CN103902785A - Structure finite element model correcting method based on multi-element uncertainty - Google Patents

Abstract

A structure finite element model correcting method based on multi-element uncertainty comprises the following steps: (1) building an initialized parameterization equivalent finite element model in finite element software; (2) screening out significance parameters; (3) obtaining sample points, and constructing an incomplete variable high-order response surface model; (4) judging validity of the response surface model, if the validity of the response surface model meets the requirement, executing the next step, and if the validity of the response surface model does not meet the requirement, executing the step (3) again; (5) building a rapid random sampling analysis model with the combination of a high-order response surface and the Monte Carlo method, and conducting statistics on a mean value and a covariance matrix of simulation output responses; (6) conducting statistics on a mean value and a covariance matrix of test output results; (7) constructing a weighting objective function of the mean values and covariances of tests and simulation; (8) reversely estimating a mean value and a covariance matrix of input parameters; (9) judging whether the mean value and the covariance matrix of the input parameters meet correction accuracy or not, if yes, stopping iteration, and if not, executing the step (8) again. According to the structure finite element model correcting method, the calculated amount of the iteration is reduced, the application range is wide, and optimization of a large-scale parameter range is achieved.

Description

A kind of based on polynary probabilistic structural finite element model modification method

Technical field:

The present invention relates to one based on polynary probabilistic structural finite element model modification method, belong to mechanical technical field.

Background technology:

In the past few decades, finite element method becomes the important method of Structural Dynamics performance prediction.Due to the simplification of model and approximate, inevitably there is the uncertainty (as how much, contact connect parameter and boundary condition and load parameter etc.) of parameter, therefore between test and finite element model result of calculation, certainly exist error.For guaranteeing finite element model precision of analysis, classic method is to utilize deterministic test figure to revise model, dwindles deviation, improves realistic model computational accuracy.But test aspect is due to noise of test specimen foozle and build-up tolerance, experimental test etc., also there is uncertainty in itself.In reality, uncertain factor is ubiquity and inevitably, considers that uncertainty derives from realistic model and test two aspects, uncertain transmission between input parameter and output response characteristic (as model frequency etc.).Only describing and have probabilistic structural system by the deterministic FEM (finite element) calculation of single, is obviously inaccurate.Therefore, consider that emulation and the probabilistic model correction of test have become current study hotspot simultaneously.

At present, model correction constantly develops as dynamic (dynamical) inverse problem, but most Model Updating Techniques belong to deterministic method.In uncertain field, Monte Carlo method is the probability analysis method that is widely used for characterising parameter uncertain problem conventionally, and cardinal principle is grab sample from the probability density function of input parameter, and repeatedly moving model obtains the probability statistical distribution of output response.Directly use Monte Carlo simulation can obtain reliable uncertain result, but need a large amount of calculating samples; Meanwhile, numerous for amendable initial uncertain parameters, need to filter out highly sensitive parameter.Therefore, counting yield problem has hindered its practical application, the labyrinth model correction of especially polynary uncertain parameters.Existing document proposes a kind of perturbation method and revises probabilistic model parameter, but need to calculate Second Order Sensitivity, and the calculated amount causing is larger.Perturbation method is to utilize the method for taylor series expansion, consider counting yield, conventionally only retain 1,2 rank Taylor expansion items, one order matrix and Second Order Sensitivity matrix are wherein comprised, but may there is morbid state in sensitivity matrix, and initial value and the parameter distribution range of the degree of accuracy of perturbation method correction result to parameter is very responsive, revises precision and need further to be improved.Another document is expanded in classical Gradient Iteration method, the correction of statistical parameter is divided into two steps and implements, and the first step is adjusted mean parameter, and second step is adjusted parametric covariance battle array.Owing to implementing correction step by step, the loaded down with trivial details and correction result of computation process lacks synchronism.Therefore when, probabilistic model correction need to be considered precision, take into account counting yield.

Summary of the invention:

1. object:

The object of the invention is to overcome the deficiencies in the prior art, provide a kind of based on polynary probabilistic structural finite element model modification method.The method is test design, the check analysis of F value, and incomplete variable higher order polynomial response surface model, Monte Carlo method, weighted mean and covariance matrix objective function that the optimized algorithm of hybrid particle swarm and simplex and test are relevant with emulation carry out integrated.Realize and reduced the calculated amount that participates in corrected parameter number and reduce iterative process; Making polynary uncertain parameters need mutual independence to be generalized to parameter can be correlated with, and has expanded usable range; Solve the optimizing of the uncertain parameters of large scale scope, synchronously revised average and covariance matrix.Finally reach and improve the correction efficiency of uncertainty structure finite element model and the object of precision of prediction.

2, technical scheme: the technical solution adopted for the present invention to solve the technical problems is:

One of the present invention is based on polynary probabilistic structural finite element model modification method, and it comprises the following steps:

Step 1: set up initial parametrization Equivalent finite element model in finite element software, the finite element model before revising;

Step 2: screen conspicuousness parameter according to test design and the check analysis of F value;

Step 3: conspicuousness parameter is re-started to test design and obtain sample point, construct incomplete variable high-order response surface model;

Step 4: the validity of response surface model judges, carries out next step if meet, if do not meet and return to step 3;

Step 5: set up the quick random sampling analytical model that high-order response surface combines with Monte Carlo method, realize uncertain forward transmission and analyze, count average and the covariance matrix of simulation data response;

Step 6: repeatedly replica test analysis, counts average and the covariance matrix of testing Output rusults;

Step 7: the correlation analysis of emulation/test, the average that structure test is relevant with emulation and the weighting objective function of covariance;

Step 8: uncertain back transfer analysis, oppositely estimates average and the covariance matrix of input parameter;

Step 9: judge whether to meet and revise precision, if meet, stop iteration; Otherwise repeat to return step 8;

Wherein, in " polynary uncertainty " described in exercise question refers to structural finite element model, there is numerous uncertain parameters, be called polynary.

Wherein, can adopt any business finite element analysis software to realize at " finite element software " described in step 1, as Nastran, ANSYS etc.

Wherein, foundation and method " the setting up initial parametrization Equivalent finite element model " described in step 1 are: according to the size of practical structures physical model and syndeton condition, utilize beam element, plate shell unit, solid element etc. to carry out finite element modeling.To occurring the position of error, the connection parameter of for example structure is carried out equivalent process, utilizes the computerese of finite element software to carry out parametrization processing simultaneously, sets up parameterized finite element model.

Wherein, in the method for " test design " described in step 2 be: test design is a kind of scientific approach of research multiple-factor and response variable relation, such as total divisor method, partial factors method, Central Composite method, Latin method, the Latin method etc. of optimizing.Utilizing test design method to carry out sample point sampling, is the basis that builds response surface model.

Wherein, in the method for " check analysis of F value " described in step 2 be: suppose the input parameter A of model to carry out significance analysis, the F value of A is:

$F_A=\frac{S_A/f_A}{S_e/f_e}~F(f_A,f_e)---\left(1\right)$

S in formula (1) _aand S _ebe respectively the sum of square of deviations of parameter A and error e; f _aand f _ebe respectively the degree of freedom of parameter A and error e.If F _a>=F _1-θ(f _a, f _e), parameter is remarkable on the impact of response; If F _a<F _1-θ(f _a, f _e), parameter is not remarkable on the impact of response; Wherein θ is the level of signifiance (as θ=0.05).The method can filter out conspicuousness parameter for revising from polynary uncertain parameters, has reduced corrected Calculation amount.

Wherein, in the method for " the incomplete variable high-order response surface model " described in step 3 be: the power of nonlinear relationship between according to input and output, can adjust response surface model.If between parameter non-linear a little less than, reduce the number of higher order term; If nonlinear relationship is stronger, increase the number of higher order term, make response surface model can describe more accurately the relation of input parameter and output response.The method has been avoided calling finite element model in each iterative process and has been calculated, and has reduced iterative computation amount and has guaranteed precision.

$\hat{y}={\mathrm{\&beta;}}_0+\underset{i=1}{\overset{k}{\mathrm{\&Sigma;}}}{\mathrm{\&beta;}}_i{x}_i+\underset{i=1}{\overset{k}{\mathrm{\&Sigma;}}}\underset{j=1(i\&NotEqual;j)}{\overset{k}{\mathrm{\&Sigma;}}}{\mathrm{\&beta;}}_{\mathrm{ij}}{x}_i{x}_j+\underset{i=1}{\overset{k}{\mathrm{\&Sigma;}}}{\mathrm{\&beta;}}_{\mathrm{ii}}{x}_i^2+\underset{i=1}{\overset{k}{\mathrm{\&Sigma;}}}{\mathrm{\&beta;}}_{3i}{x}_i^3+\underset{i=1}{\overset{k}{\mathrm{\&Sigma;}}}{\mathrm{\&beta;}}_{4i}{x}_i^4...+\underset{i=1}{\overset{k}{\mathrm{\&Sigma;}}}{\mathrm{\&beta;}}_{\mathrm{mi}}{x}_i^n---\left(2\right)$

In formula (2)

for response surface approximate function; x _i, x _jfor input parameter; K is the number of input parameter;

β ₀, β _i, β _ij, β _ii, β _3i, β _4i, β _nifor polynomial expression undetermined coefficient.

Wherein, in the method for " validity judges " described in step 4 be: response surface model can be used root-mean-square error (RMSE) relative value and coefficient of determination R to sample data degree of fitting ²two standard tests.

$\mathrm{RMSE}=\frac{1}{h\stackrel{\&OverBar;}{y}}\sqrt{\underset{i=1}{\overset{h}{\mathrm{\&Sigma;}}}{(y_i-{\hat{y}}_i)}^2}---\left(3\right)$

${\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="HDA0000490447810000022.png,HDA0000490447810000031.png"\; state="\{\{state\}\}">R</figure-callout>}}^2=1-\frac{\underset{i=1}{\overset{h}{\mathrm{\&Sigma;}}}{(y_i-{\hat{y}}_i)}^2}{\underset{i=1}{\overset{h}{\mathrm{\&Sigma;}}}{(y_i-\stackrel{\&OverBar;}{y})}^2}---\left(4\right)$

In formula (3) and (4): h is sample point number;

for response surface calculated value; y _ifor FEM (finite element) calculation value;

for the mean value of result of finite element.RMSE → 0 represents that response surface error is little; R ²→ 1 show response surface and master mould similarity high.

Wherein, in the method for " uncertain forward transmission is analyzed " described in step 5 be: the quick random sampling analysis that probabilistic input parameter process response surface model and Monte Carlo combine, obtains probabilistic output response results.

Wherein, in the method for " average and the covariance matrix " described in step 6 be: if uncertain parameters x _isample n time, sample average is:

$\stackrel{\&OverBar;}{x}=\frac{1}{n}\underset{j=1}{\overset{n}{\mathrm{\&Sigma;}}}{x}_{\mathrm{ij}},(i=1,...,m)---\left(5\right)$

The number that in formula (5), m is uncertain parameters, the average of all parameters can be expressed as the form of vector [M].

Two variable x _iand x _ksample covariance be:

$\mathrm{Cov}(x_i,x_k)=\frac{1}{n-1}\underset{j=1}{\overset{n}{\mathrm{\&Sigma;}}}(x_{\mathrm{ij}}-{\stackrel{\&OverBar;}{x}}_i)(x_{\mathrm{kj}}-{\stackrel{\&OverBar;}{x}}_k)---\left(6\right)$

Covariance matrix [C] can be expressed as:

$\left[C\right]=\left[\begin{array}{cccc}\mathrm{Var}\left(x_1\right)& \mathrm{Cov}(x_1,x_2)& ...& \mathrm{Cov}(x_1,x_m)\\ \mathrm{Cov}(x_2,x_1)& \mathrm{Var}\left(x_2\right)& ...& \mathrm{Cov}(x_2,x_m)\\ ...& ...& ...& ...\\ \mathrm{Cov}(x_m,x_1)& \mathrm{Cov}(x_m,x_2)& ...& \mathrm{Var}\left(x_m\right)\end{array}\right]---\left(7\right)$

Var (x in formula (7) _i)=Cov (x _i, x _i)=(σ (x _i)) ², σ is standard deviation; If be separate between variable, the standard deviation that diagonal line of covariance matrix is each variable square, other correlativity is 0, as the formula (8).

$\left[C\right]=\left[\begin{array}{cccc}{\left(\mathrm{\&sigma;}\left(x_1\right)\right)}^2& 0& ...& 0\\ 0& {\left(\mathrm{\&sigma;}\left(x_2\right)\right)}^2& ...& 0\\ ...& ...& ...& ...\\ 0& 0& ...& {\left(\mathrm{\&sigma;}\left(x_m\right)\right)}^2\end{array}\right]---\left(8\right)$

Average in other step and the computing method of covariance matrix are all with identical herein.

Wherein, in the method for " correlation analysis of emulation/test " described in step 7 be:

The average objective function that emulation is relevant with test findings is:

$J\left(\mathrm{\&theta;}\right)={\left|\right|z_m-z\left(\mathrm{\&theta;}\right)\left|\right|}_2^2---\left(9\right)$

Z in formula (9) _mfor the sample average vector of test findings, Z (θ) is the sample average of corresponding simulation data result, the mean vector that θ is input parameter.

The covariance matrix objective function that emulation is relevant with test findings is:

$J(\mathrm{Cov}\left(\mathrm{\&theta;}\right),\mathrm{Cov}\left(\mathrm{\&theta;}\right))={\left|\right|\mathrm{Cov}(z_m,z_m)-\mathrm{Cov}(z\left(\mathrm{\&theta;}\right),z\left(\mathrm{\&theta;}\right))\left|\right|}_F^2---\left(10\right)$

Cov (Z in formula (10) _m, Z _m) be the sample covariance matrix of test findings, Cov (Z (θ), Z (θ)) is the sample covariance matrix of simulation data result.

By weighted method, average and two objective functions of covariance matrix are transformed into the single-goal function shown in a formula (12), synchronous backward is estimated average and the covariance of uncertain input parameter, i.e. uncertain back transfer analysis.The problem that the method has avoided substep to revise.

$I(z_m,z\left(\mathrm{\&theta;}\right))=w_{\mathrm{mean}}{\left|\right|z_m-z\left(\mathrm{\&theta;}\right)\left|\right|}_2^2+w_{\mathrm{cov}}{\left|\right|\mathrm{Cov}(z_m,z_m)-\mathrm{Cov}(z\left(\mathrm{\&theta;}\right),z\left(\mathrm{\&theta;}\right))\left|\right|}_F^2---\left(11\right)$

I(z _m,z(θ))=w _meanJ(θ)+w _covJ(Cov(θ),Cov(θ))→min （12）

W in formula (11) and (12) _covand w _meanfor weighting matrix.

Wherein, method in " the probabilistic back transfer analysis " described in step 8 is: take the emulation shown in formula (12) with the relevant weighting function of test as optimization aim, the hybrid optimization algorithm that adopts population to combine with simplex is combined quick Monte Carlo random sampling and is analyzed to the probabilistic input parameter value of reverse sync correction.The method has been avoided the calculating of perturbation method medium sensitivity matrix and has been applicable to the optimizing of large scale range parameter, has improved the precision of revising.

Wherein, comprise aspect iterations and weighting objective function minimum value two at " end condition " described in step 9.If iterations has reached maximum times or objective function minimum value is no more than set-point, stop iteration; Otherwise continuation iterative process.

3, advantage and effect:

Beneficial effect compared with prior art of the present invention is:

(1) polynomial response surface model of use variable-order, can be convenient to express more accurately the relation between input parameter and output response according to actual conditions adjustment; The check analysis of F value filters out conspicuousness parameter from multiple parameters, has reduced the number of parameters that participates in correction; Monte Carlo random sampling based on response surface, compared with direct Monte Carlo Analysis, has greatly reduced the calculated amount of each iteration simultaneously; (2) represent the uncertainty of parameter with average and covariance matrix, can will between parameter, need separate being generalized to can be separate, therefore the scope of application be wider; (3) average of structure output response and the weighting objective function of covariance matrix, utilize hybrid particle swarm and simplex optimization algorithm to carry out uncertain back transfer analysis, avoided the correction of substep model.Also avoided the sensitivity matrix in perturbation method to calculate, realized the optimizing of large scale parameter area simultaneously.

Accompanying drawing explanation

Fig. 1. polynary probabilistic structural finite element model correction flow process;

Fig. 2. aircaft configuration finite element model;

Fig. 3. the root-mean-square error relative value RMSE of response surface model validity check;

Fig. 4. the coefficient of determination R of response surface model validity check ²;

Fig. 5. the mean value error contrast of the output modalities frequency before and after the correction of aircraft finite element model;

Fig. 6. the covariance matrix error contrast of the output modalities frequency before and after the correction of aircraft finite element model;

Embodiment

Below in conjunction with drawings and Examples, the present invention is elaborated

Explanation the present invention as an example of the dynamic analysis of the finite element model of aircaft configuration shown in Fig. 2 example.The aluminum beam that model aircraft is mainly rectangle by cross section forms, and comprises fuselage, wing, vertical tail, tailplane, both sides winglet composition.

See Fig. 1, one of the present invention is based on polynary probabilistic structural finite element model modification method, and the method concrete steps are as follows:

Step 1: in Nastran finite element software, aircaft configuration is simplified to processing, set up initial parameter finite element model (model before revising).Geometry and the material parameter of the parts such as initial finite element model middle machine body, wing are relatively accurate, but the model simplification of structure junction is more, error is larger.Therefore the Equivalent Modeling parameter of the concentrated junction of error all needs further to revise according to modal test data.

The line parameter processing of going forward side by side 12 separate parameters as input parameter in modeling process, is respectively: fuselage/wing linking springs torsional rigidity, fuselage bending stiffness z, fuselage bending stiffness y, vertical tail bending stiffness x, vertical tail torsional rigidity, wing bending stiffness x, wing bending stiffness z, wing horizontal stiffness element length, both sides winglet bending stiffness y, vertical tail rigid element length, wing are to fuselage vertical off setting distance, wing twist rigidity; (wherein x, y, z represents direction, and in this embodiment, between input and output parameter, is separate.）

Step 2: input parameter is determined to the probability distribution of samples points with optimizing latin square experiment method for designing, then carry out finite element analysis, calculate the i.e. front 6 rank model frequency values of output response at sample point place.Carry out subsequently the check analysis of F value, suppose the input parameter A of model aircraft to carry out significance analysis, the F value of A is:

$F_A=\frac{S_A/f_A}{S_e/f_e}~F(f_A,f_e)$

If F _a>=F _1-θ(f _a, f _e), parameter is remarkable on the impact of output modalities frequency; If F _a<F _1-θ(f _a, f _e), parameter is not remarkable on the impact of output modalities frequency.If level of signifiance θ=0.05 can filter out 7 to the significant input parameter of output modalities frequency influence according to F value result of calculation from 12 parameters, as shown in table 1.

7 conspicuousness input parameters that table 1 filters out

Label	Parameter name
			1	Fuselage/wing junction torsional rigidity
2	Fuselage bending stiffness z
		3	Vertical tail bending stiffness x
4	Vertical tail torsional rigidity
		5	Wing bending stiffness x
6	Wing is to fuselage vertical off setting distance
		7	Wing twist rigidity

Step 3: 7 conspicuousness input parameters that screen are re-started to optimization latin square experiment design and obtain sample point, obtain input parameter vector X and front 6 rank output modalities frequency vector Y.Build response surface model with incomplete 4 rank polynomial expressions and express the relation between input parameter and front 6 rank output modalities frequencies, the expression-form of its response surface model is:

$\hat{y}={\mathrm{\&beta;}}_0+\underset{i=1}{\overset{7}{\mathrm{\&Sigma;}}}{\mathrm{\&beta;}}_i{x}_i+\underset{i=1}{\overset{7}{\mathrm{\&Sigma;}}}\underset{j=1(i\&NotEqual;j)}{\overset{7}{\mathrm{\&Sigma;}}}{\mathrm{\&beta;}}_{\mathrm{ij}}{x}_i{x}_j+\underset{i=1}{\overset{7}{\mathrm{\&Sigma;}}}{\mathrm{\&beta;}}_{\mathrm{ii}}{\mathrm{<figure-callout\; id="2"\; label="x\; i"\; filenames="HDA0000490447810000022.png,HDA0000490447810000031.png"\; state="\{\{state\}\}">x</figure-callout>}}_i^{}+\underset{i=1}{\overset{7}{\mathrm{\&Sigma;}}}{\mathrm{\&beta;}}_{3i}{x}_i^3+\underset{i=1}{\overset{7}{\mathrm{\&Sigma;}}}{\mathrm{\&beta;}}_{4i}{x}_{\mathrm{<figure-callout\; id="4"\; label="i"\; filenames="HDA0000490447810000022.png,HDA0000490447810000031.png"\; state="\{\{state\}\}">i</figure-callout>}}^4$

If ε is error of fitting vector, β is polynomial expression undetermined coefficient vector, and the pass between input parameter and output modalities frequency vector is:

Y=Xβ+ε

According to least square method, solve polynomial coefficient vector β, i.e. [β ₀, β _i, β _ij, β _ii, β _3i, β _4i]

ε ^Tε=（Y-Xβ） ^T（Y-Xβ）

$\frac{\&PartialD;\left({\mathrm{\&epsiv;}}^T\mathrm{\&epsiv;}\right)}{\&PartialD;\mathrm{\&beta;}}=-2X^T(Y-\mathrm{X\&beta;})=0$

β=(X ^TX) ^-1X ^TY

Step 4: the validity of response surface model judges.Choose wherein 10 sample points, with root-mean-square error (RMSE) relative value and coefficient of determination R ²two standards are checked the precision of the model aircraft response surface of structure;

$\mathrm{RMSE}=\frac{1}{10\stackrel{\&OverBar;}{y}}\sqrt{\underset{i=1}{\overset{10}{\mathrm{\&Sigma;}}}{(y_i-{\hat{y}}_i)}^2};{\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="HDA0000490447810000022.png,HDA0000490447810000031.png"\; state="\{\{state\}\}">R</figure-callout>}}^2=1-\frac{\underset{i=1}{\overset{10}{\mathrm{\&Sigma;}}}{(y_i-{\hat{y}}_i)}^2}{\underset{i=1}{\overset{10}{\mathrm{\&Sigma;}}}{(y_i-\stackrel{\&OverBar;}{y})}^2}$

Respectively as shown in Figure 3 and Figure 4, the RMSE value of front 6 rank model frequencies is in close proximity to 0 to the response surface validity check result of model aircraft, and R ²value levels off to 1, shows that response surface model and finite element model similarity are high, and the response surface model of structure is effectively, can start next step analytical calculation.

Step 5: accurate incomplete 4 rank response surface models are combined with Monte Carlo sampling, set up the sampling analysis of quick Monte Carlo, and Output rusults is carried out to statistical study.Be input to the uncertain forward transmission of output, obtain average and the covariance matrix of simulation analysis output modalities frequency.

Step 6: build test platform, repeatedly repeated Modal Experimental Analysis discrete output modalities frequency is added up, obtains testing average and the covariance matrix of output modalities frequency.

Step 7: emulation/test-analysis correlation, the average that structure test is relevant with emulation and the weighting objective function of covariance matrix;

Step 7.1: realize realistic model consistent with the correlativity of pilot system.Make the average Z (θ) of realistic model output modalities frequency approach as far as possible test objective value Z on the one hand _m: i.e. J (θ) → min;

J(θ)=(z _m-z(θ)) ^T(z _m-z(θ))

$J\left(\mathrm{\&theta;}\right)={\left|\right|z_m-z\left(\mathrm{\&theta;}\right)\left|\right|}_2^2$

Step 7.2: the fluctuation Cov (Z (θ), Z (θ)) that also should make on the other hand the uncertain caused output modalities frequency of input parameter as far as possible with the fluctuation Cov (Z of trial value _m, Z _m) consistent: i.e. J (Cov (θ), Cov (θ)) → min.According to F matrix norm method:

$J(\mathrm{Cov}\left(\mathrm{\&theta;}\right),\mathrm{Cov}\left(\mathrm{\&theta;}\right))={\left|\right|\mathrm{Cov}(z_m,z_m)-\mathrm{Cov}(z\left(\mathrm{\&theta;}\right),z\left(\mathrm{\&theta;}\right))\left|\right|}_{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="HDA0000490447810000022.png,HDA0000490447810000031.png"\; state="\{\{state\}\}">F</figure-callout>}}^2$

Step 7.3: by weighted method, average and two objective functions of covariance matrix are transformed into a single-goal function, carry out correction model thereby needn't implement step-by-step optimization, reduced the complicacy of makeover process.

$I(z_m,z\left(\mathrm{\&theta;}\right))=w_{\mathrm{mean}}{\left|\right|z_m-z\left(\mathrm{\&theta;}\right)\left|\right|}_2^2+w_{\mathrm{cov}}{\left|\right|\mathrm{Cov}(z_m,z_m)-\mathrm{Cov}(z\left(\mathrm{\&theta;}\right),z\left(\mathrm{\&theta;}\right))\left|\right|}_{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="HDA0000490447810000022.png,HDA0000490447810000031.png"\; state="\{\{state\}\}">F</figure-callout>}}^2$

W herein _covand w _meanall be set to 1.

Step 8: uncertain back transfer analysis, by the weighting the minimization of object function of average and covariance matrix, i.e. I (z _m, z (θ)) and=w _meanj (θ)+w _covj (Cov (θ), Cov (θ)) → min, utilizes the hybrid optimization algorithm that population and simplex combine to carry out optimizing, has realized average and the covariance matrix of reverse sync correction input parameter, has avoided step-by-step optimization.Also avoided Taylor expansion medium sensitivity matrix computations and the Gradient Iteration of perturbation method, realized the accurate optimizing of large scale parameter area, increased convergence precision simultaneously.

Step 9: judge whether to meet and revise precision, if meet, stop iteration; Otherwise repeat to return step 8; This routine stopping criterion for iteration is that maximum iteration time 100 times or objective function precision are no more than 0.0001.

In this example, carry out the finite element model correction of aircaft configuration, through the average [M of the front 6 rank model frequencies of revised aircraft finite element model ₆] and covariance matrix [C ₆] represent respectively:

$\left[M_6\right]=\left[\begin{array}{c}{\mathrm{\&mu;}}_1\\ {\mathrm{\&mu;}}_2\\ ...\\ {\mathrm{\&mu;}}_6\end{array}\right],\left[C_6\right]=\left[\begin{array}{cccc}{{\mathrm{\&sigma;}}_1}^2& 0& ...& 0\\ 0& {{\mathrm{\&sigma;}}_2}^2& ...& 0\\ ...& ...& ...& ...\\ 0& 0& ...& {{\mathrm{\&sigma;}}_6}^2\end{array}\right]$

After revising, the average of model aircraft output modalities frequency and the predicated error of covariance matrix have had very large reducing compared with initial finite element model (model before revising), before wherein revising, contrast as shown in Figure 5 with revised mean value error, the error of revising front and revised covariance matrix contrasts as shown in Figure 6.Therefore, modification method of the present invention has well solved structural finite element model precision of prediction problem under polynary condition of uncertainty.

Claims (10)

1. based on a polynary probabilistic structural finite element model modification method, it is characterized in that: it comprises the following steps:

Step 1: set up initial parametrization Equivalent finite element model in finite element software, the finite element model before revising;

Step 2: screen conspicuousness parameter according to test design and the check analysis of F value;

Step 3: conspicuousness parameter is re-started to test design and obtain sample point, construct incomplete variable high-order response surface model;

Step 4: the validity of response surface model judges, carries out next step if meet, if do not meet and return to step 3;

Step 5: set up the quick random sampling analytical model that high-order response surface combines with Monte Carlo method, realize uncertain forward transmission and analyze, count average and the covariance matrix of simulation data response;

Step 6: repeatedly replica test analysis, counts average and the covariance matrix of testing Output rusults;

Step 7: the correlation analysis of emulation/test, the average that structure test is relevant with emulation and the weighting objective function of covariance;

Step 8: uncertain back transfer analysis, oppositely estimates average and the covariance matrix of input parameter;

Step 9: judge whether to meet and revise precision, if meet, stop iteration; Otherwise repeat to return step 8.

2. one according to claim 1, based on polynary probabilistic structural finite element model modification method, is characterized in that: at " finite element software " described in step 1, be to adopt any business finite element analysis software to realize; Foundation and the method for described " setting up initial parametrization Equivalent finite element model " are: according to the size of practical structures physical model and syndeton condition, utilize beam element, plate shell unit, solid element to carry out finite element modeling; To occurring the position of error, carry out equivalent process as the connection parameter of structure, utilize the computerese of finite element software to carry out parametrization processing simultaneously, set up parameterized finite element model.

3. one according to claim 1 is based on polynary probabilistic structural finite element model modification method, it is characterized in that: the method in " test design " described in step 2 is: test design is a kind of scientific approach of research multiple-factor and response variable relation, as total divisor method, partial factors method, Central Composite method, Latin method, the Latin method of optimizing; Utilizing test design method to carry out sample point sampling, is the basis that builds response surface model; The method of described " check analysis of F value " is: suppose the input parameter A of model to carry out significance analysis, the F value of A is:

$F_A=\frac{S_A/f_A}{S_e/f_e}~F(f_A,f_e)---\left(1\right)$

S in formula (1) _aand S _ebe respectively the sum of square of deviations of parameter A and error e; f _aand f _ebe respectively the degree of freedom of parameter A and error e; If F _a>=F _1-θ(f _a, f _e), parameter is remarkable on the impact of response; If F _a<F _1-θ(f _a, f _e), parameter is not remarkable on the impact of response; Wherein θ is the level of signifiance, as θ=0.05; The method can filter out conspicuousness parameter for revising from polynary uncertain parameters, has reduced corrected Calculation amount.

4. one according to claim 1 is based on polynary probabilistic structural finite element model modification method, it is characterized in that: the method at " the incomplete variable high-order response surface model " described in step 3 is: the power of nonlinear relationship between according to input and output, can adjust response surface model; If between parameter non-linear a little less than, reduce the number of higher order term; If nonlinear relationship is stronger, increase the number of higher order term, make response surface model can describe more accurately the relation of input parameter and output response; The method has been avoided calling finite element model in each iterative process and has been calculated, and has reduced iterative computation amount and has guaranteed precision;

$\hat{y}={\mathrm{\&beta;}}_0+\underset{i=1}{\overset{k}{\mathrm{\&Sigma;}}}{\mathrm{\&beta;}}_i{x}_i+\underset{i=1}{\overset{k}{\mathrm{\&Sigma;}}}\underset{j=1(i\&NotEqual;j)}{\overset{k}{\mathrm{\&Sigma;}}}{\mathrm{\&beta;}}_{\mathrm{ij}}{x}_i{x}_j+\underset{i=1}{\overset{k}{\mathrm{\&Sigma;}}}{\mathrm{\&beta;}}_{\mathrm{ii}}{x}_i^2+\underset{i=1}{\overset{k}{\mathrm{\&Sigma;}}}{\mathrm{\&beta;}}_{3i}{x}_i^3+\underset{i=1}{\overset{k}{\mathrm{\&Sigma;}}}{\mathrm{\&beta;}}_{4i}{x}_i^4...+\underset{i=1}{\overset{k}{\mathrm{\&Sigma;}}}{\mathrm{\&beta;}}_{\mathrm{mi}}{x}_i^n---\left(2\right)$

In formula (2)

for response surface approximate function; x _i, x _jfor input parameter; K is the number of input parameter;

β ₀, β _i, β _ij, β _ii, β _3i, β _4i, β _nifor polynomial expression undetermined coefficient.

5. one according to claim 1, based on polynary probabilistic structural finite element model modification method, is characterized in that: the method at " validity judges " described in step 4 is: response surface model is RMSE relative value and coefficient of determination R to sample data degree of fitting root-mean-square error ²two standard tests;

$\mathrm{RMSE}=\frac{1}{h\stackrel{\&OverBar;}{y}}\sqrt{\underset{i=1}{\overset{h}{\mathrm{\&Sigma;}}}{(y_i-{\hat{y}}_i)}^2}---\left(3\right)$

$R^2=1-\frac{\underset{i=1}{\overset{h}{\mathrm{\&Sigma;}}}{(y_i-{\hat{y}}_i)}^2}{\underset{i=1}{\overset{h}{\mathrm{\&Sigma;}}}{(y_i-\stackrel{\&OverBar;}{y})}^2}---\left(4\right)$

In formula (3) and (4): h is sample point number;

for response surface calculated value; y _ifor FEM (finite element) calculation value;

for the mean value of result of finite element; RMSE → 0 represents that response surface error is little; R ²→ 1 show response surface and master mould similarity high.

6. one according to claim 1 is based on polynary probabilistic structural finite element model modification method, it is characterized in that: the method " uncertain forward transmission is analyzed " described in step 5 is: the quick random sampling analysis that probabilistic input parameter process response surface model and Monte Carlo combine, obtains probabilistic output response results.

7. one according to claim 1, based on polynary probabilistic structural finite element model modification method, is characterized in that: the method at " average and the covariance matrix " described in step 6 is: if uncertain parameters x _isample n time, sample average is:

$\stackrel{\&OverBar;}{x}=\frac{1}{n}\underset{j=1}{\overset{n}{\mathrm{\&Sigma;}}}{x}_{\mathrm{ij}},(i=1,...,m)---\left(5\right)$

The number that in formula (5), m is uncertain parameters, the average of all parameters is expressed as the form of vector [M];

Two variable x _iand x _ksample covariance be:

$\mathrm{Cov}(x_i,x_k)=\frac{1}{n-1}\underset{j=1}{\overset{n}{\mathrm{\&Sigma;}}}(x_{\mathrm{ij}}-{\stackrel{\&OverBar;}{x}}_i)(x_{\mathrm{kj}}-{\stackrel{\&OverBar;}{x}}_k)---\left(6\right)$

Covariance matrix [C] is expressed as:

$\left[C\right]=\left[\begin{array}{cccc}\mathrm{Var}\left(x_1\right)& \mathrm{Cov}(x_1,x_2)& ...& \mathrm{Cov}(x_1,x_m)\\ \mathrm{Cov}(x_2,x_1)& \mathrm{Var}\left(x_2\right)& ...& \mathrm{Cov}(x_2,x_m)\\ ...& ...& ...& ...\\ \mathrm{Cov}(x_m,x_1)& \mathrm{Cov}(x_m,x_2)& ...& \mathrm{Var}\left(x_m\right)\end{array}\right]---\left(7\right)$

Var (x in formula (7) _i)=Cov (x _i, x _i)=(σ (x _i)) ², σ is standard deviation; If be separate between variable, the standard deviation that diagonal line of covariance matrix is each variable square, other correlativity is 0, as the formula (8):

$\left[C\right]=\left[\begin{array}{cccc}{\left(\mathrm{\&sigma;}\left(x_1\right)\right)}^2& 0& ...& 0\\ 0& {\left(\mathrm{\&sigma;}\left(x_2\right)\right)}^2& ...& 0\\ ...& ...& ...& ...\\ 0& 0& ...& {\left(\mathrm{\&sigma;}\left(x_m\right)\right)}^2\end{array}\right]---\left(8\right)$

Average in other step and the computing method of covariance matrix are all with identical herein.

8. one according to claim 1, based on polynary probabilistic structural finite element model modification method, is characterized in that: the method in " correlation analysis of emulation/test " described in step 7 is:

The average objective function that emulation is relevant with test findings is:

$J\left(\mathrm{\&theta;}\right)={\left|\right|z_m-z\left(\mathrm{\&theta;}\right)\left|\right|}_2^2---\left(9\right)$

Z in formula (9) _mfor the sample average vector of test findings, Z (θ) is the sample average of corresponding simulation data result, the mean vector that θ is input parameter;

The covariance matrix objective function that emulation is relevant with test findings is:

$J(\mathrm{Cov}\left(\mathrm{\&theta;}\right),\mathrm{Cov}\left(\mathrm{\&theta;}\right))={\left|\right|\mathrm{Cov}(z_m,z_m)-\mathrm{Cov}(z\left(\mathrm{\&theta;}\right),z\left(\mathrm{\&theta;}\right))\left|\right|}_F^2---\left(10\right)$

Cov (Z in formula (10) _m, Z _m) be the sample covariance matrix of test findings, Cov (Z (θ), Z (θ)) is the sample covariance matrix of simulation data result;

By weighted method, average and two objective functions of covariance matrix are transformed into the single-goal function shown in a following formula (12), synchronous backward is estimated average and the covariance of uncertain input parameter, i.e. uncertain back transfer analysis; The problem that the method has avoided substep to revise;

$I(z_m,z\left(\mathrm{\&theta;}\right))=w_{\mathrm{mean}}{\left|\right|z_m-z\left(\mathrm{\&theta;}\right)\left|\right|}_2^2+w_{\mathrm{cov}}{\left|\right|\mathrm{Cov}(z_m,z_m)-\mathrm{Cov}(z\left(\mathrm{\&theta;}\right),z\left(\mathrm{\&theta;}\right))\left|\right|}_F^2---\left(11\right)$

I(z _m,z(θ))=w _meanJ(θ)+w _covJ(Cov(θ),Cov(θ))→min （12）

W in formula (11) and (12) _covand w _meanfor weighting matrix.

9. one according to claim 1 is based on polynary probabilistic structural finite element model modification method, it is characterized in that: the method in " the probabilistic back transfer analysis " described in step 8 is: take the emulation shown in formula (12) with the relevant weighting function of test as optimization aim, the hybrid optimization algorithm that adopts population to combine with simplex is combined quick Monte Carlo random sampling and is analyzed to the probabilistic input parameter value of reverse sync correction; The method has been avoided the calculating of perturbation method medium sensitivity matrix and has been applicable to the optimizing of large scale range parameter, has improved the precision of revising.

10. one according to claim 1 is based on polynary probabilistic structural finite element model modification method, it is characterized in that: at " end condition " described in step 9, comprise iterations and weighting objective function minimum value two aspects, if iterations has reached maximum times or objective function minimum value is no more than set-point, stop iteration; Otherwise continuation iterative process.

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