The design and optimization of ship hull forms play a crucial role in enhancing the performance and efficiency of marine vessels. This study focuses on integrating a cylindrical central body part within a conventional ship hull to explore its impact on hydrodynamic characteristics and overall vessel performance. The research employs hydrodynamical concepts, parametric studies, and optimization algorithms to analyze the design space systematically. The aim of including the cylindrical central body is to investigate its influence on reducing resistance, improving fuel efficiency, and enhancing maneuverability. A new optimization model based on the cylindrical body inclusion in the hull form is developed. The existing generalized reduced gradient (GRG) optimization method is also adopted to determine the accuracy of the proposed methodology. It is revealed that the resistance predicted by the GRG method is much closer to the original result of the parent hull form. A container vessel is taken as a case study example. The new, simplified, approach developed here provides a greater reduction in the resistance values of the case study vessel. Hence, the adoption of a cylindrical hull form in ship design can improve hydrodynamic performance. Although the results from the GRG method and the new scheme agree within the speed range of 0–5 m/s, some deviations are noted. In conclusion, it is observed that although the inclusion of the cylindrical body together with the adoption of the optimum design scheme is capable of improving the resistance performance of a ship, further studies are necessary to understudy the effects of this approach on the other vessel performance metrics.
I. INTRODUCTION
The hull of a ship is its most noticeable structural entity. It acts as a watertight enclosure for the cargo, machinery, and accommodation spaces, protecting them from weather, flooding, and structural damage. Although this might be a non-technical or layman’s understanding of a ship’s hull, it does not encompass all of its aspects. The hull of a ship significantly influences its overall performance, efficiency, and stability. Indeed, achieving the ideal hull form in the early part of the design process is essential, since alteration during later stages of design and subsequent construction can result in significant costs and consequences.1
One of the hydrodynamic performance characteristics influenced by hull form is ship resistance. Effiong Udo and Daniel2 conducted a resistance simulation of a roll-on, roll-off passenger (RoPax) ship using an empirical approach. A computational fluid dynamics (CFD)-based numerical model has recently been used in simulations to characterize the effects of fluid–structure interaction on ship resistance for different ship hull forms.3 The resistance of a bulk carrier was simulated by comparing its optimized hull form and parent hull form, and it was concluded that a significant reduction in resistance is possible, taking account of the overall nature and size of the hull.4 Adding certain features to hull forms can also improve the resistance performance of ships. In this context, a fishing vessel has been used as a case study, and observations indicated that adding a bulbous bow to the forward part of the vessel reduced its resistance by 2%.5
From a historical perspective, there have been tremendous innovations in ship design, with a great focus on optimization of hull form. These are related to the definition and determination of mathematical hull forms initiated by Admiral David W. Taylor before World War 1. The use of analytical models to describe hull forms indicated that water lines and sectional area curves took the form of separate fifth-order curves for forebody and afterbody.6 Despite significant advances in ship hull design, however, there remains a gap in understanding the optimal integration of a cylindrical central body for enhanced hydrodynamic performance. Previous research has explored various aspects of hull optimization, but a comprehensive analysis of the impact of a cylindrical central body on factors such as resistance and stability is notably lacking.
The problem addressed in this present work is the suboptimal performance and limitations associated with conventional ship hull designs. Such designs often lead to inefficiencies in hydrodynamic performance, potential compromises in structural integrity, and challenges in maintaining stability under varying operational conditions. These limitations hinder the overall effectiveness and safety of maritime vessels. The need arises for a novel approach to ship hull design that can simultaneously enhance hydrodynamic efficiency, ensure structural robustness, and improve stability.
Several other related studies have been carried out previously. Makris and Grigoropoulos7 investigated hull form optimization procedures to improve hydrodynamic performance. Another study was carried out by Özmen,8 who analyzed the seakeeping performance of fishing vessels, with the aim of formulating a relationship between the geometrical characteristics of a series of hull forms and the seakeeping qualities of the corresponding vessels. In addition, he established a statistical relationship between these hull forms and the added resistance and calm water resistance characteristics. Wilson et al.9 carried out a hull form optimization in the context of early-stage ship design. Other works along the same lines to the present study include those on the design of surface fairings for ship hulls by Sariöz10 (who used the B-spline curves and surfaces that have been employed extensively to define hull geometries) and by Koelman.11
The starting point for the design of a ship is a given set of requirements concerning the ship’s type, speed, payload, range, and operating conditions. The overall design task is determined when the design’s definition embraces both the needs of the customer and the designer’s criteria with regard to technical acceptability. Moving forward in the design process, the scope for variation eventually decreases, while at the same time, more information becomes available about the design.
The stability of a ship’s hull is one of the critical design points that affect the overall stability of the vessel. As ships vary in displacement owing to variations in load, the corresponding variations in the vertical position of the center of gravity are of great significance, and vessels need a large ballast tank and hull to compensate for these variations. In performing a preliminary stability analysis, an estimation of the lightship displacement and the corresponding center of gravity is necessary.12
This has brought about the idea of including a cylindrical body in a vessel for purposes of stability, maintenance of hydrostatic pressure, and withstanding the maneuvering conditions experienced on the sea. If a ship has insufficient stability, certain operating scenarios could result in a disaster. That is one of the reasons why Sadik and Bekir13 emphasized that even with adequate stability, operations in beam seas or following seas must be carried out with care. Water can be shipped on deck more readily when a vessel operates in beam seas. Also, if the frequency of roll induced by beam seas is the same as the natural frequency of roll, the ship could roll beyond its range of positive stability, resulting in capsize.
This work aims to contribute to the development of mathematically defined hull forms of vessels for ship performance assessment. Its objectives are as follows:
To establish mathematical equations for the definition of hull forms.
To implement and simulate the models developed for assessment.
To investigate existing hull forms and their performance characteristics.
To compare the performances of existing hulls with those provided by the mathematically definitions.
To investigate the impact of hull form on hydrodynamic performance.
II. RESEARCH METHODOLOGY
A. Parameterization of hull form
The hull form is first parameterized into three segments: forebody, mid-body, and afterbody. This helps to decide the parts for the optimization in which the cylindrical body is included. A schematic of the design water plane is shown in Fig. 1. Here, the mid-body length LMB has been chosen to be greater than either of the lengths of the forebody LFB and afterbody LAB. The sum of the lengths of the two end parts is varied with the mid-body length. These variations are illustrated in Table I.
Variation of the lengths of the forebody, afterbody, and mid-body of the model.
| Variation of the sum of lengths | LFB | LAB | LMB | Sum |
|---|---|---|---|---|
| LFB + LAB = LMB | 0.75 | 1.5 | 0.75 | 3 |
| LFB + LAB < LMB | 0.5 | 2 | 0.5 | 3 |
| LFB + LAB > LMB | 1 | 1 | 1 | 3 |
| Variation of the sum of lengths | LFB | LAB | LMB | Sum |
|---|---|---|---|---|
| LFB + LAB = LMB | 0.75 | 1.5 | 0.75 | 3 |
| LFB + LAB < LMB | 0.5 | 2 | 0.5 | 3 |
| LFB + LAB > LMB | 1 | 1 | 1 | 3 |
B. Modeling of hull form using cylindrical equations
These mathematical models have been developed by including the cylindrical body part in the mid-body part of the hull. Continuity theory has been incorporated to ensure the absence of discontinuity in the hull. The models are segmented in terms of the lengths of the three body parts.
C. Hydrodynamic performance analysis
D. Optimization design of hull form for resistance performance analysis
E. GRG method
The GRG method is an optimization technique for solving nonlinear programming problems.15 In the present problem, resistance optimization is performed by systematically adjusting the mid-body, afterbody, and forebody while satisfying the constraints (14)–(16). The gradients are used as guides to search for the optimal solutions.
F. Demonstrative case study
The hull form used is presented in Fig. 2. A case study hull form was developed in DELFTship modeling software. Table II lists the principal parameters of this vessel, a container ship. Some of these basic parameters are used in the simulations based on the mathematical equations established above.
Principal parameters of case study ship.
| Parameter | Value |
|---|---|
| Vessel type | Container |
| Length overall LOA | 90 m |
| Length between perpendiculars | 85 m |
| Beam | 13.75 m |
| Draft | 4.2 m |
| LCB | 42.311 m |
| Cb | 0.7 |
| Cp | 0.7264 |
| Parameter | Value |
|---|---|
| Vessel type | Container |
| Length overall LOA | 90 m |
| Length between perpendiculars | 85 m |
| Beam | 13.75 m |
| Draft | 4.2 m |
| LCB | 42.311 m |
| Cb | 0.7 |
| Cp | 0.7264 |
Scaled-down parameters of model.
| Parameter | Value |
|---|---|
| Scale λ | 30 |
| Length overall LOA | 3 m |
| Length between perpendiculars | 2.83 m |
| Beam | 0.46 m |
| Draft | 0.14 m |
| LCB | 1.41 m |
| Cb | 0.7 |
| Cp | 0.7264 |
| Parameter | Value |
|---|---|
| Scale λ | 30 |
| Length overall LOA | 3 m |
| Length between perpendiculars | 2.83 m |
| Beam | 0.46 m |
| Draft | 0.14 m |
| LCB | 1.41 m |
| Cb | 0.7 |
| Cp | 0.7264 |
The mathematical models and optimization schemes discussed in this work were programmed in Excel, while the resistance and power of the parent hull form were generated in DELFTship.
III. RESULTS AND DISCUSSION
A. Resistance of case study vessel
The hydrodynamic result for the case study parent hull form is presented in Fig. 3, from which it can be seen that the resistance increases with the speed of the vessel. Also, there is a corresponding increase in the power requirement of the ship. It is generally observed that an increase in a ship’s speed leads to an increase in its power characteristics up to the third power of the speed. However, the present result does not agree with this general observation. It is, rather, in agreement with an experimental study investigating the prediction of speed–power relationship where data of about 50 000 noon reports from 88 tank vessels in operation were used.17 It was revealed that the effective power was less than the third power of the operational speed in the speed range below the design speed.17 This is demonstrated in Fig. 3 by the dotted green curve representing a second-order dependence. The dotted red curve represents the predicted third-order dependence of the resistance. This corresponds to the power prediction (shown in Sec. III C).
B. Resistance of optimized hull
The proposed optimization scheme using a cylindrical model and the GRG nonlinear solver results in a new approach that can be developed further to ensure accuracy and robustness in the prediction of resistance of a new ship. The results obtained with these methods are presented in Fig. 4. It can be observed that the GRG performs far better than the present authors’ method. However, they both tend to agree in the speed range below 5 m/s. This is because the authors’ models involve a simplification that requires further development to ensure robustness. The results also reveals that the authors’ third model performs better than the first and second models because it contains a greater part of the hull form that has been optimized with the inclusion of the cylindrical model.
C. Power prediction of parent hull form
It can be seen from Fig. 5 that the power curve changes from a quadratic curve to a higher-order curve at a speed of 8 m/s. This indicates the ship’s operational limit. As already discussed in Sec. III A, it can be seen that the parent hull form originally required about 600 kW at a speed of 8 m/s. Above this speed, the power increases drastically, with severe implications for the cost of operation.
D. Power characteristics of optimized hull form
Figure 6 illustrates the speed–power curve of the optimized hull form. As can be seen, the GRG predicted the effective power at 8 m/s to be about 489 kW. This is a huge reduction compared with the experimental result, as shown in Fig. 5.
E. Comparative characterization of case study vessel and optimized hull form based on resistance performance
Figure 7 indicates a significant reduction in resistance according to the GRG solver and the authors’ models compared with the original data, owing to the inclusion of the cylindrical body part.
Resistance performance prediction of optimized hull using the GRG method, the authors’ models, and original resistance data.
Resistance performance prediction of optimized hull using the GRG method, the authors’ models, and original resistance data.
F. Comparative characterization of case study vessel’s hull form and optimized hull form based on power prediction
In a similar vein to the resistance discussed above, Fig. 8 shows that the power requirement of the vessel is drastically reduced when the methodology developed in this paper is adopted. It further indicates a certain level of accuracy near the origin. There is, however, a dramatic deviation after 3 m/s. This can be attributed to the fact that the predicted power varies approximately as the third power of the speed, which will magnify the error terms in the GRG scheme.
Power performance prediction of optimized hull using GRG and original hull power characteristics.
Power performance prediction of optimized hull using GRG and original hull power characteristics.
IV. CONCLUSION
Hull form design has long played an essential role in the design process of a ship, and will continue to do so, with future developments aiming to simplify this complex task. The hull form has significant effects on the overall performance of a ship, with resistance, propulsion, maneuvering, and seakeeping all representing hydrodynamic characteristics that require accurate predictions. Moreover, the development of innovative ship hull forms can help achieve optimum design requirements.
This work has adopted a novel optimization approach that has led to the achievement of a great reduction in the resistance of an illustrative case study vessel. This in turn has translated into an improvement in power demand. The generalized reduced gradient (GRG) method has also been applied in this work as a solver to verify the effectiveness of the proposed approach. A continuity test has been conducted to ensure the continuity of the curved surface of the hull. This work has thus made an important contribution to ship design and has laid the foundation for further work considering additional performance metrics.
The following recommendations are made for further study:
The method developed in this work should be further investigated to assess its viability for practical application in design.
The problem of discontinuity should be further investigated.
The development of a hybrid hull form with a mix of cylindrical parts should be considered.
The employment of the GRG solver in hydrodynamic characterization has not been fully explored, and high-fidelity results could perhaps be achieved via review, implementation, and comparative analysis of new and recent optimization algorithms.
ACKNOWLEDGMENTS
The authors wish to acknowledge their various affiliates for providing an enabling environment for this work.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Anietie Effiong Udo: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Methodology (equal); Project administration (equal); Software (equal); Supervision (equal); Writing – review & editing (equal). Charles A. N. Johnson: Methodology (equal); Supervision (equal); Writing – review & editing (equal). John Pius Archibong: Conceptualization (equal); Methodology (equal); Resources (equal); Visualization (equal).
DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.
REFERENCES
