Forward Design Method of Hexahedral Mesh Model Based on Quadrilateral Meshing

Abstract

The present invention discloses a forward design method of a hexahedral mesh model based on quadrilateral mesh division, S1, constructing a two-dimensional entity on a sketch through basic graphic elements. S2, obtaining a divided quadrilateral mesh; S3, updating the coordinates of the internal vertices of the quadrilateral mesh to the average value of the coordinates of adjacent vertices; S4, constructing a hexahedral mesh from the quadrilateral mesh; S5, setting a loop on the surface of the hexahedral mesh and adjusting the surface mesh; S6, for the quadrilaterals in S5 that have been adjusted and have vertices on the S4 loop, subdividing them according to different situations; S7, using the quadrilateral meshes in S5 and S6 that are inside the loop as the new bottom surface, stretching, rotating or sweeping operations are performed in the manner of S4 to obtain the final hexahedral mesh model. The present invention can solve the two tasks of CAD design and mesh division at the same time.

Description

Technical Field

[0001] The invention relates to the technical field of hexahedral mesh models, and in particular to a forward design method of a hexahedral mesh model based on quadrilateral mesh division. Background Art

[0002] As a form of representation that contains both the surface and internal information of an object, volume mesh can provide richer information than surface mesh and has important applications in CAE. Tetrahedral mesh and hexahedral mesh are the two most commonly used volume meshes. Hexahedral mesh units have better shape regularity and symmetry, and usually have higher accuracy and stability in numerical simulation. However, the current CAD step only designs the surface of the model. In subsequent CAE finite element analysis and other tasks, the model needs to be further divided into volume meshes. According to statistics, the time for model preprocessing and mesh generation accounts for about 80% of the entire analysis time. Meshing has become its performance bottleneck.

[0003] Therefore, it is urgent to propose a forward design method for hexahedral meshes to solve the time-consuming defect of the hexahedral mesh generation algorithm, realize the design and construction of hexahedral meshes in the process of CAD forward design, and avoid the time-consuming problem of the traditional volume mesh generation method of first designing and then dividing. Summary of the invention

[0004] One purpose of the present invention is to propose a forward design method for a hexahedral mesh model based on quadrilateral mesh division. The present invention has high speed and good interactivity, supports sketch design, stretching, rotation, sweeping, and re-stretching operations, and can simultaneously solve the two tasks of CAD design and mesh division.

[0005] A forward design method for a hexahedral mesh model based on quadrilateral mesh division according to an embodiment of the present invention comprises the following steps:

[0006] S1. Construct two-dimensional entities on the sketch through basic graphic elements;

[0007] S2, using a quadrilateral meshing algorithm to mesh the two-dimensional entity to obtain a meshed quadrilateral mesh;

[0008] S3, optimizing the quadrilateral mesh, and updating the coordinates of the internal vertices of the quadrilateral mesh to the average value of the coordinates of the adjacent vertices;

[0009] S4, constructing a three-dimensional mesh from the quadrilateral mesh optimized in S3, and obtaining a set of hexahedrons from each quadrilateral in the two-dimensional sketch by stretching, rotating or sweeping, and the hexahedrons corresponding to all the quadrilateral meshes together constitute a hexahedral mesh;

[0010] S5. Using polygonal marks on the hexahedral mesh surface to indicate the boundary loop of the graphic that needs to be re-stretched, rotated or swept, finding the intersection of the quadrilateral mesh on the hexahedral mesh surface and the boundary loop, and adjusting the quadrilateral mesh on the hexahedral mesh surface to conform to the required shape according to the position of the intersection of the quadrilateral mesh and the boundary loop;

[0011] S6. For the quadrilaterals adjusted in S5 and having vertices on the loop of S4, subdivide them according to different situations;

[0012] S7. Take the quadrilateral mesh inside the loop in S5 and S6 as the new bottom surface, and perform stretching, rotation or sweeping operations in the manner of S4 to obtain the final hexahedral mesh model.

[0013] Optionally, the entity of the sketch includes an outer ring and multiple inner rings, the outer ring is the outer contour of the sketch, and the inner rings are holes on the sketch. The basic graphic elements include line segments, polylines, arcs and subdivision curves. For the two types of basic graphic elements, arcs and subdivision curves, sampling and discretization are performed with specified accuracy to obtain a set of discrete polylines on the sketch, and the two-dimensional entities on the sketch are represented by polylines.

[0014] Optionally, the S2 specifically includes:

[0015] S21, specify the grid size S to divide the 2D sketch plane into a square grid, and only keep the squares whose four vertices are inside the 2D solid;

[0016] S22, identify all corner feature vertices of the internal square grid respectively All corner feature vertices of a 2D solid , and match between the two;

[0017] Corner feature vertices of the inner square mesh The judgment method is: the vertex whose number of adjacent quadrilaterals is 1 or 3, and whose adjacent vertices do not have the number of adjacent quadrilaterals of 1 or 3 is a corner feature vertex;

[0018] Corner feature vertex of 2D solid The judgment method is as follows: the vertex whose angle between two adjacent line segments is less than the threshold is the corner feature vertex;

[0019] S23, corner feature vertices for a square grid on a two-dimensional solid , if the corner feature vertex and the corner feature vertex of the nearest square mesh The distance between , then the corner feature vertices of the square grid are The projection is set to the corner feature vertex of the 2D solid , otherwise the distance from the corner feature vertex of the two-dimensional entity will be The projection of the boundary vertices on the nearest square mesh is set as the corner feature vertex of the 2D solid ;

[0020] S24. For other vertices on the boundary of the square grid that have no projection set, set their projections to their projections on the two-dimensional entity, that is, the points on the boundary of the two-dimensional entity that are closest to them. If the projections are aligned with a corner feature vertex on the two-dimensional entity, If they overlap, their projections are changed to angular feature vertices of two-dimensional entities. midpoints of adjacent sides;

[0021] S25. Sequentially connect all the edge points and their projections in all the square grids, and all the generated quadrilaterals and the inner square grids constitute the quadrilateral grid after the division.

[0022] Optionally, the quadrilateral mesh is optimized by: Update and repeat the update process until convergence, the internal vertices of the quadrilateral mesh The updated coordinates are:

[0023] ;

[0024] in, The internal vertex coordinates of the quadrilateral mesh after the vertex is updated. The inner vertices of the quad mesh The number of adjacent vertices, For the internal vertices of the quad mesh Adjacent vertices.

[0025] Optionally, the stretching step is to calculate the average side length of the quadrilateral mesh , given a direction vector and the corresponding stretch length , calculate the most suitable number of stretching layers:

[0026] ;

[0027] Calculate the translation vector for a single step for:

[0028] ;

[0029] in, Indicates that the vector The component in the axial direction, Indicates that the vector The component in the axial direction, Indicates that the vector Component in the axial direction;

[0030] For a quad in a quad mesh , a new quadrilateral is generated in each layer during the stretching process in an iterative manner, and the corresponding vertices of the quadrilateral in the previous layer are connected with those of the quadrilateral in the next layer to obtain the hexahedron corresponding to the quadrilateral in this layer:

[0031] ;

[0032] in, For the quadrilateral in The hexahedron generated by the layer, express In the The corresponding vertices of the layer, , To construct 8 vertices into a hexahedron, the iteration of each layer of vertices is:

[0033] ;

[0034] Through an iterative process, each quadrilateral mesh is stretched n layers along direction d to form a hexahedral mesh.

[0035] Optionally, the step of rotating is to calculate the average side length of the quadrilateral mesh , given two points on the rotation axis, denoted as and The total angle of rotation is , calculate the average distance from the points on the quadrilateral mesh to the rotation axis , calculate the most suitable number of stretching layers:

[0036] ;

[0037] Calculate the rotation angle for each layer:

[0038] ;

[0039] For each quadrilateral in the quad mesh , a new quadrilateral is generated at each layer in the rotation process in an iterative manner. In each layer, each vertex is the corresponding vertex of the previous layer rotated around the axis by an angle :

[0040] ;

[0041] in, express In the The corresponding vertices of the layer, is the translation vector of the rotation step;

[0042] Connect the corresponding vertices of the quadrilateral in the previous layer and the quadrilateral in the next layer to obtain the hexahedron corresponding to the quadrilateral in this layer.

[0043] Optionally, the sweeping step is given by a series of points The path composed of each quadrilateral in the quadrilateral mesh , a new quadrilateral is generated at each layer according to the path generation, and the sweep algorithm calculates the path direction. The path direction of each layer , the path direction is from arrive Directions:

[0044] ;

[0045] Then calculate the orthogonal basis vectors and set the initial upward vector for , for each layer , calculate the right direction , and and Orthogonal:

[0046] ;

[0047] in, It is a normalization operation;

[0048] Recalculate the up vector of the next layer for:

[0049] ;

[0050] Then calculate the vertex transformation, based on the quadrilateral vertices of the previous layer Calculate the quadrilateral vertices of the next layer , connect the corresponding vertices of the quadrilaterals in the previous layer with those in the next layer to obtain the hexahedron corresponding to the quadrilaterals in this layer.

[0051] Optionally, the subdivision process specifically includes, for a quadrilateral AHGD whose vertex A is on the loop, taking the center of the quadrilateral as , respectively connect With AH, AD center, and connect The quadrilateral is divided into three parts. For the quadrilateral CDEF whose two adjacent vertices C and E are on the loop, connect the midpoints of AD and BC. The quadrilateral is divided into two parts. For the quadrilateral ABCD whose three vertices A, B, and C are on the loop, take the center of the quadrilateral as , respectively connect The midpoint of AD and the midpoint of CD , at this time the quadrilateral is divided into three;

[0052] For the subdivided surface quadrilateral, find the quadrilateral opposite to it, apply the same subdivision rule to the opposite quadrilateral, and repeat the process until reaching another boundary. When the adjusted quadrilaterals form a path, delete all hexahedrons on the path, and the corresponding vertices of the adjusted quadrilaterals form a new hexahedron.

[0053] The beneficial effects of the present invention are:

[0054] The present invention can complete the forward construction of hexahedral meshes at a relatively fast speed and ensure that no hexahedron with a negative Jacobian value will appear. It only takes a few seconds to complete the forward construction of tens of thousands of hexahedral mesh models, thus solving the problems of the prior art. BRIEF DESCRIPTION OF THE DRAWINGS

[0055] The accompanying drawings are used to provide a further understanding of the present invention and constitute a part of the specification. Together with the embodiments of the present invention, they are used to explain the present invention and do not constitute a limitation of the present invention. In the accompanying drawings:

[0056] Figure 1 A flow chart of a forward design method of a hexahedral mesh model based on quadrilateral mesh division proposed by the present invention;

[0057] Figure 2 The invention proposes a forward design method for a hexahedral mesh model based on quadrilateral mesh division, which includes a two-dimensional entity on a sketch and a square mesh inside the entity.

[0058] Figure 3 The invention proposes a forward design method for a hexahedral mesh model based on quadrilateral mesh division, which is a quadrilateral mesh after the internal square meshes are projected and connected.

[0059] Figure 4 It is a quadrilateral mesh optimized by a forward design method of a hexahedral mesh model based on quadrilateral mesh division proposed by the present invention.

[0060] Figure 5 The present invention proposes a forward design method for a hexahedral mesh model based on quadrilateral mesh division, in which the quadrilateral mesh is stretched to generate a hexahedral mesh.

[0061] Figure 6 The present invention proposes a forward design method for a hexahedral mesh model based on quadrilateral mesh division, in which the quadrilateral mesh is rotated to generate a hexahedral mesh.

[0062] Figure 7 The present invention proposes a forward design method for a hexahedral mesh model based on quadrilateral mesh division, in which the quadrilateral mesh is swept to generate a hexahedral mesh.

[0063] Figure 8 A schematic diagram of subdivision rules for optimizing quadrilateral meshes in a forward design method for a hexahedral mesh model based on quadrilateral mesh division proposed by the present invention.

[0064] Fig. 9 The present invention proposes a forward design method for a hexahedral mesh model based on quadrilateral mesh division according to the surface quadrilateral mesh adjusted as needed.

[0065] Fig.10 The invention provides a surface quadrilateral mesh optimized by a forward design method of a hexahedral mesh model based on quadrilateral mesh division.

[0066] Fig.11 This is a schematic diagram of a hexahedral mesh after re-stretching in a forward design method of a hexahedral mesh model based on quadrilateral mesh division proposed in the present invention. DETAILED DESCRIPTION

[0067] The present invention will now be described in further detail with reference to the accompanying drawings. These drawings are simplified schematic diagrams, which only illustrate the basic structure of the present invention in a schematic manner, and therefore only show the components related to the present invention.

[0068] refer to Figure 1-Figure 11 , a forward design method for a hexahedral mesh model based on quadrilateral mesh division, comprising the following steps:

[0069] S1. Construct two-dimensional entities on the sketch through basic graphic elements;

[0070] S2, using a quadrilateral meshing algorithm to mesh the two-dimensional entity to obtain a meshed quadrilateral mesh;

[0071] S3, optimizing the quadrilateral mesh, and updating the coordinates of the internal vertices of the quadrilateral mesh to the average value of the coordinates of the adjacent vertices;

[0072] S4, constructing a three-dimensional mesh from the quadrilateral mesh optimized in S3, and obtaining a set of hexahedrons from each quadrilateral in the two-dimensional sketch by stretching, rotating or sweeping, and the hexahedrons corresponding to all the quadrilateral meshes together constitute a hexahedral mesh;

[0073] S5. Using polygonal marks on the hexahedral mesh surface to indicate the boundary loop of the graphic that needs to be re-stretched, rotated or swept, finding the intersection of the quadrilateral mesh on the hexahedral mesh surface and the boundary loop, and adjusting the quadrilateral mesh on the hexahedral mesh surface to conform to the required shape according to the position of the intersection of the quadrilateral mesh and the boundary loop;

[0074] Specifically, for models that need to be re-stretched, rotated, and swept, their surface meshes need to be adjusted. Use closed curves or polygonal markers on the surface of the hexahedral mesh to indicate the boundary loop of the area that needs to be re-stretched, rotated, or swept, find the intersection of the surface quadrilateral mesh and the loop, and for each edge that intersects with the loop, move the endpoint that is closer to the intersection to the boundary. At the same time, do another scan. If both points on the diagonal of a quadrilateral are on the boundary at the same time, choose to move one of the remaining two vertices of the quadrilateral to the boundary. The vertex selected is determined based on the quality of the quadrilateral after it is moved.

[0075] S6. For the quadrilaterals adjusted in S5 and having vertices on the loop of S4, subdivide them according to different situations;

[0076] S7, taking the quadrilateral mesh inside the loop in S5 and S6 as the new bottom surface, stretching, rotating or sweeping operations are performed in the manner of S4 to obtain the final hexahedral mesh model. Fig.11 Shows a hexahedral mesh model after a cylinder is extruded from the surface of the hexahedral mesh.

[0077] In this implementation, the entity of the sketch includes an outer ring and multiple inner rings, the outer ring is the outer contour of the sketch, and the inner ring is the hole on the sketch. The basic graphic elements include line segments, polylines, arcs and subdivision curves. For the two types of basic graphic elements, arcs and subdivision curves, sampling and discretization are performed with specified accuracy to obtain a set of discrete polylines on the sketch, and the two-dimensional entities on the sketch are represented by polylines.

[0078] Attached Figure 2 The boundary of the forward-designed two-dimensional entity is given, and the two-dimensional entity is composed of line segments, polylines and sampled continuous curves. In this implementation, S2 specifically includes:

[0079] S21, specify the grid size S to divide the 2D sketch plane into a square grid, and only keep the squares whose four vertices are inside the 2D solid;

[0080] S22, identify all corner feature vertices of the internal square grid respectively All corner feature vertices of a 2D solid , and match between the two;

[0081] Corner feature vertices of the inner square mesh The judgment method is: the vertex whose number of adjacent quadrilaterals is 1 or 3, and whose adjacent vertices do not have the number of adjacent quadrilaterals of 1 or 3 is a corner feature vertex;

[0082] Corner feature vertex of 2D solid The judgment method is as follows: the vertex whose angle between two adjacent line segments is less than the threshold is the corner feature vertex;

[0083] S23, corner feature vertices for a square grid on a two-dimensional solid , if the corner feature vertex and the corner feature vertex of the nearest square mesh The distance between , then the corner feature vertices of the square grid are The projection is set to the corner feature vertex of the 2D solid , otherwise the distance from the corner feature vertex of the two-dimensional entity will be The projection of the boundary vertices on the nearest square mesh is set as the corner feature vertex of the 2D solid ;

[0084] S24. For other vertices on the boundary of the square grid that have no projection set, set their projections to their projections on the two-dimensional entity, that is, the points on the boundary of the two-dimensional entity that are closest to them. If the projections are aligned with a corner feature vertex on the two-dimensional entity, If they overlap, their projections are changed to angular feature vertices of two-dimensional entities. midpoints of adjacent sides;

[0085] S25. Sequentially connect all the edge points and their projections in all the square grids, and all the generated quadrilaterals and the inner square grids constitute the quadrilateral grid after the division.

[0086] Attached Figure 2 The quadrilateral mesh after the inner square mesh is projected and connected is given in the attached figure. Figure 4 For attachment Figure 3 The result after quadrilateral mesh optimization is that the quadrilateral mesh optimization step greatly improves the quality of the quadrilateral mesh and removes degenerate quadrilaterals and concave polygons. In this embodiment, the quadrilateral mesh optimization specifically includes: for each internal vertex of the quadrilateral mesh Update and repeat the update process until convergence, the internal vertices of the quadrilateral mesh The updated coordinates are:

[0087] ;

[0088] in, The internal vertex coordinates of the quadrilateral mesh after the vertex is updated. The inner vertices of the quad mesh The number of adjacent vertices, For the internal vertices of the quad mesh Adjacent vertices.

[0089] In this embodiment, the stretching step is to calculate the average side length of the quadrilateral mesh , given a direction vector and the corresponding stretch length , calculate the most suitable number of stretching layers:

[0090] ;

[0091] Calculate the translation vector for a single step for:

[0092] ;

[0093] in, Indicates that the vector The component in the axial direction, Indicates that the vector The component in the axial direction, Indicates that the vector Component in the axial direction;

[0094] For a quad in a quad mesh , a new quadrilateral is generated in each layer during the stretching process in an iterative manner, and the corresponding vertices of the quadrilateral in the previous layer are connected with those of the quadrilateral in the next layer to obtain the hexahedron corresponding to the quadrilateral in this layer:

[0095] ;

[0096] in, For the quadrilateral in The hexahedron generated by the layer, express In the The corresponding vertices of the layer, , To construct 8 vertices into a hexahedron, the iteration of each layer of vertices is:

[0097] ;

[0098] Through an iterative process, each quadrilateral mesh is stretched n layers along direction d to form a hexahedral mesh.

[0099] In this embodiment, the rotation step is to calculate the average side length of the quadrilateral mesh , given two points on the rotation axis, denoted as and The total angle of rotation is , calculate the average distance from the points on the quadrilateral mesh to the rotation axis , calculate the most suitable number of stretching layers:

[0100] ;

[0101] Calculate the rotation angle for each layer:

[0102] ;

[0103] For each quadrilateral in the quad mesh , a new quadrilateral is generated at each layer in the rotation process in an iterative manner. In each layer, each vertex is the corresponding vertex of the previous layer rotated around the axis by an angle :

[0104] ;

[0105] in, express In the The corresponding vertices of the layer, is the translation vector of the rotation step;

[0106] Connect the corresponding vertices of the quadrilateral in the previous layer and the quadrilateral in the next layer to obtain the hexahedron corresponding to the quadrilateral in this layer.

[0107] In this embodiment, the step of sweeping is to give a series of points The path composed of each quadrilateral in the quadrilateral mesh , a new quadrilateral is generated at each layer according to the path generation, and the sweep algorithm calculates the path direction. The path direction of each layer , the path direction is from arrive Directions:

[0108] ;

[0109] Then calculate the orthogonal basis vectors and set the initial upward vector for , for each layer , calculate the right direction , and and Orthogonal:

[0110] ;

[0111] in, It is a normalization operation;

[0112] Recalculate the up vector of the next layer for:

[0113] ;

[0114] Then calculate the vertex transformation, based on the quadrilateral vertices of the previous layer Calculate the quadrilateral vertices of the next layer , connect the corresponding vertices of the quadrilaterals in the previous layer with those in the next layer to obtain the hexahedron corresponding to the quadrilaterals in this layer.

[0115] In this embodiment, the subdivision process specifically includes taking the center of a quadrilateral AHGD with a vertex A on the loop as , respectively connect With AH, AD center, and connect The quadrilateral is divided into three parts. For the quadrilateral CDEF whose two adjacent vertices C and E are on the loop, connect the midpoints of AD and BC. The quadrilateral is divided into two parts. For the quadrilateral ABCD whose three vertices A, B, and C are on the loop, take the center of the quadrilateral as , respectively connect The midpoint of AD and the midpoint of CD , at this time the quadrilateral is divided into three;

[0116] For the subdivided surface quadrilateral, find the quadrilateral opposite to it, apply the same subdivision rule to the opposite quadrilateral, and repeat the process until reaching another boundary. When the adjusted quadrilaterals form a path, delete all hexahedrons on the path, and the corresponding vertices of the adjusted quadrilaterals form a new hexahedron.

[0117] The present invention can complete the forward construction of hexahedral meshes at a relatively fast speed and ensure that no hexahedron with a negative Jacobian value will appear. It only takes a few seconds to complete the forward construction of tens of thousands of hexahedral mesh models, thus solving the problems of the prior art.

[0118] The above description is only a preferred specific implementation manner of the present invention, but the protection scope of the present invention is not limited thereto. Any technician familiar with the technical field can make equivalent replacements or changes according to the technical scheme and inventive concept of the present invention within the technical scope disclosed by the present invention, which should be covered by the protection scope of the present invention.

Claims

1. A forward design method for a hexahedral mesh model based on quadrilateral meshing, characterized in that: The steps include: S1. Construct two-dimensional entities on the sketch through basic graphic elements; S2, using a quadrilateral meshing algorithm to mesh the two-dimensional entity to obtain a meshed quadrilateral mesh; S3, optimizing the quadrilateral mesh, and updating the coordinates of the internal vertices of the quadrilateral mesh to the average value of the coordinates of the adjacent vertices; S4, constructing a three-dimensional mesh from the quadrilateral mesh optimized in S3, and obtaining a set of hexahedrons from each quadrilateral in the two-dimensional sketch by stretching, rotating or sweeping, and the hexahedrons corresponding to all the quadrilateral meshes together constitute a hexahedral mesh; S5. Using polygonal marks on the hexahedral mesh surface to indicate the boundary loop of the graphic that needs to be re-stretched, rotated or swept, finding the intersection of the quadrilateral mesh on the hexahedral mesh surface and the boundary loop, and adjusting the quadrilateral mesh on the hexahedral mesh surface to conform to the required shape according to the position of the intersection of the quadrilateral mesh and the boundary loop; S6. For the quadrilaterals adjusted in S5 and having vertices on the loop of S4, subdivide them according to different situations; S7. Take the quadrilateral mesh inside the loop in S5 and S6 as the new bottom surface, and perform stretching, rotation or sweeping operations in the manner of S4 to obtain the final hexahedral mesh model.

2. The forward design method of a hexahedral mesh model based on quadrilateral mesh division according to claim 1, characterized in that: The entity of the sketch includes an outer ring and multiple inner rings, the outer ring is the outer contour of the sketch, and the inner rings are holes on the sketch. The basic graphic elements include line segments, polylines, arcs and subdivision curves. For the two types of basic graphic elements, arcs and subdivision curves, sampling and discretization are performed with specified accuracy to obtain a set of discrete polylines on the sketch, and the two-dimensional entities on the sketch are represented by polylines.

3. The forward design method of a hexahedral mesh model based on quadrilateral mesh division according to claim 1, characterized in that: The S2 specifically includes: S21, specify the grid size S to divide the 2D sketch plane into a square grid, and only keep the squares whose four vertices are inside the 2D solid; S22, identify all corner feature vertices of the internal square grid respectively All corner feature vertices of a 2D solid , and match between the two; Corner feature vertices of the inner square mesh The judgment method is: the vertex whose number of adjacent quadrilaterals is 1 or 3, and whose adjacent vertices do not have the number of adjacent quadrilaterals of 1 or 3 is a corner feature vertex; Corner feature vertex of 2D solid The judgment method is as follows: the vertex whose angle between two adjacent line segments is less than the threshold is the corner feature vertex; S23, corner feature vertices for a square grid on a two-dimensional solid , if the corner feature vertex and the corner feature vertex of the nearest square mesh The distance between , then the corner feature vertices of the square grid are The projection is set to the corner feature vertex of the 2D solid , otherwise the distance from the corner feature vertex of the two-dimensional entity will be The projection of the boundary vertices on the nearest square mesh is set as the corner feature vertex of the 2D solid ; S24. For other vertices on the boundary of the square grid that have no projection set, set their projections to their projections on the two-dimensional entity, that is, the points on the boundary of the two-dimensional entity that are closest to them. If the projections are aligned with a corner feature vertex on the two-dimensional entity, If they overlap, their projections are changed to angular feature vertices of two-dimensional entities. midpoints of adjacent sides; S25. Sequentially connect all the edge points and their projections in all the square grids, and all the generated quadrilaterals and the inner square grids constitute the quadrilateral grid after the division.

4. The forward design method of a hexahedral mesh model based on quadrilateral mesh division according to claim 1, characterized in that: The quadrilateral mesh is optimized by: Update and repeat the update process until convergence, the internal vertices of the quadrilateral mesh The updated coordinates are: ； in, The internal vertex coordinates of the quadrilateral mesh after the vertex is updated. The inner vertices of the quad mesh The number of adjacent vertices, For the internal vertices of the quad mesh Adjacent vertices.

5. The forward design method of a hexahedral mesh model based on quadrilateral mesh division according to claim 1, characterized in that: The stretching is to calculate the average side length of the quadrilateral mesh , given a direction vector and the corresponding stretch length , calculate the most suitable number of stretching layers: ； Calculate the translation vector for a single step for: ； in, Indicates that the vector The component in the axial direction, Indicates that the vector The component in the axial direction, Indicates that the vector Component in the axial direction; For a quad in a quad mesh , a new quadrilateral is generated in each layer during the stretching process in an iterative manner, and the corresponding vertices of the quadrilateral in the previous layer are connected with those of the quadrilateral in the next layer to obtain the hexahedron corresponding to the quadrilateral in this layer: ； in, For the quadrilateral in The hexahedron generated by the layer, express In the The corresponding vertices of the layer, , To construct 8 vertices into a hexahedron, the iteration of each layer of vertices is: ； Through an iterative process, each quadrilateral mesh is stretched n layers along direction d to form a hexahedral mesh.

6. The forward design method of a hexahedral mesh model based on quadrilateral mesh division according to claim 1, characterized in that: The rotation step is to calculate the average side length of the quadrilateral mesh , given two points on the rotation axis, denoted as and The total angle of rotation is , calculate the average distance from the points on the quadrilateral mesh to the rotation axis , calculate the most suitable number of stretching layers: ； Calculate the rotation angle for each layer: ； For each quadrilateral in the quad mesh , a new quadrilateral is generated at each layer in the rotation process in an iterative manner. In each layer, each vertex is the corresponding vertex of the previous layer rotated around the axis by an angle : ； in, express In the The corresponding vertices of the layer, is the translation vector of the rotation step; Connect the corresponding vertices of the quadrilateral in the previous layer and the quadrilateral in the next layer to obtain the hexahedron corresponding to the quadrilateral in this layer.

7. The forward design method of a hexahedral mesh model based on quadrilateral mesh division according to claim 1, characterized in that: The sweeping step is given by a series of points The path composed of each quadrilateral in the quadrilateral mesh , a new quadrilateral is generated at each layer according to the path generation, and the sweep algorithm calculates the path direction. The path direction of each layer , the path direction is from arrive Directions: ； Then calculate the orthogonal basis vectors and set the initial upward vector for , for each layer , calculate the right direction , and and Orthogonal: ； in, It is a normalization operation; Recalculate the up vector of the next layer for: ； Then calculate the vertex transformation, based on the quadrilateral vertices of the previous layer Calculate the quadrilateral vertices of the next layer , connect the corresponding vertices of the quadrilaterals in the previous layer with those in the next layer to obtain the hexahedron corresponding to the quadrilaterals in this layer.

8. The forward design method of a hexahedral mesh model based on quadrilateral mesh division according to claim 1, characterized in that: The subdivision process specifically includes taking a quadrilateral AHGD with a vertex A on the loop and taking the center of the quadrilateral as , respectively connect With AH, AD center, and connect The quadrilateral is divided into three parts. For the quadrilateral CDEF whose two adjacent vertices C and E are on the loop, connect the midpoints of AD and BC. The quadrilateral is divided into two parts. For the quadrilateral ABCD whose three vertices A, B, and C are on the loop, take the center of the quadrilateral as , respectively connect The midpoint of AD and the midpoint of CD , at this time the quadrilateral is divided into three; For the subdivided surface quadrilateral, find the quadrilateral opposite to it, apply the same subdivision rule to the opposite quadrilateral, and repeat the process until reaching another boundary. When the adjusted quadrilaterals form a path, delete all hexahedrons on the path, and the corresponding vertices of the adjusted quadrilaterals form a new hexahedron.

Images