CN110210151A - Lattice structure based on B-spline parameterizes implicit modeling and optimization method - Google Patents
Lattice structure based on B-spline parameterizes implicit modeling and optimization method Download PDFInfo
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Abstract
The invention proposes a kind of, and the lattice structure based on B-spline parameterizes implicit modeling and optimization method, several fundamental lattice forms are defined first, the basic combination dot matrix of then definition, then building covers region-wide B-spline interpolation curved surface, substantially combine this in dot matrix as the control point of B-spline surface, any point corresponding one on two-dimensional structure as combining composite lattice made of dot matrix interpolation substantially.During lattice design, the parameter by changing each control point dot matrix can achieve the purpose of design total region dot matrix.This method directlys adopt known dot matrix pattern as Basic Design element and carries out modeling and optimization, can be obtained a plurality of types of dot matrix patterns using less variable.And dot matrix pattern freely changes with locations of structures is different, and optimum results boundary is smooth, comprising explicit parameter information, can be with active computer Autocad Seamless integration-.
Description
Technical field
The present invention relates to a kind of modeling and optimization method of lattice structure, especially a kind of lattice structure based on B-spline
Parameterize implicit modeling and optimization method.
Background technique
It explores new material and designs the hot topic that new structure pattern is the new century always.The birth of 3D printing technique with
It continues to develop and provides convenience for the design and manufacture of anisotropic material and lattice structure.Meanwhile shape and topological optimization field
Experts and scholars setting for material property and internal lattice structure is increasingly focused on while optimizing structure monnolithic case
Meter.Lattice structure design is widely used sciemtifec and technical sphere ahead of the curve and engineering because its design freedom is high, fine structure degree is good
Using in the middle.Therefore, how to design practical and effective lattice structure becomes a big hot topic in structure optimization field.
Document " A modified model for concurrent topology optimization of
structures and materials.Xingjun Gao,Haitao Ma[J].Acta Mech.Sin.(2015)31(6):
890-898 " a kind of cooperative optimization method of multiple dimensioned dot matrix unit cell is disclosed.Macroscopic view uses solid respectively to same with micromodel
Property material penalty method (Solid isotropic material with penalty, SIMP), using the pseudo- density of unit as
Design variable.The method use the microstructure of single form, the rotation angle that lattice structure is added becomes as new design
Amount.But the lattice design mode that document uses lacks unified parameter characterization, the design variable Numerous that single dot matrix includes
And grid is depended on, so that lattice structure engineering characteristics are poor, it is difficult to and existing computer aided design software Seamless integration-.
Summary of the invention
In order to overcome the problems, such as that shortage unified parameters of the existing technology characterization, design variable number are more, the present invention is mentioned
A kind of implicit modeling and optimization method of lattice structure parametrization based on B-spline is gone out.
This method defines several basic latticed forms (such as matrix pattern, rice font), each fundamental lattice packet first
Containing respective characterising parameter (length and width of such as thin bar), and described by respective level set function.The basic combination of then definition
Dot matrix, the dot matrix are the boolean sum of all fundamental lattices, and characterising parameter includes the parameter of each fundamental lattice.Then, one is constructed
A region-wide B-spline interpolation curved surface of covering combines this in dot matrix as the control point of B-spline surface substantially, and therefore, two dimension is tied
Any point on structure corresponds to one as combining composite lattice made of dot matrix interpolation substantially.Homogenization method is used as connecting macro
A kind of homogeneous material can be equivalent to based on macro stiffness matrix with microcosmic bridge, any form of dot matrix by seeing
It calculates.During lattice design, the parameter by changing each control point dot matrix can achieve design total region dot matrix
Purpose.Compared to the design method of background technique, this method directly adopt known dot matrix pattern as Basic Design element into
Row modeling and optimization can be obtained a plurality of types of dot matrix patterns using less variable.And dot matrix pattern is with locations of structures
Difference freely changes, and optimum results boundary is smooth, comprising explicit parameter information, can be with active computer Autocad
Seamless integration-.
Based on the above principles, the technical solution of the present invention is as follows:
A kind of lattice structure based on B-spline parameterizes implicit modeling and optimization method, it is characterised in that: including with
Lower step:
Step 1: choosing m fundamental lattice pattern, each fundamental lattice pattern is made of several basic configurations, according to every
The characterising parameter of a basic configuration is described fundamental lattice using implicit modeling method, determines the water of each fundamental lattice
Flat set function Φi, wherein the level set function of each fundamental lattice is the level set for forming each basic configuration of the fundamental lattice
The boolean sum of function；The characterising parameter is divided into preset parameter and n design parameter；
Step 2: fundamental lattice being combined into a basic combination dot matrix, calculates the level set function Φ of basic combination dot matrix
With Equivalent Elasticity matrix DH；
The level set function Φ of basic combination dot matrix is the level set function of each fundamental lattice of the basic combination dot matrix of composition
Boolean sum；
The Equivalent Elasticity matrix D of basic combination dot matrixHIt is obtained using following procedure:
According to homogenization method, dot matrix will be combined substantially and divides finite element grid, then Equivalent Elasticity matrix DHIt can be by following formula
It calculates:
D in formula11 H~D33 HFor 9 components of Equivalent Elasticity matrix, Q11~Q33It is basic combined spot under different equivalent operating condition
The sum of each unit strain energy in battle array；I, j is enabled to take 1,2,3, Q respectivelyijBy formula
It is calculated, in formula | Y | indicate basic combination dot matrix volume, e is that the finite element grid divided is numbered, e=1,
2 ..., E, ui eWithIt is motion vector of e-th of finite element grid under i-th and the equivalent operating condition of jth kind respectively, by limited
First equation solution obtains, keIt is the element stiffness matrix of e-th of finite element grid, by formula
B is the element strain matrix of finite element grid, Ω in calculating formulaeFor the unit area of e-th of finite element grid, D0
For the material matrix of homogeneous material, α is one indivisible；Periodic boundary condition, solving finite element are applied to basic combination dot matrix
Equation, obtains motion vector, finally acquires Equivalent Elasticity matrix DH；
Step 3: for macroscopical physical model to be optimized, being divided into the finite element interpolation grid based on B-spline, net
Lattice number of nodes is Nx × Ny × Nz；A basic combination dot matrix is assigned at each grid node, and according to the method meter of step 2
Calculate the Equivalent Elasticity matrix for combining dot matrix at each grid node substantially；
Step 4: macroscopical physical model to be optimized is having a size of Lx × Ly × Lz, again to macroscopical physical model to be optimized
FEM meshing is carried out, the stiffness matrix of entire macroscopical physical model is calculated using following procedure:
Step 4.1: according to the following formula
Calculate the elastic matrix D of each gridH(X, Y, Z), wherein (X, Y, Z) is grid element center coordinate, p is B-spline
Order, Ni,p(ξ)、Nj,p(η) and Nk,p(ζ) is B-spline shape function,To be combined substantially at grid node in step 3 (i, j, k)
The Equivalent Elasticity matrix of dot matrix；
Step 4.2: according to formula
The stiffness matrix K of entire macroscopical physical model is calculated, wherein Ω is entire macroscopical physical model region；
Step 5: the design parameter for choosing all basic combination dot matrix is the design variable of topology optimization problem, each basic
Combining dot matrix includes m × n design variable, total to obtain Nx × Ny × Nz × m × n design variable；
Step 6: applying boundary condition and load on the finite element model that step 4 is established, establish the mechanics of macrostructure
Model；Choose the minimum optimization aim of macrostructure compliance, total volume fraction ratio of macrostructure with combine lattice structure substantially
Divide than being used as constraint function, sets design variable initial value and variation range, establish the Optimized model of lattice structure design problem；
Step 7: the Optimized model established to step 6 optimizes, and obtains optimum results.
Further preferred embodiment, a kind of implicit modeling and optimization method of lattice structure parametrization based on B-spline,
It is characterized by: the characterising parameter in step 1 includes the center position x of basic configurationc、yc, length l, width w, radius r and/
Or the rotation angle [alpha] around central point.
Further preferred embodiment, a kind of implicit modeling and optimization method of lattice structure parametrization based on B-spline,
It is characterized by: when basic configuration is hyperelliptic, level set function φskAre as follows:
Further preferred embodiment, a kind of implicit modeling and optimization method of lattice structure parametrization based on B-spline,
It is characterized by: when basic configuration is bowlder, level set function φckAre as follows:
Further preferred embodiment, a kind of implicit modeling and optimization method of lattice structure parametrization based on B-spline,
It is characterized by: B-spline shape function is all made of following form, wherein N in step 4I, p(ξ) is
In formula, ξi∈{ξ1,ξ2,...,ξNx+p+1By Nx-p-1 node equally distributed on (0,1) and in ξ=0
With the p+1 duplicate node composition at ξ=1；And for Nj,p(η), then replace with j for the i in above-mentioned form, and ξ replaces with η, Nx
Replace with Ny；For Nk,pI in above-mentioned form is then replaced with k by (ζ), and ξ replaces with ζ, and Nx replaces with Nz.
Further preferred embodiment, a kind of implicit modeling and optimization method of lattice structure parametrization based on B-spline,
It is characterized by: the Optimized model of the lattice structure design problem in step 6 are as follows:
find xc,yc,w,l,r,α
J is macrostructure compliance in formula, and U is the motion vector of macrostructure, and F is macrostructure load vectors, gvtWith
gVRespectively indicate basic combination lattice structure point than with the total volume fraction ratio of structure；VΩtCombine the volume of dot matrix substantially for t-th,
ΩtFor the dot matrix region of t-th of basic combination dot matrix, VΩFor entire macrostructure volume, NelFor macrostructure mechanical model
In grid number, velemFor the volume of the elem grid in macrostructure mechanical model,It respectively indicates given total
Than the upper limit, H indicates Heaviside function for the volume fraction ratio upper limit and basic combination lattice structure point.
Beneficial effect
The beneficial effect of the present invention compared with prior art is:
The method of the present invention, as optimization design basic element, passes through adjusting dot matrix using the dot matrix of several form knowns
Control parameter achievees the purpose that design dot matrix.Different from traditional parametric method based on unit or boundary, this method is direct
It is parameterized for existing dot matrix configuration, by being combined, adjusting to these configurations, is freely converted out a variety of suitable
Dot matrix, adaptation to local conditions realize the lattice design in global scope.And it can be convenient using the dot matrix for having form in engineering
With computer aided design software Seamless integration-.In addition, the dot matrix Interpolation using B-spline ensure that the height in whole region
Rank continuity realizes that dot matrix configuration carries out gradient gradual change with position difference, ensure that the connectivity between adjacent dot matrix.Table
Sign aspect, lattice structure are all made of level set function and describe outer boundary, the smooth no sawtooth in boundary.
Additional aspect and advantage of the invention will be set forth in part in the description, and will partially become from the following description
Obviously, or practice through the invention is recognized.
Detailed description of the invention
Above-mentioned and/or additional aspect of the invention and advantage will become from the description of the embodiment in conjunction with the following figures
Obviously and it is readily appreciated that, in which:
Fig. 1 is the fundamental lattice structure and its control parameter schematic diagram that the method for the present invention uses.
Fig. 2 is the schematic diagram that fundamental lattice is constituted basic combination dot matrix by the method for the present invention.
Fig. 3 is the B-spline dot matrix interpolation grid schematic diagram that the method for the present invention is located on macroscopic design domain.
Fig. 4 is the lattice structure general illustration that the method for the present invention institute interpolation obtains.It is different using 3 kinds respectively in figure
Interpolation grid, the dot matrix of available 3 kinds different patterns.
Fig. 5 is design domain, boundary condition and the operating condition of the method for the present invention parametrization lattice structure.
Fig. 6 be the method for the present invention parametrization lattice structure taken under above-mentioned operating condition fundamental lattice form, size and its
Control parameter.
Fig. 7 is the initial configuration effect picture of the method for the present invention parametrization lattice structure, indicates original state lattice structure
Overall effect.
Fig. 8 is the optimum results effect picture of the method for the present invention parametrization lattice structure, and lattice structure is whole after expression optimization
Body effect.
Specific embodiment
The embodiment of the present invention is described below in detail, the embodiment is exemplary, it is intended to it is used to explain the present invention, and
It is not considered as limiting the invention.
Referring to Fig. 5~Fig. 8.The present invention is used to parameterize the implicit modeling and optimization design of lattice structure.Consider that structure is set
Meter area size is 90mm × 30mm, and region left side horizontal direction is fixed, and lower right vertical direction is fixed.Region upper left side by
One downward concentrated force F=1000N effect.The Young's modulus and Poisson's ratio for the solid material that lattice structure uses are respectively E
=20GPa, ν=0.3.
Step 1, for optimization problem shown in fig. 5, choose m=3 kind fundamental lattice pattern, each dot matrix pattern is by several
Hyperelliptic bar composition, comprising respective width d as design variable, i.e. n=1.Basic combination dot matrix schematic diagram is as shown in Figure 6.
The hyperelliptic initial parameter that it is included is as shown in the table.
Table 1
Dot matrix is described using implicit modeling method, each hyperelliptic level set function φskMeet following relationship
The then level set function Φ of fundamental latticeiIt can be expressed as
Φi=max (φsk) (2)
Step 2: 2 fundamental lattices being combined into a basic combination dot matrix, calculate the level set of basic combination dot matrix
Function phi and Equivalent Elasticity matrix DH。
The level set function Φ of basic combination dot matrix is the level set function of each fundamental lattice of the basic combination dot matrix of composition
Boolean sum:
Φ=Φ1∪Φ2...∪Φi...∪Φm=max (Φi) (3)
The Equivalent Elasticity matrix D of basic combination dot matrixHIt is obtained using following procedure:
According to homogenization method, dot matrix will be combined substantially and divides finite element grid, then Equivalent Elasticity matrix DHIt can be by following formula
It calculates:
D in formula11 H~D33 HFor 9 components of Equivalent Elasticity matrix, Q11~Q33It is basic combined spot under different equivalent operating condition
The sum of each unit strain energy in battle array；I, j is enabled to take 1,2,3, Q respectivelyijBy formula
It is calculated, in formula | Y | indicate basic combination dot matrix volume, e is the finite element grid number divided,WithIt is motion vector of e-th of finite element grid under i-th and the equivalent operating condition of jth kind respectively, passes through
Finite element equation solves to obtain, keIt is the element stiffness matrix of e-th of finite element grid, by formula
B is the element strain matrix of finite element grid, Ω in calculating formulaeFor the unit area of e-th of finite element grid, D0
For the material matrix of homogeneous material, α is one indivisible, takes α=10 here-3；Periodic boundary item is applied to basic combination dot matrix
Part, solving finite element equation obtain motion vector, finally acquire Equivalent Elasticity matrix DH。
Step 3: for macroscopical physical model to be optimized, being divided into the finite element interpolation grid based on B-spline, net
Lattice number of nodes is Nx × Ny × Nz；A basic combination dot matrix is assigned at each grid node, and according to the method meter of step 2
Calculate the Equivalent Elasticity matrix for combining dot matrix at each grid node substantially.
It is illustrated in the present embodiment using two dimensional model, size Nx × Ny=9 × 3 of grid.
Step 4: macroscopical physical model to be optimized is having a size of Lx × Ly × Lz, again to macroscopical physical model to be optimized
FEM meshing is carried out, the stiffness matrix of entire macroscopical physical model is calculated using following procedure:
Step 4.1: according to the following formula
Calculate the elastic matrix D of each gridH(X, Y, Z), wherein (X, Y, Z) is grid element center coordinate, p is B-spline
Order, Ni,p(ξ)、Nj,p(η) and Nk,p(ζ) is B-spline shape function,To be combined substantially at grid node in step 3 (i, j, k)
The Equivalent Elasticity matrix of dot matrix；
Step 4.2: according to formula
The stiffness matrix K of entire macroscopical physical model is calculated, wherein Ω is entire macroscopical physical model region.
In the present embodiment, size Lx × Ly=90 × 30 in two dimensional model macroscopic design domain, B-spline shape function be all made of with
2 rank B-splines of lower form, wherein Ni,p(ξ) is
In formula, ξi∈{ξ1,ξ2,...,ξNx+p+1By Nx-p-1 node equally distributed on (0,1) and in ξ=0
With the p+1 duplicate node composition at ξ=1；And for Nj,p(η), then replace with j for the i in above-mentioned form, and ξ replaces with η, Nx
Replace with Ny.
Step 5: the design parameter for choosing all basic combination dot matrix is the design variable of topology optimization problem, each basic
Combining dot matrix includes m × n design variable, total to obtain Nx × Ny × Nz × m × n design variable.
Nx × Ny × m × n=81 design variable is shared in the present embodiment.
Step 6: applying boundary condition and load on the finite element model that step 4 is established, establish the mechanics of macrostructure
Model；Choose the minimum optimization aim of macrostructure compliance, total volume fraction ratio of macrostructure with combine lattice structure substantially
Divide than being used as constraint function, sets design variable initial value and variation range, establish the Optimized model of lattice structure design problem.
In the present embodiment, the Optimized model of lattice structure design problem are as follows:
find xc,yc,w,l,r,α
J is macrostructure compliance in formula, and U is the motion vector of macrostructure, and F is macrostructure load vectors, gvtWith
gVRespectively indicate basic combination lattice structure point than with the total volume fraction ratio of structure；VΩtCombine the volume of dot matrix substantially for t-th,
ΩtFor the dot matrix region of t-th of basic combination dot matrix, VΩFor entire macrostructure volume, NelFor macrostructure mechanical model
In grid number, velemFor the volume of the elem grid in macrostructure mechanical model,It respectively indicates given total
Than the upper limit, H indicates Heaviside function for the volume fraction ratio upper limit and basic combination lattice structure point.
Step 7: the Optimized model established to step 6 optimizes, and obtains optimum results.
The lattice structure schematic diagram difference of optimization front and back is as shown in Figure 7,8.It can be found that using the dot matrix based on B-spline
The implicit modeling and optimization method of structure parameterization obtains the dot matrix configuration of change of gradient, and structure explicitly includes that hyperelliptic shape is special
Sign, can be directly integrated with computer aided design software.Structure entirety compliance and the volume fraction ratio for optimizing front and back are as shown in table 2.
The result shows that the lattice structure based on B-spline, which parameterizes implicit modeling and optimization method, can design clear effective dot matrix
Structure improves the rigidity of structure, solves the problems, such as that shortage unified parameters characterization of the existing technology, design variable number are more.
Table 1
| Structure compliance (kJ) | The total volume fraction ratio of structure | |
| Before optimization | 13.88 | 0.46 |
| After optimization | 4.96 | 0.5 |
Although the embodiments of the present invention has been shown and described above, it is to be understood that above-described embodiment is example
Property, it is not considered as limiting the invention, those skilled in the art are not departing from the principle of the present invention and objective
In the case where can make changes, modifications, alterations, and variations to the above described embodiments within the scope of the invention.
Claims (6)
1. a kind of lattice structure based on B-spline parameterizes implicit modeling and optimization method, it is characterised in that: including following step
It is rapid:
Step 1: choosing m fundamental lattice pattern, each fundamental lattice pattern is made of several basic configurations, according to each base
The characterising parameter of this shape is described fundamental lattice using implicit modeling method, determines the level set of each fundamental lattice
Function phii, wherein the level set function of each fundamental lattice is the level set function for forming each basic configuration of the fundamental lattice
Boolean sum；The characterising parameter is divided into preset parameter and n design parameter；
Step 2: fundamental lattice being combined into a basic combination dot matrix, the level set function Φ of basic combination dot matrix is calculated and waits
Imitate elastic matrix DH；
The level set function Φ of basic combination dot matrix is the cloth of the level set function of each fundamental lattice of the basic combination dot matrix of composition
You and；
The Equivalent Elasticity matrix D of basic combination dot matrixHIt is obtained using following procedure:
According to homogenization method, dot matrix will be combined substantially and divides finite element grid, then Equivalent Elasticity matrix DHIt can be calculated by following formula:
D in formula11 H~D33 HFor 9 components of Equivalent Elasticity matrix, Q11~Q33It is to be combined in dot matrix substantially under different equivalent operating condition
The sum of each unit strain energy；I, j is enabled to take 1,2,3, Q respectivelyijBy formula
It is calculated, in formula | Y | indicate basic combination dot matrix volume, e is that the finite element grid divided is numbered, e=1,2 ..., E,WithIt is motion vector of e-th of finite element grid under i-th and the equivalent operating condition of jth kind respectively, is asked by finite element equation
Solution obtains, keIt is the element stiffness matrix of e-th of finite element grid, by formula
B is the element strain matrix of finite element grid, Ω in calculating formulaeFor the unit area of e-th of finite element grid, D0It is equal
The material matrix of material, α are one indivisible；Periodic boundary condition applied to basic combination dot matrix, solving finite element equation,
It obtains motion vector, finally acquires Equivalent Elasticity matrix DH；
Step 3: for macroscopical physical model to be optimized, being divided into the finite element interpolation grid based on B-spline, grid section
Points are Nx × Ny × Nz；A basic combination dot matrix is assigned at each grid node, and is calculated often according to the method for step 2
Substantially the Equivalent Elasticity matrix of dot matrix is combined at a grid node；
Step 4: macroscopical physical model to be optimized carries out macroscopical physical model to be optimized having a size of Lx × Ly × Lz again
FEM meshing calculates the stiffness matrix of entire macroscopical physical model using following procedure:
Step 4.1: according to the following formula
Calculate the elastic matrix D of each gridH(X, Y, Z), wherein (X, Y, Z) is grid element center coordinate, p is the order of B-spline,
Ni,p(ξ)、Nj,p(η) and Nk,p(ζ) is B-spline shape function,To combine dot matrix substantially at grid node in step 3 (i, j, k)
Equivalent Elasticity matrix；
Step 4.2: according to formula
The stiffness matrix K of entire macroscopical physical model is calculated, wherein Ω is entire macroscopical physical model region；
Step 5: the design parameter for choosing all basic combination dot matrix is the design variable of topology optimization problem, each basic combination
Dot matrix includes m × n design variable, total to obtain Nx × Ny × Nz × m × n design variable；
Step 6: applying boundary condition and load on the finite element model that step 4 is established, establish the mechanical model of macrostructure；
The minimum optimization aim of macrostructure compliance is chosen, total volume fraction ratio of macrostructure is compared to lattice structure point is combined substantially
For constraint function, design variable initial value and variation range are set, the Optimized model of lattice structure design problem is established；
Step 7: the Optimized model established to step 6 optimizes, and obtains optimum results.
2. a kind of lattice structure based on B-spline parameterizes implicit modeling and optimization method, feature according to claim 1
Be: the characterising parameter in step 1 includes the center position x of basic configurationc、yc, length l, width w, radius r and/or in
The rotation angle [alpha] of heart point.
3. a kind of lattice structure based on B-spline parameterizes implicit modeling and optimization method, feature according to claim 2
It is: when basic configuration is hyperelliptic, level set function φskAre as follows:
4. a kind of lattice structure based on B-spline parameterizes implicit modeling and optimization method, feature according to claim 2
It is: when basic configuration is bowlder, level set function φckAre as follows:
5. a kind of lattice structure based on B-spline parameterizes implicit modeling and optimization method, feature according to claim 1
Be: in step 4, B-spline shape function is all made of following form, wherein Ni,p(ξ) is
In formula, ξi∈{ξ1,ξ2,...,ξNx+p+1By Nx-p-1 node equally distributed on (0,1) and ξ=0 and ξ=
P+1 duplicate node composition at 1；And for Nj,p(η), then replace with j for the i in above-mentioned form, and ξ replaces with η, Nx replacement
For Ny；For Nk,pI in above-mentioned form is then replaced with k by (ζ), and ξ replaces with ζ, and Nx replaces with Nz.
6. a kind of lattice structure based on B-spline parameterizes implicit modeling and optimization method, feature according to claim 2
It is: the Optimized model of the lattice structure design problem in step 6 are as follows:
find xc,yc,w,l,r,α
J is macrostructure compliance in formula, and U is the motion vector of macrostructure, and F is macrostructure load vectors, gvtAnd gVPoint
Biao Shi not combine substantially lattice structure point than with the total volume fraction ratio of structure；For the volume of t-th of basic combination dot matrix, ΩtFor
The dot matrix region of t-th of basic combination dot matrix, VΩFor entire macrostructure volume, NelFor the net in macrostructure mechanical model
Lattice number, velemFor the volume of the elem grid in macrostructure mechanical model,Respectively indicate given total volume fraction ratio
Than the upper limit, H indicates Heaviside function for the upper limit and basic combination lattice structure point.
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| CN113345536A (en) * | 2021-05-31 | 2021-09-03 | 山东大学 | Structural topology optimization method based on extreme anisotropy lattice material |
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