Homotopy method for inverse design of the bulbous bow of a container ship

The homotopy method is utilized in the present inverse hull design problem to minimize the wave-making coefficient of a 1300 TEU container ship with a bulbous bow. Moreover, in order to improve the computational efficiency of the algorithm, a properly smooth function is employed to update the homotopy parameter during iteration. Numerical results show that the homotopy method has been successfully applied in the inverse design of the ship hull. This method has an advantage of high performance on convergence and it is credible and valuable for engineering practice.


Introduction
The traditional method to predict the ship resistance is by ship model experiment. Although CFD has been more and more applied in ship resistance prediction, ship model experiment is still an irreplaceable method. Numerical simulation and optimization is still an emerging field of numerical research. There are countless optimization theories and applications in this field. Compared with the model test, simulation and optimization based on CFD is faster, cheaper and can provide more details of the flow.
Ship hull optimization is a complex process because various parameters must be chosen to meet the demand. From the hydrodynamic point of view, reducing ship resistance is one of the most attractive goals. Resistance reduction can lead to the reducing of fuel consumption and is more environmentally friendly. And many interesting works on hull form optimization have been presented in recent years (Percival et al., 2001;Campana et al., 2006;Wilson et al., 2011;Hochkirch and Bertram, 2009;Tahara et al., 2006).
Especially, the bulbous bow form hydrodynamically affects the velocity field near the bow in the region of the ris-ing bow wave. Thus, a well designed bulbous bow can reduce the wave resistance efficiently. However, the ship hull optimization problem is a kind of strongly nonlinear problem. Efforts have been made to search for the optimal hull by using different algorithms based on CFD prediction. There are two kinds of algorithms to solve the optimization problem. One is the so-called heuristic algorithms and the other is the conventional gradient-based algorithms, the former, such as Genetic Algorithm (Dejhalla et al., 2002;Bagheri and Ghassemi, 2014), the initial solution does not need to be carefully selected and the global solution can be obtained. However, these methods are very time-consuming because they have to establish the so-called populations and thus a large number of CFD processes must be done. In contrast, the gradient-based algorithms, such as the Levenberg- Marquardt method (Huang et al., 1998), are more effective than the heuristic algorithms. However, there is no guarantee that the global optimum will be reached and the optimization results largely rely on the initial solution.
Chen and Huang (2002,2004) and Chen et al. (2006) studied the inverse design problem of ship hull optimization. The advantage of the inverse design scheme is that the optimal hull form can be obtained directly according to the preferable wash wave. The Levenberg-Marquardt method has been proved to be a powerful algorithm in inverse computations, especially in parameter estimations.
In this paper, a hydrodynamic optimization problem for a ship hull has been treated and an optimization procedure has been developed to minimize the objective function related to the flow past a ship moving with steady forward speed in calm water. The major highlight of this study is that the homotopy algorithm method has been used in the optimization process. The flow solver coupled with the homotopy algorithm method has been used to solve the problem, and a computational example of a 1300TEU container ship is presented in order to demonstrate the effectiveness of the method.

Optimization problem formulation
Optimization problem may be generally formulated as a problem of minimizing the objective function f(x) of a number of variables x 1 , x 2 , ..., x n . While in ship hull optimization problem, a number of variables also are needed to be chosen to determine the optimal ship hull. Three main parts form the ship hull optimization problem. They are algorithms used in the optimization process, numerical representation of the ship hull and the tools for resistance prediction.
In past years, many researchers have paid a lot of attention to the algorithms used in ship hull optimization. The gradient-based methods, such as the Levenberg-Marquardt method, which uses the gradient of the objective function during iteration, are generally more effective than the heuristic algorithms. However, there is no guarantee that the global optimum will be reached. In other words, a suitable initial solution may lead to the minimum result while an improper initial solution may lead to a local optimum. Since Liao (2009) proposed the homotopy analysis method, many types of nonlinear problems have been solved by homotopy. In recent years, this method has been successfully used in several areas of mathematics, especially in the area of nonlinear inverse problems (Hetmaniok et al., 2012;Wang and Wang, 2014). Furthermore, homotopy methods can reach a solution by tracing a path from a fairly arbitrary initial point and approaches derived from homotopy methods can find the global optimum in situations where other gradient-based methods cannot find solution (Kuno and Seader, 1988).
For the numerical representation of the hull surface used in optimization design, various methods can be found in the literature. In this study, the B-spline surface scheme is chosen. The B-spline surface scheme can represent the complicated hull surface with a small number of parameters (control points). Once the control points are moved, the entire hull surface can be changed correspondingly.
The desirable CFD tools for bulbous bow optimization should be accurate, easy to converge, can be submitted into a subrou tine for noninteractive calculation and should not be time-consuming. Therefore, SHIPFLOW is chosen as the flow solver integrated with the homotopy algorithm to optimize the bulbous bow.
In this study, a 1300TEU container ship is selected as the research object. The corresponding hydrodynamics data, such as the wave-making resistance coefficient C w which can be obtained by performing the CFD analysis on the hull form, have been chosen as the optimization objective. By giving a much smaller C w as the design target and then performing the inverse design process, we can obtain the bulbous bow with minimum wave-making resistance in a short time.

B-spline surface
NURBS (Non-Uniform Rational B-spline) is a very good modeling method proposed by Versprille (1975). Compared with the traditional modeling methods, it can better control the curve of the objective surface and can describe the contour more accurately.
By utilizing the NURBS surface, the hull form can be expressed as (Piegl and Tiller, 1997): (1) where d i, j (i=0, ..., n; j=0, ..., m) form a bidirectional control net, w i, j are the weights. N i, k (u), N j, q (v) are the non-uniform rational B-spline basic functions defined on the knot vectors.
When the node vectors u and v are fixed, a new ship hull can be generated by modifying the control vertex positions. A ship hull model of NURBS surface is shown in Fig. 1.

Inverse hull design problem
For the inverse hull design problem, the main hull is regarded as being unknown and is dominated by a set of control points B. The inverse design problem can be stated as: modifying the control points B to design the new hull.

Optimization model
The optimization objective, such as C w , can be expressed as a function of B:u=u(B). Then a least squares problem can be formed: where B indicates the control points of the ship hull, u(B) means the optimization objectives computed by CFD. u c is the optimization target desired by the author, in this study, it is 70%-80% of the original value.

Homotopy method
To our problem, by applying the optimality condition, an equivalent system can be obtained: and then, the homotopy method is used to solve the nonlinear equation above. The problem of solving the equation above has been transformed to the problem of solving the fixed point homotopy equation as below: (1 ¡ t) where t is an homotopy parameter; B 0 is the initial estimation of solution. Obviously, when t=1, the equation above can be solved easily, B=B 0 . when t=0, Eq. (4) is transformed to Eq. (3). Therefore, if the homotopy parameter t can track through a proper path from one to zero, the original problem of Eq. (4) can be solved. As t changes, by using Taylor series expansion at the nth iteration step, the linearized version of formulation can be obtained: The strategy of determining the homotopy parameter t n is very important. There are two criteria for updating the homotopy parameter that should be abided by Cui and Yang (2005). First, t should be diminished through a smooth path during iteration. Second, t should be terminated at a proper point close to zero, but not zero. Here we propose a function as follows: t n = (n + 0:1) ¡k ; where n is the number of iterations, k is the adjustable parameter. It is clear that this function can naturally meet the two requirements mentioned above. In our problem, we suggest that k should be chosen from 0.1 to 0.3. The finite difference method (FDM) is a common method used to estimate the derivative. Eq. (1) is the difference formula used in this study.
where F is the objective function, x is the variable, and h is a small amount of disturbance. In this study, F represents the wave-making coefficient, x represents the coordinate value of the control point in the Y direction, and h is 0.001.

Computational procedure
The iterative computational procedure for the present inverse design problem can be summarized as follows: Step 1: Choose the original hull form as the initial B at Iteration 1 to start the computation.
Step 2: Make resistance prediction to obtain the wave making coefficient.
Step 5: Check the stopping criterion; if not satisfied, go to Step 1.
The stopping criterion is defined as follows: In this study, ε is 0.0001. The computational procedure is shown in Fig. 2.

Results and discussions
A container ship with bulbous bow is chosen as an example. And the inverse design of the bulbous bow has been done. The process of optimization is performed by both the Levenberg-Marquardt method and the homotopy method. In this study, the wave-making coefficient has been chosen as the optimization objective. And the optimization has been done at the design speed, (U=2.156 m/s) and the corresponding Froude number is Fr=0.260. The optimization target is 80% of the original value, while the homotopy adjustable parameter is 0.3. Fig. 3 shows the ten control points which are chosen as the optimization parameters. Notably, in this study, only the coordinate values in the Y direction are modified because the change in the X or the Z direction may lead to smoothness of the ship hull lines. Fig. 4 (LM is short for Levenberg-Marquardt and HM is short for homotopy) shows a history plot when using the LM and HM algorithms to find the optimum bow shape. Both methods are very fast with the time-taking being 3-5 hours only. However, in contrast, the homotopy method has better behaviour in convergence.  Table 2 shows that the wave-making coefficient of the optimized hulls reduced by 18.1% and 21.6% in comparison with the initial hull when using LM and HM method, respectively. However, the displacement has risen by 3% of the initial value while the value of L cb /L wl and the wetted surface have been modified within 1% of range.
As it can be seen in Fig. 5, the breadth of the optimized hull is larger than that of the initial in fore part. The draft of the optimized hull has decreased dramatically. The changes in the variable parameters of the hull are made to reach the minimum wave-making coefficient for the container ship.
During the execution of the optimization algorithms, the coordinate values of the control points of the hull have been changed. The variations of the coordinate values of the control points before and after optimization are shown in Table 3.
The results presented in this study confirm that, despite its simplicity, the homotopy method can be a useful tool in the inverse design of the ship hull.

Conclusion
In order to avoid the difficulties of long-time consuming and local convergence in the ship hull optimization, the main contribution of this research is to apply the homotopy method into the inverse design of the ship hull. Two algorithms, the Levenberg-Marquard algorithm and the homotopy algorithm have been used in the present study. Results show that the present method needs only three to four hours to obtain the optimal hull and it has an advantage of high performance on convergence. The application of the present algorithm is not only just limited to the container ship, but also applicable to other kinds of vessels.