A method for constructing the leading and trailing edges of blades with arbitrary center angles

Abstract

The invention discloses a method for constructing the leading and trailing edges of blades with arbitrary center angles. The method controls the shape of a complementary circle by adjusting the first point or the last point of a blade back or a blade basin in a design space, obtains blade back and blade basin curves by fitting blade section type value points with a spline curve, and calculates two unit tangent vectors at the leading edge or trailing edge of the blade section and the center angle formed by them. In combination with basic knowledge of circles and conic sections, control points and weight factors are obtained according to the type of complementary circle and the size of the center angle, and any circular arc and conic section that meet the convex hull requirements are accurately constructed, so that the G1 continuity of the constructed complementary circle curve and the blade back and blade basin curves at the connection points can be ensured, and the type, slope, taper and size of the complementary circle can be controlled by a small number of parameters. By introducing a relational expression and equation for the associated conic section shape control parameters and the number of blade sections, the dynamic change of the complementary circle curve along the span direction of the blade is realized, and the complex design of the leading and trailing edges of three-dimensional blades is completed.

Description

Technical Field

[0001] The present application relates to the field of parametric design of aircraft engine blades, and in particular to a method for constructing leading and trailing edges of blade profiles with arbitrary center angles. Background Art

[0002] The compressor and turbine are one of the key components of aircraft engines, and the blades are their basic components. In order to improve the efficiency of the compressor and turbine, the aerodynamic performance of the blade is worthy of attention. Compared with other parts, the shape of the leading edge and trailing edge of the blade directly affects the development of the boundary layer along the blade surface and the downstream flow field structure, and plays an important role in the overall performance of the blade. In general, blades with elliptical leading and trailing edges have better aerodynamic performance, because the use of an elliptical leading edge can suppress the excessive expansion of the airflow on the leading edge surface, thereby reducing the adverse pressure gradient on the suction surface and avoiding laminar separation on the leading edge surface.

[0003] At present, the leading edge and trailing edge of the blade are basically arcs and are generally described by the ratio of the semi-major axis to the semi-minor axis in the form of a standard ellipse equation. In engineering design applications, after the user gives the ellipticity, the position of the endpoints of the leading edge or trailing edge of the blade basin and the blade back is iteratively adjusted to align the leading edge first point or trailing edge tail point, and then the position of the center of the circle is calculated by the affine transformation principle between the circle and the ellipse, and the coordinates of several points on the arc are calculated by the vector transformation and distance transformation from the center of the circle to the point on the arc, and then these points are added to the shape value point set of the blade basin and blade back curves, and finally the blade section curve containing the leading edge and the trailing edge is uniformly generated using a spline curve. Although this approach based on affine transformation can ensure the G1 continuity of the connection between the elliptical arc segment and the blade basin and blade back, the limited interpolation points not only have limited accuracy and variation of the arc, but also cannot generate a complementary arc for blades with a center angle greater than or equal to 180° at the leading edge or trailing edge (especially controlled diffusion blades, supercritical blades / airfoils, etc.), so it cannot meet high-difficulty design requirements.

[0004] Since the flow characteristics of the upper and lower blade surfaces are quite different, the asymmetric leading and trailing edge design plays a significant role in improving blade stall and expanding the low-loss operating range of the blade. In a compressor or turbine, the flow on the suction side and pressure side of the blade is quite different. The leading edge is the starting position of the flow. The asymmetric design can alleviate the leading edge flow acceleration effect, improve the control of the suction peak of the blade, and improve the flow characteristics on the suction side and the pressure side. The trailing edge is the end position of the flow. The asymmetric design can better control the blade wake, thereby improving the interference between the rotor and the stator and improving the efficiency of multi-stage compressors and turbines.

[0005] In addition, in rotating impeller machinery, due to the difference in axial radius between the blade root and the blade tip and the effect of centrifugal force, the spanwise flow of the blade will have a significant pressure gradient. Specifically, as the air flows along the spanwise direction of the blade, the boundary layer gradually thickens, especially in the middle to the trailing edge area, where the thickened boundary layer is more likely to separate due to instability. The flow at the connection between the blade root and the impeller is also prone to form a low-energy boundary layer, resulting in flow separation and coupling with the flow in the middle of the blade to produce secondary flow. In addition, the large pressure gradient at the confluence of the pressure surface and the suction surface will also cause separation at the trailing edge, thereby affecting the downstream flow. However, current blade design research mainly focuses on the optimization of the pressure surface and the suction surface, while the leading trailing edge adopts a consistent design method, that is, the different sections of the entire blade use a unified ellipticity, resulting in a decrease in the design freedom of the leading trailing edge of the three-dimensional blade. This design idea only realizes the control of the flow by the leading trailing edge at the two-dimensional level. Since the characteristics of the three-dimensional flow are not taken into account, the flow control ability of the three-dimensional leading trailing edge on the blade surface is reduced, resulting in a decrease in the optimization effect of the suction surface and the pressure surface. Summary of the invention

[0006] In view of the difficulties in the current parametric design of the leading and trailing edges of engine blades, the present invention proposes a method for constructing the leading and trailing edges of blades with arbitrary center angles. The method controls the shape of the supplementary circle by adjusting the first or last point of the back or basin of the blade in the design space. By using the two unit tangent vectors at the leading or trailing edge of the blade section curve and the center angle formed by them, combined with the basic knowledge of circles and conic sections, the control points and weights are solved based on the type of supplementary circle and the size of the center angle, so as to accurately construct the arc and conic section that meet the convex hull requirements. At the same time, on the premise of ensuring the G1 continuity of the supplementary circle curve and the back and basin curves at the connection point, the type, slope, taper and size of the supplementary circle are controlled by a small number of parameters. By introducing the relationship and equations that associate the supplementary circle design parameters with the number of blade sections, the dynamic change of the supplementary circle design parameters along the span direction of the blade is realized, thereby completing the complex design of the leading and trailing edges of three-dimensional blades.

[0007] The technical solution adopted by the present invention is: S1: using the first two points or the last two points of the blade basin and blade back profile point set, first use the two-point formula to construct two straight line analytical expressions at the leading edge or the trailing edge on the two-dimensional plane of the blade profile section, and calculate the slope of the angle bisector formed by the two straight line analytical expressions, and then use the angle bisector slope to calculate the slope of the perpendicular line perpendicular to the angle bisector;

[0008] S2: using the z value of the first or last point of the leaf basin and leaf back shape value points in the Cartesian coordinate system and the slope of the angle bisector, selecting the first or last point through which the perpendicular line of the angle bisector passes, and combining the slope of the perpendicular line of the angle bisector, using the first or last point through which the perpendicular line of the angle bisector passes to obtain the perpendicular line of the angle bisector;

[0009] S3: using the perpendicular line of the angle bisector to find the intersection point with the straight line where the other corresponding first point or tail point to be aligned is located, and inserting the intersection point to the front of the first point or the tail of the tail point to complete the alignment of the leading edge point or the trailing edge point, and obtaining the aligned leaf basin and leaf back shape value point set;

[0010] S4: using two sliding parameters for controlling the slope and size of the conic section, the first point or the last point of the aligned blade basin and blade back profile point set can be moved between the design space using the intersection point P0 of the straight line at the leading edge or the trailing edge and the next profile point after the first point or the previous profile point before the tail point, to obtain the final blade basin and blade back profile point set;

[0011] S5: fitting the final blade basin and blade back shape value point set to obtain a blade basin and blade back curve, and obtaining two unit tangent vectors at the leading end point or the trailing end point of the blade basin and blade back curve based on the blade basin and blade back curve;

[0012] S6: Calculate the central angle formed by the blade basin curve and the blade back curve at the leading edge or the trailing edge using the two unit tangent vectors;

[0013] S7: Select the type of conic section at the leading edge or trailing edge, set the node vector of the quadratic NURBS curve according to the type of conic section, calculate the coordinates of the control points of the conic curve in the Cartesian coordinate system and the corresponding weight factors according to different situations of the center angle, and use the quadratic NURBS curve to fit the control point set to obtain the leading edge or trailing edge complementary circle curve of the blade section;

[0014] S8: According to the relationship between the conic section type control parameter, the taper control parameter and the sliding parameter along the blade span direction, define the equation group of the leading edge or trailing edge fillet type, size, slope and taper along the blade span direction, and repeat S1 to S7 until the construction of the leading edge fillet curve and the trailing edge fillet curve of all blade sections is completed;

[0015] S9: For each blade section, the leading edge fillet curve, the trailing edge fillet curve, the blade basin curve, and the blade back curve are connected at adjacent endpoints to form a closed blade section curve, and a lofting operation is performed between all blade sections to generate a final three-dimensional blade entity.

[0016] Compared with the prior art, the present invention has the following beneficial effects:

[0017] The method of the present application can effectively solve the problems existing in the design of the leading and trailing edges of three-dimensional blades, increase the degree of freedom of the leading and trailing edge shape control, realize the adaptive construction of different types of conic sections with arbitrary central angles of the leading and trailing edges of complex blade sections, and the cross-section design of customized leading and trailing edge supplementary circle curves, so that the leading and trailing edge design of three-dimensional blades is more in line with engineering requirements and has high practical value and application prospects.

[0018] It should be understood that the foregoing general description and the following detailed description are exemplary and explanatory only and are not restrictive of the present application. BRIEF DESCRIPTION OF THE DRAWINGS

[0019] The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate embodiments consistent with the present application and, together with the description, serve to explain the principles of the present application.

[0020] Figure 1 The present invention is a flow chart showing a method for constructing a leading and trailing edge of a blade profile with an arbitrary center angle according to an exemplary embodiment.

[0021] Figure 2 The diagram is a schematic diagram showing a solution to the slope of a perpendicular line of an angle bisector at a leading edge or a trailing edge of a blade section according to an exemplary embodiment.

[0022] Figure 3 The diagram is a schematic diagram showing selection of points through which a perpendicular line of an angle bisector passes at a leading edge or a trailing edge of a blade section in different situations according to an exemplary embodiment.

[0023] Figure 4 The present invention is a schematic diagram showing the solution of the intersection point of a perpendicular line of an angle bisector at a leading edge or a trailing edge and another corresponding straight line, and the insertion of a leading edge first point or a trailing edge last point according to an exemplary embodiment.

[0024] Figure 5 It is a schematic diagram showing the control effect of the sliding parameters Shift_factor1 and Shift_factor2 on the head point or tail point of the blade back or blade basin according to an exemplary embodiment.

[0025] Figure 6 It is a schematic diagram of two unit tangent vectors at the leading end point or the trailing end point of the B-spline curve of the cross-section blade base and blade back according to an exemplary embodiment.

[0026] Figure 7 It is a schematic diagram for judging the range of angles between two unit tangent vectors at the leading edge endpoints or the trailing edge endpoints of the blade back and blade basin spline curves and the actual center of the circle according to an exemplary embodiment.

[0027] Figure 8 It is a schematic diagram of the construction of conic sections of different types, symmetry or asymmetry, and different center angles according to an exemplary embodiment.

[0028] Fig. 9 It is a schematic diagram showing the relationship between the weight factor of the control point P0 obtained by different taper control parameters and the type of conic section generated according to an exemplary embodiment.

[0029] Fig.10 It is a schematic diagram showing a process of generating a leading edge or trailing edge fillet of a single blade section in the method according to an exemplary embodiment.

[0030] Fig.11 The diagram is a schematic diagram showing a process of generating a consistent leading edge or trailing edge fillet for an initial blade profile cross section set lacking a leading edge and a trailing edge according to an exemplary embodiment.

[0031] Fig.12 The diagram is a schematic diagram showing a process of generating a non-uniform leading edge or trailing edge fillet for an initial blade profile cross section set lacking a leading edge and a trailing edge according to an exemplary embodiment.

[0032] Fig.13 It is a schematic diagram of a process of generating a three-dimensional blade through a lofting operation after generating a consistent leading edge or trailing edge filler circle for an initial blade section set lacking a leading edge and a trailing edge according to an exemplary embodiment.

[0033] Fig.14 It is a schematic diagram of a process of generating a three-dimensional blade through a lofting operation after generating a non-uniform leading edge or trailing edge filler circle for an initial blade section set lacking a leading edge and a trailing edge according to an exemplary embodiment. DETAILED DESCRIPTION

[0034] Exemplary embodiments will be described in detail herein, examples of which are shown in the accompanying drawings. When the following description refers to the drawings, the same numbers in different drawings represent the same or similar elements unless otherwise indicated. The implementations described in the following exemplary embodiments do not represent all implementations consistent with the present application. Instead, they are merely examples of devices and methods consistent with some aspects of the present application as detailed in the appended claims.

[0035] The terms used in this application are for the purpose of describing specific embodiments only and are not intended to limit this application. The singular forms of "a", "said" and "the" used in this application and the appended claims are also intended to include plural forms unless the context clearly indicates other meanings. It should also be understood that the term "and / or" used herein refers to and includes any or all possible combinations of one or more associated listed items.

[0036] The present application provides a method for constructing the leading and trailing edges of blades with arbitrary center angles. The applicable blade model data structure stores the blade basin and blade back profile value points based on the blade profile sections stacked in the span direction. In addition to aircraft engine blades, this method is also applicable to the design of the leading and trailing edges of wings designed using the wing element theory. Here, the present application is described in detail using the aircraft engine compressor blade model as an example. Figure 1 A method for constructing the leading and trailing edges of blades with arbitrary center angles in the present application comprises the following sub-steps:

[0037] S1: Using the first two points or the last two points of the blade basin and blade back profile point set, first use the two-point formula to construct two straight line analytical expressions at the leading edge or the trailing edge on the two-dimensional plane of the blade profile section, and calculate the slope of the angle bisector formed by the two straight line analytical expressions, and then use the angle bisector slope to calculate the slope of the perpendicular line perpendicular to the angle bisector;

[0038] Specifically, Figure 2 The mid-blade basin and blade back shape value point set and the partial enlargement of the leading edge and trailing edge are shown. For the points at the leading edge or trailing edge: P1 (x1, y1, z1), P2 (x2, y2, z2) and P3 (x3, y3, z3), P4 (x4, y4, z4), use the two-point formula to construct two straight line analytical expressions l1 and l2 on the two-dimensional plane as follows:

[0039]

[0040] Using the analytical expressions of l1 and l2, we can find the slope k of the angle bisector of ∠P1P0P3 as follows:

[0041]

[0042] The point P0 is the intersection of straight lines l1 and l2.

[0043] Then the slope of the perpendicular line perpendicular to the angle bisector is k p The calculation method is as follows:

[0044]

[0045] This step is designed to adapt to various complex blade shapes. By analyzing the geometry at the leading edge or trailing edge, the two-point formula is used to directly and quickly calculate the slope of the angle bisector k and the slope of the perpendicular line k to the angle bisector. p The angle bisector k will be used in the next step to determine the point through which the perpendicular line of the angle bisector needs to pass, and the coordinates of the point combined with the slope of the perpendicular line of the angle bisector are used to find the analytical expression of the angle bisector. This step has strong adaptability in engineering. For any blade type, the slope of the angle bisector k and the slope of the perpendicular line of the angle bisector k can be quickly found. p .

[0046] S2: using the z value of the first or last point of the leaf basin and leaf back shape value points in the Cartesian coordinate system and the slope of the angle bisector, select the first or last point through which the perpendicular line of the angle bisector passes, and combine the slope of the perpendicular line of the angle bisector to obtain the perpendicular line of the angle bisector using the first or last point passed; this step includes the following sub-steps:

[0047] S21: For the leading edge, if the z values ​​of the first points P1 and P3 in the leaf basin and leaf back profile point sets are different, the first point with the smaller z value is selected as the point through which the perpendicular line of the angle bisector passes; if the z values ​​of the first points P1 and P3 are the same, if the slope of the angle bisector is greater than 0, the point with the smaller y value is selected as the point through which the perpendicular line of the angle bisector passes, otherwise the point with the larger y value is selected as the point through which the perpendicular line of the angle bisector passes;

[0048] Specifically, Figure 3 As shown in the local enlarged view of the middle front edge, in the first case ①, the z values ​​of P1 and P3 are different, and P3 with a smaller z coordinate value is directly selected as the point through which the perpendicular line of the angle bisector passes; in the second case ②, the z coordinate values ​​of P1 and P3 are the same, and it is impossible to directly judge by the z value, but the slope at this time is less than 0 and the y value of P3 is greater than the y value of P1, so it is judged that the point through which the perpendicular line of the angle bisector passes is P3; in the third case ③, the z coordinate values ​​of P1 and P3 are the same, and it is impossible to directly judge by the z value, but the slope at this time is greater than 0 and the y value of P1 is less than the y value of P3, so it is judged that the point through which the perpendicular line of the angle bisector passes is P1.

[0049] This step is designed to automatically and correctly select the point through which the perpendicular line of the angle bisector at the leading edge passes in various situations. This step takes into account the diversity of different situations, and gives clear judgment rules according to the different z values, y values ​​and slopes, taking into account the possibility of different geometric positions, ensuring that the correct choice can be made under different conditions, and avoiding the misjudgment that may be caused by a single standard. This step simplifies the computational complexity by using coordinate comparison. From the perspective of geometry itself, by directly comparing the z values, the point with smaller z is selected, which is simple and direct. When the z values ​​are the same and cannot be directly judged, further judgment is made by combining the y value and the slope, which reduces complex geometric calculations, simplifies the judgment logic, and makes the algorithm easy to implement; this step reduces the dependence on complex calculations (such as distance calculations), and chooses to determine the point through which the perpendicular line passes by directly comparing the coordinate values ​​instead of using complex geometric calculations (such as distance, angle, etc.), which can improve the computational efficiency and reduce the computational cost. This is especially important for real-time application scenarios.

[0050] S22: For the trailing edge, if the z values ​​of the tail points P1 and P3 of the leaf basin and leaf back type value points are different, the tail point with the larger z value is selected as the point through which the perpendicular line of the angle bisector passes; if the z values ​​of the tail points P1 and P3 are the same, if the slope of the angle bisector is greater than 0, the point with the larger y value is selected as the point through which the perpendicular line of the angle bisector passes, otherwise the point with the smaller y value is selected as the point through which the perpendicular line of the angle bisector passes;

[0051] Specifically, Figure 3As shown in the local enlarged view of the middle front edge, in the first case ①, the z values ​​of P1 and P3 are different, and P3 with a larger z coordinate value is directly selected as the point through which the perpendicular line of the angle bisector passes; in the second case ②, the z coordinate values ​​of P1 and P3 are the same, and it is impossible to directly judge by the z value, but the slope at this time is greater than 0 and the y value of P3 is greater than the y value of P1, so it is judged that the point through which the perpendicular line of the angle bisector passes is P3; in the third case ③, the z coordinate values ​​of P1 and P3 are the same, and it is impossible to directly judge by the z value, but the slope at this time is less than 0 and the y value of P1 is less than the y value of P3, so it is judged that the point through which the perpendicular line of the angle bisector passes is P1.

[0052] This step is designed to automatically and correctly select the point through which the perpendicular line of the angle bisector passes at the trailing edge in various situations. The effect obtained by this step when selecting the point through which the perpendicular line of the angle bisector passes is the same as that of S21. Although the selection of the first point and the tail point adopts the same logic, this step distinguishes the selection logic of the first point and the tail point according to the different coordinate characteristics of the front and tail edges to avoid logical confusion in different parts.

[0053] S23: using the first point or the last point passed through, obtaining a perpendicular line l3 of the angle bisector at the leading edge or the trailing edge;

[0054] l3: z = z t -k p ·(yy t )

[0055] where k p is the slope of the perpendicular line to the angle bisector, y t With z t The y-coordinate and z-coordinate of the first or last point through which the perpendicular line of the angle bisector passes.

[0056] Specifically, Figure 3 As shown in the various situations in , combined with the slope of the perpendicular line of the angle bisector, substituting the y coordinate and z coordinate of the first or last point passed through, the analytical expression of the perpendicular line of the angle bisector at the leading edge or trailing edge can be obtained.

[0057] This step is designed to provide clear judgment and solution criteria, so as to realize the automatic solution of the perpendicular line of the angle bisector at the leading edge or trailing edge. This step establishes the analytical expression of the perpendicular line through clear steps, standardizes the calculation process, and reduces the possibility of ambiguous judgment or human error; this step is particularly conducive to programming implementation because it can avoid complex situations caused by special geometric situations (such as parallelism, coincidence, etc.); this step directly uses the slope of the perpendicular line and the coordinates of the passing point to solve the equation, avoiding complex geometric derivation, making the solution process intuitive and easy to understand; this step converts the geometric problem into algebraic calculation, simplifies the overall calculation process, and directly uses the coordinates and slope of the point to solve the equation, which helps to quickly iterate in numerical calculations. Because it relies on basic algebraic operations, it is easy to implement in different computing platforms and programming languages, which enhances the universality of the method.

[0058] S3: using the perpendicular line of the angle bisector to find the intersection point with the straight line where the other corresponding first point or tail point to be aligned is located, and inserting the intersection point to the front of the first point or the tail of the tail point to complete the alignment of the leading edge point or the trailing edge point, and obtaining the aligned leaf basin and leaf back shape value point set; this step includes the following sub-steps:

[0059] S31: When P1 is the first point in the leaf back value point set, P3 is the first point in the leaf basin value point set, and the point through which the perpendicular line of the angle bisector passes is P3, the intersection point of the perpendicular line l3 of the angle bisector and the straight line l1 where point P1 is located is obtained as P 1_new , P 1_new Insert before the first point of the leaf back type value point set; conversely, when the point through which the perpendicular line of the angle bisector passes is P1, find the intersection point of the perpendicular line l3 of the angle bisector and the straight line l2 where point P3 is located as P 3_new , P 3_new Insert before the first point in the leaf basin type value point set; update the index to obtain the leaf basin and leaf back type value point set after the leading edge is aligned. At this time, ∠P2P1P3=∠P4P3P1 at the leading edge;

[0060] Specifically, Figure 4 As shown in the enlarged area of ​​the middle front edge, the point through which the perpendicular line l3 of the angle bisector passes is P3. Find the intersection point P of l3 and l1. 1_new , P 1_new Before inserting the first point in the leaf back type value point set, update the index of the leaf back type value point set, thereby completing the alignment of the points at the leading edge. Through geometric analysis, it can be seen that at this time ∠P2P1P3=∠P4P3P1.

[0061] This step is designed to accurately align the leading edge points, reduce error accumulation, simplify the processing of complex geometric problems, and enhance geometric consistency. This step can accurately align the leading edge points by solving the intersection of the vertical lines l3 and l1 and inserting them into the blade back value point set, thereby maintaining the geometric continuity and consistency of the point set, which is very important for ensuring the smoothness of the geometric shape and the accuracy of the design requirements; this step avoids directly adjusting existing points, and by inserting new points instead of directly moving or adjusting existing points, it reduces interference with the original points, avoids error accumulation and geometric deformation. Inserting new intersections can more accurately align the point set while ensuring the rigor of the geometric relationship; this step avoids more complex geometric derivations and simplifies the processing of complex geometric problems through simple geometric operations (finding intersections and inserting them), which not only simplifies the calculation process, but also improves processing efficiency and accuracy. It is particularly suitable for engineering design and optimization scenarios that require rapid iteration.

[0062] S32: When P1 is the end point of the leaf back value point concentration, P3 is the end point of the leaf basin value point concentration, and the point through which the perpendicular line of the angle bisector passes is P3, the intersection point of the perpendicular line l3 of the angle bisector and the straight line l1 where point P1 is located is obtained as P 1_new , P 1_new Insert the tail point of the leaf back value point concentration; conversely, when the point through which the perpendicular line of the angle bisector passes is P1, find the intersection point of the perpendicular line l3 of the angle bisector and the straight line l2 where point P3 is located as P 3_new , P 3_new After inserting the tail point in the blade basin type value point set; updating the index to obtain the blade basin and blade back type value point set after aligning the leading edge, at this time, ∠P2P1P3=∠P4P3P1 at the trailing edge;

[0063] Specifically, Figure 4 As shown in the enlarged area of ​​the middle tail edge, the point through which the perpendicular line l3 of the angle bisector passes is P3. Find the intersection point P of l3 and l1. 1_new , P 1_new After inserting the tail point in the leaf back type value point set, the index of the leaf back type value point set is updated, thereby completing the alignment of the points at the trailing edge. Through geometric analysis, it can be seen that at this time ∠P2P1P3=∠P4P3P1.

[0064] This step is designed to accurately align the trailing edge points, reduce error accumulation, simplify the processing of complex geometric problems, and enhance geometric consistency. This step achieves the same effect as S31 when finding the intersection point and inserting the type value point set. Although the solution of the new tail point and the new first point adopts the same logic, due to the different positions of the first point and the tail point in the type value point set, it is necessary to distinguish the insertion logic of the new first point and the new tail point to avoid logical confusion in different parts, resulting in incorrect insertion point positions and causing curve fitting failure.

[0065] Among them, when P1 is the first point in the leaf back type value point concentration and P3 is the first point in the leaf basin type value point concentration, point P2 is the point after the first point in the leaf back type value point concentration, and point P4 is the point after the first point in the leaf basin type value point concentration; when P1 is the tail point in the leaf back type value point concentration and P3 is the tail point in the leaf basin type value point concentration, point P2 is the point before the tail point in the leaf back type value point concentration, and point P4 is the point before the tail point in the leaf basin type value point concentration.

[0066] S4: using two sliding parameters for controlling the slope and size of the conic section, the first point or the last point of the aligned blade basin and blade back profile point set can be moved between the design space using the intersection point P0 of the straight line at the leading edge or trailing edge and the next profile point after the first point or the previous profile point before the tail point, to obtain the final blade basin and blade back profile point set; this step includes the following sub-steps:

[0067] S41: For the leading edge or the trailing edge, when the sliding parameter Shift_factor1 of the blade back is 0 and the sliding parameter Shift_factor2 of the blade base is 0, the positions of the point P1 and the point P3 remain unchanged;

[0068] Specifically, Figure 5 As shown in the partial enlargement of the leading edge or trailing edge, when the sliding parameter Shift_factor1 of the blade back is 0 and the sliding parameter Shift_factor2 of the blade basin is 0, the positions of point P1 and point P3 can be kept unchanged, and point P1 and point P3 are still aligned.

[0069] This step is designed to align the first or tail point while keeping the coordinates of the first and tail points of the initial blade back and blade basin value point set unchanged to the greatest extent. This step is usually applicable to the design of symmetrical front and rear edge fill circles. If the design requires the generation of symmetrical front and rear edge fill circles, and the initial front and rear edge value points are not aligned, the blade back sliding parameter Shift_factor1 is set to 0 and the blade basin sliding parameter Shift_factor2 is set to 0, indicating that the coordinates of the first and tail points in the initial blade back and blade basin value point set are aligned and corrected, and a symmetrical front and rear edge fill circle is generated on this basis.

[0070] S42: For the leading edge or trailing edge, when Shift_factor1>0, Shift_factor2>0, P1 moves toward P0 on l1, and P3 moves toward P0 on l2;

[0071] Specifically, Figure 5As shown in the local enlargement of the leading edge or trailing edge, when the slip parameter Shift_factor1 of the blade back is greater than 0 and the slip parameter Shift_factor2 of the blade basin is greater than 0, point P1 and point P3 are aligned, and P1 moves toward P0 along the straight line l1 where it is located, and the actual moving distance is the product of the distance between P1 and P0 and the slip parameter Shifttfactor1. P3 moves toward P0 along the straight line l2 where it is located, and the actual moving distance is the product of the distance between P3 and P0 and the slip parameter Shift_factor2.

[0072] This step is designed to achieve the movement of P1 and P3 to P0 along their respective straight lines within the design space of the leading edge or trailing edge. When P1 and P3 move to P0 at the same time with the same trend, the generated leading and trailing edge filler circle will become smaller. This step accurately controls the adjustment of the leading edge or trailing edge shape by moving P1 and P3 to P0 along a straight line. In the process of adjusting P1 and P3, the geometric continuity and gradual characteristics are ensured by moving along their respective straight lines, avoiding sudden changes or discontinuities caused by point position changes, and has significant advantages in maintaining the smoothness and consistency of the overall design; this step realizes progressive geometric optimization, facilitates design adjustment, and contributes to the flexibility of design optimization. It can gradually approach the ideal design state through small step adjustments; this step can generate filler circles of different sizes to meet various design requirements by adjusting the trend and amplitude of the movement, such as enhancing or weakening the aerodynamic characteristics of a specific area; this step can simplify the control logic of the design parameters by controlling the movement of the point to move along a straight line toward P0, making this geometric adjustment easy to implement parameterized control. In parametric design, this linear movement method can be easily controlled precisely through design parameters (such as movement distance and direction), enabling rapid design iterations.

[0073] S43: For the leading edge or trailing edge, when Shift_factor1<0, Shift_factor2<0, P1 moves toward P2 on l1, and P3 moves toward P4 on l2;

[0074] Specifically, Figure 5 As shown in the local enlargement of the leading edge or trailing edge, when the slip parameter Shift_factor1 of the blade back is less than 0 and the slip parameter Shift_factor2 of the blade basin is less than 0, point P1 and point P3 are aligned, and P1 moves toward P2 along the straight line l1 where it is located, and the actual moving distance is the product of the distance between P1 and P2 and the absolute value of the slip parameter Shift_factor1. P3 moves toward P4 along the straight line l2 where it is located, and the actual moving distance is the product of the distance between P3 and P4 and the absolute value of the slip parameter Shift_factor2.

[0075] This step is designed to achieve the movement of P1 and P3 toward P2 or P4 along their respective straight lines within the design space of the leading edge or trailing edge. When P1 and P3 move to P2 and P4 at the same time with the same trend, the generated leading and trailing edge filler circle will become larger. This step precisely controls the adjustment of the leading or trailing edge shape by moving P1 and P3 toward P2 and P4 along a straight line. In the process of adjusting P1 and P3, the degree of freedom is limited by moving along their respective straight lines, ensuring the continuity and gradual characteristics of the geometry, avoiding the sudden change or discontinuity caused by the change of the point position, and having significant advantages in maintaining the smoothness and consistency of the overall design; this step realizes progressive geometric optimization, facilitates design adjustment, and contributes to the flexibility of design optimization. It can gradually approach the ideal design state through small step adjustments; this step can generate filler circles of different sizes to meet various design requirements by adjusting the trend and amplitude of the movement, such as enhancing or weakening the aerodynamic characteristics of a specific area; this step can simplify the control logic of the design parameters by controlling the movement of the points to move along a straight line toward P2 and P4, making this geometric adjustment easy to implement parameterized control. In parametric design, this linear movement method can be easily controlled precisely through design parameters (such as movement distance and direction), enabling rapid design iterations.

[0076] S44: when the point through which the perpendicular line of the angle bisector passes is P3, the sliding parameter Shift_factor1 of the leaf back is set to -1, indicating that the alignment point P1 is cancelled, and the shape value point set of the original leaf back is used to generate the supplementary circle; when the point through which the perpendicular line of the angle bisector passes is P1, the sliding parameter Shift_factor2 of the leaf basin is set to -1, indicating that the alignment point P3 is cancelled, and the shape value point set of the original leaf basin is used to generate the supplementary circle;

[0077] Specifically, Figure 5 As shown in the partial enlargement of the leading edge or trailing edge, the point through which the perpendicular line that bisects the angle passes is P3. When the sliding parameter Shift_factor1 of the blade back is -1 and the sliding parameter Shift_factor2 of the blade basin is 0, the aligned point P1 is deleted, the index is updated, and the initial blade back shape value point set is restored.

[0078] This step is designed to retain and use the initial shape value point set of the blade back or blade basin to generate the leading and trailing edge filling circle. If the initial shape value point set of the blade back or blade basin does not need to be aligned, the aligned points will be deleted and restored to the initial shape value point set of the blade back or blade basin. When the initial shape of the blade back or blade basin already meets the design requirements, additional alignment adjustments may cause morphological deviations or unnatural changes. Therefore, restoring the initial point set can maintain the original design intention and avoid design deviations caused by excessive adjustments. Therefore, this step cancels the alignment of the leading and trailing edge points through simple parameter settings. Deleting the aligned points can simplify the geometric point set structure, reduce redundant points, and make the design more flexible and precise.

[0079] S5: fitting the final blade basin and blade back shape value point set to obtain a blade basin and blade back curve, and obtaining two unit tangent vectors at the leading end point or the trailing end point of the blade basin and blade back curve based on the blade basin and blade back curve;

[0080] Specifically, Figure 6 As shown in , the leaf basin and leaf back curves are obtained by fitting the leaf basin and leaf back shape point sets. If a cubic B-spline is used for fitting, the tangent vector of the cubic B-spline at the endpoint can be obtained by fitting each basis function The derivation is:

[0081]

[0082] Among them, S(u) is the value of the spline curve at parameter u, P i is the ith control point, is the cubic basis function of the ith control point, and u is the parameter of the spline, usually in the interval [u min ,u max ] within the changes.

[0083] At each endpoint, the tangent vector can be calculated by substituting the endpoint parameter value into this derivative expression. So the first endpoint tangent vector can be calculated by substituting u min Substituting into the derivative formula, the tail endpoint tangent vector can be obtained by replacing u max Substitute into the derivative formula to obtain .

[0084] This step is designed to accurately obtain the tangent vector at the endpoint of the leading edge or trailing edge, and use the tangent vector in the subsequent process to generate the leading and trailing edge filler circle that is precisely tangent to the blade basin and blade back curve at the endpoint, so as to ensure the first-order (G1) continuity of the filler circle and the blade basin and blade back curve. This step obtains the tangent vector at the endpoint of the leading edge or trailing edge, so as to design a geometrically continuous and smooth filler circle with the blade basin and blade back curve. This method can ensure that the filler circle maintains the first-order derivative continuity with the main part of the blade, thereby maintaining the smoothness of the curve. This is very important for improving the aerodynamic performance of the blade, and can reduce fluid separation and turbulence. If the curve of the leading edge or trailing edge is not smoothly connected at the endpoint, it may cause adverse aerodynamic effects when the fluid passes through the place, such as vortex, separation and other phenomena. By using the tangent vector to generate a precisely tangent filler circle, geometric continuity can be ensured, thereby reducing the performance loss caused by discontinuity. The tangent vector is used to generate the supplementary circle. No matter how the blade back sliding parameter Shift_factor1 and the blade basin sliding parameter Shift_factor2 are taken in the previous step, or how the curve shapes of the blade basin and the blade back change, the first-order continuity of each part of the geometric design (blade basin, blade back, leading edge, and trailing edge) at the connection point can be guaranteed.

[0085] S6: Calculate the central angle formed by the blade basin curve and the blade back curve at the leading edge or the trailing edge using the two unit tangent vectors; this step includes the following sub-steps:

[0086] S61: determining the relative trend of the blade basin and the blade back curve at the head or tail according to two unit tangent vectors at the head end point or the tail end point of the blade basin and the blade back curve;

[0087] Specifically, Figure 7 As shown in the local enlargement of the middle front and trailing edge, the relative trends of the two unit tangent vectors at the leading or trailing endpoints of the blade basin and blade back curves are divided into three situations: pointing close, pointing parallel, and pointing away, which are used to determine the range of the central angle of the generated complementary circle.

[0088] This step is designed to determine the range of the center angle of the fill circle at the leading and trailing edges. In NURBS curve theory, control points and weights can be used to accurately describe the shape of the arc. Different center angles correspond to different arc segments. Therefore, when generating a fill circle, it is necessary to determine the corresponding control point and weight configuration scheme based on the size of the center angle. By accurately determining the center angle, the most suitable fill circle can be generated according to different angles to ensure that the fill circle is naturally connected with other parts of the blade geometrically, avoiding curve mutations or discontinuities. Therefore, determining the size of the center angle of the fill circle is an indispensable step.

[0089] S62: If the two unit tangent vectors of the blade basin curve and the blade back curve at the leading end or the trailing end point close to each other, the central angle is:

[0090]

[0091] in and are two vectors of the blade basin and blade back curve at the leading end point or the trailing end point.

[0092] Specifically, Figure 7 As shown in the first case ① of the partial enlargement of the front and rear edges, the blade basin and the blade back curves at the front or rear edges tend to approach each other, and the straight lines extending along the directions of the two unit tangent vectors at the leading or rear endpoints must intersect. According to the geometric analysis here, the center angle formed by the two normal vectors perpendicular to the tangent vectors at the leading or rear endpoints is less than 180°. According to the inverse cosine theorem and the supplementary angle theorem, the size of the center angle of the supplementary circle can be calculated based on the two unit tangent vectors.

[0093] This step is designed to accurately determine the size of the center angle of the complementary circle when the two unit tangent vectors point close to each other. This step determines the size of the center angle of the complementary circle when the two unit tangent vectors point close to each other by calculating the angle between the tangent vectors, which can simplify the geometric calculation process. Compared with other more complex geometric construction methods, using tangent vectors to solve the center angle is more direct and also improves the accuracy of the generated curve.

[0094] S63: If the two unit tangent vectors of the blade basin curve and the blade back curve at the leading end point or the trailing end point away from each other, the central angle is θ+π, that is, the central angle is greater than 180°;

[0095] Specifically, Figure 7 As shown in the second case ② of the partial enlargement of the front and rear edges, the blade basin and the blade back curve tend to move away from each other at the leading edge or the trailing edge, and the straight lines extending in opposite directions along the two unit tangent vectors at the leading end point or the trailing end point must intersect. According to the geometric analysis here, the central angle formed by the two normal vectors perpendicular to the tangent vector at the leading end point or the trailing end point is greater than 180°.

[0096] This step is designed to accurately determine the size of the center angle of the complementary circle when the two unit tangent vectors point away from each other. This step determines the size of the center angle of the complementary circle when the two unit tangent vectors point away from each other by calculating the angle between the tangent vectors, and distinguishes it from the situation when the two unit tangent vectors point close to each other, so as to avoid obtaining an incorrect center angle of the complementary circle.

[0097] S64: If the two unit tangent vectors of the blade basin curve and the blade back curve at the leading end or the trailing end point in parallel, that is, Then the central angle of the circle is equal to 180°.

[0098] Specifically, Figure 7 As shown in the third case ③, which is a partial enlargement of the middle front and rear edges, the blade basin and the blade back curves are parallel to each other at the leading edge or the trailing edge, and the straight lines extending in the direction or the opposite direction of the two unit tangent vectors at the leading end point or the trailing end point do not intersect. According to the geometric analysis here, the central angle formed by the two normal vectors perpendicular to the tangent vector at the leading end point or the trailing end point is equal to 180°.

[0099] This step is designed to determine the size of the center angle of the complementary circle when the two unit tangent vectors point parallel to each other. This step determines the size of the center angle of the complementary circle when the two unit tangent vectors point parallel to each other by calculating the angle between the tangent vectors, and distinguishes between the situations when the two unit tangent vectors point close to each other or away from each other, to avoid obtaining an erroneous center angle of the complementary circle.

[0100] S7: Select the type of conic section at the leading edge or trailing edge, set the node vector of the quadratic NURBS curve according to the type of conic section, calculate the coordinates of the control points of the conic curve in the Cartesian coordinate system and the corresponding weight factors according to the different situations of the center angle, and use the quadratic NURBS curve to fit the control point set to obtain the leading edge or trailing edge complementary circle curve of the blade section; this step includes the following sub-steps:

[0101] S71: Set the node vector of the rational B-spline defined by points P1, P0, and P3 as control points to U = [0, 0, 0, 1, 1, 1], and the weight factor to [ω1 = ω3 = 1, ω0 = cos(θ / 2)];

[0102] Specifically, the original control points and weight factors are expressed in the Cartesian coordinate system as follows:

[0103]

[0104] Among them, the first three values ​​in the brackets are the x, y, and z coordinates of the point in the Cartesian coordinate system, and the fourth value is the weight factor of the point.

[0105] This step is designed to use a quadratic NURBS curve to describe the front and rear edge of the complementary circular line. The advantage of using a quadratic NURBS curve to describe the front and rear edge of the complementary circular line in this step is that it does not require too many control points and weight factors as definition data, and does not require the construction of a combined curve to complete the construction of a whole section of the complementary circular line.

[0106] S72: If you need to construct a perfect circle, insert a node Get the new node vector If you need to construct other conic sections, insert the node twice Get the new node vector

[0107] Specifically, insert a node into the original rational B-spline defined by points P1, P0, P3 as control points and node vector U = [0,0,0,1,1,1]. Two new control points Q1 and Q2 will be introduced, and the new control points P1, Q1, Q2, and P3 will be used to describe the circular arc; for example, insert two nodes into the original rational B-spline defined by points P1, P0, and P3 as control points and node vector U = [0,0,0,1,1,1] Three new control points Q1, S, Q2 will be introduced, and other conic sections will be described by new control points P1, Q1, S, Q2, P3.

[0108] This step is designed to use different quadratic NURBS curve description methods according to the type of the front and rear edge filling circle. This step inserts nodes in the rational B-spline and introduces new control points Q1, Q2 or Q1, S, Q2 to more finely control the shape of the conic curve. According to different filling circle requirements, quadratic NURBS curves are used to describe the right circular arc or other conic sections. According to NURBS theory, insert a node The generated control points Q1 and Q2 can be used to accurately describe the arc and insert the nodes twice. The generated control points Q1, S, Q2 can describe other conic curves (such as elliptical arcs, hyperbolas or parabolic segments). This method can meet the generation requirements of different curves, and the generated curve segments have strong convex hull properties, which enhances the flexibility of geometric description.

[0109] S73: Using the de Casteljau curve interpolation formula, if a perfect circle needs to be constructed at the leading edge or trailing edge, the rule for inserting new control points is:

[0110]

[0111] Where ω is the weight factor of the control point. According to the properties of a perfect circle, for a perfect circular arc with a center angle less than 180°, the weight factor of the original control point P0 can be expressed by the degree of the center angle as Where e is half the distance between P1 and P3, f is the distance between P1 and P0 or between P3 and P0, and the weight factors of the original control points P1 and P3 are both 1. Therefore, the new control points and weight factors are expressed in the Cartesian coordinate system as follows:

[0112]

[0113] The first three values ​​in the brackets are the x, y, and z coordinates, and the fourth value is the weight factor. At this time, points P1 and P3 must be aligned;

[0114] If it is necessary to construct other conic sections at the leading edge or trailing edge, the de Casteljau curve interpolation formula is applied twice. The control point set and weight factor after inserting the new control point are expressed in the Cartesian coordinate system as follows:

[0115]

[0116] Similarly, the first three values ​​in the brackets are the x, y, and z coordinates, and the third value is the weight factor. The calculation formula for ω0 is as follows:

[0117]

[0118] Wherein point P is a point between point P0 and the midpoint R of point P1 and point P2. The position of point P between point P0 and point R can be controlled by the third first taper control parameter Bisect_factor input by the user. This parameter is used to control the taper of the conic curve.

[0119] Specifically, Figure 8 As shown in the figure, for the leading edge or trailing edge of the blade back and blade basin curve, based on the original control points P1, P0, and P3, insert After the node is set, the coordinates of the new control points Q1 and Q2 and the weight factor ω can be obtained according to the de Casteljau curve interpolation formula Q1 ,ω Q2 , the calculation method of the weight factor is related to the angle of the center angle, ensuring that an accurate perfect arc can be generated at any center angle less than 180°; Figure 8 As shown in other types of cones with a central angle less than 180 degrees, for the leading edge or trailing edge of the blade back and blade basin curve, two insertions are made based on the original control points P1, P0, and P3. After the node is set, the coordinates of the new control points Q1, S, Q2 and the weight factor ω can be obtained according to the de Casteljau curve interpolation formula Q1 ,ω S ,ω Q2 The calculation method of the weight factor is related to the position of point P, and the position of point P is determined by the first taper control parameter Bisect_factor, which ensures that accurate curves can be generated at any center angle less than 180°. The relationship between the types of other conic sections generated and the weight factor ω0 is as follows: Fig. 9 As shown, when When , the generated conic section is an elliptical arc; when When , the generated conic section is a parabola segment: when , the generated conic section is a hyperbola segment.

[0120] This step is designed to generate an arc or other conic section (such as an elliptical arc, a parabola segment, a hyperbola segment) less than 180° at the leading edge or trailing edge of the blade that is G1 continuous with the back and blade basin curves at points P1 and P3 by modifying the node vector and using the de Casteljau algorithm, and to convert the calculation of the control points and weights into coordinate operations in the Cartesian coordinate system. The de Casteljau algorithm used in this step is an algorithm that accurately calculates the interpolation points of a curve. It can calculate new control points by recursive interpolation to ensure that the generated curve segment has good smoothness and geometric continuity. Since the solution of the new control point in this step is combined with the center angle of the circle, the fitted leading edge or trailing edge complementary circle curve and the blade basin and blade back curve always maintain G1 continuity at points P1 and P3. By converting the interpolation calculation of the control point into a simple coordinate operation in the Cartesian coordinate system, the complex geometric generation problem is simplified. The process of calculating the new control points Q1, Q2, and S only depends on the original control points P1, P0, P3 and the inserted nodes. When generating other types of conic curves, the first taper control parameter Bisect_factor is used to control the taper of the curve, making the process of generating complex conic sections more intuitive and operable.

[0121] S74: For a positive arc with a center angle greater than 180°, the weight factor of the original control point P0 can be expressed by the center angle as At the same time, the control point formula of the perfect circle Corrected to For other conic sections with central angles greater than 180°, the weight factor of the original control point P0 is expressed as -ω0 by the weight factor of P0 in other conic sections with central angles less than 180°;

[0122] Specifically, Figure 8 As shown in the figure, for the leading edge or trailing edge of the blade back and blade basin curve, based on the original control points P1, P0, and P3, insert After the node is reached, according to the de Casteljau curve interpolation formula, the weight factor of the original control point P0 is re-expressed as At the same time, the control point formula of the perfect circle Corrected to The coordinates of the new control points Q1 and Q2 and the weight factor ω can be calculated when the central angle is greater than 180° Q1 ,ω Q2 , the calculation method of the weight factor is still related to the angle of the center angle, ensuring that an accurate arc can be generated at any center angle greater than 180°. Therefore, the new control point and weight factor are expressed in the Cartesian coordinate system as follows:

[0123]

[0124] like Figure 8 As shown in other types of cones with a central angle greater than 180 degrees, for the leading edge or trailing edge of the blade back and blade basin curve, two insertions are made based on the original control points P1, P0, and P3. After the node is reached, according to the de Casteljau curve interpolation formula, the weight factor of the original control point P0 is re-expressed as -ω0 to obtain the coordinates of the new control points Q1, S, Q2 and the weight factor ω Q1 ,ω S ,ω Q2 The calculation method of the weight factor is related to the position of point P, and the position of point P is determined by the first taper control parameter Bisect_factor, which ensures that an accurate curve can be generated at any center angle greater than 180°. The relationship between the types of other conic sections generated and the weight factor ω0 is as follows: Fig.10 As shown, when When , the generated conic section is an elliptical arc; when When , the generated conic section is a parabola segment; when When , the generated conic section is a hyperbola segment. Therefore, the new control points and weight factors are expressed in the Cartesian coordinate system as follows:

[0125]

[0126] This step is designed to generate an arc or other conic section (such as an elliptical arc, a parabola segment, a hyperbola segment) with a center angle greater than 180° at the leading edge or trailing edge of the blade, which is continuous with the back and basin curves at points P1 and P3 by inserting nodes and using the de Casteljau algorithm, and to convert the calculation of the control points and weights into coordinate operations in the Cartesian coordinate system. This step solves the design requirements of generating conic sections with a center angle greater than 180° at the leading edge or trailing edge of certain blade sections, and has good adaptability for blade sections that use supercritical airfoil designs. The main process and effect of this step are similar to generating a positive arc or other conic section with a center angle less than 180°, but by modifying the control point formula of the positive circle or taking the negative weight factor ω0, an arc or other conic section with a center angle greater than 180° can be obtained. Since this step adopts a process similar to that of generating a perfect circular arc or other conic sections less than 180°, the program can be slightly modified on the basic algorithm without designing a completely different logic, which is more flexible and consistent in implementation and application.

[0127] S75: When the central angle is 180°, the two unit tangent vectors at the leading or trailing endpoints of the blade basin and blade back curves are and There is no intersection between the straight lines l1 and l2 where they are located. The new control point set and weight factor of the perfect circle obtained by using the infinite control point are expressed in the Cartesian coordinate system as follows:

[0128]

[0129] in is the distance between point P1 and point P3;

[0130] The new control point set and weight factor of other types of conic curves obtained by using the infinite control point are expressed in the Cartesian coordinate system as follows:

[0131]

[0132] in and are the y-coordinates and z-coordinates of point Q1 and point Q2 respectively, is the distance between point P1 and point P3, Cone_ratio is the second taper control parameter input by the user, and is used to control the taper of the non-circular conic section when the center angle is 180°.

[0133] Specifically, Figure 8 The de Casteljau curve interpolation formula is no longer applicable. Therefore, for the leading or trailing edge of the blade back and blade basin curve, based on the original control points P1 and P3, the After the node, the coordinate calculation of the new control points Q1 and Q2 depends only on the two unit tangent vectors at the leading and trailing edges. Since the center angle of the circle is 180° at this time, the position of Q1 is The distance of the positive arc radius is extended in the direction, and the position of Q2 is P3 along The radius of the positive arc is the distance between P1 and P3. The weight factors of the new control points Q1 and Q2 are like Figure 8 As shown in other types of cones with a central angle of 180°, the de Casteljau curve interpolation formula is no longer applicable. Therefore, for the leading or trailing edge of the blade back and blade basin curve, two interpolations are made based on the original control points P1 and P3. After the node, the coordinate calculation of the new control points Q1 and Q2 depends only on the two unit tangent vectors at the leading and trailing edges. Since the center angle of the circle is 180° at this time, the position of Q1 is The direction extends a certain distance, and the position of Q2 is P3 along The distance of the extension is the distance between P1 and P3. The product of the second taper control coefficient, and the weight factors of the new control points Q1 and Q2 are

[0134] This step is designed to deal with the special situation where the de Casteljau curve interpolation formula cannot be used to find the new control point when the central angle of the leading edge or the trailing edge is 180°. Since the two unit tangent vectors at the leading edge or the trailing edge do not have an intersection at this time, the traditional de Casteljau interpolation formula cannot be directly applied. This geometric problem can be avoided by the infinite control point method. Reasonable control points are generated based on the distance between P1 and P3 and their directionality, ensuring the rationality of the generated right circular arc or other conic curve segments when the central angle is 180°. In this case, the position of the control points Q1 and Q2 simply depends on the distance between the two endpoints P1 and P3, avoiding complex interpolation calculations. The weight factors of the control points Q1 and Q2 are The selection of S weight factor 1 ensures from a geometric perspective that the various types of leading edge or trailing edge complementary circle curves fitted by the quadratic NURBS curve when the central angle is 180° always maintain G1 continuity with the blade basin and blade back curves at points P1 and P3, which not only meets the geometric requirements but also simplifies the curve generation process.

[0135] S8: According to the relationship between the conic section type control parameter, the taper control parameter and the sliding parameter along the blade span direction, define the equation group of the leading edge or trailing edge fillet type, size, slope and taper along the blade span direction, repeat S1 to S7 until the construction of the leading edge fillet curve and the trailing edge fillet curve of all blade sections is completed; this step includes the following sub-steps:

[0136] S81: Define the relationship between the conic section type control parameter If_circle, the first taper control parameter Bisect_factor, the second taper control parameter Cone_ratio, the blade back sliding parameter Shift_factor1 and the blade basin sliding parameter Shift_factor2 along the blade span direction section number change:

[0137]

[0138] where t s is the number of blade span sections from the blade root to the blade tip, n s =1,2,3…,t is the serial number of the current blade section;

[0139] Specifically, the relationship relates the conic section type control parameter If_circle, the first taper control parameter Bisect_factor, the second taper control parameter Cone_ratio, the blade back sliding parameter Shift_factor1, and the blade basin sliding parameter Shift_factor2 to the number of blade span sections. According to the rules in the relationship, a set of equations for the leading edge or trailing edge fill circle design parameters to vary with the number of blade span sections can be proposed. As the independent variable, the design parameters are used as the dependent variable, so as to automatically realize the use of different design parameters in different blade sections.

[0140] This step is designed to provide a relationship between the leading edge or trailing edge fillet design parameters and the number of spanwise sections of the blade. By associating these parameters with the number of spanwise sections of the blade, the designer can ensure that the fillet design parameters change smoothly across the entire span of the blade, avoiding sudden changes or discontinuities in different sections. From the root to the tip of the leading or trailing edge of the blade, the fillet is no longer just a uniform shape, but can gradually change according to design needs to meet the requirements of fluid mechanics performance or structural strength. Such a parameterized relationship can provide a continuous and consistent design solution, ensuring that the design on each section meets the design requirements of the overall blade.

[0141] S82: for each blade section constituting the blade, defining an equation for the shape control parameters of the leading edge conic section and the trailing edge conic section of the blade section changing with the number of blade sections according to the relationship;

[0142] Specifically, according to the relationship, the ratio of the number of blade sections to the total number of blade sections is used as the independent variable, and the range of the independent variable is To 1, taking the leading edge or trailing edge conic section shape control parameters as independent variables, a set of linear or nonlinear equations can be defined to describe the change process of the conic section shape control parameters at the leading edge or trailing edge from the blade root to the blade tip.

[0143] This step is designed to achieve continuous change of parameters with the number of blade sections in the span direction, provide a framework for automatic adjustment of leading edge or trailing edge fillet design parameters, support the design of complex blade geometries, optimize the design process, and reduce design time. This step uses the ratio of the number of blade sections to the total number of blade sections as the independent variable. to 1 covers the entire spanwise range of the blade, ensuring the continuity of the design parameters from the root to the tip of the blade. Through this set of linear or nonlinear equations, designers can automatically adjust the conic section shape control parameters of the leading edge or trailing edge according to the overall design requirements of the blade. This automated process reduces the complexity of manual adjustment and greatly improves design efficiency, especially when multiple blade sections need to be adjusted in a coordinated manner. The set of equations automatically optimizes and adjusts the design parameters of different sections so that the entire blade from root to tip can meet the unified design goals. Designers can choose linear or nonlinear equations to define the law of the change of the conic section shape control parameters with the number of blade spanwise sections according to specific needs: if the parameter change is relatively simple and stable, the linear equation can provide a better fitting effect. For example, when the conic section shape control parameters need to transition evenly between the root and the tip of the blade, the linear equation can accurately describe the gradual change of the parameters; when the conic section shape control parameters need to be more finely adjusted in a specific area, the nonlinear equation can provide more flexible control. For example, specific adjustments may need to be made in the middle or tip of the blade to optimize aerodynamic performance, and nonlinear equations can describe how these parameters change in more detail.

[0144] S83: repeat S1 to S7, and use different conic section shape control parameters to independently control the construction of each leading edge or trailing edge fillet curve, until the construction of the leading edge fillet curves and trailing edge fillet curves of all blade sections is completed;

[0145] Specifically, Fig.10 As shown in the process of generating the leading edge or trailing edge of a single blade section, for the leading edge or trailing edge of the blade section, the conic section type control parameter If_circle of the leading edge and trailing edge is set to 1, and the node vector Align the control points P1 and P3, calculate the size of the central angle through the two unit tangent vectors, and then select the solution method of the positive arc control point according to the size of the central angle, calculate the positions of the control points Q1 and Q2 and the corresponding weight factors, and finally use the quadratic NUBRS curve to fit the control point set to obtain the leading edge or trailing edge perfect circle supplementary curve of the blade section.

[0146] For Fig.11 As shown in the process of generating the consistent leading edge or trailing edge fillet of multiple blade sections shown in , the equations defining the leading edge and trailing edge fillet design parameters from the blade root to the blade tip as the number of blade sections varies are as follows:

[0147]

[0148] The specific effect achieved by this linear equation group is: as the number of blade sections increases, the slip parameters of the leading edge or trailing edge remain unchanged, the taper control parameters are all zero, and all the supplementary circles are positive arcs.

[0149] For Fig.12 As shown in the process of generating non-uniform leading edge or trailing edge fill circles for multiple blade sections shown in , the equations defining the shape control parameters of the leading edge conic section from the blade root to the blade tip as the number of blade sections changes are as follows:

[0150]

[0151] The specific effect achieved by this linear equation group is: as the number of blade sections increases, the proportion of the blade back leading edge control point P1 sliding toward the intersection point P0 on l1 is It can be seen that the slip ratio of point P1 at the leading edge of the blade root is the minimum The sliding ratio of point P1 at the leading edge of the blade tip is the maximum value of 0.1, and the actual sliding distance is the product of the distance between P1 and P0 and the sliding parameter. At the same time, the position of the control point P3 at the leading edge of the blade basin does not change with the number of sections. This means that from the root to the tip of the blade, the slope of the leading edge of the blade gradually increases and gradually tilts toward the blade basin; the taper control parameter is The increment of varies with the number of sections. It can be seen that the taper control parameter at the leading edge of the blade root is the minimum value. The maximum value of the taper control parameter at the leading edge of the blade tip is 0.3. This means that the taper of the leading edge of the blade increases continuously from the root to the tip of the blade. At the same time, the leading edges of the odd-numbered sections are controlled to generate a positive arc, and the leading edges of the even-numbered sections are controlled to generate other types of conic sections. The leading edge conic section shape control parameters of each blade section will be automatically obtained through the equation according to the number of sections until the leading edge supplementary circle curve of each blade section is generated.

[0152] The equations that define the shape control parameters of the conic section of the trailing edge from the blade root to the blade tip as the number of blade sections changes are as follows:

[0153]

[0154] The specific effect achieved by this linear equation group is: as the number of blade sections increases, the proportion of the blade basin trailing edge control point P3 sliding toward the intersection point P0 on l2 is It can be seen that the slip ratio of point P3 at the trailing edge of the blade root is the minimum. The sliding ratio of point P3 at the trailing edge of the blade tip is the maximum value of 0.1, and the actual sliding distance is the product of the distance between P3 and P0 and the sliding parameter. At the same time, the position of the control point P1 on the trailing edge of the blade back is kept constant with the number of sections. This means that from the root to the tip of the blade, the slope of the leading edge of the blade gradually increases and gradually tilts toward the back of the blade; the taper control parameter is The increment of changes with the number of sections. It can be seen that the taper control parameter at the trailing edge of the blade root is the minimum value. The maximum value of the taper control parameter at the trailing edge of the blade tip is 0.3. This means that the taper of the trailing edge of the blade profile increases continuously from the blade root to the blade tip; at the same time, the trailing edges of the odd-numbered sections are controlled to generate a positive arc, and the trailing edges of the even-numbered sections are controlled to generate other types of conic sections; the leading edge fillet design parameters of each blade profile section will be automatically obtained through the equation according to the number of sections until the leading edge fillet curve of each blade profile section is generated.

[0155] This step is to automatically generate appropriate fillet curves at the leading and trailing edges of the blade, and dynamically adjust the conic section type, sliding parameters and taper control parameters through a set of linear equations, so that these geometric parameters can be designed to be systematically and regularly changed with the sequence number of the spanwise section of the blade. The core of this step is to ensure that the generation process of the conic section can adapt to the design requirements on different sections, so as to achieve precise geometric control at different parts of the blade. Through the formula, the type of the conic section of the leading or trailing edge of each section can be individually controlled at the leading or trailing edge. This design diversity can meet different aerodynamic requirements on different sections of the blade. The position of the control point P1 or P3 can be moved according to a certain rule, which means that on each section, the geometric shape of the leading edge of the back of the blade gradually changes, so that the curve maintains dynamic adjustment on different sections and can adapt to the changes in the spanwise shape of the blade. The adjustment of the taper control parameter gradually increases or decreases with the dynamic progressive adjustment of the spanwise section, so that the fillet curve of each section is more adapted to specific design requirements and improves the flow performance and geometric continuity of the leading edge.

[0156] S9: For each blade section, the leading edge fillet curve, the trailing edge fillet curve, the blade basin curve, and the blade back curve are connected at adjacent endpoints to form a closed blade section curve, and a lofting operation is performed between all blade sections to generate the final three-dimensional blade entity. This step includes the following sub-steps:

[0157] Specifically, for each blade section, all endpoints of the leading edge fillet curve, trailing edge fillet curve, blade basin curve, and blade back curve are traversed to compare and detect whether there are the same endpoints between the fillet curve and the blade curve. In terms of data structure, the leading edge fillet curve or trailing edge fillet curve with the blade basin curve or blade back curve that have the same endpoints are connected at the endpoints until the four curves form a closed blade section curve with a complete topological structure. The lofting operation is performed between all closed blade sections in the order from the blade root to the blade top to generate a three-dimensional blade entity.

[0158] like Fig.13 As shown, a perfect arc is generated for the leading edge and trailing edge of each blade section of the initial blade shape, the four curves of each blade section are connected at the same endpoints to form a closed blade section curve with a complete topological structure, and a three-dimensional blade is generated after performing a lofting operation between the blade sections.

[0159] like Fig.14 The figure shows that the leading edge or trailing edge filling circle of each section is generated for the initial blade according to the equation group defined in S83, the four curves of each blade section are connected at the same endpoints to form a closed blade section curve with a complete topological structure, and a three-dimensional blade is generated after performing a lofting operation between the blade sections.

[0160] This step is designed to generate a three-dimensional blade between closed blade sections after the leading edge fillet and the trailing edge fillet are generated. By detecting and connecting the curve endpoints, each blade section is ensured to form a closed contour curve. Closed sections are crucial in geometric modeling because they ensure that the geometric topology of the blade is complete, without disconnected or unclosed curves, which helps to avoid topological discontinuities or geometric gaps in the subsequent lofting and three-dimensional modeling process. By lofting each section curve in a certain order, a three-dimensional blade entity from the root to the tip of the blade can be generated. The lofting operation can ensure a smooth transition between sections, so that the generated three-dimensional blade is geometrically continuous and smooth, meeting the requirements of aerodynamic design. This method can effectively maintain the fluid properties of the blade and avoid mutations or discontinuities in the blade shape.

[0161] The method of the present invention combines the basic knowledge of conic sections and circles within the framework of the classic NURBS representation to accurately construct arbitrary arcs and conic sections. The method emphasizes the control of the shape of the complementary circle by adjusting the first or last point of the blade back or blade basin in the design space. The two unit tangent vectors of the blade basin and blade back curves at the leading edge or trailing edge endpoints are used to solve the control points and weight factors based on the complementary circle type and the size of the central angle, so as to accurately construct arcs and conic sections that meet the convex hull requirements. While ensuring the G1 continuity of the two ends of the constructed conic section and the connection between the blade basin line and the blade back line endpoints, the shape of the conic section is controlled by a small number of parameters to achieve the control of the slope, taper and size of the leading edge or trailing edge complementary circle. By introducing the relationship and equation between the associated conic section shape control parameters and the number of blade sections in the span direction, the dynamic change of the complementary circle curve along the blade span direction is achieved, thereby completing the complex design of the leading edge and trailing edge of the three-dimensional blade.

[0162] Those skilled in the art will readily appreciate other embodiments of the present application after considering the description and practicing the contents disclosed herein. The present application is intended to cover any modification, use or adaptation of the present application, which follows the general principles of the present application and includes common knowledge or customary techniques in the art that are not disclosed in the present application. The description and examples are intended to be exemplary only, and the true scope and spirit of the present application are indicated by the claims.

[0163] It should be understood that the present application is not limited to the precise structures that have been described above and shown in the drawings, and that various modifications and changes may be made without departing from the scope thereof. The scope of the present application is limited only by the appended claims.

Claims

1. A method for constructing the leading and trailing edges of a blade with an arbitrary center angle, characterized in that: include: S1: Using the first two points or the last two points of the blade basin and blade back profile point set, first use the two-point formula to construct two straight line analytical expressions at the leading edge or the trailing edge on the two-dimensional plane of the blade profile section, and calculate the slope of the angle bisector formed by the two straight line analytical expressions, and then use the angle bisector slope to calculate the slope of the perpendicular line perpendicular to the angle bisector; S2: using the z value of the first or last point of the leaf basin and leaf back shape value points in the Cartesian coordinate system and the slope of the angle bisector, selecting the first or last point through which the perpendicular line of the angle bisector passes, and combining the slope of the perpendicular line of the angle bisector, using the first or last point through which the perpendicular line of the angle bisector passes to obtain the perpendicular line of the angle bisector; S3: using the perpendicular line of the angle bisector to find the intersection point with the straight line where the other corresponding first point or tail point to be aligned is located, and inserting the intersection point to the front of the first point or the tail of the tail point to complete the alignment of the leading edge point or the trailing edge point, and obtaining the aligned leaf basin and leaf back shape value point set; S4: Using two sliding parameters for controlling the slope and size of the conic section, the first point or the last point of the aligned blade basin and blade back profile point set can be at the intersection of the straight line at the leading edge or the trailing edge. Move between the design space of the next shape value point after the first point or the previous shape value point of the last point to obtain the final leaf basin and leaf back shape value point set; S5: fitting the final blade basin and blade back shape value point set to obtain a blade basin and blade back curve, and obtaining two unit tangent vectors at the leading end point or the trailing end point of the blade basin and blade back curve based on the blade basin and blade back curve; S6: Calculate the central angle formed by the blade basin curve and the blade back curve at the leading edge or the trailing edge using the two unit tangent vectors; S7: Select the type of conic section at the leading edge or trailing edge, set the node vector of the quadratic NURBS curve according to the type of conic section, calculate the coordinates of the control points of the conic curve in the Cartesian coordinate system and the corresponding weight factors according to different situations of the center angle, and use the quadratic NURBS curve to fit the control point set to obtain the leading edge or trailing edge complementary circle curve of the blade section; S8: According to the relationship between the conic section type control parameter, the taper control parameter and the sliding parameter along the blade span direction, define the equation group of the leading edge or trailing edge fillet type, size, slope and taper along the blade span direction, and repeat S1 to S7 until the construction of the leading edge fillet curve and the trailing edge fillet curve of all blade sections is completed; S9: For each blade section, the leading edge fillet curve, the trailing edge fillet curve, the blade basin curve, and the blade back curve are connected at adjacent endpoints to form a closed blade section curve, and a lofting operation is performed between all blade sections to generate a final three-dimensional blade entity.

2. The method for constructing the leading and trailing edges of a blade profile with an arbitrary center angle according to claim 1, characterized in that: S2 includes the following sub-steps: S21: For the leading edge, if the first point of the leaf basin and leaf back shape value point set and If the z values ​​are different, the first point with the smaller z value is selected as the point through which the perpendicular line of the angle bisector passes; if the first point and The z value is the same. If the slope of the angle bisector is greater than 0, select the point with a smaller y value as the point through which the perpendicular line of the angle bisector passes. Otherwise, select the point with a larger y value as the point through which the perpendicular line of the angle bisector passes. S22: For the trailing edge, if the leaf basin is at the end of the leaf back value point and If the z value is different, the tail point with the larger z value is selected as the point through which the perpendicular line of the angle bisector passes; if the tail point and The z value is the same. If the slope of the angle bisector is greater than 0, select the point with a larger y value as the point through which the perpendicular line of the angle bisector passes. Otherwise, select the point with a smaller y value as the point through which the perpendicular line of the angle bisector passes. S23: Using the first point or the last point passed, find the perpendicular line of the angle bisector at the leading edge or the trailing edge ; ； in is the slope of the perpendicular line to the angle bisector, and The first or last point through which the perpendicular line of the angle bisector passes Coordinates and coordinate.

3. The method for constructing the leading and trailing edges of a blade with an arbitrary center angle according to claim 2, characterized in that: S3 includes the following sub-steps: S31: When is the first point in the leaf back value point concentration, is the first point in the leaf basin value point set, and the point through which the perpendicular line of the angle bisector passes is When , find the perpendicular line of the angle bisector With point The straight line The intersection point is ,Will Insert before the first point of the leaf back value point set; conversely, when the point through which the perpendicular line of the angle bisector passes is When , find the perpendicular line of the angle bisector With point The straight line The intersection point is ,Will Insert before the first point in the leaf basin type value point set; update the index to obtain the leaf basin and leaf back type value point set after aligning the leading edge. ; S32: When It is the tail point of the leaf back value point concentration. is the tail point of the leaf basin value point concentration, and the point through which the perpendicular line of the angle bisector passes is When , find the perpendicular line of the angle bisector With point The straight line The intersection point is ,Will Insert after the tail point of the leaf back value point concentration; conversely, when the point through which the perpendicular line of the angle bisector passes is When , find the perpendicular line of the angle bisector With point The straight line The intersection point is ,Will Insert the tail point in the leaf basin type value point set; update the index to obtain the leaf basin and leaf back type value point set after aligning the leading edge. ; Among them, when is the first point in the leaf back value point concentration, When it is the first point in the leaf basin value point set, point is the point after the first point in the leaf back value point set. is the point after the first point in the leaf basin value point set; when It is the tail point of the leaf back value point concentration. When it is the tail point of the leaf basin value point concentration, point is the point before the tail point in the leaf back value point set. It is the point before the tail point in the leaf basin type value point set.

4. The method for constructing the leading and trailing edges of a blade profile with an arbitrary center angle according to claim 3, characterized in that: S4 includes the following sub-steps: S41: For the leading edge or trailing edge, when the sliding parameter Shift_factor1 of the blade back is 0 and the sliding parameter Shift_factor2 of the blade base is 0, the point With point The position remains unchanged; S42: For the leading edge or trailing edge, when Shift_factor1>0, Shift_factor2>0, exist upward move, exist upward move; S43: For the leading edge or trailing edge, when Shift_factor1<0, Shift_factor2<0, exist upward move, exist upward move; S44: When the perpendicular line to the angle bisector passes through the point When the leaf back sliding parameter Shift_factor1 is set to -1, it means canceling the alignment point , use the original leaf back type value point set to generate the supplementary circle; when the point through which the perpendicular line of the angle bisector passes is When the leaf basin's sliding parameter Shift_factor2 is set to -1, it means canceling the alignment point. , use the shape point set of the original leaf basin to generate the complementary circle.

5. The method for constructing the leading and trailing edges of a blade profile with an arbitrary center angle according to claim 4, characterized in that: S6 includes the following sub-steps: S61: determining the relative trend of the blade basin and the blade back curve at the head or tail according to two unit tangent vectors at the head end point or the tail end point of the blade basin and the blade back curve; S62: If the two unit tangent vectors of the blade basin curve and the blade back curve at the leading end or the trailing end point close to each other, the central angle is: ； in and are two vectors of the blade basin and blade back curve at the leading end point or the trailing end point; S63: If the two unit tangent vectors of the blade basin curve and the blade back curve at the leading or trailing endpoints point away from each other, then the central angle is , that is, the central angle is greater than 180°; S64: If the two unit tangent vectors of the blade basin curve and the blade back curve at the leading end or the trailing end point in parallel, that is, , then the central angle is equal to 180°.

6. The method for constructing the leading and trailing edges of an airfoil with an arbitrary center angle according to claim 5, characterized in that: S7 includes the following sub-steps: S71: Set by point , , The knot vectors of the rational B-spline defined as control points are , the weight factor is [ ]; S72: If you need to construct a perfect circle, insert a node , get the new node vector ; If you need to construct other conic sections, insert the node twice , get the new node vector ; S73: Using the de Casteljau curve interpolation formula, if a perfect circle needs to be constructed at the leading edge or trailing edge, the rule for inserting new control points is: ； in is the weight factor of the control point. According to the properties of a perfect circle, for a perfect circular arc with a central angle less than 180°, the original control point The weight factor of can be expressed by the degree of the center angle as , where e is and half of the distance between and or and The distance between the original control points and The weight factors of are all 1; therefore, the new control points and weight factors are expressed in the Cartesian coordinate system as follows: ； The first three values ​​in the brackets are the x, y, and z coordinates, and the fourth value is the weight factor. and Must be aligned; If it is necessary to construct other conic sections at the leading edge or trailing edge, the de Casteljau curve interpolation formula is applied twice. The control point set and weight factor after inserting the new control point are expressed in the Cartesian coordinate system as follows: ； Similarly, the first three values ​​in the brackets are the x, y, and z coordinates, and the third value is the weight factor, where The calculation formula is as follows: ； The point For point With point and Point midpoint The third taper control parameter Bisect_factor input by the user can control the point At the point With point The position between , this parameter is used to control the taper of the conic section; S74: For a true arc with a center angle greater than 180°, the original control point The weight factor of can be expressed by the center angle as , and at the same time, the control point formula of the perfect circle Corrected to For other conic sections with a central angle greater than 180°, the original control point The weight factor is determined by the other conic sections with a central angle less than 180°. The weight factor is expressed as ; S75: When the central angle is 180°, the two unit tangent vectors at the leading or trailing endpoints of the blade basin and blade back curves are and and the straight lines on which they lie and There is no intersection between them. The new control point set and weight factor of the perfect circle obtained by using the infinite control point are expressed in the Cartesian coordinate system as follows: ； in For point With point The distance between The new control point set and weight factor of other types of conic curves obtained by using the infinite control point are expressed in the Cartesian coordinate system as follows: ； in , and , Points With point The y-coordinate and z-coordinate of For point With point Cone_ratio is the second taper control parameter input by the user, which is used to control the taper of the non-circular cone section when the central angle is 180°.

7. The method for constructing the leading and trailing edges of an airfoil with an arbitrary center angle according to claim 6, characterized in that: S8 includes the following sub-steps: S81: Define the relationship between the conic section type control parameter If_circle, the first taper control parameter Bisect_factor, the second taper control parameter Cone_ratio, the blade back sliding parameter Shift_factor1 and the blade basin sliding parameter Shift_factor2 along the blade span direction section number change: ； in is the number of blade span sections from blade root to blade tip, is the serial number of the current blade section; S82: for each blade section constituting the blade, defining an equation for the shape control parameters of the leading edge conic section and the trailing edge conic section of the blade section changing with the number of blade sections according to the relationship; S83: Repeat S1 to S7, and use different conic section shape control parameters to independently control the construction of each leading edge or trailing edge fillet curve, until the construction of the leading edge fillet curves and trailing edge fillet curves of all blade sections is completed.

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