CN110263384A - Three-dimensional grid curved surface Varying-thickness based on Laplacian differential area deformation biases formative method - Google Patents
Three-dimensional grid curved surface Varying-thickness based on Laplacian differential area deformation biases formative method Download PDFInfo
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Abstract
The invention discloses a kind of, and the three-dimensional grid curved surface Varying-thickness based on Laplacian differential area deformation biases formative method, comprising the following steps: constructs initial bias curved surface based on notable feature information and Poisson curve reestablishing method；Using the stress condition of finite element method (fem) analysis biasing shell, wall thickness is divided according to the stress of each cell cube height and adjusts space；Wall thickness is optimized using Laplacian differential domain deformation method.The present invention constructs bias point cloud by extracting the Morse remarkable characteristic of initial surface, then a cloud is redeveloped into triangle mesh curved surface based on Poisson method, is able to maintain archetype grown form and minutia；The region of high stress is thickeied using Laplacian differential domain deformation method, low stress area is thinned, and can improve stress concentration phenomenon, the bearing capacity of lift structure.
Description
Technical field
The invention belongs to increases material manufacturing technology fields, and in particular to a kind of three-dimensional based on Laplacian differential area deformation
Grid surface Varying-thickness biases formative method.
Background technique
Nowadays, the research boom of numerical and intelligent manufacture whole world rapid development.Design of digital is with three-dimensional mould
Type scans and is modeled as representing, and using increasing material manufacturing as representative, the combination of the two enriches capture and presents true intelligent manufacturing
Grow directly from seeds method living, realizes complex three-dimensional model in the design and manufacture of real world.However, to pass through 3D printing device customizing
Process 3-D scanning curved surface, it is necessary to be translated into physical model, such as closed surface model, or have the shell mould of thickness
Type.In face of highly complex real scene, how efficiently accurate hypostazation processing, the reality made are done to 3-D scanning curved surface
Body Model not only has the shape characteristic for meeting use environment, and reaches expected functional requirement, is in a hot research
Hold.As shown in Figure 1, since the general representation of 3-D scanning curved surface is discrete grid block, while triangle mesh curved surface has become
The industry fiducial mark of 3D printing software interface data is quasi-, thus present invention is generally directed to the one of which of Discrete Surfaces, the i.e. triangulation network
Lattice curved surface.
As the key link during Discrete Surfaces hypostazation, the superiority and inferiority for biasing formative method directly affects physical model
The quality of designing quality.Operation is biased to curved surface, one side single layer of encapsulation curved surface can be converted into the entity of cavity, beat
Materials demand during print is minimized；On the other hand, single layer non-close curved surface can be converted into shell, meet customization
The moulding of product and functional requirement.
Existing biasing means can be divided mainly into directly biasing and implicit two kinds of biasing.Directly the shortcomings that biasing, is when inclined
Set apart from it is larger when tri patch be easy selfing, this will affect the order of slicing profile, and then lead to the in kind quality processed
Problem.Although and implicitly bias and be avoided that selfing, disadvantage is equally existed when offset or dish is larger, the screw rotor of generation exists
The big position of Curvature varying is easy to become excessively round blunt or sharp, and comparison original surface has apparent minutia difference.This
Two methods are primarily directed to reduction of the screw rotor to original surface pattern, but do not account for true stress condition,
Mechanical property cannot be guaranteed.Under actual working conditions, due to the influence of the factors such as external force, self structure, part stress
It is distributed usually non-uniform, it is possible to produce stress is concentrated or even fracture accident.Therefore, the research of formative method is biased not only
It needs that triangular plate is avoided to be selfed problem, as far as possible reduction original surface pattern, while the design of its structure is also required to meet required mechanics
Performance requirement, can be according to the load being applied come adaptive adjustment housings thickness, to reduce stress office during increasing material manufacturing
Increase phenomenon in portion.
Summary of the invention
The purpose of the present invention is to provide a kind of three-dimensional grid curved surface Varying-thickness based on Laplacian differential area deformation is inclined
Set formative method.
The technical solution for realizing the aim of the invention is as follows: a kind of three-dimensional grid based on Laplacian differential area deformation
Curved surface Varying-thickness biases formative method, comprising the following steps:
Step 1 constructs initial bias curved surface based on notable feature information and Poisson curve reestablishing method；
Step 2, the stress condition that shell is biased using finite element method (fem) analysis, are drawn according to the stress of each cell cube height
Bulkhead thickness adjusts space；
Step 3 optimizes wall thickness using Laplacian differential domain deformation method.
Compared with prior art, the present invention its remarkable advantage are as follows: (1) present invention is biased using vertex biasing with implicit surface
The method combined realizes the initial iso-metric offset processing of triangle mesh curved surface, avoids generating triangular plate selfing；(2) present invention is logical
The Morse remarkable characteristic for extracting initial surface is crossed, and constructs bias point cloud, then a cloud is redeveloped by triangle based on Poisson method
Grid surface is able to maintain archetype grown form and minutia；(3) present invention biases shell by finite element analysis
Stress condition divides high low stress zones according to stress value size；(4) present invention uses Laplacian differential domain deformation method
Come thicken the region of high stress, be thinned low stress area, stress concentration phenomenon, the bearing capacity of lift structure can be improved.
Present invention is further described in detail with reference to the accompanying drawing.
Detailed description of the invention
Fig. 1 is triangular gridding curve model schematic diagram.
Fig. 2 is that Morse remarkable characteristic extracts schematic diagram.
Fig. 3 is Poisson reconstruction process schematic diagram.
Fig. 4 is stress area division methods schematic diagram.
Fig. 5 is biasing shell structure optimisation technique route schematic diagram.
Fig. 6 is Non-homogeneous Variable Thickness example of Design schematic diagram.
Specific embodiment
Emphasis of the present invention is directed to the design requirement of Varying-thickness three-dimensional shells model, is assisted using geometric modeling and finite element analysis
With the method calculated, it is primarily based on notable feature information and Poisson curve reestablishing method building initial bias curved surface, secondly benefit
With the stress condition of finite element method (fem) analysis biasing shell, wall thickness is then divided according to the stress of each cell cube height and adjusts sky
Between, wall thickness is finally optimized using Laplacian differential domain deformation method, to make the biasing shell of final non-uniform wall thickness not
Only meet increasing material manufacturing standard, while also ensuring structural strength.The present invention is that the threedimensional model design towards increasing material manufacturing mentions
For a kind of new structural optimization method, there is important engineering application value.
A kind of three-dimensional grid curved surface Varying-thickness biasing formative method based on Laplacian differential area deformation, including it is following
Step:
Step 1 constructs initial bias curved surface based on notable feature information and Poisson curve reestablishing method；Specifically:
Step 1-1,3 d-dem surface model is read, Morse remarkable characteristic is extracted；
(1) the Morse functional value f (v) of all original mesh curved surface vertex vs is calculated with average curvature:
The wherein minimum and maximum principal curvatures k on each vertexmax、kmin；
(2) Gaussian filter function W is utilizedcFunction W is kept with features, calculate the bilateral filtering value B (f (v), r) of f (v):
Wherein r indicates the radius of neighbourhood size on the grid surface vertex, and N (v, 2r) is indicated apart from the one of vertex distance 2r
Group vertex, x are any point in this group of vertex；
(3) to each grid surface vertex, it is flat with the weighting of the Morse function bilateral filtering value on neighborhood vertex to calculate it
, the as saliency value s on the vertex:
(4) the sum of saliency value obtained by initial Morse functional value and upper step is calculatedIt is bent that each grid is updated with this
The characteristic value of vertex of surface.Iteration executes this step, until characteristic point quantity meets the requirements.
According to above-mentioned steps, notable feature point process is extracted as shown in Fig. 2, characteristic point quantity letter N to grid surface
It indicates, the number of iterations is indicated with letter k.As seen from the figure, it is original that the characteristic point extracted by this method is that one kind describes
The grid vertex of surface model most significant feature, and as the number of iterations increases, quantity gradually decreases.
Step 1-2, Morse remarkable characteristic is biased, initial bias point cloud is constructed
By the extracted Morse remarkable characteristic of step 1-1, opposite direction is sweared along its direction of normal or method, movement is mutually equidistant
From building biasing point set.If original surface feature point set is V={ v1, v2..., vn, the corresponding per unit system of each characteristic point
Arrow collection is combined into N={ n1, n2..., nn, an offset or dish d is given, then calculating the coordinate of characteristic point after biasing
v′i=vi±niD, 1≤i≤n
Wherein, sign represents biased direction difference.If doing addition along direction of normal, that is, outwardly biased；If being sweared along method
Direction i.e. inwardly biasing, then do subtraction.Composition biasing point set V '={ v ' after the completion of calculating1, v '2... v 'n}。
Step 1-3, it is rebuild using Poisson, initial bias point cloud is redeveloped into triangle mesh curved surface
Assuming that a given region M and its boundaryIndicator function χMIt is defined as
ReconstructThe problem of i.e. can be exchanged into reconstruct χMThe problem of, make to contact between invocation point cloud and indicator function
Get up.
(1) integral relation of cloud and indicator function is obtained by gradient relation.For arbitrary pointDefinitionFor the normal vector that the point is inside,For a smoothing filter, thenForAlong p point normal direction
The translation of amount, q are the point that p point exports after smothing filtering；In addition, withDerivative come approximate χM。
(2) vector field of point cloud, the ladder of approximate calculation indicator function are obtained using the method for divided block according to integral relation
Spend field.A cloud Ω is divided into mutually disjoint regionAbove formula can be by integrating the region of segmentation
Summation carrys out approximate calculation.And it is each it is small integral can be approximated to be normal function, replace with the corresponding filter function value of point s.p and
The product of region area.
(3) Poisson's equation is solved.Vector spaceAnd indicator functionMeet following equilibrium relationships:
Derivation is distinguished to equal sign both sides, Laplace's equation can be obtained:
Above-mentioned partial differential equation problem is solved to need to do sliding-model control to object.After Poisson algorithm for reconstructing divides space,
Defining its node set is O, function space Fo, then vector spaceCan be with approximate representation
Wherein, Ng (s) is 8 nearest neighbors of s, αo,sIt is three line interpolation weights.
Thus it does further approximation to equation to simplify, final solve obtains indicator functionReselection point cloud sample coordinate
Mean value as equivalent, the triangle mesh curved surface for calculating corresponding contour surface, and then being rebuild, as initial bias are bent
Face:
Wherein,R3Indicate three-dimensional vector space.
Step 2, the stress condition that shell is biased using finite element method (fem) analysis, are drawn according to the stress of each cell cube height
Bulkhead thickness adjusts space；Specifically:
(1) tetrahedral grid division is carried out to biasing shell using TetGen grid dividing tool；
(2) stress analysis is carried out to biasing shell using OOFEM FEM calculation library；
(3) high low stress zones division is carried out to biasing shell；The each unit node obtained according to previous step stress analysis
Stress value size, two stress threshold S of reasonable setHAnd SL(SH>SL), Shell model is divided into three parts, respectively height is answered
(stress value is higher than S in power regionH), low stress zones (stress value be lower than SL) and transitional region (stress value falls between),
As shown in Figure 5.
Step 3 optimizes wall thickness using Laplacian differential domain deformation method, specifically:
(1) mathematical model of optimization problem is established.The surface shape for being most ideally maintained optimization front and back of optimization is most
It measures similar, it is therefore desirable to correct the geometrical deviation of optimization front and back shape using penalty, could retain original to high reduction degree
The minutia on beginning surface.Optimization object function and constraint condition are defined as follows:
Min:H (x)=C (x)+λ * D (M, M0)
S.t. [K] { u }={ P }
g(M,M0)=0
|V-V0|≤ε
Wherein, x indicates that tetrahedron, C (x) indicate the flexibility of Shell model；D(M,M0) indicate to optimize the geometry of front and back shape
Deviation；λ is the coefficient of balance between flexibility and geometrical deviation；[K] indicates that the stiffness matrix of model, { u } indicate motion vector,
{ P } indicates dead load vector；g(M,M0) indicate that grid surface optimizes the geometrical relationship between the vertex of front and back；V0It is respectively indicated with V
Optimize the model volume of front and back, viThe volume of corresponding i-th of cell cube, volume change need to control in a certain range ε, to meet
Design requirement.
(2) the adaptive deformation of biasing thickness of shell is realized using Laplacian differential domain deformation method.For inner surface
Different stress divide the grid vertex in region, its deformation direction and deflection: the region of high stress are set separately, along vertex normal vector dnSide
To outside deformation, distance r1；Low stress area, then along vertex normal vector dnOpposite direction is deformed inward, distance r2；Transitional region is kept not
Become.Apex coordinate after deformation are as follows:
v′i=vi±dn·r
Corresponding strain energy of distortion function are as follows:
Wherein, δi、δi' respectively indicate the Laplacian coordinate of original and deformed grid vertex, uiIndicate obligatory point
Coordinate；M indicates the number of obligatory point, and r indicates the deflection of each iterative process in vertex, and subscript 1,2 respectively refers to answer for high and low
The vertex in power area.wi、wjIndicate the weight factor of deformation, this factor is smaller, the easier movement of corresponding vertex.Therefore, for needing
Fix position housing outer surface and other obligatory points, settable biggish weight factor, to keep the constant of its position
Property；For the region that needs deform, lesser weight factor can be set according to stress intensity, in order to optimize thickness.
Below by embodiment and attached drawing, the present invention is described in detail.
Embodiment
A kind of three-dimensional grid curved surface Varying-thickness biasing formative method based on Laplacian differential area deformation, including it is following
Step:
Step 1 reads 3 d-dem surface model, extracts Morse remarkable characteristic, as shown in Figure 2；
(1) the Morse functional value f (v) of all original mesh curved surface vertex vs is calculated with average curvature:
The wherein minimum and maximum principal curvatures k on each vertexmax、kmin, this kind of characteristic information can by fitting surface come
It solves.Following form can be used to indicate for the cubic polynomial of fitting surface:
Z=Ax3+Bx2y+Cxy2+Dy3+Ex2+Fxy+Gy2+Hx+ly
(2) Gaussian filter function W is utilizedcFunction W is kept with features, calculate the bilateral filtering value B (f (v), r) of f (v):
Wherein r indicates the radius of neighbourhood size of the grid vertex, and N (v, 2r) indicates one group of top apart from vertex distance 2r
Point, x are any point in this group of vertex.
(3) to each grid vertex, the weighted average of itself and the Morse function bilateral filtering value on neighborhood vertex is calculated, i.e.,
For the saliency value s on the vertex:
(4) the sum of saliency value obtained by initial Morse functional value and upper step is calculatedEach vertex is updated with this
Characteristic value.Iteration executes this step, until characteristic point quantity meets the requirements.
According to above-mentioned steps, notable feature point process is extracted as shown in Fig. 2, characteristic point quantity letter N to grid surface
It indicates, the number of iterations is indicated with letter k.As seen from the figure, it is original that the characteristic point extracted by this method is that one kind describes
The grid vertex of surface model most significant feature, and as the number of iterations increases, quantity gradually decreases.
Step 2, biasing Morse remarkable characteristic, construct initial bias point cloud
By the extracted remarkable characteristic of step 1, opposite direction, mobile equidistance, building are sweared along its direction of normal or method
Play biasing point set.If original surface feature point set is V={ v1, v2..., vn, the corresponding per unit system arrow collection of each characteristic point is combined into
N={ n1, n2..., nn, an offset or dish d is given, then calculating the coordinate of characteristic point after biasing
v′i=vi±niD, 1≤i≤n
Wherein, sign represents biased direction difference.If doing addition along direction of normal, that is, outwardly biased；If being sweared along method
Direction i.e. inwardly biasing, then do subtraction.Composition biasing point set V '={ v ' after the completion of calculating1, v '2... v 'n}。
Step 3 is rebuild using Poisson, initial bias point cloud is redeveloped into triangle mesh curved surface, as shown in Figure 3；
Poisson reconstruction belongs to one kind based on implicit function in a cloud method for reconstructing, and Fundamentals of Mathematics are exactly Poisson's equation.?
In the information that orientation point cloud system is included, the position of point cloud representation curved surface, the method arrow for putting cloud indicates the inward-outward direction of curved surface, this
It is the core concept that Poisson is rebuild.
Assuming that a given region M and its boundaryIndicator function χMIt is defined as
ReconstructThe problem of i.e. can be exchanged into reconstruct χMThe problem of, make to contact between invocation point cloud and indicator function
Get up.
(1) integral relation of cloud and indicator function is obtained by gradient relation.For arbitrary pointDefinitionFor the normal vector that the point is inside,For a smoothing filter, thenForAlong p point normal direction
The translation of amount, q are the point that p point exports after smothing filtering.In addition, withDerivative come approximate χM。
(2) vector field of point cloud, the ladder of approximate calculation indicator function are obtained using the method for divided block according to integral relation
Spend field.A cloud Ω is divided into mutually disjoint regionAbove formula can be by integrating the region of segmentation
Summation carrys out approximate calculation.And it is each it is small integral can be approximated to be normal function, replace with the corresponding filter function value of point s.p and
The product of region area.
(3) Poisson's equation is solved.Vector spaceAnd indicator functionMeet following equilibrium relationships:
Derivation is distinguished to equal sign both sides, Laplace's equation can be obtained:
Above-mentioned partial differential equation problem is solved to need to do sliding-model control to object.After Poisson algorithm for reconstructing divides space,
Defining its node set is O, function space Fo, then vector spaceCan be with approximate representation
Wherein, Ng (s) is 8 nearest neighbors of s, αo,sIt is three line interpolation weights.
Thus it does further approximation to equation to simplify, final solve obtains indicator functionReselection point cloud sample coordinate
Mean value as equivalent, calculate corresponding contour surface, and then obtain rebuilding curved surface:
Wherein,R3Indicate three-dimensional vector space.
Step 4, biasing Shell Finite Element Method analysis, divide high low stress area
(1) tetrahedral grid division is carried out to biasing shell using TetGen grid dividing tool
(2) stress analysis is carried out to biasing shell using OOFEM FEM calculation library
(3) high low stress zones division is carried out to biasing shell.The each unit node obtained according to previous step stress analysis
Stress value size, two stress threshold S of reasonable setHAnd SL(SH>SL), Shell model is divided into three parts, respectively height is answered
(stress value is higher than S in power regionH), low stress zones (stress value be lower than SL) and transitional region (stress value falls between),
As shown in Figure 4.
Step 5, biasing thickness of shell adaptive optimization
(1) mathematical model of optimization problem is established.The surface shape for being most ideally maintained optimization front and back of optimization is most
It measures similar, it is therefore desirable to correct the geometrical deviation of optimization front and back shape using penalty, could retain original to high reduction degree
The minutia on beginning surface.Optimization object function and constraint condition are defined as follows:
Min:H (x)=C (x)+λ * D (M, M0)
S.t [K] { u }={ P }
G (M, M0)=0
|y-y0|≤ε
Wherein, x indicates that tetrahedron, C (x) indicate the flexibility of Shell model；D(M,M0) indicate to optimize the geometry of front and back shape
Deviation；λ is the coefficient of balance between flexibility and geometrical deviation；[K] indicates that the stiffness matrix of model, { u } indicate motion vector,
{ P } indicates dead load vector；g(M,M0) indicate that grid surface optimizes the geometrical relationship between the vertex of front and back；V0It is respectively indicated with V
Optimize the model volume of front and back, viThe volume of corresponding i-th of cell cube, volume change need to control in a certain range ε, to meet
Design requirement.
(2) the adaptive deformation of biasing thickness of shell is realized using Laplacian differential domain deformation method.For inner surface
Different stress divide the grid vertex in region, its deformation direction and deflection: the region of high stress are set separately, along vertex normal vector dnSide
To outside deformation, distance r1；Low stress area, then along vertex normal vector dnOpposite direction is deformed inward, distance r2；Transitional region is kept not
Become.Apex coordinate after deformation are as follows:
v′i=vi±dn·r
Corresponding strain energy of distortion function are as follows:
Wherein, δi、δi' respectively indicate the Laplacian coordinate of original and deformed grid vertex；uiIndicate obligatory point
Coordinate；M indicates the number of obligatory point, and r indicates the deflection of each iterative process in vertex；Subscript 1,2 respectively refers to answer for high and low
The vertex in power area.wi、wjIndicate the weight factor of deformation, this factor is smaller, the easier movement of corresponding vertex.Therefore, for needing
Fix position housing outer surface and other obligatory points, settable biggish weight factor, to keep the constant of its position
Property；For the region that needs deform, lesser weight factor can be set according to stress intensity, in order to optimize thickness.
Above-mentioned steps 1-3 is 3 d-dem curved surface initial bias process, and step 4-5 is biasing shell structure optimization process,
Wherein the finite element analysis of step 4 is that optimization pre-processing data basis, the technology of adaptive thickness optimization method are provided for step 5
Route is as shown in figure 5, last Non-homogeneous Variable Thickness example of Design is as shown in Figure 6.
Claims (4)
1. a kind of three-dimensional grid curved surface Varying-thickness based on Laplacian differential area deformation biases formative method, which is characterized in that
The following steps are included:
Step 1 constructs initial bias curved surface based on notable feature information and Poisson curve reestablishing method；
Step 2, the stress condition that shell is biased using finite element method (fem) analysis, according to the stress height dividing wall of each cell cube
Thickness adjusts space；
Step 3 optimizes wall thickness using Laplacian differential domain deformation method.
2. the three-dimensional grid curved surface Varying-thickness according to claim 1 based on Laplacian differential area deformation biases moulding
Method, which is characterized in that step 1 specifically:
Step 1-1,3 d-dem surface model is read, Morse remarkable characteristic is extracted；
(1) the Morse functional value f (v) of all original mesh curved surface vertex vs is calculated using average curvature:
kmax、kminFor the minimum and maximum principal curvatures on each vertex；
(2) Gaussian filter function W is utilizedcFunction W is kept with features, calculate the bilateral filtering value B (f (v), r) of f (v):
Wherein r is the radius of neighbourhood size on the grid surface vertex, and N (v, 2r) is one group of vertex apart from vertex distance 2r, x
For any point in this group of vertex；
(3) to each grid surface vertex, the weighted average of itself and the Morse function bilateral filtering value on neighborhood vertex is calculated, i.e.,
For the saliency value s on the vertex:
(4) the sum of saliency value obtained by initial Morse functional value and upper step is calculatedEach grid surface top is updated with this
The characteristic value of point；Iteration executes this step, until characteristic point quantity meets the requirements；
Step 1-2, Morse remarkable characteristic is biased, initial bias point cloud is constructed；
By the extracted Morse remarkable characteristic of step 1-1, opposite direction, mobile equidistance, structure are sweared along its direction of normal or method
Build up biasing point set；If original surface feature point set is V={ v1, v2..., vn, the corresponding per unit system arrow set of each characteristic point
For N={ n1, n2..., nn, an offset or dish d is given, then calculating the coordinate of characteristic point after biasing:
v′i=vi±niD, 1≤i≤n
Wherein, sign represents biased direction difference；If doing addition along direction of normal, that is, outwardly biased；If along direction of normal
That is inwardly biasing, then do subtraction；Composition biasing point set V '={ v ' after the completion of calculating1, v '2... v 'n}；
Step 1-3, it is rebuild using Poisson, initial bias point cloud is redeveloped into triangle mesh curved surface；
Assuming that a given region M and its boundaryIndicator function χMIt is defined as
ReconstructThe problem of i.e. can be exchanged into reconstruct χMThe problem of, make to have contacted between invocation point cloud and indicator function
Come；
(1) integral relation of cloud and indicator function is obtained by gradient relation；For arbitrary pointDefinitionFor this
The inside normal vector of point,For a smoothing filter, thenForAlong the translation of p point normal vector, q
The point exported after smothing filtering for p point；WithDerivative come approximate χM；
(2) vector field of point cloud, the gradient fields of approximate calculation indicator function are obtained using the method for divided block according to integral relation；
A cloud Ω is divided into mutually disjoint regionS ∈ Ω, above formula can be by carrying out integral summation to the region of segmentation come approximate
It calculates；
(3) Poisson's equation is solved；Vector spaceAnd indicator functionMeet following equilibrium relationships:
Derivation is distinguished to equal sign both sides, Laplace's equation can be obtained:
After Poisson algorithm for reconstructing divides space, defining its node set is O, function space Fo, then vector spaceIt can be approximate
It is expressed as
Wherein, Ng (s) is 8 nearest neighbors of s, αO, sIt is three line interpolation weights；
Thus it does further approximation to equation to simplify, final solve obtains indicator functionReselection point cloud sample coordinate it is equal
Value is as equivalent, the triangle mesh curved surface for calculating corresponding contour surface, and then being rebuild, as initial bias curved surface:
Wherein,R3Indicate three-dimensional vector space.
3. the three-dimensional grid curved surface Varying-thickness according to claim 1 based on Laplacian differential area deformation biases moulding
Method, which is characterized in that step 2 specifically:
(1) tetrahedral grid division is carried out to biasing shell using TetGen grid dividing tool；
(2) stress analysis is carried out to biasing shell using OOFEM FEM calculation library；
(3) high low stress zones division is carried out to biasing shell；The each unit node stress obtained according to previous step stress analysis
It is worth size, sets two stress threshold SHAnd SL, SH> SL, Shell model is divided into three parts, stress value is higher than SHIt is height
Stress area, stress value are lower than SLIt is low stress zones, it is transitional region that stress value, which falls between,.
4. the three-dimensional grid curved surface Varying-thickness according to claim 1 based on Laplacian differential area deformation biases moulding
Method, which is characterized in that step 3 specifically:
(1) mathematical model of optimization problem is established；
Optimization object function and constraint condition are defined as follows:
Min:H (x)=C (x)+λ * D (M, M0)
S.t. [K] { u }={ P }
G (M, M0)=0
|V-V0|≤ε
Wherein, x indicates that tetrahedron, C (x) indicate the flexibility of Shell model；D (M, M0) indicate to optimize the geometrical deviation of front and back shape；
Coefficient of balance of the λ between flexibility and geometrical deviation；[K] indicates that the stiffness matrix of model, { u } indicate that motion vector, { P } indicate
Dead load vector；G (M, M0) indicate that grid surface optimizes the geometrical relationship between the vertex of front and back；V0Optimization front and back is respectively indicated with V
Model volume, viThe volume of corresponding i-th of cell cube；
(2) the adaptive deformation of biasing thickness of shell is realized using LaDlacian differential domain deformation method；
The grid vertex that region is divided for inner surface difference stress, is set separately its deformation direction and deflection: the region of high stress,
Along vertex normal vector dnDirection deforms outward, distance r1；Low stress area, then along vertex normal vector dnOpposite direction is deformed inward, distance r2；
Transitional region remains unchanged；Apex coordinate after deformation are as follows:
v′i=vi±dn·r
Corresponding strain energy of distortion function are as follows:
Wherein, δi、δi' respectively indicate the Laplacian coordinate of original and deformed grid vertex, uiIndicate the seat of obligatory point
Mark；M indicates the number of obligatory point, and r indicates the deflection of each iterative process in vertex, and subscript 1,2 is respectively referred to for high and low stressed zone
Vertex；wi、wjIndicate the weight factor of deformation.
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