A method for designing and manufacturing a multi-scale geometric product

Abstract

The application discloses a design and manufacturing method of a multi-scale geometric product, which comprises the following steps: a user imports a three-dimensional grid in a CAD system through an interactive interface; an initial base function is generated based on topological information of the three-dimensional grid, and a mapping from a parameter domain to a physical space is established; control coefficients of each control node are set, a unit structure expressed by a combination of the control coefficients and the base function is generated in the parameter domain of a grid unit, and the unit structure is mapped to the physical space to obtain an initial geometric product; a target grid unit and corresponding control coefficients are subdivided, and a truncated hierarchical base function is generated according to a subdivision relationship; a fine layer control coefficient is adjusted to increase geometric details of the unit structure, and finally a geometric product with multi-scale characteristics is obtained by mapping. The application can construct a truncated hierarchical base function for a control grid with an arbitrary topology, overcomes the topological limitation of a traditional B-spline base function, and can gradually introduce local geometric details through hierarchical refinement, so that the design and manufacturing of a multi-scale geometric product are realized.

Description

Technical Field

[0001] This invention belongs to the field of computer-aided design and manufacturing technology, and relates to a method for designing and manufacturing multi-scale geometric products based on arbitrary topological control grids and hierarchical basis functions. Background Technology

[0002] Computer-aided design (CAD) systems provide a mature foundation for geometric representation and modeling. With parametric geometric modeling at its core, they employ B-splines, NURBS, and other methods to accurately describe complex shapes and offer features such as modeling and Boolean computation. Due to their accuracy and ease of use, CAD has become a primary tool for engineering product design. Driven by design demands such as lightweight structures and functional gradients, geometric modeling needs to describe not only macroscopic geometry but also microscale features within local areas. In this context, multi-scale modeling has become a key technology: its goal is to express the overall shape and local geometric details at different resolutions within a unified geometric framework, thereby supporting more flexible and refined cross-scale geometric design.

[0003] Existing CAD systems are based on B-splines and NURBS parametric representations, the core of which is to construct differentiable geometric objects using control meshes and basis functions. This type of representation can guarantee local support, smoothness, and computability. However, this representation relies on a regular tensor product parametric domain, and the control mesh structure has global coupling. This means that refining local regions requires inserting nodes along the entire parametric direction, leading to an increase in the number of global control points. Therefore, traditional representations are more suitable for modeling macroscopic shapes with consistent scale, but struggle to support local refinement and the expression of micro-scale geometric features.

[0004] To overcome the global coupling limitations of traditional B-splines / NURBS in local refinement, researchers have developed various spline parameterization representations that support local refinement, among which truncated hierarchical B-splines (THB-Splines) are the most representative technique. They achieve adaptive refinement and detail enhancement of local regions by hierarchically organizing basis functions, while maintaining the smoothness and computability of the overall model, providing a more flexible framework for expressing microscale geometric features. However, truncated hierarchical B-splines still rely on regular tensor product topologies, a constraint that limits their adaptability in multi-scale geometric modeling.

[0005] In summary, traditional CAD technology struggles to support multi-scale modeling. Although methods such as truncated layered B-splines have the ability to locally refine and enhance details, they still rely on regular tensor product topologies, making it difficult to achieve effective cross-scale representation on topologically complex objects. Summary of the Invention

[0006] To address the aforementioned technical problems in existing technologies, this invention proposes a method for designing and manufacturing multi-scale geometric products. By allowing users to import a 3D control mesh through a CAD system's user interface, and by specifying control coefficients, subdivision units, and the number of subdivision layers, a multi-scale geometric product can be generated from any input topological control mesh. This product can then be directly used for processing and manufacturing via a CAM system. The specific technical solution is as follows:

[0007] A method for designing and manufacturing multi-scale geometric products includes the following steps:

[0008] Step 1: The user imports the 3D mesh through the interactive interface of the CAD system and extracts the control points and topology information of each mesh cell;

[0009] Step 2: Define the local coordinate system of the mesh element based on the element topology information, construct the initial basis functions of the element control points, and establish the geometric mapping from the mesh element parameter domain to the physical space;

[0010] Step 3: Set the control coefficients of the unit control points to generate the unit structure, and map it to the physical space to obtain the initial geometric product model;

[0011] Step 4: Subdivide the target mesh cells and their corresponding control coefficients, generate truncated layered basis functions based on the subdivision relationship, adjust the control coefficients associated with the fine layered basis functions, increase the geometric details of the cell structure, and map to obtain a multi-scale geometric product model;

[0012] Step 5: Import the multi-scale geometric product model obtained in Step 4 into the CAM system for processing and manufacturing.

[0013] Furthermore, the three-dimensional mesh is composed of the edge skeleton of an unstructured hexahedral mesh, and the control point of each mesh cell is the vertex of all 1-ring neighborhood cells of that cell; the cell topology information is defined by the connection relationship between each cell vertex and its adjacent vertices.

[0014] Furthermore, step 2 specifically includes:

[0015] Step 2.1: Obtain the topological information of each cell in Step 1, perform singularity detection on the mesh cells, construct a local coordinate system and parameter domain based on the detected singularities, and construct the initial basis functions for the control points;

[0016] Step 2.2: Establish the mapping relationship from the grid cell parameter domain to the physical space through the combination of basis functions and control points.

[0017] Further, step 2.1 specifically includes:

[0018] Step 2.1.1: Obtain cell topology information and detect the number of singular edges in the cells. If the number of singular edges is greater than 3, the mesh topology is incorrect and topology correction is required. In a regular Cartesian hexahedral mesh: each internal edge is shared by 4 hexahedral cells. If the number of adjacent hexahedral cells of an edge is ≠ 4, then the edge is a singular edge.

[0019] If the number of singular edges is less than or equal to 3, perform the following steps:

[0020] When the number of singular edges is 0, it is determined to be a non-singular element. Then, cubic B-spline basis functions are applied to the control points supporting the element, and a local parameter domain is established by selecting any vertex of the element as the origin.

[0021] If there is a singular edge, it is determined to be a singular cell, and any endpoint of the singular edge is selected as the origin;

[0022] When the number of singular edges is greater than 1, it is judged as a singular cell, and the intersection of the singular edges is selected as the origin;

[0023] Step 2.1.2: Based on the local coordinate origin, establish a local coordinate system and a unit cube parameter domain.

[0024] Furthermore, when it is determined to be a singular element in step 2.1.1, the following operations are performed:

[0025] S1: By constructing a subdivision matrix, singular units and corresponding parameter domains are subdivided to obtain singular and non-singular subunits;

[0026] S2: If the desired coordinate point does not fall within a non-singular sub-unit, continue to subdivide the singular sub-units until the desired coordinate point falls within a non-singular sub-unit, thereby obtaining the final subdivision level and constructing the selection matrix.

[0027] S3: Apply cubic B-spline basis functions to the control points of non-singular sub-elements, and backtrack them to the parent element through the subdivision matrix to obtain the singular basis functions of the parent element control points.

[0028] Furthermore, the subdivision rules are as follows: First, insert the cell center point, which is the weighted average sum of the cell vertices; second, insert the face center point of the cell face, which is the weighted average sum of the four vertices of the face and the cell center points of adjacent cells; then, insert the cell edge center point, which is the weighted average sum of the edge endpoints, the face center points of adjacent faces, and the cell center points of adjacent bodies; next, update the cell vertex coordinates, which is the weighted average sum of the vertex and the adjacent edge, face, and body center points; finally, connect the face center point and the edge center point, and connect the body center point and the face center point to update the topological relationship; the volume parameter domain is directly subdivided into an octree.

[0029] Further, step 3 specifically involves: first, setting corresponding control coefficients for all initial basis functions generated in step 2. The control coefficients are predefined by the system. Then, within the grid cell parameter domain, the cell structure is generated by combining the control coefficients and basis functions. If the cell structure needs to be redesigned, the values ​​of the control coefficients are optimized and adjusted. Finally, the designed cell structure is transformed into physical space.

[0030] Furthermore, step 4 specifically includes:

[0031] Step 4.1: Determine the mesh cells to be subdivided and the number of subdivision layers, and subdivide the mesh cells and control coefficients to obtain the sub-cells of each layer and the control coefficients of each layer's control points;

[0032] Step 4.2: Perform truncated layered basis function generation on each sub-unit, then adjust the control coefficients associated with the fine-layered basis functions to add geometric details to the unit structure, and finally map the updated unit structure to the physical space to obtain a multi-scale geometric product model.

[0033] Further, step 4.1 specifically includes:

[0034] Step 4.1.1: The user specifies the subdivision unit and the number of subdivision layers, or the relevant parameters are determined through an adaptive algorithm;

[0035] Step 4.1.2: Subdivide each unit to be subdivided into a specified number of layers, store the sub-units of each layer, and subdivide the control coefficients of the control points corresponding to the parent unit in the same subdivision method to obtain the control coefficients of the control points of each layer.

[0036] Further, step 4.2 specifically includes:

[0037] Step 4.2.1: Perform singularity detection on sub-units and establish local parameter domains. The specific execution is the same as step 2.1.

[0038] Step 4.2.2: Generate the basis functions of the fine-layer control points involved in the sub-unit according to steps S1~S3, truncate the parent layer basis functions involved in the sub-unit, and combine them to obtain the truncated layered basis functions;

[0039] The parent layer basis functions can be represented by a linear combination of the fine layer basis functions, with the combination coefficients derived from the subdivision matrix. When the support domain of a parent layer basis function crosses the fine layer parameter domain, i.e., the 2-ring element of the parent layer control point contains the fine layer region, the weights corresponding to the fine layer basis functions in the linear combination are reset to 0, which is called truncation.

[0040] Step 4.2.3: Adjust the fine-layer control coefficients obtained in step 4.1.2, and use the truncated layer basis function-control coefficient combination to obtain a unit structure with increased geometric details;

[0041] Step 4.2.4: Transform the updated unit structure to physical space using a truncated layered basis function-control point combination mapping.

[0042] The beneficial effects of this invention are: the method of this invention can construct basis functions that adapt to local structures on arbitrary topological meshes, thereby breaking through the dependence of traditional CAD on regular tensor product topology; at the same time, by organizing basis functions in a hierarchical manner, adaptive densification and detail enhancement of local regions are achieved, while maintaining the smoothness and computability of the model in the process, providing a more flexible framework for the design of multi-scale geometric products. Attached Figure Description

[0043] Figure 1 This is a flowchart illustrating the overall process framework of a multi-scale geometric product design and manufacturing method according to the present invention.

[0044] Figure 2 This is a flowchart of the basis function generation process of the present invention.

[0045] Figure 3 This is a schematic diagram of the geometric mapping of the present invention.

[0046] Figure 4 This is a schematic diagram of the layer-by-layer subdivision of the present invention.

[0047] Figure 5 This is an example diagram of the multi-scale geometric product design of the present invention. Detailed Implementation

[0048] To make the objectives, technical solutions, and technical effects of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings.

[0049] like Figure 1 As shown in the figure, a method for designing and manufacturing multi-scale geometric products based on arbitrary topological control grids and hierarchical basis functions according to an embodiment of the present invention includes the following steps:

[0050] Step 1: The user imports the 3D mesh through the interactive interface of the CAD system and extracts the control points P and element topology information of each mesh element;

[0051] ,

[0052] The three-dimensional mesh is composed of the edge skeleton of an unstructured hexahedral mesh. The control point of each mesh cell is the vertex of all 1-ring neighborhood cells of that cell. The cell topology information is defined by the connection relationship between each cell vertex and its adjacent vertices.

[0053] Step 2: Define the local coordinate system o-uvw of the mesh element based on the element topology information, construct the initial basis function N of the element control points, and establish the geometric mapping f from the mesh element parameter domain to the physical space.

[0054] Step 2 specifically includes:

[0055] Step 2.1: Obtain the topological information of each cell from Step 1, perform singularity detection on the mesh cells, construct a local coordinate system and parameter domain based on the detected singularities, and construct the initial basis functions for the control point P: .

[0056] Step 2.2: Using the aforementioned combination of initial basis functions and control points, establish the mapping relationship from the element parameter domain to the physical space: .

[0057] refer to Figure 2 Step 2.1 specifically includes:

[0058] Step 2.1.1: Obtain cell topology information and check the number of singular edges. If the number of singular edges is greater than 3, the mesh topology is incorrect and needs to be corrected. In an ideal regular Cartesian hexahedral mesh: each internal edge is shared by 4 hexahedral cells, and each internal vertex is shared by 8 hexahedral cells. An edge is a singular edge if the number of its adjacent hexahedral cells is ≠ 4, and a vertex is a singular vertex if the number of its adjacent hexahedral cells is ≠ 8. Under the subdivision rule, the number of singular edges in any hexahedral local neighborhood is restricted to a maximum of 3.

[0059] If the number of singular edges is less than or equal to 3, perform the following steps:

[0060] When the number of singular edges is 0, it is determined to be a non-singular element. Then, a cubic B-spline basis function is applied to the control points supporting the element, and any vertex of the element is selected as the origin.

[0061] If there is a singular edge, it is determined to be a singular cell, and any endpoint of the singular edge is selected as the origin;

[0062] When the number of singular edges is greater than 1, it is judged as a singular cell, and the intersection of the singular edges is selected as the origin;

[0063] Step 2.1.2: Based on the origin o, establish the local coordinate system o-uvw and the unit cube parameter domain. .

[0064] Specifically, when a singular element is identified in step 2.1.1, the following operations are performed:

[0065] S1: Singular cells and their corresponding parameter domains are subdivided by constructing a subdivision matrix A. The subdivision rules are as follows: First, insert the cell center point, which is the weighted average sum of the cell vertices; second, insert the face center point of the cell face, which is the weighted average sum of the four vertices of the face and the center points of adjacent cells; then, insert the center point of the cell edge, which is the weighted average sum of the edge endpoints, the face center point of the edge's adjacent face, and the center point of the edge's adjacent volume; next, update the cell vertex coordinates, which is the weighted average sum of the vertex and the center points of adjacent edges, faces, and volumes; finally, connect the face center point with the edge center point, and connect the volume center point with the face center point to update the topological relationships. The parameter domain of the standard cell mesh is directly subdivided using an octree.

[0066] All weight combinations constitute the subdivision matrix A, and the inserted new point Q can be represented as a linear combination of the original control points:

[0067] ,

[0068] By subdividing the singular unit using a subdivision matrix, one singular subunit and seven non-singular subunits can be obtained.

[0069] S2: If the point to be calculated in the parameter domain does not fall within a non-singular sub-unit, then continue to subdivide the singular sub-units until the coordinate point to be calculated falls within a non-singular sub-unit, thereby obtaining the final subdivision level lev and constructing the selection matrix G.

[0070] The selection matrix G is a 64×n matrix whose elements are either 0 or 1. The selection matrix is ​​used to obtain the control points of the subdivided non-singular subunits: there are 8 subunits after subdivision, and a total of n control points for the 8 subunits. The selection matrix obtains 64 control points for one of the non-singular subunits from these n control points. For a non-singular subunit, if the i-th control point among the 64 control points is the j-th control point among the n control points of the lev-th subdivision layer, then S... ij =1, the rest are 0.

[0071] S3: Apply cubic B-spline basis function b to the non-singular sub-element control points, and backtrack them to the parent element through the subdivision matrix to obtain the singular basis function N of the parent element control points. The specific calculation formula is as follows:

[0072] ,

[0073] in, It is the eigenvector matrix of the subdivision matrix. It is the eigenvalue diagonal matrix of the subdivision matrix.

[0074] Step 3: Set the control coefficient C for each unit control point to generate the unit structure. And map it to physical space to obtain the initial geometric product model S:

[0075] , ,

[0076] like Figure 3 As shown, step 3 specifically involves: first, setting corresponding control coefficients for all initial basis functions generated in step 2. The control coefficients can be predefined by the system or set by the user. Then, the element structure is generated by combining the control coefficients and basis functions within the grid element parameter domain. If the element structure needs to be redesigned, the values ​​of the control coefficients are optimized and adjusted. Finally, the designed element structure is transformed into physical space.

[0077] Step 4: Subdivide the target mesh cells selected by the user or adaptively selected by the system to obtain control points for each layer. And generate truncated hierarchical basis functions. Adjust the control coefficients of the fine-layer basis function correlation. This involves adding geometric details to the unit structure and mapping it to obtain a multi-scale geometric product model. The specific expression is as follows:

[0078] ,

[0079] ,

[0080] ,

[0081] ;

[0082] Specifically, step 4 includes the following:

[0083] Step 4.1: Determine the mesh cells to be subdivided and the number of subdivision layers, perform mesh and control coefficient subdivision, and obtain the sub-cells and control coefficients of each layer's control points, such as... Figure 4 As shown, the following sub-steps are specifically executed:

[0084] Step 4.1.1: The user specifies the subdivision unit and the number of subdivision layers, or the relevant parameters are determined through an adaptive algorithm;

[0085] Step 4.1.2: Subdivide each unit to be subdivided into a specified number of layers, store the sub-units of each layer, and subdivide the control coefficients of the control points corresponding to the parent unit in the same subdivision method to obtain the control coefficients of the control points of each layer.

[0086] Step 4.2: Truncate and layered basis functions are generated for each sub-unit. Then, the control coefficients associated with the fine-layer basis functions are adjusted to add geometric details to the unit structure. Finally, the updated unit structure is mapped to physical space to obtain a multi-scale geometric product model, such as... Figure 5 As shown, the following sub-steps are specifically executed:

[0087] Step 4.2.1: Perform singularity detection on sub-units and establish local parameter domains. The specific execution is the same as in step 2.1.

[0088] Step 4.2.2: Generate the basis functions of the fine-layer control points involved in the sub-unit according to steps S1~S3, truncate the parent layer basis functions involved in the sub-unit, and combine them to obtain the truncated layered basis functions;

[0089] The parent layer basis functions can be represented by a linear combination of the fine layer basis functions, with the combination coefficients derived from the subdivision matrix. When the support domain of a parent layer basis function crosses the fine layer parameter domain, i.e., the 2-ring element of the parent layer control point contains the fine layer region, the weights corresponding to the fine layer basis functions in the linear combination are reset to 0, a process known as truncation.

[0090] Step 4.2.3: Adjust the fine-layer control coefficients obtained in step 4.1.2, and use the truncated layer basis function-control coefficient combination to obtain a unit structure with increased geometric details;

[0091] Step 4.2.4: Transform the updated unit structure to physical space using a truncated layered basis function-control point combination mapping.

[0092] Step 5: Manufacturing of multi-scale geometric products.

[0093] The geometric product obtained in step 4 is then used for actual manufacturing. This can be done through 3D printing or other methods.

[0094] First, import the geometric product designed in the above steps into the CAM system, and then display it on the user interface for further user operations. The specific display technology and subsequent operations are consistent with existing commercial CAM software.

[0095] In summary, after a user imports a 3D mesh through the CAD system's interactive interface, this invention constructs basis functions for the mesh of arbitrary topology through subdivision, including both traditional spline basis functions and singular basis functions. Local geometric details are introduced through layer-by-layer subdivision and adjustment of control coefficients, thereby realizing the design of multi-scale geometric products. The layer-by-layer subdivision can be user-specified subdivision units and the number of layers, or it can be adaptively determined by an optimization algorithm.

[0096] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention in any way. Although the implementation process of the present invention has been described in detail above, those skilled in the art can still modify the technical solutions described in the foregoing examples or make equivalent substitutions for some of the technical features. All modifications and equivalent substitutions made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for designing and manufacturing multi-scale geometric products, characterized in that, Includes the following steps: Step 1: The user imports the 3D mesh through the interactive interface of the CAD system and extracts the control points and topology information of each mesh cell; Step 2: Define the local coordinate system of the mesh element based on the element topology information, construct the initial basis functions of the element control points, and establish the geometric mapping from the mesh element parameter domain to the physical space; Step 3: Set the corresponding control coefficients for all the initial basis functions generated in Step 2 to generate the unit structure, and map it to the physical space to obtain the initial geometric product model; Step 4: Subdivide the target mesh cells and their corresponding control coefficients, generate truncated layered basis functions based on the subdivision relationship, adjust the control coefficients associated with the fine layered basis functions, increase the geometric details of the cell structure, and map to obtain a multi-scale geometric product model; Step 5: Import the multi-scale geometric product model obtained in Step 4 into the CAM system for processing and manufacturing; Step 2 specifically includes: Step 2.1: Obtain the topological information of each cell in Step 1, perform singularity detection on the mesh cells, construct a local coordinate system and parameter domain based on the detected singularities, and construct the initial basis functions for the control points; Step 2.2: Establish the mapping relationship from the grid cell parameter domain to the physical space through the basis function-control point combination; Step 2.1 specifically includes: Step 2.1.1: Obtain cell topology information and detect the number of singular edges in the cells. If the number of singular edges is greater than 3, the mesh topology is incorrect and topology correction is required. In a regular Cartesian hexahedral mesh: each internal edge is shared by 4 hexahedral cells. If the number of adjacent hexahedral cells of an edge is ≠ 4, then the edge is a singular edge. If the number of singular edges is less than or equal to 3, perform the following steps: When the number of singular edges is 0, it is determined to be a non-singular element. Then, cubic B-spline basis functions are applied to the control points supporting the element, and a local parameter domain is established by selecting any vertex of the element as the origin. If there is a singular edge, it is determined to be a singular cell, and any endpoint of the singular edge is selected as the origin; When the number of singular edges is greater than 1, it is judged as a singular cell, and the intersection of the singular edges is selected as the origin; Step 2.1.2: Based on the local coordinate origin, establish a local coordinate system and a unit cube parameter domain.

2. The design and manufacturing method as described in claim 1, characterized in that, The three-dimensional mesh is composed of the edge skeleton of an unstructured hexahedral mesh. The control point of each mesh cell is the vertex of all 1-ring neighborhood cells of that cell. The cell topology information is defined by the connection relationship between each cell vertex and its adjacent vertices.

3. The design and manufacturing method as described in claim 1, characterized in that, If the element is identified as a singular element in step 2.1.1, perform the following operations: S1: By constructing a subdivision matrix, singular units and corresponding parameter domains are subdivided to obtain singular and non-singular subunits; S2: If the desired coordinate point does not fall within a non-singular sub-unit, then continue to subdivide the singular sub-units until the desired coordinate point falls within a non-singular sub-unit, thereby obtaining the final subdivision level and constructing the selection matrix. The selection matrix G is a 64×n matrix whose elements take values ​​of 0 or 1, and n is the number of control points. S3: Apply cubic B-spline basis functions to the control points of non-singular sub-elements, and backtrack them to the parent element through the subdivision matrix to obtain the singular basis functions of the parent element control points.

4. The design and manufacturing method as described in claim 3, characterized in that, The subdivision rules are as follows: First, insert the cell center point, which is the weighted average sum of the cell vertices; second, insert the face center point of the cell face, which is the weighted average sum of the four vertices of the face and the cell center points of adjacent cells; then insert the cell edge center point, which is the weighted average sum of the edge endpoints, the face center points of adjacent faces, and the cell center points of adjacent bodies; next, update the cell vertex coordinates, which is the weighted average sum of the vertex and the adjacent edge, face, and body center points; finally, connect the face center point and the edge center point, and connect the body center point and the face center point to update the topological relationship; the volume parameter domain is directly subdivided into an octree.

5. The design and manufacturing method as described in claim 3, characterized in that, Step 3 specifically involves: first, the system predefines control coefficients; then, within the grid cell parameter domain, the control coefficients and basis functions are combined to generate the cell structure; if the cell structure needs to be redesigned, the values ​​of the control coefficients are optimized and adjusted; and finally, the designed cell structure is transformed into physical space.

6. The design and manufacturing method as described in claim 3, characterized in that, Step 4 specifically includes: Step 4.1: Determine the mesh cells to be subdivided and the number of subdivision layers, and subdivide the mesh cells and control coefficients to obtain the sub-cells of each layer and the control coefficients of each layer's control points; Step 4.2: Perform truncated layered basis function generation on each sub-unit, then adjust the control coefficients associated with the fine-layered basis functions to add geometric details to the unit structure, and finally map the updated unit structure to the physical space to obtain a multi-scale geometric product model.

7. The design and manufacturing method as described in claim 6, characterized in that, Step 4.1 specifically includes: Step 4.1.1: The user specifies the subdivision unit and the number of subdivision layers, or the relevant parameters are determined through an adaptive algorithm; Step 4.1.2: Subdivide each unit to be subdivided into a specified number of layers, store the sub-units of each layer, and subdivide the control coefficients of the control points corresponding to the parent unit in the same subdivision method to obtain the control coefficients of the control points of each layer.

8. The design and manufacturing method as described in claim 6, characterized in that, Step 4.2 specifically includes: Step 4.2.1: Perform singularity detection on sub-units and establish local parameter domains. The specific execution is the same as in step 2.

1. Step 4.2.2: Generate the basis functions of the fine-layer control points involved in the sub-unit according to steps S1~S3, truncate the parent layer basis functions involved in the sub-unit, and combine them to obtain the truncated layered basis functions; The parent layer basis function can be represented by a linear combination of the fine layer basis functions, and the combination coefficients come from the subdivision matrix. When the support domain of a certain parent layer basis function crosses the fine layer parameter domain, that is, the 2-ring cell of the parent layer control point contains the fine layer region, the weight corresponding to the fine layer basis function in the linear combination is reset to 0, which is called truncation. Step 4.2.3: Adjust the fine-layer control coefficients obtained in step 4.1.2, and use the truncated layer basis function-control coefficient combination to obtain a unit structure with increased geometric details; Step 4.2.4: Transform the updated unit structure to physical space using a truncated layered basis function-control point combination mapping.

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