WO2025065830A1 - Isogeometric analysis parameterization migration method based on discrete geometric mapping - Google Patents

Abstract

Disclosed in the present invention is an isogeometric analysis parameterization migration method based on discrete geometric mapping. The method comprises: first, classifying a given feature constraint; then, on the basis of a classification result, if high-quality parameterization cannot be generated by means of directly moving the boundary of a patch where the constraint is located, segmenting the patch where the constraint is located into a plurality of sub-patches, such that the constraint is located at the boundary of a new sub-patch; then, performing radial basis function interpolation, and moving a constraint point or endpoints of a constraint line to the boundary of the patch; and finally, in the case where feature points or lines cause cracks in the patches, using an optimization method to perform local parameterization quality optimization during which it is guaranteed that a constraint position remains unchanged, so as to obtain a high-quality parameterization result meeting the constraint. By means of the present invention, a patch having serious quality reduction is optimized, a feature constraint is met, and a parameterization result having relatively high quality is also obtained, thereby meeting the application of subsequent analysis.

Description

A parametric migration method for isogeometric analysis based on discrete geometric mapping Technical Field

The invention relates to the technical field of geometric model processing, and in particular to an isogeometric analysis parameterization migration method based on discrete geometric mapping.

Background Art

In 2005, Professor Hughe first proposed isogeometric analysis. Its basic idea is to use the same mathematical language in modeling and analysis, and directly apply the geometric spline model designed in CAD (Computer Aided Design) software to the physical analysis in CAE (Computer Aided Engineering), so as to achieve a seamless combination of CAD and CAE. Whether it is a geometric model designed by CAD software or a model reconstructed by 3D scanning, it usually only has a spline boundary of the region. In order to obtain the physical quantities distributed inside the model, it is necessary to construct a spline representation of the internal area of the model, which is the parameterization problem in isogeometric analysis. Parameterization is an important step that plays a cornerstone role in isogeometric analysis, and its quality greatly affects the accuracy and stability of subsequent analysis.

At present, there are many computational domain parameterization methods for isogeometric analysis in academia, but these parameterization methods for isogeometric analysis all generate the overall spline representation of the computational domain from the boundary model, and do not consider the situation with internal constraints. In isogeometric analysis, for planar models with feature points and line constraints, when parameterizing, it is hoped that the feature points and lines are located at the corners or boundaries of the patch, so that constraints can be imposed on the features in subsequent isogeometric analysis. How to generate high-quality patches near the feature constraints and ensure that the boundaries of the patches are well attached to the feature constraints so that specific constraints are met in simulation analysis is a key problem that needs to be solved in the field of isogeometric analysis parameterization, which has certain theoretical value and application prospects.

Summary of the invention

In order to solve the above problems, the present invention proposes a parameterized migration method of isogeometric analysis based on discrete geometric mapping. For plane geometric models and internal point features or line features, the present invention provides a high-quality parameterization method that makes the features located at the internal facet boundary, which facilitates the subsequent imposition of constraints at the features.

In order to solve the above technical problems, the technical solution of the present invention is:

A parameterized migration method for isogeometric analysis based on discrete geometric mapping includes the following steps:

Step (1) classifying the given feature constraints;

Step (2) according to the classification result of step (1), if it is not possible to directly move the boundary of the face where the constraint is located to generate a high-quality parameterization, then split the face where the constraint is located into multiple sub-faces, so that the constraint can be located at the boundary of the new sub-face, which is convenient for applying the constraint;

Step (3) performing radial basis function (RBF) interpolation to move the constraint point or the endpoint of the constraint line to the boundary of the patch;

3-1. Generate a triangular mesh of a plane model;

3-2. Perform RBF interpolation movement;

Step (4) When a feature point or line has a small crack in the patch due to its special position, an optimization method is used to optimize the local parameterization quality to ensure that the constraint position remains unchanged, so as to obtain a high-quality parameterization result that satisfies the constraint;

4-1. Construct a quality optimization function that preserves constraints;

4-2. Fit the plane model boundary and obtain the final optimization result through interpolation movement.

Preferably, the step (1) comprises the following sub-steps:

1-1. For a given plane model, perform parameterization based on segmentation;

1-2. For feature point constraints, the corner points of the patch closest to the feature points on the plane model are used as the reference, and the feature points are divided into three categories according to their positions relative to the reference points;

1-3. For feature line constraints, the feature lines are divided into 7 categories based on the position of the feature line endpoints in the patch, taking the patch corner point closest to the feature line on the plane model as the reference.

Preferably, in step (2), the method for dividing the constrained patch into multiple sub-patches is:

Uniformly sample the patch where the feature constraint is located, calculate the sampling point closest to the feature point or the first and last endpoints of the feature line, and the node value to be inserted is the node value corresponding to the selected sampling point. Assuming that the order of the patch where the constraint is located in both directions is k, insert the node value until its repetition is k, so that the patch is divided into multiple small patches.

Preferably, in step 3-1, the method for generating a triangular mesh of a plane model is as follows: for a plane model with known parameterization results, a Delaunay triangulation (a classic triangulation algorithm that can maximize the minimum angle of the mesh triangle after triangulation) is performed to generate a background mesh according to the positions of its internal control points and boundary control polygons.

Preferably, in step 3-2, the RBF interpolation moving method is:

According to the constraint classification, the boundary control points of the patch to be moved are determined, and the corresponding m mesh vertices are marked in the background mesh. According to the positions of the m points to be moved and the target point, the offset of the m points is determined. Based on the RBF interpolation method, the positions of other control points inside the plane geometric model are determined to obtain the initial parameterization that satisfies the feature constraints.

Preferably, the step (4) comprises the following sub-steps:

4-1. For geometric models with known parametric results and internal feature point constraints or line constraints, the above process can generate parametric results that accurately meet the constraints. However, for some point and line constraints with special locations, although the above method can avoid the decline of internal parametric quality as much as possible while meeting the constraints, the patches near the features may have tiny cracks due to segmentation and discretization interpolation movement. The quality of parameterization can be improved by optimizing the following objective function:

in,
Q _l (P _i,j )＝∫ _Ω ||s _u || ² +||s _v || ² dΩ

represents the parametric line length function,
Q _u (P _i,j )＝∫ _Ω ||s _uu || ² +2||s _uv || ² +||s _vv || ² dΩ

represents the uniformity function of the parameter line,
Q _o (P _i,j )＝∫ _Ω (s _u ·s _v ) ² dΩ

represents the orthogonality function,

represents the skewness function,
Q _a (P _i,j )＝∫ _Ω |s _u ,s _v | ² dΩ

represents the area function,

represents the eccentricity function. _{w l} , w _u , w _o , w _s , w _a , w _e are non-negative weight values that determine the degree of influence of the related functions. _{P i,j} are the control points in the region C formed by all the thinned and split patches in the initial parameterization, and s is the patch in the region C. _{s v} represents the first-order partial derivative of the patch with respect to the v direction, s _u represents the first-order partial derivative of the patch with respect to the u direction, s _uu represents the second-order partial derivative of the patch with respect to the u direction, s _uv represents the mixed partial derivative of the patch with respect to the u and v directions, s _vv represents the second-order partial derivative of the patch with respect to the v direction, and s _v represents the first-order partial derivative of the patch with respect to the v direction.

4-2. Since the boundary of the geometric model is not fixed during quality optimization, the shape of the model changes. To solve this problem, after optimization, the boundary curve basis function of C is used to approximate the curve of this part before, and the final optimization result can be obtained by interpolation movement.

The present invention has the following characteristics and beneficial effects:

The present invention proposes a constrained parameterization method for a plane model for isogeometric analysis, which can obtain a parameterization result that satisfies the internal point constraint or line constraint under the premise of the known initial parameterization of the input model. For feature constraints close to the endpoints of the facet boundary, the coordinates of the facet boundary control points can be moved based on RBF interpolation to obtain a parameterization result that satisfies the constraint. For some point constraints or line constraints with special positions, the facets where the point constraints or line constraints are located are subdivided using heavy node interpolation before RBF interpolation, and then subsequent operations are performed to ensure the quality of the facets to a certain extent. The present invention includes a local area quality optimization method, which optimizes the facets with severe quality degradation, and obtains a parameterization result with higher quality while satisfying the feature constraints, which meets the application of subsequent analysis.

BRIEF DESCRIPTION OF THE DRAWINGS

In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative labor.

FIG1 is a flowchart of steps of an embodiment of the present invention;

Figure 2 shows the parameterization result when p is located inside the model and p _s is located inside the Cir _p circle with p as the center;

Figure 3 shows the parameterization result when p is located on the boundary of the model, or p _s is not located within the Cir _p circle with p as the center;

Figure 4 is the parameterization result corresponding to classification 3;

Figure 5 is the parameterization result corresponding to classification 4;

Figure 6 is the parameterization result corresponding to classification 5;

Figure 7 is the parameterization result corresponding to category 6;

Figure 8 shows the result of direct RBF interpolation movement of the constraint point located in the center of the patch;

FIG9 is a parameterized result satisfying the constraints after local refinement;

Figure 10 shows the parameterization result when the constraint line is located on the diagonal line;

FIG11 is a parameterized result satisfying the constraints after local quality optimization.

DETAILED DESCRIPTION

It should be noted that, in the absence of conflict, the embodiments of the present invention and the features in the embodiments may be combined with each other.

In the description of the present invention, it should be understood that the terms "center", "longitudinal", "lateral", "up", "down", "front", "back", "left", "right", "vertical", "horizontal", "top", "bottom", "inside", "outside" and the like indicate positions or positional relationships based on the positions or positional relationships shown in the accompanying drawings, and are only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and therefore cannot be understood as limiting the present invention. In addition, the terms "first", "second", and the like are only used for descriptive purposes, and cannot be understood as indicating or implying relative importance or implicitly indicating the number of technical features indicated. Thus, features defined as "first", "second", and the like may explicitly or implicitly include one or more of the features. In the description of the present invention, unless otherwise specified, "multiple" means two or more.

In the description of the present invention, it should be noted that, unless otherwise clearly specified and limited, the terms "installed", "connected", and "connected" should be understood in a broad sense, for example, it can be a fixed connection, a detachable connection, or an integral connection; it can be a mechanical connection or an electrical connection; it can be a direct connection, or it can be indirectly connected through an intermediate medium, or it can be the internal communication of two components. For ordinary technicians in this field, the specific meanings of the above terms in the present invention can be understood by specific circumstances.

The present invention provides an isogeometric parameterization method for a plane model with feature constraints, as shown in FIG1 , comprising the following steps:

Step (1) classifies the given feature constraints.

Specific:

1-1. For a given planar model, the method in "Constructing IGA-suitable planar parameterization from complex CAD boundary by domain partition and global/local optimization" is used to generate partition-based parameterization.

1-2. For feature point constraints, define the patch corner point closest to the feature point _ps on the plane model as p, and the arc lengths of the patch boundary curves _C1 and _C2 on both sides of p are _dis1 and _dis2 respectively. The circle with p as the center and min( _dis1 /2, _dis2 /2) as the radius is the circle _Cirp . It is divided into two categories according to whether the corner point p closest to _ps is on the outer boundary of the input model. If the corner point p is on the outer boundary of the model, moving the corner point p to meet the feature constraint will change the overall shape of the model, which does not meet the requirements of simulation analysis in actual engineering, and is marked as category 1. If the feature point _ps is located in the circle _Cirp with p as the center, directly interpolating the moving corner point can obtain a high-quality parameterized result while satisfying the constraint. Based on this, the case where the corner point p is located inside the model is subdivided into two categories, namely category 2 and category 3, according to whether _ps is located in the circle _Cirp with p as the center, as shown in Figures 2 and 3;

1-3. For the feature constraint line, its first and last endpoints are p _a and p _b , the nearest patch corners to p _a and p _b are p ₁ and p ₂ , the arc lengths of the boundary curves Ca1 _{and Ca2 on} both sides of p ₁ are dis _a1 and dis _a2 respectively, and the arc lengths of the boundary curves C _b1 and C _b2 on both sides of p ₂ are dis _b1 and dis _b2 respectively. The circle with p ₁ as the center and min (dis _a1 /2, dis _a2 /2) as the radius is Similarly, a circle with p ₂ as the center and min(dis _b1 /2,dis _b2 /2) as the radius is If the nearest corner points p ₁ , p ₂ of the first and last endpoints p _a , p _b of the feature line constraint are points on the diagonal line of the patch, directly moving the internal patch boundary of the model to meet the feature constraint will greatly reduce the quality of the patch around the constraint, and it is marked as category 1; if the corner points p ₁ , p ₂ closest to the feature line constraint are the two endpoints of the patch boundary, and p ₁ , p ₂ are both located on the external boundary of the input model, similar to the feature point constraint, directly moving the patch boundary will change the external shape of the model, and it is marked as category 2; if p ₁ , p ₂ are the two endpoints of the patch boundary and one is located at the model boundary and the other is located inside the model, assuming that p ₁ is located at the model boundary, at this time, determine whether the constraint point p _b corresponding to p ₂ is in the circle with p ₂ as the center. If it is not in the circle If p ₁ and p ₂ are the two endpoints of the patch boundary and p ₁ and p ₂ are both inside the model, the model is classified according to whether p ₁ and p ₂ are respectively in the corresponding circle. This situation is further divided into three categories, namely category 5, category 6 and category 7, as shown in Figures 4 to 7.

Step (2) divides the constrained patch into multiple sub-patches.

The specific implementation method is as follows:

Assume that the face where the constraint is located is

The node vectors are U = {u ₀ ,u ₁ ,…,un _+p+1 } and V = {v ₀ ,v ₁ ,…,v _m+q+1 }, respectively. Ni _,p (u) represents the i-th p-order spline basis function in the direction of the node vector U, Nj _,q (v) represents the j-th q-order spline basis function in the direction of the node vector V, and Pi _,j represents the control vertices of the patch. The node values to be inserted are Assume that after interpolation in the u direction, the new node vector is in:

Then the new node vector U ₁ can define n+2 p-order B-spline basis functions: The initial p×q degree B-spline surface can be obtained by the new B-spline basis function and the new control vertices after refinement express, The definition of is as follows:

in

Where u is a parameter and u _i ¹ is the node value of the node vector U ₁ .

To split the original patch f into multiple small patch units, the above process needs to be repeated p times in the u direction and q times in the v direction, so that the patch f is broken into 4 small patch units _f'i , i=1,...,4.

Step (3) performs RBF interpolation to move the constraint points or the endpoints of the constraint lines to the patch boundaries, as shown in FIG8 .

3-1. Generate a triangular mesh of the plane model.

Specifically, the method for generating a triangular mesh for a plane model with known parameterized results is: based on the Triangle library (a classic mesh processing program library), Delaunay triangulation is performed according to the parameterized internal control point positions and boundary control polygons.

3-2. Determine the boundary control points of the patch to be moved according to the constraint classification, and mark the corresponding m mesh vertices in the background mesh. Determine the offset of the m points according to the positions of the m points to be moved and the target point. Determine the positions of other control points inside the plane geometric model based on the RBF interpolation method, and obtain the initial parameterization that satisfies the feature constraints.

Specifically, RBF is composed of the sum of the results of m radial basis functions q(t _i ), i=1,…,m, t _i represents the distance between the point to be moved and the target point. RBF can provide a smooth interpolation function for the entire discrete space:

Where a _i ,i＝1,…,m and c ₀ ,c ₁ ,c ₂ ,c ₃ are (m+4) unknown quantities that need to be calculated, thus forming a certain affine transformation. RBF interpolation can calculate the unknown quantities in the above equation according to the equation system formed by the following formula, and determine the offset of other grid points except m known points.

Where A _x , A _y , A _z are (m+4)-dimensional column vectors (a ₁ , a ₂ ,…, a _m , c ₀ , c ₁ , c ₂ , c ₃ ) composed of unknown quantities. G is a (m+4)×(m+4) matrix, as shown below:

in The present invention uses the following function to determine g _ij :

Usually k=1, so (m+4) coefficients to be determined can be determined, and the RBF interpolation function can be determined. Based on the determined RBF interpolation function, the offsets of other grid points inside the model except the m known points are calculated to determine the initial parameterization results that meet the feature constraints.

Step (4) For the facets with serious quality degradation, an optimization method is used to perform local parameterized quality optimization while keeping the constraint position unchanged, and a high-quality parameterized result that satisfies the constraints is obtained, as shown in FIG9 .

4-1. For geometric models with known parametric results and internal feature point constraints or line constraints, the above process can generate parametric results that accurately meet the constraints. However, for some point and line constraints with special positions, although the above method can avoid the decline of internal parametric quality as much as possible while satisfying the constraints, the internal patches near the features will be affected. Tiny cracks may appear due to interpolation moves for segmentation and discretization. The quality of parameterization is improved by optimizing the following objective function:

in,

Q _l (P _i,j )＝∫ _Ω ||s _u || ² +||s _v || ² dΩ

represents the parametric line length function,
Q _u (P _i,j )＝∫ _Ω ||s _uu || ² +2||s _uv || ² +||s _vv || ² dΩ

represents the uniformity function of the parameter line,
Q _o (P _i,j )＝∫ _Ω (s _u ·s _v ) ² dΩ

represents the orthogonality function,

represents the skewness function,
Q _a (P _i,j )＝∫ _Ω |s _u ,s _v | ² dΩ

represents the area function,

represents the eccentricity function. _{w l} , w _u , w _o , w _s , w _a , w _e are non-negative weight values that determine the degree of influence of the related functions. _{P i,j} is the control point in the region C formed by all the thinned and split patches in the initial parameterization, s is the patch in the region C, s _v represents the first-order partial derivative of the patch with respect to the v direction, s _uu represents the second-order partial derivative of the patch with respect to the u direction, s _uv represents the mixed partial derivative of the patch with respect to the u and v directions, s _vv represents the second-order partial derivative of the patch with respect to the v direction, and s _v represents the first-order partial derivative of the patch with respect to the v direction.

4-2. The boundary of region C is recorded as The boundary of plane model A is Fixed constraint line _Cs and the boundary of region C The control points on the curve segment are optimized, and the positions of other control points inside are optimized, and the parameterization quality of the model is greatly improved. However, since the boundary of A is not fixed, the shape of the model changes. To solve this problem, the present invention uses The curve basis function of the curve is approximately optimized before the part of the curve is optimized, and the final optimization result is obtained by interpolation movement, as shown in Figures 10 and 11.

The embodiments of the present invention are described in detail above with reference to the accompanying drawings, but the present invention is not limited to the described embodiments. For those skilled in the art, various changes, modifications, substitutions and variations of these embodiments including components are made without departing from the principles and spirit of the present invention, and still fall within the scope of protection of the present invention.

Claims (7)

A parameterized migration method for isogeometric analysis based on discrete geometric mapping, characterized in that it comprises the following steps: Step (1) classifying the given feature constraints; Step (2) according to the classification result, if it is not possible to directly move the boundary of the face where the constraint is located to generate parameterization, then split the face where the constraint is located into multiple sub-faces, so that the constraint is located at the boundary of the new sub-face; Step (3) performing radial basis function (RBF) interpolation to move the constraint point or the endpoint of the constraint line to the boundary of the patch; Step (4) In the case where cracks appear in the face patch due to feature points or lines, an optimization method is used to perform local parameterization quality optimization to ensure that the constraint position remains unchanged, thereby obtaining a parameterization result that satisfies the constraints. The isogeometric analysis parameterized migration method based on discrete geometric mapping according to claim 1 is characterized in that the specific process of step (1) is as follows: 1-1. For a given plane model, perform parameterization based on segmentation; 1-2. For feature point constraints, the corner points of the patch closest to the feature points on the plane model are used as the reference, and the feature points are divided into three categories according to their positions relative to the reference points; 1-3. For feature line constraints, the feature lines are divided into 7 categories based on the position of the feature line endpoints in the patch, taking the patch corner point closest to the feature line on the plane model as the reference. According to the parameterized migration method of isogeometric analysis based on discrete geometric mapping as claimed in claim 2, it is characterized in that step 1-2 specifically comprises: for the feature point constraint, define the patch corner point closest to the feature point _ps on the plane model as p, the arc lengths of the patch boundary curves _C1 and _C2 on both sides of p are _dis1 and _dis2 respectively, and the circle with p as the center and min( _dis1 /2, _dis2 /2) as the radius is the circle _Cirp , and the corner point p closest to _ps is divided into two categories according to whether it is on the outer boundary of the input model. If the corner point p is on the outer boundary of the model, moving the corner point p satisfies the feature constraint, so that the overall shape of the model changes, which does not meet the requirements of simulation analysis in actual engineering, and is marked as category 1; if the feature point _ps is located in the circle _Cirp with p as the center, directly interpolating and moving the corner point obtains a parameterized result while satisfying the constraint, and accordingly, the situation where the corner point p is located inside the model is subdivided into two categories, namely category 2 and category 3, according to whether _ps is located in the circle _Cirp with p as the center. According to the parameterized migration method of isogeometric analysis based on discrete geometric mapping as described in claim 2, it is characterized in that steps 1-3 are specifically as follows: for the feature constraint line, its first and last endpoints are p _a and p _b , the patch corner points closest to p _a and p _b are p ₁ and p ₂ , the arc lengths of the boundary curves _Ca1 and _Ca2 on both sides of p ₁ are dis _a1 and dis _a2 respectively, and the arc lengths of the boundary curves C _b1 and C _b2 on both sides of p ₂ are dis _b1 and dis _b2 respectively, with p ₁ as the center and The circle with radius min(dis _a1 /2,dis _a2 /2,) is a circle Similarly, a circle with p ₂ as the center and min(dis _b1 /2,dis _b2 /2) as the radius is If the nearest corner points p ₁ , p ₂ of the first and last endpoints p _a , p _b of the feature line constraint are points on the diagonal line of the patch, directly moving the internal patch boundary of the model to meet the feature constraint will greatly reduce the quality of the patch around the constraint, and it is marked as category 1; if the corner points p ₁ , p ₂ closest to the feature line constraint are the two endpoints of the patch boundary, and p ₁ , p ₂ are both located on the external boundary of the input model, similar to the feature point constraint, directly moving the patch boundary will change the external shape of the model, and it is marked as category 2; if p ₁ , p ₂ are the two endpoints of the patch boundary and one is located on the model boundary and the other is located inside the model, assuming that p ₁ is located on the model boundary, at this time, determine whether the constraint point p _b corresponding to p ₂ is in the circle with p ₂ as the center. If it is not in the circle If p ₁ and p ₂ are the two endpoints of the patch boundary and p ₁ ,,p ₂ are both inside the model, classify them according to whether p ₁ ,,p ₂ are respectively in the corresponding circle This situation is further divided into three categories, namely category 5, category 6 and category 7. The isogeometric analysis parameterized migration method based on discrete geometric mapping according to claim 2 is characterized in that the specific process of dividing the face where the constraint is located into multiple sub-faces in step (2) is as follows: The patch where the feature constraint is located is uniformly sampled, and the sampling point closest to the feature point or the first and last endpoints of the feature line is calculated. The node value to be inserted is the node value corresponding to the selected sampling point. Assuming that the order of the patch where the constraint is located in both directions is k, the patch is divided into multiple small patches by inserting node values until its repetition degree is k. The isogeometric analysis parameterized migration method based on discrete geometric mapping according to claim 5 is characterized in that the specific process of step (3) is as follows: 3-1. Generate a triangular mesh of a plane model; For a plane model with known parameterized results, Delaunay triangulation is performed based on the positions of its internal control points and boundary control polygons to generate a background mesh. 3-2. Perform RBF interpolation movement; According to the constraint classification, the boundary control points of the patch to be moved are determined, and the corresponding m mesh vertices are marked in the background mesh; according to the positions of the m points to be moved and the target point, the offsets of the m points are determined; based on the RBF interpolation method, the positions of other control points inside the plane geometric model are determined to obtain the initial parameterization that satisfies the feature constraints. The isogeometric analysis parameterized migration method based on discrete geometric mapping according to claim 6 is characterized in that the specific process of step (4) is as follows: 4-1. Construct a quality optimization function that preserves constraints; The quality of parameterization is improved by optimizing the following objective function:
in,
Q _l (P _i,j )＝∫ _Ω ||s _u || ² +||s _v || ² dΩ represents the parametric line length function,
Q _u (P _i,j )＝∫ _Ω ||s _uu || ² +2||s _uv || ² +||s _vv || ² dΩ represents the uniformity function of the parameter line,
Q _o (P _i,j )＝∫ _Ω (s _u ·s _v ) ² dΩ represents the orthogonality function,
represents the skewness function,
Q _a (P _i,j )＝∫ _Ω |s _u ,s _v | ² dΩ represents the area function,
represents the eccentricity function; w _l , w _u , w _o , w _s , w _a , w _e are non-negative weight values that determine the influence of the relevant functions; Pi _,j are the control points in the region C formed by all the thinned and split patches in the initial parameterization, s is the patch in the region C, s _v represents the first-order partial derivative of the patch with respect to the v direction, s _u represents the first-order partial derivative of the patch with respect to the u direction, s _uu represents the second-order partial derivative of the patch with respect to the u direction, s _uv represents the mixed partial derivative of the patch with respect to the u and v directions, and s _vv represents the second-order partial derivative of the patch with respect to the v direction; 4-2. Fit the plane model boundary and obtain the optimization result through interpolation movement; After the optimization, the boundary curve basis function of C is used to optimize the previous part of the curve, and the optimization result is obtained by interpolation movement.