Radial basis function mesh deformation method and airfoil design method based on virtual nodes

Abstract

The present invention discloses a radial basis function mesh deformation method based on virtual nodes and an airfoil design method. Currently, generating a high-quality deformed mesh by a mesh deformation method is still an open problem. The present invention first obtains the distribution of internal virtual nodes based on the internal normal of the moving boundary point; then reduces the number of internal virtual nodes to obtain a small number of uniformly distributed internal virtual nodes; then obtains the displacement of each internal virtual point by interpolation of K adjacent boundary nodes; finally, uses the internal virtual nodes and the original boundary nodes to construct the linear equilibrium equation of the object surface, solves the coefficient matrix, and thus interpolates and calculates the displacement of the internal nodes to obtain the deformed mesh. The present invention effectively improves the quality of the deformed mesh and the degree of mesh deformation, and realizes the rapid generation of high-quality mesh units for the deformation of the aircraft airfoil section. This greatly reduces the time and computing power required for aircraft airfoil optimization.

Description

Technical Field

[0001] The present invention relates to the technical field of numerical simulation, and in particular to a radial basis function grid deformation method and an airfoil design method based on virtual nodes. Background Art

[0002] Numerical simulation has become an indispensable tool for analyzing physical characteristics in many engineering applications. In these applications, boundary deformation presents some problems. For example, free surface problems, multi-body separation problems, forced vibration problems, and fluid-structure coupling problems. In order to numerically solve these problems, the mesh covering the computational domain should change as the computational domain deforms. The commonly used processing methods in industry are mesh reconstruction methods and mesh deformation methods. Mesh reconstruction requires regenerating the mesh according to the geometric shape of the model after each deformation, resulting in huge time consumption and computational overhead. The mesh deformation method changes the internal mesh according to the movement of the boundary, maintaining the topological structure of the original mesh. For this type of problem, a better choice is to use the mesh deformation method for numerical simulation.

[0003] Mesh deformation methods can be categorized as physical analogy and interpolation analogy. Compared to physical analogy, interpolation analogy does not require attention to the connectivity between mesh nodes, resulting in high storage efficiency and better mesh deformation. Furthermore, under the same deformation scheme, its mesh computational efficiency is higher than that of physical analogy. Interpolation analogy methods based on radial basis functions transfer the displacement of boundary nodes to other nodes within the computational domain based on the basis functions. These methods are applicable to meshes of any topological shape, exhibit strong robustness, and maintain good boundary orthogonality after deformation.

[0004] However, it is not easy to generate high-quality meshes using radial basis function-based mesh deformation methods. The reason is that the traditional radial basis function interpolation method easily produces low-quality mesh units such as distorted units and intersecting units near the moving boundary due to the rapid descent rate of the basis function gradient, which causes the deformation to stop early and affects the subsequent numerical analysis. Currently, generating high-quality deformed meshes is still an open problem. Most of the existing radial basis function-based deformation methods rely solely on the displacement of known boundary points, which makes it difficult to adapt to the large-displacement mesh deformation requirements in industry. Summary of the Invention

[0005] The purpose of the present invention is to provide a radial basis function (RBF) two-dimensional mesh deformation method based on internal virtual control points to meet the needs of non-structured mesh deformation in two-dimensional space. According to the deformation of the boundary of the NACA0012 two-dimensional mesh model, high-quality mesh units are quickly generated, the degree of mesh deformation is improved, and the rapid optimization of aircraft airfoils is achieved.

[0006] In a first aspect, the present invention provides a radial basis function mesh deformation method based on virtual nodes, which comprises the following steps:

[0007] Step 1: Obtain the displacement of all boundary nodes of the deformed contour relative to the original contour. The original contour has been meshed.

[0008] Step 2: Divide all nodes on the original contour into three types: smooth boundary nodes, convex hull corner nodes, and concave hull corner nodes.

[0009] For all nodes p on all original contours mb , along p in the computational domain mb Normal direction n pb Create a virtual node p n As shown in formula (1);

[0010] p n =p mb +λn pb

[0011] Where λ is the step size.

[0012] Step 3: Reduce the number of virtual nodes created in step 2.

[0013] Step 4: For each virtual node in the selected set S obtained in step 3, obtain the K boundary nodes adjacent to each virtual node through the K-nearest neighbor algorithm; use each virtual node as an internal node in the radial basis function interpolation method to calculate the displacement of each virtual node.

[0014] Step 5: Using the virtual nodes and the boundary nodes of the original contour, radial basis function interpolation is used to construct the boundary linear equilibrium equations to obtain a coefficient matrix. This is used to interpolate the displacements of the internal nodes of the deformed contour mesh relative to the internal nodes of the original contour mesh. Based on the original contour mesh, all its mesh nodes are offset according to the corresponding displacement values to obtain the mesh of the deformed contour.

[0015] Preferably, in step 2, if the angle between the two adjacent edges of a node on the moving boundary is less than 30°, then the node is a smooth boundary node. If the angle between the two adjacent edges of a node on the moving boundary is greater than 30° and the node is on a convex boundary, then the node is a convex hull sharp corner node. If the angle between the two adjacent edges of a node on the moving boundary is greater than 30° and the node is on a concave boundary, then the node is a concave hull sharp corner node.

[0016] Preferably, in step 2, the step length λ corresponding to the convex hull sharp-cornered nodes is greater than the step length λ corresponding to the smooth boundary nodes and the concave hull sharp-cornered nodes.

[0017] Preferably, in step 2, for each convex hull corner node, multiple virtual nodes are additionally generated on both sides thereof.

[0018] As a preference, multiple virtual nodes p are additionally generated on both sides of the convex hull corner node. n The expression of (t) is as follows:

[0019]

[0020] Where n i (i=1,2) are the unit normal vectors of the two boundaries connected by the convex hull corner node at the convex hull corner node, n cv is the average unit normal vector of n1 and n2; θ is n cv and n i t=1 / T,2 / T,...,1; T is the number of virtual nodes inserted on one side of the convex hull corner node.

[0021] Preferably, in step 2, virtual nodes generated for the concave hull corner nodes are deleted. The deletion method is: calculate the distance between the virtual node and the nearest node on the moving boundary. If the shortest distance calculated for a virtual node is λ, then the virtual node is retained; otherwise, the virtual node is discarded.

[0022] Preferably, in step 3, the specific process of reducing the number of virtual nodes is as follows:

[0023] 3-1. From all virtual nodes obtained in step 1, select a number of virtual nodes less than N as the initial selected set. N is the number of nodes after the preset reduction. Set the initial selected set S = {p0, p1, ..., p m}. m is the number of nodes in the initial selected set S.

[0024] 3-2. Define the distance between any virtual node and the selected set S as the distance between the node closest to the virtual node in the selected set S and the virtual node; traverse the virtual nodes that are not selected in the selected set S and select the virtual node p with the largest distance from the selected set S new ; Set the virtual node p new Add to the selected set S.

[0025] 3-3. If the number of virtual nodes in the selected set S is less than N, go to step 3-2 and continue adding virtual nodes to the selected set S; otherwise, the reduction ends.

[0026] Preferably, the specific process of the radial basis function interpolation method is as follows:

[0027] In the first stage, the displacement of each boundary node is interpolated using the displacement of the boundary nodes to solve the coefficient vector αi By setting the boundary mesh nodes as control points and the actual displacement of each boundary control node as the function value, the linear system is constructed as follows:

[0028] S=Φα

[0029] Where S={s1,s2,…,s n} T is the actual displacement of the boundary control point, α={α1,α2,…,α n} is the coefficient matrix of the radial basis function. Φ is the radial basis function relationship between the control points, and its expression is as follows:

[0030]

[0031] In the second stage, when the coefficient matrix α of the radial basis function is ready, the internal point displacement s(r in )as follows:

[0032]

[0033] Among them, ‖r in -r i ‖ is the internal node r in The normalized Euclidean distance between the i-th radial basis function control node and the i-th radial basis function control node. is the radial basis function,

[0034] Radial Basis Function The expression is as follows:

[0035]

[0036] Where ξ is the normalized distance.

[0037] Preferably, in step 4, during the calculation process, the expression of the normalized distance ξ used by the radial basis function interpolation method is as follows:

[0038]

[0039] Where γ is the scaling factor, γ>1; R is the influence radius of the radial basis function interpolation.

[0040] Preferably, the specific process of step 5 is as follows:

[0041] 5-1. Disrupt the order of all boundary nodes and virtual nodes to form a set of reserve nodes; randomly divide each reserve node in the set into several groups

[0042] 5-2. Select a number of preliminary nodes as the initial control point set

[0043] 5-3. Use radial basis function interpolation to calculate the estimated displacement of each preliminary node that is not in the control point set; and calculate the displacement error between the estimated displacement and the actual displacement.

[0044] 5-4. Obtain the local maximum error within each group. If the maximum displacement error in the group is less than the given threshold, skip the group; otherwise, add the preliminary node with the maximum displacement error in the group to the control point set and continue traversing the next group.

[0045] 5-5. Repeat steps 5-3 to 5-4 until the maximum displacement error in all groups after each interpolation is less than a given threshold.

[0046] The final set of control points is used to obtain the displacement values of the internal nodes of the deformed contour mesh relative to the original contour; for the mesh of the original contour, all its mesh nodes are offset according to the corresponding displacement values to obtain the mesh of the deformed contour.

[0047] In a second aspect, the present invention provides an aircraft airfoil optimization method, comprising the following steps:

[0048] Step 1: Obtain the initial contour of the aircraft airfoil section and generate a mesh based on the initial contour.

[0049] Step 2: According to the design requirements of the aircraft, the shape of the aircraft airfoil section is adjusted based on the initial contour to obtain the deformed contour.

[0050] Step 3: Generate a mesh of the deformed contour based on the initial contour mesh according to the aforementioned radial basis function mesh deformation method based on virtual nodes.

[0051] Step 4: Perform numerical simulation based on the mesh of the deformed profile to determine whether the deformed profile meets the design requirements as an aircraft airfoil cross-section. If so, the deformed profile is used as the optimized aircraft airfoil. If not, repeat steps 2 and 3 until a satisfactory aircraft airfoil is obtained.

[0052] The beneficial effects of the present invention are:

[0053] 1. This method generates corresponding virtual nodes for different boundary types by moving the normals of boundary points and interpolating the displacements of the virtual nodes using adjacent nodes. The virtual nodes and boundary nodes are then used as control nodes for radial basis functions to calculate the displacements of internal mesh nodes, thereby obtaining a deformed mesh. This method achieves mesh deformation for complex motions based on the NACA0012 model.

[0054] 2. The present invention is oriented to the needs of numerical simulations such as boundary deformation, and can quickly generate efficient grids according to different motion forms. It is also applicable to other complex two-dimensional models.

[0055] 3. The present invention provides a method for optimizing aircraft airfoils, which screens out airfoils that meet design requirements by continuously adjusting the airfoil cross-section and performing numerical simulation. Since the grids of different airfoil profiles are all obtained by deforming the initial profile, the time and computing power required for airfoil optimization can be greatly reduced, significantly improving the efficiency of aircraft airfoil optimization. BRIEF DESCRIPTION OF THE DRAWINGS

[0056] Figure 1 Schematic diagram of the mesh of the original outline in Example 1 of the present invention;

[0057] Figure 2 Schematic diagram of the deformed profile set in Example 1 of the present invention;

[0058] Figure 3 A schematic diagram of virtual nodes generated in the mesh of the original outline in step 2 of embodiment 1 of the present invention;

[0059] Figure 4 This is a schematic diagram of adding additional virtual nodes at the corner nodes of the convex hull in Example 1 of the present invention;

[0060] Figure 5 This is a schematic diagram of virtual nodes after the number is reduced in step 3 of embodiment 1 of the present invention;

[0061] Figure 6 Schematic diagram of a mesh of a deformed contour generated in Example 1 of the present invention;

[0062] Figure 7a for Figure 6 A partial enlarged view of part A;

[0063] Figure 7b for Figure 6 A partial enlarged view of part B;

[0064] Figure 7c for Figure 6 A partial enlarged view of part C. DETAILED DESCRIPTION

[0065] The present invention will be further described below with reference to the accompanying drawings.

[0066] Example 1

[0067] A radial basis function mesh deformation method based on virtual nodes includes the following steps:

[0068] Step 1: Generate the mesh of the original contour; deform the original contour according to the deformation configuration requirements to obtain the displacement of all boundary nodes of the deformed contour relative to the original contour; in this embodiment, the original contour is the cross-sectional contour of the NACA0012 wing. The mesh of the original contour is as follows: Figure 1 shown.

[0069] Step 2: Obtain the internal normal of the boundary node based on the original contour, and obtain the distribution of the internal virtual nodes based on the boundary features, as follows:

[0070] like Figure 2 As shown, for the mesh model of the original contour, internal virtual nodes are generated according to the moving boundary features, as shown in Figure 2 and Figure 3 As shown. All nodes on the moving boundary are divided into three types: smooth boundary nodes, convex hull sharp-cornered nodes, and concave hull sharp-cornered nodes. If the angle between the two adjacent edges of a node on the moving boundary is less than 30°, then the node is a smooth boundary node. If the angle between the two adjacent edges of a node on the moving boundary is greater than 30°, and the node is on a convex boundary, then the node is a convex hull sharp-cornered node. If the angle between the two adjacent edges of a node on the moving boundary is greater than 30°, and the node is on a concave boundary, then the node is a concave hull sharp-cornered node.

[0071] For all nodes p mb , along p in the computational domain mb Normal direction n pb Create a virtual node p n :

[0072] p n =p mb +λn pb (1)

[0073] Where λ is the distance between the virtual node and the moving boundary. The step size λ for convex corner nodes is larger than that for smooth boundary nodes and concave corner nodes. The step size at convex corners is appropriately increased to ensure that the distribution of virtual nodes is consistent with the moving boundary.

[0074] In order to make the virtual nodes evenly distributed near the moving boundary, for each convex hull corner node, multiple virtual nodes are generated on both sides of it using linear interpolation, such as Figure 4 As shown. The virtual node p formed by interpolation n The expression of (t) is as follows:

[0075]

[0076] Where n i (λ=1,2) are the unit normal vectors of the two boundaries connected by the convex hull corner node at the convex hull corner node, ncv is the average unit normal vector of n1 and n2; θ is n cv and n i Note that the number of virtual nodes generated around the convex corner depends on the value of t. t = 1 / T, 2 / T, ..., 1; T is the number of virtual nodes inserted on one side of the convex hull corner node (i.e., the number of inserted nodes is 2T).

[0077] For the region where the concave hull corner nodes are located, the virtual nodes generated according to Equation (1) are too dense, and only some of them are selected as potential virtual nodes. To this end, the distance from each virtual node to the nearest node on the moving boundary is calculated. If the shortest distance calculated for a virtual node is λ, then the virtual node is retained; otherwise, it is discarded.

[0078] Step 3: Reduce the number of virtual nodes inside the grid to obtain N evenly distributed internal virtual nodes. Figure 4 As shown in , N is a number related to the problem size and is calculated based on a given ratio of the total potential virtual nodes. During the virtual node reduction process, a number of virtual nodes are first selected as the initial selected set, and then a new virtual node is added to the selected set each time until the number of virtual nodes in the selected set reaches N. To obtain an even distribution of virtual nodes around the moving boundary, the newly selected virtual node should have the maximum distance from each virtual node in the selected set.

[0079] The specific process of reducing the number of virtual nodes is as follows:

[0080] 3-1. From all virtual nodes obtained in step 1, select a number of virtual nodes less than N as the initial selected set. These nodes can be nodes around key features of the moving boundary, such as virtual nodes generated from convex or concave corners. Set the initial selected set S = {p0, p1, ..., p m}. m is the number of nodes in the initial selected set S.

[0081] 3-2. Define the distance between any virtual node and the selected set S as the distance between the node closest to the virtual node in the selected set S and the virtual node; traverse the virtual nodes that are not selected in the selected set S and select the virtual node p with the largest distance from the selected set S new ; Set the virtual node p new Add to the selected set S; virtual node p new The expression is as follows:

[0082]

[0083] Among them, ‖p i -p j‖ is p i and p j The distance between them.

[0084] 3-3. If the number of virtual nodes in the selected set S is less than N, go to step 3-2 and continue adding virtual nodes to the selected set S; otherwise, exit the loop and obtain the virtual node distribution after the number is reduced, such as Figure 5 shown.

[0085] Step 4: For each virtual node in the selected set S obtained in step 3, use the K-nearest neighbor algorithm to obtain the K boundary nodes adjacent to each virtual node; use each virtual node as an internal node in the radial basis function interpolation method to calculate the displacement of each virtual node. During the calculation process, the expression of the normalized distance ξ used by the radial basis function interpolation method is shown in formula (9):

[0086]

[0087] Here, γ is a scaling factor, and γ>1 makes the virtual node gain more influence from its K nearest boundary nodes in terms of displacement.

[0088] The specific process of radial basis function interpolation is as follows:

[0089] The interpolation using radial basis function is divided into two stages; the first stage is to calculate the coefficient α by displacement interpolation at the boundary nodes i ; The second stage is to interpolate the internal nodes to obtain their displacement.

[0090] The basic formula for radial basis function interpolation is:

[0091]

[0092] Where s(r) is the displacement obtained by interpolating the node; ‖rr i ‖ is the Euclidean distance between grid node r and the i-th radial basis function control node. n is the number of radial basis function control nodes. is the radial basis function, and its expression is shown in formula (5):

[0093]

[0094] Where ξ is the normalized distance and R is the influence radius of the radial basis function interpolation.

[0095] In the first stage of the radial basis function interpolation method, the displacement of each boundary node is interpolated using the displacement of the boundary node to solve the coefficient vector α i By setting the boundary mesh nodes as control points and setting the actual displacement of each boundary control node as the function value, a linear system is constructed as shown in Equation (6):

[0096] S=Φα (6)

[0097] Where S={s1,s2,…,s n} T is the actual displacement of the boundary control point, α={α1,α2,…,α n} is the coefficient matrix of the radial basis function. Φ represents the radial basis function relationship between the control points, and its expression is shown in Equation (7).

[0098]

[0099] In the second stage of the radial basis function interpolation method, when the coefficient matrix α of the radial basis function is ready, the internal point displacement s(r in ) is shown in formula (8):

[0100]

[0101] Among them, r in are the coordinates of the internal nodes.

[0102] Step 5: Use the virtual nodes and the original boundary nodes to construct the boundary linear equilibrium equation using the radial basis function interpolation method to obtain the coefficient matrix, thereby interpolating the displacement of the internal nodes in the original grid.

[0103] To achieve efficient solution, the radial basis function interpolation method selects as few control points as possible. This embodiment performs a greedy data simplification algorithm on boundary nodes and virtual nodes. The specific process is as follows:

[0104] 5-6. Disrupt the order of all boundary nodes and virtual nodes to form a set of reserve nodes; randomly divide each reserve node in the set into several groups

[0105] 5-7. Randomly select three preliminary nodes as the initial control point set

[0106] 5-8. Use radial basis function interpolation to calculate the estimated displacement of each preliminary node that is not in the control point set; and calculate the displacement error between the estimated displacement and the actual displacement.

[0107] In this step, the expression of the normalized distance ξ used by the radial basis function interpolation method is shown in formula (10):

[0108]

[0109] 5-9. Obtain the local maximum error within each group. If the maximum displacement error in the group is less than the given threshold, skip the group; otherwise, add the preliminary node with the maximum displacement error in the group to the control point set and continue traversing the next group.

[0110] 5-10. Repeat steps 5-3 to 5-4 until the maximum displacement error in all groups after each interpolation is less than a given threshold.

[0111] 5-11. Use the final set of control points to obtain the displacement values of the internal nodes of the deformed contour relative to the original contour; for the original contour mesh, all its mesh nodes are offset according to the corresponding displacement values to obtain the deformed contour mesh. Figure 6 、 7a , 7b and 7c.

[0112] Example 2

[0113] An aircraft airfoil optimization method comprises the following steps:

[0114] Step 1: Obtain the initial profile of the aircraft airfoil section, and generate a mesh based on the initial profile using a frontier advancing method; in this embodiment, the initial profile is the cross-sectional profile of a NACA0012 wing.

[0115] Step 2: According to the design requirements of the aircraft, the shape of the aircraft airfoil section is adjusted based on the initial contour to obtain the deformed contour.

[0116] Step 3: According to the radial basis function mesh deformation method based on virtual nodes provided in Example 1, a mesh of the deformed contour is generated based on the initial contour mesh.

[0117] Step 4: Perform numerical simulation based on the mesh of the deformed profile to determine whether the deformed profile meets the design requirements as an aircraft airfoil cross-section. If so, the deformed profile is used as the optimized aircraft airfoil. If not, repeat steps 2 and 3 until a satisfactory aircraft airfoil is obtained.

Claims

1. A radial basis function mesh deformation method based on virtual nodes, characterized by: The following steps are involved: Step 1: Obtain the displacement of all boundary nodes of the deformed contour relative to the original contour; the original contour has been meshed; the original contour is the cross-sectional contour of the wing; Step 2: Divide all nodes on the original contour into three types: smooth boundary nodes, convex hull corner nodes, and concave hull corner nodes; For each convex hull corner node, multiple virtual nodes are additionally generated on both sides of it; multiple virtual nodes p are additionally generated on both sides of the convex hull corner node n The expression of (t) is as follows: Where n i (i=1,2) are the unit normal vectors of the two boundaries connected by the convex hull corner node at the convex hull corner node, n cv is the average unit normal vector of n1 and n2; θ is n cv and n i The angle between the two nodes; t = 1 / T, 2 / T, ..., 1; T is the number of virtual nodes inserted on one side of the convex hull corner node; For all nodes p on all original contours mb , along p in the computational domain mb Normal direction n pb Create a virtual node p n as follows: p n =p mb +λn pb Where λ is the step size; Step 3: Reduce the number of virtual nodes created in step 2; Step 4: For each virtual node in the selected set S obtained in step 3, obtain the K boundary nodes adjacent to each virtual node using the K-nearest neighbor algorithm; use each virtual node as an internal node in the radial basis function interpolation method to calculate the displacement of each virtual node; Step 5. Use the virtual nodes and the boundary nodes of the original contour to construct the boundary linear equilibrium equation using the radial basis function interpolation method to obtain the coefficient matrix. Interpolation is used to obtain the displacement of the internal nodes of the deformed contour mesh relative to the internal nodes in the original contour mesh. Based on the original contour mesh, all its mesh nodes are offset according to the corresponding displacement values to obtain the mesh of the deformed contour.

2. The radial basis function mesh deformation method based on virtual nodes according to claim 1, characterized in that: In step 2, if the angle between the two adjacent edges of a node on the moving boundary is less than 30°, then the node is a smooth boundary node; if the angle between the two adjacent edges of a node on the moving boundary is greater than 30° and the node is on a convex boundary, then the node is a convex hull sharp-cornered node; if the angle between the two adjacent edges of a node on the moving boundary is greater than 30° and the node is on a concave boundary, then the node is a concave hull sharp-cornered node.

3. The radial basis function mesh deformation method based on virtual nodes according to claim 1, characterized in that: In step 2, the step size λ corresponding to the convex hull sharp corner nodes is larger than the step size λ corresponding to the smooth boundary nodes and the concave hull sharp corner nodes.

4. The radial basis function mesh deformation method based on virtual nodes according to claim 1, characterized in that: In step 2, the virtual nodes generated for the concave hull sharp corner nodes are deleted; the deletion method is: calculate the distance from the virtual node to the nearest node on the moving boundary; if the shortest distance calculated for a virtual node is λ, then the virtual node is retained, otherwise the virtual node is discarded.

5. The radial basis function mesh deformation method based on virtual nodes according to claim 1, characterized in that: In step 3, the specific process of reducing the number of virtual nodes is as follows: 3-1. From all virtual nodes obtained in step 1, select a number of virtual nodes less than N as the initial selected set; N is the preset number of nodes after reduction; set the initial selected set S = {p0, p1, ..., p m }; m is the number of nodes in the selected initial set S; 3-2. Define the distance between any virtual node and the selected set S as the distance between the node closest to the virtual node in the selected set S and the virtual node; traverse the virtual nodes that are not selected in the selected set S and select the virtual node p with the largest distance from the selected set S new ; Set the virtual node p new Add to the selected set S; 3-3. If the number of virtual nodes in the selected set S is less than N, go to step 3-2 and continue adding virtual nodes to the selected set S; otherwise, the reduction ends.

6. The radial basis function mesh deformation method based on virtual nodes according to claim 1, characterized in that: The specific process of the radial basis function interpolation method is as follows: In the first stage, the displacement of each boundary node is interpolated using the displacement of the boundary nodes to solve the coefficient vector α i By setting the boundary mesh nodes as control points and the actual displacement of each boundary control node as the function value, the linear system is constructed as follows: S=Φα Where S={s1,s2,…,s n } T is the actual displacement of the boundary control point, α={α1,α2,…,α n } is the coefficient matrix of the radial basis function; Φ is the radial basis function relationship between the control points, and its expression is as follows: In the second stage, when the coefficient matrix α of the radial basis function is ready, the internal point displacement s(r in )as follows: Among them, ||r in -r i || is the internal node r in The normalized Euclidean distance to the i-th radial basis function control node; is the radial basis function, Radial Basis Function The expression is as follows: Where ξ is the normalized distance.

7. The radial basis function mesh deformation method based on virtual nodes according to claim 6, characterized in that: In step 4, during the calculation process, the expression of the normalized distance ξ used by the radial basis function interpolation method is as follows: Where γ is the scaling factor, γ>1; R is the influence radius of the radial basis function interpolation.

8. The radial basis function mesh deformation method based on virtual nodes according to claim 6, characterized in that: The specific process of step 5 is as follows: 5-1. Disrupt the order of all boundary nodes and virtual nodes to form a set of reserve nodes; randomly divide each reserve node in the set into several groups 5-2. Randomly select several preliminary nodes as the initial control point set 5-3. Use radial basis function interpolation to calculate the estimated displacement of each preliminary node that is not in the control point set; and calculate the displacement error between the estimated displacement and the actual displacement; 5-4. Obtain the local maximum error within each group; if the maximum displacement error in the group is less than a given threshold, skip the group; otherwise, add the preliminary node with the maximum displacement error in the group to the control point set and continue traversing the next group; 5-5. Repeat steps 5-3 to 5-4 until the maximum displacement error in all groups after each interpolation is less than a given threshold; The final set of control points is used to obtain the displacement values of the internal nodes of the deformed contour mesh relative to the original contour; for the mesh of the original contour, all its mesh nodes are offset according to the corresponding displacement values to obtain the mesh of the deformed contour.

9. A method for designing an aircraft airfoil, characterized by: The following steps are involved: Step 1: Obtain the initial profile of the aircraft airfoil section and generate a mesh based on the initial profile; Step 2: According to the design requirements of the aircraft, the shape of the aircraft airfoil section is adjusted based on the initial outline to obtain the deformed outline; Step 3: generating a mesh of the deformed contour based on the initial contour mesh according to the radial basis function mesh deformation method based on virtual nodes as claimed in claim 1; Step 4: Perform numerical simulation based on the mesh of the deformed profile to determine whether the deformed profile meets the design requirements as the aircraft airfoil cross-section; if so, use the deformed profile as the optimized aircraft airfoil; If the design requirements are not met, repeat steps 2 and 3 until a suitable aircraft airfoil is obtained.

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