WO2024050935A1

WO2024050935A1 - Three-dimensional model cutting method for high-quality stamping and discharging

Info

Publication number: WO2024050935A1
Application number: PCT/CN2022/127503
Authority: WO
Inventors: 黄劲; 鲍虎军; 王诗怡; 陈炯; 高希峰
Original assignee: 浙江大学
Priority date: 2022-09-05
Filing date: 2022-10-25
Publication date: 2024-03-14
Also published as: CN115423979A

Abstract

The present invention relates to the field of computer-aided design. Disclosed is a three-dimensional model cutting method for high-quality stamping and discharging. The method comprises the following steps: 1) fully considering geometric information of a curved surface of a model, generating a four-symmetry direction field on an input curved surface and extracting singular points; 2) performing correlation analysis by using the number of the singular points and the length of cutting lines, and simplifying redundant singular points; 3) designing a set of curved surface cutting edges perpendicular to each other, and cutting the input curved surface into a structure homeomorphic to a circular disc while connecting the singular points; 4) performing low-distortion parameterization on the cut curved surface and optimizing the parameterization result; and 5) obtaining high-quality stamping and discharging by using a plane parameterization optimization result. The method has the following advantages: high efficiency of a polysquare in stamping and discharging is utilized, and geometric characteristics of the polysquare are fully considered, so that the operation is simple, and the robustness is high; moreover, the present invention has a very important practical application value for high-quality sheet metal stamping and discharging.

Description

A three-dimensional model cutting method for high-quality stamping and layout

Technical field

The invention belongs to the field of computer-aided design, and specifically relates to a three-dimensional model cutting method for high-quality stamping and discharging.

Background technique

Using computer graphics technology, the parametric method with high packing efficiency has wide application in stamping and discharging. In order to achieve high-quality and efficient discharge results, the industry generally hopes that stamping discharge materials have higher packing efficiency (PE, packing efficiency), shorter cutting boundary length (BL, boundary length) and smaller shape distortion ( ED, energy of shape distortion). However, obtaining results that satisfy the above three indicators often involves complex multi-objective optimization. Such optimization is usually difficult to solve efficiently due to its high degree of nonlinearity and non-convexity.

For this complex and difficult-to-solve multi-objective optimization problem, the existing technology usually breaks it down into a two-step solution: first, only considering the two indicators of cutting length (BL) and shape distortion (ED), the surface model parameters are into the plane area; then, the packing efficiency (PE) index is introduced in the plane area, and the plane packing efficiency (PE) is re-optimized while taking into account the cutting length (BL) and shape distortion (ED). Obviously, in such a two-step optimization strategy, the texture atlas generated in the second step strongly depends on the parameterization in the first step. Once the first step introduces a large number of redundant cuts or huge shape distortions, the final result will be The result falls into a local optimal solution, making it impossible to obtain high-quality stamping and discharge materials. Therefore, within the framework of two-step optimization, a parameterization technique that is conducive to plane texture re-optimization is crucial.

Many existing technologies only focus on one step of the above two-step optimization. Although they have made outstanding contributions in the single-step field, they all ignore the connection between the two steps.

Contents of the invention

The purpose of this invention is to overcome the existing deficiencies and provide a three-dimensional model cutting method for high-quality stamping and discharging, thereby bridging the gap between the two-step optimization strategies in traditional cutting technology to meet the application requirements of high quality and high efficiency. . .

In order to achieve the purpose of the present invention, the technical solutions provided are as follows:

A three-dimensional model cutting method for high-quality stamping and layout, including the following steps:

S1: According to the discrete Gaussian curvature of the input model, optimize the smooth discrete four-symmetric direction field on the three-dimensional surface of the input model and obtain the singular points of the discrete direction field;

S2: Simplify the singular points of the discrete direction field obtained in step S1;

S3: Connect the singular points simplified in step S1 to obtain a set of mutually perpendicular surface cutting edges;

S4: Cut the input model along the surface cutting edge into a model that is topologically homeomorphic to the disk, and perform isotropic low-distortion parameterization on the cut surface model;

S5: Use plane texture optimization technology to optimize the parameterization results and obtain stamping layout.

Further, the step S1 includes:

S11: Define a local coordinate system on each patch of the input model;

S12: For any common edge e _ij connecting triangle i and triangle j on the grid of the input model, calculate the smooth energy of the four-symmetry direction field:

In the formula, θ _i and θ _j respectively represent the angle between any component of the four-symmetric direction field on triangle i and triangle j relative to the x-axis, κ _ij represents the angle difference between the x-axis parallel transmission of triangle i and triangle j, and p _ij represents Alignment jump between two triangle orientation fields,

Represents the smooth energy of the four-symmetry direction field transmitted in parallel from triangle i to triangle j along the common edge e _ij ;

S13: Integrate the smooth energy of the four-symmetry direction field and obtain the corresponding θ _i , θ _j and p _ij at the minimum integral value;

S14: Calculate the corresponding integer basic index according to the angular deficit of each vertex:

Among them, N( _vi ) represents the set of triangles surrounding the vertex _vi as the center, d( _vi ) represents the angular deficit of the vertex _vi , and α _b ( _vi ) represents the integer basic index corresponding to the vertex _vi . ;Use the integer basic indexes α _b (v _i ) corresponding to all vertices to form a basic index set I _b ;

S15: According to the basic index set I _b , calculate the index of each vertex v _i under the four-symmetry direction field:

The index α(vi ₎ corresponding to all vertices is formed into an index set I(vi ₎ , and the points corresponding to the non-zero elements in the set I(vi ₎ are extracted as singular points.

Further, the calculation formula of the angle deficiency of each vertex v _i is:

in,

Represents the discrete Gaussian curvature of vertex v _i , Δ _ijk represents the triangle composed of vertices _vi , v _j and v _k , and ∠jik represents the angle value.

Further, the step S2 includes:

S21: According to the index of each singular point, the index value of the singular point is composed into an index vector α = [α ₁ , α ₂ ,..., α _i ,..., α _N ], where α _i is the i-th The index value corresponding to singular points, N represents the number of singular points of the four-symmetric direction field extracted in step S1;

S22: Enumerate all pairs of integer vectors δ _i and δ _j , i≠j, the dimensions of δ _i and δ _j are the same as the dimensions of the index vector α, and only the i-th element of δ _i is 1, and the other elements are all 0, only the jth element of δ _j is 1, and the other elements are all 0;

S23: Iteratively optimize the index vector α:

Among them, α ^k represents the index vector of the k-th iteration, the initial value α ⁰ of the iteration is the α obtained in step S21, ||.|| ₀ represents the L0 norm,

represents the index value of the i-th singular point in the index vector of the k-th iteration, S ^k represents the simplified energy function value of the k-th iteration; E _I (α ^k ) represents the light of the four-symmetric direction field at the k-th iteration degree of compliance; w _n is the weight value;

S24: When S ^k ≥ S ^k-1 , stop optimization and find all non-zero elements in the index vector α ^k-1 of the k-1 iteration, and use the points corresponding to all non-zero elements as the final singular points. .

Further, the calculation formula for the smoothness of the symmetrical direction field is:

Among them, α _g is a vector composed of discrete Gaussian curvatures of all singular points,

is the pseudo-inverse matrix of the Laplacian operator of the triangular mesh graph, and the superscript T represents the transpose.

Further, the step S3 includes:

S31: Obtain the initial cutting edge set C ⁰ according to the topological characteristics of the input model;

S32: Use an iterative method to connect all singular points in the singular point set to the cutting line formed by connecting several cutting edges in the cutting edge set end-to-end. During the process of connecting singular points, the following conditions need to be met:

All singular points fall on the cutting line;

The cutting lines intersect as perpendicularly as possible;

After the input model is cut along the cutting line, it should be homeomorphic to the disk;

The total length of the cutting line should be as short as possible.

Further, the step S31 is specifically:

Determine the genus of the input model. If the genus of the input model is greater than 0, first find all n non-shrinkable rings on the input model, and then perform n-1 connection operations recursively. Each connection uses the shortest The path connects two currently disconnected ring handles, and finally the cutting edge at the end of the recursion is used as the initial cutting edge set C ⁰ ;

If the genus of the input model is less than or equal to 0, then for each pair of singular points (σ _i , σ _j ), the cutting length is evaluated according to the following formula:

Among them, p(σ _i ,σ _j ) represents the shortest path along the grid line from point σ _i to point σ _j , d(.) represents the distance function, σ _k represents the k-th singular point, d(σ _k ,p The calculation method of (σ _i ,σ _j )) is:

stσ _m ∈p(σ _i ,σ _j ).

Among them, σ _m represents the m-th singular point;

Take the path p(σ _i ,σ _j ) corresponding to the minimum length evaluation value l(σ _i ,σ _j ) as the initial cutting edge set C ⁰ .

Further, the step S32 is specifically:

For a singular point set containing M points, when constructing the t-th cut, it is necessary to use the existing t-1 cuts to find the singular point that makes the following formula obtain the minimum value:

C ^t =p(σ _k ,C ^t-1 )∪C ^t-1

In the formula, σ _k represents the k-th singular point, C ^t-1 represents the cutting edge set corresponding to the t-1th cutting, p(σ _k ,C ^t-1 ) represents the point σ _k to the cutting edge set C ^t- The shortest path along the grid lines of ¹ ;

After obtaining the singular point σ _k corresponding to the minimum value, the shortest path p(σ _k ,C ^t-1 ) is merged into the cutting edge set to obtain the cutting edge set C ^t corresponding to the tth cut.

Compared with the prior art, the present invention has the following beneficial effects:

Based on the discovery that "parameterization of structures shaped like polysquares is beneficial to stamping and discharging", the present invention provides a high-efficiency three-dimensional model cutting technical solution. This technical solution can map the curved surface to a parameter domain shaped like a multi-connected square with low distortion, which is conducive to the subsequent optimization of plane packing efficiency, thereby achieving the purpose of high-quality and high-efficiency stamping and discharge.

Through observation, the present invention found that the corner points of multi-connected squares have internal angles that are integer multiples of 90 degrees. When mapped to the cutting points of the curved surface, they correspond to angle deficits that are integer multiples of 90 degrees. Since the singular point of the four-symmetry direction field naturally satisfies this corner deficit property, this technical solution intends to draw on the technical ideas of the four-symmetry direction field and use it to design and guide the parameterization of high-quality and high-efficiency stamping and layout.

The present invention can efficiently and robustly obtain high-quality and high-efficiency stamping and discharging on several models, fully embodying the practical effect and value of the technology of the present invention. Compared with existing cutting technology, the present invention can obtain higher quality stamping and discharge materials, and improves the packaging efficiency (PE), cutting length (BL) and shape distortion (ED) indicators.

Description of the drawings

Figure 1 is a flow chart of a three-dimensional model cutting method for high-quality stamping and discharging;

Figure 2 is a schematic diagram of the three-dimensional model cutting method process using a specific input model as an example;

Figure 3 is a statistical diagram of the relative improvement of PE, BL and ED indicators in the data set compared with the traditional method, where a-f is the comparative distribution diagram of the present invention and the VarCuts method, and g-l is the comparative distribution diagram of the present invention and the OptCuts method;

Figure 4 is a schematic diagram of the layout results of the three-dimensional model.

Detailed ways

The present invention will be further elaborated and described below in conjunction with the accompanying drawings and specific embodiments.

As shown in Figure 1-2, it is a flow chart of a three-dimensional model cutting method for high-quality stamping and discharge provided in a preferred embodiment of the present invention. The main steps include 4 steps, namely S1 ~ S4:

S1: Based on the three-dimensional geometric shape of the input model, calculate the discrete Gaussian curvature, optimize to obtain a smooth discrete four-symmetry direction field on the three-dimensional surface, and then extract the singular points of the four-symmetry direction field;

S2: While fully considering the smoothness of the four-symmetry direction field, merge and simplify the singular points of the four-symmetry direction field extracted in step S1;

S3: For the simplified singular points in step S2, optimize to obtain a set of mutually perpendicular surface cutting edges, and then cut the input model along the surface cutting edges into a model that is topologically homeomorphic to the disk;

S4: Perform isotropic low-distortion parameterization on the cut surface model;

S5: Use plane texture optimization technology to optimize the parametric results to obtain high-quality stamping layout.

The specific implementation of S1 to S4 in this embodiment and its effects will be described in detail below.

First of all, although the four-symmetry field generation technology is relatively mature, the existing four-symmetry field generation technology often considers factors such as feature alignment and grid density. Such redundant constraints will not only lead to low solution speed, but also make the solution The device is stuck at the local optimal solution and cannot obtain the global optimal solution. Therefore, the goal of step S1 should be the optimization of the four-symmetry direction field under the intrinsic geometric characteristics of the model, that is, only considering the Gaussian curvature of the model without considering the alignment of the characteristic lines, etc. Other irrelevant factors. For this purpose, the present invention introduces the evaluation energy of the smoothness of the four-symmetry direction field with respect to Gaussian curvature, thereby obtaining the four-symmetry direction field and singular points that meet the application goals.

For the discrete expression of the four-symmetric direction field, the present invention adopts the form of piecewise constants, that is, on each patch of the triangular mesh, the direction field has a fixed direction. Since any two triangles in a triangular mesh in space are often non-coplanar, a local coordinate system needs to be introduced to consider the angle difference after parallel transmission of direction fields between different patches.

In this embodiment, according to the Gaussian curvature information at each point of the model, the smooth four-symmetry direction field is designed according to S11 to S15:

S11: Define a local coordinate system on each patch of the input model, where the x-axis is the unit vector on any side of the triangle, the y-axis is obtained by rotating the x-axis counterclockwise 90° in the triangle plane, and the z-axis is the x-axis Obtained by crossing the y-axis;

S12: For any common edge e _ij on the grid connecting triangle i and triangle j, express the smooth energy of the four-symmetry direction field as

In the formula, θ _i and θ _j respectively represent the angle between any component of the four-symmetric direction field on triangle i and triangle j relative to the x-axis, κ _ij represents the angle difference between the parallel transmission of the x-axis of triangle i and triangle j, and the integer p _ij represents the aligned jump between the two triangle orientation fields,

S13: Integrate the smooth energy of the four-symmetry orientation field over the entire input model

where A _ij is the average area of triangle i and triangle j, and a mixed integer solver is used to obtain all θ _i , θ _j and p _ij corresponding to the minimum value of E _s ;

S14: For each vertex v _i on the triangle mesh, calculate its angular loss

in,

Represents the discrete Gaussian curvature of vertex v _i , N( _vi ) represents a set of triangles surrounding the vertex v _i as the center, Δ _ijk represents the triangle composed of vertices _vi , v _j , v _k , ∠jik represents the angle value; then, based on the angular deficit of each vertex v _i , further calculate the integer basic index corresponding to v _i

The integer basic indexes α _b (v _i ) corresponding to all vertices form a basic index set I _b ;

S15: Calculate the index of each vertex v _i under the four-symmetry direction field according to the basic index set I _b

The index α(vi ₎ corresponding to all vertices is formed into an index set I(vi ₎ , and the points that are not 0 in the set I(vi ₎ are extracted as singular points extracted in step S1.

When the surface geometry of the input model has many fluctuations, the number of singular points obtained in step S1 is often very large, which brings huge computational challenges to the subsequent cutting process. In addition, redundant adjacent singular points will not only cause the cutting line to have a large number of zigzag shapes, but also cause the final nesting result to be too finely packed due to the cutting line being too long. Inspired by the Gauss-Bonnet theorem, the present invention provides an idea of merging and simplifying adjacent singular point pairs with indexes (-1, +1) under a topology that ensures the global four-symmetry direction field. However, the method of simply merging the nearest (-1, +1) point pairs has two major disadvantages: on the one hand, it is difficult to determine a unified merging termination condition due to the geometric specificity of the model; on the other hand, it is difficult to determine only by the number of singular points. It is difficult to assess the degree of distortion after subsequent cuts. Therefore, the present invention provides a singular point simplification method that fully considers the inherent geometric shape of the model. Specifically, the singular point merging and simplification process in this embodiment is mainly implemented through step S2. The specific method is described in detail below:

S22: Enumerate all pairs of integer vectors δ _i and δ _j (i≠j) to participate in the calculation of step S23, where the dimensions of δ _i and δ _j are the same as the dimensions of the index vector α, and δ _i has only the The i element is 1, the other elements are all 0, δ _j is similar, only the jth element is 1;

S23: Iteratively optimize the index vector α according to the following model:

represents the index value of the i-th singular point in the index vector of the k-th iteration, S ^k represents the simplified energy function value of the k-th iteration; here E _I (α ^k ) represents the four-symmetry direction field of the k-th iteration Smoothness, the specific calculation method is as follows:

is the pseudo-inverse matrix of the Laplacian operator of the triangular mesh graph, and the superscript T represents the transpose;

In addition, w _n is the weight value, and the calculation method is:

Here, ||α ⁰ || ₀ represents the number of non-zero elements in the vector α ⁰ , and λ is the parameter that controls the number of singular points;

After the above S2 step, the simplified singular point set can finally be determined. In response to the demand for high-efficiency and high-quality stamping and discharging, the present invention needs to connect the simplified singular points through cutting lines, thereby flattening the curved surface model to a planar area shaped like a multi-connected square with low distortion. For this purpose, the following conditions need to be met in the process of connecting singular points: first, all singular points fall on the cutting line; second, the cutting lines should intersect as perpendicularly as possible; third, After the model is cut along the cutting line, it should be homeomorphic to the disc; fourth, the total length of the cutting line should be as short as possible. In this embodiment, in order to satisfy the above four conditions at the same time, the specific method of performing vertical cutting in step S3 is as follows:

S31: Obtain the initial cut according to the topological characteristics of the input model: determine the genus of the input model. If the genus of the input model is greater than 0, execute step S311 and then proceed to step S32; otherwise, execute step S312 and then proceed to step S32;

S311: First find all n non-shrinkable handles on the input model, and then perform n-1 connection operations recursively. Each connection uses the shortest path to connect the two currently disconnected handles. Finally, the cutting edge at the end of the recursion is used as the initial cutting edge set C ⁰ ;

In this embodiment, the connecting lines in each triangular grid are called cutting edges, and several cutting edges connected end to end are connected to form a cutting line.

S312: For each pair of singular points (σ _i , σ _j ), the cutting length is evaluated according to the following formula:

stσ _m ∈p(σ _i ,σ _j ).

Among them, σ _m represents the m-th singular point;

Take the path p(σ _i ,σ _j ) corresponding to the minimum length evaluation value l(σ _i ,σ _j ) as the initial cutting edge set C ⁰ ;

S32: Use an iterative method to connect all singular points in the singular point set to the cutting line. Specifically, for a singular point set containing M points, when constructing the t-th cutting, you need to use the existing t- 1 cut, looking for the singular point that makes the following formula obtain the minimum value:

C ^t =p(σ _k ,C ^t-1 )∪C ^t-1

In the formula, C ^t-1 represents the cutting edge set corresponding to the t-1th cut, and p(σ _k ,C ^t-1 ) represents the shortest path along the grid line from point σ _k to the cutting edge set C ^t-1 ; After obtaining the singular point σ _k corresponding to the minimum value, merge the obtained shortest path p(σ _k ,C ^t-1 ) into the cutting edge set to obtain the cutting edge set C ^t corresponding to the tth cut.

Thus, through the aforementioned S1 to S3, the vertical cutting of the complete three-dimensional model has been obtained, that is, the three-dimensional curved surface of any topological form can be cut along the cutting line into a surface that is homeomorphic to the disk.

The specific parameterization method of step S4 is briefly described below:

S41: Cut the model along the cutting line obtained in step S3, and use the Tutte diagram embedding algorithm to deform the cut surface model into a flat space;

S42: Optimizing symmetric Dirichlet energy in planar space:

Among them, a _i represents the area of the i-th triangle f _i , J _i represents the deformation Jacobian of the curved triangle under plane mapping,

represents the square of the F-norm.

So far, the present invention has constructed a low-distortion mapping from a three-dimensional curved surface to a polygon-like shape, in order to achieve high-quality and high-efficiency stamping and discharging effects.

Next, the parametric results are further optimized using the plane parametric optimization method:

S5: Using plane parameterized optimization method (Liu H Y, Fu 13.), and set the lower bound of the boxing efficiency to 0.8 to obtain the final stamping discharge.

Based on the above embodiment method, the following applies it to specific examples to demonstrate its effects. The specific process is as mentioned above and will not be repeated. The following mainly shows its specific parameter settings and implementation effects.

Taking the model of a public data set as an example, the present invention will be described in detail below. The specific steps are as follows:

1) Use papers from top international journals (Liu H Y, Fu 13.) Public data set. For each model in the data set, use Blender software to stitch its cracks, and eliminate non-manifold and edge-containing degenerate triangle meshes in the stitched model. Among them, there are a total of 5588 models in the public data set, and a total of 53 models are non-manifold or edge-degraded meshes during the stitching process. A total of 5588-53=5519 valid data.

2) According to the aforementioned step S1, for each model in the effective data set, establish an optimization model based on the smooth energy mentioned above, and use the mixed integer optimization solver in the libigl code library to solve it to obtain the smooth four-symmetry direction field, and extract singular points according to the method mentioned above.

3) Follow the aforementioned step S2 to iteratively simplify the singular points, and find the optimal (-1, +1) singular point pair for merging in each iteration. While reducing the number of singular points, the model after cutting is also considered. Degree of deformation. In this embodiment, the weight ratio between the shape distortion energy of the model and the number of singular points is 1:0.5, that is, λ is set to 0.5. When the total optimization energy no longer decreases, the iteration stops.

4) Connect the simplified singular points according to the aforementioned step S3. First, classify according to the topological characteristics of the input model. If it is not spherical homeomorphism, perform the initial cutting process according to the aforementioned S311. If it is spherical homeomorphism, perform the initial cutting process according to S312, and then proceed. Perform subsequent cutting design according to step S32;

5) According to the aforementioned step S4, use the cutting line in step S3 to cut the model into a disc homeomorphic surface, and then perform low-distortion planar parametric mapping.

The cutting method of the present invention is named OrthoCuts. Compared with the original classic VarCuts method (Sharp N, Crane K. Variational surface cutting [J]. ACM Transactions on Graphics (TOG), 2018, 37 (4): 1- 13.) and OptCuts (Li M, Kaufman D M, Kim V G, et al. Optcuts:Joint optimization of surface cuts and parameterization[J]. ACM Transactions on Graphics (TOG), 2018, 37(6):1- 13.) method, which is improved in terms of packing efficiency (PE), cutting length (BL) and shape distortion (ED). On the full data set, the relative improvement values of binning efficiency, cutting length and shape distortion were calculated:

Among them, the superscript our represents the quantitative index value in the present invention, and the superscript cmp represents the index value in the comparative example. The percentage statistics on the full data set are shown in Table 1 below:

Table 1

In order to further count the number of various results between the OrthoCuts method proposed by the present invention and the classic method in the comparative example, all the results in the data set are divided into four categories ABCD: A) The three indicators of the present invention are better than the comparative method; B) Two indicators of the present invention are better than the comparison method and one indicator is worse than the comparison method; C) One indicator of the present invention is better than the comparison method and two indicators are worse than the comparison method; D) Three indicators of the present invention are all worse than the comparison method method. The specific statistical results are shown in Table 2.

Table 2

In order to more intuitively display the comparative performance of methods under various indicators, the present invention provides a discrete frequency distribution diagram of the relative improvement values (r _PE , r _BL , r _ED ) of boxing efficiency, cutting length and shape distortion, as well as two relative improvement values Two-dimensional discrete distribution map under two combinations. As shown in Figure 3, where af is the comparison distribution diagram with VarCuts, gl is the comparison distribution diagram with OptCuts, a and g show the distribution of r _PE , b and h show the distribution of r _BL , c and i shows the distribution of r _ED . As can be seen from Figure 3, compared with the classic technologies VarCuts and OptCuts, most of the results in the data satisfy r _PE ≥ 0, r _BL ≥ 0, r _ED ≥ 0. Therefore, the present invention can achieve higher equipment performance at the same time. Box efficiency, shorter cutting length, and less distorted nesting effect. In addition, in order to show the discharging effect on a specific model, the present invention also shows several sample results, as shown in Figure 4. It can be seen that the present invention can flatten the model with low distortion through mutually perpendicular cuts, such as The polysquare structure achieves high-efficiency and high-quality nesting results.

The above-described embodiment is only a preferred solution of the present invention, but it is not intended to limit the present invention. Those of ordinary skill in the relevant technical fields can also make various changes and modifications without departing from the spirit and scope of the present invention. Therefore, any technical solution obtained by adopting equivalent substitution or equivalent transformation shall fall within the protection scope of the present invention.

Claims

A three-dimensional model cutting method for high-quality stamping and layout, which is characterized by including the following steps:

S1: According to the discrete Gaussian curvature of the input model, optimize the smooth discrete four-symmetric direction field on the three-dimensional surface of the input model and obtain the singular points of the discrete direction field;

S2: Simplify the singular points of the discrete direction field obtained in step S1;

S3: Connect the singular points simplified in step S1 to obtain a set of mutually perpendicular surface cutting edges;

S4: Cut the input model along the surface cutting edge into a model that is topologically homeomorphic to the disk, and perform isotropic low-distortion parameterization on the cut surface model;

S5: Use plane texture optimization technology to optimize the parameterization results and obtain stamping layout.
The three-dimensional model cutting method for high-quality stamping and discharging according to claim 1, characterized in that the step S1 includes:

S11: Define a local coordinate system on each patch of the input model;

S12: For any common edge e ij connecting triangle i and triangle j on the grid of the input model, calculate the smooth energy of the four-symmetry direction field:

In the formula, θ i and θ j respectively represent the angle between any component of the four-symmetric direction field on triangle i and triangle j relative to the x-axis, κ ij represents the angle difference between the x-axis parallel transmission of triangle i and triangle j, and p ij represents Alignment jump between two triangle orientation fields,
Represents the smooth energy of the four-symmetry direction field transmitted in parallel from triangle i to triangle j along the common edge e ij ;

S13: Integrate the smooth energy of the four-symmetry direction field and obtain the corresponding θ i , θ j and p ij at the minimum integral value;

S14: Calculate the corresponding integer basic index according to the angular deficit of each vertex:

Among them, N( vi ) represents the set of triangles surrounding the vertex vi as the center, d( vi ) represents the angular deficit of the vertex vi , and α b ( vi ) represents the integer basic index corresponding to the vertex vi . ;Use the integer basic indexes α b (v i ) corresponding to all vertices to form a basic index set I b ;

S15: According to the basic index set I b , calculate the index of each vertex v i under the four-symmetry direction field:

The index α(vi ) corresponding to all vertices is formed into an index set I(vi ) , and the points corresponding to the non-zero elements in the set I(vi ) are extracted as singular points.
The three-dimensional model cutting method for high-quality stamping and discharging according to claim 2, characterized in that the angle loss calculation formula of each vertex vi is:

in,
Represents the discrete Gaussian curvature of vertex v i , Δ ijk represents the triangle composed of vertices vi , v j and v k , and ∠jik represents the angle value.
The three-dimensional model cutting method for high-quality stamping and discharging according to claim 1, characterized in that the step S2 includes:

S21: According to the index of each singular point, the index value of the singular point is composed into an index vector α = [α 1 , α 2 ,..., α i ,..., α N ], where α i is the i-th The index value corresponding to singular points, N represents the number of singular points of the four-symmetric direction field extracted in step S1;

S22: Enumerate all pairs of integer vectors δ i and δ j , i≠j, the dimensions of δ i and δ j are the same as the dimensions of the index vector α, and only the i-th element of δ i is 1, and the other elements are all 0, only the jth element of δ j is 1, and the other elements are all 0;

S23: Iteratively optimize the index vector α:

Among them, α k represents the index vector of the k-th iteration, the initial value α 0 of the iteration is the α obtained in step S21, ||.|| 0 represents the L0 norm,
represents the index value of the i-th singular point in the index vector of the k-th iteration, S k represents the simplified energy function value of the k-th iteration; E I (α k ) represents the light of the four-symmetric direction field at the k-th iteration degree of compliance; w n is the weight value;

S24: When S k ≥ S k-1 , stop optimization and find all non-zero elements in the index vector α k-1 of the k-1 iteration, and use the points corresponding to all non-zero elements as the final singular points. .
The three-dimensional model cutting method for high-quality stamping and discharging according to claim 4, characterized in that the smoothness calculation formula of the symmetrical direction field is:

Among them, α g is a vector composed of discrete Gaussian curvatures of all singular points,
is the pseudo-inverse matrix of the Laplacian operator of the triangular mesh graph, and the superscript T represents the transpose.
The three-dimensional model cutting method for high-quality stamping and discharging according to claim 1, characterized in that the step S3 includes:

S31: Obtain the initial cutting edge set C 0 according to the topological characteristics of the input model;

S32: Use an iterative method to connect all singular points in the singular point set to the cutting line formed by connecting several cutting edges in the cutting edge set end-to-end. During the process of connecting singular points, the following conditions need to be met:

All singular points fall on the cutting line;

The cutting lines intersect as perpendicularly as possible;

After the input model is cut along the cutting line, it should be homeomorphic to the disk;

The total length of the cutting line should be as short as possible.
The three-dimensional model cutting method for high-quality stamping and discharging according to claim 6, characterized in that the step S31 is specifically:

Determine the genus of the input model. If the genus of the input model is greater than 0, first find all n non-shrinkable rings on the input model, and then perform n-1 connection operations recursively. Each connection uses the shortest The path connects two currently disconnected ring handles, and finally the cutting edge at the end of the recursion is used as the initial cutting edge set C 0 ;

If the genus of the input model is less than or equal to 0, then for each pair of singular points (σ i , σ j ), the cutting length is evaluated according to the following formula:

Among them, p(σ i ,σ j ) represents the shortest path along the grid line from point σ i to point σ j , d(.) represents the distance function, σ k represents the k-th singular point, d(σ k ,p The calculation method of (σ i ,σ j )) is:

stσ m ∈p(σ i ,σ j ).

Among them, σ m represents the m-th singular point;

Take the path p(σ i ,σ j ) corresponding to the minimum length evaluation value l(σ i ,σ j ) as the initial cutting edge set C 0 .
The three-dimensional model cutting method for high-quality stamping and discharging according to claim 6, characterized in that the step S32 is specifically:

For a singular point set containing M points, when constructing the t-th cut, it is necessary to use the existing t-1 cuts to find the singular point that makes the following formula obtain the minimum value:

C t =p(σ k ,C t-1 )∪C t-1

In the formula, σ k represents the k-th singular point, C t-1 represents the cutting edge set corresponding to the t-1th cutting, p(σ k ,C t-1 ) represents the point σ k to the cutting edge set C t- The shortest path along the grid lines of 1 ;

After obtaining the singular point σ k corresponding to the minimum value, the shortest path p(σ k ,C t-1 ) is merged into the cutting edge set to obtain the cutting edge set C t corresponding to the tth cut.