A Parametric Design Method for Unstepped Planing Hulls Using Longitudinal Functions and Shape Coefficients

Abstract

This paper proposes a specifically parametric design method for planing hulls using longitudinal functions and shape coefficients in order to meet the requirements for optimizing the hydrodynamic performance of planing hulls. To fully define the geometry of the planing hull, a series of design parameters and a set of longitudinal functions and shape coefficients are introduced to define key geometric features. The main frame curves of the hull are designed from bottom to top to ensure the priority and independence of parameters related to the planing surface. The mathematical equations of the control points of the keel curve, chine curve, sheer curve, and surface station curve of the hull framework are established and solved based on B-spline theory. This configures the basis for generating a continuous smooth surface of the hull. Finally, based on the frame curves, the hull surface was generated by using NURBS surface interpolation. The design parameters, especially the longitudinal functions and shape coefficients, can intuitively and independently control the key features of the hull form, which allow control over key geometric features that are highly relevant to the hydrodynamics of the planing hull. By utilizing this approach, rapid production of deep-V and radial planing hulls is achievable, resulting in closed and smooth hull surfaces. Case studies have provided evidence that the modeling of monohull unstepped planing hulls with diverse characteristics can be effectively accomplished through the definition of these parameters.

1. Introduction

The planing hull, extensively utilized as a specialized type of high-speed vessel, differs from conventional hull forms. Such a scheme utilizes the principle of planing on the lifting surface to reduce the wetted area and thus achieve a drag reduction effect at high speeds; hence, the planing hull exhibits superior high-speed performance. Early design efforts for planing hulls relied mainly on model testing results [1]. Up until now, extensive tests have been performed on various hull forms, and hydrodynamic performance diagrams and empirical formulas have been derived from a set of test results. Based on these findings, suitable parent hulls have been selected for modified designs, the results of which depend on the designer’s personal experience and subjective judgment. With advances in computer processing power and the development of simulation techniques, the simulation-based design (SBD) method has been broadly utilized in the hull design process due to its flexibility, efficiency, and cost-effectiveness. Parametric modeling of hull forms enables rapid generation of hull models on a computer and forms an essential foundation for hull optimization using SBD.

In general, there are two main categories of parametric design approaches for hull design. One approach involves modifying existing hull surfaces by designing transformation parameters, such as the free-form deformation (FFD) method, the so-called Lackenby transformation method, and the morphing approach [2,3]. These methodologies depend on the selection of an initial hull shape, which leads to a limited design space, and many transformation parameters have no physical meaning. Another approach is full-parametric modeling, which constructs point–curve–surface entities based on given design parameter constraints without relying on the initial hull shape. This design approach achieves a complete definition of the entire hull geometry solely through the design parameters.

In the 1970s, Kuiper introduced mathematical methods for representing ship hull surfaces [4]. Since the 1990s, numerous investigators have conducted research on parametric modeling of hull forms. Harries and coworkers [5,6,7] proposed a parametric design method and developed a corresponding design software, CAESES. CAESES is a powerful design software widely used in the field of hull design, which can assist in defining geometric features, optimizing design parameters, and generating high-quality hull models with its advanced algorithms and intuitive interface. This approach can be effectively employed to optimize the hydrodynamic performance of ship forms. Zhang et al. [8] and Zhang [9] applied the NURBS theory to determine longitudinal characteristic curves and employed optimization methods to generate waterlines, thereby obtaining the hull surface. Lu [10] utilized NURBS functions to represent the surfaces of ship hulls, focusing on key issues in ship design. Kostas et al. [11] employed T-splines in the Rhino modeling environment to construct hull surfaces with a bulbous bow that obtained second-order continuity, except for extraordinary points. Mancuso [12] applied parameter constraints to establish keel lines and waterlines for the sail hull and used B-spline surfaces for parametric modeling of the fairing. Zhou et al. [13] proposed a NURBS-based parametric method for waterline ship surfaces, which parameterizes the ship hull surface by classifying geometric feature factors and designing feature curves. The NURBS techniques were adopted for the geometric modeling of hull curves and surfaces, allowing hull deformation driven by geometric feature parameters, and a corresponding software package was also developed. Paérez [14,15,16] implemented B-spline methods to achieve parametric modeling of a variety of ships, including waterline ships, planing hulls, and small-waterline catamarans. Khan et al. [17] divided the hulls into three segments and performed parametric modeling for each segment to produce various ship forms.

The goal of parametric modeling is to use concise and physical design parameters to generate ship forms. Too few design parameters lead to insufficient design space, while too many make the method too complicated and lose the importance of parametric modeling. Therefore, in order to develop an applicable parametric method for optimizing the hydrodynamic performance of the hull, the geometric features related to the hull form are here investigated during the hull hydrodynamic study.

In previous investigations, empirical equations were established using parameters such as the Froude number, beam/length ratio, and deadrise angle to evaluate the performance of planing hulls [1]. In recent years, more scholars have focused on the finer geometric features of planing hulls and conducted research on their hydrodynamic performance. Pacuraru et al. [18] performed computational fluid dynamics (CFD) investigations on the total resistance, sinkage, and trim angle of the planing hull by establishing calculation models for the hydrodynamic performance of seven boat types. They examined the effect of various features, including spray rails and chines, on the performance of the planing hull. Ma [19] utilized a monohull planing model as the main form to establish a double chine planing hull and analyzed the effects of the deadrise angle and spray rail width between the two chines as design parameters on both the total resistance and wave resistance. Zhao et al. [20] developed a prediction model for the residual resistance of a planing hull via a random forest model with six variables, including the geometric characteristics of the hull such as the length/beam ratio (L/B), prismatic coefficient (Cp), and Froude number (Fr). Tran et al. [21] established a resistance objective function incorporating three major F-parameters—the static load coefficient, longitudinal center of gravity position relative to the beam (L_CG/B), and deadrise angle (β)—and utilized an approximate optimization model to obtain the optimal resistance hull form for a high-speed passenger boat. Matveev [22] considered non-prismatic hull forms with variable and negative deadrise angles in the hydrodynamic modeling approach for planing hulls. Schachter et al. [23] proposed a special approach, the so-called “virtual prisms”, to evaluate the dynamic equilibrium of the planing hull on the water surface and then applied it to hull forms with varying deadrise angles along the length of the boat. Kim et al. [24] conducted experimental and CFD investigations on a deep-V hull form and analyzed the effect of the bow shape on the resistance and seakeeping performance.

In general, the construction of the planing surface, particularly the distribution of deadrise angles and the profile shape, plays a vital role in the hydrodynamic performance of the planing hull. However, current design approaches for planing hulls rarely consider the deadrise angle as a design parameter. Therefore, this study aims to incorporate crucial design parameters, including longitudinal functions for the deadrise angle of the planing surface, to establish a modeling approach that focuses on key geometric features, especially the planing surface.

The main goals of this article can be summarized as follows: (i) In response to the need for optimizing the hydrodynamic performance of planing boats, appropriate design parameters with high flexibility and independence are reasonably introduced. Therefore, the designer can freely choose them for local or global optimization; (ii) a modeling approach based on B-spline theory is proposed. Using the design parameters as constraints, a set of equations is appropriately constructed to generate the main feature curves and sections (station curves) of the hull, which are solved to form the main frame of the vessel. To this end, the hull surface is constructed by employing B-spline surface interpolation. The modeling approach is implemented in a programming environment, and hull models are established to validate the practicality of the proposed approach.

2. A Brief Introduction to B-Spline Theory

The first step in the design process is curve modeling, which essentially employs the Bézier and B-spline curve methods. The B-spline curve theory is appropriately derived from the Bézier curve theory, thus allowing the extension of Bézier curves to B-spline curves. B-spline curves possess excellent geometric properties, including endpoint interpolation, affine invariance, and a convex hull. Additionally, such approaches are essentially characterized by fast algorithmic execution speed and numerical stability and can be extended to non-rational B-splines (NURBS) and extensively utilized in various computer geometric standards, such as initial graphics exchange specification (IGES) and standard for the exchange of product model data (STEP).

2.1. B-Spline Curve [25]

The pth-order B-spline curve with parameter normalization is defined as follows:

$$S(u)={\displaystyle {\sum }_{i=0}^n{N}_{(i,p)}(u)\cdot P_i},$$

(1)

where P_i signifies the control points, and N_i,p(u) represents the pth-order B-spline basis functions defined on the non-periodic node vector U. Throughout this paper, the curve parameter u in all curve definitions is assumed to be in the normalized parameter range (i.e., $0\le u\le 1$).

$$\mathit{U}=\{u_0^k,u_1^k\dots u_{n+p+1}^k\}=\{\underset{p+1}{\underbrace{0,0\dots ,0,}}{u}_1,u_2\dots u_{n-p-1},\underset{p+1}{\underbrace{1,1\dots 1}}\}$$

(2)

$$N_{i,0}=\left\{\begin{array}{c}1, u_i^k\le u<u_{i+1}^k\\ 0, others\end{array}\right.$$

(3)

$$N_{i,p}={\displaystyle \frac{u-u_i}{u_{i+p}-u_i}}{N}_{i,p-1}(u)+{\displaystyle \frac{u_{i+p+1}-u_i}{u_{i+p+1}-u_{i+1}}}{N}_{i+1,p-1}(u)$$

(4)

2.2. Bézier Curve

In addition to B-spline theory, the Bézier method is another important approach in curve and surface generation. The curve is determined by a polynomial function whose shape is influenced by the positions of several control points. One of the key features of the Bézier curve is that it always passes through the first and the last control points, and its shape is smoothly interpolated between these endpoints based on the weights assigned to the intermediate control points. By taking the values specified in Equation (5) via the node vector U_B, the basis function N_i,p(u) is transformed into a Bernstein polynomial. As a result, the pth-order B-spline curve is equivalent to the pth-order Bézier curve with p + 1 control points.

$${\mathit{U}}_B=\{\underset{p+1}{\underbrace{0,0\dots ,0}},\underset{p+1}{\underbrace{1,1\dots 1}}\}$$

(5)

At this point, the Bézier curve and its first-order derivative equation take the following form:

$$\left\{\begin{array}{l}\mathbf{S}(u)={\displaystyle \sum _{i=0}^p{B}_{i,p}}(u)\cdot P_i\\ {\mathbf{S}}^{\prime }(u)={\displaystyle \sum _{i=0}^p{B^{\prime }}_{i,p}}(u)\cdot P_i\end{array}\right.$$

(6)

in which $B_{i,p}^{\prime }(u)$ can be obtained by differentiating the corresponding basis function $B_{i,p}(u)$ with respect to u. The quadratic and cubic Bézier curve basis functions and their first derivatives are shown in

Table 1. Bézier curve basis functions.

i	$B_{\mathit{i}\mathbf{,}\mathbf{3}}(\mathit{u})$	$B_{\mathit{i}\mathbf{,}\mathbf{3}}^{\prime }(\mathit{u})$	$B_{\mathit{i}\mathbf{,}\mathbf{2}}(\mathit{u})$	$B_{\mathit{i}\mathbf{,}\mathbf{2}}^{\prime }(\mathit{u})$
0	(1 − u)3	−3 (1 − u)2	(1 − u)2	−2 (1 − u)
1	3u (1 − u)2	3 (1 − u)2 − 6u (1 − u)	2u (1 − u)	2 − 4u
2	3u2 (1 − u)	6u (1 − u) − 3u2	u2	2u
3	u3

. Bézier curves will be applied to the design of most of the curves in this paper, mainly second-order and third-order curves.

2.3. Degree Reduction of Bézier Curves

Reducing the degree of Bézier curves has significant applications in computer graphics and curve design. This can be utilized to simplify the editing and manipulation of curves, reduce the computational effort, and provide a more efficient representation of curves. Figure 1 illustrates a cubic Bézier curve with control points represented by P_s, ${\mathbf{P}}_1^3$, ${\mathbf{P}}_2^3$, and P_e. The dashed lines in the figure represent the control polygon with control points as its vertices. The equation curve is as follows:

$${\mathbf{S}}^3(u)=B_{0,3}(u)\cdot P_s+B_{1,3}(u)\cdot P_1^3+B_{2,3}(u)\cdot P_2^3+B_{3,3}(u)\cdot P_e$$

(7)

By meeting the conditions specified in Equation (8) via these control points,

$$\left\{\begin{array}{c}{P}_1^3={\displaystyle \frac{1}{3}}{P}_s+{\displaystyle \frac{2}{3}}{P}_1^2\\ P_2^3={\displaystyle \frac{1}{3}}{P}_e+{\displaystyle \frac{2}{3}}{P}_1^2\end{array}\right.$$

(8)

By substituting Equation (8) with the coefficients from Table 1, the quadratic Bézier curve with control points P_s, ${\mathbf{P}}_1^2$, and P_e would be equivalent to ${\mathbf{S}}^3(u)$, as follows:

$${\mathbf{S}}^3(u)=B_{0,2}(u)\cdot P_s+B_{1,2}(u)\cdot P_1^2+B_{2,2}(u)\cdot P_e={\mathbf{S}}^2(u)$$

(9)

Higher-order Bézier curves offer increased design space and freedom but also introduce additional design parameters. Herein, the cubic Bézier curve is employed to construct various curves, including station curves of the planing surface. Taking advantage of the characteristics of Bézier curves, the shape coefficients are appropriately designed so that when designers only need to use lower-order curves, the corresponding design parameters can be set to 0, effectively reducing the cubic Bézier curve to a quadratic one. This strategy allows for the preservation of complex design capabilities offered by higher-order curve surfaces while simultaneously reducing the complexity associated with employing lower-order curves for design purposes. In this paper, this method will be used to design certain curves with high design flexibility, such as the bow curved lift section. The goal is to obtain a unified design form that can be compatible with curves of different complexities.

3. Description of the Method

The typical body contour curves of a monohull unstepped planing hull with spray rails are depicted in Figure 2. The hull coordinate system O-xyz is utilized, where the x-axis extends from the stern to the bow, the y-axis is horizontal and points towards the left side of the craft, and the z-axis is vertical, pointing upwards. The origin is located on the vertical line, which is determined by intersecting the plane of the stern and the longitudinal section of the craft, and it is at the same height as the forefoot point at the keel. From bottom to top, the presented curves are the keel curve, inner and outer chine curves, and sheer curves.

The projection of the feature curves of the planing hull in the plane view and the lateral view are shown in Figure 3. The inner chine curve and the keel curve define the boundary of the planing surface. These curves determine the horizontal projection area, deadrise angle, maximum beam, longitudinal curvature, and other characteristics of the planing surface. Additionally, the transverse curvature of the surface is determined by the transverse section curve parameters. The inner and outer chine curves possess similar shapes and define the characteristics of the spray rail surface. The space between the outer chine curve and the sheer curve defines the hull side surface, which can be obtained by establishing the frame using the side-view station curves.

Table 2. Design parameters.

Category	Name	Description
Length	L	Hull length
	Lc	Length along the chine curve
	L0	Length of the straight line of the keel
	Lsm	Abscissa of the maximum width along the sheer curve
	xAc	Centroid abscissa of the horizontal projection of the planing surface
Height	ht	Height of the keel curve at the stern
	Hc	Height of the chine curve
	Hf	Height of the bow
	Ht	Height of the stern
Width	Bt	Width of the sheer curve at the stern
	Bsm	Maximum width of the sheer curve
	Bc0	Width of the inner chine at the stern
	Bcm	Maximum width of the sheer curve
Angle	αk	Angle at the foremost point of the keel in lateral view
	βs	Angle at the foremost point of the sheer in plan view
	βc	Angle at the foremost point of the inner chine in plan view
Area	A	Area of planing surface in plan view
Shape Parameters	λk1, λk2	Shape coefficients of keel curve in lateral view
	λs1, λs2	Shape coefficients of sheer curve in lateral view
	λe, λf	Shape coefficients of upper surface stations
Longitudinal Function	θ(x)	Longitudinal function of deadrise angle
	γk(x), γc(x)	Longitudinal function of tangent angle at the endpoints of station curves of the planing surface
	ks(x), km(x)	Longitudinal function of shape coefficients of station curves of the planing surface
	Wr(x)	Longitudinal function of the width of the spray rail

provides a complete set of design parameters required to define a planing craft.

The modeling procedure is illustrated in Figure 4. It commences with the keel curve and progresses upwards, establishing a system of equations for the control points of each feature curve and station curve. These equations are subsequently solved to form the wireframe, which is then appropriately interpolated to generate the hull surface.

3.1. Definition of the Keel Curve

The keel curve of the planing hull is shown in Figure 5. This curve can be divided into three segments based on different longitudinal features, with C¹ continuity maintained at the joint points P_k1 and P_k3. The first segment (i.e., the bottom straight section) is located near the stern of the hull. Some boat designs employ a certain longitudinal incline in this segment to facilitate the arrangement of propulsion devices, which can be constructed using straight-line segments based on the endpoints. The second segment (i.e., bow curved lift section) of the keel curve is located in the bow’s planing surface lift section. It starts at point P_k1, passes through point P_k2, and ends at point P_k3, exhibiting relatively high curvature. The final segment (i.e., the bow outward section) is positioned at the stem of the ship and shows the tangential continuity at P_k3 with the second segment.

3.1.1. Definition of the Bow Curved Lift Section

The curved lift section of the bow is mainly related to its navigation performance. The steeply sloping shape allows the bow to lift at low speeds, which is of great advantage for planing. However, from the perspective of spray performance requirements, a smaller angle between the keel curve and the waterline is desirable to improve spray performance. To balance these two requirements, the design of the bow of a planing boat usually possesses a relatively flat shape below the waterline and a steeper shape above the waterline. To represent this curve, a cubic Bézier curve is taken into account, with P_k1 and P_k3 as curve endpoints used. ${\mathbf{C}}_{k1}(x_{ck1},0,z_{ck1})$ and ${\mathbf{C}}_{k2}(x_{ck2},0,z_{ck2})$ were the control points to be solved.

$${\mathbf{S}}_{k2}\left(u\right)=B_{0,3}\left(u\right)\cdot {\mathbf{P}}_{k1}+B_{1,3}\left(u\right)\cdot {\mathbf{C}}_{k1}+B_{23}\left(u\right)\cdot {\mathbf{C}}_{k2}+B_{3,3}\left(u\right)\cdot {\mathbf{P}}_{k3}$$

(10)

As illustrated in Figure 6, the chord line P_k1P_k3 is constructed, and the point P_k2 represents the furthest point of the curve from the chord P_k1P_k3. The curve S_k2(u) satisfies the following conditions: (i) it passes through point P_k2; (ii) the slope of the curve at point P_k2 is equal to the slope of the chord line; and (iii) at point P_k1, it is tangent to the first segment. Therefore, a system of equations can be established and solved to obtain the coordinates of the control points of the curve, as follows:

$$\left[\begin{array}{cccc}{B}_{1,3}\left(u_{pk2}\right)& 0& B_{2,3}\left(u_{pk2}\right)& 0\\ 0& B_{1,3}\left(u_{pk2}\right)& 0& B_{2,3}\left(u_{pk2}\right)\\ {\eta }_1{B}_{1,3}^{\prime }& -B_{1,3}^{\prime }\left(u_{pk2}\right)& {\eta }_1{B}_{2,3}^{\prime }\left(u_{pk2}\right)& -B_{2,3}^{\prime }\left(u_{pk2}\right)\\ {\eta }_2& -1& 0& 0\end{array}\right]\left[\begin{array}{c}{x}_{Ck1}\\ z_{Ck1}\\ x_{Ck2}\\ z_{Ck2}\end{array}\right]=\left[\begin{array}{c}{x}_{Pk2}-B_{0,3}\left(u_{pk2}\right)\cdot L_0-B_{3,3}\left(u_{pk2}\right)\cdot L_c\\ z_{Pk2}-B_{3,3}\left(u_{pk2}\right)\cdot H_c\\ B_{3,3}^{\prime }\left(u_{pk2}\right)\cdot H_c-{\eta }_1{B}_{0,3}^{\prime }\left(u_{pk2}\right)\cdot L_0-{\eta }_1{B}_{3,3}^{\prime }\left(u_{pk2}\right)\cdot L_c\\ {\eta }_2\cdot L_0\end{array}\right]$$

(11)

where η₁ represents the slope of the chord line P_k1P_k3, η₂ denotes the slope of the tangent vector at point P_k1, and u_pk2 denotes the corresponding curve parameter value at point P_k2. u_pk2 can be calculated by the chord-length parameterization approach.

To express the local geometric relationships in terms of relatively independent design parameters, point P_k2 should be transformed into a relative position form using parameters λ_k1 and λ_k2, as in the following form:

$${\mathbf{P}}_{k2}={\mathbf{P}}_{k1}+\left(1+{\lambda }_{k1}\right)\cdot {\mathit{n}}_{k1}+\left(1+{\lambda }_{k2}\right)\cdot {\mathit{n}}_{k2}$$

(12)

In addition, ${\mathit{n}}_{k1}=0.5\cdot {\mathbf{P}}_{k1}{\mathbf{P}}_{k3}$, and ${\mathit{n}}_{k2}=0.5\tan ({\alpha }_{k1})\cdot \left|{\mathit{n}}_{k1}\right|{\mathit{n}}_{\perp }$ where ${\mathit{n}}_{\perp }$ denotes the unit vector perpendicular to n_k1. When both parameters λ_k1 and λ_k2 are set equal to 0, it can be readily proven that the curve would be equivalent to a quadratic B-spline curve.

3.1.2. Definition of the Bow Outward Section

The outward section of the bow is located at the end of the keel curve, which starts from point P_k3 and ends at P_k4 (see Figure 7). At point P_k3, it satisfies first-order continuity with curve S_k2(u). The tangent vector of endpoint P_k4 makes an angle α_k with the x-axis, which is appropriately expressed via a quadratic Bézier curve, as given by Equation (13). Further, ${\mathbf{C}}_{k3}(x_{Ck3},0,z_{Ck3})$ denotes the control point that should be suitably determined.

$${\mathbf{S}}_{k3}\left(u\right)=B_{0,2}\left(u\right)\cdot {\mathbf{P}}_{k3}+B_{1,2}\left(u\right)\cdot {\mathbf{C}}_{k3}+B_{2,2}\left(u\right)\cdot {\mathbf{P}}_{k4}$$

(13)

Control point C_k3 can be evaluated by solving Equation (14), which is established based on the slope constraints at both endpoints, given that the end angle at the design parameter is α_k.

$$\left[\begin{array}{cc}-{\eta }_3& 1\\ -\tan ({\alpha }_k)& 1\end{array}\right]\left[\begin{array}{c}{x}_{Ck3}\\ z_{Ck3}\end{array}\right]=\left[\begin{array}{c}{H}_c-{\eta }_3\cdot L_c\\ H_f-tan({\alpha }_k)\cdot L\end{array}\right]$$

(14)

where the slope of the tangent vector at P_k3 is denoted by η₃, which is evaluated by control point C_k2 and point P_k3.

3.2. Definition of the Stations of the Planing Surface

The planing surface plays a pivotal role in the hydrodynamic performance of the watercraft during the planing process. Depending on the vessel type and performance requirements, there are various design forms for the planing surface station curves (see Figure 8). The angle between the chord line connecting the two endpoints of the planing surface station curve and the horizontal plane is defined as the deadrise angle θ. Moreover, it is possible to incorporate multiple cross-sectional types and deadrise angles at different longitudinal positions for the same planing hull model. In this way, a series of longitudinal coefficients can be utilized to define the main characteristics of the planing surface.

3.2.1. Definition of the Inner Chine Curve in Plan View

The inner chine curve in plan view is illustrated in Figure 9, extending from P_c0 at the stem to P_c1 at the transom through the widest point P_cm. A cubic Bézier curve is utilized to represent the chine curve, where C_c2 and C_c3 represent the control points to be determined.

$$\mathbf{S}\left(u\right)=B_{0,3}\left(u\right)\cdot {\mathbf{P}}_{c0}+B_{1,3}\left(u\right)\cdot {\mathbf{C}}_{c1}+B_{23}\left(u\right)\cdot {\mathbf{C}}_{c2}+B_{3,3}\left(u\right)\cdot {\mathbf{P}}_{c1}$$

(15)

The main parameters used for the design geometry are as follows: stern width (B_t), maximum width of inner chine curve (B_cm), longitudinal length of inner chine curve (L_c), end tangent angle (β_c), horizontal projection area of the planing surface (A), and longitudinal coordinate of the centroid of horizontal projection (x_Ac). Since the area and centroid require some integrals to be calculated, it is not convenient to construct linear equations to get control points. By introducing the x-coordinate values x_cm and corresponding curve parameter u_cm values, as well as combining the constraints of terminal tangent angle and maximum width of chine, a system of equations for curve control points can be constructed, as follows:

$$\left[\begin{array}{cccc}{B}_{1,3}\left(u_{cm}\right)& 0& B_{2,3}& 0\\ 0& B_{1,3}\left(u_{cm}\right)& 0& B_{2,3}\left(u_{cm}\right)\\ 0& -B_{1,3}^{\prime }\left(u_{cm}\right)& 0& -B_{2,3}^{\prime }\left(u_{cm}\right)\\ 0& 0& \tan \left({\beta }_c\right)& -1\end{array}\right]\left[\begin{array}{c}{x}_{Cc1}\\ y_{Cc1}\\ x_{Cc2}\\ y_{Cc2}\end{array}\right]=\left[\begin{array}{c}{x}_{cm}-B_{3,3}\left(u_{cm}\right)\cdot L_c\\ B_{cm}-B_{0,3}\left(u_{cm}\right)\cdot B_{c0}\\ B_{0,3}^{\prime }\left(u_{cm}\right)\cdot B_{c0}\\ \tan \left({\beta }_c\right)\cdot L_c\end{array}\right]$$

(16)

By solving Equation (16), the control points C_c2 and C_c3 can be obtained. By substituting C_c2 and C_c3 into Equation (15), curve equation S(u) could be appropriately calculated, where $X\left(u,x_{pcm},u_{cm}\right)$ and $Y\left(u,x_{pcm},u_{cm}\right)$ represent the x and y components of S(u). Therefore, both the area $A^{\prime }$ and centroid x-coordinates $x_{AC}^{\prime }$ can be stated by x_cm and u_cm, as follows:

$$A^{\prime }(x_{cm},u_{cm})=2{{\displaystyle \int }}_0^{1}Y\left(u,x_{cm},u_{cm}\right)X^{\prime }\left(u,x_{cm},u_{cm}\right)du$$

(17)

$$x_{AC}^{\prime }(x_{cm},u_{cm})={\displaystyle \frac{{{\displaystyle 2\int }}_0^{1}X\left(u,x_{cm},u_{cm}\right)\cdot Y\left(u,x_{cm},u_{cm}\right)\cdot X^{\prime }\left(u,x_{cm},u_{cm}\right)du}{A(x_{cm},u_{cm})}}$$

(18)

Considering the design parameters of A and x_Ac, the objective function is constructed as follows:

$$\left\{\begin{array}{c}target1=A(x_{pcm},u_{cm})-A\\ target2=x_{Ac}(x_{pcm},u_{cm})-x_{Ac}\end{array}\right.$$

(19)

The non-dominated sorting genetic algorithm II (NSGA-II) [26] is a widely used multi-objective optimization algorithm that can effectively handle multiple conflicting objectives simultaneously. The NSGA-II is implemented to search for x_cm and u_cm, aiming to minimize the objective function in Equation (19) and satisfy the area and centroid conditions. By substituting the obtained solutions into Equation (16), the numerical solution of the curve control points C_c2 and C_c3 can be obtained by satisfying the area and centroid requirements.

3.2.2. Equations and Solution of the Stations of the Planing Surface

Along the longitudinal range [0, Lc] of the keel curve, equidistant coordinate values are selected. Within each segment, the cubic Bézier curves are utilized to construct the sliding surface station curves as shown by the orange curve in Figure 10.

$${\mathbf{S}}_i\left(u\right)=B_{0,3}\left(u\right)\cdot {\mathbf{Q}}_{ki}+B_{1,3}\left(u\right)\cdot {\mathbf{C}}_{1,i}+B_{23}\left(u\right)\cdot {\mathit{C}}_{2,i}+B_{3,3}\left(u\right)\cdot {\mathbf{Q}}_{ci}$$

(20)

For the analysis of the i-th station, by substituting x_i into the segmented keel curve in Equation (20), the intersection point Q_ki of the keel curve and the cross-section can be determined. By substituting x_i into the horizontal projection of the inner chine curve in Equation (15), the corresponding abscissa y_Qci of Q_ci is evaluated. Hence, the chord length can be evaluated by the following: l_i = y_Qci/cos(θ_i). For ease of analysis, a local coordinate system Q_ki-x_iy_iz_i is established, with Q_ki as the origin. In this local coordinate system, the x-axis aligns with the hull coordinate system, and the y-axis aligns with the chord line. Let the rotation matrix around the x-axis be denoted as R_θi. The relationship between the coordinates of an arbitrary point P in the hull coordinate system and its coordinates ${\mathbf{P}}^{\prime }$ in the local coordinate system Q_ki-x_iy_iz_i is as follows:

$$\left\{\begin{array}{l}P=P^{\prime }\cdot R_{\theta i}+Q_{ki}\\ P^{\prime }=(P-Q_{ki})\cdot R_{\theta i}^{-1}\end{array}\right.$$

(21)

The i-th station curve is analyzed in the local coordinate system Q_ki-x_iy_iz_i. In this case, with Q_ki as the origin, the coordinates of Q_ci are (0,l_i,0). The control points C_1,i and C_2,i are sought to satisfy the following parameter conditions: at the extremum point P_mi, the first derivative of the curve’s Z component, Z′(u), is equal to zero, and the z-coordinate at the extremum point P_mi is denoted as z_pmi. Additionally, the angle between the tangent vectors at both ends and the chord line is denoted by γ_ci and γ_ki. The system of equations can be derived as follows:

$$\left[\begin{array}{cccc}\tan ({\gamma }_{ki})& -1& 0& 0\\ 0& 0& \tan ({\gamma }_{ci})& 1\\ 0& B_{1,3}^{\prime }\left(u_{pmi}\right)& 0& B_{2,3}^{\prime }\left(u_{pmi}\right)\\ 0& B_{1,3}\left(u_{pmi}\right)& 0& B_{2,3}\left(u_{pmi}\right)\end{array}\right]\left[\begin{array}{c}{y}_{C1,i}\\ z_{C1,i}\\ y_{C2,i}\\ z_{C2,i}\end{array}\right]=\left[\begin{array}{c}0\\ \tan ({\gamma }_{ci})\cdot l_i\\ 0\\ z_{pmi}\end{array}\right]$$

(22)

where the shape coefficients k_s,i and k_m,i are introduced to define u_pmi and z_pmi since u_pmi cannot be obtained via the chord-length parameterization method.

$$\left\{\begin{array}{c}{u}_{pmi}=k_{s,i}+0.5\\ z_{pmi}={\displaystyle \frac{1}{2}}\cdot \left(1+k_{m,i}\right)\cdot {\displaystyle \frac{\tan ({\gamma }_{ki})\left|\tan ({\gamma }_{ci})\right|}{\left|\tan ({\gamma }_{ci})\right|+\left|\tan ({\gamma }_{ki})\right|}}{L}_i\end{array}\right.$$

(23)

The transverse curvature of the planing surface is determined by the shape coefficients (k_s,i and k_m,i) and the angles (γ_ci and γ_ki). By solving Equation (22), the local coordinate values of the control points are evaluated. By introducing these values to Equation (21), the global coordinate values of C_1,i and C_2,i are obtained. It can be proven that when the curved shape is either concave or convex, the value of k_s,i and k_m,i can be set equal to 0, resulting in an equivalent quadratic B-spline curve.

3.2.3. Longitudinal Function

In the process of generating the station curves of the planing surface, five groups of parameters are introduced for each station, including the deadrise angle, the endpoint tangent angles (γ_ki and γ_ci), and the shape coefficients (k_si and k_mi). These variables can be obtained by evaluating five longitudinal functions θ(x), γ_k(x), γ_c (x), k_s(x), and k_m(x) at the corresponding x_i values. The longitudinal function θ(x) for the deadrise angle of the planing surface is then established using piecewise functions (see Figure 11). The deadrise angle of the planing surface moderately changes smoothly along the longitudinal direction in the middle and rear sections, which are constructed using linear functions. In the middle and front sections, the deadrise angle of the planing surface exhibits an S-shaped distribution, which is constructed with reference to an S-shaped function: f(t) = (1 − t^m)ⁿ (t = 0~1). This allows us to obtain the deadrise angle, as follows:

$$\theta (x)=\left\{\begin{array}{c}{\displaystyle \frac{x}{L_0}}\times {\theta }_b+(1-{\displaystyle \frac{x}{L_c}}){\theta }_a (x<L_0)\\ {\left[1-{\left({\displaystyle \frac{L_c-x}{L_c-L_0}}\right)}^n\right]}^m\cdot ({\theta }_c-{\theta }_b)+{\theta }_b (x\ge L_0)\end{array}\right.$$

(24)

The relative shape of the station curves is mainly influenced by γ_k(x), γ_c (x), k_s(x), and k_m(x), in addition to θ(x). To meet the design requirements for the lateral bending shape of the planing surface, constant values or simple functions can be utilized for construction. By using constant values, a planing surface with a constant cross-sectional shape can be achieved. Alternatively, the use of straight lines or other simple functions usually results in a gradual change in the planing surface.

3.3. Definition of the Chine Curves

3.3.1. Definition of the Inner Chine

Fitting the discrete points Q_ci that were calculated in Section 3.2.2 can establish the inner chine curve. There are two main methods for fitting 3D curves with discrete points: approximation and interpolation. The approximation method constructs curves with fewer nodes, establishes an overdetermined system of equations, and utilizes algorithms such as least squares to evaluate the curve control points. In the context of this approach, the curve does not entirely pass through the discrete points; hence, this is regarded as an approximate approach. The other method is interpolation, where a specified number of nodes are employed to construct a curve that satisfies the requirement of passing through the discrete points. Therefore, it represents a relatively accurate analytical method. In the present study, a cubic B-spline curve interpolation is employed to establish the inner chine curve in the following form:

$$\mathbf{S}(u)={\displaystyle \sum _{i=0}^n{N}_{i,3}(u)P_i}, 0\le u\le 1$$

(25)

In the above relation, the curve is interpolated at the nodes Q_ci. In the case of n_k discrete points, the number of nodes on the curve is determined as n_k + 2. The node vector U = {u₀, u₁, … u_n+4} is obtained via the chord-length parameterization method. The interpolation curve interpolates at all nodes with the discrete points Q_ci. The nodes at both the beginning and the end should satisfy the following condition:

$$\left\{\begin{array}{c}{P}_0=Q_0\\ P_1={\displaystyle \frac{u_4}{3}}{D}_0+P_0\\ P_n=Q_{n-2}\\ P_{n-1}=P_n-{\displaystyle \frac{1-u_{n+1}}{3}}{D}_n\end{array}\right.$$

(26)

where D₀ and D_n represent the first derivative vectors at the beginning and end points of the curve.

By obtaining the diagonal equation system based on the discrete point conditions, as displayed by Equation (27), the control points for the 3D curve of the chine line can be obtained by solving the following set of equations:

$$\left[\begin{array}{c}{Q}_1-a_1{P}_1\\ Q_2\\ ⋮\\ Q_{n-4}\\ Q_{n-3}-c_{n-3}{P}_{n-1}\end{array}\right]=\left[\begin{array}{ccccccc}{b}_1& c_1& 0& \cdots & 0& 0& 0\\ a_2& b_2& c_2& \cdots & 0& 0& 0\\ ⋮& ⋮& ⋮& & ⋮& ⋮& ⋮\\ 0& 0& 0& \cdots & a_{n-4}& b_{n-4}& c_{n-4}\\ 0& 0& 0& \cdots & 0& a_{n-3}& b_{n-3}\end{array}\right]\left[\begin{array}{c}{P}_2\\ P_3\\ ⋮\\ P_{n-3}\\ P_{n-2}\end{array}\right]$$

(27)

3.3.2. Definition of the Outer Chine

The outer chine curve is established with the inner chine curve as the reference (Figure 12). Let us denote n_i as the unit normal vector at the position of the chine curve in the horizontal plane. To start from the discrete point Q_ci, these points on the outer chine curve can be evaluated as follows:

$${\mathbf{Q}}_{ri}=Q_{ci}+W_r(x)\cdot n_i$$

(28)

The longitudinal function, denoted as W_r(x), can be utilized to calculate the variation of the width of the splash guard along the ship’s length. The spray rail can be designed with a uniform width in the mid-aft section of the hull while gradually tapering and narrowing towards the bow. To construct this function, the multi-segment function composed of straight lines and quadratic functions can be utilized as follows:

$$W_r(x)=\left\{\begin{array}{c}{W}_{r\mathrm{m}} x\le L_0\hfill \\ W_{r\mathrm{m}}\times (1-{({\displaystyle \frac{x-L_0}{L_c-L_0}})}^2) x>L_0\end{array}\right.$$

(29)

The discrete points Q_ri on the outer chine curve can be appropriately calculated based on Equation (28). As a result, the control equation for the outer chine curve of the spray rail can be then constructed via the curve interpolation method (as explained in Section 3.3.1) in the following form:

$${\mathbf{S}}_r(u)={\displaystyle \sum _{i=0}^n{N}_{i,3}(u)P_{ri}}, 0\le u\le 1$$

(30)

where P_ri represents the control points of the outer chine curve. Due to the similarity between the inner and the outer chine curves, the node vectors of the inner chine curves are employed for interpolating the outer chine curve. This approach is capable of effectively facilitating the subsequent construction of the spray rail utilizing a ruled surface. For the design of a planing hull with spray rails, the additional outer chine curve is needed. If there are no spray rails, only the inner chine is needed.

3.4. Definition of the Sheer Curve

3.4.1. Definition of the Sheer Curve in the Plan View

The plan view of the sheer curve on the planing hull is shown in Figure 13. A cubic Bézier curve is employed to represent the horizontal projection of the chine curve.

$${\mathbf{S}}_s(u)=B_{0,3}(u)\cdot {\mathbf{P}}_{s0}+B_{1,3}(u)\cdot {\mathbf{C}}_{s1}+B_{2,3}(u)\cdot {\mathbf{C}}_{s2}+B_{3,3}(u)\cdot {\mathbf{P}}_{s1}$$

(31)

The points P_s0 and P_s1, when projected onto the horizontal plane as (0,B_t,0) and (L,0,0), are used to define the Bézier curve endpoints. For this purpose, the coordinates of the control points C_s1 and C_s2 should be determined, and the curve S_s(u) must satisfy the following requirements: passing through the widest point P_sm and having a tangential angle β_s with the x-axis at point P_s1. By establishing a system of linear equations based on these conditions, the control points for the horizontal projection of the sheer curve can be obtained by solving the following equations:

$$\left[\begin{array}{cccc}{B}_{1,3}\left(u_{sm}\right)& 0& B_{2,3}\left(u_{sm}\right)& 0\\ 0& B_{1,3}\left(u_{sm}\right)& 0& B_{2,3}\left(u_{sm}\right)\\ 0& -B_{1,3}^{\prime }\left(u_{sm}\right)& 0& -B_{2,3}^{\prime }\left(u_{sm}\right)\\ 0& 0& \tan \left({\beta }_s\right)& -1\end{array}\right]\left[\begin{array}{c}{x}_{Cs1}\\ y_{Cs1}\\ x_{Cs2}\\ y_{Cs2}\end{array}\right]=\left[\begin{array}{c}{L}_{sm}-B_{3,3}\left(u_{sm}\right)\cdot L\\ B_{sm}-B_{0,3}\left(u_{sm}\right)\cdot B_t\\ B_{0,3}^{\prime }\left(u_{sm}\right)\cdot B_t\\ \tan \left({\beta }_s\right)\cdot L\end{array}\right]$$

(32)

where u_sm can be calculated by the chord-length parameterization approach.

3.4.2. Definition of the Sheer Curve in the Lateral View

According to Figure 14a, the lateral view curve of the chine could be either a straight line or a slightly curved line. By the way, such a curve can be represented via a quadratic Bézier curve, as presented in Equation (33). The coordinates of the control point C_s3 are to be determined. According to the properties of the tangents at the endpoints of a quadratic B-spline curve, if the angular direction of the tangents at both ends of the curve is employed as a constraint, C_s3 can be obtained as the intersection point of the two tangents. However, improper values of the tangential angles γ₀ and γ₁ may result in the tangents intersecting at an unreasonable position, as illustrated in Figure 14b.

$${\mathbf{S}}_{\mathrm{s}}(u)=B_{0,2}(u)\cdot {\mathbf{P}}_{s0}+B_{1,2}(u)\cdot {\mathbf{C}}_{s3}+B_{2,2}(u)\cdot {\mathbf{P}}_{s1}$$

(33)

On the other hand, designers are more concerned with the distribution of the height of the side projection of the chine curve along the longitudinal direction of the hull and the curvature of the curve. To this end, the coefficients λ_s1 and λ_s2 are introduced as shape coefficients, and Equation (33) is adopted to construct the relative position relationship of the control point coordinates C_s3, as demonstrated in Figure 15.

$$C_{s3}=P_{s0}+(0.5+{\lambda }_{s2})(P_{s1}-P_{s0})+{\lambda }_{s1}(P_{s1}-P_{s0})\cdot R_y$$

(34)

where R_y represents the matrix for rotating 90° around the y-axis.

3.4.3. Definition of the 3D Curve of the Sheer

Along the longitudinal range of the hull [0, L], n coordinate values are sampled equally. First, the value of the parameter u_i corresponding to Equation (31) is calculated. The parameter equation for the x-component of the lateral projection curve is as follows:

$$x_i={\left(1-u_i\right)}^3\cdot {\mathit{x}}_{ps0}+3u_i{(1-u_i)}^2\cdot {\mathit{x}}_{cs1}+3u_i^2(1-u_i)\cdot {\mathit{x}}_{cs2}+u_i^3\cdot {\mathit{x}}_{ps1}$$

(35)

Equation (34) can be solved numerically using Newton’s iteration method. After obtaining the value of the parameter u_i, it is replaced in the lateral projection curve in the horizontal plane, and the y-coordinate of the point can be calculated as follows:

$$y_i={\left(1-u_i\right)}^3\cdot y_{ps0}+3u_i{(1-u_i)}^2\cdot y_{cs1}+3u_i^2(1-u_i)\cdot y_{cs2}+u_i^3\cdot y_{ps1}$$

(36)

Similarly, the corresponding coordinates z_i can be obtained, resulting in a set of discrete points Q_si(x_i,y_i,z_i) on the chine curve. By employing a 3D curve interpolation approach, a 3D cubic B-spline curve is constructed to represent the sheer curve.

$$S_s(u)={\displaystyle \sum _{i=0}^{n+2}{N}_{i,3}(u)P_i}$$

(37)

3.5. Definition of the Stations of the Upper Surface

The outer chine curve and sheer curve should be discretely sampled as the starting points for each station curve of the upper surface. The B-spline surface essentially represents an interpolation function defined on a uv grid of quadrilaterals. The discretization approach determines the distribution of uv grid lines in the upper hull surface. Figure 16a illustrates the distribution of uv grid lines formed using the cross-sectional method, where distortions and C¹ discontinuities are present at the bow. Within the range [0, 1], n parameter values of u_i are uniformly sampled, and then these are respectively substituted into the 3D curve expressions in Equations (30) and (36) for the outer chine curve and the sheer curve. The resulting discrete points Q_si and Q_ri serve as the endpoints for the station curves, forming the trend of the uv grid, as illustrated in Figure 16b. The obtained uv grid exhibits a reasonable distribution in the bow region.

As shown in Figure 17, in most cases, when the points Q_si and Q_ri are not located within the same cross-section, a local coordinate system Q_ri-x_iy_iz_i is established, with point Q_ri as the origin. The orientation of the local coordinate system is specified as follows: the corresponding Z-axis is aligned with the hull coordinate system, the Y-axis is pointed towards the horizontal projection point of Q_si, and the X-axis is determined by the right-hand rule. The relationship between the coordinates of point P in the hull coordinate system and its coordinates P’ in the local coordinate system Q_ri-x_iy_iz_i can be stated as follows:

$$P=P^{\prime }{\mathbf{R}}_{\phi }+Q_{ri}$$

(38)

where R_φ represents the coordinate transformation matrix for the rotation around the Z-axis, with the rotation angle φ being the angle between the local coordinate system’s Y-axis and the hull coordinate system.

Within the local coordinate system Q_ri-x_iy_iz_i, the station curve (Figure 17) can be constructed via a quadratic Bézier curve in the y_i-z_i plane, as follows:

$$S_{t,i}(u)=B_{0,2}(u)\cdot Q_{ri}+B_{1,2}(u)\cdot C_{m,i}+B_{0,1}(u)\cdot Q_{si}, 0\le u\le 1$$

(39)

Following the approach outlined in Section 3.4.2, we are able to convert the control point C_mi into expressions for the parameters λ_e and λ_f. Utilizing the coordinate relationship presented in Equation (3), we can then derive the coordinates of C, as presented in Equation (40). Notably, d1 governs the distribution of the bulge positions along the station curve, while d2 impacts the overall curvature of the station curve. The specific C’s coordinates are given as follows:

$$C_{mi}=P_1+(0.5+{\lambda }_e)(P_2-P_1)+{\lambda }_f(P_2-P_1)\cdot R_{\phi }{R}_y{R}_{\phi }^{-1}$$

(40)

where R_y represents the matrix for rotating 90° around the y-axis.

3.6. Surface Generation

Upon establishing the station curves, a set of (n + 1) × (m + 1) discrete points on the surface can be generated based on the equations of station curves, k = 1, 2, …, n and l = 1, 2, …, m. A (p,q)-order B-spline tensor product surface is then established to interpolate these discrete points in the following form:

$$Q_{k,l}=F\left(u_k, v_l\right)={\displaystyle \sum }_{i=0}^n{\displaystyle \sum }_{j=0}^m{N}_{i,p}\left(u_k\right)N_{j,q}\left(v_l\right)P_{i,j}={\displaystyle \sum }_{i=0}^n{N}_{i,p}\left(u_k\right)\left({\displaystyle \sum }_{j=0}^m{N}_{j,q}\left(v_l\right)P_{i,j}\right)$$

(41)

By establishing that $R_{i,l}={\displaystyle {\displaystyle \sum }_{j=0}^m}{N}_{j,q}\left(v_l\right)P_{i,j}$, the following is obtainable:

$$Q_{k,l}={\displaystyle \sum }_{i=0}^n{N}_{i,p}\left(u_k\right)R_{i,l}$$

(42)

where the surface interpolation problem is transformed into a curve interpolation problem. By employing curve interpolation methods, n + 1 curve interpolations can be performed to solve R_i,l, and similarly, m + 1 interpolations can be applied to R_i,l to obtain the surface control points. Substituting these control points into Equation (41) yields the appropriate interpolated surface.

For the construction of the transom, spray rail, and deck using governing surfaces, since the two boundary curves used to construct these surfaces are of the same order and share the same node vector, the usual process of increasing the order and refining the curves can be omitted. Therefore, the governing surface can be directly constructed using the control points of the boundary curves, as in the following form:

$$F(u,v)={\displaystyle \sum _{i=0}^{n+1}{\displaystyle \sum _{j=0}^1{N}_{i,3}(u)}}{N}_{j,1}(u)P_{i,j}$$

(43)

where P_i,0 and P_i,1 each denote the control points of the two boundary curves of the surface. The parameter vector u is identical to the boundary curves, while the node vector in the v direction is set as V = {0,0,1,1}.

4. Discussion and Example

4.1. Parameters’ Influence on Planing Surface

The influence of changing the shape coefficients λ_k1 and λ_k2 on the bow curve can be observed in Figure 18. λ_k1 determines the relative position of point P_k2 in the direction parallel to the chord line, mainly affecting the distribution of the convex positions of the curve. When the value of λ_k1 increases, the convex position moves forward, and when the value decreases, the convex position moves backward. On the other hand, λ_k1 determines the perpendicular distance between point P_k2 and the chord line, influencing the overall curvature of the curve. The larger the value of λ_k1, the greater the overall bending degree of the curve will be. When the value of λ_k2 is −1, the curve degenerates into a straight line.

The transverse shape of the planing surface is primarily influenced by γ_c, γ_k, k_s, and k_m. When γ_c and γ_k take a value of 0, the corresponding cross-sectional shape represents a flat-bottomed planing surface, as presented in Figure 19. When both γ_c and γ_k are positive, the planing surface becomes a concave curve, as illustrated in Figure 20. In this case, k_s affects the overall curvature, whereas k_m affects the symmetry of the curve. The same principles apply to the convex surface. When γ_c and γ_k have opposite signs, the station curve forms a waveform curve, as illustrated in Figure 21. The effects of k_s and k_m on the curve can be observed in this figure. As can be seen, k_m influences the overall curvature of the curve, while k_s primarily affects the symmetry of the curve. Therefore, through the shape coefficients, various transverse bending shapes of the planing surface can be designed and modified.

Three sets of planing surface styles generated by the longitudinal functions of deadrise angles are presented in Figure 22. Figure 22a demonstrates a planing hull with a small deadrise angle, whereas Figure 22b exhibits a substantial variation in the deadrise angle with increasing vessel length. According to Figure 22c, the planing hull demonstrates a larger deadrise angle and a sharper bow. These results demonstrate that the modification of the deadrise angle function could effectively design and modify the distribution of deadrise angles for the planing hull.

4.2. Example

Using the design method presented in this study, programming modeling is performed for planing hulls. A deep-V hull form with spray rails and a radiused chine hull form have been generated. The design parameters are presented in

Table 3. Parameters of the examples.

Name	Unit	Deep-V	Radiused Chine
ht	m	0	0.16
L0	m	9.96	6.64
Lc	m	20.16	13.44
Hc	m	1.56	1.02
L	m	21	14
Hf	m	2.76	1.84
Bt	m	2.2	1.3
Ht	m	2.3	1.42
Lsm	m	6.78	4.52
Bsm	m	2.5	1.7
Bcm	m	1.86	1.24
Bc0	m	1.78	1.19
αk	°	35	35
βs	°	30	30
βc	°	25	25
A	m2	31.32	13.92
xAc	m	8.784	5.856
λk1		−0.37	−0.36
λk2		−0.06	−0.05
λs1		0.87	0.85
λs2		0.13	0.13
λe		0	0.1
λf		0.07	0.06
θa	°	7	12.5
θb	°	17	17
θc	°	45	45
(m,n)		(2, 3.2)	(2, 3.2)
γk(x),	°	10	8 + 3 (1 − x/L)
γc(x),	°	10	8 + 3 (1 − x/L)
ks		0	0.12
km		0	−0.06
WRM	m	0.09	0

, and the models are demonstrated in Figure 23 and Figure 24. The deep-V hull form has a significant deadrise angle and exhibits minimal variation in its cross-sectional shape. On the other hand, the radiused chine hull form exhibits a relatively complex planing surface, with transverse curvature and varying deadrise angles along the longitudinal axis, resulting in the cross-section curves of the planing surface at a single point. By modifying the design parameters, a complete definition of the shape of the planing hull can be achieved, which allows rapid production of the model. Taking into account the various design feasibilities of the planing surfaces described in Section 4.1, based on the method presented in this paper, there is a relatively large design space for the design of planing boats, especially for the planing surfaces. This can meet the design requirements of different forms of monohull unstepped planing hulls.

5. Conclusions

This paper presents a parametric design approach for monohull unstepped planing hulls. In the design process, longitudinal functions and shape coefficients are used for the design of the shapes of key areas. The curve and surface equations of the ship are appropriately evaluated and analyzed based on B-spline theory, resulting in a smoothly continuous surface for the hull. The key findings of the present investigation are summarized as follows:

• When conducting multi-parameter optimization of the hydrodynamic performance of planing hulls, increasing the independence of design parameters can reduce the complexity of the problem. In this study, a detailed analysis and formal transformation of the design parameters have been carried out. A series of non-dimensional shape coefficients are derived, which can independently control the geometric characteristics of the hull. These design parameters represent well-defined physical interpretations and hydrodynamic concepts, specifically targeting geometric characteristics that exhibit a significant impact on the performance of planing hulls. The design of shape coefficients takes advantage of the decimation characteristics of Bézier curves and provides flexibility in the selection of shape coefficients.
• A bottom-up approach to hull frame design is developed. To this end, a system of equations is established based on the design parameters to solve the control points of the curves. Appropriate curve node vectors and curve discretization methods are set, which enhance the efficiency and quality of surface production. The resulting hull surface is of high accuracy and smoothness, which indicates the suitability of the B-spline approach for the parametric modeling of planing hulls.
• An appropriate design program has been developed based on the parametric design method proposed in this article, which allows the rapid production of smooth and complete hull surfaces. By modifying the design parameters, the dimensions of the hull can be appropriately adjusted, or different shapes of the hull can be produced. The generation of different hull forms and the analysis of design parameters indicate that this design method provides ample design space and high degrees of freedom for the forms of monohull unstepped planing hulls.
• In this study, all the curves and surfaces used can be expressed in the form of NURBS, which indicates their high universality. For other designers, they can either implement the design work entirely through programming languages or perform secondary development based on existing commercial geometric design software so as to achieve the batch generation or modification of the geometric models of planing hulls. This is of great significance in today’s context, where a large number of calculation samples are required, or iterative optimization of the hull forms is to be realized through program control.
• The next step involves the exploitation of this design approach as a basis for further research on optimizing the hydrodynamic performance of planing hulls. In addition, parameterized modeling research can be extended to hull types that have additional hull features, such as chines and steps.