Isogeometric analysis parameterization migration method based on discrete geometric mapping

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

[0001] The present 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

[0002] In 2005, Professor Hughe first proposed isogeometric analysis. Its basic idea is to use the same mathematical language for 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), thus achieving a seamless combination of CAD and CAE. Whether it is a geometric model designed using CAD software or a model reconstructed using three-dimensional 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 region inside the model. This 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.

[0003] Currently, there are many computational domain parameterization methods for isogeometric analysis in academia. However, these parameterization methods for isogeometric analysis all generate a global spline representation of the computational domain from the boundary model, without considering the situation where there are internal constraints. In isogeometric analysis, for planar models with feature point 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 issue that needs to be solved in the field of isogeometric analysis parameterization, which has certain theoretical value and application prospects.

[0004] Summary of the Invention

[0005] To solve the above problems, the present invention proposes a parameterized migration method for 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 locates the features at the internal patch boundary, which facilitates the subsequent imposition of constraints at the features.

[0006] In order to solve the above technical problems, the technical solution of the present invention is:

[0007] A parameterized migration method for isogeometric analysis based on discrete geometric mapping includes the following steps:

[0008] Step (1) classify the given feature constraints;

[0009] Step (2) based on the classification result of step (1), if it is not possible to directly move the boundary of the constraint patch to generate a high-quality parameterization, then split the constraint patch into multiple sub-patches so that the constraint can be located at the boundary of the new sub-patches, which is convenient for applying the constraint;

[0010] Step (3) performing radial basis function (RBF) interpolation to move the constraint point or the endpoint of the constraint line to the patch boundary;

[0011] 3-1. Generate a triangular mesh of the plane model;

[0012] 3-2. Perform RBF interpolation movement;

[0013] Step (4) For the case where the feature points or lines have small cracks in the patch due to their special positions, an optimization method is used to optimize the local parameterization quality while ensuring that the constraint positions remain unchanged, thereby obtaining a high-quality parameterization result that satisfies the constraints.

[0014] 4-1. Construct a quality optimization function that preserves constraints;

[0015] 4-2. Fit the plane model boundary and obtain the final optimization result through interpolation movement.

[0016] Preferably, the step (1) includes the following sub-steps:

[0017] 1-1. For a given plane model, perform parameterization based on segmentation;

[0018] 1-2. For feature point constraints, the corner point closest to the feature point on the plane model is used as the reference, and the feature points are divided into three categories according to their positions relative to the reference point;

[0019] 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.

[0020] Preferably, the method for dividing the constrained patch into multiple sub-patches in step (2) is:

[0021] Uniformly sample the patch containing the feature constraint and calculate the sampling point closest to the feature point or the endpoints of the feature line. The node value to be inserted is the node value corresponding to the selected sampling point. Assuming that the degree of the constraint patch is k in both directions, the patch is divided into multiple small patches by inserting node values ​​until the degree of repetition is k.

[0022] Preferably, in step 3-1, the method for generating a triangular mesh of a plane model is: for a plane model with known parameterization results, Delaunay triangulation (a classic triangulation algorithm that can maximize the minimum angle of the mesh triangle after triangulation) is performed on the plane model according to the positions of its internal control points and boundary control polygons to generate a background mesh.

[0023] Preferably, in step 3-2, the RBF interpolation movement method is:

[0024] Based on the constraint classification, the control points on the patch boundary to be moved are identified, and the corresponding m mesh vertices are marked in the background mesh. Based on the positions of the m points to be moved and the target point, the offset of the m points is determined. Using RBF interpolation, the positions of the remaining control points within the plane geometric model are determined, resulting in an initial parameterization that satisfies the feature constraints.

[0025] Preferably, the step (4) includes the following sub-steps:

[0026] 4-1. For a geometric model with known parameterized results and internal feature point constraints or line constraints, the above process can generate a parameterized result that accurately satisfies the constraints. However, for some point or line constraints with special locations, although the above method can minimize the degradation of internal parameterization quality while satisfying the constraints, the internal patches near the features may move due to segmentation and discretization interpolation, resulting in tiny cracks. The parameterization quality is improved by optimizing the following objective function:

[0027] Among them, Q l (P i,j )=∫ Ω ||s u || 2 +||s v || 2 dΩ

[0028] represents the parametric line length function, Q u (P i,j )=∫ Ω ||s uu || 2 +2||s uv || 2 +||s vv || 2 dΩ

[0029] represents the uniformity function of the parameter line, Q o (P i,j )=∫ Ω (s u ·s v ) 2 dΩ

[0030] represents the orthogonality function,

[0031] represents the skewness function, Q a (P i,j )=∫ Ω |s u ,s v | 2 dΩ

[0032] represents the area function,

[0033] represents the eccentricity function. w l ,w u ,w o ,w s ,w a ,w e Is a non-negative weight value that determines the influence of the related function. 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 region C. 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 about the u direction, s uv Represents the mixed partial derivatives 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, s v Represents the first-order partial derivative of the patch with respect to the v direction.

[0034] 4-2. Because the boundaries of the geometric model are not fixed during quality optimization, the model shape 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 is obtained through interpolation movement.

[0035] The present invention has the following characteristics and beneficial effects:

[0036] The present invention proposes a constrained parameterization method for planar models for isogeometric analysis, which can obtain parameterized results that meet internal point constraints or line constraints under the premise of knowing the initial parameterization of the input model. For feature constraints close to the endpoints of the patch boundary, the coordinates of the patch boundary control points can be moved based on RBF interpolation to obtain parameterized results that meet the constraints. For some point constraints or line constraints with relatively special positions, the patch where the point constraints or line constraints are located is subdivided using heavy node interpolation before RBF interpolation, and then subsequent operations are performed to ensure the quality of the patch to a certain extent. The present invention includes a local area quality optimization method to optimize the patch with serious quality degradation, and obtain a parameterized result with higher quality while meeting the feature constraints, which can meet the application of subsequent analysis. BRIEF DESCRIPTION OF THE DRAWINGS

[0037] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. 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 any creative work.

[0038] FIG1 is a flowchart of steps of an embodiment of the present invention;

[0039] Figure 2 shows that p is located inside the model, p s Located at Cir with p as the center p Parameterized results within the circle;

[0040] Figure 3 shows that p is on the boundary of the model, or p s Not located in the circle centered on p p Parameterized results within the circle;

[0041] Figure 4 shows the parameterization results corresponding to classification 3;

[0042] Figure 5 shows the parameterization results corresponding to category 4;

[0043] Figure 6 shows the parameterization results corresponding to category 5;

[0044] Figure 7 shows the parameterization results corresponding to category 6;

[0045] Figure 8 shows the result of direct RBF interpolation movement of the constraint point located in the center of the patch;

[0046] Figure 9 shows the parameterized result that satisfies the constraints after local refinement;

[0047] Figure 10 shows the parameterization results when the constraint line is located on the diagonal line;

[0048] Figure 11 shows the parameterized result that satisfies the constraints after local quality optimization. DETAILED DESCRIPTION

[0049] 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.

[0050] 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 orientations or positional relationships based on the orientations 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", etc. are only used for descriptive purposes and cannot be understood as indicating or implying relative importance or implicitly indicating the number of the indicated technical features. Therefore, features defined as "first", "second", etc. 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.

[0051] In the description of the present invention, it should be noted that, unless otherwise expressly specified or limited, the terms "mounted," "connected," and "connected" should be understood in a broad sense. For example, they may refer to fixed connections, detachable connections, or integral connections; mechanical connections or electrical connections; direct connections or indirect connections through an intermediate medium; and internal communication between two components. Those skilled in the art will understand the specific meanings of the above terms in the present invention based on specific circumstances.

[0052] The present invention provides an isogeometric parameterization method for a plane model with feature constraints, as shown in FIG1 , comprising the following steps:

[0053] Step (1) classifies the given feature constraints.

[0054] Specifically:

[0055] 1-1. For a given planar model, generate a partition-based parameterization using the method described in "Constructing IGA-suitable planar parameterization from complex CAD boundary by domain partition and global / local optimization";

[0056] 1-2. For feature point constraints, define the distance from the feature point p on the plane model. s The nearest patch corner is 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 circle Cir p According to the distance from p s The nearest corner point p is divided into two categories based on 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 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. s Located on the circle Cir with p as the center p Direct interpolation of moving corner points can obtain high-quality parameterized results while satisfying the constraints. Based on this, when the corner point p is located inside the model, according to p s Is it located in the circle Cir with p as the center? p It is subdivided into two categories, namely Classification 2 and Classification 3, as shown in Figures 2 and 3;

[0057] 1-3. For the characteristic constraint line, its starting and ending points are p a ,p b , from p a ,p b The nearest patch corners are p1 and p2, and the boundary curves C on both sides of p1 a1 ,C a2 The arc lengths are dis a1 ,dis a2 , boundary curves C on both sides of p2 b1 ,C b2 The arc lengths are dis b1 ,dis b2 , with p1 as the center and min(dis a1 / 2,dis a2 / 2) is a circle with a radius of Similarly, with p2 as the center and min(dis b1 / 2,dis b2 / 2) is a circle with a radius of If the leading and trailing endpoints p of the feature line constraint a ,p bThe nearest corner points p1 and p2 are points on the diagonal line of the patch. Directly moving the internal patch boundary of the model to meet the feature constraint will cause the quality of the patch around the constraint to drop significantly, and it is marked as category 1. If the corner points p1 and p2 closest to the feature line constraint are the two endpoints of the patch boundary, and p1 and p2 are both located at 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 p1 and p2 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 p1 is located at the model boundary, at this time, determine the constraint point p corresponding to p2. b Is it in the circle with p2 as the center? Inside, if it is not in the circle If p1 and p2 are the two endpoints of the patch boundary and both p1 and p2 are inside the model, the model is classified as category 3, otherwise it is classified as category 4. This situation is further divided into three categories, namely category 5, category 6 and category 7, as shown in Figures 4 to 7.

[0058] Step (2) divides the constrained patch into multiple sub-patches.

[0059] The specific implementation method is as follows:

[0060] Assume that the face where the constraint is located is

[0061] The node vectors are U={u0,u1,…,u n+p+1} and V={v0,v1,…,v m+q+1}, N i,p (u) represents the i-th p-th spline basis function in the direction of the node vector U, N j,q (v) represents the j-th q-order spline basis function in the direction of the node vector V, P i,j Indicates 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:

[0062] Then the new node vector U1 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:

[0063] in

[0064] Where u is a parameter, u i 1 is the node value of the node vector U1.

[0065] To split the original face f into multiple small face units, the above process needs to be repeated p times in the u direction and q times in the v direction, so that the face f is broken into 4 small face units f' i ,i=1,…,4。

[0066] 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 .

[0067] 3-1. Generate a triangular mesh of the plane model.

[0068] Specifically, a method for generating a triangular mesh for a plane model with known parameterized results is to perform Delaunay triangulation based on the parameterized internal control point positions and boundary control polygons based on the Triangle library (a classic mesh processing program library).

[0069] 3-2. Based on the constraint classification, determine the boundary control points of the patch to be moved and mark the corresponding m mesh vertices in the background mesh. Determine the offset of the m points based on the positions of the m points to be moved and the target point. Using RBF interpolation, determine the positions of the remaining control points within the plane geometric model to obtain an initial parameterization that satisfies the feature constraints.

[0070] Specifically, RBF consists 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:

[0071] where a i , i = 1, …, m, and c0, c1, c2, and c3 are the (m + 4) unknowns that need to be calculated, thus forming a specific affine transformation. RBF interpolation can calculate the unknowns in the above equation using the following system of equations, determining the offset of all grid points other than the m known points.

[0072] Among them A x ,A y ,A z is an (m+4)-dimensional column vector (a1, a2, ..., a m,c0,c1,c2,c3). G is a (m+4)×(m+4) matrix, as shown below:

[0073] in The present invention uses the following function to determine g ij :

[0074] Typically, k = 1, which allows us to determine (m+4) coefficients to be determined and the RBF interpolation function to be determined. Based on the determined RBF interpolation function, the offsets of other grid points within the model, excluding the m known points, are calculated to determine the initial parameterization results that meet the feature constraints.

[0075] In step (4), for the facets with serious quality degradation, an optimization method is used to perform local parameterization quality optimization while keeping the constraint position unchanged, and a high-quality parameterization result that meets the constraints is obtained, as shown in FIG9 .

[0076] 4-1. For a geometric model with known parameterized results and internal feature point constraints or line constraints, the above process can generate a parameterized result that accurately satisfies the constraints. However, for some point or line constraints with special locations, although the above method can minimize the degradation of internal parameterization quality while satisfying the constraints, the internal patches near the features may move due to segmentation and discretization interpolation, resulting in tiny cracks. The parameterization quality is improved by optimizing the following objective function:

[0077] in,

[0078] Q l (P i,j )=∫ Ω ||s u || 2 +||s v || 2 dΩ

[0079] represents the parametric line length function, Q u (P i,j )=∫ Ω ||s uu || 2 +2||s uv || 2 +||s vv || 2 dΩ

[0080] represents the uniformity function of the parameter line, Q o (P i,j )=∫ Ω (s u ·sv ) 2 dΩ

[0081] represents the orthogonality function,

[0082] represents the skewness function, Q a (P i,j )=∫ Ω |s u ,s v | 2 dΩ

[0083] represents the area function,

[0084] represents the eccentricity function. w l ,w u ,w o ,w s ,w a ,w e Is a non-negative weight value that determines the influence of the related function. 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 region C, and 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 about the u direction, s uv Represents the mixed partial derivatives 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, s v Represents the first-order partial derivative of the patch with respect to the v direction.

[0085] 4-2. The boundary of region C is recorded as The boundary of plane model A is Fixed constraint line C s 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. 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.

[0086] 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. It will be apparent to those skilled in the art that various changes, modifications, substitutions, and variations of these embodiments, including components, without departing from the principles and spirit of the present invention are still within the scope of protection of the present invention.

Claims

1. A parameterized migration method for isogeometric analysis based on discrete geometric mapping, characterized in that: The steps include: 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.

2. The isogeometric analysis parameterization 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.

3. The isogeometric analysis parameterization migration method based on discrete geometric mapping according to claim 2 is characterized in that: Steps 1-2 are as follows: For feature point constraints, define the distance from the feature point p on the plane model. s The nearest patch corner point is p. 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 Cir. p , according to the distance p s Whether the nearest corner point p is on the outer boundary of the input model is divided into two categories. If the corner point p is on the outer boundary of the model, moving the corner point p satisfies the feature constraint, which changes the overall shape of the model and does not meet the requirements of simulation analysis in actual engineering. It is marked as category 1. s Located on the circle Cir with p as the center p Direct interpolation moves the corner point to obtain a parameterized result while satisfying the constraints. Based on this, the corner point p is located inside the model. s Is it located in the circle Cir with p as the center? p It is divided into two categories, namely category 2 and category 3.

4. The isogeometric analysis parameterization migration method based on discrete geometric mapping according to claim 2 is characterized in that: Steps 1-3 are as follows: For the feature constraint line, its first and last endpoints are p a ,p b , from p a ,p b The nearest patch corners are p1,,p2, and the boundary curves C on both sides of p1 a1 ,C a2 The arc lengths are dis a1 ,dis a2 , the boundary curves C on both sides of p2 b1 ,C b2 The arc lengths are dis b1 ,dis b2 , with p1 as the center and min(dis a1 / 2,dis a2 / 2,) is a circle with a radius of Similarly, with p2 as the center and min(dis b1 / 2,dis b2 / 2) is a circle with a radius If the leading and trailing endpoints p of the feature line constraint a ,p b The nearest corner points p1 and p2 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 p1 and p2 closest to the feature line constraint are the two endpoints of the patch boundary, and p1 and p2 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 p1 and p2 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 p1 is located at the model boundary, at this time, determine the constraint point p corresponding to p2. b Is it in the circle with p2 as the center? If it is not in the circle If p1 and p2 are the two endpoints of the patch boundary and p1,,p2 are both inside the model, it is marked as category 3, otherwise it is marked as category 4; if p1,,p2 are both inside the model, it is marked as category 3 according to whether p1,,p2 are respectively in the corresponding circle This situation is further divided into three categories, namely category 5, category 6 and category 7.

5. The isogeometric analysis parameterization 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.

6. The isogeometric analysis parameterization 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.

7. The isogeometric analysis parameterization 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 is a non-negative weight value that determines the influence of the related function; 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 region C, and 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 about the u direction, s uu represents the second-order partial derivative of the patch about the u direction, s uv represents the mixed partial derivatives of the patch about the u and v directions, 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 curve of the previous part, and the optimization result is obtained by interpolation movement.

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