CN109063355A - Near-optimal method based on particle group optimizing Yu Kriging model - Google Patents
Near-optimal method based on particle group optimizing Yu Kriging model Download PDFInfo
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Abstract
Near-optimal method disclosed by the invention based on particle group optimizing Yu Kriging model, belongs to Optimum design of engineering structure technical field.Implementation method of the present invention is as follows: being evolved using population and guides simple sample point to global optimum's disaggregation, cluster is carried out to simple sample point using FuzzycMeans Clustering method and carries out interest domain identification, realizes and sufficiently excavates design space information on the basis of balancing local search with global explore.By increasing high-precision sample point newly in interest regional sequence, Kriging model is constantly updated, guides to optimize and is quickly restrained to globally optimal solution.The present invention can effectively promote the global exploring ability and local search ability of optimization method, reduce the calculating cost in complex engineering system optimisation process, avoid omitting globally optimal solution as far as possible, improve the reliability of design result.Present invention is suitably applied to the complex engineering system design optimization fields comprising high accuracy analysis model, are able to solve corresponding correlation engineering problem.
Description
Technical field
The near-optimal method based on particle group optimizing Yu Kriging model that the present invention relates to a kind of, belongs to engineering optimization
Design field.
Background technique
To promote designing quality, design iterations, multidisciplinary design optimization (Multidisciplinary Design are reduced
Optimization, MDO) it is widely used to the design of complex engineering system.However, being needed in complex engineering system design process
It calls High Precision Simulation analysis model (such as finite element model) to improve design confidence level, optimization cost is caused to sharply increase.
In addition, MDO problem usually requires to ask by multidisciplinary analysis (Multidisciplinary Analysis, MDA) process iteration
Solution, has been further exacerbated by computational complexity.In order to alleviate the computational complexity problem that spacecraft MDO is faced, based on acting on behalf of mould
The optimization method (Metamodel-based Design and Optimization, MBDO) of type has obtained both at home and abroad in recent years
The common concern of scholar.MBDO method be intended to by building calculation amount it is small and with the comparable agent model of master mould precision, substitution
Former high-precision model (or MDA process) is used for design optimization.Optimisation strategy (also known as adaptive M BDO based on dynamic proxy model
Method) it has a clear superiority in terms of improving global convergence and optimization efficiency, become current research hot spot.
Agent model updates and management strategy is the core of adaptive M BDO method, mainly has and is adopted based on space reduction sequence
Sample and based on space padding sequence sample two kinds.It, gradually will sampling based on space reduction sequential sampling method as iteration carries out
Space converges to the region there may be globe optimum, only carries out sequential sampling in this lesser design space, usually
Local search ability with higher, but disadvantage is possible omit real globe optimum.It is adopted based on space padding sequence
Quadrat method does not change original design space, and by constructing certain target function, the process of guiding to optimize focuses on that there may be close
Sample point is increased newly like the region of optimum point.
Gunz bionic optimization algorithm is randomness algorithm based on probability, and robustness is preferable, and Typical Representative includes that heredity is calculated
Method (Genetic Algorithm, GA) and particle swarm optimization algorithm (Particle Swarm Optimization, PSO) etc..
But since objective function call number is higher in optimization process, it is seldom individually used for optimization MDO problem.In recent years, some scholars
Agent model auxiliary gunz bionic Algorithm (Metamodel-assisted Heuristic Algorithm) is proposed, is had preferably
Optimization efficiency and convergence.
Further to promote the efficiency and convergence that adaptive M BDO method optimizes complex engineering system, set forth herein one kind
Near-optimal method based on particle group optimizing Yu Kriging model: evolving using population and fuzzy space clustering method
Sequence Kriging agent model optimization method (Kriging Metamodel Assisted Global Optimization
Method Using Particle Swarm Evolution and Fuzzy Clustering,PSFC-KRG)。
Technical solution in order to better illustrate the present invention does certain introduction to related related Fundamentals of Mathematics below.
(1) Kriging agent model
Kriging model expression is as follows:
F (x)=g (x)+Z (x) (1)
In formula, g (x) is multinomial overall situation approximate model；Z (x) be mean value be zero, variance σ2, covariance is not zero
Gaussian random process indicates the partial deviations of global approximate model.The usually desirable constant value β of g (x), then formula (1) can convert are as follows:
F (x)=β+Z (x) (2)
The covariance matrix of Z (x) may be expressed as:
Cov[Z(xi),Z(xj)]=σ2R[R(xi,xj)] (3)
In formula, R is correlation matrix, and R is related coefficient, i, j=1,2 ..., ns, nsFor sample point quantity.R is symmetrical square
Battle array, diagonal entry 1, R expression formula are as follows:
In formula, nvFor design variable number, parameter P mainly influences the flatness of correlation coefficient value, usually takes constant 2.θkFor
Unknown correlation parameter vector, θkCorrelation between bigger each sample point of explanation is stronger, generally for different designs variable, θkIt is fixed to take
Value θ.
Formula (4) is converted into as a result,
Another associated vector r (x) is introduced, for indicating the correlation between future position and known sample point, such as formula (6) institute
Show.
Kriging prediction model may be expressed as:
In formula, y is the column vector being made of sample point response, and 1 is the vector that each element is 1,For location parameter,
Shown in its calculation method such as formula (8).
σ2And R is the function of θ, σ2Calculation method such as formula (9) shown in.
Related coefficient θ can be acquired by Maximum-likelihood estimation, shown in calculation method such as formula (10).
In addition, Kriging model can provide the variance s of predicted value at the x of arbitrary point by formula (11)2(x), to assess
The approximate error of Kriging model.
(2) particle group optimizing
The basic thought of particle swarm optimization algorithm, which is derived from, looks for food to flock of birds and finds the simulation of Bird's Nest social behavior, and from
It gains enlightenment in this biotic population behavior for solving optimization problem.Each particle indicates that optimization problem solution is empty in PSO algorithm
Between in an alternative solution, the fitness function of all particles obtains by the objective function of optimization problem.I-th of particle is tieed up in n
Position in space is denoted as xi=(xi1,xi2,…,xin), speed is denoted as vi=(vi1,vi2,…,vin).In the evolution of PSO algorithm
In iterative process, each particle remembers (the optimal location p that i-th of particle searches so far according to itselfi) and population
Memory (the optimal location q) that entire population searches so far updates the position of itself, to search for design space most
Excellent solution.The particle position that PSO algorithm is taken is with speed more new formula
In formula, t is the current algebra of population；r1And r2For the random number between [0,1]；c1Most for Particle tracking itself history
The weight coefficient of the figure of merit；c2For the weight coefficient of Particle tracking group optimal value.0 ω≤1 < is inertia weight coefficient.Inertia power
Weight coefficient is bigger, and particle more tends to global search；Inertia weight coefficient is smaller, and particle more tends to local search, therefore inertia
Weight coefficient reduces with the increase of the number of iterations, and more new formula is
ωt+1=ωt×ωd (13)
Wherein ωdFor inertia weight attenuation coefficient.
(3) FuzzycMeans Clustering method
FuzzycMeans Clustering method (Fuzzy C-means Clustering Method, FCM) is same by minimizing
The dissimilar degree for clustering lower data carries out data clusters.After specified cluster numbers c, cluster is determined by solving following optimization problem
Sample point in center and each Cluster space
In formula: U is m sample point xjThe subordinated-degree matrix of (j=1,2 ..., m, x ∈ R)；V=(v1,v2,…,vnc) in vi
Indicate ith cluster center, 1≤i≤c；N is greater than 1 constant, usually takes n=2；μijIndicate j-th of sample point for i-th
The degree of membership of a Cluster space.The calculation formula of standard Euclidean distance norm is
dij=| | xj-vi|| (15)
Shown in the constrained optimization problem construction Lagrangian such as formula (16) indicated formula (14).
It is rightIt optimizes and optimal subordinated-degree matrix U can be obtained*With cluster centre v*, by each sample point
It is divided into the maximum Cluster space of sample point degree of membership.
Summary of the invention
For the prior art, there are following technical problems: (1) being difficult to take in local search ability and global exploring ability
It obtains well balanced；(2) it is easy to omit globally optimal solution；(3) convergence rate is slow.It is disclosed by the invention based on particle group optimizing with
The near-optimal method technical problems to be solved of Kriging model are: based on particle group optimizing and Kriging model, flat
Design space information is sufficiently excavated on the basis of weighing apparatus local search and global exploration, is realized fast to guarantor's precision of complex engineering system
Response modeling improves complex engineering system design optimization quality.Have the advantages that (1) avoids omitting globally optimal solution；(2)
Fast convergence rate.Present invention is suitably applied to the complex engineering system design optimization fields comprising high accuracy analysis model, can
Solve the problems, such as corresponding correlation engineering.
The purpose of the present invention is what is be achieved through the following technical solutions.
Near-optimal method disclosed by the invention based on particle group optimizing Yu Kriging model, is evolved using population
It guides magnanimity simple sample point to global optimum's disaggregation, simple sample point is clustered using FuzzycMeans Clustering method
Carry out interest domain identification, that is, realizes and sufficiently excavate design space information on the basis of balancing local search and the overall situation is explored.
By increasing high-precision sample point newly in interest regional sequence, Kriging model is constantly updated, is guided to optimize quickly to global optimum
Solution convergence.Near-optimal method based on particle group optimizing and Kriging model can effectively promote the global of optimization method and visit
Suo Nengli and local search ability reduce the calculating cost in complex engineering system optimisation process, while avoiding omitting as far as possible
Globally optimal solution improves the reliability of design result.
Near-optimal method disclosed by the invention based on particle group optimizing Yu Kriging model, includes the following steps:
Step 1: generating initial sample point, initial sample point by the super side's experimental design of Latin in initial designs space
Number NiniIt is found out by formula (1), n in formulavFor design variable number.
Step 2: high accuracy analysis model is called to obtain objective function and constraint function response at sample point, and by sample
This point and its response are saved in sample point database.
Step 3: constructing augmented objective function based on all sample points and its response in step 2 sample point database.
Step 3 concrete methods of realizing are as follows:
Step 3.1: as k=1, i.e., first time iteration when, take penalty factor initial value μ(1)=1, as k > 1, according to
The penalty factor μ of k-1 iteration(k-1), kth time iteration maximum constrained degree of violating ψmax, penalty factor growth factor a, constraint degree of violating hold
Poor ψtol, obtain the penalty factor μ of kth time iteration(k)As shown in formula (2)；
Step 3.2: constructing penalty function, p in formula according to formula (3)dIt (x) is design variable boundary constraint penalty term, pg(x) it is
Constraint condition penalty term, gjIt (x) is j-th of constraint function, xiFor i-th of design variable,WithRespectively i-th design
The lower bound of variable and the upper bound, above-mentioned i and j are counting variable；
Step 3.3: the penalty factor μ found out according to formula (2)(k)The penalty function P (x) found out with formula (3) constructs augmentation target
Function F (x, μ(k)) as follows.
F(x,μ(k))=f (x)+μ(k)P(x) (4)
Step 4: the augmented objective function based on step 3 construction, all samples in two sample point database of obtaining step
The augmented objective function response of point, based on all sample points in step 2 sample point database and its augmented objective function response
Value constructs Kriging agent model.
Step 5: the classical global optimization approach of application optimizes the Kriging model constructed in step 4, acquisition is worked as
Preceding potential optimal solution x(k).High accuracy analysis model is called to obtain x(k)The real goal function and constraint function response at place, and
By x(k)And its true response is saved in sample point database described in step 2.
Classical global optimization approach described in step 5 includes genetic algorithm (Genetic Algorithm, GA), simulates and move back
Fiery algorithm (Simulated Annealing, SA), sequential quadratic programming (Sequential Quadratic Programming,
SQP) etc..
Step 6: whether the near-optimal method based on particle group optimizing and Kriging model of inspection restrains, it is specific to receive
It holds back shown in criterion such as formula (5).It is approximate excellent with Kriging model based on particle group optimizing if meeting optimization convergence criterion
Change method terminates, by current optimal solution x(k)Optimal solution as optimization exports；If being unsatisfactory for optimization convergence criterion, turn to
Step 7.In formula (5)WithIndicate kth time iteration and kth -1 time real goal function response, ε is according to analysis mould
Depending on type precision.
Step 7: choosing the smallest n of target function value in sample point databasebestA sample point is evolved as population
Initial sample point set is carried out population using Kriging model predication value as objective function and evolved, and it is pre- that constraint condition, which is arranged,
Survey variance s2(x) it is no more thanThe number of iterations that population is evolved takes niter, store whole niterSample point in secondary iteration.
Step 8: the sample point stored in step 7 is divided into n using FuzzycMeans Clustering methodcA cluster is empty
Between S(j)(j=1 ..., nc) in, obtain each Cluster space S(j)In optimal sample point as the optimal sample point of son.Take ncHeight
Current optimum point x of the optimum point as kth time iteration in optimal sample point(k), by x(k)And its true response is stored in sample
Point data base.Delete ncCluster in the optimal sample point of height where the maximum optimal sample point of son of augmented objective function value is empty
Between, remaining Cluster space is merged into new design space, i.e. the sampling interest region of kth time iteration.K=k+1 is enabled, is returned
Step 2.
It further include step 9: will be approximate with Kriging model based on particle group optimizing described in step 1 to step 8
Optimization method is applied to be exported comprising the high complex engineering system design optimization field for calculating time consuming analysis model according to step 6
Optimal solution instruct complex engineering system to design, and solve the problems, such as corresponding correlation engineering.
The optimization of engineering design described in step 9 field includes the Optimal Structure Designing containing extensive finite element analysis, contains
There is the multidisciplinary design optimization of the Aerodynamic optimization design, complex engineering system of high-precision flow dynamics analysis.
The multidisciplinary design optimization field of the complex engineering system includes aircraft, automobile, ship domain.
The utility model has the advantages that
1, the near-optimal method disclosed by the invention based on particle group optimizing Yu Kriging model, using population into
Change guidance magnanimity simple sample point to global optimum's disaggregation, simple sample point is gathered using FuzzycMeans Clustering method
Class carries out interest domain identification, that is, realizes that design space is sufficiently excavated on the basis of balancing local search and the overall situation is explored believes
Breath can effectively promote the global exploring ability and local search ability of optimization method, reduce complex engineering system optimisation process
In calculating cost, while as far as possible avoid omit globally optimal solution, improve the reliability of design result.
2, the near-optimal method disclosed by the invention based on particle group optimizing Yu Kriging model, is suitably applied packet
Containing the high complex engineering system design optimization field for calculating time consuming analysis model, such as structure containing extensive finite element analysis is excellent
Change design, the Aerodynamic optimization design containing high-precision flow dynamics analysis, aircraft/automobile/ship domain complex engineering system
Multidisciplinary design optimization.
Detailed description of the invention
Fig. 1 is the near-optimal method flow diagram based on particle group optimizing Yu Kriging model；
Fig. 2 is pressure vessel appearance schematic diagram.
Specific embodiment
Objects and advantages in order to further illustrate the present invention, in the following with reference to the drawings and specific embodiments to the present invention do into
The explanation of one step, and by with efficient global optimization method (Efficient Global Optimization, EGO), chase after peak and adopt
Quadrat method (Mode Pursuing Sampling, MPS) and its derivative algorithm constraint chase after the peak method of sampling (Constraint
Importance Mode Pursuing Sampling, CiMPS) results of three kinds of classical approximation optimization algorithms is compared, and it is right
Comprehensive performance of the invention carries out verifying analysis.
Embodiment 1
Illustrate specific implementation process as embodiment below by design of pressure vessels example.
Design of pressure vessels (Pressure Vessel Design, PVD) problem the purpose is to reduce include welding, material
Be molded over interior design cost.Pressure vessel needs store 750ft under the pressure of 3000psi3Air, outside shaped like Fig. 1
It is shown.PVD problem includes four design variables, x1For the thickness of ballhead, x2For thickness of shell, x3For radius, x4For length.
Optimized model expression formula is as follows:
Design variable n is taken in the present embodimentv=4.The present embodiment is disclosed based on particle group optimizing and Kriging model
Near-optimal method, specific implementation step are as follows:
Step 1: generating initial sample point by the super side's experimental design of Latin in initial designs space, the number of iterations k is enabled
=1.According to design variable number nv=4, take initial sample point number Nini=12.
Step 2: high accuracy analysis model is called to obtain objective function and constraint function response at sample point, and by sample
This point and its response are saved in sample point database.The objective function response at sample point x, i.e. pressure are indicated with f (x)
Vessel Design cost；The constraint function response at sample point x is indicated with g (x).
Step 3: constructing augmented objective function F based on all sample points and its response in step 2 sample point database
(x,μ(k))。
Step 3 concrete methods of realizing are as follows:
Step 3.1: as k=1, i.e., first time iteration when, take penalty factor initial value μ(1)=1, as k > 1, according to
The penalty factor μ of k-1 iteration(k-1), kth time iteration maximum constrained degree of violating ψmax, penalty factor growth factor a, constraint degree of violating hold
Poor ψtol, show that the penalty factor such as formula (2) of kth time iteration is shown；
Step 3.2: constructing penalty function, p in formula according to formula (3)dIt (x) is design variable boundary constraint penalty term, pg(x) it is
Constraint condition penalty term, gjIt (x) is j-th of constraint function, xiFor i-th of design variable,WithRespectively i-th design
The lower bound of variable and the upper bound, above-mentioned i and j are counting variable；
Step 3.3: the penalty factor μ found out according to formula (2)(k)The penalty function P (x) found out with formula (3) constructs augmentation target
Function F (x, μ(k)) as follows.
F(x,μ(k))=f (x)+μ(k)P(x) (4)
Step 4: based on the augmented objective function that step 3 constructs, all samples in two sample point database of obtaining step
The augmented objective function response of point, based on all sample points in step 2 sample point database and its augmented objective function response
Value constructs Kriging agent model.
Step 5: using genetic algorithm (Genetic Algorithm, GA) to the Kriging model constructed in step 4
Optimize the current potential optimal solution x of acquisition(k).Analysis model is called to obtain x(k)The real goal function f (x at place(k)) and constraint
Function response g (x(k)), and by x(k)、f(x(k))、g(x(k)) be saved in sample point database.
Step 6: whether the near-optimal method based on particle group optimizing and Kriging model of inspection restrains, it is specific to receive
It holds back shown in criterion such as formula (5).It is approximate excellent with Kriging model based on particle group optimizing if meeting optimization convergence criterion
Change method Optimizing Flow terminates, by current optimal solution x(k)Optimal solution as optimization exports；If it is quasi- to be unsatisfactory for optimization convergence
Then, then step 7 is turned to.In formula (5)WithIndicate kth time iteration and kth -1 time real goal function response.ε takes
0.001。
Step 7: choose sample point database in the smallest 16 sample points of target function value as population evolve at the beginning of
Beginning sample point set is carried out population using Kriging model predication value as objective function and evolved, and constraint condition is arranged as prediction
Variance s2(x) it is no more than 1000.The number of iterations that population is evolved takes 10, stores the sample point in all 10 iteration.
Step 8: the sample point stored in step 7 is divided into 3 Cluster spaces using FuzzycMeans Clustering method
S(j)(j=1 ..., nc) in, obtain each Cluster space S(j)In optimal sample point as the optimal sample point of son.Take 3 sons most
Current optimum point x of the optimum point as kth time iteration in excellent sample point(k), by x(k)And its true response is stored in sample point
Database.The Cluster space in 3 optimal sample points of son where the maximum sub- optimum point of augmented objective function value is deleted, it will be remaining
Cluster space merge into new design space.Enable k=k+1, return step two.
It is right using the near-optimal strategy (PSFC-KRG) proposed by the present invention based on particle group optimizing and Kriging model
Design of pressure vessels problem optimizes, and result and classical approximation optimization algorithm are chased after the peak method of sampling (MPS) and its derived
Algorithm constraint chases after the peak method of sampling (CiMPS) comparison.It is influenced to exclude enchancement factor, every kind of algorithm is respectively to design of pressure vessels
Problem carries out 10 suboptimization, feasible optimal solution minimum value, average value and objective function call number obtained by 10 suboptimization of statistics
Average value and constraint function call number average value.The results are shown in Table 1.
Table 1 PSFC-KRG, MPS, CiMPS are to design of pressure vessels problem optimum results
From the data in the table, the feasible optimal solution of PSFC-KRG optimization gained and MPS and CiMPS are substantially suitable.MPS and
Objective function call number (number of function evaluations, nfe) needed for CiMPS optimizes is significantly less than
PSFC-KRG, but the constraint function call number (number of constraint evaluations, nce) of PSFC-KRG is bright
It is aobvious to be less than MPS and CiMPS.Comprehensively consider total call number of objective function and constraint function, the calculating cost tool of PSFC-KRG
It has a clear superiority.
In order to which the advantage of PSFC-KRG is better described, further chooses 7 numerical value test problems and optimize, with classics
It the efficient global optimization strategy (EGO) of near-optimal strategy and chases after the peak method of sampling (MPS) and compares.Numerical value test problem includes
GP,SC,BR,GF,RS,HN,F16.For above-mentioned 7 test problems, the approximate optimal solution obtained when by comparing iteration ends
Size carry out measure algorithm efficiency.It is influenced to exclude accidentalia, every kind of algorithm distinguishes Filled function to each test test problem
10 times.Shown in the mathematical model of 7 test problems such as formula (6)~(12).
Goldstein-Price function (GP, nv=2)
Six-hump Camel function (SC, nv=2)
Brainin function (BR, nv=2)
Generalized Polynomial function (GF, nv=2)
Rastrigin function (RS, nv=2)
Hartmann function (HN, nv=6)
α in formulaij、ciAnd pijValue as shown in table 2 and table 3.
2 factor alpha of tableijAnd ciValue
3 coefficient p of tableijValue
F16 function (F16, nv=16)
α in formulaijValue is as follows
For PSFC-KRG, population initial population scale N is takenbest=Nini, evolutionary generation niter=10, Cluster space
Number nc=3, objective function improves degree tolerance ε=0.001.For MPS, simple sample points N is takencheap=104, initial sample
Scale Nini=(nv+1)(nv+2)/2-1+nv, objective function improvement degree tolerance ξ1=0.001, approximation quality tolerance ξ2=
0.01.For EGO, N is selectedini=(nv+1)(nv+2)/2-1+nvA initial sample architecture Kriging agent model, remaining calculation
Method parameter keeps default setting.
For EGO is arranged and optimizes numerical value test according to PSFC-KRG and MPS test problem optimum results convenient for comparative study
When problemAs shown in table 4.
The setting of 4 maximum target function call number of table
The optimum results of PSFC-KRG, EGO, MPS are as shown in table 5.
The comparison of 5 PSFC-KRG, EGO, MPS numerical value test problem optimum results of table
From the data in table 5, it can be seen that objective function calling time when PSFC-KRG, EGO, MPS solve above-mentioned 7 numerical value test example
Number it is almost the same, i.e., Different Optimization algorithm consumption calculating cost it is almost the same, can as comparison optimization obtained by optimal solution come
Compare the global optimizing ability of Different Optimization algorithm.
For GP function, SC function, BR function, HN function, F16 function, the middle position of PSFC-KRG optimization gained optimal solution
Several and variation range all has clear superiority compared with EGO and MPS.For GP function, EGO and MPS optimize gained optimal solution
Median respectively may be about 6 times and 8 times of theoretical optimal solution 3.000, and the median of PSFC-KRG optimization gained optimal solution is only
3.079, it is very close with theoretical optimal solution, while the minimum value for optimizing gained optimal solution can achieve 3.000.For SC letter
The median of number, EGO and PSFC-KRG optimization gained optimal solution is -1.032, consistent with theoretical optimal solution, and MPS optimizes institute
The median for obtaining optimal solution is -0.980, and optimality is poor；In addition, PSFC-KRG optimization gained optimal solution variation range be [-
1.032, -1.031], varied less near theoretical optimal solution.For BR function, algorithms of different optimum results comparative situation with
SC function is similar, and the optimality of PSFC-KRG result is similar with EGO, and the two is substantially better than MPS, and the robust of PSFC-KRG result
Property be better than MPS.For HN function, the median and minimum value of PSFC-KRG optimization gained optimal solution are -3.322, with theory
Optimal solution is consistent, and maximum value -3.203 is even less than the median and MPS optimization gained optimal solution of EGO optimization gained optimal solution
Minimum value.For F16 function, EGO and MPS optimization gained optimal solution and theoretical 25.875 gap of optimal solution are larger, this reflects
The scarce capacity of EGO and MPS optimization higher-dimension problem out, and the median of PSFC-KRG optimization gained optimal solution is 26.292, with
EGO is compared with the result of MPS to have clear improvement, and the minimum value of PSFC-KRG optimization gained optimal solution is 25.879, close to theory
Optimal solution.
For GF function and RS function, PSFC-KRG optimization gained optimal solution is not better than EGO and MPS comprehensively.For GF
The median of function, PSFC-KRG optimization gained optimal solution is 0.316,1.915 of the 0.748 and MPS better than EGO, and is optimized
The variation range of the variation range ratio MPS optimization gained optimal solution of gained optimal solution is small, illustrates the Shandong of PSFC-KRG optimum results
Stick is preferable.For RS function, the median of PSFC-KRG optimization gained optimal solution is between EGO and MPS, and optimization gained is most
The variation range of excellent solution is larger compared with MPS and optimality is poor.
By above-mentioned comparison it can be easily recognized that PSFC-KRG is in the process of optimization for solving complex engineering system
In, it can be improved the optimizing ability and optimization efficiency of optimum design method, while the robustness of optimum design method can be enhanced.
PSFC-KRG method is suitable for the huge engineering design of various operands and optimizes field, such as contains the work of extensive finite element analysis
Journey Optimal Structure Designing, it is multiple containing the hydromechanical Aerodynamic optimization design of high precision computation and aircraft, automobile, ship etc.
The multidisciplinary design optimization of miscellaneous engineering system.
Above-described specific descriptions have carried out further specifically the purpose of invention, technical scheme and beneficial effects
It is bright, it should be understood that the above is only a specific embodiment of the present invention, the protection model being not intended to limit the present invention
It encloses, all within the spirits and principles of the present invention, any modification, equivalent substitution, improvement and etc. done should be included in the present invention
Protection scope within.
Claims (5)
1. the near-optimal method based on particle group optimizing Yu Kriging model, it is characterised in that include the following steps,
Step 1: generating initial sample point, initial sample point number by the super side's experimental design of Latin in initial designs space
NiniIt is found out by formula (1), n in formulavFor design variable number；
Step 2: high accuracy analysis model is called to obtain objective function and constraint function response at sample point, and by sample point
And its response is saved in sample point database；
Step 3: constructing augmented objective function based on all sample points and its response in step 2 sample point database；
Step 3.1: as k=1, i.e., first time iteration when, take penalty factor initial value μ(1)=1, as k > 1, according to kth -1 time
The penalty factor μ of iteration(k-1), kth time iteration maximum constrained degree of violating ψmax, penalty factor growth factor a, constraint degree of violating tolerance
ψtol, obtain the penalty factor μ of kth time iteration(k)As shown in formula (2)；
Step 3.2: constructing penalty function, p in formula according to formula (3)dIt (x) is design variable boundary constraint penalty term, pgIt (x) is constraint
Condition penalty term, gjIt (x) is j-th of constraint function, xiFor i-th of design variable,WithRespectively i-th of design variable
Lower bound and the upper bound, above-mentioned i and j are counting variable；
Step 3.3: the penalty factor μ found out according to formula (2)(k)The penalty function P (x) found out with formula (3) constructs augmented objective function F
(x,μ(k)) as follows；
F(x,μ(k))=f (x)+μ(k)P(x) (4)
Step 4: the augmented objective function based on step 3 construction, all sample points in two sample point database of obtaining step
Augmented objective function response is based on all sample points and its augmented objective function response in step 2 sample point database,
Construct Kriging agent model；
Step 5: the classical global optimization approach of application optimizes the Kriging model constructed in step 4, obtain current latent
In optimal solution x(k)；High accuracy analysis model is called to obtain x(k)The real goal function at place and constraint function response, and by x(k)
And its true response is saved in sample point database described in step 2；
Step 6: whether the near-optimal method based on particle group optimizing and Kriging model of inspection restrains, specific convergence is quasi-
Then as shown in formula (5)；If meeting optimization convergence criterion, the near-optimal side based on particle group optimizing Yu Kriging model
Method terminates, by current optimal solution x(k)Optimal solution as optimization exports；If being unsatisfactory for optimization convergence criterion, step is turned to
Seven；In formula (5)WithIndicate kth time iteration and kth -1 time real goal function response, ε is according to analysis model essence
Depending on degree；
Step 7: choosing the smallest n of target function value in sample point databasebestA sample point is evolved initial as population
Sample point set is carried out population using Kriging model predication value as objective function and evolved, and it is prediction side that constraint condition, which is arranged,
Poor s2(x) it is no more thanThe number of iterations that population is evolved takes niter, store whole niterSample point in secondary iteration；
Step 8: the sample point stored in step 7 is divided into n using FuzzycMeans Clustering methodcA Cluster space S(j)
(j=1 ..., nc) in, obtain each Cluster space S(j)In optimal sample point as the optimal sample point of son；Take ncHeight is optimal
Current optimum point x of the optimum point as kth time iteration in sample point(k), by x(k)And its true response is stored in sample points
According to library；Delete ncCluster space in the optimal sample point of height where the maximum optimal sample point of son of augmented objective function value, will
Remaining Cluster space merges into new design space, i.e. the sampling interest region of kth time iteration；Enable k=k+1, return step
Two.
2. the near-optimal method based on particle group optimizing Yu Kriging model as described in claim 1, it is characterised in that:
It further include step 9, by the near-optimal method described in step 1 to step 8 based on particle group optimizing Yu Kriging model
Applied to the complex engineering system design optimization field comprising high calculating time consuming analysis model, the optimal solution exported according to step 6
It instructs complex engineering system to design, and solves the problems, such as corresponding correlation engineering.
3. the near-optimal method based on particle group optimizing Yu Kriging model as claimed in claim 2, it is characterised in that:
The optimization of engineering design described in step 9 field includes the Optimal Structure Designing containing extensive finite element analysis, containing high-precision
The multidisciplinary design optimization of the Aerodynamic optimization design, complex engineering system of flow dynamics analysis.
4. the near-optimal method based on particle group optimizing Yu Kriging model as claimed in claim 3, it is characterised in that:
The multidisciplinary design optimization field of the complex engineering system includes aircraft, automobile, ship domain.
5. the near-optimal method based on particle group optimizing Yu Kriging model as described in claim 1,2,3 or 4, special
Sign is: classical global optimization approach described in step 5 includes genetic algorithm, simulated annealing, sequential quadratic programming.
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