CN110245410A - Heterogeneous material thermoelastic structure method of topological optimization design based on multi-parameter variable - Google Patents
Heterogeneous material thermoelastic structure method of topological optimization design based on multi-parameter variable Download PDFInfo
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Abstract
The present invention proposes a kind of heterogeneous material thermoelastic structure Topology Optimization Method based on multi-parameter variable, design domain is modeled and described using the B-spline space of multiple parameters, the control point in each B-spline space is by the design variable as optimization problem, the corresponding phase material in each B-spline space, between Control point mesh where design variable, it is mutually indepedent between Control point mesh and structural context grid, therefore the corresponding finer background grid of less design variable can be used.The design variable that this method uses is independent of background grid, with high-order continuity, and the high-order continuity in B-spline space makes this method itself be provided with preferable global convergence, can avoid the numerical problems such as intermediate materials and checkerboard patterns automatically without other means.
Description
Technical field
The present invention relates to a kind of heterogeneous material thermoelastic structure method of topological optimization design, especially a kind of to be based on B-spline
The heterogeneous material thermoelastic structure Topology Optimization Method of form multi-parameter variable.
Background technique
Heterogeneous material structure is the structure collectively formed by the material of a variety of different characteristics and function.In engineering design,
The heterogeneous material structure constituted using a variety of solid materials and cavity is very common.Heterogeneous material Topology Optimization Method refer to
Under fixed boundary condition and architecture quality or volume constraint, to seek structure optimum performance (such as overall stiffness maximum) as target,
Method to constitute rational deployment is designed to the dosage of each phase material, shape and empty occupation rate.Thermoelastic structure is
Refer under Thermal Load, carry out expansion or shrinkage because of material self attributes, thus in the internal structure for generating mechanical stress.
Heterogeneous material thermoelastic structure topological optimization has higher flexibility and complexity compared to the topological optimization of homogenous material, from
More dimensions improve the performance of structure.And the Thermal-mechanical Coupling effect that thermoelastic structure generates when by mechanical load and thermal force
Should also to topological optimization more stringent requirements are proposed and challenge.
Document " A mass constraint formulation for structural topology
optimization with multiphase materials.Tong Gao,Weihong
Zhang.Int.J.Numer.Meth.Engng 2011；88:774-796 " discloses a kind of based on unit puppet density theory and more
The method of topological optimization design of phase material interpolation model.Document is using multiple pseudo- density variables based on unit to heterogeneous material
Attribute, such as Young's modulus, volume, density carry out interpolation, by gradient optimal method to the shape of structure and material properties into
Row optimization.This method analyzes two kinds of typical material interpolation models: recurrence material interpolation model (RMMI) and reciprocity material are inserted
It is worth model (UMMI), and its computational accuracy is compared under the operating condition of quality constraint and volume constraint respectively.Pass through comparison
Show that reciprocity material interpolation model (UMMI) can obtain more preferably structure, advantage is more significant under quality constraint.However
Document the method uses the discrete design variable based on unit, poor continuity between unit, Numerous, and is easy to appear
Intermediate materials and checkerboard patterns, it is necessary to which being handled by additional numerical value is improved.
Summary of the invention
In order to overcome variable poor continuity existing for existing heterogeneous material Topology Optimization Method, be easy to appear intermediate materials and
The problem of checkerboard patterns, and be applied in thermoelastic structure, the invention proposes a kind of based on multi-parameter variable
Heterogeneous material thermoelastic structure Topology Optimization Method.
This method is modeled and is described to design domain using the B-spline space of multiple parameters.With conventional pseudo- density
Method is different, and the discrete heat sources based on unit are replaced with the continuous variable of high-order by this method.The control in each B-spline space
Point is by the design variable as optimization problem, the corresponding phase material in each B-spline space.Control point mesh where design variable
Between, it is mutually indepedent between Control point mesh and structural context grid.Therefore it is more smart that less design variable correspondence can be used
Thin background grid.For heterogeneous material optimization problem, this method is by B-spline space and reciprocity material interpolation model (UMMI) phase
In conjunction with, corresponding volume constraint and quality constraint under optimize.In order to accelerate to restrain, this method joined traditional SIMP
The numerical value penalty factor of form and RAMP form.Compared to the design method of background technique, the design variable that this method uses is disobeyed
Rely in background grid, there is high-order continuity, and the high-order continuity in B-spline space is provided with this method itself preferably
Global convergence can avoid the numerical problems such as intermediate materials and checkerboard patterns without other means automatically.
Based on the above principles, the technical solution of the present invention is as follows:
A kind of heterogeneous material thermoelastic structure method of topological optimization design based on multi-parameter variable, feature exist
In: the following steps are included:
Step 1: determining the size l of design domainx×ly, the number m of heterogeneous material, and determine the number p in m B-spline space
With respective Control point mesh size nxk×nyk, k=1 ..., m；Establish parameter region corresponding with design domain, setting parameter
Region (ξ, η), design domain (x, y) ∈ [0, lx]×[0,ly], then have
Step 2: establishing m B-spline space, the pseudo- density values ρ of arbitrary point in space(k)(ξ, η) is by formula
It obtains, wherein Pij (k)Represent the pseudo- density values at k-th of grid control point, PminFor one setting it is indivisible,
Ni,p(ξ), Nj,p(η) is p B-spline shape function；
Step 3: by the rectangular area Ω of design domain embedding method, and the quadrilateral mesh of division rule, establish finite element
Model；The level set function φ in given design domain, calculates the stiffness matrix of each unit:
Wherein ΩeIndicate unit region, H (φ) is Heaviside function, and B is strain-transposed matrix, and D (ρ) is
Unitary elasticity matrix；It is calculated by reciprocity material interpolation model:
D in formulaiIndicate that the elastic matrix of i-th kind of material, χ (ρ) indicate pseudo- density penalty；
For thermoelastic structure, it is also necessary to calculate thermal force FTh:
FTh=Δ T ∫ΩBT·β·H(φ)dΩ
Wherein Δ T indicates the temperature change of structure, and β is the thermal stress coefficient vector of structure, by reciprocity material interpolation model
It is calculated:
Wherein βi=αi·DiΦ, αiFor the thermal expansion coefficient of i-th kind of material, Φ is a constant vector: Φ in two dimension=
[1 1 0]T；Φ=[1 1100 0] in three-dimensionalT；
Step 4: the stiffness matrix of each unit being assembled into structure Bulk stiffness matrix K, is applied on finite element model
Boundary condition and load, wherein load includes the mechanical load F of settingMeThe thermal force F obtained with step 3Th, establish macroscopic view knot
Mechanical model K (the ρ of structure(k)) U=FMe+FTh, and solution node motion vector U；
Step 5: choosing the control point data in m B-spline space as design variable, selecting structure compliance C is optimization
Target, the volume fraction performance indicator of structure set design variable initial value and variation range, establish multiphase as constraint function
The Optimized model of material topology optimization problem:
G in formulaVkIt indicates volume fraction, enablesIt indicates the given volume fraction upper limit, then has
Step 6: the Optimized model established to step 5 optimizes, and obtains optimum results.
Further preferred embodiment, a kind of heterogeneous material thermoelastic structure topological optimization based on multi-parameter variable
Design method, it is characterised in that: B-spline shape function is all made of following form in step 2, wherein Ni,p(ξ) is
In formula, ξi∈{ξ1,ξ2,...,ξnxk+p+1It is the equally distributed nx on (0,1)k- p-1 nodes and in ξ=0
And have p+1 duplicate node at ξ=1 respectively；And for Nj,pI in above-mentioned form is then replaced with j by (η), and ξ replaces with η,
nxkReplace with nyk。
Further preferred embodiment, a kind of heterogeneous material thermoelastic structure topological optimization based on multi-parameter variable
Design method, it is characterised in that: pseudo- density penalty takes two kinds of forms of SIMP and RAMP according to operating condition in step 3:
Wherein pn is penalty coefficient.
Beneficial effect
The beneficial effect of the present invention compared with prior art is: the method for the present invention is made using several independent B-spline spaces
For the Basic Design element of topological optimization, design variable is made to be no longer dependent on background grid, realized through less Variable Control essence
Refined net.And the continuous field energy in B-spline space enough guarantees that structure possesses continuous shape and material distribution.In addition, of the invention
Due to using continuous variable, discrete variable bring such as gray shade unit, intermediate materials and checkerboard patterns can be avoided automatically
Equal numerical value defect, has better convergence, can obtain more preferably result.
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 B-spline space multi-parameter schematic diagram of the method for the present invention.
Fig. 2 is the method for the present invention Control point mesh (left side) and B-spline space schematic diagram (right side).
Fig. 3 is the design section and operating condition schematic diagram of the method for the present invention.It is a MBB girder construction in figure.
Fig. 4 is the optimum results figure of the B-spline multi-parameter of the method for the present invention, shows three kinds of different materials in figure altogether
Expect, the interface between material is clear.
Fig. 5 is three B-spline surfaces in the method for the present invention optimum results.Three curved surfaces respectively represent point of three kinds of materials
Cloth range, curved surface white portion represent material, and black portions represent cavity.
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. 3~Fig. 5.Heterogeneous material Structural Topology Optimization Design is carried out for a MBB girder construction in the present embodiment, if
Meter area size is 60mm × 20mm, and the region left side is constrained by horizontal direction, and the lower right corner is constrained by vertical direction.The region upper left corner
By a concentrated force F=100N effect straight down, not by Thermal Load.It is designed using three kinds of different materials,
The Young's modulus and Poisson's ratio of three kinds of candidate materials are respectively E1=70Pa, E2=120Pa, E3=210Pa, ν1=ν2=0.34,
ν3=0.3.
Step 1: determining that the B-spline space number needed is m=3, the number for choosing B-spline is p=5, each B-spline control
Dot grid size processed is 60 × 20；Referring to shown in attached drawing 1, parameter region corresponding with design domain, setting parameter region are established
(ξ, η) ∈ [0,1] × [0,1], design domain (x, y) ∈ [0,60] × [0,20], then have
Step 2: establishing 3 B-spline spaces, set Pmin=10-5For the lower bound at control point, it is intended to pseudo- density be avoided to be equal to
Caused numerical problem when zero, the pseudo- density values ρ of arbitrary point in space(k)(ξ, η) is by formula
It obtains, wherein Pij (k)Represent the pseudo- density values at k-th of grid control point, Ni,p(ξ), Nj,p(η) is p B sample
Shape function, using following form, wherein Ni,p(ξ) is
In formula, ξi∈{ξ1,ξ2,...,ξnxk+p+1It is the equally distributed nx on (0,1)k- p-1 nodes and in ξ=0
And have p+1 duplicate node at ξ=1 respectively；And for Nj,pI in above-mentioned form is then replaced with j by (η), and ξ replaces with η,
nxkReplace with nyk。
Step 3: by the rectangular area Ω of design domain embedding method, and the quadrilateral mesh of division rule, establish finite element
Model；The level set function φ in given design domain.For the present embodiment since design domain itself is very regular, so there is no need to calculate level set
Function phi.
Calculate the stiffness matrix of each unit:
Wherein ΩeIndicate unit region, H (φ) is Heaviside function, and B is strain-transposed matrix, and D (ρ) is
Unitary elasticity matrix；It is calculated by reciprocity material interpolation model:
D in formulaiIndicate that the elastic matrix of i-th kind of material, χ (ρ) indicate pseudo- density penalty；In the present embodiment are as follows:
D (ρ)=χ (ρ(1))(1-χ(ρ(2)))(1-χ(ρ(3)))+(1-χ(ρ(1)))χ(ρ(2))(1-χ(ρ(3)))+(1-χ
(ρ(1)))(1-χ(ρ(2)))χ(ρ(3))
Pseudo- density penalty takes two kinds of forms of SIMP and RAMP according to operating condition:
Wherein pn is penalty coefficient.In view of operating condition is static(al) operating condition in the present embodiment, therefore choose penalty coefficient pn=3's
SIMP form χ (ρ)=ρ3.And variable ρ(1)-ρ(3)It can be obtained by step 2.
Step 4: the stiffness matrix of each unit being assembled into structure Bulk stiffness matrix K, is applied on finite element model
Boundary condition and load F establish mechanical model K (ρ) U=F of macrostructure, and solution node motion vector U.
Step 5: choosing the control point data in m B-spline space as design variable, selecting structure compliance C is optimization
Target, for the volume fraction performance indicator of structure as constraint function, setting design variable initial value is ρ(1)=ρ(2)=ρ(3)=
0.16, variable bound is respectively 10-5With 1, the Optimized model of heterogeneous material topology optimization problem is established:
G in formulaV1,gV2And gV3Respectively indicate material 1, the volume fraction of material 2 and material 3.Limit the volume of three kinds of materials
Score must not exceed 16.67%, then has
Step 6: in optimization design platform BOSS-QuattroTMThe interior optimization that step 5 is established using GCMMA optimization algorithm
Model optimizes, and obtains optimum results.
The material distribution of structure and convergence curve are as shown in Figure 4 after optimization.The corresponding B-spline surface top view of three kinds of materials
As shown in Figure 5.The heterogeneous material method of topological optimization design based on multi-parameter variable that the present invention uses can obtain clearly
Smooth material boundary and structural configuration, the corresponding B-spline surface of each material has high-order continuity, and convergence process is steady
Rapidly, there are not the numerical problems such as gray shade unit in entire optimization process.The result shows that the multiphase material based on multi-parameter variable
Material thermoelasticity method of topological optimization design can obtain clear smooth material boundary and structure structure while improving the rigidity of structure
The problem of type solves the variable poor continuity of background technique design method, is easy to appear intermediate materials and checkerboard patterns.
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 (3)
1. a kind of heterogeneous material thermoelastic structure method of topological optimization design based on multi-parameter variable, it is characterised in that: packet
Include following steps:
Step 1: determining the size l of design domainx×ly, the number m of heterogeneous material, and determine m B-spline space number p and respectively
From Control point mesh size nxk×nyk, k=1 ..., m；Establish parameter region corresponding with design domain, setting parameter region
(ξ, η), design domain (x, y) ∈ [0, lx]×[0,ly], then have
Step 2: establishing m B-spline space, the pseudo- density values ρ of arbitrary point in space(k)(ξ, η) is by formula
It obtains, wherein Pij (k)Represent the pseudo- density values at k-th of grid control point, PminFor indivisible, the N of a settingi,p
(ξ), Nj,p(η) is p B-spline shape function；
Step 3: by the rectangular area Ω of design domain embedding method, and the quadrilateral mesh of division rule, establish finite element model；
The level set function φ in given design domain, calculates the stiffness matrix of each unit:
Wherein ΩeIndicate unit region, H (φ) is Heaviside function, and B is strain-transposed matrix, and D (ρ) is unit bullet
Property matrix；It is calculated by reciprocity material interpolation model:
D in formulaiIndicate that the elastic matrix of i-th kind of material, χ (ρ) indicate pseudo- density penalty；
For thermoelastic structure, it is also necessary to calculate thermal force FTh:
Wherein Δ T indicates the temperature change of structure, and β is the thermal stress coefficient vector of structure, is calculated by reciprocity material interpolation model
It obtains:
Wherein βi=αi·DiΦ, αiFor the thermal expansion coefficient of i-th kind of material, Φ is a constant vector: Φ=[1 1 in two dimension
0]T；Φ=[1 1100 0] in three-dimensionalT；
Step 4: the stiffness matrix of each unit being assembled into structure Bulk stiffness matrix K, applies boundary on finite element model
Condition and load, wherein load includes the mechanical load F of settingMeThe thermal force F obtained with step 3Th, establish macrostructure
Mechanical model K (ρ(k)) U=FMe+FTh, and solution node motion vector U；
Step 5: the control point data in m B-spline space is chosen as design variable, selecting structure compliance C is optimization aim,
The volume fraction performance indicator of structure sets design variable initial value and variation range, establishes heterogeneous material as constraint function
The Optimized model of topology optimization problem:
G in formulaVkIt indicates volume fraction, enablesIt indicates the given volume fraction upper limit, then has
Step 6: the Optimized model established to step 5 optimizes, and obtains optimum results.
2. a kind of heterogeneous material thermoelastic structure topology optimization design side based on multi-parameter variable according to claim 1
Method, it is characterised in that: B-spline shape function is all made of following form in step 2, wherein Ni,p(ξ) is
In formula, ξi∈{ξ1,ξ2,...,ξnxk+p+1It is the equally distributed nx on (0,1)k- p-1 nodes and in ξ=0 and ξ
There is p+1 duplicate node at=1 respectively；And for Nj,p(η), then replace with j for the i in above-mentioned form, and ξ replaces with η, nxkIt replaces
It is changed to nyk。
3. a kind of heterogeneous material thermoelastic structure topology optimization design side based on multi-parameter variable according to claim 1
Method, it is characterised in that: pseudo- density penalty takes two kinds of forms of SIMP and RAMP according to operating condition in step 3:
Wherein pn is penalty coefficient.
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