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An innovative method for parametric design of planing tunnel vessel Hull form
Ocean Engineering 60 (2013) 14–27
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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
An innovative method for parametric design of planing tunnel vessel Hull form Morteza Ghassabzadeh, Hassan Ghassemi n Department of Ocean Engineering, Amirkabir University of Technology, Hafez Ave., 15875-4413 Tehran, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 17 February 2012 Accepted 25 November 2012 Available online 16 January 2013
In this work, a new mathematical procedure has been developed to automatically generate the hull form of planing tunnel vessels. Due to the complex and time consuming nature of the hull form generation, it is vital to use a nimble and precise method in designing and optimization of hull form of these vessels. In the present method, the minimum number of input control parameters has been employed to design the hull form geometry. In the first step of modeling procedure, four longitudinal guideline functions are defined that use two different approaches, namely, polynomial functions and Non-Uniform Rational B-Spline (NURBS) curves. In the second step, using specified key points, the section curves are generated via three different models including; the parabolic, the elliptic and the NURBS curves. Then, the fair transverse sections, the fair surfaces and the solid models could be generated. In a case study, the generation of a number of hull forms based on three different models, implies that the method is capable of designing tunnel vessels precisely and quickly. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Planing tunnel vessel Hull form Parametric design NURBS
1. Introduction In the hull form design process, creation of a geometric model of the vessel is an indispensible step determining the hydrodynamic characteristics and structural analysis. In the case of highly interactive design factors, the optimum hull should be achieved through spiral design to get ahead of all constraints. Thus, it is needed to repeat the design process plenty of times to reach optimum mode. Designing a geometrical model and its mesh generation are pre-processors of the software, which should be conjugated with solver. In complex bodies like multi-hull shape and planing tunnel vessel, the presented method is efficient enough and applicable to formulate the hull geometrical models. Advanced high speed marine vehicles (AHSMV) have four categories based on their physical support: hydrodynamic, aerodynamic, air power lift, and hydrostatic (buoyancy) forces (Papanikolaou, 2002). Fig. 1 presents various types of the AHSMV based on physical support on the hull form. Tunnel vessels are an especial type of the advanced vessels that only a few researches have studied their hull form design. These vessels are usually classified in planing vessels category according to the volumetric Froude number. The tunnel vessel with consideration of lift force has a transient behavior that switches between three case of aerodynamic, hydrodynamic, and hydrostatic. Finally, based on the hull form shape categorization, these vessels are considered as multi-hull ones. The speed and resistance are two key parameters in the hydrodynamic performance of the planing tunnel vessel. Thus, the generation of the hull form modeling is a major step in optimizing hull form in the both numerical and experimental works. As this process is costly and time consuming, the access to an automated method in generating the hull form is an essential task based on the least input control parameters. Some researchers have worked on the generating of different simple hull forms. Min & Kang (1998) developed a theoretical hull form design method for displacement-type super-high-speed ships. They used a hull form design method to prepare a series of 60 hull forms with systematic variations of the most important design variables, and conducted the model tests for the ship models. A parametric and mathematical modeling approach was presented by Abt et al. (2001) to design the ship hull forms based on friend ship method which allows creation and variation of ship’s hulls, quickly and efficiently. Hinatsu (2004) created a method to generate the surface of vessel
n
Corresponding author. Tel.: þ98 21 64543112; fax: þ98 21 66412495. E-mail address: [email protected] (H. Ghassemi).
0029-8018/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.oceaneng.2012.11.015
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Fig. 1. Types of the advanced marine vehicles based on the hull form.
hull with the least parameters using the Fourier series and non-uniform B-Spline. Kim and Nowaki (2005) proposed a parametric design method using four basic mathematic operators to create complex hull forms of vessel with B-Spline curves. An algorithm to generate the shape of the hull form to limit the perturbation range of the control points and acquiring a better fair body presented by Liu et al. (2005). In addition, Perez-Arribas et al. (2006) carried out presented a method to automatic generation of the hull form of vessel using of the Spline curves. The B-Spline surfaces method was used by two groups of researchers (Mancuso, 2006; Sarioz, 2006) in hull form design at the same time. Another alternative work was a method to create a quasi-developable B-Spline surface between two limit curves. The boundary curves such as center line, chins and sheer lines of a vessel were modeled by B-Spline curves (Perez and Suarez, 2007). Also, Wang and Zou (2008) have generated the vessel geometry based on a mixed method composed of non-uniform B-Spline together with an iterative procedure. The parametric generation of geometry in curved bilge fishing vessels was reported based on the non-uniform B-Spline surfaces by Perez and Clemente (2008). Recently, Ventura and Guedes Soares (2012) presented a methodology for the computation of the intersection of parametric surfaces using NURBS with special attention to some specific requirements which is applicable in ship hull modeling. Calkins et al. (2001) developed a computational automated method to define the hull form of a planing vessel in the conceptual design step. They focused on the mono-hull and catamaran planing vessels hull form design using a mathematical procedure. Most of these researches have worked on the simple hull form generation, not complex multi-hull tunnel vessels. It seems that there is a lack of quick and developed method in the planing tunnel vessels modeling studies. This paper developed an innovative method to generate the hull form automatically in the conceptual design stage of these vessels. This method is based on the geometry creation guidelines with the least control and hull form adjustment input parameters. The modeling procedure consists of two main steps. At first, four longitudinal guideline functions are defined based on the position of sections along the vessel length by using two different ways of polynomial functions and NURBS curves. In the transverse section, the coordinates of four key points play an important role to define the tunnel form. So, considering these points, the section curves can be generated using three different models; parabolic form, elliptic form and the NURBS curves. Finally, the fair transverse sections, the fair surfaces and the solids model can be generated. The following sections are organized as follows. The methodology details and definition of the model are described in Section 2. The governing equations are defined in Section 3. The generated various model results are discussed in Section 4. The effect of the main geometry parameters and sub-parameters are illustrated in the planing tunnel hull form. Finally, in Section 5 conclusions have been presented.
2. Methodology 2.1. Background The simplest body form of a tunnel vessel is shown in Fig. 2. The body of tunnel vessels is composed of three general parts. The first part is the central part which provides the hydrodynamic lift force. The front of this part has been curved up in order to distribute the dynamic pressure force from mixed air and water during the increasing vessel speed. The moderate pressure generated at the bottom of the hull prevents porpoising instability. The second part is the planing tunnel which produces the hydrodynamic and aerostatic lift forces by imprisonment of air and breast waves. The third part is a rigid skirt on the outer sides of tunnels. Its role is air imprisonment, preventing air escape and breasting the waves, continuously. It is obvious that vessels with better performance can be obtained by the addition of hydrodynamic hull form elements or with increasing the number of tunnels. Two feasible vessels are shown in Fig. 3.
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Fig. 2. A simple schema of the components of tunnel vessel body.
Fig. 3. Samples of commercial tunnel vessels.
Fig. 4. Body plan of tunnel vessel. The prismatic part and tunnel are demonstrated by solid line and dash line, respectively.
The optimum hull form design is one of the most important problems that are viable by changing the form of the body and the hydrodynamic analysis iteratively. Unfortunately, this takes a long time. Therefore, it is essential to develop a quick way to generate and modify the form automatically. In the following, the basic geometrical hydrodynamic parameters and our innovative generation method are described. 2.2. Model description In the present method, the tunnel vessels hull form is divided into two main parts as can be seen in Fig. 4. These two parts, that are considered to have the greatest effect on the hydrodynamics behavior independently, are defined as a) Prismatic body: in this part, the main parameters are the width of the prismatic body, the deadrise angle at each transverse section, and their longitudinal variations profiles. b) Tunnel: the most important parameters of the tunnel are the height of tunnel, width of tunnel, and their longitudinal variations profiles. An additional factor that significantly affects the tunnel is the form of the cross section. It means that the section form may change by the transverse movement of control point 3. These parameters are shown in Fig. 5 and introduced in rows 1–4 of Table 1. In this type of vessel, total breadth of the hull usually changes a little, so it is assumed as a constant value. Therefore, the summation of the prismatic width and tunnel parts is always constant and equals the total width of the vessel. The height and width of the tunnel part in each transverse section might change in respect with length variations. As a result, the cross section of tunnel form can vary between different shapes like, circle, ellipse, parabola, etc. continuously and fairly. In addition, the variations of width and height of the tunnel can cause twisting, shrinking or enlargement of the tunnel cross section. Decrease or increase in deadrise angle can change the shape of the prismatic part and lead to changing the hydrodynamic lift force.
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Fig. 5. Different parameters of tunnel vessel section at the length of x.
Table 1 Definition of the employed parameters. Row 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Symbol
Value
Description (unit)
bT(x) bP(x) hT(x) b(x) B L H Bv D11 D20 D31 D30 D41 D40 N m1 m2 m3 C11 C21 C31 C41 Cb32 Zi(x), Yi(x)
– – – – 3 12 1.5 1.5 0.9 0.25 1.15 0.5 0.75 0.25 15 5 30 30 50 45 40 35 50 –
Tunnel width (m) Prismatic part width (m) Tunnel height (m) Deadrise angle (m) Vessel width (m) Vessel length (m) Vessel height (m) Maximum Prismatic part width (m) Height of the fore point of guideline 1 (m) Height of the aft point of guideline 2 (m) Height of the fore point of guideline 3 (m) Height of the aft point of guideline 3 (m) Height of the fore point of guideline 4 (m) Height of the aft point of guideline 4 (m) Number of transverse sections (dimensionless) Number of points between guidelines 1 and 2 in each transverse section (dimensionless) Number of points between guidelines 2 and 3 in each section (dimensionless) Number of points between guidelines 3 and 4 in each section (dimensionless) Percentage of length of fixed part of guideline 1 from aft vessel (%) Percentage of length of the fixed part of guideline 2 from aft vessel (%) Percentage of length of the fixed part of guideline 3 from aft vessel (%) Percentage of length of the fixed part of guideline 4 from aft vessel (%) Percentage of closing of the guideline 3 to guideline 2 Width and height functions of guideline I (dimesionless)
In this configuration, each of the mentioned parameters is calculated at the optimum state and the tunnel vessel is ultimately designed. So, the main parameters that affect the design include: the deadrise angle, prismatic body width, tunnel cross section shape (circle, ellipse, parabola, etc.). The tunnel width and height in each transverse section should be determined. The best fairing way for the created volume from joining transverse sections is to define the continuous variation of these parameters based on the vessel’s length. Thus, in order to generate a smooth volume, the longitudinal function of each parameter should be specified. Four different lines are distinguished in the half-body of the hull form with two tunnels as illustrated in Figs. 6 and 7. Line 1 is the axis of symmetry of the vessel body and prismatic part. Line 2 is the boundary between the prismatic part and the tunnel part. Lines 3 and 4 are the maximum height location of the tunnel and the side boundary of the tunnel and vessel, respectively. These four lines give four key points in each transverse section that can be used as the creating points for the given section form. As a case in point, Fig. 5 shows the four points in a transverse section. As can be seen in this figure, the prismatic part is formed by straight connections between points 1 and 2. The tunnel is identified by connecting points 2, 3 and 4. The form of cross section depends on the position of the four points and any changes in the location of one point can change the cross section form. Indeed: 1. When the height of the prismatic part varies (means point 2) the deadrise angle has changed. 2. Since the vessel’s width is constant, division of the width between the prismatic part and the tunnel is achieved by changing the width of point 2. 3. The relative displacement of points 2, 3 and 4 changes the tunnel cross section form. 4. One of the most important hull form characteristics is point 3. If point 3 moves to the left or right, causes the generation of circulating flow over the tunnel.
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Fig. 6. Front view of the generated body by use of guidelines.
Fig. 7. Bottom view of the generated body by use of guidelines.
5. These changes can be applied to each transverse cross section. By relative displacement of points of two adjacent cross sections, the longitudinal curve of each of the parameters, mentioned above, can be controlled.
2.3. Definition of model parameters The proposed method in this article is based on the use of the before mentioned four lines that henceforth are named the ‘‘guidelines’’, as shown in Figs. 6 and 7. If we define these guidelines by functions of the length position (x), the coordinates of the four points will be specified in each cross section (given the x value). Then, the cross section is created by connecting these points and the 3-D model of the vessel is obtained by joining produced sections. Guideline 1 is the axis of symmetry (center line) of the prismatic part and vessel. The height of this guideline varies along the vessel length only as z1 ¼z(x) and its transverse position is fixed at y1 ¼ 0. Guideline 2 is the side line of the prismatic part and tunnel. It connects these two parts. The height and transverse position of this guideline changes along the vessel length (z2 ¼ z(x), y2 ¼y(x)). Guideline 3 is the middle line of the tunnel. The height and transverse position of this guideline also changes along the vessel length (z3 ¼z(x), y3 ¼y(x)) and the maximum height of the tunnel is associated with this guideline. Guideline 4 is the side line of the vessel and the tunnel. The height of this guideline varies along the vessel length only as z4 ¼z(x) and its transverse position is fixed at y4 ¼B/2. With controlling these four guidelines, all the parameters mentioned in the previous section can be adjusted as following: 1. 2. 3. 4. 5. 6.
Vessel width ¼twice the transverse distance between guidelines 1 and 4. Tunnel width¼ transverse distance between guidelines 2 and 4 in each transverse section. Prismatic part width ¼transverse distance between guidelines 1 and 2 in each transverse section. Deadrise angle¼the ratio of height difference of guidelines 1 and 2 to their width difference in each transverse section. Tunnel height¼the height difference between guidelines 1 and 3 in each transverse section. The transverse section form of tunnel ¼connection of the three points of guidelines 2, 3 and 4 by an arbitrary curve (NURBS, ellipse or parabola) in each transverse section. 7. The longitudinal profile of tunnel¼ an integration of longitudinal guidelines 2, 3 and 4 curves (quadratic and cubic polynomial functions or NURBS). 8. The longitudinal profile of the prismatic part ¼an integration of the longitudinal guidelines 1 and 2 curves (quadratic and cubic polynomial functions or NURBS). Thus, using the defined four guidelines, all parameters of the hull form can be determined and a change in guidelines (curves) influences the value of hull form parameters. A significant advantage of this method is the continuity of variations along the vessel length. It also guarantees the surface smoothness of the produced 3D model. The control parameters are used as input values to prepare, adjust and modify the four guidelines. The overall length, width and height of the vessel (rows 5–7 in Table 1) are considered as the first input values by the user. The maximum prismatic body width (row 8), which happens at the end of the vessel, is assumed as another input data. Afterwards, parameters of the guidelines are defined based on the
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given data in Table 1. The height of the beginning and end points of the guidelines (rows 9–14 in Table 1) are defined in terms of longitudinal position (x) to obtain the constant coefficients of guidelines equations. Guidelines 1 and 2 have the same start points due to their intersection at the fore of the vessel. The width of guideline 3, which is between guidelines 2 and 4, is defined based on its adjacency distance to guideline 2 (row 23 at Table 1). Guidelines 1 and 4 are only functions of the height while guidelines 2 and 3 are functions of both height and width. Considering the type of the body form, all changes along these guidelines occur on the approximate distance of (1/3 1/2) L form the forward of the vessel whereas the rates of variations are almost small or zero at the aft part of the vessel. Therefore, the parallel parts of the hull of guidelines are defined as some percentage of the vessel length (rows 19–22 of Table 1).
3. The governing equations In this paper, two different methods are employed to delineate the four guidelines. In the first method, a second order polynomial is used to define guidelines (except for guideline 3 where a third order polynomial is used because of the bulge change). In the second one, NURBS curve relations are used to define guidelines. Three different models have been used to describe each transverse section with four points of guidelines. Three different curves (parabolic, elliptic and NURBS curves) are applied to define the tunnel form between points 2, 3 and 4. In all three models, the prismatic body is produced by direct connection (straight line) of point 1 to point 2 in each section (for instance see Fig. 4). The summary of how to define these three different models are described in Table 2. In order to generate models 1 and 2, four guidelines are defined which have quadratic curves. The width and height functions in each of these guidelines are defined by the length position (x). To determine the constant coefficients of the quadratic relations of each guideline, three boundary conditions are needed to define each guideline function. These conditions are obtained based on the beginning and end points of each guideline which are entered by the user as an input data. Indexes and values of these parameters are given In Table 1, rows 9–14. Since most of the planing tunnel hull forms have a uniform cross section in the aft of vessel, the variations of guidelines should also be constant in some percentage of the length vessel from aft. In order to cover this purpose, a number of coefficients are assumed in the equations which their definitions, indexes and values can be observed in rows 19–22 in Table 1. These coefficients are used in the definition of boundary conditions of the guidelines. For simplicity reduction of input parameters, the transverse position of the guideline 3 has been defined based on its closing percentage coefficient to guideline no. 2 as expressed in row 23 in Table 1. The guidelines are described by polynomial functions in the first and second models. To minimize input parameters and because of the negligible effect of higher orders polynomials, guidelines no. 1, 2, 4 are defined by square polynomial equations. The keel, chine and side guidelines (no. 1, 2, 4) can be generally represented with a square polynomial function as the following equation: ( Z i ðXÞ ¼ C 2 X 2 þ C 1 X þ C 0 ð1Þ Y i ðXÞ ¼ B2 X 2 þ B1 X þ B0 Concave form of these three guidelines is constant along the vessel length, but concave curvature of guideline no. 3 changes along the vessel, therefore it is defined by a cubic polynomial function as Eq. (2). ( Z 3 ðXÞ ¼ C 3 X 3 þ C 2 X 2 þ C 1 X þ C 0 ð2Þ Y 3 ðXÞ ¼ B3 X 3 þ B2 X 2 þ B1 X þB0 where, Zi and Yi are the height and width of guideline no. i. These two parameters are defined based on length (X) and the constant values (Cj, Bj) that are obtained imposing boundary conditions. The guideline functions are described as follows: – Guideline 1 definition This guideline is defined at the center line of the vessel where the origin of the coordinate system is located at the aft point. Some boundary conditions are firstly required at the aft and fore. Since the aft part of the guideline has the lowest height, i.e. the aft height is set to be zero. At the fore part of the vessel, it is defined based on the input height of the start point (D11). The general equation for the height (Z1(X)) of the guideline and their boundary conditions are expressed as follows: dZ 1 Boundary conditions : Z 1ðX ¼ LC 11 Þ ¼ 0, ¼ 0, Z 1ðX ¼ LÞ ¼ D11 dx ðX ¼ LnC 11 Þ
8 < Z 1 ðX Þ ¼
2 D11 11 D11 11 D11 X 2 þ L C2C X þ C C2 2C L2 ðC 11 2 2C 11 þ 1Þ ð 11 2 2C 11 þ 1Þ ð 11 11 þ 1Þ
ð3Þ
: Y ðXÞ ¼ 0 1
Table 2 The conditions used in three different models. Model no.
Definition type of guideline
Definition type of prismatic body
Definition type of tunnel
1 2 3
Second order polynomial Second order polynomial Third order NURBS
Straight line Straight line Straight line
Parabolic Elliptic Third order NURBS
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– Guideline 2 Guideline 2 changes in both longitudinal and transverse directions. Therefore, it should be defined by two equations (Z2(X), Y2(X)). Both equations are defined as quadratic. To determine the coefficients of the equation in each direction, three boundary conditions are required. The height of the aft has been set based on the input data (D20). At the stem of the vessel, the guidelines of 1 and 2 coincide with the same data, (Z2(X ¼ L) ¼Z1(X ¼ L) ¼D11). The guideline width (Y2) at the aft vessel length has been set equal to the maximum prismatic width (Bv). The general equation of the height and the width of a given guideline and their boundary conditions are expressed as follows: 2 Z 2ðX ¼ L C 21 Þ ¼ D20 , dZ ¼ 0, Z 2 ðX ¼ LÞ ¼ D11 dx ðX ¼ L C Þ 21 B:C: : 2 Y 2ðX ¼ L C 21 Þ ¼ Bv , dY ¼ 0, Y 2ðX ¼ LÞ ¼ 0 dx ðX ¼ L C 21 Þ
Z 2 ðX Þ ¼
D11 D20 X 2 þ L 2 CC 212 ð2D20C Dþ111Þ X þ D11 þ ð1 C2 C2212Þ CðD20þD111 Þ L2 ðC 21 2 2 C 21 þ 1Þ ð 21 Þ ð 21 Þ 21 21
Y 2 ðX Þ ¼
2 C 21 Bv Bv X2 þ X þ ð1 22 C 21 Þ Bv L2 ðC 21 2 2 C 21 þ 1Þ L ðC 21 2 2 C 21 þ 1Þ ðC 21 2 C 21 þ 1Þ
ð4Þ
– Guideline 3 As already defined, the order of the polynomial function of this guideline equals to 3 and it has four constant coefficients. To obtain these constant values, the height variations in the aft percentage of the length, are set to zero and the height of the aft and fore of the guideline are assumed based on the input data (D30, D31). The guideline width is defined based on its closeness ratio to guideline 2 (Cb32) compared with guideline 4. The function of height (Z3(X)) and width (Y3(X)) and the boundary conditions of guideline 3 are expressed as follows: 3 3 B:C: : Z 3ðX ¼ LC 31 Þ ¼ D30 , dZ ¼ 0, Z 3ðX ¼ LÞ ¼ D31 , dZ ¼0 dx dx ðX ¼ LnC 31 Þ
8 < Z 3 ðX Þ ¼
ðX ¼ LÞ
2ðD31 D30 Þ 6ðD31 D30 ÞC 31 X 3 þ 2 3ðD2 31 D30 Þð1 þ C 31 Þ X2 þ X þ D31 þ ðD231 D30 Þð1 þ 3C 31 Þ LðC 31 2 2C 31 þ 1Þð1C 31 Þ L3 ðC 31 2 2C 31 þ 1Þð1C 31 Þ L ðC 31 2C 31 þ 1Þð1C 31 Þ ðC 31 2C 31 þ 1Þð1C 31 Þ
: Y 3 ðXÞ ¼ C b32 Y 2 ðXÞ þ ð1C b32 Þ B=2
ð5Þ
– Guideline 4 This guideline is only a function of the height of the side part of the vessel. The input data (D40, D41) are given for the height of the fore and aft, respectively. The function of guideline 4 and its boundary conditions are expressed as 4 ¼ 0, Z 4ðX ¼ LÞ ¼ D41 B:C: : Z 4ðX ¼ LC 41 Þ ¼ D40 , dZ dx ðX ¼ LC 41 Þ
8 < Z 4 ðX Þ ¼
ÞðD40 D41 Þ D41 D40 41 ðD40 D41 Þ X 2 þ 2C X þ D41 þ ð2CC41 1 LðC 41 2 2C 41 þ 1Þ L2 ðC 41 2 2C 41 þ 1Þ ð 41 2 2C 41 þ 1Þ
: Y 4 ðX Þ ¼
B 2
ð6Þ
The tunnel shape can be determined by the definition of the quadruplet points in each transverse section. In the tunnel with a parabolic curve (model 1), if the width values vary between points 2 and 4, the height of the points on the parabolic curve can be calculated using the following matrix equations: 8 9 2 2 3 2 3 3ð1Þ 8 9 2 3 2 2 2 3 a a a Y2 Y2 1 Y2 1 Y2 > > = = < Z2 > < Z2 > h i 6 7 6 7 6 7 6 7 6 7 2 2 2 Z3 ¼ 4 Y 3 Y i 1 4 b 5 Y 2 rY i r Y 4 ) Zi ¼ Y i Y3 1 5 4 b 5 ) 4 b 5 ¼ 4 Y3 Y3 1 5 Z3 ð7Þ > > ; ; :Z > :Z > c c c Y42 Y 4 1 Y 42 Y 4 1 4 4 In case of the ellipsoid shape (model 2) for the tunnel cross section, the following relations between points 2, 3 and 4 can be used (Eq. (8)). By assuming an arbitrary width between the width values of points 2 and 4 for ellipsoid points (Yi), the height of these points (Zi) could be obtained. In this type of cross section, this limitation should be noticed that the transverse position of the guideline 3 is always in the middle of the width of guidelines 2 and 4 because of the ellipsoid shape. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Y i Y 4 þ2 Y 2 Z 4 þZ 2 Y 4 Y 2 2 Z 4 Z 2 2 Y 4 þY 2 2 Z4 þ Z2 2 a¼ b¼ Y 3 ) Z i ¼ b 1 þ þ Z3 þ 2 2 2 2 a 2
Y 2 rY i r Y 4
ð8Þ
To produce guidelines and tunnel sections in model 3, the third order NURBS curve is used. The relations applied for this form are expressed as follows (Piegl and Tiller, 1997): Pn 0 N i,p ðuÞwi P i C ðuÞ ¼ Pi ¼ , n i ¼ 0 N i,p ðuÞwi
U ¼ fu0 ,u1 ,:::,ur1 ,ur g
ð9Þ
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In the Eq. (9), p, the order of curve, equals to 3; n þ1, the number of control points, is 6 for guidelines and 3 for sections in each part of tunnel; r þ1, the size of knot vectors, equals to 10 for guidelines and 8 for sections in each part of the tunnel. The knot vector values (ui), the control points (Pi ) and their weights (wi) are assumed as optional values for each guideline or section curve of the tunnel. The shape of the curve can be properly adjusted by controlling these parameters. To control the shape, the number of control points is fixed at 6 and also uniform knot vectors are used. The knot vectors values of guidelines and sections are shown in the following equation: 1 2 U Guidelines ¼ 0, 0, 0, 0, , , 1, 1, 1, 1 3 3 U Sections ¼ f0, 0, 0, 0, 1, 1, 1,1g ð10Þ
4. Result and discussion 4.1. Comparison of the developed models The three created models for the same input data are plotted and compared to each other. In the first model, longitudinal guideline curves are polynomial and transverse tunnel forms are parabolic. In the second model, the guidelines were as the first one and tunnel form with elliptic shape was used. The guidelines and tunnel form were defined by NURBS curves in the third model. The consequent cross sections are shown in Figs. 8–10 for the first, second and third models, respectively. In these figures, the front view of the body plan lines has been shown. The parameters of the length, width, and maximum prismatic body’s width are assumed as 15, 4 and 1.5 m, respectively for all three models. The other parameters and their values are given in Table 1. Based on these values and Eqs. (1–4), the functions of guidelines for first and second model were obtained. The coefficients of these curves are expressed by the following equation: Z1 ¼ 0:0250X 2 -0:3000X þ0:9000,
Y 1 ¼ 0,
Z2 ¼ 0:0149X2 0:1612X þ1:1149,
Y 2 ¼ 0:0344X 2 þ0:3719X þ0:4959,
Z3 ¼ 0:0035X 3 þ 0:0878X 2 -0:6018X þ1:7518,
Y 3 ¼ 0:5X 2 þ 0:75,
Z4 ¼ 0:0082X 2 0:0690X þ 0:3950,
Y 4 ¼ 1:5,
ð11Þ
In the initial study of these three figures (Figs. 8–10), it can be seen that the generated model with the NURBS curves (model 3) is more similar to elliptic model (model 2). But the parabolic model (model 1) is different from others. On these figures, red part is the prismatic body, the blue part is the tunnel and the pink part is the rigid edge. In fact, the section model is generated by the use of the point model. The number of points in each section is selected based on three parameters of m1, m2, m3. Parameter m1 is the number of points between key points of guidelines 1 and 2 in the prismatic body. Parameter m2 is the number of points between key points of guidelines 2 and 3 in the tunnel part. Parameter m3 is the number of points between key points of guidelines 3 and 4 in opposite side of the tunnel. According to the defined parameters, the generated model can is compatible with numerical analysis software. By changing the number of points in each section, the model becomes smoother and the consequent curves and surfaces used for grid computation become more suitable. Two views of the generated points of the three models, front and isometric, are shown in Figs. 11–13. Using the generated points, firstly, the curves and then the surfaces are made. As an example, the lastly produced surface of the third model has been shown in Fig. 14 which is produced by a macro in commercial CAD software.
Fig. 8. The first model based on quadratic curves.
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Fig. 9. The second model based on elliptic and quadratic curves.
Fig. 10. The third model based on NURBS curves.
The second model (elliptic) is subject to a constraint in which guideline 3 can only be located at the middle distance between guidelines 2 and 4. So, the value of parameter Cb32 is 50%. 4.2. The effect of the main geometry parameters The changes in the geometry of the model have been studied by changing the models’ parameters for two case studies, A and B. The cross sections obtained by the three developed models are shown in Figs. 15–17. Input parameters of two cases are summarized in Table 3. The prismatic body width (row 4) and the parallel parts of the four guidelines (rows 11–14) in case B in comparison with those of case A are greater. Besides, some changes have been applied at the height of beginning points. The end points of guidelines (rows 5–10) have also changed. The height of the front points of guidelines 1 and 3, and the end point of guideline 3 have been reduced from case A to case B. The height of the end point of guideline 2 is greater in case A than case B. The beginning and end points of guideline 4 and the transverse position of guideline 3 are the same in two case studies (row 15). It is observed that by applying these changes, the deadrise angle increases, the tunnel section shape changes and the height and width of it decrease for case B compared with case A. 4.3. The effect of the sub-parameters The geometric parameters for three developed models that control the form of the body are presented in Table 3. In addition, for the third model, that uses the NURBS to define guidelines and tunnel sections’ curvature, the knot vectors, the control points and the weight as well as the main geometry parameters are effective enough to change the hull form. In the other words, this model can be more easily
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Fig. 11. The points of first model based on quadratic curves.
Fig. 12. The points of second model based on the quadratic and elliptic curves.
modified and controlled than the other two models. Although, when the number of parameters is increasing, their consonance and fitting process becomes more difficult and time consuming. Using the third model, with the same basic parameters according to Table 1, two cases have been generated. The difference between these two cases, which are both based on NURBS curves, is due to the diversity in the control points of these cases. These points’ coordinate differences are described in Table 4.
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Fig. 13. The points of third model based on NURBS curves.
Fig. 14. The different views of generated surfaces from third model.
As it may be noticed, 6 control points are treated to define the third order NURBS curve. In this study, changes of the control points 4 and 5 are studied in all 4 guidelines. The displacement of the control points changes the guidelines curve and as a result, the body form changes. As it can be seen in Fig. 18, the volume of the prismatic body in case D is larger than case C and it has a large width on the front of the vessel. On the other hand, in case C, this part is thinner and smaller in volume. These changes also affect the shape of the tunnel and especially the front of the vessel so that the curvature of case D has been larger than case C. In the third model, the number of control points and the size of knot vectors are increasable capability. Thus, the controlling and achievement of the desired hull form can be done more accurately. This way has the disadvantage of the additional input data and assumptions which increase the complexity and time consumption in the geometric model generation. The advanced multi-tunnel high speed craft hull form can be designed using the third model for a hull form generation that has individual hydrodynamic ability. For example, the cross section view of a 4 tunnel vessel is shown in Fig. 19, where the points and sections of semi hull form are shown. Using this section model and the created surface between the two consecutive sections, the solid model of the hull form is generated. This solid model is shown in Fig. 20.
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Fig. 15. The result of changes in main parameters of first model. Left side is Case A and right side is Case B.
Fig. 16. The result of changes in main parameters of second model. Left side is Case A and right side is Case B.
Fig. 17. The result of changes in main parameters of third model. Left side is Case A and right side is Case B.
Table 3 The values of geometry parameters of tow cases and their variations. Row 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Parameter (unit)
Case A
Case B
B L H Bv D11 D20 D31 D30 D41 D40 C11 C21 C31 C41 Cb32
3 12 1.5 1 1.05 0.20 1.40 0.85 0.75 0.25 30 30 30 30 50
3 12 1.5 1.5 0.9 0.25 1.15 0.5 0.75 0.25 50 45 40 35 50
(m) (m) (m) (m) (m) (m) (m) (m) (m) (m) (%) (%) (%) (%) (%)
26
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Table 4 The values of coordinates (in meters) of control points in two cases.
Guideline X Y Z W Guideline X Y Z W Guideline X Y Z W Guideline X Y Z W n
C. P. 1
C. P. 2
C. P. 3
C. P. 4n
C. P. 5n
C. P. 6
0 0 0 1
5.7 0 0 1
6 0 0 1
8/11 0 0.03/0.675 1.05/0.95
9/12 0 0.15/0.75 1.05/0.95
12 0 0.9 1
0 0.75 0.25 1
5.13 0.75 0.25 1
5.4 0.75 0.25 1
7.6/9.8 0.625/0.612 0.423/0.662 1.15/0.95
8.7/10.9 0.375/0.544 0.597/0.721 1.15/0.95
12 0 0.9 1
0 1.125 0.5 1
4.56 1.125 0.5 1
4.8 1.125 0.5 1
7.2 1.062/1.056 0.673/0.933 1.05/0.95
8.4 0.812/0.988 0.847/1.042 1.05/0.95
12 0.75 1.15 1
0 1.5 0.25 1
3.99 1.5 0.25 1
4.2 1.5 0.25 1
6.8 1.5 0.383/0.333 1
8.1 1.5 0.517/0.417 1
12 1.5 0.75 1
1
2
3
4
Value of case C/value of case D.
Fig. 18. The effect of control points coordinates change in two NURBS cases. Left side is Case C and right side is Case D.
Fig. 19. The points model of 4 tunnels vessel. Right side is key points view and left side is sections view.
5. Conclusion In this article, the geometric model of the planing tunnel vessel is generated using a new mathematical method in which the number of input parameters is minimized. The method is based on the creation of four longitudinal guidelines and the quadruplet key points. First, the key points are obtained from guideline functions in each length position. The hull form sections are generated using these
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Fig. 20. The solid and section models of 4 tunnels vessel.
points; then the surfaces and the solid model are produced. The generated sections are fair curves with sufficient and suitable points. Therefore, the surface or solid model created in the 3-D model generation software does not need further fairing. This leads to a quick and automatic generation of the hull form. Furthermore, this facilitates the generation of a completely smooth model that can be directly transferred to the mesh generation software for computational fluid dynamics analysis. Note that the vessel width along its length should be constant in the present method. In the case study, three different models are used for hull form generation. Comparing the models shows that the elliptic model has the most limitations, whereas, the parabolic model is more flexible. In this model, there is a relative balance between the restrictions and model’s ability to control the geometry. Although, NURBS model is complex and consumes more time because of the sub-parameters (knot vectors, control points, etc.), its more flexible and controllable than the other two models. In addition, the distances between generated points are adjustable at each section so their density can be controlled. The process of generation of these models indicated that the proposed method is quick and accurate. Obviously, the generated model can be used for primary study and optimization in conceptual design. In addition, the other details of hull forms such as spray-rail and transverse steps can be added to the model for improvement of the vessel’s hydrodynamics behavior. The capability of the method to generate advanced similar vessels’ hull form such as four tunnel crafts can be examined in the future works.
Acknowledgments This research was supported by MRC (Marine Research Centre) at Amirkabir University of Technology. The authors wish to thank the reviewers for their valuable comments and suggestion. References Abt, C., Bade, S.D., Birk, L., Harries S., 2001. Parametric hull form design – a step towards one week ship design, 8th International Symposium on Practical Design of Ships and Other Floating Structures, PRADS 2001, Shanghai. Calkins, D., Schachter, R.D., Oliveira, L.T., 2001. An automated computational method for planing hull form definition in concept design. Ocean Eng. 28 (3), 297–327. Hinatsu, M., 2004. Fourier NUBS method to express ship hull form. J. Mar. Sci. Technol. 9, 43–49. Kim, H.-C., Nowaki, H., 2005. Parametric design of complex hull forms. Ship & Technol. 9 (1), 47–63. Liu, Y.-J., Zhu, X.-L., Ji, Z.-Sh., 2005. Ship hull plate processing surface fairing with constraints based on B-Spline. J. Mar. Sci. Appl. 4 (3). Mancuso, A., 2006. Parametric design of sailing hull shapes. Ocean Eng. 33 (2), 234–246. Min, K.S., Kang, S.H., 1998. Systematic study on the hull forms design and resistance prediction of displacement-type super-high-speed ships. J. Mar. Sci. Technol. 3, 63–75. Papanikolaou, A., 2002. Development and potential of advanced marine vehicles concepts. Bull. Kansai Soc. Naval Arch. Jpn. 55, 50–54. Perez, F.L., Clemente, J.A., 2008. Parametric generation, modeling, and fairing of simple hull lines with the use of non-uniform rational B-Spline surfaces. J. Ship Res. 52 (1), 1–15. Perez, F., Suarez, J.A., 2007. Quasi-developable B-Spline surfaces in ship hull design. Comput. Aided Des. 39, 853–862. Perez-Arribas, F., Suarez-Suarez, J.A., Fernandez-Jambrina, L., 2006. Automatic surface modeling of a ship hull. Comput. Aided Des. 38, 584–594. Piegl, L., Tiller, W., 1997. The NURBS Book. Springer, Berlin. Sarioz, E., 2006. An optimization approach for fairing of ship hull forms. Ocean Eng. 33 (13), 2105–2118. Ventura, M., Guedes Soares, C., 2012. Surface intersection in geometric modeling of ship hulls. J. Mar. Sci. Technol. 17 (1), 114–124. Wang, H., Zou, Z.-J., 2008. Geometry modeling of ship hull based on non-uniform B-Spline. J. Shanghai Jiao tong Univ. 13 (2), 189–192.
Contents lists available at SciVerse ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
An innovative method for parametric design of planing tunnel vessel Hull form Morteza Ghassabzadeh, Hassan Ghassemi n Department of Ocean Engineering, Amirkabir University of Technology, Hafez Ave., 15875-4413 Tehran, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 17 February 2012 Accepted 25 November 2012 Available online 16 January 2013
In this work, a new mathematical procedure has been developed to automatically generate the hull form of planing tunnel vessels. Due to the complex and time consuming nature of the hull form generation, it is vital to use a nimble and precise method in designing and optimization of hull form of these vessels. In the present method, the minimum number of input control parameters has been employed to design the hull form geometry. In the first step of modeling procedure, four longitudinal guideline functions are defined that use two different approaches, namely, polynomial functions and Non-Uniform Rational B-Spline (NURBS) curves. In the second step, using specified key points, the section curves are generated via three different models including; the parabolic, the elliptic and the NURBS curves. Then, the fair transverse sections, the fair surfaces and the solid models could be generated. In a case study, the generation of a number of hull forms based on three different models, implies that the method is capable of designing tunnel vessels precisely and quickly. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Planing tunnel vessel Hull form Parametric design NURBS
1. Introduction In the hull form design process, creation of a geometric model of the vessel is an indispensible step determining the hydrodynamic characteristics and structural analysis. In the case of highly interactive design factors, the optimum hull should be achieved through spiral design to get ahead of all constraints. Thus, it is needed to repeat the design process plenty of times to reach optimum mode. Designing a geometrical model and its mesh generation are pre-processors of the software, which should be conjugated with solver. In complex bodies like multi-hull shape and planing tunnel vessel, the presented method is efficient enough and applicable to formulate the hull geometrical models. Advanced high speed marine vehicles (AHSMV) have four categories based on their physical support: hydrodynamic, aerodynamic, air power lift, and hydrostatic (buoyancy) forces (Papanikolaou, 2002). Fig. 1 presents various types of the AHSMV based on physical support on the hull form. Tunnel vessels are an especial type of the advanced vessels that only a few researches have studied their hull form design. These vessels are usually classified in planing vessels category according to the volumetric Froude number. The tunnel vessel with consideration of lift force has a transient behavior that switches between three case of aerodynamic, hydrodynamic, and hydrostatic. Finally, based on the hull form shape categorization, these vessels are considered as multi-hull ones. The speed and resistance are two key parameters in the hydrodynamic performance of the planing tunnel vessel. Thus, the generation of the hull form modeling is a major step in optimizing hull form in the both numerical and experimental works. As this process is costly and time consuming, the access to an automated method in generating the hull form is an essential task based on the least input control parameters. Some researchers have worked on the generating of different simple hull forms. Min & Kang (1998) developed a theoretical hull form design method for displacement-type super-high-speed ships. They used a hull form design method to prepare a series of 60 hull forms with systematic variations of the most important design variables, and conducted the model tests for the ship models. A parametric and mathematical modeling approach was presented by Abt et al. (2001) to design the ship hull forms based on friend ship method which allows creation and variation of ship’s hulls, quickly and efficiently. Hinatsu (2004) created a method to generate the surface of vessel
n
Corresponding author. Tel.: þ98 21 64543112; fax: þ98 21 66412495. E-mail address: [email protected] (H. Ghassemi).
0029-8018/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.oceaneng.2012.11.015
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Fig. 1. Types of the advanced marine vehicles based on the hull form.
hull with the least parameters using the Fourier series and non-uniform B-Spline. Kim and Nowaki (2005) proposed a parametric design method using four basic mathematic operators to create complex hull forms of vessel with B-Spline curves. An algorithm to generate the shape of the hull form to limit the perturbation range of the control points and acquiring a better fair body presented by Liu et al. (2005). In addition, Perez-Arribas et al. (2006) carried out presented a method to automatic generation of the hull form of vessel using of the Spline curves. The B-Spline surfaces method was used by two groups of researchers (Mancuso, 2006; Sarioz, 2006) in hull form design at the same time. Another alternative work was a method to create a quasi-developable B-Spline surface between two limit curves. The boundary curves such as center line, chins and sheer lines of a vessel were modeled by B-Spline curves (Perez and Suarez, 2007). Also, Wang and Zou (2008) have generated the vessel geometry based on a mixed method composed of non-uniform B-Spline together with an iterative procedure. The parametric generation of geometry in curved bilge fishing vessels was reported based on the non-uniform B-Spline surfaces by Perez and Clemente (2008). Recently, Ventura and Guedes Soares (2012) presented a methodology for the computation of the intersection of parametric surfaces using NURBS with special attention to some specific requirements which is applicable in ship hull modeling. Calkins et al. (2001) developed a computational automated method to define the hull form of a planing vessel in the conceptual design step. They focused on the mono-hull and catamaran planing vessels hull form design using a mathematical procedure. Most of these researches have worked on the simple hull form generation, not complex multi-hull tunnel vessels. It seems that there is a lack of quick and developed method in the planing tunnel vessels modeling studies. This paper developed an innovative method to generate the hull form automatically in the conceptual design stage of these vessels. This method is based on the geometry creation guidelines with the least control and hull form adjustment input parameters. The modeling procedure consists of two main steps. At first, four longitudinal guideline functions are defined based on the position of sections along the vessel length by using two different ways of polynomial functions and NURBS curves. In the transverse section, the coordinates of four key points play an important role to define the tunnel form. So, considering these points, the section curves can be generated using three different models; parabolic form, elliptic form and the NURBS curves. Finally, the fair transverse sections, the fair surfaces and the solids model can be generated. The following sections are organized as follows. The methodology details and definition of the model are described in Section 2. The governing equations are defined in Section 3. The generated various model results are discussed in Section 4. The effect of the main geometry parameters and sub-parameters are illustrated in the planing tunnel hull form. Finally, in Section 5 conclusions have been presented.
2. Methodology 2.1. Background The simplest body form of a tunnel vessel is shown in Fig. 2. The body of tunnel vessels is composed of three general parts. The first part is the central part which provides the hydrodynamic lift force. The front of this part has been curved up in order to distribute the dynamic pressure force from mixed air and water during the increasing vessel speed. The moderate pressure generated at the bottom of the hull prevents porpoising instability. The second part is the planing tunnel which produces the hydrodynamic and aerostatic lift forces by imprisonment of air and breast waves. The third part is a rigid skirt on the outer sides of tunnels. Its role is air imprisonment, preventing air escape and breasting the waves, continuously. It is obvious that vessels with better performance can be obtained by the addition of hydrodynamic hull form elements or with increasing the number of tunnels. Two feasible vessels are shown in Fig. 3.
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Fig. 2. A simple schema of the components of tunnel vessel body.
Fig. 3. Samples of commercial tunnel vessels.
Fig. 4. Body plan of tunnel vessel. The prismatic part and tunnel are demonstrated by solid line and dash line, respectively.
The optimum hull form design is one of the most important problems that are viable by changing the form of the body and the hydrodynamic analysis iteratively. Unfortunately, this takes a long time. Therefore, it is essential to develop a quick way to generate and modify the form automatically. In the following, the basic geometrical hydrodynamic parameters and our innovative generation method are described. 2.2. Model description In the present method, the tunnel vessels hull form is divided into two main parts as can be seen in Fig. 4. These two parts, that are considered to have the greatest effect on the hydrodynamics behavior independently, are defined as a) Prismatic body: in this part, the main parameters are the width of the prismatic body, the deadrise angle at each transverse section, and their longitudinal variations profiles. b) Tunnel: the most important parameters of the tunnel are the height of tunnel, width of tunnel, and their longitudinal variations profiles. An additional factor that significantly affects the tunnel is the form of the cross section. It means that the section form may change by the transverse movement of control point 3. These parameters are shown in Fig. 5 and introduced in rows 1–4 of Table 1. In this type of vessel, total breadth of the hull usually changes a little, so it is assumed as a constant value. Therefore, the summation of the prismatic width and tunnel parts is always constant and equals the total width of the vessel. The height and width of the tunnel part in each transverse section might change in respect with length variations. As a result, the cross section of tunnel form can vary between different shapes like, circle, ellipse, parabola, etc. continuously and fairly. In addition, the variations of width and height of the tunnel can cause twisting, shrinking or enlargement of the tunnel cross section. Decrease or increase in deadrise angle can change the shape of the prismatic part and lead to changing the hydrodynamic lift force.
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Fig. 5. Different parameters of tunnel vessel section at the length of x.
Table 1 Definition of the employed parameters. Row 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Symbol
Value
Description (unit)
bT(x) bP(x) hT(x) b(x) B L H Bv D11 D20 D31 D30 D41 D40 N m1 m2 m3 C11 C21 C31 C41 Cb32 Zi(x), Yi(x)
– – – – 3 12 1.5 1.5 0.9 0.25 1.15 0.5 0.75 0.25 15 5 30 30 50 45 40 35 50 –
Tunnel width (m) Prismatic part width (m) Tunnel height (m) Deadrise angle (m) Vessel width (m) Vessel length (m) Vessel height (m) Maximum Prismatic part width (m) Height of the fore point of guideline 1 (m) Height of the aft point of guideline 2 (m) Height of the fore point of guideline 3 (m) Height of the aft point of guideline 3 (m) Height of the fore point of guideline 4 (m) Height of the aft point of guideline 4 (m) Number of transverse sections (dimensionless) Number of points between guidelines 1 and 2 in each transverse section (dimensionless) Number of points between guidelines 2 and 3 in each section (dimensionless) Number of points between guidelines 3 and 4 in each section (dimensionless) Percentage of length of fixed part of guideline 1 from aft vessel (%) Percentage of length of the fixed part of guideline 2 from aft vessel (%) Percentage of length of the fixed part of guideline 3 from aft vessel (%) Percentage of length of the fixed part of guideline 4 from aft vessel (%) Percentage of closing of the guideline 3 to guideline 2 Width and height functions of guideline I (dimesionless)
In this configuration, each of the mentioned parameters is calculated at the optimum state and the tunnel vessel is ultimately designed. So, the main parameters that affect the design include: the deadrise angle, prismatic body width, tunnel cross section shape (circle, ellipse, parabola, etc.). The tunnel width and height in each transverse section should be determined. The best fairing way for the created volume from joining transverse sections is to define the continuous variation of these parameters based on the vessel’s length. Thus, in order to generate a smooth volume, the longitudinal function of each parameter should be specified. Four different lines are distinguished in the half-body of the hull form with two tunnels as illustrated in Figs. 6 and 7. Line 1 is the axis of symmetry of the vessel body and prismatic part. Line 2 is the boundary between the prismatic part and the tunnel part. Lines 3 and 4 are the maximum height location of the tunnel and the side boundary of the tunnel and vessel, respectively. These four lines give four key points in each transverse section that can be used as the creating points for the given section form. As a case in point, Fig. 5 shows the four points in a transverse section. As can be seen in this figure, the prismatic part is formed by straight connections between points 1 and 2. The tunnel is identified by connecting points 2, 3 and 4. The form of cross section depends on the position of the four points and any changes in the location of one point can change the cross section form. Indeed: 1. When the height of the prismatic part varies (means point 2) the deadrise angle has changed. 2. Since the vessel’s width is constant, division of the width between the prismatic part and the tunnel is achieved by changing the width of point 2. 3. The relative displacement of points 2, 3 and 4 changes the tunnel cross section form. 4. One of the most important hull form characteristics is point 3. If point 3 moves to the left or right, causes the generation of circulating flow over the tunnel.
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Fig. 6. Front view of the generated body by use of guidelines.
Fig. 7. Bottom view of the generated body by use of guidelines.
5. These changes can be applied to each transverse cross section. By relative displacement of points of two adjacent cross sections, the longitudinal curve of each of the parameters, mentioned above, can be controlled.
2.3. Definition of model parameters The proposed method in this article is based on the use of the before mentioned four lines that henceforth are named the ‘‘guidelines’’, as shown in Figs. 6 and 7. If we define these guidelines by functions of the length position (x), the coordinates of the four points will be specified in each cross section (given the x value). Then, the cross section is created by connecting these points and the 3-D model of the vessel is obtained by joining produced sections. Guideline 1 is the axis of symmetry (center line) of the prismatic part and vessel. The height of this guideline varies along the vessel length only as z1 ¼z(x) and its transverse position is fixed at y1 ¼ 0. Guideline 2 is the side line of the prismatic part and tunnel. It connects these two parts. The height and transverse position of this guideline changes along the vessel length (z2 ¼ z(x), y2 ¼y(x)). Guideline 3 is the middle line of the tunnel. The height and transverse position of this guideline also changes along the vessel length (z3 ¼z(x), y3 ¼y(x)) and the maximum height of the tunnel is associated with this guideline. Guideline 4 is the side line of the vessel and the tunnel. The height of this guideline varies along the vessel length only as z4 ¼z(x) and its transverse position is fixed at y4 ¼B/2. With controlling these four guidelines, all the parameters mentioned in the previous section can be adjusted as following: 1. 2. 3. 4. 5. 6.
Vessel width ¼twice the transverse distance between guidelines 1 and 4. Tunnel width¼ transverse distance between guidelines 2 and 4 in each transverse section. Prismatic part width ¼transverse distance between guidelines 1 and 2 in each transverse section. Deadrise angle¼the ratio of height difference of guidelines 1 and 2 to their width difference in each transverse section. Tunnel height¼the height difference between guidelines 1 and 3 in each transverse section. The transverse section form of tunnel ¼connection of the three points of guidelines 2, 3 and 4 by an arbitrary curve (NURBS, ellipse or parabola) in each transverse section. 7. The longitudinal profile of tunnel¼ an integration of longitudinal guidelines 2, 3 and 4 curves (quadratic and cubic polynomial functions or NURBS). 8. The longitudinal profile of the prismatic part ¼an integration of the longitudinal guidelines 1 and 2 curves (quadratic and cubic polynomial functions or NURBS). Thus, using the defined four guidelines, all parameters of the hull form can be determined and a change in guidelines (curves) influences the value of hull form parameters. A significant advantage of this method is the continuity of variations along the vessel length. It also guarantees the surface smoothness of the produced 3D model. The control parameters are used as input values to prepare, adjust and modify the four guidelines. The overall length, width and height of the vessel (rows 5–7 in Table 1) are considered as the first input values by the user. The maximum prismatic body width (row 8), which happens at the end of the vessel, is assumed as another input data. Afterwards, parameters of the guidelines are defined based on the
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given data in Table 1. The height of the beginning and end points of the guidelines (rows 9–14 in Table 1) are defined in terms of longitudinal position (x) to obtain the constant coefficients of guidelines equations. Guidelines 1 and 2 have the same start points due to their intersection at the fore of the vessel. The width of guideline 3, which is between guidelines 2 and 4, is defined based on its adjacency distance to guideline 2 (row 23 at Table 1). Guidelines 1 and 4 are only functions of the height while guidelines 2 and 3 are functions of both height and width. Considering the type of the body form, all changes along these guidelines occur on the approximate distance of (1/3 1/2) L form the forward of the vessel whereas the rates of variations are almost small or zero at the aft part of the vessel. Therefore, the parallel parts of the hull of guidelines are defined as some percentage of the vessel length (rows 19–22 of Table 1).
3. The governing equations In this paper, two different methods are employed to delineate the four guidelines. In the first method, a second order polynomial is used to define guidelines (except for guideline 3 where a third order polynomial is used because of the bulge change). In the second one, NURBS curve relations are used to define guidelines. Three different models have been used to describe each transverse section with four points of guidelines. Three different curves (parabolic, elliptic and NURBS curves) are applied to define the tunnel form between points 2, 3 and 4. In all three models, the prismatic body is produced by direct connection (straight line) of point 1 to point 2 in each section (for instance see Fig. 4). The summary of how to define these three different models are described in Table 2. In order to generate models 1 and 2, four guidelines are defined which have quadratic curves. The width and height functions in each of these guidelines are defined by the length position (x). To determine the constant coefficients of the quadratic relations of each guideline, three boundary conditions are needed to define each guideline function. These conditions are obtained based on the beginning and end points of each guideline which are entered by the user as an input data. Indexes and values of these parameters are given In Table 1, rows 9–14. Since most of the planing tunnel hull forms have a uniform cross section in the aft of vessel, the variations of guidelines should also be constant in some percentage of the length vessel from aft. In order to cover this purpose, a number of coefficients are assumed in the equations which their definitions, indexes and values can be observed in rows 19–22 in Table 1. These coefficients are used in the definition of boundary conditions of the guidelines. For simplicity reduction of input parameters, the transverse position of the guideline 3 has been defined based on its closing percentage coefficient to guideline no. 2 as expressed in row 23 in Table 1. The guidelines are described by polynomial functions in the first and second models. To minimize input parameters and because of the negligible effect of higher orders polynomials, guidelines no. 1, 2, 4 are defined by square polynomial equations. The keel, chine and side guidelines (no. 1, 2, 4) can be generally represented with a square polynomial function as the following equation: ( Z i ðXÞ ¼ C 2 X 2 þ C 1 X þ C 0 ð1Þ Y i ðXÞ ¼ B2 X 2 þ B1 X þ B0 Concave form of these three guidelines is constant along the vessel length, but concave curvature of guideline no. 3 changes along the vessel, therefore it is defined by a cubic polynomial function as Eq. (2). ( Z 3 ðXÞ ¼ C 3 X 3 þ C 2 X 2 þ C 1 X þ C 0 ð2Þ Y 3 ðXÞ ¼ B3 X 3 þ B2 X 2 þ B1 X þB0 where, Zi and Yi are the height and width of guideline no. i. These two parameters are defined based on length (X) and the constant values (Cj, Bj) that are obtained imposing boundary conditions. The guideline functions are described as follows: – Guideline 1 definition This guideline is defined at the center line of the vessel where the origin of the coordinate system is located at the aft point. Some boundary conditions are firstly required at the aft and fore. Since the aft part of the guideline has the lowest height, i.e. the aft height is set to be zero. At the fore part of the vessel, it is defined based on the input height of the start point (D11). The general equation for the height (Z1(X)) of the guideline and their boundary conditions are expressed as follows: dZ 1 Boundary conditions : Z 1ðX ¼ LC 11 Þ ¼ 0, ¼ 0, Z 1ðX ¼ LÞ ¼ D11 dx ðX ¼ LnC 11 Þ
8 < Z 1 ðX Þ ¼
2 D11 11 D11 11 D11 X 2 þ L C2C X þ C C2 2C L2 ðC 11 2 2C 11 þ 1Þ ð 11 2 2C 11 þ 1Þ ð 11 11 þ 1Þ
ð3Þ
: Y ðXÞ ¼ 0 1
Table 2 The conditions used in three different models. Model no.
Definition type of guideline
Definition type of prismatic body
Definition type of tunnel
1 2 3
Second order polynomial Second order polynomial Third order NURBS
Straight line Straight line Straight line
Parabolic Elliptic Third order NURBS
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– Guideline 2 Guideline 2 changes in both longitudinal and transverse directions. Therefore, it should be defined by two equations (Z2(X), Y2(X)). Both equations are defined as quadratic. To determine the coefficients of the equation in each direction, three boundary conditions are required. The height of the aft has been set based on the input data (D20). At the stem of the vessel, the guidelines of 1 and 2 coincide with the same data, (Z2(X ¼ L) ¼Z1(X ¼ L) ¼D11). The guideline width (Y2) at the aft vessel length has been set equal to the maximum prismatic width (Bv). The general equation of the height and the width of a given guideline and their boundary conditions are expressed as follows: 2 Z 2ðX ¼ L C 21 Þ ¼ D20 , dZ ¼ 0, Z 2 ðX ¼ LÞ ¼ D11 dx ðX ¼ L C Þ 21 B:C: : 2 Y 2ðX ¼ L C 21 Þ ¼ Bv , dY ¼ 0, Y 2ðX ¼ LÞ ¼ 0 dx ðX ¼ L C 21 Þ
Z 2 ðX Þ ¼
D11 D20 X 2 þ L 2 CC 212 ð2D20C Dþ111Þ X þ D11 þ ð1 C2 C2212Þ CðD20þD111 Þ L2 ðC 21 2 2 C 21 þ 1Þ ð 21 Þ ð 21 Þ 21 21
Y 2 ðX Þ ¼
2 C 21 Bv Bv X2 þ X þ ð1 22 C 21 Þ Bv L2 ðC 21 2 2 C 21 þ 1Þ L ðC 21 2 2 C 21 þ 1Þ ðC 21 2 C 21 þ 1Þ
ð4Þ
– Guideline 3 As already defined, the order of the polynomial function of this guideline equals to 3 and it has four constant coefficients. To obtain these constant values, the height variations in the aft percentage of the length, are set to zero and the height of the aft and fore of the guideline are assumed based on the input data (D30, D31). The guideline width is defined based on its closeness ratio to guideline 2 (Cb32) compared with guideline 4. The function of height (Z3(X)) and width (Y3(X)) and the boundary conditions of guideline 3 are expressed as follows: 3 3 B:C: : Z 3ðX ¼ LC 31 Þ ¼ D30 , dZ ¼ 0, Z 3ðX ¼ LÞ ¼ D31 , dZ ¼0 dx dx ðX ¼ LnC 31 Þ
8 < Z 3 ðX Þ ¼
ðX ¼ LÞ
2ðD31 D30 Þ 6ðD31 D30 ÞC 31 X 3 þ 2 3ðD2 31 D30 Þð1 þ C 31 Þ X2 þ X þ D31 þ ðD231 D30 Þð1 þ 3C 31 Þ LðC 31 2 2C 31 þ 1Þð1C 31 Þ L3 ðC 31 2 2C 31 þ 1Þð1C 31 Þ L ðC 31 2C 31 þ 1Þð1C 31 Þ ðC 31 2C 31 þ 1Þð1C 31 Þ
: Y 3 ðXÞ ¼ C b32 Y 2 ðXÞ þ ð1C b32 Þ B=2
ð5Þ
– Guideline 4 This guideline is only a function of the height of the side part of the vessel. The input data (D40, D41) are given for the height of the fore and aft, respectively. The function of guideline 4 and its boundary conditions are expressed as 4 ¼ 0, Z 4ðX ¼ LÞ ¼ D41 B:C: : Z 4ðX ¼ LC 41 Þ ¼ D40 , dZ dx ðX ¼ LC 41 Þ
8 < Z 4 ðX Þ ¼
ÞðD40 D41 Þ D41 D40 41 ðD40 D41 Þ X 2 þ 2C X þ D41 þ ð2CC41 1 LðC 41 2 2C 41 þ 1Þ L2 ðC 41 2 2C 41 þ 1Þ ð 41 2 2C 41 þ 1Þ
: Y 4 ðX Þ ¼
B 2
ð6Þ
The tunnel shape can be determined by the definition of the quadruplet points in each transverse section. In the tunnel with a parabolic curve (model 1), if the width values vary between points 2 and 4, the height of the points on the parabolic curve can be calculated using the following matrix equations: 8 9 2 2 3 2 3 3ð1Þ 8 9 2 3 2 2 2 3 a a a Y2 Y2 1 Y2 1 Y2 > > = = < Z2 > < Z2 > h i 6 7 6 7 6 7 6 7 6 7 2 2 2 Z3 ¼ 4 Y 3 Y i 1 4 b 5 Y 2 rY i r Y 4 ) Zi ¼ Y i Y3 1 5 4 b 5 ) 4 b 5 ¼ 4 Y3 Y3 1 5 Z3 ð7Þ > > ; ; :Z > :Z > c c c Y42 Y 4 1 Y 42 Y 4 1 4 4 In case of the ellipsoid shape (model 2) for the tunnel cross section, the following relations between points 2, 3 and 4 can be used (Eq. (8)). By assuming an arbitrary width between the width values of points 2 and 4 for ellipsoid points (Yi), the height of these points (Zi) could be obtained. In this type of cross section, this limitation should be noticed that the transverse position of the guideline 3 is always in the middle of the width of guidelines 2 and 4 because of the ellipsoid shape. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Y i Y 4 þ2 Y 2 Z 4 þZ 2 Y 4 Y 2 2 Z 4 Z 2 2 Y 4 þY 2 2 Z4 þ Z2 2 a¼ b¼ Y 3 ) Z i ¼ b 1 þ þ Z3 þ 2 2 2 2 a 2
Y 2 rY i r Y 4
ð8Þ
To produce guidelines and tunnel sections in model 3, the third order NURBS curve is used. The relations applied for this form are expressed as follows (Piegl and Tiller, 1997): Pn 0 N i,p ðuÞwi P i C ðuÞ ¼ Pi ¼ , n i ¼ 0 N i,p ðuÞwi
U ¼ fu0 ,u1 ,:::,ur1 ,ur g
ð9Þ
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In the Eq. (9), p, the order of curve, equals to 3; n þ1, the number of control points, is 6 for guidelines and 3 for sections in each part of tunnel; r þ1, the size of knot vectors, equals to 10 for guidelines and 8 for sections in each part of the tunnel. The knot vector values (ui), the control points (Pi ) and their weights (wi) are assumed as optional values for each guideline or section curve of the tunnel. The shape of the curve can be properly adjusted by controlling these parameters. To control the shape, the number of control points is fixed at 6 and also uniform knot vectors are used. The knot vectors values of guidelines and sections are shown in the following equation: 1 2 U Guidelines ¼ 0, 0, 0, 0, , , 1, 1, 1, 1 3 3 U Sections ¼ f0, 0, 0, 0, 1, 1, 1,1g ð10Þ
4. Result and discussion 4.1. Comparison of the developed models The three created models for the same input data are plotted and compared to each other. In the first model, longitudinal guideline curves are polynomial and transverse tunnel forms are parabolic. In the second model, the guidelines were as the first one and tunnel form with elliptic shape was used. The guidelines and tunnel form were defined by NURBS curves in the third model. The consequent cross sections are shown in Figs. 8–10 for the first, second and third models, respectively. In these figures, the front view of the body plan lines has been shown. The parameters of the length, width, and maximum prismatic body’s width are assumed as 15, 4 and 1.5 m, respectively for all three models. The other parameters and their values are given in Table 1. Based on these values and Eqs. (1–4), the functions of guidelines for first and second model were obtained. The coefficients of these curves are expressed by the following equation: Z1 ¼ 0:0250X 2 -0:3000X þ0:9000,
Y 1 ¼ 0,
Z2 ¼ 0:0149X2 0:1612X þ1:1149,
Y 2 ¼ 0:0344X 2 þ0:3719X þ0:4959,
Z3 ¼ 0:0035X 3 þ 0:0878X 2 -0:6018X þ1:7518,
Y 3 ¼ 0:5X 2 þ 0:75,
Z4 ¼ 0:0082X 2 0:0690X þ 0:3950,
Y 4 ¼ 1:5,
ð11Þ
In the initial study of these three figures (Figs. 8–10), it can be seen that the generated model with the NURBS curves (model 3) is more similar to elliptic model (model 2). But the parabolic model (model 1) is different from others. On these figures, red part is the prismatic body, the blue part is the tunnel and the pink part is the rigid edge. In fact, the section model is generated by the use of the point model. The number of points in each section is selected based on three parameters of m1, m2, m3. Parameter m1 is the number of points between key points of guidelines 1 and 2 in the prismatic body. Parameter m2 is the number of points between key points of guidelines 2 and 3 in the tunnel part. Parameter m3 is the number of points between key points of guidelines 3 and 4 in opposite side of the tunnel. According to the defined parameters, the generated model can is compatible with numerical analysis software. By changing the number of points in each section, the model becomes smoother and the consequent curves and surfaces used for grid computation become more suitable. Two views of the generated points of the three models, front and isometric, are shown in Figs. 11–13. Using the generated points, firstly, the curves and then the surfaces are made. As an example, the lastly produced surface of the third model has been shown in Fig. 14 which is produced by a macro in commercial CAD software.
Fig. 8. The first model based on quadratic curves.
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Fig. 9. The second model based on elliptic and quadratic curves.
Fig. 10. The third model based on NURBS curves.
The second model (elliptic) is subject to a constraint in which guideline 3 can only be located at the middle distance between guidelines 2 and 4. So, the value of parameter Cb32 is 50%. 4.2. The effect of the main geometry parameters The changes in the geometry of the model have been studied by changing the models’ parameters for two case studies, A and B. The cross sections obtained by the three developed models are shown in Figs. 15–17. Input parameters of two cases are summarized in Table 3. The prismatic body width (row 4) and the parallel parts of the four guidelines (rows 11–14) in case B in comparison with those of case A are greater. Besides, some changes have been applied at the height of beginning points. The end points of guidelines (rows 5–10) have also changed. The height of the front points of guidelines 1 and 3, and the end point of guideline 3 have been reduced from case A to case B. The height of the end point of guideline 2 is greater in case A than case B. The beginning and end points of guideline 4 and the transverse position of guideline 3 are the same in two case studies (row 15). It is observed that by applying these changes, the deadrise angle increases, the tunnel section shape changes and the height and width of it decrease for case B compared with case A. 4.3. The effect of the sub-parameters The geometric parameters for three developed models that control the form of the body are presented in Table 3. In addition, for the third model, that uses the NURBS to define guidelines and tunnel sections’ curvature, the knot vectors, the control points and the weight as well as the main geometry parameters are effective enough to change the hull form. In the other words, this model can be more easily
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Fig. 11. The points of first model based on quadratic curves.
Fig. 12. The points of second model based on the quadratic and elliptic curves.
modified and controlled than the other two models. Although, when the number of parameters is increasing, their consonance and fitting process becomes more difficult and time consuming. Using the third model, with the same basic parameters according to Table 1, two cases have been generated. The difference between these two cases, which are both based on NURBS curves, is due to the diversity in the control points of these cases. These points’ coordinate differences are described in Table 4.
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Fig. 13. The points of third model based on NURBS curves.
Fig. 14. The different views of generated surfaces from third model.
As it may be noticed, 6 control points are treated to define the third order NURBS curve. In this study, changes of the control points 4 and 5 are studied in all 4 guidelines. The displacement of the control points changes the guidelines curve and as a result, the body form changes. As it can be seen in Fig. 18, the volume of the prismatic body in case D is larger than case C and it has a large width on the front of the vessel. On the other hand, in case C, this part is thinner and smaller in volume. These changes also affect the shape of the tunnel and especially the front of the vessel so that the curvature of case D has been larger than case C. In the third model, the number of control points and the size of knot vectors are increasable capability. Thus, the controlling and achievement of the desired hull form can be done more accurately. This way has the disadvantage of the additional input data and assumptions which increase the complexity and time consumption in the geometric model generation. The advanced multi-tunnel high speed craft hull form can be designed using the third model for a hull form generation that has individual hydrodynamic ability. For example, the cross section view of a 4 tunnel vessel is shown in Fig. 19, where the points and sections of semi hull form are shown. Using this section model and the created surface between the two consecutive sections, the solid model of the hull form is generated. This solid model is shown in Fig. 20.
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Fig. 15. The result of changes in main parameters of first model. Left side is Case A and right side is Case B.
Fig. 16. The result of changes in main parameters of second model. Left side is Case A and right side is Case B.
Fig. 17. The result of changes in main parameters of third model. Left side is Case A and right side is Case B.
Table 3 The values of geometry parameters of tow cases and their variations. Row 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Parameter (unit)
Case A
Case B
B L H Bv D11 D20 D31 D30 D41 D40 C11 C21 C31 C41 Cb32
3 12 1.5 1 1.05 0.20 1.40 0.85 0.75 0.25 30 30 30 30 50
3 12 1.5 1.5 0.9 0.25 1.15 0.5 0.75 0.25 50 45 40 35 50
(m) (m) (m) (m) (m) (m) (m) (m) (m) (m) (%) (%) (%) (%) (%)
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Table 4 The values of coordinates (in meters) of control points in two cases.
Guideline X Y Z W Guideline X Y Z W Guideline X Y Z W Guideline X Y Z W n
C. P. 1
C. P. 2
C. P. 3
C. P. 4n
C. P. 5n
C. P. 6
0 0 0 1
5.7 0 0 1
6 0 0 1
8/11 0 0.03/0.675 1.05/0.95
9/12 0 0.15/0.75 1.05/0.95
12 0 0.9 1
0 0.75 0.25 1
5.13 0.75 0.25 1
5.4 0.75 0.25 1
7.6/9.8 0.625/0.612 0.423/0.662 1.15/0.95
8.7/10.9 0.375/0.544 0.597/0.721 1.15/0.95
12 0 0.9 1
0 1.125 0.5 1
4.56 1.125 0.5 1
4.8 1.125 0.5 1
7.2 1.062/1.056 0.673/0.933 1.05/0.95
8.4 0.812/0.988 0.847/1.042 1.05/0.95
12 0.75 1.15 1
0 1.5 0.25 1
3.99 1.5 0.25 1
4.2 1.5 0.25 1
6.8 1.5 0.383/0.333 1
8.1 1.5 0.517/0.417 1
12 1.5 0.75 1
1
2
3
4
Value of case C/value of case D.
Fig. 18. The effect of control points coordinates change in two NURBS cases. Left side is Case C and right side is Case D.
Fig. 19. The points model of 4 tunnels vessel. Right side is key points view and left side is sections view.
5. Conclusion In this article, the geometric model of the planing tunnel vessel is generated using a new mathematical method in which the number of input parameters is minimized. The method is based on the creation of four longitudinal guidelines and the quadruplet key points. First, the key points are obtained from guideline functions in each length position. The hull form sections are generated using these
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Fig. 20. The solid and section models of 4 tunnels vessel.
points; then the surfaces and the solid model are produced. The generated sections are fair curves with sufficient and suitable points. Therefore, the surface or solid model created in the 3-D model generation software does not need further fairing. This leads to a quick and automatic generation of the hull form. Furthermore, this facilitates the generation of a completely smooth model that can be directly transferred to the mesh generation software for computational fluid dynamics analysis. Note that the vessel width along its length should be constant in the present method. In the case study, three different models are used for hull form generation. Comparing the models shows that the elliptic model has the most limitations, whereas, the parabolic model is more flexible. In this model, there is a relative balance between the restrictions and model’s ability to control the geometry. Although, NURBS model is complex and consumes more time because of the sub-parameters (knot vectors, control points, etc.), its more flexible and controllable than the other two models. In addition, the distances between generated points are adjustable at each section so their density can be controlled. The process of generation of these models indicated that the proposed method is quick and accurate. Obviously, the generated model can be used for primary study and optimization in conceptual design. In addition, the other details of hull forms such as spray-rail and transverse steps can be added to the model for improvement of the vessel’s hydrodynamics behavior. The capability of the method to generate advanced similar vessels’ hull form such as four tunnel crafts can be examined in the future works.
Acknowledgments This research was supported by MRC (Marine Research Centre) at Amirkabir University of Technology. The authors wish to thank the reviewers for their valuable comments and suggestion. References Abt, C., Bade, S.D., Birk, L., Harries S., 2001. Parametric hull form design – a step towards one week ship design, 8th International Symposium on Practical Design of Ships and Other Floating Structures, PRADS 2001, Shanghai. Calkins, D., Schachter, R.D., Oliveira, L.T., 2001. An automated computational method for planing hull form definition in concept design. Ocean Eng. 28 (3), 297–327. Hinatsu, M., 2004. Fourier NUBS method to express ship hull form. J. Mar. Sci. Technol. 9, 43–49. Kim, H.-C., Nowaki, H., 2005. Parametric design of complex hull forms. Ship & Technol. 9 (1), 47–63. Liu, Y.-J., Zhu, X.-L., Ji, Z.-Sh., 2005. Ship hull plate processing surface fairing with constraints based on B-Spline. J. Mar. Sci. Appl. 4 (3). Mancuso, A., 2006. Parametric design of sailing hull shapes. Ocean Eng. 33 (2), 234–246. Min, K.S., Kang, S.H., 1998. Systematic study on the hull forms design and resistance prediction of displacement-type super-high-speed ships. J. Mar. Sci. Technol. 3, 63–75. Papanikolaou, A., 2002. Development and potential of advanced marine vehicles concepts. Bull. Kansai Soc. Naval Arch. Jpn. 55, 50–54. Perez, F.L., Clemente, J.A., 2008. Parametric generation, modeling, and fairing of simple hull lines with the use of non-uniform rational B-Spline surfaces. J. Ship Res. 52 (1), 1–15. Perez, F., Suarez, J.A., 2007. Quasi-developable B-Spline surfaces in ship hull design. Comput. Aided Des. 39, 853–862. Perez-Arribas, F., Suarez-Suarez, J.A., Fernandez-Jambrina, L., 2006. Automatic surface modeling of a ship hull. Comput. Aided Des. 38, 584–594. Piegl, L., Tiller, W., 1997. The NURBS Book. Springer, Berlin. Sarioz, E., 2006. An optimization approach for fairing of ship hull forms. Ocean Eng. 33 (13), 2105–2118. Ventura, M., Guedes Soares, C., 2012. Surface intersection in geometric modeling of ship hulls. J. Mar. Sci. Technol. 17 (1), 114–124. Wang, H., Zou, Z.-J., 2008. Geometry modeling of ship hull based on non-uniform B-Spline. J. Shanghai Jiao tong Univ. 13 (2), 189–192.