CN105303253A - Multidisciplinary reliability design optimization method based on CSSO and optimization models of different precisions - Google Patents

Abstract

The invention relates to a multidisciplinary reliability design optimization method based on CSSO and optimization models of different precisions. The method comprises the following steps that 1) a low-precision model is constructed in a response surface method and the like for the optimization problem; 2) the low-precision model is limited by utilizing a multiplication scale function to ensure the convergence consistency between a low-precision approximation model and a high-precision model; 3) parallel subspace optimization strategies are integrated to an initialized optimization model to carry out multidisciplinary analysis; 4) a global sensitivity equation (GSE) is used to analyze the sensitivity of the system; 5) subspaces are optimized in parallel; 6) system level coordinated optimization is carried out; 7) a credible region is detected, and the size of the credible region is adjusted; 8) MPP search is carried out on the basis of an advance mean value method; 9) the reliability is determined, a most possible failure point MPP is solved and used to calculate g(uk) which is lower than 0; 10) the convergence is determined; and 11) a determinacy design optimization model is reconstructed.

Description

A kind of multidisciplinary reliability design optimization method based on CSSO and many precision optimizings model

Technical field

The invention belongs to the reliability design optimisation technique field of complex product, relate to a kind of multidisciplinary reliability design optimization method of serializing.Particularly relate to a kind of multidisciplinary reliability design optimization method based on CSSO and many precision optimizings model.

Background technology

The deterministic optimization of tradition does not consider uncertain factor, and along with the progress of science and technology, uncertain impact in engineering design more and more receives publicity.Such as large-scale weaponry, nuclear power facility, boats and ships, electronic product and the aerospace flight vehicles etc. of some large scaled complex product are often very high to the requirement of reliability.Because these product safety are supreme, once the often consequence that lost efficacy is serious.Therefore, in the design of complex product, manufacture, maintenance, researcher has carried out a series of research to its reliability.Research shows that the impact of Design Stage on product reliability is huge, and it is 70% ~ 80% to the contribution rate of product quality according to statistics.Practical experience both domestic and external shows, product reliability is determined by design, ensured by manufacture, setup and manage.Incorporate fail-safe analysis and design optimization technology just can make product can have optimizing structure of economy in the design phase of complex product, enough reliabilities can be obtained again.So the Research Significance of the reliability design optimization method of related products is great.

But, current multidisciplinary reliability design optimization method be all by multidisciplinary design optimization strategy and analysis method for reliability directly integrated, define typical three layers of nested circulation, efficiency is extremely low and may cause the consequence that result optimizing result cannot restrain, for system multiple coupled on a large scale, problem is particularly outstanding.Therefore the present invention attempts the efficiency improving multidisciplinary reliability design optimization from three aspects.On the one hand, concurrent subspace optimization strategy CSSO and analysis method for reliability are combined, allow every sub spaces can carry out design optimization simultaneously, to reduce systematic analysis number of times, realize the thought of Parallel Design.And propose to improve for the problem that the moving range of CSSO algorithm subspace design variable is narrow.Secondly, the present invention uses the sequence optimisation of improvement to carry out decoupling zero with reliability assessment framework to nested multidisciplinary design optimization, make multidisciplinary optimization and fail-safe analysis be able to order to carry out, thus reach minimizing recurring series, improve the object of multidisciplinary reliability design optimization efficiency.Last in order to ensure the precision of multidisciplinary design optimization, present invention uses a kind of many precision optimizings based on inter-trust domain model, use low accuracy model to carry out concrete optimization to calculate, high-precision model is used for analyzing and correcting low accuracy model, thus ensures to optimize precision.For the design of complex product, carry out the multidisciplinary reliability design optimization method taking into account optimization precision and optimization efficiency and there is important theoretical direction and practical value.

Summary of the invention

The object of the invention is to provide a kind of multidisciplinary reliability design optimization method based on CSSO and many precision optimizings model, the concurrent subspace optimization strategy of improvement and single subject analysis method for reliability carry out effectively integrated by the method, and use the sequence optimisation of improvement and reliability assessment framework that decoupling zero is carried out in determinacy multidisciplinary design optimization and multidisciplinary fail-safe analysis, and a kind of many precision optimizings based on inter-trust domain model is for ensureing computational accuracy.Invention gives particular content and the flow process of the method, for scientifically ensureing that the reliability of Complex Product Design and design optimization efficiency and precision provide new instrument.Below content of the present invention is introduced in detail respectively:

In order to consider precision and the efficiency of multidisciplinary reliability design optimization as a whole, the present invention is under the sequence optimisation improved and reliability assessment framework, and integrated concurrent subspace optimization strategy and many precision optimizings model carry out multidisciplinary reliability design optimization.As shown in Figure 1, its concrete steps are as follows for the flow process of method:

Step one, low accuracy model is constructed for optimization problem, can the acquisitions such as response phase method be utilized.

Step 2, in order to ensure the convergence consistance of low precision approximate model and high-precision model, multiplication scaling function is used to carry out the low accuracy model of limit structure.At certain Optimum Points x _kplace, multiplication scaling function can be expressed as:

F _h(x _k) represent kth time loop optimization point x _kthe analysis result of place's high-precision model, f _l(x _k) represent kth time loop optimization point x _klocate the analysis result of low accuracy model.At x _kneighbouring multiplication scaling function can be defined as:

β'(x)＝β(x _k)+Δβ(x _k)(x-x _k)(2)

Wherein Δ β (x _k) represent at Optimum Points x _kthe derivative of place's gradient function, then can be expressed as in the result optimizing xth step proximate analysis in circulation: f _h(x)=β (x _k) f _l(x).

In order to ensure counting yield, propose the interval method of approximation of scaling function herein, scaling function value can upgrade after several times circulation.The cycle index at interval is

N _kit is right to represent the value rounded up.Wherein, α is attenuation coefficient, and m is self-defining initial adjustment value.Work as n _kwhen being less than 1, all n _p=1 (p > k).X _kfor the design parameter value that epicycle circulation obtains.When cycle index increases, upgrade interval more and more less, near convergence point, interval is approximately 1, namely more less until n close to convergence point interval _k=1 namely continuously every.

Step 3, multidisciplinary analysis is carried out for the integrated concurrent subspace optimization strategy of initialization Optimized model, there to be the Optimized model of two subsystems, y ₁and y ₂for state variable, by the multidisciplinary analysis of its simultaneous as shown in the formula

The functional value of limit state function can be obtained according to the epicycle iterative value solving state variable value and the design variable obtained.In addition, state variable value is also the basis of next step system sensitivity analysis.

Step 4, employing global sensitivity equation (GSE) carry out system sensitivity analysis, and there to be the multidisciplinary problem of two subsystems, its global sensitivity equation is shown below

Dy _i/ dx _jbe i-th state variable to design parameter x _jsensitivity value.Limit state function g is to a jth design variable x _jsusceptibility information can be obtained by following formula:

Wherein n represents the number of state variable.

Step 5, subspace parallel optimization, the task that the subspace of Concurrent Subspace method is optimized is under the condition that the change of system goal function value is very little, reduce the violation amount of subspace giant ties.Can be expressed as the optimization problem of kth sub spaces:

Wherein, F is system goal function, but only optimizes the design variable of this subspace.S ^p, r _k ^p, t _k ^pfor subspace cross influence coefficient.Here be constant.C ^pfor the giant ties of p subspace.

Step 6, system-level coordination optimization, the optimization aim of each subspace is all system goal function, and the minimum F that system-level coordination optimization obtains just becomes the function of r and t.System-level coordination optimization problem can be expressed as follows:

Step 7, detection inter-trust domain, and adjust inter-trust domain size, concrete steps are:

Step1: calculate trusted zones discriminant function

Wherein P (x) is penalty, and it can be defined as P (x)=f (x)+λ _k∑ max (0, g _i(x)), penalty factor

Step2: according to the value adjustment trusted zones of the trusted zones discriminant function calculated in step1.

Step 8, carry out MPP search based on advanced averaging method (AdvancedMeanValue, AMV).Advanced average point method steepest descent direction constantly more new standardized direction vector with alternative non-linear constrain, thus obtain convergence result more efficiently.Wherein the testing site of kth+1 circulation is calculated as follows:

u _k+1＝β _t·e _k(10)

Wherein, represent the gradient direction vector of kth time circulation time function of state.U represents the design vector of normed space, β _trepresent fiduciary level requirement.

Step 9, judging reliability, calculating g (u according to obtaining the most probable failure point (MPP) _k), g (u _k) < 0?, be that then this takes turns the discontented sufficient reliability requirement of design; Otherwise, meet reliability requirement.

Step 10, convergence judge, (u _k+1-u _k+1)/u _k< ε? then terminate; Otherwise go to step 11

Step 11, reconstruct the deterministic design Optimized model.

Step1: the optimal design point x utilizing the deterministic design optimization to obtain ^kwith the most probable failure point that i-th probabilistic constraints is obtained by fail-safe analysis determine the motion-vector of this constraint condition

Step2: build new constraint function based on convex linearization approximation technique: suppose g _ifor i-th constraint condition of certain subject, in conjunction with the motion-vector built in the shift strategy in SORA, then at x ^(k-1)the constraint condition that place rebuilds can be expressed as:

represent the motion-vector of kth-1 time circulation, herein, can by shown in formula (8) based on the constraint condition in the determinacy multidisciplinary design optimization that convex linearization approximation technique builds near MPP point:

In formula (12), be respectively the functional value of i-th probability constraints at MPP point place and sensitivity information.The present invention can adopt method of finite difference to obtain for the probability constraints with implied expression formula.

Step3: combine the most reliability failures point corresponding with stochastic parameter variable build new the deterministic design Optimized model thus, go to step one.

Advantage and effect: the design optimization problem that the present invention be directed to the large scaled complex product such as weaponry, nuclear power facility, boats and ships, electronic product and aerospace flight vehicle.The present invention is directed to the design optimization problem of complex product, from the viewpoint of computational accuracy and counting yield two, concurrent subspace optimization strategy, sequence optimisation and reliability assessment framework and many precision Approximate Model Method are carried out integrated.Establish one more rationally efficient complex product reliability design optimization method.Many precision approximate model is in order to avoid the insecure shortcoming of low, the low accuracy model result of calculation of high-precision model counting yield, utilize low accuracy model to be optimized the accuracy of calculating, the low accuracy model of high-precision model adjunct test, while precision is optimized in guarantee, improve counting yield.And the RBMDO of nested for tradition circulation has been decoupled into two modules that order performs by the sequence optimisation improved with reliability assessment framework, and ensure the convergence of optimization by convex linearization method.The method is more convenient for engineering staff to carry out the optimal design of different product, and the design for complex product provides rational foundation, improves designing quality and the design efficiency of product.

Accompanying drawing explanation

Fig. 1 is based on the multidisciplinary reliability design Optimizing Flow figure of CSSO and inter-trust domain many precision optimizings model.

Fig. 2 undercarriage impact damper subject coupling figure.

In figure, symbol description is as follows:

S in Fig. 1 _irepresent the motion-vector of i-th constraint condition; x ^krepresent the optimal design point of kth wheel circulation time, represent the most probable failure point that i-th probabilistic constraints is obtained by fail-safe analysis.

D in Fig. 2 _irepresent the deterministic design parameter; p _irepresent probabilistic design parameter; x _srepresent and share design variable; x _irepresent subject i independently design variable; z _irepresent the output of i-th subject; Y _ijrepresent that subject i is to the input of subject j.The implication of all the other parameters is in table 2.

Embodiment

See Fig. 1-2, below in conjunction with undercarriage impact damper, the present invention is further illustrated.It is typical MDO problem that the design of the undercarriage buffer of aircraft designs multiple subject.The subject related to has three: structure, calorifics, fluid mechanics.Minimum for optimization aim with the weight of the undercarriage buffer of aircraft, its constraint of constraint mainly in each subject and constraint of damping efficiency.Here consider undercarriage buffer design, processing, assembling in error and the multiple uncertainty such as material and atmospheric pressure.The method of inventing is adopted to carry out the multidisciplinary reliability design optimization of undercarriage impact damper.

Design starting condition: undercarriage bumper material adopts 30CrMnSiNi2A, and its damping efficiency needs to be greater than 65%.Other starting condition are in table 1.Optimized model is such as formula (13).

Table 1 undercarriage impact damper initial designs condition

s.t.g ₁＝S-500＜0

g ₂＝F _max/(M ₀+M _u)-2.5＜0

Pr{g ₆＝2.75r _p-l _co＜0}＞R ₁(3)

Pr{g ₇＝PV ^γ-6850＜0}＞R ₂(1)

Pr{g ₈＝6750-PV ^γ＜0}＞R ₂(2)

Pr{g ₉＝Δh/(2r _b)-0.5＜0}＞R ₃(1)

Pr{g ₁₀＝r _p/r _b-7＜0}＞R ₃(2)

R ₁＝R ₂＝R ₃＝Φ(β)＝0.9987

All parameters are explained and are distributed respectively as shown in table 2, table 3.The Optimized model of undercarriage impact damper has 10 constraints, in table 4

Table 2 parameter is explained

The distribution situation of table 3 parameter

Variable	Average	COV *	Span/mm	Distribution
					R c	μ Rc	0.1	[80,120]	Normal state
r c	μ rc	0.1	[50,100]	T distributes
					l co	μ co	0.05	[250,500]	Normal state
R p	μ Rp	0.1	[50,100]	Normal state
					r p	μ rp	0.1	[20,100]	T distributes

l p	μ lp	0.05	[900,1200]	Normal state
					r b	μ rb	0.1	[5,15]	Normal state
△h	μ △h	0.05	[5,10]	Normal state
					S	—	—	[400,550]	—
r bn	—	—	[4,8]	—
					σ b	1600	0.1	—	Normal state
P 0	2.4	0.1	—	F distributes

Note: * is the coefficient of variation

The constraint of the Optimized model of table 4 undercarriage impact damper

Expression formula	Constrained interpretation	Expression formula	Constrained interpretation
				g1	Draft gear travel retrains	g6	The overlapping minimum length constraint of cylinder piston
g2	The maximum overload coefficient of impact damper	g7	The adiabatic compression process constraints of buffer-empty air cavity
				g3	The damping efficiency constraint of impact damper	g8	The adiabatic compression process constraints of buffer-empty air cavity
g4	The maximum static strength constraint of cylinder	g9	Aperture coefficient of flow and constraint
				g5	The maximum static strength constraint of piston	g10	Aperture flow equation retrains

R ₁=R ₂=R ₃=Φ (β)=0.9987, Low confidence limit=3.Between the subject of undercarriage buffer optimization model, coupled relation as shown in Figure 2.The concrete implementation step using the present invention to carry out multidisciplinary design optimization to this impact damper is as follows:

Step one, set up corresponding high precision and low accuracy model respectively for the limit state function in example, here conveniently understand, we open as high-precision model at the second order Taylor Expansion at average point place by ultimate service state function, open as low accuracy model by the first order Taylor exhibition formula of limit state function at average point.With g ₁₀for example, its high precision agent model can be expressed as: low precision agent model is

The multiplication scaling function of step 2, use formula (1) ensures the convergence consistance of low precision approximate model and high-precision model.

Step 3, utilization formula (9) carry out multidisciplinary analysis, and by couple state function Simultaneous Equations, obtain state variable value, for impact damper example, multidisciplinary analysis equation is as follows.

Step 4, use formula (10) carry out the system sensitivity analysis based on global sensitivity equation, and this example is the multidisciplinary problem having three subsystems, and the change of its global sensitivity equation is as follows:

Obtain dy/dx _iafter, the probabilistic constraints in gear drive fail-safe analysis, its limit state function g is about i-th Random Design variable x _igradient information dg/dx _i, just can obtain by through type (10):

Step 5, subspace parallel optimization, for structure storage subsystem, only optimize the design parameter relevant with this subsystem in subsystem: S, σ _b, p ₀, r _p, R _c, r _c, R _p, the optimization of subsystem can be carried out as follows:

Circulation time, makes s for the first time ^p, optimal value S=461.3342, r of the design parameter of structure subsystem can be obtained _p=38.5416, R _c=90.5632, r _c=61.8725, R _p=55.8522

Step 6, system-level optimization, the individual parameter value that stator system optimization obtains, to the cross influence coefficient r of subspace _k ^p, t _k ^psystem-level coordination optimization is carried out with following formula.Obtain the cross influence coefficient value that can be applied to next round circulation.

Step 7, calculating trusted zones discriminant function value wherein P (x)=f (x)+λ _k∑ max (0, g _i(x)), to circulate first, obtain trusted zones ρ ₁=0.14.Then formula (4) is utilized to adjust inter-trust domain Δ ²=0.14 Δ ¹.Enter next round iteration.

Step 8, carry out fail-safe analysis based on advanced averaging method, search the most probable failure point (MPP).With limit state function g ₁₀=r _p/ r _b-7 be example first time cyclic search MPP point (r _p=20, r _b=15, remaining variables is all in average point place value)

Step 9, judge reliability, first time cyclic search MPP point (r _p=20, r _b=15, remaining variables is all in average point place value) g now ₁₀=1.33 > 0 meet fiduciary level requirement.

Step 10, according to formula (u _k+1-u _k+1)/u _k< ε judges convergence, is, then terminate; Otherwise go to step 11.

Step 11, renewal MPP point and mean data reconstruct deterministic optimization model.

Step1: the optimal design point x utilizing the deterministic design optimization to obtain ^kwith the most probable failure point that i-th probabilistic constraints is obtained by fail-safe analysis determine the motion-vector of this constraint condition first time optimizes (r in season _p=20, r _b=15, remaining variables all gets average); For first time circulation s ₁=(0,0,0 ,-0.24 ,-0.37,0,0,0,0,0,0,0).

Step2: build new constraint function based on convex linearization approximation technique:

Step3: combine the most reliability failures point corresponding with stochastic parameter variable build new the deterministic design Optimized model thus, go to step one.

For the example of impact damper, whole multidisciplinary reliability design optimizing process obtains convergence through 19 (k=19) iteration.The specifying information of the optimum results finally obtained and probability constraints and systematic analysis number of times is respectively as shown in following table 5, table 6.

Table 5 buffer design optimum results

Design variable	R c	r c	l co	R p	r p	Target function value
							Initial value	90.1527	59.2800	345.6227	55.4163	37.9152
CSSO-RBMDO	92.7552	63.5210	351.8313	57.6522	40.2402
							Design variable	l p	r b	△h	S	r bn
Initial value	1101.3307	7.5570	5.5527	459.7726	5.0143	143865.9775
							CSSO-RBMDO	1106.3049	7.6760	5.2502	471.9804	5.1600	147558.4603

Table 6 probability constraints and systematic analysis number of times

Claims (1)

1., based on a multidisciplinary reliability design optimization method for CSSO and many precision optimizings model, it is characterized in that: the method concrete steps are as follows:

Step one, low accuracy model is constructed for optimization problem, utilize the acquisitions such as response phase method;

Step 2, in order to ensure the convergence consistance of low precision approximate model and high-precision model, multiplication scaling function is used to carry out the low accuracy model of limit structure; At certain Optimum Points x _kplace, multiplication scaling function is expressed as:

$\mathrm{\&beta;}\left(x_k\right)=\frac{f_H\left(x_k\right)}{f_L\left(x_k\right)}---\left(1\right)$

F _h(x _k) represent kth time loop optimization point x _kthe analysis result of place's high-precision model, f _l(x _k) represent kth time loop optimization point x _klocate the analysis result of low accuracy model, at x _kneighbouring multiplication scaling function is defined as:

β'(x)＝β(x _k)+Δβ(x _k)(x-x _k)(2)

Wherein, Δ β (x _k) represent at Optimum Points x _kthe derivative of place's gradient function, be then expressed as in the result optimizing xth step proximate analysis in circulation: f _h(x)=β (x _k) f _l(x);

In order to ensure counting yield, propose the interval method of approximation of scaling function, scaling function value upgrades after several times circulation, and the cycle index at interval is

N _kit is right to represent the value rounded up, wherein, α is attenuation coefficient, and m is self-defining initial adjustment value, works as n _kwhen being less than 1, all n _p=1 (p > k); x _kfor the design parameter value that epicycle circulation obtains, when cycle index increases, upgrade interval more and more less, near convergence point, interval is approximately 1, namely more less until n close to convergence point interval _k=1 namely continuously every;

Step 3, multidisciplinary analysis is carried out for the integrated concurrent subspace optimization strategy of initialization Optimized model, have the Optimized model of two subsystems, y ₁and y ₂for state variable, by the multidisciplinary analysis of its simultaneous as shown in the formula

$\left\{\begin{array}{c}{y}_1=y_1(x,y_{21})\\ y_2=y_2(x,y_{12})\end{array}\right.---\left(3\right)$

Namely obtain the functional value of limit state function according to the epicycle iterative value solving state variable value and the design variable obtained, in addition, state variable value is also the basis of next step system sensitivity analysis;

Step 4, employing global sensitivity equation GSE carry out system sensitivity analysis, and have the multidisciplinary problem of two subsystems, its global sensitivity equation is shown below

$\left[\begin{array}{cc}I& -\frac{\&part;y_1}{\&part;y_2}\\ -\frac{\&part;y_2}{\&part;y_1}& I\end{array}\right]\left[\begin{array}{c}\frac{{\mathrm{dy}}_1}{{\mathrm{dx}}_i}\\ \frac{{\mathrm{dy}}_2}{{\mathrm{dx}}_i}\end{array}\right]=\left[\begin{array}{c}\frac{{\mathrm{dy}}_1}{{\mathrm{dx}}_i}\\ \frac{{\mathrm{dy}}_2}{{\mathrm{dx}}_i}\end{array}\right]---\left(4\right)$

Dy _i/ dx _jbe i-th state variable to design parameter x _jsensitivity value, limit state function g is to a jth design variable x _jsusceptibility information obtained by following formula:

$\frac{dg}{{\mathrm{dx}}_j}=\frac{\&part;g}{\&part;x_j}+\underset{n}{\overset{1}{\&Sigma;}}\frac{\&part;g}{\&part;y_i}\frac{{\mathrm{dy}}_i}{{\mathrm{dx}}_j}---\left(5\right)$

Wherein n represents the number of state variable;

Step 5, subspace parallel optimization, the task that the subspace of Concurrent Subspace method is optimized is under the condition that the change of system goal function value is very little, reduce the violation amount of subspace giant ties, is expressed as the optimization problem of kth sub spaces:

$\{\begin{array}{cc}\min & F\left(X_k\right)\\ s.t.& C^p\&le;C^{p0}\&lsqb;s^p(1-{r_k}^p)+(1-s^p){t_k}^p\&rsqb;\end{array}---\left(6\right)$

Wherein, F is system goal function, but only optimizes the design variable of this subspace, s ^p, r _k ^p, t _k ^pfor subspace cross influence coefficient, be constant here, C ^pfor the giant ties of p subspace;

Step 6, system-level coordination optimization, the optimization aim of each subspace is all system goal function, and the minimum F that system-level coordination optimization obtains just becomes the function of r and t; System-level coordination optimization problem is expressed as follows:

$\left\{\begin{array}{cc}\min & F({r_k}^p,{t_k}^p)\\ s.t.& \underset{k}{\&Sigma;}{r_k}^p=1;\underset{k}{\&Sigma;}{t_k}^p=0;\\ & 0\&le;{r_k}^p\&le;1\end{array}\right.---\left(7\right)$

Step 7, detection inter-trust domain, and adjust inter-trust domain size, concrete steps are:

Step1: calculate trusted zones discriminant function

${\mathrm{\&rho;}}_k=\frac{P_H\left(x\right)-P_H\left({x_k}^{*}\right)}{P_S\left(x\right)-P_S\left({x_k}^{*}\right)}---\left(8\right)$

Wherein P (x) is penalty, and it is defined as P (x)=f (x)+λ _k∑ max (0, g _i(x)), penalty factor

Step2: according to the value adjustment trusted zones of the trusted zones discriminant function calculated in step1;

$\left\{\begin{array}{c}{\mathrm{\&Delta;}}^{k+1}={\mathrm{\&rho;}}_k{\mathrm{\&Delta;}}^k,{\mathrm{\&rho;}}_k\&le;0.25\\ {\mathrm{\&Delta;}}^{k+1}={\mathrm{\&Delta;}}^k,0.25\&le;{\mathrm{\&rho;}}_k\&le;0.75\\ {\mathrm{\&Delta;}}^{k+1}=\frac{2}{{\mathrm{\&rho;}}_k}{\mathrm{\&Delta;}}^k,0.25\&le;{\mathrm{\&rho;}}_k\&le;0.75\end{array}\right.---\left(9\right)$

Step 8, carry out MPP search based on advanced averaging method and AMV; Advanced average point method steepest descent direction constantly more new standardized direction vector with alternative non-linear constrain, thus obtain convergence result more efficiently, the testing site of wherein kth+1 circulation is calculated as follows:

u _k+1＝β _t·e _k(10)

Wherein, represent the gradient direction vector of kth time circulation time function of state, U represents the design vector of normed space, β _trepresent fiduciary level requirement;

Step 9, judging reliability, according to obtaining the most probable failure point MPP, calculating g (u _k), g (u _k) < 0?, be that then this takes turns the discontented sufficient reliability requirement of design; Otherwise, meet reliability requirement;

Step 10, convergence judge, (u _k+1-u _k+1)/u _k< ε? then terminate; Otherwise go to step 11;

Step 11, reconstruct the deterministic design Optimized model;

Step1: the optimal design point x utilizing the deterministic design optimization to obtain ^kwith the most probable failure point that i-th probabilistic constraints is obtained by fail-safe analysis determine the motion-vector of this constraint condition

Step2: build new constraint function based on convex linearization approximation technique: suppose g _ifor i-th constraint condition of certain subject, in conjunction with the motion-vector built in the shift strategy in SORA, then at x ^(k-1)the constraint condition that place rebuilds is expressed as:

$g_i(x-s_i^{(k-1)}){|}_{x=x^{(k-1)}}=g_i(x-(x_M^{(k-1)}-x_{MPPi}^{(k-1)}\left)\right){|}_{x=x^{(k-1)}}=g_i\left(x_{MPPi}^{(k-1)}\right)---\left(11\right)$

represent the motion-vector of kth-1 circulation, based on the constraint condition in the determinacy multidisciplinary design optimization that convex linearization approximation technique builds near MPP point by shown in formula (8):

$\begin{array}{c}{g}_i(x-s_i^{(k-1)}){|}_{x\&ap;x^{(k-1)}}=g_i(x-s_i^{(k-1)}){|}_{x\&ap;x^{(k-1)}}=g_i\left(x_{MPPi}^{(k-1)}\right){|}_{x_{MPPi}=x_{MPPi}^{(k-1)}}\\ =g_i\left(x_{MPPi}^{(k-1)}\right)+\underset{+}{\mathrm{\&Sigma;}}\frac{\&part;g_i}{\&part;x_i}{|}_{x_{MPPi}=x_{MPPi}^{(k-1)}}(x_i-x_i^{(k-1)})+\underset{-}{\mathrm{\&Sigma;}}\frac{\&part;g_i}{\&part;x_i}{|}_{x_{MPPi}=x_{MPPi}^{(k-1)}}\frac{x_i^{(k-1)}}{x_i}(x_i-x_i^{(k-1)})\end{array}---\left(12\right)$

In formula (12), with be respectively the functional value of i-th probability constraints at MPP point place and sensitivity information; Method of finite difference is adopted to obtain for the probability constraints with implied expression formula;

Step3: combine the most reliability failures point corresponding with stochastic parameter variable build new the deterministic design Optimized model thus, go to step one.