A pleated paper modeling method based on multi-objective optimization

Abstract

The present invention discloses a pleated paper modeling method based on multi-objective optimization, comprising the following steps: S1, by processing data, interactively inputting the subdivision number of the model length and width, obtaining all points, generating triangular faces according to the points, and finally generating a plane model to simulate the plane paper; S2, adding random perturbations to the points on the plane model m, obtaining a three-dimensional model M, simulating the plane paper after pleating; S3, adding expandable constraints to the three-dimensional model M, minimizing the expandable residual amount; S4, adding area constraints to the three-dimensional model M, minimizing the area residual amount; S5, adding expandable constraints and area constraints to the three-dimensional model M at the same time. The present invention realizes a pleated paper modeling method based on multi-objective optimization by adding expandable constraints and area constraints to the three-dimensional model. The method can be used in the engineering field to study and realize a multi-objective optimization modeling design structure.

Description

Technical Field

[0001] The invention relates to the field of three-dimensional modeling in computer graphics, and is a pleated paper modeling method based on multi-objective optimization. Background Art

[0002] Pleated paper is a process of pressing paper into various shapes in an artistic form, that is, a flat paper is pleated to obtain a target shape. The random variability of the shape and the rationality of the structure of pleated paper make it practical for engineers and artists to study the design and practical application of pleated paper in the engineering field. In the computer modeling process, if multiple constraints are added to the target model at the same time, the best solution may be found for one of the optimization constraints. Multi-objective optimization usually has several or more objective functions, which usually compete and conflict with each other. It is difficult to find an optimal solution that satisfies multiple objective functions at the same time. A piece of paper is pleated to obtain a target shape. The paper must remain rigid and foldable. Then the paper is transformed from the target shape to the flat paper. It requires that the target shape is expandable and that the area of ​​the target shape after expansion is equal to the area of ​​the flat paper. The present invention adds expandable constraints and area constraints to the three-dimensional model after pleating, and proposes a pleated paper modeling method based on multi-objective optimization. Summary of the invention

[0003] The present invention proposes a pleated paper modeling method based on multi-objective optimization. A plane model is generated by computer modeling, and a three-dimensional model is obtained through a random perturbation process to simulate pleats. The three-dimensional model should maintain the expandable characteristics after unfolding, and at the same time, the area after unfolding is equal to the area of ​​the plane model, that is, restrictive constraints are added to the three-dimensional model, expandable constraints and area constraints are added to the three-dimensional model, and a pleated paper modeling method based on multi-objective optimization is realized. The modeling structure of the method can be used in the engineering field, for example, to realize an arch bridge structure with excellent rigidity performance, and as an automatically folding micro container structure, etc.

[0004] In order to achieve the above purpose, the technical solution provided by the present invention is as follows:

[0005] A pleated paper modeling method based on multi-objective optimization, the method comprising:

[0006] S1. Process the data according to the three-dimensional coordinate axis, interactively input the number of subdivisions of the model length and width in the control panel, obtain all the points, and then generate triangular faces based on the points to generate a plane model in the 3D space, recorded as model m, simulating a flat piece of paper;

[0007] S2, adding random perturbations to the points on the plane model m in the x-direction, the y-direction and the z-direction to obtain an uneven three-dimensional model, recorded as model M, simulating the flat paper after being folded;

[0008] S3, adding expandable constraints to the three-dimensional model M, minimizing the expandable residual, so that each internal vertex of the three-dimensional model M meets the expandable constraints, that is, the wrinkled paper meets the expandable constraints;

[0009] S4, adding area constraints to the three-dimensional model M, minimizing the area residual, so that the area of ​​the three-dimensional model M after random perturbation and the area residual of the plane model m are minimized, that is, the area of ​​the wrinkled paper is equal to the area of ​​the plane paper;

[0010] S5. Add expandable constraints and area constraints to the three-dimensional model M at the same time to realize a pleated paper modeling method based on multi-objective optimization.

[0011] To further illustrate, the step S1 is specifically as follows:

[0012] S11, establishing three-dimensional coordinate axes, respectively recorded as x-axis, y-axis and z-axis, processing the data, and obtaining the values ​​of the length and width of the input model;

[0013] S12, interactively controlling the number of subdivisions of the length and width of the input model on the control panel;

[0014] S13, obtaining the position information of all points by interactively inputting the number of subdivisions;

[0015] S14, after obtaining the position information of all points, generate a surface according to the points; establish a data relationship for all points, map them into a two-dimensional grid map, and form a triangular surface ΔABC from the two-dimensional grid map in accordance with the data relationship;

[0016] S15, generating a plane model in the 3D space according to the triangular surface, denoted as model m, the plane model m simulates a flat piece of paper;

[0017] To further illustrate, the step S2 is specifically as follows:

[0018] S21. Based on the plane model m, introduce variable N v , N v It refers to the number of vertices on the plane model m;

[0019] S22, the point on the plane model m is denoted as p i (x i ,y i ,z i ), where i = 1, 2, ..., N v ;

[0020] S23, introducing a variable δ, δ is the value of random noise interference, and its value can be positive or negative;

[0021] S24, using formula (1) to perform random noise interference on the position coordinates of all points on the plane model m in the x direction, y direction and z direction, so as to achieve the folding of a plane paper;

[0022] S25. After adding random noise interference to all points on the plane model m, an uneven three-dimensional model M is obtained. The points on the three-dimensional model M are denoted as P i (x i ′,y i ′,z i ′), where i = 1, 2, ..., N v , use the obtained three-dimensional model M to simulate a wrinkled paper;

[0023]

[0024] S26, after the random disturbance, the position information of the point changes, and expandability constraints and area constraints are added to the three-dimensional model M;

[0025] Add an expandable constraint to the three-dimensional model M so that the sum of the angles around the internal vertices of the three-dimensional model M is equal to 2π, that is, the crumpled paper is expandable;

[0026] Add area constraints to the three-dimensional model M so that the residual area between the three-dimensional model M and the two-dimensional plane model m after random perturbation is minimized, that is, the area of ​​the wrinkled paper is equal to the area of ​​the flat paper;

[0027] Add expandable constraints and area constraints to the three-dimensional model M at the same time, so that the sum of the angles around the internal vertices of the three-dimensional model M is equal to 2π and the residual amount of the area of ​​the three-dimensional model M and the area of ​​the plane model m after random perturbation is minimized. That is, the crumpled paper is expandable and its area is equal to that of the plane paper.

[0028] To further illustrate, the step S3 is specifically as follows:

[0029] S31. Define the expandable constraint conditions: introduce variable α i,k , α i,k is the kth angle of the i-th vertex of the three-dimensional model M. When α i,1 +α i,2 +α i,3 +α i,4 +α i,5 +α i,6 =2π, that is, when the sum of the angles around the internal vertices of the three-dimensional model M is equal to 2π, the three-dimensional model M satisfies the expandable constraint;

[0030] S32, introduce variables It refers to the number of vertices around the i-th vertex on the three-dimensional model M;

[0031] S33, introduce variable P i and P i,k , where P i refers to the i-th vertex on the three-dimensional model M, P i,k refers to the surrounding vertices of the i-th vertex, where

[0032] S34, adding expandable constraints to the internal vertices of the three-dimensional model M; defining expandable folding residuals, deriving an expandable constraint objective function according to the expandable folding residuals, and minimizing the expandable constraint objective function;

[0033] S35. The expandable constraint objective function is solved by formula (2), where the expandable folded residual is defined by formula (3).

[0034]

[0035]

[0036] where f i is the expandable residual defined, α i,k is the kth angle of the i-th vertex of the three-dimensional model M. When the angle α i,k is a vector With vector The angle between When the angle α i,k is a vector With vector The angle between them is given by formula (4);

[0037]

[0038] S36, using an optimization algorithm to solve the expandable constraints to obtain a gradient in the descending direction;

[0039] Solve using formula (2), formula (3) and formula (4) Get the gradient in the x direction;

[0040]

[0041] Solve using formula (2), formula (3) and formula (4) Get the gradient in the y direction;

[0042]

[0043] Solve using formula (2), formula (3) and formula (4) Get the gradient in the z direction;

[0044]

[0045] S37, using the gradients obtained by solving formula (5), formula (6) and formula (7), and using the gradients based on the Levenberg–Marquardt optimization algorithm to iteratively solve the expandable constraint objective function, and minimizing the objective function so that the three-dimensional model M satisfies the expandable constraint, that is, the sum of the angles around the internal vertices of the three-dimensional model M is equal to 2π, and the crumpled paper is expandable;

[0046] To further illustrate, the step S4 is specifically as follows:

[0047] S41, based on the plane model m, after adding random noise interference, the position of the point changes, and an uneven three-dimensional model M is obtained, and an area constraint is added to the three-dimensional model M;

[0048] S42. Based on the three-dimensional model M, introduce variable N f , N f It refers to the number of triangular faces of the three-dimensional model M;

[0049] S43. Let the area of ​​a single triangle of the plane model m be S, then the area of ​​the entire two-dimensional plane model is N f S;

[0050] S44, adding an area constraint to the three-dimensional model M; defining an area residual, obtaining an area constraint objective function according to the area residual, and minimizing the area constraint objective function;

[0051] S45, the area constraint objective function is proposed to be solved by formula (8), wherein the area residual is defined by formula (9);

[0052]

[0053]

[0054] where g j is the defined area residual, refers to the area of ​​the triangular face ΔABC generated after random perturbation, and are the vectors in the triangular face ΔABC respectively;

[0055] S46, solving the area constraint based on finite differences to obtain a gradient in the descending direction;

[0056] Using formula (8) and formula (9) to solve Get the gradient in the x direction;

[0057]

[0058] Using formula (8) and formula (9) to solve Get the gradient in the y direction;

[0059]

[0060] Using formula (8) and formula (9) to solve Get the gradient in the z direction;

[0061]

[0062] S47. Based on the gradients obtained by solving formula (10), formula (11) and formula (12), the area constraint objective function is iteratively solved using the gradients based on the Levenberg–Marquardt optimization algorithm, and the objective function is minimized so that the three-dimensional model M satisfies the area constraint, that is, the residual amount of the area of ​​the three-dimensional model M and the area of ​​the plane model m after random perturbation is minimized, and the area of ​​the wrinkled paper is equal to the area of ​​the plane paper;

[0063] To further illustrate, the step S5 is specifically as follows:

[0064] S51, adding expandable constraints and area constraints to the three-dimensional model M at the same time, solving the multi-objective optimization problem of the three-dimensional model M, and minimizing the multi-objective constraint objective function;

[0065] S52, propose a multi-objective constraint objective function and solve it by formula (13);

[0066] E=λ1E dev +λ2E area , (13)

[0067] where λ1 and λ2 refer to weights; E dev It is to add expandable constraints to the internal vertices of the three-dimensional model M and define the expandable constraint formula; E area It is to add area constraint to the three-dimensional model M and define the area constraint formula;

[0068] S53, solving the multi-objective constraints on the three-dimensional model M based on finite differences to obtain the gradient of the descent direction;

[0069] Solve using formula (13), formula (2) and formula (8) Get the gradient in the x direction;

[0070]

[0071] Solve using formula (13), formula (2) and formula (8) Get the gradient in the y direction;

[0072]

[0073] Solve using formula (13), formula (2) and formula (8) Get the gradient in the z direction;

[0074]

[0075] S54. Based on the gradients obtained by solving formulas (14), (15) and (16), the gradients are used to iteratively solve the multi-objective constraint objective function of the three-dimensional model M based on the Levenberg–Marquardt optimization algorithm, and the multi-objective constraint objective function is minimized so that the sum of the angles around the internal vertices of the three-dimensional model M is equal to 2π and the residual difference between the area of ​​the three-dimensional model M and the area of ​​the plane model m after random perturbation is minimized. These two constraints are satisfied at the same time, that is, the crumpled paper is expandable and its area is equal to the area of ​​the plane paper.

[0076] The beneficial effects of the present invention are:

[0077] 1. Construct a pleated paper modeling structure composed of triangles;

[0078] 2. Implement a multi-objective optimization method for target shape;

[0079] 3. This method can be used in engineering and art fields to study and implement modeling design and solve multi-objective constraints. BRIEF DESCRIPTION OF THE DRAWINGS

[0080] Figure 1 Schematic diagram of the process of the present invention; DETAILED DESCRIPTION

[0081] The present invention will be further described below in conjunction with the accompanying drawings.

[0082] The present invention will be described in detail below in conjunction with the various embodiments shown in the accompanying drawings. However, these embodiments do not limit the present invention, and any structural, methodological, or functional changes made by a person skilled in the art based on these embodiments are all within the scope of protection of the present invention.

[0083] like Figure 1 As shown, the present invention is a pleated paper modeling method based on multi-objective optimization, comprising the following steps:

[0084] S1. Process the data according to the three-dimensional coordinate axis, interactively input the number of subdivisions of the model length and width in the control panel, obtain all the points, and then generate triangular faces based on the points to generate a plane model in the 3D space, recorded as model m, simulating a flat piece of paper;

[0085] S2, adding random perturbations to the points on the plane model m in the x-direction, the y-direction and the z-direction to obtain an uneven three-dimensional model, recorded as model M, simulating the flat paper after being folded;

[0086] S3, adding expandable constraints to the three-dimensional model M, minimizing the expandable residual, so that each internal vertex of the three-dimensional model M meets the expandable constraints, that is, the wrinkled paper meets the expandable constraints;

[0087] S4, adding area constraints to the three-dimensional model M, minimizing the area residual, so that the area of ​​the three-dimensional model M after random perturbation and the area residual of the plane model m are minimized, that is, the area of ​​the wrinkled paper is equal to the area of ​​the plane paper;

[0088] S5. Add expandable constraints and area constraints to the three-dimensional model M at the same time to realize a pleated paper modeling method based on multi-objective optimization.

[0089] To further illustrate, the step S1 is specifically as follows:

[0090] S11, establishing three-dimensional coordinate axes, respectively recorded as x-axis, y-axis and z-axis, processing the data, and obtaining the values ​​of the length and width of the input model;

[0091] S12, interactively controlling the number of subdivisions of the length and width of the input model on the control panel;

[0092] S13, obtaining the position information of all points by interactively inputting the number of subdivisions;

[0093] S14, after obtaining the position information of all points, generate a surface according to the points; establish a data relationship for all points, map them into a two-dimensional grid map, and form a triangular surface ΔABC from the two-dimensional grid map in accordance with the data relationship;

[0094] S15, generating a plane model in the 3D space according to the triangular surface, denoted as model m, the plane model m simulates a flat piece of paper;

[0095] To further illustrate, the step S2 is specifically as follows:

[0096] S21. Based on the plane model m, introduce variable N v , N v It refers to the number of vertices on the plane model m;

[0097] S22, the point on the plane model m is denoted as p i (x i ,y i ,z i ), where i = 1, 2, ..., N v ;

[0098] S23, introducing a variable δ, δ is the value of random noise interference, and its value can be positive or negative;

[0099] S24, using formula (1) to perform random noise interference on the position coordinates of all points on the plane model m in the x direction, y direction and z direction, so as to achieve the folding of a plane paper;

[0100] S25. After adding random noise interference to all points on the plane model m, an uneven three-dimensional model M is obtained. The points on the three-dimensional model M are denoted as P i (x i ′,y i ′,z i ′), where i = 1, 2, ..., N v , use the obtained three-dimensional model M to simulate a wrinkled paper;

[0101]

[0102] S26, after the random disturbance, the position information of the point changes, and expandability constraints and area constraints are added to the three-dimensional model M;

[0103] Add an expandable constraint to the three-dimensional model M so that the sum of the angles around the internal vertices of the three-dimensional model M is equal to 2π, that is, the crumpled paper is expandable;

[0104] Add area constraints to the three-dimensional model M so that the residual area between the three-dimensional model M and the two-dimensional plane model m after random perturbation is minimized, that is, the area of ​​the wrinkled paper is equal to the area of ​​the flat paper;

[0105] Add expandable constraints and area constraints to the three-dimensional model M at the same time, so that the sum of the angles around the internal vertices of the three-dimensional model M is equal to 2π and the residual amount of the area of ​​the three-dimensional model M and the area of ​​the plane model m after random perturbation is minimized. That is, the crumpled paper is expandable and its area is equal to that of the plane paper.

[0106] To further illustrate, the step S3 is specifically as follows:

[0107] S31. Define the expandable constraint conditions: introduce variable α i,k , α i,k is the kth angle of the i-th vertex of the three-dimensional model M. When α i,1 +αi,2 +α i,3 +α i,4 +α i，5 +α i，6 =2π, that is, when the sum of the angles around the internal vertices of the three-dimensional model M is equal to 2π, the three-dimensional model M satisfies the expandable constraint;

[0108] S32, introduce variables It refers to the number of vertices around the i-th vertex on the three-dimensional model M;

[0109] S33, introduce variable P i and P i,k , where P i refers to the i-th vertex on the three-dimensional model M, P i，k refers to the surrounding vertices of the i-th vertex, where

[0110] S34, adding expandable constraints to the internal vertices of the three-dimensional model M; defining expandable folding residuals, deriving an expandable constraint objective function according to the expandable folding residuals, and minimizing the expandable constraint objective function;

[0111] S35. The expandable constraint objective function is solved by formula (2), where the expandable folded residual is defined by formula (3).

[0112]

[0113]

[0114] where f i is the defined expandable residual. For example, when the number of internal vertices is equal to 6, the difference between the sum of the angles of the six angles around the vertex and 2π is calculated. When the number of a certain vertex inside is not equal to 6, f i =0; α i,k is the kth angle of the i-th vertex of the three-dimensional model M. When the angle α i,k is a vector With vector The angle between When the angle α i,k is a vector With vector The angle between them is given by formula (4);

[0115]

[0116] S36, using an optimization algorithm to solve the expandable constraints to obtain a gradient in the descending direction;

[0117] Solve using formula (2), formula (3) and formula (4) Get the gradient in the x direction;

[0118]

[0119] Solve using formula (2), formula (3) and formula (4) Get the gradient in the y direction;

[0120]

[0121] Solve using formula (2), formula (3) and formula (4) Get the gradient in the z direction;

[0122]

[0123] S37, using the gradients obtained by solving formula (5), formula (6) and formula (7), and using the gradients based on the Levenberg–Marquardt optimization algorithm to iteratively solve the expandable constraint objective function, and minimizing the objective function so that the three-dimensional model M satisfies the expandable constraint, even if the sum of the angles around the internal vertices of the three-dimensional model M is equal to 2π, the crumpled paper is expandable;

[0124] To further illustrate, the step S4 is specifically as follows:

[0125] S41, based on the plane model m, after adding random noise interference, the position of the point changes, and an uneven three-dimensional model M is obtained, and an area constraint is added to the three-dimensional model M;

[0126] S42. Based on the three-dimensional model M, introduce variable N f , N f It refers to the number of triangular faces of the three-dimensional model M;

[0127] S43. Let the area of ​​a single triangle of the plane model m be S, then the area of ​​the entire two-dimensional plane model is N f S;

[0128] S44, adding an area constraint to the three-dimensional model M; defining an area residual, obtaining an area constraint objective function according to the area residual, and minimizing the area constraint objective function;

[0129] S45, the area constraint objective function is proposed to be solved by formula (8), wherein the area residual is defined by formula (9);

[0130]

[0131]

[0132] where g j is the defined area residual, refers to the area of ​​the triangular face ΔABC generated after random perturbation, and are the vectors in the triangular face ΔABC respectively;

[0133] S46, solving the area constraint based on finite differences to obtain a gradient in the descending direction;

[0134] Using formula (8) and formula (9) to solve Get the gradient in the x direction;

[0135]

[0136] Using formula (8) and formula (9) to solve Get the gradient in the y direction;

[0137]

[0138] Using formula (8) and formula (9) to solve Get the gradient in the z direction;

[0139]

[0140] S47. Based on the gradients obtained by solving formula (10), formula (11) and formula (12), the area constraint objective function is iteratively solved using the gradients based on the Levenberg–Marquardt optimization algorithm, and the objective function is minimized so that the three-dimensional model M satisfies the area constraint, that is, the residual amount of the area of ​​the three-dimensional model M and the area of ​​the plane model m after random perturbation is minimized, and the area of ​​the wrinkled paper is equal to the area of ​​the plane paper;

[0141] To further illustrate, the step S5 is specifically as follows:

[0142] S51, adding expandable constraints and area constraints to the three-dimensional model M at the same time, solving the multi-objective optimization problem of the three-dimensional model M, and minimizing the multi-objective constraint objective function;

[0143] S52, propose a multi-objective constraint objective function and solve it by formula (13);

[0144] E=λ1E dev +λ2E area , (13)

[0145] where λ1 and λ2 refer to weights; E dev It is to add expandable constraints to the internal vertices of the three-dimensional model M and define the expandable constraint formula; e areaIt is to add area constraint to the three-dimensional model M and define the area constraint formula;

[0146] S53, based on finite difference, solve the above multi-objective constraints on the three-dimensional model M to obtain the gradient of the descent direction; use formula (13), formula (2) and formula (8) to solve Get the gradient in the x direction;

[0147]

[0148] Solve using formula (13), formula (2) and formula (8) Get the gradient in the y direction;

[0149]

[0150] Solve using formula (13), formula (2) and formula (8) Get the gradient in the z direction;

[0151]

[0152] S54. Based on the gradients obtained by solving formulas (14), (15) and (16), the gradients are used to iteratively solve the multi-objective constraint objective function of the three-dimensional model M based on the Levenberg–Marquardt optimization algorithm, and the multi-objective constraint objective function is minimized so that the sum of the angles around the internal vertices of the three-dimensional model M is equal to 2π and the residual difference between the area of ​​the three-dimensional model M and the area of ​​the plane model m after random perturbation is minimized. These two constraints are satisfied at the same time, that is, the crumpled paper is expandable and its area is equal to the area of ​​the plane paper.

[0153] It should be noted that this specification is explained according to the above-mentioned implementation mode, and this explanation method is used to allow readers to better understand the design steps and various methods of application of the present invention. People in the relevant field need to take the entire specification as a whole, and various technical solutions can be appropriately combined to form other implementation modes that can be understood by people in the relevant field.

[0154] The series of detailed descriptions listed above are only specific descriptions of feasible implementation methods of the present invention. They are not intended to limit the scope of protection of the present invention. All equivalent methods or changes that do not deviate from the technical creation of the present invention should be included in the scope of protection of the present invention.

Claims

1. A pleated paper modeling method based on multi-objective optimization, characterized in that: The steps include: S1. Process the data according to the three-dimensional coordinate axis, interactively input the number of subdivisions of the model length and width in the control panel, obtain all the points, and then generate triangular faces based on the points to generate a plane model in the 3D space, recorded as model m, simulating a flat piece of paper; S2, adding random perturbations to the points on the plane model m in the x-direction, the y-direction and the z-direction to obtain an uneven three-dimensional model, recorded as model M, simulating the flat paper after being folded; S3, adding expandable constraints to the three-dimensional model M, minimizing the expandable residual, so that each internal vertex of the three-dimensional model M meets the expandable constraints, that is, the wrinkled paper meets the expandable constraints; The specific implementation of step S3 includes: S31. Define the expandable constraint conditions: introduce variable α i,k , α i,k is the kth angle of the i-th vertex of the three-dimensional model M. When α i,1 +α i,2 +α i,3 +α i,4 +α i,5 +α i,6 =2π, that is, when the sum of the angles around the internal vertices of the three-dimensional model M is equal to 2π, the three-dimensional model M satisfies the expandable constraint; S32, introduce variables It refers to the number of vertices around the i-th vertex on the three-dimensional model M; S33, introduce variable P i and P i,k , where P i refers to the i-th vertex on the three-dimensional model M, P i,k refers to the surrounding vertices of the i-th vertex, where S34, adding expandable constraints to the internal vertices of the three-dimensional model M; defining expandable folding residuals, deriving an expandable constraint objective function according to the expandable folding residuals, and minimizing the expandable constraint objective function; The implementation of S34 includes: S341, the expandable constraint objective function is defined by formula (2), and the expandable folded residual is defined by formula (3), Among them, f i is the expandable residual defined, α i,k is the kth angle of the i-th vertex of the three-dimensional model M. When the angle α i,k is a vector With vector The angle between When the angle α i,k is a vector With vector The angle between them is given by formula (4); S342, using an optimization algorithm to solve the expandable constraints to obtain a gradient in a descending direction; Solve using formula (2), formula (3) and formula (4) Get the gradient in the x direction; Solve using formula (2), formula (3) and formula (4) Get the gradient in the y direction; Solve using formula (2), formula (3) and formula (4) Get the gradient in the z direction; S343, using the gradients obtained by solving formula (5), formula (6) and formula (7), and using the gradients based on the Levenberg-Marquardt optimization algorithm to iteratively solve the expandable constraint objective function, and minimizing the objective function so that the three-dimensional model M satisfies the expandable constraint, that is, the sum of the angles around the internal vertices of the three-dimensional model M is equal to 2π, and the crumpled paper is expandable; S4, adding area constraints to the three-dimensional model M, minimizing the area residual, so that the area of ​​the three-dimensional model M after random perturbation and the area residual of the plane model m are minimized, that is, the area of ​​the wrinkled paper is equal to the area of ​​the plane paper; The specific implementation of step S4 includes: S41, based on the plane model m, after adding random noise interference, the position of the point changes, and an uneven three-dimensional model M is obtained, and an area constraint is added to the three-dimensional model M; S42. Based on the three-dimensional model M, introduce variable N f , N f It refers to the number of triangular faces of the three-dimensional model M; S43. Let the area of ​​a single triangle of the plane model m be S, then the area of ​​the entire two-dimensional plane model is N f S; S44, adding an area constraint to the three-dimensional model M; defining an area residual, obtaining an area constraint objective function according to the area residual, and minimizing the area constraint objective function; The implementation of S44 includes: S441, the area constraint objective function is solved by formula (8), and the area residual is defined by formula (9); where g j is the defined area residual, refers to the area of ​​the triangular face ΔABC generated after random perturbation, and are the vectors in the triangular face ΔABC respectively; S442, solving the area constraint based on finite differences to obtain a gradient in the descending direction; Using formula (8) and formula (9) to solve Get the gradient in the x direction; Using formula (8) and formula (9) to solve Get the gradient in the y direction; Using formula (8) and formula (9) to solve Get the gradient in the z direction; S443. Based on the gradients obtained by solving formula (10), formula (11) and formula (12), the area constraint objective function is iteratively solved using the gradients based on the Levenberg-Marquardt optimization algorithm, and the objective function is minimized so that the three-dimensional model M satisfies the area constraint, that is, the area residual of the three-dimensional model M and the plane model m after random perturbation is minimized, and the area of ​​the wrinkled paper is equal to the area of ​​the plane paper; S5. Add expandable constraints and area constraints to the three-dimensional model M at the same time to realize pleated paper modeling based on multi-objective optimization; The specific implementation of step S5 includes: S51, adding expandable constraints and area constraints to the three-dimensional model M at the same time, solving the multi-objective optimization problem of the three-dimensional model M, and minimizing the multi-objective constraint objective function; S52, propose a multi-objective constraint objective function and solve it by formula (13); E=λ1E dev +λ2E area , (13) where λ1 and λ2 refer to weights; E dev Add expandable constraints to the internal vertices of the three-dimensional model M; E area It is to add area constraints to the three-dimensional model M; S53, solving the multi-objective constraints on the three-dimensional model M based on finite differences to obtain the gradient of the descent direction; Solve using formula (13), formula (2) and formula (8) Get the gradient in the x direction; Solve using formula (13), formula (2) and formula (8) Get the gradient in the y direction; Solve using formula (13), formula (2) and formula (8) Get the gradient in the z direction; S54. Based on the gradients obtained by solving formulas (14), (15) and (16), the gradients are used to iteratively solve the multi-objective constraint objective function of the three-dimensional model M based on the Levenberg-Marquardt optimization algorithm, and the multi-objective constraint objective function is minimized so that the sum of the angles around the internal vertices of the three-dimensional model M is equal to 2π and the residual difference between the area of ​​the three-dimensional model M and the area of ​​the plane model m after random perturbation is minimized. These two constraints are satisfied at the same time.

2. The pleated paper modeling method based on multi-objective optimization according to claim 1, characterized in that: The specific implementation of step S1 includes: S11, establishing three-dimensional coordinate axes, respectively recorded as x-axis, y-axis and z-axis, processing the data, and obtaining the values ​​of the length and width of the input model; S12, interactively controlling the number of subdivisions of the length and width of the input model on the control panel; S13, obtaining the position information of all points by interactively inputting the number of subdivisions; S14, after obtaining the position information of all points, generate a surface according to the points; establish a data relationship for all points, map them into a two-dimensional grid map, and form a triangular surface ΔABC from the two-dimensional grid map in accordance with the data relationship; S15. Generate a plane model in 3D space according to the triangular surface, denoted as model m. The plane model m simulates a flat piece of paper.

3. The pleated paper modeling method based on multi-objective optimization according to claim 1, characterized in that: The specific implementation of step S2 includes: S21. Based on the plane model m, introduce variable N v , N v It refers to the number of vertices on the plane model m; S22. Let the point on the plane model m be denoted as p i (x i ,y i ,z i ), where i = 1, 2, ..., N v ; S23, introducing a variable δ, δ is the value of random noise interference, and its value can be positive or negative; S24, using formula (1) to perform random noise interference on the position coordinates of all points on the plane model m in the x direction, y direction and z direction, so as to achieve the folding of a plane paper; S25. After adding random noise interference to all points on the plane model m, an uneven three-dimensional model M is obtained. The points on the three-dimensional model M are denoted as P i (x i ′,y i ′,z i ′), where i = 1, 2, ..., N v , use the obtained three-dimensional model M to simulate a wrinkled paper;

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