A Quadrilateral Mesh Generation Method Based on Poincare Theorem

Abstract

The present invention discloses a quadrilateral mesh generation method based on Poincare theorem, which relates to the field of numerical simulation of fluid mechanics, including: S1, in the process of generating triangulation of complex shape components of spacecraft, based on the topological characteristics of Poincare theorem, the complex anisotropic topological structure is converted into isotropic topology, and then an unstructured mesh is generated; S2, the shortest distance is combined with the coordinates of the grid point to obtain the shortest distance field, and the gradient of the shortest distance field is solved to obtain the corresponding gradient field; S3, a number of initial points covering the entire geometric area are selected on the boundary of the complex shape components of the spacecraft, and the initial points are used as the starting points of streamline tracing to extract streamlines; S4, a closed polygon is formed based on the streamline, the wall boundary point and the intersection of the streamline and the far field to complete the generation of the quadrilateral mesh. The present invention provides a quadrilateral mesh generation method based on Poincare theorem, which can effectively ensure the overall quality of the mesh and the stability of the generation process.

Description

Technical Field

[0001] The present invention relates to the field of numerical simulation of fluid mechanics, and more specifically, to a method for generating quadrilateral meshes based on the Poincare theorem. Background Art

[0002] Computational Fluid Dynamics (CFD) uses computers to simulate fluid flow processes and is widely used in aerospace, automobile manufacturing, marine engineering and other fields. In CFD applications, the control equations of fluid flow need to be solved on the discretized calculation area, and this discretization process is achieved by dividing the complex geometric area into a large number of basic units (such as quadrilateral or triangular units). The discrete points defined on the vertices of each basic unit are called grid nodes, and the relevant physical quantities of fluid flow such as velocity and pressure are numerically calculated at these nodes.

[0003] Mesh generation technology is crucial in the entire CFD simulation process because it directly affects the discretization quality of the computational space, thus determining the computational accuracy and efficiency. A high-quality mesh should have characteristics such as regular unit shapes, reasonable node distribution, and good adaptation to complex geometric shapes. Otherwise, insufficient mesh quality will lead to increased numerical errors and may even cause simulation failure.

[0004] The lack of automation in traditional mesh generation methods, especially for mesh generation of complex shapes (such as complex shape parts of spacecraft), is one of the problems that need to be solved in the current mesh generation process. Many existing methods still rely on a lot of manual intervention and parameter adjustment when dealing with complex shapes, making it difficult to achieve full automation. This not only increases the workload, but also easily leads to poor controllability and consistency of the generated results. Secondly, when dealing with complex shapes such as sharp corners and drastic changes in curvature, traditional methods often find it difficult to avoid local mesh intersections or deformities. Such problems are common in areas with complex geometric shapes, such as places where streamlines are drastically curved or areas with complex fluid boundary conditions. Self-intersection and mesh distortion not only affect the stability and accuracy of numerical calculations, but may also cause computational divergence and affect the credibility of the final simulation results. For complex geometric shapes, traditional methods require a large number of pre-processing steps to generate high-quality meshes, and density division is usually required in different geometric feature areas. This manual adjustment not only increases the workload in the early stages, but also leads to inefficiency in the entire simulation process, especially when dealing with large-scale calculations. This problem is particularly prominent. Summary of the invention

[0005] An object of the present invention is to solve at least the above problems and / or disadvantages and to provide at least the advantages which will be described hereinafter.

[0006] In order to achieve these objectives and other advantages of the present invention, a method for generating a quadrilateral mesh based on the Poincare theorem is provided, comprising:

[0007] S1. In the process of triangulating the complex shape components of spacecraft, based on the topological characteristics of Poincare's theorem, the complex anisotropic topological structure is transformed into an isotropic topology, and then an unstructured grid is generated;

[0008] S2, the shortest distance SDF ( g j ) is combined with the corresponding grid point coordinates to obtain the Laplace field SDF. By solving the gradient of the Laplace field SDF, the corresponding gradient field is obtained v ;

[0009] S3. Select several initial points covering the entire geometric area on the boundary of the complex shape components of the spacecraft r 0, the initial point r 0 is used as the starting point of streamline tracing, from the generated gradient field v Extract streamlines from

[0010] S4, based on the streamlines, wall boundary points and the intersection points of the streamlines and the far field, a closed polygon is formed to complete the quadrilateral mesh generation of the complex shape parts of the spacecraft;

[0011] Among them, the shortest distance SDF ( g j ) represents each grid point in the unstructured grid g j The shortest distance to the geometry boundary.

[0012] Preferably, in S1, the unstructured grid is generated for the complex-shaped components of the spacecraft using the Delaunay triangulation algorithm, and during the generation process, the density and distribution of the unstructured grid are adjusted based on the local geometric characteristics of the complex-shaped components of the spacecraft.

[0013] Preferably, in S2, the shortest distance SDF ( g j ) is obtained by the following formula:

[0014]

[0015] In the above formula, D ji Represents a grid point g j To the border point b j The distance j =1, 2 …m, i=1, 2 …n , m is the number of grid points, n is the number of boundary points.

[0016] Preferably, in S2, a vector field is obtained by solving the Laplace field to characterize the direction and velocity information of each point. Then, for a two-dimensional field, the gradient field is The weight is:

[0017]

[0018] The final vector for:

[0019]

[0020] In the above formula, is a scalar field, representing a certain physical field; is the gradient, which indicates the direction and rate of change of the scalar field; is the rate of change of the scalar field in the x direction, that is, the partial derivative of ∅ with respect to x; is the rate of change of the scalar field in the y direction, that is, the partial derivative of ∅ with respect to y; and Used to represent the values ​​at (i+1,j) and (i-1,j); and Used to represent the values ​​at (i,j+1) and (i,j-1); is the distance between adjacent grid points in the x-axis direction; is the distance between adjacent grid points in the y-axis direction.

[0021] Preferably, in S3, for any initial point r 0, from the initial point r Starting from 0, along the gradient field v The direction of the flow is gradually moved to obtain the path of the streamline;

[0022] The streamline is generated by iterating the following formula:

[0023]

[0024] In the above formula, r n is the current streamline position, r n+1 It is the position of the next flow line. v ( r n ) is the gradient direction of the current grid point, is the step length.

[0025] The present invention has at least the following beneficial effects:

[0026] First, the present invention overcomes the defects of traditional mesh generation technology in dealing with complex geometric shapes, such as insufficient automation and local self-intersection problems. Through the topological characteristics of the Poincare theorem, the automation level of the mesh generation process is improved, and the dependence on manual intervention and parameter adjustment in the mesh generation process is greatly reduced, achieving a high degree of automation. Compared with the existing technology, users no longer need to perform tedious manual adjustments on complex shape components of spacecraft.

[0027] Secondly, the present invention utilizes the natural non-intersecting characteristics of streamlines during streamline tracing, which means that streamlines will not intersect or overlap at sharp corners or curvature changes of complex spacecraft components, effectively avoiding the occurrence of local self-intersection, overlap or deformity of the grid, thereby ensuring the overall quality of the grid and the stability of the generation process.

[0028] Other advantages, objectives and features of the present invention will be embodied in part through the following description, and in part will be understood by those skilled in the art through study and practice of the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0029] Figure 1 Schematic diagram of the boundary of the NACA0012 airfoil in Example 1 of the present invention;

[0030] Figure 2 It is a schematic diagram of the triangulated airfoil boundary in Example 1 of the present invention;

[0031] Figure 3 A schematic diagram of the Laplace field constructed in Example 1 of the present invention;

[0032] Figure 4 Schematic diagram of the gradient field constructed in Example 1 of the present invention;

[0033] Figure 5 This is a schematic diagram of the streamline constructed in Example 1 of the present invention;

[0034] Figure 6 A schematic diagram of a quadrilateral grid constructed in Example 1 of the present invention;

[0035] Figure 7 This is a schematic diagram of an unstructured grid generated after triangulation of a snowflake shape in Example 2 of the present invention;

[0036] Figure 8 A schematic diagram of a Laplace field constructed for Example 2 of the present invention;

[0037] Fig. 9 Schematic diagram of the gradient field constructed in Example 2 of the present invention;

[0038] Fig.10 This is a streamline diagram constructed in Example 2 of the present invention;

[0039] Fig.11 A schematic diagram of a second-order snowflake grid constructed in Example 2 of the present invention;

[0040] Fig.12 This is a schematic diagram of the streamlines of the third-order snowflake grid constructed in Example 2 of the present invention. DETAILED DESCRIPTION

[0041] The present invention is further described in detail below in conjunction with the accompanying drawings so that those skilled in the art can implement the invention with reference to the description.

[0042] S1. Triangulation of complex shapes

[0043] The Poincare conjecture puts forward an important conjecture about three-dimensional manifolds. Its core idea is: in three-dimensional space, any simply connected, closed three-dimensional manifold must be homeomorphic to a three-dimensional sphere. According to this idea of ​​"topological transformation", the complex anisotropic topological structure is transformed into an isotropic topology. Using Delaunay triangulation to triangulate complex shapes is one of the key steps to generate high-quality meshes. In this process, an unstructured mesh is generated based on the given shape contour. The density and distribution of the mesh are adjusted according to the local geometric characteristics of the shape. For areas with large curvature at the boundary, a denser triangular mesh is generated to better capture the details and features of the geometry.

[0044] S2, isotropic field (Laplace field)

[0045] For each grid point g j , calculate its distance to all boundary points b j The Euclidean distance of .

[0046] Euclidean distance The calculation formula is:

[0047]

[0048] In the above formula, x j , x i Represent the horizontal coordinates of two grid points, y j , y i Respectively represent the ordinates of two grid points;

[0049] Therefore, a distance matrix can be constructed D , whose elements Dji Represents a grid point g j To the border point b j Distance:

[0050]

[0051] in, D The dimension is m×n , m is the number of grid points, n is the number of boundary points.

[0052] For each grid point g j , find the shortest distance of all its boundary points (it should be noted that SDF ( g j ) refers to the representation of a certain distance, and the shortest distance field is the SDF, which is the shortest distance of all points in the entire field), that is:

[0053]

[0054] The calculated SDF values ​​are combined with the corresponding grid point coordinates to form a new data set. The SDF calculated on an unstructured grid is considered as a Laplace field.

[0055] The SDF describes the shortest distance from each grid point to the geometry boundary, with positive values ​​indicating that the grid point is outside the geometry and negative values ​​indicating that it is inside the geometry.

[0056] SDF satisfies the harmonic characteristics similar to the Laplace field: in areas far from the boundary, its gradient hardly changes and presents smooth continuity. This is highly consistent with the mean characteristic of the Laplace field (that is, the value at any point is equal to the average value of its neighboring points).

[0057] The SDF field value does not have local extreme points in the internal area, which is consistent with the maximum principle of the Laplace field (the maximum and minimum values ​​only appear at the boundary), indicating that the uniqueness of its solution is also consistent with the Laplace field.

[0058] S3. Solving the Laplace field

[0059] The shortest distance field (SDF) is actually a Laplace field. By calculating the gradient of the shortest distance field (SDF), a vector field can be obtained to represent the direction and velocity information of each point. For a two-dimensional field, the gradient field The weight is:

[0060]

[0061] The final vector for:

[0062]

[0063] In the above formula, is a scalar field, representing a certain physical field; is the gradient, which indicates the direction and rate of change of the scalar field; is the rate of change of the scalar field in the x direction, that is, the partial derivative of ∅ with respect to x; is the rate of change of the scalar field in the y direction, that is, the partial derivative of ∅ with respect to y; and Used to represent the values ​​at (i+1,j) and (i-1,j); and Used to represent the values ​​at (i,j+1) and (i,j-1); is the distance between adjacent grid points in the x-axis direction; is the distance between adjacent grid points in the y-axis direction.

[0064] The magnitude of the gradient indicates how fast the change is, and the direction of the gradient provides the direction of the fastest path from any point to the boundary.

[0065] In mesh generation, the SDF gradient provides the direction of the fastest path from any point to the boundary, thus laying the foundation for streamline tracing.

[0066] S4, take streamlines to form a four-sided octagonal column

[0067] A streamline is a curve whose tangential direction at each point is consistent with the local velocity field, and its trajectory is given by the differential equation:

[0068]

[0069] The gradient field describes the local change of the physical field in space, and the direction of the gradient determines the direction of the streamline tracking. Select several initial points on the boundary of the complex shape r 0 is used as the starting point for streamline tracing, and streamlines are extracted from the generated gradient field. These initial points cover the entire geometric area to ensure that the generated streamlines can be evenly distributed in the entire area. r Starting from 0, along the gradient field v The streamline generation is a step-by-step iterative process. At each step, the direction of the gradient field at the current position is calculated by numerical methods and moved to the next position:

[0070]

[0071] in, r nis the current streamline position, r n+1 It is the position of the next flow line. v ( r n ) is the gradient direction of the current grid point, is the step length.

[0072] S5. Forming a Grid

[0073] A closed polygon is formed by using streamlines, wall boundary points, and the intersection points of streamlines and far fields. To ensure the uniformity and smoothness of the mesh, the interpolation of streamlines uses a nonlinear method, making the mesh denser when close to the boundary layer and gradually sparse when away from it; finally, a regular quadrilateral mesh is generated between the streamlines and the boundary.

[0074] The above scheme of the present invention overcomes the defects of insufficient automation and local self-intersection problems in traditional grid generation technology when dealing with complex geometric shapes. By introducing the Poincare theorem, the present invention greatly reduces the dependence on manual intervention and parameter adjustment in the grid generation process, and achieves a high degree of automation. Compared with the existing technology, users no longer need to make tedious manual adjustments to complex shapes. By utilizing the natural non-intersecting nature of streamlines, the self-intersection, overlap or deformity of the grid common in traditional methods is avoided.

[0075] Embodiment 1:

[0076] In order to verify the method in this paper, the mesh generation of NACA0012 airfoil in complex shape parts of spacecraft is taken as an example.

[0077] Figure 1 shows the boundary mesh points of this example, where the boundary of the airfoil (NACA0012) has 33 nodes. In the mesh generation process of the NACA0012 airfoil shape, Figure 2 shows the result of the triangulation of the NACA0012 airfoil. An unstructured mesh is generated in the far field, which consists of 306,051 mesh elements and 153,546 points. Figure 3 The Laplace field formed by the transformation of the anisotropic topological structure of the wall boundary into an isotropic topological structure is shown. The second is to solve the Laplace field formation Figure 4 The gradient field shown in Figure 1 is traced according to the gradient direction of each grid point to form a streamline as shown in Figure 1. Figure 5 The streamlines shown in Figure 6. This streamline structure divides the area from the airfoil to the far field into 6 different topological regions, fills the quadrilateral mesh in different topological regions, and performs encryption on the boundary layer. Furthermore, Figure 6 shows the result of generating 500 layers of meshes for the airfoil shape. Figure 6 It can be seen that the present invention generates a grid with excellent orthogonality at the leading edge and trailing edge of the airfoil.

[0078] Example 2

[0079] Generally speaking, it is more challenging to generate boundary meshes for concave boundaries than for convex boundaries. Therefore, in order to increase the complexity of the shape, a two-dimensional snowflake is taken as an example in the actual task to further verify the method of this paper. First, Delaunay meshing is used to generate an unstructured mesh for the snowflake. There are 34144 points and 65871 mesh cells in total, which can accurately capture the geometric details of the snowflake shape. The results are shown in Figure 2. Figure 7 shown.

[0080] Based on the unstructured grid, the corresponding Signed Distance Function (SDF) solution model is constructed using the generated grid point information. The model determines whether the point is inside, outside, or on the boundary of the geometry by calculating the distance and sign of each grid point to the surface of the target geometry. The results of SDF can not only accurately describe the spatial distribution of the geometry, but also provide high-quality numerical support for subsequent applications such as geometry processing, physical simulation, and scene understanding. Based on the new data set obtained by this method, the results are as follows Figure 8 shown.

[0081] Based on the calculated Signed Distance Function (SDF), we further solve the corresponding Laplace field to obtain richer geometric gradient information. In this process, by applying the Laplace operator to the SDF, we can reveal the local variation characteristics of the distance field in space. By accurately solving the Laplace field, we can not only obtain the gradient of the SDF, but also conduct in-depth analysis of the local variation trend in the field. The results are as follows: Fig. 9 shown.

[0082] Select several grid points on the boundary of the snowflake as the starting points of the streamlines, and trace the streamlines according to the direction of the gradient. The results are as follows: Fig.10 shown.

[0083] A closed polygon is formed by the streamlines, the wall boundary points, and the intersection points of the streamlines and the far field. In order to ensure the uniformity and smoothness of the mesh, the streamlines are interpolated in a nonlinear way, making the mesh denser when it is close to the boundary layer and gradually sparse when it is far away; finally, a regular quadrilateral mesh is generated between the streamlines and the boundary. The result is shown in Fig.11 , Fig.12 As shown. Figure 11-Figure 12 It can be seen that the snowflake shape generates regular grids at both the concave and sharp corners, and eventually gradually tends to a circle.

[0084] The above solution is only an illustration of a preferred embodiment, but is not limited thereto. When implementing the present invention, appropriate replacement and / or modification can be performed according to user needs.

[0085] Although the embodiments of the present invention have been disclosed above, they are not limited to the applications listed in the specification and the embodiments. It can be fully applied to various fields suitable for the present invention. For those familiar with the art, additional modifications can be easily realized. Therefore, without departing from the general concept defined by the claims and equivalent scope, the present invention is not limited to the specific details and the illustrations shown and described here.

Claims

1. A quadrilateral mesh generation method based on Poincare theorem, characterized in that: include: S1. In the process of triangulating the complex shape components of spacecraft, based on the topological characteristics of Poincare's theorem, the complex anisotropic topological structure is transformed into an isotropic topology, and then an unstructured grid is generated to generate the Laplace field. S2, the shortest distance SDF ( g j ) is combined with the corresponding grid point coordinates to obtain the Laplace field associated with the shortest distance SDF , by solving the gradient of the Laplace field SDF to obtain the corresponding gradient field v ; S3. Select several initial points covering the entire geometric area on the boundary of the complex shape components of the spacecraft r 0, the initial point r 0 is used as the starting point of streamline tracing, from the generated gradient field v Extract streamlines from S4, based on the streamlines, the wall boundary points and the intersection points of the streamlines and the far field, a closed polygon is formed to complete the quadrilateral mesh generation; Among them, the shortest distance SDF ( g j ) represents each grid point in the unstructured grid g j Shortest distance to the geometry boundary; In S2, the shortest distance SDF ( g j ) is obtained by the following formula: In the above formula, D ji Represents a grid point g j To the border point b j The distance j =1, 2 … m , i=1, 2… n , m is the number of grid points, n is the number of boundary points; In S2, a vector field is obtained by solving the Laplace field to represent the direction and velocity information of each point. For a two-dimensional field, the gradient field is The weight is: The final vector for: In the above formula, is a scalar field, representing a certain physical field; is the gradient, which indicates the direction and rate of change of the scalar field; is the rate of change of the scalar field in the x direction, that is, the partial derivative of ∅ with respect to x; is the rate of change of the scalar field in the y direction, that is, the partial derivative of ∅ with respect to y; and Used to represent the values ​​at (i+1,j) and (i-1,j); and Used to represent the values ​​at (i,j+1) and (i,j-1); is the distance between adjacent grid points in the x-axis direction; is the distance between adjacent grid points in the y-axis direction.

2. The method for generating a quadrilateral mesh based on the Poincare theorem according to claim 1, wherein: In S1, the Delaunay triangulation algorithm is used to generate an unstructured grid for the complex shape, and during the generation process, the density and distribution of the unstructured grid are adjusted based on the local geometric characteristics of the complex shape.

3. The method for generating a quadrilateral mesh based on the Poincare theorem according to claim 1, wherein: In S3, for any initial point r 0, from the initial point r Starting from 0, along the gradient field v The direction of the flow is gradually moved to obtain the path of the streamline; The streamline is generated by iterating the following formula: In the above formula, r n is the current streamline position, r n+1 It is the position of the next flow line. v ( r n ) is the gradient direction of the current grid point, is the step length.

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