Multi-objective optimization modeling method for fitting free-form surface based on torpedo origami

Abstract

The invention discloses a multi-objective optimization modeling method based on a torpedo origami fitting free-form surface, and the method comprises the following steps: S1, generating a target surface with any curvature through a parameter equation or a parameter-based surface difference method, and then laying a classical six-fold torpedo origami unit on the target surface, and obtaining a basic torpedo grid model; s2, defining unfoldable and flatly foldable constraints, and realizing a three-dimensional torpedo paper folding structure meeting geometric constraints; s3, defining an area homogenization constraint, and obtaining a torpedo origami structure meeting an equal-area torpedo unit; s4, a multi-objective optimization function is constructed, and a torpedo origami structure meeting the geometric constraint (S2) and the area homogenization constraint (S3) is achieved; according to the method, the target surface with any curvature can be generated by utilizing the parameter equation and the parameter-based curved surface difference method, and wide design freedom degree and flexibility are provided for subsequent origami modeling.

Description

Technical Field

[0001] The invention relates to the technical field of computer graphics and three-dimensional modeling, and in particular to a multi-objective optimization modeling method based on mine origami fitting free-form surfaces. Background Art

[0002] Origami is well known as an ancient art of paper folding. By introducing universal symbol systems and mathematics, the research on origami has shifted from the original artistic aspect to the scientific and engineering aspects, and origami engineering has developed rapidly. With its unique design methods and flexibility, the application of the ancient art of origami to solve various problems is increasingly favored by researchers. Based on the folding and unfolding principles of rigid origami, people have designed structures with rigidity, stability and deformability. These structures have significant applications in spacecraft, foldable buildings, and medical equipment. Secondly, engineering origami research promotes innovative design and efficient structures, achieves lightweight, foldability and resource optimization, and promotes technological progress in multiple fields.

[0003] The six-fold mine unit is a classic origami pattern, which is composed of four valley folds and two mountain folds intersecting at the center point. It has a unique folding method and strong surface fitting ability. In view of the flexibility of the mine origami unit, the present invention establishes a basic model based on the six-fold mine unit fitting free surface, and adds multi-objective optimization constraints to the entire mine grid to obtain a free surface mine origami structure modeling method that meets specific constraints. Summary of the invention

[0004] In the present invention, a multi-objective optimization modeling method based on mine folding fitting free-form surface is proposed, and a target surface with arbitrary curvature is determined by using a parametric equation or a surface interpolation method based on parameters. In order to approximate the target surface, the mine unit is flattened on the target surface. Then, the mine base mesh with a partially folded state is generated by adjusting the mine unit vertices. Finally, multi-objective optimization constraints are added to the mine mesh to realize the mine origami structure modeling with expandable and flat foldable constraints using equal-area mine units.

[0005] In order to achieve the above object, the technical solution provided by the present invention is as follows:

[0006] A multi-objective optimization modeling method based on mine fold fitting free-form surface includes the following steps:

[0007] S1: Use parametric equations or parameter-based surface interpolation methods to generate a target surface with arbitrary curvature, and then flatten the classic six-fold mine origami unit onto the target surface to obtain a basic mine grid model;

[0008] S2: Define expandable and flat foldable constraints to achieve a three-dimensional mine origami structure that satisfies geometric constraints;

[0009] S3: Define area homogenization constraints to obtain the mine origami structure that satisfies the equal-area mine unit;

[0010] S4: Construct a multi-objective optimization function to achieve a mine origami structure that satisfies the geometric constraints (S2) and the area homogenization constraints (S3).

[0011] Furthermore, the above step S1 is specifically as follows:

[0012] S11. First, determine the target surface. For a general free-form surface, it can be described by a parametric equation, namely, the function S(u,v), with u and v as input parameters. Figure 1 , modeling process of a quarter vase (free-form surface with parametric equations).

[0013] More generally, complex surfaces do not have fixed parametric equations to represent them. In this case, we can use NURBS surfaces, which are mathematical models that accurately represent complex shapes based on a combination of control points, weights, and basis functions. The formula is as follows:

[0014]

[0015] Where S(u,v) represents the point on the surface under the parameters (u,v); P i,j is the control point, i and j represent the index of the control point in the u and v directions respectively, defining the shape of the surface; N i,p (u) and N j,q (v) are B-spline basis functions with respect to parameters u and v respectively; p and q are the orders of basis functions, which determine the smoothness of the surface; and the basis functions in NURBS are usually B-spline basis functions, which are defined as follows:

[0016]

[0017] where u i Represents a node value in a node vector, the node vector {u0,u1,u2,…,u n+p+1} is a non-decreasing sequence of values ​​that is used to partition the parameter domain and affect the shape of the basis function; n is the number of control points minus 1 (the index of the control points ranges from 0 to n); p is the order of the B-spline (p = 3 for cubic B-spline);

[0018] Where u is a parameter. According to the above formula, as long as the parameters u and v are given, the target surface without fixed parameter equation can be determined. Figure 2 As shown, the modeling steps for creating a free-form surface in Rhino are given.

[0019] S12, then tile the mine units, and use S to map each node (u, v) on the mine grid to the vertex P (x, y, z) on its corresponding mine grid model, so as to achieve the mine unit tiling the target surface, that is, calculate the vertex coordinates of a single mine unit according to the uv parameter equation, and tile the mine units one by one along the u and v directions until the entire target surface is covered, as shown in Figure 1 (b), the effect shown in 2(c).

[0020] S13. In order to generate the mine base mesh in a partially folded state, we need to adjust the mine unit vertices. This step corresponds to Figure 1 (b)-(c), Figure 2 (c)-(d) process. Figure 3 , P1, P2, P3, P4, P u , P d is the edge vertex of the mine unit, P c is the center vertex of the mine unit; it shows the adjustment process of the mine unit vertex, and changes P1, P2, P u Move -δ distance along v direction, P3, P4, P d Move δ distance along the v direction; then the center point P c Move σ distance along the plane normal vector direction of the mine unit plane, that is, P1P2P3P4, and let the unit normal vector of the plane P1P2P3P4 be n. Since it is known that P c Coordinates, regarded as a vector, so calculate the modified P′ c The coordinate formula is as follows:

[0021] P′ c =P c +σn. (3)

[0022] Furthermore, the above step S2 specifically includes the following contents:

[0023] S21. Use the angle relationship of the internal vertices of the mine grid to define the expandable constraint and the flat foldable constraint. Let α i,k (k=1,2,…,6) is the kth angle of the ith vertex of the mine grid. i When the sum of the surrounding angles is equal to 2π, the expandability constraint is satisfied, that is, α i,1 +α i,2 +α i,3 +α i,4 +α i,5 +α i,6=2π. At the same time, according to Kawasaki's theorem, for an origami model that can be folded completely flat, along the folds around any vertex, if you start from any fold line and calculate the angles and values ​​of valley folds and mountain folds alternately in clockwise or counterclockwise order, for an even number of folds, around the inner vertex P i The absolute value of the difference between the alternating angles and is equal to 0, that is, when α i,1 -α i,2 +α i,3 -α i,4 +α i,5 -α i,6 = 0, the model is flat and foldable.

[0024] S22, the objective function of the expandable constraint and its residual formula are as follows:

[0025]

[0026]

[0027] in, is the developable residual defined, N in S22 and S23 v represents the number of vertices of the mine origami mesh, is the kth angle of the ith vertex of the mine grid; Indicates P i The number of adjacent vertices of When the internal point P i The number of adjacent vertices is 6; when When P i is the boundary point. Since the position change of the boundary point does not affect the expandable constraint and the flat foldable constraint, it is set to 0.

[0028] S23, the objective function of the flat foldable constraint and its residual formula are as follows:

[0029]

[0030]

[0031] where r i ff is the flattenable residual;

[0032] Among them, N v represents the number of vertices of the mine origami mesh, is the kth angle of the ith vertex of the mine grid. When the internal point P i The number of adjacent vertices is 6; when When P iIt is the boundary point. Since the position change of the boundary point does not affect the expandable constraint and the flat foldable constraint, it is set to 0.

[0033] S24, the above formula about angle α i,k The calculation method of , for example, the angle between vectors v and w is as follows:

[0034] ∠[v,w]=2atan2(||v×w||,||v||||w||+v·w), (8)

[0035] Among them, ||·|| is the Euclidean 2-norm, and atan2 is based on the range The inverse tangent function of .

[0036] S25, introduce the maximum expandable residual and the maximum flattenable foldable residual At the same time, users can set the corresponding threshold as the termination condition of the iterative optimization process.

[0037] Furthermore, the above step S3 specifically includes the following contents:

[0038] S31. Area homogenization constraint is to construct a three-dimensional model by unifying the triangular area of ​​the mine unit and thus restricting the shape of the mine unit.

[0039] S32. The mine unit is actually composed of triangles. A mine unit has six triangular faces. Therefore, these six faces are classified according to Figure 4 The colors in a are divided into three types of triangles, and their areas are represented by S I ,S II ,S III Then calculate the average area of ​​each type of triangle in the mine origami Let the area of ​​each face converge to the average area of ​​its corresponding class to unify the area of ​​the mine unit.

[0040] S33, the area homogenization constraint function and its corresponding residual formula are defined as follows:

[0041]

[0042]

[0043] Where N f is the number of mine mesh model faces, S j is the area of ​​the jth face of the mine grid. According to the color, when the face type is green, When the face type is red face, When the face type is a blue face,

[0044] S34, Introduction Represents the maximum area normalized residual, which is used as the iteration termination condition. When it is less than a given threshold, the mine unit satisfies the area homogeneity constraint.

[0045] Furthermore, the above step S4 specifically includes the following contents:

[0046] S41. Propose a multi-objective optimization problem: construct an expandable and flat foldable mine origami structure using equal-area mine units.

[0047] S42. Define the multi-objective optimization function formula E:

[0048] E=ω dev E dev +ω ff E ff +ω area E area , (11)

[0049] where ω dev ,ω ff and ω area are the corresponding constraint function weights, E dev With E ff is the expandable and flattenable folding constraint imposed on the mine grid, and E area The function is to impose area homogenization constraints on the mine unit triangles, so as to construct a free-form surface mine origami model using equal-area mine units.

[0050] S43. Use the Levenberg-Marquardt Method iterative optimization algorithm to adjust the parameter values ​​to minimize the objective function so as to minimize the residual between the objective function and the actual observed value.

[0051] S44, iterative optimization termination condition: when the maximum expandable residual in formula (5) The maximum flattenable foldable residual in formula (7) The maximum area normalized residual of formula (10) is less than a given threshold (set as 1e in this paper). -12 ), the iterative optimization process terminates. At the same time, if the number of iterations reaches the specified number (set as 100 in this paper), the iterative optimization process terminates.

[0052] The beneficial effects of the present invention are:

[0053] The present invention can generate a target surface with arbitrary curvature by utilizing parametric equations and a parameter-based surface interpolation method, providing a wide range of design freedom and flexibility for subsequent origami modeling. By defining expandable and flat folding constraints, it is ensured that the generated three-dimensional mine origami structure can accurately meet the geometric requirements. The introduction of area homogenization constraints ensures the uniformity of the generated mine origami units in area, thereby improving the consistency of the origami structure and reducing differences. The method of the present invention can be widely used in the field of engineering origami, and has significant application prospects in medical equipment, construction, aerospace and other fields that require surface modeling, such as vascular stents, deformable robots, etc. BRIEF DESCRIPTION OF THE DRAWINGS

[0054] Figure 1 The origami modeling process of the mine on the quarter vase surface (surface with parametric equations); a represents the target surface of the quarter vase; b represents the state of flattening the mine unit (unfolded state) onto the target surface; c represents the basic mine grid of the quarter vase;

[0055] Figure 2 The process of modeling the origami mine using NURBS surface (parameter-free surface) in Rhino; a represents the target surface; b represents the state of flattening the mine unit (unfolded state) onto the target surface; c represents the basic mine grid;

[0056] Figure 3 Mine unit vertex adjustment process;

[0057] a: The mine unit to be adjusted consists of six folds intersecting at the center point, including a center point P c , six edge points P1, P2, P3, P4, P u and P d ;

[0058] b: The mine units after preliminary adjustment, where P1, P2 and P u Along the v direction, increasing -δ distance, P3, P4 and P d Increase -δ distance along v direction;

[0059] c: center point P of the mine unit c The adjustment process of P c Move σ distances in the direction of the found quantity along the plane P1P2P3P4, and get P′ c ;

[0060] Figure 4 Six-fold mine unit base and mine origami folding diagram;

[0061] a: The mine units are divided into three types of surfaces and marked with colors, namely green, red and blue surfaces. Their areas are respectively represented by S I , SII , S III express;

[0062] b: By N r ×N c The mine grid crease diagram consists of mine units, where N r =3, N c =3;P i is the ith vertex of the mine grid, α i,k (k=1,2,…,6) is P i The kth angle of the vertex;

[0063] Figure 5 Schematic diagram of the process for implementing the present invention. DETAILED DESCRIPTION

[0064] 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.

[0065] like Figure 5 As shown, the present invention is a three-dimensional surface inverse modeling method based on mine origami and its variants, comprising the following steps:

[0066] S1: Use parametric equations or parameter-based surface interpolation methods to generate a target surface with arbitrary curvature, and then flatten the classic six-fold mine origami unit onto the target surface to obtain a basic mine grid model;

[0067] S2: Define expandable and flat foldable constraints to achieve a three-dimensional mine origami structure that satisfies geometric constraints;

[0068] S3: Define area homogenization constraints to obtain the mine origami structure that satisfies the equal-area mine unit;

[0069] S4: Construct a multi-objective optimization function to achieve a mine origami structure that satisfies the geometric constraints (S2) and the area homogenization constraints (S3);

[0070] The specific contents of step S1 of the present invention include:

[0071] S11. First, determine the target surface. For a general free-form surface, it can be described by a parametric equation, namely, the function S(u,v), with u and v as input parameters. Figure 1 , modeling process of a quarter vase (free-form surface with parametric equations).

[0072] More generally, complex surfaces do not have fixed parametric equations to represent them. In this case, we can use NURBS surfaces, which are mathematical models that accurately represent complex shapes based on a combination of control points, weights, and basis functions. The formula is as follows:

[0073]

[0074] Where S(u,v) represents the point on the surface under the parameters (u,v); P i,j is the control point, i and j represent the index of the control point in the u and v directions respectively, defining the shape of the surface; N i,p (u) and N j,q (v) are B-spline basis functions with respect to parameters u and v respectively; p and q are the orders of basis functions, which determine the smoothness of the surface; and the basis functions in NURBS are usually B-spline basis functions, which are defined as follows:

[0075]

[0076] Where u is a parameter. According to the above formula, as long as the parameters u and v are given, the target surface without fixed parameter equation can be determined. Figure 2 As shown, the modeling steps for creating a free-form surface in Rhino are given.

[0077] S12, then tile the mine units, and use S to map each node (u, v) on the mine grid to the vertex P (x, y, z) on its corresponding mine grid model, so as to achieve the mine unit tiling the target surface, that is, calculate the vertex coordinates of a single mine unit according to the uv parameter equation, and tile the mine units one by one along the u and v directions until the entire target surface is covered, as shown in Figure 1 (b), the effect shown in 2(c).

[0078] S13. In order to generate the mine base mesh in a partially folded state, we need to adjust the mine unit vertices. This step corresponds to Figure 1 (b)-(c), Figure 2 (c)-(d) process. Figure 3 , showing the mine unit vertex adjustment process, P1, P2, P u Move -δ distance along v direction, P3, P4, P d Move δ distance along the v direction; then the center point P c Move σ distance along the plane normal vector direction of the mine unit plane, that is, P1P2P3P4, and let the unit normal vector of the plane P1P2P3P4 be n. Since it is known that P c Coordinates, regarded as a vector, so calculate the modified P′ c The coordinate formula is as follows:

[0079] P′c =P c +σn. (3)

[0080] The above step S2 specifically includes the following contents:

[0081] S21. Use the angle relationship of the internal vertices of the mine grid to define the expandable constraint and the flattenable foldable constraint. i When the sum of the surrounding angles is equal to 2π, the expandability constraint is satisfied, that is, α i,1 +α i,2 +α i,3 +α i,4 +α i,5 +α i,6 =2π. At the same time, according to Kawasaki's theorem, for an origami model that can be folded completely flat, along the folds around any vertex, if you start from any fold line and alternately calculate the angles and values ​​of valley folds and mountain folds in a clockwise or counterclockwise order, for an even number of folds, around the inner vertex P i The absolute value of the difference between the alternating angles and is equal to 0, that is, when α i,1 -α i,2 +α i,3 -α i,4 +α i,5 -α i,6 = 0, the model is flat and foldable.

[0082] S22, the objective function of the expandable constraint and its residual formula are as follows:

[0083]

[0084]

[0085] S23, the objective function of the flat foldable constraint and its residual formula are as follows:

[0086]

[0087]

[0088] Among them, N v represents the number of vertices of the mine origami mesh, is the kth angle of the ith vertex of the mine grid. When the internal point P i The number of adjacent vertices is 6; when When P i It is the boundary point. Since the position change of the boundary point does not affect the expandable constraint and the flat foldable constraint, it is set to 0.

[0089] S24, the above formula about angle α i,k The calculation method of , for example, the angle between vectors v and w is as follows:

[0090] ∠[v,w]=2atan2(||v×w||,||v||||w||+v·w), (8)

[0091] Among them, ||·|| is the Euclidean 2-norm, and atan2 is based on the range The inverse tangent function of .

[0092] S25, introduce the maximum expandable residual and the maximum flattenable foldable residual At the same time, users can set the corresponding threshold as the termination condition of the iterative optimization process.

[0093] The specific contents of step S3 of the present invention include:

[0094] S31. Area homogenization constraint is to construct a three-dimensional model by unifying the triangular area of ​​the mine unit and thus restricting the shape of the mine unit.

[0095] S32. The mine unit is actually composed of triangles. A mine unit has six triangular faces. Therefore, these six faces are classified according to Figure 4 The colors in a are divided into three types of triangles, and their areas are represented by S I ,S II ,S III Then calculate the average area of ​​each type of triangle in the mine origami Let the area of ​​each face converge to the average area of ​​its corresponding class to unify the area of ​​the mine unit.

[0096] S33, the area homogenization constraint function and its corresponding residual formula are defined as follows:

[0097]

[0098]

[0099] Where N f is the number of mine mesh model faces, S j is the area of ​​the jth face of the mine grid. According to the color, when the face type is green, When the face type is red face, When the face type is a blue face,

[0100] S34, Introduction Represents the maximum area normalized residual, which is used as the iteration termination condition. When it is less than a given threshold, the mine unit satisfies the area homogeneity constraint.

[0101] The step S4 specifically includes the following contents:

[0102] S41. Propose a multi-objective optimization problem: construct an expandable and flat foldable mine origami structure using equal-area mine units.

[0103] S42. Define the multi-objective optimization function formula E:

[0104] E=ω dev E dev +ω ff E ff +ω area E area , (11)

[0105] where ω dev ,ω ff and ω area are the corresponding constraint function weights, E dev With E ff is the expandable and flattenable folding constraint imposed on the mine grid, and E area The function is to impose area homogenization constraints on the mine unit triangles, so as to construct a free-form surface mine origami model using equal-area mine units.

[0106] S43. Use the Levenberg-Marquardt Method iterative optimization algorithm to adjust the parameter values ​​to minimize the objective function so as to minimize the residual between the objective function and the actual observed value.

[0107] S44, iterative optimization termination condition: when the maximum expandable residual in formula (5) The maximum flattenable foldable residual in formula (7) The maximum area normalized residual of formula (10) is less than a given threshold (set as 1e in this paper). -12 ), the iterative optimization process terminates. At the same time, if the number of iterations reaches the specified number (set as 100 in this paper), the iterative optimization process terminates.

[0108] 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 multi-objective optimization modeling method based on free-form surface fitting of mine origami, characterized in that: The steps include: S1: Use parametric equations or parameter-based surface interpolation methods to generate a target surface with arbitrary curvature, and then flatten the classic six-fold mine origami unit onto the target surface to obtain a basic mine grid model; S2: Define expandable and flat foldable constraints to achieve a three-dimensional mine origami structure that satisfies geometric constraints; S3: Define area homogenization constraints to obtain the mine origami structure that satisfies the equal-area mine unit; S4: Construct a multi-objective optimization function to achieve a mine origami structure that satisfies the geometric constraints (S2) and the area homogenization constraints (S3).

2. The multi-objective optimization modeling method based on mine origami fitting free-form surface according to claim 1 is characterized in that: The steps of generating the basic mesh model in S1 are as follows: S11. First, determine the target surface. For a general free-form surface, it can be described by a parametric equation, namely, the function S(u,v), with u and v as input parameters; the modeling process of a quarter vase (free-form surface with a parametric equation); More generally, complex surfaces do not have fixed parametric equations to represent them. In this case, non-uniform rational B-spline NURBS surfaces are used. It is a mathematical model based on a combination of control points, weights, and basis functions to accurately represent complex shapes. The formula is as follows: Where S(u,v) represents the point on the surface under the parameters (u,v); P i,j is the control point, i and j represent the index of the control point in the u and v directions respectively, defining the shape of the surface; N i,p (u) and N j,q (v) are B-spline basis functions with respect to parameters u and v respectively; p and q are the orders of basis functions, which determine the smoothness of the surface; and the basis functions in NURBS are usually B-spline basis functions, which are defined as follows: where u i Represents a node value in a node vector, the node vector {u0,u1,u2,…,u n+p+1 } is a non-decreasing numerical sequence used to divide the parameter domain and affect the shape of the basis function; n is the number of control points minus 1 (the index range of control points is 0 to n); p is the order of the B-spline, and p = 3 for cubic B-spline; According to the above formula, as long as the parameters u and v are given, the target surface with no fixed parameter equation can be determined, and the modeling steps for establishing a free-form surface in Rhino are given; S12, then tiling the mine units, mapping each node (u, v) on the mine grid to the vertex P (x, y, z) on its corresponding mine grid model through S, so as to achieve tiling the target surface with the mine units, that is, calculating the vertex coordinates of a single mine unit according to the uv parameter equation, and tiling the mine units one by one along the u and v directions until the entire target surface is covered; S13. In order to generate the mine base mesh in a partially folded state, we need to adjust the mine unit vertices. The mine unit vertex adjustment process is shown. P1, P2, P3, P4, P u , P d is the edge vertex of the mine unit, P c is the center vertex of the mine unit; P1, P2, P u Move -δ distance along v direction, P3, P4, P d Move δ distance in the v direction; then move the center point p c Move σ distance along the plane normal vector direction of the mine unit plane, that is, P1P2P3P4, and let the unit normal vector of the plane P1P2P3P4 be n. Since it is known that P c Coordinates, regarded as a vector, so calculate the modified P′ c The coordinate formula is as follows: P′ c =P c +σn. (3)。 3. The multi-objective optimization modeling method based on mine origami fitting free-form surface according to claim 1 is characterized in that: The specific method of S2 is: S21. Use the angle relationship of the internal vertices of the mine grid to define the expandable constraint and the flat foldable constraint. Let α i,k (k=1,2,…,6) is the kth angle of the ith vertex of the mine grid. i When the sum of the surrounding angles is equal to 2π, the expandability constraint is satisfied, that is, α i,1 +α i,2 +α i,3 +α i,4 +α i,5 +α i,6 =2π; At the same time, according to Kawasaki's theorem, for an origami model that can be folded completely flat, along the folds around any vertex, if you start from any fold line and calculate the angles and values ​​of valley folds and mountain folds alternately in clockwise or counterclockwise order, for an even number of folds, around the inner vertex P i The absolute value of the difference between the alternating angles and is equal to 0, that is, when α i,1 -α i,2 +α i,3 -α i,4 + α i ,5 -α i,6 = 0, the model is flat and foldable; S22, the objective function of the expandable constraint and its residual formula are as follows: in, is the developable residual defined, N in S22 and S23 v represents the number of vertices of the mine origami mesh, is the kth angle of the ith vertex of the mine grid; Indicates P i The number of adjacent vertices of When the internal point P i The number of adjacent vertices is 6; when When P i is the boundary point. Since the position change of the boundary point does not affect the expandable constraint and the flat foldable constraint, it is set to 0. S23, the objective function of the flat foldable constraint and its residual formula are as follows: among them is the flattenable residual; S24, the above formula about angle α i,k The calculation method of , for example, the angle between vectors v and w is as follows: ∠[v,w]=2 atan2(||v×w||,||v||||w||+v·w), (8) Among them, ||·|| is the Euclidean 2-norm, and atan2 is based on the range The inverse tangent function of ; S25, introduce the maximum expandable residual and the maximum flattenable foldable residual At the same time, users can set the corresponding threshold as the termination condition of the iterative optimization process.

4. The multi-objective optimization modeling method based on mine origami fitting free-form surface according to claim 1 is characterized in that: The steps for defining the area homogenization constraint in S3 are as follows: S31. Area homogenization constraint is to construct a three-dimensional model by unifying the triangular area of ​​the mine unit and thus restricting the shape of the mine unit; S32. The mine unit is actually composed of triangles. A mine unit has six triangular faces. Therefore, these six faces are classified and the colors are divided into three types of triangles. Their areas are expressed as S I ,S II ,S III Represents; then calculate the average area of ​​each type of triangle in the mine origami The area of ​​each face is made to converge to the average area of ​​its corresponding class, thus unifying the area of ​​the mine unit; S33, the area homogenization constraint function and its corresponding residual formula are defined as follows: Where N f is the number of mine mesh model faces, S j is the area of ​​the jth face of the mine grid. According to the color, when the face type is green, When the face type is red face, When the face type is a blue face, S34, Introduction Represents the maximum area normalized residual, which is used as the iteration termination condition. When it is less than a given threshold, the mine origami model satisfies the area homogenization constraint.

5. The multi-objective optimization modeling method based on mine origami fitting free-form surface according to claim 1 is characterized in that: The specific method of adding a multi-objective optimization function to the generated basic grid in S4 is as follows: S41. Propose a multi-objective optimization problem: construct an expandable and flat foldable mine origami structure using equal-area mine units; S42. Define a multi-objective optimization function E: E=ω dev E dev +oh ff E ff +oh area E area , (11) where ω dev ,ω ff and ω area are the corresponding constraint function weights, E dev With E ff is the expandable and flattenable folding constraint imposed on the mine grid, and E area The function is to impose area uniformity constraints on the mine unit triangles, so as to realize the construction of a free-form surface mine origami model using equal-area mine units; S43, using the Levenberg-Marquardt Method iterative optimization algorithm to adjust the parameter value to minimize the objective function so as to minimize the residual between the objective function and the actual observed value; S44, iterative optimization termination condition: when the maximum expandable residual in formula (5) The maximum flattenable foldable residual in formula (7) When the maximum area normalized residual of formula (10) is less than a given threshold, the iterative optimization process is terminated. At the same time, if the number of iterations reaches the specified number, the iterative optimization process is terminated.

6. The multi-objective optimization modeling method based on mine origami fitting free-form surface according to claim 5 is characterized in that: Given a threshold value of 1e -12 , the number of iterations is set to 100.

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