CN109408845A - A kind of instable more material radiator structure Topology Optimization Methods of consideration increasing material manufacturing - Google Patents
A kind of instable more material radiator structure Topology Optimization Methods of consideration increasing material manufacturing Download PDFInfo
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Abstract
The invention discloses a kind of instable more material radiator structure Topology Optimization Methods of consideration increasing material manufacturing.Influence of this method by the unstability during increasing material manufacturing to structural behaviour, in the case where having uncertain in view of material thermal conductivity, structure thermal force and safe temperature etc., by the interpolation model based on more material radiator structures, more material radiator structure topological optimization mathematical models are constructed under temperature Reliability Constraint.Then by unit puppet density as design variable, by architecture quality as objective function, the optimum results of more material radiator structures under prescribed conditions are obtained by the iterative calculation of MMA algorithm.The present invention considers uncertain and material quantity the different influences to optimum results in topological optimization, guarantees that structure has better balance between economy and reliability.
Description
Technical field
The present invention relates to radiator structure topological optimization technology fields, and in particular to a kind of consideration increasing material manufacturing is instable
More material radiator structure Topology Optimization Methods, this method consider the not true of material thermal conductivity, structure thermal force and allowable temperature
It is qualitative, based on the interpolation model of more material radiator structures, to optimize characteristic distance as constraint condition, to more material radiator structures
Carry out topology optimization design.
Background technique
Although increasing material manufacturing is a current very powerful and exceedingly arrogant manufacturing technology, since it still locates as an emerging technology
In incomplete stage, at many aspects, there is also some critical problems, it may be said that this field of increasing material manufacturing also
There is very big research space.
The micro-structured inner of increasing material manufacturing product is there is some protrusions, folds and hole, present in these structures
It is the high-temp liquid or powder rapid curing that will be melted that problem, which is due to the principle of increasing material manufacturing, then the side by successively printing
Formula causes the product processed from microcosmic point to construct the object of three-dimensional so being difficult to control its cooling and shaping in solidification
On see and will appear these problems, these problems can further result in the dispersibility or uncertainty of material property again, and everything
It can be attributed to the unstability of preparation process.
Why to pay attention to by increases material manufacturing technology bring unstability, concrete reason there are following three points:
First, institute's inherent instability not can avoid during material machine-shaping.For example, coming to coaxial powder-feeding technology
It says, unstability existing for sweep speed and powder feeding rate is not enough or dilutes going out for the problems such as excessive for will lead to powder feeding
It is existing, influence the quality of machine-shaping；When using Stereo Lithography Apparatus Rapid Prototyping technology, because unstable existing for spot diameter
Property will will lead to the soft edge of the model in model micro-structure, and since basis material difference divides shakiness existing for layer thickness
Qualitative meeting so that interlayer residual stress appearance；When being sintered using selective laser melting forming technique to powder, by
The unstability existing for laser threat warner speed can make the size of microscopical structure certain deviation occur, and melting temperature and pre-
Existing unstability can lead to powder again and cannot equably melt between hot temperature.Above-mentioned objective fact shows in structure preparation
In the process there is dispersibility in material property, thin angle of seeing has dimensional discrepancy.
Structure is pacified brought by preparation process unstability second, transmitting of the material property across scale equally exacerbates
The hidden danger of full property.There is scholar once to do systematic research to space lattice structure, finds the unstability meeting of preparation process
Leading to the thin sight size of testpieces, there is uncertainties, differ the actual value of elasticity modulus with theoretical value and reach
63%.This is because certain structural materials are there are nonlinear constitutive relation, macroscopic property are complex, in preparation process not
Dimensional discrepancy caused by stability can be built up, and finally can generate the influence that can not ignore to the macro property of structure.
Third, certain materials cannot be given full play to by traditional optimum design method in the case where preparation process is unstable
The performance advantage of material.The U.S. when designing reusable launch vehicle, the body scheme originally proposed can meet simultaneously thermal protection,
The requirement of structural behaviour and propulsion capability, but because the unstable heat-insulating capability for leading to practical structures of manufacturing process and prediction are tied
Fruit difference is excessive, forced to have selected more conservative design scheme.Current optimum design method fails in view of preparation
Influence of the various uncontrollable factors to structure in the process, it is infeasible in practice to may result in final design scheme, can only
The design scheme abandoned the optimization to structural behaviour and use other cost performances not high.
Therefore, the unstability for studying increasing material manufacturing process has certain theory significance and engineering value.And prepared by structure
The unstability of process can further cause the dispersibility of material property again, and this dispersibility or uncertainty are widely present in fact
In the structured design process of complication system, main source has " people, machine, material, method, ring, survey " six broad aspects, can be to the work of structure
Make performance and safety generation seriously affects.Wherein, it is interior in terms of this just to belong to " machine " for the unstability of increasing material manufacturing process
Hold.
Closely for centuries, people are constantly deepening probabilistic understanding, successively have found in Practical Project
Uncertainty present in problem have objective unpredictability, the ambiguity of Subjective, and in complex situations by
The not perfect property caused by poor information.Reliability of structure analysis in, uncertainty can be divided into stochastic uncertainty,
Fuzzy uncertainty (or cognition is uncertain) and bounded-but-unknown uncertainty (or uncertain but bounded).
Wherein, it is not true that the dispersibility for the material property as caused by preparation unstability that the present invention is studied just belongs to bounded
It is qualitative.This kind of uncertainty is the scarcity due to sample information in complex situations, so in caused subjective understanding can not
Intellectual.Uncertain but bounded is widely present in fact in Practical Project problem, and the influence to structural reliability can not be ignored.
It is worth noting that, these uncertain factors also have accumulation and aggravation effect, when small uncertain amount is tired
Product superposition can make the result for predicting to obtain originally and reality generate when especially multiple small uncertainty amounts accumulate superposition
Sizable error, or even distortion completely.Nowadays, the structural concept design stage still remains serious security risk,
The either phase of product design process all cannot only consider certainty factor and ignore the influence of uncertain factor.Therefore, exist
Increases material manufacturing technology is widely used under this overall background of aerospace field at present, considers in structural analysis and design by increasing
It is probabilistic caused by the unstability of material manufacture to influence to be very necessary and significant.
Simultaneously as poly-material structure often manufactures (such as increasing material manufacturing) using complicated technique, these techniques can not be kept away
The meeting exempted from impacts the characteristic of material.In addition, being usually on active service using high design objective and the poly-material structure of manufacturing cost
In complex load environment, load is difficult to precise quantification, has certain dispersibility.Probabilistic presence will be to the peace of structure
Full property causes very important influence, in this case, will be difficult to obtain using deterministic more material Topology Optimization Methods
Meet design requirement as a result, the safety and reliability of structure must just can guarantee using the method for reliability topological optimization.
The problems in practical for engineering, since operating condition is complicated and sample data is less, related probabilistic information is difficult to accurately obtain
, but the boundary of uncertain factor is easy to determine, thus proposes the non-probability decision using salient rate as theoretical basis
The concept of property.
However the research work of Multidisciplinary systems topological optimization correlation theory just starts to walk, in the field grinding at present
Study carefully that achievement is also less, especially the research in terms of the Multidisciplinary systems topological optimization based on more material radiator structures or even
Blank out.Based on this, in the case where considering increasing material manufacturing unstability, more materials are carried out using Multidisciplinary systems method
The Topology Optimization of aviation radiator structure not only has certain theory significance, moreover it is possible to provide relevant engineering problem effective
Method.
Summary of the invention
The present invention provides a kind of instable more material radiator structure Topology Optimization Methods of consideration increasing material manufacturing.It is led
Wanting content is: the uncertainty of the material property as caused by increasing material manufacturing unstability, structural bearing etc. is considered, based on more
Material radiator structure interpolation model, to optimize characteristic distance d as constraint condition, the performance for giving full play to poly-material structure is excellent
Gesture enables topological optimization result to guarantee the requirement of structure economics and reliability simultaneously.
The technical solution adopted by the present invention are as follows: a kind of instable more material radiator structure topologys of consideration increasing material manufacturing are excellent
The step of change method, the method, is as follows:
Step 1: it for considering probabilistic more material radiator structure topology optimization problems, seeks to consider that material is led
The uncertainty of hot coefficient, structure thermal force and safe temperature.Using interval variable KIIndicate overall thermal rigidity interval matrix, PI
Indicate overall thermal load interval vector,It indicates temperature range vector, and original certainty is substituted with this
Amount, wherein subscript I indicates that variable is interval variable, and N indicates the number of temperature freedom degree, it is as follows to obtain finite element equation:
KItI=PI
In view of governing equation is linear equation, can be acquired using the vertex combined method for considering sensitivity
In a certain temperature componentsThe upper bound and lower bound.
Step 2: it is theoretical according to Multidisciplinary systems after the bound for finding out temperature range using vertex combined method,
A new Multidisciplinary systems index can be constructed on the basis of non-probability decision degree R, i.e. optimization characteristic distance d.Work as d > 0
When, corresponding reliability R < Rtarg, do not meet design requirement；And as d≤0, corresponding reliability R > Rtarg, meet and set
Meter requires.
Step 3: density-stiffness interpolation schemes are based on, more material radiator structure interpolation models are established:
Wherein λiThe material thermal conductivity of i-th of unit, λ after expression interpolation1Indicate the thermal coefficient of material 1, λ2Indicate material
The thermal coefficient of material 2, λ3Indicate the thermal coefficient of material 3, x1,i、x2,iAnd x3,iRespectively indicate i-th of unit design variable 1,
Design variable 2 and design variable 3, p (p > 1) indicate penalty factor.
Step 4: on the basis of classical topologies optimized mathematical model, by optimization characteristic distance d as constraint condition, and
In conjunction with above-mentioned more material interpolation models, the Multidisciplinary systems topological optimization mathematical modulo based on more material radiator structures can be established
Type are as follows:
Wherein, M indicates the architecture quality in design section, ViIndicate the volume of i-th of unit, n is indicated in design section
Unit sum, ρ1、ρ2And ρ3The density of material 1, material 2 and material 3 is respectively indicated,x 1、x 2Withx 3Respectively indicate design variable
1, the lower bound of design variable 2 and design variable 3,WithRespectively indicate design variable 1, design variable 2 and design variable 3
The upper bound.djIndicate that the Multidisciplinary systems of j-th of constraint, m indicate the sum of constraint.
Step 5: constraint function and objective function are obtained for the local derviation of design variable by the use of adjoint vector method
Number carries out sensitivity analysis, in order to the solution of subsequent gradients optimization algorithm.
Step 6: by the iterative calculation of MMA algorithm, while considering Reliability Constraint and relative variation, if a certain change
The result ridden instead of walk meets the constraint condition of non-probability decision degree d≤0, and the sum of design variable variable quantity of two steps is small before and after iteration
When a preset value ε, then iterative process terminates, using the result of present topology optimization as final optimum results.
Wherein, the uncertain of material thermal conductivity, structure thermal force and safe temperature is considered in the step one
Property, and original certainty amount is substituted in the form of interval variable.
Wherein, judge index of the optimization characteristic distance d as structure Multidisciplinary systems is used in the step two.
Wherein, the interpolation model based on more material radiator structures is used in the step three.
Wherein, the feelings of bi-material layers and three materials are considered in topological optimization model constructed in the step four
Condition.
Wherein, constraint function obtained by when carrying out sensitivity analysis in the step five and objective function become design
The partial derivative of amount.
Wherein, consider that Reliability Constraint and relative variation are used as simultaneously in the step six and judge that iterative process is
The standard of no end.
Compared with current existing method, present invention is different in that:
The present invention considers the uncertain but bounded as caused by increasing material manufacturing unstability in radiator structure topological optimization
Influence, with optimize characteristic distance for constraint carried out reliability topological optimization, ensure that optimization gained configuration safety.Together
When, by introducing the concept of poly-material structure, the performance advantage of more materials is given full play in optimization design, mitigates knot as far as possible
The quality of structure, economy has been taken into account in the case where meeting reliability requirement.
Detailed description of the invention
Fig. 1 is the flow chart for considering the instable more material radiator structure Topology Optimization Methods of increasing material manufacturing；
Fig. 2 is Multidisciplinary systems model two dimension interference schematic diagram；
Fig. 3 is optimization characteristic distance schematic diagram；
Fig. 4 is model schematic used in example；
Fig. 5 is loading method schematic diagram in example；
Fig. 6 is optimum results schematic diagram in example, wherein Fig. 6 (a) is single material certainty topological optimization result；Fig. 6
It (b) is bi-material layers certainty topological optimization result；Fig. 6 (c) is three material certainty topological optimization results；Fig. 6 (d) is single material
Reliability topological optimization result；Fig. 6 (e) is bi-material layers reliability topological optimization result；Fig. 6 (f) is three reliabilities of material topology
Optimum results.
Specific embodiment
A kind of instable more material radiator structure topologys of consideration increasing material manufacturing of the present invention described below are excellent
Whole processes of change method:
1, it for considering probabilistic more material radiator structure topology optimization problems, seeks to consider material conducts heat system
The uncertainty of number, structure thermal force and safe temperature.And engineering in practice, it is difficult to obtain the accurate of uncertain factor
Information, but its boundary relatively easily determines, it is contemplated that and this case, specific practice are using interval variable KIIndicate that structure is whole
Body heat rigidity interval matrix, PIIndicate overall thermal load interval vector,Indicate temperature range vector, and with this
Original certainty amount is substituted, wherein subscript I indicates that variable is interval variable, and N indicates the number of temperature freedom degree, can must have
Limit first equation are as follows:
KItI=PI (1)
Wherein, it is contemplated that governing equation is linear equation, therefore can be acquired with the following vertex combined method for considering sensitivityIn a certain temperature components tj IThe upper bound and lower bound.
Consider the vertex combined method of sensitivity are as follows:If function f (x1,x2,…,xn) to any independent variable xi(i=1,2 ...,
N) be all it is dull, argument list can be shown as interval variable at this time, it may be assumed that
Wherein,x iWithRespectively parameter lower bound and the upper bound.
It can be obtained by monotonicity again, the value interval of f is:
R therein be known as vertex combination ordinal number, r=1,2 ..., 2n,ki=1,2 respectively refer to
The lower bound of parameter and the upper bound, i.e.,
Thus according to vertex combined method, temperature value interval corresponding to j-th of constraint can be obtainedAre as follows:
Wherein For the actual temperature section of j-th of temperature restraint,
Subscript ki=1,2, work as kiIndicate that corresponding value removes boundary, works as k when=1iIndicate that corresponding value takes the upper bound when=2, it may be assumed that
In actual calculating process, it is big for response each uncertain variables can be calculated with the mode of difference
Small influence.When sensitivity of the response to variable is greater than 0, then response is maximum when variable takes the upper bound, response when removing boundary
It is minimum；And when response to the sensitivity of variable less than 0 when, then response is minimum when variable takes the upper bound, and response is most when removing boundary
Greatly.Accordingly, for having the case where n uncertain variables, can be analyzed by the positive and negative analysis of n times difference sensitivity and 2 secondary responses, most
The upper bound and the lower bound of response are acquired eventually.
2, theoretical based on Multidisciplinary systems after the bound for finding out temperature range using vertex combined method, Ke Yijian
Non-probabilistic set-based reliability model under Liru under temperature restraint.
If tj,aFor the actual temperature of j-th of temperature restraint, tj,sFor the safe temperature of j-th of temperature restraint, consider not true
Qualitatively effect, is expressed as interval variable for temperature, it may be assumed that
Wherein t andLower bound and the upper bound for respectively referring to temperature, represent in above-mentioned two section on the same number axis simultaneously
When coming, it is possible that the case where interference.
If the limit state function M (t of structurej,s,tj,a) are as follows:
M(tj,s,tj,a)=tj,s-tj,a (8)
Then its limiting condition plane or failure plane are as follows:
M(tj,s,tj,a)=tj,s-tj,a=0 (9)
As M (tj,s,tj,aIndicate that the structure meets given constraint condition when) >=0, as M (tj,s,tj,aIt is indicated when) < 0
Given constraint condition is not met.To actual temperature and safe temperature interval variableIt is standardized transformation
Processing:
WhereinReferred to as section radius.After standardization, there is δ
tj,a∈ [- 1,1], δ tj,s∈[-1,1].Formula (10) is updated in limiting condition plane equation, can be obtained:
Thus δ t can be derivedj,sWith δ tj,aBetween relational expression are as follows:
Above formula can be indicated in plane right-angle coordinate, and mark δ uj,sWith δ uj,aValue interval, as shown in Figure 2.
By the area S of safety zoneABFEDWith the area and S of safety zone and failed areasABCDThe ratio between be known as non-probability can
By spending R.The situation intersected for variable region and limiting condition plane is solved into R below.Limiting condition plane is solved first
With straight line δ tj,s=-1 intersection point enables the δ t in formula (12)j,s=-1 can obtain δ tj,aAre as follows:
It enablesIt can solveThen limiting condition plane and straight line are found out
δtj,a=1 intersection point enables the δ t in formula (13)j,a=1, δ t can be obtainedj,sAre as follows:
It enablesIt can solveThus, it is possible to obtain non-probability decision degree R
Expression formula are as follows:
Result before is substituted into (16), R can be indicated are as follows:
It similarly can be in the hope of the expression formula of R under remaining five kinds of situation are as follows:
It is constant that it is permanent in some cases, which to can be seen that reliability R from formula (17), be would become hard to for MMA algorithm
It is optimized towards being correctly oriented, a new Multidisciplinary systems index is thus had also been proposed on the basis of reliability R, i.e.,
Optimize characteristic distance d.As shown in figure 3, being d by the distance definition of former limiting condition plane to target limit state plane.Wherein,
Target limit state plane is parallel with former limiting condition plane, and its non-probability decision degree RtargIt is a given value.
And the reliability R of targettargGenerally all close to 1, therefore target limit state plane is generally all in variable region
The lower right corner, first two kinds of special circumstances in consideration variable region and target limit state plane intersection situation.It first calculates critical
In the case of limiting condition plane slope.For k1, there is (2 × 2/k1× 1/2)/4=1-Rtarg, k can be solved1=1/2 (1-
Rtarg), similarly available k2=2 (1-Rtarg).When the slope of limiting condition plane takes different values respectively, it is contemplated that its
With the relationship between critical slope, by the expression formula of the available d of range formula between two parallel lines are as follows:
As d > 0, limiting condition plane is in the reliability R with targettargAbove corresponding target limit state plane, and
At this time since target value is greater than the area of security domain, corresponding reliability R < Rtarg, do not meet design requirement.As d≤0,
Limiting condition plane is in the reliability R with targettargBelow corresponding target limit state plane, and at this time since target value is small
In the area for being equal to security domain, corresponding reliability R > Rtarg, meet design requirement.
3, the interpolation model based on single material density and rigidity is given in classical SIMP model, it may be assumed that
E=xpE0 (19)
Wherein E0The elasticity modulus of presentation-entity material, E indicate that the elasticity modulus after interpolation, x indicate design variable (unit
Relative density), p indicates penalty factor (being often taken as 3).It can guarantee the mistake in topological optimization by the interpolation model of single material
The continuity of elasticity modulus of materials in journey, and the main function of penalty factor is then the centre significantly reduced in optimum results
Density Units.
And since studied is more material radiator structures to the present invention, the elastic modulus E in above formula should be become material
Thermal coefficient λ, it can thus be concluded that:
λ=xpλ0, single material (20) wherein λ0The thermal coefficient of presentation-entity material, λ indicate the thermally conductive system after interpolation
Number, x indicate design variable (unit relative density), and p indicates penalty factor (being often taken as 3).
Above-mentioned single material interpolation model is referred again to, more material interpolation models based on radiator structure are constructed are as follows:
Wherein λiThe material thermal conductivity of i-th of unit, λ after expression interpolation1Indicate the thermal coefficient of material 1, λ2Indicate material
The thermal coefficient of material 2, λ3Indicate the thermal coefficient of material 3, x1,i、x2,iAnd x3,iRespectively indicate i-th of unit design variable 1,
Design variable 2 and design variable 3, p (p > 1) indicate penalty factor.By the interpolation model, two or three of material can be realized
Continuity interpolation model.
4, it on the basis of forefathers' topological optimization mathematical model, by optimization characteristic distance d as constraint condition, and combines upper
Single material/more materials interpolation model is stated, can be established excellent based on single material/more materials radiator structure Multidisciplinary systems topology
Change mathematical model are as follows:
Wherein, V indicates the structural volume in design section, ViIndicate the volume of i-th of unit, n is indicated in design section
Unit sum,xIndicate the lower bound of design variable,Indicate the upper bound of design variable.djIndicate that the non-probability of j-th of constraint can
By property, m indicates the sum of constraint.
Wherein, M indicates the architecture quality in design section, ViIndicate the volume of i-th of unit, n is indicated in design section
Unit sum, ρ1、ρ2And ρ3The density of material 1, material 2 and material 3 is respectively indicated,x 1、x 2Withx 3Respectively indicate design variable
1, the lower bound of design variable 2 and design variable 3,WithRespectively indicate design variable 1, design variable 2 and design variable 3
The upper bound.djIndicate that the Multidisciplinary systems of j-th of constraint, m indicate the sum of constraint.
When solving topology optimization problem by mobile asymptote optimization method (MMA), since MMA algorithm is a kind of gradient
Optimization algorithm will specifically obtain constraint function and objective function and design is become so needing to carry out sensitivity analysis
The partial derivative of amount.But since the number of the design variable in the Optimized model is far more than the number of constraint, if passing through difference
Mode is solved, it will huge calculation amount is brought in sensitivity analysis.According to this feature, adjoint vector can be passed through
The use of method carries out sensitivity analysis.
Due to the sensitivity solution procedure of bi-material layers and three materials be it is similar, below by taking bi-material layers topological optimization as an example,
Provide the basic process of sensitivity analysis.By the chain type Rule for derivation of compound function, it is a to jth (j=1,2 ..., m) constraint and
Speech optimizes characteristic distance djTo design variable xi(i=1,2 ..., n) (xiFor x1,iOr x2,i) sensitivity are as follows:
Wherein,
WhereinWithThis two parts can be obtained by (23) direct solution, butWithIt but can not be direct
It is calculated, Yao Liyong augmented vector approach is solved:
Wherein, λj(j=1,2 ..., m) is expressed as adjoint vector, the Lagrange multiplier that is otherwise known as vector.And due to PI-
KItI=0, thereforeAsk formula (27) to design variable xiPartial derivative can obtain:
Wherein,
For any λjFor formula (28) set up, it is possible thereby to by selecting suitable λjSo thatIt is full
Foot, even,
Since hot stiffness matrix is symmetrical matrix, transposition can be carried out to formula (30) both ends and obtained:
Shown by formula (31) by applying virtual thermal force to modelAfterwards, obtained temperature is λj.It solves
λ outjLater, the bound of temperature is to the partial derivative of design variable at obligatory point are as follows:
WhereinRespectively indicate correspondenceAdjoint vector, the hot stiffness matrix of unit and temperature column vector,λ j、K j、tRespectively indicate corresponding tj,aAdjoint vector, the hot stiffness matrix of unit and temperature column vector.In this model, since heat carries
Lotus PIDo not change with design variable, i.e.,Formula (32) can then be rewritten are as follows:
It can be obtained by acquired results before (2) again:
Wherein K1,jIndicate hot stiffness matrix of j-th of unit with respect to design variable 1, K2,jIt indicates that j-th of unit is opposite to set
The hot stiffness matrix for counting variable 2, so having:
So formula (33) may finally rewrite are as follows:
Same sensitivity of the available optimization aim M for design variable are as follows:
In an iterative process simultaneously consider Reliability Constraint and relative variation, if the result of a certain iteration step meet it is non-
The constraint condition of probability decision degree d≤0, and before and after iteration two steps the sum of design variable variable quantity less than a preset value ε when,
Then determine that iterative process terminates, resulting optimum results are the preferred configuration under specified criteria.
Instance analysis
In order to verify the feasibility and validity of the method, this example has done one referring to the radiator structure of engineering in practice
A simplified model, design domain, boundary condition and institute's heating load copy actual conditions to be designed, while in order to guarantee to tie
There is structure certain optimization space to have done certain approximate processing.Main structure body as shown in Figure 4 be a 30mm × 30mm ×
The cuboid of 20mm is equally spacedly dispersed with five 30mm × 20mm × 2mm fins on it, takes a length of 1mm of element sides, most
Total is divided into 24000 8 node hexahedral elements at last.Wherein, at five points as shown in Figure 5 of cuboid bottom
It is applied with the concentration heat stream loading of P=10W, and given t=200 DEG C of temperature restraint, is applied with pair on the surface of five fins
Stream heat exchange load is as boundary condition, wherein surface coefficient of heat transfer h=130W/ (m2K), environment temperature t0=25 DEG C.Using leading
Hot coefficient is respectively λ1=160W/ (mK), λ2=90W/ (mK) and λ3It is excellent that three kinds of materials of=25W/ (mK) carry out topology
Change, the density of these three materials is followed successively by ρ1=2800kg/m3、ρ2=1500kg/m3And ρ3=400kg/m3.It is assumed that concentrating
Thermal force, thermal coefficient and safe temperature are all that bounded is uncertain, and specific value is as shown in the table.
Fig. 6 (a), Fig. 6 (b), Fig. 6 (c), Fig. 6 (d), Fig. 6 (e), Fig. 6 (f) are to be radiated in this example based on practical respectively
The simplified model of structure design carries out single material/bi-material layers/tri- materials, certainty/reliability topological optimization result, wherein
Grey represents the preferable material 1 of heating conduction, and Dark grey represents the material 2 of moderate performance, and light gray represents the poor material of performance
Material 3.
The mass fractions relative of optimization gained configuration is as shown in the table:
It can be seen that: the mass fractions relative highest of single material topological optimization result, bi-material layers and three materials topology
The case where optimization gained configuration compares single material has apparent loss of weight, and the result of three material topological optimizations is slightly better than double materials
The case where material, this shows the configuration that can be more optimized for the model in this example by the use of three kinds of materials.
Meanwhile the mass fractions relative of reliability topological optimization result also greater than corresponding certainty topological optimization the case where, can be with
Find out and inevitably sacrifices certain economy under the premise of guaranteeing that structural reliability requires.In addition, being opened up from more materials
Flutter in the result of optimization it can also be seen that with high-termal conductivity 1 integrated distribution of material on the main heat transfer path of structure, and
The poor material 2 of heating conduction and material 3 are then distributed in the lesser region of carrying, primarily serve auxiliary heat transfer and mitigate structure weight
The effect of amount.By the verifying of this example, applicability of the Topology Optimization Method in engineering structure is illustrated, can be correlation
The optimization design problem of engineering in practice provides some enlightenments and certain technical support.
In conclusion the invention proposes a kind of instable more material radiator structure topological optimizations of consideration increasing material manufacturing
Method.This method is considering the material thermal conductivity as caused by increasing material manufacturing unstability, structure thermal force and safe temperature etc.
In the case that aspect is uncertain, by the interpolation model based on more material radiator structures, constructed under temperature Reliability Constraint
More material radiator structure topological optimization mathematical models.Then by unit puppet density as design variable, by architecture quality as
Optimization aim finally obtains the preferred configuration of more material radiator structures under prescribed conditions by the iterative calculation of MMA algorithm.
The above content is the detailed process of the method for the invention, are based on this, the scope of the present invention includes: to examine
Consider the instable technical solution for more material radiator structure topology optimization designs of increasing material manufacturing.
Claims (7)
1. a kind of instable more material radiator structure Topology Optimization Methods of consideration increasing material manufacturing, it is characterised in that: including such as
Lower step:
Step 1: it for considering probabilistic more material radiator structure topology optimization problems, seeks to consider material conducts heat system
The uncertainty of number, structure thermal force and safe temperature, using interval variable KIIndicate overall thermal rigidity interval matrix, PIIt indicates
Overall thermal load interval vector,It indicates temperature range vector, and original certainty amount is substituted with this,
Middle subscript I indicates that variable is interval variable, and N indicates the number of temperature freedom degree, it is as follows to obtain finite element equation:
KItI=PI
In view of governing equation is linear equation, can be acquired using the vertex combined method for considering sensitivityIn certain
One temperature componentsThe upper bound and lower bound；
Step 2: after the bound for finding out temperature range using vertex combined method, according to Multidisciplinary systems theory, non-general
A new Multidisciplinary systems index can be constructed on the basis of rate reliability R, i.e. optimization characteristic distance d, as d > 0, institute
Corresponding reliability R < Rtarg, do not meet design requirement；And as d≤0, corresponding reliability R > Rtarg, meet design and want
It asks；
Step 3: density-stiffness interpolation schemes are based on, more material radiator structure interpolation models are established:
Wherein λiThe material thermal conductivity of i-th of unit, λ after expression interpolation1Indicate the thermal coefficient of material 1, λ2Indicate material 2
Thermal coefficient, λ3Indicate the thermal coefficient of material 3, x1,i、x2,iAnd x3,iIt respectively indicates the design variable 1 of i-th of unit, set
Variable 2 and design variable 3 are counted, p (p > 1) indicates penalty factor；
Step 4: it on the basis of classical topologies optimized mathematical model, by optimization characteristic distance d as constraint condition, and combines
Above-mentioned more material interpolation models can establish the Multidisciplinary systems topological optimization mathematical model based on more material radiator structures
Are as follows:
Wherein, M indicates the architecture quality in design section, ViIndicate the volume of i-th of unit, n indicates the unit in design section
Sum, ρ1、ρ2And ρ3The density of material 1, material 2 and material 3 is respectively indicated,x 1、x 2Withx 3Respectively indicate design variable 1, design
The lower bound of variable 2 and design variable 3,WithRespectively indicate the upper bound of design variable 1, design variable 2 and design variable 3.
djIndicate that the Multidisciplinary systems of j-th of constraint, m indicate the sum of constraint；
Step 5: obtaining constraint function and objective function for the partial derivative of design variable by the use of adjoint vector method, into
Line sensitivity analysis, in order to the solution of subsequent gradients optimization algorithm；
Step 6: by the iterative calculation of MMA algorithm, while considering Reliability Constraint and relative variation, if a certain iteration step
Result meet the constraint condition of non-probability decision degree d≤0, and before and after iteration the sum of design variable variable quantity of two steps less than one
When a preset value ε, then iterative process terminates, using the result of present topology optimization as final optimum results.
2. a kind of instable more material radiator structures topological optimization side of consideration increasing material manufacturing according to claim 1
Method, it is characterised in that: the uncertain of material thermal conductivity, structure thermal force and safe temperature is considered in the step one
Property, and original certainty amount is substituted in the form of interval variable.
3. a kind of instable more material radiator structures topological optimization side of consideration increasing material manufacturing according to claim 1
Method, it is characterised in that: optimization characteristic distance d is used to refer in the step two as the judgement of structure Multidisciplinary systems
Mark.
4. a kind of instable more material radiator structures topological optimization side of consideration increasing material manufacturing according to claim 1
Method, it is characterised in that: the interpolation model based on more material radiator structures is used in the step three.
5. a kind of instable more material radiator structures topological optimization side of consideration increasing material manufacturing according to claim 1
Method, it is characterised in that: consider the feelings of bi-material layers and three materials in constructed topological optimization model in the step four
Condition.
6. a kind of instable more material radiator structures topological optimization side of consideration increasing material manufacturing according to claim 1
Method, it is characterised in that: constraint function and objective function obtained by when carrying out sensitivity analysis in the step five are for design
The partial derivative of variable.
7. a kind of instable more material radiator structures topological optimization side of consideration increasing material manufacturing according to claim 1
Method, it is characterised in that: consider that Reliability Constraint and relative variation are used as simultaneously in the step six and judge iterative process
The standard whether terminated.
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