CN110287534A - Consider the robustness Topology Optimization Method of the probabilistic shell class formation of thickness - Google Patents
Consider the robustness Topology Optimization Method of the probabilistic shell class formation of thickness Download PDFInfo
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- G06F30/15—Vehicle, aircraft or watercraft design
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Abstract
A kind of robustness Topology Optimization Method considering the probabilistic shell class formation of thickness disclosed by the invention, comprising the following steps: determine the random field probability distribution of thickness t first；Establish the robustness topological optimization mathematical model of structure；It is discrete to random field progress using series optimum linearity technique of estimation (EOLE) method, reduce the number of stochastic variable；Determine the sample point of polynomial chaos expression (PCE)；The structural response and sensitivity information at sample point are calculated based on finite element method；Calculate the mean value and variance of structural response；Calculating target function value and objective function sensitivity information；By sensitivity and objective function information input MMA, design variable is updated；Judging result convergence, convergence then export optimum results, do not restrain, continue iteration.
Description
Technical field
The present invention relates to vehicles, aircraft Thin-shell Optimal Structure Designing field, and in particular to a kind of shell class formation
Robustness Topology Optimization Method.
Background technique
Vehicle, Flight Vehicle Structure generally use the thin-wall construction of lightweight, due to plate raw material, processing and manufacturing etc.
The reason of, be constantly present certain deviation between the thickness and design value of thin-wall construction formal product so that structural behaviour with set
It is had differences between meter, performance is also different between batch and batch.The space industry vehicle stringent for quality requirement,
The demand of Aircraft structure minute design is also continuously improved, in the case where thickness design is more fine, the change of very little
Change the large change that can also cause structural behaviour, influences reliability of structure.
Existing Thin-shell infrastructure product is designed generally according to design thickness lower variation of tolerance or nominal size, is existed
Below three aspect deficiency:
(1) it is designed according to design thickness lower variation of tolerance, be easy to cause shell major part actual (real) thickness to be above design value, make
It is overweight at actual product, influence overall performance；
(2) it is designed according to nominal value, be easy to cause practical local thickness less than normal, production reliability reduces, under serious conditions
Infrastructure product may directly be scrapped, and the overall quality of model is seriously affected, and product average unit cost is caused to improve；
(3) existing design method does not consider caused localized design Parameters variation in technical process, causes design and craft
Between actual separation, design-technique Coupling Design can not be carried out, product design-production iteration cycle is long.
Summary of the invention
Problem to be solved by this invention is: breaking through the existing method by determining thickness value design, considers that technical process is drawn
The deviation of the Thin-shell structural thickness risen, it is practical according to manufacturing process, the probability distribution of thickness deviation is obtained, by thickness deviation
The mean value and variance of the structural response of statistical property and shell structure establish quantitative relationship, consider that thickness is uncertain to realize
Shell class formation robust optimization design, establish between design and craft coupling connection.
The invention is realized by the following technical scheme: it is a kind of consider the probabilistic shell class formation of thickness robustness open up
Flutter optimization method, comprising the following steps:
S1: the finite element model of shell structure is established, and determines the random field probability distribution of thickness of shell t；
S2: the robustness topological optimization mathematical model of shell structure flexibility response is established:
min:μ(f(x,ξ))+ασ(f(x,ξ))
S.t.K (ξ) U=P
fV<f0
Wherein, f (x, ξ) indicates structural object response；X is design variable relevant to thickness of shell, x=t/ μt, μtFor shell
The mean value of body thickness；ξ is the stochastic variable of thickness of shell；μ (f (x, ξ)) is the average value of target response；σ (f (x, ξ)) is mesh
Mark the variance of response；α is the mean value of target response and the weight coefficient of variance；K (ξ) is the rigidity of structure；U is the position of structure
It moves；P is the external force of structure；fvIndicate optimization structural volume ratio；f0Indicate the constraint volume fraction ratio of structure；
S3: it is carried out using random field of the series optimum linearity technique of estimation to normal distribution discrete:
Wherein, x' indicates the unit midpoint coordinates of finite element model；T (x', ω) is element thickness random field, and ω is all
The a subset in the formed space of element thickness stochastic regime；ξi(ω) is i-th of stochastic variable；U (x') is element thickness
Mean value；ΨiFor the ith feature vector of covariance matrix；λiFor the ith feature value of covariance matrix；I is positive integer, and n is
Overall cell number；
In the case where it is ζ that precision, which is truncated, characteristic value is ranked up by size, obtains Discrete Stochastic field truncation item number s:
Wherein,Before indicating s characteristic values with；Indicate the sum of all characteristic values；
S4: structural response is expressed as chaos multinomial, the chaos polynomial f (ξ) of structural response is expressed as follows:
Wherein, Φi(ξ) is orthogonal polynomial basis function；giFor i-th of coefficient of chaos multinomial；NPCE+ 1 indicates that chaos is more
The total item of formula, total item determine by the number s and the polynomial order p of chaos of the discrete obtained stochastic variable of random field,
Relational expression are as follows:
Wherein, p >=3；
S5: the sample point of polynomial chaos expression is obtained using Kronrod-Patterson Integral Rule；
S6: the structural response and sensitivity information at each sample point obtained in S5, sensitivity computing method are calculated
It is as follows:
Wherein, f (x, ξq) be q-th of sample point target function value；ξqIndicate the random variable values of q-th of sample point, xe
=te/μtFor the design variable of e-th of unit at q-th of sample point, teFor the thickness of e-th of unit at q-th of sample point；ue
For e-th of Displacement of elemental node vector at q-th of sample point；keFor the Stiffness Matrix of e-th of unit at q-th of sample point；Q is positive
Integer；
S7: the polynomial coefficient of chaos is solved:
Wherein:
Indicate the mean value of i-th of basic function square；
E[fΦi] indicate structural response and i-th of basis function product mean value.
E[fΦi] obtained with following integral form:
Wherein, ρ (ξ) is the weight function of chaos polynomial basis function；Ω is optimization design domain；f(ξq) indicate by
The structural object for q-th of sample point that finite element analysis obtains responds；NqFor total sample point number；Φi(ξq) it is q-th of sample
I-th of basis function values at point, wqThe weight at each point to calculate acquisition in S5；
S8: the mean value and variance of structural object response are solved:
Mean μ (f (ξ))=g0,
Variance
Wherein, g0For chaos polynomial constant item；For calculated in S7 acquisition i-th of basic function square it is equal
Value
S9: calculating target function value and objective function sensitivity information, objective function sensitivity computing method are as follows:
Φ0(ξq) it is constant basic function item at q-th sample point；E(Φ0 2) it is constant base letter at q-th sample point
Several squares of mean value；
S10: sensitivity and target function value input optimization algorithm are updated into design variable；
S11: judging result convergence, convergence then export optimum results, do not restrain, return to S6 and continue to iterate to calculate.
The advantages of the present invention over the prior art are that:
(1) method of the invention can be considered because plate raw material, processing and manufacturing etc. the reason of caused by thickness not
It determines, obtains the more stable practical shell structure product of performance；
(2) method of the invention is to consider the uncertain shell structure design method of thickness, reduces the report of actual product
Useless rate, improves the overall quality of model, reduces the average unit cost of infrastructure product；
(3) method of the invention establish the quantization between design and craft connection, can be designed-technique coupling set
Meter, improves design efficiency.
Detailed description of the invention
Fig. 1 is a pedestal shell tank structural finite element model schematic diagram of the invention；
Fig. 2 is a pedestal shell tank structural robustness optimum results schematic diagram of the invention；
Fig. 3 is flow chart of the method for the present invention.
Specific embodiment
Elaborate below with reference to embodiment to the present invention, the present embodiment under the premise of the technical scheme of the present invention into
Row is implemented, and gives detailed embodiment, but protection scope of the present invention is not limited to following embodiments.
A kind of robustness Topology Optimization Method considering the probabilistic shell class formation of thickness, comprising the following steps:
S1: the finite element model of shell structure is established, and determines the random field probability distribution of thickness of shell t；
S2: the robustness topological optimization mathematical model of shell structure flexibility response is established:
min:μ(f(x,ξ))+ασ(f(x,ξ))
S.t.K (ξ) U=P
fV<f0
Wherein, f (x, ξ) indicates structural object response；X is design variable relevant to thickness of shell, x=t/ μt, μtFor shell
The mean value of body thickness；ξ is the stochastic variable of thickness of shell；μ (f (x, ξ)) is the average value of target response；σ (f (x, ξ)) is mesh
Mark the variance of response；α is the mean value of target response and the weight coefficient of variance；K (ξ) is the rigidity of structure；U is the position of structure
It moves；P is the external force of structure；fvIndicate optimization structural volume ratio；f0Indicate the constraint volume fraction ratio of structure；
S3: it is carried out using random field of the series optimum linearity technique of estimation to normal distribution discrete:
Wherein, x' indicates the unit midpoint coordinates of finite element model；T (x', ω) is element thickness random field, and ω is all
The a subset in the formed space of element thickness stochastic regime；ξi(ω) is i-th of stochastic variable；U (x') is element thickness
Mean value；ΨiFor the ith feature vector of covariance matrix；λiFor the ith feature value of covariance matrix；I is positive integer, and n is
Overall cell number；
In the case where it is ζ that precision, which is truncated, characteristic value is ranked up by size, obtains Discrete Stochastic field truncation item number s:
Wherein,Before indicating s characteristic values with；Indicate the sum of all characteristic values；
S4: structural response is expressed as chaos multinomial, the chaos polynomial f (ξ) of structural response is expressed as follows:
Wherein, Φi(ξ) is orthogonal polynomial basis function；giFor i-th of coefficient of chaos multinomial；NPCE+ 1 indicates that chaos is more
The total item of formula, total item determine by the number s and the polynomial order p of chaos of the discrete obtained stochastic variable of random field,
Relational expression are as follows:
Wherein, p >=3；
S5: the sample point of polynomial chaos expression is obtained using Kronrod-Patterson Integral Rule；
S6: the structural response and sensitivity information at each sample point obtained in S5, sensitivity computing method are calculated
It is as follows:
Wherein, f (x, ξq) be q-th of sample point target function value；ξqIndicate the random variable values of q-th of sample point, xe
=te/μtFor the design variable of e-th of unit at q-th of sample point, teFor the thickness of e-th of unit at q-th of sample point；ue
For e-th of Displacement of elemental node vector at q-th of sample point；keFor the Stiffness Matrix of e-th of unit at q-th of sample point；Q is positive
Integer；
S7: the polynomial coefficient of chaos is solved:
Wherein:
Indicate the mean value of i-th of basic function square；
E[fΦi] indicate structural response and i-th of basis function product mean value.
E[fΦi] obtained with following integral form:
Wherein, ρ (ξ) is the weight function of chaos polynomial basis function；Ω is optimization design domain；f(ξq) indicate by
The structural object for q-th of sample point that finite element analysis obtains responds；NqFor total sample point number；Φi(ξq) it is q-th of sample
I-th of basis function values at point, wqThe weight at each point to calculate acquisition in S5；
S8: the mean value and variance of structural object response are solved:
Mean μ (f (ξ))=g0,
Variance
Wherein, g0For chaos polynomial constant item；For calculated in S7 acquisition i-th of basic function square it is equal
Value
S9: calculating target function value and objective function sensitivity information, objective function sensitivity computing method are as follows:
Φ0(ξq) it is constant basic function item at q-th sample point；E(Φ0 2) it is constant base letter at q-th sample point
Several squares of mean value；
S10: sensitivity and target function value input optimization algorithm are updated into design variable；
S11: judging result convergence, convergence then export optimum results, do not restrain, return to S6 and continue to iterate to calculate.
Embodiment:
As shown in figure 3, a kind of robustness topology for considering the probabilistic shell class formation of thickness provided by the invention is excellent
Change method, it is more detailed, comprising the following steps:
S1: establishing the finite element model of a pedestal thin spherical shell tank structure, as shown in Figure 1 and Figure 2, for the present invention
One pedestal shell tank structural finite element model schematic diagram of embodiment, radius of sphericity R=20, sphere thickness t with
Airport probability distribution is normal distribution, and mean value and variance are (1.0,0.1)；
S2: the robustness topological optimization mathematical model of structure flexibility response, column are established are as follows:
min:μ(f(x,ξ))+ασ(f(x,ξ))
S.t.K (ξ) U=P
fV<f0
Wherein, f (x, ξ) indicates the target response of structure flexibility；X is design variable relevant to thickness, x=t/ μt；ξ is
The stochastic variable of thickness；μ is the average value of target response；σ is the variance of target response；α is the mean value and variance of target response
Weight coefficient, this example α=4；K (ξ) is the rigidity of structure；U is the displacement of structure；P is the external force of structure；fvIndicate optimization knot
Structure volume ratio；f0Indicate the constraint volume fraction ratio of structure, f in this example0<0.5；
S3: it is carried out using random field of series optimum linearity technique of estimation (EOLE) method to normal distribution discrete:
Wherein, x' indicates the unit midpoint coordinates of finite element model；T (x', ω) is element thickness random field, and ω is all
The a subset in the formed space of element thickness stochastic regime；ξi(ω) is i-th of stochastic variable；U (x') is element thickness
Mean value；ΨiFor the ith feature vector of covariance matrix；λiFor the ith feature value of covariance matrix；I is positive integer, and n is
Overall cell number；
Preferably, when thickness t is distributed as normal distribution, the form of covariance function takes C (x1,x2)=σ2exp(-||
x1-x2||2/l2), wherein l is the correlation length of thickness t random field, x1、x2Indicate two unit midpoint coordinateies in finite element model；
In the case where it is ζ that precision, which is truncated, characteristic value size is ranked up, obtains Discrete Stochastic field truncation item number s:
Wherein,Before indicating s characteristic values with；Indicate the sum of all characteristic values；
In ζ=95%, take covariance function C (x1,x2)=σ2exp(-||x1-x2||2/l2), the feelings of correlation length l=20
Under condition, s=3, wherein x1、x2Indicate two unit midpoint coordinateies of finite element model；
S4: structural response is expressed as chaos multinomial, the chaos polynomial expression of structural response is as follows:
Wherein, Φi(ξ) is orthogonal polynomial basis function, and this example basic function takes Hermite form；F (ξ) is the mesh of structure
Mark response；giFor i-th of coefficient of chaos multinomial；NPCE+ 1 indicates that the polynomial total item of chaos, total item are discrete by random field
The polynomial order p (p >=3) of the number s and chaos of obtained stochastic variable determines, is therebetween are as follows:
S5: in the case where s=3 p=3, using the polynomial chaos of Kronrod-Patterson Integral Rule acquisition
The sample point of (PCE), number 19 is unfolded；
S6: calculating the structural response and sensitivity information at each sample point, and sensitivity computing method is as follows:
Wherein, f (x, ξq) be q-th of sample point target function value；ξqIndicate the random variable values of q-th of sample point, xe
=te/μtFor the design variable of e-th of unit at q-th of sample point, teFor the thickness of e-th of unit at q-th of sample point；ue
For e-th of Displacement of elemental node vector at q-th of sample point；keFor the Stiffness Matrix of e-th of unit at q-th of sample point；Q is positive
Integer；
S7: the polynomial coefficient of chaos is solved:
Wherein:
It indicates the mean value of i-th of basic function square, can be acquired according to the expression formula of Hermite basic function；
E[fΦi] can be obtained with following integral form:
Wherein, ρ (ξ) is the weight function of chaos polynomial basis function；Ω is optimization design domain；f(ξq) indicate by
The structural object for the q sample point that finite element analysis obtains responds；NqFor total sample point number 19；Φi(ξq) it is at sample point
Basis function values, wqThe weight at each point to calculate acquisition in S5；
S8: the mean value and variance of structural object response are solved:
Mean μ (f (ξ))=g0
Variance
Wherein, g0For chaos polynomial constant item；NPCE=19；For i-th of basic function for calculating acquisition in S7
Square mean value
S9: calculating target function value and objective function sensitivity information, objective function sensitivity computing method are as follows:
Φ0(ξq) it is constant basic function item at q-th sample point, Hermite form constant basic function item is 1；E
(Φ0 2) be constant basic function square at q-th sample point mean value；
S10: sensitivity and target function value are inputted into optimization algorithm, this example uses mobile asymptote optimization algorithm (MMA)
Update design variable；
S11: judging result convergence, convergence then export optimum results, do not restrain, and return to S6 and continue iteration.
It is obvious to a person skilled in the art that invention is not limited to the details of the above exemplary embodiments, Er Qie
In the case where without departing substantially from spirit or essential attributes of the invention, the present invention can be realized in other specific forms.Therefore, no matter
From the point of view of which point, the present embodiments are to be considered as illustrative and not restrictive, and the scope of the present invention is by appended power
Benefit requires rather than above description limits, it is intended that all by what is fallen within the meaning and scope of the equivalent elements of the claims
Variation is included within the present invention.Any reference signs in the claims should not be construed as limiting the involved claims.
Unspecified part of the present invention belongs to technology well known to those skilled in the art.
Claims (10)
1. it is a kind of consider the probabilistic shell class formation of thickness robustness Topology Optimization Method, which is characterized in that including with
Lower step:
S1: the finite element model of shell structure is established, and determines the random field probability distribution of thickness of shell t；
S2: the robustness topological optimization mathematical model of shell structure flexibility response is established；
S3: it is carried out using random field of the series optimum linearity technique of estimation to normal distribution discrete；
In the case where it is ζ that precision, which is truncated, characteristic value is ranked up by size, obtains Discrete Stochastic field truncation item number s；
S4: structural response is expressed as chaos multinomial；
S5: the sample point of polynomial chaos expression is obtained using Kronrod-Patterson Integral Rule；
S6: the structural response and sensitivity information at each sample point obtained in S5 are calculated；
S7: the polynomial coefficient of chaos is solved；
S8: the mean value and variance of structural object response are solved；
S9: calculating target function value and objective function sensitivity information；
S10: sensitivity and target function value input optimization algorithm are updated into design variable；
S11: judging result convergence, convergence then export optimum results, do not restrain, return to S6 and continue to iterate to calculate.
2. a kind of robustness topological optimization side for considering the probabilistic shell class formation of thickness according to claim 1
Method, it is characterised in that: in the S2, the robustness topological optimization mathematical model of shell structure flexibility response,
min:μ(f(x,ξ))+ασ(f(x,ξ))
S.t.K (ξ) U=P
fV<f0
Wherein, f (x, ξ) indicates structural object response；X is design variable relevant to thickness of shell, x=t/ μt, μtFor shell thickness
The mean value of degree；ξ is the stochastic variable of thickness of shell；μ (f (x, ξ)) is the average value of target response；σ (f (x, ξ)) rings for target
The variance answered；α is the mean value of target response and the weight coefficient of variance；K (ξ) is the rigidity of structure；U is the displacement of structure；P is
The external force of structure；fvIndicate optimization structural volume ratio；f0Indicate the constraint volume fraction ratio of structure.
3. a kind of robustness topological optimization for considering the probabilistic shell class formation of thickness according to claim 1 or 2
Method, it is characterised in that:
In S3, it is discrete after normal distribution random field are as follows:
Wherein, x' indicates the unit midpoint coordinates of finite element model；T (x', ω) is element thickness random field, and ω is all units
The a subset in the formed space of thickness stochastic regime；ξi(ω) is i-th of stochastic variable；U (x') is the mean value of element thickness；
ΨiFor the ith feature vector of covariance matrix；λiFor the ith feature value of covariance matrix；I is positive integer, and n is overall single
First number.
4. a kind of robustness topological optimization side for considering the probabilistic shell class formation of thickness according to claim 3
Method, it is characterised in that:
In S3, the calculation formula of item number s is truncated in Discrete Stochastic field are as follows:
Wherein,Before indicating s characteristic values with；Indicate the sum of all characteristic values.
5. a kind of robustness topological optimization side for considering the probabilistic shell class formation of thickness according to claim 4
Method, it is characterised in that:
In S4, the chaos polynomial f (ξ) of structural response is expressed as follows:
Wherein, Φi(ξ) is orthogonal polynomial basis function；giFor i-th of coefficient of chaos multinomial；NPCE+ 1 indicates chaos multinomial
Total item, total item determines by the number s and the polynomial order p of chaos of the discrete obtained stochastic variable of random field, relationship
Formula are as follows:
Wherein, p >=3.
6. a kind of robustness topological optimization for considering the probabilistic shell class formation of thickness according to claim 4 or 5
Method, it is characterised in that:
In S6, the sensitivity computing method of q-th of sample point is as follows:
Wherein, f (x, ξq) be q-th of sample point target function value；ξqIndicate the random variable values of q-th of sample point, xe=te/
μtFor the design variable of e-th of unit at q-th of sample point, teFor the thickness of e-th of unit at q-th of sample point；ueFor q
E-th of Displacement of elemental node vector at a sample point；keFor the Stiffness Matrix of e-th of unit at q-th of sample point；Q is positive integer.
7. a kind of robustness topological optimization side for considering the probabilistic shell class formation of thickness according to claim 6
Method, it is characterised in that:
In S7, the polynomial coefficient of chaos
Wherein:Indicate the mean value of i-th of basic function square；
E[fΦi] indicate structural response and i-th of basis function product mean value.
8. a kind of robustness topological optimization side for considering the probabilistic shell class formation of thickness according to claim 7
Method, it is characterised in that:
E[fΦi] obtained with following integral form:
Wherein, ρ (ξ) is the weight function of chaos polynomial basis function；Ω is optimization design domain；f(ξq) indicate by finite element
Analyze the structural object response of q-th obtained of sample point；NqFor total sample point number；Φi(ξq) it is at q-th of sample point
I-th of basis function values, wqThe weight at each point to calculate acquisition in S5.
9. a kind of robustness topological optimization side for considering the probabilistic shell class formation of thickness according to claim 8
Method, it is characterised in that:
In S8, the mean value and variance of structural object response are as follows:
Mean μ (f (ξ))=g0,
Variance
Wherein, g0For chaos polynomial constant item；For calculated in S7 acquisition i-th of basic function square mean value
10. a kind of robustness topological optimization side for considering the probabilistic shell class formation of thickness according to claim 9
Method, it is characterised in that:
In S9, objective function sensitivity computing method is as follows:
Φ0(ξq) it is constant basic function item at q-th sample point；E(Φ0 2) it is constant basic function square at q-th sample point
Mean value.
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| CN108491578A (en) * | 2018-02-13 | 2018-09-04 | 东南大学 | A kind of Random dynamic loads recognition methods based on perturbative matrix |
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| CN108491578A (en) * | 2018-02-13 | 2018-09-04 | 东南大学 | A kind of Random dynamic loads recognition methods based on perturbative matrix |
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