An anisotropic triangular mesh generation method

Abstract

The present application belongs to the technical field of computer graphics geometry processing, and particularly relates to an anisotropic triangular mesh generation method, which comprises the following steps: obtaining triangular mesh data of an object model, and expanding the triangular mesh data from a three-dimensional space to a two-dimensional plane by using a mesh parameterization method; inputting set data of each vertex into a trained deep neural network model based on the expanded triangular mesh data, so as to obtain anisotropy measurement parameters and predicted sampling point coordinates 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, so as to obtain optimized triangular mesh data on the two-dimensional plane; and mapping the optimized triangular mesh data from the two-dimensional plane back to the three-dimensional space through barycentric coordinates; and the present application can ensure the quality of generated meshes and improve the operation efficiency.

Description

TECHNICAL FIELD

[0001] The present application belongs to the technical field of computer graphics geometry processing, and particularly relates to an anisotropic triangular mesh generation method. BACKGROUND

[0002] Mesh generation and optimization techniques play an important role in the fields of mechanical analysis, fluid simulation, etc. The triangular mesh of an object model, as the basis for 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 components and the diversification of application requirements put forward higher requirements for triangular mesh generation technology.

[0003] Most of the previous triangular mesh optimization methods are mainly isotropic distribution. Although this method is simple to calculate, it has obvious limitations in dealing with complex geometry and high curvature areas. Uniformly distributed triangular meshes are difficult to accurately describe geometric details, resulting in large errors in finite element calculation results, while in flat or low stress areas, the over-dense distribution of uniform meshes also causes 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 finer mesh distribution in key feature areas and coarser mesh distribution in flat areas. This optimization method not only improves the analysis accuracy, but also significantly reduces the computational cost. Therefore, anisotropic mesh optimization technology has important application value in the fields of stress analysis of aerospace components, structural optimization of complex mechanical parts, and 3D printing slicing processing, etc.

[0004] Existing anisotropic mesh generation methods still face many challenges, one of the main problems being that the ACVT anisotropic optimization technology needs to rely on a large amount of manual intervention, making it difficult to achieve complete automation. SUMMARY

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

[0006] S1. Obtain the 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 expanded triangular mesh data, input the set data of each vertex into a trained deep neural network model to obtain the anisotropy measurement parameter and the predicted sampling point coordinates of each vertex; the set data includes the coordinates of the vertex, the set of 1-neighborhood triangular face normal vectors of the vertex, and the set of 1-neighborhood triangular face distances of the vertex;

[0008] S3. Construct ACVT energy function according to anisotropy metric parameter and predicted sampling point coordinates and minimize to obtain optimized triangular mesh data on two-dimensional plane;

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

[0010] Advantages of the present application:

[0011] The present application proposes an anisotropic triangular mesh generation method combining deep learning and mesh parameterization. The triangular mesh in three-dimensional space is converted to two-dimensional plane using mesh parameterization, and then ACVT is used to re-mesh using Euclidean distance, avoiding time-consuming geodesic distance calculation in three-dimensional space. Deep learning technology is combined to automatically calculate anisotropy metric, avoiding manual intervention. BRIEF DESCRIPTION OF DRAWINGS

[0012] Figure 1 Flow chart of the anisotropic triangular mesh generation method based on deep learning and mesh parameterization according to the present application;

[0013] Figure 2 Two-dimensional plane diagram of the LSCM mesh parameterization according to the embodiment of the present application;

[0014] Figure 3 Two-dimensional plane diagram of the isotropic mesh converted to anisotropic mesh by ACVT according to the embodiment of the present application;

[0015] Figure 4 Three-dimensional surface diagram of the isotropic mesh converted to anisotropic mesh by ACVT according to the embodiment of the present application. DETAILED DESCRIPTION

[0016] The technical solutions in the embodiments of the present application will be described clearly and completely below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are only part of the embodiments of the present application, not all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative labor are within the scope of protection of the present application.

[0017] With the gradual maturity of the application of deep learning in the field of geometric processing, the anisotropy of the triangular mesh surface can be automatically set by a deep learning method, so that the complex feature area has a fine triangular surface distribution, and the flat area has a sparse triangular surface distribution.

[0018] Based on this, the application provides an anisotropic triangular mesh generation method, as shown in the accompanying drawings, comprising the following steps: Figure 1

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

[0020] Specifically, step S1 uses an LSCM (Least Square Conformal Maps) mesh parameterization method to expand the triangular mesh data from three-dimensional space to a two-dimensional plane, as shown in the accompanying drawings. Figure 2

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

[0022]

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

[0024] Where St represents the area of the triangle, which is converted into a complex number form.

[0025]

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

[0027]

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

[0029] ​​S13. In order to avoid zero solutions to the optimization problem, let the boundary point coordinates of the mesh surface be the 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, Indicates the j-th vertex weight of the i-th triangle, v j represents the jth vertex of the i-th triangle, represents twice the area of ​​the triangle, that is t i Represents the i-th triangle;

[0034] S14. Obtain a linear system based on the complex form of all triangle energy functions, and thus obtain 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 matrix.

[0038] S2. Based on the unfolded triangular mesh data, input the set data of each vertex into the trained deep neural network model to obtain the anisotropic metric 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. Expanding the anisotropic triangular mesh dataset from three-dimensional space to two-dimensional plane by using mesh parameterization method;

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

[0043] S24. Taking the set data of the vertex as input, the anisotropy metric parameters of the vertex and the initial sampling point coordinates as output, training the graph convolutional neural network GCN (Graph Convolutional Network) by using the mean square error loss function until the model parameters converge to obtain the trained deep neural network model; wherein the mean square error loss function L is expressed as

[0044]

[0045] Wherein, N represents the number of vertices, f i represents the 1-neighborhood triangle face normal vector set of the i-th vertex in the original mesh data M o ; d i represents the 1-neighborhood triangle distance set of the i-th vertex in the original mesh data M o ; v i represents the coordinates of the i-th vertex in the original mesh data M o ; 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.

[0046] {θ i ,σ i ,μ i} represents the anisotropy metric parameters 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; y i represents the coordinates of the i-th vertex in the anisotropic mesh data M e , which is taken as the real label.

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

[0048] Based on the expanded triangular mesh data, the coordinates of the vertex are determined, and all 1-neighborhood triangle faces of the vertex are searched; for each 1-neighborhood triangle face, the normal vector and the circumcircle center are obtained, and the Euclidean distance from the circumcircle center to the vertex is calculated.

[0049] The normal vectors of all 1-neighborhood triangular faces form a set of vertex 1-neighborhood triangular face normal vectors, and all Euclidean distances form a set of vertex 1-neighborhood triangular face distances.

[0050] S3. Construct an ACVT energy function according to the anisotropy metric parameter and the predicted sampling point and minimize it to obtain the triangular mesh data optimized on a two-dimensional plane, as shown in the left figure of FIG. 3. Figure 3 The left figure of FIG. 3 is a two-dimensional plane diagram of an isotropic triangular mesh, which becomes the right figure of FIG. 3 after ACVT, that is, an anisotropic triangular mesh two-dimensional plane, and the triangular faces are more densely distributed in the area with holes near the boundary.

[0051] Specifically, step S3 specifically includes:

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

[0053]

[0054] where H(θ i ,σ i ,μ i ) represents the anisotropy metric of the i-th vertex, θ i represents the curvature angle of the i-th vertex, σ i represents the horizontal scaling coefficient of the i-th vertex, μ i represents the vertical scaling coefficient of the i-th vertex; and Rot(·) represents a rotation matrix.

[0055] S32. Construct an ACVT energy function E(Z) according to the anisotropy metric and the coordinates, which is expressed as

[0056]

[0057] where Ω i represents the Voronoi cell region of the i-th 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 anisotropy metric distance, p i represents the set of circumcircle center coordinates of the 1-neighborhood triangular face of the i-th vertex; and m represents the number of 1-neighborhood triangular faces of the i-th vertex.

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

[0059]

[0060] wherein p denotes a probability density function, c i denotes the target centroid of the Voronoi cell region of the i-th vertex, p i,j denotes the circumcircle center of the j-th 1-neighbor triangle of the i-th vertex;

[0061] S34. Update the coordinates of the prediction sampling point according to the gradient so as to approximate the target centroid position, denoted as

[0062]

[0063] wherein a is the gradient descent rate;

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

[0065] S4. Map the optimized triangular mesh data from the two-dimensional plane back to the three-dimensional space through the barycentric coordinates. As shown in the left figure of FIG. 4, the three-dimensional surface graph of the isotropic triangular mesh is shown, and after ACVT, the right figure shows the anisotropic triangular mesh, in which the triangular faces are more densely distributed in the areas with holes near the boundary, the triangular faces are more sparsely distributed in the flat areas, and the triangular faces tend to be equilateral triangles. Figure 4

[0066] Specifically, step S4 specifically includes:

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

[0068] S42. For each vertex v c , find a vertex v c with the same index in the unfolded triangular mesh data; b Similarly, find a vertex v c with the same index in the triangular mesh data; o

[0069] S43. Obtain a triangular face B containing vertex v b in the unfolded triangular mesh data; obtain a triangular face O containing vertex v o in the triangular mesh data;

[0070] S44. Calculate the area of the triangular face B, calculate the areas of the new triangular faces formed by the three vertices of the triangular face B and v c two by two, and calculate the weights according to the areas, denoted as

[0071] ​​

[0072] wherein B1, B2, B3 are three vertices of triangle B, denotes the area of triangle B, denotes the area of the new triangle formed by vertices B2, B3 and vertex v c denotes the area of the new triangle formed by vertices B2, B3 and vertex v denotes the area of the new triangle formed by vertices B1, B3 and vertex v c denotes the area of the new triangle formed by vertices B1, B3 and vertex v denotes the area of the new triangle formed by vertices B1, B2 and vertex v c denotes the area of the new triangle formed by vertices B1, B2 and vertex v(δ, β, γ) denotes the barycentric coordinates;

[0073] S45. obtaining vertex v c mapped back to the space from the barycentric coordinates new , is denoted as

[0074] v new = δ x v o1 + β x v o2 + γ x v o3

[0075] wherein v o1 , v o2 , v o3 denote the coordinates of the three vertices of triangle O respectively;

[0076] S46. repeating 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 application, unless otherwise explicitly specified and limited, the terms “mounting”, “setting”, “connecting”, “fixing”, “rotating” and the like should be understood in a broad sense, for example, can be fixedly connected, or can be detachably connected, or can be integrated; can be mechanically connected, or can be electrically connected; can be directly connected, or can be indirectly connected through an intermediate medium; can be the internal communication of two elements or the interaction relationship between two elements, unless otherwise explicitly limited, the above-mentioned terms in the present application can be understood according to the specific meaning according to the specific circumstances by the ordinary skilled in the art.

[0078] Although the embodiments of the present application have been shown and described, it can be understood by the ordinary skilled in the art that various changes, modifications, replacements and variations can be made to the embodiments without departing from the principles and spirits of the present application, the scope of the present application is defined by the appended claims and their equivalents.

Claims

1. A method of anisotropic triangle mesh generation, characterized by, The method comprises the following steps: S1. Obtain triangle mesh data of an object model, and expand the triangle mesh data from a three-dimensional space to a two-dimensional plane by using a mesh parameterization method; S2. Based on the expanded triangle mesh data, input set data of each vertex into a trained deep neural network model to obtain anisotropy measurement parameters and predicted sampling point coordinates of each vertex; the set data comprises coordinates of the vertex, a 1-neighborhood triangle face normal vector set of the vertex, and a 1-neighborhood triangle face distance set of the vertex; S3. Construct an ACVT energy function according to the anisotropy measurement parameters and the predicted sampling point coordinates and minimize the ACVT energy function to obtain optimized triangle mesh data in the two-dimensional plane; Step S3 specifically comprises: S31. Calculate an anisotropy measurement according to the anisotropy measurement parameters, and represent the anisotropy measurement as where H(θ i ,σ i ,μ i ) denotes the anisotropy measure of the i-th vertex, θ i denotes the curvature angle of the i-th vertex, σ i denotes the horizontal scaling factor of the i-th vertex, μ i denotes the vertical scaling factor of the i-th vertex; Rot(·) denotes the rotation matrix; S32. Construct an ACVT energy function E(Z) according to the anisotropy measurement and the coordinates, and represent the ACVT energy function as wherein Ω i represents the Voronoi cell region of the i-th vertex, z i represents the predicted sampling point coordinate 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 1-neighborhood triangle face circumscribed circle center coordinate set of the i-th vertex; m represents the number of 1-neighborhood triangle faces of the i-th vertex; S33. Minimizing the ACVT energy function E(Z) to obtain the gradient is represented as where p denotes a probability density function, c i denotes the target centroid of the Voronoi cell region of the i-th vertex, p i,j denotes the circumcircle center of the j-th 1-neighborhood triangle face of the i-th vertex; S34. Update the coordinates of the predicted sampling point according to the gradient; S35. Update the coordinates of the vertex to the updated coordinates of the predicted sampling point, and repeat steps S32-S35 until the ACVT energy function is minimized to a threshold value; S4. Map the optimized triangle mesh data from the two-dimensional plane back to the three-dimensional space through barycentric coordinates.

2. The method of claim 1, wherein, The process of training the deep neural network model in step S2 comprises: S21. Obtain an anisotropic triangle mesh data set, wherein each anisotropic triangle mesh data comprises original mesh data and anisotropic mesh data; S22. Expand the anisotropic triangle mesh data set from a three-dimensional space to a two-dimensional plane by using a mesh parameterization method; S23. Obtain the expanded anisotropic triangle mesh data set to obtain set data of each vertex; S24. Train a graph convolutional neural network by taking the set data of the vertex as input and taking anisotropy measurement parameters and predicted sampling point coordinates of the vertex as output, until model parameters converge to obtain a trained deep neural network model; wherein a mean square error loss function L is represented as where N denotes the number of vertices, f i denotes the 1-neighborhood triangle face normal vector set of the i-th vertex; d i denotes the 1-neighborhood triangle face distance set of the i-th vertex; v i denotes the coordinate of the i-th vertex, GCN(·) denotes the graph convolutional neural network; D i denotes the output of the graph convolutional neural network, z i denotes the predicted sampling point coordinate of the i-th vertex; {θ i ,σ i ,μ i} denotes the anisotropy metric parameters of the i-th vertex, θ i denotes the curvature angle of the i-th vertex, σ i denotes the horizontal scaling coefficient of the i-th vertex, μ i denotes the vertical scaling coefficient of the i-th vertex; y i denotes the real label of the i-th vertex.

3. The method of claim 1, wherein, The process of obtaining set data of each vertex comprises: Determine the coordinates of the vertex based on the expanded triangle mesh data, and find all 1-neighborhood triangle faces of the vertex; for each 1-neighborhood triangle face, obtain a normal vector and a circumcircle center, and calculate a Euclidean distance from the circumcircle center to the vertex; Assemble the normal vectors of all 1-neighborhood triangle faces into a 1-neighborhood triangle face normal vector set of the vertex, and assemble all Euclidean distances into a 1-neighborhood triangle face distance set of the vertex.

4. The method of claim 1, wherein, Step S4 specifically comprises: S41. In the triangle mesh data, each vertex corresponds to an index, and the index of the vertex is unchanged in subsequent processing; S42. For each vertex v in the optimized triangular mesh data in the two-dimensional plane c , find vertex v c with the same index in the unfolded triangular mesh data b ; also find vertex v c with the same index in the unfolded triangular mesh data o ; S43. In the unfolded triangular mesh data, obtain a triangular face B containing vertex v b ; in the triangular mesh data, obtain a triangular face O containing vertex v o ; S44. Calculate the area of the triangular face B, calculate the area of the new triangular face formed by the three vertices of the triangular face B respectively in pairs, according to the area calculation weight, represented as c S44. Calculate the area of the triangular face B, calculate the area of the new triangular face formed by the three vertices of the triangular face B respectively in pairs, according to the area calculation weight, represented as where B1, B2, B3 are the three vertices of the triangular face B, denotes the area of the triangular face B, denotes the area of the new triangular face formed by the vertices B2, B3 and the vertex v c denotes the area of the new triangular face formed by the vertices B1, B2 and the vertex v denotes the area of the new triangular face formed by the vertices B1, B3 and the vertex v c denotes the area of the new triangular face formed by the vertices B1, B2 and the vertex v denotes the area of the new triangular face formed by the vertices B1, B2 and the vertex v c denotes the area of the new triangular face formed by the vertices B1, B2 and the vertex v S45. Get vertex v from triangle O and barycentric coordinates c vertex coordinates v mapped back into space new is represented as v new = δ x v o1 + β x v o2 + γ x v o3 where v o1 , v o2 , v o3 denote the coordinates of the three vertices of the triangular face O, respectively. S46. Repeat steps S42-S45 to obtain a result of mapping the optimized triangle mesh data from the two-dimensional plane back to the three-dimensional space.

5. The method of claim 1, wherein, Step S1 expands the triangle mesh data from a three-dimensional space to a two-dimensional plane by using an LSCM mesh parameterization method.

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