CN116611285A - Axisymmetric mine paper folding structure modeling method based on multi-objective optimization and expandable solar cell panel structure - Google Patents

Abstract

The invention discloses an axisymmetric water mine paper folding structure modeling method based on multi-objective optimization and an expandable solar cell panel structure, which comprises the following steps: s1, designing a two-dimensional contour curve by adopting an interactive modeling mode, generating a three-dimensional target curved surface by rotating the curve around a common axis, and then tiling a mine unit to obtain a basic grid model B _a The method comprises the steps of carrying out a first treatment on the surface of the S2, defining expandable constraint, and realizing a three-dimensional mine paper folding structure meeting geometric constraint; s3, defining a distance constraint, and realizing a three-dimensional mine paper folding structure meeting the distance constraint; s4, defining a uniformity constraint to obtain a mine paper folding structure meeting the uniformity constraint; s5, pair B _a Implement multiple functionsTarget constraint, solving two types of optimization problems; s6, pair B _a And simultaneously implementing three constraints to obtain a result model. The invention is described in the section B _a Two kinds of constraints or three kinds of constraints are added, and a modeling method of the axisymmetric torpedo folding paper structure is achieved. The resulting structure can be used in foldable houses, foldable pipes, foldable robot wheels, etc.

Description

Axisymmetric mine paper folding structure modeling method based on multi-objective optimization and expandable solar cell panel structure

Technical Field

The invention relates to the technical field of computer three-dimensional graph modeling, in particular to an axisymmetric mine paper folding structure modeling method based on multi-objective optimization and an expandable solar cell panel structure.

Background

The paper folding structure is formed by folding a plane paper serving as a material through folds. The rationality and shape variability of the paper folding structure makes it possible for engineers and scientists to study the design and function of paper folding in the engineering field. The paper folding structure is widely applied to the fields of science and engineering, and is as small as nano materials and as large as aviation grade. In application, the invention is applicable to the simulation of automatic folding micro-containers for controlling drug delivery, self-expanding folded paper stent grafts based on mine folding paper, self-folding centimeter-level crawling robots composed of shape memory composite materials, and expandable solar arrays based on folded paper for spacecraft applications.

Mine folding paper is one of the most widely used paper folding patterns, and the central vertex of a mine unit intersected by six folds consists of two mountain folds and four valley folds, which point out the folding direction of each fold in a plane. Due to the mirror image replication characteristic of the mine units and the fact that the degree of freedom of the mine paper folding is high in the practical application process, the six-fold mine units can flexibly and reversely design the mine paper folding model due to the fact that the unfolding ratio of the six-fold mine units between the unfolding state and the sealing state is large. Based on the symmetrical characteristic of the mine folding paper, the invention uses the basic grid model generated based on the intersection of six folds, and adds multi-objective constraint to the left half part of a single strip of the basic grid model, so as to obtain the mine folding paper modeling method with the axisymmetrical characteristic.

Disclosure of Invention

The invention provides a modeling method of an axisymmetric mine paper folding structure based on multi-objective optimization and an expandable solar cell panel structure. In computer programming software, firstly, a two-dimensional contour curve is designed in an interactive modeling mode, a three-dimensional target curved surface is generated by rotating the contour curve around a public axis, then a mine unit is tiled, a basic grid model is obtained, expandable constraint, distance constraint and uniformity constraint are defined, a three-dimensional mine folding paper structure meeting optimization constraint is obtained, then multi-target constraint is implemented on the basic grid model, two types of optimization problems are solved, and finally three optimization constraints are added to the basic grid model at the same time, so that a mine folding paper capable of being folded flatly fits a target surface more accurately and crease patterns of the mine folding paper are uniform.

In order to achieve the above purpose, the technical scheme provided by the implementation of the invention is as follows:

an axisymmetric mine paper folding structure modeling method based on multi-objective optimization, the method comprising:

s1, adopting an interactive modeling mode to design a two-dimensional contour curve, generating a three-dimensional target curved surface by rotating the contour curve around a common axis, and then tiling a mine unit to obtain a basic grid model B _a ；

S2, defining expandable constraint, and realizing a three-dimensional mine paper folding structure meeting geometric constraint;

s3, defining a distance constraint, and realizing a three-dimensional mine paper folding structure meeting the distance constraint;

s4, defining a uniformity constraint to obtain a mine paper folding structure meeting the uniformity constraint;

s5, pair B _a Implementing multi-objective constraint to solve two types of optimization problems;

s6, pair B _a And simultaneously implementing three optimization constraints to obtain an ideal result model.

Further describing, the step S1 specifically includes:

s11, firstly modeling a three-dimensional target curved surface: generating a three-dimensional target curved surface by rotating the 2D contour curve Γ around the z-axis using axisymmetric characteristics;

s12, a NURBS curve is adopted to represent a two-dimensional contour curve Γ: the control points of NURBS are specified by the user, and the contour shape can be interactively designed by adding, deleting and moving these control points to generate different three-dimensional target surfaces.

S13, tiling the mine units according to axisymmetric characteristics to generate a basic grid model; the method comprises the following steps:

as shown in FIG. 1, set parameter P _i Representing vertices, N, of a finger mine base mesh model _s Representing the number of mesh model strips, N _b Representing the number of mine units in a stripe, Θ represents the angle of rotation about the z-axis,for example by vertex P _i The left part of the axisymmetric mine base mesh model strip consisting of (i=1, 2., 11) is noted +.>

Mirror replication with respect to the x-z plane: the unit vector n= (0, 1, 0) perpendicular to the x-z plane, from which the mirror matrix M of the three-dimensional space with respect to the x-z plane is deduced, is represented in equation (1):

will beMirror image copying transformation is carried out on the x-z plane for one time, so that a strip of the mine paper folding model is obtained;

rotational replication around the z-axis: the resulting swathe was rotationally replicated around the z-axis as shown in equation (2):

V′＝R(Θ)V， (2)

where V represents the coordinates before a point rotates around the z-axis, V' is the coordinates after solving for the rotation angle, and R (Θ) is the rotation matrix, defined in equation (3):

that is, to construct a base mesh model with axisymmetric properties, the strip is rotated Θ about the z-axis by N copies _s -1 time;

the mine paper folding unit is tiled on the target curved surface after mirror image copying and rotation copying to generate a basic grid model, and the basic grid model is used as the primary approximation of the mine paper folding unit on the target curved surface and is used as the initial state of the following optimization process as shown in fig. 2. Based on the symmetry characteristics, the optimization is performed on the left part of the mine basic grid model stripPerformed above.

Further describing, the step S2 specifically includes:

s21, defining that the expandable constraint condition is met: introducing the variable alpha _i，k ，α _i，k The kth angle, which is the ith vertex of the three-dimensional basic mesh model, is shown in FIG. 1 (a), when alpha _i，1 +α _i，2 +α _i，3 +α _i，4 +α _i，5 +α _i，6 When =2pi, i.e. when the internal vertices P of the mine base mesh model _i When the sum of the angles of the surroundings is equal to 2 pi, the expandable constraint is satisfied, and the mine paper folding model is expandable; since the mine basic grid model strip is of a plane symmetrical structure, alpha can be deduced _i，1 ＝α _i，6 、α _i，2 ＝α _i，5 and α_i，3 ＝α _i，4 In the followingAdding expandable constraints thereto, i.e. when alpha _i，1 +α _i，2 +α _i，3 When pi, the water mine paper folding model is expandable;

s22, defining that the flat folding condition is met: when surrounding the internal apex P _i The sum of the alternating angles of (a) is equal to pi, i.e. when alpha _i，1 +α _i，3 +α _i，5 ＝α _i，2 +α _i，4 +α _i，6 Pi, and crease patterns should avoid self-intersection, the water mine paper folding model is flat foldable；

S23, an objective function E capable of expanding constraint _f Defined by equation (4), the corresponding expandable constraint residual is defined by equation (5):

wherein Is defined as the expandable constraint residual quantity, N _v Left part of the strip of the mine foundation grid model>Alpha, alpha _i，k Is the kth fan angle of vertex i, +.>When->At the time, vertex P _i The number of adjacent vertices is 4, < >>When->When (I)>

S24, calculating an included angle alpha by using a formula (6) _i，k For example, the angle α between vectors v and w;

α＝2atan 2(||v×w||，||v||||w||+v·w)， (6)

wherein, I.I are Euclidean 2-norms.

S25, utilize and />Respectively represent the vertexes P _i The maximum and average expandable constraint residual amounts of (2) and minimizing the expandable constraint residual amounts, wherein the sum of angles around the internal vertices of the basic mesh model is equal to 2 pi, and the torpedo folding paper model is expandable;

further describing, the step S3 specifically includes:

s31, defining that the distance constraint condition is met: when the vertex P of the basic grid model _i When the distance from the projection point on the corresponding contour curve is smaller than a threshold value given by a user, the distance constraint is met, and the basic grid model is considered to be more accurately fitted to the target surface;

s32, rotating theta around the z axis according to the profile curve gamma on the x-z plane to obtain a curve gamma _a Classifying the vertexes, classifying the left part of the strip of the mine basic grid modelThe points on the graph are divided into a center point, a point on the Γ curve, and Γ _a Three classes of points on the curve, e.g., point P in FIG. 3 ₃ 、P ₆ and P₉ Is the center point, P ₁ 、P ₅ 、P ₇ and P₁₁ Is the point on the Γ curve, P ₂ 、P ₄ 、P ₈ and P₁₀ Is Γ _a Points on the curve, vertices P on the contour curve Γ _i Marked as->Curve Γ _a Vertex P on _i Marked as->

S33, vertex P _i To a corresponding curveThe projection point on the plane is denoted as P' _i Calculate the vertex P _i (x _i ，y _i ，z _i ) And projection point P' _i (x _i ′，y _i ′，z _i ' distance d between _i As shown in formula (7):

s34, objective function E of distance constraint _d Defined by equation (8), the amount of distance residual corresponding to it is defined by equation (9):

wherein Is the distance residual quantity of the defined ith point, L _B The length of the diagonal line of the boundary box of the three-dimensional mine grid model is used for realizing dimensionless folding paper of the mine.

S35, because of the concave-convex feeling presented by the mine paper folding model, the point and the center point on the curve are respectively on the upper surface and the lower surface, and the distance constraint only considers the point on the upper surface, so the point P is regarded as the vertex _i When the type of (c) is a center point,when the vertex P _i Is of the type (I)When d _i Refers to vertex->To the projection point on the contour curve ΓDistance, as the vertex P _i The type of (2) is +.>When d _i Refers to the vertexTo curve Γ _a The distance of the projection points on the model is +.> and />Are all on the corresponding curves, so +.>Approaching 0.

S36, utilize and />Respectively represent the vertexes P _i The maximum and average distance constraint residual quantity of the corresponding contour curves are calculated, the distance between the vertex of the basic grid model and the projection point on the corresponding contour curves is smaller than a threshold value given by a user, and the mine paper folding model meets the distance constraint;

further describing, the step S4 specifically includes:

s41, defining a uniformity constraint condition: the three-dimensional mine folding paper is constructed by using uniform mine units, the plane crease patterns of the three-dimensional mine folding paper display the mine units with the same size, and the special mine units can be uniform rectangles or squares;

s42, firstly, the left part of the mine basic grid model stripThe upper edges are classified into oneAltogether, there are 4 classes: as shown in FIG. 1 (b), i.e., long side, short side, transverse side and oblique side, their lengths are denoted by L _L 、L _s 、L _h and L_b Representation, then calculate +.>Average length of each class of edges on +.> and />The length of each edge is made to approach the average length of the corresponding classification, so as to achieve the effect of uniform mine units.

S43, homogenizing constrained objective function E _u Defined by equation (10), the amount of uniformization residue corresponding to it is defined by equation (11):

wherein Is defined as the amount of uniformized residue, N _e Is the left part of the mine grid model +.>The number of edges, L _j Refers to->The length of the j-th side of the upper part;

s44 when the type of edge is a long edge,when the type of edge is short side, +.>When the type of edge is a lateral edge, +.>When the type of edge is hypotenuse, +.>A uniform mine unit can be obtained at this time. In particular whenWhen the mine units are unified rectangles; when-> and />At the same time can be pushed outA uniform square mine unit can be finally obtained.

S45, utilize and />The maximum and average uniformity constraint residual amounts are respectively represented, and the uniformity constraint residual amount is minimized, at this time, the mine folding paper meets the uniformity constraint, and the mine units of the corresponding plane folding pattern are represented as uniform shapes;

further describing, the step S5 specifically includes:

s51, implementing multi-objective constraint restriction on a basic grid model, and solving two types of optimization problems: (I) Constructing a flat foldable mine paper to more accurately fit the target surface; (II) constructing a flat foldable 3D mine paper using the homogenized mine units;

s52, defining an optimization problem I: simultaneously adding expandable constraint and distance constraint to the basic grid model, and solving a multi-objective optimization problem I related to the basic grid model;

s53, providing a multi-objective constraint objective function E _I Solving by equation (12);

E _I ＝ω _f E _f +ω _d E _d ， (12)

wherein ω_f and ω_d Refers to the weight, E _f Is to the left part of the mine grid model stripAdding an extensible constraint to the internal vertices of E _d Is to->Adding a distance constraint, and finally realizing the flat folding mine paper folding which is more accurately fitted with the axisymmetric target surface;

s54, defining an optimization problem II: simultaneously adding expandable constraint and homogenizing constraint to the basic grid model, and solving a multi-objective optimization problem II related to the basic grid model;

s55, providing a multi-objective constraint objective function E _II Solving by equation (13):

E _II ＝ω _f E _f +ω _u E _u ， (13)

wherein ω_f and ω_u Refers to the weight, E _f Is to the left part of the mine grid model stripAdding an extensible constraint to the internal vertices of E _u Is to->The upper edge adds a uniformity constraint. Implementation of constructing axisymmetry using unified mine unitsA mine folding paper. Namely, on the premise that the 3D mine folding paper can be folded flatly, the plane crease patterns of the 3D mine folding paper display mine units with the same size;

s56, solving an objective function: solving the class I optimization and the class II optimization based on finite difference to obtain gradient in the descending direction;

s57, optimizing termination conditions: in class I optimization as in equation (5)Maximum expandable residual amount of (2) and +.>When the maximum distance residual amount of (2) is smaller than the given threshold, the class I optimization process is terminated. In class II optimization when +.>Maximum expandable residual amount of (2) and +.>When the maximum amount of uniformization residuals is less than a given threshold, the class II optimization process is terminated. Meanwhile, when the iteration times reach the designated times or the variation of the residual quantity in the iteration process is small, and no obvious difference exists before and after, the class I optimization process and the class II optimization process are terminated;

further describing, the step S6 specifically includes:

s61, after the optimization of the class I and the optimization of the class II are terminated, the minimization of the residual quantity has a better result, and the expandable constraint, the distance constraint and the uniformity constraint are simultaneously applied to the basic grid model, so that the flat foldable mine paper is more accurately fitted to the target surface, and the crease pattern of the flat foldable mine paper shows uniform mine units, as shown in fig. 4;

s62, proposing an objective function E using 3 constraints simultaneously _III Defined in equation (14):

E _III ＝ω _f E _f +ω _d E _d +ω _u E _u ， (14)

wherein ω_f 、ω _d and ω_u Is the weight, E _f Is to the left part of the mine grid model stripAdding an extensible constraint to the internal vertices of E _d Is to->Adding distance constraints, E _u Is to->Adding a uniformity constraint to the upper edge;

s63, solving an objective function: solving the objective function of the 3 constraints, and minimizing the multi-objective residual quantity. Solving an objective function based on finite difference to obtain a descent gradient;

s64, termination conditions of 3 constraints:

when (when) and />If the threshold value is smaller than the given threshold value of the user, the optimization is terminated; further improving the limiting condition to optimize, the smaller the possibility of obtaining an ideal result is, so that when the iteration times reach the designated times or the variation of the residual quantity in the iteration process is small, the optimization process is terminated when no obvious difference exists between the front and the back; an axisymmetric mine paper folding model which simultaneously implements 3 optimization constraints and a crease pattern corresponding to the axisymmetric mine paper folding model are shown;

further described, the solving objective function is specifically: the two types of optimization problems and the optimization problem with 3 constraints added simultaneously are optimized by using the same framework, and an objective function of the optimization problem is called as E. As the vertexes can keep symmetrical structure only when moving on the plane where they initially lie, polar coordinates are introduced for axisymmetric mine paper and r_i Is defined in formula (15) and formula (16) to represent +.>According to-> and />To construct a gradient; vertex P _i Is moved in the plane of the original place, so +.>Then +.>And +.about.of formula (19)>To construct a gradient.

Solving two types of optimization problems and simultaneously adding 3 kinds of constraint optimization problems, and calculating gradient in the descending direction, wherein:

according to the chain law, involved in the optimization processThe definition is as shown in formula (20):

generating the mine folding paper with axisymmetric characteristics: the left part of the obtained optimized water mine folding paper strip is recorded asBased on the properties of the mine paper folding, use +.>And generating the final three-dimensional mine folding paper. Specifically: by->Mirror image replication with respect to the x-z plane to construct a stripe S _O Strip S _O Rotation replication N about the z-axis _s -1 time, producing a torpedo paper with axisymmetric properties +.>

By utilizing the structural modeling method, the invention also provides an expandable solar cell panel structure, which is obtained by the structural modeling method, as shown in fig. 5.

The invention has the beneficial effects that:

1. adding two different types of optimization to the basic grid model, solving the two types of optimization problems, and constructing a mine paper folding structure with axisymmetric characteristics;

2. realizing a multi-objective optimization method aiming at a generated basic grid model, and simultaneously adding expandable constraint, distance constraint and uniformity constraint;

3. the structure obtained by the method is widely applied to the engineering field, and expands the exploration of axisymmetric variable structures, such as foldable houses, foldable tubular materials, foldable robot wheels, DNA paper folding fields and the like.

Drawings

FIG. 1 is a diagram of a mine unit and a mine crease pattern where six creases intersect;

fig. 1 (a): a mine unit, wherein alpha _i，k (k=1, 2.., 6) is P < th > _i The kth angle of the vertex;

fig. 1 (b): the mine units comprise 4 types of sides, namely long side, short side, transverse side and oblique side, the lengths of which are respectively L _l 、L _s 、L _h and L_b A representation;

fig. 1 (c): crease pattern of mine folding paper composed of four strips (one strip contains 3 mine units), wherein N _s ＝4、N _b =3; in (c), P ₂ P ₄ 、P ₅ P ₇ and P₈ P ₁₀ Is a long side, P ₁ P ₃ 、P ₃ P ₅ 、P ₄ P ₆ 、P ₆ P ₈ 、P ₇ P ₉ and P₉ P ₁₁ Is short side, P ₁ P ₂ 、P ₄ P ₅ 、P ₇ P ₈ and P₁₀ P ₁₁ Is a transverse edge, P ₂ P ₃ 、P ₃ P ₄ 、P ₅ P ₆ 、P ₆ P ₇ 、P ₈ P ₉ and P₉ P ₁₀ Is a bevel edge;

FIG. 2 is an overview of the generation of axisymmetric water-mine paper folding;

fig. 2 (a): a 2D contour curve Γ of the user-interactive design;

fig. 2 (b): rotating the generated target surface about a common axis z-axis using the 2D profile Γ;

fig. 2 (c): tiling the mine units to generate a basic grid model with axisymmetric characteristics for approximating the target surface shown in fig. 2 (b);

FIG. 3 is a contour curve corresponding to a strip and vertex showing an axisymmetric 3D mine paper folding;

FIG. 4 is a diagram of an axisymmetric mine paper folding structure satisfying 3 constraints and a crease pattern corresponding to the axisymmetric mine paper folding structure, wherein the expandable constraint, the distance constraint and the uniformity constraint are added to a basic grid model at the same time;

FIG. 5 is a schematic view of an expandable solar cell panel structure;

FIG. 6 is a flow chart of the present invention.

Detailed Description

The present invention will be described in detail below with reference to the embodiments shown in the drawings. These embodiments are not intended to limit the invention and structural, methodological, or functional modifications of these embodiments that may be made by one of ordinary skill in the art are included within the scope of the invention.

The invention relates to an axisymmetric water mine paper folding structure modeling method based on multi-objective optimization, wherein one embodiment of modeling is realized by utilizing a computer IDEA development tool and Java language programming, and the production and the manufacture of a three-dimensional structure can adopt 3D printing or other industrial manufacturing modes.

As shown in fig. 6, the modeling process of the present invention includes the steps of:

s1, in computer programming software, adopting an interactive modeling mode to design a two-dimensional contour curve, rotating the contour curve around a common axis to generate a three-dimensional target curved surface, and then tiling a mine unit to obtain a basic grid model B _a ；

S2, defining expandable constraint, and realizing a three-dimensional mine paper folding structure meeting geometric constraint;

s3, defining a distance constraint, and realizing a three-dimensional mine paper folding structure meeting the distance constraint;

s4, defining a uniformity constraint to obtain a mine paper folding structure meeting the uniformity constraint;

s5, pair B _a Implementing multi-objective constraint to solve two types of optimization problems;

s6, pair B _a And simultaneously implementing three optimization constraints to obtain a model.

Further describing, the step S1 specifically includes:

s11, firstly modeling a three-dimensional target curved surface: generating a three-dimensional target curved surface by rotating the 2D contour curve Γ around the z-axis using axisymmetric characteristics;

s12, a NURBS curve is adopted to represent a two-dimensional contour curve Γ: the control points of NURBS are specified by the user, and the contour shape can be interactively designed by adding, deleting and moving these control points to generate different three-dimensional target surfaces.

S13, tiling the mine units according to axisymmetric characteristics to generate a basic grid model; the method comprises the following steps:

as shown in FIG. 1, set parameter P _i Representing vertices, N, of a finger mine base mesh model _s Representing the number of mesh model strips, N _b Representing the number of mine units in a stripe, Θ represents the angle of rotation about the z-axis,for example by vertex P _i The left part of the axisymmetric mine base mesh model strip consisting of (i=1, 2., 11) is noted +.>

Mirror replication with respect to the x-z plane: the unit vector n= (0, 1, 0) perpendicular to the x-z plane, from which the mirror matrix M of the three-dimensional space with respect to the x-z plane is deduced, is represented in equation (1):

will beMirror image copying transformation is carried out on the x-z plane for one time, so that a strip of the mine paper folding model is obtained;

rotational replication around the z-axis: the resulting swathe was rotationally replicated around the z-axis as shown in equation (2):

V′＝R(Θ)V， (2)

where V represents the coordinates before a point rotates around the z-axis, V' is the coordinates after solving for the rotation angle, and R (Θ) is the rotation matrix, defined in equation (3):

that is, to construct a base mesh model with axisymmetric properties, the strip is rotated Θ about the z-axis by N copies _s -1 time;

spreading the mine paper folding unit on the target curved surface after mirror image replication and rotation replication to generate a basic grid model, wherein the basic grid model is used as the primary approximation of the mine paper folding unit on the target curved surface as shown in figure 2 and is used as the initial state of the subsequent optimization process, and the optimization is performed on the left part of the mine basic grid model strip according to the symmetrical characteristicPerformed above.

Further describing, the step S2 specifically includes:

s21, defining that the expandable constraint condition is met: introducing the variable alpha _i，k ，α _i，k The kth angle, which is the ith vertex of the three-dimensional basic mesh model, is shown in FIG. 1 (a), when alpha _i，1 +α _i，2 +α _i，3 +α _i，4 +α _i，5 +α _i，6 When =2pi, i.e. when the internal vertices P of the mine base mesh model _i When the sum of the angles of the surroundings is equal to 2 pi, the expandable constraint is satisfied, and the mine paper folding model is expandable; since the mine basic grid model strip is of a plane symmetrical structure, alpha is deduced _i，1 ＝α _i，6 、α _i，2 ＝α _i，5 and α_i，3 ＝α _i，4 In the followingAdding expandable constraints thereto, i.e. when alpha _i，1 +α _i，2 +α _i，3 When pi, the water mine paper folding model is expandable;

s22, defining that the flat folding condition is met: when surrounding the internal apex P _i The sum of the alternating angles of (a) is equal to pi, i.e. when alpha _i，1 +α _i，3 +α _i，5 ＝α _i，2 +α _i，4 +α _i，6 Pi, and crease patterns should avoid self-intersection, the torpedo paper folding model is flat foldable;

s23, an objective function E capable of expanding constraint _f Defined by equation (4), the corresponding expandable constraint residual is defined by equation (5):

wherein Is defined as the expandable constraint residual quantity, N _v Left part of the strip of the mine foundation grid model>Alpha, alpha _i，k Is the kth fan angle of vertex i, +.>When->At the time, vertex P _i The number of adjacent vertices is 4, < >>When->When (I)>

S24, calculating an included angle alpha by using a formula (6) _i，k For example, the angle α between vectors v and w;

α＝2atan 2(||v×w||，||v||||w||+v·w)， (6)

wherein, I.I is Euclidean 2-norm;

s25, utilize and />Respectively represent the vertexes P _i The maximum and average expandable constraint residual amounts of (2) and minimizing the expandable constraint residual amounts, wherein the sum of angles around the internal vertices of the basic mesh model is equal to 2 pi, and the torpedo folding paper model is expandable;

further describing, the step S3 specifically includes:

s31, defining that the distance constraint condition is met: when the vertex P of the basic grid model _i When the distance from the projection point on the corresponding contour curve is smaller than a threshold value given by a user, the distance constraint is met, and the basic grid model is considered to be more accurately fitted to the target surface;

s32, rotating theta around the z axis according to the profile curve gamma on the x-z plane to obtain a curve gamma _a Classifying the vertexes, classifying the left part of the strip of the mine basic grid modelThe points on the graph are divided into a center point, a point on the Γ curve, and Γ _a Three classes of points on the curve, as shown in FIG. 3, e.g., point P ₃ 、P ₆ and P₉ Is the center point, P ₁ 、P ₅ 、P ₇ and P₁₁ Is the point on the Γ curve, P ₂ 、P ₄ 、P ₈ and P₁₀ Is Γ _a Points on the curve, vertices P on the contour curve Γ _i Marked as->Curve Γ _a Vertex P on _i Marked as->

S33, vertex P _i The projection point on the corresponding curve is marked as P' _i Calculate the vertex P _i (x _i ，y _i ，z _i ) And projection point P' _i (x _i ′，y _i ′，z _i ' distance d between _i As shown in formula (7):

s34, objective function E of distance constraint _d Defined by equation (8), the amount of distance residual corresponding to it is defined by equation (9):

wherein Is the distance residual quantity of the defined ith point, L _B The length of the diagonal line of the boundary box of the three-dimensional mine grid model is used for realizing dimensionless folding paper of the mine;

s35, because of the concave-convex feeling presented by the mine paper folding model, the point and the center point on the curve are respectively on the upper surface and the lower surface, and the distance constraint only considers the point on the upper surface, so the point P is regarded as the vertex _i When the type of (c) is a center point,when the vertex P _i The type of (2) is +.>When d _i Refers to vertex->The distance to the projection point on the contour curve Γ, when the vertex P _i The type of (2) is +.>When d _i Refers to vertex->To curve Γ _a The distance of the projection points on the model is +.> and />Are all on the corresponding curves, so +.>Approaching 0;

s36, utilize and />Respectively represent the vertexes P _i The maximum and average distance constraint residual quantity of the corresponding contour curves are calculated, the distance between the vertex of the basic grid model and the projection point on the corresponding contour curves is smaller than a threshold value given by a user, and the mine paper folding model meets the distance constraint;

further describing, the step S4 specifically includes:

s41, defining a uniformity constraint condition: the three-dimensional mine folding paper is constructed by using uniform mine units, the plane crease patterns of the three-dimensional mine folding paper display the mine units with the same size, and the special mine units can be uniform rectangles or squares;

s42, dividing the left part of the mine basic grid model stripThe upper edges are classified into 4 categories: as shown in FIG. 1 (b), i.e., long side, short side, transverse side and oblique side, their lengths are respectively denoted by L _l 、L _s 、L _h and L_b Representing, calculate->Average length of each class of edges on +.> and />The length of each side is made to approach the average length of the corresponding classification so as to achieve the effect of uniform mine units;

s43, homogenizing constrained objective function E _u Defined by equation (10), the amount of uniformization residue corresponding to it is defined by equation (11):

wherein Is defined as the amount of uniformized residue, N _e Is the left part of the mine grid model +.>The number of edges, L _j Refers to->The length of the j-th side of the upper part;

s44 when the type of edge is a long edge,when the type of edge is short side, +.>When the type of edge is a lateral edge, +.>When the type of edge is hypotenuse, +.>A uniform mine unit can be obtained at this time; in particular whenWhen the mine units are unified rectangles; when-> and />At the same time can be pushed outFinally, uniform square mine units can be obtained;

s45, utilize and />The maximum and average uniformity constraint residual amounts are respectively represented, and the uniformity constraint residual amount is minimized, at this time, the mine folding paper meets the uniformity constraint, and the mine units of the corresponding plane folding pattern are represented as uniform shapes;

further describing, the step S5 specifically includes:

s51, implementing multi-objective constraint restriction on a basic grid model, and solving two types of optimization problems: (I) Constructing a flat foldable mine paper to more accurately fit the target surface; (II) constructing a flat foldable 3D mine paper using the homogenized mine units;

s52, defining an optimization problem I: simultaneously adding expandable constraint and distance constraint to the basic grid model, and solving a multi-objective optimization problem I related to the basic grid model;

s53, providing a multi-objective constraint objective function E _I Solving by equation (12);

E _I ＝ω _f E _f +ω _d E _d ， (12)

wherein ω_f and ω_d Refers to the weight, E _f Is to the left part of the mine grid model stripAdding an extensible constraint to the internal vertices of E _d Is to->Adding a distance constraint, and finally realizing the flat folding mine paper folding which is more accurately fitted with the axisymmetric target surface;

s54, defining an optimization problem II: simultaneously adding expandable constraint and homogenizing constraint to the basic grid model, and solving a multi-objective optimization problem II related to the basic grid model;

s55, providing a multi-objective constraint objective function E _II Solving by equation (13):

E _II ＝ω _f E _f +ω _u E _u ， (13)

wherein ω_f and ω_u Refers to the weight, E _f Is to the left part of the mine grid model stripAdding an extensible constraint to the internal vertices of E _u Is to->The upper edge is added with uniformity constraint, so that the axisymmetric mine folding paper is constructed by using uniform mine units, namely, the 3D mine folding paper displays the mine units with the same size in a plane crease pattern on the premise of being capable of being folded flatly;

s56, solving an objective function: solving the class I optimization and the class II optimization based on finite difference to obtain gradient in the descending direction;

s57, optimizing termination conditions: in class I optimization as in equation (5)Maximum expandable residual amount of (2) and +.>When the maximum distance residual amount of (2) is smaller than a given threshold value, terminating the class I optimization process; in class II optimization when +.>Maximum expandable residual amount of (2) and +.>When the maximum uniform residual quantity of the (2) is smaller than a given threshold value, the class II optimization process is terminated, and when the iteration times reach the appointed times or the variation quantity of the residual quantity in the iteration process is small and there is no difference before and after, the class I optimization process and the class II optimization process are terminated;

further describing, the step S6 specifically includes:

s61, after the optimization of the class I and the optimization of the class II are terminated, the minimization of the residual quantity has a better result, and the expandable constraint, the distance constraint and the uniformity constraint are simultaneously applied to the basic grid model, so that the flat foldable mine paper is more accurately fitted to the target surface, and the crease pattern of the flat foldable mine paper shows uniform mine units, as shown in fig. 4;

s62, proposing an objective function E using 3 constraints simultaneously _III Defined in equation (14):

E _III ＝ω _f E _f +ω _d E _d +ω _u E _u ， (14)

wherein ω_f 、ω _d and ω_u Is the weight, E _f Is to the left part of the mine grid model stripAdding an extensible constraint to the internal vertices of E _d Is to->Adding distance constraints, E _u Is to->Adding a uniformity constraint to the upper edge;

s63, solving an objective function: solving the objective function of the 3 constraints, and minimizing the multi-objective residual quantity. Solving an objective function based on finite difference to obtain a descent gradient;

s64, termination conditions of 3 constraints:

when (when) and />If the threshold value is smaller than the given threshold value of the user, the optimization is terminated; further improving the limiting condition to optimize, the smaller the possibility of obtaining an ideal result is, so that when the iteration times reach the designated times or the variation of the residual quantity in the iteration process is small, the optimization process is terminated when no obvious difference exists between the front and the back; exhibiting at the same timeAn axisymmetric mine paper folding model implementing 3 optimization constraints and a crease pattern corresponding to the axisymmetric mine paper folding model;

further described, the solving objective function is specifically:

the solution objective function of claims 6-7: the two types of optimization problems and the optimization problem with 3 constraints added simultaneously are optimized by using the same framework, and an objective function of the optimization problem is called as E. As the vertexes can keep symmetrical structure only when moving on the plane where they initially lie, polar coordinates are introduced for axisymmetric mine paperAnd ri, defined in equation (15) and equation (16), to represent +.>According to-> and />To construct a gradient; vertex P _i Is moved in the plane of the original place, so +.>Then +.>And +.about.of formula (19)>To construct a gradient.

Solving two types of optimization problems and simultaneously adding 3 kinds of constraint optimization problems, and calculating gradient in the descending direction, wherein:

according to the chain law, involved in the optimization processThe definition is as shown in formula (20):

generating the mine folding paper with axisymmetric characteristics: the left part of the obtained optimized water mine folding paper strip is recorded asBased on the properties of the mine paper folding, use +.>And generating the final three-dimensional mine folding paper. Specifically: by->Mirror image replication with respect to the x-z plane to construct a stripe S _O Strip S _O Rotation replication N about the z-axis _s -1 time, producing a torpedo paper with axisymmetric properties +.>

By utilizing the structural modeling method, the invention also provides an expandable solar cell panel structure, which is obtained by the structural modeling method, as shown in fig. 5.

It should be noted that, the present disclosure is explained in terms of embodiments, but not every specific implementation step has an independent technical solution, and this explanation mode is used to make the reader more aware of the design steps and various methods of operation of the present disclosure. The person skilled in the relevant art needs to take the whole specification as a whole, and various technical solutions can be combined appropriately to form other embodiments which can be understood by the person skilled in the relevant art.

The above list of detailed descriptions is only specific to practical embodiments of the present invention, and they are not intended to limit the scope of the present invention, and all equivalent manners or modifications that do not depart from the technical scope of the present invention should be included in the scope of the present invention.

Claims (10)

1. The axisymmetric mine paper folding structure modeling method based on multi-objective optimization is characterized by comprising the following steps of:

s1, adopting an interactive modeling mode to design a two-dimensional contour curve, generating a three-dimensional target curved surface by rotating the contour curve around a common axis, and then tiling a mine unit to obtain a basic grid model B _a ；

S2, defining expandable constraint, and realizing a three-dimensional mine paper folding structure meeting geometric constraint;

s3, defining a distance constraint, and realizing a three-dimensional mine paper folding structure meeting the distance constraint;

s4, defining a uniformity constraint to obtain a mine paper folding structure meeting the uniformity constraint;

s5, pair B _a Implementing multi-objective constraint to solve two types of optimization problems;

s6, pair B _a And simultaneously implementing three optimization constraints to obtain a model.

2. The method for modeling an axisymmetric torpedo paper structure based on multi-objective optimization according to claim 1, wherein the step Sl specifically comprises:

s11, firstly modeling a three-dimensional target curved surface: generating a three-dimensional target curved surface by rotating the 2D contour curve Γ around the z-axis using axisymmetric characteristics;

s12, a NURBS curve is adopted to represent a two-dimensional contour curve Γ: the control points of NURBS are specified by the user, and the contour shape can be interactively designed by adding, deleting and moving these control points to generate different three-dimensional target surfaces.

S13, tiling the mine units according to axisymmetric characteristics to generate a basic grid model; the method comprises the following steps:

as shown in FIG. 1, set parameter P _i Representing vertices, N, of a finger mine base mesh model _s Representing the number of mesh model strips, N _b Representing the number of mine units in a stripe, Θ represents the angle of rotation about the z-axis,for example by vertex P _i The left part of the axisymmetric mine base mesh model strip consisting of (i=1, 2., 11) is noted +.>

Mirror replication with respect to the x-z plane: the unit vector n= (0, 1, 0) perpendicular to the x-z plane, from which the mirror matrix M of the three-dimensional space with respect to the x-z plane is deduced, is represented in equation (1):

will beMirror copy transformation once with respect to the x-z planeObtaining a strip of the mine paper folding model;

rotational replication around the z-axis: the resulting swathe was rotationally replicated around the z-axis as shown in equation (2):

V′＝R(Θ)V， (2)

where V represents the coordinates before a point rotates around the z-axis, V' is the coordinates after solving for the rotation angle, and R (Θ) is the rotation matrix, defined in equation (3):

that is, to construct a base mesh model with axisymmetric properties, the strip is rotated Θ about the z-axis by N copies _s -1 time;

spreading the mine paper folding unit on the target curved surface after mirror image replication and rotation replication to generate a basic grid model, wherein the basic grid model is used as the primary approximation of the mine paper folding unit on the target curved surface and is used as the initial state of the subsequent optimization process, and the optimization is performed on the left part of the mine basic grid model strip according to the symmetrical characteristicPerformed above.

3. The method for modeling an axisymmetric torpedo paper structure based on multi-objective optimization according to claim 1, wherein the step S2 is specifically:

s21, defining that the expandable constraint condition is met: introducing the variable alpha _i，k ，α _i，k The kth angle, alpha, of the ith vertex of the three-dimensional underlying mesh model _i，1 +α _i，2 +α _i，3 +α _i，4 +α _i，5 +α _i，6 When =2pi, i.e. when the internal vertices P of the mine base mesh model _i When the sum of the angles of the surroundings is equal to 2 pi, the expandable constraint is satisfied, and the mine paper folding model is expandable; since the mine basic grid model strip is of a plane symmetrical structure, alpha is deduced _i，1 ＝α _i，6 、α _i，2 ＝α _i，5 and α_i，3 ＝α _i，4 In the followingAdding expandable constraints thereto, i.e. when alpha _i，1 +α _i，2 +α _i，3 When pi, the water mine paper folding model is expandable;

s22, defining that the flat folding condition is met: when surrounding the internal apex P _i The sum of the alternating angles of (a) is equal to pi, i.e. when alpha _i，1 +α _i，3 +α _i，5 ＝α _i，2 +α _i，4 +α _i，6 Pi, and crease patterns should avoid self-intersection, the torpedo paper folding model is flat foldable;

s23, an objective function E capable of expanding constraint _f Defined by equation (4), the corresponding expandable constraint residual is defined by equation (5):

wherein Is defined as the expandable constraint residual quantity, N _v Left part of the strip of the mine foundation grid model>Alpha, alpha _i，k Is the kth fan angle of vertex i, +.>When->At the time, vertex P _i The number of adjacent vertices is 4, < >>When->When r is _i ^f ＝0；

S24, calculating an included angle alpha by using a formula (6) _i，k For example, the angle α between vectors v and w;

α＝2atan 2(||v×w||，||v||||w||+v·w)， (6)

wherein, I.I is Euclidean 2-norm;

s25, utilize and />Respectively represent the vertexes P _i The maximum and average expandable constraint residual amounts of (2) are minimized, and at this time, the sum of angles around the internal vertices of the basic mesh model is equal to 2 pi, and the torpedo folding paper model is expandable.

4. The method for modeling an axisymmetric torpedo paper structure based on multi-objective optimization according to claim 1, wherein the step S3 is specifically:

s31, defining that the distance constraint condition is met: when the vertex P of the basic grid model _i When the distance from the projection point on the corresponding contour curve is smaller than a threshold value given by a user, the distance constraint is met, and the basic grid model is considered to be more accurately fitted to the target surface;

s32, rotating theta around the z axis according to the profile curve gamma on the x-z plane to obtain a curve gamma _a Classifying the vertexes, classifying the left part of the strip of the mine basic grid modelThe points on the graph are divided into a center point, a point on the Γ curve, and Γ _a Three classes of points on the curve, as shown in FIG. 3, e.g., point P ₃ 、P ₆ and P₉ Is the center point, P ₁ 、P ₅ 、P ₇ and P₁₁ Is the point on the Γ curve, P ₂ 、P ₄ 、P ₈ and P₁₀ Is Γ _a Points on the curve, vertices P on the contour curve Γ _i Marked as->Curve Γ _a Vertex P on _i Marked as->

S33, vertex P _i The projection point on the corresponding curve is marked as P' _i Calculate the vertex P _i (x _i ，y _i ，z _i ) And projection point P' _i (x _i ′，y _i ′，z _i ' distance d between _i As shown in formula (7):

s34, objective function E of distance constraint _d Defined by equation (8), the amount of distance residual corresponding to it is defined by equation (9):

wherein Is the distance residual quantity of the defined ith point, L _B The length of the diagonal line of the boundary box of the three-dimensional mine grid model is used for realizing dimensionless folding paper of the mine;

s35, because of the concave-convex feeling presented by the mine paper folding model, the point and the center point on the curve are respectively on the upper surface and the lower surface, and the distance constraint only considers the point on the upper surface, so the point P is regarded as the vertex _i When the type of (c) is a center point,when the vertex P _i The type of (2) is +.>When d _i Refers to vertex->The distance to the projection point on the contour curve Γ, when the vertex P _i The type of (2) is +.>When d _i Refers to vertex->To curve Γ _a The distance of the projection points on the model is +.> and />Are all on the corresponding curves, so +.>Approaching 0;

s36, utilize and />Respectively represent the vertexes P _i And (3) minimizing the distance constraint residual quantity, wherein the distance from the top point of the basic grid model to the projection point on the corresponding contour curve is smaller than a threshold given by a user, and the mine paper folding model meets the distance constraint.

5. The method for modeling an axisymmetric torpedo paper structure based on multi-objective optimization according to claim 1, wherein the step S4 is specifically:

s41, defining a uniformity constraint condition: the three-dimensional mine folding paper is constructed by using uniform mine units, the plane crease patterns of the three-dimensional mine folding paper display the mine units with the same size, and the special mine units can be uniform rectangles or squares;

s42, dividing the left part of the mine basic grid model stripThe upper edges are classified into 4 categories: namely a long side, a short side, a transverse side and a bevel side, the lengths of which are respectively L _l 、L _s 、L _h and L_b Representing, calculate->Average length of each class of edge on and />The length of each edge is close to the average length of the corresponding classification to achieve the effect of uniform mine units；

S43, homogenizing constrained objective function E _u Defined by equation (10), the amount of uniformization residue corresponding to it is defined by equation (11):

wherein Is defined as the amount of uniformized residue, N _e Is the left part of the mine grid model +.>The number of edges, L _j Refers to->The length of the j-th side of the upper part;

s44 when the type of edge is a long edge,when the type of edge is short side, +.>When the type of edge is a lateral edge,when the type of edge is hypotenuse, +.>A uniform mine unit can be obtained at this time; specially when->When the mine units are unified rectangles; when-> and />At the same time +.>Finally, uniform square mine units can be obtained;

s45, utilize and />The maximum and average uniformity constraint residual amounts are respectively represented, and the uniformity constraint residual amount is minimized, at this time, the mine folding paper satisfies the uniformity constraint, and the mine units of the corresponding plane folding pattern are represented as uniform shapes.

6. The method for modeling an axisymmetric torpedo paper structure based on multi-objective optimization according to claim 3, wherein the step S5 specifically comprises:

s51, implementing multi-objective constraint restriction on a basic grid model, and solving two types of optimization problems: (I) Constructing a flat foldable mine paper to more accurately fit the target surface; (II) constructing a flat foldable 3D mine paper using the homogenized mine units;

s52, defining an optimization problem I: simultaneously adding expandable constraint and distance constraint to the basic grid model, and solving a multi-objective optimization problem I related to the basic grid model;

s53, providing a multi-objective constraint objective function E _I Solving by equation (12);

E _I ＝ω _f E _f +ω _d E _d ， (12)

wherein ω_f and ω_d Refers to the weight, E _f Is to the left part of the mine grid model stripAdding an extensible constraint to the internal vertices of E _d Is to->Adding a distance constraint, and finally realizing the flat folding mine paper folding which is more accurately fitted with the axisymmetric target surface;

s54, defining an optimization problem II: simultaneously adding expandable constraint and homogenizing constraint to the basic grid model, and solving a multi-objective optimization problem II related to the basic grid model;

s55, providing a multi-objective constraint objective function E _II Solving by equation (13):

E _II ＝ω _f E _f +ω _u E _u ， (13)

wherein ω_f and ω_u Refers to the weight, E _f Is to the left part of the mine grid model stripAdding an extensible constraint to the internal vertices of E _u Is to->The upper edge is added with uniformity constraint, so that the axisymmetric mine folding paper is constructed by using uniform mine units, namely, the 3D mine folding paper displays the mine units with the same size in a plane crease pattern on the premise of being capable of being folded flatly;

s56, solving an objective function: solving the class I optimization and the class II optimization based on finite difference to obtain gradient in the descending direction;

s57, optimizing termination condition: in class I optimization as in equation (5)Maximum expandable residual amount of (2) and +.>When the maximum distance residual amount of (2) is smaller than a given threshold value, terminating the class I optimization process; in class II optimization when +.>Maximum expandable residual amount of (2) and +.>When the maximum uniform residual quantity of the method is smaller than a given threshold value, the class II optimization process is terminated, and meanwhile, when the iteration times reach the designated times or the variation quantity of the residual quantity in the iteration process is small, the class I optimization process and the class II optimization process are terminated when no difference exists before and after the iteration process.

7. The method for modeling an axisymmetric torpedo paper structure based on multi-objective optimization according to claim 1, wherein the step S6 is specifically:

s61, after the class I optimization and the class II optimization are terminated, the minimization of residual quantity has a better result, and the expandable constraint, the distance constraint and the uniformity constraint are simultaneously applied to the basic grid model, so that the flat foldable mine folding paper can be more accurately fitted to the target surface, and the crease pattern of the flat foldable mine folding paper shows uniform mine units;

s62, proposing an objective function E using 3 constraints simultaneously _III Defined in equation (14):

E _III ＝ω _f E _f +ω _d E _d +ω _u E _u ， (14)

wherein ω_f 、ω _d and ω_u Is the weight, E _f Is to the left side of the mine grid model stripPart of theAdding an extensible constraint to the internal vertices of E _d Is to->Adding distance constraints, E _u Is to->Adding a uniformity constraint to the upper edge;

s63, solving an objective function: solving 3 constrained objective functions, minimizing multi-objective residual quantity, and solving the objective functions based on finite difference to obtain a descent gradient;

s64, termination conditions of 3 constraints:

when (when) and />If the threshold value is smaller than the given threshold value of the user, the optimization is terminated; or when the iteration times reach the designated times or the variation of the residual quantity in the iteration process is small, and no obvious difference exists before and after, the optimization is terminated.

8. The method for modeling an axisymmetric water-mine paper folding structure based on multi-objective optimization according to claim 6 or 7, wherein the solving objective function in claim 6 or 7: the two kinds of optimization problems and the optimization problem with the simultaneous addition of 3 kinds of constraints are optimized by using the same framework, and an objective function of the optimization problem is called by E, and as vertexes can keep a symmetrical structure only when moving on a plane where the vertexes initially exist, polar coordinates are introduced for an axisymmetric mine paper folding structure and r_i In the formula(15) And the expression +.>According to-> and />To construct a gradient; vertex P _i Is moved in the plane of the original place, so +.>Then +.>And +.about.of formula (19)>To construct a gradient;

solving two types of optimization problems and simultaneously adding 3 kinds of constraint optimization problems, and calculating gradient in the descending direction, wherein:

according to the chain law, involved in the optimization processThe definition is as shown in formula (20):

9. the method for modeling an axisymmetric water mine paper folding structure based on multi-objective optimization according to claim 1, wherein the left part of the obtained optimized water mine paper folding strip is recorded asBased on the properties of the mine paper folding, use +.>Generating final three-dimensional mine folded paper, specifically: by->Mirror image replication with respect to the x-z plane to construct a stripe S _O Strip S _O Rotation replication N about the z-axis _s -1 time, generating a torpedo paper folding structure with axisymmetric properties +.>

10. An expandable solar panel structure, characterized in that it is obtained by the structural modeling method according to claims 1-9.