CN103413175A - Closed non-uniform rational B-spline curve fairing method based on genetic algorithm - Google Patents

Abstract

The invention discloses a closed non-uniform rational B-spline (NURBS) curve fairing method based on a genetic algorithm to solve a technical problem that the speed of the existing closed non-uniform rational B-spline curve fairing method is slow. The technical scheme is that a data point is added between two data points which show reverse curvature to carry out curve interpolation again, the position of the newly added data point is determined under the constraint of a fairing principle by using a genetic algorithm, and thus the shaped of a curve is adjusted to smooth the curve. According to the method, the curve with interpolation again passes through an original data point strictly, the reverse curvature problem of the original curve is solved, and the curve is smoother than curve disclosed by the background technology. At the same time, a phenomenon that the closed section line at a splicing part of a back edge with a leaf back and a leaf basin shows depressions is effectively solved, and thus the leaf section line is smoother. The method is suitable for a plurality of 3D CAD software platforms, thus the deficiency of the existing NURBS curve interpolation research is effectively made up, and the fairing speed is raised.

Description

Closed non-uniform rational B-spline curve fairing method based on genetic algorithm

Technical Field

The invention relates to a closed non-uniform rational B-spline curve fairing method, in particular to a closed non-uniform rational B-spline curve fairing method based on a genetic algorithm.

Background

Complex products, such as three-dimensional model construction of aircraft engine blades or generator blades, typically utilize contour points derived from pneumatic (or other computed) data to construct a blade airfoil profile, i.e., a blade airfoil profile is constructed by "section curve interpolation contour points — curved surface cross-section curves. At present, the front/rear edge of the blade mostly gives the circle center and the radius position of the front/rear edge as known conditions, and a section line is created through the first-order geometric continuous connection of the front/rear edge, a blade back and a blade basin, so that the curved surface modeling is completed. The trend of the section line molding method is to use the molding value points to directly represent the shape, and a second-order geometrically continuous section line is created through direct interpolation of the molding value points. However, since the curvature change of the trailing edge of the blade body is large and the shape is very sensitive to aerodynamic performance, the shape points near the trailing edge are very dense, and the shape points on the blade back and the blade basin are relatively sparse, so that a closed section line at the joint of the trailing edge and the blade back and the blade basin is dimpled, and the profile of the blade body is affected. Therefore, how to treat the pits to make the pits smooth is a real problem, and the existing three-dimensional CAD software system cannot solve the problem.

The Non-Uniform Rational B-Spline method represents that curve surfaces are already the standard expression form of CAD systems, which not only solves the incompatibility problem of free curve surfaces and elementary analytic curve surfaces and overcomes the defects of Bezier and B-Spline methods, but also enables the weight factors and Non-Uniform node vectors to more effectively control the shape of the curve surfaces, and strictly defines the product geometry by a Uniform mathematical model in a CAD system, so that the system is simplified, data management is easy, and the method is convenient for engineers to use, and simultaneously improves the surface modeling capability.

For NURBS curves generated by interpolation, the curvature direction is reversed at the transition from the denser points of the type value points to the thinner points of the type value points, so that the curves are not smooth. At present, for the non-smooth condition of the NURBS interpolation curve, there are mainly smooth methods such as dead pixel modification, weight factor adjustment and node vector adjustment, but these methods all have disadvantages. If the type value point is accurate, the originally correct type value point can be artificially damaged by adopting a bad point modification method; in engineering application, model points interpolated by the NURBS curve often have no weight factors, so that the NURBS curve cannot be widely applied by modifying the weight factors to adjust the shape of the NURBS curve; experiments prove that adjusting the node vector is not the most effective method for solving the problem of curve reverse curvature.

The document "research of curve fairing based on genetic algorithm, volume 13, phase 13 of Chinese mechanical engineering, and the first half of 2 months in 2002" discloses a curve fairing method based on genetic algorithm. On the basis of the traditional fairing, the method provides one of the standards for measuring the fairing of the curve by taking the variance of the extreme value of the curvature of the curve as a measure, and introduces a genetic algorithm and a fuzzy mathematical control mechanism. The method mainly researches the fairing problem after grating vectorization, has certain limitation, and is low in speed by using a curve curvature extreme value variance method under the condition that a plurality of data points exist.

Disclosure of Invention

In order to overcome the defect that the existing closed non-uniform rational B-spline curve fairing method is low in speed, the invention provides a closed non-uniform rational B-spline curve fairing method based on a genetic algorithm. The method is characterized in that a type value point is added between two type value points with reverse curvature to perform curve interpolation again, the position of the newly added type value point is determined under the constraint of a fairing criterion by adopting a genetic algorithm, and then the shape of the curve is adjusted to be fairing. The method ensures that the re-interpolated curve not only strictly passes through the original model value point, but also solves the problem of reverse curvature of the original curve, and is smoother than the curve disclosed by the background technology. Meanwhile, the phenomenon that closed section lines at the splicing positions of the trailing edge, the blade back and the blade basin are concave is effectively solved, the section lines of the blades are smoother, and the method is suitable for various three-dimensional CAD software platforms, so that the defects of the existing NURBS curve interpolation research are effectively overcome, and the speed is high.

The technical scheme adopted by the invention for solving the technical problems is as follows: a closed non-uniform rational B-spline curve fairing method based on genetic algorithm is characterized by comprising the following steps:

and step one, determining a closed NURBS curve interpolation method, and performing interpolation by adopting a three-time closed NURBS curve. Is provided with m +1 type value points q₀,q₁,q₂,...q_mAnd q is₀＝q_mThe point of the shape value is the connection point of the segment in the curve, i.e. q_iHas a node value u_i+3. The NURBS curve consists of n +1 control vertices d₀，d₁，d₂，....d_nAnd node vector U ═ U₀,u₁,...,u_n+4]To define the domain as u e [ u ∈ [ [ u ]₃,u_n+1]＝[0,1]. Where n is m +2, there are m +3 unknown control vertices.

1.1 calculating parameter value sequence t by cumulative chord length method₀,t₁,t₂,...t_mDefining an intra-domain node value as u₃＝t₀,u₄＝t₁,u₅＝t₂,...u_m＝u_m+3Nodes outside the domain of definition are determined as u₀＝u_n-2-1，u₁＝u_n-1-1，u₂＝u_n-1，u_n+2＝u₄+1，u_n+3＝u₅+1，u_n+4＝u₆+1。

1.2 inverse for interpolating m +1 type value points q₀,q₁,q₂,...q_mAnd q is₀＝q_mThe equation of the three-time closed NURBS curve is expressed as

$p\left(u\right)=\underset{j=0}{\overset{n}{\mathrm{\Σ}}}{d}_j{R}_{j,3}\left(u\right)=\underset{j=i-3}{\overset{i}{\mathrm{\Σ}}}{d}_j{R}_{j,3}\left(u\right),u\∈[u_i,u_{i+1}]\⋐[u_3,u_{n+1}],$

Wherein, $R_{i,3}\left(u\right)=\frac{w_j{N}_{j,k}\left(u\right)}{\underset{j=0}{\overset{n}{\mathrm{\Σ}}}{w}_j{N}_{j,k}\left(u\right)}$ is a cubic rational basis function.

Defining a curve into a domain

The node value in the equation is substituted into the equation to satisfy the interpolation condition, i.e.

$p\left(u_i\right)=\underset{j=i-3}{\overset{i}{\mathrm{\Σ}}}{d}_j{R}_{j,3}\left(u_i\right)=q_{i-3},i=\mathrm{3,4},\·\·\·,n$

The above formula contains n-2 equations in total. Coincidence of the first and the last three control vertexes d_n-2＝d₀，d_n-1＝d₁，d_n＝d₂The number of unknown control vertices is reduced to n-2. Solving n-2 unknown control vertexes by a catch-up method from a linear equation set consisting of n-2 equations.

1.3 solving for control vertex d_iBefore, d is obtained_iCorresponding weight factor w_iI is 0, 1. If the value point q of each type is known_iWeight factor of

i is 0, 1.. m, then

$\left\{\begin{array}{c}\underset{j=i-3}{\overset{i}{\mathrm{\&Sigma;}}}{w}_j{R}_{j,3}\left(u_i\right)=\stackrel{\&OverBar;}{w_{i-3}},i=\mathrm{3,4},...,n\\ w_{n-2}=w_0,w_{n-1}={\mathrm{<figure-callout\; id="1"\; label="w"\; filenames="HDA00003491160900011.png"\; state="\{\{state\}\}">w</figure-callout>}}_1,w_n=w_2\end{array}\right.$

Simultaneously establishing the above equation set to obtain the control vertex d_iWeight factor w of_i。

Step two, determining a point q_kAnd point q_k+1Wherein at q_kAnd q is_k+1The inverse curvature between the two model points is not smooth.

Step three, calculating a point q by adopting a genetic algorithm_kAnd point q_k+1The position of the new addition value point in between.

3.1 suppose that q is_kAnd q is_k+1Between two type value points, there appears inverse curvature irregularity, where q_kAnd q is_k+1Is manually selected.

Is q_k(x_k,y_k) And q is_k+1(x_k+1,y_k+1) A new type value point therebetween, Δ x is q_k+0.5(x_k+0.5,y_k+0.5) Edge of

Bias value of direction, Δ y, perpendicular toBias values in the directions. Through calculation, the following results are obtained:

$\left\{\begin{array}{c}{x}_{k+0.5}=x_k+\mathrm{\&Delta;x}\&CenterDot;\cos \mathrm{\&theta;}+\mathrm{\&Delta;y}\&CenterDot;\cos \mathrm{\&eta;}\\ y_{k+0.5}=y_k+\mathrm{\&Delta;x}\&CenterDot;\sin \mathrm{\&theta;}+\mathrm{\&Delta;y}\&CenterDot;\sin \mathrm{\&eta;}\end{array}\right.$

wherein, $\mathrm{\&theta;}=\arctan \frac{y_{k+1}-y_k}{x_{k+1}-x_k},$ $\mathrm{\&eta;}=\arctan (-\frac{x_{k+1}-x_k}{y_{k+1}-y_k}),$ $0<\mathrm{\&Delta;x}<\left|\stackrel{\&RightArrow;}{q_k{q}_{k+1}}\right|.$

calculating delta x and delta y to obtain a newly added value point q_k+0.5(x_k+0.5,y_k+0.5) Thus, Δ x and Δ y serve as two genes on the chromosome of the genetic algorithm.

The energy of the 3.2 curve is defined as E ═ k-²ds, where k is the curvature and s is the arc length; a variable Q is defined representing the maximum value of the curvature change,

the objective function of curve fairing is min (f), where f is α · E + β · Q, α is the weight factor of the curve strain energy variation value, β is the weight factor of the curve maximum curvature variation value, and α + β is 1.

The objective function of the curve fairing is min (f (Δ x, Δ y)); the fitness is determined as

Wherein

Δ x and Δ y are two genes on the chromosome.

3.3, encoding the two genes delta x and delta y on the chromosome by adopting an encoding mode as floating point numbers, and respectively reserving three bits after decimal point; the size of the population was 10, and the initial population was generated randomly for each individual.

3.4 genetic operators include selection, crossover and mutation. The selection method adopts a roulette method; the crossing method adopts single-point crossing, and crossing points are randomly generated; the mutation method is a Gaussian mutation method.

3.5 starting the genetic algorithm to calculate the position of the new incremental value point.

And fourthly, interpolating a new NURBS curve through the original model value points and the new model value points.

If the curve has reverse curvature, jumping to the second step, and smoothing the curve again; and if the curve meets the fairing requirement, outputting the current interpolation curve as a final result.

The weight factor alpha of the strain energy change value of the curve is 0.7.

The weight factor beta of the maximum curvature change value of the curve is 0.3.

The invention has the beneficial effects that: the method adds the type value points between the two type value points with the reverse curvature to perform curve interpolation again, and the positions of the newly added type value points are determined by adopting a genetic algorithm under the constraint of a fairing criterion, so that the shape of the curve is adjusted to be fairing. The method ensures that the re-interpolated curve not only strictly passes through the original model value point, but also solves the problem of reverse curvature of the original curve, and is smoother than the curve disclosed by the background technology. Meanwhile, the phenomenon that closed section lines at the splicing positions of the trailing edge, the blade back and the blade basin are concave is effectively solved, the section lines of the blades are smoother, and the method is suitable for various three-dimensional CAD software platforms, so that the defects of the existing NURBS curve interpolation research are effectively overcome, and the fairing speed is improved.

The present invention will be described in detail below with reference to the accompanying drawings and examples.

Drawings

FIG. 1 is a diagram of new addition value points.

FIG. 2 is a schematic representation of blade body section line data points.

FIG. 3 is an enlarged partial view of the airfoil section trailing edge portion data points.

FIG. 4 is an initial airfoil section line constructed from airfoil section line data points.

Fig. 5 shows a curvature comb where the cross-sectional lines of the blade body show reverse curvature.

FIG. 6 is a schematic view of the shape point where the reverse curvature occurs in the section line of the blade body, where the shape point p is located₁And p₂Between which there appears a negative curvature irregularity at the type value point q₁And q is₂The reverse curvature is not smooth.

FIG. 7 is a schematic cross-sectional view of a smooth blade body.

Figure 8 is a curved smooth curvature comb.

Detailed Description

Referring to fig. 1 to 8, taking section line interpolation of a blade body of a certain type of blade as an example, taking visual studio2010 as a development tool, and developing and implementing the invention in detail on a design software NX7.5 platform by using NXOpen API.

Step 1: and determining a closed NURBS curve interpolation method, and performing interpolation by adopting a three-time closed NURBS curve. Is provided with m +1 type value points q₀,q₁,q₂,...q_mAnd q is₀＝q_mThe point of the shape value is the connection point of the segment in the curve, i.e. q_iHas a node value u_i+3. The NURBS curve consists of n +1 control vertices d₀，d₁，d₂，....d_nAnd node vector U ═ U₀,u₁,...,u_n+4]To define the domain as u e [ u ∈ [ [ u ]₃,u_n+1]＝[0,1]. Where n is m +2, i.e. the number of control vertices is 2 more than the number of model value points, and there are m +3 unknown control vertices.

1.1 determine the node vector. Calculating a parameter value sequence t by adopting a method of accumulating chord lengths₀,t₁,t₂,...t_mDefining an intra-domain node value as u₃＝t₀,u₄＝t₁,u₅＝t₂,...u_m＝u_m+3Nodes outside the domain of definition are determined as u₀＝u_n-2-1，u₁＝u_n-1-1，u₂＝u_n-1，u_n+2＝u₄+1，u_n+3＝u₅+1，u_n+4＝u₆+1。

1.2 back-computing the control vertices. For interpolating m +1 pattern value points q₀,q₁,q₂,...q_mAnd q is₀＝q_mThe equation of the three-time closed NURBS curve is expressed as

$p\left(u\right)=\underset{j=0}{\overset{n}{\mathrm{\&Sigma;}}}{d}_j{R}_{j,3}\left(u\right)=\underset{j=i-3}{\overset{i}{\mathrm{\&Sigma;}}}{d}_j{R}_{j,3}\left(u\right),u\&Element;[u_i,u_{i+1}]\&Subset;[u_3,u_{n+1}],$

Wherein $R_{i,3}\left(u\right)=\frac{w_j{N}_{j,k}\left(u\right)}{\underset{j=0}{\overset{n}{\mathrm{\&Sigma;}}}{w}_j{N}_{j,k}\left(u\right)}$ Is a cubic rational basis function.

Defining a curve into a domain

The node value in the equation is substituted into the equation to satisfy the interpolation condition, i.e.

$p\left(u_i\right)=\underset{j=i-3}{\overset{i}{\mathrm{\&Sigma;}}}{d}_j{R}_{j,3}\left(u_i\right)=q_{i-3},i=\mathrm{3,4},\&CenterDot;\&CenterDot;\&CenterDot;,n$

The above formula contains n-2 equations in total. Coincidence of the first and the last three control vertexes d_n-2＝d₀，d_n-1＝d₁，d_n＝d₂The number of unknown control vertices is reduced to n-2. Solving n-2 unknown control vertexes by a catch-up method from a linear equation set consisting of n-2 equations.

1.3 determine control vertex weights. At solving for control vertex d_iBefore, d is obtained_iCorresponding weight factor w_iI is 0, 1. If the value point q of each type is known_iWeight factor of

i is 0, 1.. m, then

$\left\{\begin{array}{c}\underset{j=i-3}{\overset{i}{\mathrm{\&Sigma;}}}{w}_j{R}_{j,3}\left(u_i\right)=\stackrel{\&OverBar;}{w_{i-3}},i=\mathrm{3,4},...,n\\ w_{n-2}=w_0,w_{n-1}={\mathrm{<figure-callout\; id="1"\; label="w"\; filenames="HDA00003491160900011.png"\; state="\{\{state\}\}">w</figure-callout>}}_1,w_n=w_2\end{array}\right.$

Simultaneously establishing the above equation set to obtain the control vertex d_iWeight factor w of_i。

Step 2: self-determining point q_kAnd point q_k+1Wherein at q_kAnd q is_k+1The inverse curvature between the two model points is not smooth.

And step 3: calculating point q using a genetic algorithm_kAnd point q_k+1The position of the new addition value point in between.

3.1 determining chromosomes for genetic algorithms. Suppose that q is_kAnd q is_k+1Between two type value points, there appears inverse curvature irregularity, where q_kAnd q is_k+1Is manually selected. q. q.s_k+0.5(x_k+0.5,_yk+0.5) Is q_k(x_k,y_k) And q is_k+1(x_k+1,y_k+1) A new type value point therebetween, Δ x is q_k+0.5(x_k+0.5,y_k+0.5) Edge of

Bias value of direction, Δ y, perpendicular toBias values in the directions. Through calculation, the following results are obtained:

$\left\{\begin{array}{c}{x}_{k+0.5}=x_k+\mathrm{\&Delta;x}\&CenterDot;\cos \mathrm{\&theta;}+\mathrm{\&Delta;y}\&CenterDot;\cos \mathrm{\&eta;}\\ y_{k+0.5}=y_k+\mathrm{\&Delta;x}\&CenterDot;\sin \mathrm{\&theta;}+\mathrm{\&Delta;y}\&CenterDot;\sin \mathrm{\&eta;}\end{array}\right.$

wherein, $\mathrm{\&theta;}=\arctan \frac{y_{k+1}-y_k}{x_{k+1}-x_k},$ $\mathrm{\&eta;}=\arctan (-\frac{x_{k+1}-x_k}{y_{k+1}-y_k}),$ $0<\mathrm{\&Delta;x}<\left|\stackrel{\&RightArrow;}{q_k{q}_{k+1}}\right|.$

calculating delta x and delta y to obtain a newly added value point q_k+0.5(x_k+0.5,y_k+0.5) Thus, Δ x and Δ y serve as two genes on the chromosome of the genetic algorithm.

3.2 calculating the fitness of the genetic algorithm. A smooth curve generally meets the following conditions that the second order of the curve is geometrically continuous; no singularities and unwanted inflection points; the curvature change is uniform; the strain energy is small. A reasonable fairing criterion is provided by comprehensively considering two aspects of curvature change and strain energy.

The energy of the curve is defined as E ═ k²ds, where k is the curvature and s is the arc length; a variable Q is defined representing the maximum value of the curvature change,

i.e. the curve is discretized into 1000 points, and E and Q are derived by the function of analyzing the curvature of the curve. The objective function for determining the curve smoothness is min (f), where f is α · E + β · Q, α and β are the weighting factors of the curve strain energy and the maximum curvature change value, respectively, in this example, α is 0.7 and β is 0.3. Fitness of genetic algorithm $\mathrm{fitness}=\frac{1}{f(\mathrm{\&Delta;x},\mathrm{\&Delta;y})}.$

In the genetic algorithm, the degree of superiority and inferiority of an individual is evaluated based on the degree of fitness of the individual, thereby determining the magnitude of genetic chance. The objective function of the curve smoothness is min (f (Δ x, Δ y)), and the smaller the value of f (Δ x, Δ y), the more excellent the individual is, and therefore, the fitness is determined as

Wherein

Δ x and Δ y are two genes on the chromosome.

3.3 determining the code and initial population. Encoding floating point numbers by adopting an encoding mode, and respectively reserving three bits after decimal points for two genes delta x and delta y on a chromosome; the size of the population was 10, i.e. the population consisted of 10 individuals, with the initial population being generated randomly per individual.

3.4 determining the genetic operator. Genetic operators include selection, crossover and mutation. The selection method adopts a roulette method; the crossing method adopts single-point crossing, and crossing points are randomly generated; the mutation method is a Gaussian mutation method.

And evaluating the population by utilizing the fitness of the genetic algorithm. If the genetic algebra is satisfied, ending the program and outputting a final curve; otherwise, the genes of the chromosome are subjected to selection, crossing and mutation operations, wherein the selection method adopts a roulette method; the crossing method adopts single-point crossing, the crossing probability Pc is 0.8, and the crossing points are randomly generated; the mutation method adopts a Gaussian mutation method, and the mutation probability Pm is 0.8.

3.5 starting the genetic algorithm to calculate the position of the new incremental value point.

And 4, step 4: and interpolating a new NURBS curve through the original model value points and the new model value points.

And 5: if the curve has a place with reverse curvature, jumping to the step 2, and smoothing the curve again; and if the curve meets the fairing requirement, outputting the current interpolation curve as a final result.

Claims (3)

1. A closed non-uniform rational B-spline curve fairing method based on a genetic algorithm is characterized by comprising the following steps:

step one, determining a closed NURBS curve interpolation method, and performing interpolation by adopting a three-time closed NURBS curve; is provided with m +1 type value points q₀,q₁,q₂,...q_mAnd q is₀＝q_mThe point of the shape value is the connection point of the segment in the curve, i.e. q_iHas a node value u_i+3(ii) a The NURBS curve consists of n +1 control vertices d₀，d₁，d₂，....d_nAnd node vector U ═ U₀,u₁,...,u_n+4]To define the domain as u e [ u ∈ [ [ u ]₃,u_n+1]＝[0,1](ii) a Wherein n is m +2, and m +3 unknown control vertexes are total;

1.1 calculating parameter value sequence t by cumulative chord length method₀,t₁,t₂,...t_mDefining an intra-domain node value as u₃＝t₀,u₄＝t₁,u₅＝t₂,...u_m＝u_m+3Nodes outside the domain of definition are determined as u₀＝u_n-2-1，u₁＝u_n-1-1，u₂＝u_n-1，u_n+2＝u₄+1，u_n+3＝u₅+1，u_n+4＝u₆+1；

1.2 inverse for interpolating m +1 type value points q₀,q₁,q₂,...q_mAnd q is₀＝q_mThe equation of the three-time closed NURBS curve is expressed as

$p\left(u\right)=\underset{j=0}{\overset{n}{\mathrm{\&Sigma;}}}{d}_j{R}_{j,3}\left(u\right)=\underset{j=i-3}{\overset{i}{\mathrm{\&Sigma;}}}{d}_j{R}_{j,3}\left(u\right),u\&Element;{[u}_i,u_{i+1}]\&Subset;[u_3,u_{n+1}],$

Wherein,

is a cubic rational basis function;

defining a curve into a domain

The node value in the equation is substituted into the equation to satisfy the interpolation condition, i.e.

$p\left(u_i\right)=\underset{j=i-3}{\overset{i}{\mathrm{\&Sigma;}}}{d}_j{R}_{j,3}\left(u_i\right)=q_{i-3},$ i＝3,4,…,n

The above formula contains n-2 equations in total; coincidence of the first and the last three control vertexes d_n-2＝d₀，d_n-1＝d₁，d_n＝d₂The number of unknown control vertexes is reduced to n-2; solving n-2 unknown control vertexes by a catch-up method from a linear equation set consisting of n-2 equations;

1.3 solving for control vertex d_iBefore, d is obtained_iCorresponding weight factor w_iIf the value of each type q is known, i is 0,1, …, n_iWeight factor of

I is 0,1, …, m, then

$\left\{\begin{array}{c}\underset{j=i-3}{\overset{i}{\mathrm{\&Sigma;}}}{w}_j{R}_{j,3}\left(u_i\right)=\stackrel{\&OverBar;}{w_{i-3}},i=\mathrm{3,4},...,n\\ w_{n-2}=w_0,w_{n-1}=w_1,w_n=w_2\end{array}\right.$

Simultaneously establishing the above equation set to obtain the control vertex d_iWeight factor w of_i；

Step two, determining a point q_kAnd point q_k+1Wherein at q_kAnd q is_k+1The two model points have inverse curvature and incompliance;

step three, calculating a point q by adopting a genetic algorithm_kAnd point q_k+1The position of the new addition value point;

3.1 suppose that q is_kAnd q is_k+1Between two type value points, there appears inverse curvature irregularity, where q_kAnd q is_k+1Is manually selected; q. q.s_k+0.5(x_k+0.5,y_k+0.5) Is q_k(x_k,y_k) And q is_k+1(x_k+1,y_k+1) A new type value point therebetween, Δ x is q_k+0.5(x_k+0.5,y_k+0.5) Edge of

Bias value of direction, Δ y, perpendicular to

A bias value in a direction; through calculation, the following results are obtained:

$\left\{\begin{array}{c}{x}_{k+0.5}=x_k+\mathrm{\&Delta;x}\&CenterDot;\cos \mathrm{\&theta;}+\mathrm{\&Delta;y}\&CenterDot;\cos \mathrm{\&eta;}\\ y_{k+0.5}=y_k+\mathrm{\&Delta;x}\&CenterDot;\sin \mathrm{\&theta;}+\mathrm{\&Delta;y}\&CenterDot;\sin \mathrm{\&eta;}\end{array}\right.$

wherein, $\mathrm{\&theta;}=\arctan \frac{y_{k+1}-y_k}{x_{k+1}-x_k},$ $\mathrm{\&eta;}=\arctan (-\frac{x_{k+1}-x_k}{y_{k+1}-y_k}),$ $0<\mathrm{\&Delta;x}<\left|\stackrel{\&RightArrow;}{q_k{q}_{k+1}}\right|;$

calculating delta x and delta y to obtain a newly added value point q_k+0.5(x_k+0.5,y_k+0.5) Thus, Δ x and Δ y act as two genes on the chromosome of the genetic algorithm;

the energy of the 3.2 curve is defined as E ═ k-²ds, where k is the curvature and s is the arc length; a variable Q is defined representing the maximum value of the curvature change,

the target function of the curve fairing is min (f), wherein f is alpha.E + beta.Q, alpha is a weight factor of the change value of the strain energy of the curve, beta is a weight factor of the maximum curvature change value of the curve, and alpha + beta is 1;

the objective function of the curve fairing is min (f (Δ x, Δ y)); the fitness is determined as

Wherein

Δ x and Δ y are two genes on the chromosome;

3.3, encoding the two genes delta x and delta y on the chromosome by adopting an encoding mode as floating point numbers, and respectively reserving three bits after decimal point; the size of the population is 10, and each individual of the initial population is randomly generated;

3.4 genetic operators including selection, crossover and mutation; the selection method adopts a roulette method; the crossing method adopts single-point crossing, and crossing points are randomly generated; the mutation method adopts a Gaussian mutation method;

3.5 starting a genetic algorithm to calculate the position of the new incremental value point;

step four, interpolating a new NURBS curve through the original model value points and the newly added model value points;

if the curve has reverse curvature, jumping to the second step, and smoothing the curve again; and if the curve meets the fairing requirement, outputting the current interpolation curve as a final result.

2. The closed non-uniform rational B-spline curve fairing method based on genetic algorithm as recited in claim 1, characterized in that: the weight factor alpha of the strain energy change value of the curve is 0.7.

3. The closed non-uniform rational B-spline curve fairing method based on genetic algorithm as recited in claim 1, characterized in that: the weight factor beta of the maximum curvature change value of the curve is 0.3.

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