Three-dimensional anisotropic grid generation method based on streamline tracking topological structure

Abstract

The invention discloses a three-dimensional anisotropic grid generation method based on a streamline tracking topological structure, and relates to the field of numerical simulation of fluid mechanics, and the method comprises the steps: S1, carrying out the tetrahedral subdivision processing of a computational domain of an aircraft through employing a triangulation algorithm Delaunay, and achieving the isotropic grid subdivision of the computational domain; s2, using isotropic grid point information generated in the isotropic topological structure to construct a corresponding sign distance SDF resolving model so as to obtain a scalar field satisfying a Laplace equation; s3, calculating the gradient of the scalar field by adopting an SDF gradient solving method based on local least square fitting to obtain a corresponding gradient field; s4, performing streamline tracking in the gradient field to obtain a corresponding grid topological structure; and S5, constructing a structured hexahedral mesh covering the whole computational domain according to the mesh topological structure and the corresponding surface mesh. According to the method, theoretical unification and engineering feasibility of two-dimensional to three-dimensional grid structure mapping are realized.

Description

Technical Field

[0001] The present invention relates to the field of numerical simulation of fluid mechanics, and more particularly to a method for generating three-dimensional anisotropic grids based on streamline tracing topological structures. Background Art

[0002] In computational fluid dynamics (CFD) numerical simulations, mesh generation is the most fundamental and critical step, directly impacting the accuracy and efficiency of subsequent simulations. High-quality meshes not only more accurately describe geometric boundaries and physical field distributions, but also significantly accelerate solution convergence. Therefore, automated mesh generation technology has long been a key and challenging area in CFD research.

[0003] Currently, generating high-quality structured or unstructured meshes in complex geometric regions remains challenging. To address this, a quadrilateral mesh generation method based on the Poincare theorem (patent application number: 202411671207.1) has been proposed. This method combines Delaunay triangulation, SDF field construction, and Laplace field solution to automatically generate high-quality quadrilateral meshes in two-dimensional regions through streamline tracing. This method demonstrates excellent geometric adaptability and automation capabilities for processing two-dimensional anisotropic topological structures, and demonstrates excellent performance in meshing complex two-dimensional shapes. The core of this method is the construction of a Laplace field and gradient field derivation based on a two-dimensional field. In particular, the streamline tracing process relies on the stability of the two-dimensional vector field and the principle of non-intersection of streamlines. However, this method cannot be directly extended to three-dimensional processing objects. Specifically, when processing three-dimensional objects with multi-scale features, high-curvature surfaces, or complex topological structures, traditional methods often suffer from low efficiency and poor quality. However, directly extending the quadrilateral mesh generation method based on the Poincare theorem to three dimensions presents the following technical challenges: 1. Streamlines in a three-dimensional field have a higher probability of crossing, making it difficult to maintain a clear and consistent topological structure; 2. The construction and gradient calculation complexity of SDF in three-dimensional structures increase significantly, making it difficult to directly reuse traditional two-dimensional methods; 3. The lack of a strict 3D topological continuity control mechanism makes the generated 3D mesh prone to distortion and degradation.

[0004] Therefore, a two-dimensional method of quadrilateral mesh generation based on the Poincare theorem cannot be directly applied to the mesh generation of three-dimensional complex structures, especially in the gradient calculation, where there are significant technical bottlenecks. Summary of the Invention

[0005] An object of the present application is to solve at least the above problems and / or drawbacks and to provide at least the advantages set forth hereinafter.

[0006] To achieve these objects and other advantages and in accordance with the purpose of the application, as embodied and broadly described herein, there is provided a method for generating a three-dimensional anisotropic mesh based on a streamline tracking topology, comprising: S1, based on the closed manifold homeomorphism theory in the Poincare conjecture, adopting the tetrahedral partitioning algorithm Delaunay to perform tetrahedral partitioning processing on the calculation domain of the aircraft, converting the anisotropic topology into an isotropic topology, so as to realize isotropic mesh partitioning of the calculation domain; S2, based on the unstructured mesh, using the isotropic mesh point information generated in the isotropic topology, constructing a corresponding signed distance SDF solving model, the SDF solving model calculates the shortest distance from each mesh element center to the geometric boundary, so as to obtain a scalar field satisfying the Laplace equation; S3, using a SDF gradient solving method based on local least square fitting to calculate the gradient of the scalar field, so as to obtain a corresponding gradient field; S4, performing streamline tracking in the gradient field to obtain a corresponding mesh topology; S5, generating a final anisotropic mesh according to the mesh topology and the corresponding surface mesh, so as to construct a structured hexahedral mesh covering the entire calculation domain.

[0007] Preferably, in S3, the SDF gradient solving method based on local least square fitting comprises: S30, determining the neighborhood of each grid point p i by K-neighbor searching p j , and i =1,2,… m , j =1,2,… k , k is the number of neighboring grid points of each grid point, m is the total number of grid points; S31, constructing the following linear equation based on the neighborhood p j and the corresponding SDF value : A × c = b In the above formula, A is the matrix constructed by the neighborhood p j , b is the SDF value of the neighborhood p jCorresponding SDF value The constructed vector set c is the gradient vector, and , is the gradient component corresponding to the SDF, c 0 is a constant term; S32, in solving the gradient vector c Introducing the regularization term in the process λI , to obtain the following normal equation: In the above formula, A T is the transpose of matrix A.

[0008] Preferably, in S4, the streamline tracing refers to solving the streamline equation using a numerical integration method to complete the tracing of the streamline along the gradient direction in three-dimensional space; The streamline equation is represented by the following formula: In the above formula, x n is the current grid point, x n+1 is the next grid point, v ( x n ) is the current grid point x n The gradient direction at , and , is the step length.

[0009] Preferably, in S5, the structured hexahedral grid is generated as follows: Taking the given surface mesh and the starting point of the streamline as the initial patch, each node of the initial patch is advanced along the specified direction by streamline tracing, thereby generating a series of equidistant layered nodes. Each layer of nodes corresponds to the cross-section of the streamline at different step sizes, thus constructing a structured hexahedral mesh covering the entire computational domain.

[0010] The present invention has at least the following beneficial effects: First, the present invention breaks through the limitations of two-dimensional solutions and achieves a natural expansion to three dimensions: existing two-dimensional grid generation methods are based on streamline tracing and topological mapping of plane fields, which are difficult to directly extend to three-dimensional space. In particular, there are technical obstacles in maintaining grid topological consistency and controlling cell distortion on complex three-dimensional manifolds. The present invention introduces the theory of closed manifold homeomorphism, combined with the Laplace properties of SDF fields in three dimensions, to construct a spatially topologically continuous hexahedral grid through three-dimensional streamline tracing, achieving for the first time the theoretical unification and engineering feasibility of mapping two-dimensional to three-dimensional grid structures.

[0011] Secondly, a potential and irrotational three-dimensional vector field is constructed to realize a three-dimensional topological skeleton with naturally non-intersecting streamlines: Different from the use of the non-intersecting characteristics of streamlines in a two-dimensional field in Patent 1, the present invention constructs a vector field (gradient field) in a three-dimensional SDF scalar field, and its streamlines still maintain the non-intersecting nature, and realizes structural consistent advancement in the streamline tracking direction in three dimensions, avoiding the common streamline kinking, self-intersection and structural disorder problems in three-dimensional space, and ensuring the topological clarity and geometric consistency of the generated grid.

[0012] Third, the present invention combines three-dimensional local least squares fitting with K-nearest neighbor search to accurately capture high curvature gradient changes in complex three-dimensional geometry: the present invention uses local least squares method to obtain SDF gradient, avoiding the problem of low accuracy of traditional differential calculation under irregular grids. It is particularly suitable for three-dimensional engineering models containing complex features such as sharp corners and grooves, thereby constructing more accurate and stable three-dimensional streamline paths to support the subsequent topological growth of structured grids.

[0013] Fourthly, the present invention realizes automatic, high-quality three-dimensional structured hexahedral mesh construction under complex boundaries: based on the existing surface mesh, structured layered nodes are gradually generated along the three-dimensional streamlines, which significantly improves the orthogonality, smoothness and unit quality of the hexahedral mesh.

[0014] Fifth, the present invention realizes automatic, high-quality three-dimensional structured hexahedral mesh construction under complex boundaries: on the basis of the existing surface mesh, layered nodes are gradually generated along the three-dimensional streamlines, and adaptive layering and multi-scale refinement strategies are introduced to enable high-density meshes to be generated in the wall area to ensure simulation accuracy, and sparse meshes to be generated in the far-field area to improve computational efficiency, thereby achieving a balance between computational accuracy and efficiency while ensuring the orthogonality, smoothness and unit quality of the mesh.

[0015] Other advantages, objectives and features of the present invention will be reflected in part from the following description and will be understood by those skilled in the art through study and practice of the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] Figure 1 It is a schematic diagram of the process of the present invention; Figure 2 Schematic diagram of an unstructured grid generated for a DLR-F11 wing-body assembly in one embodiment of the present invention; Figure 3 A shortest distance field diagram generated for a DLR-F11 wing-body assembly in one embodiment of the present invention; Figure 4 A gradient field map generated for a DLR-F11 wing-body assembly in one embodiment of the present invention; Figure 5 A grid topology diagram generated when performing streamline tracing on a DLR-F11 wing-body assembly in one embodiment of the present invention; Figure 6 A grid diagram generated for a DLR-F11 wing-body assembly under one viewing angle in one embodiment of the present invention; Figure 7 This is a grid diagram generated for the DLR-F11 wing-body assembly at another viewing angle in one embodiment of the present invention; Figure 8 Schematic diagram of homeomorphic transformation of different closed manifolds. DETAILED DESCRIPTION

[0017] The present invention will be described in further detail below in conjunction with the accompanying drawings so that those skilled in the art can implement the invention with reference to the description.

[0018] This paper proposes a method for generating anisotropic three-dimensional meshes based on streamline tracing topological structures. Starting from the closed manifold homeomorphism theory revealed by the Poincare conjecture, this method combines Delaunay tetrahedralization with the SDF gradient field construction mechanism of local least-squares fitting to construct a structured hexahedral mesh structure with topological continuity and directional consistency in three-dimensional space. By controlling the streamline tracing and node advancement processes, high-quality mesh generation is automatically achieved within complex three-dimensional geometric domains, significantly improving geometric consistency, topological conformality, and automation, providing a higher-quality mesh foundation for CFD simulations.

[0019] like Figure 1 As shown, a method for generating anisotropic 3D mesh based on streamline tracing topological structure includes the following specific steps: Step 1: Use Delaunay meshing to transform the complex aircraft shape from anisotropic topology to isotropic topology.

[0020] In 1904, French mathematician Henri Poincaré proposed a conjecture about three-dimensional manifolds, which was later proved by mathematician Grigoriy Perelman. It is one of the important problems in the field of topology, as stated in Lemma 1.

[0021] Lemma 1. Any simply connected closed three-dimensional manifold must be homeomorphic to a three-dimensional sphere.

[0022] The Poincaré conjecture reveals how a closed manifold in three-dimensional space can be gradually simplified through topological changes, ultimately reducing it to a three-dimensional sphere. In this process, regardless of the initial shape of the three-dimensional manifold, after a series of topological transformations, it tends to converge to a regular structure. Figure 8The topological transformation of different three-dimensional closed manifolds is given, and the process of this topological transformation has strong similarity with the phenomenon in the generation of grid topology structure. With the gradual increase of the number of grid layers, the outermost structure of the grid tends to be regularized, and finally approaches to the spherical geometry. Through the Delaunay subdivision method, the tetrahedral subdivision processing is carried out on the calculation domain, which ensures that the generated tetrahedron is as close to the regular tetrahedron as possible, so as to realize the isotropic grid subdivision of the calculation domain. This indirectly proves the feasibility of the homeomorphism transformation in grid generation.

[0023] Step 2: scalar field (i.e. SDF) that satisfies Laplace equation.

[0024] It is worth noting that the signed distance (SDF) is a widely used technique in computational geometry, which describes the shortest distance from each grid point to the geometric boundary, and can well represent the objects in space. On the one hand, SDF can be regarded as a Laplace field under certain boundary conditions, mainly for two reasons: (i) the gradient of SDF hardly changes in the area far from the boundary, showing smoothness and continuity, which is highly consistent with the mean value property of Laplace field (i.e. the value at any point is equal to the average value of its neighboring points); (ii) SDF field value does not appear local extreme point in the internal area, which is consistent with the maximum principle of Laplace field (the maximum and minimum values only appear on the boundary), and the construction of SDF satisfies the source-free Poisson equation, i.e. Laplace equation. On the other hand, the isotropic grid subdivision of the calculation domain can be quickly realized by using the Delaunay subdivision method, and the corresponding SDF can be obtained by calculating the shortest distance from the center of each grid cell to the geometric boundary, and this SDF has an explicit expression.

[0025] Step 3: calculate the gradient of the scalar field to get the gradient field.

[0026] Therefore, this study adopts the method of stream tracing SDF gradient field to generate grid topology structure. The gradient field describes the local change of physical field in space, and the direction of gradient is the shortest path direction from the point to the geometric boundary, and the tangent direction of stream line always keeps consistent with the local SDF gradient direction. Therefore, accurate calculation of SDF gradient is crucial for constructing reasonable grid topology.

[0027] This paper adopts the local least square fitting method to calculate the gradient of SDF. Specifically, the neighborhood of each grid point is determined by K nearest neighbor (KNN) search, and an over-determined linear system is constructed based on these neighborhood grid points to fit the change of SDF value in the least square way. Let be the coordinates of a given grid point, be the SDF value of the grid point, and its neighborhood grid points and their corresponding SDF values are denoted as and , construct the following linear equation: A × c = b (1) Among them, the matrix A and vector b The definition is as follows: (2) vector The first three components of That is the SDF gradient: (3) To solve this gradient vector c , using the neighborhood points to construct the linear equation (1). However, since the equation is usually overdetermined (there are more equations than unknowns) and may have ill-conditioned problems, in order to enhance numerical stability, a regularization term is added during the calculation process. λI , transform it into the normal equation, and then solve the following normal equation: (4) By solving the normal equation, stable fitting coefficients are obtained c , from which the gradient component These gradient information not only reflects the changing trend of the local SDF, but also provides a direction basis for the subsequent geometric structure generation.

[0028] It should be noted that the present invention differs from an existing quadrilateral mesh generation method based on the Poincare theorem (referred to as Existing Method 1) in calculating the gradient of the signed distance function (SDF). (Table 1 shows a comparison of the specific differences between Existing Method 1 and the present invention.) Existing Method 1 is based on the traditional central difference method, which approximates the gradient by calculating the difference between two adjacent grid points in the coordinate direction. This method is simple to implement and computationally efficient, making it suitable for regular grids and locally smooth scenes. However, its gradient estimation relies on a very small number of neighboring points and is susceptible to local noise or numerical errors. Near boundaries, the lack of sufficient symmetric neighboring points reduces the accuracy of the difference calculation, and gradient estimation may even be impossible. In three dimensions, the applicability and accuracy of this method are further reduced.

[0029] The present invention proposes a method for solving the SDF gradient based on local least squares fitting. This method determines the neighborhood of each target point through K-nearest neighbor (KNN) search, and then constructs an overdetermined linear equation system based on the spatial coordinates of all points in the neighborhood and the corresponding SDF values. The ill-conditioned matrix is ​​corrected by introducing a regularization term, and finally the coefficients of the local linear function are solved. The spatial gradient of this function is the SDF gradient at the target point. This method not only utilizes more neighborhood information, has higher noise resistance and numerical stability, but also can naturally adapt to three-dimensional scenes, and performs better in subsequent operations such as streamline tracing and surface generation. Therefore, the local least squares fitting based on the present invention has significant advantages and is more suitable for high-quality modeling and application of SDF gradients in complex geometric scenes.

[0030] Table 1 Step 4: Tracing streamlines in the gradient field.

[0031] In order to track the streamline along the gradient direction in three-dimensional space, the numerical integration method is used to solve the streamline equation. Given the initial grid points (control points) x 0, the calculation of streamlines can be expressed as: (5) in, is the current grid point x n The gradient direction is obtained by solving formula (4), is the step length.

[0032] Step 5: Generate the final anisotropic mesh based on the mesh topology.

[0033] This step introduces an adaptive layering strategy in the hexahedral mesh generation process. Specifically, when constructing a three-dimensional structured hexahedral mesh, the given surface mesh and the starting point of the streamline are used as the initial facet, and each node of the initial facet is advanced along the specified direction by streamline tracing. An adaptive layering strategy is introduced in the advancement process, that is, near the wall boundary layer, a smaller step size is used for streamline advancement to generate denser layered nodes and obtain a finer hexahedral mesh to accurately capture the boundary layer flow characteristics; while in the far field area, a larger step size is used for streamline advancement to generate sparse layered nodes and obtain a sparse mesh to ensure computational efficiency. This adaptive layering strategy takes into account both boundary layer capture accuracy and overall computational efficiency.

[0034] Furthermore, the present invention combines a multi-scale refinement strategy on the basis of an adaptive layering strategy, first generating an overall consistent coarse hexahedral mesh, and then for high curvature areas or geometric feature areas, wherein the high curvature area refers to the area in the calculation domain where the local curvature value of the surface exceeds a preset threshold, which can be quantified by indicators such as the maximum principal curvature, the minimum principal curvature, the average curvature or the Gaussian curvature; the geometric feature area refers to the area in the calculation domain where there are sharp corners, edges, thin walls or narrow gaps and other geometric structures, which can be determined by the local boundary normal change rate or the ratio of the local geometric size to the overall characteristic scale. In the above-mentioned area, the mesh can be locally encrypted to improve the local mesh fineness while maintaining the stability and computational efficiency of the global mesh, so that the final mesh has a higher local fineness while ensuring global stability, and finally constructs a structured hexahedral mesh covering the entire calculation domain, with uniform quality and excellent orthogonality.

[0035] Example: In the actual task, taking the DLR-F11 wing-body assembly as an example, the Delaunay mesh is first used to generate an unstructured grid for the DLR-F11, such as Figure 2 As shown in Figure 3, the entire computational domain is hemispherical, with a total of 5,994,227 points and 15,151,978 grid cells, which can accurately capture the geometric details of DLR-F11.

[0036] Based on the unstructured grid, the corresponding Signed Distance Function (SDF) solution model is constructed using the generated grid point information. This 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 target geometry surface. Figure 3 As shown in Figure 3, the results of SDF can not only accurately describe the spatial distribution of geometry, but also provide high-quality numerical support for subsequent applications such as geometry processing, physical simulation, and scene understanding.

[0037] Based on the calculated Signed Distance Function (SDF), the gradient of the scalar field that satisfies the Laplace equation is calculated, and the following is obtained: Figure 4 The gradient field shown in Figure 3 is used to obtain richer geometric gradient information.

[0038] According to the control points selected on the surface grid and the gradient information of each grid point in the gradient field, streamline tracing is performed to form the following Figure 5 The network topology corresponding to the streamline tracing shown.

[0039] Based on this topological structure and the corresponding surface mesh, the following Figure 6-Figure 7 The DLR-F11 volume mesh is shown. Figure 6-Figure 7It can be seen that the DLR-F11 volume grid is evenly distributed overall, the shape of each unit is regular, and there is no unit distortion or intersection, which ensures the reliability of the grid and the stability of subsequent calculations.

[0040] The above solution is only an illustration of a preferred embodiment, but is not limited thereto. When implementing the present invention, appropriate replacements and / or modifications can be made according to user needs.

[0041] Although the embodiments of the present invention have been disclosed above, they are not limited to the applications listed in the specification and exemplary embodiments. They can be applied to a variety of fields suitable for the present invention. Further modifications will be readily apparent to those skilled in the art. Therefore, the present invention is not limited to the specific details and illustrations shown and described herein without departing from the general concept defined by the claims and their equivalents.

Claims

1. A three-dimensional anisotropic mesh generation method based on streamline tracing topological structure, characterized in that: include: S1. Based on the closed manifold homeomorphism theory in the Poincare conjecture, the computational domain of the aircraft is tetrahedronized using the Delaunay triangulation algorithm, converting the anisotropic topology into an isotropic one to achieve isotropic meshing of the computational domain. S2. Based on the unstructured grid, the corresponding signed distance SDF solution model is constructed using the isotropic grid point information generated in the isotropic topology. The SDF solution model calculates the shortest distance from the center of each grid cell to the geometric boundary to obtain a scalar field that satisfies the Laplace equation; S3, using the SDF gradient solution method based on local least squares fitting to calculate the gradient of the scalar field to obtain the corresponding gradient field; S4. tracing streamlines in the gradient field to obtain the corresponding grid topology; S5. Generate the final anisotropic mesh based on the mesh topology and the corresponding surface mesh to construct a structured hexahedral mesh covering the entire computational domain.

2. The method for generating a three-dimensional anisotropic grid based on a streamline tracing topological structure according to claim 1, wherein: In S3, the SDF gradient solving methods based on local least squares fitting include: S30, determine the grid point through K-nearest neighbor search p i Neighborhood p j ,and i =1,2,… m , j =1,2,… k , k is the number of grid points adjacent to each grid point, m is the number of all grid points; S31, based on the neighborhood p j And the corresponding SDF value Construct the following linear equation: A × c = b In the above formula, A Neighborhood p j The constructed matrix, b For the neighborhood p j Corresponding SDF value The constructed vector set, c is the gradient vector, and , is the gradient component corresponding to the SDF, c 0 is a constant term; S32, in solving the gradient vector c Introducing the regularization term in the process λI , to obtain the following normal equation: In the above formula, A T is the transpose of matrix A.

3. The method for generating a three-dimensional anisotropic grid based on a streamline tracing topological structure according to claim 2, wherein: In S4, the streamline tracing refers to solving the streamline equation using a numerical integration method to complete the tracing of the streamline along the gradient direction in three-dimensional space; The streamline equation is represented by the following formula: In the above formula, x n is the current grid point, x n+1 is the next grid point, v ( x n ) is the current grid point x n The gradient direction at , and , is the step length.

4. The method for generating a three-dimensional anisotropic grid based on a streamline tracing topological structure according to claim 1, wherein: In S5, the structured hexahedral mesh is generated as follows: Using a given surface mesh and the starting point of a streamline as the initial patch, each node of the initial patch is advanced in a specified direction by tracing the streamlines, gradually generating layered nodes along the 3D streamlines. An adaptive layering strategy and a multi-scale refinement strategy are introduced during the advancement process to construct a structured hexahedral mesh that covers the entire computational domain, has uniform quality, and has excellent orthogonality. The adaptive layering strategy refers to using a smaller step size to advance the streamlines in the wall area to generate denser layered nodes, and using a larger step size to advance the streamlines in the far field area to generate sparser layered nodes. The multi-scale refinement strategy is to first generate an overall consistent coarse hexahedral mesh, and then locally encrypt the high curvature area or geometric feature area.

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