A three-dimensional anisotropic mesh generation method based on streamline tracing topology

Abstract

The application discloses a three-dimensional anisotropic grid generation method based on streamline tracing topology structure and relates to the field of numerical simulation and simulation of fluid mechanics, and comprises the following steps: S1, adopting a triangular partitioning algorithm Delaunay to perform tetrahedral partitioning processing on the calculation domain of an aircraft, so as to realize isotropic grid partitioning of the calculation domain; S2, using the isotropic grid point information generated in the isotropic topology structure, constructing a corresponding symbolic distance SDF solving model, so as to obtain a scalar field satisfying Laplace equation; S3, adopting an 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 tracing in the gradient field, so as to obtain a corresponding grid topology structure; and S5, constructing structured hexahedral grids covering the whole calculation domain according to the grid topology structure and the corresponding surface grid. The application realizes the theoretical unification and engineering feasibility of two-dimensional to three-dimensional grid structure mapping.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of numerical simulation of fluid mechanics. More specifically, the present application relates to a three-dimensional anisotropic grid generation method based on streamline tracking topology. BACKGROUND

[0002] In the numerical simulation of computational fluid dynamics (CFD), grid generation is the most basic and critical step, which directly affects the accuracy and efficiency of subsequent simulation calculation. High-quality grids not only can more accurately describe the geometric boundary and physical field distribution, but also can significantly improve the convergence speed of solution. Therefore, the automatic grid generation technology has been the focus and difficulty in CFD research for a long time.

[0003] Currently, there are still many challenges in realizing structured or high-quality unstructured grid generation in complex geometric regions. For this reason, a quadrilateral grid generation method based on Poincare theorem (patent application number: 202411671207.1) is proposed in the prior art, which combines Delaunay triangulation, SDF field construction and Laplace field solution, and automatically generates high-quality quadrilateral grids in two-dimensional regions through streamline tracking. It can be seen that this method has good geometric adaptability and automation capability in the processing of two-dimensional anisotropic topology structure, and exhibits excellent performance in two-dimensional complex shape grid division. It can be seen that the core of this method is to construct the Laplace field and gradient field based on the two-dimensional field, especially in the streamline tracking process, which 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, i.e. when processing objects are three-dimensional objects with multi-scale characteristics, high-curvature surfaces or complex topology structures, the traditional method often has the problems of low efficiency and poor quality. However, if the quadrilateral grid generation method based on Poincare theorem is directly extended to three dimensions, the following technical difficulties will be encountered:

[0004] 1. The streamlines in three-dimensional fields have a higher intersection probability, making it difficult to maintain a clear and consistent topology structure;

[0005] 2. The construction of SDF and gradient calculation in three-dimensional structures significantly increases the complexity, and the traditional two-dimensional method cannot be directly reused;

[0006] 3. There is a lack of strict three-dimensional topology continuity control mechanism, which leads to distortion and degeneration of the generated three-dimensional grid.

[0007] Therefore, the two-dimensional method of the quadrilateral grid generation method based on Poincare theorem cannot be directly applied to the grid generation of three-dimensional complex structures, especially in the gradient calculation aspect, which has a significant technical bottleneck. SUMMARY

[0008] An object of the present application is to solve at least the above problems and / or drawbacks and to provide at least the advantages later described.

[0009] 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 grid based on streamline tracking topology, comprising:

[0010] S1, based on the closed manifold homeomorphism theory in 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 grid partitioning of the calculation domain;

[0011] S2, based on the unstructured grid, using the isotropic grid point information generated in the isotropic topology, constructing a corresponding signed distance SDF solving model, the SDF solving model calculates the shortest distance from the center of each grid element to the geometric boundary, so as to obtain a scalar field satisfying Laplace equation;

[0012] S3, using the SDF gradient solving method based on local least square fitting to calculate the gradient of the scalar field, so as to obtain the corresponding gradient field;

[0013] S4, performing streamline tracking in the gradient field, so as to obtain the corresponding grid topology;

[0014] S5, generating the final anisotropic grid according to the grid topology and the corresponding surface grid, so as to construct a structured hexahedral grid covering the entire calculation domain.

[0015] Preferably, in S3, the SDF gradient solving method based on local least square fitting comprises:

[0016] 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;

[0017] S31, constructing the following linear equation based on the neighborhood p j and the corresponding SDF value

[0018] A ×​c = b

[0019] In the above formula, A is a neighborhood p j The constructed matrix, b is a neighborhood p j The corresponding SDF value The constructed vector set c is a gradient vector, and , is the gradient component corresponding to the SDF, c 0 is a constant term;

[0020] S32, in the process of solving the gradient vector c Regularization term λI is introduced to obtain the following normal equation:

[0021]

[0022] In the above formula, A T is the transpose of the matrix A.

[0023] Preferably, in S4, the streamline tracking refers to solving the streamline equation by using a numerical integration method to complete the tracking of the streamline in the gradient direction in the three-dimensional space;

[0024] Wherein, the streamline equation is characterized by the following formula:

[0025]

[0026] In the above formula, x n is the current grid point, x n+1 is the next grid point, v ( x n ) is the gradient direction at the current grid point x n , and , is the step size.

[0027] Preferably, in S5, the structured hexahedral mesh is generated in the following manner:

[0028] With the given surface mesh and the starting point of the streamline as the initial patch, each node of the initial patch is pushed in the specified direction by the streamline tracking method, thereby generating a series of equidistant layered nodes, and each layer of nodes corresponds to the cross section of the streamline under different step sizes, thereby constructing a structured hexahedral mesh covering the entire calculation domain.

[0029] The present application at least includes the following advantages:

[0030] Firstly, the present application breaks through the limitation of two-dimensional scheme and realizes natural expansion to three dimensions: the existing two-dimensional grid generation method is based on planar field streamline tracking and topological mapping, which is difficult to be directly extended to three-dimensional space, especially in maintaining grid topological consistency and controlling cell distortion on three-dimensional complex manifold. The present application introduces the theory of closed manifold homeomorphism, combines the Laplace property of SDF field in three dimensions, and constructs a hexahedral grid with spatial topological continuity through three-dimensional streamline tracking, which realizes the theoretical unity and engineering feasibility of two-dimensional to three-dimensional grid structure mapping for the first time.

[0031] Secondly, the present application constructs a potential and non-rotational three-dimensional vector field and realizes the natural non-intersection three-dimensional topological skeleton of streamlines: unlike the use of the non-intersection property of streamlines in two-dimensional field in patent 1, the present application constructs a vector field (gradient field) in three-dimensional SDF scalar field, which still maintains the non-intersection property of streamlines, and realizes the consistent propulsion of the structure in the direction of streamline tracking in three dimensions, avoiding the common problems of streamline kinking, self-intersection and structural disorder in three-dimensional space, and ensuring the topological clarity and geometric consistency of the generated grid.

[0032] Thirdly, the present application combines three-dimensional local least square fitting and K-nearest neighbor search to accurately capture the high curvature gradient change in complex three-dimensional geometry: the present application uses local least square fitting to obtain SDF gradient, which avoids the problem of low accuracy of traditional difference calculation under irregular grid, and is especially suitable for three-dimensional engineering models containing complex features such as sharp corners and grooves, so as to construct more accurate and stable three-dimensional streamline path to support the topological growth of subsequent structured grid.

[0033] Fourthly, the present application realizes automatic and high-quality three-dimensional structured hexahedral grid construction under complex boundary: on the basis of the existing surface grid, structured layered nodes are gradually generated along the three-dimensional streamlines, which significantly improves the orthogonality, fairness and cell quality of the hexahedral grid.

[0034] Fifthly, the present application realizes automatic and high-quality three-dimensional structured hexahedral grid construction under complex boundary: on the basis of the existing surface grid, layered nodes are gradually generated along the three-dimensional streamlines, and adaptive layering and multi-scale refinement strategies are introduced, so that high-density grid can be generated in the wall region to ensure simulation accuracy, and sparse grid can be generated in the far field region to improve calculation efficiency, so as to realize the balance between calculation accuracy and efficiency while ensuring the orthogonality, fairness and cell quality of the grid.

[0035] Other advantages, objects and features of the present application will be partly embodied by the following description, and partly understood by those skilled in the art through research and practice of the present application. BRIEF DESCRIPTION OF DRAWINGS

[0036] Figure 1 A flowchart of the present application;

[0037] Figure 2 A non-structured grid generated for the DLR-F11 wing-body combination in an embodiment of the present application;

[0038] Figure 3 A shortest distance field map generated for the DLR-F11 wing-body combination in an embodiment of the present application;

[0039] Figure 4 A gradient field map generated for the DLR-F11 wing-body combination in an embodiment of the present application;

[0040] Figure 5 A grid topology map generated when streamline tracing is performed for the DLR-F11 wing-body combination in an embodiment of the present application;

[0041] Figure 6 A grid map from one perspective generated for the DLR-F11 wing-body combination in an embodiment of the present application;

[0042] Figure 7 A grid map from another perspective generated for the DLR-F11 wing-body combination in an embodiment of the present application;

[0043] Figure 8 A schematic diagram of homeomorphism transformation of different closed manifolds. DETAILED DESCRIPTION

[0044] The present application will be further described below with reference to the accompanying drawings so as to enable those skilled in the art to implement the present application according to the description in the specification.

[0045] The present application proposes an anisotropic three-dimensional grid generation method based on streamline tracing topology. The method starts from the closed manifold homeomorphism theory revealed by Poincare conjecture, combines the SDF gradient field construction mechanism of Delaunay tetrahedral division and local least square fitting, and constructs a structured hexahedral grid structure with topological continuity and directional consistency in three-dimensional space. By controlling the streamline tracing and node advancing process, high-quality grid generation in complex three-dimensional geometric domain is automatically realized, which significantly improves the geometric consistency, topological shape preservation and automation degree, and provides a better grid basis for CFD simulation.

[0046] As shown in Figure 1 , the anisotropic three-dimensional grid generation method based on streamline tracing topology specifically includes the following operation steps:

[0047] Step 1: The complex aircraft shape is converted from an anisotropic topology to an isotropic topology using Delaunay tessellation.

[0048] In 1904, French mathematician Henri Poincaré proposed a conjecture about three-dimensional manifolds, which was later proven by mathematician Grigori Perelman. It is one of the important problems in topology, as stated in Lemma 1.

[0049] Lemma 1. Any simply connected closed three-dimensional manifold is homeomorphic to a three-dimensional sphere.

[0050] Poincaré's conjecture reveals how a closed manifold in three-dimensional space can be gradually simplified through topological changes, ultimately reducing to a three-dimensional sphere. In this process, regardless of the initial shape of the three-dimensional manifold, it will tend to a regularized structure after a series of topological transformations. Figure 8 The schematic diagram of the homeomorphism transformation of different three-dimensional closed manifolds is given. The process of topological transformation has strong similarity with the phenomenon in the generation of grid topology structure, and as the number of grid layers increases, the outermost structure of the grid tends to be regularized and eventually approaches a spherical geometric shape. Through the Delaunay tessellation method, the computational domain is tetrahedralized to ensure that the generated tetrahedrons are as close to regular tetrahedrons as possible, achieving isotropic grid tessellation of the computational domain. This indirectly proves the feasibility of homeomorphism transformation in grid generation.

[0051] Step 2: Scalar field satisfying Laplace's equation (i.e., SDF).

[0052] It is worth noting that the signed distance (Signed Distance Field, SDF), as a widely used technique in computational geometry, describes the shortest distance from each grid point to the geometric boundary, which 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) SDF has almost no gradient change in areas 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 values do not appear local extreme points in the internal region, which is consistent with the maximum principle of Laplace field (the maximum and minimum values only appear on the boundary). SDF satisfies the source-free Poisson equation, i.e., Laplace equation. On the other hand, using the Delaunay tessellation method can quickly achieve isotropic grid tessellation of the computational domain, and by calculating the shortest distance from each grid cell center to the geometric boundary, the corresponding SDF can be obtained, and this SDF has an explicit expression.

[0053] Step 3: Calculate the gradient of the scalar field to obtain the gradient field.

[0054] Therefore, this study adopts the method of streamlines tracing the SDF gradient field to generate the mesh topology. The gradient field describes the local change of the physical field in space, and the direction of the gradient is the shortest path direction from the point to the geometric boundary. The tangent direction of the streamline always keeps consistent with the local SDF gradient direction. Therefore, accurate calculation of the SDF gradient is crucial for constructing a reasonable mesh topology.

[0055] This paper adopts the local least squares fitting method to calculate the SDF gradient. 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 a least squares manner. 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 be denoted as and , the following linear equations are constructed:

[0056] A × c = b (1)

[0057] where the matrix A and the vector b are defined as follows:

[0058] (2)

[0059] The first three components of the vector , i.e. , are the SDF gradient:

[0060] (3)

[0061] In order to solve this gradient vector c , linear equations (1) are constructed using neighborhood points. However, since the equations are usually over-determined (more equations than unknowns) and may have ill-conditioned problems, in order to enhance numerical stability, a regularization term λI is added during the calculation process, which is converted into a normal equation, so as to solve the following normal equation:

[0062] (4)

[0063] By solving this normal equation, the stable fitting coefficient c is obtained, from which the gradient components are derived. These gradient information not only reflects the local SDF change trend, but also provides a directional basis for subsequent geometric structure generation.

[0064] It should be noted that the present application is different from an existing quadrilateral mesh generation method based on Poincare theorem (referred to as existing method 1) in the calculation of signed distance function (SDF) gradient (Table 1 is a specific difference comparison result of existing method 1 and the present application), and the existing method 1 is based on the traditional central difference method, and the gradient is approximately solved by calculating the difference value of two adjacent points in the coordinate direction. This method is simple to implement, high in calculation efficiency, and suitable for regular grids and local smooth scenarios. However, the gradient estimation depends on a small number of neighboring points, which is easily disturbed by local noise or numerical error. Near the boundary, the difference calculation accuracy decreases due to the lack of enough symmetric neighboring points, and even the gradient estimation cannot be performed. In three dimensions, the applicability and accuracy of the method are further reduced.

[0065] The present application proposes an SDF gradient solving method based on local least square fitting. The method determines the neighborhood of each target point through K nearest neighbor (KNN) search, and then constructs an over-determined linear equation set based on the spatial coordinates and corresponding SDF values of all points in the neighborhood, and finally solves the coefficients of the local linear function by introducing a regularization term to correct the ill-conditioned matrix. The spatial gradient of the function is the SDF gradient at the target point. This method not only uses more neighborhood information, has higher noise resistance and numerical stability, but also naturally adapts to three-dimensional scenes, and performs better in subsequent operations such as streamline tracking and surface generation. Therefore, the local least square fitting based on the present application has significant advantages and is more suitable for high-quality modeling and application of SDF gradient in complex geometric scenes.

[0066] Table 1

[0067]

[0068] Step 4: Perform streamline tracking in the gradient field.

[0069] 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 point (control point) x 0, the calculation of the streamline can be represented as:

[0070] (5)

[0071] Wherein, is the gradient direction of the current grid point x n obtained by formula (4), is the step size.

[0072] Step 5: Generate the final anisotropic mesh according to the mesh topology structure.

[0073] This step introduces an adaptive layering strategy in the process of hexahedral mesh generation. Specifically, when constructing a three-dimensional structured hexahedral mesh, the given surface mesh and the starting point of the streamlines are used as the initial patches. Each node of the initial patch is pushed along the specified direction through streamline tracking. During the pushing process, an adaptive layering strategy is introduced. In the near-wall boundary layer, a smaller step size is used for streamline pushing to generate denser layered nodes and obtain finer hexahedral meshes to accurately capture the flow characteristics of the boundary layer. In the far-field region, a larger step size is used for streamline pushing to generate sparse layered nodes and obtain sparse meshes to ensure computational efficiency. This adaptive layering strategy balances the boundary layer capture accuracy and overall computational efficiency.

[0074] Further, the present application combines a multi-scale refinement strategy based on the adaptive layering strategy. First, a consistent coarse hexahedral mesh is generated, and then the high-curvature regions or geometric feature regions are refined. The high-curvature region refers to a region in the calculation domain where the local curvature value of the surface exceeds a predetermined threshold. It can be quantified by the maximum principal curvature, minimum principal curvature, average curvature, or Gaussian curvature. The geometric feature region refers to a region in the calculation domain where there are sharp corners, edges, thin walls, or narrow gaps. It can be determined by the local boundary normal change rate or the ratio of the local geometric size to the overall feature size. In the above regions, the mesh can be locally refined to improve the local mesh fineness while maintaining the stability and computational efficiency of the global mesh. The final mesh has higher local fineness while maintaining global stability, and a structured hexahedral mesh with uniform quality and excellent orthogonality is constructed to cover the entire calculation domain.

[0075] Embodiment:

[0076] In actual tasks, taking the DLR-F11 wing-body combination as an example, first, Delaunay subdivision is used to generate unstructured meshes for DLR-F11, as shown in Figure 2 The entire calculation domain is hemispherical, with a total of 5994227 points and 15151978 mesh elements, which can accurately capture the geometric details of DLR-F11.

[0077] Based on the unstructured mesh, the corresponding Signed Distance Function (SDF) solving model is constructed using the generated mesh point information. This model determines whether a point is inside, outside, or on the boundary of a geometric shape by calculating the distance and sign of each mesh point to the target geometric surface. As shown in Figure 3 The results of SDF not only accurately describe the spatial distribution of the geometry, but also provide high-quality numerical support for subsequent geometric processing, physical simulation, and scene understanding applications.

[0078] Based on the calculated Signed Distance Function (SDF), the gradient of the scalar field satisfying Laplace equation is calculated, and the gradient field as shown in Figure 4 is obtained to obtain more abundant geometric gradient information.

[0079] According to the selected control points on the surface mesh and the gradient information of each mesh point in the gradient field, streamline tracing is performed to form the network topology corresponding to the streamline tracing as shown in Figure 5

[0080] According to the topology structure and the corresponding surface mesh, the DLR-F11 volume mesh as shown in Figure 6-7 is generated. Figure 6-7 It can be seen that the DLR-F11 volume mesh is uniformly distributed as a whole, each element is regular in shape, and there is no element distortion or intersection phenomenon, which ensures the reliability of the mesh and the stability of the subsequent calculation.

[0081] The above scheme is only a description of a preferred example, but is not limited thereto. In the implementation of the present application, appropriate replacement and / or modification can be made according to the user's needs.

[0082] Although the embodiments of the present application have been disclosed as above, it is not limited to the application listed in the specification and the embodiments. It can be fully applied to various fields suitable for the present application. Additional modifications can be easily realized by those skilled in the art. Therefore, the present application is not limited to specific details and the figures shown and described herein without departing from the general concept defined by the claims and the equivalent scope.​

Claims

1. A method for generating three-dimensional anisotropic meshes based on streamline tracing topology, characterized in that, include: S1. Based on the homeomorphism of closed manifolds in the Poincaré conjecture, the computational domain of the aircraft is tetrahedralized using the Delaunay triangulation algorithm to transform the anisotropic topology into an isotropic topology, thereby achieving isotropic mesh partitioning of the computational domain. S2. Based on the unstructured mesh, the corresponding symbolic distance SDF solution model is constructed by utilizing the information of isotropic mesh points generated in the isotropic topology. The SDF solution model calculates the shortest distance from the center of each mesh cell to the geometric boundary to obtain the scalar field that satisfies the Laplace equation. S3. The gradient of the scalar field is calculated using the SDF gradient solution method based on local least squares fitting to obtain the corresponding gradient field. S4. Perform streamline tracing in the gradient field to obtain the corresponding mesh 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 three-dimensional anisotropic meshes based on streamline tracing topology as described in claim 1, characterized in that, In S3, the SDF gradient solution method based on local least squares fitting includes: S30. Determine grid points through K-nearest neighbor search. p i neighborhood p j ,and i =1,2,… m , j =1,2,… k , k The number of neighboring grid points for each grid point. m This represents the total number of grid points. S31, Neighborhood-based p j and the corresponding SDF value Construct the following linear equation: A × c = b In the above formula, A For the neighborhood p j The constructed matrix b For the neighboring area p j Corresponding SDF value The constructed vector set, c Let be the gradient vector, and , These are the gradient components corresponding to SDF. c 0 is a constant term; S32. Solving for the gradient vector c Introducing regularization terms during the process λI Thus, the following normal equation is obtained: In the above formula, A T Let A be the transpose of matrix A.

3. The method for generating three-dimensional anisotropic meshes based on streamline tracing topology as described in claim 2, characterized in that, In S4, streamline tracing refers to solving the streamline equation using numerical integration methods to complete the tracing of streamlines along the gradient direction in three-dimensional space. The streamline equation is characterized by the following formula: In the above formula, x n For the current grid point, x n+1 For the next grid point, v ( x n ) represents the current grid point x n The gradient direction at that point, and , The step size.

4. The method for generating three-dimensional anisotropic meshes based on streamline tracing topology as described in claim 1, characterized in that, In S5, the structured hexahedral mesh is generated as follows: Using a given surface mesh and the starting point of the streamlines as the initial patch, each node of the initial patch is advanced along a specified direction by streamline tracing, gradually generating layered nodes along the three-dimensional streamlines. An adaptive layering strategy and a multi-scale refinement strategy are introduced in the advancement process to construct a structured hexahedral mesh that covers the entire computational domain, has uniform quality, and excellent orthogonality. The adaptive layering strategy refers to using a smaller step size for streamline advancement in the wall region to generate denser layered nodes, and using a larger step size for streamline advancement in the far field region to generate sparser layered nodes. The multi-scale refinement strategy refers to first generating a coarse hexahedral mesh that is consistent throughout, and then locally refining the mesh in areas with high curvature or geometric features.

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