CN108763668A - The model of gear region parameter method replaced with boundary based on subdivision technology - Google Patents
The model of gear region parameter method replaced with boundary based on subdivision technology Download PDFInfo
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Abstract
The invention discloses the model of gear region parameter methods replaced with boundary based on subdivision technology.Graphical model with complex region is difficult the parametrization for accomplishing high quality.The present invention reduces the concave-convex degree of each section of curve by curve segmentation method, ensures the network of quadrilaterals lattice model that not will produce overturning, to optimize the quality of model of gear parametrization, and meets it in engineer application for waiting validity of geometrical analysis to require；Pass through the modification to model of gear Polygonal Boundary average distance value, the distance between uniform boundary vertex, to improve the production rate of quadrilateral mesh later；Ensure that the shape of the model of gear will not change by using the method that boundary is replaced, the use of method for fairing and C-C divided methods is then optimized the model of gear on smoothness and continuity, to meet smoothness and robustness requirement of the engineer application for equal geometrical analysis.
Description
Technical field
The invention belongs to computer aided design and engineering application fields, and in particular to be replaced based on subdivision technology and boundary
Model of gear region parameter method.
Background technology
CAD (CAD) technology and computer-aided engineering (CAE) technology today computer application and
In development, there is very far-reaching influence, they are all obtained in fields such as aerospace, automobile making, architectural design, mechano-electronics
To being widely applied.In cad technique, body form is indicated by boundary model, and boundary model contains
The information of geometrical model curved inner surface.Equal geometrical analysis are frequently necessary to carry out the parametrization inside geometrical model region, therefore are
The equal geometrical analysis that allow that CAD model is more applicable for, it is necessary to the method that research builds region parameter by giving data boundary
Technology.
Equal geometrical analysis are a kind of new techniques of the precise geometrical expression progress analog simulation based on CAD model, it
It proposes to realize that the seamless fusion of CAD/CAE opened up a new way, while being also that the geometric modeling field for having become ripe is filled with
New vigor.In equal geometrical analysis, the parametrization of computational domain is still current one of research hotspot, frequent in finite element analysis
The parameterized model for needing high-precision and validity, to be applied in the research and design of high-level CAD or CAE models.And it is
Reach these requirements, it is necessary to being optimized for the grid of parametrization and some special areas, specific to some
The complexity of method, region largely affects the validity and quality of parametrization.
The parametrization in a region is needed in equal geometrical analysis to meet three requirements：1) it does not generate selfing, namely
Mapping from parameter field to physical domain is interior penetrates；2) isoparametric element element should as much as possible uniformly；3) isoparametric element element should to the greatest extent can
Energy ground is orthogonal.Currently, the boundary information indicated by B-spline curves can be handled there are many method, they include variation
Harmonic wave method, technology of dividing and ruling, reflection method, the decomposition method based on profile, multi-panel piece parametric method and non-standard B-spline
Parametric method.But the parametrization of some simple regions can only be handled in existing method mostly, and for complex region
Some graphical models, be but difficult to accomplish the parametrization of high quality.Such as a kind of situation be handle some have concave-convex degree compared with
When big net region, it is easy to form the grid of overturning, the serious quality for affecting region parameter at this.
Invention content
The purpose of the present invention is the deficiencies for existing method, propose a kind of gear replaced with boundary based on subdivision technology
Model area parametric method, this method have mainly used the thought that a kind of boundary replaces and C-C are combined to segment, realized to by curve
The parametrization that the model of gear of boundary composition works well, and obtain final parameterized model.
The present invention is specific as follows：
Step 1 samples the inside and outside boundary of the model of gear, on inside and outside boundary since first sampled point,
4 sampled points are taken to be fitted to a cubic uniform B-splines successively every time, in adjacent two cubic uniform B-splines, after
When first sampled point used when one cubic uniform B-splines fitting is fitted for previous cubic uniform B-splines
The last one sampled point used；After fitting, inner boundary is made of 4 cubic uniform B-splines, outer boundary by 24 three times
Uniform B-Spline Curve forms；Cubic uniform B-splines expression formula is：
Wherein, PiFor the control point position for building cubic uniform B-splines, i takes 0,1,2 or 3, Ni,3(t) it is B
Spline base function, P (t) are the match point position on the cubic uniform B-splines formed, parameter t ∈ node intervals [0,1].
Step 2 passes through the inside and outside boundary through step 1 treated the model of gear curvilinear characteristic point and carries out curve minute
It cuts, then converts the inside and outside boundary of the model of gear to Polygonal Boundary geometrical model, it is specific as follows：
Step a, by the method for parabola interpolation, the inside and outside borderline curvilinear characteristic point of the model of gear, curve are detected
Characteristic point includes the point with extreme curvature and inflection point of curve, and curve segmentation is carried out at curvilinear characteristic point.
The process that parabola interpolation detects point with extreme curvature is specific as follows：By the inside and outside every borderline of the model of gear
The node interval [0,1] of cubic uniform B-splines is divided into 20 equal portions section [tn-1,tn], n=1,2 ..., 20, then according to
Secondary traversal [tn-1,tn+1], computation interval [tn-1,tn+1] in three match point curvature k (tn-1)、k(tn) and k (tn+1), if full
Sufficient condition k (tn)>k(tn-1) and k (tn)>k(tn+1), then current interval is section where bent curvature of a curve maximum point, and if
Meet condition k (tn)<k(tn-1) and k (tn)<k(tn+1), then current interval is section where curvature minimum point.Curvature maximum
Section where section where point or curvature minimum point passes through three point (t of interpolationn-1,k(tn-1)), (tn,k(tn)) and (tn+1,
k(tn+1)) obtain parabola f (m)=a2m2+a1m+a0, wherein a0,a1,a2For coefficient, then by setting leading for f (m)
Number f ' (m)=0 acquire the m values of the point in parabola on symmetry axis, are denoted as t '：
Using parabola peak or the limit curvature of minimum point point t ' as cubic uniform B-splines in section [tn-1,
tn] in curvature maximum point or curvature minimum point.
The process of parabola interpolation detection curve inflection point is specific as follows：By inside and outside borderline every three of the model of gear
The node interval [0,1] of secondary Uniform B-Spline Curve is divided into 20 equal portions section [tn-1,tn], [t is then traversed successivelyn-1,tn], meter
Calculate section [tn-1,tn] two endpoints curvature value, meet k (tn-1)*k(tn)<When 0, there are points of inflexion on a curve in the section.So
Midpoint (the t in the parameter section is calculated afterwardsn-0.5,k(tn-0.5)), wherein tn-0.5=1/2* (tn-1+tn).If | k (tn-0.5)-0
|<0.0001, just directly by tn-0.5As point of inflexion on a curve, otherwise, still through (tn-1,k(tn-1)), (tn-0.5,k(tn-0.5))
(tn,k(tn)) one parabola f (m)=b of three point interpolation2m2+b1m+b0, wherein b0,b1,b2For coefficient, finds out f (m)=0 and exist
Section [tn-1,tn] in unique root, and using root as cubic uniform B-splines in section [tn-1,tn] in inflection point.
Step b, using the curvilinear characteristic point of cubic uniform B-splines in step a as cut-point, to the curve weight of segmentation
New distribution control point；By node interval [ta, tb] two-end-point taAnd tbThe t substituted into formula (1) is found out at segmentation curve two-end-point
Control vertex PaAnd Pb.The curve of segmentation and the match point position correspondence of cubic uniform B-splines are as follows：
In formula, Pab(t ") is segmentation curve matching point position, parameter t " ∈ node intervals [ta, tb]；According to the song of segmentation
The match point position correspondence of line and cubic uniform B-splines calculates segmentation curve residue intermediate control point position difference
For：
Wherein, P0And P1For the control point position of former cubic uniform B-splines, Pa1And Pa2It is two among segmentation curve
Control point position, to be that segmentation curve is assigned with new control point position according to former cubic uniform B-splines, respectively
Pa, Pa1, Pa2, Pb。
Step 3 first traverses the segmentation point sequence of acquisition, calculates the distance between adjacent segmentation point, and calculate
The average value h of each adjacent segmentation point distance.When the distance between adjacent segmentation point is less than 0.5 λ h, wherein 0 < λ < 1, it will be adjacent
The latter divides point deletion in cut-point；The distance between adjacent segmentation point be more than 1.5 λ h when, by between adjacent segmentation point away from
From subtracting one after the result rounding plus 0.5 λ h resulting values divided by λ h as the quantity for being inserted into cut-point, thus to the adjacent segmentation
Section [t belonging to pointn-1,tn] divided equally, to realize the segmentation of corresponding B-spline Curve, then according in step 2
The method that step b redistributes control point redistributes control point to the curve of segmentation.Finally, it is bent to be sequentially connected every segmentation
The control point at line both ends, to convert curved boundary to obtain the Polygonal Boundary geometrical model of the model of gear；Use quadrangle
Subdivision algorithm generates the network of quadrilaterals lattice model of the model of gear, and carries out SmoothNumerical TechniqueandIts to grid using iteration method for fairing.
Step 4 controls grid model to the quadrilateral mesh model construction of the model of gear and carries out C-C sub-dividing operations, together
When by replacing boundary Control point and curve node be inserted into that treated keeping step 1 is fitted through cubic uniform B-splines
Model of gear curved boundary shape.
Step a, the control point of network of quadrilaterals lattice model each quadrilateral surface sheet border and inside is calculated using linear interpolation,
Four vertex including quad patch generate the control point sequence of a 4*4 in each quadrilateral surface on piece, then will pass through
The control point on segmentation curved boundary in step 3 after curve segmentation be substituted into corresponding quadrilateral mesh model boundary pass through it is linear
On the control point that interpolation obtains, to construct the control grid model for parametrization.
Step b, the internal vertex of control grid model is handled using C-C divided methods, generates C-C subdivided meshes models.
Step c, the method for being inserted into node and control point using B-spline curves increases every in step 3 after curve segmentation
The number of control points of boundary curve, it is specific as follows to match the number on C-C subdivided meshes model boundaries vertex：
After generating C-C subdivided meshes models, the number of vertices corresponding to the boundary of each quad patch increases to 7
It is a, still ensureing that knot vector is uniform, three times uniformly by every in the boundary curve after curve segmentation in step 3
B-spline curves increase by 3 nodes, while changing every cubic uniform B-spline song in the boundary curve in step 3 after curve segmentation
The control point position of centre two of line, and increase by 3 control points, the position at 7 new control points calculate separately for：
P′0=P0
P′6=P3
Wherein, Pr', r=0,1 ..., 6 be the position at new control point, Ps, s=0, after 1,2,3 is curve segmentation in step 3
Boundary curve in every cubic uniform B-splines original 4 control point positions.
It is inserted into after node, the new control point of each edge circle curve is substituted into corresponding quadrilateral mesh model boundary successively
On control point after C-C is segmented.
Step 5, the C-C subdivided meshes model for obtaining the step c in step 4 are as the control net for drawing B-spline surface
Lattice compare the network of quadrilaterals lattice model generated in step 3, and each quad patch, which corresponds to drafting one, 7 rows 7 row control point
Cubic uniform B-spline curved surface completes the parametrization of plane domain.
Further, the calculation formula of curvature is：
In formula, det is determinant operation, and P ' (t) are the first derivative of P (t), and P ＂ (t) are the second dervative of P (t)；x
(t), y (t) is the coordinate value at P (t) points, and x ' (t), x ＂ (t) are respectively the first derivative and second dervative of x (t), y ' (t), y ＂
(t) be respectively y (t) first derivative and second dervative, | | P ' (t) | | be P ' (t) modular arithmetic.
The device have the advantages that：
The present invention proposes a kind of complex region parametric method of high quality, passes through curve segmentation method, curve segmentation
The concave-convex degree for reducing each section of curve afterwards ensures not will produce the network of quadrilaterals lattice model of overturning, to optimize with this
The quality of model of gear parametrization, and it is met in engineer application for waiting validity of geometrical analysis to require.Later
By the modification to model of gear Polygonal Boundary average distance value, the distance between uniform border vertices, to improve
The production rate of quadrilateral mesh.In addition, ensure that the shape of the model of gear will not change by using the method that boundary is replaced, light
Use along method and C-C divided methods is then optimized the model of gear on smoothness and continuity, to meet
Smoothness and robustness requirement of the engineer application for equal geometrical analysis.
Description of the drawings
Fig. 1 is a model of gear used in the embodiment of the present invention；
Fig. 2 is the Polygonal Boundary geometrical model of the model of gear after curved boundary conversion；
Fig. 3 is the schematic diagram of the limit curvature point and inflection point of B-spline curves；
Fig. 4 is segmentation and the control point reallocation schematic diagram of B-spline curves；
Fig. 5 is that Fig. 2 carries out the quadrilateral mesh illustraton of model after quadrangle grids；
Fig. 6 is deletion and the insertion schematic diagram of curve segmentation point；
Fig. 7 is substituted into corresponding quadrilateral mesh model boundary for the control point on the segmentation curved boundary after curve segmentation and leads to
Cross the schematic diagram on the control point that linear interpolation obtains；
Fig. 8 is the final parameterized model of Fig. 1 middle gear models.
Specific implementation mode
The curved boundary model that a gear shape is provided below in conjunction with practical application carries out region using the present invention to it
Parametrization allows that the model of gear is more applicable for and waits geometrical analysis to obtain the parameterized model needed in engineer application.
It is specific as follows based on the model of gear region parameter method that subdivision technology is replaced with boundary：
Step 1, Fig. 1 are a model of gear, are sampled to the inside and outside boundary of the model of gear, on inside and outside boundary
Since first sampled point, 4 sampled points is taken to be fitted to a cubic uniform B-splines successively every time, adjacent two three
In secondary Uniform B-Spline Curve, first sampled point that when latter cubic uniform B-splines fitting uses for previous item three times
The last one sampled point that Uniform B-Spline Curve uses when being fitted；Control point is adjusted, cubic uniform B-splines is made to be fitted tooth
Take turns the curved boundary of model；After fitting, inner boundary is made of 4 cubic uniform B-splines, and outer boundary is uniform three times by 24
B-spline curves form, and every cubic uniform B-splines are containing there are four control points；Cubic uniform B-splines expression formula is：
Wherein, PiFor the control point position for building cubic uniform B-splines, i takes 0,1,2 or 3, Ni,3(t) it is B
Spline base function, P (t) are the match point position on the cubic uniform B-splines formed, parameter t ∈ node intervals [0,1].
Step 2 passes through the inside and outside boundary through step 1 treated the model of gear curvilinear characteristic point and carries out curve minute
It cuts, then converts the inside and outside boundary of the model of gear to Polygonal Boundary geometrical model, it is specific as follows：
Step a, by the method for parabola interpolation, the inside and outside borderline curvilinear characteristic point of the model of gear, curve are detected
Characteristic point includes the point with extreme curvature and inflection point (such as Fig. 3) of curve, and curve segmentation is carried out at curvilinear characteristic point.The song of curve
Rate extreme point is the more apparent place of the concave-convex degree of curve itself, and the concave-convex degree of each section of curve is reduced after curve segmentation,
And it thereby guarantees that in the network of quadrilaterals lattice model being subsequently generated without there is the case where overturning grid (such as Fig. 5).
The process that parabola interpolation detects point with extreme curvature is specific as follows：By the inside and outside every borderline of the model of gear
The node interval [0,1] of cubic uniform B-splines is divided into 20 equal portions section [tn-1,tn], n=1,2 ..., 20, then according to
Secondary traversal [tn-1,tn+1], computation interval [tn-1,tn+1] in three match point curvature k (tn-1)、k(tn) and k (tn+1), if full
Sufficient condition k (tn)>k(tn-1) and k (tn)>k(tn+1), then current interval is section where bent curvature of a curve maximum point, and if
Meet condition k (tn)<k(tn-1) and k (tn)<k(tn+1), then current interval is section where curvature minimum point.Curvature maximum
Section where section where point or curvature minimum point passes through three point (t of interpolationn-1,k(tn-1)), (tn,k(tn)) and (tn+1,
k(tn+1)) obtain parabola f (m)=a2m2+a1m+a0, wherein a0,a1,a2For coefficient, then by setting leading for f (m)
Number f ' (m)=0 acquire the m values of the point in parabola on symmetry axis, are denoted as t '：
Using parabola peak or the limit curvature of minimum point point t ' as cubic uniform B-splines in section [tn-1,
tn] in curvature maximum point or curvature minimum point.
The process of parabola interpolation detection curve inflection point is specific as follows：By inside and outside borderline every three of the model of gear
The node interval [0,1] of secondary Uniform B-Spline Curve is divided into 20 equal portions section [tn-1,tn], [t is then traversed successivelyn-1,tn], meter
Calculate section [tn-1,tn] two endpoints curvature value, meet k (tn-1)*k(tn)<When 0, illustrate turning there are curve in the section
Point.Then midpoint (the t in the parameter section is calculatedn-0.5,k(tn-0.5)), wherein tn-0.5=1/2* (tn-1+tn).If | k
(tn-0.5)-0|<0.0001, just directly by tn-0.5As point of inflexion on a curve, otherwise, still through (tn-1,k(tn-1)), (tn-0.5,
k(tn-0.5)) and (tn,k(tn)) one parabola f (m)=b of three point interpolation2m2+b1m+b0, wherein b0,b1,b2For coefficient, f is found out
(m)=0 in section [tn-1,tn] in unique root, and using root as cubic uniform B-splines in section [tn-1,tn] in
Inflection point.
The calculation formula of curvature is：
In formula, det is determinant operation, and P ' (t) are the first derivative of P (t), and P ＂ (t) are the second dervative of P (t)；x
(t), y (t) is the coordinate value at P (t) points, and x ' (t), x ＂ (t) are respectively the first derivative and second dervative of x (t), y ' (t), y ＂
(t) be respectively y (t) first derivative and second dervative, | | P ' (t) | | be P ' (t) modular arithmetic.
Step b, as shown in figure 4, using the curvilinear characteristic point of cubic uniform B-splines in step a as cut-point, to dividing
The curve cut redistributes control point；By node interval [ta, tb] two-end-point taAnd tbIt is bent that the t substituted into formula (1) finds out segmentation
Control vertex P at line two-end-pointaAnd Pb.The curve of segmentation and the match point position correspondence of cubic uniform B-splines are such as
Under：
In formula, Pab(t ") is segmentation curve matching point position, parameter t " ∈ node intervals [ta, tb]；According to the song of segmentation
The match point position correspondence of line and cubic uniform B-splines calculates segmentation curve residue intermediate control point position difference
For：
Wherein, P0And P1For the control point position of former cubic uniform B-splines, Pa1And Pa2It is two among segmentation curve
Control point position, so just according to former cubic uniform B-splines be segmentation curve be assigned with new control point position, respectively
Pa, Pa1, Pa2, Pb。
Step 3, as shown in fig. 6, traversed first to the segmentation point sequence of acquisition, calculate between adjacent segmentation point away from
From, and calculate the average value h of each adjacent segmentation point distance.(λ is in 0 < λ < 1 when the distance between adjacent segmentation point is less than 0.5 λ h
In take a value), by the latter cut-point in adjacent segmentation point (the P points in such as Fig. 6) delete；The distance between adjacent segmentation point
When more than 1.5 λ h, subtract one as slotting after the distance between adjacent segmentation point to be added to the result rounding of 0.5 λ h resulting values divided by λ h
Enter the quantity of cut-point, to the section [t belonging to the adjacent segmentation pointn-1,tn] divided equally, to realize corresponding B three times
The segmentation of spline curve redistributes the curve of segmentation then according to the method that step b redistributes control point in step 2
Control point (the Q points in such as Fig. 6).Finally, it is sequentially connected the control point at every segmentation curve both ends, to convert curved boundary
The Polygonal Boundary geometrical model of the model of gear being made of 110 vertex is obtained, as shown in Fig. 2, its vertex distance reaches
The effect of relatively uniform distribution, while the quantity of former boundary curve has risen to 110 from 28.
The network of quadrilaterals lattice model of the model of gear is generated using quadrangle grids algorithm, and using iteration method for fairing to net
Lattice carry out SmoothNumerical TechniqueandIts, to obtain network of quadrilaterals lattice model that is uniform and smooth and having preferable robustness.It is gear as shown in Figure 5
The quadrilateral mesh model result that model generates, wherein quadrilateral mesh quantity are 196.
Step 4 controls grid model to the quadrilateral mesh model construction of the model of gear and carries out C-C sub-dividing operations, together
When by replacing boundary Control point and curve node be inserted into that treated keeping step 1 is fitted through cubic uniform B-splines
Model of gear curved boundary shape.
Step a, as shown in fig. 7, calculating each quadrilateral surface sheet border of network of quadrilaterals lattice model and interior using linear interpolation
The control point in portion includes four vertex (d of quad patch0,0、d0,3、d3,0And d3,3, wherein d0,0As P0, d0,3As P3)
Inside, the control point sequence (i.e. 4 rows 4 arrange control point) that a 4*4 is generated in each quadrilateral surface on piece, then will pass through step
The control point on segmentation curved boundary in 3 after curve segmentation is substituted into corresponding quadrilateral mesh model boundary and passes through linear interpolation
Obtained control point d0,1、d0,2On, to construct the control grid model for parametrization.
Step b, the internal vertex of control grid model is handled using C-C divided methods, generates C-C subdivided meshes models,
The effect of C-C subdivisions is tessellated mesh, improves the continuity and smoothness of grid.
Step c, the method for being inserted into node and control point using B-spline curves increases every in step 3 after curve segmentation
The number of control points of boundary curve, to match the number on C-C subdivided meshes model boundaries vertex.
After generating C-C subdivided meshes models, the number of vertices corresponding to the boundary of each quad patch increases to 7
It is a, still ensureing that knot vector is uniform, three times uniformly by every in the boundary curve after curve segmentation in step 3
B-spline curves increase by 3 nodes, while changing every cubic uniform B-spline song in the boundary curve in step 3 after curve segmentation
The control point position of centre two of line, and increase by 3 control points, the position at 7 new control points calculate separately for：
P′0=P0
P′6=P3
Wherein, Pr', r=0,1 ..., 6 be the position at new control point, Ps, s=0, after 1,2,3 is curve segmentation in step 3
Boundary curve in every cubic uniform B-splines original 4 control point positions.
It is inserted into after node, the new control point of each edge circle curve is substituted into corresponding quadrilateral mesh model boundary successively
On control point after C-C is segmented.It can realize step 1 in this way treated and be fitted through cubic uniform B-splines
The effect of model of gear curved boundary shape invariance can equally retain on the parameterized model after parametrization through uniform three times
The model of gear curved boundary shape of B-spline curves fitting.
Step 5, as shown in figure 8, the C-C subdivided meshes model that the step c in step 4 is obtained as drawing B-spline song
The control grid in face compares the network of quadrilaterals lattice model generated in step 3, and each quad patch, which corresponds to drafting one, 7 rows 7
The cubic uniform B-spline curved surface at control point is arranged to complete the parametrization of plane domain, thus obtains the parametrization of final high quality
As a result.Fig. 8 is the final parameterized model of Fig. 1 middle gear models.
We have just obtained the high quality parameterized model of gear in this way, substantially met CAD or CAE application in for
The demand of parameterized model slickness and robustness.
The present invention reduces the concave-convex degree of each section of curve by curve segmentation method after curve segmentation, ensured with this
The network of quadrilaterals lattice model that not will produce overturning to optimize the quality of model of gear parametrization, and meets it in work
For the validity requirement of equal geometrical analysis in Cheng Yingyong.Later by being repaiied to model of gear Polygonal Boundary average distance value
Change, the distance between uniform border vertices, to improve the production rate of quadrilateral mesh.In addition, being replaced by using boundary
Method ensure that the shape of the model of gear will not change, the use of method for fairing and C-C divided methods is then in smoothness and continuous
Property on the model of gear is optimized, to meet smoothness and robustness requirement of the engineer application for equal geometrical analysis.
To sum up, the present invention is satisfied high-precision and is had by the combined use of subdivision technology and boundary replacement method
Effect property require Gear Parameterization model, it is adapted to the parametrization of complex region, so as to be applied to high-level CAD or
In the research and design of CAE models.
Claims (2)
1. the model of gear region parameter method replaced with boundary based on subdivision technology, it is characterised in that：This method is specifically such as
Under：
Step 1 samples the inside and outside boundary of the model of gear, on inside and outside boundary since first sampled point, every time
4 sampled points are taken to be fitted to a cubic uniform B-splines successively, in adjacent two cubic uniform B-splines, latter item
First sampled point that cubic uniform B-splines use when being fitted uses when being fitted for previous cubic uniform B-splines
The last one sampled point；After fitting, inner boundary is made of 4 cubic uniform B-splines, and outer boundary is uniform three times by 24
B-spline curves form；Cubic uniform B-splines expression formula is：
Wherein, PiFor the control point position for building cubic uniform B-splines, i takes 0,1,2 or 3, Ni,3(t) it is B-spline base
Function, P (t) are the match point position on the cubic uniform B-splines formed, parameter t ∈ node intervals [0,1]；
Step 2 carries out curve segmentation to the inside and outside boundary through step 1 treated the model of gear by curvilinear characteristic point, so
Convert the inside and outside boundary of the model of gear to Polygonal Boundary geometrical model afterwards, it is specific as follows：
Step a, by the method for parabola interpolation, the inside and outside borderline curvilinear characteristic point of the model of gear, curvilinear characteristic are detected
Point includes the point with extreme curvature and inflection point of curve, and curve segmentation is carried out at curvilinear characteristic point；
The process that parabola interpolation detects point with extreme curvature is specific as follows：Three times by inside and outside borderline every of the model of gear
The node interval [0,1] of Uniform B-Spline Curve is divided into 20 equal portions section [tn-1,tn], n=1,2 ..., 20, then successively time
Go through [tn-1,tn+1], computation interval [tn-1,tn+1] in three match point curvature k (tn-1)、k(tn) and k (tn+1), if meeting item
Part k (tn)>k(tn-1) and k (tn)>k(tn+1), then current interval is section where bent curvature of a curve maximum point, and if met
Condition k (tn)<k(tn-1) and k (tn)<k(tn+1), then current interval is section where curvature minimum point；Curvature maximum point institute
Pass through three point (t of interpolation in section where section or curvature minimum pointn-1,k(tn-1)), (tn,k(tn)) and (tn+1,k
(tn+1)) obtain parabola f (m)=a2m2+a1m+a0, wherein a0,a1,a2For coefficient, then pass through the derivative of setting f (m)
F ' (m)=0 acquires the m values of the point in parabola on symmetry axis, is denoted as t '：
Using parabola peak or the limit curvature of minimum point point t ' as cubic uniform B-splines in section [tn-1,tn] in
Curvature maximum point or curvature minimum point；
The process of parabola interpolation detection curve inflection point is specific as follows：Three times by inside and outside borderline every of the model of gear
The node interval [0,1] of even B-spline curves is divided into 20 equal portions section [tn-1,tn], [t is then traversed successivelyn-1,tn], calculate area
Between [tn-1,tn] two endpoints curvature value, meet k (tn-1)*k(tn)<When 0, there are points of inflexion on a curve in the section；Then it counts
Calculate the midpoint (t in the parameter sectionn-0.5,k(tn-0.5)), wherein tn-0.5=1/2* (tn-1+tn)；If | k (tn-0.5)-0|<
0.0001, just directly by tn-0.5As point of inflexion on a curve, otherwise, still through (tn-1,k(tn-1)), (tn-0.5,k(tn-0.5)) and
(tn,k(tn)) one parabola f (m)=b of three point interpolation2m2+b1m+b0, wherein b0,b1,b2For coefficient, f (m)=0 is found out in area
Between [tn-1,tn] in unique root, and using root as cubic uniform B-splines in section [tn-1,tn] in inflection point；
Step b, using the curvilinear characteristic point of cubic uniform B-splines in step a as cut-point, the curve of segmentation is divided again
With control point；By node interval [ta, tb] two-end-point taAnd tbThe t substituted into formula (1) finds out the control at segmentation curve two-end-point
Vertex P processedaAnd Pb；The curve of segmentation and the match point position correspondence of cubic uniform B-splines are as follows：
In formula, Pab(t ") is segmentation curve matching point position, parameter t " ∈ node intervals [ta, tb]；According to the curve of segmentation and three
The match point position correspondence of secondary Uniform B-Spline Curve calculates segmentation curve residue intermediate control point position：
Wherein, P0And P1For the control point position of former cubic uniform B-splines, Pa1And Pa2For two controls among segmentation curve
Point position, to be that segmentation curve is assigned with new control point position, respectively P according to former cubic uniform B-splinesa, Pa1,
Pa2, Pb；
Step 3 first traverses the segmentation point sequence of acquisition, calculates the distance between adjacent segmentation point, and calculate each phase
The average value h of adjacent cut-point distance；When the distance between adjacent segmentation point is less than 0.5 λ h, wherein 0 < λ < 1, by adjacent segmentation
The latter divides point deletion in point；When the distance between adjacent segmentation point is more than 1.5 λ h, the distance between adjacent segmentation point is added
Subtract one after the result rounding of upper 0.5 λ h resulting values divided by λ h as the quantity for being inserted into cut-point, to the adjacent segmentation point institute
Section [the t of categoryn-1,tn] divided equally, to realize the segmentation of corresponding B-spline Curve, then according to step b in step 2
The method for redistributing control point redistributes control point to the curve of segmentation；Finally, every segmentation curve both ends are sequentially connected
Control point, so that curved boundary be converted to obtain the Polygonal Boundary geometrical model of the model of gear；It is calculated using quadrangle grids
Method generates the network of quadrilaterals lattice model of the model of gear, and carries out SmoothNumerical TechniqueandIts to grid using iteration method for fairing；
Step 4 controls grid model to the quadrilateral mesh model construction of the model of gear and carries out C-C sub-dividing operations, leads to simultaneously
It crosses and replaces boundary Control point and the insertion of curve node to keep step 1 tooth that treated is fitted through cubic uniform B-splines
Take turns model curved boundary shape；
Step a, the control point of network of quadrilaterals lattice model each quadrilateral surface sheet border and inside is calculated using linear interpolation, including
Four vertex of quad patch generate the control point sequence of a 4*4 in each quadrilateral surface on piece, then will pass through step
The control point on segmentation curved boundary in 3 after curve segmentation is substituted into corresponding quadrilateral mesh model boundary and passes through linear interpolation
On obtained control point, to construct the control grid model for parametrization；
Step b, the internal vertex of control grid model is handled using C-C divided methods, generates C-C subdivided meshes models；
Step c, the method for being inserted into node and control point using B-spline curves increases each edge circle after curve segmentation in step 3
The number of control points of curve, it is specific as follows to match the number on C-C subdivided meshes model boundaries vertex：
After generating C-C subdivided meshes models, the number of vertices corresponding to the boundary of each quad patch increases to 7,
In the case of still ensureing that knot vector is uniform, by every cubic uniform B-spline in the boundary curve after curve segmentation in step 3
Curve increases by 3 nodes, while changing in the boundary curve in step 3 after curve segmentation every cubic uniform B-splines
Intermediate two control point positions, and increase by 3 control points, the position at 7 new control points calculate separately for：
P′0=P0
P′6=P3
Wherein, Pr', r=0,1 ..., 6 be the position at new control point, Ps, s=0,1,2,3 is the side after curve segmentation in step 3
Original 4 control point positions of every cubic uniform B-splines in boundary's curve；
It is inserted into after node, the new control point of each edge circle curve is substituted into corresponding quadrilateral mesh model boundary successively to be passed through
On control point after C-C subdivisions；
Step 5, the C-C subdivided meshes model for obtaining the step c in step 4 are right as the control grid for drawing B-spline surface
According to the network of quadrilaterals lattice model generated in step 3, each quad patch, which corresponds to drafting one, has 7 rows 7 row control point three times
Uniform B-spline surfaces complete the parametrization of plane domain.
2. the model of gear region parameter method according to claim 1 replaced with boundary based on subdivision technology, special
Sign is：The calculation formula of curvature is：
In formula, det is determinant operation, and P ' (t) are the first derivative of P (t), and P ＂ (t) are the second dervative of P (t)；X (t), y
(t) it is the coordinate value at P (t) points, x ' (t), x ＂ (t) are respectively the first derivative and second dervative of x (t), and y ' (t), y ＂ (t) divide
Not Wei y (t) first derivative and second dervative, | | P ' (t) | | be P ' (t) modular arithmetic.
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