Chi YANG, Fuxin HUANG

Center for Computational Fluid Dynamics, George Mason University, Fairfax, Virginia, USA,

E-mail: cyang@gmu.edu

An overview of simulation-based hydrodynamic design of ship hull forms*

Chi YANG, Fuxin HUANG

Center for Computational Fluid Dynamics, George Mason University, Fairfax, Virginia, USA,

E-mail: cyang@gmu.edu

This review paper presents an overview of simulation-based hydrodynamic design optimization of ship hull forms. A computational tool that is aimed to accomplishing early-stage simulation-based design in terms of hydrodynamic performance is discussed in detail. The main components of this computational tool consist of a hydrodynamic module, a hull surface modeling module, and an optimization module. The hydrodynamic module includes both design-oriented simple CFD tools and high-fidelity CFD tools. These integrated CFD tools are used for evaluating hydrodynamic performances at different design stages. The hull surface modeling module includes various techniques for ship hull surface representation and modification. This module is used to automatically produce hull forms or modify existing hull forms in terms of hydrodynamic performance and design constraints. The optimization module includes various optimization algorithms and surrogate models, which are used to determine optimal designs in terms of given hydrodynamic performance. As an illustration of the computational tool, a Series 60 hull is optimized for reduced drag using three different modification strategies to outline the specific procedure for conducting simulation-based hydrodynamic design of ship hull forms using the present tool. Numerical results show that the present tool is well suited for the hull form design optimization at early design stage because it can produce effective optimal designs within a short period of time.

simulation based design, ship hull form optimization, radial basis function, NURBS, hydrodynamics

Introduction

Hydrodynamic design of ships involves several stages, from preliminary and early-stage design to latestage and final design. Hydrodynamic optimization is an important aspect of ship design. The computational fluid dynamics (CFD) based simulation tools can thus be used for evaluating the hydrodynamic performance of a design, or design alternatives during a design process.

In order to compare the merit of different designs quantitatively, the multi-objective functions that measure ship hydrodynamic performances are defined. These objective functions can be evaluated using CFD-based simulation tools for a given design, i.e., a hull form associated with a set of design parameters/variables. An optimal hull form that exhibits the best hydrodynamic performance can then be obtained via an optimization technique. Therefore, the simulation-basedhydrodynamic design/optimization tool usually consists of a CFD module that can be used to compute the flow field and evaluate the objective functions, a hull surface modeling module that can be used to produce hull forms via given sets of design parameters/variables, and an optimization module that can be used to minimize the objective functions under given constraints.

Preliminary and early hydrodynamic design requires the evaluation of large number of objective functions during iterative procedures. The use of high fidelity models in design optimization, especially at early design stage, can be prohibitively expensive. In order to reduce the computational cost and still being able to rank different designs, it is essential to have computational tools that account for essential (but not necessarily all) relevant physics, and are highly efficient (with respect to CPU and user input time) and robust. Thus, linear potential flow assumptions may be in order for this stage of the design. As the design progresses, the level of physical realism needs to be upgraded, leading to Euler, RANS, and perhaps VLES computations for the final stage of the design. Therefore, it is necessary to develop both potential-flowbased simple CFD tools and Euler/RANS/Navier-Stokes based high-fidelity CFD tools. These different CFD tools can be used at different stages of the design. The simple CFD tools can be used to evaluate the hydrodynamic performance, such as drag and seakeeping, in the optimization process and the high-fidelity CFD tools can be used for the validation of the hydrodynamic performance of the final optimal hull forms obtained.

In order to perform simulation-based hydrodynamic design optimization of ship hull forms, it is very important to develop accurate and effective hull-surface representation and modification techniques to ensure: (1) Only a small number of parameters (design variables) are required to represent an existing hull form or produce a new hull form from scratch, thus minimizing the number of objective function evaluations, (2) Large variation of hull forms can be obtained by modifying given parameters (design variables), thus producing different type of hull forms, (3) Modified region can join the original design smoothly without discontinuities when only a part of the hull can be modified, (4) Practical hull form (both three-dimensional fairness and manufacture practicability) can be preserved and various geometrical constraints can be easily implemented in the optimization process.

Various optimization techniques can be adopted to minimize the objective functions. It is essential to include optimization techniques that are capable of finding local and/or global minimum for either single objective or multi-objective functions. These optimization techniques can be integrated into the optimization module. A suitable optimization technique can be selected from the optimization module to meet design needs.

In summary, the simulation-based hydrodynamic design/optimization of ship hull forms generally requires the following steps: (1) define an initial hull form with a set of design parameters/variables using hull surface modeling technique, (2) simulate the flow for the given hull form using CFD tools and evaluate the objective functions in terms of the flow solutions obtained, (3) minimize the objective functions using a optimization technique and update the design variables. Steps (1) to (3) are repeated until the solutions are converged, thus generating an optimal hull form with the best hydrodynamic performance. It is clear that extensive computing time is required to conduct simulation-based hydrodynamic design/optimization of ship hull forms if the flow for the given hull form is simulated by high-fidelity CFD tools during the iterations.

With the rapid development of computer hardware and numerical techniques in recent years, CFD-based computational tools have become more and more popular in analyzing hydrodynamic performance of ships. Some of these tools have been used to conduct hull form optimization studies[1-15], which have demonstrated that CFD-based hydrodynamic design optimization is extremely valuable. However, most of these tools cannot be used in the early-stage design of ship hull forms because they are very computing intensive, and hard to use for non-CFD experts. An efficient and effective simulation-based computational tool for the early-stage design of ship hull forms has been developed by the authors and their colleagues over the years[10,12,13,16-28]. An overview of this integrated computational tool and its application to the innovative hydrodynamic design of ship hull forms at the earlystage design are presented in details in this review paper.

The main components of this simulation-based hydrodynamic design and optimization tool consist of a hydrodynamic module, a hull surface modeling module, and an optimization module. The hydrodynamic module includes both design-oriented simple CFD tools and high-fidelity CFD tools. These integrated CFD tools are used for evaluating hydrodynamic performances at different design stages. The hull surface modeling module includes various techniques for ship hull surface representation and modification. This module is used to automatically produce hull forms or modify existing hull forms in terms of hydrodynamic performance and design constraints. The optimization module includes various optimization algorithms and surrogate models, which are used to determine optimal designs in terms of given hydrodynamic performance.

Many ship hull geometry modeling techniques have been reported in the literature[29-39]for the representation and modification of hull forms. These modeling techniques can be classified into two categories: conventional modeling and parametric modeling[29]. Conventional modeling techniques build on a lowlevel definition of geometry. For example, points are used to define curves, and curves are used to define surfaces. NURBS representation of the hull surface can be regarded as one of the conventional modeling techniques of the hull form, and some of the NURBS’control points can be taken as design variables for hull form modification in the CFD-based hydrodynamic optimization of the hull form. Although conventional modeling techniques offer the great flexibility with regard to geometry and topology, special cares are necessary to ensure the generation of the reasonable and practical hull forms if conventional modeling techniques are used for the hull form optimization. In addition, the conventional modeling of the hull form requires a large number of design variables. Parametric modeling techniques, on the other hand, build on highlevel entities. These entities are called form parameters in geometric modeling. The major advantage of parametric modeling techniques is that small to intermediate modifications can be produced very efficiently and only a small number of design variables are required. However, the parametric modeling of the hull sur-face does not allow for large variation of the hull form during optimization cycles. The advantages and disadvantages of each class of modeling techniques have been well documented in the literature[29]. It should be noted that many of these geometry modeling techniques are available in existing popular computer aided design (CAD) system, such as CASES from Friendship system[29], NAPA[37], Rhino and others. The hull surface modification techniques are usually developed in close relation to the hull surface modeling techniques adopted.

In order to benefit from both conventional modeling and parametric modeling techniques, a number of techniques have been developed for the hull-surface representation and modification. Specifically, a parametric hull form representation and modification technique associated with the sectional area curve is developed to modify the hull form globally during optimization cycles. A radial basis function (RBF) based approach is developed to modify the hull form locally and/or globally during optimization cycles. In order to allow for both local and global variation of the hull form, the RBF-based hull form modification technique can be combined with the parametric hull form modification technique, or can be used alone. In comparison to other hull form modification technique, these techniques are both flexible and efficient. They only require a small number of design variables defined in terms of the form parameters and movable control nodes of the radial basis function. These modification techniques can be applied to the hull form represented by discrete surfaces or NURBS (Non-Uniform Rational B-Spline) surfaces. They can be used to produce optimal hull forms that have superior hydrodynamic performance efficiently and effectively in the CFD-based hydrodynamic optimization of hull forms[12-23,35-36]. The other advantage of these hull form representation and modification techniques is that the effect of every design parameter on the hydrodynamic performance can be investigated before performing the optimization. For a baseline hull represented by NURBS surfaces, this RBF-based modification technique has been further developed recently to generate a bulbous bow automatically or to modify an existing bow in terms of the hydrodynamic performance and given geometry constraints during the optimization process[19].

A RBF-based modification technique, a combined RBF and NURBS surface modification technique, and a parametric representation and modification technique are reviewed in this paper. In addition to hull surface modeling, a number of multi-objective optimization techniques have been developed and implemented in the optimization module, which includes genetic algorithms[12], a new improved artificial bee colony optimization algorithm[10], differential evolutionary optimization algorithm, and several surrogate based optimization methods[19,22,24]. A number of CFD tools have been integrated in the hydrodynamic module for evaluating hydrodynamic performance. A suitable optimization algorithm, CFD tool, and hull form modification technique can be selected in the simulation-based hydrodynamic design optimization tool to meet the design needs at different stages of the design.

The present simulation-based multi-objective hydrodynamic optimization tool has been used for the design of mono-hull ships[16,19-22,24,26,27]and multi-hull ships[13,17,18]. In order to validate the computational tool, various model tests have been conducted for the initial hulls and optimal hulls obtained[13,26,28]. For the purpose of illustration, several innovative optimal hull forms generated by the present multi-objective hydrodynamic optimization tool will be discussed in details to demonstrate the effectiveness of the present tool in ship design, especially at the early stage of the design.

1. CFD Tools for hydrodynamic performance evaluation

Hydrodynamic design of ships involves several stages, from preliminary and early-stage design to late-stage and final design. Simulation-based hull form hydrodynamic optimization relies on computational fluid dynamics (CFD) solvers to evaluate the objective functions in terms of the flow around a ship hull. There may be thousands of potential candidate hulls at the early-stage design. Therefore, it is essential to develop highly efficient and robust CFD tools for this stage of the design.

A simple CFD tool (computer code SSF) is developed based on the Neumann-Michell (NM) theory of the potential flow and the ITTC 1957 Model-Ship Correlation Line. This simple CFD tool has been used to evaluate the wave drag and the total drag for various monohull ships and multihull ships. The detailed numerical implementation and validation of this design oriented simple CFD tool can be found in authors’ previous work[40-42]. It should be noted that this simple CFD tool, SSF, is highly efficient and very robust. Numerical predictions by SSF can capture the solution trends pretty well, which is essential for the optimization. Thus, this simple CFD tool is well suited for the hydrodynamic optimization for reduced resistance at the early stage of the design. In fact, it has been used widely for the hydrodynamic optimization of both monohull and multihull ships for reduced resistance

In the case of evaluating the seakeeping performance, a ship motion program (SMP) based on the strip theory is integrated to the present simulationbased design optimization tool. The validations of the simple CFD tool, SMP, were reported in the literature[43]. It has been improved several times since its first version was developed in 1981. The latest one isSMP95. Its manual can be found in the document[44]. SMP is extremely efficient, and is well suited for seakeeping evaluations at the early stage of the design. The hull form optimization for improved seakeeping performance using simple CFD tool, SMP, has been demonstrated in authors’ previous work[24].

As the design progresses, the level of physical realism needs to be upgraded, leading to Euler, RANS, and perhaps VLES computations for the final stage of the design. Thus, in addition to the potential-flow based simple CFD tools, the hydrodynamic module in the present tool also includes Euler/RANS/Navier-Stokes based high-fidelity CFD tools. One of them is FEFLO, a generic in-house CFD code that is based on the finite element method and unstructured grids. The other is a tool customized from open source CFD package OpenFOAM. Both tools can be used to evaluate hydrodynamic performance of the optimal hull forms in the late design stage. Validations and applications of these tools can be found in authors’ previous work[45-47].

In summary, it is essential to include both design oriented simple CFD tools and high-fidelity CFD tools in the hydrodynamic module of the simulation-based design optimization tool, so that they can meet design needs at different stages of the design. The simple CFD tools can be used to evaluate the hydrodynamic performance, such as drag and seakeeping, in the optimization process during the early stage of the design and the high-fidelity CFD tools can be used for the validation of the hydrodynamic performance of the final optimal hull forms obtained at the final stage of the design. In addition, simple and high-fidelity CFD tools can also be used together via a variable fidelity model if necessary to accelerate optimization process without sacrificing the accuracy in hydrodynamic performance evaluations[48].

2. Ship hull surface representation and modificatoin

Ship hull forms are represented by non-uniform rational B-spline (NURBS) surfaces in most of the modern computer aided ship design environments. A NURBS surface is mathematically simple as it is a function of two parameters mapping to a surface in the three-dimensional space, where the mapping is defined by the B-spline basis functions, knot vectors of parameters, NURBS control points and weights of control points. A hull form is usually composed of one or several NURBS surface patches of curvature continuity C2. Compared with the representation of discrete surfaces, hull forms represented by NURBS surfaces are visually fair and perfectly smooth. In order to enhance simulation-based design optimization of the hull form, a NURBS-based hull surface modification technique is developed to produce new hull forms that are still represented by NURBS surfaces.

An initial hull form represented by the NURBS surfaces can be modified by relocating the control points of the NURBS surfaces. Therefore, an optimal hull form can be obtained if the control points of NURBS surfaces can be used as design variables. However, this approach can be very computational expensive because a large number of control points are required in the NURBS representation of a typical hull form. In addition, unrealistic hull form can be produced easily in the optimization process due to the fact that excessive free-form designs are allowed by a large number of design variables. This approach also makes it hard to enforce the necessary geometry constraints. In order to take advantage of the NURBS representation of hull forms and allow for sufficient free-form design, a number of modification methods are developed to relocate the control points of the NURBS surfaces without using all of them as design variables, thus obtaining practical new hull forms that satisfy the required geometry constraints. These modification methods include: an affine transformation method, a free-form deformation method, a NURBS based free-form deformation method, a shifting method, and a radial basis function method.

The affine transformation method scales the main dimensions (i.e. ship length, draft, and beam) of the hull in three orthogonal directions with certain constraints, such as displacement and block coefficient. This method is very simple and it is especially useful at the early stage design when large variations of the hull form are allowed.

The free-form deformation (FFD) method was proposed by Sederberg and Parry[49]and has been used widely in ship hull form optimization. Some of the FFD application examples can be found in these literatures[9,15]. The classical FFD method is implemented in the present computational tool, in which a set of free control nodes are defined as design variables. The control points of the NURBS surfaces are moved according to the movement of the free control nodes, thus obtaining new hull forms in the optimization process. It should be noted that the number of free control nodes associated with the free-form deformation is much smaller than the number of the control points associated with the NURBS surface representation. By selecting the position and the number of the free-form control lattices appropriately, both local and global modification of ship hull surface can be achieved.

The classical FFD method provides a powerful modeling tool for the hull form modification. However, it is not easy to control the shape and satisfy the given constraints in some cases. As a result, there are many variations in FFD method in addition to the classical FFD method. Among them, a particular useful one is called NURBS based free-form deformation[50](NFFD) method. Compared with the classical FFD method, theNFFD method adopts the non-uniform B-spline solid with non-uniform divisions and variations of basis order to provide greater flexibility in deforming the 3-D control lattices. The deformation of these 3-D lattices is used to obtain the modification of the control points of the NURBS surfaces. Therefore, the NFFD method can allow for greater flexibility in ship hull form deformation in comparison to classical FFD method. The main advantages of using NFFD instead of FFD have been well documented in the literature[51]. It is especially more advantageous to use the NFFD method when only two or more separated small parts of the hull form are allowed to be changed since the classical FFD method may require two or more independent boxes for the control lattices, and the NFFD method only require one box for the control lattices. Furthermore, the continuity of the NFFD method is satisfied automatically. The NURBS based free-form deformation method is implemented in the present computational tool as well to meet different design needs. More discussions about the classical FFD method and the NFFD method will be presented elsewhere.

Fig.1 Comparisons of body plans and sectional area curves between the initial (original) hull and the modified hull obtained from shifting method

The shifting method[27]is a hull form modification method based on classical Lackenby[36]method. In the shifting method, the hull form variation is achieved by the translation of section profiles along the longitudinal direction. Specifically, the longitudinal position of sections is shifted to modify the prismatic coefficient, the longitudinal center of buoyancy, and the parallel mid-body of the initial hull. In the CFD-based hull form optimization, the sectional area curve of the initial hull form is modified during the optimization process. The new hull form is obtained by moving the stations of the initial hull form along the longitudinal direction. The amount of the movement is determined by comparing the modified sectional area curve and the original one. When the hull surface is defined by NURBS surfaces, the new hull surface can be obtained by moving the control points of the NURBS surfaces according to the movement defined at given stations. Application of the shifting method to the ship hull form optimization can be found in authors’ previous work[12,22,27]. A demonstration of the shifting method for the modification of a given ship hull form is shown in Fig.1, where the body plans and sectional area curves between the initial (original) hull and the modified hull obtained from the shifting method are plotted. It can be seen from Fig.1 that the shifting method redistributes the volume of the hull in longitudinal direction.

The shifting method implemented in the present computational tool is based on the variation of the ship sectional area curve. Any small changes of the form parameters associated with the sectional area curve generally cause the modification of the hull form in a large portion of the hull area. Therefore, the shifting method can be used to perform global modification of ship hull form, in which the form parameters are used to define the sectional area curve. In order to perform the local modification of the hull form, it is necessary to either introduce new form parameters or develop a new modification approach.

Fig.2 Fixed and movable RBF control nodes (blue dots and red dots) used for modifying part of the hull

A surface modification technique based on radial basis function (RBF) interpolation has been developed to define the local variation of the hull surface during the optimization process. Specifically, two types of RBF control nodes are required. One is called fixed control nodes selected in order to keep the surface unchanged near the control nodes, thus satisfying given geometry constraints. The other type is movable control nodes that can be considered as design variables in shape optimization. During optimization process, these movable RBF control nodes are relocated by agiven optimization algorithm in order to minimize the objective functions. The displacement of the NURBS control points can be obtained using RBF interpolation method accordingly, thus producing a new hull form. It should be noted that the fixed and movable RBF control nodes in the RBF-based hull form modification method can be defined in terms of the design needs to allow local or global modification of the hull form with given geometry constraints.

Fig.3 Comparisons of body plans and sheer plans between the original hull and the modified hull obtained usng RBF-based hull form modification method

The RBF-based hull form modification method has been used successfully to modify various types of hull forms including monohulls[21,23,26-28], catamaranns[17], and trimarans[13,18], in hydrodynamic design optimization of ship hull forms. Detailed descriptions of the method can be found in authors’ previous work[21,23,27]. As an example, Fig.2 shows the fore body of a ship hull represented by the NURBS surface, the NURBS control points, and the RBF control nodes. All dots in Fig.2 denote the NURBS control points. The blue NURBS control points in Fig.2 are also used as fixed RBF control nodes to keep the hull form unchanged in the region that is above and behind the blue dots. The red NURBS control points in Fig.2 are also used as movable RBF control nodes. Specifically, the red dot denoted as P1can be moved in both longitudinal and vertical directions, and the red dots denoted as P2and P3can be moved in transverse direction only. These three red movable RBF control nodes can serve as four design variables according to their degrees of freedoms of motion. Their displacements can be obtained from the given optimization algorithm during the optimization process. The movement of the cyanNURBS control points are obtained by the RBF interpolation in terms of the fixed and movable RBF control nodes. The hull form in the bow region that is bounded by the blue dots can thus be modified using the RBF-based modification method.

Fig.4 Comparisons of body plans between the original hull and the modified hull obtained using RBF-based hull form modification method with fixed waterline constraint (a) and fixed station constraint at x =0.3(b)

Fig.5 Comparisons of body plans and sheer plans between the original hull that does not have a bulbous bow and the modified hull that has a bulbous bow obtained using RBF-based hull form modification method

As a demonstration, a modified hull form is generated using the RBF-based hull form modification method, where fixed and movable RBF control nodes are shown in Fig.2. The comparison of body plans and sheer plans between the initial (original) hull form and the modified hull form obtained using the RBF-based hull form modification method are shown in Fig.3, respectively. It can be seen from Fig.3 that the RBF-based modification method can produce a smooth new hull form that satisfies the geometry constraints, i.e., keeps the hull form unchanged in the region above and behind the blue dots shown in Fig.2.

To further demonstrate the capability of enforcing geometry constraints, the Series 60 hull is used as an initial hull for conducting two types of specified hull form modifications. Figure 4 shows the comparison of body plans between the initial (original) hull and the modified hulls obtained with fixed waterline constraint and fixed station constrain at x=0.3, respectively, using the RBF-based hull form modification method. Moreover, the present method can also allow for a large free-form deformation, including the generation of a bulbous bow for a ship hull that does not have a bulbous bow. This capability has been demonstrated in authors’ previous work[21,23]. For the purpose of illustrations, the Series 60 hull is again used as an initial hull, a bulbous bow is generated using the RBF-based modification method. Figure 5 shows the comparison of body plans and sheer plans between the original Series 60 hull and the modified hull that has a bulbous bow obtained using the RBF-based hull form modification method.

Fig.6 Flow chart of the optimization procedure

Table 1 Main particulars of the Series 60 hull(C B=0.6)

Fig.7 Three-dimensional view of the original Series 60 hull

3. Optimization algorithms and surrogate models

Hydrodynamic optimization of ship hull forms can be formulated as follows

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where fiare objective functions that measure hydrodynamic performances, such as resistance and seakeeping,Xis a vector of design variables inddimensions that are linked to the parameters for the hull form modification,n is the number of objective functions,S is the feasible solution set resulting from removing portion of the space Rdprohibited by the constraints, such as displacement, ship length, beam, draft, block coefficient, and lower and upper bound of design variables. The problem described by Eq.(1) is a multi-objective optimization problem when nis greater than one. It becomes a single-objective optimization problem whenn is equal to one.

Various optimization techniques and procedures can be used to solve the optimization problem described by Eq.(1). The design-oriented simple CFD tools are usually well suited for evaluating the flows about the ship, thus obtaining the objective functions during the optimization process. For some optimization problems, the low-fidelity simple CFD tools may not be adequate for evaluating the ship hydrodynamic performance. Instead of using complex and computationally expensive high-fidelity CFD solvers with standard optimization methods, a variable fidelity method can be adopted. This method uses low-fidelity models and a scaling function to approximate the high-fidelity models to reduce computational cost. The variable fidelity method aims to maximize the use of low-fidelity, cheaper models in iterative procedures with occasional, but systematic, recourse to high-fidelity, and more expensive models for monitoring the progress of the algorithm. It is globally convergent to the solution of the original, high fidelity problem[48].

In order to accelerate the design process and further reduce the computing time, a surrogate based optimization method can be adopted. For most of the hydrodynamic optimization problems, the simple CFD tools can be used to construct the surrogate model for each objective function. Surrogate models can then be used for evaluating the objective functions in the optimization process. The flow chart of a surrogate-based optimization procedure is shown in Fig.6. An initial hull form is first input, the objective functions and design constraints are then defined. The appropriate modification methods are selected according to the design constraints and requirements. Design variables and their variation ranges are also determined for given modification methods during this step. A series of candidate hull forms required for constructing the surrogate models are generated using the selected hull form modification methods. The flows about all candidate hull forms are then evaluated using simple CFD tools, and the surrogate model for each objective function are constructed afterwards. Optimizations can then be performed using a given optimization technique, in which objective functions are evaluated by the surrogate models.

A number of optimization algorithms have been implemented in the optimization module of the present simulation-based hydrodynamic design optimization tool to meet different design needs. The available optimization algorithms in the present computational tool include gradient based optimization algorithm, genetic optimization algorithm, particle swarm optimization algorithm, artificial bee colony algorithm, and differential evolution algorithm. In addition, two surrogate models are implemented in the present tool, which are radial basis function surrogate model and Kriging model. More details of the optimization algorithms and surrogate models can be found in authors’ previous work[10,12,15,18,24,25]. With the help of the surrogate model, it is very efficient to find a set of hull forms that have better hydrodynamic performance in most cases. If the surrogate model based optimization fails, such as the constructed surrogate models are not accurate enough, it is necessary to evaluate objective functions with simple CFD tools directly during the optimization process. If the optimization requires highfidelity CFD tools, the variable fidelity method[48]can be adopted to achieve the same accuracy as the highfidelity CFD tools but with much reduced computing time.

Fig.8 Distribution of RBF control modes used for bulbous bow generation in Case 1

Fig.9 Distribution of RBF control nodes used for entire hull modification in Case 2

Fig.10 Distribution of RBF control nodes used for entire hull modification and bulbous bow generation in Case 3

Fig.11 Cross validation of six surrogate models for hull form optimization at two design speeds in three optimization cases

Fig.12 Comparison of 3-D view between the original hull and the optimal hulls obtained from Case 1-Case 3

4. Illustrative examples

The main particulars of the Series 60 hull (CB= 0.6) are listed in Table 1 and the three dimensional view of the ship hull is shown in Fig.7.

The objective function fiis defined as follows:

The bulbous bow generation method is briefly introduced here, more details can be found in authors’ previous work[21]. The four bulb parameters (bl, bw, bh, bt)are defined as follows:

(1) Bulb length parameter (bl): a ratio of the bulb protruding length to the length of the ship between perpendiculars.

(2) Bulb width parameter(bw): a ratio of the maximum bulb breadth to the beam of the ship.

(3) Bulb height scale parameter(bh): a nondimensional parameter for adjusting the height of the bulbous bow.

(4) Bulb lateral scale factor(bt): a non-dimensional parameter for adjusting the fullness of the lateral profile of the bulbous bow.

The generation of the bulbous bow is achieved via the movement of the six moveable RBF control nodes plotted in red dots in Fig.8. The blue dots in Fig.8 are the fixed RBF control nodes, which can keep the hull form unchanged in the region outside of the blue dots. The movements of these six red RBF control nodes are determined by above four bulb parameters (bl, bw, bh, bt). Therefore, these four bulb parameters can be used as design variables. The displacement of the cyan NURBS control points can be obtained via RBF interpolation, thus generating a bulbous bow. Details on bulbous bow generation can be found in authors’ previous work[21].

Fig.13 Comparison of body plans between the original hull and the optimal hulls obtained from Case 1-Case 3

Fig.14 Comparison of sheer plans between the original hull and the optimal hulls obtained from Case 1-Case 3

In the case of modifying entire hull form, four movable RBF control nodes plotted in red dots in Fig.9 are chosen as design variables. These four design variables are only allowed to move in transverse directions. The fixed RBF control nodes are plotted in blue dots in Fig.9, which are used as constraints to keep the stem and stern profiles, waterline and mid-body of the hull unchanged. The hull form modification in the entire hull body except the stem and stern profiles, waterline and mid-body of the hull can thus be achieved by computing the displacement of the cyan NURBS control points using RBF interpolation method based on these fixed and movable RBF control nodes. The optimal hull form obtained in such a way meets the design requirement specified in Case 2.

Case 3 combines the bulbous bow generation considered in Case 1 and the modification of the entire hull form except the stem and stern profiles, waterline and mid-body of the hull considered in Case 2. There are a total of seven design variables. Four of them are bulb parameters for bulbous bow generation, and three of them denoted as P1-P3in Fig.10 are the movable RBF control nodes that are allowed to change in transverse directions only.

Surrogate based optimization method is employed to perform three optimization cases. Specifically, radial basis function surrogate model is selected to approximate the objective functions and multi-objective artificial bee colony algorithm is used to determine optimal solution set. More details of the surrogate models and optimization algorithms can be found in authors’ previous work[10,24,26]. In summary, six surrogate models are constructed to predict the total drag at two speeds for three optimization cases. The constructed models are validated using cross validation, which is shown in Fig.11. In the cross validation, each sample point is evaluated from the RBF surrogate model that is constructed by the rest of the sample points. It can be observed from Fig.11 that the estimated total resistance (fE)obtained from the surrogate model shows a fairly good agreement with the total resistance(fC) calculated by the simple CFD tool directly for the variants of all sample hulls at two given speeds in three optimization cases.

Once the optimization is finished, three sets of optimal solutions are obtained. Based on the criteria of the best average performance at two design speeds, three optimal designs are selected from three sets of optimal solutions. The comparison of 3-D view between the original and three optimal hulls is shown in Fig.12. In addition, the comparisons of body plan and sheer plan between the original hull and three optimal hulls are shown in Figs.13 and 14, respectively.

In order to check the performance of three optimal hulls, the wave drag coefficients, total drag coefficients, wave drag and total drag are evaluated for the original and three optimal hulls for the speed range Fr=0.2 to 0.4. The comparisons of wave drag coefficients, total drag coefficients, wave drag and total drag are shown in Fig.15. As it can be seen from Fig.15, three optimal hulls have better performance in the speed range close to the design speeds. The optimal hull from Case 1 has better performance than that from Case 2 at high speed. The optimal hull from Case 2 has better performance at low speed range. The optimal hull from Case 3 has the best overall performance.

Fig.15 Comparisons of drag coefficients and drag between the original hull and three optimal hulls obtained from Case 1-Case 3

It should be noted that optimal designs obtained from the present simulation-based optimization tool hav e been v alidat ed b y m odel te sts in the towing tank forvarioustypesofhullformsinauthors’previous work[13,26,28]. In the case of the Series 60 hull, the validation of the optimal hull form obtained using a slightly different modification method has been conducted using the high-fidelity CFD tool and towing tank experiments, respectively. It has demonstrated that the optimal hull form obtained using the present computational tool can achieve significant resistance reduction[26]. Therefore, the optimal hull forms obtained in this review paper can be expected to have the similar drag reduction since the same computational tool is used except that a slightly different modification method is applied in the optimization process.

5. Conclusion

A computational tool has been developed for the simulation-based hydrodynamic design of ship hull forms. Many innovative hull forms have been produced by this tool over the past several years for reduced drag, or for reduced drag and improved seakeeping. Experimental validations have been performed for the optimal hull forms generated by this simulation-based hydrodynamic design optimization tool, and substantial drag reductions have been achieved for various types of ships, including monohulls, catamarans and trimarans. A review of past studies have demonstrated that the present simulation-based hydrodynamic design optimization tool is invaluable to ship design. It is well suited for the innovative hydrodynamic design of ship hull forms at the early stage of the design.

Acknowledgement

This work was sponsored by the Office of Naval Research. Ms. Kelly Cooper was the technical monitor.

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(Received October 15, 2016, Revised October 30, 2016)

* Biography:Chi YANG, Female, Ph. D., Professor