CN108763668B - Gear model region parameterization method based on subdivision technology and boundary replacement - Google Patents

Abstract

The invention discloses a gear model region parameterization method based on subdivision technology and boundary replacement. High quality parameterization of graphical models with complex regions is difficult to achieve. The invention reduces the concave-convex degree of each section of curve by a curve segmentation method, ensures that a turnover quadrilateral mesh model cannot be generated, thereby optimizing the parameterization quality of the gear model and meeting the validity requirement of the gear model on isogeometric analysis in engineering application; then, the distances between the vertexes of the boundaries are homogenized by modifying the average distance value of the polygonal boundaries of the gear model, so that the generation rate of the quadrilateral meshes is improved; the shape of the gear model is ensured not to be changed by using a boundary replacement method, and the gear model is optimized on the aspects of smoothness and continuity by using a fairing method and a C-C subdivision method, so that the requirements of engineering application on the smoothness and robustness of isogeometric analysis are met.

Description

Gear model region parameterization method based on subdivision technology and boundary replacement

Technical Field

The invention belongs to the field of computer aided design and engineering application, and particularly relates to a gear model region parameterization method based on subdivision technology and boundary replacement.

Background

Computer Aided Design (CAD) technology and Computer Aided Engineering (CAE) technology have a profound impact in today's computer applications and development, and they are widely used in aerospace, automotive manufacturing, architectural design, mechatronics, and other fields. In CAD techniques, the shape of an object is represented by a boundary model, which contains information about the internal surface of the geometric model. Isogeometric analysis often requires parameterization within the geometric model region, and therefore, in order to make a CAD model more suitable for isogeometric analysis, it is necessary to study the technique of constructing a parameterization of the region from given boundary data.

The isogeometric analysis is a novel technology for carrying out analog simulation based on the accurate geometric representation of a CAD model, opens up a new way for realizing the seamless fusion of CAD/CAE, and simultaneously injects new vitality into the mature geometric modeling field. In isogeometric analysis, parameterization of a computational domain is still one of the current research hotspots, and a high-precision and efficient parameterized model is often required in finite element analysis so as to be applied to research design of a high-level CAD or CAE model. To meet these requirements, the mesh used for parameterization and some special areas need to be optimized, and for a certain method, the complexity of the area greatly influences the effectiveness and quality of parameterization.

Parameterization of a region in isogeometric analysis needs to satisfy three requirements: 1) it does not self-cross, i.e. the mapping from the parameter domain to the physical domain is injective; 2) the isoparametric should be as homogeneous as possible; 3) the isoparametric elements should be as orthogonal as possible. Currently, there are many methods that can process boundary information represented by B-spline curves, including a variation harmonic method, a divide and conquer technique, a mapping method, a contour-based decomposition method, a multi-surface piece parameterization method, and a non-standard B-spline parameterization method. However, most of the existing methods can only process parameterization of some simple regions, and for some graphic models with complex regions, high-quality parameterization is difficult to achieve. For example, when some grid regions with large unevenness are processed, an inverted grid is easily formed at the grid regions, and the quality of regional parameterization is seriously affected.

Disclosure of Invention

The invention aims to provide a gear model region parameterization method based on a subdivision technology and boundary replacement aiming at the defects of the existing method.

The invention specifically comprises the following steps:

step 1, sampling an inner boundary and an outer boundary of a gear model, wherein the inner boundary and the outer boundary are sequentially provided with 4 sampling points from a first sampling point each time to be fitted into a cubic uniform B spline curve, and in two adjacent cubic uniform B spline curves, the first sampling point adopted in the fitting of the next cubic uniform B spline curve is the last sampling point adopted in the fitting of the previous cubic uniform B spline curve; after fitting, the inner boundary consists of 4 cubic uniform B spline curves, and the outer boundary consists of 24 cubic uniform B spline curves; the cubic uniform B-spline curve expression is:

wherein, P_iFor the control point positions for constructing cubic uniform B-spline curves, i is 0,1,2 or 3, N_i,3(t) is a B spline basis function, P (t) is the position of a fitting point on a formed cubic uniform B spline curve, and a parameter t belongs to a node interval [0,1 ∈]。

Step 2, performing curve segmentation on the inner boundary and the outer boundary of the gear model processed in the step 1 through curve characteristic points, and then converting the inner boundary and the outer boundary of the gear model into a polygonal boundary geometric model, which is concretely as follows:

step a, detecting curve characteristic points on the inner boundary and the outer boundary of the gear model by a parabolic interpolation method, wherein the curve characteristic points comprise curvature extreme points and inflection points of a curve, and performing curve segmentation at the curve characteristic points.

The process of detecting curvature extreme points by parabolic interpolation is specifically as follows: the node interval [0,1 ] of each cubic uniform B-spline curve on the inner boundary and the outer boundary of the gear model is divided into three]Are equally divided into 20 equal intervals [ t ]_n-1,t_n]N is 1,2, …, 20, and then sequentially traversed [ t ═ t_n-1,t_n+1]Calculating the interval [ t_n-1,t_n+1]Inner three fitting point curvatures k (t)_n-1)、k(t_n) And k (t)_n+1) If the condition k is satisfied (t_n)>k(t_n-1) And k (t)_n)>k(t_n+1) The current interval is the interval where the curvature maximum point of the curve is located, and if the condition k (t) is satisfied_n)<k(t_n-1) And k (t)_n)<k(t_n+1) And if so, the current interval is the interval where the curvature minimum value point is located. The interval of the curvature maximum value point or the interval of the curvature minimum value point passes through three interpolation points (t)_n-1,k(t_n-1))，(t_n,k(t_n) Are a and (t)_n+1,k(t_n+1) A parabola f (m) ═ a) is obtained₂m²+a₁m+a₀Wherein a is₀,a₁,a₂For the coefficients, then by setting the derivative f '(m) of f (m) to 0, the value of m, denoted t', for the point on the axis of symmetry in the parabola is found:

taking the limit curvature point t' of the highest point or the lowest point of the parabola as a cubic uniform B-spline curve in the interval [ t_n-1,t_n]An inner curvature maxima point or a curvature minima point.

The process of detecting the curve inflection point by parabolic interpolation specifically comprises the following steps: the node interval [0,1 ] of each cubic uniform B-spline curve on the inner boundary and the outer boundary of the gear model is divided into three]Are equally divided into 20 equal intervals [ t ]_n-1,t_n]Then sequentially traverse [ t_n-1,t_n]Calculating the interval [ t_n-1,t_n]Curvature values of both end points, satisfy k (t)_n-1)*k(t_n)<At 0, there is an inflection point in the curve in this interval. Then, the midpoint (t) of the parameter interval is calculated_n-0.5,k(t_n-0.5) In which t) is_n-0.5＝1/2*(t_n-1+t_n). If | k (t)_n-0.5)-0|<0.0001, directly adding t_n-0.5As an inflection point of the curve, otherwise, still passes (t)_n-1,k(t_n-1))，(t_n-0.5,k(t_n-0.5) Are a and (t)_n,k(t_n) A parabola f (m) ═ b of three-point interpolation₂m²+b₁m+b₀Wherein b is₀,b₁,b₂For the coefficient, find f (m) 0 in the interval [ t-_n-1,t_n]The only root value in the interval [ t ] is used as a cubic uniform B spline curve_n-1,t_n]An inner inflection point.

Step B, taking the curve characteristic points of the cubic uniform B-spline curve in the step a as segmentation points, and redistributing control points to the segmented curve; will node interval t_a，t_b]Two end points t_aAnd t_bSubstituting t in formula (1) to obtain control vertexes P at two end points of the division curve_aAnd P_b. The corresponding relation of the fitting point positions of the segmented curve and the cubic uniform B-spline curve is as follows:

in the formula, P_ab(t ') is the fitting point position of the segmentation curve, and the parameter t' belongs to the node interval [ t_a，t_b](ii) a Calculating the positions of the remaining intermediate control points of the segmented curve according to the corresponding relation of the fitting point positions of the segmented curve and the cubic uniform B-spline curve as follows:

wherein, P₀And P₁Is the control point position, P, of the original cubic uniform B-spline curve_a1And P_a2Two control point positions in the middle of the segmentation curve are allocated, so that new control point positions are allocated to the segmentation curve according to the original cubic uniform B spline curve, and the new control point positions are respectively P_a，P_a1，P_a2，P_b。

Step 3, firstly traversing the obtained segmentation point sequence, calculating the distance between adjacent segmentation points, and calculating the average of the distances between the adjacent segmentation pointsThe mean value h. When the distance between the adjacent segmentation points is less than 0.5 lambdah, wherein 0 is more than lambdah and less than 1, deleting the next segmentation point in the adjacent segmentation points; when the distance between adjacent division points is greater than 1.5 lambdah, the distance between adjacent division points is added with 0.5 lambdah, the result of dividing the result by lambdah is rounded and then is subtracted by one to obtain the number of inserted division points, thereby the interval [ t ] of the adjacent division points is_n-1,t_n]And (c) carrying out equipartition so as to realize the segmentation of the corresponding cubic B-spline curve, and then reallocating the control points to the segmented curve according to the method for reallocating the control points in the step (B) in the step (2). Finally, the control points at the two ends of each segmentation curve are connected in sequence, so that the curve boundary is converted to obtain a polygonal boundary geometric model of the gear model; and generating a quadrilateral mesh model of the gear model by using a quadrilateral subdivision algorithm, and carrying out smooth treatment on the mesh by using an iterative smooth treatment method.

And 4, constructing a control grid model for the quadrilateral grid model of the gear model, performing C-C subdivision operation, and simultaneously keeping the curve boundary shape of the gear model subjected to cubic uniform B-spline curve fitting and processed in the step 1 by replacing boundary control points and inserting curve nodes.

Step a, calculating each quadrilateral patch boundary and internal control points of the quadrilateral mesh model by using linear interpolation, wherein the control points comprise four vertexes of the quadrilateral patch, generating a 4 x 4 control point sequence on each quadrilateral patch, and then replacing the control points on the segmentation curve boundary divided by the curve in the step 3 to the control points obtained by linear interpolation of the corresponding quadrilateral mesh model boundary, thereby constructing the control mesh model for parameterization.

And b, processing the internal vertex of the control mesh model by using a C-C subdivision method to generate a C-C subdivision mesh model.

Step C, using a method of inserting nodes and control points by using a B spline curve, increasing the number of the control points of each boundary curve after curve segmentation in the step 3 to match the number of the boundary vertexes of the C-C subdivision grid model, and specifically as follows:

after a C-C subdivision grid model is generated, the number of vertexes corresponding to the boundary of each quadrilateral surface patch is increased to 7, 3 nodes are added to each cubic uniform B spline curve in the boundary curve obtained after curve segmentation in the step 3 under the condition that the node vectors are still ensured to be uniform, meanwhile, the positions of two control points in the middle of each cubic uniform B spline curve in the boundary curve obtained after curve segmentation in the step 3 are changed, 3 control points are added, and the positions of 7 new control points are respectively calculated as:

P′₀＝P₀

P′₆＝P₃

wherein, P_r' r is 0,1, …,6 is the position of the new control point, P_sAnd s is 0,1,2 and 3, which are the original 4 control point positions of each cubic uniform B-spline in the boundary curve after curve segmentation in the step 3.

And after the nodes are inserted, sequentially replacing the new control points of each boundary curve to the control points of the corresponding quadrilateral mesh model boundary after C-C subdivision.

And 5, taking the C-C subdivision mesh model obtained in the step 4 as a control mesh for drawing the B-spline surface, and drawing a cubic uniform B-spline surface with 7 rows and 7 columns of control points corresponding to each quadrilateral surface patch to complete parameterization of the plane area by contrasting the quadrilateral mesh model generated in the step 3.

Further, the formula for calculating the curvature is:

where det is the determinant operation, P '(t) is the first derivative of P (t), and P' (t) is the second derivative of P (t); x (t), y (t) is the coordinate value at point P (t), x '(t), x "(t) is the first derivative and the second derivative of x (t), y' (t), y" (t) is the first derivative and the second derivative of y (t), and | P '(t) | | is the modulo operation of P' (t).

The invention has the following beneficial effects:

the invention provides a high-quality complex region parameterization method, which reduces the concave-convex degree of each section of curve after curve segmentation by a curve segmentation method so as to ensure that a turnover quadrilateral mesh model cannot be generated, thereby optimizing the parameterization quality of a gear model and meeting the requirement of the parameterization on isogeometric analysis in engineering application. And then, the distances between the vertexes of the boundaries are uniformed by modifying the average distance value of the polygon boundaries of the gear model, so that the generation rate of the quadrilateral meshes is improved. In addition, the shape of the gear model is guaranteed not to be changed by using a boundary replacement method, and the gear model is optimized on the aspects of smoothness and continuity by using a fairing method and a C-C subdivision method, so that the requirements of engineering application on the smoothness and robustness of isogeometric analysis are met.

Drawings

FIG. 1 is a gear model employed in embodiments of the present invention;

FIG. 2 is a polygonal boundary geometry model of the gear model after curve boundary translation;

FIG. 3 is a schematic diagram of the extreme curvature points and inflection points of a B-spline curve;

FIG. 4 is a schematic diagram of segmentation and control point redistribution of a B-spline curve;

FIG. 5 is a diagram of a quadrilateral mesh model after quadrilateral subdivision in FIG. 2;

FIG. 6 is a schematic diagram of deletion and insertion of curve split points;

FIG. 7 is a schematic diagram of replacement of control points on a segmented curve boundary after curve segmentation onto control points obtained by linear interpolation of a corresponding quadrilateral mesh model boundary;

fig. 8 is a final parameterized model of the gear model of fig. 1.

Detailed Description

A curve boundary model of a gear shape is given by combining practical application, and the gear shape is subjected to regional parameterization by using the method, so that a parameterized model required in engineering application is obtained, and the gear model is more suitable for isogeometric analysis.

A gear model region parameterization method based on subdivision technology and boundary replacement specifically comprises the following steps:

step 1, as shown in fig. 1, a gear model is adopted, an inner boundary and an outer boundary of the gear model are sampled, the inner boundary and the outer boundary are respectively provided with a first sampling point, 4 sampling points are sequentially taken each time to be fitted into a cubic uniform B spline curve, and in two adjacent cubic uniform B spline curves, the first sampling point adopted in the fitting of the latter cubic uniform B spline curve is the last sampling point adopted in the fitting of the former cubic uniform B spline curve; adjusting the control points to enable the cubic uniform B-spline curve to fit the curve boundary of the gear model; after fitting, the inner boundary consists of 4 cubic uniform B spline curves, the outer boundary consists of 24 cubic uniform B spline curves, and each cubic uniform B spline curve contains four control points; the cubic uniform B-spline curve expression is:

wherein, P_iFor the control point positions for constructing cubic uniform B-spline curves, i is 0,1,2 or 3, N_i,3(t) is a B spline basis function, P (t) is the position of a fitting point on a formed cubic uniform B spline curve, and a parameter t belongs to a node interval [0,1 ∈]。

Step 2, performing curve segmentation on the inner boundary and the outer boundary of the gear model processed in the step 1 through curve characteristic points, and then converting the inner boundary and the outer boundary of the gear model into a polygonal boundary geometric model, which is concretely as follows:

step a, detecting curve characteristic points on the inner boundary and the outer boundary of the gear model by a parabolic interpolation method, wherein the curve characteristic points comprise curvature extreme points and inflection points (shown in figure 3) of a curve, and performing curve segmentation at the curve characteristic points. The curvature extreme point of the curve is a place where the concave-convex degree of the curve is obvious, the concave-convex degree of each section of the curve is reduced after the curve is divided, and therefore the situation that the turnover grids do not exist in a subsequently generated quadrilateral grid model is guaranteed (as shown in fig. 5).

The process of detecting curvature extreme points by parabolic interpolation is specifically as follows: the node interval [0,1 ] of each cubic uniform B-spline curve on the inner boundary and the outer boundary of the gear model is divided into three]Are equally divided into 20 equal intervals [ t ]_n-1,t_n]N is 1,2, …, 20, and then sequentially traversed [ t ═ t_n-1,t_n+1]Calculating the interval [ t_n-1,t_n+1]Inner three fitting point curvatures k (t)_n-1)、k(t_n) And k (t)_n+1) If the condition k (t) is satisfied_n)>k(t_n-1) And k (t)_n)>k(t_n+1) The current interval is the interval where the curvature maximum point of the curve is located, and if the condition k (t) is satisfied_n)<k(t_n-1) And k (t)_n)<k(t_n+1) And if so, the current interval is the interval where the curvature minimum value point is located. The interval of the curvature maximum value point or the interval of the curvature minimum value point passes through three interpolation points (t)_n-1,k(t_n-1))，(t_n,k(t_n) Are a and (t)_n+1,k(t_n+1) A parabola f (m) ═ a) is obtained₂m²+a₁m+a₀Wherein a is₀,a₁,a₂For the coefficients, then by setting the derivative f '(m) of f (m) to 0, the value of m, denoted t', for the point on the axis of symmetry in the parabola is found:

taking the limit curvature point t' of the highest point or the lowest point of the parabola as a cubic uniform B-spline curve in the interval [ t_n-1,t_n]An inner curvature maxima point or a curvature minima point.

The process of detecting the curve inflection point by parabolic interpolation specifically comprises the following steps: the node interval [0,1 ] of each cubic uniform B-spline curve on the inner boundary and the outer boundary of the gear model is divided into three]Are equally divided into 20 equal intervals [ t ]_n-1,t_n]Then sequentially traverse [ t_n-1,t_n]Calculating the interval [ t_n-1,t_n]Curvature values of both end points, satisfy k (t)_n-1)*k(t_n)<When 0, it indicates that there is an inflection point in the curve in this interval. Then, the midpoint (t) of the parameter interval is calculated_n-0.5,k(t_n-0.5) In which t) is_n-0.5＝1/2*(t_n-1+t_n). If | k (t)_n-0.5)-0|<0.0001, directly adding t_n-0.5As an inflection point of the curve, otherwise, still passes (t)_n-1,k(t_n-1))，(t_n-0.5,k(t_n-0.5) Are a and (t)_n,k(t_n) A parabola f (m) ═ b of three-point interpolation₂m²+b₁m+b₀Wherein b is₀,b₁,b₂For the coefficient, find f (m) 0 in the interval [ t-_n-1,t_n]The only root value in the interval [ t ] is used as a cubic uniform B spline curve_n-1,t_n]An inner inflection point.

The formula for calculating the curvature is:

where det is the determinant operation, P '(t) is the first derivative of P (t), and P' (t) is the second derivative of P (t); x (t), y (t) is the coordinate value at point P (t), x '(t), x "(t) is the first derivative and the second derivative of x (t), y' (t), y" (t) is the first derivative and the second derivative of y (t), and | P '(t) | | is the modulo operation of P' (t).

Step b, as shown in FIG. 4, the three times of the step a are averagedTaking curve characteristic points of the uniform B spline curve as dividing points, and redistributing control points to the divided curves; will node interval t_a，t_b]Two end points t_aAnd t_bSubstituting t in formula (1) to obtain control vertexes P at two end points of the division curve_aAnd P_b. The corresponding relation of the fitting point positions of the segmented curve and the cubic uniform B-spline curve is as follows:

in the formula, P_ab(t ') is the fitting point position of the segmentation curve, and the parameter t' belongs to the node interval [ t_a，t_b](ii) a Calculating the positions of the remaining intermediate control points of the segmented curve according to the corresponding relation of the fitting point positions of the segmented curve and the cubic uniform B-spline curve as follows:

wherein, P₀And P₁Is the control point position, P, of the original cubic uniform B-spline curve_a1And P_a2Two control point positions in the middle of the segmentation curve are allocated, so that new control point positions are allocated to the segmentation curve according to the original cubic uniform B spline curve, and are respectively P_a，P_a1，P_a2，P_b。

And step 3, as shown in fig. 6, traversing the obtained segmentation point sequence, calculating the distance between adjacent segmentation points, and calculating the average value h of the distances between the adjacent segmentation points. Deleting the next division point (such as the point P in the figure 6) in the adjacent division points when the distance between the adjacent division points is less than 0.5 lambdah (the value of the lambda is taken from 0 < lambada < 1); when the distance between adjacent division points is greater than 1.5 lambdoh, the value obtained by adding 0.5 lambdoh to the distance between adjacent division points is divided by lambdoh, and the result is rounded and then is reduced by oneAs the number of inserted division points, the interval [ t ] to which the adjacent division point belongs_n-1,t_n]And (c) carrying out equipartition so as to realize the segmentation of the corresponding cubic B-spline curve, and then reallocating control points (such as points Q in figure 6) to the segmented curve according to the method of reallocating control points in the step B in the step 2. Finally, the control points at the two ends of each segmentation curve are connected in sequence, so that the curve boundary is converted to obtain a polygonal boundary geometric model of the gear model, wherein the polygonal boundary geometric model is composed of 110 vertexes, as shown in fig. 2, the vertex distances of the polygonal geometric model are relatively uniformly distributed, and the number of the original boundary curves is increased from 28 to 110.

And generating a quadrilateral mesh model of the gear model by using a quadrilateral subdivision algorithm, and performing smooth treatment on the mesh by using an iterative smooth method, thereby obtaining a uniform and smooth quadrilateral mesh model with better robustness. The result of the quadrilateral mesh model generated for the gear model is shown in fig. 5, in which the number of quadrilateral meshes is 196.

And 4, constructing a control grid model for the quadrilateral grid model of the gear model, performing C-C subdivision operation, and simultaneously keeping the curve boundary shape of the gear model subjected to cubic uniform B-spline curve fitting and processed in the step 1 by replacing boundary control points and inserting curve nodes.

Step a, as shown in fig. 7, calculating the control points at the boundary and inside of each quadrilateral patch of the quadrilateral mesh model by using linear interpolation, including four vertices (d) of the quadrilateral patch_0,0、d_0,3、d_3,0And d_3,3Wherein d is_0,0Is namely P₀，d_0,3Is namely P₃) In the method, a 4 x 4 control point sequence (namely 4 rows and 4 columns of control points) is generated on each quadrilateral surface, and then the control points on the boundary of the segmentation curve segmented by the curve in the step 3 are replaced to the control points d obtained by linear interpolation of the boundary of the corresponding quadrilateral mesh model_0,1、d_0,2And constructing a control mesh model for parameterization.

And b, processing the internal vertex of the control grid model by using a C-C subdivision method to generate a C-C subdivision grid model, wherein the C-C subdivision has the function of refining the grid and improving the continuity and smoothness of the grid.

And step C, increasing the number of the control points of each boundary curve after the curve is segmented in the step 3 by using a method of inserting the nodes and the control points by using the B spline curve so as to match the number of the boundary top points of the C-C subdivision grid model.

After a C-C subdivision grid model is generated, the number of vertexes corresponding to the boundary of each quadrilateral surface patch is increased to 7, 3 nodes are added to each cubic uniform B spline curve in the boundary curve obtained after curve segmentation in the step 3 under the condition that the node vectors are still ensured to be uniform, meanwhile, the positions of two control points in the middle of each cubic uniform B spline curve in the boundary curve obtained after curve segmentation in the step 3 are changed, 3 control points are added, and the positions of 7 new control points are respectively calculated as:

P′₀＝P₀

P′₆＝P₃

wherein, P_r' r is 0,1, …,6 is the position of the new control point, P_sAnd s is 0,1,2 and 3 which are the original 4 control point positions of each cubic uniform B spline curve in the boundary curve after the curve segmentation in the step 3And (4) placing.

And after the nodes are inserted, sequentially replacing the new control points of each boundary curve to the control points of the corresponding quadrilateral mesh model boundary after C-C subdivision. Therefore, the effect that the shape of the gear model curve boundary subjected to the cubic uniform B-spline curve fitting and processed in the step 1 is unchanged can be achieved, and the shape of the gear model curve boundary subjected to the cubic uniform B-spline curve fitting can be retained on the parameterized model.

And 5, as shown in fig. 8, taking the C-C subdivision grid model obtained in the step 4 as a control grid for drawing the B-spline surface, and drawing a cubic uniform B-spline surface with 7 rows and 7 columns of control points corresponding to each quadrilateral surface patch to complete parameterization of the plane area by contrasting the quadrilateral grid model generated in the step 3, thereby obtaining a final high-quality parameterization result. Fig. 8 is the final parameterized model of the gear model of fig. 1.

In this way we have a high quality parameterized model of the gear that substantially meets the requirements for smoothness and robustness of the parameterized model in CAD or CAE applications.

The invention reduces the concave-convex degree of each section of curve after curve segmentation by a curve segmentation method so as to ensure that a turnover quadrilateral mesh model cannot be generated, thereby optimizing the quality of gear model parameterization and meeting the requirement of the gear model on effectiveness of isogeometric analysis in engineering application. And then, the distances between the vertexes of the boundaries are uniformed by modifying the average distance value of the polygon boundaries of the gear model, so that the generation rate of the quadrilateral meshes is improved. In addition, the shape of the gear model is guaranteed not to be changed by using a boundary replacement method, and the gear model is optimized on the aspects of smoothness and continuity by using a fairing method and a C-C subdivision method, so that the requirements of engineering application on the smoothness and robustness of isogeometric analysis are met.

In conclusion, the gear parameterization model meeting the requirements of high precision and effectiveness is obtained by combining the subdivision technology and the boundary replacement method, and the gear parameterization model can adapt to parameterization of a complex region, so that the gear parameterization model can be applied to research and design of a high-level CAD (computer-aided design) or CAE (computer aided engineering) model.

Claims (2)

1. A gear model region parameterization method based on subdivision technology and boundary replacement is characterized by comprising the following steps: the method comprises the following specific steps:

step 1, sampling an inner boundary and an outer boundary of a gear model, wherein the inner boundary and the outer boundary are sequentially provided with 4 sampling points from a first sampling point each time to be fitted into a cubic uniform B spline curve, and in two adjacent cubic uniform B spline curves, the first sampling point adopted in the fitting of the next cubic uniform B spline curve is the last sampling point adopted in the fitting of the previous cubic uniform B spline curve; after fitting, the inner boundary consists of 4 cubic uniform B spline curves, and the outer boundary consists of 24 cubic uniform B spline curves; the cubic uniform B-spline curve expression is:

wherein, P_iFor the control point positions for constructing cubic uniform B-spline curves, i is 0,1,2 or 3, N_i,3(t) is a B spline basis function, P (t) is the position of a fitting point on a formed cubic uniform B spline curve, and a parameter t belongs to a node interval [0,1 ∈]；

Step 2, performing curve segmentation on the inner boundary and the outer boundary of the gear model processed in the step 1 through curve characteristic points, and then converting the inner boundary and the outer boundary of the gear model into a polygonal boundary geometric model, which is concretely as follows:

step a, detecting curve characteristic points on the inner boundary and the outer boundary of the gear model by a parabolic interpolation method, wherein the curve characteristic points comprise curvature extreme points and inflection points of a curve, and performing curve segmentation at the curve characteristic points;

the process of detecting curvature extreme points by parabolic interpolation is specifically as follows: the node interval [0,1 ] of each cubic uniform B-spline curve on the inner boundary and the outer boundary of the gear model is divided into three]Are equally divided into 20 equal intervals [ t ]_n-1,t_n]N is 1,2, …, 20, and then sequentially traversed [ t ═ t_n-1,t_n+1]Calculating the interval [ t_n-1,t_n+1]Inner three fitting point curvatures k (t)_n-1)、k(t_n) And k (t)_n+1) If the condition k (t) is satisfied_n)>k(t_n-1) And k (t)_n)>k(t_n+1) The current interval is the interval where the curvature maximum point of the curve is located, and if the condition k (t) is satisfied_n)<k(t_n-1) And k (t)_n)<k(t_n+1) If the current interval is the interval where the curvature minimum value point is located; the interval of the curvature maximum value point or the interval of the curvature minimum value point passes through three interpolation points (t)_n-1,k(t_n-1))，(t_n,k(t_n) Are a and (t)_n+1,k(t_n+1) A parabola f (m) ═ a) is obtained₂m²+a₁m+a₀Wherein a is₀,a₁,a₂For the coefficients, then by setting the derivative f '(m) of f (m) to 0, the value of m, denoted t', for the point on the axis of symmetry in the parabola is found:

taking the limit curvature point t' of the highest point or the lowest point of the parabola as a cubic uniform B-spline curve in the interval [ t_n-1,t_n]An inner curvature maximum point or curvature minimum point;

the process of detecting the curve inflection point by parabolic interpolation specifically comprises the following steps: the node interval [0,1 ] of each cubic uniform B-spline curve on the inner boundary and the outer boundary of the gear model is divided into three]Are equally divided into 20 equal intervals [ t ]_n-1,t_n]Then sequentially traverse [ t_n-1,t_n]Calculating the interval [ t_n-1,t_n]Curvature values of both end points, satisfy k (t)_n-1)*k(t_n)<When 0, the inflection point of the curve exists in the interval; then, the midpoint (t) of the parameter interval is calculated_n-0.5,k(t_n-0.5) In which t) is_n-0.5＝1/2*(t_n-1+t_n) (ii) a If | k (t)_n-0.5)-0|<0.0001, directly adding t_n-0.5As an inflection point of the curve, otherwise, still passes (t)_n-1,k(t_n-1))，(t_n-0.5,k(t_n-0.5) Are a and (t)_n,k(t_n) A parabola f (m) ═ b of three-point interpolation₂m²+b₁m+b₀Wherein b is₀,b₁,b₂For the coefficient, find f (m) 0 in the interval [ t-_n-1,t_n]The only root value in the interval [ t ] is used as a cubic uniform B spline curve_n-1,t_n]An inner inflection point;

step B, taking the curve characteristic points of the cubic uniform B-spline curve in the step a as segmentation points, and redistributing control points to the segmented curve; will node interval t_a，t_b]Two end points t_aAnd t_bSubstituting t in formula (1) to obtain control vertexes P at two end points of the division curve_aAnd P_b(ii) a The corresponding relation of the fitting point positions of the segmented curve and the cubic uniform B-spline curve is as follows:

in the formula, P_ab(t ') is the fitting point position of the segmentation curve, and the parameter t' belongs to the node interval [ t_a，t_b](ii) a Calculating the positions of the remaining intermediate control points of the segmented curve according to the corresponding relation of the fitting point positions of the segmented curve and the cubic uniform B-spline curve as follows:

wherein, P₀And P₁Is the control point position, P, of the original cubic uniform B-spline curve_a1And P_a2Two control point positions in the middle of the segmentation curve are allocated, so that new control point positions are allocated to the segmentation curve according to the original cubic uniform B spline curve, and the new control point positions are respectively P_a，P_a1，P_a2，P_b；

Step 3, firstly traversing the obtained segmentation point sequence, calculating the distance between adjacent segmentation points, and calculating the average value h of the distance between each adjacent segmentation point; when the distance between the adjacent segmentation points is less than 0.5 lambdah, wherein 0 is more than lambdah and less than 1, deleting the next segmentation point in the adjacent segmentation points; when the distance between adjacent division points is greater than 1.5 lambdah, the distance between adjacent division points is added with 0.5 lambdah, the result of dividing the result by lambdah is rounded and then is subtracted by one to obtain the number of inserted division points, thereby the interval [ t ] of the adjacent division points is_n-1,t_n]Carrying out equipartition so as to realize the segmentation of the corresponding cubic B-spline curve, and then reallocating control points to the segmented curve according to the method of reallocating control points in the step B in the step 2; finally, the control points at the two ends of each segmentation curve are connected in sequence, so that the curve boundary is converted to obtain a polygonal boundary geometric model of the gear model; generating a quadrilateral mesh model of the gear model by using a quadrilateral subdivision algorithm, and carrying out smooth treatment on the mesh by using an iterative smooth treatment method;

step 4, constructing a control grid model for the quadrilateral grid model of the gear model, performing C-C subdivision operation, and simultaneously keeping the curve boundary shape of the gear model which is processed in the step 1 and is subjected to cubic uniform B-spline curve fitting through replacing boundary control points and curve node insertion;

step a, calculating each quadrilateral patch boundary and internal control points of the quadrilateral mesh model by using linear interpolation, wherein the control points comprise four vertexes of the quadrilateral patch, generating a 4 x 4 control point sequence on each quadrilateral patch, and then replacing the control points on the segmentation curve boundary divided by the curve in the step 3 to the control points obtained by linear interpolation of the corresponding quadrilateral mesh model boundary, thereby constructing a control mesh model for parameterization;

b, processing the internal vertex of the control mesh model by using a C-C subdivision method to generate a C-C subdivision mesh model;

step C, using a method of inserting nodes and control points by using a B spline curve, increasing the number of the control points of each boundary curve after curve segmentation in the step 3 to match the number of the boundary vertexes of the C-C subdivision grid model, and specifically as follows:

after a C-C subdivision grid model is generated, the number of vertexes corresponding to the boundary of each quadrilateral surface patch is increased to 7, 3 nodes are added to each cubic uniform B spline curve in the boundary curve obtained after curve segmentation in the step 3 under the condition that the node vectors are still ensured to be uniform, meanwhile, the positions of two control points in the middle of each cubic uniform B spline curve in the boundary curve obtained after curve segmentation in the step 3 are changed, 3 control points are added, and the positions of 7 new control points are respectively calculated as:

P′₀＝P₀

P′₆＝P₃

wherein, P_r' r is 0,1, …,6 is the position of the new control point, P_sS is 0,1,2,3, which is the original 4 control point positions of each cubic uniform B-spline curve in the boundary curve after curve segmentation in step 3;

after inserting the nodes, sequentially replacing the new control points of each boundary curve to the control points of the corresponding quadrilateral mesh model boundary after C-C subdivision;

and 5, taking the C-C subdivision mesh model obtained in the step 4 as a control mesh for drawing the B-spline surface, and drawing a cubic uniform B-spline surface with 7 rows and 7 columns of control points corresponding to each quadrilateral surface patch to complete parameterization of the plane area by contrasting the quadrilateral mesh model generated in the step 3.

2. The subdivision technique and boundary replacement based gear model region parameterization method according to claim 1, characterized in that: the formula for calculating the curvature is:

where det is the determinant operation, P '(t) is the first derivative of P (t), and P' (t) is the second derivative of P (t); x (t), y (t) is the coordinate value at point P (t), x '(t), x "(t) is the first derivative and the second derivative of x (t), y' (t), y" (t) is the first derivative and the second derivative of y (t), and | P '(t) | | is the modulo operation of P' (t).

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