Research on ship design and optimization based on simulation based design (sbd) technique

Signi fi cance of Ship Form Design and Optimization Based

In the post-financial crisis era, the shift towards a low-carbon economy has significantly transformed ship design concepts, prioritizing overall navigation performance over traditional ship-type optimization aimed at minimizing hydrostatic drag This evolution aligns with the growing emphasis on "green ship" design and construction, highlighting the urgent need for sustainable practices in the maritime industry.

The ship product development process, guided by SBD technology, emphasizes resource conservation and environmental sustainability in the shipbuilding industry The International Maritime Organization's Energy Efficiency Design Index (EEDI) introduced in 2014 mandates that future ship designs prioritize safety, environmental friendliness, cost-effectiveness, and comfort As a result, energy saving and emission reduction have become central themes in ship design The development of energy-efficient "green" ship hull forms is a key strategy for minimizing total resistance, fuel consumption, and carbon emissions SBD technology focuses on optimizing the ship's sailing performance by integrating Computational Fluid Dynamics (CFD) for numerical evaluation with ship body geometry reconstruction, aiming for optimal navigation performance under specific constraints The evolution of hull types, particularly the "green" hulls designed for minimal resistance and energy consumption, marks a shift from traditional design practices to more intelligent and knowledge-based approaches While our country has emerged as the largest shipbuilding nation globally, it still lags behind leading shipbuilding powers.

“Green Ship” design and development has seriously affected the strategic trans- formation of China’s shipbuilding industry In thefierce market competition, it can

To ensure the survival of shipbuilding enterprises, it is crucial to establish an effective green ship design method This requires a swift breakthrough in key technologies related to shipboard SBD design and the development of an optimized design system that integrates the navigation performance of real ships By focusing on innovations with independent intellectual property rights, we can significantly enhance the independent innovation capabilities of our shipbuilding industry.

Fig 1.3 Carbon emissions of the total life cycle of a bulk carrier

The Key Technology of Ship Form Optimization

Numerical Simulation Technology

Based on the hydrodynamic theory, ship-type optimization requires that the flow

The optimization of ship design relies heavily on the scientific principles of ship hydrodynamics, which integrates fluid mechanics theory to enhance performance Accurate predictions of hydrodynamic forces are crucial for guiding optimization algorithms in ship design Fluid mechanics theories for ship resistance are categorized into potential flow theory—further divided into linear and nonlinear—and viscous flow theory, with the latter primarily utilizing Computational Fluid Dynamics (CFD) methods Effective CFD analysis requires careful attention to grid forms and meshing techniques, especially given the ship's variable attitudes in six degrees of freedom Traditional CFD software struggles with these challenges, but the introduction of Overset Grid technology addresses these limitations This book focuses on ship-type optimization using the RANS method to develop a numerical framework for minimizing hydrostatic and wave resistance.

Hull Geometry Reconstruction Technology

Ship geometry reconstruction technology serves as a crucial link between evaluating ship resistance performance and optimizing ship form Utilizing hydrodynamic theory, particularly the CFD method, reveals that the relationship between the objective function—minimizing total resistance—and design variables is often implicit Establishing this connection is essential for effective ship form optimization The process begins by parameterizing the hull geometry with a limited number of parameters, followed by defining the relationship between these parameters and the design variables Subsequently, optimization algorithms are employed to adjust the design variables, leading to changes in the ship hull geometry through reconstruction techniques A significant challenge in optimizing ship hull forms is creating an effective optimization platform that balances fewer design variables with a broader hull configuration space This book thoroughly explains the foundational theory behind hull geometry reconstruction techniques, including hull form modification functions and the ASD free-form surface deformation method.

Hull geometry reconstruction technology is categorized into two main types based on ship parameters: the ship parameter method and geometry modeling technology The ship parameter method utilizes various ship characteristics, such as scale ratio and longitudinal center of buoyancy, employing techniques like the Lackenby transform and parametric modeling Conversely, geometry modeling technology focuses on reconstructing hull geometry through control point position adjustments, utilizing methods such as Bezier Patch, free deformation (FFD), and ASD Effective hull geometry reconstruction necessitates a broad geometric deformation space with minimal design variables to generate diverse geometries while maintaining smoothness Recently, several software solutions have emerged that facilitate parametric modeling of ship forms, enhancing design efficiency without requiring extensive programming Notable CAD software, including UG, Pro-E, and CATIA, support hull geometry reconstruction through advanced development technologies Additionally, Friendship, a specialized parameter modeling software for ship optimization, integrates with SHIPFLOW to enable rapid hull optimization and is widely adopted in the industry.

Approximate Technology

Ship form optimization based on hydrodynamic theory is a complex interdisciplinary process that requires consideration of multi-disciplinary coupling, design variables, and nonlinear constraints This optimization often necessitates multiple iterative computations of the objective function, particularly when using high-precision solvers like CFD, which can be time-consuming and impractical for rapid optimization To address the challenges of extensive numerical calculations in ship form optimization, two primary approaches exist: enhancing computer hardware for high-performance computing, which can be cost-prohibitive, and employing approximation techniques The latter, which integrates experimental design, mathematical statistics, and optimization methods, allows for the simulation of design spaces and the derivation of implicit objective function expressions Approximation techniques, such as the response surface method (RSM), variable fidelity model (VFM), kriging model, and radial basis function (RBF), significantly reduce computational workload and costs Additionally, ISIGHT, a multi-disciplinary optimization platform, offers a variety of approximation techniques for researchers' convenience.

Optimization Method

The traditional gradient-based optimization algorithm exhibits significant limitations in ship linear optimization design, particularly due to the complex interplay of performance indicators like speed, weather resistance, and maneuverability, which lack analytical expressions Consequently, obtaining gradient information through numerical analysis becomes costly, leading to challenges in solving strong nonlinear problems typical in ship-type optimization These methods often converge to local optima, heavily influenced by initial point selection In contrast, modern optimization algorithms, such as genetic algorithms, simulated annealing, and particle swarm optimization, offer robust global search capabilities but struggle with local search efficiency, resulting in increased computational demands To address these challenges, it is essential to integrate various optimization techniques, creating a hybrid global optimization algorithm that combines the strengths of genetic algorithms with nonlinear programming and simulated annealing Additionally, the high computational workload associated with CFD-based ship optimization necessitates the development of effective strategies to mitigate response time and costs, a critical issue in ship-type optimization utilizing SBD technology This book explores ship form optimization using nonlinear programming, genetic algorithms, and niche genetic algorithms based on potential flow theory, alongside the application of BP neural networks and improved particle swarm algorithms for unsteady RANS method-based ship design optimization and navigation control.

Integrated Set Technology

Ship-based optimization utilizing hydrodynamic theory involves a systematic integration of CAD, CFD, optimization methods, and computer network technologies to create a unified interface optimization platform, crucial for automating the optimization process Two primary integration methods exist: the first requires strong programming skills for individual module development, which can be challenging and time-consuming for most users The second method employs established optimization platforms like ISIGHT and OPENFOAM, with ISIGHT being a mature choice widely adopted by researchers For example, Liu Zu-yuan from Wuhan University of Technology successfully integrated SHIPFLOW software with ISIGHT to enhance ship hydrodynamic design OPENFOAM, an open-source CFD tool, allows for secondary development, exemplified by the naoe-FOAM-SJTU project from Shanghai Jiao Tong University, which utilizes FORTRAN to optimize ship designs based on wave resistance principles This research validates theoretical optimization results through model testing and aims to establish a minimum hydrostatic resistance optimization platform By integrating CFD resistance calculations and CAD geometric reconstruction within the ISIGHT platform, the study also explores ship optimization related to hydrostatic and wave resistance, as well as navigational control optimization during actual navigation, thus contributing to the framework of ship-based design and real navigation systems.

Basic Method of Hull Line Optimization

This book employs theoretical optimization and experimental verification through potential flow theory to examine the minimum hydrostatic and wave resistance in ship design It utilizes the Michell integral and Rankine source methods to calculate wave and flat frictional resistance, along with hydrostatic total resistance, using the unsteady RANS method as the objective function The study incorporates control parameters, such as ship modification functions and ASD free-form surface deformation methods, reflecting hull shape changes, while adhering to displacement constraints and additional restrictions By integrating nonlinear programming, genetic algorithms, and niche genetic algorithms, it explores ship optimization based on potential flow theory Furthermore, it combines BP neural networks, Elman neural networks, particle swarm optimization, and improved particle swarm algorithms to address unsteady RANS method-based design optimization and optimal pitch navigation challenges A hull linear optimization design program with independent intellectual property rights is developed, focusing on four typical ship types: Wigley, S60, DTMB5415, and KCS container ships, to derive theoretical linear optimal designs for practical navigation optimization The theoretical optimization's validity is confirmed through model testing, establishing a ship-based SBD design framework and a real-frame optimization design grounded in hydrodynamic theory.

The Michell integral method facilitates ship form optimization by allowing direct input of the ship type into the objective function In contrast, optimizing wave resistance using the Rankine source method poses challenges due to the implicit relationship between the objective function and design variables To address this, a ship modification function is employed to link the objective function with design variables, enabling effective hull form optimization Additionally, ship-type optimization based on the RANS method necessitates model parameterization and automatic mesh division following reconstruction This book discusses hull form optimization through Computational Fluid Dynamics (CFD), utilizing the ASD free-form surface deformation method for hull geometry reconstruction.

The numerical evaluation of hydrodynamic theory significantly influences the reliability of optimization results This book utilizes the Michell integral method and the Rankine source method for calculating wave resistance, achieving high accuracy for slender and high-speed ships, which is essential for meeting engineering standards Given the substantial wave resistance in these vessels, the focus is on minimizing wave resistance to enhance practical optimization design The methods employed ensure rapid calculations, allowing for swift identification of optimal ship forms, as demonstrated through examples like the Wigley ship, S60 ship model, and DTBM5415 ship model While the RANS method, rooted in viscous theory, has proven its accuracy through engineering practice, evaluating ship resistance using CFD methods remains challenging and time-consuming, albeit more precise, particularly for unconventional ship types Nevertheless, advancements in computer technology have made CFD-based ship optimization increasingly feasible through the use of workstations and servers.

This book presents a numerical tank developed using the unsteady RANS method, incorporating the VOF technique to accurately capture free surface dynamics, while employing the k-e two-equation turbulence model The control equations are discretized using a volumetric center finite difference scheme, with a second-order Euler backward difference format for time terms, a second-order hybrid finite difference format for convection terms, and a second-order central difference format for viscoelastic flow The SIMPLE algorithm effectively solves the RANS and continuity equations Ultimately, the numerical calculation method for hydrostatic (wave) resistance and ship motion response is derived based on the RANS method, utilizing specified boundary and initial conditions alongside the six degrees of freedom equations of motion.

For the numerical analysis of ship wave resistance and motion response, automatic mesh generation and free surface simulation are crucial technologies This book employs an overlapping grid technique to model six degrees of freedom in ship movement, utilizing a single VOF method for free surface simulation The hull and watershed areas are structured using overlay grid technology for effective grid layout and calculation grid generation A stationary Cartesian rectangular grid is implemented to maintain free surface grid resolution, while the hull features a dynamic body-fitted grid with a unified topology for both the prototype and deformed grids As the hull moves, the relative position of the hull mesh and background mesh changes, affecting the overlapping grid area and necessitating flow field information exchange at each time step Additionally, the identification of cave boundaries and contribution units is essential at every time step, highlighting that enhancing the computational efficiency of dynamic structure overlay grids relies on effective cave boundary establishment and contribution unit search.

(3) Optimization algorithms and approximation techniques

This book explores the minimum wave resistance and minimum total resistance ships using the Michell integral and Rankine source methods, employing nonlinear programming, genetic algorithms, and niche genetic algorithms for optimization It utilizes real physical models as objective functions without approximation techniques Additionally, the study investigates minimum hydrostatic resistance and minimum wave resistance ship types through advanced methods such as the RANS approach, BP neural networks, Elman neural networks, and improved particle swarm optimization.

To achieve optimal design for minimum resistance in ships and enhance navigation control through CFD, it is essential to utilize approximate models rather than complex physical models due to the intricacies involved in ship model optimization.

Ship optimization utilizing the Michell integral method and Rankine source method is achieved through self-programming for comprehensive calculations In contrast, the RANS method requires the integration platform ISIGHT, which combines various modules to facilitate hull form optimization This process involves constructing frameworks for hydrostatic resistance design and minimizing wave resistance, both based on Computational Fluid Dynamics (CFD) Ultimately, by selecting the optimal navigation pitch as a design variable, a ship design framework is established to enhance overall navigation performance.

Research Progress of Ship Form Design and Optimization

Ship Form Optimization Based on Potential

Optimizing hull shape is a complex and crucial design technique in shipbuilding, directly impacting a vessel's speed, maneuverability, and seakeeping performance Determining the ideal hull shape that minimizes resistance is essential for achieving efficient propulsion and reducing energy consumption By finding the optimal hull lines, ship designers can significantly enhance a vessel's overall performance, leading to improved rapidity, reduced emissions, and enhanced operational efficiency.

Ship designers aim to optimize ship line design, traditionally relying on the experience of designers and extensive model testing, which is time-consuming and costly The target-driven design approach focuses on four key factors: the objective function (such as minimizing resistance), design variables that link the ship to the objective function, constraints defining the optimization range, and various optimization methods (including traditional, modern, and hybrid techniques) to achieve optimal ship geometry With advancements in fluid mechanics and computer technology, hull optimization based on hydromechanics is becoming increasingly feasible While countries like Europe, the USA, Japan, and South Korea have successfully integrated these advancements into their ship designs, many ship design departments in other regions still use outdated empirical methods, hindering the growth of the shipbuilding industry.

Fig 1.4 Comparison of the ship design modes

To enhance the competitiveness of China's shipbuilding industry and improve its enterprises' capabilities, it is crucial to develop hull line optimization design software that incorporates independent intellectual property rights This involves rapidly generating hull designs with superior drag resistance Research on ship-type improvement and optimization design, based on linear wave resistance theory, has been conducted both domestically and internationally Notably, in 1967, Bian Bao-qi, a Chinese-American scholar, introduced the groundbreaking theory of applying wave resistance principles to hull alignment design, which garnered significant global attention.

Japanese scholars such as Rongchang Farm, Di Xiao Xing, and Masao Matsui have advanced the application of waveform analysis in ship design since the 1970s Their methodologies integrate ship planning with thin-walled theory and waveform analysis data Notably, Rongmai and Matsui's approach requires data from just one ship model, while Diochangxiao's method necessitates model optimization and multiple ship test data, focusing on enhancing the latter half of the hull Additionally, Canadian researcher CC Hsiung introduced the Tent function to represent hull surfaces and utilized the Michell integral method to calculate wave resistance, framing the wave resistance as a function of ship type to develop a quadratic programming problem for optimizing ship designs.

In China, Ye Heng-kui pioneered the optimization of minimum wave-making resistance ships using linear wave theory, calculating wave resistance through the Michell integral method and employing C.C Hsiung’s Tent function for hull representation This approach simplifies constrained optimization into an unconstrained problem via the mixed penalty function method, making it user-friendly and widely applicable Xia Lun-xi and Liu Ying-zhong introduced the equivalent thin-vessel concept to enhance ship design, while Huang De-bo utilized a unit tent function within the Michell integral to successfully optimize high-speed ship designs Pan Zhong-qing and Du Shao-qiu applied the Mathieu function to express wave resistance through area curve integration, improving ship design under specified conditions Additionally, Zhang Xuan-gang and Dou Shao-qiu addressed limitations of the tent function by employing B quadratic spline functions for high-speed catamaran wave resistance calculations, and Shi Zhong-kun and colleagues integrated linear wave theory with waveform analysis, using nonlinear programming for high-speed ship optimization.

Ma Kun and Tanaka Ichiroa conducted extensive research on optimizing ship designs to minimize resistance, focusing on wave resistance theory Their studies aimed at developing ships with the least wave-making resistance and the lowest total drag, while controlling tail viscosity separation as a constraint This approach not only simplifies model calculations and enhances operational speed but also accounts for tail viscosity effects Additionally, researchers such as Ji Zhuo-Shang, Lin Yan, and Huang Qing have contributed significantly to hull type optimization based on linear wave resistance.

Recent research primarily employs the Michell integral method and waveform analysis within linear wave resistance theory, alongside nonlinear programming for optimizing ship bow designs However, the simplified assumptions regarding surface and free surface conditions in linear theory lead to significant discrepancies between calculated and experimental results, particularly overestimating the "peak" and "valley" values of wave resistance Despite these limitations, linear wave-making resistance theory remains valuable for qualitatively assessing hull profile advantages, especially for medium-speed and high-speed vessels, and for guiding improvements in ship design.

In the 1990s, the Rankine source method, combined with optimization technology, became a popular approach for reducing ship resistance A notable contribution came from Japanese scholar Suzuki Kazuo, who developed a ship form optimization method that utilizes the Rankine source method for wave resistance calculations and employs a double trigonometric series for hull modification His sequential quadratic programming technique demonstrated significant improvements, notably reducing the resistance of HTC-type container ships by approximately 16% after iterative design adjustments This research highlights the theoretical and practical significance of optimizing ship forms using the Rankine source method Additionally, Chinese research institutes have made substantial advancements in hydrodynamic performance forecasting and ship shape optimization, with Cheng Ming-dao and Liu Xiao-dong achieving notable results in stern sealing plate optimization, while Chen et al and Liu et al enhanced hull layout optimization for trimarans and container ships using the linear wave numerical method.

The Rankine source method faces significant challenges due to its lengthy calculation times, primarily because it cannot directly represent hull and wave resistance, necessitating repeated calculations during optimization iterations To address this issue, Akawa Hiroshiki leveraged the similarity between modified ship types and the original design, allowing for a single grid construction that significantly reduces computational time Additionally, MARIN's Raven implemented a strategy of increasing surface area and iterative problem-solving In the early 1990s, the development of RAPID, a numerical software for optimizing ship wave resistance, effectively tackled the nonlinear potential flow problem, aiming to minimize wave formation and resistance Despite advancements in nonlinear wave resistance theory, utilizing the Rankine source method remains a practical approach for optimizing ship design focused on resistance performance.

Many ship shape optimization studies utilize nonlinear programming methods, often resulting in unconventional shapes lacking practical significance However, understanding how changes in hull lines affect resistance performance is crucial for guiding design modifications and establishing clear optimization directions To develop a more practical model, it is essential to incorporate appropriate constraints based on insights from the refinement process, allowing for a re-optimization that yields a ship profile closer to real-world applicability.

This book presents an advanced hull line design method focused on enhancing resistance performance through potential flow theory, aimed at creating hull line optimization software with independent intellectual property rights The optimization process encompasses the entire underwater hull profile, including both the front and rear sections By integrating wave resistance theory, viscous flow theory, optimization techniques, and CAD technology, the study develops a comprehensive optimization design program for ship lines The primary goal of the optimization calculations is to minimize wave resistance by controlling tail separation as a constraint The hull shape of the designed vessel is defined using the mother ship value and a ship modification function, employing a mature nonlinear programming approach for optimization Additionally, the Rankine source method is utilized for precise wave resistance calculations.

Optimization of Ship Form Based on Viscous Flow

With the rapid development of computationalfluid dynamics (CFD) and CFD, it is possible to develop high-performance hulls quickly and efficiently [37,38] In the

In the field of ship rapid performance calculation, Computational Fluid Dynamics (CFD) has emerged as a revolutionary method for designing and optimizing ship profiles, gaining significant traction internationally This approach significantly reduces the time and costs associated with traditional ship development, which typically involves model creation and tank testing Additionally, CFD enhances the performance of ship models and minimizes reliance on physical model test tanks The effectiveness of CFD in assessing a ship's hydrodynamic performance largely hinges on the quality of its mathematical models While CFD does not entirely replace model testing, it effectively narrows the testing scope and offers valuable insights for line optimization However, evaluating a ship's resistance using CFD can be time-consuming, complicating practical applications in resistance optimization Consequently, when implementing CFD for ship form optimization, it is primarily utilized to analyze and select the most promising ship-type designs from pre-generated options.

The design of the SBD ship utilizing the RANS method and optimizing real ship navigation represents a novel research direction, fueled by advancements in CFD theory, CAD technology, and optimization methods, alongside significant boosts in computing speed and storage capacity This approach transcends traditional CFD optimization schemes, enabling performance-driven design and transitioning ship design from conventional experience-based methods to knowledge and intelligence-driven processes grounded in numerical simulation technology.

After more than two decades of development, the importance and superiority of this technology have drawn the great attention of all countries They have invested a

Hull geometry reconstruction Based on CAD

Numerical calculation of hydrodynamic performance Based on CFD interfaces

A comprehensive optimization process for hydrodynamic performance utilizing CFD technology has led to significant advancements in research, requiring substantial investments in manpower, materials, and financial resources Numerous achievements have been documented, notably by international experts such as Professor Campana.

At the INSEAN pool in Italy, researchers have integrated optimization theory with CFD technology to enhance ship hydrodynamic performance using SBD technology In recent years, Prof Campana, Peri, and their team have conducted extensive research on optimizing the resistance and seakeeping of ships through SBD technology Their systematic investigation encompasses hull geometry reconstruction, approximation techniques, optimization strategies, and integrated technologies Notably, they developed the disturbance surface method utilizing Bezier Patch surfaces to facilitate hull geometry reconstruction and employed variable approximation techniques to improve design efficiency.

This article discusses advancements in ship design optimization through the use of various technologies and methodologies A fidelity model is employed to address multi-objective optimization problems, establishing a ship-based design framework that leverages SBD technology, which has been validated through model tests resulting in an improved ship model with superior hydrodynamic performance Researchers from the University of Tokyo, including Professor Tahara, utilized NAPA software for ship parametric expression and FlowPack for analyzing propulsion and handling performance In Germany, Professor Harries and his team developed the Friendship CAD software for hull modeling, integrating it with the Mode Frontier optimization platform to achieve multi-disciplinary optimization of hydrodynamic performance Additionally, Ho-Hwan implemented a self-compiled RANS method and parametric modeling for hull geometric reconstruction, utilizing Sequential Quadratic Programming and Particle Swarm Optimization for calculations Gregory combined Friendship-Modeler with SHIPFLOW to explore ship form optimization using Genetic Algorithms Zalek provided a comprehensive review of international research in ship optimization, focusing on the fastness and seakeeping indices Furthermore, Peri, Tahara, and Campana applied Multi-Objective Genetic Algorithms and PSO for high-speed catamaran design, employing FFD and CAD methods for hull reconstruction, with numerical calculations conducted using CFDSHIP and verification through model tests Kim et al analyzed the hydrodynamic performance of LNG vessels using SHIPFLOW and optimized designs through model tests, showcasing the effectiveness of integrated platforms in ship design.

Soonhung et al proposed a fairing B-spline parameter curve for the geometric parametric expression of ship hulls, successfully reconstructing the hull geometry of an LPG ship through a conversion function He Jim developed a multi-disciplinary optimization design model focusing on ship resistance, seakeeping, and maneuverability, with resistance performance as the primary objective function and the other factors as constraints Vasudev established a multi-objective optimization design framework for ships, utilizing model geometric parameters as design variables and employing viscous drag calculations from CFD software SHIPFLOW as the objective function, optimizing the design of an intelligent water robot using the Nondominated Sorting Genetic Algorithm (NSGA-II).

Currently, there is a lack of formal publications on wave resistance optimization using the CFD method, although significant overseas research exists that can inform future ship design Notable studies include Orihara and Miyata's work on the RANS equation through the finite volume method, which simulated a container ship's wave resistance in regular waves using overlapping grid techniques, and Carrica et al.'s calculations of the DTMB5512's motion response at varying speeds with the same method Additionally, Tezdogan et al employed unsteady RANS methods to assess wave resistance during low-speed sailing These advancements highlight key technological breakthroughs in ship design optimization, including hull geometry reconstruction and high-precision CFD simulations, leading to the development of commercial software that applies these findings to real-world ship design However, existing optimization objectives primarily focus on hydrostatic resistance, neglecting the impact of wave resistance, ship movement, propulsion, and maneuverability on overall navigation performance.

Since the late 1990s, China has seen rapid advancements in ship design and optimization using the RANS method, primarily aimed at predicting hydrostatic resistance and enhancing ship design for specific types Chen utilized ShipFlow software to optimize a small waterline catamaran by analyzing various hull elements, including draft, shape, spacing, and pillar length Additionally, Xu employed the CAD-CFD integrated platform Friendship-Framework for automatic deformation, followed by optimization using ShipFlow software.

In recent years, the integration of optimization theory and parametric expression technology in shipbuilding has led to advancements in minimizing ship resistance through CFD numerical simulation software and ship geometry reconstruction Liang utilized the OPTIMUS 5.2 platform, integrating GAMBIT and Fluent software to create a ship-based optimization strategy using response surface models, demonstrating the effectiveness of automated submarine headline optimization Studies by Chang et al., Xie et al., Huang and Feng, and Su focused on linear multidisciplinary optimization of hull designs, combining CFD software SHIPFLOW with the ISIGHT platform, which enhanced optimization efficiency and yielded practical engineering results Qian et al further advanced this by employing ISIGHT to integrate CFD, ship form transformation, and hybrid optimization algorithms to achieve minimal resistance in ship design Additionally, Shahid explored ball-head optimization using CFD, where Fluent calculated total resistance as the objective function, achieving effective shape transformation and meshing through GAMBIT, complemented by genetic algorithm optimization in MATLAB.

Li [75] constructed an optimized design framework of ship hydrodynamic con-

This article explores a configuration utilizing SBD technology, emphasizing the Bezier Patch local geometric reconstruction method and the FFD global geometric reconstruction method, alongside the PSO optimization algorithm It addresses the challenge of automatically generating complex meshes.

In optimizing comprehensive ship navigation performance, researchers are increasingly linking navigation efficiency to ship-type parameters through regression analysis of model test results Wang et al proposed a multi-objective optimization system aimed at minimizing wave resistance and drag, utilizing CFD software and potential flow theory to achieve their goals, with successful application demonstrated on an oil tanker Zhang focused on multi-disciplinary optimization for container ships using the Energy Efficiency Design Index (EEDI), integrating navigation performance metrics within a full parametric model through the ISIGHT platform Zhou's research highlighted the optimization of ship energy efficiency, employing a multi-objective genetic algorithm to enhance the EEDI by adjusting speed and load capacity during the design phase Lastly, Sheng-Zhong developed a multi-objective optimization platform combining SBD technology and global optimization algorithms, successfully reducing total resistance by 5% in a bulk carrier, thereby significantly enhancing navigation performance.

Despite the absence of published research on ship-type optimization focused on minimizing wave resistance, advancements in wave resistance calculations offer valuable technical support for ship form optimization Shen et al developed the naoe-FOAM-SJTU solver using OPENFOAM, employing the VOF method to capture free surfaces alongside the SST K-W turbulence model for viscous flow, and the PISO method for velocity-pressure coupling Their work includes calculations of the heave and pitching motion of the DTMB 5512 ship in various wave conditions, as well as an analysis of wave drag Similarly, Zhao et al created a RANS-based CFD hydrodynamic performance calculation system utilizing overlapping grids to effectively simulate ship resistance and response during movement Additionally, Shi et al established a three-dimensional numerical wave-making tank to simulate wave motion of a ship model in irregular waves based on viscous theory.

The article discusses the development of a commercial CFD numerical simulation software aimed at calculating total resistance for international maritime applications It employs parametric modeling and geometric reconstruction techniques, utilizing optimization algorithms through platforms like ISIGHT to create an SBD-based ship-type optimization system Recent advancements focus on enhancing navigation performance by establishing regression formulas for optimization Additionally, the wave resistance and energy efficiency index (EEDI) serve as objective functions for optimization design, conducted on commercial platforms based on wave resistance standards.

The calculation of wave resistance is confined to potential flow theory and does not assess the energy-saving benefits post-optimization of actual ship designs Research indicates that international scholars have developed a theoretical framework for optimizing ship navigation performance using SBD technology, achieving significant advancements in performance-driven design for boat-type optimization This includes transitioning from single resistance analysis to integrated optimization of multiple performance metrics such as seakeeping and maneuverability, applicable to both simple mathematical models like the Wigley ship and more complex practical vessels While ship resistance optimization based on the RANS method has been largely implemented in China, it primarily focuses on single-objective or basic multi-speed target optimization, lacking comprehensive multi-speed resistance optimization or evaluations of navigation effects in waves Furthermore, there is insufficient analysis of energy-saving impacts under varying load conditions for optimized speeds, highlighting a gap in real ship navigation optimization This work addresses these issues by exploring key technologies in ship SBD design and real navigation optimization systems based on the unsteady RANS method, including a comprehensive performance evaluation system utilizing unstructured RANS methods, ship hull geometric reconstruction via ASD surface deformation, and optimization mechanisms through BP neural networks, Elman neural networks, and particle swarm optimization, culminating in a robust real ship navigation optimization design system that features independent evaluation and optimization through numerical valuation.

The unsteady RANS method is employed to assess hydrostatic and wave resistance as key objective functions, utilizing hull shape parameters as design variables while maintaining displacement as a fundamental constraint Additional considerations include propulsion factors, such as propeller nonuniformity, and maneuverability indices, integrated with an optimization algorithm for ship-type design Furthermore, the study explores navigation control optimization aimed at achieving minimal trim at design speed, ultimately enhancing the ship's navigational performance and providing a theoretical basis for optimizing real ship navigation systems.

Research Project

The general research plan of this book is shown in Fig.1.6.

The input data from the mother ship, which includes the type table, main scale, and hull geometry parameters, is essential for parametric expression and modeling Additionally, the Michell integral method allows for the direct use of the hull type value in calculations.

The overall research program integrates ship shape values and wave resistance expressions through a tent function, eliminating the need for reconstructing the ship hull geometry Utilizing the Rankine source method, the boat shape modification function serves to define the hull's shape, with its parameters acting as design variables In contrast, the RANS method employs the ASD free surface deformation technique to parametrically express and reconstruct both the modified hull and its constant components Throughout the optimization process, meshing is conducted, allowing for controlled deformation of the ASD free surface, thereby achieving the desired free deformation.

After reconstruction, the hull values are input into the Rankine source and RANS numerical calculations, which involve computational modeling and grid generation based on boundary conditions and domain size The process includes automatic grid division to create a calculation grid file To ensure numerical accuracy and stability, the impact of varying mesh densities on the results is analyzed, leading to the determination of the final mesh file This numerical simulation assesses the design speed of wave resistance, calculating both hydrostatic total resistance and wave resistance as the objective function.

This study employs a response surface model for ship-type optimization, focusing on minimizing wave-making and total resistance through nonlinear programming, genetic algorithms, and niche-based genetic algorithms Additionally, it utilizes BP neural networks, Elman neural networks, and both standard and improved particle swarm optimization algorithms to optimize ship hydroforming based on CFD methods, aiming to reduce hydrostatic and wave resistance The convergence and reliability of these algorithms are validated through test functions.

The integration of comprehensive functional modules using FORTRAN enables the development of a data interface between various program components, including the Michell integral method and Rankine source method, which are essential for establishing optimized mathematical models By employing the RANS method alongside CFD resistance-solving software, CAD modeling software, and meshing software on the ISIGHT platform, the optimization of ship models is achieved The wave resistance and total resistance, derived from hydrostatic and wave resistance calculations, serve as objective functions reflecting hull shape changes based on design variables, with displacement constraints as a basic requirement An optimal design framework is constructed focusing on minimizing wave resistance, total resistance, and optimizing trim Through the optimization analysis of four typical hulls—Wigley ship, S60 ship model, DTMB5415 ship, and KCS container ship—key metrics such as theoretical minimum wave resistance and total drag are obtained This research provides a theoretical foundation and technical support for designing "green ships" and developing new ship types aimed at energy conservation and emission reduction.

1 Zhao F, Wu C-S, Zhang Z-R (2015) Preliminary an a lysis of key issues in the development of numerical tank J Ship Mech 19(10): 1210 – 1220

2 Gong M, Zhang Y, Wang X (1999) Simulation-based design and its application in auto- fl ight system design Journal of Beijing University of Aeronautics and Astronautics 25(3): 288 – 291

3 Zakerdoost H, Ghassemi H, Ghiasi M (2013) Ship hull form optimization by evolutionary algorithm in order to diminish the drag J Marine Sci Appl (12): 170 – 179

4 Li SZ (2012) Research on hull form design optimization based on SBD technique Wu xi: China Ship Scienti fi c Research Center

5 Wang J-H, Wan D-C (2016) Numerical simulation of pure yaw motion using dynamic overset grid technology Chinese J Hydrodyn 31(5): 567 – 574

6 Liu Z, Feng Bai W, Zhan C (2010) Multidisciplinary design optimization of ship hull from. National Defense Industry Press

7 Cao HJ, Wan De-C (2013) Three-dimensional numerical wave tank based on naoe-FOAM-SJTU solver J Fudan Univ (Natl Sci) 52(5): 627 – 634

8 Zhang BJ (2009) Research on optimization design of hull lines and minimum Resistance hull form Dalian: Dalian University of Technology

9 Li ZZ (2005) Optimizaiton research of ship form based on the numerical value calculation of wave resistance Dalian: Dalian University of Technology

10 Liu YZ (2003) Theory of ship wave making resisitance BeiJing: National Defense Industry Press

11 Xia LX (1982) Equivalent thin ship method and its application in ship type modi fi cation. Shang Hai: Shanghai Jiaotong University

12 Hsiung CC (1981) Optimal ship forms for minimum wave resistance J Ship Res 25(2):95 – 116

13 Hsiung CC (1984) Optimal ship forms for minimum total resistance J Ship Res 28(3):163 – 172

14 Ye HK (1985) The wave resistance calculation and optimization of ship form with the tent function Shipbuilding of China, 28 – 39

15 Xia L, Liu YZ (1984) Equivalent thin ship and ship form improvement Shipbiulding of China, 1 – 12

16 Huang D, Mu J (1997) Interpretation, modi fi cation and calculation of linearized wave resistance theory J Harbin Eng Univ 18(5): 18 – 14

17 Pan ZQ, Cai RQ, Du SQ (1989) A method of hull form improvement by using mathieu function Shipbiulding of China, 44 – 53

18 Zhang XG, Du S, Cai R (1990) A method for the calculation of wave resistance of catamaran and hull from improvement by using biquadratic spline functions Shipbiulding of China, 1 – 15

19 Shi ZK, Zheng JM, Huang Y (1991) Shipform optimization and a ship form with minimum wave tesistance J Huazhong univ Sci Tech 19(5): 121 – 128

20 Ma K, Ichiro T (1997) A study of optimal hull form for minimum resistance J Hydrodyn 12 (1): 114 – 122

21 Ma K, Ichiro, T (1994) A study of minimum resistance hull form with consideration of separation (1nd Report) J Kansai Soc Naval Archit Jpn (221), 9 – 15

22 Ma K, Ichiro, T (1994) A study of minimum resistance hull form with consideration of separation (2nd Report) J Kansai Soc Naval Archit Jpn (222), 41 – 47

23 Ichiro T, Ma, K (1995) Low resistance and high ef fi ciency of ship type determination.

J Kansai Soc Naval Archit Jpn (223), 49 – 57

24 Ji ZS, Lishufan, GCJ (1982) An applied mixed integer programming method in ship design.

25 Lin Y, Ji Z, Dai Y S (1996) Mathematical description and computer method for b splnie hull surface Shipbiulding of China 4: 83 – 86

26 Lin Y, Zhu ZH, Jizhuo S (1997) Form description with function parameter Shipbiulding of China 3: 74 – 78

27 Lin Y, Jizhuo S, Li TL (1999) Optimum approach to bulbous bow design to reduce resistance and improve propulsion J Dalian Univ Technol 39(6): 785 – 791

28 Huang Q (1990) Estimation of wave-pattern resistance to ships and optimization of hull forms J Huazhong Univ Technol 18(5): 69 – 76

29 Saha GK, Suzuki K, Kai H (2004) Hydrodynamic optimization of ship hull forms in shallow water J Mar Sci Technol 9:51 – 62

30 Masut S, Suzuki K (2001) Experimental Veri fi cation of optimized hull form based on rankine source method J Kansai Soc NA Jpn 236, 27 – 32

31 Cheng DM, Liu XD, He SL (1999) Wave making resistance calculation of transom ship J Ship Mech 3(1): 6 – 12

32 Liu XD, Li BJ, Zhu D (2003) Numerical simulation of the fl ow of the stern ship Hydrodyn Res Prog A 18(2): 168 – 175

33 Chen J-P, Zhu D-X, HE S-L (2006) Resear ch on numer ical pr ediction method for wave makingr esistance of catamar an/tr imar an J Ship Mech 10 (2): 23 – 29

34 Chen W-M, Chen X-P (2007) Application of CFD in ship hull lines optimization J SSSRI 30 (1): 30 – 32

35 Raven HC (1993) Nonlinear ship wave calculations using the PARPID method In: Proceeding 6th international conference on numerical ships hydrodynamics, Iowa City

36 Raven HC, Valkhof HH (1995) Application of nonlinear ship wave calculations in design, symp, on practical design of ships and fl oating structures Seoul, Korea

37 Heimann J (2005) CFD Based optimization of the wave-making characteristics of ship hulls. Technical University Berlin, 23 – 42

38 Wang XF, Zhou S, Chen Z (1992) Computational ship fl uid mechanics shang hai: Shanghai Jiaotong University Press

39 Deng R, Huang D-B, Zhou G-L (2008) Numerical calculation of resistance of trimarans.

40 Tahara Y, Saitoh Y, Himeno Y (1999) CFD-aided optimization of tanker stern form -1st report: minimization of viscous resistance J Kansai Soc NA, Jpn.231, 29 – 36

41 Peri D, Campana EF (2003) Multidisciplinary design optimization of a naval surface combatant J Ship Res 47(1):1 – 12

42 Peri D, Campana EF (2005) High- fi delity models and multi objective global optimization algorithms in simulation-based design J Ship Res 49(3):159 – 175

43 Campana EF, Peri D, Tahara Y, Stern F (2004) Comparison and validation of CFD based local optimization methods for surface combatant bow In: The 25th symposium on naval hydrodynamics, Canada

44 Campana EF, Peri D, Tahara Y et al (2009) Numerical optimization methods for ship hydrodynamic design SNAME Annual Meeting

45 Campana EF, Liuzzi D, Lucidi S, Peri D et al (2009) New global optimization methods for ship design problems Optim Eng 10(4): 533 – 555

46 Tahara Y, Sugimoto S, Murayama S, et al (2003) Development of CAD/CFD/ optimizerinte grated hull-form design system Kansai Soc NA, Jpn, (240): 29 – 36

47 Tahara Y, Tohyama S (2006) CFD-Based multi-objective optimization method for ship design Int J Num Methods Fluids 52:499 – 527

48 Tahara Y, Peri D, Campana EF, Stern F (2008) Computational fl uid dynamics based multiobjective optimization of a surface combatant Marine Sci Technol 13(2):95 – 116

49 Tahara Y, Hino T, Kandasamy M et al (2011) CFD-based multiobjective optimization of waterjet propelled high speed ships In: 11th international conference on fast sea transportation FAST 2011, Honolulu, Hawaii, USA

50 Tahara Y, Peri D, Campana EF, Stern F (2011) Single- and multiobjective design optimization of a fast multihull ship: numerical and experimental results J Marine Sci Technol 16(4):412 – 433

51 Harries S, Valdenazzi F, Abt C et al (2001) Investigation on optimization strategies for the hydrodynamic design of fast ferries In: 5th International conference on fast sea transportation, Southhampton, UK

52 Ho-Hwan C (2010) Hull form parameterization technique with local and global optimization algorithms In: Proceedings of MARTEC 2010 the international conference on marine technology, BUET, Dhaka, Bangladesh

53 Gregory J, Grigoropoulos, Dimitris S et al (2010) Hull-form optimization in calm and rough water Comput Aided Des 42: 977 – 984

54 Zalek SF (2007) Multi-criterion evolutionary optimization of ship hull forms for propulsion and seakeeping Michigan University

55 Peri D, Campana EF (2006) Simulation based design of fast multihull ship In: 26th Symposium on naval hydrodynamics, Rome, Italy

56 Campana EF, Peri D (2006) Shape optimization in ship hydrodynamics using computational fl uid dynamics Comput Methods Appl Mech Eng 196:634 – 651

57 Tahara Y, Peri D, Campana EF, Stern F (2008) Single and multiobjective design optimization of a fast multihull ship numerical and experimental results In: 27th Symposium on naval hydrodynamics Seoul, Korea

58 Kim KT, Bath fi eld F, Nicolas et al (2014) Hydrodynamic optimization of twin-skeg LNG ships by CFD and model testing Int J Naval Archit Ocean Eng 6(2): 392 – 405

59 Soonhung H, Yeon-Seung L, Young BC (2012) Hydrodynamic hull form optimization using parametric models J Marine Sci Technol 17(1):1 – 17

60 Jim H, Shari H, Nickolas V (2012) Multidisciplinary design optimization of ship hull forms using metamodels In: ASME Internationl design engineering technical conferences/ computers and information in engineering conference, Washington, DC, 847 – 856

61 Vasudev KL, Sharma R, Bhattacharyya SK (2014) A multi-objective optimization design framework integrated with CFD for the design of AUVs Methods Oceanography 10:138 – 165

62 Orihara H, Miyata H (2003) Evaluation of added resistance in regular incident waves by computational fl uid dynamics motion simulation using an overlapping grid system J Marine Sci Technol 8(2):47 – 60

63 Carrica PM, Wulson RV, Noack RW et al (2007) Ship motions using single-phase level set with dynamic overset grids Comput Fluids 36(9):1415 – 1433

64 Tezdogan T, Demirel YK, Kellett P et al (2015) Full-scale unsteady RANS CFD simulations of ship behaviour and performance in head seas due to slow steaming Ocean Eng 97:186 – 206

65 Chen XN (2007) Research on optimization of ship form for swath Shanghai: Shanghai Jiao Tong University

66 Xu L (2012) Optimization of ship hull resistance based on CFD method Shanghai: Shanghai Jiao Tong University

67 Liang J (2008) Ship hull lines optimization based on hydrodynamic performance Shanghai: Shanghai Jiao Tong University

68 Chang H, Feng B, Liu Z (2012) Research on application of approximate model in hull form optimization Shipbuilding of China 53(1): 88 – 98

69 Xie L, Feng BW, Liu ZY (2011) Automatic optimization of high-speed hull forms using CFD.

J Huazhong Univ Sci Tech (Ntl Sci Edition), 2011 39(6): 129 – 132

70 Huang J-F,Feng B-W (2012) Investigation on optimization strategies for the hydrodynamic design of DTMB 5415 In: The 3rd International conference on manufacturing science and engineering Xiamen, Peoples R China (479), 1950 – 1954

71 Su Z-D (2014) CFD-based hull form resistance and fl ow fi eld multiobjective optimization research In: Proceedings of the twenty-fourth (2014) international ocean and polar engineering conference busan, Korea, 15 – 20

72 Qian J Comprehensive hydrodynamics optimization for hull form Via MDO Wu han: Wuhan University of Technology

73 Qian J-K, Mao X-F, Wang X-Y (2012) Ship hull automated optimization of minimum resistance via CFD and RSM technique J Ship Mech 16(1 – 2): 36 – 43

74 Shahid M (2012) A research on the ship hull form optimization using viscous CFD and Genetic algorithm Harbin: Harbin Engineering University2012

75 LI S-Z, Zhao F, Yang L (2010) Multi-objective optimization for airfoil hydrodynamic performance design based on CFD techniques J Ship Mech 14(11): 1441 – 1448

Wang S, Chen JP, and Wei JF (2013) presented research on an integrated optimization system that addresses both calm water resistance and additional resistance caused by waves This work was showcased at the 25th National Hydrodynamics Seminar and the 12th National Hydrodynamics Academic Conference held in Zhoushan, Zhejiang, where they discussed its implications for hydrodynamic studies (pp 928-933).

77 Zhang WX (2012) Comprehensive optimization of hull form for containership in wave based on EEDI Wu Han: Wuhan University of Technology

78 Zhou Y (2012) Research on optimization method for high performance vessel Research on optimization method for high performance vessel Dalian:Dalian Maritime University

79 Sheng-Zhong Li, Feng Zhao, Qi-Jun Ni (2014) Bow and stern shape integrated optimization for a full ship by a simulation-based design technique J Ship Res 58(2):83 – 96

80 Shen Z-R, Ye H-X, Wan D-C (2014) URANS simulations of ship motion responses in long-crest irregular waves J Hydrodyn 26(3): 436 – 446

81 Zhao F-M, Gao C-J, Qiong XIA (2011) Overlap grid research on the application of ship CFD.

82 Shi B-W, Liu Z-J, Wu M (2014) Numerical simulation of ship motions in irregular head waves J Ship Mech 18(8): 906 – 915

Overview

The evaluation of ship hydrodynamic performance is crucial for optimizing ship designs using SBD technology, as it establishes a mathematical model for the optimization problem The precision of this performance evaluation significantly influences the quality of the optimization outcomes During the optimization process, the algorithm relies on performance predictions to determine the next search direction, making the reliability of these predictions essential for guiding the optimization effectively and determining the overall success of the design.

The precision of numerical predictions in ship hydrodynamics significantly impacts the effectiveness of ship model optimization Consequently, fundamental requirements for ship optimization, grounded in hydrodynamic theory and numerical simulation technologies, must be established.

In ship-type optimization using hydrodynamic theory, rapid numerical methods are essential for calculating a vessel's resistance performance The optimization process relies on continuous resistance calculations based on hull shapes modified through geometry reconstruction technology, necessitating advancements in numerical simulation speed Potential flow theory has historically fulfilled these requirements, gaining popularity in the 1990s However, recent advancements in computer technology have enabled CFD-based ship-type optimization, which, while more accurate, remains slower than potential flow theory and is heavily reliant on computer hardware In today's competitive market, the ability to quickly optimize ship designs is crucial for success.

B.-J Zhang and S.-L Zhang, Research on Ship Design and Optimization

Based on Simulation-Based Design (SBD) Technique, https://doi.org/10.1007/978-981-10-8423-2_2

Effective ship design methods are crucial for market competitiveness, with 27 well-optimized approaches identified A key challenge in Computational Fluid Dynamics (CFD) ship-type optimization is enhancing the accuracy of numerical simulations related to ship hydrodynamics.

The accuracy of numerical simulation technology significantly influences the quality of optimized ships While potential flow theory methods, like the Michell integral and Rankine source methods, provide high accuracy for evaluating high-speed, slender vessels, they often fall short for specialized ship forms, leading to distorted optimization results In contrast, Computational Fluid Dynamics (CFD) offers more precise resistance values and comprehensive flow field information, making it increasingly favored for ship resistance evaluations Despite the advantages of CFD, potential flow theory remains superior in certain calculations, suggesting that a hybrid approach utilizing different theories tailored to specific ship types is most effective for hydrodynamic optimization.

The numerical evaluation method for ship-type optimization, grounded in hydrodynamic theory, effectively identifies how minor alterations in hull geometry impact results This high sensitivity to design changes is crucial during the optimization process, where adjustments to design variables are made based on numerical simulations to develop new ship models Subsequently, a CAD module reconstructs these models for further numerical evaluation Consequently, the method demands exceptional sensitivity due to the minimal variations in ship type being analyzed.

Currently, various methods exist for predicting a ship's resistance performance, primarily categorized into three types: linear wave resistance theory, approximate nonlinear wave resistance theory, and Computational Fluid Dynamics (CFD) This book focuses on key representative methods within these categories, specifically the Michell integral method, the Rankine source method, and the Reynolds-Averaged Navier-Stokes (RANS) method.

Michell Integral Method

Use the Tent Function to Express the Ship Type

The numerical calculation of wave resistance through Michelle’s integral method hinges on accurately representing the hull function Typically, the hull surface is defined using discrete point values, while the tent function effectively correlates hull shape values with wave-making resistance To utilize the tent function, the hull surface is segmented into rectangular grids, defined by a specific number of waterlines and stations The initial station line is positioned at the forefront of the hull, with the final station line located at the rear, as illustrated in Fig 2.1.

The first waterline serves as the baseline, while the last waterline is designated as the design waterline The rectangular cells located at the grid points (xi, zj) are formed by the station number lines (i–1) and (i + 1), along with the waterlines (j–1) and (j + 1) Additionally, a unit tent function is defined, which attains a value of 1 at the grid point (xi, zj) and a value of 0 at the boundaries of the cell, as illustrated in Fig 2.2.

The unit tent function \( h(i,j)(x,z) \) associated with the grid point \( (x_i, z_j) \) can be expressed mathematically as follows: \[h(i,j)(x,z) = \left(1 - \frac{x - x_i}{x_{i+1} - x_i}\right)\left(1 - \frac{z - z_j}{z_{j+1} - z_j}\right) \quad \text{for } x_i \leq x < x_{i+1} \text{ and } z_j \leq z < z_{j+1}\]\[+ \left(1 - \frac{x - x_i}{x_{i+1} - x_i}\right)\left(\frac{z - z_j}{z_{j+1} - z_j}\right) \quad \text{for } x_i \leq x < x_{i+1} \text{ and } z_{j} \leq z < z_{j+1}\]\[+ \left(\frac{x - x_i}{x_{i+1} - x_i}\right)\left(1 - \frac{z - z_j}{z_{j+1} - z_j}\right) \quad \text{for } x_{i} \leq x < x_{i+1} \text{ and } z_j \leq z < z_{j+1}\]\[+ \left(\frac{x - x_i}{x_{i+1} - x_i}\right)\left(\frac{z - z_j}{z_{j+1} - z_j}\right) \quad \text{for } x_{i} \leq x < x_{i+1} \text{ and } z_j \leq z < z_{j+1}.\] This formula captures the behavior of the unit tent function at the specified grid points.

Looking closely at the expression, although the unit tent function h (i,j) (x, z) is not a linear function, in each quadrant of a cell, h (i,j) (x, z) is a linear function of x for a

fixed z, or for afixed x, h (i,j) (x, z) is a linear function of z According to this feature

The hull grid arrangement of the tent function allows for the creation of a function that approximates the hull surface when combined with the hull value Specifically, if the hull value at the coordinates (x i, z j) is denoted as y ij, the approximate hull function can be defined accordingly.

X j yijh ði ; jị ðx;zị ð2:2ị

According to the tent function, we can see that at the grid point (xi, zj), h ði ; jị ðx;zị ẳ1, so

Or^hðx;zị ẳh ði ; jị ðx;zị

Equation (2.3) serves to approximate the hull's surface function, with the accuracy of this approximation being influenced by the mesh size As illustrated in Figure 2.3, a family of tent functions effectively approximates the hull surface within a rectangular unit The grid points (xi, zj) and their adjacent points—(xi-1, zj), (xi, zj+1), (xi+1, zj), and (xi, zj-1)—form straight lines that help in modeling the waterline and station line of the hull surface Consequently, a smaller mesh results in a more precise representation of the hull surface when applying Equation (2.3).

Derivation of Michell Integral Formula

In the study of wave interactions with a uniform flow, the basic flow is established as uniform, with wave potential satisfying the linear free surface condition Michell's approach, under conditions of a thin hull, infinite water depth, and symmetrical flow, employs the method of separation of variables to derive the relevant velocity potential and the formula for wave-making resistance.

In the right-hand rectangular coordinate system fixed on the hull, the origin is positioned at the undisturbed stationary surface at the bow of the load waterline The x-axis and y-axis align with the still water surface, while the ship moves uniformly along the x-axis in the negative direction at a speed of -U By applying the principle of motion conversion, the scenario can be interpreted as the ship navigating through a uniform flow with a velocity of U.

L = 2 fxðx;zịe K 0 z sec 2 h ỵ iK 0 xsec h dx

L = 2 fxðx;zịsinðK0xsechịdx Fig 2.3 Hull part of tent function family

In the formula,K0 is wave number, andK0 ẳ c g 2, unit: 1/m; cis ship speed, unit: m/s; gis gravitational acceleration, unit: m/s 2 ; qis water density, unit: kg/m 3 ; yẳ fðx;zịis surface equation of ship form;

Let kẳsech dkẳtanhsechdh dhẳ cos 2 h sin h dk sec 3 hdhẳ sec h sin hdkẳ k

Then, the wave-making resistance formula becomes:

The dimensionless wave number \( c_0 \) is defined as \( L^2 K_0 = 2v_g L^2 = 2Fr^{1/2} \), leading to the relationship \( K_0 = 2c L_0 \), where \( X \) and \( Z \) are also dimensionless By setting \( m = Tz + 1 \) and \( z = (1m) T, n = Lx, k = u^2 + 1 \) (to eliminate the singularity of \( k = 1 \)), the wave-making resistance formula is expressed as \( 8qg c_0 pL \).

0 ðI 2 ỵJ 2 ị ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðu 2 ỵ 1ị 2 ðu 2 ỵ 1ị 2 1 q 2udu ẳ16qgc 0 pL

0 ðI 2 ỵJ 2 ị ðu 2 ffiffiffiffiffiffiffiffiffiffiffiffiffi ỵ 1ị 2 u 2 þ 2 p du ð2:6ị

1 = 2 fxðx;yị cosð2c 0 ðu 2 ỵ1ị xịdx

1 = 2 fxðx;yị sinð2c 0 ðu 2 ỵ1ị xịdx

Introduce tent function below, and then derive Michell points:

Successively transformed kẳsech;uẳ ffiffiffiffiffiffiffiffiffiffiffi k1 p , and introduce the following dimensionless variables: xẳn=L; yẳg=b; zẳf=T; bẳB=2

The dimensionless hull functionhðx;zị ẳ 1 b Hðn;fị

The dimensionless hull slope functionhxðx;zị ẳ L bHnðn;fị

The nondimensional wave numberc 0 ẳ 2v gL 2ẳ 2Fr 1 2ẳ L 2 K0

The formula of wave-making resistance of Michell integral becomes

0 ðPðuị 2 ỵQðuị 2 ị ðu 2 ffiffiffiffiffiffiffiffiffiffiffiffiffi ỵ 1ị 2 u 2 þ 2 p du ð2:7ị

D S ij h ði x ; jị ðx;zị cosð2c 0 kxị e 2c 0 T L k 2 ð1zị dxdz ð2:8ị

D S ij h ði x ; jị ðx;zị sinð2c 0 kxị e 2c 0 T L k 2 ð1zị dxdz ð2:9ị

Introduce the tent function h ði;jị x ðx,z)ẳ

1 x i x i1 ð1z j z z j z j1ị x i1 \x\xi; z j1 \z\zj 1 x i x i1 ð1z j z z j z jỵ1ị x i1 \x\xi; zj\z\z jỵ 1 1 x i x iỵ1 ð1z j z z j z j1ị xi\x\xi ỵ 1; z j1 \z\zj 1 x i x iỵ1 ð1z j z z j z jỵ1ị x i \x\x i ỵ 1 ; z j \z\z j ỵ 1

TransformPðkịand Qðkị, and get the following two formulas:

1 x iỵ1 x iẵsinð2c 0 kx i ỵ 1 ị sinð2c 0 kx i ị

1 x iỵ1 x iẵcosð2c 0 kx iỵ 1 ị cosð2c 0 kx i ị

Ejðk;c 0 ; T L ị ẳZ z jỵ1 z j e 2c 0 T L k 2 ð1zị 1 z j z z j z jþ 1 dzþ Z z j z j1 e 2c 0 T L k 2 ð1zị 1 z j z z j z j1 dz ẳ 1

The procedures in the program are as follows:

2integralẵðPðukị 2 ỵQðukị 2 ị ðu 2 k ffiffiffiffiffiffiffiffiffiffiffiffiffi ỵ 1ị 2 u 2 k þ 2 q ; uk2 ẵ0;2 ð2:13ị

X j yijCiðkk;c 0 ị Ejðkk;c 0 ; T L ị; kkẳu 2 kỵ1; xi2 ẵ0;1; zj2 ẵ0;1; yij2 ẵ0;1 Qðkkị ẳX i

X j yijSiðkk;c 0 ị Ejðkk;c 0 ; T L ị; k k ẳu 2 kỵ1 Among them:

1 x iỵ1 x iẵsinð2c 0 kxi ỵ 1ị sinð2c 0 kxiị

1 x iỵ1 x iẵcosð2c 0 kx iỵ 1ị cosð2c 0 kxiị

The wave-making resistance coefficient and wave-making resistance formula are as follows:

2integralẵðPðukị 2 ỵQðukị 2 ị ðu 2 k ffiffiffiffiffiffiffiffiffiffiffiffiffi ỵ 1ị 2 u 2 k þ 2 q ; uk2 ẵ0;2 ð2:14ị

Rankine Source Method

Basic Equation

The Rankine source method calculates wave-making resistance by substituting uniform flow in thin-boat theory with stacked flow around the hull In this approach, a Cartesian coordinate system is fixed on the hull, where the X-axis and Y-axis align with the undisturbed hydrostatic surface The X-axis points toward the stern of the ship, representing uniform flow, while the Y-axis extends to the starboard side, and the Z-axis is oriented vertically upward.

The velocity around the hull consists of two components: the velocity potential (U) from the flow around the double model and the wave velocity potential (u), which accounts for the effects of the free surface.

Assume that thefluid is a perfect fluid with no spin The ship’s wave problem satisfies the Laplace equation and the following boundary conditions: r 2 ðUỵuị ẳ0 ð2:17ị

(1) Hull boundary conditions: The normal component of the hull surface velocity component in the direction is zero, which means that streamline cannot pene- trate the interior of the ship.

In the formula,n ^ ẳn x ^ i ỵn y ^ j ỵn z k ^ means pointing to the normal direction of the hull.

(2) Free surface condition: The velocity potential of the free surface satisfies the dynamic conditions and kinematic conditions. gfþ1

2U 1 2 on zẳfðx;yị ð2:19ị / x f x ỵ/ y f y / z ẳ0 on zẳfðx;yị ð2:20ị

Eliminate the wave height from Eqs (2.19) and (2.20) to get the following equation

2/ y ðr/ r/ị y ỵg/ z ẳ0 on zẳfðx;yị ð2:21ị

The equation must adhere to wave-free radiation conditions at the hull's far front, ensuring that upstream disturbances propagate solely downstream, while downstream disturbances do not influence upstream physical phenomena.

Fig 2.5 Coordinate system of turbulence fl ow

Linearization of Free Surface Conditions

In the context of the free surface condition, the nonlinear term of the potential function is satisfied at the free surface By disregarding higher-order terms in the velocity potential, the overmold solution can derive the linearized free surface condition The perturbation potential is considered a small quantity compared to the superimposed potential.

It can be obtained from formula (2.21) that

2/ y ð/ 2 x ỵ/ 2 y ỵ/ 2 z ị y ỵg/ z ẳ0 ð2:23ị Substituting formula (2.6) into formula (2.23), we get

2ðUỵuị y fðUỵuị 2 y ỵ ðUỵuị 2 y ỵ ðUỵuị 2 z g y ỵgðUỵuị z ẳ0: ð2:24ị

Using Eq (2.22), and neglecting the higher order term of the wave potentialuin free surface condition (2.24), the overmold solutionUcan be linearized into the following form

2UyðU 2 xỵU 2 yị y ỵUxðUxu x ỵUyu y ị x ỵUyðUxu x ỵUyu y ị y þ1

2u y ðU 2 x ỵU 2 y ị y ỵgu z ẳ0 onZẳ0 ð2:25ị For any equation F (x, y) = 0, there is

The subscript indicates the velocity gradient of the double model potential U along the streamline direction in the z = 0 planes of symmetry Consequently, the equation can be reformulated accordingly.

To further simplify the above equation, we get

U 2 lu ll ỵ2UlUllu l ỵgu z ẳ U 2 lUllonZ ẳ0 ð2:28ị

Solution of Free Surface Conditions

Rankine sources are used to express the velocity potentialsUand u, respectively, on the double model surface and the undisturbed free surface.

In the formula, rẳ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxnị 2 ỵ ðygị 2 ỵ ðzfị 2 q r 0 ẳ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxnị 2 ỵ ðygị 2 ỵz 2 q

The perturbation potential \( u \) in Eq (2.30) addresses the interaction between the hull and the free surface The current numerical algorithm modifies the double model equation based on the second combination in formula (2.30) The solution for the double model flow is derived from numerically solving the boundary value problem, which adheres to the Neumann-type boundary condition of the double model hull Consequently, results can be obtained from formulas (2.18) and (2.29).

To approximate the solution of the equation, the surface \( S_B \) of the superimposed molded body is segmented into \( N_B \) facets It is assumed that the source intensity \( r_B \) at the center of each facet remains constant Consequently, the equation for the \( i \)-th facet can be reformulated accordingly.

This equation is the second kind of Fredholm integral equation over the entire hull surfaceS B , and after simplification, the above equation can be written as

2prBðiị ỵX N B jẳ1 j6ẳi rBðjị nxiVxijỵnyiVyijỵnziVzij ẳn ^ ðiị:U 1 ;iẳ1NB ð2:34ị

The double model solution B, derived from the Hess–Smith (1964) method, involves calculating the velocity potential components Vx, Vy, and Vz In scenarios where the free surface is rigid and the Froude number is zero, this solution closely resembles that of the free surface Once the double model velocity potential U is determined using formula (2.29), the corresponding double model streamlines are traced along the water surface These streamlines do not penetrate the hull and are utilized to generate free surface grids.

Equation (2.28) outlines the free surface boundary conditions, highlighting that \( l \) represents the velocity gradient along the streamline direction This is distinct from the streamlines of the double model The velocity potential flow can be calculated using the formula \( u_l = Ux \).

The distinction between these methods lies in the fact that the free flow direction is largely substituted by the flow direction of the double model When the free surface is divided into N_F surface elements, the expressions for u_l and u_ll in Eq (2.28) can be formulated for the ith element of the free surface as follows: u_l(i) = Σ(N_F) r_F(j) L_F(i) + Σ(N_B).

DrBðjịLBðijị ð2:36ị u ll ðiị ẳX N F jẳ1 r F ðjịCLFðijị ỵX N B jẳ1

CLBðijị ẳX N 1 nẳ1 enLBðin;jị

CLFðijị ẳX N1 nẳ1 enLFðin;jị

In the above equation,e n is thefinite difference operator upstream of pointNand is calculated as follows.

To meet radiation conditions, the finite difference operator is employed to represent the two derivative phases along the double-body streamline velocity potential On the free surface in the streamline direction, the derivative phase of the function f(x, y, z) is expressed as df(i,j)/dl = f(i)(i,j) = Uxij.

U 2 xijþU 2 yij q f x ði;jị ỵ Uyij

Xin lỗi, nhưng tôi không thể xử lý văn bản mà bạn đã cung cấp Nó có vẻ như là một chuỗi ký tự không có nghĩa rõ ràng Bạn có thể cung cấp một đoạn văn khác hoặc thông tin cụ thể hơn để tôi có thể giúp bạn viết lại hoặc tóm tắt nội dung một cách chính xác không?

The introduction of this function is calculated by a single-pointfinite difference upstream operator dx dn ẳ xðiỵ1ị xðiị xðiị xði1ị ð3xðiị4xði1ị ỵ xði2ịị ð11xðiị18xði1ị ỵ9xði2ị2xði3ịị 2

: ðiẳ1ị ðiẳ2ị ðiẳ3ị ðiẳ4ị ð2:41ị dy dnẳ yðiỵ1ị yðiị yðiị yði1ị ð 3y ð i ị 4y ð i 1 ị ỵ y ð i 2 ị ð11yðiị18yði1ị ỵ9yði2ị2yði3ịị 2

: ðiẳ1ị ðiẳ2ị ðiẳ3ị ðiẳ4ị ð2:42ị df dnẳ f i þ 1 ; j f i ; j f i ; j f i1 ; j ð3f i;j 4f i1;j þ f i2;j ð11f i;j 18f i1;j ỵ9f 2 i2;j 2f i3;j ịị

Similarly,dx/dz, dy/dz,anddf/dzcan also be obtained The relation between the associated coordinate system(x, y, z)and(n,η,f)on the free surface is given below

Then, the vertical velocity component on the free surface can be expressed as:

By substituting Eqs (2.36), (2.37), and (2.45) into (2.28), we get a linear system of equations forr F and △r B

U 2 lðiị X N F jẳ1 rFðjịCLFðijị ỵX N B jẳ1

" # ỵ2UlðiịUllðiị X N F jẳ1 rFðjịL F ðijị ỵX N B jẳ1

2pgrFðiị ẳ U 2 lðiịUllðiị ð2:46ị Rearranging the above equation, we can get

DrBðjịABðijị 2pgrFðiị ẳBðiị;iẳ1NF; ð2:47ị

A F ðijị ẳU 2 lðiịCL F ðijị ỵ2UlðiịUllðiịL F ðijị;

Bðiị ẳ U 2 lðiịUllðiị Substituting Eq (2.30) into (2.18) gives us

The solution of Eqs (2.46) and (2.48) is obtained by iterative method, so

Eq (2.46) can be written as

X N F jẳ1 r F ðjịAFðijị 2pgr F ðiị ẳBðiị X N B jẳ1

For this initial value problem, the source distribution over the entire hull surface can be approximated by the double model solution

In order to find the first-order approximate solution of r F , we substitute

Substituting the solutionr ð1ị F of Eq (2.51) into Eq (2.48), we get

After obtaining thefirst-order approximate solution ofM ð1ị rBfrom formula (2.52), the second-order approximate solutionr F can be obtained from Eq (2.49)

X N F jẳ1 r ð2ị F ðjịAFðijị 2pgr ð2ị F ðiị ẳBðiị X M 1 jẳ1

By substituting the solution r ð2ị F of Eq (2.53) into Eq (2.48), we obtain the second-order approximate solution of△r B

Calculate the Wave Resistance

The hull surface pressure is determined by the velocity potential surrounding the hull and can be represented using the Bernoulli equation, which adheres to the free surface boundary conditions.

2qrðUỵuị rðUỵuị ð2:56ị After expansion, pp 1 ẳ1

2qhU 2 1 2gzU 2 xU 2 yU 2 z2Uxu x 2Uyu y 2Uzu z i ð2:57ị The pressure coefficient can be expressed as

U 1 2 hU 2 2gzU 2 x U 2 y U 2 z 2Uxu x 2Uyu y 2Uzu z i ð2:58ị

Assuming that the pressure inside the hull is a constant pressure, the wave-making resistance can be calculated by the following formula

The hull surface element area is represented by △Si, while nxi denotes the x-direction component of the surface element unit's normal direction The waveform can subsequently be derived using equation (2.19).

Mesh Classi fi cation by Rankine Source Method

Mesh classification plays a crucial role in calculating wave resistance using the Rankine source method, as different mesh shapes significantly impact the results Numerical dispersion in the Rankine source method occurs not only at the hull-fluid interface but also across the entire free surface Typically, the free surface is treated as infinite, with numerical dispersion focused on areas near the free surface affected by the ship Regions of the free surface further upstream are generally regarded as unaffected by the ship's interference, particularly at the far downstream and side edges.

The Rankine source method illustrates mesh classification, highlighting that the influence of a ship on the free surface diminishes with increasing distance Consequently, in the downstream direction and on the free side, the impact of the free surface on the flow field around the ship can be analyzed effectively, provided a sufficiently large area is considered during numerical discretization Figure 2.6 exemplifies a grid division on both the hull and the free surface.

Dawson's meshing experience indicates that the free area mesh should have a half-width of approximately 3L/8, with an upstream stretch of L/4 and a downstream extension of 0.25 wavelengths, while the grid surface is inclined at a 45-degree angle downstream Additionally, it is recommended to encrypt the grid near the bow and stern Musker (1989) proposed that the free surface grid should extend 1.0L forward, 1.5L at the stern, and 1.5L across the boat's breadth This section consolidates the insights from previous research to effectively divide the free surface grid and hull grid based on the specific conditions of the selected vessel.

The Dawson method utilizes hull meshing to divide the hull surface into quadrilateral four-node units, employing triangular elements at the bow, stern, and bottom of the ship as needed The Hess–Smith method is applied for calculating the double model solution, beginning with the specification of streamlines, point counts, initial coordinates, and distances between streamline starting positions Subsequently, the coordinates of unknown points on the streamlines are determined Once all streamline points are established, free surface meshes are created by connecting adjacent points, categorized into right-angle and back-swept areas A four-point streamlined windward difference format is implemented to maintain the radiation condition of the far front without wave interference, while a two-point difference format is used for the last few rows of the free surface mesh to mitigate excessive tail amplitude This approach reflects Dawson's expertise in mesh generation.

Calculation Procedure of Rankine Source Method

To analyze ship hull performance, first input essential ship data, including hull value, captain details, fluid mass density, speed, and the free surface calculation area, following the specified format for the S60 ship The hull grid is then automatically generated Utilizing the Boundary Element Method, the Hess-Smith program module calculates the potential flow field, offering the Rankine source module a streamlined distribution over the velocity field and free surface around the hull, without accounting for wave or viscous effects The free surface streamline is derived using the Runge-Kutta method, while the free surface mesh is created through streamline tracking The Rankine program module combines boundary elements and finite differences to deliver the velocity field, pressure field, wave-making resistance, and waveform around the ship in a non-tacky, wave-assumed manner Ultimately, the calculation results, including velocity, pressure, wave resistance, waveform, and flow field streamlines around the hull, are outputted as data files.

Inputting data of hull form and speed

Generating grid of hull and free surface

Hess-Smith method is used to calculate the flow field of superposition, the calculation of element geometry and the calculation of influence coefficient

Arranging source in hull form and free surface

Solving the algebraic equation set, obtaining the source strength, thus obtaining the velocity potential and the wave making resistance

Velocity distribution of hull form and free surface

Correction of geometric quantity of surface

Fig 2.7 Program fl owchart of Rankine source method

1 0 1 2 8 7 Node number on the unit

Hull grid nodes Hull grid units

1 -1.63972 0.00000 Coordinate values along the first streamline of the hull x,y

Coor dinates along the first Streamline of the hull

1 -1.63972 0.06250 The starting point of the first streamline coordinate value x,y

The coordinates of the starting point of each streamline in the free-surface grid

This section evaluates the reliability of the program by calculating the wave-making resistance of the Wigley mathematical model and the S60 ship model, while also plotting free surface waveforms The results show a strong correlation with experimental data from Osaka University, Japan Additionally, the findings are compared with both experimental values and previous calculations, reinforcing the accuracy of the program.

Examples

The Wigley ship is frequently utilized in experimental and numerical hull calculations, with its hull surface represented by specific mathematical equations Key parameters include a length (L) of 2.0, a beam-to-length ratio (B/L) of 0.1, and a draft-to-length ratio (T/L) of 0.0625 The hull surface equation is defined as y = 0.1 (1 - n²)(1 - f²), where n = x/L and f = Z/T The hull surface is divided into 160 surface elements, while 690 surface elements are allocated to the free surfaces Visual representations include the cross profile and waterline of the hull, along with the free surface mesh and waveform Additionally, wave-making resistance coefficient curves and a comparison of calculated versus experimental wave-making resistance coefficients are presented.

The wave-making resistance coefficient curve for the Wigley ship, illustrated in Fig 2.12, demonstrates that Rankine’s method yields a smooth and stable curve, with results closely aligning to experimental values, despite some discrepancies due to the neglect of viscosity and nonlinear factors In contrast, the Michell integral method exhibits significant fluctuations, diverging notably from the experimental value curve Additionally, high Fourier numbers effectively reproduce Kelvin wave shapes in the free surface waveforms.

The comparison of calculated and experimental values of the wave-making resistance coefficient, illustrated in Fig 2.13, shows that the trends identified by various authors align well with the experimental curve Notably, the wave-making resistance coefficient derived from the method discussed in this section closely matches the experimental results for Fr values between 0.25 and 0.34 Therefore, this method is deemed a reliable approach for evaluating the resistance performance of elongated vessels, such as the Wigley ship model.

The design parameters of the S60 ship are detailed in Table 2.1, featuring a hull surface divided into 120 bins and a free surface segmented into 752 bins The hull's cross section and waterlines are illustrated in Figures 2.14 and 2.15, while the free surface mesh is depicted in Figure 2.16 The waveform of the free surface is presented in Figure 2.17, and the wave resistance coefficient curve is shown in Figure 2.18 Additionally, Figure 2.19 compares the calculated wave resistance coefficient with the test values.

The wave resistance coefficient curve for the S60 ship, as illustrated in Fig 2.18, reveals that the trends of the three analyzed methods are largely consistent While the Rankine source method yields results that closely align with test values, there remains a discrepancy with the experimental data due to the exclusion of nonlinear factors Notably, when the Fourier number exceeds 0.24, the Michell integral method exhibits significant fluctuations; this is attributed to its basis in thin ship theory, which does not account for the actual thickness of the ship, leading to considerable deviation in results Additionally, the free surface waveforms generated by the ship's traveling waves display distinct characteristics.

Fig 2.8 Body lines of Wigley ship hull

Fig 2.9 Waterlines of Wigley ship hull

Fig 2.10 Free surface mesh generation of Wigley ship

Fig 2.11 Wave contour map of Wigley ship

Kelvin wave shapes, transverse waves and scattered waves, and wave regimes limited to±19º28’.

The comparison of calculated and experimental wave resistance coefficients, as illustrated in Fig 2.19, reveals a consistent fluctuation trend between the two sets of data This alignment indicates that the wave resistance coefficient curves produced by various authors closely match the experimental results.

Fig 2.12 Comparison of wave resistance coef fi cient curves of Wigley ship

Fig 2.13 Comparison between the calculated value and the experimental value of wave-making resistance coef fi cient of Wigley ship

Table 2.1 Main design parameters of S60 ship

Length (L) Beam (B) Design draft (T) Block coef fi cient (C b ) Design Fr

The calculated curves closely align with those derived by Suzuki Kazuo, whose findings have been validated by numerous theories and experiments This consistency highlights the effectiveness of this method for calculating wave resistance values and optimizing ship design in the future.

Fig 2.14 Body lines of S60 ship

Fig 2.16 Free surface mesh generation of S60 ship

The DTMB5415 ship serves as a widely recognized international numerical simulation standard model, supported by comprehensive experimental data Key dimensions and parameters of the vessel are detailed in Table 2.2 The hull is segmented into 2,664 grids, while the free surface consists of 4,100 grids Body lines and waterlines are illustrated in Figs 2.20 and 2.21, with the free surface mesh depicted in Fig 2.22 Additionally, the waveform contour is presented in Fig 2.23, and Fig 2.24 compares the wave resistance coefficient with the corresponding test values.

Fig 2.17 Wave contour map of S60 ship Fr = 0.30

Fig 2.18 Comparison of wave-making resistance coef fi cient curves of S60 ship

The comparison of calculated and experimental values of wave resistance coefficients reveals that the Rankine source method aligns closely with experimental results, particularly at the design speed point.

The value of Fr is 0.28, indicating a close relationship between the speeds; however, there is a significant disparity at lower speeds The Kelvin waveform observed at the free surface is distinctly pronounced and aligns closely with the recorded waveform data.

Fig 2.19 Comparison between calculated value and experimental value of wave-making resistance coef fi cient of S60 ship

Table 2.2 Principal dimensions of DTMB5415 ship

Fig 2.20 Body lines of DTMB5415 ship

Fig 2.21 Waterlines of DTMB5415 ship

Fig 2.22 Free surface mesh generation

Fig 2.23 Waveform contour map, Fr = 0.28

Basic Theory of CFD

Mass Conservation Equation

The law states that the increase in mass per unit time within a fluid micro-body is equivalent to the net increase of the micro-body over the same time interval This relationship leads to the formulation of the Mass Conservation Equation.

If the fluid is incompressible, the density is constant and the above equation becomes:

So the Mass Conservation Equation is also called continuity equation.

Fig 2.24 Comparison between wave-making resistance coef fi cient curve and experimental values

Momentum Conservation Equation (N-S Equation)

The law states that the momentum of a fluid within a micro-body is directly proportional to the total external forces acting on it, aligning with Newton's second law This relationship is mathematically represented by a governing equation.

In the formula, X, Y, Z, respectively, represent the unit mass force that the micro-body receives in three directions, p represents thefluid pressure, and v is the fluid kinematic viscosity coefficient.

Reynolds Equation

The equations outlined are universal for Newtonian fluids in both laminar and turbulent flow However, utilizing Direct Numerical Simulation (DNS) and Large Eddy Simulation (LES) is impractical under current computational conditions Therefore, in practical applications, the primary method for turbulence calculation is solving the Reynolds-Averaged Navier-Stokes (RANS) equations, focusing on the mean turbulence characteristics.

In the Cartesian coordinate system, the variables are defined as xi (i = 1, 2, 3), with g1 and g2 equal to 0, and g3 representing -g, where g indicates the acceleration due to gravity in the positive z-direction This additional term introduced is known as Reynolds stress or turbulent stress.

Turbulence Model

The k-Ɛ turbulence model effectively addresses the transport of both turbulence pulsation velocity and length It is more widely utilized than the zero-equation and one-equation turbulence models, having undergone extensive testing The fundamental equations for turbulence energy transport and energy dissipation are integral to its application.

@Xj þqC 1 EeqC 2 e 2 kỵpffiffiffiffive ð2:69ị whereC1ẳmax 0h :43; g ỵ g 5 i

Wall Function Method

The wall function method simplifies turbulence flow analysis by employing a high Reynolds number turbulence model for the core region, while using semiempirical formulas to connect wall physical quantities to unknowns in the turbulence core This approach allows for the direct determination of node variables in the control volume near the wall surface, eliminating the need for dense meshing in the wall area Consequently, only the first interior node must be positioned in a well-developed turbulence region.

The wall function method is essential for identifying the boundary between the logarithmic law and the viscous sublayer In this calculation, the parameter c y+ is chosen as the defining point for this demarcation.

When the control volume node near the wall meets the condition y + < y c +, the flow within the control volume occurs at the base of the adhesive layer In this scenario, the velocity exhibits a linear distribution in the direction normal to the wall, represented by the equation u = y.

When the control volume node next to the wall meets the condition of c y+ > y +, the flow within the control volume is characterized by a logarithmic distribution of velocity along the wall's normal direction This relationship can be expressed as u ỵ ẳ1 klnEy ỵ ð2:71ị, where k represents the Karman constant and E is a constant that pertains to wall roughness For smooth walls, the Karman constant k is 0.4, while the roughness constant E is 9.8.

Meanwhile, in the CFD software, it is recommended that y + is calculated by the following formula: y ỵ ẳDyðC 1 l = 4 k 1 = 2 ị l ð2:72ị where k is the turbulent kinetic energy of the node.

Boundary Condition

The Dirichlet and Neumann conditions are the most prevalent types of boundary conditions applied to closed boundaries Dirichlet boundary conditions, classified as the first type, specify the values of a function on the boundary, while Neumann boundary conditions, known as the second type, define the values of the function's derivative on the boundary.

Thefirst type of boundary condition describes the number of variables on the boundary or partial boundary of the calculation region, which is:

/ẳ/ 0 ; on the boundary ð2:73ị where/ 0 indicates the number of a physical quantity/on the boundary.

The second type of boundary condition describes the normal component of the gradient of the variable on the boundary, which is:

Velocity inlet boundary conditions are essential for defining the flow velocity and related scalar variables at the inlet In the case of Dirichlet boundary conditions, the entrance velocity is set in advance, typically under uniform flow conditions If the uniform flow velocity is denoted as !u c, it establishes a consistent flow profile at the inlet.

The turbulent kinetic energy k and turbulent dissipation rate e are usually derived from the experimental data or given by the following formula: kẳ3

2ðuIị 2 ð2:76ị whereuis the average velocity and I is the intensity of turbulence, according to the following formula:

Outflow boundary conditions are essential for simulating exit boundaries where the velocity and pressure are not predetermined These conditions are applicable when the flow at the exit is fully developed, but they cannot be utilized for compressible flow or in conjunction with pressure inlet boundaries within the same flow field.

Symmetric boundary conditions are applied when the physical shape and flow solutions exhibit mirror symmetry, which is often the case with ships, as they are typically symmetrical about their mid-longitudinal section This symmetry allows the mid-longitudinal section to be designated as the plane of symmetry in simulations On this symmetrical plane, there is no exchange of physical quantities, such as mass and heat, resulting in a normal velocity of zero, expressed mathematically as un · n = 0, where n represents the unit normal vector of the symmetrical plane.

In viscous fluids, the wall surface acts as a nonboundary condition, meaning that the fluid's velocity at the interface matches the velocity of the solid boundary.

At the solid boundary, if the velocity of solid boundary is!u c, then theflowing solid boundary condition is:

This is called a slip-free condition.

Free Surface Simulation

At present, based on the numerical calculation of RANS-based ship’sflowfield, the free surface simulation can be divided into interface tracking and interface capturing.

The interface tracking method involves tracking the free surface by dynamically adjusting the grid, treating the free surface as a coordinate surface This approach allows for precise boundary condition management at the free surface and is commonly utilized in maritime applications However, it faces challenges with wave breaking and overlapping interfaces, necessitating mesh updates at each time step, which can lead to significant computational demands.

The interface capture method utilizes the Euler viewpoint to describe moving interfaces within a fixed computational mesh that encompasses the entire fluid domain, eliminating the need for mesh movement during calculations This approach tracks the free surface using various techniques, such as equivalence functions and particle labeling, effectively managing complex free interfaces like wave rolling, stern wave breaking, and deck waves Several algorithms, including MAC, level-set, and VOF, can be applied to this problem, with this book primarily focusing on the Volume of Fluid (VOF) method to accurately capture the interface between air and water.

Volume of Fluid (VOF) is an interface tracking technique that operates on a fixed Euler grid to simulate various flow models by solving the momentum equation and volume fractions of multiple fluids In this method, water and air are treated as a single medium, defined by a fluid volume function, u, which represents the ratio of the volume of the target fluid to the mesh volume A value of u=1 indicates that the cell is completely filled with the target fluid, while u=0 signifies that it is filled with a different fluid The region where u transitions rapidly from 0 to 1 represents the free surface.

In the equation X n p ẳ 1 ðm_ pq m_ qp ị ð2:82ị, a1 and a2 represent the volume fractions of water and air, respectively The value of aq is set at 0.5, indicating the interface between water and air When q equals 0, the unit is entirely filled with water, while at q equals 1, the unit is completely filled with air.

Numerical Solution Method

Solving the Reynolds equation directly is complex and challenging Existing CFD algorithms typically discretize the differential equations to derive approximate values computationally Common numerical discretization methods are widely employed in this process.

finite difference method,finite element method,finite volume method, and so on. Refer to the relevant reference books for details.

The SIMPLE algorithm is a semi-implicit method designed for solving pressure-coupled equations in both incompressible and compressible flow fields It employs a "guess-correction" process on a staggered grid to compute the pressure field, facilitating the resolution of the Navier–Stokes equations This method is fundamental in computational fluid dynamics for accurately modeling fluid behavior.

To determine the velocity field, the discrete form of the momentum equation is solved using a given pressure field Since the pressure field may be assumed or inaccurate, the resulting velocity field typically does not satisfy the continuous equation and requires correction The correction principle states that the velocity field corresponding to the adjusted pressure field can fulfill the continuous equation at that iteration level By substituting the relationship between pressure and velocity from the discrete momentum equation into the discrete continuous equation, the pressure correction can be derived A new velocity field is then calculated based on this corrected pressure field The convergence of the velocity field is checked; if it is not convergent, the corrected pressure is used as the new pressure field, and the calculation process is repeated until a convergent solution is achieved.

Meshing

An effective grid layout and robust encryption are essential for enhancing calculation accuracy and identifying local flow details The quality of mesh generation significantly impacts simulation and calculation outcomes Consequently, general grid design should adhere to specific principles.

For optimal mesh distribution, it is essential to maintain a sparse yet reasonable configuration The mesh density should be increased near the free surface to accurately capture wave forms, while a sparser mesh is adequate for the basin's bottom, where fluctuations have minimal impact Additionally, due to the significant parameter gradients near the object plane, a denser mesh is necessary, whereas areas further from the object can utilize a sparser mesh due to minor parameter variations.

(2) Grid lines should be as orthogonal as possible, and curves should be as smooth as possible The grid line should be consistent with theflow direction.

For optimal accuracy in simulations, discrete meshes must be positioned as close to the object body as possible If mesh nodes are not directly attached to the object's surface, interpolation methods can introduce errors in the boundary conditions Additionally, the flow field parameters are influenced by the number of boundary parameters, leading to corresponding errors in the results.

In computational fluid dynamics (CFD), ensuring that the mesh quality meets calculation requirements is crucial, as negative volume must be avoided The first step in numerical simulation of the flow field involves discretizing the computational area by dividing the mesh Mesh types include structural, unstructured, dynamic, and overlapping meshes, each with unique advantages and disadvantages This article will specifically focus on the benefits and challenges of overlapping mesh.

Recent advancements in ship research have highlighted the importance of accurately calculating ship resistance, particularly concerning ship movement Traditional mesh techniques, both structured and unstructured, struggle with simulating moving objects effectively In contrast, overlapping mesh technology has emerged as a superior solution, facilitating easier simulation of hull states and enhancing the resolution of large-scale movement problems This innovative approach divides the model into distinct meshes, which are then integrated into a background mesh, resulting in higher-quality grids capable of accurately addressing significant motion The process involves marking hold and interpolated points, removing overlapping units through excavation, and conducting interpolation for data exchange at interfaces, ultimately leading to comprehensive flow field calculations Notably, the flow dynamics between the acceptor mesh surface and the nearest activated mesh surface are closely aligned, ensuring precise data representation across the mesh network.

In this equation, ai represents the interpolation weight factor, ф1 denotes the dependent variable of the variable supply unit, and i indicates the node for all variables illustrated by the green triangle An algebraic equation for a C unit is formed using three adjacent cells (N1–N3) and three overlapping cells (N4–N6) To ensure the equation is solved correctly, the coefficient matrix of the solution equation must be updated to satisfy the residuals, whether for the separation solution or the coupled solution.

This section discusses the use of overlapping meshes to partition computational domains Among various interpolation methods, linear interpolation employs shape functions to link the centers of variable grids, facilitating the transfer of the receiver mesh center across interpolation units While this method may be less efficient, it offers greater accuracy, as illustrated in the specific meshing example provided in Fig 2.5.4.

The Establishment of Numerical Wave Tank

Wave Making at Velocity Boundary

The velocity boundary wave method utilizes wave-making techniques to establish a specific wave velocity at the entrance boundary This approach is more cost-effective, simpler to implement, and offers greater accuracy and slower decay rates than traditional physical test tanks Additionally, it provides a more straightforward alternative to replicating physical testing methods.

fixed velocity of the ship at the entrance boundary as well as to avoid the difficulties caused by the moving boundary [10].

The wave front equation is:

The velocity field in the overlapping mesh is described by the equations u = a * e^(kz) * cos(kx * e^t) + U, v(x, y, z) = 0, and w = a * e^(kz) * sin(kx * e^t) Here, k represents the wave number, calculated using the formula k = 2π/λ, while λ denotes the natural frequency of the wave, which can be determined by the expression λ = √(

Numerical Wave Cancelation

Damping of waves can be achieved by incorporating resistance to vertical motion The method developed by Choi and Sung involves adding a resistance term to the vertical velocity equation, effectively enhancing wave damping.

S d z ẳqðf1ỵf 2 jbjịe j 1 e 1 1b ð2:88ị wherejẳ x x ed x x sd sd n d

,x sd is the starting point of absorbing region,x ed is the outlet boundary of the wave tank,f 1 ,f 2 andn d are the parameters of the model, andbis the vertical velocity component.

The Six Degrees of Freedom (SDOF) Motion Equation

While establishing the equation of motion of the ship, two reference coordinate systems are established, as shown in Fig.2.26: One is a fixed coordinate system

The moving coordinate system, denoted as GXYZ, is anchored at the hull's center of gravity (G) In this system, Gx, Gy, and Gz represent the middle cross section, the middle longitudinal section, and the water plane intersecting at the center of gravity, respectively.

With the ship’s coordinate system, the positive direction of X-axis, Y-axis, and Z-axis is, respectively, bow, starboard, and bottom.

The force formula of Newton’s second law shows that:

Fẳ^iXỵ^jYỵ^kZẳmaẳmd dtð^iuỵ^jvỵ^kwị ẳmðud^i dtþ^idu dt þvd^j dtþ^jdv dt þwd^k dt þ^kdw dtị ð2:89ị

The moment formula of Newton’s second law shows that:

M1ẳ^iKỵ^jMỵ^kNẳd dtð^iIxpỵ^jIyqỵ^kIzrị ẳIxpd^i dtỵ^idðIxpị dt þIyqd^j dtỵ^jdðIyqị dt þIzrd^k dt ỵ^kdðIzrị dt ð2:90ị where d^i dtẳ^jr^kq; d^j dtẳ^kp^ir; d^k dt ẳ^iq^jp

The motion equation of a ship, derived from the aforementioned formulas and incorporating coordinate transformation, is defined with the origin of the moving coordinate system positioned at the center of gravity.

Fig 2.26 Fixed coordinates and coordinates with the ship

The equation NẳI z wỵ ðIyI x ịpq ð2:91ị defines the hull mass in kilograms, with variables u, v, and w representing the hull's velocities in meters per second Additionally, p, q, and r denote the hull's angular velocities in radians per second External forces acting on the hull are represented by X, Y, and Z, measured in newtons (N) The moments of force applied to the hull's center of gravity, outside the body, are indicated by L, M, and N, with units in newton-meters (N-m) The derivatives of velocity and angular velocity are also represented by the variables u, v, w, p, q, and r, respectively.

(a) The entire computational domain meshing

(b) The profile of mesh near the free surface

Fig 2.27 Meshing of Wigley type

Examples

This section demonstrates the use of the Wigley, S60, and DTMB5415 models to calculate total resistance in static water using CFD methods The results are compared with experimental values to validate the accuracy of the CFD analysis, providing a foundation for future ship-type optimizations For detailed ship parameters and line drawings, refer to the preceding sections The resistance calculations were conducted using STRA-CCM+ software.

This study utilizes the Wigley ship model from Section 2.3.7 for numerical simulation, leveraging its comprehensive experimental data An overlapping mesh technique is employed to separate the hull and free surface grids, resulting in a total of 1.18 million meshes The hull grid is illustrated in Figure 2.28, while the free surface grid is depicted in Figure 2.29 The calculations are performed on a DELL workstation equipped with a 12-core processor running at 3.4 GHz and 64 GB of RAM.

Fig 2.28 Meshing of the hull of Wigley type

Fig 2.29 Waveform about free surface of Wigley type, Fr = 0.30

Fig 2.30 Comparison of total resistance coef fi cient curve and experimental values of Wigley type

(a) Meshing in whole calculation domain

(b) The profile of mesh near the free surface

The S60 type memory system, featuring a 256G hard disk and an M4000 video card, demonstrates a close alignment between numerical calculations and experimental results, as illustrated in Fig 2.30.

The main dimensions and parameters of the S60 type are shown in Table2.1. The overlapping grid technique is used to divide the hull and free surface grid, as is

Fig 2.32 Meshing of the hull of S60 type

Fig 2.33 Waveform about free surface of S60 type, Fr = 0.285

The comparison of the total resistance coefficient curve for the S60 type, illustrated in Fig 2.34, reveals that the numerical calculations closely match the experimental values With a total of 1.32 million meshes utilized, the hull grid is depicted in Fig 2.32, while the free surface grid is shown in Fig 2.33 This analysis highlights the accuracy of the computational results in relation to the experimental data.

The DTMB5415 type's main dimensions and parameters are detailed in Table 2.2 An overlapping mesh technique has been employed to segment the hull and free surface grid, resulting in a total of 2.23 million meshes, as illustrated in Fig 2.35 The hull grid is depicted in Fig 2.36, while Fig 2.37 shows the free surface grid Additionally, Fig 2.38 presents a comparison between numerical calculations and experimental values, demonstrating that the calculated results closely align with the experimental findings.

The CFD method offers a more accurate analysis of resistance performance for three typical ship types, outperforming traditional potential flow theory by aligning more closely with experimental values.

(a) Meshing in whole calculation domain

(b) The profile of mesh near the free surface

Fig 2.35 Meshing of DTMB5415 type

Fig 2.38 Comparison of total resistance coef fi cient curve and experimental values of DTMB5415 type

Fig 2.37 Waveform about free surface of DTMB5415 type, Fr = 0.28

The meshing of the DTMB5415 hull, utilizing an optimal number of grids and advanced machine configurations, enhances ship-based optimization by targeting CFD-calculated resistances, ultimately leading to more reliable outcomes.

Study on the Uncertainty of CFD Affecting the Calculation

Resistance Calculation

The cutting volume mesh for the numerical simulation tank features the front, upper, and lower surfaces designated as velocity inlets, with the ship's speed defined at the front surface boundary The rear surface functions as a pressure outlet, while the hull is modeled as a rigid body surface, and the sides are treated as symmetry boundaries.

To create an effective overlapping mesh model, two components are essential: the background body and the overlapping body The overlapping body is derived by subtracting the hull from the cuboid In the background body, the boundary conditions must align precisely with the cutting volume mesh settings For the overlapping body, the left side of the cuboid is defined as symmetry, while the remaining outer surfaces are configured as an overset mesh, and the hull surface is designated as a rigid body surface.

Taking DTMB5415 ship as an example, CFD method is used to calculate the total resistance of the hull in the static water.

(1) Calculation pretreatment: Establish geometric model, mesh, grid quality inspection, and set the boundary conditions.

The calculation process involves utilizing a three-dimensional nonstationary separation implicit solver, where the continuous equation and motion equation serve as the control equations for the entire model An appropriate turbulence model is selected to analyze the flow field, while the VOF two-phase flow model effectively captures the free surface Additionally, the SIMPLE method is employed to couple pressure and velocity for accurate results.

fields The detailed calculation process is shown in Fig.2.39.

2.6.2 Analysis of CFD In fl uencing Factors

The CFD method for calculating hull resistance often encounters errors due to discrepancies between hydrostatic and experimental values, with variations in calculation conditions leading to different error magnitudes These errors stem from uncertainties throughout the modeling process, including resistance calculation flow meshing, calculation methods, and iterative computations Key error types include mathematical model errors, iterative errors, rounding errors, truncation errors, and calculation errors To enhance resistance prediction accuracy, it is crucial to identify the primary and secondary influencing factors in the resistance calculation Addressing these factors individually can be time-consuming; thus, this section proposes an experimental design to effectively analyze the factors impacting CFD resistance calculations.

Experimental design efficiently selects representative test points from a system model to analyze the relationships within extensive data sets, aiming to achieve optimal results with minimal tests and cycles This method is both fast and cost-effective Among various experimental design techniques, the Latin square, developed by the renowned mathematician and physicist Euler, organizes different design elements into squares with edges, ensuring that each row forms a complete unit group and that each process appears only once per column.

According to previous research experience, the main factors affecting the CFD calculation results include three factors: (1) the first boundary layer thickness,

The study utilizes a Latin square design to analyze three key factors: the turbulence model, the number of grids, and the cutting volume mesh A four-level design model was established, but due to errors in the RST model calculations on overlapping meshes, a simplified three-level design model was created, as detailed in Table 2.3 The Latin square design list, presented in Table 2.4, features a 16/9 test design The total drag coefficient at a speed of Fr = 0.281, calculated using the CFD method, is included in this table Additionally, Figure 2.40 illustrates the discrepancies between the calculated total resistance and the experimental values.

The hydrodynamic performance of the DTMB5415 vessel in hydrostatic water was numerically simulated using 16 types of cutting body meshes and 9 different calculation methods The results indicate that the error |Ɛ| between the total resistance calculated by the CFD method and the experimental values ranges from 2.5% to 18.5% for cutting body meshes, with case 9 yielding the closest results and case 15 showing the largest error For overlapping meshes, the error |Ɛ| ranges from 1.25% to 14.5%, with working condition 2 providing results closest to the experimental values and working condition 1 exhibiting the largest error These findings highlight the significant impact of different calculation methods on resistance results, underscoring the need for a systematic analysis of the factors influencing these calculations.

The regression analysis results, as presented in Table 2.5, reveal the influence of various factors on cutting volume mesh The factors are ranked by their degree of impact, with turbulence mode B having the greatest effect, followed by mesh number C and the boundary layer thickness of the first floor A In the overlapping mesh, the order of influence shifts, with mesh quantity C leading, followed by turbulence mode B.

Table 2.3 Four levels of three factors

The thickness of the fi rst boundary layer A

The mesh number (relative ratio) C C 1 49.984 35.35

Table 2.4 Design list and drag calculation results of Latin square array

No A B C Cutting volume mesh A B C Overlapping mesh

The boundary layer thickness on the first floor has a minimal impact on the calculation results, while the selection of the turbulence model significantly influences outcomes Additionally, the number of grids in the overlapping meshes plays a crucial role in the accuracy of the calculations, as indicated by the error between the CFD and EFD results.

According to the data presented in Table 2.4 and illustrated in Fig 2.41, the k-e turbulence model demonstrates the lowest mean resistance error of 3.25% for cutting volume meshes, while the SA model shows the highest error at 9.55% In the case of overlapping meshes, the SST k-ω turbulence model achieves the smallest average resistance error of 5.51%, whereas the SA model again exhibits the largest error at 10.03% This indicates that the k-e model is more effective for simulating cutting volume mesh calculations, while the SST k-ω model is preferable for overlapping grid calculations, provided that the appropriate mesh count and first-layer boundary thickness are utilized.

This section presents a multi-speed numerical simulation of the DTMB5415 using both cut volume mesh and overlapping mesh techniques The analysis demonstrates that the calculation methods for working conditions 9 and 2 effectively predict hull resistance To enhance calculation accuracy, the same meshing and computational approach as the working conditions is employed to forecast hull resistance The CFD results are then compared with experimental values, as illustrated in Fig 2.42.

The calculation results indicate that the total resistance trends of the two meshes align closely with experimental values, exhibiting minimal errors Specifically, the mean error for the cutting mesh's calculated resistance is 2.5%, while the overlapping mesh shows an average error of 2.78% At the design speed of Fr = 0.281, the calculated errors are 2.56% for the cutting mesh and 1.28% for the overlapping mesh This analysis demonstrates that the calculation method employed can accurately predict hull resistance, with the overlapping mesh providing particularly high accuracy for drag predictions at the design speed and its vicinity.

Factor Deviation square sum ss

The error associated with overlapped grid calculations in design speed is minimal; however, the average error is slightly elevated due to the higher number of overlapping meshes compared to the total cutting volume mesh, resulting in increased calculation time The cutting volume mesh demonstrates distinct advantages in accurately predicting the resistance of multi-speed hulls By appropriately increasing the number of meshes, the accuracy of resistance calculations can be enhanced Table 2.6 illustrates the flow field distribution around DTMB5415 and the stern pressure distribution at a speed of Fr = 0.281, highlighting the red region as the background area and the blue region as the overlap area in the overlapping grid flow field.

The reliability of CFD simulation results is crucial for their applicability, as varying evaluation methods complicate comparisons between different researchers' findings Consequently, conducting CFD uncertainty analysis has emerged as a vital aspect of CFD research and application This section focuses on assessing CFD uncertainty based on ITTC recommendations, specifically through the processes of verification and validation.

This section analyzes the uncertainty of the total resistance of DTMB5415 in static water using the cutting volume mesh as an example The study employs three sets of meshes for analysis, with a mesh ratio of r G = 1.414 Additionally, the free surface mesh and waveform diagram are presented in Table 2.7.

Overview

Hull shape optimization utilizing the Michell integral method leverages the hull shape value embedded in the wave resistance expression, allowing direct use of this value as a design variable without the need for hull geometry parameterization Unlike traditional ship-type optimization methods, which require geometry reconstruction to link the objective function with design variables, the advancements in computational speed and graphics have made CFD-based ship optimization feasible This process involves a series of CFD numerical simulations and CAD graphic design tools, where an optimization algorithm iteratively adjusts design parameters to generate new hull geometries The CFD tool evaluates these designs and provides feedback to the optimization platform until the optimal hydrodynamic performance is achieved, all without human intervention This high degree of automation enhances the prospects for CFD ship optimization, contrasting with traditional methods that rely on manual adjustments, which limit the efficiency and scope of ship form transformations The emergence of automatic ship geometry reconstruction technology is pivotal in overcoming these limitations, enabling broader optimization capabilities.

B.-J Zhang and S.-L Zhang, Research on Ship Design and Optimization

Based on Simulation-Based Design (SBD) Technique, https://doi.org/10.1007/978-981-10-8423-2_3

Reconstruction technology 85 offers a powerful tool for the swift generation of diverse ship types and transformations through CFD ship optimization, enabling genuine "ship optimization." This advancement holds significant importance for achieving enhanced maritime efficiency and performance.

“digital shipbuilding,” “green shipbuilding”and promoting the design of ship form from the traditional experience mode to the knowledge-based mode.

Research Progress of Hull Linear Expression

Overseas Research Situation

Jochen Harries in Germany developed a comprehensive parametric design method for ships in his doctoral dissertation, leading to the creation of the commercial CAD software Friendship This software enables the direct generation of ship forms based on specific characteristic parameters, which are utilized as optimization design variables Utilizing the fully parametric model system Friendship-Modeler, Harries and Abt expressed hull shape parameters for design variables, facilitating ship form optimization through SHIPFLOW software to minimize wave resistance Abt et al optimized the DTMB5415 model via parametric modeling to reconstruct hull geometry, selecting six parameters for ship generation Additionally, Tahara and Kim employed the Lackenby transformation alongside a radial basis function-based method for local geometry reconstruction of the hull.

In the optimization of total resistance, Kim utilized the Bezier patch method for the geometric reconstruction of bulbous bows Similarly, Campana and Para optimized wave amplitudes for surface ships using this method Between 2003 and 2009, multiple studies focused on the DTMB5415 model, targeting wave resistance, seakeeping, and stern flow field as optimization objectives Detailed investigations into hull geometric reconstruction employed both the Bezier patch and CAD-based methods Furthermore, Peri, Tahara, and Campana explored two multi-objective global optimization algorithms to enhance high-speed catamarans, utilizing free-form deformation and CAD-based methods for hull geometry reconstruction.

Fig 3.2 Fully parametric model system Friendship-Modeler expressed of the ship hull

Recent research highlights significant advancements by foreign scholars in the automatic geometric reconstruction of hull structures Various commercial software with independent intellectual property rights has been developed and effectively utilized for ship-type optimization The achievements in these countries can be attributed to well-established experimental conditions, a concentrated pool of researchers, and sustained long-term efforts.

Domestic Research

Hull line optimization in China is rooted in the mathematical representation of ship types The advent of Bezier and B-spline curves has significantly advanced interaction design By the mid-1980s, the application of B-spline surfaces and rational spline curves became prevalent in hull surface design, marking a pivotal development in the field, as highlighted by Tahara et al in 1981.

In 1985, researchers first utilized B-spline surfaces to define hull surfaces, building on earlier work that described hull surfaces using Bezier surfaces These methods leverage spline concepts for design and expression, initially requiring original value points and subsequently refining the surface through human-computer interaction to achieve the desired design specifications.

Since the mid-1990s, NURBS (Non-Uniform Rational B-Splines) curves and surfaces have emerged as a key focus in computational geometry, becoming a leading method for representing curves and surfaces The application of NURBS in modeling hull surfaces has gained significant attention, as shipbuilders strive to create hull line shapes based on specific ship shape parameters Although many domestic researchers have extensively investigated the representation of curves and surfaces in ship design, an effective and practical parametric design technology for ship types has yet to be developed, and there is a lack of design software with independent intellectual property rights.

The application of uniform B-spline curves to ship design, as proposed by Harries, led to the introduction of NURBS curves for modeling sterns and bulbous bows In her doctoral dissertation, a comprehensive study on parametric ship representation was presented Ping et al analyzed the geometrical characteristics of circular hulls, establishing geometric expressions for key lines and points Utilizing NAPA BASIC language, a macro-program was developed for profile parametric design, aimed at enhancing digital design efficiency for round-bilge vessels Researchers including Xie, Zhang, and Bao-Ji explored the least resistance ship design method based on the Rankine source method, optimizing nonlinear designs while adhering to displacement constraints Zhang's doctoral dissertation detailed three-dimensional parametric design methodologies for ships and offshore platforms, while Yu employed the Framework module in Friendship software to modify a 3100TEU prototype.

To optimize hull lines, Fu and Chen, along with Baiwei et al., introduced two distinct parametric modeling methods utilizing NURBS expressions for ship-type optimization within CAD/CFD frameworks One method involves manipulating NURBS control vertex coordinates directly to modify ship parameters, while the other employs a mother-based development approach to create a parametric fusion module for parameter transformation By utilizing ISIGHT software, both methods were applied to enhance the bulbous bow of a container ship, demonstrating that the ship-type modification and fusion approach is a valuable engineering parameterized modeling technique.

Our research on automatic geometric reconstruction of hulls has matured, particularly in the study of NURBS curves and surfaces, achieving international advanced levels However, we still face gaps compared to leading shipbuilding nations like Japan, South Korea, and Europe, primarily in several key areas.

The parametric representation of hull geometry primarily focuses on basic hull types, such as mathematical models, which restricts the design space Consequently, this limitation makes it challenging to achieve optimal hull performance under specific conditions.

(2) Domestic research is relatively fragmented, did not form a unified, shared design platform, but did not form a commercial software with independent intellectual property rights.

Currently, there is a lack of an effective and transparent framework for ship-type optimization grounded in hydrodynamics theory The integration and convergence of the objective function, design variables representing hull geometry, and optimization methods remain unresolved challenges in this field.

Basic Connotation of Hull Geometric Reconstruction

Hull geometric reconstruction is a crucial ship formation technology that restores the relationship between hull geometry and topological structure during transformations This method plays a significant role in ship conceptual design, overall design, and hull form optimization As a comprehensive technology, hull geometrical representation encompasses various research fields and serves as a core component in hull design and performance calculations, directly impacting a ship's navigation performance Additionally, hull geometric reconstruction technology is essential for CFD-based ship optimization, where design variables are adjusted through optimization algorithms, resulting in changes to the hull geometry.

(1) Enter the model values and major dimensions of the mother ship for geometric modeling (shape design), generate interfacefiles for common CFD software: such as IGES format (geometric description);

(2) Use CFD software for numerical simulation (shape analysis) to show flow

(3) Evaluate the objective function (shape evaluation);

(4) Generate the optimal hull form (shape transform) through optimization strategy.

Fundamental Principles of Hull Geometry Reconstruction

The hull geometry reconstruction technique effectively connects CFD assessment with optimization algorithms, allowing for the iterative design of ship hulls By solving design variables through optimization algorithms, the governing parameters of the hull's geometry are determined, leading to the regeneration of a new hull form This new geometry is then evaluated using CFD software to assess its performance The results of this evaluation inform further adjustments to the hull parameters, creating a feedback loop that continues until the optimal hull design is achieved Consequently, hull geometric reconstruction technology serves as a foundational element for ship optimization and is recognized as a critical area of research.

(1) To ensure the smoothness of hull geometry reconstruction

To ensure the continuity of the second derivative in hull geometry representation, it is essential to connect both the modified and fixed sections using polynomial functions or double trigonometric series.

The hull's shape can be refined to such a degree of smoothness that it eliminates any discrepancies in optimization outcomes, with the hull's optimal performance being solely attributed to variations in geometric shapes.

CFD-based hull form optimization is a complex nonlinear process that necessitates continuous iteration to achieve a ship design that fulfills specific requirements To enhance efficiency and practicality in this optimization process, it is essential to represent hull geometry using a minimal number of design parameters, such as ship modification functions or polynomial function parameters, thereby reducing calculation time.

(3) To ensure that the design space is as wide as possible

To achieve optimal ship performance, it is essential to explore a broad design space that includes various hull geometries This requires determining hull geometry reconstruction parameters to generate diverse hull forms However, this need for variety contradicts the goal of minimizing computational time, presenting a challenge that hull geometry reconstruction technology must address effectively.

Hull Geometric Reconstruction Method

Hull Form Modi fi cation Function Method

The geometry of the hull is expressed by the parametric method, and the opti- mization of the ship design is carried out by the mathematical optimization method.

Since the 1960s, Japanese researchers have played a significant role in the application of parametric techniques for ship generation, leading to substantial advancements in the field From the 1980s onward, countries worldwide have heavily invested in manpower and financial resources, resulting in numerous published achievements Notably, Suzuki and Iokamori developed a minimum wave resistance ship model using the Rankine source method and introduced a modified trigonometric series function to represent hull shapes This foundational work has inspired subsequent scholars to refine and adapt the ship modification function to better articulate both local and overall hull shapes.

The ship modification function method utilizes a series of numbers, either trigonometric or polynomial, to represent changes in ship type, with hull shape alterations determined by these parameters This approach allows design parameters to serve as direct variables for optimization, enabling both the entire ship and its components to be expressed parametrically with fewer design variables However, its limitations include a lack of flexibility, a constrained geometric space, and a restricted trend in the shape of the modified ship type, which is entirely governed by the ship modification function Essentially, the modified ship's shape, y(x, z), is defined using a ship-type modification function, w(x, z), based on the original ship type, f0(x, z).

X n amnsin pð xx0 x min x 0 ị mỵ 2 sin pðz0z z 0 ỵTị n ỵ 2 m;n ẳ1;2;3; .

X n amnsin pð xx0 x max x 0 ị mỵ 2 sin pðz0z z 0 ỵTị n ỵ 2 m;nẳ1;2;3; .;0xL=2

Fig 3.4 Application of ship-type modi fi cation function

In the given equation, L represents the primary longitudinal coordinates of the ship's bow, including the bulbous bow, while T denotes the maximum depth of coordinates to be modified, which corresponds to the draft if the baseline remains unchanged With fixed values for m and n ranging from 1 to 5, there are a total of 25 design variables (25amn) This approach streamlines the selection of the ship modification function, effectively reducing the number of design variables and enhancing the optimization speed.

Polynomial Expansion Method

The ship-type function of a design ship is defined as the sum of the parent ship-type function and the variation function relative to the parent ship This relationship can be mathematically expressed as y(x,z) = y0(x,z) + Δy(x,z).

Dyðx;zị ẳDyðxị ZẳWL1 ỵDyðxị ZẳWL2 ỵDyðxị ZẳWL3 ỵDyðxị ZẳWL4 ỵ ỵDyðxị ZẳWLN

To optimize the model, fix the z-coordinate throughout its depth, allowing the unit change function to solely depend on the x-variable Subsequently, expand this unit change function into a polynomial form along the x-direction.

Dyðxị ZẳWL1 ẳa01ỵa11xỵa21x 2 ỵa31x 3 ỵ ỵak1x k Dyðxị ZẳWL2 ẳa02ỵa12x a22x 2 ỵa32x 3 ỵ ỵak2x k Dyðxị Z ẳ WL3 ẳa03ỵa13xỵa23x 2 ỵa33x 3 ỵ ỵak3x k Dyðxị ZẳWLN ẳa0Nỵa1Nxỵa2Nx 2 ỵa3Nx 3 ỵ ỵakNx k ð3:3ị

In the formula, [A] = a 01 , a 11 , …,a kN are the parameters of each unit change function to be expanded If the parameter [A] is given, the value of each unit

The ship-type modification transformation function allows for the determination of variations in ship width at different waterline positions (WL1, WL2, WL3, WLN) along the x-direction By utilizing the unit change function values for each waterline position, the unit change function for any given waterline can be derived through cubic spline interpolation This interpolation is grounded in the functions N mi (f) and N mj (n), which serve as the basis for the depth direction of the interpolation.

In the formula, N mi (f) and N mj (n) represent standard B-spline functions, where n and k denote the number of internal nodes (excluding endpoints) in the modified range for the directions of n and f, respectively Additionally, m indicates the order of the B-spline function.

Here take m = 4, n = 3, k = 2 n 3 ẳn 2 ẳn 1 ẳn 0 \n 1 \n 2 \ \n n ỵ 1 ẳn n ỵ 2 ẳn n ỵ 3 ẳn n ỵ 4 f 3 ẳf 2 ẳf 1 ẳf 0 \f 1 \f 2 \ :\f n ỵ 1 ẳf n ỵ 2 ẳf n ỵ3 ẳf n ỵ 4

Taking B-spline function parameters as design variables, and the total number of design variables is 12, as shown in Table3.1.

Spline Function Method

Two primary methods exist for expressing changes in hull shape using spline functions: the B-spline curve and nonuniform rational B-spline (NURBS) curves or surfaces The B-spline method defines the hull surface through a complex set of parameters that serve as design variables, resulting in a more intricate design process In contrast, NURBS allows for direct use of control vertices as design variables for optimization, making it a more versatile option with broad applications in hull design.

Table 3.1 B-spline function parameters as design variables Cij i=1 2 3 4 5 6 7 j=6 0.0 0.0 0.0 0.0 0.0 0.0 0.0

The hull is segmented into n sections along the end direction, with each section further divided into m points in the f direction The B-spline basis functions, denoted as Nmiðfị and Nmjðnị, are computed based on this division, as illustrated in Fig 3.6.

Jun-ichi et al explored tail optimization for ships to minimize viscous resistance using nonlinear programming and represented hull shapes with B-spline functions Meanwhile, Masuda and Suzuki employed polynomial functions and spline interpolation to define hull shapes, investigating minimum thrust deduction and tail optimization to reduce secondary flow energy through potential flow theory, which resulted in an expanded solution space.

Geometric Modeling Technique

Geometric modeling technology integrates computer-aided geometric design with computer graphics, focusing on mathematical methods that create shapes suitable for computer processing while meeting geometric design requirements This technology enhances shape information transfer and product data exchange In ship design, popular geometric modeling techniques include the free-form deformation (FFD) approach, Bezier patch method, and NURBS surface method.

3.5.4.1 Free-Form Deformation Method (FFD)

Free-forming technology, an important branch in computer graphics, was first proposed by Sederberg and Parry in 1986 [46] and has achieved rapid growth in the

The B-spline curve expression has evolved significantly in ship-type applications over the last two decades, utilizing a grid formed by control vertices to transform internal targets through vertex manipulation This versatile method is applicable to solid modeling systems for any surface type, enabling both local and global transformations while ensuring geometric continuity of the deformed surface Its compatibility with various object representations allows for seamless integration into existing software modeling systems, enhancing interactivity and controllability in object animation However, challenges remain in controlling deformation accurately, as achieving precise results can be difficult due to the numerous design variables involved in representing complex surfaces.

The FFD method employs the Bernstein basis function to define the functional relationship between lattice nodes and the position of any point within the lattice Its mathematical expression is represented as: x(s, t, u) = Dx(s, t, u).

DPi ; j ; k ð3:7ị where x(s, t, u)-the global coordinate of any point in the control frame

P ijk -coordinate matrix of control grid points(i, j, k)

DPijk-displacement matrix of control grid points(i, j, k) lnm-control grid points

(s, t, u)-local coordinates in the control lattice

B i1 l1 ðsị-an i-1 of l-1-order Bernstein polynomials, the expression of which is:

B i l1 1 ðsị ẳ ðl1ị! ði1ị!ðl1ị!s i 1 ð1sị l i ð3:8ị The above formula is written in the form of matrix:

Based on the FFD method to generate the geometry of the hull and the defor- mation of the grid, the deformations of the specific steps are as follows:

To construct a parameter body, deformation is defined within a closed three-dimensional lattice by utilizing control vertices and a corresponding set of parameters in the basic function Consequently, each point in the three-dimensional space, represented as (x, y, z), is mapped to a specific set of parameter coordinates (u, v, w).

The three-dimensional grid consists of an ordered grid of control points:

Vi ; j ; kẳ ðxi ; j ; k;yi ; j ; k;zi ; j ; kị ð3:10ị

In the context of control parameters, V_ijk represents the vertex for each control, while W_ijk denotes the initial setting unit associated with each vertex The variables a, b, and c correspond to the number of divisions along the u, v, and w directional parameters, respectively.

After the grid points are created, an ordered B-spline basis function (u, v, w) is assigned, and the sequence of each parameter variable may be different as follows:

2n ðcỵ1ị ð3:12ị where p, m, n are the order of the parameters of the basic functions u, v, w The corresponding node vector is:

To create a nonuniform node vector for each endpoint of a three-dimensional grid volume, it is essential to maintain equal order for seamless insertion of new values The expression for the node vector U is defined as ui, with V and W following a similar structure.

0 0ip i ðp1ị pi ðqpị a ðqpị iq

Assign a random B-spline basis function to each node vector, and the basis function is measured by the W standard recursive formula.

Bi ; rðtị ẳ tti ti þ r 1ti

Bi ; r1ðtị ỵ ti ỵ rt ti þ rti þ 1

In the formula, i2 f0;1;2;3 .qg, q is the number of nodes, B is a B-spline basis function; except W, it is conventionally interpreted as zero.

The points on the object are calculated based on the simple extension formula of B-spline The three-variable B-spline formula is:

P s kẳ0Bi ; pðuịBj ; mðvịBk ; nðwịWi ; j ; kVi ; j ; k

P s kẳ0Bi ; pðuịBj ; mðvịBk ; nðwịWi ; j ; k ð3:16ị where P is a point on the Cartesian coordinate vector (x ,y, z)model.

The inverse point problem effectively addresses the description of an embedded object by determining the parameter coordinates (u, v, w) corresponding to each point in three-dimensional space (x, y, z).

In solid objects, the identification of deformed shapes relies on a set of parameter coordinate points, which are divided into H parts along the grid's orthogonality and alignment with the data axis To locate these coordinates, the Golden Numerical Search method is employed, establishing the search range based on the bounding points of each parameter variable (u, v, and w) Each meaningful vector point interval corresponds to a solid spline span, with boundaries defined by evaluating the appropriate knot point values, facilitating the identification of the span object for each point For numerical evaluation, the midpoint of various node segments serves as the initial estimate, enhancing the accuracy of the search process.

(3) Deformation of the parameter body This process is usually replaced by the vertices of the 3D mesh.

The closed solid can be modified by substituting the lattice control points, offering greater flexibility compared to the previous FFD method Additionally, the use of B-spline basis functions allows for the artificial maintenance of continuity between solids, enabling an almost limitless deformation process.

The evaluation of deformation effects on embedded objects involves utilizing parameter coordinate points alongside a deformation control lattice This process allows for the assessment of the new positions of the embedded point set Subsequently, the topology of the original model is employed to accurately reconstruct the deformed object.

Various algorithms effectively evaluate B-spline basis functions, enabling a straightforward assessment of distortion effects on embedded objects This approach allows for real-time evaluation of how deformation influences an embedded object grid However, for complex objects undergoing multiple transformations, it is often more efficient to complete all grid deformations prior to computing their effects.

Using the DTMB5415 ship as a reference, the hull surface control points are established, particularly noting that points 1-3 near the bulbous bow are adjustable while the others remain fixed This configuration ensures a seamless connection between the bulbous bow and the main hull By altering the distance and direction of these control points, the shape of the bulbous bow's surface is modified, resulting in a newly smoothed surface.

In 1971, Bezier from Renault introduced a method for defining curves through polygon control, allowing designers to easily modify curve shapes by adjusting control vertices, which made the process highly predictable and widely adopted To enhance this method's capabilities, Gordon and Riesenfeld modified it for surface design, particularly in shipbuilding By superimposing Bezier surfaces onto a parent hull and adjusting curve node positions, designers can achieve various surface shapes for geometric reconstruction of the hull This approach simplifies the design process, as fewer design variables lead to smoother results, although it limits modifications to local areas of the hull.

The shape of a Bezier curve is determined solely by the placement of its control points, as illustrated in Fig 3.8 Specifically, Bezier curve 1 is defined by four control points, with the first two points lying on the curve itself, while the other two control points, Q1 and Q3, are positioned off the curve By adjusting the positions of Q1 and Q3, the original Bezier curve 1 can be transformed into a new Bezier curve 2.

The geometric reconstruction of the bulbous bow involves altering its geometry to achieve a specific design By utilizing a position vector Qj (where j = 0, 1, 2,…k) that represents n + 1 points in space, the interpolation formula for the coordinates of each point on the Bezier curve can be expressed as: r(u) = Σ (from j = 0 to k) Xj.

QjNj ; kðuị u2 ẵ0;1 ð3:17ị where Q i is the control vertex of Bezier parametric curve.

N j ; k ðuị ẳC n j u j ð1uị nj ẳ k! j!ðkjị!u j ð1uị kj ðjẳ0;1;2 .kị

NURBS is an excellent surface modeling methods that has been widely used in CAD/CAM and computational geometry and computer graphics in recent years

Traditional Optimization Methods

The Basic Idea of Nonlinear Programming

The general model of nonlinear programming is [2]: ðPị min x2S fðxị ð4:1ị

In the formula,Sis a subset ofR n , andfðxịis defined onS orR n

In optimization, when dealing with a set of real numbers \( \mathbb{R}^n \), the associated plan is referred to as the unconstrained problem Conversely, when the set is a subset of \( \mathbb{R}^n \), it is termed a constrained problem The maximization problem \( \max_{x \in S} f(x) \) can be transformed into an equivalent minimization problem \( \min_{x \in S} f(x) \), allowing us to focus solely on the minimization aspect, which is defined by the feasible set \( S \).

In the context of problem (P), a feasible point is defined as point x within the feasible set The objective function, denoted as f(x), is evaluated to identify the optimal solution, which is the minimal point in set S The corresponding value of the objective function at this optimal solution is referred to as the optimal value of the problem.

Definition 1 LetSbe a nonempty set inR n ,f :S!R Forx 2S, if existe[0, such that fðxị fðx ị;8x2S\Oðx ;eị ð4:2ị

Among them, Oðx ;eị is the open ball R n with x as the center and e as the radius, which

Oðx ;eị ẳfx2R n :kxx k\eg ð4:3ị

A point \( x \) is considered a local minimum of a function \( f \) within a set \( S \) if \( f(x) \leq f(y) \) for all \( y \) in \( S \) Furthermore, \( x \) is classified as a global minimum of \( f \) in \( S \) if this condition holds for all \( x \) in \( S \) If the conditions outlined in formulas 4.1 or 4.2 are strictly satisfied for \( x \neq x \), then \( x \) is identified as a strictly local or strictly global minimum of \( f \) in \( S \).

In particular, we call the problem (NP) minfðxị s:t: hiðxị ẳ0;iẳ1; .;m gjðxị 0;jẳ1; .;p standard nonlinear programming Theref;h i ;g j is all real-valued function onR n ,and

“s:t:”means“restricted.”hiðxị ẳ0 is called the equality constraint, andgiðxị 0 is called the inequality constraint.

The development of nonlinear programming theories is significantly linked to convex sets and functions, as the extension of linear programming concepts to nonlinear contexts necessitates a thorough exploration of convex functions This area of study, known as convex analysis, emerged as a distinct discipline in the 1950s A key concept in this field is "feasible directions," which plays a crucial role in understanding optimality.

Definition 2 Letx2SR n To say that d2R n is a feasible direction for the x (aboutS), if exista[0, such thatxþad2S;8a2b0;ac, as shown in graph4.1, d 1 ;d 2 are possible directions forx, andd 3 is not.

The system uses nonlinear programming method to solve the problem along the feasible direction from the feasible point.

Gradient Method

The gradient method is a straightforward analytical technique rooted in an intuitive concept: the function f(X) decreases most rapidly in the direction of the negative gradient, ∇f(X(k)) Therefore, pursuing this direction is a logical approach to finding optimal solutions.

In the formula,akis the step in the negative gradient direction.

Generally, the optimal step factor ak *is determined by one-dimensional mini- mization in this direction, that fðX ðkị a k rfðX ðkị ịị ẳmin a k fðX ðkị akrfðX ðkị ịị ð4:5ị

The gradient method, also known as the steepest descent method, involves determining a new design point X(k + 1) and calculating the gradient of the function f(X) at that point This process is repeated iteratively until the gradient approaches zero, indicating that the minimum point has been reached, as the search progresses in the direction of the negative gradient.

Sequential Unconstrained Optimization Method

Many optimization problems are constrained, but they can often be transformed into unconstrained optimization problems, allowing the use of unconstrained optimization methods for solutions The Sequential Unconstrained Minimization Techniques (SUMT) leverage this transformation by introducing additional terms into the objective function that account for the constraints, thereby creating a new objective function for unconstrained optimization By carefully selecting these additional terms, the sequence of optimal points from the new function can converge to the optimal solution of the original constrained problem SUMT can be categorized into different methods based on the additional terms used, including the penalty function method, barrier function method, and mixed penalty function method.

The penalty factor, denoted as Mk[0, plays a crucial role in the penalty function method for constrained optimization problems The penalty term, represented as pðxị, is a function defined on R n To effectively solve these optimization challenges, specific computational steps must be followed, utilizing the penalty function approach.

(1) SelectingM1[0, the accuracy ise[0,c2 and the initial point isx ð0ị , order kẳ1.

(2) Taking x ð k 1 ị as the initial point to solve the unconstrained optimization problems minFðx;Mkị ẳfðxị ỵMk

P L i ẳ 1 g i ỵ ðxị, let x ðkị ẳxðMkịbe the optimal solution.

(4) If s\e, the iteration ends and take x ẳx ð k ị ; otherwise, let

M kỵ 1ẳcMk;kẳkỵ1, and then back to the second step.

The above algorithm end criterions\ecan also be changed to: IfMkpðx ðkị ị\e, take x ẳx ðkị and the iteration end; otherwise, let Mk ỵ 1ẳcMk;kẳkỵ1 and continue iterating.

The barrier function method is suitable for the constrained optimization problem of minfðxị s:t:x2S

The process begins with a feasible starting point and iterates through various feasible points To maintain the iterative point within the feasible set, a "wall" is constructed along the boundary of the constraint set S, preventing the iterative points from exiting this feasible region The calculation follows specific steps to ensure adherence to these constraints.

(1) Letr 1 [0;c2, and the accuracy ise[0.

(2) Finding an interior point of the feasible set S,x ð0ị 2intS; letkẳ1.

(3) Takex ðk1ị as the initial point, and solve problems using the method of solving unconstrained optimization problems. minFðx;r k ị ẳfðxị ỵr k Bðxị s:t:x2intS ð4:7ị

Let its optimal solution bex ð k ị ẳxðrkị.

(4) Checking whether x ðkị meets the termination criterion, ifx ðkị is satisfied, the iteration ends; otherwise, takerk ỵ1, andrk ỵ1\rk, letkẳkỵ1, then back to the third step.

The iterative process in optimization involves a continuous increase of the external method parameter M_k and a decrease of the interior point method parameter r_k, complicating the resolution of unconstrained minimum problems The choice of r_k and M_k significantly affects convergence speed Solutions derived from the external point method often lack feasibility and may only provide approximate satisfaction, while the interior point method faces challenges in feasible regions and struggles with optimization problems that include equation constraints Starting from an initial point x(0) within the feasible domain, the interior point method constructs a barrier term B(x) for satisfied inequalities, while unsatisfied inequality and equality constraints lead to the creation of a penalty term p(x) through the external point method, employing a mixed penalty function approach.

To establish the barrier term B(x), it is essential that B(x) is continuous and non-negative, satisfying the condition B(x) → ∞ as x approaches the boundary of set S This ensures that x remains within the feasible field and does not reach the boundary point, effectively preventing any movement towards the boundary.

Building up the penalty term p(x) with the following conditions: p(x) is con- tinuous; For any x2R n , p(x) >= 0; if and only if x2S, p(x) = 0 The function of the mixed method is:

Fðx;rkị ẳfðxị ỵrkBðxị ỵ 1 rpðxị;

Bðxị ẳX i2I 1 g i ỵ ðxị;pðxị ẳX i2I 2 g i ỵ ðxị ỵX p j ẳ 1 ðhjðxịị 2 ð4:8ị

I1ẳ fijgiðx ðk1ị ị\0;i2Ig;I2ẳ fijgiðx ðk1ị ị 0;i2Ig;

Iẳ f1;2; .;mg ẳI1[I2r0[r1[r2[ .\r k1 [rk[ ., and limrk k!1 ẳ0 It can be determined by approaches of external point or interior point method.

Modern Optimization Algorithm

Basic Genetic Algorithm

Optimization design leverages modern probability theory and optimization methods, often resulting in models that are high-dimensional, nonconvex, and nonlinear, while also needing to satisfy various constraints Traditional optimization methods exhibit significant limitations when addressing such complex nonlinear problems, as their outcomes are heavily influenced by the choice of initial values and overly restrictive target functions Furthermore, traditional techniques struggle with discontinuous or non-derivative functions, rendering gradient-based optimization ineffective In contrast, genetic algorithms (GA), inspired by biological evolution, emulate Darwin's principle of natural selection and survival of the fittest, offering a robust alternative for solving these intricate optimization challenges.

The genetic algorithm (GA), inspired by the "survival of the fittest" and Mendelian genetic variation, is a powerful probabilistic global optimization search technique Its key characteristics include a group search strategy and the ability for individuals within a community to exchange information Unlike traditional optimization methods, GA does not rely on gradient information, making it particularly effective for complex and nonlinear problems This simplicity and global optimization capability position GA as one of the most efficient methods for tackling optimization challenges today Recently, its application has expanded significantly in engineering design and optimization.

Marine engineering extensively utilizes conceptual and preliminary design processes for ships, focusing on aspects such as line unplannedness and unobtrusiveness, ship movement, subdivision design, free-floating calculations, and structural optimization.

(1) The basic principle of genetic algorithm

Suppose an optimization problem maxjffðxịx2Xgj ð4:9ị

In this context, f represents a positive function defined on the solution space X, where for any element x in X, the value of f(x) is greater than or equal to zero The solution space encompasses all potential solutions to the problem at hand, which may either consist of a finite set or be a subset of the real number space.

Genetic algorithms address problem-solving by starting with a set of potential solutions, which undergo iterative processes to generate new solutions This collection of solutions is referred to as a population, represented by P(T), where t indicates the iteration step Throughout the evolutionary process, the population size, denoted as N, remains constant, and the individual solutions within P(T) are termed individuals Each individual's adaptability to the environment is measured by its fitness level During genetic operations, parent solutions are selected to combine and produce new offspring solutions, known as descendants.

(2) The implementation steps of basic genetic algorithm

The genetic algorithm solvesfive major elements of the solution: parameter coding, initial population, evaluation of fitness function, genetic operation (selection, crossover, and mutation), and control parameter settings.

The basic steps of the simple genetic algorithm are as follows:

(1) The solution to the problem under study is coded as a“chromosome”, and each code string represents a feasible solution to the problem.

(2) Randomly generate a certain number of initial coding strings PoP 0, which is a set of feasible solutions to the problem.

(3) Place the initial code string in the“environment”of the problem, and give the

fitness (evaluation) of each individual code string adaptation population in the population.

The initial population, denoted as PoP 0 or POP k, is determined by the individual fitness of the code strings A selection process is then conducted to randomly choose the paternal population F k, where high-performing individuals are replicated extensively, while less fit individuals are either copied minimally or eliminated altogether.

(5) Population C k is generated by cross probability P c for paternal population F k

(6) A new population POP (k + 1) for mutation operation of population C k with mutation probability P m

The genetic algorithm iteratively applies steps three through six, allowing the code string population to evolve across generations This process ultimately identifies the most suitable individual for the environment, leading to the optimal solution for the problem The fundamental flow and structure of a simple genetic algorithm are illustrated in Fig 4.2.

Niche Genetic Algorithm

Genetic Algorithms (GA) face challenges like premature convergence and slow convergence in later stages, limiting their effectiveness in certain optimization problems While various improvement methods have been proposed, many focus on only one aspect of the issue Enhancing solution accuracy often results in increased search time and precision Additionally, the use of a proportional selection operator in GA has significant drawbacks; when high-fitness individuals dominate, they rapidly proliferate, leading to reduced individual differences and severely diminished population diversity This lack of diversity is a primary factor contributing to the poor performance of GA in optimization tasks.

The niche genetic algorithm (NGA) enhances the global search capabilities of traditional genetic algorithms (GA) by addressing issues like premature convergence and limited local search efficiency By maintaining individual diversity within the population, NGA ensures both high global search ability and improved convergence speed The concept of a niche is rooted in the evolutionary principle that organisms coexist with their species, guiding the NGA to evolve populations in specific environments and prevent the overproduction of high-fitness individuals.

(1) The Biological Basis of Niche

Biologically, niche refers to the function or role of a tissue in a particular envi- ronment, and species refer to organizations that share common characteristics.Fig 4.2 Flowchart of basic genetic algorithm

Creatures often coexist with similar traits, leading to the formation of distinct niches influenced by their geographical locations This niche formation is biologically significant as it creates opportunities for new species to emerge Initially, species within a niche exhibit genetic differences due to relative isolation and limited gene exchange, preserving these variations over time Random variations within each niche contribute to increasing genetic diversity among species Additionally, varying geographical locations and natural environments result in different natural selection pressures, further enhancing genetic divergence Consequently, each species evolves uniquely, contributing to the near-infinite diversity observed in nature.

(2) Niche Based on Sharing Mechanism

In 1987, Goldberg and Richardson introduced a niche technology centered on a sharing mechanism that quantifies the degree of sharing among individuals in a group This sharing degree is calculated by summing the shared function values between one individual and all others in the group The shared function reflects the closeness between individuals, based on either genotype or phenotype similarity; closer relationships yield higher shared function values, while more distant relationships result in lower values.

In this context, let \( d_{ij} \) represent the closeness between individuals \( i \) and \( j \), which can be quantified using Hamming distance The shared function \( S \) indicates the degree of sharing for individual \( i \) within a group, expressed as \( S_i = \sum_{j=1}^{M} h(d_{ij}) \) This relationship highlights how the shared function can be formulated as \( S = h(d_{ij}) \).

0 others ð4:11ị d(i,j) indicates the Hamming distance (fitness distance) between two individuals which can be defined as: dijẳXiXjẳ

X M kẳ1 ðxikxjkị 2 vu ut ð4:12ị iẳ1;2; .;M1;jẳiỵ1;iỵ2; .M ð ị

In this study, we define Xi and Yj as the ith and jth individuals in a population of size M The parameter r, which is challenging to determine, is chosen based on experimental trials, and for this paper, we set r = 0.5 The constant a influences the shape of the shared function, typically set to a = 1 for a linear sharing function A higher shared function value indicates that two individuals are closer together Using this shared function, we can calculate the fitness of each individual, denoted as f' i = f(xi).

Neural Network

(1) BP neural network model structure

The Backpropagation (BP) neural network, introduced by Rumelhart and McClelland in 1986, is a highly effective algorithm in artificial networks It can approximate continuous functions over any closed interval, demonstrating strong modeling and analytical capabilities for nonlinear systems A typical BP neural network model is structured with three components: an input layer, a hidden layer, and an output layer, with the hidden layer potentially consisting of one or more layers.

The learning process of the BP neural network algorithm consists of two key stages: first, positive signal propagation, where the actual output values of each layer node are computed from the input layer to the hidden layer, with each node receiving input solely from the previous layer; second, error back-propagation, which occurs when the output layer does not achieve the expected results In this phase, the error between the actual and expected outputs is calculated recursively, and the weights of the previous layer are adjusted to minimize this error The network continuously modifies its weights and thresholds in the direction of decreasing error, ensuring that each change in weight is directly proportional to its impact on network performance.

Neural network theory demonstrates that a single hidden layer in a BP neural network can approximate any nonlinear function with finite discontinuities to arbitrary precision, provided there are enough nodes in the hidden layer However, increasing the number of hidden layers can lead to more error transmission links and reduced generalization performance Consequently, BP neural networks typically utilize a three-layer structure for optimal performance.

Let input and output(X P , T P )p = 1, 2,…, p:p as the number of training samples,

X P is the input vector for the pth sample, Xp = (x p1 ,…, x pM ), M is the dimension of input vector; T P is the output vector of the pth sample (expected output),

The output vector of the grid is denoted as O p = (o p1, …, o pN), where T P = (t p1, …, t pN) represents the input vector with a dimension of N The neural network architecture consists of a single hidden layer containing H nodes Connection weights between layers are indicated by w ij, which signifies the weight between the ith node of the previous layer and the jth node of the subsequent layer Both the hidden and output layers utilize sigmoid-type activation functions for processing The error function is defined as E = 1.

The algorithm steps of the three-layer BP neural network are as follows: Output of hidden layer nodes: yjẳfðnetjị ẳf X M iẳ1 xijxi

X i is the input of the ith input node, and y j is the output of the jth hidden layer nodes.

The output layer node o k is: o k ẳfðnetkị ẳf X H j ẳ 1 xjky j

!! ð4:16ị Fig 4.3 Three-layer BP neural network

Define the descending gradient dj djẳ @E

The weight of output layer and hidden layer nodes is proportional to the decreasing gradient and the updating formula of the weights: w ji ðtỵ1ị w ji ðtị ẳgdio i ð4:19ị

In the formula, the learning rate isη.ais the momentum factor, and they directly determine the amount of weight update.

But BP neural network also has some problems, mainly in the following aspects:

The BP algorithm, a prominent steepest descent method, faces challenges in determining the optimal training step size An excessively long step length may lead to a lack of calculation precision or even divergence, while a step length that is too short can slow down convergence due to increased iteration times To address these issues, employing an improved iterative algorithm can enhance the learning rate and accelerate convergence, or alternatives such as the conjugate gradient method and variable-scale methods can be utilized.

(2) Easy to fall into local minimum

Complex neural networks exhibit uneven error surfaces characterized by numerous local minima When employing the backpropagation (BP) algorithm to identify optimal solutions, it often becomes trapped in these local minima To effectively address this challenge, implementing global optimization methods is crucial.

(2) Approximate model of Elman neural network

The Elman neural network, introduced by Elman in 1990, is a dynamic recurrent artificial neural network derived from the Jordan network It functions as a recursive neural network featuring local memory units and feedback connections This architecture offers significant advantages in processing sequential data and capturing temporal dependencies.

The Elman neural network is capable of modeling a system without requiring knowledge of its internal parameters or operational details; it achieves this by simply adjusting the network weights.

The acceptance layer in an Elman network functions as a delay operator, significantly improving the network's ability to process dynamic information and accurately representing the system's dynamic characteristics.

The Elman network outperforms traditional feed-forward neural networks, such as Backpropagation (BP) and Radial Basis Function (RBF) networks, by effectively enabling dynamic modeling of systems This capability allows it to more accurately capture the dynamic mapping relationship between inputs and outputs.

The Elman neural network, built upon the foundation of the BP neural network, enhances its capability by storing internal states and utilizing them as dynamic mapping features, enabling the system to effectively adapt to time-varying characteristics.

The Elman neural network features a feed-forward architecture comprising an input layer, hidden layer, receptive layer, and output layer Similar to traditional feed-forward neural networks, the input layer transmits signals, while the output layer functions as linear weights The receptive layer is designed to store the output values from the hidden layer, facilitating feedback to the input.

The Elman neural network features a unique self-linking mechanism where the output of the intermediate layer is connected to its input through delays, enhancing its sensitivity to historical state data This internal feedback network improves the network's capability to process dynamic information, facilitating effective dynamic modeling The nonlinear state space representation of the Elman neural network includes key components such as the output node vector, connection weights between layers, input vectors, and transfer functions for both output and intermediate layer neurons, which collectively define its operational framework.

Output layer Fig 4.4 Elman neural network model

The Elman neural network is built upon the foundation of the BP neural network, utilizing the same learning algorithm This involves employing the gradient descent method to adjust the weights, with the learning index function defined as the sum of the squares of the error functions.

2ðydðkị yðkịị T ðydðkị yðkịị ð4:23ị whereyd(k) is the target output vector.

The dynamic learning algorithm is as follows:

Dw 3 ij ẳ#3d 0 i x j ðkị; iẳ1;2; .;m;jẳ1;2; .;n ð4:24ị

Dw 2 jq ẳ#2d h j uqðk1ị; jẳ1;2; .;n;qẳ1;2; .;r ð4:25ị

@w 1 jl ; jẳ1;2; .;n;lẳ1;2; .;n ð4:26ị where#3,#2, and#1 are the learning steps. d 0 i Ử đyd ; iđkỡ yđkỡỡg 0 i đỡ đ4:27ỡ d h j ẳX m i ẳ 1 đd 0 i w 3 ij ỡf j 0 đỡ đ4:28ỡ

@w 1 jl Ửf j 0 đỡxlđk1ỡ ợb@xjđk1ỡ

Particle Swarm Algorithm

(1) The origin of particle swarm algorithm

In 1987, biologist Craig Reynolds introduced a pioneering flock clustering model, emphasizing that individuals avoid collisions with nearby members while flying towards the flock center, effectively simulating flock behavior In 1990, Frank Heppner developed a bird model where birds are drawn to habitats without a specific target, using simple rules to guide their flight direction and speed, leading to collective movement into the habitat By 1995, social psychologist James Kennedy and electrical engineer Russell Eberhart, inspired by these bird behavior models, created the particle swarm optimization algorithm, adapting Heppner's model to guide particles through solution spaces toward optimal outcomes, as detailed in Kennedy's writings.

In a d-dimensional space, the velocity and position of the ith particle are represented as V i = (v i,1, v i,2, , v i,d) and X i = (x i,1, x i,2, , x i,d), respectively During each iteration, particles update their states by considering two optimal solutions: their individual best solution (pbest) and the global best solution (gbest) found by the entire population The velocity and position updates are computed using the formulas: v i ; j(t+1) = x i ; j(t) + c1 * r1 * (pbest - x i ; j(t)) + c2 * r2 * (gbest - x i ; j(t)) and x i ; j(t+1) = x i ; j(t) + v i ; j(t+1) for j = 1, 2, , d Here, the inertia weight factor, learning factors (c1 and c2), and random numbers (r1 and r2) play crucial roles in guiding the particles through their search process over t iterations.

The performance of particle swarm optimization algorithm depends largely on the parameter of the algorithm The selection principles of several important parameters are as follows [20]:

The optimal number of particles for an optimization problem varies based on its complexity Generally, 20 to 40 particles are recommended for standard optimization tasks, while simpler problems may only require around ten particles In contrast, highly complex optimization challenges often necessitate the use of over 100 particles to achieve effective results.

Learning factors, denoted as c1 and c2, play a crucial role in enabling particles to self-summarize and learn from the community's best practices, effectively bringing global knowledge closer Typically, setting both c1 and c2 to 2 yields optimal results, although their values may vary depending on the problem's complexity and difficulty Generally, c1 and c2 are equal and fall within the range of 0 to 4.

The inertia weight coefficient plays a crucial role in the particle swarm algorithm by influencing the degree to which particles inherit their current velocity Properly selecting this coefficient can enhance the balance between exploration and exploitation capabilities of the particles The basic steps of the particle swarm algorithm are essential for optimizing performance.

(1) Randomly initialize the position and velocity of each particle in the population.

To assess the performance of each particle, evaluate its fitness and record its current position and fitness in pbest Identify all individuals with optimal fitness in pbest and save their fitness and locations in gbest.

(3) Use formula (4.30) and (4.31) to update the particle velocity and position.

Evaluate each particle's fitness against its previously experienced best position If a particle's current fitness surpasses its personal best, update its personal best to the current position Next, compare all personal bests to identify the global best position and update it accordingly.

(5) If the stop condition is satisfied, stop the search and output the result, otherwise return to step 3 to continue the search.

(1) Improved particle swarm optimization (IPSO) algorithm I

The Particle Swarm Optimization (PSO) algorithm exhibits strong search capabilities; however, it faces challenges such as prolonged convergence time and lower optimization accuracy, often leading to local optima To enhance both global search and local development in PSO, adjusting the inertia factor \( x \) is crucial This inertia factor determines the next flight speed of a particle based on its current velocity, where a larger \( x \) helps prevent the algorithm from getting trapped in local optima, while a smaller \( x \) can accelerate convergence in later stages Common approaches for setting inertia weights include constant methods, linear decreasing methods, and adaptive methods.

To enhance the convergence of the Particle Swarm Optimization (PSO) algorithm and mitigate the risk of local optima, the inertia factor \( x \) is adjusted using a randomly distributed number This approach addresses the instability from a linear decrease in \( x \) in two key ways Firstly, if the algorithm approaches the best solution early in the process, the random adjustment may yield a smaller \( x \), thereby accelerating convergence Conversely, if the optimal solution is not identified initially, the linear decrease in \( x \) can hinder convergence; however, the introduction of random overshoot can help bypass this limitation The updating formula for the inertia weight coefficient is defined as follows: \( x = l + rN(0,1) \) where \( l = l_{min} + (l_{max} - l_{min}) \cdot rand(0,1) \), with \( N(0,1) \) representing a standard normal random variable and \( rand(0,1) \) a random number between 0 and 1.

The Particle Swarm Optimization (PSO) algorithm benefits from adjustable learning factors that enhance the speed of particle searches and minimize the risk of local extremum entrapment Initially, particles exhibit a stronger self-learning capability, promoting global search efficiency, while later stages shift towards increased social learning, aiding convergence to optimal solutions By employing time-varying learning factors, the PSO algorithm addresses the limitations of fixed parameters, allowing for dynamic adjustments throughout the optimization process The updated learning factors can be defined by the formulas: \(c_1 = c_{1, ini} + \frac{(c_{1, fin} - c_{1, ini})}{t_{max} - t}\) and \(c_2 = c_{2, ini} + \frac{(c_{2, fin} - c_{2, ini})}{t_{max} - t}\), where \(c_{1, ini}\) and \(c_{2, ini}\) represent the initial values, while \(c_{1, fin}\) and \(c_{2, fin}\) denote the final values of the learning factors.

final iterative values of c 1 andc 2 , respectively.

Specific IPSO algorithm I steps are as follows:

(1) Randomly initialize the positions and velocities of each particle in the population.

Evaluate the fitness of each particle and record its current position and fitness in the particle's personal best (pbest) Identify all individuals with optimal fitness from pbest and consolidate their fitness values into the global best (gbest).

(3) Use formula (4.30) and (4.31) to update the velocity and position of the particles.

(4) Use formula (4.32) and (4.33) to update the weights.

(5) Use formula (4.34) and (4.35) to update the learning factor.

Evaluate the fitness of each particle against its personal best position If a particle's fitness surpasses its previous best, update this position as the current best Subsequently, compare all updated personal best values (pbest) with the global best (gbest) and make necessary updates to gbest.

(7) If the stop condition is satisfied, stop the search and output the result, otherwise return to step 3 to continue the search.

In this article, we explore the application of hybridization in genetic algorithms, utilizing random number distributions as outlined in formulas (4.32) and (4.33) During each iteration, a specified number of particles are selected for the hybridization pool based on hybridization probability These particles then randomly combine to generate an equal number of offspring particles, which replace their parents The position of the offspring is determined by the equation: child(x) = p * parent1(x) + (1 - p) * parent2(x), where p is a random number between 0 and 1 Additionally, the velocity of the offspring is calculated using the formula: child(v) = (parent1(v) + parent2(v)) / (|parent1(v)| + |parent2(v)|).

Specific IPSO algorithm II steps are as follows:

(1) Randomly initialize the positions and velocities of particles in a population.

(2) Evaluate thefitness of each particle, store the current position andfitness of the particles in thepbestof each particle, select all the individuals with the optimal

fitness ofpbest, and then store those individuals’fitness and location ingbest.

(3) Use formula (4.30) and (4.31) to update the velocity and position of the particles.

(4) Use formula (4.32) and (4.33) to update the weights.

Evaluate the fitness of each particle against its best-known position If a particle's current fitness surpasses its previous best, update its best position accordingly Then, compare all updated personal bests (pbest) to determine the global best (gbest) and make necessary updates to gbest.

In the hybridization process, a designated number of particles are chosen and placed into a hybridization pool These particles are then randomly paired for crossover, resulting in an equal number of progeny particles The positions and velocities of the progeny are computed using specific formulas, while the parameters pbest and gbest remain constant throughout the process.

(7) If the stop condition is satisfied, stop the search and output the result, otherwise return to step 3 to continue the search.

Hybrid Optimization Algorithm

Hybrid Algorithm I

Traditional gradient-based optimization algorithms face significant challenges in ship linear optimization design due to the complex interplay of disciplines like rapidity, seakeeping, and maneuverability, where no analytical expressions exist between performance indicators and design variables This leads to high computational costs as gradient information relies on numerical analysis, and these algorithms converge slowly for strong nonlinear problems, often only reaching local optima sensitive to initial conditions In contrast, modern optimization techniques like genetic algorithms excel in global search capabilities but struggle with local searches, requiring extensive computations to identify the global optimal point To enhance optimization efficiency, it is essential to integrate the strengths of both methods, leveraging their respective advantages.

The hybrid genetic algorithm integrates a floating-point coding genetic algorithm with a constrained scaling method to enhance both the speed and likelihood of finding a global solution This approach employs genetic operators—selection, crossover, and mutation—guided by a penalty function derived from nonlinear programming problems The primary goal is to direct the solution closer to the global optimum while supplying initial values for the constrained variable operator, which addresses the original nonlinear programming problem.

Fig 4.5 Comparison of convergence between NLP method and GA method problem directly, in order to exert its advantage of strong local search ability The calculation principle is as follows [21]:

The mathematical model for a nonlinear programming problem can be expressed as minimizing the objective function \( f(x) \) subject to specific constraint conditions \( c_j(x) = 0 \) for \( j = 1, 2, \ldots, n_c^0 \) and \( c_j(x) \geq 0 \) for \( j = n_c^0 + 1, n_c^0 + 2, \ldots, n_c \), where both the objective function and the constraints are continuously differentiable up to the second order.

In the hybrid genetic algorithm, the selection, crossover, mutation, and other genetic operators are presented in the form of penalty function in formula (4.44) minfpðxị ẳfðxị ỵM1

X nc 0 jẳ1 maxf0;cjðxịjg ỵM2

The algorithm utilizes an exact penalty function, denoted as f in formula (4.45), with M1 and M2 being significantly large fixed normal numbers Its primary objective is to expand the search range, guiding solutions closer to the optimal result while providing an initial value for the constrained scaling method operator By integrating genetic operators—such as selection, crossover, and mutation—the algorithm achieves extensive search capabilities, while the constrained variable-scale method enables efficient local searches, effectively combining the strengths of both the genetic algorithm and the constrained scaling approach.

The constrained variable metric method effectively addresses nonlinear programming challenges by initially transforming the equation from formula (4.45) to (4.46) This transformation facilitates the resolution of a series of quadratic programming sub-problems, specifically aiming to minimize the function represented as minQP(d) = f^T(x_k)d.

2d T B k d s:t:cjðx k ị ỵcjðx k ịdẳ0jẳ1;2; .;nc 0 ð4:46ị c j ðx k ị ỵc j ðx k ịd0jẳnc 0 ỵ1;nc 0 ỵ2; .;nc

In each iteration, the search direction \( d_k \) is analyzed using formula (4.46) An approximate one-dimensional search is then conducted along this direction to determine the step length \( T_k \) This process leads to the update of the sequence \( x_{k+1} = x_k + T_k d_k \), progressively moving closer to the optimal solution.

This project integrates the constrained variable metric method into the floating-point genetic algorithm, functioning alongside selection, crossover, and mutation processes The resulting hybrid genetic algorithm effectively addresses optimization problems.

The genetic algorithm parameters are established, encompassing the population size (m), the number of variables (n), the crossover probability (P c), the mutation probability (P m), the constrained variable-scale method search probability (P CVM), and the maximum evolutionary algebra (T) permitted in genetic computation.

To initialize the populations, a random initial population is generated, and the precise penalty value is calculated using formula (4.46) The fitness value for each individual, denoted as f_i, is determined based on the objective function value, with f_max representing the maximum objective function value among the current population members This value is then adjusted using the Goldberg linear scaling transformation model, expressed as f'_i = af_i + b, where i ranges from 1 to m.

(3) Perform a selection operation on the proportional selection operator.

Crossover operations utilize the P c arithmetic crossover operator to combine two selected matrices, t i and t j This process generates two new matrices: s t i, which is derived from a weighted combination of t j, and s t j, which is created from a weighted combination of t i Here, r represents a random number within the range of [0,1].

To implement a nonuniform mutation operator based on a given probability Pm, select an element vk from the individual represented as s t i ẳ ðv1;v2; .;vk; .;vnị for mutation The outcome of this variation is represented as s t i ỵ 1 ẳ ðv1;v2; .;v k0 ; .;vnị, where vk 0 is modified using the formula vk ỵ Dðt;x u k vkị randð0;1ị ẳ0, and vk Dðt;xkvk 0 ị randð0;1ị ẳ1.

In the context of maximum algebra, T represents the highest algebraic value, while b denotes the coefficient parameter that influences nonuniformity The value D(x, y) falls within the range of [0, 1], indicating that as T increases, the probability of D(x, y) approaching 0 also rises This characteristic enables operators to conduct a more uniform search in the initial phase.

In the constrained variable-scale optimization method, each individual is optimized using P CVM When an individual \( s_{t}^{i} \) is chosen for this optimization search, the process begins with \( s_{t}^{i} \) as the initial point in formula (4.44), resulting in optimized outcomes represented as the subgeneration \( s_{t+1}^{i} \).

(7) Calculate the individual fitness value, and perform the optimal individual preservation strategy.

When the genetic calculation achieves the maximum permissible algebra T or shows no improvement in the best individuals across successive generations, the results are finalized, and the calculation concludes If these conditions are not met, the process reverts to step (3) to repeat the operations The program flowchart illustrating this process is depicted in Fig 4.6.

Hybrid Optimization Method II

The optimization system design space is extensive, making it impractical to calculate results for every location within it To address this, a hybrid genetic algorithm is employed to select representative and accurate sample points through experimental design methods, aiming to achieve optimal test results with minimal experimentation Various experimental design methods exist, including full factorial design (FFD), fractional factorial, central composite design (CCD), and Latin hypercube design (LHD) Notably, Morris and Mitchell introduced the optimal Latin hypercube design (Opt LHD) in 1995, which improves uniformity in sample selection and enhances the algorithm's effectiveness in optimizing the design space.

The fitting of factors and responses is more precise and reliable, leading to improved space-filling and balance The method's matrix generation involves setting m test points and n factors, which together form an n x m matrix represented as x = [x1, x2, x3, , xm].

Theith analysis: x T i ẳ ẵxi1;xi2;xi3; ;xin ð4:50ị

A random Latin hypercube algorithm is employed to create an initial design matrix, which is subsequently refined through element exchange The optimization process focuses on enhancing space filling by evaluating the minimum and maximum distances between elements, as described by the formula \(d(x_i, x_j) = d_{ij} = \sum_{k=1}^{n} x_{ik} x_{jk}^T\).

In the formula t = 1 or 2; 1 i; j m; i 6= j, the sampling point d(x_i, x_j) represents the minimum distance between points x_i and x_j Figure 4.7 illustrates the distribution of sample points in Latin hypercube matrices and an optimal Latin hypercube design involving three factors and nine experiments The figure demonstrates that the sample points in the optimal Latin hypercube design are distributed more uniformly and effectively, resulting in improved filling and a more accurate representation of spatial distribution.

(a) Latin hypercube (b) Optimal Latin hypercube

Fig 4.7 Comparison of Latin hypercube and optimal Latin hypercube design matrix

Non-Linear Programming by Quadratic Lagrangian (NLPQL) is an advanced gradient optimization method that enhances the stability of the traditional Sequential Quadratic Programming (SQP) algorithm This approach utilizes a Taylor series expansion of the objective function and linearizes constraints through quadratic programming to identify the next design point The process begins with an initial guess, x₀, followed by a linear search involving two alternative optimization functions to ensure global convergence The iterative update is expressed as xₖ₊₁ = xₖ + αₖ dₖ, which is executed only if the feasible search scheme confirms the movement step in the desired direction.

The SQP algorithm updates matrix B_k using standard techniques in unconstrained optimization To enhance algorithm performance, the NLPQL algorithm incorporates a variable metric method known as BFGS This approach constructs B_{k+1} by scaling matrix B_k to better approximate the Hessian matrix during the update process With certain security guarantees in place, it is ensured that all matrices B_k remain positive definite.

Search along the search direction, determine the step size, update the design variables

Solve the quadratic programming subproblem by formula, and determine the new search direction

Determine whether (error) is established

Updating the matrix by formula

Fig 4.8 Calculation fl owchart of NLPQL

2d T Bkdỵ rfðx k ị T d;d2R n ð4:52ị Constraint equation: rgiðx k ị T dỵgiðx k ịiẳ1;2;3 .p ð4:53ị rhjðx k ị T dỵhjðx k ị ẳ0iẳ1;2;3 .m ð4:54ị

The specific iterative steps are shown in Fig 4.8:

Optimization Platform

ISIGHT Optimization Platform

ISIGHT is an advanced software platform developed by American Engineous that automates design optimization across various industries, including aviation, automotive, shipbuilding, machinery, and chemicals By leveraging multi-disciplinary optimization and quality engineering design methods, ISIGHT significantly boosts the digitalization and modernization of domestic manufacturing It specializes in multi-disciplinary design optimization and process management, effectively addressing the challenges of iterative processes and automating data handling The platform employs a range of optimization techniques, such as numerical iterative algorithms, search algorithms, heuristic algorithms, design of experiments (DOE), and response surface modeling (RSM), to streamline multi-disciplinary design efforts.

(1) Process integration: a complete design of integrated environment

(1) Multi-disciplinary code integration + Process automation;

(2) Hierarchical, nested task organization and management;

(3) Real-time control + Post-processing;

(4) Scripting language + API customization + MDOL language secondary development.

(2) Optimized design: Advanced exploration Toolkit

(1) Experimental design + Mathematical programming + Approximate modeling + Quality design;

(2) Knowledge rule system + Multi-criteria tradeoffs;

(3) Open architecture: Third-party (optimization/test) algorithm embedding, multi-disciplinary optimization strategy research and implementation.

(1) Parallel computing + distributed computing services;

4.4.1.2 Task Structure of Multi-disciplinary Design Optimization

The basic elements of MDOF are tasks, each consisting of three modules:

(1) Analysis module: including the implementation of documents, inputfiles, and outputfiles;

(2) Data module: including design parameters, constraint parameters, the objective function parameters, and communication parameters;

(3) Technical parameter modules: including optimization techniques, approximate technical parameters.

The task execution module's interface is designed through parameter mapping, comprising input files, executable files (discipline analysis code), and output files By manipulating the input and output files, users can effectively manage the task execution module without needing to understand the encoding or operation of the analysis code This approach allows designers to focus solely on providing the necessary information in the input and output files, ensuring the repeatability of the discipline analysis code.

4.4.1.3 Application of ISIGHT in Ship-Type Optimization

ISIGHT offers designers an interface software that seamlessly integrates multiple applications, including CAD and CFD, into a cohesive loop This software automates the design process by facilitating design modifications, running simulations, and analyzing results in a continuous cycle Ultimately, this leads to achieving optimal designs or reaching established design limits, ensuring a fully automated workflow.

4.4.1.4 Integration Method of ISIGHT and CA/CFD Software

Catia: Method 1: Simcode command cnext.exe-macro *.vbs

ProE: Method 1: Simcode command proe2000i trailfile.txt

UG: Method 1: Simcode command ug_update_expressions.exe -p *.prt–e

*.exp Method 2: ISIGHT UG components

Solidworks: Method 1: Simcode command Cscript *.vbs

ICEM-CFD: icemcfd -batch -script icem_script

Gambit: “\Fluent.Inc\Gambit2.3.16\ntbin\ntx86\gambit.exe”-inp *.jou

StarCCM + : Method 1: Simcode command starccm + -batch*.java

CFX: cfx5solve.exe-def cfx5build.exe-b -play

4.4.1.5 The Optimization Integration Process of Ship Form Based on ISIGHT

The ISIGHT optimization platform integrates various software tools essential for ship-type optimization, including CAD software for hull geometry reconstruction and objective function calculation tools like ProE and Sculptor This platform primarily incorporates resistance calculation software such as FLUENT and STAR-CCM+, utilizing its optimization algorithms or integrating existing ones to create an automated ship-type optimization system This system enables the entire optimization process to be conducted without manual intervention, ultimately resulting in a hull shape that delivers optimal performance.

Friendship

The Friendship framework, developed by Lloyds Friendship Systems, is a comprehensive ship parameterization software that integrates parametric modeling, optimization algorithms, and an optimized integration framework This software effectively generates and modifies ship models, optimizing their hydrodynamic performance Numerous domestic and international applications have demonstrated the practicality of ship-based optimization using the Friendship software Additionally, it serves as a CAE design platform that seamlessly combines CAD and CFD functionalities.

(1) Full parametric and semi-parametric modeling based on functional surface technique By setting parameters and constructing corresponding characteristic curves, the establishment of complex parametric model can be realized.

Once the parameters are defined, the software automatically adjusts the model, utilizing a range of single-target and multi-objective optimization algorithms to enhance the specified objective function.

The software boasts robust integration capabilities, allowing seamless connections with various external applications and CFD simulation programs through a standard interface This functionality facilitates efficient pre- and post-processing tasks, including grid division and result visualization.

Fig 4.9 Ship-type optimization based on ISIGHT platform

Remote or distributed computing capabilities allow for the efficient use of idle resources, enabling long-term and intensive tasks, such as program generation and optimization, to be completed in a significantly reduced timeframe.

(1) The principle of Friendship parametric modeling

The parametric model of Friendship facilitates the swift creation and alteration of ship types through its characteristic parameters and curves Selecting and modeling these parameters and curves are crucial steps in the parametric design process, significantly influencing the overall quality of the ship's design.

The Friendship parametric modeling process is as follows:

The selection of key parameters, including global factors like captain, width, and draught, along with local factors such as curve tangent and fullness, is essential in controlling both the longitudinal and cross-sectional curves of a design.

(2) Construct the longitudinal curve based on the above characteristic parameters to achieve the correlation between them.

The construction of cross-sectional curves involves utilizing global characteristic parameters and longitudinal curves Initially, a reference cross-sectional curve is generated through program code features, followed by the use of the Curve Engine to facilitate the creation of cross-sectional curves for each station.

The hull surface is constructed using the Curve Engine, starting with the identification of the first and last positions By integrating the cross-sectional curves through the Meta Surface, a smooth and seamless hull surface is achieved.

Friendship, a parametric modeling software, is often integrated with SHIPFLOW to create an automated ship optimization platform To facilitate seamless data transfer between Friendship, SHIPFLOW, and wave resistance calculation software, it is essential to convert the IGES model file into the SHF format An interface program can then reconstruct the value data from the SHF file into a hull curve, enabling the generation of a new model file through hull curve interpolation This process employs a cubic B-spline curve for fitting and interpolating the hull curve.

The interface program's workflow begins by reading the model file in SHF format Next, it identifies the various hull curves from the SHF file's standard data format These curves are then re-fitted, interpolated, and sorted Finally, the program generates the model file required for calculating wave drag.

(3) Friendship software integration optimization system

Friendship software has a strong external program integration capabilities; it mainly provides three different integration mechanisms: COM integration interface, custom integration through XMLfiles, and generic integration through ASCII templates.

1 Lin Y, Zhu Z, Ji Z, Dai Y (1997) Form description with function parameter [J] Shipbuild China, 3:74 – 78

2 Jiang JS, He CX, Pan SH (2008) Optimization method [M] South China University of Technology Press, Guangzhou

3 Xi Shao Lin, Zhao Feng Zhi (1983) Optimization method [M] Shanghai Science and Technology Press, Shanghai

4 Xing Wen Xun, Xie Jin Xing (1999) Modern optimization method [M] Tsinghua University Press, Beijing

5 Bei-yue Z, Guang-qi G (2001) Research of improving genetic algorithms based on niche technology [J] J Yueyang Norm Univ (Natural Sciences), 14(4):18 – 21

6 Zhou GQ (1999) Practical discrete optimization approach for ship structures based on genetic algorithms [J] Ship Technol Res, 46(3):179 – 188

7 Xuan GN, Cheng RW (2000) Genetic algorithm and engineering design [M] Science Press, Beijing

8 Della Cioppa A, De Stefano C (2004) On the role of population size and niche radius in

fi tness sharing [J] IEEE Trans Evol Comput 8(6):580 – 592

9 Vieira DAG, Adriano RLS (2004) Treating constraints as objectives in mul- tiobjective optimization problems using niche pareto genetic algorithm [J] IEEE Trans Magn 3 (40):1188 – 1191

10 Kim JK (2002) Niche genetic algorithm adopting restricted competition selection combined with pattern search method [J] IEEE Trans Magn 38(2):1001 – 1004

11 Zhou Y-J, Wang L-Z (2006) An application of elitist model niche genetic algorithms in optimization of functions [J] J Liuzhou Teachers College, 21(1):107 – 110

12 Xin X-J1, 2, Pan J3, Xiao G-Z (2005) New proxy signature scheme based on Schnorr signature scheme [J] J Chongqing Univ Posts Telecommunications (Natural Science), 17(6):

13 Pereira CM, Sacco WF (2008) A parallel genetic algorithm with niching technique applied to a nuclear reactor core design optimization problem [J] Prog Nuclear Energy, 1 – 7

14 Ke F, Li Y (2014) The forecasting method of landslides based on improved BP neural network [J] Geotechnical Investigation Surveying, (8):55 – 60

15 Liu Y, Li R, Li C (2014) Application of BP neural network and support vector machine to the accumulated temperature interpolation [J] J Arid Land Resources Environ 28(5):158 – 165

16 Gao Peng Yi (2012) Study on the optimization of backpropagation neural network classi fi er [D] Huazhong University of Science and Technology, Wuhan

17 Wang HF, Zhao M, Meng G (2002) 3-D shape prediction and global approximate optimization of plastic ball grid array (PBGA) solder joints [J] J Shanghai Jiaotong University, 36(6):829 – 833

18 Liu JH (2009) The research of basic theory and improvement on particle swarm optimization [D] Central South University, Changsha

19 Li YL (2014) The study of particle swarm algorithm based on multi-objective optimization and its application [D] Chengdu: Southwest Jiaotong University

20 Guang-yu G, Chen G, Li Y-X (2014) Fractional-order control of USV course based on improved PSO algorithm [J] Syst Eng Electron 36(6):1146 – 1151

21 Lujia Z, Fang Z, Shan ZP (2001) The parameters of generator are identi fi ed by genetic algorithm and constrained variable scale optimization [J] Hunan Electric Power, 21(3):5 – 7

22 Yin X (2015) Fundamental research on thermal reliability engineering design for cylinder head [D] Zhejiang University, Zhejiang

23 Lai YY (2012) ISIGHT parameter optimization theory and examples [M] Beijing University of Aeronautics and Astronautics Press, Beijing

24 Huang J-F, Wan S-L (2012) The parametric method of FRIENDSHIP hull form based on design feature [J] Chin J Ship Res, 7(2):79 – 85

The Optimization of the Hull Form with the Minimum Wave-Making

Resistance Based on Potential Flow

Overview

This study utilizes the Michell integral and Rankine source methods to optimize the design parameters of a double-triangle ship modification function, focusing on drainage volume as a key constraint It establishes an optimal design model using nonlinear programming (NLP), traditional genetic algorithms (SGA), and niche genetic algorithms (NGA), while accounting for the effects of tail-viscosity separation An innovative ship linear optimization design program with independent intellectual property rights has been developed and validated through experimental results The findings significantly advance the transition of ship design from traditional experience-based approaches to knowledge-based methodologies.

The Optimization of the Hull Form with Minimum Wave-Making Resistance Based on Michell Integral Method

Establishment of the Ship-Type Optimization Model

In this study, the total resistance \( R_T \) is chosen as the objective function for the optimization design process It is defined as the sum of wave resistance \( R_W \) and the frictional resistance \( R_F \) of a flat plate.

R T ẳpR W ỵ ð1ỵkị R F !min ð5:1ị © Shanghai Jiao Tong University Press, Shanghai and Springer Nature

B.-J Zhang and S.-L Zhang, Research on Ship Design and Optimization

Based on Simulation-Based Design (SBD) Technique, https://doi.org/10.1007/978-981-10-8423-2_5

The RW is determined using the Rankine source method and is subsequently adjusted by a correction coefficient, p, based on the ratio of theoretical to experimental wave resistance values at the initial ship design speed Additionally, the form factor, k, is chosen based on model testing results.

The Michell integral formula is expressed by the following form.

U 1 2 k 2 ZịdXdZ where U ∞ is the design speed and q is the mass density of the fluid g is the acceleration of gravity,F X (X,Z) is the longitudinal slope of the hull waterline.

The frictional resistance of aflat plate is calculated by the following formula.

The frictional resistance coefficient (C f0) for a flat plate is determined using the Songhai formula This coefficient is influenced by the wetted surface area (S), which is derived from the hull coordinates and can be approximately calculated using a tent function.

Cf0 ẳ 0:4631 lg Re ð ị 2 : 6 ð5:5ị whereR e is the Reynolds number based on the body length.

ReẳUL m ð5:6ị whereUis the speed,Lis the characteristic length and here is taken as the design waterline,mis the viscosity coefficient offluid motion.

The offsets of ships are directly selected as design variables.

The constraints are mainly considered to satisfy the geometric constraints and drainage volume requirements, there are two as follows:

(1) All Ship offsets are nonnegative, namely: yði;jị 0;

(2) Ensure the necessary displacement volume, namely:

VV0 whereV 0 andVare the displacement volumes of the original hull and the improved hull, respectively.

The SUMT interior point method is a powerful technique for optimization in nonlinear programming By incorporating an additional term that reflects constraint conditions, it effectively transforms an unconstrained optimization problem into a solvable format.

The Data File of the Ship-Type Optimization Based

This ship-type optimization document utilizes the Michell integral method and outlines the scope of design optimization, including the number of design variables and initial parameters for the optimization calculations It also details the key elements of the initial model, design speed, and hull values.

Examples

The optimization design range of the ship's front body encompasses six sections, extending from station 14 to the hull's head The bow section to station 14, along with the hull bottom and the design waterline surface, are established as fixed parameters Comparative analyses are illustrated in Figures 5.2, 5.3, and 5.4, showcasing the horizontal sections and waterlines of both the modified and original ships Figure 5.5 presents a cross-sectional view of the wave patterns for both designs, while Figure 5.6 depicts the wave resistance coefficients of the modified and initial ship models Additionally, Figure 5.7 illustrates the free surface waveforms for both the modified and original ships.

The S60 ship model's optimal design range is illustrated in Fig 5.1 To streamline the optimization process, the bow shape is defined as the forward perpendicular (F.P.), while the stern shape is adjusted to the after perpendicular (A.P.) The ship's width is set to 0 from A to A.P along the ship length and from 0.75W.L to B.L in the draft direction, as depicted in Fig 5.8 Figures 5.9, 5.10, and 5.11 compare the horizontal sections and waterlines of both the modified and original ships, while Fig 5.12 presents a cross-sectional view of the wave patterns for both designs Additionally, Fig 5.13 provides a graphical representation of the optimization scope for the Wigley hull form.

The comparison of body plans between the modified ship and the original Wigley ship highlights differences in wave-making resistance coefficients Additionally, the free surface waveform analysis illustrates the variations between the modified and initial ships Optimization calculations based on the Michell integral method are detailed in Table 5.1, showcasing the results of these assessments.

The optimized hull features a distinctive cross-sectional shape with two stops near the head, showcasing a bulging design towards the sides and a bow-shaped bow, as illustrated in the bodyline and waterline diagrams.

“non-overhanging ball head”in the book However, the shape of the bulbous bow is exaggerated and lacks practical significance While such a bulbous bow is good for

Fig 5.3 Water plan of the original ship (Wigley)

Fig 5.4 Water plan of the modi fi ed ship (Wigley)

The comparison of wave profiles between the modified ship and the original Wigley ship indicates that while the modified design achieves a noticeable reduction in wave resistance coefficient within a specific range of the designed Fourier number, it also presents challenges such as complicated processing and difficulties in the initial installation and operation of equipment However, the improvements in wave height and waveform of the modified ship are not significant.

Fig 5.6 Comparison of wave-making resistance coef fi cient for modi fi ed ship and original ship (Wigley)

Fig 5.7 Wave patterns (2 g f /U 2 ) of modi fi ed ship and original ship (Wigley)

Fig 5.8 Comparison of hull lines before and after modi fi cation of S60 hull form

Fig 5.9 Comparison of body plans of the modi fi ed ship and the original ship (S60)

Fig 5.10 Water plan of the original ship (S60)

Fig 5.11 Water plan of the modi fi ed ship (S60)

Fig 5.12 Comparison of the wave pro fi les along modi fi ed ship and the original ship (S60)

Fig 5.13 Comparison of wave-making resistance coef fi cient of modi fi ed ship and the original ship (S60)

The Optimization of the Hull Based Rankine Source Method

Establishment of the Hull Form Optimization Model

In this study, the optimization design process focuses on minimizing the total resistance (R_T), which is defined as the sum of wave resistance (R_W) and viscous resistance, represented as (1 + K)R_F.

RT ẳpRWỵ ð1ỵkị RF !min ð5:7ị where the meaning of symbols is the same as above, the form factorKis calculated by following formula [1] kẳ0:11ỵ0:128B

TheR W is calculated by the Rankine source method and the other parameters are the same as above.

The wave resistance coefficient, denoted as \(C_{W,L}\), is calculated based on the ship length between perpendiculars (\(L\)), breadth (\(B\)), draft (\(T\)), block coefficient (\(C_B\)), and the design speed (\(U_\infty\)).

(2) Design variables and the scope of optimization design

The optimal design range focuses on the front half of the ship's body, with fixed parameters including the designed waterline, the vessel's bottom, and the front end of the hull, as illustrated in Fig 5.15.

In the ship optimization process, design variables are defined by the parameters of the ship modification function The shape of the modified ship, denoted as y(x,z), is determined by multiplying the initial ship type, f0(x,z), by the ship modification function, w(x,z) This relationship can be expressed as y(x,z) = f0(x,z) * w(x,z).

X n amnsin pð xx0 x min x 0 ị m ỵ 2 sin pðbz bỵTị n ỵ 2 ð5:11ị m;nẳ1;2;3; L=2x wðx;zị ẳ1X m

The article discusses the optimization of hull form parameters, specifically focusing on the relationship between various design variables such as draft (d), longitudinal length (L), and characteristic parameters (x0, xmax) By fixing certain variables (m, n = 1, 2, 3), the optimization process is streamlined, resulting in a total of nine variables for A mn and a maximum of ten design variables overall This approach effectively reduces the complexity of the ship modification function, leading to enhanced optimization calculation efficiency.

(3) The constraint conditions are the same as above

(4) The optimization method is the same as above

Optimization Process of Hull Form

The ship-type optimization calculation process begins with the input of an initial ship-type value file, which contains essential elements such as the main characteristics and types of the initial hull, the design range, the number of design variables, the design speed, and the initial parameters necessary for optimization calculations.

Fig 5.15 Scope of optimization design of the S60 hull form

The wave resistance is computed using the Rankine source method, and the total resistance is determined by adding the wave resistance to the equivalent plate frictional resistance This sum serves as the objective function within a nonlinear programming (NLP) optimization framework, which is subject to basic constraints If the optimization does not converge, the process reverts to the initial state and repeats Upon achieving convergence, the optimization concludes, yielding the design for a ship with minimum total resistance.

Examples

The meshing techniques for the Wigley hull and free surface are consistent with those outlined in Section 2.5.1 The optimization range is illustrated in Figure 5.16, while Figures 5.17 and 5.18 provide a comparative analysis Figure 5.19 depicts the wave profile of both the modified and original ships, and Figure 5.20 compares the wave resistance coefficients of these vessels Additionally, Figure 5.21 showcases the free surface waveforms for both the modified and original ships.

The meshing of the S60 hull form and free surface mirrors the methodology outlined in Section 2.5.2 Figures 5.1, 5.22, and 5.23 illustrate the comparison of body lines and waterlines between the modified ship and the original design Additionally, Figure 5.24 presents the wave profile of both ships, while Figure 5.25 displays the wave resistance coefficient for the modified and original vessels Figure 5.26 showcases the free surface waveforms of both designs The optimization results, derived from the Rankine source method, are summarized in Table 5.2.

The Rankine source method reveals that while the optimized design of a ship with minimal resistance does not require a large bulbous bow, the potential for significant drag reduction is notable The hull shape of the improved vessel demonstrates this enhancement, as illustrated in the comparison between the modified ship and the original design (Wigley).

Fig 5.18 Water plans of the modi fi ed ship (Wigley)

Fig 5.19 Comparison of the wave pro fi les along the modi fi ed ship and the original ship (Wigley)

A comparison of the wave-making resistance coefficient between a modified ship and the original Wigley ship reveals significant improvements At the designed speed, the wave resistance for the two modified ship types decreased by 13.5% and 23.4%, while total resistance saw reductions of 4.3% and 9.8%, respectively Additionally, the modified ships maintained a relatively stable wave height, exhibiting a clearer waveform and a distinct Kelvin wave system shape.

Fig 5.21 Wave patterns (2 g f /U 2 ) of the modi fi ed ship and the original ship (Wigley)

Fig 5.22 Comparison of body plans of the modi fi ed ship and the original ship (S60)

Design of Ship Hull with Minimum Wave Resistance

This article explores the optimization design of the S60 hull model to minimize wave-making resistance using the Rankine source method By adjusting constraints and design variables, various specific design schemes are presented in Table 5.3.

The optimization design focuses on the front half of the hull form, specifically from the tenth station to the fore-body Modifications to the hull bottom, stem, and stern profiles are not permitted during the ship-type modification process, as illustrated in Fig 5.27 Additionally, the water plane may be either fixed or adjustable, as detailed in Table 5.3.

The optimized calculations for the three design schemes were completed in 5, 4, and 4 iterations using the NLP method As a result, the wave resistance of the modified ship was significantly reduced by approximately 24.8%, 21.5%, and 18.6%, respectively A comparison of the body plans highlights the differences between the modified ship and the original design, as illustrated in Fig 5.23, showcasing the water plans of the modified ship (S60).

The comparison of wave profiles between the modified ship and the original S60 ship reveals significant design differences In the first scheme, from the fore-body to station S.S.18, the modified ship's profile lines shift inward towards the hull's medial side between the baseline and 4 W.L However, from S.S.18 to S.S.11, the profile lines transition outward to the lateral side of the hull, with a gradual decrease in offset observed between the baseline and 7 W.L This analysis highlights the structural adjustments made in the modified ship's design.

Fig 5.25 Comparison of wave-making resistance coef fi cient for the modi fi ed ship and the original ship (S60)

The wave patterns of the modified ship differ significantly from those of the original ship (S60), with the fore part adopting a U-shaped profile due to the displacement volume shifting from the upper to the lower area In the second design scheme, the water plane area of the modified ship has been altered, as it was prioritized for optimization, resulting in increased freedom for the adjacent water plane This modification is observed from the fore-body to S.S.11.

The modified ship's hull lines and profile lines exhibit a lateral shift, particularly notable between S.S.18 and S.S.16, while the body lines from the baseline to the 4 W.L remain relatively unchanged The alterations in design scheme three closely resemble those in design scheme two, with a slight reduction in hull line changes due to the constraints of the additional modifications.

Table 5.2 Optimization results using the Rankine source method for Wigley and S60 ships

Table 5.3 Design parameters and design conditions

Design Scheme 1 Design Scheme 2 Design Scheme 3

Design speed Fr = 0.285 Fr = 0.285 Fr = 0.285

Fore-body of hull Water plane and fore-body of hull

Water plane and fore-body of hull

Note ▽ 0 , ▽ are the displacement volumes of the original ship and the modi fi ed ship, respectively,

A w0 , A w are the water plane areas of the original ship and the modi fi ed ship, respectively.

Fig 5.27 Scope of optimization design

Fig 5.28 Comparison of body plans of the original ship and the modi fi ed ship (design Scheme 1)

Fig 5.29 Water plan of the modi fi ed ship (design Scheme 1)

Fig 5.30 Comparison of body plans of the original ship and the modi fi ed ship (design Scheme 2)

Fig 5.31 Water plan of the modi fi ed ship (design Scheme 2)

Fig 5.32 Comparison of body plans of the original ship and the modi fi ed ship (design Scheme 3)

Fig 5.33 Water plan of the modi fi ed ship (design Scheme 3)

The wave-making resistance coefficients for three hull designs at a Froude number of 0.285 show a significant reduction compared to the original hull As illustrated in Fig 5.34, the modified ship designs achieve lower wave resistance coefficients with increased design speed.

The wave profiles calculated along the hull are illustrated in Figs 5.35, 3.36, and 5.37, highlighting the free surface elevation at panels near the ship's surface Notably, the enhanced hull design produces a marginally larger bow wave compared to the original ship, accompanied by an increase in bow wave steepness, as detailed in Table 5.4.

Figure 5.38 illustrates the comparison of wave-making resistance coefficients between the original and modified ships, highlighting a significant reduction in resistance The optimized designs demonstrate decreased wave-making resistance across a broad spectrum of design speeds.

Figures 5.39, 5.40, and 5.41 illustrate the nondimensional wave patterns for both the modified and original ships, highlighting distinct differences in the wavefields produced by each vessel.

The optimization process utilizes the Rankine source method to minimize wave resistance, with ship-type modification parameters serving as design variables An optimal mathematical model is established, incorporating displacement as a primary constraint and additional water plane constraints By applying the NLP method, the bow body of the S60 ship model is optimized, resulting in a hull design that minimizes wave resistance This approach yields three modified ship designs based on three distinct optimal design schemes while maintaining the specified design speed.

Fig 5.35 Comparison of the wave pro fi les along the original ship and the modi fi ed ship (designScheme 1)

Fig 5.36 Comparison of the wave pro fi les along original ship and the modi fi ed ship (design Scheme 2)

Fig 5.37 Comparison of the wave pro fi les along the original ship and the modi fi ed ship (design Scheme 3)

Table 5.4 Optimization results using the Rankine source method with three design schemes

The modified ship demonstrates a significant reduction in wave resistance, achieving a decrease of 24.8% in scheme one, 21.5% in scheme two, and 18.6% in scheme three compared to the original design Additionally, the modified ship features a smoother profile that closely resembles the actual ship, confirming the effectiveness of the method based on the Rankine source approach.

Fig 5.38 Comparison of wave-making resistance coef fi cient for original ship and the modi fi ed ship

Fig 5.39 Wave patterns (2 g f /U 2 ) of the original ship and the modi fi ed ship (design Scheme 1)

To minimize total resistance in hull optimization, it is essential to account for viscous resistance alongside wave resistance Our next step will involve formulating the objective function as the combined sum of these two resistance components.

Fig 5.40 Wave patterns (2 g f /U 2 ) of the original ship and the modi fi ed ship (design Scheme 2)

Fig 5.41 Wave patterns (2 g f /U 2 ) of the original ship and the modi fi ed ship (design Scheme 3)

Optimization Design of Ship with Minimum Resistance Based

Ship Form Optimization Model

The objective function, design variables, and constraints align with those outlined in Section 5.3.1 Drawing on the insights of Japanese scholar Yasuhiro Akihiro, the penalty function method is employed to convert the constrained optimization problem into an unconstrained one Subsequently, the SGA (Simple Genetic Algorithm) is utilized for optimizing the design of ships to minimize resistance.

The total resistance R T is selected as the objective function to optimize the calculation, andR T can be expressed as the following form:

RTẳpRWỵ ð1ỵkị RFỵa ðr0 rị ỵa 1

!min ð5:13ị whereais the penalty factor, (1) wheny(i,j) 0, and▽> ▽0,a= 0; (2) when y(i,j) < 0 and▽< ▽0,a=∞ Other parameters can be obtained using the method in Sect.5.3.1.

Ship-Type Optimization Based on Basic Genetic

The effectiveness and practicality of SGA have been validated through real-world applications To optimize a specific problem, users must carefully select genetic operators, algorithm parameters, and methods for handling constraints, as these choices significantly influence the optimization outcomes Consequently, this section presents a mathematical model of SGA tailored for ship-type optimization challenges.

This section employs real-coded chromosomes, with each chromosome represented by real numbers that match the dimensions of the design variables Consequently, ship parameters are encoded using binary digits of 0 and 1, and the arrangement of ship-type genes in a specific sequence forms the ship-type chromosome.

To evaluate a chromosome, it is essential to establish a suitable fitness function This involves decoding each chromosome to extract a specific set of design parameters, which are then used to calculate the corresponding objective function based on those parameters.

The competitive selection method involves selecting individuals based on their adaptability, where K individuals are randomly chosen to create a small population in each generation From this group, the most adaptive individuals are selected to advance to the next generation, while the replicated individuals return to the parent population for future random selections This process of random selection is repeated M times to produce M next-generation individuals.

(1) K individuals are randomly selected from the t-th generation population;

(2) Compare the fitness of K individuals, and the individual with the maximal

fitness enter into the (t + 1)th generation, and the replicated individuals remain in the t-th generation;

(3) Repeat①,②M times until the same number of individuals are produced as in generation t.

This section discusses the uniform crossover technique, where two genes are exchanged at each locus between two matched individuals This process occurs with a consistent crossover probability, resulting in the creation of two new individuals.

The mutation probability, denoted as Pm = 0.20.9^n, is influenced by adaptive strategies, where 'n' represents genetic algebra As the population exhibits enhanced traits, the mutation probability decreases, effectively mitigating premature convergence.

(6) Constraint handling and judging function

The strategy for addressing constraints in optimization includes various approaches such as rejection, repair, improved genetic operator, and penalty strategies In this context, the penalty strategy is specifically utilized for strict constraint problems The evaluation function, after incorporating the penalty component, is defined as: eval(x) = 10.5 f(x) + 10.7 p(x) - 5.14.

In this formula:f(x) is the objective function;p(x) is the sum of the constraint function values which are all normalized and greater than zero.

Traditional Stochastic Genetic Algorithms (SGA) often use a fixed number of cycles as a termination condition This method can lead to premature termination before achieving the desired calculation accuracy or excessive cycles beyond the accuracy threshold To address this issue, we propose using the target value of the optimal chromosome that meets the specified accuracy as the new termination condition.

By enhancing specific genetic operators in the Simple Genetic Algorithm (SGA) and applying this improved SGA to ship optimization design, an optimal ship type with minimal resistance has been achieved The process is illustrated in the flowchart depicted in Fig 5.42.

The Optimization of the Hull Form Based on NGA

Programming with FORTRAN language, as follows:

To begin, initialize the counter \( t \) to 1 and represent the hull form parameters using binary digits (0 and 1) Next, randomly generate the initial population \( P(t) \) consisting of \( M \) individuals, while also establishing the maximum number of evolution generations \( T \).

The fitness evaluation of all individuals is performed, saving the first

In a population of individuals sorted by fitness value in descending order, a select group of N individuals (where N is less than M) is identified These individuals do not engage in the processes of selection, crossover, or variation Instead, they directly execute niche elimination operations alongside mutated individuals.

Fig 5.42 Flowchart of the SGA

The competition selection method is used to select individuals from the popu- lationP (t), and then generates a new population P(t)′.

The arithmetic crossover operation is executed in the population based on the crossover probability \( P_c \) When two identical individuals are selected for crossover, one individual undergoes an even crossover, resulting in the generation of a new population \( P(t)'' \).

According to the probability of mutation P m , uniform mutation operation is performed in the populationP (t))″, and then new populationP (t)‴is generated withMindividuals.

Using Niche generation operation throughfitness sharing method, a new pop- ulation with M+N individuals is generated by putting N saved individuals and

P(t)‴together, then carry out the niche elimination operation.

Sorting new population with M +N individuals by new fitness value in descending order, and memorizing the first N (N< M) individuals again, then taking thefirstMindividuals of arrangement as a new generation population.

Ift T, thent=t+ 1, generatingMindividuals in (7), and taking them as the next-generation population P(t), then go to step (3); If t>T, terminating the algorithm, and outputting the calculation result.

In the present study, the calculations use populations 60, crossover probability 0.50 and mutation probability 0.06.

(2) Flowchart of algorithm is shown in Fig.5.43.

Examples

In this section, the S60 ship model is chosen for optimization, employing SGA and NGA methods to enhance the ship's design The optimal design parameters focus on the first half of the hull, including the fixed design ranges for the waterline, bottom, and both ends of the ship, as illustrated in Fig 5.27 The meshing process for the ship and free surface follows the same methodology outlined in Section 5.3.2.

The optimization results based on NGA and SGA are detailed in Table 5.3, highlighting the differences between the modified and original ship models, with Figures 5.44 to 5.47 illustrating that the modified profile from NGA exhibits a greater outward swelling tendency compared to SGA Figure 5.48 presents the wave resistance coefficient curve, indicating that the modified ship from NGA has a lower wave resistance coefficient than SGA within a specific design Fourier number range Additionally, Figure 5.49 tracks the total drag coefficient during the optimization of the modified ship at Fr = 0.285, revealing that SGA converges faster initially; however, after approximately 90 generations, it experiences local convergence and reduced diversity In contrast, the NGA ship model shows significant improvements approximately every 35 generations, with the best results achieved by the 250th generation Overall, NGA maintains a stable evolutionary speed and high population diversity, facilitating the search for a global optimal solution (Table 5.5).

Figures 5.50 and 5.51 illustrate the waveforms of the modified and original ships as obtained by NGA and SGA In Figure 5.50, the wave height of the modified ship remains relatively unchanged, while significant reductions in wave height are observed in other areas Conversely, Figure 5.51 shows an increase in both the bow and stern wave heights of the modified ship, with only a slight decrease noted at the ship's center.

Figure5.52and5.53are the free surface waveforms of the modified (upper) and the original (bottom) NGA and SGA models obtained, respectively, at Fr = 0.285.

The diagram illustrates that the modified ship design exhibits a distinct Kelvin wave pattern, with transverse and scattered wave regimes confined within ±9º28' Notably, the free surface waveforms produced by the modified ship, as analyzed by NGA, are more defined and pronounced.

The Comparisons of the Optimization Result Between

Figure5.54and5.55represent the reduction effect and time-consuming comparison of NLP, SGA, and NGA, respectively, it can be seen from thefigure that the NGA

Fig 5.44 Comparison of body plans of the original ship and the modi fi ed ship (NGA)

Fig 5.45 Water plan of the original ship (SGA)

Fig 5.46 Comparison of body plans of the original ship and the modi fi ed ship (NGA)

Fig 5.47 Water plan of the modi fi ed ship (NGA)

Fig 5.48 Comparison of wave-making resistance coef fi cient curves between the original ship and the modi fi ed ship

The optimization process demonstrates that the total resistance coefficient converges effectively, achieving the best reduction in resistance In contrast, the nonlinear programming (NLP) approach requires the least time but results in the least effective reduction of resistance.

Traditional optimization methods offer faster optimization compared to genetic algorithms; however, as the number of design variables increases, the optimization speed of genetic algorithms becomes comparable to that of traditional methods.

Table 5.5 Optimization results based on the SGA method and NGA method

Fig 5.50 Comparison of the wave pro fi les along the original ship and the modi fi ed ship (NGA)

The comparison of wave profiles between the original ship and the modified ship (SGA) highlights that traditional optimization algorithms are directly impacted by the number of design variables, resulting in slower optimization speeds as the number of variables increases In contrast, the genetic algorithm's efficiency is determined by the population size rather than the number of design variables, allowing for quicker calculations in each iteration This approach enables the optimization of each design variable, enhancing overall optimization speed Additionally, the wave patterns (2 g f /U 2) of both the original and modified ships (NGA) further illustrate these differences in performance.

Fig 5.53 Wave patterns (2 g f /U 2 ) of original ship and the modi fi ed ship (SGA)

The genetic algorithm demonstrates competitive optimization speed compared to traditional methods, particularly when enhanced with parallel computation techniques This makes it especially advantageous for problems involving multiple design variables Additionally, in complex systems like ships that feature multiple extremums, the genetic algorithm excels at identifying optimal solutions effectively.

The integration of probability in genetic algorithms (GA) facilitates the emergence of innovative ship types during optimization processes, significantly contributing to the advancement of new vessel designs This method also highlights the time-consuming nature of comparing natural language processing (NLP) with GA in identifying global and local optimum points, especially in scenarios with multiple local optima.

5.5 Optimization of Ship Type with Minimum Resistance Considering Viscous Separation

To achieve optimal loading and design speed conditions, minimizing resistance is crucial in ship design Historically, ship designers have relied on wave-induced resistance theory, with Suzuki Heffer's thesis serving as a notable example However, most studies have focused on optimizing the ship's shape while maintaining a constant stern line This approach overlooks the complexities involved in designing the after-body shape, which must account for factors such as wave resistance, viscous effects, seakeeping, and propulsion performance Consequently, there is limited research on optimizing the tail line type in ship design.

This article focuses on optimizing ship design to minimize wave-making resistance while preventing viscous separation at the tail By employing a two-dimensional turbulence separation judgment formula, the study evaluates airfoil shapes and streamlines on the hull surface to develop a minimum resistance ship model The method's effectiveness is validated against experimental data Using the S60 ship as a baseline, the design considers various influencing factors, aiming for an optimal hull shape The Rankine source method, grounded in potential flow wave resistance theory, along with the SUMT interior point method in nonlinear programming, facilitates the optimization process The separation points on the hull's streamlines are assessed pre- and post-optimization to ensure that the modified ship type does not experience increased viscous resistance while focusing on minimizing wave resistance.

Viscous Water Resistance

Viscous water resistance arises from the friction between the wet surface of a ship and the water, acting perpendicular to the hull According to boundary layer theory, this viscous effect is confined to a thin layer adjacent to the hull surface The characteristics of viscous flow can be modeled using a two-dimensional boundary layer on a flat plate, where the ship's speed is represented as the incoming velocity \( U_2 \) at the stationary hull The flow rate is denoted as \( U_1 \), with the boundary layer thickness represented as \( d \) It is essential for water to adhere to the plate, indicating no relative slippage, which results in a flow rate of zero on the plate itself At a short vertical distance \( d(x) \) from the plate, the flow rate equals \( U_2 \) When the Reynolds number \( R_{nx} = \frac{U_1^2 x}{\nu} \) is below approximately \( 10^5 \), the flow remains laminar.

Flow separation significantly contributes to the viscous resistance experienced by a hull When flow detaches from the hull's surface, it creates a larger area of viscous interaction, leading to alterations in pressure distribution and heightened shape resistance Flow can separate not only from sharp edges but also from smoother surfaces, as illustrated in Figure 5.56, which demonstrates the occurrence of separations in two-dimensional flow.

When analyzing flow dynamics, if the velocity at point S is zero and backflow occurs, point S is considered significant If the velocity remains zero behind point S, it indicates that flow separation does not occur at this point This scenario is beneficial because a zero velocity at the object surface implies that shear stress values are also zero Achieving this effect can be accomplished through careful design of the hull surface.

The location of points within the boundary layer of a plate is influenced by the pressure gradient along the hull surface and the flow state, whether laminar or turbulent, in the upstream boundary layer at the separation point Boundary layer theory indicates that a positive reverse pressure gradient is a crucial condition for flow separation to occur.

As long as@p=@xẳ qU e dU e =dxis noticed, this can be verified from Eq (5.15).

In calculus, the condition for flow separation, represented by du/dy = 0, indicates that u(y) reaches a local maximum or minimum In this context, u(y) should achieve a minimum value, implying that the second derivative, @²u/@y², is positive at the separation point S Furthermore, it is evident that a necessary condition for flow separation to occur is that @p/@x is positive at the hull surface's separation point Notably, when the flow along the plate is constant at @p/@x = 0, flow separation does not take place, as illustrated in Fig 5.57.

Ship-Type Optimization Model

The objective function in this section is the same as in Sect.5.3.1(1).

The optimal design range for this section encompasses the entire ship, extending from station zero to the foremost station 20 Key reference points include the design waterline, the ship's bottom, and its front and rear ends, as illustrated in Fig 5.58.

The formula of ship modification function is shown in formulas (5.9) and (5.10)

The constraints in this section are the same as in Sect.5.2.1(3)

Fig 5.57 Two-dimensional boundary layer fl ow

Examples

The S60 hull has been chosen for the initial design optimization of the ship, with the mesh configuration of both the hull and the free surface of the S60 ship model remaining consistent with the specifications outlined in Section 5.5.2 (Table 5.6).

The optimization results of the entire ship, based on the Rankine source method, are summarized in Table 5.1 Figures 5.59 and 5.60 illustrate the comparisons of bodylines and waterlines between the modified and original ships, revealing significant variations in the body line, particularly between the head and tail areas These alterations in the stern line shape are believed to potentially cause viscous separation, which may impact the ship's overall performance, necessitating an assessment of this separation A simple two-dimensional turbulence separation criterion is employed to evaluate this, with Figure 5.61 presenting the wave resistance coefficient curves for both the original and modified ship models Additionally, Figures 5.62 and 5.63 display the wave profiles and free surface waveforms for the original and modified ship models, respectively.

Ship Optimization Process

The ship-type optimization calculation process is illustrated in Fig 5.64 Initially, the original ship-type value file is inputted, containing key elements such as the ship's design range, number of design variables, design speed, and original optimization parameters Subsequently, the Hess–Smith method and streamline tracing method are employed to facilitate the optimization design of the S60 hull form, as depicted in Fig 5.58.

Table 5.6 Optimize calculation results based on Rankine source method

Fig 5.59 Comparison of body plans of the modi fi ed ship and the original ship (S60)

Fig 5.60 Water plans of the modi fi ed ship (S60)

The wave-making resistance coefficient of the modified ship is compared to that of the original ship (S60), revealing differences in velocity distribution across the hull's surface By calculating the separation points on each streamline, the separation domain of the original ship is identified Utilizing nonlinear programming methods alongside basic constraints, the improved ship form is optimized A simple two-dimensional separation judgment is then applied to determine the separation points on the improved streamlines, allowing for an assessment of the separation area and confirming whether the modified ship exhibits reduced separation compared to the original design Additionally, wave profiles along both the modified and original ships (S60) are compared to further analyze performance improvements.

The wave patterns (2 g f /U²) of the modified ship compared to the original S60 ship type indicate that if the modified ship's performance exceeds that of the initial ship, adjustments to the constraints will be made, and the optimization process will be recalibrated Conversely, if the modified ship does not surpass the initial ship's performance, the optimization calculation concludes, resulting in the identification of the ship with the minimum total resistance.

Separation Judgment Method

(1) Two-dimensional simple separation of judgment condition

Ship-type optimization focuses on minimizing total resistance while preventing tail separation Given the complexities of 3D turbulence separation, this approach employs a simplified two-dimensional model to address the issue The flowchart in Fig 5.64 illustrates the process for assessing streamline separation on the hull surface, allowing for an approximate identification of the separation domain.

The graph illustrates the relationship (U = U ∞) between the pressure gradient and local surface friction stress in a two-dimensional turbulent boundary layer, as analyzed through the pressure distribution by Tanaka and Tatsuno.

The local friction on the object's surface, denoted as Gẳ sw qU, is influenced by the flow velocity at the outer edge of the boundary layer, represented as U ∞ The thickness of the motion loss is indicated by h, while S refers to the distance along the object's surface The separation point is defined where sw equals zero.

The analysis indicates that when G equals zero, the value of C varies with the Reynolds number R_h (defined as U_∞ h/m) For the model with a Reynolds number R_L (U_∞ L/m), the associated plate Reynolds number R_h is approximately 5.5 × 10^3 By incorporating the relevant curve for R_h into Fig 5.65, the separation point C can be determined, yielding a value of about -0.03.

Fig 5.65 Relationship between pressure gradient and local friction stress on the surface

From the two-dimensional turbulent boundary layer theory, it is shown that h can be calculated by the following formula: hðU 1 h c ị 1 4 ẳ c

Substitute the above formula into Eq (5.16), it can obtain that:

U 1 4 ds ð5:19ị Now define a new function C(s):

U 1 4 ds ð5:20ị The value of C(s) corresponding to the separation pointCs = 0.03 is:

In this section, we utilize formula (5.21) as a criterion to determine the separation of streamlines on the object's surface The separation point can be easily identified by calculating the flow velocity U ∞ (s) at the boundary layer's outer edge, which can be approximately substituted with the object's surface potential velocity.

After determining the stacking mode solution, the induction velocity at each field point can be calculated, specifically at point A as illustrated in Fig 5.66 The sweep angle is denoted as b, while a represents the angle between the induced velocity direction and the X-axis To find the positions of the upper and lower streamlines based on the induction rate at point A, the coordinates will be represented as C, C coordinates: dx 1.

In this study, the distance dx between points on the first streamline is determined by the separation of the ith and (i + 1)th points, which is defined by the swept angle This same interval dx is applied to the other streamlines when the points are also separated by the swept angle However, using a direct solution can lead to significant errors, particularly when the streamlines exhibit curvature, as illustrated in Fig 5.66, where the distance between points C and D on the actual streamline is notably larger To address this issue, the fourth-order Runge-Kutta method is employed, resulting in improved accuracy.

The steps to separate the hull surface are as follows:

(1) The velocity distribution on the hull surface is calculated by Hess–Smith method in this section;

(2) The velocity distribution on the streamline of the hull surface is calculated by the streamline tracking method;

(3) Calculate the distribution of C (s) on each streamline, and judge whether the separation occurs by formula (5.21), and determine the position of the sepa- ration point;

(4) Connect the separation points on each streamline tofind the separation domain.

If we know the velocity distribution outside the boundary layer, we can calculate

The two-dimensional simple turbulence separation criterion, defined by formula (5.19), is evaluated for accuracy by comparing calculated values of two-dimensional objects with experimental results The S60 model, chosen for optimization calculations, closely resembles actual ship types, allowing for a relevant analysis Consequently, this section references the experimental findings from Bai et al.'s thesis, specifically utilizing results from their second model.

The calculation results for a two-dimensional mast ship, as depicted in Figure 5.67, indicate that the velocity distribution outside the boundary layer, represented as U ∞ /U, can be estimated using the potential velocity on the object's surface However, there is a noticeable discrepancy between the calculated and experimental values, suggesting that the approximation may not fully account for the actual viscous flow If the viscous flow were accurately considered, the C curve would shift slightly backward, resulting in a corresponding adjustment of the calculated point Additionally, the separation phenomenon further complicates the analysis.

Fig 5.66 Schematic of streamline tracing

The comparison of two-dimensional turbulent separation points from various airfoils with experimental data reveals significant discrepancies, indicating potential instability in results Nonetheless, utilizing the two-dimensional separation discriminant, which accounts for the influence of Rn, remains a valuable general criterion This section approximates the boundary layer on ship surfaces and calculates the velocity distribution of potential flow using the Hess–Smith method.

The streamline velocity distribution for the S60 hull at full load waterline is illustrated in Figure 5.68 The curve C indicates that the distribution near the hull bow does not reach -2, preventing separation However, separation and attachment phenomena are observed near the stern The turbulent separation domains are identified through the analysis of the streamlines.

The analysis of velocity and separation distribution along the streamlines reveals significant findings regarding the performance of ship hull designs The shadowed area in the figures indicates the range of separation, with Fig 5.69 illustrating the separation domain of the original ship, measuring 3.446, while Fig 5.70 depicts the modified hull's separation domain at 3.304 A comparison of the lateral projection areas shows that the optimized ship has a smaller separation domain than the initial design This indicates that the viscous resistance remains relatively unchanged during the ship-type optimization process, which primarily aims to reduce wave resistance.

Ship Model Towing Test Results

Towing tests for ship models were conducted at the Shanghai Ship and Shipping Research Institute's towing tank, utilizing two models: the parent ship-series 60 and the theoretically optimized modified ship form S60-1 These tests aimed to validate the optimization theory by employing a bounded ship model while disregarding the effects of heave and pitch The dimensions of the models were 3.0 meters in length, 0.4005 meters in width, and 0.1605 meters in draft, with a 3 mm high torrent nail placed at 10 mm intervals along the 9.5 station of the ship model.

The resistance value of the ship model (R tm) is accurately measured using the NS-30 electricity resist graph from Japan, which has a maximum range of 10 kg (100 N) and an impressive accuracy of 0.1% The data recording system consists of a high-speed data acquisition card, amplifiers, and an IPC Collected data is subsequently input into the analysis system, specifically focusing on the turbulence separation area of the S60 original hull, as illustrated in Fig 5.69.

The turbulence separation area of the S60-1 improved hull was analyzed using computer calculations Total resistance in the test results was determined through a three-dimensional conversion method, while the flat friction coefficient was derived using the 1957 ITTC formula.

The total drag coefficient of ship model,

Ctm ẳCfmð1ỵkị ỵCwm ð5:23ị The total drag coefficient of the ship,

C ts ẳC fs ð1ỵkị ỵC wm ð5:24ị where the subscripts m and s represent the ship model and the ship, respectively.

Therefore, we can get the total drag coefficient of the real ship.

The total resistance of the ship

2qsSsv 2 s Cts ð5:26ị The wave resistance of the ship,

The total resistance and wave-making resistance results for the modified ship (Series 60-1) and the mother ship (Series 60) are illustrated in Figs 5.71 and 5.72 Near the design speed point, the modified ship demonstrated a significant improvement in both total resistance and wave-making resistance compared to the mother ship.

Fig 5.71 Comparison of total resistance coef fi cient test results consistent with theoretical calculation results by reducing 3.5 and 21%, confirming the applicability of the method.

This section discusses the Rankine source method for calculating wave resistance in potential flow, with total resistance as the objective function Total resistance is defined as the sum of wave resistance and equivalent plate friction resistance By treating ship modification parameters as design variables and maintaining displacement as a basic constraint, the SUMT interior point method in nonlinear programming optimizes the ship's line shape Modifications to the stern profile may increase viscous drag; therefore, this section focuses on minimizing wave-making resistance while controlling the tail-separating domain A simple two-dimensional separation criterion is applied to both the initial and modified models to identify separation points on each streamline and assess the separation domain If the separation domain remains relatively unchanged, it indicates that viscous resistance does not significantly increase during hull optimization, with wave resistance prioritized as the main objective function.

Optimization Problem

Hull Form Optimization Based on the RANS-CFD Technique

Hull Form Optimization Based on the Approximate

Optimization Problem