A Parametrization and Surface Generation Method for Propeller Blades Based on B-Splines

Abstract

The present invention belongs to the technical field of engineering design and optimization, specifically, a method for parameterizing and generating surfaces of propeller blades based on B-splines: first, based on the B-spline curve method, the least squares method is adopted to fit the blade section value points at different radii, and the parameterization of the blade section and the outer contour is realized by using methods such as proportion, offset, and coordinate transformation; secondly, based on the B-spline surface method, the propeller blade surface, blade back, blade root and blade top surfaces are efficiently generated; finally, the relevant program is written based on the Python language, and a UML class diagram of the program is given; the present invention programs the entire process through the language Python to realize the parameterization of propeller blades and the automatic generation of surfaces, effectively improving the propeller modeling efficiency and the flexibility of parameterization variable selection in the optimization process.

Description

Technical Field

[0001] The present invention belongs to the technical field of engineering design and optimization, and in particular, relates to a propeller blade parameterization and surface generation method based on B-splines. Background Art

[0002] Propellers are the primary form of modern propulsion, and their design performance directly determines the speed, maneuverability, and noise performance of marine equipment. When optimizing propeller shape, it is necessary to define and represent the propeller shape using a set of design parameters through a certain functional relationship. This allows propellers with different geometric shapes to be automatically generated by adjusting the parameters, thus transforming the propeller shape optimization process into an optimization process for this set of design parameters. The propeller shape primarily consists of a hub and blades. The hub shape is typically simple and easy to describe parametrically; however, the blades are typical free-form surfaces with complex and variable geometries. Except in a few cases, they are difficult to describe using simple shape parameters and mathematical expressions. Therefore, parametric modeling and surface generation of propeller blades are prerequisites for achieving propeller optimization design.

[0003] Traditional propeller blade parameterization methods mainly rely on secondary development of commercial software to achieve automatic generation of blades. This consumes a lot of resources, and commercial software is equivalent to a black box for designers. It is difficult to obtain and operate parameterized geometric information, and it is not possible to directly change the shape according to physical meaning. In addition, from the perspective of optimization design, it is hoped that the parameterization method of propeller blades can accurately describe the blade shape and have a large deformation space, and it is also hoped that the number of parameterized variables can be flexibly controlled. Otherwise, too many variables will lead to long optimization time and low optimization efficiency. The parameterization method based on secondary development of commercial software is not flexible enough and is not easy to meet the requirements. Therefore, it is urgent to invent a propeller blade parameterization and surface generation method based on B-splines to effectively improve the flexibility of propeller blade parameterization and the efficiency of blade surface generation. Summary of the Invention

[0004] In order to solve the problem of establishing a parametric model and flexible deformation of propeller blades, the present invention proposes a propeller blade parameterization and surface generation method based on B-splines.

[0005] The specific technical solutions adopted in the present invention are as follows:

[0006] A propeller blade parameterization and surface generation method based on B-spline, specifically comprising the following steps:

[0007] Step 1: Find a B-spline curve to fit the type value points, fit the remaining data points except the first and last endpoints in the form of least squares, and solve the control point variables to obtain the B-spline blade section curve fitted at any radius;

[0008] Step 2: Transform the corresponding control point coordinates by defining the width scale factor, thickness scale factor, and offset factor. Calculate the new control point coordinates using the coordinate transformation formula. Use the defined local and global parametric variables to manipulate the control points of each blade section to achieve parametric deformation of the propeller blade shape.

[0009] Step 3: Calculate the control points, node vectors U and V on the blade surface and blade back surface meshes, generate the blade surface and blade back surface of the propeller, and give the corresponding mathematical expressions;

[0010] Step 4: Calculate the control points, node vectors U and V on the blade root and blade tip surface meshes, generate the blade root and blade tip surfaces of the propeller, and give the corresponding mathematical expressions;

[0011] Step 5: Combining the proposed blade parameterization method and blade surface generation method, the entire process is programmed using the Python language to achieve parameterization of the propeller blade and automatic surface generation.

[0012] In the above technical solution, the B-spline curve in step 1 is expressed as the following mathematical expression:

[0013]

[0014] Where, P i is the control point of the curve, N i，p (u) is the p-order B-spline basis function defined on the curve node vector U(m+1). Unless otherwise stated, a=0, b=1. The order p, the number of control points n+1 and the number of nodes m+1 satisfy the relationship: m=n+p+1.

[0015] The specific process of step 1 is as follows:

[0016] (3) Let {Q k (x k ,y k , z k )|0≤k≤l} is a radius r of the initial propeller blade extension profile j The l+1 two-dimensional leaf section value points at , and are arranged in sequence on the leaf section curve with the leaf section edge as the starting and ending points;

[0017] (4) Find the B-spline curve's corresponding value point {Q k (x k ,y k , z k )|0≤k≤l} for fitting, and satisfy the conditions: Q0=C(0), Q l = C(1), for the remaining data points Q except the first and last endpoints k, approximate in the form of least squares, and solve for n+1 control point variables P i To minimize the error, the specific expression is:

[0018]

[0019] Where, is each type value point Q k There are usually three ways to determine the corresponding parameter values There are three methods: uniform parameterization, chord length parameterization, and centripetal parameterization. Considering that the curvature of the shape value point at the leading edge of the blade section is large, and the centripetal parameterization can obtain better results when the data points change sharply, the centripetal parameterization method is selected here. The specific expression of the centripetal parameterization method is described as:

[0020]

[0021] (3) To ensure that each node interval contains at least one The node vector U(m+1) is selected as follows:

[0022]

[0023] Solving the least squares problem can be transformed into solving the following linear equations with (n-1) unknowns:

[0024] (N T N)P=R

[0025] Where N is a (m-1)×(n-1) matrix consisting of scalars, and R is a column vector consisting of n-1 points. Solving this linear equation system gives the radius r. i B-spline blade section curve fitted at .

[0026] In step 2, the x direction is set as the width direction of the leaf section, the y direction is set as the thickness direction of the leaf section, and the z direction is set as the radius direction of the leaf section. Represents the radius r j The coordinate value of the i-th control point on the leaf section defines the width scaling factor s j Change radius r j The leaf section width at , defines the thickness scaling factor t j Change radius r j The thickness of the leaf section at , defines the offset factor Change radius r j The coordinates of the control points on the leaf section are scaled and offset to obtain the radius r j The coordinate value of the i-th control point on the leaf section It can be expressed by the following formula:

[0027]

[0028] Assuming control point coordinates The three components in the Cartesian coordinate system are expressed as Envelope it to a radius of r j On the cylindrical surface, perform the corresponding coordinate transformation and calculate the new control point coordinates Define the radius r j The pitch angle is (The pitch angle of the variable pitch propeller is different at different radii), the longitudinal angle is θ, and the specific coordinate conversion formula can be described as follows:

[0029]

[0030] By using defined local and global parametric variables to manipulate the control points of each blade section, parametric deformation of the propeller blade shape can be achieved.

[0031] In the step 3,

[0032] (1) The B-spline surface is mainly defined by the control point grids in two directions, two node vectors and univariate B-spline basis functions. The specific equation is:

[0033]

[0034] Where, P i，j is the control point on the surface mesh; N i，p (u) and N j，p (v) are the p-order B-spline basis functions on the node vectors U(r+1) and V(s+1) respectively; the order p, the number of control points n+1 and the number of nodes r+1 satisfy the relationship: r=n+p+1; the order q, the number of control points m+1 and the number of nodes s+1 satisfy the relationship: s=m+q+1.

[0035] (2) Using the B-spline curve splitting method, the leaf section curve is divided into two independent curves without changing the curvature of the split point curve. The specific splitting point is taken at the middle component of the B-spline curve node vector U(m+1), which is obtained by dividing m+1 by 2. After splitting, the leaf surface curves with different radii r are obtained. i Control points of the leaf section curve and node vector Get different radius r on the back surface of the leaf j Control points of the leaf section curve and node vector Leaf surfaces use all split curve node vectors The average value of the node vector v is used as the node vector; the back surface of the leaf uses all the split curve node vectors The average value of is taken as the node vector U, and the node vectors of the corresponding fitting curves are taken as the node vector V for both the leaf surface and the leaf back surface.

[0036] (3) The mathematical expression of the blade surface can be described as:

[0037]

[0038] The mathematical expression of the leaf back surface can be described as:

[0039]

[0040] In the step 4,

[0041] (1) Select the control point of the leaf section curve at the radius r0 (i.e. the leaf root) on the leaf surface and the control point of the leaf section curve at radius r0 on the leaf back surface As the upper and lower rows of control points F of the blade root surface i0 and F i1 , using the node vector U of the blade surface or the blade back surface as the node vector U of the blade root surface, and selecting {0, 0, 1, 1} as the node vector v of the blade root surface, the mathematical expression of the blade root surface can be described as:

[0042]

[0043] (2) The blade top surface needs to take into account the hydrodynamic characteristics, and four rows of control points are used to generate the required arc surface: directly select the radius r on the blade surface m The control point of the leaf section curve at (i.e., at the leaf top) As the first row of control points G i1 , select the radius r on the back surface of the leaf m The control point of the leaf section curve at As the 4th row control point G i4 Then, the second and third rows of control points are calculated using the following expressions:

[0044]

[0045] The node vector U of the blade surface or the blade back surface is used as the node vector U of the blade top surface, and {0, 0, 0, 0, 1, 1, 1, 1} is selected as the node vector V of the blade top surface. The mathematical expression of the blade top surface can be described as:

[0046]

[0047] In the step 4: combining the proposed blade parameterization method and blade surface generation method, the entire process is programmed using the Python language to achieve parameterization of the propeller blade and automatic surface generation.

[0048] The entire process is programmed using the Python programming language, with a detailed description of the internal structure and interrelationships of the classes within the program. The Curve module contains basic methods for B-spline curves, the Surface module contains basic methods for B-spline surfaces, the BladeSection module implements parameterization of propeller blade sections, the Parameterizaton module implements parameterization of blade profiles, and the PropellerShape module generates propeller blade face, back, root, and tip surfaces.

[0049] The beneficial effects of the present invention are as follows: the present invention uses B-spline surfaces to perform parametric modeling on the geometric shape of a three-dimensional propeller blade; according to the mathematical description of the B-spline curve, the B-spline curve is found to fit the type value points, the control point variables are solved, and the B-spline blade section curve fitted at any radius is generated; the new control point coordinates are calculated through the coordinate transformation formula to realize the parametric deformation of the propeller blade shape; based on the mathematical expression of the B-spline surface, the blade surface, blade back, blade root and blade top surfaces of the propeller are generated respectively; the entire process is programmed by the language Python to realize the parameterization of the propeller blade and the automatic generation of the surface, which effectively improves the propeller modeling efficiency and the flexibility of parameterized variable selection during the optimization process. BRIEF DESCRIPTION OF THE DRAWINGS

[0050] Figure 1 is the blade extension profile radius r in the present invention j Schematic diagram of leaf section type value points at .

[0051] Figure 2 is the blade radius r in the present invention j Schematic diagram of the leaf section B-spline fitting curve and control points at .

[0052] Figure 3 It is a schematic diagram of the blade surface and blade back curved surface and their control points in the present invention.

[0053] Figure 4 It is a schematic diagram of the blade root surface and its control points in the present invention.

[0054] Figure 5 It is a schematic diagram of the blade top surface and its control points in the present invention.

[0055] Figure 6 It is a UML class diagram of the propeller blade parameterization and surface generation program of the present invention. DETAILED DESCRIPTION

[0056] In order to deepen the understanding of the present invention, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. The embodiments are only used to explain the present invention and do not limit the scope of protection of the present invention.

[0057] Embodiment: A method for parameterizing and generating a curved surface of a propeller blade based on B-splines, comprising the following steps:

[0058] Step 1: Describe the mathematical expression of the B-spline curve and achieve flexible control of the curve shape through the B-spline curve control points:

[0059] The B-spline curve can be expressed as the following mathematical expression:

[0060]

[0061] Where, P i is the control point of the curve, N i，p (u) is the p-order B-spline basis function defined on the curve knot vector U(m+1). Unless otherwise specified, a=0 and b=1. The order p, the number of control points n+1, and the number of knots m+1 satisfy the relationship: m=n+p+1;

[0062] Step 2: Find a B-spline curve to fit the type value points. Approximate the remaining data points except the first and last endpoints in the least squares form and solve the control point variables to obtain the B-spline blade section curve fitted at any radius:

[0063] (1) Let {Q k (x k ,y k , z k )|0≤k≤l} is a radius r of the initial propeller blade extension profile j The l+1 two-dimensional leaf section value points at , and are arranged in sequence on the leaf section curve with the leaf section edge as the starting and ending points, such as Figure 1 As shown,

[0064] (2) Find the B-spline curve's corresponding value point {Q k (x k ,y k , z k )|0≤k≤l} for fitting, and satisfy the conditions: Q0=C(0), Q l = C(1), for the remaining data points Q except the first and last endpoints k , approximate in the form of least squares, and solve for n+1 control point variables P i To minimize the error, the specific expression is:

[0065]

[0066] Where, is each type value point Q k There are usually three ways to determine the corresponding parameter values Methods: uniform parameterization, chord length parameterization and centripetal parameterization. Considering that the curvature of the shape value point at the leading edge of the blade section is large, and the centripetal parameterization can obtain better results when the data points change sharply, the centripetal parameterization method is selected here. The expression of the centripetal parameterization method is specifically described as:

[0067]

[0068] (3) To ensure that each node interval contains at least one The node vector U(m+1) is selected as follows:

[0069]

[0070] Solving the least squares problem can be transformed into solving the following linear equations with (n-1) unknowns:

[0071] (N T N)P=R

[0072] Where N is a (m-1)×(n-1) matrix consisting of scalars, and R is a column vector consisting of n-1 points. Solving this linear equation system gives the radius r. i The B-spline blade section curve fitted at Figure 2 As shown;

[0073] Step 3: Transform the corresponding control point coordinates by defining the width scale factor, thickness scale factor, and offset factor, and calculate the new control point coordinates through the coordinate transformation formula. Use the defined local and global parametric variables to manipulate the control points of each blade section to achieve parametric deformation of the propeller blade shape:

[0074] (1) Press Figure 1 In the coordinate system, the x direction is the width direction of the leaf section, the y direction is the thickness direction of the leaf section, and the z direction is the radius direction of the leaf section. Represents the radius r j The coordinate value of the i-th control point on the leaf section defines the width scaling factor s j Change radius r j The leaf section width at , defines the thickness scaling factor t j Change radius r j The thickness of the leaf section at the position of the offset factor δ is defined. j =[x δj ,y δj ,0] T Change radius rj The coordinates of the control points on the leaf section are scaled and offset to obtain the radius r j The coordinate value of the i-th control point on the leaf section It can be expressed by the following formula:

[0075]

[0076] (2) Assume the coordinates of the control points The three components in the Cartesian coordinate system are expressed as Envelope it to a radius of r j On the cylindrical surface, perform the corresponding coordinate transformation and calculate the new control point coordinates Define the radius r j The pitch angle is (The pitch angle of the variable pitch propeller is different at different radii), the longitudinal angle is θ, and the specific coordinate conversion formula can be described as follows:

[0077]

[0078] (3) By manipulating the control points of each blade section using defined local and global parameterized variables (as listed in Table 1 below), the parametric deformation of the propeller blade shape can be achieved;

[0079] Table 1 Parameterized variables of blade shape

[0080]

[0081] Step 4: Calculate the control points, node vectors U and V on the blade surface mesh and the blade back surface mesh, generate the blade surface and blade back surface of the propeller, and give the corresponding mathematical expressions:

[0082] (1) The B-spline surface is mainly defined by the control point grids in two directions, two node vectors and univariate B-spline basis functions. The specific equation is:

[0083]

[0084] Where, P i，j is the control point on the surface mesh; N i，p (u) and N j，p (b) are the p-order B-spline basis functions on the knot vectors U(r+1) and V(s+1), respectively; the order p, the number of control points n+1, and the number of knots r+1 satisfy the relationship: r=n+p+1; the order q, the number of control points m+1, and the number of knots s+1 satisfy the relationship: s=m+q+1,

[0085] (2) Using the B-spline curve splitting method, the leaf section curve is divided into two independent curves without changing the curvature of the splitting point curve. The specific splitting point is taken at the middle component of the B-spline curve node vector U(m+1), which is obtained by dividing m+1 by 2 and taking the integer. After splitting, the leaf surface curves with different radii r are obtained. i Control points of the leaf section curve and node vector Get different radius r on the back surface of the leaf j Control points of the leaf section curve and node vector Leaf surfaces use all split curve node vectors The average value of is used as the node vector V, and the back surface of the leaf uses all the split curve node vectors The average value of is taken as the node vector U, and the node vectors of the corresponding fitting curves are taken as the node vector V for both the leaf surface and the leaf back surface.

[0086] (3) The mathematical expression of the blade surface can be described as:

[0087]

[0088] The mathematical expression of the leaf back surface can be described as:

[0089]

[0090] The generated blade surface is as follows Figure 3 As shown;

[0091] Step 5. Calculate the control points, node vectors U and V on the blade root and blade tip surface meshes, generate the blade root and blade tip surfaces of the propeller, and give the corresponding mathematical expressions:

[0092] (1) Select the control point of the leaf section curve at the radius r0 (i.e. the leaf root) on the leaf surface and the control point of the leaf section curve at radius r0 on the leaf back surface As the upper and lower rows of control points F of the blade root surface i0 and F i1 , using the node vector U of the blade surface or the blade back surface as the node vector U of the blade root surface, and selecting {0, 0, 1, 1} as the node vector V of the blade root surface, the mathematical expression of the blade root surface can be described as:

[0093]

[0094] The generated blade root surface is as follows Figure 4 As shown,

[0095] (2) The blade top surface needs to take into account the hydrodynamic characteristics, and 4 rows of control points are used to generate the required arc surface. Directly select the radius r on the blade surface m The control point of the leaf section curve at (i.e., at the leaf top) As the first row of control points G i1 , select the radius r on the back surface of the leaf m The control point of the leaf section curve at As the 4th row control point G i4 , then, calculate the second and third rows of control points using the following expressions:

[0096]

[0097] The node vector U of the blade surface or the blade back surface is used as the node vector U of the blade top surface, and {0, 0, 0, 0, 1, 1, 1, 1} is selected as the node vector V of the blade top surface. The mathematical expression of the blade top surface can be described as:

[0098]

[0099] The generated blade top surface is as follows Figure 5 As shown;

[0100] Step 6. Combining the proposed blade parameterization method and blade surface generation method, the entire process is programmed using the Python language to achieve parameterization and automatic surface generation of the propeller blade: Use the glue language Python to program the entire process. Figure 6 This paper presents a UML class diagram for the propeller blade parameterization and surface modeling program, detailing the internal structure and interrelationships of the classes. The Curve module includes basic methods for B-spline curves, the Surface module includes basic methods for B-spline surfaces, the BladeSection module implements parameterization of propeller blade sections, the Parameterizaton module implements parameterization of blade profiles, and the PropellerShape module generates propeller blade face, back, root, and tip surfaces.

[0101] The basic principles, main features, and advantages of the present invention are shown and described above. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The above embodiments and descriptions are merely illustrative of the principles of the present invention. Various changes and modifications may be made to the present invention without departing from the spirit and scope of the present invention. Such changes and modifications are intended to fall within the scope of the present invention. The scope of protection claimed in the present invention is defined by the appended claims and their equivalents.

Claims

1. A propeller blade parameterization and surface generation method based on B-spline, characterized in that: The specific steps include: Step 1: Find a B-spline curve to fit the type value points, fit the remaining data points except the first and last endpoints in the form of least squares, and solve the control point variables to obtain the B-spline blade section curve fitted at any radius; Step 2: Transform the corresponding control point coordinates by defining the width scale factor, thickness scale factor, and offset factor. Calculate the new control point coordinates using the coordinate transformation formula. Use the defined local and global parametric variables to manipulate the control points of each blade section to achieve parametric deformation of the propeller blade shape. Step 3: Calculate the control points, node vectors U and V on the blade surface and blade back surface meshes, generate the blade surface and blade back surface of the propeller, and give the corresponding mathematical expressions; Step 4: Calculate the control points, node vectors U and V on the blade root and blade tip surface meshes, generate the blade root and blade tip surfaces of the propeller, and give the corresponding mathematical expressions; Step 5: Combining the proposed blade parameterization method and blade surface generation method, the entire process is programmed using Python to achieve parameterization and automatic surface generation of propeller blades. The B-spline curve in step 1 is expressed as the following mathematical expression: Where, P i is the control point of the curve, N i，p (u) is a p-order B-spline basis function defined on the curve knot vector U(m+1), a=0, b=1, and the order p, the number of control points n+1, and the number of knots m+1 satisfy the relationship: m=n+p+1; The specific process of step 1 is as follows: (1) Let {Q k (x k ,y k , z k )|0≤k≤l} is a radius r of the initial propeller blade extension profile j The l+1 two-dimensional leaf section value points at , and are arranged in sequence on the leaf section curve with the leaf section edge as the starting and ending points; (2) Find the B-spline curve's corresponding value point {Q k (x k ,y k , z k )|0≤k≤l} for fitting, and satisfy the conditions: Q0=C(0), Q l = C(1), for the remaining data points Q except the first and last endpoints k , approximate in the form of least squares, and solve for n+1 control point variables P i To minimize the error, the specific expression is: w.r.t.P i Where, is each type value point Q k The corresponding parameter values are determined using the centripetal parameterization method The expression is specifically described as: (3) Select the node vector U(m+1) as follows: Solving the least squares problem is transformed into solving the following linear equations with (n-1) unknowns: (N T N)P=R Where N is a (m-1)×(n-1) matrix consisting of scalars, and R is a column vector consisting of n-1 points. Solving this linear equation system gives the radius r. i B-spline blade section curve fitted at .

2. The method for parameterizing and generating a curved surface of a propeller blade based on B-splines according to claim 1, characterized in that: In step 2, the x direction is set as the width direction of the leaf section, the y direction is set as the thickness direction of the leaf section, and the z direction is set as the radius direction of the leaf section. Represents the radius r j The coordinate value of the i-th control point on the leaf section defines the width scaling factor s j Change radius r j The leaf section width at , defines the thickness scaling factor t j Change radius r j The thickness of the leaf section at the position of the offset factor δ is defined. j =[χ δj ,y δj ,0] T Change radius r j The coordinates of the control points on the leaf section are scaled and offset to obtain the radius r j The coordinate value of the i-th control point on the leaf section It can be expressed by the following formula: Assuming control point coordinates The three components in the Cartesian coordinate system are expressed as Envelope it to a radius of r j On the cylindrical surface, perform the corresponding coordinate transformation and calculate the new control point coordinates Define the radius r j The pitch angle is The vertical oblique angle is θ, and the specific coordinate conversion formula can be described as follows: By using defined local and global parametric variables to manipulate the control points of each blade section, parametric deformation of the propeller blade shape can be achieved.

3. The method for parameterizing and generating a curved surface of a propeller blade based on B-splines according to claim 2, characterized in that: In step 3, the B-spline surface is defined by a control point grid in two directions, two node vectors, and a univariate B-spline basis function. The specific equation is: Where, P i，j is the control point on the surface mesh; N i，p (u) and N j，p (v) are the p-order B-spline basis functions on the node vectors U(r+1) and V(s+1) respectively; the order p, the number of control points n+1 and the number of nodes r+1 satisfy the relationship: r=n+p+1; the order q, the number of control points m+1 and the number of nodes s+1 satisfy the relationship: s=m+q+1.

4. The method for parameterizing and generating a curved surface of a propeller blade based on B-splines according to claim 3, characterized in that: In step 3, the B-spline curve splitting method is used to split the leaf section curve into two independent curves without changing the curvature of the splitting point curve. The specific splitting point is taken at the middle component of the B-spline curve node vector U(m+1), which is obtained by dividing m+1 by 2 and taking the integer. After splitting, the leaf surface curves with different radii r are obtained. i Control points of the leaf section curve and node vector Get different radius r on the back surface of the leaf j Control points of the leaf section curve and node vector Leaf surfaces use all split curve node vectors The average value of the node vector V is used as the node vector; the back surface of the leaf uses all the split curve node vectors The average value of is taken as the node vector U, and the node vectors of the corresponding fitting curves are taken as the node vector V for both the leaf surface and the leaf back surface.

5. The method for parameterizing and generating a curved surface of a propeller blade based on B-splines according to claim 4, characterized in that: In the step 3, The mathematical expression of the leaf surface can be described as: The mathematical expression of the leaf back surface can be described as:

6. The method for parameterizing and generating a curved surface of a propeller blade based on B-splines according to claim 5, characterized in that: Step 4 includes the following process: Process 1: Select the control point of the leaf section curve at radius r0 on the leaf surface and the control point of the leaf section curve at radius r0 on the leaf back surface As the upper and lower rows of control points F of the blade root surface i0 and F i1 , using the node vector U of the blade surface or the blade back surface as the node vector U of the blade root surface, and selecting {0, 0, 1, 1} as the node vector V of the blade root surface, the mathematical expression of the blade root surface can be described as: Process 2: Use 4 rows of control points to generate the required arc surface: directly select the radius r on the blade surface m The control point of the leaf section curve at As the first row of control points G i1 , select the radius r on the back surface of the leaf m The control point of the leaf section curve at As the 4th row control point G i4 , then, calculate the second and third rows of control points using the following expressions: The node vector U of the blade surface or the blade back surface is used as the node vector U of the blade top surface, and {0, 0, 0, 0, 1, 1, 1, 1} is selected as the node vector V of the blade top surface. The mathematical expression of the blade top surface is described as:

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