Embedded domain isogeometric analysis method and system based on model boundary adaptive division

Abstract

The invention belongs to the technical field of computational mechanics, and particularly discloses an embedded domain isogeometric analysis method and system based on model boundary adaptive division, and the method comprises the steps: embedding a patch model constructed according to a CAD model into a background grid, and constructing the background grid based on a B-spline function; based on a background grid, performing adaptive division on the patch model: combining coplanar patches with the same normal vector direction to form a polygon, and performing topological division on the polygon with a complex configuration to obtain polygon units; intersection points of the polygon units and the grid units are determined, a Delaunay subdivision method is adopted to divide the patches again, it is guaranteed that each new patch unit only belongs to a single grid unit, and the normal vector direction of each new patch unit is consistent with the normal vector direction of an original patch unit; and carrying out geometric analysis such as an embedded domain based on the patch model. According to the method, the problem of grid division in analysis methods such as finite elements is solved, meanwhile, compared with other embedded domain methods, the problem complexity is simplified, and the analysis efficiency and universality are improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of computational mechanics, and more specifically, relates to a geometric analysis method and system based on embedding domains and adaptive partitioning of model boundaries. Background Art

[0002] Isogeometric Analysis (IGA) is an analytical method that uses the same spline basis functions to represent both geometric and analytical models. It aims to address the time-consuming and computationally expensive translation between CAD and CAE models, which differ in their representation. Compared to traditional finite element methods, IGA offers the advantages of lower performance consumption and higher accuracy. However, because traditional CAD models use B-rep facets, it is difficult to construct parameterized analytical models based on them. Consequently, IGA cannot be directly applied to engineering models of complex structures.

[0003] Therefore, there is an urgent need for a simulation analysis method based directly on the patch model, which can directly analyze the CAD engineering model with complex geometric topological structure. Summary of the Invention

[0004] In response to the above-mentioned defects or improvement needs of the prior art, the present invention provides a geometric analysis method and system such as an embedded domain based on adaptive partitioning of model boundaries, the purpose of which is to achieve direct and efficient analysis of engineering structures with complex geometric topological structures represented by patch models.

[0005] To achieve the above object, according to one aspect of the present invention, a geometric analysis method based on embedding domain and adaptive partitioning of model boundaries is proposed, comprising the following steps:

[0006] S1. Completely embedding the patch model into a background grid; the patch model is converted from a CAD model of the structure to be analyzed, and the background grid is constructed based on a B-spline function;

[0007] S2. Adaptively divide and update the patch units in the patch model based on the background grid:

[0008] The coplanar patch elements with the same normal vector direction are merged to form polygons, and then the polygons with complex configurations are topologically divided to obtain polygonal units. The intersection points of the polygonal units and the grid cells in the background grid are determined, and the patch elements are re-divided using the Delaunay subdivision method to ensure that each new patch element belongs to only a single grid cell and its normal vector is consistent with the normal vector direction of the corresponding original patch element.

[0009] S3. Perform geometric analysis such as embedding domain based on the patch model to obtain displacement cloud map and stress cloud map.

[0010] As a further preferred method, geometric analysis such as embedding domain is performed based on the patch model to obtain displacement cloud maps and stress cloud maps, including:

[0011] According to the positional relationship between the patch units updated after adaptive division and the grid units in the background grid, the clipping units and the solid units are screened out from the grid units;

[0012] Based on the equilibrium equation, the element stiffness matrices of the trimmed element and the solid element are solved separately, and the element stiffness matrices are assembled into the overall stiffness matrix;

[0013] The displacement cloud map and stress cloud map are obtained based on the overall stiffness matrix.

[0014] As a further preferred embodiment, the clipping unit and the entity unit are determined as follows:

[0015] If a patch unit exists in a grid unit, the grid unit is a clipping unit; based on the position of each clipping unit, a grid unit that can represent an entity is determined as an entity unit.

[0016] As a further preferred embodiment, an integration rule on the clipping unit is constructed based on the moment fitting equation and the point elimination algorithm, and the unit stiffness matrix of the clipping unit is solved according to the integration rule.

[0017] As a further preferred embodiment, the integration rule on the clipping unit is specifically as follows:

[0018] Reconstruct the patch unit inside the clipping unit into a closed patch model;

[0019] Scatter enough initial integration points inside the reconstructed patch model;

[0020] Based on the moment fitting equation, the non-negative least squares method is used to iteratively remove redundant integration points.

[0021] As a further preference, Gaussian integral is used to solve the unit stiffness matrix of the solid unit.

[0022] As a further preference, the equilibrium equation is constructed based on the Nitche variational method.

[0023] As a further preferred method, the displacement cloud map and the stress cloud map are obtained based on the global stiffness matrix, including:

[0024] Based on the overall stiffness matrix, the displacement field and the corresponding strain field are obtained, which are converted into the displacement and stress values of the original patch unit nodes. Then, the node information of the original patch model is interpolated and written into the vtk file to obtain a visual displacement cloud map and stress cloud map.

[0025] As a further preference, the facet unit is a triangular facet, and the shape of the mesh unit in the background mesh is a regular hexahedron.

[0026] According to another aspect of the present invention, a geometric analysis system of an embedded domain based on adaptive partitioning of model boundaries is provided, comprising a processor for executing the above-mentioned geometric analysis method of an embedded domain based on adaptive partitioning of model boundaries.

[0027] In general, the above technical solutions conceived by the present invention have the following technical advantages compared with the existing technology:

[0028] 1. The embedded domain and other geometric analysis methods designed in this invention realize direct numerical analysis of patch files, overcoming the difficulty of mesh division in finite element and other analysis methods. At the same time, compared with other embedded domain methods, it greatly simplifies the complexity of the problem and can directly analyze engineering structures with complex geometric topological structures represented by patch models, thereby improving the efficiency and versatility of the analysis.

[0029] 2. The present invention designs a model boundary adaptive partitioning method based on the background grid, which effectively reduces the number of reconstructed triangular facets, reduces the complexity of the boundary, and improves the efficiency of calculation.

[0030] 3. The present invention realizes the visualization of the numerical analysis results of the triangular face model through the node interpolation method based on the vtk file. BRIEF DESCRIPTION OF THE DRAWINGS

[0031] Figure 1 This is a flow chart of a geometric analysis method based on an embedded domain and adaptive partitioning of a model boundary according to an embodiment of the present invention;

[0032] Figure 2 Schematic diagram of a triangular patch model according to an embodiment of the present invention;

[0033] Figure 3 (a) and (b) are schematic diagrams of embedding a triangular face model into a background mesh according to an embodiment of the present invention;

[0034] Figure 4 (a)-(c) are schematic diagrams of the re-divided patch model according to an embodiment of the present invention;

[0035] Figure 5 (a) and (b) are schematic diagrams of triangular facet models reconstructed according to an embodiment of the present invention;

[0036] Figure 6 This is a schematic diagram of a three-dimensional simulation of a steering knuckle according to an embodiment of the present invention;

[0037] Figure 7 This is a schematic diagram of a steering knuckle triangle patch file according to an embodiment of the present invention;

[0038] Figure 8 This is a displacement cloud diagram of the model after being subjected to force according to an embodiment of the present invention;

[0039] Figure 9 von-Mises stress cloud diagram of the model according to the embodiment of the present invention. DETAILED DESCRIPTION

[0040] In order to make the objectives, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely for the purpose of explaining the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.

[0041] The embodiment of the present invention provides a geometric analysis method based on the embedded domain and adaptive partitioning of the model boundary, such as Figure 1 As shown, the following steps are included:

[0042] S1. Based on the CAD model of the structure to be analyzed in a standard format, a patch model (a triangular patch model in this embodiment) that meets certain geometric accuracy requirements is constructed, and geometric information such as points, patches, and normal vectors of the triangular patch model is obtained, such as Figure 2 shown.

[0043] Construct a background grid based on B-spline function, where each grid unit is a regular hexahedron; embed the triangular patch model completely into the background grid, such as Figure 3 As shown; define the material properties, load magnitude and direction of the model, and define the Dirichlet boundary and Neumann boundary.

[0044] S2. Adaptively divide and update the boundaries of the triangular face model based on the background grid, split the triangular face units that originally existed in multiple grid units at the same time, and each triangle of the re-divided triangular face model belongs to only a single grid unit, and ensure that the normal vector of the divided triangle faces points to the outside of the model.

[0045] Specifically, the following steps are included:

[0046] The triangular facets of the triangular facet model are classified according to their position on the plane to which they belong, and the facet units that are coplanar and have the same normal vector direction are merged into a polygon; the complex configuration with multiple polygons on the same plane is topologically divided to obtain multiple polygonal units with topological relationships between them;

[0047] Process each polygon unit separately to obtain three types of points: polygon vertices, intersections of polygons and grid lines, and intersections of polygon edges and grid surfaces, and construct a point set;

[0048] Perform Delaunay subdivision based on the point set, re-divide the triangles, and remove the triangles outside the polygon to ensure that the normal vector direction of the re-divided triangles is consistent with the original triangles, and complete the adaptive division of the triangles, such as Figure 4 shown.

[0049] S3. Perform geometric analysis such as embedding domain based on the patch model, including the following steps:

[0050] S31. According to the positional relationship between the boundary of the triangular face model and the grid cells in the background grid, the grid cells are divided into clipping cells and solid cells.

[0051] Specifically, the adaptively divided triangular facets are distributed to each grid cell according to the center position; the grid cell with a triangular facet is the clipping cell, and the solid cell is divided according to the position of the clipping cell (that is, the entire grid cell is inside the model and has no intersection with the model boundary); the remaining grid cells are virtual cells.

[0052] S32. Based on the Nitche variational method, the equilibrium equation is constructed, and the integral terms are solved to calculate the stiffness matrix of each unit; then the overall stiffness matrix is assembled.

[0053] Specifically, the following equilibrium equation is constructed based on the Nitche variational method:

[0054]

[0055] Where u is the displacement function, δu is the variation of the displacement function, ε is the strain, σ is the stress, C is the elastic matrix, Ω is the physical domain, Γ N is the Newman boundary, Γ D is the Dirichlet boundary, n is the outer normal vector of the boundary, b is the body force, is the external load applied on the Newman boundary, is the displacement constraint on the Dirichlet boundary, β is the stability coefficient; is the weighted residual term of the boundary conditions, is the weighted residual term of the unknown reaction force, is a stable term.

[0056] When solving the equilibrium equation, within each clipped element, the element sub-stiffness matrix is solved according to the clipped element integration rule, the element stiffness matrix is solved using Gaussian integral on the solid element, and the remaining terms of the equilibrium equation are solved by integration on the Dirichlet boundary and Neumann boundary.

[0057] Furthermore, the cropping unit integration rule is constructed based on the moment fitting equation and the point elimination algorithm, as follows:

[0058] Reconstruct the model boundary inside the clipping unit into a closed triangular face model, such as Figure 5 As shown;

[0059] Scatter enough initial integration points inside the reconstructed patch model;

[0060] Based on the moment fitting equation, the non-negative least squares method is used to iteratively remove redundant integration points; the moment fitting equation is as follows:

[0061]

[0062] Among them, f1~f m are all basis functions defined inside the cell, is the parameter space, refers to n q points, The weight corresponding to each integral point.

[0063] S33. Visualization based on triangular patch model.

[0064] Specifically, the corresponding strain field is obtained by solving the equilibrium equation to obtain the displacement field, and then the values of the triangular patch nodes of the original patch model in the displacement field and strain field are obtained. The visual image of the triangular patch model is obtained by interpolating the triangular patch based on the node displacement value and stress value, and saved as a vtk format file to obtain the overall visual image (displacement cloud map and stress cloud map), which can be opened in visualization software such as ParaView.

[0065] To demonstrate the effectiveness of the method of the present invention, a specific example is used for verification below.

[0066] like Figure 6 The figure shows a three-dimensional simulation example of a steering knuckle. Surface D of the knuckle is a Dirichlet boundary and is completely fixed. T1, T2, and T3 are all Neumann boundaries. They are subjected to uniformly distributed loads of 2 MPa, 1 MPa, and 0.3 MPa, respectively. The load direction is perpendicular to the plane and outward. The material's Young's modulus is E = 210 GPa, and the Poisson's ratio μ = 0.3.

[0067] Use CAD software to draw the model and export the triangular patch file, such as Figure 7 shown.

[0068] In this example, the NURBS basis functions of order r and q were both quadratic, and the background mesh size was 50 × 50 × 50. Geometric analysis methods, such as embedding domains based on adaptive model boundary partitioning, were used to obtain the displacement information of the control mesh nodes, which was then visualized by writing it into a VTK file.

[0069] like Figure 8The displacement cloud diagram of the model after being subjected to force is shown as follows: Figure 9 The figure shows the von-Mises stress cloud diagram of the model. The displacement and stress cloud diagrams are continuous and smooth. The results of the numerical analysis are accurate and consistent with the actual situation. The analysis process is simple and efficient, and no manual intervention in meshing is required, which greatly improves the efficiency of the simulation.

[0070] In summary, the geometric analysis methods constructed in this paper, such as the embedded domain method based on adaptive model boundary partitioning, can directly perform numerical analysis on facet models without the geometric reconstruction process of meshing similar to the finite element method. Therefore, complex models can be directly analyzed, greatly improving design efficiency. At the same time, the model boundary adaptive partitioning method proposed in this paper reduces the number of reconstructed triangles, effectively improving computational efficiency.

[0071] It will be easily understood by those skilled in the art that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A geometric analysis method based on embedding domain and adaptive partitioning of model boundaries, characterized in that: The steps include: S1. Completely embedding the patch model into a background grid; the patch model is converted from a CAD model of the structure to be analyzed, and the background grid is constructed based on a B-spline function; S2. Adaptively divide and update the patch units in the patch model based on the background grid: The coplanar patch elements with the same normal vector direction are merged to form polygons, and then the polygons with complex configurations are topologically divided to obtain polygonal units. The intersection points of the polygonal units and the grid cells in the background grid are determined, and the patch elements are re-divided using the Delaunay subdivision method to ensure that each new patch element belongs to only a single grid cell and its normal vector is consistent with the normal vector direction of the corresponding original patch element. S3. Perform geometric analysis such as embedding domain based on the patch model to obtain displacement cloud map and stress cloud map.

2. The geometric analysis method based on embedding domain and adaptive partitioning of model boundaries according to claim 1, characterized in that: Based on the patch model, geometric analysis such as embedded domain is performed to obtain displacement cloud maps and stress cloud maps, including: According to the positional relationship between the patch units updated after adaptive division and the grid units in the background grid, the clipping units and the solid units are screened out from the grid units; Based on the equilibrium equation, the element stiffness matrices of the trimmed element and the solid element are solved separately, and the element stiffness matrices are assembled into the overall stiffness matrix; The displacement cloud map and stress cloud map are obtained based on the overall stiffness matrix.

3. The geometric analysis method based on embedding domain and adaptive model boundary partitioning according to claim 2, characterized in that: The clipping unit and the entity unit are determined as follows: If a patch unit exists in a grid unit, the grid unit is a clipping unit; based on the position of each clipping unit, a grid unit that can represent an entity is determined as an entity unit.

4. The geometric analysis method based on embedding domain and adaptive model boundary partitioning according to claim 2, characterized in that: An integration rule on the tailored element is constructed based on the moment fitting equation and the point elimination algorithm, and the element stiffness matrix of the tailored element is solved according to the integration rule.

5. The geometric analysis method based on embedding domain and adaptive model boundary partitioning according to claim 4, characterized in that: The integration rules on the clipping unit are as follows: Reconstruct the patch unit inside the clipping unit into a closed patch model; Scatter enough initial integration points inside the reconstructed patch model; Based on the moment fitting equation, the non-negative least squares method is used to iteratively remove redundant integration points.

6. The geometric analysis method based on embedding domain and adaptive model boundary partitioning according to claim 2, characterized in that: Gaussian integral is used to solve the element stiffness matrix of the solid element.

7. The geometric analysis method based on embedding domain and adaptive model boundary partitioning according to claim 2, characterized in that: The equilibrium equation is constructed based on the Nitche variational method.

8. The geometric analysis method based on embedding domain and adaptive partitioning of model boundaries according to claim 2, characterized in that: Based on the global stiffness matrix, displacement cloud maps and stress cloud maps are obtained, including: Based on the overall stiffness matrix, the displacement field and the corresponding strain field are obtained, which are converted into the displacement and stress values of the original patch unit nodes. Then, the node information of the original patch model is interpolated and written into the vtk file to obtain a visual displacement cloud map and stress cloud map.

9. The geometric analysis method based on embedding domain and adaptive partitioning of model boundaries according to any one of claims 1 to 8, characterized in that: The patch unit is a triangular patch, and the shape of the mesh unit in the background mesh is a regular hexahedron.

10. A geometric analysis system based on embedding domain and adaptive partitioning of model boundaries, characterized in that: The method comprises a processor, wherein the processor is used to execute the geometric analysis method such as the embedded domain based on the adaptive partitioning of the model boundary as described in any one of claims 1 to 9.

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