Sequential Design-Space Reduction and Its Application to Hull-Form Optimization

Abstract

Hull-form optimization is a complex engineering problem. Owing to the several numerical simulations and complex design-performance spaces, hull-form optimization is considered an inefficient process, which makes determining the global optimum difficult. This study used rough set theory (RST) to acquire knowledge and reduce the design space for hull-form optimization. Furthermore, we studied one of the hull-form optimization problems by practically applying RST to the appropriate number of sampling points. To solve this problem, we proposed the RST-based sequential design-space reduction (SDSR) method that uses interval theory to calculate subspace intersections and unions, as well as test calculations to choose an appropriate stopping criterion. Finally, SDSR was used to optimize a KRISO container ship to minimize the wave-making resistance. The results were compared to those of direct optimization and one-time design-space reduction, thus proving the feasibility of this method.

1. Introduction

Owing to the increasing need for energy resources and the growing ubiquity of marine resource exploitation, many countries and international organizations have imposed stringent environmental requirements on shipping industries. The Energy Efficiency Design Index (EEDI), which was made mandatory by the International Maritime Organization (IMO), has directly affected the core competitiveness and international market share of the shipbuilding industry, resulting in new challenges in the development of green ships. Therefore, extensive efforts must be made to develop energy-efficient ships to address this challenge.

Considering the rapid development of computational fluid dynamics (CFD) and computer technology over the past few decades, CFD-based hull-form optimization has become an effective method for developing energy-efficient ships. Gammon [1] used a multiobjective genetic algorithm to perform multiobjective optimization of the resistance performance, seakeeping performance, and stability of fishing boats. Feng et al. [2,3] used the potential flow method to optimize the resistance performance of a variety of hull forms. Cheng et al. [4] used Shipflow to optimize an S60 hull form. Serani et al. [5] optimized the hull form of the DTMB 5415 model to reduce the resistance for Fn (Froude number) = 0.25. Yang et al. [6] used response surface models and multiobjective optimization algorithms. Kim and Yang [7] used radial basis function (RBF) to vary the hull forms during the optimization process. Zhang et al. [8,9] presented a ship hull-form optimization loop using the surrogate model to reduce the wave-making resistance of the Wigley ship. Jafaryeganeh et al. [10] applied a multiobjective genetic algorithm to deal with the optimization of the internal layout of oil tankers under uncertainties. Feng et al. [11] selected the Wendland ψ3,1 function as the basis functions of RBF interpolation, and the modification method was used to optimize a trimaran model.

In hull-form optimization, techniques such as parametric hull-surface modification and CFD numerical simulation and optimization are directly used to obtain optimal ship designs that satisfy a given set of constraints. However, considering hull-form optimization is a complex engineering problem, it involves several numerical simulations and has a complex design-performance space. Consequently, the hull-form optimization process is highly inefficient, making it difficult to determine the global optimum. The most commonly used approaches to solve these problems are as follows:

(1) High-efficiency optimization algorithms: The approach is used to develop an algorithm that can determine the global optimum using a small number of iterations. (2) Approximate modeling: This method uses a set of numerically simulated samples to construct an approximate model that can replace CFD computations, significantly reducing computational costs. (3) Using high-performance computing clusters: Modern computing hardware and parallel computing technologies can be used to increase computational speed and decrease computing time [12].

Although significant progress has been made in the aforementioned techniques, their efficiency and the quality of the resulting solutions are insufficient for practical application.

Therefore, many studies have been performed on design-space reduction. In studies conducted by Wang [13,14,15], Chu [16], Cheng et al. [17], Zhou et al. [18], Li et al. [19], Chi [20], Tseng et al. [21], Qiu et al. [22], Ye et al. [23], and D’Agostino et al. [24], design-space reduction was combined with optimization theory, which significantly improved the optimization efficiency and product quality.

This study applies design-space reduction to hull-form optimization by simulating a set of samples prior to hull-form optimization and using rough set theory (RST) to study the internal features of the design space to identify the design space that contains the optimal ship. Figure 1 shows an example of design-space reduction, where circle represents sample. Figure 1a shows the initial design space. The optimization space was large and exhibited complex internal characteristics. Therefore, the initial design space was sampled to obtain a uniform distribution of simulated sample points in the design space. Design-space reduction was then used to analyze and delineate the design space to locate attractive subspaces (i.e., R₁, R₂, and R₃ in Figure 1b), which significantly reduced the range of values considered by the design variables.

RST has various applications in data preprocessing and design-space reduction and is often combined with other methods. However, several limitations hinder the use of design-space reduction in design optimization. For example, the number of sampling points are usually chosen using a subjective, experience-based approach; the larger the number of samples, the greater the likelihood that it captures the characteristics of the design space. However, a large number of sampling points significantly increase the required number of numerical simulations and computational costs. While reducing the number of sampling points might reduce the computational costs, it increases the risk that the sampled points may not be representative of the design space. To solve these problems, we propose an RST-based sequential design-space reduction (SDSR) method for ship design optimization. First, interval theory is used to compute the intersection and union of subspaces. Then, a stopping criterion suitable for hull-form optimization is proposed to determine the termination conditions of the iteration.

The remainder of this paper is organized as follows. Section 1 presents the introduction. Section 2 introduces RST-based design-space reduction. Section 3 describes the proposed RST-based SDSR method and discusses its application to some problems. In Section 4, a KCS ship is optimized using the genetic algorithm, RST-based design-space reduction, and RST-based SDSR. The results are then analyzed and summarized in Section 5.

2. Design-Space Reduction Based on Rough Set Theory

RST, a mathematical tool that requires no a priori information other than samples that represent the problem at hand, was proposed by Pawlak in 1982. Therefore, RST is an objective method that characterizes or processes a problem. The aim of RST is to extract the rules that govern a set of elements through element classification and set-logic operations. In particular, RST uses indiscernibility relations, lower approximations, upper approximations, and reductions to characterize and express an information system. Interested readers can refer to the works by Pawlak [25] for a more detailed explanation of RST.

Figure 2 describes the procedure for RST-based design-space reduction.

Step 1: Construction of the initial information system. First, the design variables and their range of values are ascertained depending on the design optimization problem to determine the initial design space. Then, the design space is sampled and the target value of each sample is calculated to construct the data sample set.

Step 2: Discretization of continuous data. All continuous target values and variables in the decision table are transformed into discrete data. First, the target values are discretized using defining boundary values based on subject area knowledge, experience, or rules to classify the target values into two to three grades, which are expressed using different numbers or symbols. Then, the variables are discretized by defining a number of cuts followed by partitioning the attribute space into finite subspaces based on these cuts to ensure the values of the objects in all the subspaces are divided into the same intervals.

Step 3: Reduction in design variables. The discretized decision table is simplified by removing attributes that do not affect the classification power and knowledge of the decision table.

Step 4: Identifying attractive spaces. After attribute discretization and variable reduction, the simplified decision table is used to derive causal relations between the decision attributes and condition attributes. The rules that contain the minimum decision value (target value) are particularly important, considering the range of values of their condition attributes (variables) constitute an attractive space. The sum of all attractive spaces corresponds to the new design space obtained from design-space reduction.

3. RST-Based Sequential Design-Space Reduction

RST is applied to ship simulation samples in this application of RST-based design-space reduction. The larger the number of samples, the greater the probability of the characteristics space to be captured by RST. However, a larger number of samples significantly increase the computational cost associated with numerical simulations. Conversely, a low number of samples could fail to represent the design space despite being computationally efficient. Therefore, a sequential sampling strategy, called the sequential design-space reduction (SDSR) method, was used.

3.1. Optimization with Sequential Design-Space Reduction

In sequential sampling, the sampling process is divided into a number of iterations. In each iteration, the current to-be-sampled region is determined from the previous iteration. The sampling points are then added accordingly to provide the maximum amount of new information about the original model.

Research about sequential sampling began in the 1990s when Jones et al. [26] proposed the DIRECT (Dividing Rectangles) algorithm. In this algorithm, the search space is partitioned into hyperrectangles and the centerpoint of each hyperrectangle is evaluated. The hyperrectangle that potentially contains the optimum solution is selected based on the previously evaluated centerpoints, before being divided into another set of hyperrectangles to create the next search space. This process continues until the convergence criterion is satisfied.

SDSR is a technique that uses a relatively small number of sampling points to reflect the characteristics of the design space. In this technique, the number of sampling points and search space in each iteration are determined based on the information from the previous sampling iteration. In this study, SDSR was used to progressively shrink the design space while adding sampling points in the subdesign spaces to ultimately identify the optimal ship parameters.

Figure 3 shows the SDSR optimization procedure. Module 1 constructs the sample set of the design space, whereas Module 2 performs the design-space reduction according to the sample set to obtain the attractive spaces Sa. If the stopping condition is satisfied, the iteration stops. This step is followed by the optimization or selection of the optimal solution. If the stopping condition is not satisfied, it indicates that the number of sampled points are insufficient. In this case, more sampling points are added to the design space (while retaining all existing sampling points), and the simulated sample set is updated accordingly by repeating the iterative process (Modules 1 and 2).

During this iterative process, the attractive design spaces Sa_i+1 and Sa gradually overlap each other while approaching the optimal design space So, as shown in Figure 4. Therefore, the relationship between Sa_i+1 and Sa_i (especially the intersection between these subspaces) can be used to adjudge whether the design space is sufficiently reduced. This relationship will be analyzed in Appendix A.

3.2. Solution for Subspace Intersections

The BIAS for Matlab (b4m) interval arithmetic toolbox was used to solve subspace intersections and calculate the overlap coefficient C. The procedure is as follows: (1) The minimum interval of each attribute (as defined by its cuts) is expressed in vector form. (2) The minimum intervals of each attribute are merged to form minimum subspaces. (3) The volume of all minimum subspaces is computed. (4) Containment relations between the minimum subspaces and Sa_i are adjudged, and Sa_i is expressed as a set of minimum subspaces to determine its volume Vol (Sa_i). (5) The intersection and union vectors of Sa_i and Sa_i+1 are obtained by adjudging the equivalence of their minimum subspaces. (6) Finally, the overlap coefficient C (the stopping criterion) is computed using the union and intersection volumes of Sa_i and Sa_i+1.

$$C=\frac{Vol(S_{a_i}\cap S_{a_{(i+1)}})}{Vol(S_{a_i}\cup S_{a_{(i+1)}})}$$

(1)

where Vol is the volume of the design space.

Appendix B shows the exact procedure of the aforementioned process.

3.3. Setting the Stopping Condition for Sequential Design-Space Reduction

The stopping condition (i.e., C > C0) is a criterion that determines whether the iteration should stop or continue. In other words, it is an indirect solution to the question of “how many sampling points should be sampled”. As the iteration progresses, the Sa gradually approaches the optimal design space. However, owing to the high computational cost, it is not viable to allow Sa to be infinitely close to So. Herein, we use common test functions to study how C changes when Sa_i and Sa_i+1 are reduced during the iteration to determine C0. Appendix A shows the validation work to determine C0.

By analyzing the effect of changes in the design subspaces during SDSR on the SC and Hartman functions, it was found that each additional iteration reduced the design space and increased the overlap coefficient C up to or close to 1, as shown in Equation (2).

$$\lim C=\underset{i\to \infty }{{\displaystyle \lim }}\frac{Vol(S_{a_i}\cap S_{a_{(i+1)}})}{Vol(S_{a_i}\cup S_{a_{(i+1)}})}=1$$

(2)

However, the total number of samples increased when SDSR was applied to problems of greater complexity. Considering the complexity and computational cost of hull-form optimization problems, C > 0.85 was chosen as the stopping criterion for these problems to ensure SDSR was reliably performed within a reasonable amount of time.

4. Application of SDSR to Hull-Form Optimization and Design

Herein, a KRISO container ship (KCS) (see Figure 5) model is optimized to minimize its wave-making resistance. The main parameters of the KCS model are shown in

Table 1. Main dimensions and parameters of the standard KCS model.

Length between Perpendiculars Lpp/m	Waterline Length Lwl/m	Breadth at Loaded Waterline Bwl/m	Moulded Depth D/m	Draught T/m	Block Coefficient Cb	$\mathbf{Displacement} \nabla$/m3
7.2786	7.2944	1.0194	0.6013	0.3418	0.6505	1.6478

.

4.1. Optimization Object and Constraints

• Optimization Object

The aim of the optimization is to minimize the wave-making resistance at a Froude number (Fr) of 0.26, as shown below.

$$\min f=R_w,Fr=0.26$$

(3)

Note: Fr = 0.26 is the design speed of a KCS. The wave-making resistance was calculated using the Shipflow fluid dynamics software suite and the optimization was performed in the fully constrained conditions, without considering the heave and trim. The computational domain is 1 times the length (Lpp) in width, 0.5 times the length in height forward of the bow, and 1.2 times the length in height behind the stern. There are a total of 7783 surface elements in the mesh of the free surface and hull surface. Figure 6 shows the hull surface and free-flow surface element mesh.

• Constraints

To prevent excessively large changes in deadweight and buoyancy, the hydrostatic condition was imposed, where the displacement $\nabla$ and longitudinal center of buoyancy Lcb were restricted to changes within ±1%. The grid nodes at the deck edge line and keel and near the midship cross-section were fixed.

4.2. Method for Hull-Surface Modification and To-Be-Optimized Variables

In this study, radial basis function (RBF) interpolation was used to perform hull-surface modification [4,7,11,27]. Five representative nonuniform rational basis spline (NURBS) control nodes (numbered 1–5) were used as movable control nodes (see Figure 7) on the hull. The direction and range of movement of the control nodes were configured based on our experience.

As shown in Figure 7, control node 1 was movable along the x (length of the ship) and z (draught) directions, whereas nodes 2–5 were movable along the y (breadth of the ship) direction. Control node 1 determines the length and height of the bulbous bow, whereas control node 2 sets the breadth of the bulbous bow and the shape of the bilge. Control nodes 3 and 5 adjust the entrance angle and shape of the entrance at the water plane, respectively, whereas control nodes 2 and 4 adjust the shape of the bow and bilge, respectively. Control nodes 2–5 are also used to adjust the degree of U- or V-shape of the hull.

Therefore, there are a total of six variables: X₁, Z₁, Y₂, Y₃, Y_4, and Y₅, and the value range of these variables (shown in

Table 2. Initial range of variable values (in units of m).

Variable	Y2	Y3	Y4	Y5	X1	Z1
Lower bound	0.06000	0.08200	0.10000	0.24500	−0.08000	0.17000
Initial value	0.07358	0.09447	0.13299	0.26038	−0.00256	0.19463
Upper bound	0.08500	0.10600	0.14500	0.27500	0.00000	0.23500

) constitutes the initial design space for hull-form optimization. These value ranges were chosen based on our experience to preclude excessively large changes in the hull. Based on our analysis, the chosen value ranges were sufficient to cause significant changes in the shape of the hull and minimize its wave-making resistance.

4.3. Results and Analysis

4.3.1. Case 1: CFD-Based Hull-Form Optimization

The multi-island genetic algorithm was used to directly optimize the wave-making resistance within the initial design space. Figure 8 shows the convergence graph of the algorithm over 1000 CFD calculations.

Table 3. Variable values, hydrostatic parameters, and target values of the optimized ship.

	Y2/m	Y3/m	Y4/m	Y5/m	X1/m	Z1/m	Lcb	$\nabla$/kg	Rw/N	Rt/N
Optimized ship	0.0836	0.1057	0.1411	0.2587	−0.0785	0.2265	0.508	1.697	11.048	80.155
Original	0.0736	0.0945	0.1330	0.2604	−0.0026	0.1946	0.509	1.691	12.237	80.476

shows the parameters giving the lowest wave-making resistance, whereas Figure 9 shows the lines plan of the optimized ship design.

As shown in Table 3, displacement increased by 0.4%, whereas the wave-making resistance decreased by 9.7% in the optimized ship. Considering the wave-making resistance only accounts for a small proportion of the total resistance, the total resistance decreased by 0.4%. In the optimized ship, the bulbous bow extended forward and upward in a “wing-like” shape, effectively increasing the ship length below the waterline and the entrance length, thereby reducing the wave-making resistance. Furthermore, the changes made to control nodes 2, 3, and 4 increased in breadth at the bow. However, the breadth decreased around control node 5.

According to the dynamic pressure distributions shown in Figure 10, the area of the low-pressure area behind the bulbous bow became smaller after optimization, which reduced the steepness of the pressure gradient across the bow. The longitudinal wave cuts at different parts of the KCS before and after optimization are shown in Figure 11. The wave amplitude at the first 1/4 of the ship (from the bow) decreased significantly after optimization, indicating that the optimized ship was wasting less energy on making waves and undertaking less work to overcome the wave-making resistance, thus explaining the improved wave-making resistance in the optimized ship.

4.3.2. Case 2: RST-Based Design-Space Reduction

First, an initial information system was constructed based on the aforementioned to-be-optimized variables and their value ranges. Then, the system was discretized to facilitate the extraction of attractive spaces using RST. Finally, gradient optimization was used to determine the optimal solution within each of the subspaces and produce the optimal ship design.

Firstly, a uniform design was used to generate a uniform distribution of sampling points in the initial design space (61 sampling points). Then, RBF interpolation was used to generate the corresponding ship models. Finally, CFD was used to compute the drag performance of each ship design. Using the procedure described in Section 2, 15 rules were obtained, as shown in

Table 4. Decision rules.

No.	Decision Rule	Support
1	(Y2 = “(−Inf,0.0719)”)&(Y3 = “(0.097,Inf)”)&(Y4 = “(0.13415,Inf)”)&(X1 = “(−Inf,−0.06335)”) = >(d = 0[1])	1
2	(Y2 = “(−Inf,0.0719)”)&(Y3 = “(0.097,Inf)”)&(Y4 = “(−Inf,0.13415)”)&(X1 = “(−Inf,−0.06335)”) = >(d = 0[3])	3
3	(Y2 = “(0.0719,Inf)”)&(Y3 = “(0.097,Inf)”)&(Y4 = “(0.13415,Inf)”)&(X1 = “(−0.06335,Inf)”) = >(d = 0[3])	3
4	(Y2 = “(0.0719,Inf)”)&(Y3 = “(0.097,Inf)”)&(Y4 = “(0.13415,Inf)”)&(X1 = “(−Inf,−0.06335)”) = >(d = 0[1])	1
5	(Y2 = “(0.0719,Inf)”)&(Y3 = “(0.097,Inf)”)&(Y4 = “(−Inf,0.13415)”)&(X1 = “(−Inf,−0.06335)”) = >(d = 0[1])	1
6	(Y2 = “(−Inf,0.0719)”)&(Y3 = “(−Inf,0.097)”)&(Y4 = “(0.13415,Inf)”)&(X1 = “(−0.06335,Inf)”) = >(d = 1[4])	4
7	(Y2 = “(0.0719,Inf)”)&(Y3 = “(0.097,Inf)”)&(Y4 = “(−Inf,0.13415)”)&(X1 = “(−0.06335,Inf)”) = >(d = 1[6])	6
…	…	…
15	(Y2 = “(0.0719,Inf)”)&(Y3 = “(−Inf,0.097)”)&(Y4 = “(−Inf,0.13415)”)&(X1 = “(−0.06335,Inf)”) = >(d = 1[14])	14

. In this table, the “support” column shows the number of objects supporting the rule. Rules may be interpreted as a certain target value being obtained when the variables take values in certain ranges.

For example, Rule 1 may be interpreted as follows: if Y₂, Y₃, Y_4, and X₁ take values within (−Inf, 0.0719), (0.097, Inf), (0.13415, Inf), and (−Inf, −0.06335), respectively, the decision attribute is then 0.

Y₅ and Z₁ did not appear in any of the rules. According to RST, this can be interpreted as Y₅ and Z₁ not affecting the rules, regardless of their values. In theory, Y₅ and Z₁ can be treated as initial coordinates (i.e., the values of the original ship) in the hull-form optimization problem. However, if these coordinates are fixed, the variations in the other coordinates could result in unsmooth hull surfaces. Therefore, the optimization ranges of both these variables were still used as their initial range of values in the optimization.

As shown in Table 4, there are five target values with the rule “d = 0” (rules 1–5); hence, five attractive spaces were generated after design-space reduction (

Table 5. Subspaces after design-space reduction.

Subspace	Variable	Y2	Y3	Y4	Y5	X1	Z1
1	Lower bound	0.06	0.097	0.13415	0.245	−0.08	0.170
	Upper bound	0.0719	0.106	0.145	0.275	−0.06335	0.235
2	Lower bound	0.06	0.097	0.1	0.245	−0.08	0.170
	Upper bound	0.0719	0.106	0.13415	0.275	−0.06335	0.235
3	Lower bound	0.0719	0.097	0.13415	0.245	−0.06335	0.170
	Upper bound	0.085	0.106	0.145	0.275	0	0.235
4	Lower bound	0.0719	0.097	0.13415	0.245	−0.08	0.170
	Upper bound	0.085	0.106	0.145	0.275	−0.06335	0.235
5	Lower bound	0.0719	0.097	0.1	0.245	−0.08	0.170
	Upper bound	0.085	0.106	0.13415	0.275	−0.06335	0.235

).

If any pair of subspaces have a variable with adjacent value ranges and all their other variables have the same value ranges, this pair of subspaces can be merged into one subspace. For example, subspaces 1 and 2 have adjacent Y₄ value ranges and all their other variables have identical value ranges. Therefore, subspaces 1 and 2 can be merged by combining the value ranges of Y₄ to create a new subdesign space. Two subdesign spaces were obtained by merging the five subspaces in Table 5, as shown in

Table 6. Merged subspaces and the effectiveness of design-space reduction.

Subspace	Variable	Y2	Y3	Y4	Y5	X1	Z1	Volume (×10−10)	Percentage
1	Lower bound	0.06	0.097	0.1	0.245	−0.08	0.170	3.287	7.80%
	Upper bound	0.085	0.106	0.145	0.275	−0.06335	0.235
2	Lower bound	0.0719	0.097	0.13415	0.245	−0.06335	0.170	1.580	3.75%
	Upper bound	0.085	0.106	0.145	0.275	0	0.235

. Furthermore, the last two columns in this table show the volume of each subspace and its percentage of the initial design space, indicating that the total volume of the two subspaces is 4.867 × 10⁻¹⁰, i.e., 11.55% of the size of the initial design space. Therefore, there was a significant reduction in the design space.

The nonlinear programming by quadratic lagrangian (NLPQL) method was used to directly search the two subspaces. The convergence graph is shown in Figure 12. A total of 347 CFD calculations were performed, wherein 286 calculations were performed during the optimization and 61 CFD calculations were performed to obtain the initial designs. The solution that minimized the target value was selected as the optimal solution, which is analyzed below.

Figure 13 shows the lines plan before and after optimization. The bulbous bow was extended forward and upward. Furthermore, the surfaces at stations 19 and 20 expanded outward, while the surfaces at stations 17 and 18 became more concaved. Therefore, a ship that has Fr = 0.26, a slightly raised bulbous bow, slightly convexed surfaces at stations 18 and 19, and slightly concaved surface at station 16 can significantly reduce the wave-making resistance.

Based on the dynamic pressure distribution shown in Figure 14, the low-pressure area at the bulbous bow became smaller after optimization, and the maximum pressure decreased. As a result, the pressure gradient across the bow became more uniform. As shown in Figure 15, the wave amplitudes decreased in the first 1/2 of the ship (from the bow) after optimization, indicating that less energy was wasted on wave-making on the optimized ship.

4.3.3. Case 3: RST-Based SDSR

1st reduction: Two subspaces were obtained by reducing the initial set of simulated samples (see

Table 7. Two subspaces obtained from the first reduction.

1st Reduction	Variable	Y2	Y3	Y4	Y5	X1	Z1
Subspace 1	Lower bound	0.06	0.097	0.1	0.245	−0.08	0.170
	Upper bound	0.085	0.106	0.145	0.275	−0.06335	0.235
Subspace 2	Lower bound	0.0719	0.097	0.13415	0.245	−0.06335	0.170
	Upper bound	0.085	0.106	0.145	0.275	0	0.235

). To facilitate comparison, the initial sample set in Case 3 was kept identical to that in Case 2. As a result, the subspaces obtained after one reduction were identical to those in Case 2.

2nd reduction: Uniform design was used to add new ship-type sample points in subspaces 1 and 2 (from the 1st reduction). CFD calculations were then performed to calculate their wave-making resistances and form two sets of sample points with the pre-existing sample points in each subspace. Data discretization, variable simplification, and rule extraction were performed on these sample points to finish the second round of design-space reduction, which resulted in two subspaces (see

Table 8. Two subspaces obtained after the 2nd reduction.

2nd Reduction	Variable	Y2	Y3	Y4	Y5	X1	Z1
Subspace 3	Lower bound	0.06	0.097	0.1201	0.245	−0.08	0.170
	Upper bound	0.085	0.106	0.145	0.2675	−0.06335	0.235
Subspace 4	Lower bound	0.0719	0.097	0.13415	0.245	−0.06335	0.19705
	Upper bound	0.085	0.106	0.145	0.275	0	0.235

).

3rd reduction: First, a set number of sample points were added to subspaces 3 and 4 (from the 2nd reduction), which were then combined with the original sample points in these subspaces to create two sets of sample points. RST was then used to reduce the two subspaces based on these sample point sets. Thus, subspace 3 was reduced and divided into “subspace 5” and “subspace 6”, whereas subspace 4 was reduced to “subspace 7”. The three subspaces obtained from the 3rd reduction are shown in

Table 9. Three subspaces obtained from the 3rd reduction.

3rd Reduction	Variable	Y2	Y3	Y4	Y5	X1	Z1
Subspace 5	Lower bound	0.06	0.097	0.1201	0.245	−0.08	0.2229
	Upper bound	0.085	0.106	0.145	0.2675	−0.06335	0.235
Subspace 6	Lower bound	0.06	0.097	0.1201	0.245	−0.08	0.170
	Upper bound	0.085	0.106	0.145	0.2675	−0.0675	0.2229
Subspace 7	Lower bound	0.0719	0.097	0.13415	0.245	−0.06335	0.19705
	Upper bound	0.085	0.106	0.145	0.275	−0.0196	0.235

.

4th reduction: A set number of sample points were added to the subspaces obtained from the 3rd reduction to create three sets of sample points, which were then combined with the pre-existing sample points in these subspaces to construct three sets of sample points. RST was then used to perform design-space reduction on these sample point sets. During this process, subspace 5 was divided into “subspace 8” and “subspace 9”, while subspaces 6 and 7 were reduced to “subspace 10” and “subspace 11”, respectively. The four subspaces obtained from the 4th reduction are shown in

Table 10. Three subspaces obtained from the 4th reduction.

4th Reduction	Variable	Y2	Y3	Y4	Y5	X1	Z1
Subspace 8	Lower bound	0.06	0.097	0.1201	0.245	−0.08	0.2296
	Upper bound	0.085	0.106	0.145	0.2675	−0.06335	0.235
Subspace 9	Lower bound	0.0662	0.097	0.1201	0.245	−0.08	0.2229
	Upper bound	0.085	0.106	0.145	0.2675	−0.06335	0.2296
Subspace 10	Lower bound	0.06125	0.097	0.1201	0.245	−0.08	0.17
	Upper bound	0.085	0.106	0.145	0.2675	−0.0675	0.2229
Subspace 11	Lower bound	0.0719	0.097	0.13415	0.245	−0.06335	0.19705
	Upper bound	0.085	0.106	0.145	0.269	−0.0196	0.235

.

Table 11. Effect of the design-space reductions and changes in the overlap coefficient with each iteration.

	Initial Space	1st Reduction	2nd Reduction	3rd Reduction	4th Reduction
Number of points	60	22	18	13	——
Number of design spaces	1	2	2	3	4
Volume × 10−10	42.12	4.86758	2.28686	1.72468	1.52069
Number of good samples in the subspaces	——	9	19	31	48
Reduction efficacy	100.00%	11.56%	5.43%	4.09%	3.61%
Overlap coefficient	——	0.1156	0.4698	0.7542	0.8817

shows how the reduction efficacy, overlap coefficient, and other relevant factors changed with each iteration in the SDSR method. Herein, “reduction efficacy” refers to the ratio of the reduced design space to the initial design space in terms of percentage, which is negatively correlated with the overlap coefficient. Figure 16 shows how the overlap coefficient and reduction efficacy changed with each iteration. The overlap coefficient increased with each iteration, and considering the overlap coefficient was already 0.88 by the 4th iteration (i.e., greater than 0.85), no further reductions were performed. Considering the volume of the final design space was only 3.61% of the initial design space, the reduction in the design space was significant.

A total of 113 CFD calculations were performed in Case 3. The ship with the smallest target value among these 113 samples was selected as the optimal ship, and its hydrostatic parameters and wave-making resistance values are shown in

Table 12. Optimal solution for Case 3.

	Y2/m	Y3/m	Y4/m	Y5/m	X1/m	Z1/m	Lcb	$\nabla$/kg	Rw/N	Rt/N
Case 3	0.085	0.106	0.145	0.255	−0.0689	0.235	0.508	1.697	10.999	80.210
Original	0.0736	0.0945	0.1330	0.2604	−0.0026	0.1946	0.509	1.691	12.237	80.476

. Considering the wave-making resistance of the optimal ship was 10.12% lower than the original, we can conclude that the optimization was effective.

Figure 17 shows how the lines plan of the optimal ship differs from the original. In the optimal ship, the bulbous bow extended forward and upward in a “wing-like” shape. The surface at stations 19 and 20 expanded outward, whereas that at station 17 became concaved. Therefore, in a KCS with a ship speed of Fr = 0.26, a slightly raised bulbous bow, slight convexing at the hull surfaces adjacent to the bulbous bow, and some degree of concaving at the surface just behind the bow can help the wave-making resistance.

Figure 18 shows the distribution of hydrodynamic pressures on the bow of the original ship and the optimal ship in Case 3. After hull-form optimization, the low-pressure region at the bulbous bow became smaller and the maximum pressure decreased. As a result, the pressure distribution at the bow became more uniform after optimization. Figure 19 shows the longitudinal wave cuts at different parts of the KCS before and after optimization. The wave amplitude decreased significantly in the first 1/2 of the optimal ship, indicating that less energy was wasted on wave-making in the optimal ship.

4.3.4. Comparative Analysis

All optimal ships from Cases 1–3 (see

Table 13. Comparison between the optimal solutions from Cases 1–3.

	Y2/m	Y3/m	Y4/m	Y5/m	X1/m	Z1/m	Rw/N	Improvement in Wave-Making Resistance	Number of CFD Calculations
CASE1	0.0836	0.1057	0.1411	0.2587	−0.0785	0.2265	11.048	−9.71%	1000
CASE2	0.0850	0.1060	0.1450	0.2602	−0.0633	0.2350	10.976	−10.30%	346
CASE3	0.0850	0.1060	0.1450	0.2550	−0.0689	0.2350	10.999	−10.12%	113

) showed approximately a 10% decrease in the wave-making resistance. Therefore, we can conclude that the effectiveness of hull-form optimization was roughly the same in all three cases. In terms of efficiency, Case 1 required 1000 CFD calculations, the highest of all cases. Case 2 required a total of 346 CFD calculations, considering the optimizations were performed immediately after one reduction. Multiple reductions were performed in Case 3, reducing the total number of CFD calculations to 113 (the lowest of all cases). Therefore, hull-form optimization was least efficient in Case 1 and most efficient in Case 3. Therefore, using SDSR in hull-form optimization gives the same optimization efficacy while significantly reducing the computational cost.

By comparing Cases 1–3, it was observed that the design-space reduction (Case 2) reduced computational complexity compared to direct optimization, allowing for the use of gradient-based optimization methods to directly search for the optimal solution, which significantly reduced the computational cost. However, the one-time space reduction did not provide a natural answer to the number of samples that should be considered. As a result, the samples did not adequately represent the design space, and the reduction was insignificant. Furthermore, the optimization converged to local optima. SDSR, involving multiple reductions, effectively solved all of the problems above and gave smaller optimal design spaces. Therefore, considering SDSR makes hull-form optimization much more feasible and allows for the optimal ship design to be selected without any further optimization, it is considered an excellent method for reducing the computational cost of hull-form optimization.

5. Conclusions

Hull-form optimization is an effective method for improving ship speeds and cost efficiency, making it an important tool for green ship design. With the advent of hull-surface-modification techniques, CFD numerical simulations, optimization techniques, and continuously improving computing hardware, CFD-based hull-form optimization has become very common. However, this approach exhibits significant practical flaws owing to its low computational efficiency and difficulty in identifying the global optimum. To solve these issues, this study proposed the RST-based SDSR technique that uses interval theory to compute the intersections and unions of the subdesign spaces. Furthermore, a suitable stopping criterion for hull-form optimization was proposed to facilitate the application of RST-based SDSR to hull-form optimization problems. Lastly, a KCS model was optimized using our method to minimize the wave-making resistance. The results were then compared to those of direct optimization and one-time design-space reduction, thus proving the feasibility of our method. Cases 1–3 showed approximately a 10% decrease in the wave-making resistance. In terms of efficiency, direct optimization (Cases 1) required 1000 CFD calculations. One-time design-space reduction (Case 2) required a total of 346 CFD calculations. Multiple reductions (Case 3) reduced the total number of CFD calculations to 113 (the lowest of all cases). Therefore, hull-form optimization was least efficient in Case 1 and most efficient in Case 3. Therefore, using SDSR in hull-form optimization gives the same optimization efficacy while significantly reducing the computational cost.