CN110263384B - Three-dimensional mesh curved surface variable-thickness bias modeling method based on Laplacian differential domain deformation - Google Patents

Three-dimensional mesh curved surface variable-thickness bias modeling method based on Laplacian differential domain deformation Download PDF

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CN110263384B
CN110263384B CN201910451010.XA CN201910451010A CN110263384B CN 110263384 B CN110263384 B CN 110263384B CN 201910451010 A CN201910451010 A CN 201910451010A CN 110263384 B CN110263384 B CN 110263384B
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杨璐
张长东
刘婷婷
施昕
王聪
管志方
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Nanjing University of Science and Technology
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Abstract

The invention discloses a three-dimensional mesh curved surface variable-thickness bias modeling method based on Laplacian differential domain deformation, which comprises the following steps of: constructing an initial offset curved surface based on the significant characteristic information and a Poisson curved surface reconstruction method; utilizing finite element calculation to analyze the stress condition of the offset shell, and dividing a wall thickness adjusting space according to the stress level of each unit body; and optimizing the wall thickness by adopting a Laplacian differential domain deformation method. According to the method, the Morse significant feature points of the original surface are extracted, the offset point cloud is constructed, and the point cloud is reconstructed into the triangular mesh curved surface based on the Poisson method, so that the basic form and the detail features of the original model can be maintained; the Laplacian differential domain deformation method is adopted to thicken the high stress region and thin the low stress region, so that the stress concentration phenomenon can be improved, and the bearing capacity of the structure is improved.

Description

Three-dimensional mesh curved surface variable-thickness bias modeling method based on Laplacian differential domain deformation
Technical Field
The invention belongs to the technical field of additive manufacturing, and particularly relates to a three-dimensional mesh curved surface variable-thickness bias modeling method based on Laplacian differential domain deformation.
Background
Nowadays, research hot tide of digital and intelligent manufacturing is rapidly developing all over the world. The digital design is represented by three-dimensional model scanning and modeling, and the intelligent manufacturing is represented by additive manufacturing, so that the combination of the three methods enriches the methods for capturing and presenting real life, and the design and manufacturing of the complex three-dimensional model in the real world are realized. However, to customize a three-dimensional scanning curved surface by a 3D printing device, it is necessary to convert it into a solid model, such as a closed curved surface model, or a shell model with thickness. In the face of a highly complex real scene, how to perform efficient and accurate materialization processing on a three-dimensional scanning curved surface enables an obtained entity model to have the morphological characteristics which meet the use environment and meet the expected functional requirements, and the method is a hot research content. As shown in fig. 1, since a general representation form of a three-dimensional scanning surface is a discrete mesh, and a triangular mesh surface has become an industry standard for 3D printing software interface data, the present invention is mainly directed to one of the discrete surfaces, i.e., a triangular mesh surface.
As a key link in the process of materializing the discrete curved surface, the quality of the offset modeling method directly influences the quality of the design quality of the entity model. The curved surface is subjected to offset operation, so that on one hand, the single-layer closed curved surface can be converted into an entity with a cavity, and the material requirement in the printing process is reduced; on the other hand, the single-layer non-closed curved surface can be directly converted into the shell, and the modeling and function requirements of the customized product are met.
The existing biasing methods can be mainly divided into direct biasing and implicit biasing. The direct offset has the disadvantage that the triangular patch is easy to self-cross when the offset distance is large, which affects the ordering of the slice outline and further causes quality problems of the real object processing. Although the implicit offset can avoid self-crossing, the implicit offset also has the defect when the offset distance is larger, and the generated offset curved surface is easy to become over-rounded or sharp at the position with large curvature change, and has obvious detail characteristic difference compared with the original curved surface. The two methods mainly aim at the reduction of the original curved surface appearance by the offset curved surface, but do not consider the real stress condition and cannot ensure the mechanical property. Under actual conditions, due to the influence of factors such as external force action and self structure, the stress distribution of parts is usually uneven, and stress concentration and even fracture accidents are likely to occur. Therefore, the research of the offset modeling method needs not only to avoid the selfing problem of the triangular plate and restore the original curved surface morphology as much as possible, but also to meet the required mechanical property requirement in the structural design, and the thickness of the shell can be adaptively adjusted according to the applied load so as to reduce the local stress increase phenomenon in the material increase manufacturing process.
Disclosure of Invention
The invention aims to provide a three-dimensional mesh curved surface variable-thickness offset modeling method based on Laplacian differential domain deformation.
The technical solution for realizing the purpose of the invention is as follows: a three-dimensional mesh curved surface variable-thickness bias modeling method based on Laplacian differential domain deformation comprises the following steps:
step 1, constructing an initial offset curved surface based on the significant characteristic information and a Poisson curved surface reconstruction method;
step 2, utilizing finite element calculation to analyze the stress condition of the offset shell, and dividing a wall thickness adjusting space according to the stress level of each unit body;
and 3, optimizing the wall thickness by adopting a Laplacian differential domain deformation method.
Compared with the prior art, the invention has the following remarkable advantages: (1) the method combines vertex bias and implicit curved surface bias to realize initial equidistant bias processing of the triangular mesh curved surface, and avoids generating triangular plate selfing; (2) according to the method, the Morse significant feature points of the original surface are extracted, the offset point cloud is constructed, and the point cloud is reconstructed into the triangular mesh curved surface based on the Poisson method, so that the basic form and the detail features of the original model can be maintained; (3) the stress condition of the offset shell is analyzed through finite elements, and high and low stress areas are divided according to the stress values; (4) according to the invention, a Laplacian differential domain deformation method is adopted to thicken the high stress region and thin the low stress region, so that the stress concentration phenomenon can be improved, and the bearing capacity of the structure is improved.
The present invention is described in further detail below with reference to the attached drawing figures.
Drawings
FIG. 1 is a schematic diagram of a triangular mesh surface model.
FIG. 2 is a schematic diagram of Morse salient feature point extraction.
Fig. 3 is a schematic representation of the poisson reconstruction process.
Fig. 4 is a schematic diagram of a stress region dividing method.
Fig. 5 is a schematic diagram of an optimization technical route of the structure of the offset shell.
Fig. 6 is a schematic view showing an example of the design of the non-uniform variable thickness structure.
Detailed Description
The method mainly aims at the design requirement of a variable-thickness three-dimensional shell model, adopts a method of geometric modeling and finite element analysis cooperative calculation, firstly constructs an initial offset curved surface based on significant characteristic information and a Poisson curved surface reconstruction method, secondly analyzes the stress condition of the offset shell by utilizing finite element calculation, secondly divides a wall thickness adjusting space according to the stress of each unit body, and finally optimizes the wall thickness by adopting a Laplacian differential domain deformation method, so that the final offset shell with non-uniform wall thickness not only meets the material increase manufacturing standard, but also can ensure the structural strength. The invention provides a new structure optimization method for the design of the three-dimensional model for additive manufacturing, and has important engineering application value.
A three-dimensional mesh curved surface variable-thickness bias modeling method based on Laplacian differential domain deformation comprises the following steps:
step 1, constructing an initial offset curved surface based on the significant characteristic information and a Poisson curved surface reconstruction method; the method specifically comprises the following steps:
step 1-1, reading a three-dimensional discrete surface model, and extracting Morse significant feature points;
(1) calculating the Morse function value f (v) of the vertex v of the original mesh surface by using the average curvature:
Figure GDA0002311998050000021
wherein the maximum and minimum principal curvatures k of each vertexmax、kmin；
(2) Using a Gaussian filter function WcAnd a feature retention function WsCalculating a bilateral filtered value of f (v), B (f (v), r):
Figure GDA0002311998050000022
wherein r represents the size of the neighborhood radius of the vertex of the curved surface of the mesh, N (v,2r) represents a group of vertexes which are 2r away from the vertex, and x is any point in the group of vertexes;
(3) for each mesh surface vertex, calculating the weighted average of the mesh surface vertex and the Morse function bilateral filter value of the neighborhood vertex, namely the weighted average is the significant value s of the vertex:
Figure GDA0002311998050000031
(4) calculating the sum of the initial Morse function value and the significant value obtained in the previous stepSo as to update the characteristic value of each mesh surface vertex. This step is iteratively performed until the number of feature points meets the requirements.
The process of extracting the salient feature points from the mesh surface according to the steps is shown in FIG. 2, and the number of the feature points is represented by the letter N
The number of iterations is indicated by the letter k. It can be seen from the figure that the feature points extracted by this method are a kind of mesh vertices describing the most significant features of the original surface model, and the number of the feature points gradually decreases as the number of iterations increases.
Step 1-2, biasing the Morse significant feature points and constructing an initial biased point cloud
And (3) moving the Morse significant characteristic points extracted in the step (1-1) for equal distance along the normal vector direction or the reverse direction of the normal vector, and constructing a bias point set. Setting the characteristic point set of the original curved surface as V ═ V1，v2，…，vnAnd the unit normal vector set corresponding to each feature point is N ═ N1，n2，…，nnGiven an offset distance d, calculating the coordinates of the offset feature points
v′i＝vi±nid，1≤i≤n
Wherein the signs represent different bias directions. If the offset is outward along the normal vector direction, adding; if the bias is inward along the normal vector direction, the subtraction is performed. Composition bias point set V ' ═ V ' after calculation is completed '1，v′2，…v′n}。
Step 1-3, reconstructing initial offset point cloud into triangular mesh curved surface by Poisson reconstruction
Suppose that a region M and its boundaries are given
Figure GDA0002311998050000033
The index function χ is defined as
Figure GDA0002311998050000034
ReconstructionThe problem of (a) is converted into the problem of reconstructing χ, so that the point cloud and the indicative function can be associated with each other.
(1) And obtaining an integral relation of the point cloud and the indicating function according to the gradient relation. For any point
Figure GDA0002311998050000036
Definition ofIs the normal vector of the point inwards
Figure GDA0002311998050000038
Is a smoothing filter, then
Figure GDA0002311998050000039
Is composed of
Figure GDA00023119980500000310
Translating along a p-point normal vector, wherein q is a point output after the p-point is subjected to smooth filtering; in addition, χ is approximated by the derivative of χ x F.
Figure GDA00023119980500000311
(2) And obtaining a vector field of the point cloud by using a partitioning method according to an integral relation, and approximately calculating a gradient field of the indicating function. Segmenting a point cloud S into mutually disjoint regions AsWith AsHas an area of approximately AsThe integral of (c) can be approximated by summing the integrals of the divided regions. And each small integral can be approximated as a constant function and replaced by the product of the filter function value and the area corresponding to point p.
Figure GDA00023119980500000312
(3) And solving the poisson equation. Vector space
Figure GDA0002311998050000041
And an indication function
Figure GDA0002311998050000042
The following equation relationship is satisfied:
Figure GDA0002311998050000043
the two sides of the equal number are respectively derived to obtain a Laplace equation:
Figure GDA0002311998050000044
solving the above partial differential equation problem requires discretization of the object. After a Poisson reconstruction algorithm partitions a space, a node set is defined as O, and a function space is defined as FoThen vector space
Figure GDA0002311998050000045
Can be approximately expressed as
Figure GDA0002311998050000046
Where ng(s) is the nearest 8 neighbors of s, αo,sIs the three-line interpolation weight, Fo(q) F for node o close to qoLinear and function of (c).
Therefore, the equation is further approximately simplified, and finally, an indication function is obtained by solving
Figure GDA0002311998050000047
And then selecting the mean value of the point cloud sample coordinates as an equivalent, calculating a corresponding equivalent surface, and further obtaining a reconstructed triangular mesh curved surface which is an initial offset curved surface:
Figure GDA0002311998050000048
wherein
Figure GDA0002311998050000049
R3Representing a three-dimensional vector space.
Step 2, utilizing finite element calculation to analyze the stress condition of the offset shell, and dividing a wall thickness adjusting space according to the stress level of each unit body, wherein the method specifically comprises the following steps:
(1) tetrahedral meshing is carried out on the offset shell by utilizing a TetGen meshing tool;
(2) stress analysis is carried out on the offset shell by utilizing an OOFEM finite element calculation library;
(3) dividing high and low stress areas of the offset shell; reasonably setting two stress thresholds S according to the stress value of each unit node obtained by the stress analysis in the previous stepHAnd SL(SH>SL) Dividing the shell model into three parts, namely high stress areas (the stress value is higher than S)H) Low stress region (stress value lower than S)L) And transition regions (stress values in between) as shown in fig. 5.
Step 3, optimizing the wall thickness by adopting a Laplacian differential domain deformation method, which specifically comprises the following steps:
(1) and establishing a mathematical model of the optimization problem. The optimal situation of optimization is to keep the surface shapes before and after optimization as similar as possible, so a penalty function is needed to correct the geometric deviation of the shapes before and after optimization, and the detailed characteristics of the original surface can be kept with high reduction degree. The optimization objective function and the constraint condition are defined as follows:
min:H(x)＝C(x)+λ*D(M,M0)
s.t.[K]{u}＝{P}
g(M,M0)＝0
Figure GDA00023119980500000410
|V-V0|≤ε
wherein x represents tetrahedron, C (x) represents compliance of the shell model; d (M, M)0) Representing the geometric deviation of the shape before and after optimization; λ is the balance coefficient between compliance and geometric deviation; [ K ]]A stiffness matrix representing a model, { u } represents a displacement vector, { P } represents a static load vector; g (M, M)0) Representing the geometrical relationship between the vertices before and after the optimization of the mesh surface; v0And V represents the model volume before and after optimization, x, respectivelyiCorresponding to the i-th unit cell, viThe volume change of the unit body of the ith unit needs to be controlled within a certain range epsilon so as to meet the design requirement.
(2) And realizing the self-adaptive deformation of the thickness of the offset shell by adopting a Laplacian differential domain deformation method. And respectively setting the deformation direction and the deformation amount of the grid vertexes of the inner surface in different stress division regions: high stress region along vertex normal vector dnDeformed outwardly in the direction of the distance r1(ii) a Low stress region, along vertex normal vector dnDeformed inward in the opposite direction by a distance r2(ii) a The transition region remains unchanged.
The transformed vertex coordinates are:
v′i＝vi±dn·r
the corresponding deformation energy function is:
Figure GDA0002311998050000051
wherein, deltai、δi' Laplacian coordinates, u, representing the original and deformed mesh vertices, respectivelyiCoordinates representing the constraint points; m represents the number of constraint points, r represents the deformation of the vertex in each iteration process, and subscripts 1 and 2 respectively represent the vertexes of the high and low stress regions. w is ai、wjA weight factor representing the deformation, the smaller this factor, the easier the corresponding vertex moves. Therefore, for the outer surface of the shell needing to be fixed in position and other constraint points, a larger weight factor can be set to keep the position of the shell unchanged; for the region needing deformation, a smaller weight factor can be set according to the stress magnitudeIn order to optimize the thickness.
The present invention will be described in detail below with reference to examples and the accompanying drawings.
Examples
A three-dimensional mesh curved surface variable-thickness bias modeling method based on Laplacian differential domain deformation comprises the following steps:
step 1, reading a three-dimensional discrete surface model, and extracting Morse significant feature points, as shown in FIG. 2;
(1) calculating the Morse function value f (v) of the vertex v of the original mesh surface by using the average curvature:
Figure GDA0002311998050000052
wherein the maximum and minimum principal curvatures k of each vertexmax、kminSuch characteristic information can be solved by fitting a surface. The cubic polynomial used to fit a surface can be represented in the form:
z＝Ax3+Bx2y+Cxy2+Dy3+Ex2+Fxy+Gy2+Hx+ly
(2) using a Gaussian filter function WcAnd a feature retention function WsCalculating a bilateral filtered value of f (v), B (f (v), r):
where r represents the neighborhood radius size of the vertex of the mesh, N (v,2r) represents a set of vertices that are 2r away from the vertex, and x is any point in the set.
(3) For each grid vertex, calculating the weighted average of the grid vertex and the Morse function bilateral filter value of the neighborhood vertex, namely the weighted average is the significant value s of the vertex:
Figure GDA0002311998050000061
(4) calculating the sum of the initial Morse function value and the significant value obtained in the previous step
Figure GDA0002311998050000062
Thereby updating the feature value of each vertex. This step is iteratively performed until the number of feature points meets the requirements.
According to the steps, the process of extracting the salient feature points from the mesh surface is shown in fig. 2, the number of the feature points is represented by a letter N, and the iteration times is represented by a letter k. It can be seen from the figure that the feature points extracted by this method are a kind of mesh vertices describing the most significant features of the original surface model, and the number of the feature points gradually decreases as the number of iterations increases.
Step 2, biasing the Morse significant feature points and constructing an initial biased point cloud
And (3) moving the salient feature points extracted in the step (1) for equal distance along the normal vector direction or the reverse direction of the normal vector, and constructing a bias point set. Setting the characteristic point set of the original curved surface as V ═ V1，v2，…，vnAnd the unit normal vector set corresponding to each feature point is N ═ N1，n2，…，nnGiven an offset distance d, calculating the coordinates of the offset feature points
v′i＝vi±nid，1≤i≤n
Wherein the signs represent different bias directions. If the offset is outward along the normal vector direction, adding; if the bias is inward along the normal vector direction, the subtraction is performed. Composition bias point set V ' ═ V ' after calculation is completed '1，v′2，…v′n}。
Step 3, reconstructing the initial offset point cloud into a triangular mesh curved surface by using Poisson reconstruction, as shown in FIG. 3;
poisson reconstruction belongs to a category of point cloud reconstruction methods based on implicit functions, and the mathematical basis of the Poisson reconstruction is the Poisson equation. In the information contained in the directional point cloud system, the point cloud represents the position of a curved surface, and the normal vector of the point cloud represents the inside and outside directions of the curved surface, which is the core idea of poisson reconstruction.
Suppose that a region M and its boundaries are givenThe index function χ is defined as
Figure GDA0002311998050000064
Reconstruction
Figure GDA0002311998050000065
The problem of (a) is converted into the problem of reconstructing χ, so that the point cloud and the indicative function can be associated with each other.
(1) And obtaining an integral relation of the point cloud and the indicating function according to the gradient relation. For any point
Figure GDA0002311998050000066
Definition of
Figure GDA0002311998050000067
Is the normal vector of the point inwards
Figure GDA0002311998050000068
Is a smoothing filter, thenIs composed of
Figure GDA00023119980500000610
And (5) translating along a p-point normal vector, wherein q is a point output after the p point is subjected to smooth filtering. In addition, χ is approximated by the derivative of χ x F.
Figure GDA00023119980500000611
(2) And obtaining a vector field of the point cloud by using a partitioning method according to an integral relation, and approximately calculating a gradient field of the indicating function. Segmenting a point cloud S into mutually disjoint regions AsWith AsHas an area of approximately AsThe integral of (c) can be approximated by summing the integrals of the divided regions. And each small integral can be approximated as a constant function and replaced by the sum of filter function values corresponding to the point pThe product of the area of the regions.
Figure GDA0002311998050000071
(3) And solving the poisson equation. Vector space
Figure GDA0002311998050000072
And an indication function
Figure GDA0002311998050000073
The following equation relationship is satisfied:
Figure GDA0002311998050000074
the two sides of the equal number are respectively derived to obtain a Laplace equation:
Figure GDA0002311998050000075
solving the above partial differential equation problem requires discretization of the object. After a Poisson reconstruction algorithm partitions a space, a node set is defined as O, and a function space is defined as FoThen vector spaceCan be approximately expressed as
Figure GDA0002311998050000077
Where ng(s) is the nearest 8 neighbors of s, αo,sIs the three-line interpolation weight, Fo(q) F for node o close to qoLinear and function of (c).
Therefore, the equation is further approximately simplified, and finally, an indication function is obtained by solving
Figure GDA0002311998050000078
Then selecting the mean value of the point cloud sample coordinates as the equivalence, and calculating the corresponding equivalent surfaceAnd further obtaining a reconstructed curved surface:
Figure GDA0002311998050000079
whereinR3Representing a three-dimensional vector space.
Step 4, bias shell finite element analysis is carried out, and high and low stress regions are divided
(1) Tetrahedral meshing of offset enclosures using a TetGen meshing tool
(2) Stress analysis of offset casing using OOFEM finite element computation library
(3) And dividing the bias shell into high and low stress areas. Reasonably setting two stress thresholds S according to the stress value of each unit node obtained by the stress analysis in the previous stepHAnd SL(SH>SL) Dividing the shell model into three parts, namely high stress areas (the stress value is higher than S)H) Low stress region (stress value lower than S)L) And transition regions (stress values in between) as shown in fig. 4.
Step 5, self-adaptive optimization of the thickness of the offset shell
(1) And establishing a mathematical model of the optimization problem. The optimal situation of optimization is to keep the surface shapes before and after optimization as similar as possible, so a penalty function is needed to correct the geometric deviation of the shapes before and after optimization, and the detailed characteristics of the original surface can be kept with high reduction degree. The optimization objective function and the constraint condition are defined as follows:
min:H(x)＝C(x)+λ*D(M，M0)
s.t·[K]{u}＝{P}
g(M，M0)＝0
Figure GDA00023119980500000711
|y-y0|≤ε
wherein x represents a tetrahedronAnd C (x) represents the compliance of the shell model; d (M, M)0) Representing the geometric deviation of the shape before and after optimization; λ is the balance coefficient between compliance and geometric deviation; [ K ]]A stiffness matrix representing the model, { u } represents a displacement vector,
{ P } represents the static load vector; g (M, M)0) Representing the geometrical relationship between the vertices before and after the optimization of the mesh surface; v0And V represents the model volume before and after optimization, respectively, ViThe volume change of the unit body of the ith unit needs to be controlled within a certain range epsilon so as to meet the design requirement.
(2) And realizing the self-adaptive deformation of the thickness of the offset shell by adopting a Laplacian differential domain deformation method. And respectively setting the deformation direction and the deformation amount of the grid vertexes of the inner surface in different stress division regions: high stress region along vertex normal vector dnDeformed outwardly in the direction of the distance r1(ii) a Low stress region, along vertex normal vector dnDeformed inward in the opposite direction by a distance r2(ii) a The transition region remains unchanged.
The transformed vertex coordinates are:
v′i＝vi±dn·r
the corresponding deformation energy function is:
wherein, deltai、δi' Laplacian coordinates representing the original and deformed mesh vertices, respectively; u. ofiCoordinates representing the constraint points; m represents the number of the constraint points, and r represents the deformation of the vertex in each iteration process; subscripts 1, 2 refer to the vertices of the high and low stress regions, respectively. w is ai、wjA weight factor representing the deformation, the smaller this factor, the easier the corresponding vertex moves. Therefore, for the outer surface of the shell needing to be fixed in position and other constraint points, a larger weight factor can be set to keep the position of the shell unchanged; for the area needing deformation, a smaller weight factor can be set according to the stress magnitude so as to optimize the thickness.
The steps 1-3 are an initial biasing process of the three-dimensional discrete curved surface, the steps 4-5 are an optimization process of the biasing shell structure, the finite element analysis of the step 4 is to provide an optimization pretreatment data base for the step 5, the technical route of the adaptive thickness optimization method is shown in fig. 5, and finally, the non-uniform variable thickness structure design example is shown in fig. 6.

Claims (2)

1. A three-dimensional mesh curved surface variable-thickness bias modeling method based on Laplacian differential domain deformation is characterized by comprising the following steps:
step 1, constructing an initial offset curved surface based on the significant characteristic information and a Poisson curved surface reconstruction method;
step 2, utilizing finite element calculation to analyze the stress condition of the offset shell, and dividing a wall thickness adjusting space according to the stress level of each unit body;
step 3, optimizing the wall thickness by adopting a Laplacian differential domain deformation method;
the step 1 specifically comprises the following steps:
step 1-1, reading a three-dimensional discrete surface model, and extracting Morse significant feature points;
(1) calculating the Morse function value f (v) of the vertex v of the curved surface of all the original meshes by using the average curvature:
Figure FDA0002311998040000012
kmax、kminmaximum and minimum principal curvatures for each vertex;
(2) using a Gaussian filter function WcAnd a feature retention function WsCalculating a bilateral filtered value of f (v), B (f (v), r):
Figure FDA0002311998040000013
wherein r is the size of the neighborhood radius of the vertex of the curved surface of the mesh, N (v,2r) is a group of vertexes which are 2r away from the vertex, and x is any point in the group of vertexes;
(3) for each mesh surface vertex, calculating the weighted average of the mesh surface vertex and the Morse function bilateral filter value of the neighborhood vertex, namely the weighted average is the significant value s of the vertex:
(4) calculating the sum of the initial Morse function value and the significant value obtained in the previous step
Figure FDA0002311998040000015
Updating the characteristic value of each mesh surface vertex; iteratively executing the step until the number of the characteristic points meets the requirement;
step 1-2, biasing the Morse significant feature points, and constructing an initial biased point cloud;
moving the Morse significant characteristic points extracted in the step 1-1 for equal distance along the normal vector direction or the reverse direction of the normal vector, and constructing a bias point set; setting the characteristic point set of the original curved surface as V ═ V1，v2，…，vnAnd the unit normal vector set corresponding to each feature point is N ═ N1，n2，…，nnGiven an offset distance d, the coordinates of the offset feature points are calculated:
v′i＝vi±nid，1≤i≤n
wherein, the signs represent different bias directions; if the offset is outward along the normal vector direction, adding; if the offset is inward along the normal vector direction, subtracting; composition bias point set V ' ═ V ' after calculation is completed '1，v′2，…v′n}；
1-3, reconstructing the initial offset point cloud into a triangular mesh curved surface by using Poisson reconstruction;
suppose that a region M and its boundaries are given
Figure FDA0002311998040000016
The index function χ is defined as
Figure FDA0002311998040000011
ReconstructionThe problem of (2) can be converted into the problem of reconstructing chi, so that the point cloud and the indicating function can be associated;
(1) obtaining an integral relation between the point cloud and the indicating function according to the gradient relation; for any point
Figure FDA0002311998040000027
Definition of
Figure FDA0002311998040000028
Is the normal vector of the point inwards
Figure FDA00023119980400000211
Is a smoothing filter, thenIs composed of
Figure FDA00023119980400000210
Translating along a p-point normal vector, wherein q is a point output after the p-point is subjected to smooth filtering; approximating χ with a derivative of χ x F;
Figure FDA0002311998040000021
(2) obtaining a vector field of the point cloud by using a partitioning method according to an integral relation, and approximately calculating a gradient field of an indication function; segmenting a point cloud S into mutually disjoint regions AsWith AsHas an area of approximately AsThe integral of (a) can be approximately calculated by integrating and summing the divided regions;
(3) solving a Poisson equation; vector space
Figure FDA00023119980400000212
And an indication function
Figure FDA00023119980400000213
The following equation relationship is satisfied:
Figure FDA00023119980400000214
the two sides of the equal number are respectively derived to obtain a Laplace equation:
Figure FDA00023119980400000215
after a Poisson reconstruction algorithm partitions a space, a node set is defined as O, and a function space is defined as FoThen vector space
Figure FDA00023119980400000217
Can be approximately expressed as:
Figure FDA0002311998040000023
where ng(s) is the nearest 8 neighbors of s, αo，sIs the three-line interpolation weight, Fo(q) F for node o close to qoA linear sum function of;
therefore, the equation is further approximately simplified, and finally, an indication function is obtained by solving
Figure FDA00023119980400000216
And then selecting the mean value of the point cloud sample coordinates as an equivalent, calculating a corresponding equivalent surface, and further obtaining a reconstructed triangular mesh curved surface which is an initial offset curved surface:
Figure FDA0002311998040000024
wherein
Figure FDA0002311998040000025
R3Representing a three-dimensional vector space;
the step 3 specifically comprises the following steps:
(1) establishing a mathematical model of an optimization problem;
the optimization objective function and the constraint condition are defined as follows:
min：H(x)＝C(x)+λ*D(M，M0)
s.t.[K]{u}＝{P}
g(M，M0)＝0
Figure FDA0002311998040000026
|V-V0|≤ε
wherein x represents tetrahedron, C (x) represents compliance of the shell model; d (M, M)0) Representing the geometric deviation of the shape before and after optimization; λ is the balance coefficient between compliance and geometric deviation; [ K ]]A stiffness matrix representing a model, { u } represents a displacement vector, { P } represents a static load vector; g (M, M)0) Representing the geometrical relationship between the vertices before and after the optimization of the mesh surface; v0And V represents the model volume before and after optimization, x, respectivelyiCorresponding to the i-th unit cell, viThe volume corresponding to the ith unit body;
(2) the adaptive deformation of the thickness of the offset shell is realized by adopting a LaDlacian differential domain deformation method;
and respectively setting the deformation direction and the deformation amount of the grid vertexes of the inner surface in different stress division regions: high stress region along vertex normal vector dnDeformed outwardly in the direction of the distance r1(ii) a Low stress region, along vertex normal vector dnDeformed inward in the opposite direction by a distance r2(ii) a The transition area remains unchanged; the transformed vertex coordinates are:
v′i＝vi±dn·r
the corresponding deformation energy function is:
Figure FDA0002311998040000031
wherein, deltai、δi' Laplacian coordinates, u, representing the original and deformed mesh vertices, respectivelyiCoordinates representing the constraint points; m represents the number of the constraint points, r represents the deformation of the vertex in each iteration process, and subscripts 1 and 2 respectively represent the vertexes of the high and low stress regions; w is ai、wjA weight factor representing the deformation.
2. The Laplacian differential domain deformation-based three-dimensional mesh surface variable-thickness bias modeling method as claimed in claim 1, wherein the step 2 specifically comprises:
(1) tetrahedral meshing is carried out on the offset shell by utilizing a TetGen meshing tool;
(2) stress analysis is carried out on the offset shell by utilizing an OOFEM finite element calculation library;
(3) dividing high and low stress areas of the offset shell; setting two stress thresholds S according to the stress value of each unit node obtained by the stress analysis in the previous stepHAnd SL，SH＞SLDividing the shell model into three parts, wherein the stress value is higher than SHIs a high stress region with a stress value lower than SLIs a low stress region and between the two is a transition region.
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