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
- Publication number
- CN109408845A CN109408845A CN201810919278.7A CN201810919278A CN109408845A CN 109408845 A CN109408845 A CN 109408845A CN 201810919278 A CN201810919278 A CN 201810919278A CN 109408845 A CN109408845 A CN 109408845A
- Authority
- CN
- China
- Prior art keywords
- reliability
- optimization
- design
- representing
- variable
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Granted
Links
- 239000000463 material Substances 0.000 title claims abstract description 131
- 238000005457 optimization Methods 0.000 title claims abstract description 83
- 238000000034 method Methods 0.000 title claims abstract description 64
- 238000004519 manufacturing process Methods 0.000 title claims abstract description 34
- 238000013461 design Methods 0.000 claims abstract description 71
- 238000004422 calculation algorithm Methods 0.000 claims abstract description 9
- 238000013178 mathematical model Methods 0.000 claims abstract description 8
- 238000004364 calculation method Methods 0.000 claims abstract description 6
- 239000000654 additive Substances 0.000 claims description 28
- 230000000996 additive effect Effects 0.000 claims description 28
- 230000017525 heat dissipation Effects 0.000 claims description 28
- 230000008569 process Effects 0.000 claims description 14
- 239000011159 matrix material Substances 0.000 claims description 9
- 238000010206 sensitivity analysis Methods 0.000 claims description 8
- 238000002360 preparation method Methods 0.000 description 9
- 230000004044 response Effects 0.000 description 9
- 230000006870 function Effects 0.000 description 8
- 230000035945 sensitivity Effects 0.000 description 8
- 238000005516 engineering process Methods 0.000 description 6
- 239000000843 powder Substances 0.000 description 6
- 238000011160 research Methods 0.000 description 6
- 239000006185 dispersion Substances 0.000 description 5
- 238000004458 analytical method Methods 0.000 description 4
- 238000010586 diagram Methods 0.000 description 4
- 230000000694 effects Effects 0.000 description 4
- 238000012545 processing Methods 0.000 description 4
- 238000012546 transfer Methods 0.000 description 4
- 239000000243 solution Substances 0.000 description 3
- 238000012938 design process Methods 0.000 description 2
- 239000010410 layer Substances 0.000 description 2
- 239000004576 sand Substances 0.000 description 2
- 239000011343 solid material Substances 0.000 description 2
- 239000013585 weight reducing agent Substances 0.000 description 2
- NAWXUBYGYWOOIX-SFHVURJKSA-N (2s)-2-[[4-[2-(2,4-diaminoquinazolin-6-yl)ethyl]benzoyl]amino]-4-methylidenepentanedioic acid Chemical group C1=CC2=NC(N)=NC(N)=C2C=C1CCC1=CC=C(C(=O)N[C@@H](CC(=C)C(O)=O)C(O)=O)C=C1 NAWXUBYGYWOOIX-SFHVURJKSA-N 0.000 description 1
- 102220628160 FERM and PDZ domain-containing protein 4_P10W_mutation Human genes 0.000 description 1
- 230000003190 augmentative effect Effects 0.000 description 1
- 230000005540 biological transmission Effects 0.000 description 1
- 230000019771 cognition Effects 0.000 description 1
- 230000001149 cognitive effect Effects 0.000 description 1
- 238000001816 cooling Methods 0.000 description 1
- 230000001186 cumulative effect Effects 0.000 description 1
- 238000009795 derivation Methods 0.000 description 1
- 238000010790 dilution Methods 0.000 description 1
- 239000012895 dilution Substances 0.000 description 1
- 230000009977 dual effect Effects 0.000 description 1
- 230000004927 fusion Effects 0.000 description 1
- 238000009413 insulation Methods 0.000 description 1
- 239000011229 interlayer Substances 0.000 description 1
- 239000007788 liquid Substances 0.000 description 1
- 238000005259 measurement Methods 0.000 description 1
- 238000002844 melting Methods 0.000 description 1
- 230000008018 melting Effects 0.000 description 1
- 238000010606 normalization Methods 0.000 description 1
- 238000000016 photochemical curing Methods 0.000 description 1
- 230000005855 radiation Effects 0.000 description 1
- 238000005245 sintering Methods 0.000 description 1
- 238000007711 solidification Methods 0.000 description 1
- 230000008023 solidification Effects 0.000 description 1
- 238000012916 structural analysis Methods 0.000 description 1
- 238000012360 testing method Methods 0.000 description 1
- 230000009466 transformation Effects 0.000 description 1
- 238000012795 verification Methods 0.000 description 1
Classifications
-
- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F30/00—Computer-aided design [CAD]
- G06F30/20—Design optimisation, verification or simulation
- G06F30/23—Design optimisation, verification or simulation using finite element methods [FEM] or finite difference methods [FDM]
-
- Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
- Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
- Y02P—CLIMATE CHANGE MITIGATION TECHNOLOGIES IN THE PRODUCTION OR PROCESSING OF GOODS
- Y02P10/00—Technologies related to metal processing
- Y02P10/25—Process efficiency
Landscapes
- Engineering & Computer Science (AREA)
- Physics & Mathematics (AREA)
- Theoretical Computer Science (AREA)
- Computer Hardware Design (AREA)
- Evolutionary Computation (AREA)
- Geometry (AREA)
- General Engineering & Computer Science (AREA)
- General Physics & Mathematics (AREA)
- Management, Administration, Business Operations System, And Electronic Commerce (AREA)
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 invention relates to the technical field of heat dissipation structure topology optimization, in particular to a multi-material heat dissipation structure topology optimization method considering additive manufacturing instability.
Background
Although additive manufacturing is a popular manufacturing technology, there are critical problems in many aspects due to its being an emerging technology which is still imperfect, and it can be said that there is a lot of research space in this field of additive manufacturing.
The problems of the additive manufacturing method are that the molten high-temperature liquid or powder is rapidly solidified, and then a three-dimensional object is constructed by layer-by-layer printing, so that the cooling and forming of the object are difficult to control during solidification, and the processed product has problems in a microscopic view, and the problems further cause dispersion or uncertainty of material performance, which can be attributed to instability of the preparation process.
The instability caused by the additive manufacturing technology is emphasized for the following three reasons:
firstly, the instability inherent in the material processing and forming process cannot be avoided. For example, for the coaxial powder feeding technology, the instability of the scanning rate and the powder feeding rate can cause the problems of insufficient powder feeding or excessive dilution and the like, and the quality of processing and forming is affected; when a three-dimensional photocuring rapid prototyping technology is adopted, the contour of a model is fuzzy when a microstructure is modeled due to instability of the diameter of a light spot, and interlayer residual stress occurs due to instability of thicknesses of different layers of a base material; when sintering powders using selective laser fusion forming techniques, there is some deviation in the size of the microscopic structures due to the instability in the laser advance speed, which in turn can lead to non-uniform melting of the powder. The above objective facts indicate that the material properties are dispersed and have dimensional deviations from the microscopic angles during the structure preparation process.
Secondly, the cross-scale transmission of material properties also aggravates the hidden trouble of structural safety caused by the instability of the preparation process. The learners have systematically studied the spatial lattice structure and found that the instability of the preparation process can cause uncertainty in the microscopic dimension of the test piece, and finally the difference between the actual value and the theoretical value of the elastic modulus reaches 63%. The reason is that some structural materials have nonlinear constitutive relation and complicated macroscopic properties, and the dimensional deviation caused by instability in the preparation process is gradually accumulated, so that the macroscopic properties of the structure can be affected non-negligibly.
Thirdly, the performance advantages of certain materials cannot be fully exerted by the traditional optimization design method under the condition of unstable preparation process. When the reusable spacecraft is designed in the united states, the originally proposed airframe scheme can simultaneously meet the requirements of thermal protection, structural performance and propulsion capability, but the difference between the thermal insulation capability of the actual structure and the prediction result is too large due to the instability of the manufacturing process, so that a more conservative design scheme is inevitably selected. The influence of various uncontrollable factors on the structure in the preparation process cannot be considered in the current optimization design method, so that the final design scheme is possibly infeasible in practice, and the optimization of the structure performance can only be abandoned and other design schemes with low cost performance can only be adopted.
Therefore, the research on the instability of the additive manufacturing process has certain theoretical significance and engineering value. The instability of the structure preparation process can further cause the dispersion of the material performance, and the dispersion or uncertainty is actually and widely existed in the structure design process of a complex system, and the main sources include six aspects of human, machine, material, method, ring and measurement, which can seriously affect the working performance and safety of the structure. The instability of the additive manufacturing process belongs to the aspect of machine.
Over the last centuries, people have increasingly become more aware of uncertainties, and have discovered that uncertainties in practical engineering problems have objective unpredictability, ambiguity in subjective cognition, and incompleteness due to lack of information in complex situations. In a reliability analysis of a structure, uncertainties can be classified into random uncertainties, fuzzy uncertainties (or cognitive uncertainties), and interval uncertainties (or bounded uncertainties).
Among these, the dispersion of material properties due to the preparation instability studied in the present invention is subject to bounded uncertainties. Such uncertainty is in complex cases due to the lack of sample information, which in turn leads to subjective insights. Bounded uncertainties are actually present in practical engineering problems in a wide range, and their impact on structural reliability is not negligible.
It is noted that these uncertainty factors also have cumulative and aggravating effects, and when a small uncertainty amount is cumulatively added, especially when a plurality of small uncertainty amounts are cumulatively added, the originally predicted result may be substantially incorrect or even completely distorted. At present, serious potential safety hazards still exist in the structural concept design stage, and the influence of uncertainty factors cannot be ignored only by considering the certainty factors at any stage of the product design process. Therefore, in the large context of the wide application of additive manufacturing technology in the field of aerospace, it is necessary and meaningful to consider the effects of uncertainty caused by instability of additive manufacturing in structural analysis and design.
Meanwhile, since the multi-material structure is often manufactured by using complex processes (such as additive manufacturing), the processes inevitably affect the characteristics of the material. In addition, multi-material structures with high design specifications and manufacturing costs are often used in complex load environments, where the loads are difficult to quantify accurately and have a certain dispersion. The existence of uncertainty has a non-negligible influence on the safety of the structure, in this case, it is difficult to obtain a result meeting the design requirement by using a deterministic multi-material topological optimization method, and the safety and the reliability of the structure can be ensured only by using a reliable topological optimization method. For the problems in engineering practice, due to the fact that the working condition is complex and sample data is few, information related to uncertainty is difficult to obtain accurately, but the limit of uncertainty factors is easy to determine, and therefore the concept of non-probability reliability based on the convex set model is provided.
However, research work on related theories of non-probabilistic reliability topological optimization just starts, and the current research results in the field are few, and especially, the research on the aspect of non-probabilistic reliability topological optimization based on a multi-material heat dissipation structure is even blank. Based on the method, under the condition of considering the instability of additive manufacturing, the topological optimization research of the multi-material aviation heat dissipation structure by adopting a non-probabilistic reliability method not only has certain theoretical significance, but also can provide an effective method for related engineering problems.
Disclosure of Invention
The invention provides a multi-material heat dissipation structure topology optimization method considering additive manufacturing instability. The main contents are as follows: uncertainty in material performance, structural bearing and the like caused by additive manufacturing instability is considered, an interpolation model of a multi-material heat dissipation structure is used, an optimized characteristic distance d is used as a constraint condition, performance advantages of the multi-material structure are fully exerted, and the requirements of structure economy and reliability can be simultaneously met by a topological optimization result.
The technical scheme adopted by the invention is as follows: a method of topological optimization of a multi-material heat dissipating structure taking into account additive manufacturing instabilities, the steps of the method being as follows:
the method comprises the following steps: for the topological optimization problem of the multi-material heat dissipation structure considering uncertainty, the uncertainty of the material heat conductivity coefficient, the structure heat load and the safe temperature is considered. Using interval variables KIRepresenting the overall thermal stiffness interval matrix, PIRepresenting the overall heat load interval vector and,and representing a temperature interval vector, and replacing the original deterministic quantity with the temperature interval vector, wherein the superscript I represents that the variable is an interval variable, and N represents the number of temperature degrees of freedom, and obtaining a finite element equation as follows:
KItI＝PI
considering that the control equation is a linear equation, the sensitivity-considered vertex combination method can be used for solvingA certain temperature componentUpper and lower bounds.
Step two: after the upper and lower bounds of the temperature interval are solved by using a vertex combination method, a new non-probability reliability index, namely an optimized characteristic distance d, can be constructed on the basis of the non-probability reliability R according to the non-probability reliability theory. When d is>At 0, the corresponding reliability R<RtargDoes not meet the design requirements; when d is less than or equal to 0, the corresponding reliability R>RtargAnd meets the design requirements.
Step three: based on the density-rigidity interpolation model, a multi-material heat dissipation structure interpolation model is established:
wherein λiIndicating the thermal conductivity of the material of the ith cell after interpolation，λ1Denotes the thermal conductivity, λ, of the material 12Denotes the thermal conductivity, λ, of the material 23Denotes the thermal conductivity, x, of the material 31,i、x2,iAnd x3,iDesign variables 1,2 and 3, p (p) of the ith cell>1) A penalty factor is indicated.
Step four: on the basis of a classical topological optimization mathematical model, the optimization characteristic distance d is used as a constraint condition, and the multi-material interpolation model is combined, so that a non-probability reliability topological optimization mathematical model based on a multi-material heat dissipation structure can be established as follows:
where M represents the structural mass in the design region, ViDenotes the volume of the ith cell, n denotes the total number of cells in the design area, p1、ρ2And ρ3Respectively representing the densities of material 1, material 2 and material 3,x 1、x 2andx 3representing the lower bounds of design variable 1, design variable 2, and design variable 3,andrepresenting the upper bounds of design variable 1, design variable 2, and design variable 3, respectively. djRepresents the non-probabilistic reliability of the jth constraint and m represents the total number of constraints.
Step five: and obtaining the partial derivatives of the constraint function and the objective function to the design variable through the use of the adjoint vector method, and carrying out sensitivity analysis so as to facilitate the solution of a subsequent gradient optimization algorithm.
Step six: and (3) through iterative calculation of an MMA algorithm, simultaneously considering reliability constraint and relative variation, if the result of a certain iteration step meets the constraint condition that the non-probability reliability d is less than or equal to 0, and the sum of the design variable variations of the two steps before and after the iteration is less than a preset value epsilon, ending the iteration process, and taking the current topology optimization result as the final optimization result.
In the first step, uncertainty of material heat conductivity coefficient, structural heat load and safe temperature is considered, and original certainty quantity is replaced by interval variable.
And in the second step, the optimized characteristic distance d is used as a judgment index of the non-probability reliability of the structure.
Wherein, an interpolation model based on a multi-material heat radiation structure is adopted in the third step.
Wherein, the topology optimization model constructed in the fourth step considers the case of dual materials and triple materials.
And step five, performing sensitivity analysis on the obtained constraint function and the obtained partial derivative of the objective function to the design variable.
And step six, simultaneously considering the reliability constraint and the relative variation as the standard for judging whether the iteration process is ended or not.
Compared with the existing method, the invention has the following differences:
according to the invention, the influence of bounded uncertainty caused by additive manufacturing instability is considered during topological optimization of the heat dissipation structure, reliability topological optimization is carried out by taking the optimized characteristic distance as constraint, and the safety of the configuration obtained by optimization is ensured. Meanwhile, by introducing the concept of a multi-material structure, the performance advantages of multiple materials are fully exerted during optimal design, the quality of the structure is reduced as much as possible, and the economical efficiency is considered under the condition of meeting the requirement of reliability.
Drawings
FIG. 1 is a flow diagram of a multi-material heat dissipating structure topology optimization method that accounts for additive manufacturing instability;
FIG. 2 is a two-dimensional interference diagram of a non-probabilistic reliability model;
FIG. 3 is a schematic of an optimized feature distance;
FIG. 4 is a schematic diagram of the model used in the example;
FIG. 5 is a schematic illustration of an example loading pattern;
FIG. 6 is a schematic diagram of an example optimization result, wherein FIG. 6(a) is a single-material deterministic topological optimization result; FIG. 6(b) is a two-material deterministic topology optimization result; FIG. 6(c) is a three-material deterministic topology optimization result; FIG. 6(d) is the single material reliability topology optimization result; FIG. 6(e) is the bi-material reliability topology optimization result; fig. 6(f) is the three-material reliability topology optimization result.
Detailed Description
The following details the overall process of the multi-material heat dissipation structure topology optimization method considering additive manufacturing instability according to the present invention:
1. for the topological optimization problem of the multi-material heat dissipation structure considering uncertainty, the uncertainty of the material heat conductivity coefficient, the structure heat load and the safe temperature is considered. In engineering practice, it is generally difficult to obtain accurate information of uncertain factors, but the limit is relatively easy to determine, and in consideration of the situation, the specific method is to adopt an interval variable KIRepresents the structural integral thermal stiffness interval matrix, PIRepresenting the overall heat load interval vector and,and representing a temperature interval vector to replace the original deterministic quantity, wherein the superscript I represents that the variable is an interval variable, N represents the number of temperature degrees of freedom, and the obtained finite element equation is as follows:
KItI＝PI(1)
in which, considering that the governing equation is a linear equation, it can be obtained by the vertex combination method considering sensitivity as followsA certain temperature component tj IUpper and lower bounds.
The vertex combination method considering the sensitivity is as follows:let a function f (x)1,x2,…,xn) For any independent variable xi(i ═ 1,2, …, n) is monotonic, and in this case the argument can be expressed as a range variable, i.e.:
wherein,x iandthe lower and upper bounds of the parameter are respectively.
And the monotonicity can obtain that the value interval of f is as follows:
wherein r is called vertex combination ordinal number, r is 1,2, …,2n，ki1,2 refer to the lower and upper bounds of the parameter, respectively, i.e.
Therefore, according to the vertex combination method, the temperature value range corresponding to the jth constraint can be obtainedComprises the following steps:
wherein For the actual temperature interval of the jth temperature constraint, superscript ki1,2, when kiWhen the value is 1, the lower bound is expressed, and when k isiWhen 2, the corresponding value is upper bound, that is:
in the actual calculation process, the influence of each uncertain variable on the magnitude of the response value can be calculated in a differential mode. When the sensitivity of the response value to the variable is greater than 0, the response value is the largest when the variable is taken as an upper bound, and the response value is the smallest when the variable is taken as a lower bound; and when the sensitivity of the response value to the variable is less than 0, the response value is minimum when the variable is taken as an upper bound, and the response value is maximum when the variable is taken as a lower bound. Thus, for the case of n uncertain variables, the upper and lower bounds of the response value can be finally obtained by n times of differential sensitivity positive-negative analysis and 2 times of response analysis.
2. After the upper and lower bounds of the temperature interval are obtained by using a vertex combination method, a non-probability set reliability model under temperature constraint can be established based on a non-probability reliability theory.
Let tj,aActual temperature, t, for jth temperature constraintj,sConstrained for jth temperatureSafe temperature, taking into account the effect of uncertainty, represents temperature as an interval variable, namely:
wherein t andwhich respectively refer to the lower and upper temperature limits, and when these two ranges are simultaneously represented on the same axis, interference may occur.
Setting the ultimate state function M (t) of the structurej,s,tj,a) Comprises the following steps:
M(tj,s,tj,a)＝tj,s-tj,a(8)
the limit state plane or failure plane is then:
M(tj,s,tj,a)＝tj,s-tj,a＝0 (9)
when M (t)j,s,tj,a) When the value is more than or equal to 0, the structure meets the given constraint condition, and when M (t)j,s,tj,a)<A value of 0 indicates that it does not comply with the given constraints. For variation of actual temperature and safety temperature intervalAnd (3) carrying out standardized transformation processing:
whereinReferred to as the interval radius. After normalization, there is δ tj,a∈[-1,1]，δtj,s∈[-1,1]. Substituting formula (10) into the poleIn the finite state plane equation, one can obtain:
from this δ t can be derivedj,sAnd δ tj,aThe relationship between them is:
the above formula can be expressed in a rectangular plane coordinate system, and δ u is markedj,sAnd δ uj,aThe value interval of (2) is shown in fig. 2.
Area S of safety zoneABFEDArea sum S with safety area and failure areaABCDThe ratio is called the non-probabilistic reliability R. R will be solved for the case where the variable region and the extreme state plane intersect. Firstly, solving the limit state plane and the straight line delta tj,sLet δ t in equation (12) be the intersection of-1j,sGet δ t ═ 1j,aComprises the following steps:
order toCan be solved to obtainThen, the limit state plane and the straight line deltat are obtainedj,aLet δ t in equation (13) be the intersection of 1j,aδ t can be obtained as 1j,sComprises the following steps:
order toCan be solved to obtainThus, the expression of the non-probability reliability R can be obtained as:
substituting (16) the previous result, R can be expressed as:
in the same way, the expression of R in the other five cases can be obtained as follows:
as can be seen from equation (17), the reliability R is constant in some cases, and it is difficult for the MMA algorithm to optimize in the correct direction, so that a new non-probabilistic reliability index, i.e., the optimized feature distance d, is proposed based on the reliability R. As shown in fig. 3, the distance from the original limit state plane to the target limit state plane is defined as d. Wherein the target extreme state plane is parallel to the original extreme state plane and its non-probability reliability RtargIs a given value.
And the reliability R of the targettargGenerally close to 1, therefore, the target extreme state plane is generally positioned at the lower right corner of the variable region, and two special cases of the intersection situation of the variable region and the target extreme state plane are considered firstly. The slope of the extreme state plane in the critical case is first calculated. For k1Has (2X 2/k)1×1/2)/4＝1-RtargCan beTo solve to obtain k1＝1/2(1-Rtarg) In the same way, k can be obtained2＝2(1-Rtarg). When the slopes of the extreme state planes respectively take different values, considering the relationship between the slopes and the critical slope, the expression of d can be obtained by the distance formula between two parallel straight lines as follows:
when d is>Reliability R of extreme state plane with target at 0targThe corresponding reliability R above the target extreme state plane, since the target value is larger than the area of the security domain<RtargAnd does not meet the design requirements. When d is less than or equal to 0, the reliability R between the limiting state plane and the targettargCorresponding reliability R below the target extreme state plane, where the target value is less than or equal to the area of the security domain>RtargAnd meets the design requirements.
3. Interpolation models based on individual material density and stiffness are given in the classical SIMP model, namely:
E＝xpE0(19)
wherein E0Representing the elastic modulus of the solid material, E representing the interpolated elastic modulus, x representing the design variable (cell relative density), and p representing a penalty factor (often taken to be 3). The continuity of the elastic modulus of the material in the process of topology optimization can be ensured through an interpolation model of a single material, and the main effect of the penalty factor is to greatly reduce the intermediate density unit in the optimization result.
Since the invention is concerned with a multi-material heat dissipation structure, the elastic modulus E in the above formula should be changed into the thermal conductivity λ of the material, so that:
λ＝xpλ0a single material (20) wherein0Denotes the thermal conductivity of the solid material, λ denotes the interpolated thermal conductivity, xRepresenting the design variables (cell relative density) and p representing a penalty factor (often taken to be 3).
And then, referring to the single-material interpolation model, constructing a multi-material interpolation model based on the heat dissipation structure as follows:
wherein λiDenotes the material thermal conductivity, λ, of the ith cell after interpolation1Denotes the thermal conductivity, λ, of the material 12Denotes the thermal conductivity, λ, of the material 23Denotes the thermal conductivity, x, of the material 31,i、x2,iAnd x3,iDesign variables 1,2 and 3, p (p) of the ith cell>1) A penalty factor is indicated. By the interpolation model, a continuous interpolation model of two or three materials can be realized.
4. On the basis of a prior topological optimization mathematical model, the optimization characteristic distance d is taken as a constraint condition, and the single-material/multi-material interpolation model is combined, so that a non-probability reliability topological optimization mathematical model based on a single-material/multi-material heat dissipation structure can be established as follows:
wherein V represents the structural volume within the design area, ViRepresenting the volume of the ith cell, n representing the total number of cells in the design area,xrepresents the lower bound of the design variable,representing the upper bound of the design variable. djRepresents the non-probabilistic reliability of the jth constraint and m represents the total number of constraints.
Where M represents the structural mass in the design region, ViDenotes the volume of the ith cell, n denotes the total number of cells in the design area, p1、ρ2And ρ3Respectively representing the densities of material 1, material 2 and material 3,x 1、x 2andx 3representing the lower bounds of design variable 1, design variable 2, and design variable 3,andrepresenting the upper bounds of design variable 1, design variable 2, and design variable 3, respectively. djRepresents the non-probabilistic reliability of the jth constraint and m represents the total number of constraints.
When solving the topology optimization problem by using a mobile asymptote optimization method (MMA), since the MMA algorithm is a gradient optimization algorithm, sensitivity analysis is required, and specifically, the partial derivatives of the constraint function and the objective function with respect to the design variable are obtained. However, since the number of design variables in the optimization model is much larger than the number of constraints, if the solution is performed in a differential manner, a huge amount of calculation is caused during sensitivity analysis. According to this feature, the sensitivity analysis can be performed by using the concomitant vector method.
Since the sensitivity solution processes for bimaterials and tri-materials are similar, the basic process of sensitivity analysis is given below with bi-material topology optimization as an example. The optimization characteristic distance d of j (j ═ 1,2, …, m) constraint is determined by the chain derivation rule of complex functionjFor design variable xi(i＝1,2,…,n)(xiIs x1,iOr x2,i) The sensitivity of (a) is:
wherein,
whereinAndthese two parts can be obtained by solving directly (23), butAndbut the calculation cannot be directly carried out, and an augmented Lagrange multiplier method is used for solving:
wherein λ isj(j ═ 1,2, …, m) is denoted as an accompanying vector, also known as the lagrange multiplier vector. Due to PI-KItI0, soThe equation (27) is calculated for the design variable xiThe partial derivative of (a) can be found:
wherein,
for any lambdajEquation (28) holds, and thus, by selecting an appropriate λjSo thatThe method meets the requirements of ordering the food,
since the thermal stiffness matrix is a symmetric matrix, the two ends of equation (30) can be transposed:
as shown by equation (31), by applying a virtual thermal load to the modelThen, the obtained temperature is λj. Solve for lambdajThen, the partial derivatives of the upper and lower bounds of the temperature at the constraint points on the design variables are:
whereinRespectively represent correspondencesThe adjoint vector, the cell thermal stiffness matrix, and the temperature column vector,λ j、K j、trespectively represent corresponding tj,aThe adjoint vector, the cell thermal stiffness matrix, and the temperature column vector. In this model, P is due to the thermal loadINot varied by design variables, i.e.Equation (32) can be rewritten as:
from the results (2) obtained previously, it is again possible:
wherein K1,jThermal stiffness matrix, K, representing the j-th cell relative design variable 12,jThe thermal stiffness matrix representing the jth cell relative to design variable 2, so there is:
therefore, equation (33) can be finally rewritten as:
it can also be obtained that the sensitivity of the optimization objective M to the design variables is:
and simultaneously considering the reliability constraint and the relative variation in the iteration process, if the result of a certain iteration step meets the constraint condition that the non-probability reliability d is less than or equal to 0 and the sum of the design variable variations of the two steps before and after the iteration is less than a preset value epsilon, judging that the iteration process is ended, and obtaining an optimized result, namely the optimal configuration under the given condition.
Example analysis
In order to verify the feasibility and the effectiveness of the method, a simplified model is made by referring to a heat dissipation structure in engineering practice, a design domain, boundary conditions and applied heat load are designed according to the actual situation, and meanwhile, certain approximate processing is made in order to ensure that the structure has certain optimization space. The main structure body shown in fig. 4 is a cuboid with the size of 30mm × 30mm × 20mm, five ribs with the size of 30mm × 20mm × 2mm are distributed on the main structure body at equal intervals, the unit side is 1mm, and finally the whole structure is divided into 24000 hexahedral units with 8 nodes. Wherein, a concentrated heat flow load of P10W is applied at five points on the bottom of the cuboid as shown in figure 5, a temperature constraint of t 200 ℃ is given, and convective heat transfer loads are applied on the surfaces of the five fins as boundary conditions, wherein the surface heat transfer coefficient h is 130W/(m)2K) ambient temperature t0At 25 ℃. With respective thermal conductivity of lambda1＝160W/(m·K)、λ290W/(m.K) and λ3Carrying out topology optimization on three materials of 25W/(m.K), wherein the densities of the three materials are rho1＝2800kg/m3、ρ2＝1500kg/m3And ρ3＝400kg/m3. It is assumed here that the concentrated thermal load, the thermal conductivity, and the safe temperature are bounded and uncertain, and the specific values are shown in the following table.
Fig. 6(a), 6(b), 6(c), 6(d), 6(e), and 6(f) are the results of single material/bi-material/tri-material, deterministic/reliable topological optimization of the simplified model based on the actual heat dissipation structure design in this example, where gray represents material 1 with better thermal conductivity, dark gray represents material 2 with moderate performance, and light gray represents material 3 with poorer performance, respectively.
The relative mass fractions for the configuration obtained by optimization are shown in the following table:
it can be seen that: the relative mass fraction of the single-material topological optimization result is the highest, the configurations obtained by the double-material topological optimization and the three-material topological optimization have obvious weight reduction compared with the single-material condition, and the results of the three-material topological optimization are slightly better than those of the double-material condition, which shows that for the model in the present example, a more optimized configuration can be obtained by using the three materials. Meanwhile, the relative quality fraction of the reliability topological optimization result is also larger than the corresponding situation of the deterministic topological optimization, so that certain economic efficiency is inevitably sacrificed on the premise of ensuring the structural reliability requirement. Furthermore, it can be seen from the result of the topological optimization of the multi-material that the material 1 with high thermal conductivity is distributed on the main heat transfer path of the structure, while the material 2 and the material 3 with poor thermal conductivity are distributed on the area with smaller load, which mainly plays the roles of auxiliary heat transfer and structure weight reduction. The verification of the example shows the applicability of the topology optimization method in the engineering structure, and can provide some inspiration and certain technical support for the optimization design problem in the relevant engineering practice.
In summary, the present invention provides a topology optimization method for a multi-material heat dissipation structure considering instability of additive manufacturing. According to the method, under the condition that uncertainty in the aspects of material heat conductivity coefficient, structure heat load, safety temperature and the like caused by additive manufacturing instability is considered, a multi-material heat dissipation structure topological optimization mathematical model is constructed under the temperature reliability constraint through an interpolation model based on a multi-material heat dissipation structure. And then, taking the unit pseudo density as a design variable, taking the structure quality as an optimization target, and finally obtaining the optimal configuration of the multi-material heat dissipation structure under the given condition through iterative computation of an MMA algorithm.
The above is a detailed flow chart of the method of the present invention, and based on this, the scope of protection of the right of the present invention includes: the technical scheme for the topological optimization design of the multi-material heat dissipation structure in consideration of additive manufacturing instability.
Claims (7)
1. A multi-material heat dissipation structure topology optimization method considering additive manufacturing instability is characterized in that: the method comprises the following steps:
the method comprises the following steps: for the topological optimization problem of the multi-material heat dissipation structure considering uncertainty, namely the uncertainty of the heat conductivity coefficient of the material, the structure heat load and the safe temperature is considered, an interval variable K is adoptedIRepresenting the overall thermal stiffness interval matrix, PIRepresenting the overall heat load interval vector and,and representing a temperature interval vector, and replacing the original deterministic quantity with the temperature interval vector, wherein the superscript I represents that the variable is an interval variable, and N represents the number of temperature degrees of freedom, and obtaining a finite element equation as follows:
KItI＝PI
considering that the control equation is a linear equation, the sensitivity-considered vertex combination method can be used for solvingA certain temperature componentUpper and lower bounds of (a);
step two: after the upper and lower bounds of the temperature interval are solved by using a vertex combination method, according to a non-probability reliability theory, a new non-probability reliability index, namely an optimized characteristic distance d, can be constructed on the basis of the non-probability reliability R, and when d is reached>At 0, the corresponding reliability R<RtargDoes not meet the design requirements; when d is less than or equal to 0, the corresponding reliability R>RtargThe design requirements are met;
step three: based on the density-rigidity interpolation model, a multi-material heat dissipation structure interpolation model is established:
wherein λiDenotes the material thermal conductivity, λ, of the ith cell after interpolation1Denotes the thermal conductivity, λ, of the material 12Denotes the thermal conductivity, λ, of the material 23Denotes the thermal conductivity, x, of the material 31,i、x2,iAnd x3,iDesign variables 1,2 and 3, p (p) of the ith cell>1) Representing a penalty factor;
step four: on the basis of a classical topological optimization mathematical model, the optimization characteristic distance d is used as a constraint condition, and the multi-material interpolation model is combined, so that a non-probability reliability topological optimization mathematical model based on a multi-material heat dissipation structure can be established as follows:
where M represents the structural mass in the design region, ViDenotes the volume of the ith cell, n denotes the total number of cells in the design area, p1、ρ2And ρ3Respectively representing the densities of material 1, material 2 and material 3,x 1、x 2andx 3representing the lower bounds of design variable 1, design variable 2, and design variable 3,andrepresenting the upper bounds of design variable 1, design variable 2, and design variable 3, respectively. djRepresenting the non-probabilistic reliability of the jth constraint, m representing the total number of constraints;
step five: obtaining a constraint function and a partial derivative of a target function to a design variable through the use of a adjoint vector method, and carrying out sensitivity analysis so as to solve a subsequent gradient optimization algorithm;
step six: and (3) through iterative calculation of an MMA algorithm, simultaneously considering reliability constraint and relative variation, if the result of a certain iteration step meets the constraint condition that the non-probability reliability d is less than or equal to 0, and the sum of the design variable variations of the two steps before and after the iteration is less than a preset value epsilon, ending the iteration process, and taking the current topology optimization result as the final optimization result.
2. A method of topological optimization of a multi-material heat dissipating structure taking into account additive manufacturing instabilities according to claim 1, characterized by: in the first step, uncertainty of material heat conductivity coefficient, structural heat load and safe temperature is considered, and original certainty quantity is replaced by interval variable.
3. A method of topological optimization of a multi-material heat dissipating structure taking into account additive manufacturing instabilities according to claim 1, characterized by: and in the second step, the optimized characteristic distance d is used as a judgment index of the non-probability reliability of the structure.
4. A method of topological optimization of a multi-material heat dissipating structure taking into account additive manufacturing instabilities according to claim 1, characterized by: in the third step, an interpolation model based on a multi-material heat dissipation structure is adopted.
5. A method of topological optimization of a multi-material heat dissipating structure taking into account additive manufacturing instabilities according to claim 1, characterized by: the topological optimization model constructed in the fourth step takes the situations of two materials and three materials into consideration.
6. A method of topological optimization of a multi-material heat dissipating structure taking into account additive manufacturing instabilities according to claim 1, characterized by: and performing sensitivity analysis in the fifth step to obtain partial derivatives of the constraint function and the objective function to the design variables.
7. A method of topological optimization of a multi-material heat dissipating structure taking into account additive manufacturing instabilities according to claim 1, characterized by: and step six, simultaneously considering the reliability constraint and the relative variation as a standard for judging whether the iteration process is ended.
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201810919278.7A CN109408845B (en) | 2018-08-14 | 2018-08-14 | Multi-material heat dissipation structure topology optimization method considering additive manufacturing instability |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201810919278.7A CN109408845B (en) | 2018-08-14 | 2018-08-14 | Multi-material heat dissipation structure topology optimization method considering additive manufacturing instability |
Publications (2)
| Publication Number | Publication Date |
|---|---|
| CN109408845A true CN109408845A (en) | 2019-03-01 |
| CN109408845B CN109408845B (en) | 2022-10-28 |
Family
ID=65464293
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CN201810919278.7A Active CN109408845B (en) | 2018-08-14 | 2018-08-14 | Multi-material heat dissipation structure topology optimization method considering additive manufacturing instability |
Country Status (1)
| Country | Link |
|---|---|
| CN (1) | CN109408845B (en) |
Cited By (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN112163291A (en) * | 2020-09-17 | 2021-01-01 | 华中科技大学 | Forming method of multi-material mechanical functional part in additive manufacturing and product |
| CN112528435A (en) * | 2020-12-07 | 2021-03-19 | 西安交通大学 | High-temperature heat pipe design optimization method |
| CN112989671A (en) * | 2021-04-09 | 2021-06-18 | 上海交通大学 | Convection heat dissipation structure topology optimization method capable of avoiding boundary material adhesion |
Citations (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| WO1998006550A1 (en) * | 1996-08-08 | 1998-02-19 | Bridgestone Corporation | Method of designing multicomponent material, optimization analyzer and storage medium on which multicomponent material optimization analysis program is recorded |
| CN106021734A (en) * | 2016-05-23 | 2016-10-12 | 北京航空航天大学 | Multi-objective reliability optimum design method for forehold thermal protection system of reusable launch vehicle |
| CN106650147A (en) * | 2016-12-30 | 2017-05-10 | 北京航空航天大学 | Continuum structure non-probability topologicaloptimization method based on bounded uncertainty |
| CN107515963A (en) * | 2017-07-17 | 2017-12-26 | 北京航空航天大学 | A kind of bi-material layers Continuum Structure Multidisciplinary systems Topology Optimization Method based on uncertain but bounded |
-
2018
- 2018-08-14 CN CN201810919278.7A patent/CN109408845B/en active Active
Patent Citations (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| WO1998006550A1 (en) * | 1996-08-08 | 1998-02-19 | Bridgestone Corporation | Method of designing multicomponent material, optimization analyzer and storage medium on which multicomponent material optimization analysis program is recorded |
| CN106021734A (en) * | 2016-05-23 | 2016-10-12 | 北京航空航天大学 | Multi-objective reliability optimum design method for forehold thermal protection system of reusable launch vehicle |
| CN106650147A (en) * | 2016-12-30 | 2017-05-10 | 北京航空航天大学 | Continuum structure non-probability topologicaloptimization method based on bounded uncertainty |
| CN107515963A (en) * | 2017-07-17 | 2017-12-26 | 北京航空航天大学 | A kind of bi-material layers Continuum Structure Multidisciplinary systems Topology Optimization Method based on uncertain but bounded |
Non-Patent Citations (3)
| Title |
|---|
| C.JIANG 等: "《Structural reliability analysis using non-probabilistic convex model》", 《COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING》 * |
| 尤芳等: "稳态热传导结构非概率可靠性拓扑优化设计", 《振动与冲击》 * |
| 蔡逸如 等: "《考虑宏细观等效的双材料连续体结构可靠性拓扑优化方法》", 《中国力学大会-2017暨庆祝中国力学学会成立60周年大会论文集（C）》 * |
Cited By (5)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN112163291A (en) * | 2020-09-17 | 2021-01-01 | 华中科技大学 | Forming method of multi-material mechanical functional part in additive manufacturing and product |
| CN112528435A (en) * | 2020-12-07 | 2021-03-19 | 西安交通大学 | High-temperature heat pipe design optimization method |
| CN112528435B (en) * | 2020-12-07 | 2022-12-09 | 西安交通大学 | High-temperature heat pipe design optimization method |
| CN112989671A (en) * | 2021-04-09 | 2021-06-18 | 上海交通大学 | Convection heat dissipation structure topology optimization method capable of avoiding boundary material adhesion |
| CN112989671B (en) * | 2021-04-09 | 2022-05-27 | 上海交通大学 | Convection heat dissipation structure topology optimization method capable of avoiding boundary material adhesion |
Also Published As
| Publication number | Publication date |
|---|---|
| CN109408845B (en) | 2022-10-28 |
Similar Documents
| Publication | Publication Date | Title |
|---|---|---|
| Vasinonta et al. | A process map for consistent build conditions in the solid freeform fabrication of thin-walled structures | |
| Somashekara et al. | Investigations into effect of weld-deposition pattern on residual stress evolution for metallic additive manufacturing | |
| Wang et al. | Physics-based multivariable modeling and feedback linearization control of melt-pool geometry and temperature in directed energy deposition | |
| Doumanidis et al. | Geometry modeling and control by infrared and laser sensing in thermal manufacturing with material deposition | |
| Mertens et al. | Optimization of scan strategies in selective laser melting of aluminum parts with downfacing areas | |
| CN109408845B (en) | Multi-material heat dissipation structure topology optimization method considering additive manufacturing instability | |
| Ponche et al. | A novel methodology of design for Additive Manufacturing applied to Additive Laser Manufacturing process | |
| US9939394B2 (en) | Process mapping of cooling rates and thermal gradients | |
| Zhang et al. | Adaptive slicing for a multi-axis laser aided manufacturing process | |
| Gorguluarslan et al. | A multilevel upscaling method for material characterization of additively manufactured part under uncertainties | |
| CN107515963B (en) | A kind of bi-material layers Continuum Structure Multidisciplinary systems Topology Optimization Method based on uncertain but bounded | |
| CN105718690A (en) | Laser 3D printing molten bath solidification behavior numerical simulation method based on time and space active tracking | |
| Wang et al. | Cylindrical slicing and path planning of propeller in wire and arc additive manufacturing | |
| Shi et al. | T-GMAW based novel Multi-node trajectory planning for fabricating grid stiffened panels: An efficient production technology | |
| Wu et al. | The virtual node polygonal element method for nonlinear thermal analysis with application to hybrid laser welding | |
| Abdulshahed et al. | Comparative study of ANN and ANFIS prediction models for thermal error compensation on CNC machine tools | |
| Ye et al. | Numerical modeling of metal-based additive manufacturing using level set methods | |
| Bhatt et al. | Automated planning for robotic multi-resolution additive manufacturing | |
| Moreira et al. | A multi-criteria h-adaptive finite-element framework for industrial part-scale thermal analysis in additive manufacturing processes | |
| CN112199881A (en) | Direct metal deposition additive simulation method and system | |
| Bosetti et al. | Identification of johnson–cook and tresca's parameters for numerical modeling of aisi-304 machining processes | |
| Ren et al. | Digital twin for the transient temperature prediction during coaxial one-side resistance spot welding of Al5052/CFRP | |
| Cao | Numerical investigation on molten pool dynamics during multi-laser array powder bed fusion process | |
| US10035220B2 (en) | Process mapping of transient thermal response due to value changes in a process variable | |
| Dantin et al. | Toward a physical basis for a predictive finite element thermal model of the LENS™ process leveraging dual-wavelength pyrometer datasets |
Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| PB01 | Publication | ||
| PB01 | Publication | ||
| SE01 | Entry into force of request for substantive examination | ||
| SE01 | Entry into force of request for substantive examination | ||
| GR01 | Patent grant | ||
| GR01 | Patent grant |