A method for optimizing mesh quality of multi-block structures of rotating machinery

Abstract

The present invention discloses a method for optimizing the quality of multi-block structural grids of rotating machinery, comprising the following steps: obtaining and reorganizing multi-block structural grid data of rotating machinery blades in the m′‑θ coordinate system; optimizing the internal grid nodes of the grid surface by using the elliptical partial differential control equation; optimizing the internal grid nodes of the internal grid edge by using the Laplace control equation; optimizing the internal endpoints by using the weighted Laplace optimization method; optimizing the periodic boundary points; optimizing the inlet / outlet boundary points; repeating until the iterative process converges or reaches a specified number of iteration steps, and outputting the optimized multi-block structural grid data of the rotating blades in the m′‑θ coordinate system. The present invention solves the technical difficulties in optimizing the quality of multi-block structural grids in the field of rotating machinery in China, accelerates the automatic generation of high-quality structural grids of various complex blade configurations of rotating machinery in China, and effectively promotes the improvement of the calculation accuracy in the numerical simulation of the whole aircraft engine in China.

Description

Technical Field

[0001] The present invention belongs to the technical field of numerical simulation of computational fluid dynamics, and specifically relates to a method for optimizing the quality of multi-block structure grids of rotating machinery based on elliptic partial differential equations in the "pre-processing" of numerical simulation of aero-engines. Background Art

[0002] The rotating machinery structural mesh generation method creates a three-dimensional mesh of a single rotating machinery blade channel by stacking multiple structural meshes with the same topological structure from blade to blade along the blade span direction. Rotating machinery blades are usually defined on multiple flow segments, which are located on the three-dimensional flow surface f in the cylindrical coordinate system. i (x, r, θ), where i is the flow segment index. The streamlines start with the same radius r(x) and flow position x and rotate to form a flow surface. Once the radius change along the flow position is known, the flow surface is uniquely defined.

[0003] The flow surface can be represented by two parameter coordinates m′, θ, where

[0004]

[0005] Where x0 is the reference position of the flow surface. If x0 is the flow position at the entrance of the blade passage, then m′∈[0,+∞), a flow surface can be defined using the m′-θ coordinate system as:

[0006]

[0007] According to the above relationship, the three-dimensional surface meshing problem can be simplified to the plane meshing problem in the parameter m′-θ coordinate system. First, the mesh is created in the m-θ coordinate, and then it is projected back to the three-dimensional flow surface to form a three-dimensional surface mesh with conformal characteristics. Finally, the three-dimensional surface mesh is stacked along the blade span direction to create a three-dimensional volume mesh of the blade channel.

[0008] The commonly used grid types in the field of numerical simulation technology of computational fluid dynamics of rotating machinery are block structured grid, unstructured grid, overlapping grid, hybrid grid and Cartesian grid. At present, the block structured grid generation method is the best method for meshing the blade channel of rotating machinery. By properly dividing the fluid flow area into grid topology, the grid contour roughly follows the blade contour, and a high-quality grid can be obtained. When using TFI (transfinite interpolation) to generate the initial grid in a single grid block, there may be folds in the generated grid, and the grid size transition may not be uniform at the interface of different blocks.

[0009] Thompson et al. first proposed the method based on partial differential equations. The mapping relationship between parameter space and physical space can be obtained by solving partial differential equations by numerical calculation. The grid generation method based on partial differential equations can be specifically divided into three categories: elliptical, parabolic and hyperbolic. The most commonly used are the Poisson equation and Laplace equation grid generation methods. The partial differential grid generation method is mainly used in the field of computational fluid dynamics to generate body-fitting grids. It can be used to remove the folds in the grids generated by the algebraic method and smooth the size transition between different block grids.

[0010] The accuracy of numerical simulation depends greatly on the quality of the mesh. For regions with simpler geometric shapes, semi-automatic mesh generation can be used to directly control the mesh quality while meshing. As the complexity of the analyzed problems increases, engineers tend to use automatic mesh generation tools to divide the mesh. However, the initial mesh obtained by automatic generation technology is usually of poor quality. Directly using the solver to solve the problem will make it difficult for the calculation results to converge. Therefore, mesh quality optimization is very important. At present, mesh optimization has become an important research topic in the field of mesh generation. Summary of the invention

[0011] The initial grid in the m′-θ coordinate system of the rotating blade is generated in each grid block by using TFI after topological construction. There are usually cells with poor quality and cannot meet the requirements of numerical simulation. Therefore, it is necessary to optimize the quality of the initial grid to remove the folds that may exist in the initial grid and improve its smoothness and orthogonality. The technical problem to be solved by the present invention is to provide a method for optimizing the quality of the grid of a multi-block structure of a rotating machinery, which can effectively improve the accuracy of numerical simulation by optimizing the grid quality.

[0012] In order to solve the above technical problems, the present invention is implemented in the following ways:

[0013] A method for optimizing mesh quality of a rotating machinery multi-block structure comprises the following steps:

[0014] S1, obtaining and reorganizing multiple structural grid data of rotating blades in the m-θ coordinate system;

[0015] S2, using elliptic partial differential governing equations to optimize the internal mesh nodes of the mesh surface;

[0016] S3, based on the optimization in step S2, optimizing the internal mesh nodes of the internal mesh edges by using the Laplace control equation;

[0017] S4, based on the optimization in step S3, optimizing the internal endpoints using a weighted Laplace optimization method;

[0018] S5. Based on the optimization in step S4, optimize the periodic boundary points;

[0019] S6. Based on the optimization in step S5, optimize the entry / exit boundary points;

[0020] S7, repeating steps S2 to S6 until the iterative process converges or reaches a specified number of iteration steps, and outputting multiple structural grid data of the optimized rotating blade in the m-θ coordinate system.

[0021] Furthermore, the step S1 specifically includes the following sub-steps:

[0022] S11, directly obtain the mesh surface nodes of all single mesh blocks in the m-θ coordinate system as the optimization data of the internal mesh nodes of the mesh surface;

[0023] S12, traversing all mesh edge nodes in the m′-θ coordinate system, obtaining internal mesh edge nodes and adjacent mesh nodes on both sides as optimization data of the internal mesh edge;

[0024] S13, traversing all grid line endpoints in the m′-θ coordinate system, obtaining an internal endpoint and a grid node adjacent to the internal endpoint as optimization data of the internal endpoint;

[0025] S14. The acquisition of periodic boundary point optimization data is divided into two cases: periodic boundary point mismatch and periodic boundary point match. If the periodic boundary point does not match, the periodic boundary point in the m′-θ coordinate system and the grid nodes on one side adjacent to the periodic boundary point are searched as the optimization data of the periodic boundary point; if the periodic boundary point matches, the upper periodic boundary point, the grid nodes adjacent to the upper periodic boundary point and the grid nodes adjacent to the lower periodic boundary point are obtained as the optimization data of the upper periodic boundary point;

[0026] S15. Find the inlet / outlet boundary point and the grid nodes on the adjacent side thereof in the m′-θ coordinate system as the optimization data of the inlet / outlet boundary point.

[0027] Furthermore, the step S2 specifically includes the following sub-steps:

[0028] S21. Given the Dirichlet boundary conditions of the elliptic partial differential control equation as the mesh surface boundary mesh nodes of a single mesh block in the m′-θ coordinate system, the Poisson equation is used to transform the coordinates from the physical domain to the parameter domain. The specific expression is as follows:

[0029]

[0030] Among them, ξ xx represents the second-order partial derivative of ξ with respect to x, ξ yy represents the second-order partial derivative of ξ with respect to y, η xxrepresents the second-order partial derivative of η with respect to x, η yy represents the second-order partial derivative of η with respect to y, P(ξ,η) and Q(ξ,η) represent the source terms controlling the grid spacing and orthogonality;

[0031] definition The control equation for mesh optimization of elliptic partial differential equations can be obtained. The specific expression is as follows:

[0032]

[0033] in, express The second-order partial derivative with respect to ξ is express The second-order partial derivative with respect to η is represents the second-order mixed partial derivative, Jacobian of coordinate transformation J = x ξ y η -x η y ξ , express The first-order partial derivative with respect to η is express The first-order partial derivative with respect to ξ, x ξ represents the first-order partial derivative of x with respect to ξ, x η represents the first-order partial derivative of x with respect to η, y η represents the first-order partial derivative of y with respect to η, y ξ represents the first-order partial derivative of y with respect to ξ;

[0034] S22, use the Sorenson source term method to calculate the source term of the control equation (2), and then use Substituting P and Q in the formula respectively, the equivalent equation expression is as follows:

[0035]

[0036] If there is orthogonality on the boundary and the second-order derivative across the boundary is 0, we get the source term The expression of ψ is as follows:

[0037]

[0038]

[0039] S23, using the point-by-point super-relaxation method to solve the formula (3) in step S22.

[0040] Furthermore, the step S3 specifically includes the following sub-steps:

[0041] S31. Given the Dirichlet boundary conditions of the Laplace governing equation for the mesh nodes on both sides of the internal mesh edge, the Laplace governing equation for the internal mesh nodes of the internal mesh edge is optimized, and its expression is as follows:

[0042]

[0043] S32, using a point-by-point super-relaxation method to solve formula (6) in step S31.

[0044] Furthermore, the specific method of step S4 is as follows:

[0045] The weighted Laplace optimization method is used to optimize the internal endpoints. The optimized internal endpoint position is set to E and the adjacent grid node position is set to E. i (i=1,2,…n), then the expression for optimizing the internal endpoint position is as follows:

[0046]

[0047] in, Represents the internal endpoint E to the adjacent grid node E i distance.

[0048] Furthermore, the step S5 specifically includes the following sub-steps:

[0049] S51. If the periodic boundary points do not match, the Neumann boundary orthogonal control equation is optimized for the periodic boundary points. The specific expression is as follows:

[0050] (X η ) i，j ·(X i+1，j -X i，j )=0 (8)

[0051] Among them, (X η ) i，j represents the first-order derivative of a grid node G at the periodic boundary in the η direction, (X i+1，j -X i，j ) represents the first-order derivative of the grid node G in the ξ direction. Formula (8) is discretized into a difference equation using the forward difference format and solved to obtain the optimized position of the periodic boundary grid node;

[0052] S52. If the periodic boundary points match, the periodic boundary points are optimized as the internal grid nodes of the internal grid edges in step S3. The Dirichlet boundary conditions of the given Laplace control equations are the grid nodes adjacent to the upper periodic boundary points in step S14, and the θ coordinates in the grid nodes adjacent to the lower periodic boundary points are translated in the positive direction of θ by a periodic grid node. The control equation (6) is discretized into a difference equation using the central difference format and solved using the point-by-point super-relaxation method to obtain the optimized position coordinates of the upper periodic boundary points; the optimized position coordinates of the upper periodic boundary points are assigned to the lower periodic boundary points, and the θ coordinates of the lower periodic boundary points are translated in the negative direction of θ by a periodicity as the optimized position coordinates of the lower periodic boundary points.

[0053] Furthermore, the step S6 specifically includes the following sub-steps:

[0054] S61, using Neumann boundary orthogonal control equation formula (8) to optimize the inlet / outlet boundary points, and obtain the optimized positions of the inlet / outlet boundary points except the first and last grid nodes;

[0055] S62. Calculate the positions of the first and last grid nodes at the inlet / outlet boundary that meet the orthogonality condition according to the first and last grid nodes at the inlet / outlet boundary.

[0056] Compared with the prior art, the present invention has the following beneficial effects:

[0057] The present invention discloses a multi-block structure mesh quality optimization method based on elliptic partial differential equations: using elliptic partial differential control equations to optimize the internal mesh nodes of a single mesh surface; using Laplace control equations to optimize the internal mesh nodes of internal mesh edges; using weighted Laplace optimization method to optimize internal endpoints; optimizing periodic boundary points; optimizing inlet / outlet boundary points; iteratively executing the above steps to optimize the mesh nodes in the m-θ coordinate system until convergence or reaching a specified number of optimization steps to obtain the best mesh quality. The technology disclosed in the present invention solves the current technical difficulties in optimizing the quality of multi-block structure meshes in the field of domestic rotating machinery, accelerates the automated high-quality structural mesh generation of various complex blade configurations of domestic rotating machinery, and makes important contributions to the research and engineering application of multi-block structure mesh quality optimization algorithms for domestic rotating machinery. BRIEF DESCRIPTION OF THE DRAWINGS

[0058] Figure 1 It is a schematic diagram of the process of the present invention;

[0059] Figure 2 It is a schematic diagram of multi-block structure grid data in the initial rotating blade m-θ coordinate system;

[0060] Figure 3This is a schematic diagram of the multi-block structure grid data in the n-θ coordinate system of the reorganized rotating blade;

[0061] Figure 4 Schematic diagram of data acquisition optimization for periodic boundary point matching;

[0062] Figure 5 Schematic diagram of the initial structure grid in the m′-θ coordinate system with mismatched periodic boundary points of Rotor37;

[0063] Figure 6 The schematic diagram of the structure grid in the m′-θ coordinate system with mismatched periodic boundary points of the optimized Rotor37;

[0064] Figure 7 Schematic diagram of the initial structure grid in the m-θ coordinate system for matching the periodic boundary points of Rotor37;

[0065] Figure 8 Schematic diagram of the structural mesh in the m-θ coordinate system for matching the optimized Rotor37 periodic boundary points;

[0066] Fig. 9 Schematic diagram of the initial structure grid in the m′-θ coordinate system for matching the periodic boundary points of the multi-stage compressor;

[0067] Fig.10 Schematic diagram of the structural grid in the m-θ coordinate system for periodic boundary point matching of the optimized multi-stage compressor. DETAILED DESCRIPTION

[0068] The specific implementation of the present invention is further described in detail below with reference to the accompanying drawings and specific examples.

[0069] like Figure 1 As shown, a method for optimizing the mesh quality of a rotating machinery multi-block structure comprises the following steps:

[0070] S1, obtaining and reorganizing multiple structural grid data of rotating blades in the m-θ coordinate system, specifically including the following sub-steps:

[0071] like Figure 2 As shown in the figure, the initial multi-block structure grid data in the m-θ coordinate system of the rotating blade only has grid faces, grid edges and grid line endpoints, which are inconsistent with the data required for grid optimization. The reorganized data can facilitate the implementation of the subsequent optimization algorithm. When optimizing the multi-block structure grid in the m-θ coordinate system of the rotating blade, as shown in the figure, Figure 3 As shown, the nodes in the multi-block structured grid are divided into: internal grid nodes of the surface, internal grid nodes of the internal grid edge, internal endpoints, periodic boundary points and inlet / outlet boundary points.

[0072] S11, directly obtain the mesh surface nodes of all single mesh blocks in the m′-θ coordinate system as the optimization data of the internal mesh nodes of the mesh surface;

[0073] S12, traversing all mesh edge nodes in the m-θ coordinate system, obtaining internal mesh edge nodes and adjacent mesh nodes on both sides as optimization data of the internal mesh edge;

[0074] S13, traversing all grid line endpoints in the m-θ coordinate system, obtaining an internal endpoint and a grid node adjacent to the internal endpoint as optimization data of the internal endpoint;

[0075] S14. The acquisition of periodic boundary point optimization data is divided into two cases: periodic boundary point mismatch and periodic boundary point match. To solve the structural grid of periodic boundary point mismatch, CFD (Computational Fluid Dynamics) software is usually required to have high flexibility and functions to effectively handle this complex situation. Some CFD software may have limited processing capabilities for such situations. At this time, it is necessary to consider generating a structural grid with periodic boundary point matching. If the periodic boundary point does not match, the periodic boundary point in the m-θ coordinate system and the grid nodes on one side adjacent to the periodic boundary point are searched as the optimization data of the periodic boundary point; if the periodic boundary point matches, such as Figure 4 As shown, the upper cycle boundary point L2, the grid node L4 adjacent to the upper cycle boundary point, and the grid node L3 adjacent to the lower cycle boundary point are obtained as the optimization data of the upper cycle boundary point;

[0076] S15. Find the inlet / outlet boundary point and the grid nodes on the adjacent side thereof in the m′-θ coordinate system as the optimization data of the inlet / outlet boundary point.

[0077] S2. Optimizing the internal mesh nodes of the mesh surface using the elliptic partial differential control equation, specifically including the following sub-steps:

[0078] S21. Given the Dirichlet boundary conditions of the elliptic partial differential control equation as the mesh surface boundary mesh nodes of a single mesh block in the m-θ coordinate system, the Poisson equation is used to transform the coordinates from the physical domain (xy coordinate system) to the parameter domain (ξ-η coordinate system). The specific expression is as follows:

[0079]

[0080] Where ξ represents the function of x and y ξ = ξ(x, y), where x and y are equivalent to the m′ and θ coordinates in the m′-θ coordinate system, ξ xx represents the second-order partial derivative of ξ with respect to x, ξ yy represents the second-order partial derivative of ξ with respect to y, η represents the function of x and y η = η(x, y), η xxrepresents the second-order partial derivative of η with respect to x, η yy represents the second-order partial derivative of η with respect to y, P(ξ,η) and Q(ξ,η) represent the source terms that control the grid spacing and orthogonality, and similarly, x and y also represent functions of ξ and η, respectively;

[0081] definition The control equation for mesh optimization of elliptic partial differential equations can be obtained. The specific expression is as follows:

[0082]

[0083] in, express The second-order partial derivative with respect to ξ is express The second-order partial derivative with respect to η is represents the second-order mixed partial derivative, P and Q are source terms, Jacobian of coordinate transformation J = x ξ y η -x η y ξ , express The first-order partial derivative with respect to η is express The first-order partial derivative with respect to ξ, x ξ represents the first-order partial derivative of x with respect to ξ, x η represents the first-order partial derivative of x with respect to η, y η represents the first-order partial derivative of y with respect to η, y ξ represents the first-order partial derivative of y with respect to ξ;

[0084] S22, use the Sorenson source term method to calculate the source term of the control equation (2), and then use Substituting P and Q in the formula respectively, the equivalent equation expression is as follows:

[0085]

[0086] If there is orthogonality on the boundary and the second-order derivative across the boundary is 0, we get the source term The expression of ψ is as follows:

[0087]

[0088] in, Directly find the point on the boundary. By the orthogonality on the boundary And the first layer grid spacing away from the boundary is obtained by combining: The calculation involves the grid points of the internal field, and the internal field source terms are obtained by exponential interpolation from the source terms on the boundary;

[0089] S23, using the point-by-point super-relaxation method to solve the formula (3) in step S22.

[0090] S3, based on the optimization of step S2, the internal mesh nodes of the internal mesh edges of step S1 are optimized using the Laplace control equation, which specifically includes the following sub-steps:

[0091] S31. Given the Dirichlet boundary conditions of the Laplace governing equation for the mesh nodes on both sides of the internal mesh edge, the Laplace governing equation for the internal mesh nodes of the internal mesh edge is optimized, and its expression is as follows:

[0092]

[0093] S32, using a point-by-point super-relaxation method to solve formula (6) in step S31.

[0094] S4. Based on the optimization of step S3, the weighted Laplace optimization method is used to optimize the internal endpoints of step S1. The specific method is as follows:

[0095] The weighted Laplace optimization method is used to optimize the internal endpoints. The optimized internal endpoint position is set to E and the adjacent grid node position is set to E. i (i=1,2,…n), then the expression for optimizing the internal endpoint position is as follows:

[0096]

[0097] in, Represents the internal endpoint E to the adjacent grid node E i distance.

[0098] S5. Based on the optimization of step S4, the periodic boundary points in step S1 are optimized, which specifically includes the following sub-steps:

[0099] S51. If the periodic boundary points do not match, the Neumann boundary orthogonal control equation is optimized for the periodic boundary points. The specific expression is as follows:

[0100] (X η ) i，j ·(X i+1，j -X i，j )=0 (8)

[0101] Among them, (X η ) i，j represents the first-order derivative of a grid node G at the periodic boundary in the η direction, (X i+1，j -X i，j) represents the first-order derivative of the grid node G in the ξ direction. Formula (8) is discretized into a difference equation using the forward difference format and solved to obtain the optimized position of the periodic boundary grid node;

[0102] S52. If the periodic boundary points match, the periodic boundary points are optimized as the internal grid nodes of the internal grid edges in step S3. The Dirichlet boundary conditions of the given Laplace control equation are the grid nodes L4 adjacent to the upper periodic boundary point L2 in step S14, and the θ coordinates in the grid nodes L3 adjacent to the lower periodic boundary point are translated in the positive direction of θ by a periodic grid node L5. The control equation for optimizing the periodic boundary points is the Laplace control equation (6). The control equation (6) is discretized into a difference equation using the central difference format and solved using the point-by-point super-relaxation method to obtain the optimized position coordinates of the upper periodic boundary point; the optimized position coordinates of the upper periodic boundary point are assigned to the lower periodic boundary point L1, and the θ coordinates of the lower periodic boundary point are translated in the negative direction of θ by a periodicity as the optimized position coordinates of the lower periodic boundary point.

[0103] S6. Based on the optimization in step S5, the inlet / outlet boundary points are optimized, which specifically includes the following sub-steps:

[0104] S61, using Neumann boundary orthogonal control equation formula (8) to optimize the inlet / outlet boundary points, and obtain the optimized positions of the inlet / outlet boundary points except the first and last grid nodes;

[0105] S62. Calculate the positions of the first and last grid nodes at the inlet / outlet boundary that meet the orthogonality condition according to the first and last grid nodes at the inlet / outlet boundary.

[0106] S7, repeating steps S2 to S6 until the iterative process converges or reaches a specified number of iteration steps, and outputting multiple structural grid data of the optimized rotating blade in the m-θ coordinate system.

[0107] In this patent application, if Figure 5 The figure shows the initial structure grid diagram in the m-θ coordinate system where the Rotor37 periodic boundary points do not match. Figure 7 The figure shows the initial structure grid schematic diagram in the m′-θ coordinate system of the Rotor37 periodic boundary point matching. Fig. 9 The figure shows the schematic diagram of the initial structure grid in the m-θ coordinate system for matching the periodic boundary points of the multi-stage compressor. The three sets of initial structure grids in the m-θ coordinate system are optimized by using the technology described in the patent. The optimized effect diagrams are shown as follows: Figure 6 , Figure 8 , Fig.10As shown in the figure. After optimization, the orthogonality and smoothness of the structured grids under the above three sets of m-θ coordinates are greatly improved, and the grid line folding existing in the initial structured grid is removed, which greatly improves the quality of the structured grid under the m-θ coordinates of the rotating blade.

[0108] The above description is only an implementation mode of the present invention. It is stated again that for ordinary technicians in this technical field, several improvements can be made to the present invention without departing from the principle of the present invention. These improvements are also included in the protection scope of the claims of the present invention.

Claims

1. A method for optimizing the quality of a rotating machinery multi-block structure grid, characterized in that: The following steps are involved: S1, obtaining and reorganizing multiple structural grid data of the rotating blade in the m′-θ coordinate system; S2, using elliptic partial differential governing equations to optimize the internal mesh nodes of the mesh surface; S3, based on the optimization in step S2, optimizing the internal mesh nodes of the internal mesh edges by using the Laplace control equation; S4, based on the optimization in step S3, optimizing the internal endpoints using a weighted Laplace optimization method; S5. Based on the optimization in step S4, optimize the periodic boundary points; S6. Based on the optimization in step S5, optimize the entry / exit boundary points; S7, repeating steps S2 to S6 until the iterative process converges or reaches a specified number of iteration steps, and outputting multiple structural grid data of the optimized rotating blade in the m′-θ coordinate system; The step S1 specifically includes the following sub-steps: S11, directly obtain the mesh surface nodes of all single mesh blocks in the m′-θ coordinate system as the optimization data of the internal mesh nodes of the mesh surface; S12, traversing all mesh edge nodes in the m′-θ coordinate system, obtaining internal mesh edge nodes and adjacent mesh nodes on both sides as optimization data of the internal mesh edge; S13, traversing all grid line endpoints in the m′-θ coordinate system, obtaining an internal endpoint and a grid node adjacent to the internal endpoint as optimization data of the internal endpoint; S14. The acquisition of periodic boundary point optimization data is divided into two cases: periodic boundary point mismatch and periodic boundary point match. If the periodic boundary point does not match, the periodic boundary point in the m′-θ coordinate system and the grid nodes on one side adjacent to the periodic boundary point are searched as the optimization data of the periodic boundary point; if the periodic boundary point matches, the upper periodic boundary point, the grid nodes adjacent to the upper periodic boundary point and the grid nodes adjacent to the lower periodic boundary point are obtained as the optimization data of the upper periodic boundary point; S15, searching for the inlet / outlet boundary point and the grid nodes on the adjacent side thereof in the m′-θ coordinate system as the optimization data of the inlet / outlet boundary point; The step S2 specifically includes the following sub-steps: S21. Given the Dirichlet boundary conditions of the elliptic partial differential control equation as the mesh surface boundary mesh nodes of a single mesh block in the m′-θ coordinate system, the Poisson equation is used to transform the coordinates from the physical domain to the parameter domain. The specific expression is as follows: Among them, ξ xx represents the second-order partial derivative of ξ with respect to x, ξ yy represents the second-order partial derivative of ξ with respect to y, η xx represents the second-order partial derivative of η with respect to x, η yy represents the second-order partial derivative of η with respect to y, P(ξ,η) and Q(ξ,η) represent the source terms controlling the grid spacing and orthogonality; definition The control equation for mesh optimization of elliptic partial differential equations can be obtained. The specific expression is as follows: in, express The second-order partial derivative with respect to ξ is express The second-order partial derivative with respect to η is represents the second-order mixed partial derivative, Jacobian of coordinate transformation J = x ξ y η -x η y ξ , express The first-order partial derivative with respect to η is express The first-order partial derivative with respect to ξ, x ξ represents the first-order partial derivative of x with respect to ξ, x η represents the first-order partial derivative of x with respect to η, y η represents the first-order partial derivative of y with respect to η, y ξ represents the first-order partial derivative of y with respect to ξ; S22, use the Sorenson source term method to calculate the source term of the control equation (2), and then use Substituting P and Q in the formula respectively, the equivalent equation expression is as follows: If there is orthogonality on the boundary and the second-order derivative across the boundary is 0, we get the source term The expression of ψ is as follows: S23, solving formula (3) in step S22 using a point-by-point super-relaxation method; The step S3 specifically includes the following sub-steps: S31. Given the Dirichlet boundary conditions of the Laplace governing equation for the mesh nodes on both sides of the internal mesh edge, the Laplace governing equation for the internal mesh nodes of the internal mesh edge is optimized, and its expression is as follows: S32, solving formula (6) in step S31 using a point-by-point super-relaxation method; The specific method of step S4 is as follows: The weighted Laplace optimization method is used to optimize the internal endpoints. The optimized internal endpoint position is set to E and the adjacent grid node position is set to E. i (i=1,2,…n), then the expression for optimizing the internal endpoint position is as follows: in, Represents the internal endpoint E to the adjacent grid node E i distance; The step S5 specifically includes the following sub-steps: S51. If the periodic boundary points do not match, the Neumann boundary orthogonal control equation is optimized for the periodic boundary points. The specific expression is as follows: (X η ) i，j ·(X i+1，j- X i，j )=0 (8) Among them, (X η ) i，j represents the first-order derivative of a grid node G at the periodic boundary in the η direction, (X i+1，j -X i，j ) represents the first-order derivative of the grid node G in the ξ direction. Formula (8) is discretized into a difference equation using the forward difference format and solved to obtain the optimized position of the periodic boundary grid node; S52, if the periodic boundary points match, then the periodic boundary points are optimized as the internal grid nodes of the internal grid edges in step S3, the Dirichlet boundary conditions of the given Laplace control equations are the grid nodes adjacent to the upper periodic boundary points in step S14, and the θ coordinates in the grid nodes adjacent to the lower periodic boundary points are translated in the positive direction of θ by a periodic grid node, the control equation (6) is discretized into a difference equation using the central difference format, and the point-by-point super-relaxation method is used to solve it, and the position coordinates of the upper periodic boundary points after optimization are obtained; the position coordinates of the upper periodic boundary points after optimization are assigned to the lower periodic boundary points, and the θ coordinates of the lower periodic boundary points are translated in the negative direction of θ by a periodicity as the position coordinates of the lower periodic boundary points after optimization; The step S6 specifically includes the following sub-steps: S61, using Neumann boundary orthogonal control equation formula (8) to optimize the inlet / outlet boundary points, and obtain the optimized positions of the inlet / outlet boundary points except the first and last grid nodes; S62. Calculate the positions of the first and last grid nodes at the inlet / outlet boundary that meet the orthogonality condition according to the first and last grid nodes at the inlet / outlet boundary.

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