CN115455588A - Turbine blade precision casting mold surface reversible deformation design method - Google Patents

Abstract

A turbine blade precision casting mold molded surface reversible deformation design method relates to the field of machinery. 1) Encrypting discrete points of a curve; 2) B, fitting a design curve by a spline; 3) Searching a corresponding point of the measured data based on a curve subdivision method; 4) Constructing a free deformation grid: deforming the object by manipulating the design grid control points; 5) And (3) iteratively calculating displacement deformation: establishing a series of fitting curves by iteratively adjusting control points of the design curve, wherein in each iteration, a difference vector of each control point is a weighted sum of data points of the target curve and some difference vectors of corresponding points on the fitting curve, and the weighted sum of the difference vectors is iteration calculation displacement deformation; 6) And performing inverse deformation on the design curve by adopting the iterative deformation. On the basis of keeping the design intention, the molded surface of the turbine blade precision casting mold is subjected to inverse deformation optimization design according to the deviation of the measuring point data, the curved surface reconstruction precision and the practicability of the molded surface of the turbine blade precision casting mold are improved, and the cavity inverse deformation optimization of the precision casting turbine blade based on the B spline characteristic is realized.

Description

Design method of anti-deformation surface of precision casting mold for turbine blade

technical field

The invention relates to the mechanical field, in particular to a B-spline curve subdivision-based anti-deformation design method for a precision casting mold surface of a turbine blade.

Background technique

Turbine blades are generally cast using directional crystallization or single crystal net shape investment casting. In the design and manufacture process of precision casting turbine blades for aero-engines in my country, the problems of low surface accuracy, unstable quality and high scrap rate of precision casting blades have not been solved due to the unreasonable size of the mold design. Major foreign engine companies have established directional solidification and precision casting production lines for single crystal turbine blades. The casting process is relatively mature, and the problem of "precise shape control" for turbine blade castings has to be solved urgently. During the precision casting process of the turbine blade, after the high-temperature liquid alloy is injected into the shell, deformation will occur as the temperature decreases. The structure and shape of the turbine blade is complex, resulting in uneven heat dissipation when the casting is cooled, and the deformation of each point of the blade is also uneven. Therefore, the actual deformation of the casting is nonlinear, which leads to the size of the casting being out of tolerance, and it is impossible to cast a qualified turbine blade. Condition. This has become a major pain point in the field of net-shape precision casting of turbine blades. Therefore, it is necessary to propose a set of anti-deformation optimization design methods for turbine blade precision casting mold surfaces that can combine design intent and measurement data.

The anti-deformation design of the precision casting mold surface needs to compensate the deformation of the casting during solidification and cooling. In China, the surface scaling method is firstly used to compensate, which is divided into uniform scaling method, chord length scaling method, mid-arc scaling method and shrinking center scaling method. The shrinkage rates of these four scaling methods are still constant, and different The difference is that the selection of shrinkage center and direction is inconsistent. Although this type of method is simple, there are obvious deficiencies: first, the uniform shrinkage approximation of the turbine blade, that is, assuming that the shrinkage rate is the same at different parts; Thickening or shrinking proportionally along the normal direction realizes the compensation of the mold cavity, ignoring the non-rigid deformation of the turbine blade. Aiming at the shortcomings of the surface scaling method, some researchers proposed to assign different shrinkage rates along the x, y and z directions, and achieved certain results. But the disadvantage is that the shrinkage rate still adopts constant or linear assignment in each direction, and the casting model needs to be constantly corrected according to the shape and size of the casting. With the maturity of the numerical simulation technology of the casting process, a method of reversely superimposing the deformation obtained by the numerical simulation on the nodes is proposed, and through multiple iterations, the shape of the turbine blade after contraction and deformation is very close to the ideal design shape. The disadvantage is that the quality of the grid is reduced where the deformation of the casting is large, which affects the convergence of numerical simulation calculations and the correctness of simulation results. Later, someone proposed a precision casting mold surface anti-deformation method based on displacement field simulation and characteristic parameter extraction. Through the extraction of characteristic parameters reflecting the blade shape, the restoration and adjustment of the blade shape, compensation for nonlinear shrinkage deformation is realized. The disadvantage is that the reverse adjustment adopts the linear reverse superposition of the deviation before and after solidification, ignoring the deformation caused by the superposition itself; for the simultaneous reverse adjustment of several characteristic parameters, ignoring the coupling relationship between the parameters; and for the geometry of different sections Reverse adjustment of feature parameters can also easily lead to difficulties in the final surface reconstruction.

Contents of the invention

The purpose of the present invention is to address the above deficiencies in the prior art, to provide a B-spline curve subdivision method for the reverse deformation design of the turbine blade precision casting mold surface, based on the deviation of the measurement point data on the basis of retaining the design intention The anti-deformation optimization design of the surface of the turbine blade precision casting mold is carried out to optimize the design size of the precision casting mold based on coupling deformation, to ensure that the casting size does not appear out of tolerance, and to improve the surface reconstruction accuracy of the turbine blade precision casting mold surface And practicability, to achieve net shape precision casting of turbine blades.

The present invention comprises the following steps:

1) Encrypting the discrete points of the curve: Grid the fitting area of the curve to be encrypted, determine the size of the influence area of the grid point x and the nodes contained in the influence area, and calculate the node value at the grid point x after determining the shape function , perform the above processing on each grid point, and perform curve encryption;

2) B-spline fitting design curve: Parametrize the discrete points on the design curve according to the cumulative chord length parameterization method, requiring the first and last points of the design curve to pass through the first and last points of the B-spline curve, and the rest of the discrete points on the design curve Use the least squares idea to fit the B-spline curve, and reduce the fitting deviation through multiple iterations, so as to determine the number of control points, so as to complete the use of B-spline to fit the design curve;

3) Find the corresponding points of the measured data points on the design curve: find the corresponding points based on the method of curve subdivision, and use the node insertion algorithm of the B-spline curve to subdivide the B-spline curve into a group of Bezier curves; in the binary tree search process Subdivide each Bezier curve again, and judge whether the condition is met. If the condition is met, then subdivide again until the curve segment is smaller than the set threshold, and judge the minimum value among the candidate points to obtain the correspondence of the measured data points on the design curve. point;

4) Constructing a free-form deformation grid: by manipulating the control points of the design grid to deform the object, local free deformation can be realized by using a locally supported B-spline basis function; considering that the object this time is a two-dimensional B-sample strips, defining the simplest grid to design as an axial grid with a grid of control points, expressed as a B-spline surface;

5) Iterative calculation of displacement and deformation: A series of fitting curves are constructed by iteratively adjusting the control points of the design curve. In each iteration, the difference vector of each control point is some difference between the data point of the target curve and the corresponding point on the fitting curve The weighted sum of the vectors and the weighted sum of the difference vectors are the iteratively calculated displacement and deformation;

6) Use iterative deformation to inversely deform the design curve: in the case where the deformed B-spline control points correspond to the control points before deformation, the accumulation of the weighted difference vectors of the data points is used as the inverse deformation parameter, and the calculation embedded in Deformation matrix of the control points in the mesh, which inversely deforms the design curve by manipulating the mesh.

In step 3), the specific method of obtaining the corresponding point of the measured data point on the design curve is an example (hereinafter referred to as the measured point) on the measured curve, and the distance from the measured point to the corresponding point on the design curve can be transformed into is the distance between the measured point and a set of Bezier curves, and then the problem of minimizing the distance function is transformed into the zero point problem of the polynomial function of the parameter t. In the binary tree search process, the Bezier curve is subdivided with depth to determine whether the polynomial function of the parameter t is on the same side of the t-axis, the same side is excluded, and the opposite side is marked as a candidate point, and finally the minimum value of the candidate points is taken as the design The closest point to the measured point on the curve.

In step 4), the construction of the freely deformable grid includes the construction of a two-dimensional freely deformable grid containing the model, the deformation operation does not directly act on the object, but acts on the embedded deformation space, and constructs a two-dimensional freely deformable grid containing the model The free deformable grid mainly includes the following two steps: construct a local two-dimensional coordinate system, then calculate the local coordinates corresponding to each vertex coordinate of the model, and embed the curve model in a frame; based on the ternary Bernstein multivariate power function, the movement control Points, using the local coordinates of model vertices, world coordinates of control points and Bernstein polynomials to recalculate the world coordinates of each vertex of the model, the framework "pulls" the model to achieve deformation.

Compared with prior art, the beneficial effect of the present invention is:

The invention provides a B-spline curve subdivision method for reverse deformation design of turbine blade precision casting mold surface. The anti-deformation optimization of the cavity with B-spline characteristics provides a convenient pre-foundation for the subsequent transition from 2D superposition to 3D reconstruction, and saves a lot of smoothing costs and time compared with discrete point 3D reconstruction. Since the adopted free-form deformation (FFD) algorithm is closely related to the parametric design method of the turbine blade profile, all relevant technologies such as turbine blade design, modeling, and processing can be easily integrated into any CAD platform, and the turbine blade can be realized efficiently. Parametric operation of blade profile.

Description of drawings

Fig. 1 is the encryption effect diagram of the blade back curve of the leaf pot with sparse point cloud of the present invention;

Fig. 2 is a schematic diagram of curve fitting of the design model leaf back of the present invention;

Fig. 3 is the curve subdivision effect figure of the present invention;

Fig. 4 is the effect diagram of free deformation mesh of the present invention;

Fig. 5 is a schematic diagram of the iterative calculation fitting error vector of the present invention.

detailed description

The following examples can enable those skilled in the art to understand the present invention more fully, but do not limit the present invention in any way.

The method for realizing the present invention comprises:

(1) Considering that discrete points have similar complexity in preserving the design intent of turbine blades, it is usually reduced to a series of 2D cross-section superposition problems. The present invention uses two sections of B-splines and two sections of arcs to characterize the blade basin and the blade back of the turbine blade section. When the point cloud at the blade basin and the blade back is relatively sparse, if the discrete point data has no noise point interference, its When the calculated implicit function curve passes through the original data points, the fitting error is zero. For noise point data, the influence of noise points is weakened by moving least squares with influence domain. Using the moving least squares method for curve and surface fitting is to grid the fitting area first, and then use the formula to find the node value on each grid point, and perform curve encryption;

(2) Fit the dorsal curve of the blade pot and blade after the section encryption of the turbine blade with B-splines, parameterize the cumulative chord length of the discrete points on the curve, pre-calculate the parameter values and node vectors, and use the least square method to solve the control Points and basis functions satisfy the curve as close as possible to the data points, and the number of control points is obtained iteratively according to the user-defined error limit;

(3) In order to avoid the shortcomings of Newton’s iteration that need to provide good initial values to get correct results and traditional numerical methods, a method based on curve subdivision is proposed to find the closest point, that is, to find the closest point on the curve for a given point. This method No iterative processing is involved. First, subdivide the B-spline of the leaf back and leaf pot curve into Bezier curves, then select the qualified curves according to whether the construction curve intersects with the t-axis, and finally subdivide the Bezier curves, and use binary tree decomposition to search for the nearest measured data point on the design curve points, the search depth is defined by user-given error limits;

(4) The deformation operation of FFD does not act directly on the object, but acts on the embedded deformation space. If the deformation space is changed, the embedded object will naturally change accordingly. The FFD algorithm mainly has two steps: embed the curve model into a frame composed of control points; when the position of the control points changes, the frame will "pull" the model to achieve deformation;

(5) Given a set of bases and a set of discrete data points to be fitted, and marked as the initial control vertices, an initial fitting curve is generated, and then the fitting error vector from each initial control vertex to the initial curve is calculated, along the Move the control vertices in the direction of the fitting error vector to generate new control vertices. This cycle can obtain a curve series with increasing fitting accuracy, and when the fitting base is the NTP base, it can ensure that the fitting error can be arbitrarily small. That is, the fitting limit curve will interpolate the initial control vertices. Using this method, a very large data set can be fitted efficiently and robustly. At the same time, during the incremental data fitting process of this method, the fitting results of the previous iteration Start a new round of iterations, thus saving a lot of calculations;

(6) In the case where the deformed B-spline control points correspond to the control points before deformation, the accumulation of the weighted difference vectors of the data points is used as the anti-deformation parameter to calculate the deformation matrix of the control points embedded in the grid, by Manipulate the mesh to undeform the design curves.

The specific implementation steps are given below in conjunction with the accompanying drawings.

Step 1: Encrypt the discrete points of the curve

的三次样条函数曲线：Taking the line laser scanning model of the aero-engine turbine blade casting as an example, aiming at the sparse point cloud of the blade pot and blade back of the turbine blade section, moving least squares is introduced to encrypt the discrete points on the curve. The basic idea of using the moving least squares method for curve encryption is to grid the fitting area first, determine the size of the influence area of the grid point x and the nodes contained in the influence area, and calculate the shape function of the grid point x Node value, the above processing is performed on each grid point, and the curve is encrypted. The encryption effect is shown in Figure 1. Among them, the shape function is expressed as φ=P ^T (P ^T ωP) ^-1 P ^T ω, the basis function P=[1,x,x ² ] ^T , P ^T is the transpose of P, and P ^-1 is the inverse of P matrix. The weight function ω is about the influence domain radius

A cubic spline curve for :

Step 2: B-spline fitting design curve

其中b表示叶背，ls表示第l个截面的曲线，t是归一化后的参数，表达式为

其中P_i ^b为该叶背曲线第i个控制顶点，N_i,3(t)为三次B样条对应第i个控制顶点的基函数。满足

其余点q_k(k＝1,…,m-1)用最小二乘拟合近似，即

是对n+1个变量P_i ^b的最小值，其中t_r是q_r(r＝0,…,m)通过累计弦长参数化的预先计算的参数值，节点矢量t_j＝{0,0,0,0,t_r,1,1,1,1}。拟合效果如图2所示。Fitting the dorsal curve of the blade basin after the section of the turbine blade is encrypted with B-splines, taking the design model of the dorsal curve as an example, the dorsal curve exists in the form of discrete points q _r (r=0,...,m), The leaf back curve is fitted by B-splines of degree 3, and the leaf back curve of the l-th section design model is denoted as

Where b represents the leaf back, ls represents the curve of the lth section, t is the parameter after normalization, and the expression is

Where P _i ^b is the i-th control vertex of the leaf back curve, N _i,3 (t) is the basis function of the cubic B-spline corresponding to the i-th control vertex. Satisfy

The remaining points q _k (k=1,...,m-1) are approximated by least squares fitting, namely

is the minimum value of n+1 variables P _i ^b , where t _r is the pre-calculated parameter value of q _r (r=0,…,m) parameterized by cumulative chord length, node vector t _j ={0, 0,0,0,t _r ,1,1,1,1}. The fitting effect is shown in Figure 2.

Step 3: Find the corresponding points of the measured data points on the design curve

其中P_i为该子曲线第i个控制点，B_i,3为对应第i个控制顶点的三次Bernstein基函数。对应点可以描述为：求涡轮叶片铸件线激光扫描模型实测点q_r(r＝0,…,m)在设计子曲线C_i(t)上的对应点p_r(r＝0,…,m)，q_r与其对应点p_r之间的距离是最小的，求最小化的函数是：

可以表示为t轴上的Bezier曲线,最小化求解问题即可转化为求零点问题。对

使用deCasteljau算法进行递归细分，同时检查细分后子曲线的控制点是否都在t轴同一侧(即子曲线与t轴无交点)，若同侧则对应区间标记为排除最近点的区间。在排除不合格曲线的同时，利用二叉树方法搜索给定深度的未标记节点区间内的候选点。最后取候选点内最小值作为对应点p_r。The design curve B-spline is subdivided into Bezier curves by de Boor recursive algorithm, the subdivision effect is shown in Figure 3, and the jth subcurve after subdivision is expressed

Where P _i is the i-th control point of the sub-curve, and B _i,3 is the cubic Bernstein basis function corresponding to the i-th control vertex. The corresponding point can be described as: find the corresponding point p _r (r=0,...,m) of the measured point q _r (r=0,...,m) on the design sub-curve C _i (t) of the turbine blade casting line laser scanning model ), the distance between q _r and its corresponding point p _r is the smallest, and the function to find the minimization is:

It can be expressed as a Bezier curve on the t-axis, and the minimization problem can be transformed into a zero-point problem. right

Use the deCasteljau algorithm for recursive subdivision, and at the same time check whether the control points of the subdivided sub-curves are on the same side of the t-axis (that is, the sub-curves do not intersect with the t-axis), and if they are on the same side, the corresponding interval is marked as the interval excluding the nearest point. While excluding unqualified curves, the binary tree method is used to search for candidate points within the unmarked node interval of a given depth. Finally, take the minimum value in the candidate point as the corresponding point p _r .

Step 4: Construct a Free Deformable Mesh

不管控制点世界坐标如何变化，局部坐标

都是固定不变的，假设p₀是局部坐标系原点，p是模型顶点坐标。移动控制点，利用模型顶点局部坐标、控制点世界坐标和Bernstein多项式B_i,m重新计算模型每个顶点的世界坐标：

实现利用网格空间带动曲线变形的效果，如图4所示；其中，u/v是横/纵向网格参数，B_i,m为对应第i个m次的Bernstein基函数，B_j,n为对应第j个n次的Bernstein基函数,P_i,j为(n+1)×(m+1)的向量矩阵，内置序列为(i,j)的网格节点。Construct a two-dimensional local coordinate system, embed the design curve and the measured curve into the coordinate system, and then calculate the local coordinates corresponding to the coordinates of each vertex of the model

Regardless of how the world coordinates of the control points change, the local coordinates

All are fixed, assuming p ₀ is the origin of the local coordinate system, and p is the model vertex coordinates. Move the control point, and use the local coordinates of the model vertices, the world coordinates of the control points and the Bernstein polynomial B _i,m to recalculate the world coordinates of each vertex of the model:

Realize the effect of using the grid space to drive the curve deformation, as shown in Figure 4; where u/v is the horizontal/vertical grid parameter, B _i,m is the Bernstein basis function corresponding to the i-th m order, B _j,n is the Bernstein basis function corresponding to the jth n order, P _i,j is a vector matrix of (n+1)×(m+1), and the built-in sequence is a grid node of (i,j).

Step 5: Iterative calculation of displacement and deformation

如图5所示，第j个参数与其对应点首次偏差

其中t_j是q_r对应参数，则设计曲线B样条上第i个控制点的首个调整向量为

嵌在网格S(u,v)中的一次迭代后曲线控制点P^k可以表示为

其中，u/v是横/纵向网格参数，B_i',m为对应第i’个m次的Bernstein基函数，B_j',n为对应第j’个n次的Bernstein基函数,P_i',j'为(n+1)×(m+1)的向量矩阵，内置序列为(i’,j’)的网格节点；拟合曲线的变形可以用对其控制点操作代替，上式可以转化为

其中s代表控制点个数。利用步骤四的网格变形方法，可以获得新的控制点和新曲线；Taking the measured point q _r (r=0,...,m) of the line laser scanning model of the aero-engine turbine blade casting as an example, at the beginning of the iteration, the design curve B-spline function is used as the first (denoted as the 0th) fitting curve

As shown in Figure 5, the first deviation of the jth parameter from its corresponding point

Where t _j is the corresponding parameter of q _r , then the first adjustment vector of the i-th control point on the design curve B-spline is

The curve control point P ^k embedded in the mesh S(u,v) after one iteration can be expressed as

Among them, u/v is the horizontal/vertical grid parameter, B _{i', m} is the Bernstein basis function corresponding to the i'th m order, B _{j', n} is the Bernstein basis function corresponding to the j'th n order, P _{i', j'} is a vector matrix of (n+1)×(m+1), and the built-in sequence is (i', j') grid nodes; the deformation of the fitting curve can be replaced by the operation on its control points, The above formula can be transformed into

where s represents the number of control points. Using the grid deformation method in step 4, new control points and new curves can be obtained;

该方法依赖于数据点与相同参数曲线上对应点之间的参数距离，迭代运算的参数t为线激光扫描模型实测点计算得到的弦长参数，迭代的是设计曲线B样条的控制点，因此需要使用步骤三的方法使得变形曲线的B样条控制点和实测点对应。Similarly, after the kth iteration, the kth curve f ^k (t) satisfies

This method relies on the parameter distance between the data point and the corresponding point on the same parameter curve. The parameter t of the iterative operation is the chord length parameter calculated from the measured point of the line laser scanning model, and the iterative is the control point of the design curve B-spline. Therefore, the method of step three needs to be used to make the B-spline control points of the deformation curve correspond to the measured points.

Step 6: Use iterative deformation to reverse the deformation of the design curve

利用

对设计模型叶盆叶背曲线进行反变形，以设计模型叶盆曲线进行w次迭代逼近为例，变形后叶盆曲线表达式为

From step 5, the weighted sum of the difference vectors of each iteration approximation can be obtained

use

Inversely deform the leaf back curve of the leaf basin of the design model, and take w times iterative approximation of the leaf basin curve of the design model as an example, the expression of the leaf basin curve after deformation is

The above description is only a preferred embodiment of the present invention and an illustration of the applied technical principles. Those skilled in the art should understand that the scope of disclosure involved in the present invention is not limited to the technical solution formed by the specific combination of the above-mentioned technical features, but also covers other technical solutions formed by any combination of the above-mentioned technical features or their equivalent features. Technical solutions. For example, a technical solution formed by replacing the above-mentioned features with the technical features with similar functions disclosed in the present invention.

Claims (6)

1. The method for designing the molded surface reverse deformation of the turbine blade precision casting mold is characterized by comprising the following steps of:

1) Encrypting discrete points of a curve;

2) B spline fitting a design curve;

3) Solving the corresponding points of the measured data points on the design curve: searching corresponding points based on a curve subdivision method, and subdividing a B spline curve into a group of Bezier curves by using a node insertion algorithm of the B spline curve; subdividing each Bezier curve again in the binary tree searching process, judging whether the condition is met, if so, subdividing again until the curve segment is smaller than a set threshold value;

4) Constructing a free deformation grid: the object is deformed by manipulating the control points of the design grid, and local free deformation can be realized by adopting a B spline basis function with local support;

5) And (3) iteratively calculating displacement deformation: establishing a series of fitting curves by iteratively adjusting control points of the design curve, wherein in each iteration, a difference vector of each control point is a weighted sum of difference vectors of data points of the target curve and corresponding points on the fitting curve, and the weighted sum of the difference vectors is the iterative computation displacement deformation;

6) Performing inverse deformation on the design curve by adopting iterative deformation: under the condition that B spline control points of the design curve correspond to control points of the measured data curve, the accumulated sum of weighted difference vectors of the data points is used as an inverse deformation parameter, a deformation matrix of the control points embedded in the grid is calculated, and the design curve is subjected to inverse deformation through manipulating the grid.

2. The method for designing the reverse deformation of the profile of the turbine blade precision casting mold according to claim 1, wherein in the step 1), the specific method for encrypting the discrete points of the curve is as follows: and gridding a fitting area of a curve to be encrypted, determining the size of an influence area of a grid point x and a node contained in the influence area, calculating a node value at the grid point x after determining a shape function, performing the above processing on each grid point, and encrypting the curve.

3. The turbine blade precision casting mold surface reverse deformation design method as claimed in claim 1, wherein in step 2), the specific method of B-spline fitting the design curve is as follows: and (3) parameterizing discrete points on the design curve according to an accumulative chord length parameterizing method, requiring the head and tail end points of the design curve to pass through the head and tail end points of the B spline curve, fitting the rest of the discrete points on the design curve through the B spline curve by using a least square thought, and iteratively reducing the fitting deviation for multiple times so as to determine the number of control points, thereby completing the fitting of the design curve by using the B spline.

4. The method for designing the reverse deformation of the turbine blade precision casting mold according to claim 1, wherein in the step 3), the specific method for determining the corresponding point of the measured data point on the design curve is as follows: converting the distance from the measured data point to the corresponding point on the design curve into the distance from the measured data point to a group of Bezier curves, and converting the problem of the minimized distance function into the zero point problem of the polynomial function related to the parameter t; and in the binary tree searching process, along with depth subdivision of a Bezier curve, judging whether a polynomial function of a parameter t is on the same side of a t axis, if so, excluding the polynomial function, if not, marking the polynomial function as a candidate point, and finally, taking the minimum value in the candidate points as a corresponding point of a measured data point on a design curve.

5. The method for designing the reverse deformation of the profile of a turbine blade casting mold according to claim 1, wherein in the step 4), the free deformation mesh is constructed, and the design simple mesh is defined as an axial mesh with a control point mesh based on a two-dimensional B spline and is expressed as a B spline surface.

6. The method for designing the reverse deformation of the profile of the turbine blade precision casting mold according to claim 1, wherein in the step 4), the constructing of the free deformation mesh comprises constructing a two-dimensional free deformation mesh containing a model, and the specific steps are as follows:

(1) Constructing a local two-dimensional coordinate system, calculating local coordinates corresponding to each vertex coordinate of the model, and embedding the curve model into a frame;

(2) And moving the control point based on the ternary Bernstein multivariate power function, recalculating the world coordinate of each vertex of the model by using the local coordinates of the vertices of the model, the world coordinates of the control point and the Bernstein polynomial, and pulling the model by the framework to realize deformation.

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