Optimization Design of Minimum Total Resistance Hull Form Based on CFD Method

In order to reduce the resistance and improve the hydrodynamic performance of a ship, two hull form design methods are proposed based on the potential flow theory and viscous flow theory. The flow fields are meshed using body-fitted mesh and structured grids. The parameters of the hull modification function are the design variables. A three-dimensional modeling method is used to alter the geometry. The Non-Linear Programming (NLP) method is utilized to optimize a David Taylor Model Basin (DTMB) model 5415 ship under the constraints, including the displacement constraint. The optimization results show an effective reduction of the resistance. The two hull form design methods developed in this study can provide technical support and theoretical basis for designing green ships.


Introduction
Minimum total resistance directly impacts a ship's navigational ability and safety, and consequently, the hull form design is a crucial concern of ship designers. The shipbuilding industry must achieve innovation-driven development and strategic transformation (Chen and Huang, 2004). In the past, the optimization of a ship depended on experiments or CFD tools. However, both methods are time-consuming (Mahmood and Huang, 2012). Therefore, it is necessary to find an effective optimization method to reduce a ship's resistance and improve its performance. Recently, the ship hull form optimization based on the potential flow theory has become an established method. For example, to design an optimal hull form with the minimum wave making resistance, a ship hull form optimization method was developed by Suzuki Kazuo (Suzuki and Iokamori, 1999;Suzuki et al., 2005), integrating optimization techniques and numerical theory. This method was further used as a basis by many scholars to explore new optimization techniques to improve a ship's hydrodynamic performance. Lan (2012) used the Dawson method to calculate the wave making resistance and design an optimal hull form with the minimum wave making resistance. We have been studying the hull form design using the Rankine source method since 2009 , using Non-Linear Programming (NLP), System Genetic Algorithm (SGA) and Niche Genetic Algorithm (NGA) methods to optimize Wigley and S60 ships. The ship hull form optimization using the potential flow theory has been widely applied in preliminary designs owing to its high speed of calculations. In recent years, with the rapid development of the computer technology and mathematical knowledge, CFD tools have further improved the hull form optimization design. Shipbuilding powerhouses such as in Europe, Japan, and Korea have a head start over that in other countries in the ship design industry, and have already used this technology to design full-scale ships. Campana (Peri and Campana, 2003;Campana et al., 2004) first studied the optimization of ship hydrodynamic performance using the SBD technique (combining optimization theory with CFD technology) at the INSEAN pool in Italy. Lately, Peri and Campana (2005), and Campana et al. (2016) have carried out a lot of research work on the ships' optimization design (resistance and seakeeping) using the CFD technology. They conducted the systematic research on the hull geometry reconstruction, approximation, optimization strategy and integration. In this optimization, the disturbance surface method based on the Bezier Patch surface was developed for the hull geometry reconstruction. The variable fidelity model approximated multi-objective optimiza-tion. After optimization, the model tests were conducted to verify the results, showing the reliability of the method. Following this, Chinese scholars began researching the hull form optimization based on the CFD technique (Li et al., 2014;Feng et al., 2016;Huang et al., 2015;Wu et al., 2016). Although they have made considerable progress, China faces barriers such as low integration and the lack of wellestablished commercial software.
The aim of this paper is to present two ship hull form optimization methods. The total resistance (objective function) is calculated using the potential flow theory and the RANS method. The parameters of the hull form modification function are selected as the design variables. Under the displacement constraint, the NLP method is adopted to establish the mathematical model. The hull form optimization method based on the potential flow theory is accomplished by self-programming. The hull form optimization method based on the RANS method is implemented on the ISIGHT tool. Both of the optimization methods are automatic. It can be proved that the optimal hulls have a better hydrodynamic performance with a reduction of the total resistance compared with the parent hull.

Rankine source method
The Rankine source method is a numerical calculation method for the wave making resistance, which uses double model flow instead of uniform flow of the thin-ship theory. The Cartesian coordinate system is fixed in the ship hull (Tarafder and Suzuki, 2007). The x-axis and y-axis are fixed on the undisturbed still water surface; the x-axis is along the uniform flow towards the aft, the z-axis points upwards as shown in Fig. 1.
The total velocity potential ϕ around the ship hull consists of two parts: the double-body velocity potential ϕ 0 and the perturbed wave potential ϕ 1 representing the effect of the free surface. where, where r 0 represents the distance between the field point (x, y, z) and the source point (x′, y′, z′), and r 1 denotes the distance between the field point (x, y, z) and the source point (x′, y′, 0), S 0 is the hull surface of the double model, and S 1 is the undisturbed free surface. The Hess-Smith method is used to calculate ϕ 0 (Suzuki et al., 2005). The free surface equation can be expressed as: z = ζ(x, y).

(6)
Boundary conditions on the free surface are listed below.
Dynamic conditions: kinematic conditions: ζ Eliminating the wave height from Eq. (7) and Eq. (8) can obtain Eq. (9). The linearized free surface condition is: where the subscript l represents the differentiation along a streamline of double-body velocity potential on the panel z=0, ϕ 0l denotes the velocity gradient of the double-body velocity potential along the streamline direction l on the panel z=0, and ϕ 1l represents the velocity gradient of the wave velocity potential along the streamline direction l on the panel z=0.
The boundary condition on the ship hull is: ∂ϕ The hull surface of double model hull is divided into M 0 panels, and the undisturbed free surface is divided into M 1 panels, combined with boundary conditions Eq. (8) and Eq. (9). After discretization the following equations can be obtained.
On the hull surface: On the undisturbed free surface: In Eq. (8) and Eq. (11), the streamline derivative is solved by four points along the streamline upwind difference scheme (developed by Dawson) to satisfy the radiation conditions on a ship's far ahead (Lan, 2012). The upstream disturbance only flows downstream, and the downstream disturbance does not directly affect anything upstream. To simulate actual fluid flow, a lower order one-sided difference scheme is used to increase the numerical viscosity near the stern and the boundaries of the computational domain.
By combining Eq. (10) and Eq. (11) and the upstream conditions, a set of equations can be obtained. The Gauss elimination method is employed to obtain the discrete source of the body surface σ 1 and undisturbed still water surface Δσ 0 . Then, the perturbed wave potential ϕ 1 can be obtained. Following this, the pressure around the hull can be expressed by ϕ 1 ignoring the high-order terms.
The wave height of the free surface can be expressed as: The wave making resistance obtained by the integration of the pressure on the hull surface is: where ΔS 0l is the area of a panel on the hull surface, and n xl are the direction cosines of the normal to the panels.

Control equation
The whole flow field uses the continuity equation and RANS equations as the governing equations (Lin and Liu, 1998): is the velocity component in the x i =(x, y, z) direction, and , , , and are the fluid density, static pressure, fluid viscosity, Reynolds stresses and body forces per unit volume, respectively.

Turbulence model
The turbulence model adopts the RNG k-ε model, and the forms of the turbulence energy transport equation and energy dissipation transport equation are as follows: where μ eff is the effective dynamic viscosity, k and ε are the turbulent kinetic energy and the turbulent dissipation rate. the quantities α k and α ε are the inverse effective Prandtl numbers for k and ε, respectively; G k is the generation of turbulent kinetic energy by the mean velocity gradients; G b is the generation of turbulent kinetic energy by buoyancy; Y M represents the contribution of the fluctuating dilatation in compressible turbulence; C 1ε , C 3ε and C 2ε are empirical constants.

Discrete equation
The governing equations are discretized using the finite difference scheme of the volume center. The two-order Euler backward difference scheme is as follows: 3.4 Free surface capturing method The VOF method (Simonsen et al., 2013) is used to capture the free surface. It is a surface tracking method fixed under the Euler grid and simulates the multiphase flow model by solving the momentum equation and the volume fraction of one or more fluids. Within each control volume, the sum of the volume fractions of all the phases is one. As to Phase q, its equation is: where q=0 means that the unit is filled with water, q=1 means that the unit is filled with air, a 0 and a 1 are respectively the volume fractions of air and water, and a q = 0.5 is the interface of water and air.

Estimation object function
The total resistance R T in calm water is selected as the objective function.
For the optimization method using the potential flow theory, R T is calculated by: where R W is the wave making resistance, which is obtained using the Rankine source method, and R F is the equivalent flat plate frictional resistance: where is the plate frictional drag coefficient, which is calculated by: where Re is the Reynolds number, L is the ship length, ν is the kinematic viscosity of the fluid, U ∞ is the speed, and S is the wetted surface area. For the optimization method using the viscous flow theory, R T is calculated by: where R W is the wave making resistance, and R V is the viscous resistance. R W and R V are obtained by using the RANS method.

Design variables
In this paper, the bow hull form is to be optimized, whereas the design waterline, the hull bottom, and the stern are fixed. The modified hull form y(x, z) is calculated by: where f 0 (x, z) represents the parent hull, and w(x, z) is the hull form modification function. The hull form modification function in the first half of the ship is as follows: (29) The hull form modification function in the second half of the ship is as follows: (30) where x 0 , x max and x min are the characteristic parameters, d is the draft, α mn and β mn are the design variables, and m=n=1, 2, 3, … The number of the design variables is 18. The hull modification function method reduces the number of the design variables, improving the optimization calculation speed effectively. Fig. 2 shows the modified region of the ship.

Constraints
(1) All offsets are nonnegative, namely: is the coordinate of the hull form surface.
(2) Displacement constraint: ∇ ∇ 0 where and are the displacements of the modified hull and the parent hull.

Optimization method
The SUMT interior point method is introduced into the NLP method (Ma et al., 2003) to add the constraints in the objective function, constructing an unconstrained optimization problem. The gradient method of the direct search method is used to find an optimized hull with the minimum resistance for an optimization space.

Optimization of a ship
First, the parent ship hull lines are input. Then, the optimization code is written by using the hull form modification function to automatically build the hull lines in the optimization range. Following this, the obtained data points are imported into CAD modeling software. Next, the modified and fixed parts of the ship are merged into several new ship types. The body-fitted mesh and the structured grids are used to mesh the computational domain. The total resistance of the ship is calculated according to the method described in Section 2 and Section 3. The design variables are changed by using the NLP method. If the stopping condition is met, the optimized hull is output. Otherwise, new ships must be recalculated until the results converge. The hull form optimization method based on the potential flow theory is accomplished by self-programming. The hull form optimization method based on the RANS method is implemented on the ISIGHT tool. The flow chart of the ship hull form optimization is shown in Fig. 3.

Optimization example
The DTMB5415 ship is used within this study. The main properties of the DTMB5415 ship are presented in Table 1. The hull surface is divided into 3600 panels, and the free surface is divided into 5400 panels as shown in Fig. 4.

Rankine source method
For the first optimization approach developed in this pa- per, the total resistance (objective function) is obtained by using the Rankine source method. After the completion of the optimization, the optimization results are summarized in Table 2, where C Topt and C Torg are the total resistance coefficients of the modified hull and the parent hull. The total resistance coefficient of the optimized hull decreases by 18% at the design speed. Fig. 5 shows the relationship between the ratio of the optimized total hull resistance coefficients to total parent hull resistance coefficients and the number of iterations. The calculation is typically stable at 20 steps. Fig. 6 shows the comparison of the total resistance coefficients at different Fn values for the optimized hull and the parent hull. Fig. 7 shows the comparison of the cross section for the optimized hull and the parent hull. It can be seen from Fig. 7 that the hull lines of the optimized hull has a slight bulge on the sonar dome, which results in an increase in the displacement and wetted surface. For this optimization, the wave making resistance is selected as the objective function, and frictional resistance is obtained using the frictional resistance of a flat plate formula. The main cause of      ZHANG Bao-ji et al. China Ocean Eng., 2018, Vol. 32, No. 3, P. 323-330 the reduction of total resistance is the decrease of the wave making resistance, not the frictional resistance or any other resistance. Fig. 8 shows the comparison of the wave contours for the optimized hull and the parent hull. It can be observed that bow waves and shoulder waves of the optimized hull are reduced compared with those of the parent hull. Fig. 9 shows the comparison of the wave profiles at y/L=0.105 for the optimized hull and the parent hull. It can be seen that the amplitude of waves for the optimized hull has been reduced.
6.2 RANS method A three-dimensional numerical tank is built to predict ship performance of the DTMB5415 ship. The locations of the boundaries are presented in Fig. 10. The inlet boundary is positioned L away from the hull, and the outlet boundary is located 4L downstream. The water depth is set as L, and the width of the numerical tank is selected as L.
The computational mesh is shown in Fig. 11. The mesh is clustered on the bow, stern and free surface. The region away from the hull adopts large structured grids to reduce the number of the cells and improve the calculation efficiency. The cells of the equisize skew (grid skew) between 0 and 0.5 occupy 99.5% of the total cells. The maximum grid equisize skew is 0.62. The mesh quality meets the calculation requirement. Finally, the number of the grids is about 650 thousand in total, and the time step is set to be 0.001 s.
The optimization results are summarized in Table 3. Fig.  12 shows the evolution history of the total resistance coefficients. It can be seen that the calculation is typically stable at 36 steps. Fig. 13 shows the comparison of the total resistance coefficients C T for the optimized hull and the parent hull. It can be seen that the total resistance coefficient of the optimized hull reduces by 8.48% at the design speed. Fig.  14 shows the comparison of the cross section for the optimized hull and parent hull. Fig. 15 shows the comparison of the wave contours for the optimized hull and the parent hull. It can be seen that bow waves of the optimized hull have been reduced compared with those of the parent hull. Fig.  16 shows the pressure distribution on the hull surface for the     optimized hull and parent hull. It can be found that the surface pressure of the bow section undergoes a significant change.

Conclusions
Based on the potential flow theory and the viscous flow theory, two ship hull form optimization methods are built. In these two methods, the total resistance is taken as the objective function, and the parameters of the hull form modification function are taken as the design variables. These two methods are employed to optimize the DTMB5415 ship with the same displacement. The results are shown as follows.
(1) In the optimization method of the potential flow theory, the wave making resistance of the optimized hull decreases by 18%. Although large variations in the hull lines may affect the layout and performance of a ship, it is an important method for the preliminary stages of new ship design because of its optimization efficiency.
(2) For the optimization method using the viscous flow theory, the wave making resistance of the optimized hull decreases by 8.48%. The new hull lines change minimally, and the Kelvin wake can be captured clearly. This is a practical optimization method for the ship design industry.
Wind and waves are very important for a ship's navigation. Future work will focus on the hull form optimization in wind and waves.   ZHANG Bao-ji et al. China Ocean Eng., 2018, Vol. 32, No. 3, P. 323-330 329