An automatic block partitioning method for three-dimensional surface meshes

Abstract

The present invention discloses an automatic blocking method for three-dimensional surface meshes. The method uses Morse theory to extract three-dimensional point cloud features to obtain feature points and feature lines of geometric surfaces, and then realizes the transformation from three-dimensional physical space to two-dimensional parameter space based on least squares conformal mapping. The blocking of planes in the two-dimensional parameter space is completed in combination with cross-field theory, and finally the blocks are mapped back to three-dimensional physical space, thereby realizing the automation of surface division into multiple structural meshes. The method improves the retention of geometric characteristics of the surface based on least squares conformal parameterized mapping, reduces distortion and deformation, and effectively improves the accuracy and stability of surface parameterization. The method uses cross-field theory to efficiently and quickly generate structural blocks, so as to improve the automation and efficiency of multi-block structural mesh division.

Description

Technical Field

[0001] The invention relates to the field of fluid mechanics, and in particular to a three-dimensional curved surface grid automatic block division method based on least squares conformal mapping and a cross field method. Background Art

[0002] Multi-block mesh generation technology can divide complex geometric surfaces into multiple regions in a block manner, and generate independent mesh blocks for each region. The advantage of this method is that the mesh can be refined according to the characteristic requirements of different regions, improving the accuracy of local areas while maintaining the structural consistency of the global mesh. In numerical calculations, multi-block structured meshes are often divided manually. On the one hand, manual block division takes a lot of time and requires users to have considerable experience and skills. On the other hand, unreasonable mesh block division will lead to problems such as too large a total number of structural meshes and poor quality. Therefore, it is necessary to study methods and techniques to improve the efficiency of multi-block structured mesh division.

[0003] Surface parameterization is an important issue in computer graphics and computer-aided design. As surface parameterization gradually matures, a new perspective is provided for the processing of complex surfaces by establishing a mapping relationship between three-dimensional surfaces and specific parameter domains, parameterizing three-dimensional meshes into parameter domain space, and processing the parameterized meshes.

[0004] The plane cross field method is widely used in the identification of singularities in multi-block structured grids. Cross is a set of vectors with four orthogonal directions, which represents the direction of the associated edge at this position in the structured grid; the cross field is the field about these vector sets, also called the orthogonal frame field and the 4-symmetric direction field. In the quadrilateral grid generation, it describes the direction of the grid. Reasonable cross field distribution can make the generated grid meet specific unit size and direction requirements. The plane cross field method is derived from the three-dimensional frame field and directly applied to the generation of multi-block structured grids, which has a large amount of calculation and high complexity. Summary of the invention

[0005] In order to solve the above problems, the present invention provides a highly efficient automatic block segmentation method for three-dimensional surface meshes.

[0006] In order to achieve the above-mentioned purpose, the present invention provides an automatic blocking method for three-dimensional surface meshes, comprising: step S1, obtaining the point cloud geometry of the fluid, using the Delaunay method to form a surface represented by Delaunay triangles, and storing it according to an unstructured grid; step S2, calculating the Morse index function of the three-dimensional geometric points, judging the maximum points and saddle points, and extracting geometric feature lines; step S3, mapping the surface geometric points to the plane using the least squares conformal method, and retaining the triangle connection relationship, so as to obtain the two-dimensional coordinates of the feature lines and the plane triangle mesh; step S4, calculating the intersection, looping all units, judging whether there are singular points therein, and determining their types; step S5, calculating the positions and directions of all singular points, and generating two-dimensional structural blocks; step S6, using the inverse mapping from two-dimensional to three-dimensional to achieve the surface blocking of the fluid.

[0007] Optionally, in step S2:

[0008] The average value of the sum of the angles between each vertex of the triangular mesh and the corresponding neighboring point’s unit normal vector is used as the Morse function index, which is expressed as:

[0009] ;

[0010] In the formula, Indicates is the total number of triangles with vertices; Represents a vertex With neighboring points The normal vector angle of .

[0011] Optionally, step S2 includes:

[0012] Step S21, when the Morse index function of vertex P is greater than the Morse index function values ​​of all its adjacent points, the adjacent point is marked as a minus point. If all the adjacent points of vertex P are minus points, it is a maximum point;

[0013] Step S22: When the Morse index function of vertex P is less than the value of the Morse index function of its adjacent point, the adjacent point is marked as a plus point. If all the adjacent points of vertex P are plus points, it is a minimum point.

[0014] Step S23, defining the edge connecting the plus point and the minus point as a transition edge, when the number of transition edges formed by all adjacent points of the vertex P is 4, it is a saddle point;

[0015] Step S24: Starting from the saddle point, using the neighbor point comparison method, stopping the path search when encountering the maximum point, constructing the descending Morse complex, and connecting to generate characteristic lines.

[0016] Optionally, step S3 includes:

[0017] Each triangular mesh element is a locally orthogonal basis, where is the coordinate of the vertex in the basis. The local coordinate basis of two triangle units sharing an edge has the same direction. Considering the constraints on triangle T, the consistency criterion is applied to the inverse mapping. , Wait , The tangent vector of the curve; for each triangle the energy equation is defined:

[0018] ;

[0019] in is the area of ​​the triangle, yes Model;

[0020] The sum of the energy functions for all triangular elements:

[0021] ;

[0022] For each vertex Set a plural So that every triangle satisfies the Cauchy-Riemann equations;

[0023] The complex form of the Cauchy-Riemann equation can be written as:

[0024] ;

[0025] in, , , , , the energy equation changes the variables to:

[0026] ;

[0027] The energy function of each triangle is:

[0028] ; The triangle The vertex of is the index, where ,

[0029] Is plural The quadratic form of is expressed as a matrix ,in yes The Hermitian matrix of yes The Hermitian conjugate matrix of writing , yes A sparse matrix with rows as triangle cell indices and columns as triangle vertex indices, where ;

[0030] Without any additional constraints, the solution to the above optimization problem is 0;

[0031] Sure Point, Decompose into , similarly Can be decomposed into , yes The matrix of yes The matrix of , so the energy equation matrix form can be written as:

[0032] ;

[0033] in yes The inner product of yes conjugation of;

[0034] Finally, the linear equation is ,in ;

[0035] because exist When the time is full, ; The optimal value of is 2, and the surface is fully conformal for the extensible surface mapping. Therefore, the two farthest boundary points are fixed to maximize the diameter of the graph and ensure the consistency of the triangle direction.

[0036] Optionally, step S4 includes:

[0037] Cross at an angle express:

[0038] ;

[0039] To solve the intersection on the boundary, for a node on the boundary, the size of the internal angle formed by it and its two adjacent points is , define an integer ;when When is an even number, the crossing direction is consistent with the direction of the interior angle bisector; when When it is an odd number, the crossing direction is the angle bisector direction rotated 45° counterclockwise;

[0040] Loop through all the triangle units, and the intersections on their three nodes form a local intersection field. Use the continuity of the intersection field to determine whether there is a singular point in a local intersection field. In the local intersection field, the three vertices of the triangle are arranged in counterclockwise order, and the corresponding intersections are , and , then the type of singularity is calculated by the following formula,

[0041] ;

[0042] In the formula, Represents the minimum angle difference between two intersections, and its range is ; There are only two types of singularities identified by this formula: and ;when When , the local cross field is considered to be continuous and there is no singularity; when When , it is considered that there is a 5-valent singular point in the local cross field, that is, the number of adjacent edges of the singular point is 5; when When , it is considered that there is a 3-valent singular point in the local cross field, that is, the number of adjacent edges of the singular point is 3;

[0043] Loop through all triangular cells and find the cell that contains the singularity.

[0044] Optionally, step S5 includes:

[0045] For a triangle containing a singular point, the coordinates of its vertices are , the coordinates of the singular point Calculated as

[0046] ;

[0047] For Type The singularity of directions; the calculation of the singularity direction must first determine an initial direction , the other directions there are calculated as,

[0048] ;

[0049] For the initial direction , first determine the side with the smallest intersection difference in the triangle, and the midpoint of this side is ;but That is the vector direction, i.e.

[0050] ;

[0051] According to the information of singular points, the two-dimensional computational domain is divided into multiple blocks of structural topology with four edges.

[0052] Optionally, step S6 includes:

[0053] If the boundary point of the parameter space structure block is a triangle unit node, the three-dimensional coordinates are returned directly;

[0054] If the boundary points of the parameter space structure block are strictly on the triangle edge, linear interpolation is used to map the points of the two-dimensional line segment to the three-dimensional line segment;

[0055] If the boundary points of the parameter space structure block are strictly inside the triangle unit, the relative position and barycentric coordinates of the calculated points inside the source triangle are used to map the points into the triangle using the same barycentric coordinates, thereby converting the two-dimensional structure block into a three-dimensional structure block.

[0056] The beneficial effects of the present invention compared with the prior art are as follows: the present invention utilizes Morse theory to extract three-dimensional point cloud features, obtains feature points and feature lines of geometric surfaces, and then realizes the transformation from three-dimensional physical space to two-dimensional parameter space based on least squares conformal mapping, combines cross-field theory to complete the block division of the plane in the two-dimensional parameter space, and finally maps back to three-dimensional physical space, thereby realizing the automation of surface division into multiple structural grids and improving the division efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0057] Figure 1 It is a flow chart of an automatic block division method of a three-dimensional curved surface mesh provided by the present invention;

[0058] Figure 2 is a point cloud image of a wing section provided by an embodiment of the present invention;

[0059] Figure 3 yes Figure 2 Generate a surface plot represented by Delaunay triangles;

[0060] Figures 4a-4d It is a characteristic point classification and characteristic line path finding diagram provided by the present invention;

[0061] Figure 5 is a feature line extraction diagram provided by an embodiment of the present invention;

[0062] Figure 6a is a grid parameterization graph provided by an embodiment of the present invention;

[0063] Figure 6b is a characteristic line parameterized graph provided by an embodiment of the present invention;

[0064] Figure 7 is a schematic diagram of the angle of the intersection provided by the present invention;

[0065] Figure 8 The present invention provides a corresponding Figure 6a Schematic diagram of the singularity position, direction and multi-block structure grid division;

[0066] Fig. 9 The present invention provides a corresponding Figure 2 Schematic diagram of the singularity position, direction and multi-block structure grid division. DETAILED DESCRIPTION

[0067] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative work are within the scope of protection of the present invention.

[0068] Reference Figure 1 This embodiment provides a method for automatically dividing a three-dimensional surface mesh into blocks, comprising the following steps:

[0069] Step S1, obtaining the point cloud geometry of the fluid, using the Delaunay method to form a surface represented by Delaunay triangles, and storing it as an unstructured grid.

[0070] Specifically, a triangular mesh is generated by giving a three-dimensional space surface point cloud model. The generated mesh must ensure connectivity and consistency, there should be no overlap or interlacing between triangles, the mesh boundary should accurately fit the geometric boundary, and the number and distribution of boundary points should accurately describe the boundary shape.

[0071] In this embodiment, refer to Figure 2 , the shaded area is the geometric calculation domain, first read the geometric point cloud data and triangle connection relationship. The Delaunay method is used to form a surface represented by Delaunay triangles, such as Figure 3 As shown, Figure 3 It is a NACA0018 wing section surface mesh, and the surface is extendable.

[0072] Step S2: Calculate the Morse index function of the three-dimensional geometric points, determine the maximum points and saddle points, and extract the geometric feature lines.

[0073] Specifically, the average value of the sum of the angles between each vertex of the triangular mesh and the corresponding neighboring point's unit normal vector is used as the Morse function index, so it can be expressed as:

[0074] ;

[0075] In the formula, Indicates is the total number of triangles with vertices; Represents a vertex With neighboring points The normal vector angle of .

[0076] When the Morse index function of vertex P is greater than the Morse index function values ​​of all its adjacent points, the adjacent point is marked as a "-" point. If all the adjacent points of vertex P are "-" points, it is a maximum point, such as Figure 4a As shown; when the Morse index function of vertex P is less than the value of the Morse index function of its adjacent point, the adjacent point is marked as a "+" point. If the adjacent points of vertex P are all "+" points, it is a minimum point, such as Figure 4b As shown; define the edge connecting the "+" point and the "-" point as a transition edge. When the number of transition edges formed by all adjacent points of vertex P is 4, it is a saddle point as shown in Figure 4c As shown. Starting from the saddle point, the neighbor point comparison method is used. When the maximum point is encountered, the path search is stopped to construct the descending Morse complex and connect to generate characteristic lines, as shown in Figure 4d As shown, the types of points and feature segments can be distinguished by different colors or shapes.

[0077] In this embodiment, the neighbor point comparison method is used starting from the saddle point, and the path search is stopped when the maximum point is encountered to construct the descending Morse complex, connect the generated characteristic lines, and finally extract the characteristic lines as follows: Figure 5 shown.

[0078] Step S3: Map the surface geometric points to the plane using the least squares method and preserve the triangle connection relationship, thereby obtaining the two-dimensional coordinates of the feature lines and the plane triangle mesh.

[0079] Specifically, in this embodiment, it is assumed that each triangular mesh unit is a locally orthogonal basis, where is the coordinate of the vertex in the basis. The local coordinate basis of two triangle units sharing an edge has the same direction. Considering the constraints on triangle T, the consistency criterion is applied to the inverse mapping. , Wait , The tangent vector of the curve; for each triangle the energy equation is defined:

[0080] ;

[0081] in is the area of ​​the triangle, yes Model;

[0082] The sum of the energy functions for all triangular elements:

[0083] ;

[0084] For each vertex Set a plural So that every triangle satisfies the Cauchy-Riemann equations;

[0085] The complex form of the Cauchy-Riemann equation can be written as:

[0086] ;

[0087] in, , , , , the energy equation changes the variables to:

[0088] ;

[0089] The energy function of each triangle is:

[0090] ; The triangle The vertex of is the index, where ,

[0091] Is plural The quadratic form of is expressed as a matrix ,in yes The Hermitian matrix of yes The Hermitian conjugate matrix of writing , yes A sparse matrix with rows as triangle cell indices and columns as triangle vertex indices, where ;

[0092] Without any additional constraints, the solution to the above optimization problem is 0;

[0093] Sure Point, Decompose into , similarly Can be decomposed into , yes The matrix of yes The matrix of , so the energy equation matrix form can be written as:

[0094] ;

[0095] in yes The inner product of yes conjugation of;

[0096] Finally, the linear equation is ,in ;

[0097] because exist When the time is full, ; The optimal value of is 2, and the surface is fully conformal for the extensible surface mapping. Therefore, the two farthest boundary points are fixed to maximize the diameter of the graph and ensure the consistency of the triangle direction.

[0098] The plane diagram obtained in this embodiment is as follows Figure 6a As shown in the figure, the characteristic line parameterization results are shown in Figure 6b shown.

[0099] Step S4, calculate the crossover, loop through all cells, determine whether there is a singular point, and determine its type.

[0100] Specifically, Figure 7 As shown, the intersection is at an angle express:

[0101] ;

[0102] To solve the intersection on the boundary, for a node on the boundary, the size of the internal angle formed by it and its two adjacent points is , define an integer ;when When is an even number, the crossing direction is consistent with the direction of the interior angle bisector; when When it is an odd number, the crossing direction is the angle bisector direction rotated 45° counterclockwise.

[0103] Loop through all the triangle units, and the intersections on their three nodes form a local intersection field. Use the continuity of the intersection field to determine whether there is a singular point in a local intersection field. In the local intersection field, the three vertices of the triangle are arranged in counterclockwise order, and the corresponding intersections are , and , then the type of singularity is calculated by the following formula,

[0104] ;

[0105] In the formula, Represents the minimum angle difference between two intersections, and its range is ; There are only two types of singularities identified by this formula: and ;when When , the local cross field is considered to be continuous and there is no singularity; when When , it is considered that there is a 5-valent singular point in the local cross field, that is, the number of adjacent edges of the singular point is 5; when When , it is considered that there is a 3-valent singular point in the local intersection field, that is, the number of adjacent edges of the singular point is 3.

[0106] Loop through all triangular elements to find the element containing the singular point. For the plane mesh provided in this embodiment, one 5-valent singular point and three 3-valent singular points were found. The results are as follows Figure 8 shown.

[0107] Step S5: Calculate the positions and directions of all singular points and generate a two-dimensional structure block.

[0108] For a Delaunay triangle containing a singular point, the singular point is located at the geometric center of the triangle. Specifically, for a triangle containing a singular point, the coordinates of its vertices are , the coordinates of the singular point Calculated as

[0109] ;

[0110] For Type The singularity of The calculation of the singularity direction must first determine an initial direction , the other directions there are calculated as,

[0111] ;

[0112] Furthermore, for the initial direction , first determine the side with the smallest intersection difference in the triangle, and the midpoint of this side is .but That is the vector direction, i.e.

[0113] ;

[0114] The two-dimensional structure blocks generated in this embodiment are as follows Figure 8 As shown, Figure 8 In the figure, the arrows mark the direction of the singularity, and the dotted lines mark the multi-block structure grid blocks supplemented by the singularity.

[0115] Step S6: Use two-dimensional to three-dimensional inverse mapping to achieve surface segmentation of the fluid.

[0116] Since the geometric point cloud is mapped linearly from the surface represented by the Delaunay triangle to the plane, the points involved in the plane structure block are judged. The points are divided into three types: on the vertex of the triangle unit, strictly on the edge of the triangle unit, and strictly inside the triangle unit, and mapped to the surface mesh respectively, thereby converting the two-dimensional structure block into a three-dimensional structure block.

[0117] Specifically, (1) if the point is a triangle unit node, the 3D coordinates are directly returned; (2) if the point is strictly on the edge of the triangle, the linear interpolation method can be used to map the point of the 2D line segment to the 3D line segment. Assume On the side On, calculate exist The proportion , geometrically mapped to (3) If the point is strictly inside the triangle element, calculate the point In the source triangle Internal relative position, center of gravity coordinates satisfy ,and , use the same centroid coordinates to point Mapping to triangle Get ,in . Thus, the two-dimensional structural block is transformed into a three-dimensional structural block.

[0118] The three-dimensional structure blocks generated in this embodiment are as follows Fig. 9 As shown, Fig. 9 In the figure, the arrows mark the direction of the singularity, and the dotted lines mark the multi-block structure grid blocks supplemented by the singularity.

[0119] The automatic block partitioning method provided by the present invention adopts the least squares conformal parameterization method to map the surface to the plane, and after reducing the dimension, the plane cross field method is used to realize automatic block partitioning. Due to the conversion from three-dimensional to two-dimensional, the amount of calculation and process complexity are greatly reduced compared to directly using the three-dimensional frame field method. Based on the least squares conformal parameterization mapping, the geometric characteristics of the surface are retained, while the distortion and deformation are reduced, and the accuracy and stability of the surface parameterization are effectively improved; the cross field theory is used to efficiently and quickly generate structural blocks to improve the automation and efficiency of multi-block structure mesh division.

[0120] The present invention has many specific application paths. The above is only a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements can be made without departing from the principle of the present invention. These improvements should also be regarded as the protection scope of the present invention.

Claims

1. A method for automatic segmentation of three-dimensional surface meshes, characterized in that: include: Step S1, obtaining the point cloud geometry of the fluid, using the Delaunay method to form a surface represented by Delaunay triangles, and storing it according to an unstructured grid; Step S2, calculating the Morse index function of the three-dimensional geometric point, determining the maximum point and the saddle point, and extracting the geometric feature line; Step S3, mapping the surface geometric points to the plane using the least squares method and retaining the triangle connection relationship, thereby obtaining the two-dimensional coordinates of the feature lines and the plane triangle mesh; Step S4, calculate the crossover, loop through all cells, determine whether there is a singular point, and determine its type; Step S5, calculating the positions and directions of all singular points and generating a two-dimensional structure block; Step S6, using two-dimensional to three-dimensional inverse mapping to achieve surface segmentation of the fluid; In step S2: The average value of the sum of the angles between each vertex of the triangular mesh and the corresponding neighboring point’s unit normal vector is used as the Morse function index, which is expressed as: ； In the formula, Indicates is the total number of triangles with vertices; Represents a vertex With neighboring points The normal vector angle of The step S2 comprises: Step S21, when the Morse index function of vertex P is greater than the Morse index function values ​​of all its adjacent points, the adjacent point is marked as a minus point. If all the adjacent points of vertex P are minus points, it is a maximum point; Step S22: When the Morse index function of vertex P is less than the value of the Morse index function of its adjacent point, the adjacent point is marked as a plus point. If all the adjacent points of vertex P are plus points, it is a minimum point. Step S23, defining the edge connecting the plus point and the minus point as a transition edge, when the number of transition edges formed by all adjacent points of the vertex P is 4, it is a saddle point; Step S24, starting from the saddle point, using the neighbor point comparison method, stopping the path search when encountering the maximum point to construct a descending Morse complex, and connecting to generate characteristic lines; The step S4 comprises: Cross at an angle express: ； To solve the intersection on the boundary, for a node on the boundary, the size of the internal angle formed by it and its two adjacent points is , define an integer ;when When is an even number, the crossing direction is consistent with the direction of the interior angle bisector; when When it is an odd number, the crossing direction is the angle bisector direction rotated 45° counterclockwise; Loop through all the triangle units, and the intersections on their three nodes form a local intersection field. Use the continuity of the intersection field to determine whether there is a singular point in a local intersection field. In the local intersection field, the three vertices of the triangle are arranged in counterclockwise order, and the corresponding intersections are , and , then the type of singularity is calculated by the following formula, ； In the formula, Represents the minimum angle difference between two intersections, and its range is ; There are only two types of singularities identified by this formula: and ;when When , the local cross field is considered to be continuous and there is no singularity; when When , it is considered that there is a 5-valent singular point in the local cross field, that is, the number of adjacent edges of the singular point is 5; when When , it is considered that there is a 3-valent singular point in the local cross field, that is, the number of adjacent edges of the singular point is 3; Loop through all triangular cells and find the cell that contains the singularity.

2. The automatic block segmentation method according to claim 1, characterized in that: The step S3 comprises: Each triangular mesh element is a locally orthogonal basis, where is the coordinate of the vertex in the basis. The local coordinate basis of two triangle units sharing an edge has the same direction. Considering the constraints on triangle T, the consistency criterion is applied to the inverse mapping. , Wait , The tangent vector of the curve; for each triangle the energy equation is defined: ； in is the area of ​​the triangle, yes Model; The sum of the energy functions for all triangular elements: ； For each vertex Set a plural So that every triangle satisfies the Cauchy-Riemann equations; The complex form of the Cauchy-Riemann equation can be written as: ； in, , , , , the energy equation changes the variables to: ； The energy function of each triangle is: ; The triangle The vertex of is the index, where , ; Is plural The quadratic form of is expressed as a matrix ,in yes The Hermitian matrix of yes The Hermitian conjugate matrix of writing , yes A sparse matrix with rows as triangle cell indices and columns as triangle vertex indices, where ; Without any additional constraints, the solution to the above optimization problem is 0; Sure Point, Decompose into , similarly Can be decomposed into , yes The matrix of yes The matrix of , so the energy equation matrix form can be written as: ； in yes The inner product of yes conjugation of; Finally, the linear equation is ,in ; because exist When the time is full, ; The optimal value of is 2, and the surface is fully conformal for the extensible surface mapping. Therefore, the two farthest boundary points are fixed to maximize the diameter of the graph and ensure the consistency of the triangle direction.

3. The automatic block segmentation method according to claim 1, characterized in that: The step S5 comprises: For a triangle containing a singular point, the coordinates of its vertices are , the coordinates of the singular point Calculated as ； For Type The singularity of directions; the calculation of the singularity direction must first determine an initial direction , the other directions there are calculated as, ； For the initial direction , first determine the side with the smallest intersection difference in the triangle, and the midpoint of this side is ;but That is the vector direction, i.e. ； According to the information of singular points, the two-dimensional computational domain is divided into multiple blocks of structural topology with four edges.

4. The automatic block segmentation method according to claim 1, characterized in that: The step S6 comprises: If the boundary point of the parameter space structure block is a triangle unit node, the three-dimensional coordinates are returned directly; If the boundary points of the parameter space structure block are strictly on the triangle edge, linear interpolation is used to map the points of the two-dimensional line segment to the three-dimensional line segment; If the boundary points of the parameter space structure block are strictly inside the triangle unit, the relative position and barycentric coordinates of the calculated points inside the source triangle are used to map the points into the triangle using the same barycentric coordinates, thereby converting the two-dimensional structure block into a three-dimensional structure block.

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