CN103324798A - Random model updating method based on interval response surface model - Google Patents

Abstract

The invention relates to random model updating method based on an interval response surface model. The method is characterized by including the steps of firstly, building a second-order polynomial response surface model without cross terms according to experiment design and regression analysis; secondly, using a square completing method to convert a polynomial response surface expression into perfect square; thirdly, substituting interval parameters into the response surface expression to allow the definite response surface model to be changed into the interval response surface model; fourthly, performing interval calculation on the interval response surface model to obtain predicted structural response intervals, and combining the predicted structural response intervals with actual response intervals to build a target function; fifthly, building a optimization inversion problem to identify interval distribution of parameters. By the method, the expansion problem of interval calculation is avoided, fast calculation of structural response intervals is considered, finite element analyzing calculation and sensitivity matrix building during (interval) random model updating are avoided, a large amount of calculation time and cost is saved, and ill-conditioned optimization is avoided as much as possible.

Description

Probabilistic model modification method based on interval response surface model

Technical field

The present invention relates to the Finite Element Model Updating field, particularly a kind of probabilistic model modification method based on interval response surface model.

Background technology

Finite Element Model Updating has been widely used in the engineering fields such as Aero-Space, building, machinery.Yet, traditional model modification method ^[1-2]Be based upon on the deterministic hypothesis of the parameter basis.Uncertain factor in the engineering structure but is ubiquity and inevitable, thereby causes the uncertainty of structural parameters such as material discreteness, measuring error, processing and manufacturing error etc.If still adopt deterministic theory and method that finite element model is revised, must cause the unreliable of correction result, have greater difference with actual conditions, so considered that the probabilistic model modification method of parameter uncertainty begins to be paid close attention to recent years.

Current, probabilistic method, blur method and Novel Interval Methods are the quantifications of structural uncertainty and propagate three kinds of main method analyzing ^[3]Wherein, bar structure parameter that the most frequently used is is considered as stochastic variable to be come problem is carried out modeling and analysis, namely adopts probabilistic method.This moment, the joint probability density distribution function of parameter should be known.But when not having abundant data to verify the correctness of above-mentioned distribution function, the precision of probabilistic method just is difficult to be guaranteed ^[4]And the employing blur method ^[5]When describing uncertainty, need the membership function of parameter.But under many circumstances, determine membership function even more more difficult than probability distribution function, namely the researchist often has to choose corresponding subordinate function with very large subjectivity, so that the reliability of analysis result is also doubtful.Therefore, in order to reflect the interference of objective reality, minimizing subjective factor, in the sufficient not situation of structured testing information, the uncertain parameters in the engineering problem is considered as the interval variable of bounded, adopts Novel Interval Methods ^[6-10]It is very necessary revising finite element model.

At present, the inverse problem based on interval analysis can adopt three kinds of methods: interval algorithm ^[6], vertex scheme ^[7]And global optimization method ^[4,8]Wherein, interval algorithm has defined a series of computing (interval adds, subtracts, multiplication and division method), can be used for the interval of computation structure response.Yet interval arithmetic can't be considered the correlativity between the variable, be easy to when directly adopting interval arithmetic to find the solution structural response to cause respond interval serious expansion, and the degree of this expansion often is difficult to quantize, and makes it be difficult to be applied to engineering reality.The application of vertex scheme need to be satisfied three major premise conditions: the total quality matrix of (1) structure and stiffness matrix are the linear function of corrected parameter; (2) above-mentioned matrix can be decomposed into nonnegative definite mega-structure mass and stiffness matrix; (3) the output response is the system features value.For the engineering structure of reality, above-mentioned condition often is difficult to satisfy simultaneously; The proper vector that comprises simultaneously system when output response, or the total quality of structure and stiffness matrix be when being the nonlinear function of corrected parameter, and vertex scheme easily lost efficacy.Document [9] is theoretical based on global optimization, uses the Kriging model to substitute finite element model, adopts deterministic Model Updating Technique, optimizes to obtain and every group of structural parameters that experimental data is corresponding, and then obtains the interval of structural parameters.Said process is actual to be repeatedly deterministic models makeover process, has certain limitation: the optimized algorithm that (1) the method need to have ability of searching optimum, and this type of algorithm more complicated, optimum results easily is absorbed in local optimum, and the structural parameters interval that obtains thus is not necessarily accurate; When the statistics that (2) obtains when test was more, the method needed a large amount of double countings, will expend a large amount of assessing the cost; (3) for the engineering structure of reality, the researchist often more pays close attention to maximum, the minimum value of structural response, i.e. the interval upper bound and the lower bound of response, and therefore in the situation that only know that response is interval, the method lost efficacy possibly.

Summary of the invention

The purpose of this invention is to provide a kind of probabilistic model modification method based on interval response surface model, the present invention has not only avoided the dilatancy problem of interval arithmetic, taken into account simultaneously the quick calculating in structural response interval, so that the model makeover process needn't carry out the structure of finite element analysis computation and sensitivity matrix, thereby saved a large amount of computing times and expended, also avoided as far as possible the appearance of ill optimization problem.Do not having under the prerequisite of a large amount of statisticss, by the interval model makeover process, realized the inverse process by the interval identified parameters of structural response interval, quantized the uncertainty of structural parameters, the finite element model that makes correction is the quiet dynamic response of predict more accurately.

The present invention realizes in the following ways: a kind of probabilistic model modification method based on interval response surface model is characterized in that may further comprise the steps:

Step S01: make up the second order polynomial response surface model that does not comprise cross term based on test design and regretional analysis;

Step S02: adopt method of completing the square that the polynomial response surface expression formula is converted into the perfect square form;

Step S03: with interval parameter substitution response surface expression formula, so that the determinacy response surface model becomes interval response surface model;

Step S04: carry out the structural response interval that interval arithmetic obtains predicting at interval response surface model, and set up objective function in conjunction with actual measurement response interval;

Step S05: make up the Optimization inversion problem and come the interval of identification parameter to distribute.

In an embodiment of the present invention, the implementation of described step S01 is as follows:

Step S011: seek the required design point of match response surface based on the Central Composite design method in the test design method, be used for setting up the quadratic polynomial response surface that does not comprise cross term:

$y={\mathrm{\β}}_0+\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{\mathrm{\β}}_i{x}_i+\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{\mathrm{\β}}_{\mathrm{ii}}{x}_i^2+\mathrm{\ϵ}---\left(1\right)$

In the formula: the response of y representative structure, x _iRepresent i parameter, m is number of parameters, β ₀, β _i, β _IiBe respectively constant term, once and the undetermined coefficient of quadratic term, ε is error of fitting;

Step S012: after determining design point, obtain the corresponding response of each design point by numerical evaluation, to obtain a sample;

Step S013: based on all samples, can estimate polynomial undetermined coefficient by the least square regression analysis; Response surface model after the match is expressed as:

$y={\widehat{\mathrm{\β}}}_0+\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{\widehat{\mathrm{\β}}}_i{x}_i+\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{\widehat{\mathrm{\β}}}_{\mathrm{ii}}{x}_i^2---\left(2\right)$

In the formula

For constant term, once, the regression coefficient of quadratic term.

In an embodiment of the present invention, the implementation of described step S02 and S03 is as follows:

After the quadratic polynomial response surface model that obtains as the formula (2), adopt method of completing the square, response surface model is changed into the perfect square form, design parameter is only occurred once in function expression, the form after the conversion is as follows:

$y={\widehat{\mathrm{\β}}}_0+\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{\widehat{\mathrm{\β}}}_i{x}_i+\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{\widehat{\mathrm{\β}}}_{\mathrm{ii}}{x}_i^2$ （3）

$=\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{\widehat{\mathrm{\β}}}_{\mathrm{ii}}{(x_i+\frac{{\widehat{\mathrm{\β}}}_i}{2{\widehat{\mathrm{\β}}}_{\mathrm{ii}}})}^2+{\widehat{\mathrm{\β}}}_0-\underset{i=1}{\overset{m}{\mathrm{\Σ}}}\frac{{\widehat{\mathrm{\β}}}_i^2}{4{\widehat{\mathrm{\β}}}_{\mathrm{ii}}}$

Then with parameter x _iAdopt interval number x ^IExpression namely obtains interval response surface model:

$y^I=\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{\widehat{\mathrm{\β}}}_{\mathrm{ii}}{(x_i^I+\frac{{\widehat{\mathrm{\β}}}_i}{2{\widehat{\mathrm{\β}}}_{\mathrm{ii}}})}^2+{\widehat{\mathrm{\β}}}_0-\underset{i=1}{\overset{m}{\mathrm{\Σ}}}\frac{{\widehat{\mathrm{\β}}}_i^2}{4{\widehat{\mathrm{\β}}}_{\mathrm{ii}}}---\left(4\right).$

In an embodiment of the present invention, the implementation of described step S04 and S05 is as follows:

The interval model makeover process can be converted into following optimization indirect problem:

$\min \left[R\left(x^I\right)\right],R\left(x^I\right)=\underset{i=1}{\overset{P}{\mathrm{\Σ}}}{\left(\frac{\underset{\‾}{y_{\mathrm{ia}}}-\underset{\‾}{y_{\mathrm{ie}}}}{\underset{\‾}{y_{\mathrm{ie}}}}\right)}^2+\underset{i=1}{\overset{P}{\mathrm{\Σ}}}{\left(\frac{\stackrel{\‾}{y_{\mathrm{ia}}}-\stackrel{\‾}{y_{\mathrm{ie}}}}{\stackrel{\‾}{y_{\mathrm{ie}}}}\right)}^2---\left(5\right)$

st.VLB≤x≤VUB

X in the formula ^IBe the structure interval parameter;

With Be respectively structure i rank are theoretical and actual measurement responds lower bound, the upper bound; P is the number of structural response; St. represent constraint condition; VLB, VUB are the design space scopes; R is residual error, here as objective function; Then by seeking the minimum value of residual error R, the interval of finding the solution parameter distributes.

There is inevitably uncertainty in the material of engineering structure, geometric parameter, boundary condition and the condition of contact, considers in the model makeover process that therefore the uncertainty of above-mentioned parameter is very necessary, can make model modification method more realistic.For the engineering structure of complexity, because experimentation cost is very high, so it is unpractical often with the probability density function of description scheme response and parameter to obtain abundant statistics by a large amount of tests.Adopt this moment the probabilistic model modification method of Based on Probability statistics may have the larger error of calculation, or even infeasible.Therefore, in the situation that measurement data is less, has preferably advantage based on the probabilistic model modification method of interval analysis.The invention has the advantages that (1) ignored the interval extension problem that the interval response surface model of cross term (mutual effect between parameter) can avoid interval arithmetic to cause, thereby guarantee the computational accuracy between parameter region; (2) makeover process need not to make up sensitivity matrix, has greatly simplified optimization problem and has avoided the appearance of ill convergence process; (3) makeover process directly carries out Optimized Iterative at interval response surface model, need not to carry out complicated numerical analysis, has greatly improved the efficient of interval model correction; (4) go for parameter uncertainty (interval distribution range) situation that degree is higher; (5) can be applicable to simultaneously the more and less situation of test figure; (6) the model makeover process has reduced the interference of subjective factor, and acquired results is more true and reliable.

Description of drawings

Fig. 1 is the inventive method schematic flow sheet.

Fig. 2 is embodiment of the invention interval model correction process flow diagram.

Embodiment

The present invention will be further described below in conjunction with drawings and Examples.

As shown in Figure 1, a kind of probabilistic model modification method based on interval response surface model is characterized in that may further comprise the steps:

Step S01: make up the second order polynomial response surface model that does not comprise cross term based on test design and regretional analysis;

Step S02: adopt method of completing the square that the polynomial response surface expression formula is converted into the perfect square form;

Step S03: with interval parameter substitution response surface expression formula, so that the determinacy response surface model becomes interval response surface model;

Step S04: carry out the structural response interval that interval arithmetic obtains predicting at interval response surface model, and set up objective function in conjunction with actual measurement response interval;

Step S05: make up the Optimization inversion problem and come the interval of identification parameter to distribute.

In interval arithmetic, parametric variable is processed as independent variable, easily causes the interval extension of result of calculation.From the functional relation of structural response and parameter, if parameter repeatedly occurs in function expression, interval algorithm causes responding interval expansion possibly.Therefore, dilatancy for fear of interval arithmetic, the present invention proposes the concept of " interval response surface model ", namely adopt the quadratic polynomial response surface do not comprise cross term to come relation between approximate representation structural response and the uncertain parameters, by method of completing the square response surface model is converted into the perfect square form again, and parameter represented with interval number, thereby obtain interval response surface model.Because structural parameters only occur once, therefore, will can not cause interval extension when adopting interval arithmetic to find the solution the response interval in interval response surface model.The present invention has not only avoided the dilatancy problem of interval arithmetic, taken into account simultaneously the quick calculating in structural response interval, so that the model makeover process needn't carry out finite element analysis computation and make up sensitivity matrix, save a large amount of computing times and expended, and avoided as far as possible the appearance of ill optimization problem.Do not having under the prerequisite of a large amount of statisticss, by the interval model makeover process, realized the inverse process by the interval identified parameters of structural response interval, quantized the uncertainty of structural parameters, the finite element model that makes correction is the quiet dynamic response of predict more accurately.

Concrete, see also Fig. 2, Fig. 2 is the present embodiment interval model correction process flow diagram, the technical scheme that the present embodiment adopts mainly comprises the foundation of quadratic polynomial response surface, the structure of interval response model, the interval arithmetic of response, and based on the probabilistic model makeover process of interval response surface model.Concrete steps are as follows:

1, sets up the quadratic polynomial response surface

Response surface model ^[11]Be in fact a kind of polynomial expression mathematics expression formula, can be used for setting up the relation between system's input (parameter) and the output (response), this expression formula obtains by the match design sample.The Central Composite design method that the present invention is based in the test design method is sought the required design point of match response surface (design points), has set up the quadratic polynomial response surface that does not comprise cross term (mutual effect):

$y={\mathrm{\β}}_0+\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{\mathrm{\β}}_i{x}_i+\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{\mathrm{\β}}_{\mathrm{ii}}{x}_i^2+\mathrm{\ϵ}---\left(1\right)$

In the formula: the response of y representative structure, x _iRepresent i parameter, m is number of parameters, β ₀, β _i, β _IiBe respectively constant term, once and the undetermined coefficient of quadratic term, ε is error of fitting;

After determining design point, can obtain the corresponding response of each design point by numerical evaluation, to obtain a sample.Then based on all samples, can estimate polynomial undetermined coefficient by the least square regression analysis.Response surface model after the match can be expressed as:

$y={\widehat{\mathrm{\β}}}_0+\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{\widehat{\mathrm{\β}}}_i{x}_i+\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{\widehat{\mathrm{\β}}}_{\mathrm{ii}}{x}_i^2---\left(2\right)$

In the formula

For constant term, once, the regression coefficient of quadratic term.

2, the structure of interval response surface model

After the quadratic polynomial response surface model that obtains as the formula (2), can adopt method of completing the square, response surface model is changed into the perfect square form, design parameter is only occurred once, to avoid as far as possible the interval extension problem of interval arithmetic process in function expression.Form after the conversion is as follows:

$y={\widehat{\mathrm{\β}}}_0+\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{\widehat{\mathrm{\β}}}_i{x}_i+\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{\widehat{\mathrm{\β}}}_{\mathrm{ii}}{x}_i^2$ （3）

$=\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{\widehat{\mathrm{\β}}}_{\mathrm{ii}}{(x_i+\frac{{\widehat{\mathrm{\β}}}_i}{2{\widehat{\mathrm{\β}}}_{\mathrm{ii}}})}^2+{\widehat{\mathrm{\β}}}_0-\underset{i=1}{\overset{m}{\mathrm{\Σ}}}\frac{{\widehat{\mathrm{\β}}}_i^2}{4{\widehat{\mathrm{\β}}}_{\mathrm{ii}}}$

Then with parameter x _iAdopt interval number x ^IExpression namely obtains interval response surface model:

$y^I=\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{\widehat{\mathrm{\β}}}_{\mathrm{ii}}{(x_i^I+\frac{{\widehat{\mathrm{\β}}}_i}{2{\widehat{\mathrm{\β}}}_{\mathrm{ii}}})}^2+{\widehat{\mathrm{\β}}}_0-\underset{i=1}{\overset{m}{\mathrm{\Σ}}}\frac{{\widehat{\mathrm{\β}}}_i^2}{4{\widehat{\mathrm{\β}}}_{\mathrm{ii}}}---\left(4\right)$

3, the interval arithmetic of structural response

After obtaining interval response surface model, just can directly carry out interval arithmetic based on the interval arithmetic rule, the interval that obtains structural response distributes.

4, interval model makeover process

At last, the interval model makeover process can be converted into following optimization indirect problem:

$\min \left[R\left(x^I\right)\right],R\left(x^I\right)=\underset{i=1}{\overset{P}{\mathrm{\Σ}}}{\left(\frac{\underset{\‾}{y_{\mathrm{ia}}}-\underset{\‾}{y_{\mathrm{ie}}}}{\underset{\‾}{y_{\mathrm{ie}}}}\right)}^2+\underset{i=1}{\overset{P}{\mathrm{\Σ}}}{\left(\frac{\stackrel{\‾}{y_{\mathrm{ia}}}-\stackrel{\‾}{y_{\mathrm{ie}}}}{\stackrel{\‾}{y_{\mathrm{ie}}}}\right)}^2---\left(5\right)$

st.VLB≤x≤VUB

X in the formula ^IBe the structure interval parameter;

With

Be respectively structure i rank are theoretical and actual measurement responds lower bound, the upper bound; P is the number of structural response; St. represent constraint condition; VLB, VUB are the design space scopes; R is residual error, is objective function here.

Then by seeking the minimum value of residual error R, the interval of finding the solution parameter distributes.

The above only is preferred embodiment of the present invention, and all equalizations of doing according to the present patent application claim change and modify, and all should belong to covering scope of the present invention.

List of references

[1]M.I.Friswell,J.E.Mottershead,Finite Element Model Updating in Structural Dynamics,Kluwer Academic Press,Dordrecht,1995.

[2] Li Hui, the fourth birch. Progress In Model Updating For Structural Dynamics [J]. " Proceedings of Mechanics ", 2005,35 (2): 170 180.

[3]D.Moens,D.Vandepitte,A survey of non‐probabilistic uncertainty treatment in finite element analysis,Computer Methods in Applied Mechanics and Engineering194(12–16)(2005)1527–1555.

[4]I.Elishakoff.Essay on uncertainties in elastic and viscoelastic structures:from A.M.Freudenthal’s criticisms to modern convex modeling[J].Computers and Structures,1995,56(6):871‐895.

[5]Li Chen,S.S.Rao.Fuzzy finite‐element approach for the vibration analysis of imprecisely‐defined systems.Finite Elements in Analysis and Design27(1997)69‐83.

[6]R.Moore,R.Kearfott,M.Cloud,Introduction to Interval Analysis,Society for Industrial and Applied Mathematics,Philadelphia,2009.

[7]Qiu Zhi‐ping,Wang Xiao‐jun,M.I Friswell.Eigenvalue bounds of structures with uncertain‐but‐bounded parameters[J].Journal of Sound and Vibration,2005,282(1):297‐312.

[8]E.Hansen and G.W.Walster,Global Optimization Using Interval Analysis.Marcel Dekker,New York,2003.

[9]H.H.Khodaparast,J.E.Mottershead,K.J.Badcock.Interval model updating with irreducible uncertainty using the Kriging predictor,Mech.Syst.Sig.Process.25(2011),1204–1226.

[10]Y.S Erdogan,P.G Bakir,Inverse propagation of uncertainties in finite element model updating through use of fuzzy arithmetic,Engineering applications of artificial intelligence 26(2013)357‐367.

D.C.Montgomery,Design and analysis of experiments,3rd ed,New York:John Wiley&Sons,Inc;2006。

Claims (4)

1. probabilistic model modification method based on interval response surface model is characterized in that may further comprise the steps:

Step S01: make up the second order polynomial response surface model that does not comprise cross term based on test design and regretional analysis;

Step S02: adopt method of completing the square that the polynomial response surface expression formula is converted into the perfect square form;

Step S03: with interval parameter substitution response surface expression formula, so that the determinacy response surface model becomes interval response surface model;

Step S04: carry out the structural response interval that interval arithmetic obtains predicting at interval response surface model, and set up objective function in conjunction with actual measurement response interval;

Step S05: make up the Optimization inversion problem and come the interval of identification parameter to distribute.

2. the probabilistic model modification method based on interval response surface model according to claim 1, it is characterized in that: the implementation of described step S01 is as follows:

Step S011: seek the required design point of match response surface based on the Central Composite design method in the test design method, be used for setting up the quadratic polynomial response surface that does not comprise cross term:

$y={\mathrm{\β}}_0+\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{\mathrm{\β}}_i{x}_i+\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{\mathrm{\β}}_{\mathrm{ii}}{x}_i^2+\mathrm{\ϵ}---\left(1\right)$

In the formula: the response of y representative structure, x _iRepresent i parameter, m is number of parameters, β ₀, β _i, β _IiBe respectively constant term, once and the undetermined coefficient of quadratic term, ε is error of fitting;

Step S012: after determining design point, obtain the corresponding response of each design point by numerical evaluation, to obtain a sample;

Step S013: based on all samples, can estimate polynomial undetermined coefficient by the least square regression analysis; Response surface model after the match is expressed as:

$y={\widehat{\mathrm{\β}}}_0+\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{\widehat{\mathrm{\β}}}_i{x}_i+\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{\widehat{\mathrm{\β}}}_{\mathrm{ii}}{x}_i^2---\left(2\right)$

In the formula

For constant term, once, the regression coefficient of quadratic term.

3. the probabilistic model modification method based on interval response surface model according to claim 2, it is characterized in that: the implementation of described step S02 and S03 is as follows:

After the quadratic polynomial response surface model that obtains as the formula (2), adopt method of completing the square, response surface model is changed into the perfect square form, design parameter is only occurred once in function expression, the form after the conversion is as follows:

$y={\widehat{\mathrm{\β}}}_0+\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{\widehat{\mathrm{\β}}}_i{x}_i+\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{\widehat{\mathrm{\β}}}_{\mathrm{ii}}{x}_i^2$ （3）

$=\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{\widehat{\mathrm{\β}}}_{\mathrm{ii}}{(x_i+\frac{{\widehat{\mathrm{\β}}}_i}{2{\widehat{\mathrm{\β}}}_{\mathrm{ii}}})}^2+{\widehat{\mathrm{\β}}}_0-\underset{i=1}{\overset{m}{\mathrm{\Σ}}}\frac{{\widehat{\mathrm{\β}}}_i^2}{4{\widehat{\mathrm{\β}}}_{\mathrm{ii}}}$

Then with parameter x _iAdopt interval number x ^IExpression namely obtains interval response surface model:

$y^I=\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{\widehat{\mathrm{\β}}}_{\mathrm{ii}}{(x_i^I+\frac{{\widehat{\mathrm{\β}}}_i}{2{\widehat{\mathrm{\β}}}_{\mathrm{ii}}})}^2+{\widehat{\mathrm{\β}}}_0-\underset{i=1}{\overset{m}{\mathrm{\Σ}}}\frac{{\widehat{\mathrm{\β}}}_i^2}{4{\widehat{\mathrm{\β}}}_{\mathrm{ii}}}---\left(4\right).$

4. the probabilistic model modification method based on interval response surface model according to claim 2, it is characterized in that: the implementation of described step S04 and S05 is as follows:

The interval model makeover process can be converted into following optimization indirect problem:

$\min \left[R\left(x^I\right)\right],R\left(x^I\right)=\underset{i=1}{\overset{P}{\mathrm{\Σ}}}{\left(\frac{\underset{\‾}{y_{\mathrm{ia}}}-\underset{\‾}{y_{\mathrm{ie}}}}{\underset{\‾}{y_{\mathrm{ie}}}}\right)}^2+\underset{i=1}{\overset{P}{\mathrm{\Σ}}}{\left(\frac{\stackrel{\‾}{y_{\mathrm{ia}}}-\stackrel{\‾}{y_{\mathrm{ie}}}}{\stackrel{\‾}{y_{\mathrm{ie}}}}\right)}^2---\left(5\right)$

st.VLB≤x≤VUB

X in the formula ^IBe the structure interval parameter;

With

Be respectively structure i rank are theoretical and actual measurement responds lower bound, the upper bound; P is the number of structural response; St. represent constraint condition; VLB, VUB are the design space scopes; R is residual error, here as objective function; Then by seeking the minimum value of residual error R, the interval of finding the solution parameter distributes.

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