An improved radial basis function mesh deformation method based on modal space and greedy algorithm

Abstract

The present invention proposes an improved radial basis function grid deformation method based on modal space and greedy algorithm, and uses the greedy algorithm to solve the support point set M of each order modal vibration function. i , and taking their union, a set M of support points that meets all modal requirements is found. The support points in this set meet the physical displacement reconstruction requirements at all times, and these points can be used to update the positions of all grid points within the fluid computational domain. The support point set determined according to the present invention meets the physical displacement reconstruction requirements at all time steps, eliminating the need to repeatedly screen the support point set and calculate the RBF interpolation matrix at each time step. This significantly improves the efficiency of solving RBF mesh deformations and reduces the computational complexity of mesh deformations, thereby improving the efficiency of CFD / CSD coupled time-domain calculations using RBFs for mesh deformations.

Description

Technical Field

[0001] The present invention relates to the field of grid deformation calculation, and in particular to an improved radial basis function grid deformation method based on modal space and greedy algorithm. Background Art

[0002] During the CFD / CSD coupled solution process, structural deformation alters the physical boundary conditions of the CFD calculation, requiring the computational mesh of the flow field to be updated at every physical time step. Therefore, in addition to the CFD and CSD solvers, mesh deformation is also a key technology in CFD / CSD coupled calculations, significantly impacting their accuracy and efficiency. In recent years, mesh deformation methods based on radial basis functions have attracted widespread attention.

[0003] A. de Boer et al. from the Department of Aerospace Engineering at Delft University of Technology in the Netherlands first proposed the use of RBF interpolation for CFD mesh deformation in "Mesh deformation based on radial basis function interpolation" (DOI: 10.1016 / j.compstruc.2007.01.013). Lin Yanzhong et al. from the School of Astronautics at Beihang University compared the RBF method with the spring analogy method in "Application of Radial Basis Function Interpolation in Dynamic Meshing Technology" (DOI: 10.19596 / j.cnki.1001-246x.2012.02.005), using the flexible deformation of the F104 three-dimensional rhombus wing as an example. The results showed that the spring analogy method exhibited negative volume at slightly larger deformations, and mesh distortion became increasingly severe as the deformation amplitude increased. The RBF method, however, maintained very high mesh quality even at the highest deformation amplitudes. However, the traditional RBF method requires all surface grid points as the RBF support point set, which is computationally intensive and memory intensive, resulting in low computational efficiency. Therefore, many researchers have devoted themselves to optimizing the RBF method.

[0004] In their paper "Efficient mesh motion using radial basis functions with data reduction algorithms" (DOI: 10.1016 / j.jcp.2009.05.013), TCS Rendall and CBA Allen of the University of Bristol's Department of Aerospace Engineering introduced a greedy algorithm, effectively reducing the number of support points while maintaining the accuracy of dynamic mesh calculations, thereby improving computational efficiency. In their paper "A technique for unstructured hybrid mesh deformation based on radial basis functions," Wang Gang et al. of Northwestern Polytechnical University optimized the RBF method using a greedy algorithm, introducing the concept of function space subsets. This method selects multiple surface nodes each time and adds them to the set of support points, significantly speeding up the computation. However, this type of optimization The method still needs to repeatedly screen the support points at each moment However, for CFD / CSD coupling calculations with a large number of surface nodes, the amount of computation is still too large when repeatedly performing support point screening and radial basis function interpolation matrix solution operations at each time step. Summary of the Invention

[0005] Technical issues to be solved:

[0006] During coupled CFD / CSD calculations, the CFD grid requires updating grid point positions at every moment. Existing greedy RBF interpolation grid deformation methods require selecting a set of support points at each time step and then using radial basis function interpolation to solve for the positions of all nodes within the fluid computational domain. However, the support point selection and radial basis function grid interpolation matrix inversion process at each time step consumes a long computational time.

[0007] In order to avoid repeated support point screening operations at each time step and improve the computational efficiency of RBF mesh deformation, the present invention creatively optimizes the radial basis function mesh interpolation method and the RBF interpolation mesh deformation method based on the greedy algorithm. By introducing the modal space, an improved radial basis function mesh deformation method based on the modal space and the greedy algorithm is proposed. Before the CFD / CSD coupling calculation, the greedy algorithm and the modal vibration shape function are used to find the radial basis function support point set that meets all time steps. There is no need to repeatedly use the greedy algorithm to find the support point set and calculate the support point radial basis interpolation matrix at each time step. This solves the problem of excessive amount of mesh deformation calculation using the radial basis function interpolation method and improves the computational efficiency.

[0008] The present invention is based on the principle that, according to the modal expansion theorem of structural dynamic response, the physical displacement of the structure can be represented by a combination of the products of modal vibration mode functions of each order and the modal displacement. Therefore, the set of support points that meets the accuracy requirements of the radial basis function interpolation reconstruction of the modal vibration mode is the same as the set of support points that meets the requirements of the physical displacement reconstruction at all times.

[0009] Assume that there are N nodes on the surface, and the value of the i-th mode vibration function of each node is recorded as First, randomly select K (usually the initial value is K = 3) object surface nodes to form a node set Using Collections Construct the RBF interpolation matrix using the node coordinates and the i-th order mode vibration function, and solve The weight coefficient of the midpoint about the i-th order mode. Then, according to the interpolation relationship, the i-th order mode vibration function of all the object surface nodes is solved, and the actual i-th order mode vibration function value of all the object surface nodes is obtained. Make a difference and find the maximum error of the i-th order mode vibration function of all object surface nodes and its corresponding object surface node Will Compare with Δ, where Δ is the set accuracy coefficient (for example, Δ is 0.0001). If it is greater than Δ, the object surface node corresponding to the maximum error will be added to the node set Forming a node collection Repeat the above process until the maximum error value of the modal vibration function of the node i is less than the set accuracy coefficient Δ, and finally obtain the support point set M that meets the accuracy requirements under the i-th order mode i .

[0010] In CFD / CSD coupling calculations, in order to ensure calculation accuracy, multiple modes are generally selected for response solution. Therefore, it is necessary to use a greedy algorithm to solve the support point set M of each mode vibration function. i , and finding their union, we ultimately find a set of support points M that meets all modal requirements. The support points in this set satisfy the physical displacement reconstruction requirements at all times, and we can then use these points to update the positions of all grid points within the fluid computational domain. Therefore, we don't need to repeatedly use the greedy algorithm to find the support point set and calculate the support point RBF interpolation matrix at each time step.

[0011] The technical solution of the present invention is:

[0012] The improved radial basis function grid deformation method based on modal space and greedy algorithm comprises the following steps:

[0013] Step 1: Randomly select K object surface nodes on the surface of the structure that needs CFD / CSD coupling calculation and combine them into an RBF interpolation support point set use Construct a radial basis function interpolation matrix for the i-th order modal vibration function of the wing structure and calculate the corresponding weight coefficient

[0014] Step 2: According to the weight coefficient of the i-th order modal vibration function obtained in step 1, interpolate the modal vibration function values of all the object surface nodes on the surface of the structure; interpolate the i-th order modal vibration function value and

[0015] The actual value of the i-th mode vibration function Make a difference and select the maximum absolute value of the error of the i-th order mode vibration function and its corresponding object surface node

[0016] If the absolute value of the maximum error is greater than the set accuracy coefficient Δ, add the node corresponding to the maximum error to the node set in step 1, return to step 1, and enter the k+1th iteration. The node set Contains K+1 nodes; on the contrary, it means that the absolute value of the maximum error meets the accuracy requirements, and the set of support points that meet the interpolation accuracy of the i-th order mode is obtained, which is recorded as M i , terminate the corresponding loop of order i;

[0017] Step 3: Repeat steps 1 and 2 to solve the modal vibration function of each order of the structure and obtain the support point set M corresponding to the modal vibration function of each order that meets the accuracy requirements. i ; For all sets M i Find the union and finally get the set of support points that satisfy all the modal vibration functions;

[0018] Step 4: Use the support point displacements in the support point set that satisfies all order modal vibration functions obtained in step 3 to reconstruct the displacements of all object surface nodes, and realize RBF interpolation mesh deformation based on the greedy algorithm.

[0019] In addition, the present invention also provides a computer-readable storage medium storing computer-executable instructions, which are used to implement the above method when executed.

[0020] And a computer system comprising: one or more processors, and the above-mentioned computer-readable storage medium, which is used to store one or more programs, and when the one or more programs are executed by the one or more processors, the one or more processors implement the above-mentioned method.

[0021] Beneficial effects

[0022] The support point set determined according to the present invention meets the physical displacement reconstruction requirements at all time steps. There is no need to repeatedly screen the support point set and calculate the RBF interpolation matrix at each time step. This greatly improves the efficiency of solving RBF mesh deformation and reduces the amount of mesh deformation calculation, thereby improving the efficiency of CFD / CSD coupled time-domain calculations using RBF for mesh deformation.

[0023] Additional aspects and advantages of the present invention will be set forth in part in the description which follows and, in part, will be obvious from the description which follows, or may be learned by practice of the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0024] The above and / or additional aspects and advantages of the present invention will become apparent and readily understood from the following description of the embodiments with reference to the accompanying drawings, in which:

[0025] Figure 1 Schematic diagram of the calculation process of one time step of CFD / CSD coupling calculation in the present invention;

[0026] Figure 2 This is a flowchart of the existing RBF interpolation grid deformation method based on the greedy algorithm;

[0027] Figure 3 1 is a schematic diagram of the support point set screening process based on the modal vibration shape function of the present invention;

[0028] Figure 4 This is a schematic diagram of the planform and geometric parameters of the AGARD445.6 wing model;

[0029] Figure 5 Schematic diagram of the first four natural vibration modes of the AGARD445.6 wing structure; (a) first-order mode (first-order bending), (b) second-order mode (first-order torsion), (c) third-order mode (second-order bending), (d) fourth-order mode (second-order torsion);

[0030] Figure 6 This figure compares the CFD response results obtained by performing mesh deformation calculations at different Mach numbers using the traditional RBF interpolation method (abbreviated as RBF in the figure), the existing RBF interpolation mesh deformation method based on the greedy algorithm (abbreviated as Greedy in the figure), and the improved RBF mesh deformation method based on modal space and greedy algorithm proposed in this invention (abbreviated as Present in the figure).

[0031] (a) Ma = 0.678, comparison of the calculation results of each method for each mode in the convergence state;

[0032] (b) Ma = 0.901, comparison of the calculation results of each method for each mode in the divergent state;

[0033] (c) Ma = 0.960, comparison of the calculation results of each method for each mode under periodic motion state;

[0034] It is worth noting that the image shows some local details to highlight the differences in the results of each method: the differences between the calculation results of the improved RBF mesh deformation method based on modal space and greedy algorithm proposed in this invention and the traditional RBF interpolation method and the existing RBF interpolation mesh deformation method based on greedy algorithm meet the set accuracy requirements of this invention, which shows that the method used in this invention does not affect the accuracy of the wing CFD / CSD coupling calculation. DETAILED DESCRIPTION

[0035] In CFD / CSD coupled calculations, for a specific three-dimensional structure, such as a wing structure, a CFD calculation is first performed for one time step at the initial structural state to obtain the aerodynamic forces on the structural surface. Then, the structural displacement response in the next time step is calculated through CSD to update the flow field mesh. Figure 1 As shown by t i Time to t i+1 The CFD / CSD coupling calculation steps at time t are repeated to obtain the CFD / CSD coupling response over time. At each time step, the grid node coordinates within the fluid computational domain need to be continuously updated to complete the coupled calculation between the fluid and solid domains.

[0036] The implementation process of the existing RBF interpolation grid deformation method based on the greedy algorithm is as follows:

[0037] Assume that the total number of mesh nodes in the fluid computational domain is A, and the number of mesh nodes on the surface of the wing structure being measured is N. Randomly select K (generally K = 3) of all the surface mesh nodes (N) to form a node set M0. Use this node set to construct the radial basis interpolation function:

[0038]

[0039] Where: F r is the displacement interpolation of the node in M0, w i is the displacement weight coefficient of node i in M0, φ(r) is the radial basis function interpolation matrix, and the specific expression is shown in formula (2):

[0040]

[0041] Where: r i is the position vector of node i in M0, ||r j -r i || is the position vector distance between nodes i and j in M0, φ is the "Wendland's C 2"Function, its specific expression is shown in formula (3):

[0042]

[0043] Where η = ||rr i || / d represents the spatial coordinate relationship between each node, and d is the radius of action of the radial basis function. When η>1, it is mandatory to set This limits mesh deformation to a radius around the support point.

[0044] After establishing the K-order radial basis function interpolation matrix, invert it to find the displacement weight coefficient corresponding to each node in M0 Interpolate all surface nodes with the nodes in the node set M0, and multiply the interpolation matrix with the displacement weight coefficient to obtain the displacement of all surface nodes The actual surface node displacement of the wing structure under test is used and Perform the subtraction to find the maximum error Δ0 and its corresponding object surface node p0. Compare the maximum error Δ0 with Δ, where Δ is the set accuracy coefficient (for example, Δ is 0.0001). If Δ0 is greater than Δ, add the object surface node p0 corresponding to the maximum error to set M0, resulting in node set M1. If Δ0 is less than Δ, then set M0 is the desired support point set.

[0045] Repeat the above operation until the maximum error value of the node displacement is less than the set accuracy coefficient Δ, and finally obtain the support point set M that meets the accuracy requirements under the calculation time step. The interpolation function constructed by the nodes in M can be used to fit the spatial node displacement that meets the accuracy requirements. Figure 2 The figure shows the process of the k+1th cycle. The RBF interpolation mesh deformation method based on the greedy algorithm reconstructs the displacement of all object surface nodes by displacing a small number of support points, which reduces the number of support points and the computational complexity of the dynamic mesh deformation process. However, this method needs to repeatedly use the greedy algorithm to screen the support points at each time step. The computational complexity is still large .

[0046] To solve the above problems, the present invention creatively proposes to avoid repeating the support point screening operation at each time step, that is, to find the radial basis function support point set that meets all time steps before the CFD / CSD coupling calculation, and there is no need to repeatedly use the greedy algorithm to find the support point set and calculate the support point radial basis interpolation matrix at each time step.

[0047] To this end, the present invention proposes an improved radial basis function mesh deformation method based on modal space and greedy algorithm. According to the modal expansion theorem of structural dynamic response, physical displacement can be represented by the product combination of each order modal vibration function and modal displacement. Therefore, to meet the physical displacement interpolation accuracy requirements, it is only necessary to ensure that each order modal vibration shape meets the interpolation accuracy requirements. Therefore, the problem of finding a set of physical displacement support points can be transformed into finding a set of RBF interpolation support points for each order modal vibration shape. The specific steps are:

[0048] Step 1: Select the object surface nodes and construct the radial basis function interpolation relationship:

[0049] In the wing structure surface for CFD / CSD coupling calculation, K (the initial value is generally taken as 3) surface nodes are randomly selected and combined into a set of RBF interpolation support points. use Construct a radial basis function interpolation matrix for the i-th order modal vibration function of the wing structure and calculate the corresponding weight coefficient

[0050] Step 2: Solve the modal vibration function and select the node with the maximum error:

[0051] According to the weight coefficient of the i-th order modal vibration function obtained in step 1, the modal vibration function values of all the object surface nodes on the wing surface are interpolated. The i-th order modal vibration function value obtained by interpolation is and the actual value of the i-th order mode vibration function Make a difference and select the maximum absolute value of the error of the i-th order mode vibration function and its corresponding object surface node

[0052] If the absolute value of the maximum error is greater than the set accuracy coefficient Δ, add the node corresponding to the maximum error to the node set in step 1, return to step 1, and enter the k+1th iteration. The node set Contains K+1 nodes; on the contrary, it means that the absolute value of the maximum error meets the accuracy requirements, and the set of support points that meet the interpolation accuracy of the i-th order mode is obtained, which is recorded as M i , terminating the loop.

[0053] Step 3: Repeat each mode to construct a set of support points:

[0054] Repeat steps 1 and 2 to solve the modal vibration function of each order (the total modal order is I) of the wing structure in turn, and obtain the support point set M corresponding to the modal vibration function of each order that meets the accuracy requirements. i For all sets M i Find the union and finally get the set of support points that satisfies all modal vibration functions.

[0055] This set of support points meets the physical displacement reconstruction requirements of the wing under test at all time steps of the CFD / CSD coupled solution, and can be used as support points for the RBF mesh deformation calculation at each time step, avoiding repeated support point screening and radial basis function interpolation matrix solution at each time step.

[0056] The following describes in detail embodiments of the present invention. The embodiments are exemplary and intended to explain the present invention, but are not to be construed as limiting the present invention.

[0057] Take the transonic flutter standard example - AGARD445.6 wing model as an example. The model uses NACA 65A004 symmetrical airfoil. The wing planform and geometric parameters are as follows: Figure 4 As shown, the first four natural vibration modes of the wing structure are as follows Figure 5 Before applying the method of the present invention to perform CFD / CSD coupling calculations, 263 support points (accounting for only 9.8% of the total number of surface nodes) were selected from all 2788 surface nodes, and the support point set was used to perform mesh deformation calculations based on RBF interpolation. Figure 6 As shown, a comparison of the calculation results of the improved radial basis function mesh deformation method based on modal space and greedy algorithm proposed in this invention, the traditional RBF interpolation method, and the existing RBF interpolation mesh deformation method based on greedy algorithm under different Mach numbers and convergence and divergence characteristics shows that the CFD response curves calculated by the three methods under different conditions are almost identical (with an error of less than 0.0001). The comparison of the calculation time of each method under different conditions shown in Table 1 proves that the present invention can greatly reduce the calculation time while ensuring accuracy, significantly improving calculation efficiency.

[0058] Table 1 shows the statistics of the time taken for mesh motion calculation at different Mach numbers using the traditional RBF interpolation method (abbreviated as RBF in the figure), the existing RBF interpolation mesh deformation method based on the greedy algorithm (abbreviated as Greedy in the figure), and the improved RBF mesh deformation method based on modal space and greedy algorithm proposed in this invention (abbreviated as Present in the figure).

[0059] Table 1

[0060]

[0061] The improved radial basis function mesh deformation method based on modal space and greedy algorithm proposed in the present invention uses the greedy algorithm and modal vibration shape function to find a set of support points that meets the physical displacement reconstruction requirements at all time steps before the CFD / CSD coupling calculation. There is no need to repeatedly use the greedy algorithm to find the support point set and calculate the radial basis interpolation matrix of the support points at each time step, which reduces the calculation amount of mesh deformation.

[0062] Although the embodiments of the present invention have been shown and described above, it will be understood that the above embodiments are illustrative and are not to be construed as limitations on the present invention. A person skilled in the art may change, modify, replace and modify the above embodiments within the scope of the present invention without departing from the principles and purpose of the present invention.

Claims

1. An improved radial basis function mesh deformation method based on modal space and greedy algorithm, characterized by: The following steps are involved: Step 1: Randomly select K object surface nodes on the surface of the structure that needs CFD / CSD coupling calculation and combine them into an RBF interpolation support point set use Construct a radial basis function interpolation matrix for the i-th order modal vibration function of the wing structure and calculate the corresponding weight coefficient Step 2: According to the weight coefficient of the i-th order modal vibration function obtained in step 1, interpolate the modal vibration function values of all the object surface nodes on the surface of the structure; interpolate the i-th order modal vibration function value and the actual value of the i-th order mode vibration function Make a difference and select the maximum absolute value of the error of the i-th order mode vibration function and its corresponding object surface node If the absolute value of the maximum error is greater than the set accuracy coefficient Δ, add the node corresponding to the maximum error to the node set in step 1, return to step 1, and enter the k+1th iteration. The node set Contains K+1 nodes; on the contrary, it means that the absolute value of the maximum error meets the accuracy requirements, and the set of support points that meet the interpolation accuracy of the i-th order mode is obtained, which is recorded as M i , terminate the corresponding loop of order i; Step 3: Repeat steps 1 and 2 to solve the modal vibration function of each order of the structure and obtain the support point set M corresponding to the modal vibration function of each order that meets the accuracy requirements. i ; For all sets M i Find the union and finally get the set of support points that satisfy all the modal vibration functions; Step 4: Use the support point displacements in the support point set that satisfies all order modal vibration functions obtained in step 3 to reconstruct the displacements of all object surface nodes, and realize RBF interpolation mesh deformation based on the greedy algorithm.

2. The improved radial basis function mesh deformation method based on modal space and greedy algorithm according to claim 1, characterized in that: In step 1, the value of K is 3.

3. The improved radial basis function mesh deformation method based on modal space and greedy algorithm according to claim 1, characterized in that: In step 2, the precision coefficient Δ is set to 0.0001.

4. A computer-readable storage medium storing computer-executable instructions, characterized in that: When the instructions are executed, they are used to implement the method according to any one of claims 1 to 3.

5. A computer system comprising: One or more processors, the computer-readable storage medium according to claim 4, is used to store one or more programs, characterized in that: when the one or more programs are executed by the one or more processors, the one or more processors implement the method according to any one of claims 1 to 3.

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