CN113591356B - Construction method of non-uniform irregular spline basis function retaining sharp features - Google Patents

Abstract

The embodiment of this specification discloses a method of constructing a non-uniform irregular spline basis function that retains sharp features, including: step S1, according to the shape of the surface to be processed; step S2, based on the quadrilateral control grid, using the Bezier surface method Extract and define the initial C0 basis function; Step S3, calculate the tangent plane; Step S4, calculate the connection function; Step S5, local step-by-step optimization; S51, optimize the vertex control points; S52, optimize the edge control points; S53, optimize the surface control points ; Solve a least squares problem with linear equality constraints during step-by-step solution, that is: stMP=b transforms the least squares problem into a problem of solving a system of linear equations: The present invention proposes a construction method of a spline basis function, so that the generated spline can generate a global G1 continuous surface on a control grid of any topology.

Description

A construction method for non-uniform irregular spline basis functions that preserves sharp features

Technical field

The present application relates to the technical field of numerical control machining, and in particular to a method of constructing a non-uniform irregular spline basis function that retains sharp features.

Background technique

Catmull-Clark surfaces are widely used in the field of animation, while NURBS dominates CAD industrial design. Each surface format has its own unique advantages: Catmull-Clark surfaces can generate smooth surfaces on control networks of any topology, which is very advantageous for animation design. NURBS surfaces can be modified locally and are more suitable for high-precision industrial models. Therefore, people have developed many sets of surface expression forms, which modify the surface shape by assigning control mesh edge node distances. If there are no singular points, these surfaces are consistent with NURBS surface expressions; if all node distances are 1, the surfaces are Catmull-Clark surfaces.

The subdivision scheme is suitable for rendering applications, but not for CAD design. Because the CAD design analysis process usually requires transferring models between numerous software packages, and most of these software are based on NURBS. However, the subdivision method is not backward compatible with NURBS because near the singular point, the subdivision method produces an infinite sequence of bicubic surfaces, while NURBS can only import finite truncation. This compatibility issue is avoided by hole-patch-based methods, which replace infinite sequences near singular points with a small number of patches. Although the previous methods are all for traditional Catmull-Clark surfaces, that is, the node distance ratio is 1, it is very simple to modify these methods to handle non-uniform node distances. However, the modified surface exhibits the same problem as the subdivision result near singular points with a node distance ratio greater than 3.

Isogeometric analysis method IGA is an emerging surface analysis technology. It has the ability to analyze directly on spline models and has high accuracy compared with the finite element method. However, since the usual NURBS basis functions are rational polynomials and maintain tensor properties, the boundary intersection problem is difficult to solve.

A spline generation method is needed so that the generated spline can generate a global G1 continuous surface on the control grid of any topology.

Contents of the invention

The embodiments of this specification provide a method of constructing a non-uniform irregular spline basis function that retains sharp features, so that the generated spline has the ability to generate a globally G1 continuous surface on a control grid of any topology.

In order to solve the above technical problems, the embodiments of this specification are implemented as follows: The embodiments of this specification provide a method for constructing non-uniform irregular spline basis functions that retain sharp features, including:

Step S1: According to the shape of the surface to be processed, input a quadrilateral control mesh with arbitrary topology;

Step S2: Based on the quadrilateral control grid, use the Bezier surface method to extract and define the initial C0 basis function. Each surface of the quadrilateral control grid is represented by a bicubic Bezier surface; for the quadrilateral control grid The surface point F _i on each surface is represented by a linear combination of the four control points _Pi on each surface; the edge point E _i and the vertex V of the quadrilateral control grid are the surface points F Linear combination of _i ;

Step S3: Calculate the tangent plane;

For a singular point with degree n, the control points adjacent to the singular point are E _i , F _i , and the node distance length is _di , a _i ;

Calculate new control points Let P＝[V,E ₀ ,…,E _n-1 ,F ₀ ,…,F _n-1 ] ^T ,/> Write the segmentation rule as/>

Define the NURSS format subdivision matrix as M, and the Eigen-polyhedron-based subdivision matrix as N;

Define a circle of Bezier control points around the singular point as It is the linear combination of point V, {E _i }, {F _i }:

Step S31: Calculate the limit point C based on Eigen-polyhedron subdivision;

Use L ⁰ to represent the unitization of the eigenvector corresponding to the eigenvalue of matrix N being 1, and the limit point is defined as C=L ⁰ M ^T P;

Step S32: Calculate the tangent plane based on Eigen-polyhedron subdivision;

Define two matrices of size 2nx2n Order/> λ is the matrix/> The main eigenvalues of , then/> Written as/> Λ is a diagonal matrix composed of singular values; assume that i ₁ and i ₂ are indicators such that Λ(i ₁ ,i ₁ )=Λ(i ₂ ,i ₂ )=λ, let/> In addition to/> A diagonal matrix with zeros in other positions is obtained/> Then define the vector set/>

Step S33: Define Bezier control points around the singular point;

Bezier control points around singular points

Step S4: Calculate the connection function;

S41. For a singular point with degree n, define a weight k _i for each corner:

S42. Define the angle according to the weight;

Sum θ _i . If the summation result is not equal to 2π, normalize the angle to 2π;

Definition (i,j)=sin(θ _i )sin(θ _i+2 )…sin(theta _j )(i<j);

If n=2k+1 is an odd number:

If n=2k is an even number:

Suppose the set S is n numbers The set of , let/> is the value that occurs most often in the set. If each number appears only once, then let/>

Step S5, local step-by-step optimization;

Divide the constraints on each edge into two parts, the first two equations become vertex constraints, and the rest are called edge constraints; solve the new basis function in the following order;

S51. Optimize vertex control points;

Each singular point vertex constraint with degree n contains 3n+1 control points and 2n linear constraint equations. First, solve to obtain 3n+1 control points that satisfy the vertex constraints;

S52. Optimize edge control points;

After the vertex control points are determined, the remaining control points of the edge constraint design are controlled on each edge, which becomes a local problem. This is to solve the control points on each edge that satisfy the edge constraints;

S53. Optimize surface control points;

Solve a least squares problem with linear equality constraints during step-by-step solution, that is:

s.t.MP＝b

Convert the least squares problem into a problem of solving a system of linear equations:

S54. Design the shape of the curved surface to be processed based on the results obtained in step S53.

One embodiment of this specification achieves the following beneficial effects:

The present invention proposes a construction method of spline basis function, so that the generated spline can generate a global G1 continuous surface on the control grid of any topology, and can be directly used for CAD design and CAE analysis, thereby accelerating processing design analysis. Integrated process.

Description of the drawings

In order to more clearly explain the embodiments of this specification or the technical solutions in the prior art, the drawings needed to be used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings in the following description are only These are some of the embodiments recorded in this application. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without exerting any creative effort.

Figure 1 is a schematic diagram explaining the control grid in the embodiment of this specification;

Figure 2 is a schematic diagram explaining edge points and vertices in the embodiment of this specification;

Figure 3 is a schematic diagram for explaining points in the embodiment of this specification;

Figure 4 is a schematic diagram explaining the NURSS subdivision format in the embodiment of this specification;

Figure 5 is a schematic diagram explaining the tangent plane in the embodiment of this specification;

Figure 6 is a schematic diagram illustrating the definition of angles near singular points in the embodiment of this specification;

Figure 7 is a schematic diagram illustrating an embodiment of this specification for upgrading a singular surface to a biquintic Bezier surface to satisfy the G1 continuity condition;

Figure 8 is a schematic diagram illustrating the case of isolated singular points in the technical solution of the embodiment of this specification;

Figure 9 is the first graph of the test results of the embodiment of this specification;

Figure 10 is a second graph of the test results of the embodiment of this specification;

Figure 11 is the third graph of the test results of the embodiment of this specification;

Figure 12 is a fourth graph of the test results of the embodiment of this specification.

Detailed ways

In order to make the purpose, technical solutions and advantages of one or more embodiments of this specification more clear, the technical solutions of one or more embodiments of this specification will be clearly and completely described below in conjunction with specific embodiments of this specification and the corresponding drawings. . Obviously, the described embodiments are only some of the embodiments of this specification, but not all of the embodiments. Based on the embodiments in this specification, all other embodiments obtained by those of ordinary skill in the art without creative efforts fall within the scope of protection of one or more embodiments of this specification.

Catmull-Clark surfaces are widely used in the field of animation, while NURBS dominates CAD industrial design. Each surface format has its own unique advantages: Catmull-Clark surfaces can generate smooth surfaces on any topological control network, which is very advantageous for animation design. NURBS surfaces can be modified locally and are more suitable for high-precision industrial models. Therefore, people have developed many sets of surface expression forms, which modify the surface shape by assigning control mesh edge node distances. If there are no singular points, these surfaces are consistent with NURBS surface expressions; if all node distances are 1, the surfaces are Catmull-Clark surfaces.

The node distance ratio is defined as the maximum node distance ratio of the minimum node distance of the edges connected to each singular point. Resolved a problem that occurred with all published Catmull-Clark surfaces that support arbitrary node spacing. This problem first appears in non-uniform Catmull-Clark surfaces and NURCCs. When the node distance ratio of a singular point increases, the quality of the surface near the singular point will decrease. Cashman et al. thinned in one direction until the node distance ratio was less than 2, and then obtained a curved surface by thinning in both directions. The extended subdivision surface forces the node distance ratio of all singular points to be 1, which achieves good results in simple cases, but cannot obtain better results in other cases.

The subdivision scheme is suitable for rendering applications, but not for CAD design. Because the CAD design analysis process usually requires transferring models between numerous software packages, and most of these software are based on NURBS. However, the subdivision method is not backward compatible with NURBS because near the singular point, the subdivision method produces an infinite sequence of bicubic surfaces, while NURBS can only import finite truncation. This compatibility issue is avoided by hole-patch-based methods, which replace infinite sequences near singular points with a small number of patches. Although the previous methods are all for traditional Catmull-Clark surfaces, that is, the node distance ratio is 1, it is very simple to modify these methods to deal with non-uniform node distances. However, the modified surface exhibits the same problem as the subdivision result near singular points with a node distance ratio greater than 3.

Isogeometric analysis IGA is an emerging surface analysis technology. It has the ability to analyze directly on spline models and has higher accuracy than the finite element method. However, since the usual NURBS basis functions are rational polynomials and maintain tensor properties, the boundary intersection problem is difficult to solve. The proposed spline has the ability to generate a global G1 continuous surface on the control grid of any topology, which can be directly used in CAD design and CAE analysis, speeding up the integrated process of machining design and analysis.

The technical solution of the embodiment of the present invention proposes a framework for generating acceptable surfaces near singular points with a large node distance ratio. The framework is based on the hole-filling method, so it is backward and forward compatible with NURBS. It can be directly applied in current CAD software and transferred between various CAD analysis and design software packages without loss. It can directly support isogeometric analysis methods and connect CAD design and CAE analysis.

The technical ideas of the technical solutions of the embodiments of the present invention will be described below.

The input of the technical solution of the embodiment of the present invention is a quadrilateral control mesh with arbitrary topology, in which singular points can be at any position except the boundary, singular points support any degree and singular points can be directly connected. The method allows for non-uniform control networks, node lengths can be unequal, and supports specifying arbitrary edges, including edges that pass through singular points as continuous or discontinuous creases.

output

The output of the algorithm is a polynomial parameter surface, which is C2 continuous in the normal region and G1 continuous on the singular edges. Surfaces maintain good shaping capabilities in non-uniform situations and can create creases along specified sharp edges.

The technical solutions of the embodiments of the present invention will be described in detail below.

Overall process

Use and promote the Bezier extraction algorithm to obtain a surface consistent with the B-spline definition in the regular area. A common tangent plane is defined near the singular point. Then the G1 continuous connection function on each singular edge is defined according to the node distance around the singular point. Finally, the local distribution solves the constrained optimization problem, calculates the basis function that satisfies the G1 constraint, and obtains the final surface.

(1)Bezier extraction

As shown in Figure 1, Bezier extraction is used to define the initial C0 basis function, and each surface of the control mesh is represented by a bicubic Bezier surface.

As shown in Figure 2, calculate the point F _i on each surface:

F _2i+j = (1-α _i )[1-γ _j )P _0,0 +γ _j P _0,1 ]+α _i [(1-γ _j )P _1,0 +γ _j P _1,1 ]

in

As shown in Figure 3, calculate for each edge point E _i and vertex V:

For singular points with degree n, the calculation of the surface point F _i and the edge point E _i is the same as that of the regular area. For the vertex V:

(2) Definition of tangent plane

When the proportion of node distances around singular points is very large, the basis function obtained by the generalized Bezier extraction will appear multi-peaked. Here, the subdivision format is used to define the surface points and edge points around the singular point so that they are on the same tangent plane.

NURSS subdivision format:

As shown in Figure 4, for a singular point V with degree n, the adjacent control points are E _i , F _i , and the node distance length is _di , a _i . To calculate new control points Order/> Then the subdivision rule can be written as/>

Pastry:

Edge points:

M _i =ω _i E _i +(1-ω _i )V

vertex:

f _i =d _i-1 d _i+2 ,m _i =f _i +f _i-1

Eigen-polyhedron subdivision format

definition Define point set in R ² />

Eigen-polyhedron based segmentation can be written in the form vertex

The calculation of vertex coordinates is the same as the NURSS subdivision rule, let Is to use V,E _i ,F _i /> Substitute calculated points.

Pastry

Among them, α _i,1 and α _i,2 are the only solutions to the following equation:

edge point

in

P _i,1 = (1-α _i-1,1 )V+α _i-1,1 E _i-1 ,P _i,2 = (1-α _i,2 )V+α _i,2 E _{i+ 1}

P _i,3 = (1-α _i-1,1 )E _i +α _i-1,1 F _i-1 ,P _i,4 = (1-α _i,2 )E _i +α _i,2 E _i

β _i,1 ,β _i,2 are the only solutions to the following equations:

here Is to use V,E _i ,F _i /> Replacement is calculated.

tangent plane calculation

As shown in Figure 5, is defined as a linear combination of points V, {E _i }, {F _i }, using the eigen-polyhedron subdivision format as a guide.

Calculate the limit point C based on eigen-polyhedron subdivision

Use L ⁰ to represent the unitization of the eigenvector corresponding to the eigenvalue of matrix N being 1, and the limit point is defined as C=L ⁰ M ^T P;

Compute tangent planes based on eigen-polyhedron subdivisions

Define two matrices of size 2nx2n Order/> λ is the matrix/> The main eigenvalues of , then/> It can be written as/> Λ is a diagonal matrix composed of singular values. Assume that i ₁ and i ₂ are indicators such that Λ(i ₁ ,i ₁ )=Λ(i ₂ ,i ₂ )=λ, let/> In addition to/> A diagonal matrix with zeros in other positions, this gives Then define the vector set/>

Define Bezier control points around singular points

Bezier control points around singular points

Scale optimization

The surface points and edge points around the singular point need to be on the given tangent plane, so the distance between each point and the singular point needs to be determined. The final control point is defined as follows:

Expand to get:

It is hoped that all basis functions are non-negative, so s _i is defined as follows:

Angle calculation

As shown in Figure 6, in order to obtain a good quality surface in a non-uniform situation, the angle is now defined near the singular point. The sum of the included angles on each singular point should be equal to 2π, and the angles corresponding to the larger sides should be closer

Define weight k _i :

Define angles based on weights:

Under normal circumstances, the sum of θ _i is not equal to 2π. In this case, the angle is normalized to 2π.

Connection function definition

As shown in Figure 7, the singular surface is upgraded to a biquintic Bezier surface to satisfy the G1 continuity condition, and parameters b _i are defined for each edge. Suppose the singular point degree is n.

Definition (i,j)=sin(θ _i )sin(θ _i+2 )…sin(theta _j )(i<j)

If n=2k+1 is an odd number:

If n=2k is an even number:

Suppose the set S is n numbers The set of , let/> is the value that occurs most frequently in the set, if each number occurs only once, then let/>

The G1 continuity constraint is not the same in the case of isolated singular points and connected singular points.

As shown in Figure 8, when the singular point is isolated:

Define α(s)=a _i *a _i-1 , β(s)=a _i *ai ₊₁ , γ(s)=b _i (1-s) ² , and obtain the G1 constraint conditions as follows:

When singular points are connected:

definition The G1 constraints are obtained as follows:

Local step-by-step optimization

In order to handle the situation where there are multiple singular points on a surface and avoid solving the constraint conditions through global optimization, a local step-by-step optimization algorithm is adopted.

The constraints on each edge are divided into two parts, the first two equations become vertex constraints, and the remaining equations are called edge constraints. Solve for the new basis functions in the following order:

Optimize vertex control points

Each singular point vertex constraint with degree n contains 3n+1 control points and 2n linear constraint equations. The 3n+1 control points that satisfy the vertex constraints are first solved.

Optimize edge control points

Once the vertex control points are determined, the remaining control points of the edge constraint design are controlled on each edge, which becomes a local problem. This is to solve the control points on each edge that satisfy the edge constraints.

Optimize surface control points

Surface control points have no effect on the continuity between singular edges, so in order to make the basis function smoother, surface control points on the singular surface can minimize the integral of the second derivative, making the surface smoother.

When solving distribution, you need to solve a least squares problem with linear equality constraints, that is:

s.t.MP＝b

This optimization problem can be transformed into a solution problem of a system of linear equations:

Creases

For regular areas, the spline definition is consistent with the B-spline. In this case, the node insertion method can be used to insert heavy nodes along the specified edge, so that creases can be constructed along this edge.

For creases passing through singular points, as long as (1) the G1 constraint equation corresponding to the singular edge is eliminated, the continuity of the basis function on this edge can be reduced by one order. (2) At the same time, in order to control sharp features, the edge control points corresponding to the specified singular edges are separated. That is to say, these control points were originally linear combinations of the control points in the grid. Now each edge control point is added to the grid control points. , you can directly control and thereby control the shape of the crease. (3) Modify the basis function of the singular point and change the basis function to a linear combination of newly added control points in the surrounding circle.

If there are multiple creases passing through the same singular point, when specifying that two creases are continuous, modify the direct angles θ _i ... θ _j of these two sides so that Keep/> If these creases are not continuous, the angle remains unchanged.

Test results of the technical solutions of the embodiments of the present invention.

The constructed basis function still has good modeling ability under non-uniform conditions, and the surface near the singular point remains single-peak smooth.

For singular points with degree 5, set some edge-node distances to 6 and the rest to 1. The algorithm can obtain the unimodal G1 continuous basis function as shown in the figure.

In the actual model, the node distance of some edges in the ring model is set to 2, and the rest is 1. The edges of some nodes are set to continuous creases, and the results as shown in the figure can be obtained.

The present invention proposes a construction method of spline basis function, so that the generated spline can generate a global G1 continuous surface on the control grid of any topology, and can be directly used for CAD design and CAE analysis, thereby accelerating processing design analysis. Integrated process.

The above embodiments only illustrate the principles and effects of the present invention, but are not intended to limit the present invention. Anyone familiar with this technology can modify or change the above embodiments without departing from the spirit and scope of the invention. Therefore, all equivalent modifications or changes made by those with ordinary knowledge in the technical field without departing from the spirit and technical ideas disclosed in the present invention shall still be covered by the claims of the present invention.

Claims (1)

1. A construction method for non-uniform irregular spline basis functions that retains sharp features, characterized in that the construction method is applied in the field of CNC machining technology; the construction method includes a frame, and the frame is used to have large Near the singular point of the node distance ratio, the hole filling method is used to generate an acceptable surface; the framework is backward and forward compatible with NURBS, can be applied to current CAD software, and can be transferred between various CAD analysis and design software packages without loss; support The isogeometric analysis method connects CAD design and CAE analysis to accelerate the integrated processing design and analysis process; the construction method includes the following steps: Step S1: According to the shape of the surface to be processed, input a quadrilateral control mesh with arbitrary topology; Step S2: Based on the quadrilateral control grid, use the Bezier extraction algorithm to obtain a surface consistent with the spline definition, which specifically includes: based on the quadrilateral control grid, use the Bezier surface method to extract the initially defined C0 basis function, as described Each surface of the quadrilateral control grid is represented by a bicubic Bezier surface; for the surface point F _i on each surface of the quadrilateral control grid, the four control points _Pi on each surface are used Represented by a linear combination; the edge point E _i and the vertex V of the quadrilateral control grid are linear combinations of the face points F _i ; Step S3: Calculate the tangent plane, which specifically includes: using the NURSS subdivision format and the Eigen-polyhedron subdivision format to define the surface points and edge points around the singular point so that they are in the same tangent plane; For a singular point with degree n, the control points adjacent to the singular point are E _i , F _i , and the distance between nodes is _di , a _i ; Calculate new control points Let P＝[V,E ₀ ,…,E _n-1 ,F ₀ ,…,F _n-1 ] ^T ,/> Write the segmentation rule as/> Define the NURSS format subdivision matrix as M, and the Eigen-polyhedron-based subdivision matrix as N; Define a circle of Bezier control points around the singular point as is a linear combination of points V, E _i , F _i ; use the Eigen-polyhedron subdivision format as a guide: Step S31: Calculate the limit point C based on Eigen-polyhedron subdivision; Use L ⁰ to represent the unitization of the eigenvector corresponding to the eigenvalue of matrix N being 1, and the limit point is defined as C=L ⁰ M ^T P; Step S32: Calculate the tangent plane based on Eigen-polyhedron subdivision; Define two matrices of size 2nx2n Order/> λ is the matrix/> The main eigenvalues of , then/> Written as/> Λ is a diagonal matrix composed of singular values; i ₁ , i ₂ are indicators such that Λ (i ₁ , i ₁ ) = Λ (i ₂ , i ₂ ) = λ, let In addition to/> Diagonal matrix with zeros in other positions, get/> Then define the vector set/> Step S33: Define Bezier control points around the singular point; Bezier control points around singular points Step S4: Calculate the connection function. The connection function specifically includes: upgrading the singular surface to a biquintic Bezier surface that satisfies the G1 continuity condition; S41. For a singular point with degree n, define a weight k _i for each corner: S42. Define the angle according to the weight; Sum θ _i . If the summation result is not equal to 2π, normalize the angle to 2π; Definition (i,j)=sin(θ _i )sin(θ _i+2 )…sin(θ _j )(i<j); If n=2k+1 is an odd number: If n=2k is an even number: The set S is n numbers The set of , let/> is the value that occurs most often in the set. If each number appears only once, then let/> Step S5, local step-by-step optimization; Divide the constraints on each edge into two parts, the first two equations become vertex constraints, and the remaining equations are called edge constraints; solve the new basis functions in the following order; the new basis functions include those that are compatible with NURBS The hole-filling method connects CAD design and CAE analysis based on the NURBS transfer model; S51. Optimize vertex control points; Each singular point vertex constraint with degree n contains 3n+1 control points and 2n linear constraint equations. First, solve to obtain 3n+1 control points that satisfy the vertex constraints; S52. Optimize edge control points; After the vertex control points are determined, the remaining control points of the edge constraint design are controlled on each edge, and then the control points on each edge that satisfy the edge constraints are solved; S53. Optimize surface control points; Solve a least squares problem with linear equality constraints during step-by-step solution, that is: s.t.MP=b; Convert the least squares problem into a problem of solving a system of linear equations: S54. Design the shape of the surface to be processed according to the results obtained in step S53, and generate the spline. The spline has the ability to generate a global G1 continuous surface on the control grid of any topology, which is used for CAD design and CAE analysis.