Anisotropic triangular mesh generation method

Abstract

The invention belongs to the technical field of computer graphics geometric processing, and particularly relates to an anisotropic triangular mesh generation method, which comprises the following steps of: acquiring triangular mesh data of an object model, and expanding the triangular mesh data from a three-dimensional space to a two-dimensional plane by adopting a mesh parameterization method; based on the expanded triangular mesh data, inputting the set data of each vertex into a trained deep neural network model to obtain an anisotropy measurement parameter and a predicted sampling point coordinate of each vertex; constructing an ACVT energy function according to the anisotropy measurement parameters and the predicted sampling points, and minimizing the ACVT energy function to obtain optimized triangular grid data on a two-dimensional plane; mapping the optimized triangular mesh data from a two-dimensional plane to a three-dimensional space through barycentric coordinates; the method can guarantee the quality of the generated grid and improve the operation efficiency at the same time.

Description

Technical Field

[0001] The invention belongs to the technical field of computer graphics geometry processing, and in particular relates to an anisotropic triangle mesh generation method. Background Art

[0002] Mesh generation and optimization technology plays an important role in mechanical analysis, fluid simulation and other fields. The triangular mesh of the object model, as the basis of finite element analysis, is one of the decisive factors for calculation accuracy and efficiency, and its quality directly affects the accuracy and reliability of the analysis results. In the fields of industrial manufacturing, mechanical design, aerospace, etc., the geometric characteristics of complex parts and the diversified application requirements have put forward higher requirements for triangular mesh generation technology.

[0003] Most of the previous triangular mesh optimization methods are based on isotropic distribution. Although this method is simple to calculate, it has obvious limitations when dealing with complex geometries and high curvature areas. Uniformly distributed triangular meshes are difficult to accurately describe geometric details, resulting in large errors in finite element calculation results. In flat or low stress areas, the over-dense distribution of uniform meshes causes a waste of computing resources. Anisotropic triangular mesh optimization technology provides an effective solution to the above problems. Unlike traditional uniform meshes, anisotropic meshes can adjust the directionality and density of mesh vertex distribution according to geometric features, with a finer mesh distribution in key feature areas and a relatively coarse mesh distribution in flat areas. This optimization method can not only improve the analysis accuracy, but also significantly reduce the computational cost. Therefore, anisotropic mesh optimization technology has important application value in the fields of force analysis of aviation parts, structural optimization of complex mechanical parts, and 3D printing slice processing.

[0004] Existing anisotropic mesh generation methods still face many challenges. One of the main problems is that ACVT anisotropic optimization technology requires a lot of manual intervention and is difficult to achieve full automation. Summary of the invention

[0005] To solve the above problems, the present invention provides an anisotropic triangle mesh generation method, comprising the following steps:

[0006] S1. Obtain triangular mesh data of the object model, and use a mesh parameterization method to expand the triangular mesh data from a three-dimensional space to a two-dimensional plane;

[0007] S2. Based on the unfolded triangular mesh data, the set data of each vertex is input into the trained deep neural network model to obtain the anisotropic metric parameters of each vertex and the predicted sampling point coordinates; the set data is the coordinates of the vertex, the set of 1-neighborhood triangle normal vectors of the vertex, and the set of 1-neighborhood triangle distances of the vertex;

[0008] S3. Construct and minimize the ACVT energy function based on the anisotropic metric parameters and the predicted sampling point coordinates to obtain optimized triangular mesh data on a two-dimensional plane;

[0009] S4. Map the optimized triangular mesh data from the two-dimensional plane back to the three-dimensional space through the barycentric coordinates.

[0010] Beneficial effects of the present invention:

[0011] The present invention proposes a method for generating anisotropic triangular meshes by combining deep learning and mesh parameterization. The triangular mesh in three-dimensional space is transformed into a two-dimensional plane by using mesh parameterization, and then the ACVT re-meshing is performed using Euclidean distance, avoiding the time-consuming calculation of geodesic distance in three-dimensional space, and combining deep learning technology to automatically calculate anisotropic metrics, avoiding the problem of manual intervention. BRIEF DESCRIPTION OF THE DRAWINGS

[0012] Figure 1 This is a flow chart of the anisotropic triangular mesh generation method based on deep learning and mesh parameterization according to the present invention;

[0013] Figure 2 It is a two-dimensional plane schematic diagram of an embodiment of the present invention after being expanded through LSCM grid parameterization;

[0014] Figure 3 It is a two-dimensional plane schematic diagram of an isotropic grid converted into an anisotropic grid through ACVT according to an embodiment of the present invention;

[0015] Figure 4 It is a schematic diagram of a three-dimensional surface in which an isotropic mesh is transformed into an anisotropic mesh through ACVT according to an embodiment of the present invention. DETAILED DESCRIPTION

[0016] 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 ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0017] As deep learning applications in the field of geometric processing mature, deep learning methods can be used to automatically set the anisotropic metric of the triangular mesh surface, so that complex feature areas have fine triangular face distributions, while flat areas have sparse triangular face distributions. Graph convolutional neural networks can handle the relationship between graphs and are suitable for processing the topological relationship between triangular mesh vertices. In addition, mesh parameterization methods can reduce the dimensionality of triangular meshes in three-dimensional space to two-dimensional planes, avoiding the calculation of geodesic distances during mesh optimization. Using Euclidean distance can improve computational efficiency.

[0018] Based on this, the present invention provides an anisotropic triangle mesh generation method, such as Figure 1 As shown, the following steps are included:

[0019] S1. Obtain triangular mesh data of the object model, and use a mesh parameterization method to expand the triangular mesh data from a three-dimensional space to a two-dimensional plane.

[0020] Specifically, step S1 uses the LSCM (Least Square Conformal Maps) mesh parameterization method to expand the triangular mesh data from three-dimensional space to a two-dimensional plane, such as Figure 2 As shown, the specific process includes:

[0021] S11. For each triangle of the triangular mesh data, since only the coordinates of the x-axis and y-axis are considered after mapping to the two-dimensional plane, the coordinates of the three vertices in the triangle are set to (x1, y1), (x2, y2), and (x3, y3), respectively, and a complex number is set for each vertex, namely u1, u2, and u3; based on this, the following relationship exists

[0022]

[0023] d t =(x1y2-y1x2)+(x1y2-y1x2)+(x1y2-y1x2)=2S t

[0024] Among them, St represents the area of ​​the triangle, which is converted into a complex form:

[0025]

[0026] S12. Calculate the sum of all triangle energy functions according to the complex form:

[0027]

[0028] C(t) represents a triangle energy function, C(T) represents the sum of all triangle energy functions, and T represents the triangle set;

[0029] S13. In order to avoid zero solutions to the optimization problem, let the boundary point coordinates of the mesh surface be constraints, and set U to be the complex set of all vertices. Then the complex set The subscript f is a real number, and the coordinates of the vertices inside the mesh are real numbers. The subscript p is an imaginary number, and the coordinates of the vertices on the mesh boundary are imaginary numbers. Then, all triangle energy functions become complex numbers:

[0030] C(U)=‖Ma f U f +Ma p U p ‖ 2

[0031] And the matrix Ma=(Ma f ,Ma p )=(mass ij ) is a matrix whose size is the number of triangles times the number of vertices:

[0032]

[0033] in, represents the jth vertex weight of the ith triangle, v j represents the jth vertex of the i-th triangle face, represents twice the area of ​​the triangle, that is t i Represents the i-th triangle;

[0034] S14. A linear system is obtained from all the triangular energy functions in complex form, which results in the new vertex coordinates of the two-dimensional plane:

[0035] C(x)=‖Ax-b‖ 2

[0036]

[0037] Among them, A represents the weight matrix and b represents the offset value matrix.

[0038] S2. Based on the unfolded triangular mesh data, the set data of each vertex is input into the trained deep neural network model to obtain the anisotropic measurement parameters and predicted sampling point coordinates of each vertex; the set data is the coordinates of the vertex, the set of 1-neighborhood triangle normal vectors of the vertex, and the set of 1-neighborhood triangle distances of the vertex.

[0039] Specifically, the process of training the deep neural network model in step S2 includes:

[0040] S21. Acquire anisotropic triangular mesh data sets, wherein each anisotropic triangular mesh data M includes original mesh data M oand anisotropic mesh data M e ;

[0041] S22. Expand the anisotropic triangular mesh data set from three-dimensional space to two-dimensional plane using a mesh parameterization method;

[0042] S23. Obtain the expanded anisotropic triangular mesh data set to obtain the collection data of each vertex;

[0043] S24. Taking the vertex set data as input, the vertex anisotropic measurement parameters and the initial sampling point coordinates as output, the mean square error loss function is used to train the graph convolutional neural network GCN (Graph Convolutional Network) until the model parameters converge to obtain a trained deep neural network model; the mean square error loss function L is expressed as

[0044]

[0045] Where N represents the number of vertices, f i Represents the original grid data M o The set of normal vectors of the 1-neighborhood triangles of the i-th vertex in ; d i Represents the original grid data M o The set of 1-neighborhood triangle distances of the i-th vertex; v i Represents the original grid data M o The coordinates of the i-th vertex in D, GCN(·) represents the graph convolutional neural network; i represents the output of the graph convolutional neural network, z i Represents the predicted sampling point coordinates of the i-th vertex;

[0046] {θ i ,σ i ,μ i} represents the anisotropic metric parameter of the i-th vertex, θ i represents the curvature angle of the i-th vertex, σ i represents the horizontal scaling factor of the i-th vertex, μ i Indicates the vertical scaling factor of the i-th vertex; y i Represents anisotropic mesh data M e The coordinates of the i-th vertex in are taken as the true label.

[0047] Specifically, the process of obtaining the set data of each vertex includes:

[0048] Determine the coordinates of the vertex based on the unfolded triangular mesh data, and find all 1-neighboring triangular faces of the vertex; for each 1-neighboring triangular face, obtain the normal vector and the center of the circumscribed circle, and calculate the Euclidean distance from the center of the circumscribed circle to the vertex;

[0049] The normal vectors of all 1-neighborhood triangles are combined into a set of 1-neighborhood triangle normal vectors of the vertex, and all Euclidean distances are combined into a set of 1-neighborhood triangle distances of the vertex.

[0050] S3.According to the anisotropic metric parameters and the predicted sampling points, the ACVT energy function is constructed and minimized to obtain the optimized triangular mesh data on the two-dimensional plane, such as Figure 3 The left image is a two-dimensional plane image of an isotropic triangular mesh. After ACVT, it becomes a two-dimensional plane image of an anisotropic triangular mesh in the right image. The triangular faces are more densely distributed in areas with holes near the boundary.

[0051] Specifically, step S3 includes:

[0052] S31. Calculate the anisotropy metric according to the anisotropy metric parameter, expressed as

[0053]

[0054] Among them, H(θ i ,σ i ,μ i ) represents the anisotropy measure of the i-th vertex, θ i represents the curvature angle of the i-th vertex, σ i represents the horizontal scaling factor of the i-th vertex, μ i represents the vertical scaling factor of the i-th vertex; Rot(·) represents the rotation matrix;

[0055] S32.According to the anisotropic metric and coordinates, the ACVT energy function E(Z) is constructed and expressed as

[0056]

[0057] Among them, Ω i represents the Voronoi cell area of ​​the ith vertex, z i represents the predicted sampling point coordinates of the i-th vertex, G(z i ,p i ,θ i ,σ i ,μ i ) represents the anisotropic metric distance, p i represents the coordinate set of the center of the circumscribed circle of the 1-neighborhood triangle of the i-th vertex; m represents the number of 1-neighborhood triangles of the i-th vertex;

[0058] S33. Minimize the ACVT energy function E(Z) to obtain the gradient Expressed as

[0059]

[0060] Where ρ represents the probability density function, c i represents the target centroid of the Voronoi cell area of ​​the i-th vertex, p i,j represents the center of the circumscribed circle of the jth 1-neighborhood triangle of the i-th vertex;

[0061] S34. Update the coordinates of the predicted sampling points according to the gradient to make them close to the target centroid position, expressed as

[0062]

[0063] Where α is the gradient descent rate;

[0064] S35. Update the coordinates of the vertex to the updated predicted sampling point coordinates, and repeat steps S32-35 until the ACVT energy function is minimized to the threshold.

[0065] S4. Map the optimized triangular mesh data from the two-dimensional plane back to the three-dimensional space through the barycentric coordinates. Figure 4 The left image is a three-dimensional surface image of an isotropic triangular mesh. After ACVT, it becomes an anisotropic triangular mesh in the right image. The triangular faces are more densely distributed in areas with holes near the boundaries, and more sparsely distributed in flat areas, and the triangular faces tend to be equilateral triangles.

[0066] Specifically, step S4 includes:

[0067] S41. In the triangular mesh data, each vertex corresponds to an index, and the vertex index remains unchanged in subsequent processing;

[0068] S42. For each vertex v in the optimized triangle mesh data on the two-dimensional plane c , find the vertex v in the expanded triangle mesh data c Vertices v with the same index b ; Similarly, find the vertex v in the triangle mesh data c Vertices v with the same index o ;

[0069] S43. Get the vertex v in the expanded triangle mesh data. b triangular face B; get the vertex v in the triangular mesh data o The triangular face O;

[0070] S44. Calculate the area of ​​triangle B, and calculate the area of ​​the three vertices of triangle B after combining them in pairs and adding them to v c The area of ​​the new triangle is calculated based on the area weight, expressed as

[0071]

[0072] Among them, B1, B2, and B3 are the three vertices of triangle face B. represents the area of ​​triangle B, Represents vertices B2, B3 and vertex v c The area of ​​the new triangle is Represents vertices B1, B3 and vertex v c The area of ​​the new triangle is Represents vertices B1, B2 and vertex v c The area of ​​the new triangular face; (δ, β, γ) represents the barycentric coordinates;

[0073] S45. Get vertex v based on triangular surface O and barycentric coordinates c The vertex coordinates v mapped back to space new , expressed as

[0074] v new =δ×v o1 +β×v o2 +γ×v o3

[0075] Among them, v o1 、v o2 、v o3 Respectively represent the coordinates of the three vertices of the triangular surface O;

[0076] S46. Repeat steps S42-S45 to obtain the result of mapping the optimized triangular mesh data from the two-dimensional plane back to the three-dimensional space.

[0077] In the present invention, unless otherwise clearly stipulated and limited, the terms such as "installation", "setting", "connection", "fixation" and "rotation" should be understood in a broad sense. For example, it can be a fixed connection, a detachable connection, or an integral one; it can be a mechanical connection or an electrical connection; it can be directly connected or indirectly connected through an intermediate medium; it can be the internal connection of two elements or the interaction relationship between two elements. Unless otherwise clearly defined, ordinary technicians in this field can understand the specific meanings of the above terms in the present invention according to the specific circumstances.

[0078] Although embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that various changes, modifications, substitutions and variations may be made to the embodiments without departing from the principles and spirit of the present invention, and that the scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. A method for generating anisotropic triangular meshes, characterized in that: The following steps are involved: S1. Obtain triangular mesh data of the object model, and use a mesh parameterization method to expand the triangular mesh data from a three-dimensional space to a two-dimensional plane; S2. Based on the unfolded triangular mesh data, the set data of each vertex is input into the trained deep neural network model to obtain the anisotropic metric parameters of each vertex and the predicted sampling point coordinates; the set data is the coordinates of the vertex, the set of 1-neighborhood triangle normal vectors of the vertex, and the set of 1-neighborhood triangle distances of the vertex; S3. Construct and minimize the ACVT energy function based on the anisotropic metric parameters and the predicted sampling point coordinates to obtain optimized triangular mesh data on a two-dimensional plane; S4. Map the optimized triangular mesh data from the two-dimensional plane back to the three-dimensional space through the barycentric coordinates.

2. The anisotropic triangular mesh generation method according to claim 1, characterized in that: Step S2 The process of training the deep neural network model includes: S21. Acquire anisotropic triangular mesh data sets, wherein each anisotropic triangular mesh data set includes original mesh data and anisotropic mesh data; S22. Expand the anisotropic triangular mesh data set from three-dimensional space to two-dimensional plane using a mesh parameterization method; S23. Obtain the expanded anisotropic triangular mesh data set to obtain the collection data of each vertex; S24. Taking the vertex set data as input, the vertex anisotropic measurement parameters and predicted sampling point coordinates as output, the graph convolutional neural network is trained using the mean square error loss function until the model parameters converge to obtain a trained deep neural network model; the mean square error loss function L is expressed as Where N represents the number of vertices, f i represents the set of normal vectors of the 1-neighborhood triangles of the i-th vertex; d i represents the 1-neighborhood triangle distance set of the i-th vertex; v i represents the coordinates of the i-th vertex, GCN(·) represents the graph convolutional neural network; D i represents the output of the graph convolutional neural network, z i represents the predicted sampling point coordinates of the i-th vertex; {θ i ,σ i ,μ i } represents the anisotropic metric parameter of the i-th vertex, θ i represents the curvature angle of the i-th vertex, σ i represents the horizontal scaling factor of the i-th vertex, μ i Indicates the vertical scaling factor of the i-th vertex; y i represents the true label of the i-th vertex.

3. The anisotropic triangular mesh generation method according to claim 1, characterized in that: The process of obtaining the set data of each vertex includes: Determine the coordinates of the vertex based on the unfolded triangular mesh data, and find all 1-neighboring triangular faces of the vertex; for each 1-neighboring triangular face, obtain the normal vector and the center of the circumscribed circle, and calculate the Euclidean distance from the center of the circumscribed circle to the vertex; The normal vectors of all 1-neighborhood triangles are combined into a set of 1-neighborhood triangle normal vectors of the vertex, and all Euclidean distances are combined into a set of 1-neighborhood triangle distances of the vertex.

4. The anisotropic triangular mesh generation method according to claim 1, characterized in that: Step S3 specifically includes: S31. Calculate the anisotropy metric according to the anisotropy metric parameter, expressed as Among them, H(θ i ,σ i ,μ i ) represents the anisotropy measure of the i-th vertex, θ i represents the curvature angle of the i-th vertex, σ i represents the horizontal scaling factor of the i-th vertex, μ i represents the vertical scaling factor of the i-th vertex; Rot(·) represents the rotation matrix; S32.According to the anisotropic metric and coordinates, the ACVT energy function E(Z) is constructed and expressed as Among them, Ω i represents the Voronoi cell area of ​​the ith vertex, z i represents the predicted sampling point coordinates of the i-th vertex, G(z i ,p i ,θ i ,σ i ,μ i ) represents the anisotropic metric distance, p i represents the coordinate set of the center of the circumscribed circle of the 1-neighborhood triangle of the i-th vertex; m represents the number of 1-neighborhood triangles of the i-th vertex; S33. Minimize the ACVT energy function E(Z) to obtain the gradient Expressed as Where ρ represents the probability density function, c i represents the target centroid of the Voronoi cell area of ​​the i-th vertex, p i,j represents the center of the circumscribed circle of the jth 1-neighborhood triangle of the i-th vertex; S34. Update the coordinates of the predicted sampling points according to the gradient; S35. Update the coordinates of the vertex to the updated predicted sampling point coordinates, and repeat steps S32-35 until the ACVT energy function is minimized to the threshold.

5. The anisotropic triangular mesh generation method according to claim 1, characterized in that: Step S4 specifically includes: S41. In the triangular mesh data, each vertex corresponds to an index, and the vertex index remains unchanged in subsequent processing; S42. For each vertex v in the optimized triangle mesh data on the two-dimensional plane c , find the vertex v in the expanded triangle mesh data c Vertices v with the same index b ; Similarly, find the vertex v in the triangle mesh data c Vertices v with the same index o ; S43. Get the vertex v in the expanded triangle mesh data. b triangular face B; get the vertex v in the triangular mesh data o The triangular face O; S44. Calculate the area of ​​triangle B, and calculate the area of ​​the three vertices of triangle B after combining them in pairs and adding them to v c The area of ​​the new triangle is calculated based on the area weight, expressed as Among them, B1, B2, and B3 are the three vertices of triangle face B. represents the area of ​​triangle B, Represents vertices B2, B3 and vertex v c The area of ​​the new triangle is Represents vertices B1, B3 and vertex v c The area of ​​the new triangle is Represents vertices B1, B2 and vertex v c The area of ​​the new triangular face; (δ, β, γ) represents the barycentric coordinates; S45. Get vertex v based on triangular surface O and barycentric coordinates c The vertex coordinates v mapped back to space new , expressed as in new =δ×v o1 +β×v o2 +γ×v o3 Among them, v o1 、v o2 、v o3 Respectively represent the coordinates of the three vertices of the triangular surface O; S46. Repeat steps S42-S45 to obtain the result of mapping the optimized triangular mesh data from the two-dimensional plane back to the three-dimensional space.

6. The anisotropic triangular mesh generation method according to claim 1, characterized in that: Step S1 uses the LSCM mesh parameterization method to expand the triangular mesh data from three-dimensional space to a two-dimensional plane.

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