 0.4125 m (16.5 m in full scale), respectively. The speed is 1.179 m/s (14.5 knots in the model).

The initial grid used for the JBC resistance calculation is illustrated in Figure 7. The number of cells is 2,402,361. The Y+ of first layer thickness is approximately 50 and the number of boundary layers is 4. The expansion ratio of the boundary layer is approximately 1.3. The running attitude of the ship is fixed as even keel condition. The simulation was conducted using interFoam, a standard solver of OpenFOAM. kOmegaSST and nutUSpaldingWallFunction were used as the turbulence model and wall function, respectively.

(**a**)

(**b**)

**Figure 7.** Grid shape for Japan bulk carrier (JBC) resistance calculation. (**a**) whole domain (side view), (**b**) surface mesh of fore body (side view).

Three points on the forward perpendicular (F.P.) line were moved by 5 mm (0.2 m in full scale) to make an alternative hull form. The hull surface is split by the yellow dashed line in Figure 8. The hull surface in front of the line was set as deformable patch, while that behind the yellow line was set as fixed patch.

**Figure 8.** Definition sketch of control points for hull form variation.

The deformed hull is depicted in Figure 9. The red lines indicate the JBC station lines, whereas the blue lines denote the station lines of the deformed hull. Figure 10 displays a slice of the deformed mesh on the F.P. Because of the small deformation, the variation in mesh quality is small enough to be ignored. The time to deform a mesh with 2 million cells is approximately 118–120 s with a core of Intel Xeon CPU E5-2630 v3 2.4 GHz. The turnaround time is reasonably small. The mesh deformations and CFD simulations are conducted by shell script.

**Figure 10.** A slice of the deformed mesh at forward perpendicular forward perpendicular (F.P.) (red grid line: original mesh, blue grid line: deformed mesh).

The resistance histories of the JBC and deformed hulls are compared in Figure 11. The calculations of deformed hulls converged much faster than the JBC calculation because the result of the JBC calculation was used as the initial condition for the deformed hull resistance calculations. The calculation of the JBC resistance took approximately 60 s, whereas the calculation of deformed hulls took 20 s in flow time. The variations in the resistances are small because the hull form variation is small. The resistance coefficients are compared in Table 4.

**Figure 11.** Comparisons of resistance convergence histories. (**a**) whole time, (**b**) zoomed.

**Table 4.** Comparison of resistances of the JBC and deformed hull.


The pressure distributions and wave height contours are compared in Figures 12 and 13, respectively. It was found that the result of the initial hull form can be used as the initial condition for the deformed hull resistance calculation.

**Figure 12.** Comparison of pressure distributions around the bow (**left**: JBC only, **right**: JBC and deformed hull).

**Figure 13.** Comparison of wave height contour around the bow.

#### **4. Conclusions**

In this study, two methods for mesh deformation, namely, the RBF and IDW methods, were compared. Moreover, an improved RBF method was proposed for a largely deformed mesh. The RBF method was much faster than the IDW method, but the quality of the deformed mesh using the IDW method was better than that by the RBF method. The quality of the deformed mesh by the RBF method was improved by adding the centroids of boundary cells to the control points. The displacements of the centroids were calculated using the IDW method. The deformable region was limited for the problem involving the free surface. The limitation also reduced the calculation time.

The improved RBF method was applied to the mesh for the JBC resistance calculation to validate its applicability. The resistance was calculated by varying the bow shape with three control points. It took approximately 120 s for the mesh to deform, which is short enough to apply to practical problems. The calculation result of the initial hull form was used as the initial condition for the deformed hull form, which reduced the calculation time to approximately one-third of that of the initial hull form. Thus, the improved RBF method proposed in this study is effective and efficient for hull form variation.

In the future, the CFD-based hull form optimization will be conducted using the proposed mesh deformation method together with an optimization algorithm such as sequential quadratic programming or an adjoint variable method.

**Author Contributions:** Conceptualization, K.-L.J. and S.-M.J.; methodology, K.-L.J.; validation K.-L.J.; formal analysis, K.-L.J.; investigation, K.-L.J. and S.-M.J.; resources, K.-L.J. and S.-M.J.; writing—original draft preparation, K.-L.J. and S.-M.J.; writing—review and editing, S.-M.J.; visualization, K.-L.J.; supervision, K.-L.J. and S.-M.J.; and funding acquisition, S.-M.J. All authors have read and agreed to the published version of the manuscript.

**Funding:** This study was supported by a research fund from Chosun University (K207177004).

**Conflicts of Interest:** The authors declare no conflicts of interest.
