CN107341279A - A kind of quick near-optimal method of aircraft for high time-consuming constraint - Google Patents

A kind of quick near-optimal method of aircraft for high time-consuming constraint Download PDF

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CN107341279A
CN107341279A CN201611027519.4A CN201611027519A CN107341279A CN 107341279 A CN107341279 A CN 107341279A CN 201611027519 A CN201611027519 A CN 201611027519A CN 107341279 A CN107341279 A CN 107341279A
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龙腾
史人赫
刘莉
刘建
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Beijing Institute of Technology BIT
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Abstract

The present invention relates to a kind of quick near-optimal method of aircraft for high time-consuming constraint, belong to multidisciplinary optimization technical field.The present invention is in the problems of engineering design optimization process of processing belt restraining, by in each iterative process according to known sample point information, constraint corresponding to sample point and the penalty factor that updates every time construct agent model, by the way that a sample point is optimized and increased newly to agent model, then call functional value and binding occurrence corresponding to true analysis model acquisition, reduce penalty factor so that Optimized Iterative lays particular emphasis on improvement optimality next time if satisfaction constraint at optimization gained sample point, increase penalty factor if constraint is unsatisfactory for obtained by optimization at sample point so that Optimized Iterative is more likely to improve feasibility next time, so iterate until meeting stop criterion.The present invention is it can be considered that Local approximation precision and global approximation quality, while the ability that there is processing to constrain, and can effectively improve optimization efficiency, save aircraft optimization design cost.

Description

A kind of quick near-optimal method of aircraft for high time-consuming constraint
The present invention relates to a kind of quick near-optimal method of aircraft for high time-consuming constraint, belong in Flight Vehicle Design Multidisciplinary optimization technical field.
Background technology
With the development of science and technology engineering optimization becomes increasingly complex, many analyses and simulation software are all applied to set In meter and research, but these analyses and simulation problems are all high accuracy analysis model mostly, such as used in structural analysis Fluid Mechanics Computation (CFD) analysis model used in finite element analysis (FEA) model, aerodynamic analysis etc..High accuracy analysis mould Type also brings the problem of calculating is time-consuming while analysis precision and confidence level is improved, although computer nowadays software and hardware technology Very big progress is had been achieved for, but it is still extremely time-consuming to call the completion of high accuracy analysis model once to analyze, such as using CFD model completes an aerodynamics simu1ation analysis and stills need a few hours even tens of hours；Secondly, modern engineering design problem is often It is related to multiple subjects to intercouple.For example, Spacecraft guidance and control is related to structure, gesture stability, Orbit Transformation, position holding, electricity The subjects such as source, each subject influence each other, and mutually restrict, and the performance of aircraft is the comprehensive embodiment of each subject coupling.Due to subject Between coupled relation, the network analysis of problems of engineering design shows as multidisciplinary analysis.Substantially, multidisciplinary analysis process is One typical nonlinear solution processes, each multidisciplinary analysis are required for carrying out successive ignition, calculate and take, if each subject High accuracy analysis model is all used, amount of calculation will be very huge.Again, needed during Optimum design of engineering structure by iterating Locally or globally optimal solution can be just converged to, and iteration is required for carrying out the multidisciplinary analysis of multiple problems of engineering design every time, Cost is calculated to will be further increased.Traditional gradient algorithm is often only able to find the locally optimal solution of problem analysis, does not possess complete Office's search capability.
In order to obtain the globally optimal solution of problems of engineering design, usually directly calculated using the optimization with ability of searching optimum Method, such as genetic algorithm (GA), simulated annealing (SA) etc..But compared with traditional gradient algorithm, global optimization approach Required amount of calculation is bigger.Usually require to call thousands of times for example, optimizing an analysis model using genetic algorithm Analysis system.Complicated modern engineering design problem for largely using high-precision subject analysis model, traditional global optimization The calculating cost of strategy is excessive or even is difficult to receive.In addition, most high accuracy analysis models are all built using business CAE software at present Vertical black-box model (Black-box Model), it is extremely difficult with the interface of optimized algorithm (optimizer).
In order to reduce traditional global optimization approach to high calculating involved in present problems of engineering design optimization process Amount, is gradually studied based on the optimization method of agent model by people.It is substantially exactly construction and high accuracy analysis mould Type is approximate, but calculates the lower mathematics agent model of cost, and the model is used to optimize.Because high accuracy analysis model calculates Magnitude the time required to once is hour, and the magnitude that agent model calculates once the time used is only second even millisecond, therefore Compared with the calculating time of high accuracy analysis model, construct agent model and based on agent model optimization the calculating time often It can be ignored.In nearest 10 years, many companies all begin one's study and promoted approximate agent model technology design and The application in optimization field, such as：Engineous software companys research and develop iSIGHT, the research and development of Vanderplaats R＆D companies The Optimus of Visual DOC, LMS international organizations research and development, the ModelCenter of Phoenix companies research and development and Boeing grind The Design Explorer of hair.
Agent model can be divided into static agent model and dynamic proxy model.Static agent model is by once taking foot Enough sample points construct agent model, and agent model is kept constant in optimization process；And dynamic proxy model is sequence adopts Take sample point, progressively improved according to Given information during each Optimized Iterative with renewal agency model, until optimization convergence. Compared with static agent model, dynamic proxy model is more advantageous in optimization efficiency and result precision aspect.
Ke Lijin (Kriging) model is one of conventional agent model method, and can preferably compromise computational efficiency And approximation quality, so as to obtain extensive use.Jones etc. proposes efficient global optimization method (Efficient Global Optimization, EGO).It obtains a small amount of sample point to construct Kriging models in design space, and larger in error Place's increase sample point renewal agency model is until finding approximate globally optimal solution.EGO needs only to less model calling time Number just can find preferable solution, so as to substantially increase the efficiency of optimization.Nevertheless, also there is certain deficiency in EGO --- Constrained optimization problem can not be handled.The present invention is combined adaptive multiuser detection method with EGO, it is proposed that one kind is for high time-consuming The quick near-optimal method of aircraft of constraint, so as to efficiently solve the problems, such as constrained optimization.
In order to better illustrate technical scheme, below to the theoretical related Fundamentals of Mathematics of the physical layout being applied to It is specifically introduced：
Kriging models：
Kriging models are that a kind of estimate variance proposed by South Africa geologician Danie Krige nineteen fifty-one is minimum Unbiased Optimum Estimation Model, is formed by stacking by world model and partial deviations.For from the statistical significance, Kriging models are From correlation of variables and variability, unbiased, the one of optimal estimation are carried out to the value of regionalized variable in finite region Kind method；Said from interpolation angle, Kriging models are that the data of spatial distribution are asked with linear optimal, the one of the estimation of unbiased interpolation Kind method.
The mathematic(al) representation of Kriging models is represented by
F (x)=g (x)+Z (x) (1)
Wherein, g (x) not functions on x, are the global approximate models in the range of design space, Z (x) be average be zero, Variance is σ2, the random process that is not zero of covariance, Z (x) provides the partial deviations on the basis of global approximate model.
Generally, g (x) can use constant value β, then formula (1) is changed into
Y (x)=β+Z (x) (2)
Z (x) covariance matrix is represented by
Cov[Z(xi),Z(xj)]=σ2R[R(xi,xj)] (3)
In formula, R is correlation matrix, and R is correlation function, i=1,2 ..., ns, j=1,2 ..., ns(nsFor sample point Number).R is symmetrical matrix, and its diagonal entry is 1.
Generally, R takes Gauss correlation function to be represented by
Wherein, nvIt is design variable number, θkFor unknown relevant parameter vector, in order to reduce complexity, constant value θ is generally taken, So, formula (4) is represented by：
Introducing another associated vector r (x) is
According to knowledge of statistics, Kriging models are represented by
F (x)=β+rT(x)R-1(y-gβ) (7)
In formula, β is unknown parameter, σ2All it is θ function with R, the n that y is made up of sampled point responsesDimensional vector, when When g (x) takes constant value, g is the n dimensional vectors that element is all 1, i.e.,
G=[1,1 ..., 1]T (8)
β and σ2Least-squares estimation can be obtained by formula (9).
Relevant parameter θ can be solved in formula (10) by Maximum-likelihood estimation (Maximumlikelihoodestimation) One Dimension Optimization Problems obtain
Efficient global optimization method：
Efficient global optimization method (Efficient Global Optimization, EGO) is that a kind of Bayes's overall situation is excellent Change algorithm, its core is to increase sample point renewal agency model until finding approximate globally optimal solution in error larger part.EGO The basic procedure of optimization is：Initial sample point is generated using experimental design method and constructs Kriging agent models, with EI functions It is worth for target and using maximum point as check point, check point is assessed using high-precision model, by check point and its response Value adds sample point database, repeats this process until convergence.The mathematic(al) representation of wherein Kriging models is
F (X)=S (X)+Z (X) (11)
In formula：S (X) is X function, is the global approximate model in the range of design space；Z (X) be average be zero, variance The Gaussian random process being not zero for, covariance, there is provided the partial deviations on the basis of global approximate model.EI functions are defined as
In formula：φ is Standard Normal Distribution；φ is the probability density function of standardized normal distribution；FminFor target letter Number minimum value；It is respectively the predicted value and standard deviation of agent model at sample point X with s.The Section 1 of formula (12) will currently most The difference of Small object functional value and predicted value is multiplied by the probability that predicted value is less than current minimum target functional value, works as when predicted value is less than During preceding minimum target functional value, Section 1 can become big so that search focus on it is strong near current minimum target functional value Change local search ability.The Section 2 of formula (12) is predicted value standard deviation and the product of probability density function, when predicted value and currently When minimum target function is close, Section 2 can become big, so that search focuses on the poor region of precision of prediction and strengthens the overall situation Search capability.Therefore the method for check point is determined using EI functions while considers local and global search capability, taken into account Convergence rate and ability of searching optimum.
The content of the invention
For calculating time-consuming, Yi Jibiao during to high accuracy analysis model optimization using traditional global optimization approach The defects of accurate efficient global optimization method can not solve restricted problem.It is disclosed by the invention a kind of for the winged of high time-consuming constraint The quick near-optimal method technical problems to be solved of row device are to improve the quick near-optimal of aircraft for high time-consuming constraint Method, to the computational efficiency of high accuracy analysis model optimization process, and can solve the problem that the efficient complete of standard in global optimization approach The defects of office's optimization method can not solve restricted problem.
A kind of quick near-optimal method of aircraft for high time-consuming constraint disclosed by the invention, using based on The agent model of Kriging models, while be using adaptive multiuser detection method processing constraints, its design principle：This hair It is bright processing belt restraining problems of engineering design optimization process in, by each iterative process according to known sample point believe Breath, constraint corresponding to sample point and the penalty factor that updates every time construct agent model, by being optimized to agent model And increase a sample point newly, functional value and binding occurrence corresponding to true analysis model acquisition are then called, if in sample obtained by optimization This point place meets that constraint then reduces penalty factor so that Optimized Iterative lays particular emphasis on improvement optimality next time, if obtained by optimization Constraint is unsatisfactory at sample point and then increases penalty factor so that Optimized Iterative is more likely to improve feasibility next time, so progress is anti- For multiple iteration until meeting default maximum model call number, final output optimum results, completion is directed to the winged of high time-consuming constraint The quick near-optimal of row device.
A kind of quick near-optimal method of aircraft for high time-consuming constraint disclosed by the invention, comprises the following steps：
Step 1, initial input parameter is defined.
According to practical problem, defined analysis model, optimized variable design space, constraint number and initial sample point Number.
Step 2, initial sample point and computation model response are generated, and is saved in design sample point data base.
According to the Variational Design space defined in step 1 and initial sample point number, generated using the method for sampling corresponding Sample point, by calling true model to calculate true model response and binding occurrence corresponding to each initial sample point, And true model response and binding occurrence corresponding to above-mentioned sample point are saved in design sample point data base.
The super side's method of sampling of the preferred Latin of the method for sampling described in step 2 can make sample point be evenly distributed in design In space, so as to improve the precision of tectonic model.
Step 3, normalization factor is calculated.
According to the sample point and its corresponding true model response and binding occurrence calculating target function obtained in step 2 Normalization factor fAvgAnd the normalization factor c of constraintAvg, calculation expression is as follows：
fAvg=fSum/nPoints (13)
cAvg=cSum/nPoints (14)
Wherein nPoints be current sample point number, fSumFor current all sample point target function value absolute values With cSumFor the sum of current all sample point binding occurrence absolute values.
Step 4, pseudo- target function value is calculated.
In the design sample point data base that step 2 is obtained all sample points and its corresponding true model response and Binding occurrence extracts, and constraint is normalized, its calculation expression is as follows：
cNorm(x)=c (x)/cAvg (15)
Wherein c (x) be sample point x at binding occurrence, cNorm(x) it is the binding occurrence after normalization.Complete to constraint Processing after, object function is handled, its expression formula is as follows：
fNorm(x)=f (x)/fAvg (16)
Wherein f (x) be sample point x at target function value, fNorm(x) it is the target function value after normalization.Complete Into after the normalization of target function value and binding occurrence, construction includes the pseudo- target function value of constraint value information.As binding occurrence c (x) when being less than or equal to zero, target function value keeps constant；When binding occurrence c (x) is more than zero, to target function value processing such as Under：
Min F (x)=fNorm(x)+λmax(cNorm(x),0) (17)
Wherein, λ is penalty factor, and initial value 1, F (x) is the pseudo- target constructed according to normalization target function value and binding occurrence Functional value.All sample points are carried out with normalized as implied above.
Step 5, EI functions are constructed and optimize to obtain sample point xmin。
The sample point used used of extraction step 2 and the pseudo- target function value constructed, construct EI functions and with EI letters Numerical value is up to target and optimized, and the expression formula of EI functions is as follows：
In formula：φ is Standard Normal Distribution；φ is the probability density function of standardized normal distribution；FminFor current institute There is the minimum value of pseudo- target function value；It is respectively the predicted value and standard deviation of agent model at sample point x with s.EI functions are entered Sample point x is obtained after row optimizationmin。
Step 6, sample point x is calculatedminFunctional value and binding occurrence.
True analysis model is called to calculate xminThe object function response f at placeminAnd binding occurrence cmin, by sample point and Its corresponding object function response and binding occurrence are saved in sample point database.
Step 7, penalty factor is updated.
First, will be to cminIt is normalized, its expression formula is as follows：
cNorm=cmin/cAvg (19)
Wherein cNormFor the binding occurrence after normalization, c is obtainedNormIn maximum i.e. corresponding to maximum constrained degree of running counter to SmaxIf maximum constrained degree of running counter to is more than constraint tolerance Stol, then according to pre-defined growth factor μ1>1 increase penalty factor, So that Optimized Iterative is more likely to improve feasibility next time；Hold on the contrary, if maximum constrained degree of running counter to is less than or equal to Difference, then according to the descending factors μ of original definition2≤ 1 reduces penalty factor, so that Optimized Iterative lays particular emphasis on improvement most next time Dominance.In addition, introduce penalty factor bound constraint (λmin,λmax), on the one hand avoid penalty factor is too small from causing optimum results can not OK；On the other hand, avoid excessive the brought numerical value of penalty factor difficult in optimization process.Adaptive penalty factor λ(k+1)Calculation formula It is as follows：
Wherein, growth factor μ1, descending factors μ2, the upper limit constraint λmax, lower limit constraint λmin, constraint tolerance StolTo be adaptive Penalty factor λ(k+1)Calculate the coefficient that need to be set.
Preferably, growth factor μ1Value is 2；Descending factors μ2Value is 0.8；The upper limit constrains λmaxValue is 50；Under Limit constraint λminValue is 1；Constrain tolerance StolValue is 0.001.
Step 8, pseudo- target function value is generated.
Object function response and binding occurrence in the sample point database of extraction step 6, it is according to penalizing of being obtained in step 7 Number λ(k+1)Target function value is updated, more new formula is as follows：
Min F (x)=f (x)/fAvg+λmax(c(x)/cAvg,0) (21)
In formula, F (x) be sample point x place pseudo- target function value, f (x) be sample point x at real goal functional value, c (x) it is the binding occurrence at sample point x.
Step 9, on the basis of step 8, k=k+1 is made, step 4 progress next iteration is transferred to and is changed until reaching maximum Generation number, final output optimum results, complete the quick near-optimal of aircraft for high time-consuming constraint.
Also include step 10：By the quick near-optimal side of aircraft for high time-consuming constraint described in step 1 to step 9 Method is applied in engineering optimization, can effectively reduce optimization and calculate cost, improve optimization efficiency, helps to shorten engineering optimization The cycle of design.
The quick near-optimal method of aircraft for high time-consuming constraint described in step 1 to step 9 is applied to engineering Optimization includes：In terms of standard engineering example, meeting the application of constraints decline low pressure vessel design cost, meeting about Compression spring quality optimization application is reduced under the conditions of beam；In terms of satellite optimization design problem, the reduction pair in the case where meeting constraints Ground observation satellite quality optimization application, satellite structure quality optimization application is reduced in the case where meeting frequency constraints；Guided missile optimizes In terms of design problem, increase guided missile payload optimization application under the conditions of strength constraint is met, meeting fuel constraints Lower increase scope optimization application, the present invention have preferable application prospect in aerocraft system optimizing engineering design.
Beneficial effect：
1st, a kind of quick near-optimal method of aircraft for high time-consuming constraint disclosed by the invention, is a kind of dynamic generation Model method is managed, compared to static agent model method, this method increases a sample point newly in optimization process each time, from And construction agent model is constantly updated, improve agent model and the approximation quality of true analysis model；It is efficient compared to traditional Global optimization method, this method introduce adaptive multiuser detection method to handle constrained optimization problem, in optimization process each time It is middle that penalty factor is constantly updated according to optimization gained sample point and is used for next suboptimization, so as to guide whole optimization process to be expired The optimal solution constrained enough, that is, the defects of can solve the problem that the efficient global optimization method of standard can not solve restricted problem, expand high Imitate the scope of application of global optimization method.
2nd, a kind of quick near-optimal method of aircraft for high time-consuming constraint disclosed by the invention, can overcome tradition Global optimization method it is existing in engineering optimization calculate the shortcomings that time-consuming, and compared with static agent model, this The optimization method of invention can find the globe optimum of analysis model by calling fewer number analysis model, can be effective Reduction calculate cost, improve optimization efficiency, contribute to shorten the engineering optimization cycle, aerocraft system optimization etc. engineering design In have preferable application prospect.
3rd, a kind of quick near-optimal method of aircraft for high time-consuming constraint disclosed by the invention, by efficiently global EI functions in optimization method consider Local approximation precision and global approximation quality, while the ability that there is processing to constrain, and have Ability of searching optimum.
4th, a kind of quick near-optimal method of aircraft for high time-consuming constraint disclosed by the invention, suitable for various bands The complicated modern project optimization design problem of constraint, for many engineering designs and analysis software, such as：Flight vehicle aerodynamic analysis is soft Finite element analysis software FEA etc. in part CFD, structural analysis of flight vehicle software Nastran, Flight Vehicle Structure, carrying out, engineering is excellent In change design process, often optimization design needs to consume several hours even several days, and uses the present invention to engineering design and divide Analyse the true model in software and carry out approximation, while using adaptive multiuser detection processing constraints, to such approximate model Optimizing design only needs seconds or tens of seconds, so as to greatly shorten the Optimum design of engineering structure cycle, saves design cost, significantly Design efficiency is improved, and the optimization design scheme for meeting constraint can be obtained.
Brief description of the drawings
Fig. 1 is the quick near-optimal method flow diagram of aircraft for high time-consuming constraint；
Fig. 2 is design of pressure vessels problem schematic diagram；
Fig. 3 is penalty factor numerical value change figure herewith；
Fig. 4 is earth observation satellite Design Structure Model figure；
Fig. 5 is that earth observation satellite multidisciplinary optimization problem optimum results contrast box traction substation, wherein：Fig. 5 (a) is target letter Numerical value box traction substation, Fig. 5 (b) are target and constraint call number box traction substation, Fig. 5 (c) are maximum constrained degree of running counter to box traction substation.
Embodiment
In order to better illustrate the purpose of the present invention and advantage, below by the test example of engineering specification and one Earth observation satellite Multidisciplinary Optimization example, with reference to accompanying drawing, the present invention will be further described, and by with once sampling Construction RBF agent model technical results compare, and checking analysis is carried out to the combination property of the present invention.
Embodiment 1：By exemplified by meeting that constraints declines the application of low pressure vessel design cost.
The purpose of design of pressure vessels problem be reduced on the premise of certain constraint is met include welding, material and into Cost including type, it is assumed that pressure vessel model is that time-consuming high accuracy analysis model is calculated in engineering design, by solving phase Minimum value of the function in design space is answered, verifies C-EGO performance.The purpose of the present embodiment is to improve optimization design efficiency, Reduce the number for solving the function.
Given design of pressure vessels problem is as follows：
In order to obtain globally optimal solution, the optimization method with global optimizing ability is used.Herein using the pin of the present invention Optimizing is carried out to the design problem to the quick near-optimal design method of aircraft of high time-consuming constraint.Meanwhile in order to compare this The efficiency of method, the problem is optimized using the peak method of sampling (CIMPS) that chases after of genetic algorithm (GA) and processing constraint.
A kind of quick near-optimal method of aircraft for high time-consuming constraint disclosed in the present embodiment, implements step It is as follows：
Step 1, initial input parameter is defined.
M language definition analysis models are used according to design of pressure vessels problem, optimized variable design space lower limit for [1.0, 0.625,25,25], the upper limit is [1.375,1.0,150,240], and constraint number is 4, and initial sample point number is 20.
Step 2, initial sample point and computation model response are generated.
According to the Variational Design space defined in step 1 and initial sample point number, given birth to using the super side's method of sampling of Latin Into corresponding sample point, by call true model calculate true model response corresponding to each initial sample point and Binding occurrence, and true model response and binding occurrence corresponding to these sample points are saved in design sample point data base.
Step 3, normalization factor is calculated.
According to the sample point and its corresponding true model response and binding occurrence calculating target function obtained in step 2 Normalization factor fAvgAnd the normalization factor c of constraintAvg, its result of calculation is as follows：
fAvg=fSum/ nPoints=50850/20=2542.5
cAvg=cSum/ nPoints=[14.54,6.05,14.52,2150]/20=[0.727,0.303,0.726, 107.5]
Wherein nPoints be current sample point number, fSumFor current all sample point target function value absolute values With cSumFor the sum of current all sample point binding occurrence absolute values.
Step 4, pseudo- target function value is calculated.
In the design sample point data base that step 2 is obtained all sample points and its corresponding true model response and Binding occurrence extracts, and constraint is normalized, its calculation expression is as follows：
cNorm(x)=c (x)/cAvg (23)
Wherein c (x) be sample point x at binding occurrence, cNorm(x) it is the binding occurrence after normalization.Complete to constraint Processing after, object function is handled, its expression formula is as follows：
fNorm(x)=f (x)/fAvg (24)
Wherein f (x) be sample point x at target function value, fNorm(x) it is the target function value after normalization.Complete Into after the normalization of target function value and binding occurrence, construction includes the pseudo- target function value of constraint value information.As binding occurrence c (x) when being less than or equal to zero, target function value keeps constant；When binding occurrence c (x) is more than zero, to target function value processing such as Under：
Min F (x)=fNorm(x)+λmax(cNorm(x),0) (25)
Wherein, λ is penalty factor, and initial value 1, F (x) is the pseudo- target constructed according to normalization target function value and binding occurrence Functional value.All sample points are carried out with normalized as implied above.
Step 5, EI functions are constructed and are optimized.
The sample point used used of extraction step 2 and the pseudo- target function value constructed, construct EI functions and with EI letters Numerical value is up to target and optimized, and the expression formula of EI functions is as follows：
In formula：φ is Standard Normal Distribution；φ is the probability density function of standardized normal distribution；FminFor current institute There is the minimum value of pseudo- target function value；It is respectively the predicted value and standard deviation of agent model at sample point x with s.EI functions are entered Sample point x is obtained after row optimizationmin。
Step 6, sample point x is calculatedminFunctional value and binding occurrence
True analysis model is called to calculate xminThe object function response f at placeminAnd binding occurrence cmin, by sample point and Its corresponding object function response and binding occurrence are saved in sample point database.
Step 7, penalty factor is updated.
First, will be to cminIt is normalized, its expression formula is as follows：
cNorm=cmin/cAvg (27)
Wherein cNormFor the binding occurrence after normalization, c is obtainedNormIn maximum i.e. corresponding to maximum constrained degree of running counter to SmaxIf maximum constrained degree of running counter to is more than constraint tolerance Stol, then according to pre-defined growth factor μ1>1 increase penalty factor, So that Optimized Iterative is more likely to improve feasibility next time；Hold on the contrary, if maximum constrained degree of running counter to is less than or equal to Difference, then according to the descending factors μ of original definition2≤ 1 reduces penalty factor, so that Optimized Iterative lays particular emphasis on improvement most next time Dominance.In addition, introduce penalty factor bound constraint (λmin,λmax), on the one hand avoid penalty factor is too small from causing optimum results can not OK；On the other hand, avoid excessive the brought numerical value of penalty factor difficult in optimization process.Adaptive penalty factor calculation formula is as follows It is shown：
Wherein μ1Value is 2；μ2Value is 0.8；λmaxValue is 50；λminValue is 1；StolValue is 0.001.In this reality Apply in example, penalty factor constantly changes with the change of target function value, and its change procedure figure is as shown in Figure 3.By the Tu Ke get： When C-EGO obtains feasible solution, penalty factor can reduce therewith, so that iteration next time more focuses on changing for object function It is kind；When C-EGO obtains infeasible solution, penalty factor can increase therewith so that next suboptimization be limited in feasible zone or Near feasible zone.
Step 8, pseudo- target function value is generated.
Object function response and binding occurrence in the sample point database of extraction step 6, it is according to penalizing of being obtained in step 7 Several that target function value is updated, more new formula is as follows：
Min F (x)=f (x)/fAvg+λmax(c(x)/cAvg,0) (29)
In formula, F (x) be sample point x place pseudo- target function value, f (x) be sample point x at real goal functional value, c (x) it is the binding occurrence at sample point x.
Step 9, on the basis of step 8, k=k+1 is made, step 4 progress next iteration is transferred to and is changed until reaching maximum Generation number 100 times, final output optimum results complete optimization process.
Efficient global optimization method, genetic algorithm and processing constraint based on adaptive multiuser detection chase after the peak method of sampling Optimum results are as shown in the table.With object function call number (Nfe), constraints call number (Nce) and model calling is always Number (Ne) three indexs weigh the efficiency of optimized algorithm.Respectively using genetic algorithm (GA), CiMPS and C-EGO to optimization Problem carries out continuous 10 suboptimization.CiMPS uses the default parameter of algorithmic tool bag；GA uses the acquiescence of ga functions in MATLAB Set.The best result that listed result is obtained by three kinds of suboptimization of algorithm 10 in table 1, it is worth mentioning at this point that ten suboptimization results It is satisfied by constraining.
The design of pressure vessels problem optimum results of table 1 contrast
As seen from the results in Table 1, in terms of result optimality, GA is worst, and CiMPS is best, and C-EGO is substantially better than GA but slightly inferior In CiMPS (difference is less than 7.9%).In terms of optimization efficiency, although CiMPS only needs less object function call number, CiMPS needs to call substantial amounts of constraints to find feasible solution, causes the optimization efficiency of the algorithm relatively low.C-EGO methods institute Need the call number of object function suitable with CiMPS, but constraints call number is considerably less than CiMPS needed for C-EGO.C- Model needed for EGO calls total degree to be significantly less than CiMPS, shows that C-EGO has significant odds for effectiveness.GA efficiency is minimum, No matter model call number will be far more than other two kinds of algorithms or constrain call number.
By above-mentioned optimization process, the adaptive multiuser detection method that the present embodiment introduces is asked to handle constrained optimization Topic, the sample point according to obtained by optimization constantly updates penalty factor and is used for next suboptimization in optimization process each time, so as to draw Lead whole optimization process and obtain the optimal solution for meeting constraint, can solve the problem that the efficient global optimization method of standard can not solve constraint The defects of problem, expand the scope of application of efficient global optimization method.Meanwhile the optimization method of the present embodiment can pass through calling Fewer number analysis model and find the globe optimum of analysis model, can effectively reduce calculating cost, improve optimization effect Rate, help to shorten the engineering optimization cycle, have preferable application prospect in the engineering designs such as aerocraft system optimization.
Embodiment 2：Exemplified by the application of earth observation satellite quality optimization is reduced under meeting constraints.
The design object of earth observation satellite is the acquisition ground as much as possible in the case where ensureing resolution ratio and the distortion factor Face remote sensing images and the covering for realizing earth surface region, while satellite cost is reduced as far as possible.This paper multi-subject design is excellent Change modelling structure matrix (Design Structure Matrix, DSM) as shown in figure 4, wherein to represent subject defeated for horizontal line Go out, vertical line represents subject and inputted, coupling parameter between symbol table dendrography section at node.The multidisciplinary analysis (Multi- Disciplinary Analysis, MDA) process is made up of five subjects such as track, control, payload, power supply and structure, The coupled relation of different subjects is contained simultaneously.For the MDO problems, the present embodiment carries out coordination using fixed point iteration method and asked Solution.
The present embodiment earth observation satellite design objective be camera earth observation best performance, consider ground resolution and Covered ground size, design can integrate the function for weighted type
F (X)=M (G/G0)+N·(ψ0/ψ) (30)
In formula：M and N is weight coefficient, takes M=0.6, N=0.4；G0And ψ0The fixed value introduced for nondimensionalization, value Respectively 20m and 12 °.This paper Optimized models include 10 design variables and 14 constraints, and then earth observation satellite is multidisciplinary Design optimization problem can be described as
In formula：X is design variable, including orbit altitude h, flywheel angular momentum Hwh, solar array area Asa, southbound node when Carve DT, body height hs, body width hw, Bearing cylinder diameter R, motor power Fpro, accumulator capacity CsAnd camera Focal length f, XLBAnd XUBThe respectively lower boundary of design variable and coboundary, giConstrained for each subject.
Constraints involved by this example is as shown in table 2.Using GA, CiMPS and C-EGO to satellite multi-subject design Optimization problem carries out continuous 10 suboptimization.GA uses the default setting of ga functions in Matlab；The initial sample points of C-EGO are 100 It is individual, iterations 200.When finding to solve the problem in research, CiMPS can not normally be restrained using the convergence criterion of acquiescence, Therefore maximum iteration is used as CiMPS convergence of algorithm criterions, when MDA invocation of procedure numbers reach 300 times, CiMPS optimizations terminate and export current best result.Multidisciplinary analysis is carried out using fixed point iteration method (Multidisciplinary Analysis, MDA) ensures the uniformity of couple state variable between subject.
The quick near-optimal method specific implementation step of the aircraft for high time-consuming constraint disclosed in the present embodiment is such as Under：
Step 1, initial input parameter is defined.
Matlab m language definition analysis models, optimized variable design are used according to satellite MDO Problem Space lower limit is [600,0.18,3.0,8.0,1.5,1.2,0.24,10,30,0.2], the upper limit for [800,4.0,20.0,11.5, 5.0,2.475,0.825,200,100,1], it is 14 to constrain number, and initial sample point number is 100.
Step 2, initial sample point and computation model response are generated.
According to the Variational Design space defined in step 1 and initial sample point number, given birth to using the super side's method of sampling of Latin Into corresponding sample point, by call true model calculate true model response corresponding to each initial sample point and Binding occurrence, and true model response and binding occurrence corresponding to these sample points are saved in design sample point data base.
Step 3, normalization factor is calculated.
According to the sample point and its corresponding true model response and binding occurrence calculating target function obtained in step 2 Normalization factor fAvgAnd the normalization factor c of constraintAvg, its result of calculation is as follows：
fAvg=fSum/ nPoints=95/100=0.95
cAvg=cSum/ nPoints=[1.27e5,90.1,275,5.76e4,6.49e4,20.32,17.95,202.77, 0.14,2.32,8.06e4,6.3
1e3,466,1.09]/100
=[1.27e3,0.90,27.5,576,649,0.20,0.18,0.02,0.0014,0.023,806,63.1, 46.6,0.01]
Wherein nPoints be current sample point number, fSumFor current all sample point target function value absolute values With cSumFor the sum of current all sample point binding occurrence absolute values.
Step 4, pseudo- target function value is calculated.
In the design sample point data base that step 2 is obtained all sample points and its corresponding true model response and Binding occurrence extracts, and constraint is normalized, its calculation expression is as follows：
cNorm(x)=c (x)/cAvg (32)
Wherein c (x) be sample point x at binding occurrence, cNorm(x) it is the binding occurrence after normalization.Complete to constraint Processing after, object function is handled, its expression formula is as follows：
fNorm(x)=f (x)/fAvg (33)
Wherein f (x) be sample point x at target function value, fNorm(x) it is the target function value after normalization.Complete Into after the normalization of target function value and binding occurrence, construction includes the pseudo- target function value of constraint value information.As binding occurrence c (x) when being less than or equal to zero, target function value keeps constant；When binding occurrence c (x) is more than zero, to target function value processing such as Under：
Min F (x)=fNorm(x)+λmax(cNorm(x),0) (34)
Wherein, λ is penalty factor, and initial value 1, F (x) is the pseudo- target constructed according to normalization target function value and binding occurrence Functional value.All sample points are carried out with normalized as implied above.
Step 5, EI functions are constructed and are optimized.
The sample point used used of extraction step 2 and the pseudo- target function value constructed, construct EI functions and with EI letters Numerical value is up to target and optimized, and the expression formula of EI functions is as follows：
In formula：φ is Standard Normal Distribution；φ is the probability density function of standardized normal distribution；FminFor current institute There is the minimum value of pseudo- target function value；It is respectively the predicted value and standard deviation of agent model at sample point x with s.EI functions are entered Sample point x is obtained after row optimizationmin。
Step 6, sample point x is calculatedminFunctional value and binding occurrence
True analysis model is called to calculate xminThe object function response f at placeminAnd binding occurrence cmin, by sample point and Its corresponding object function response and binding occurrence are saved in sample point database.
Step 7, penalty factor is updated.
First, will be to cminIt is normalized, its expression formula is as follows：
cNorm=cmin/cAvg (36)
Wherein cNormFor the binding occurrence after normalization, c is obtainedNormIn maximum i.e. corresponding to maximum constrained degree of running counter to SmaxIf maximum constrained degree of running counter to is more than constraint tolerance Stol, then according to pre-defined growth factor μ1>1 increase penalty factor, So that Optimized Iterative is more likely to improve feasibility next time；Hold on the contrary, if maximum constrained degree of running counter to is less than or equal to Difference, then according to the descending factors μ of original definition2≤ 1 reduces penalty factor, so that Optimized Iterative lays particular emphasis on improvement most next time Dominance.In addition, introduce penalty factor bound constraint (λmin,λmax), on the one hand avoid penalty factor is too small from causing optimum results can not OK；On the other hand, avoid excessive the brought numerical value of penalty factor difficult in optimization process.Adaptive penalty factor calculation formula is as follows It is shown：
According to authors experience, μ in this research1Value is 2；μ2Value is 0.8；λmaxValue is 50；λminValue is 1；Stol Value is 0.001.
Step 8, pseudo- target function value is generated.
Object function response and binding occurrence in the sample point database of extraction step 6, it is according to penalizing of being obtained in step 7 Several that target function value is updated, more new formula is as follows：
Min F (x)=f (x)/fAvg+λmax(c(x)/cAvg,0) (38)
In formula, F (x) be sample point x place pseudo- target function value, f (x) be sample point x at real goal functional value, c (x) it is the binding occurrence at sample point x.
Step 8, on the basis of step 7, k=k+1 is made, step 4 progress next iteration is transferred to and is changed until reaching maximum Generation number 200 times, final output optimum results complete optimization process.
Binding occurrence corresponding to continuous ten suboptimization best result is as shown in table 2, corresponding to best result obtained by three kinds of algorithms The averaging model call number of design variable value, target function value and algorithms of different is as shown in table 3.Box traction substation in Fig. 4 is given The resultful contrast of continuous ten suboptimization institute is gone out.As shown in Table 2, during iteration ends GA, CiMPS and C-EGO preferably optimization As a result it is satisfied by constraining.
The constraints and optimal value of the earth observation satellite optimization problem of table 2
The design variable span of table 3 and optimum results contrast
It can be obtained by table 3 and Fig. 5 (a), the optimal objective function value that GA, CiMPS and C-EGO are obtained is closer to, and in mesh CiMPS is best in the optimality of offer of tender numerical value, C-EGO takes second place, GA is worst, illustrates that C-EGO solution techniques are correctly feasible.With reference to figure 5 (c) is understood, is preferably solved although CiMPS can be obtained, and its optimum results has a risk violated and constrained, and GA and C-EGO Optimum results are in restriction range.It can be obtained by Fig. 5 (b), the CiMPS on total MDA and constraint call number>GA>C-EGO, And CiMPS is more than nearly the 2000 of C-EGO, GA is more than 20 times of C-EGO, thus C-EGO has obvious odds for effectiveness.It is comprehensive with Upper analysis can obtain, although C-EGO is slightly poorer to CiMPS in optimality, compared to GA and CiMPS, C-EGO calls in model There is larger advantage, and acquired results disclosure satisfy that constraint, illustrate that C-EGO has very strong engineering practicability on number.
The result of consolidated statement 2 and table 3 understands that preliminary design scheme is infeasible, fails to meet satellite gross weight and intrinsic frequency Constraint.Compared to preliminary design scheme, the Bearing cylinder diameter increase by 83.3% of prioritization scheme obtained by C-EGO, body height 31.8% is reduced, body width increase by 49.2%, so as to improve intrinsic frequency.In addition, other design variables also have different journeys The change of degree, whole star quality is effectively reduced, finally obtains feasible optimizing design scheme.
To sum up gained, C-EGO solution efficiency is substantially better than GA and CiMPS, and can obtain preferably feasible solution.Cause This, C-EGO optimisation strategies can improve the efficiency of satellite master-plan to a certain extent, shorten the design cycle, be satellite conception Design provides strong support.Again it can be seen that the present embodiment considers Local approximation precision and global approximation quality, have simultaneously The ability for having processing to constrain, has ability of searching optimum, and can obtain the optimization design scheme for meeting constraint.
Above-described specific descriptions, the purpose, technical scheme and beneficial effect of invention are carried out further specifically It is bright, the specific embodiment that the foregoing is only the present invention is should be understood that, for explaining the present invention, is not used to limit this The protection domain of invention, within the spirit and principles of the invention, any modification, equivalent substitution and improvements done etc. all should Within protection scope of the present invention.

Claims (6)

  1. A kind of 1. quick near-optimal method of aircraft for high time-consuming constraint, it is characterised in that：Comprise the following steps,
    Step 1, initial input parameter is defined；
    According to practical problem, defined analysis model, optimized variable design space, constraint number and initial sample point number；
    Step 2, initial sample point and computation model response are generated, and is saved in design sample point data base；
    According to the Variational Design space defined in step 1 and initial sample point number, corresponding sample is generated using the method for sampling Point, by calling true model to calculate true model response and binding occurrence corresponding to each initial sample point, and will True model response and binding occurrence are saved in design sample point data base corresponding to above-mentioned sample point；
    Step 3, normalization factor is calculated；
    According to the sample point and its normalizing of corresponding true model response and binding occurrence calculating target function obtained in step 2 Change factor fAvgAnd the normalization factor c of constraintAvg, its calculation expression is as follows：
    fAvg=fSum/nPoints (1)
    cAvg=cSum/nPoints (2)
    Wherein nPoints be current sample point number, fSumFor the sum of current all sample point target function value absolute values, cSum For the sum of current all sample point binding occurrence absolute values；
    Step 4, pseudo- target function value is calculated；
    All sample points and its corresponding true model response and constraint in the design sample point data base that step 2 is obtained Value extracts, and constraint is normalized, calculation expression is as follows：
    cNorm(x)=c (x)/cAvg (3)
    Wherein c (x) be sample point x at binding occurrence, cNorm(x) it is the binding occurrence after normalization；Complete the processing to constraint Afterwards, object function is handled, expression formula is as follows：
    fNorm(x)=f (x)/fAvg (4)
    Wherein f (x) be sample point x at target function value, fNorm(x) it is the target function value after normalization；Complete target After the normalization of functional value and binding occurrence, construction includes the pseudo- target function value of constraint value information；When binding occurrence c (x) is less than Or during equal to zero, target function value keeps constant；It is as follows to target function value processing when binding occurrence c (x) is more than zero：
    Min F (x)=fNorm(x)+λmax(cNorm(x),0) (5)
    Wherein, λ is penalty factor, and initial value 1, F (x) is the pseudo- object function constructed according to normalization target function value and binding occurrence Value；All sample points are carried out with normalized as implied above；
    Step 5, EI functions are constructed and optimize to obtain sample point xmin；
    The sample point used used of extraction step 2 and the pseudo- target function value constructed, construct EI functions and with EI functional values It is up to target to optimize, the expression formula of EI functions is as follows：
    <mrow> <mi>E</mi> <mrow> <mo>(</mo> <mi>I</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mi>F</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>-</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mo>)</mo> </mrow> <mi>&amp;phi;</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>F</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>-</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> </mrow> <mi>s</mi> </mfrac> <mo>)</mo> </mrow> <mo>+</mo> <mi>s</mi> <mi>&amp;phi;</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>F</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>-</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> </mrow> <mi>s</mi> </mfrac> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>
    In formula：φ is Standard Normal Distribution；φ is the probability density function of standardized normal distribution；FminFor current all puppets The minimum value of target function value；It is respectively the predicted value and standard deviation of agent model at sample point x with s；EI functions are carried out Sample point x is obtained after optimizationmin；
    Step 6, sample point x is calculatedminFunctional value and binding occurrence；
    True analysis model is called to calculate xminThe object function response f at placeminAnd binding occurrence cmin, by sample point and its right The object function response and binding occurrence answered are saved in sample point database；
    Step 7, penalty factor is updated；
    First, will be to cminIt is normalized, its expression formula is as follows：
    cNorm=cmin/cAvg (7)
    Wherein cNormFor the binding occurrence after normalization, c is obtainedNormIn maximum i.e. corresponding to maximum constrained degree of running counter to Smax, If maximum constrained degree of running counter to is more than constraint tolerance Stol, then according to pre-defined growth factor μ1>1 increase penalty factor, so as to So that Optimized Iterative is more likely to improve feasibility next time；On the contrary, if maximum constrained degree of running counter to is less than or equal to tolerance, According to the descending factors μ of original definition2≤ 1 reduces penalty factor, so that Optimized Iterative lays particular emphasis on improvement optimality next time； In addition, introduce penalty factor bound constraint (λmin,λmax), on the one hand avoid penalty factor is too small from causing optimum results infeasible；Separately On the one hand, avoid excessive the brought numerical value of penalty factor difficult in optimization process；Adaptive penalty factor λ(k+1)Calculation formula is as follows It is shown：
    <mrow> <msup> <mi>&amp;lambda;</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </msup> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;mu;</mi> <mn>1</mn> </msub> <msup> <mi>&amp;lambda;</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </msup> <mo>,</mo> <msub> <mi>&amp;lambda;</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msub> <mi>S</mi> <mi>max</mi> </msub> <mo>&gt;</mo> <msub> <mi>S</mi> <mrow> <mi>t</mi> <mi>o</mi> <mi>l</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;mu;</mi> <mn>2</mn> </msub> <msup> <mi>&amp;lambda;</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </msup> <mo>,</mo> <msub> <mi>&amp;lambda;</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msub> <mi>S</mi> <mi>max</mi> </msub> <mo>&amp;le;</mo> <msub> <mi>S</mi> <mrow> <mi>t</mi> <mi>o</mi> <mi>l</mi> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>
    Wherein, growth factor μ1, descending factors μ2, the upper limit constraint λmax, lower limit constraint λmin, constraint tolerance StolIt is for adaptive penalize Number λ(k+1)Calculate the coefficient that need to be set；
    Step 8, pseudo- target function value is generated；
    Object function response and binding occurrence in the sample point database of extraction step 6, according to the penalty factor obtained in step 7 λ(k+1)Target function value is updated, more new formula is as follows：
    Min F (x)=f (x)/fAvg+λmax(c(x)/cAvg,0) (9)
    In formula, F (x) is the pseudo- target function value at sample point x, and f (x) is the real goal functional value at sample point x, and c (x) is Binding occurrence at sample point x；
    Step 9, on the basis of step 8, k=k+1 is made, step 4 is transferred to and carries out next iteration until reaching greatest iteration time Number, final output optimum results, complete the quick near-optimal of aircraft for high time-consuming constraint.
  2. 2. a kind of quick near-optimal method of aircraft for high time-consuming constraint according to claim 1, its feature exist In：Also include step 10, should by the quick near-optimal method of aircraft for high time-consuming constraint described in step 1 to step 9 Cost is calculated for optimization in engineering optimization, can be effectively reduced, improves optimization efficiency, helps to shorten Optimum design of engineering structure Cycle.
  3. 3. a kind of quick near-optimal method of aircraft for high time-consuming constraint according to claim 1 or 2, its feature It is：It is empty that the super side's method of sampling of the preferred Latin of the method for sampling described in step 2 can make sample point be evenly distributed in design Between in, so as to improve the precision of tectonic model.
  4. 4. a kind of quick near-optimal method of aircraft for high time-consuming constraint according to claim 1 or 2, its feature It is：Growth factor μ in described step 71Value is 2；Descending factors μ2Value is 0.8；The upper limit constrains λmaxValue is 50；Under Limit constraint λminValue is 1；Constrain tolerance StolValue is 0.001.
  5. 5. a kind of quick near-optimal method of aircraft for high time-consuming constraint according to claim 2, its feature exist In：Include applied to engineering optimization：In terms of standard engineering example, meeting constraints decline low pressure vessel design cost Using, in the case where meeting constraints reduce compression spring quality optimization application；In terms of satellite optimization design problem, meeting to constrain Under the conditions of reduce the application of earth observation satellite quality optimization, reduce satellite structure quality optimization in the case where meeting frequency constraints should With；In terms of guided missile optimization design problem, increase guided missile payload optimization application under the conditions of strength constraint is met, meeting to fire Expect to increase scope optimization application under constraints.
  6. 6. a kind of quick near-optimal method of aircraft for high time-consuming constraint according to claim 2, its feature exist In：There is preferable application prospect in aerocraft system optimizing engineering design.
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