A Random Generation Method for Rotating Rigid Body Structures Based on Interpolation Boundaries

Abstract

The present invention discloses a method for randomly generating a rotating rigid body structure based on an interpolation boundary, which relates to the technical field of rotating rigid body processing, and includes the following steps: S1. Generate a random domain according to the size of the rotating rigid body and determine a number of limit points; S2. Generate an interpolation function according to the number of limit points; S3. Use a number of fine points to draw an interpolation curve corresponding to the interpolation function, and transform the interpolation curve to generate a flexible hinge and its surrounding local structure as well as a rotating rigid body unit; S4. Generate a three-dimensional model of the rotating rigid body according to the flexible hinge and its surrounding local structure as well as the rotating rigid body unit. The present invention can quickly generate a large number of structures, reduce intermediate labor and does not require prior knowledge and design experience; compared with the local topology generation method, the present invention has stronger randomness and globality, and does not need to consider complex constraint conditions and design objectives.

Description

Technical Field

[0001] The present invention relates to the technical field of rotating rigid body processing, and specifically relates to a method for randomly generating a rotating rigid body structure based on an interpolation boundary. Background Art

[0002] A rotating rigid body structure is a type of metamaterial with peculiar mechanical properties, consisting of rigid units connected by flexible hinges. Due to the non-collinearity of the applied loads between the upper and lower hinges, the unit body rotates, resulting in a negative Poisson's ratio effect close to -1. Predicting and controlling the motion of large-scale rotating rigid body structures and enhancing the mechanical properties of rotating rigid body structures to prevent damage have important strategic significance in the applications of aerospace, biosensors, transportation, and other fields. The simulation and calculation research methods for rotating rigid bodies are mainly divided into two categories. One is the traditional method based on finite elements or homogenization, and the other is the data-driven machine learning method, such as neural networks (generative adversarial networks, variational autoencoders, graph networks), random forests, support vector machines, etc.

[0003] Currently, some researchers have studied the complex motion of rotating rigid body structures through machine learning and proved the feasibility and effectiveness of the method. However, the known rotating rigid body structures are all generated through artificial design or local topology optimization, which is not conducive to the establishment of a large-scale dataset. There are two major problems: insufficient data volume and insufficient randomness, making it impossible to achieve the diversified development of rotating rigid body structures, and thus causing difficulties for data-driven methods to work. Summary of the Invention

[0004] To solve the above problems, the present invention proposes a method for randomly generating a rotating rigid body structure based on an interpolation boundary.

[0005] The technical solution of the present invention is as follows: A method for randomly generating a rotating rigid body structure based on an interpolation boundary includes the following steps:

[0006] S1. Generate a random domain according to the size of the rotating rigid body and determine a number of limit points;

[0007] S2. Generate an interpolation function according to the number of limit points;

[0008] S3. Use a number of fine points to draw an interpolation curve corresponding to the interpolation function, and transform the interpolation curve to generate a flexible hinge and its surrounding local structure as well as a rotating rigid body unit;

[0009] S4. Generate a three-dimensional model of the rotating rigid body according to the flexible hinge and its surrounding local structure as well as the rotating rigid body unit.

[0010] Further, S1 includes the following sub-steps:

[0011] S11. Determine the design domain of the rotating rigid body element and the design domain of the flexible hinge;

[0012] S12. Generate a random domain according to the design domain of the rotating rigid body element and the design domain of the flexible hinge;

[0013] S13. Determine a number of limit points in the random domain.

[0014] Further, in S11, the design domain of the rotating rigid body element is a square, and the design domain of the flexible hinge is a rectangle;

[0015] In S12, the range expression of the random domain is:

[0016] ;

[0017] In the formula, represents the abscissa of the random domain, represents the ordinate of the random domain, represents the side length of the square, represents the length of the rectangle, represents the width of the rectangle.

[0018] Further, S2 includes the following sub-steps:

[0019] S21. Generate an index value and divide the index value into several parts;

[0020] S22. Map a number of limit points to a number of fine points by using several parts of the index value;

[0021] S23. Generate an interpolation function by using several fine points.

[0022] Further, in S22, the expression of the fine point coordinate is:

[0023] ;

[0024] In the formula, represents the abscissa of the fine point coordinate, represents the ordinate of the fine point coordinate, represents the index value.

[0025] Further, in S3, the calculation formula for transforming the interpolation curve is:

[0026] ;

[0027] In the formula, represents the abscissa of the interpolation curve, represents the ordinate of the interpolation curve, represents the curve after intercepting, rotating 90° clockwise and translating the interpolation curve The abscissa of represents the curve after intercepting, rotating 90° clockwise and translating the interpolation curve The ordinate of represents the curve after rotating 90° clockwise and translating the curve. The abscissa of represents the curve after rotating 90° clockwise and translating the curve. The ordinate of represents the abscissa of the curve after rotating 90° clockwise and translating the curve. The ordinate of the curve after rotating 90° clockwise and translating the represents the transpose matrix, represents the side length of the square, represents the width of the rectangle.

[0028] Furthermore, S4 includes the following sub-steps:

[0029] S41. Determine the length of the smooth curve according to the flexible hinge and its surrounding local structure and the fine point coordinates corresponding to the rotating rigid body unit;

[0030] S42. Use the cubic spline curve to determine the geometric contour of the rotating rigid body unit and connect them through the interpolation function to form a smooth curve;

[0031] S43. Triangulate the smooth curve to generate a number of triangular patches;

[0032] S44. Select the triangular patches whose centroids are inside the contour curve to generate a number of internal triangular patches;

[0033] S45. Use a number of internal triangular patches to generate a geometric mesh;

[0034] S46. Based on the geometric mesh, traverse the boundary coordinates of a number of internal triangular patches to generate a three-dimensional model.

[0035] Furthermore, in S45, the expression of the geometric mesh is ; In the formula, represents the abscissa of the first vertex of the triangular patch, represents the abscissa of the second vertex of the triangular patch, represents the abscissa of the third vertex of the triangular patch, represents the ordinate of the first vertex of the triangular patch, represents the ordinate of the second vertex of the triangular patch, Represents the ordinate of the third vertex of the triangular patch.

[0036] Furthermore, in S46, the number of triangular patches on the top surface of the three-dimensional model The calculation formula is:

[0037] ;

[0038] In the formula, Represents the number of internal triangular patches;

[0039] The number of triangular patches on the bottom surface of the three-dimensional model The calculation formula is:

[0040] ;

[0041] The number of triangular patches on the side surface of the three-dimensional model The calculation formula is:

[0042] ;

[0043] In the formula, Represents the length of the fine point coordinate array;

[0044] The size of the three-dimensional model The calculation formula is:

[0045] .

[0046] The beneficial effects of the present invention are:

[0047] (1) Compared with the existing manual design method, the present invention can quickly generate a large number of structures, reducing intermediate manpower and requiring no prior knowledge and design experience; compared with the local topology generation method, the present invention has stronger randomness and globality, and does not need to consider complex constraint conditions and design goals;

[0048] (2) The present invention can generate a large-scale random structure by controlling the boundary of the rotating rigid body through an interpolation function, thereby generating the necessary data set for the data-driven method and promoting the application of the method in the field of predicting the behavior and optimizing the performance of the rotating rigid body structure; the present invention can interpolate and generate a function in the entire random domain to achieve global randomness, prevent the perspective limitation caused by local randomness, and meet the quality requirements of the data set;

[0049] (3) The present invention can generate a large number of rotating rigid body structures with different characteristics in a short time, instead of designing them one by one, meeting the quantity requirements of the data set; it can be completely driven by Python code to achieve automated structure generation, and two types of structures, namely rotating rigid body units and flexible hinges, are generated respectively, which can be widely used in different researches. Description of the Drawings

[0050] Figure 1 It is a flowchart of a random generation method for a rotating rigid body structure based on an interpolation boundary;

[0051] Figure 2 It is a dimension diagram of a rotating rigid body unit;

[0052] Figure 3 It is a schematic diagram of a flexible hinge and its surrounding local structure;

[0053] Figure 4 It is a schematic diagram of a rotating rigid body unit;

[0054] Figure 5 It is a schematic diagram of the triangulation of a flexible hinge and its surrounding local structure;

[0055] Figure 6 It is a schematic diagram of the triangulation of the boundary of a rotating rigid body unit. Detailed Implementation Manner

[0056] The embodiments of the present invention will be further described below with reference to the accompanying drawings.

[0057] As Figure 1 shown, the present invention provides a random generation method for a rotating rigid body structure based on an interpolation boundary, including the following steps:

[0058] S1. Generate a random domain according to the size of the rotating rigid body and determine a number of limit points;

[0059] S2. Generate an interpolation function according to a number of limit points;

[0060] S3. Use a number of fine points to draw an interpolation curve corresponding to the interpolation function, and transform the interpolation curve to generate a flexible hinge and its surrounding local structure as well as a rotating rigid body unit;

[0061] S4. Generate a three-dimensional model of the rotating rigid body according to the flexible hinge and its surrounding local structure as well as the rotating rigid body unit.

[0062] The present invention innovatively invented a global random scheme for generating the structure boundary based on an interpolation function for the diversified generation of rotating rigid bodies and the construction of their data sets. It is driven by Python scripts to generate 2D / 3D rotating rigid body structures of any scale with boundary differences and output them in the form of a mesh as a.stl file, thus assisting in the construction of a data-driven method data set and promoting the neural network to learn simple geometric information and mechanical properties and achieve rapid prediction and design.

[0063] The rigid unit of a rotating rigid body structure is a centrosymmetric figure, that is, its geometric contour can be obtained by multiple rotations and translations of an edge, and the form of the edge can be specified by a certain functional form. The present invention takes the interpolation curve as an example and demonstrates the random generation technology of the rotating rigid body unit matrix and its flexible hinge. Three main tasks, namely two-dimensional geometric contour specification, two-dimensional / three-dimensional geometric mesh drawing and output, and large-scale data set generation, are respectively achieved, breaking through the bottlenecks such as low randomness, low versatility, and difficult optimization in the artificial design of rotating rigid body structures, solving the problems of small quantity and low quality of data under the background of data-driven design, and laying a good foundation for the training of high-fidelity machine learning models.

[0064] In an embodiment of the present invention, S1 includes the following sub-steps:

[0065] S11. Determine the design domain of the rotating rigid body unit and the design domain of the flexible hinge;

[0066] S12. Generate a random domain according to the design domain of the rotating rigid body unit and the design domain of the flexible hinge;

[0067] S13. Determine a number of limit points in the random domain.

[0068] In an embodiment of the present invention, in S11, the design domain of the rotating rigid body unit is a square, and the design domain of the flexible hinge is a rectangle;

[0069] As Figure 2 shown, in S12, the range expression of the random domain is:

[0070] ;

[0071] In the formula, represents the abscissa of the random domain, represents the ordinate of the random domain, represents the side length of the square, represents the length of the rectangle, represents the width of the rectangle.

[0072] are the coordinates of the two-dimensional random domain. A certain number of limit points are scattered in the random domain, and a unique interpolation curve is determined by multiple limit points. The number of limit points is preferably [5, 15]. Since the two-dimensional coordinates of the data points are used as features in the subsequent process of generating the data set, too many features will lead to difficulties in network training, causing the curse of dimensionality, and exponentially increasing the data volume requirement; while insufficient data points will lead to insufficient randomness of the curve and reduce the quality of the data set. In addition, to further reduce the features of the data set and prevent the problem that the interpolation curve values are too large / small due to the randomly generated limit points being too close, it is recommended to fix the value of each (or several) limit points and only randomly Value, this method greatly improves the representability of the interpolation function at the cost of sacrificing a small amount of randomness, and reduces the risks of self-intersection, excessive size, and too small size of the function.

[0073] In an embodiment of the present invention, S2 includes the following sub-steps:

[0074] S21, generate an index value and divide the index value into several parts;

[0075] S22, use several parts of the index value to map several limit points into several fine points;

[0076] S23, generate an interpolation function using several fine points.

[0077] In an embodiment of the present invention, in S22, the coordinates of the fine points The expression is:

[0078] ;

[0079] In the formula, represents the abscissa of the coordinates of the fine points, represents the ordinate of the coordinates of the fine points, represents the index value.

[0080] The interpolation function is uniquely determined by the limit points , denote the number of limit points as , and the sequence of limit points as , that is: . After the process of generating the interpolation function through the limit points, use the fine points to draw the curve. Introduce the index value , there is , subdivide the index within the range into parts, that is, the number of fine points, denote the sequence of fine points as , at this time, and are respectively the mappings of , that is .

[0081] In addition, for the mapping of , denote the corresponding to the last mapping value less than 0 as ; denote the corresponding to the last mapping value less than 5 as . To prevent significant stress concentration in the generated structure during mechanical simulation, the function curve is required to be smooth and continuous. Therefore, alternative functions include polynomial interpolation, spline interpolation, Bézier curves, B-spline curves, and Catmull-Rom splines. The meanings and mathematical representations of these functions are as follows.

[0082] Polynomial interpolation: ; where is the polynomial interpolation function, and the coefficients are determined by a system of linear equations. The polynomial interpolation method is simple to calculate but has a high computational complexity, and high-degree polynomials may exhibit the Runge phenomenon, resulting in a large deviation between the values of the interpolation polynomial at the interval edges and the actual values. It is suitable for scenarios with fewer data points and when precise passing through all data points is required.

[0083] Spline interpolation (taking cubic splines as an example): ; where is the spline function at the th point, and the coefficients are determined by ensuring the continuity of the first and second derivatives at the points. The spline interpolation method uses low-degree polynomials between every two data points, effectively reducing oscillation phenomena and being able to control the smoothness of the curve, but the calculation is relatively complex and additional constraint conditions are required to ensure the smoothness of the curve. It is suitable for scenarios with more data points and when a smooth curve is needed, such as in engineering and scientific calculations where continuous derivatives are required.

[0084] Bézier curves: ; where is the Bernstein basis function.

[0085] B-spline curves: ; where is the B-spline function, is the B-spline basis function, defined recursively, is the order of the spline. The B-spline curve method has the characteristic of local control. Changing one control point only affects a part of the curve, and control points can be flexibly added or reduced to adjust the curve, but the computational complexity is relatively high and the implementation is relatively complex. It is suitable for complex geometric modeling in CAD / CAM systems and scenarios that require high flexibility and local control.

[0086] Catmull-Rom splines: ; where is the Catmull-Rom spline function. The Catmull-Rom spline method is simple and easy to implement, passes through all control points, and maintains the endpoint slopes of the curve, but overshoot phenomena may occur, and for non-uniformly distributed control points, the curve may not be smooth enough. It is suitable for simple curve design and smoothing processing.

[0087] In the embodiment of the present invention, in S3, the calculation formula for transforming the interpolation curve is as follows:

[0088] ;

[0089] In the formula, represents the abscissa of the interpolation curve, represents the ordinate of the interpolation curve, represents the abscissa of the curve after intercepting, rotating 90° clockwise and translating the interpolation curve ; represents the ordinate of the curve after intercepting, rotating 90° clockwise and translating the interpolation curve ; represents the curve after rotating 90° clockwise and translating the curve abscissa; represents the curve after rotating 90° clockwise and translating the curve ordinate; represents the abscissa of the curve after rotating 90° clockwise and translating the curve; represents the curve after rotating 90° clockwise and translating the curve ordinate; represents the transpose matrix, represents the side length of the square, represents the width of the rectangle.

[0090] In the embodiment of the present invention, S4 includes the following sub-steps:

[0091] S41. Determine the length of the smooth curve according to the flexible hinge and its surrounding local structure and the fine point coordinates corresponding to the rotating rigid body unit;

[0092] S42. Use the cubic spline curve to determine the geometric contour of the rotating rigid body unit and connect them through the interpolation function to form a smooth curve;

[0093] S43. Triangulate the smooth curve to generate a number of triangular patches;

[0094] S44. Screen out the triangular patches whose centroids are inside the contour curve to generate a number of internal triangular patches;

[0095] S45. Use a number of internal triangular patches to generate a geometric mesh;

[0096] S46. Based on the geometric mesh, traverse the boundary coordinates of a number of internal triangular patches to generate a three-dimensional model.

[0097] In the embodiment of the present invention, in S45, the expression of the geometric grid is ; in the formula, represents the abscissa of the first vertex of the triangular patch, represents the abscissa of the second vertex of the triangular patch, represents the abscissa of the third vertex of the triangular patch, represents the ordinate of the first vertex of the triangular patch, represents the ordinate of the second vertex of the triangular patch, represents the ordinate of the third vertex of the triangular patch.

[0098] In the embodiment of the present invention, in S46, the number of top surface triangles of the three-dimensional model is calculated by the formula:

[0099] ;

[0100] In the formula, represents the number of internal triangular patches;

[0101] The number of bottom surface triangles of the three-dimensional model is calculated by the formula:

[0102] ;

[0103] The number of side surface triangles of the three-dimensional model is calculated by the formula:

[0104] ;

[0105] In the formula, represents the length of the fine point coordinate array;

[0106] The size of the three-dimensional model is calculated by the formula:

[0107] .

[0108] In the embodiment of the present invention, as Figure 3 shown, the flexible hinge and its surrounding local structure are composed of 8 segments of lines in total. Among them, the curves ①②③④ are obtained by intercepting, translating, and rotating the interpolation curve, and the line segments ⑤⑥⑦⑧ are artificially defined to generate a closed geometric structure. The function representations of each line segment are shown in Table 1, and the coordinate origin is the (0,0) point of the random domain when generating the interpolation function. It is agreed that all in the following text represents the interpolated function after fitting, represents the coordinates of any fine point on the function, Represents any fine point on the function coordinate, and Indicates the set value.

[0109] Table 1

[0110]

[0111] like Figure 4 As shown in Figure 1, the rotating rigid body unit is composed of 8 segments, where curve ① is the original interpolation function, curve ② is obtained by intercepting, rotating and translating the interpolation curve ①, curve ③ is obtained by rotating and translating the interpolation curve ②, and curve ④ is obtained by rotating and translating the interpolation curve ③. They represent curves ①, ②, ③, and ④ respectively. is the clockwise rotation matrix. In addition, line segments ⑤⑥⑦⑧ are artificially drawn straight lines used to generate closed geometric structures, which are automatically generated when generating the mesh.

[0112] In the embodiment of the present invention, the .stl file is a triangular representation of the three-dimensional surface geometry, and the generation of the .stl file relies on mesh drawing and three-dimensional modeling.

[0113] like Figure 5 and Figure 6 As shown, the fine point coordinate array given by the flexible hinge and its surrounding local structure and rotating rigid body unit , whose length is defined as , use cubic spline curves to define the geometric outline of the rotating rigid body structure, and connect them through spline interpolation functions to form a smooth curve. Use the Lawson algorithm and Bowyer-Watson algorithm to implement the Delaunay triangulation network, triangulate these points, and generate a series of triangular facets. This algorithm can ensure that the generated triangles are as close to equilateral as possible, thereby improving the quality of the mesh.

[0114] The Lawson algorithm is as follows:

[0115] a) Build includes The super triangle of all points in △P O1 P O2 P O3 , forming the initial triangulation.

[0116] b) All points are inserted one by one into the existing triangulation In the , locate the point containing The triangle △P i P j Pk If is located inside △P i P j P k inside, connect it to the three vertices of the triangle to generate △P i P j P r , △P i P r P k , and △P r P j P k three triangles. If is located on a certain side of △P i P j P k (assumed to be P i P j ), find the fourth vertex of P i P j that forms another triangle , and connect to , respectively, so as to divide the two triangles with P i P j as the side into four triangles.

[0117] c) Check whether the newly generated edges satisfy the empty circle property respectively. If not, perform the edge flipping operation (flip the bad edge P i P j to P r P k ), and perform the empty circle detection on the newly generated edges. If there are bad edges, continue to flip until there are no bad edges, and complete the insertion. Repeat the operation to insert all points in .

[0118] d) Delete the vertices and edges of the triangle △P O1 P O2 P O3 . The remaining triangles form the Delaunay triangulation T of the point set .

[0119] The Bowyer-Watson algorithm is specifically as follows:

[0120] a) Construct a super triangle containing all points in △P O1 P O2 P O3 , form the initial triangulation, and place it into the triangle linked list 。

[0121] b) Insert each point one by one into the existing triangulation, and find the triangle in it whose circumcircle contains the inserted point, which is called the affected triangle of All points are inserted one by one into the existing triangulation, and the triangle whose circumcircle contains the inserted point is found in it, and the common edges of the affected triangle are deleted. the inserted point, which is called the affected triangle of the inserted point, and the common edges of the affected triangle are deleted.

[0122] c) Connect the inserted point and all vertices of the affected triangle to complete the insertion in the Delaunay triangle linked list Repeat the operation to insert all points in it.

[0123] d) Delete the vertices and edges of the super triangle △P O1 P O2 P O3 P The remaining triangles form the Delaunay triangulation T of the point set

[0124] In the embodiment of the present invention, the edges of the 3D model are defined by the coordinates in the two-dimensional array. Among them, each internal triangular patch corresponds to a top surface and a bottom surface triangle, and each edge of the internal triangular patch corresponds to two side surface triangles. Initialize the 3D mesh object, and the size is the sum of the number of top surface, bottom surface, and side surface triangles , , .

[0125] Traverse the boundary coordinates of the internal triangular patches , generate the top surface of each patch and generate two triangles that respectively connect the two adjacent points on the bottom surface and the corresponding points on the top surface for each edge of the patch, that is, △P1P2P3 and △P2P4P3, to form the side surface. The coordinates of , , and are represented as follows:

[0126] Those of ordinary skill in the art will realize that the embodiments described herein are provided to assist the reader in understanding the principles of the present invention, and it should be understood that the scope of protection of the present invention is not limited to such specific statements and embodiments. Those of ordinary skill in the art can make various other specific deformations and combinations that do not depart from the essence of the present invention based on these technical revelations disclosed in the present invention, and these deformations and combinations are still within the scope of protection of the present invention.

Claims

1. A method for randomly generating a rotating rigid body structure based on an interpolation boundary, characterized in that: The following steps are involved: S1. Generate a random domain according to the size of the rotating rigid body and determine several limit points; S2, generating an interpolation function according to a number of limit points; S3, using a number of fine points to draw an interpolation curve corresponding to the interpolation function, and transforming the interpolation curve to generate a flexible hinge and its surrounding local structure and a rotating rigid body unit; S4, generating a three-dimensional model of a rotating rigid body according to the flexible hinge and its surrounding local structure and the rotating rigid body unit; The S4 comprises the following sub-steps: S41, determining the length of the smooth curve according to the fine point coordinates corresponding to the flexible hinge and its surrounding local structure and the rotating rigid body unit; S42, using a cubic spline curve to determine the geometric contour of the rotating rigid body unit, and connecting it through an interpolation function to form a smooth curve; S43, triangulating the smooth curve to generate a plurality of triangular facets; S44, filtering out triangular face patches whose centroids are located inside the contour curve, and generating a plurality of internal triangular face patches; S45, generating a geometric mesh using a plurality of internal triangular facets; S46. Based on the geometric mesh, traverse the boundary coordinates of several internal triangular facets to generate a three-dimensional model.

2. The method for randomly generating a rotating rigid body structure based on an interpolation boundary according to claim 1, characterized in that: The S1 comprises the following sub-steps: S11. Determine the design domain of the rotating rigid body unit and the design domain of the flexible hinge; S12, generating a random domain according to the design domain of the rotating rigid body unit and the design domain of the flexible hinge; S13. Determine several limit points in the random domain.

3. The method for randomly generating a rotating rigid body structure based on an interpolation boundary according to claim 2, characterized in that: In S11, the design domain of the rotating rigid body unit is a square, and the design domain of the flexible hinge is a rectangle; In S12, the range expression of the random domain is: ； In the formula, represents the horizontal axis of the random field, represents the ordinate of the random field, represents the side length of the square, Represents the length of the rectangle, Indicates the width of the rectangle.

4. The method for randomly generating a rotating rigid body structure based on an interpolation boundary according to claim 1, characterized in that: The S2 comprises the following sub-steps: S21, generate an index value, and divide the index value into several parts; S22, using a plurality of index values, mapping a plurality of limit points to a plurality of fine points; S23. Generate an interpolation function using a number of fine points.

5. The method for randomly generating a rotating rigid body structure based on an interpolation boundary according to claim 4, characterized in that: In S22, the fine point coordinates The expression is: ； In the formula, The horizontal coordinate represents the fine point coordinate, The ordinate represents the fine point coordinates, Indicates the index value.

6. The method for randomly generating a rotating rigid body structure based on an interpolation boundary according to claim 1, characterized in that: In S3, the calculation formula for transforming the interpolation curve is: ； In the formula, represents the abscissa of the interpolation curve, represents the ordinate of the interpolation curve, Indicates that the interpolation curve is intercepted, rotated 90° clockwise, and translated. The horizontal axis of Indicates that the interpolation curve is intercepted, rotated 90° clockwise, and translated. The vertical coordinate of Express The curve is rotated 90° clockwise and translated. The horizontal axis of Express The curve is rotated 90° clockwise and translated. The vertical coordinate of Express The horizontal coordinate of the curve after the curve is rotated 90° clockwise and translated. Express The vertical coordinate of the curve after the curve is rotated 90° clockwise and translated. represents the transposed matrix, represents the side length of the square, Indicates the width of the rectangle.

7. The method for randomly generating a rotating rigid body structure based on an interpolation boundary according to claim 1, characterized in that: In S45, the expression of the geometric grid is: ; In the formula, Represents the horizontal coordinate of the first vertex of the triangle patch, Represents the horizontal coordinate of the second vertex of the triangle patch, Represents the abscissa of the third vertex of the triangle patch, Represents the ordinate of the first vertex of the triangle patch. Represents the ordinate of the second vertex of the triangle patch, Represents the ordinate of the third vertex of the triangle patch.

8. The method for randomly generating a rotating rigid body structure based on an interpolation boundary according to claim 1, characterized in that: In S46, the number of top triangles of the three-dimensional model The calculation formula is: ； In the formula, Indicates the number of internal triangle faces; The number of base triangles of the 3D model The calculation formula is: ； The number of side triangles of the 3D model The calculation formula is: ； In the formula, Indicates the length of the fine point coordinate array; The size of the three-dimensional model The calculation formula is: 。

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