CN103455689B - Based on the interval model modification method of Taylor series expansion - Google Patents
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Abstract
The present invention relates to a kind of interval model modification method based on Taylor series expansion, first based on Taylor series expansion, set up the interval intermediate value of the interval intermediate value of structural response, radius and structural parameters, the function corresponding relation of radius, as the theoretical foundation of interval model correction, then interval model correction is carried out as follows: 1, according to the interval intermediate value of structure actual measurement response, adopt deterministic models correction technique, the interval intermediate value of correcting principle parameter; 2, in order to simplify calculating, the partial derivative of method of difference computation structure parameter at revised interval intermediate value place is adopted, to obtain the approximate expression of the function corresponding relation of interval radius; 3, according to the interval radius of structure actual measurement response, deterministic models correction technique is adopted, in the enterprising row iteration optimization in basis of the approximate expression that step 2 obtains, the interval radius of correcting principle parameter.The method theoretical foundation is clear and definite, and optimizing process is simple, easy, high to the correction precision of interval parameter.
Description
Technical field
The present invention relates to Finite Element Model Updating field, particularly a kind of interval model modification method based on Taylor series expansion.
Background technology
Finite Element Model Updating has been widely used in the engineering fields such as Aero-Space, building, machinery.But traditional model modification method is based upon on the deterministic hypothesis basis of parameter.Uncertain factor in engineering structure is but ubiquity and inevitably, such as material discrete, measuring error, processing and manufacturing error etc. thus cause the uncertainty of structural parameters.If still adopt deterministic Theories and methods to revise finite element model, the unreliable of correction result must be caused, have greater difference with actual conditions, because contemplated that the stochastic response of parameter uncertainty obtains the concern of numerous researchist.
Current, probabilistic method, blur method and Novel Interval Methods are the quantifications of structural uncertainty and propagate the three kinds of main method analyzed.Wherein, the most frequently used is structural parameters are considered as stochastic variable carry out modeling and analysis to problem, namely adopts probabilistic method.Now 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 is just difficult to be guaranteed.And when adopting blur method to describe uncertainty, need the membership function of parameter.But under many circumstances, determine that membership function is even more difficult than probability distribution function, namely researchist often has to choose corresponding subordinate function with very large subjectivity, makes the reliability of such analysis result also doubtful.Therefore, in order to reflect objective reality, reduce the interference of subjective factor, under structured testing information not fully situation, the uncertain parameters in engineering problem is considered as the interval variable of bounded, it is very necessary for adopting Novel Interval Methods to revise finite element model.
At present, based on the probabilistic model correction of interval analysis mainly in conjunction with three kinds of methods: interval algorithm, vertex scheme and global optimization method.Wherein, interval algorithm defines a series of computing (interval adds, subtract, multiplication and division method), can be used for the interval of computation structure response.But interval arithmetic cannot consider the correlativity between variable, be easy to cause responding interval severe dilation when directly adopting interval arithmetic to solve structural response, and the degree of this expansion is often difficult to quantize, and makes it be difficult to be applied to engineering reality.The application demand fulfillment three major premise conditions of vertex scheme: 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) response is exported for system features value.For the engineering structure of reality, above-mentioned condition is often difficult to meet simultaneously, and simultaneously when exporting respond packet containing the proper vector of system, or when the total quality of structure and stiffness matrix are the nonlinear function of corrected parameter, vertex scheme easily lost efficacy.Meanwhile, although global optimization approach has ability of searching optimum, in conjunction with the algorithm after interval analysis often more complicated, optimum results still may be absorbed in local optimum, thus cause predict parameter region between not necessarily accurate.If simultaneously the parameter that relates to of interval correction problem is more, then algorithm needs a large amount of iterative computation, assessing the cost at substantial.In addition, in probability or non-probabilistic framework, solve the general more complicated of corresponding Modifying model problem, if Solve problems in deterministic framework can be tried one's best, then simplification correction problem, raising counting yield are of great benefit to, are therefore necessary to study this.
Summary of the invention
The object of the present invention is to provide a kind of interval model modification method based on Taylor series expansion, the method theoretical foundation is clear and definite, and optimizing process is simple, easy, and the correction precision of interval parameter is high.
The technical solution used in the present invention is: a kind of interval model modification method based on Taylor series expansion, first based on Taylor series expansion, set up the interval intermediate value of the interval intermediate value of structural response, radius and structural parameters, the function corresponding relation of radius, as the theoretical foundation of interval model correction, then carry out interval model correction as follows:
Step 1: according to the interval intermediate value of structure actual measurement response, adopt deterministic models correction technique, the interval intermediate value of correcting principle parameter;
Step 2: in order to simplify calculating, adopts the partial derivative of method of difference computation structure parameter at revised interval intermediate value place, with the approximate expression of the function corresponding relation of the interval radius of the interval radius and structural parameters that obtain structural response;
Step 3: according to the interval radius of structure actual measurement response, adopt deterministic models correction technique, in the enterprising row iteration optimization in basis of the approximate expression that step 2 obtains, the interval radius of correcting principle parameter.
Further, set up the interval intermediate value of the interval intermediate value of structural response, radius and structural parameters, the function corresponding relation of radius, carry out as follows:
The range format of note structural parameters x is
,
be called structure interval parameter, wherein,
, n represents the dimension of structure interval parameter, and subscript I represents interval number; Interval of definition number
, then
by the interval upper bound
, lower bound
or interval intermediate value
, radius
can be expressed as:
(1)
1) by the interval intermediate value x of structural response y at structural parameters
cnear carry out Taylor series expansion, now do not consider interval radius, be reduced to deterministic parameter by parameter x, obtain:
(2)
When the degree of uncertainty of parameter x is less, formula (2) is blocked, only retention constant item and once item, can obtain:
(3)
In order to simplify calculating, utilize the partial derivative of finite difference method structural parameters, that is:
(4)
In formula
structural parameters
small perturbation;
2) in order to further consider interval radius, x is adopted complete range format x
irepresent, then the interval type of structural response is:
(5)
For symmetric interval
perseverance has:
(6)
Therefore formula (5) is expressed as further:
(7)
Obtain the interval upper bound of structural response thus
, lower bound
and interval intermediate value
, radius
be respectively:
(8)
(9)
(10)
(11)
Formula (10) establishes the interval intermediate value of structural response and the function corresponding relation of the interval intermediate value of structural parameters, and formula (11) establishes the function corresponding relation of the interval radius of structural response and the interval radius of structural parameters; Therefore, follow-uply just deterministic models correction technique can be adopted, iteration optimization on the basis of formula (10) and (11), with this interval intermediate value of correcting principle parameter and radius respectively; During the interval intermediate value of wherein correcting principle parameter, structural parameters can be reduced to deterministic parameter correction; And during the radius of corrected parameter interval, in order to simplify calculating, adopt the partial derivative of method of difference computation structure parameter at revised interval intermediate value place, to obtain the approximate expression of formula (11), then in this approximate expression enterprising row iteration optimization in basis, the interval radius of correcting principle parameter.
Further, first according to the interval intermediate value of said structure response and the function corresponding relation of interval intermediate value of structural parameters, be deterministic optimization inverse problem by interval intermediate value correction problem reduction:
According to the interval intermediate value of structure actual measurement response, the interval intermediate value by following formula correcting principle parameter:
(12)
In formula
with
be respectively the theoretical interval intermediate value responded with actual measurement in structure jth rank; M is the number of structural response; S.t. constraint condition is represented; UB, LB represent the Lower and upper bounds of structural parameters; R
1for interval intermediate value residual error, be used as the objective function of interval intermediate value here; Min [] represents the function minimized in bracket.
Further, in order to simplify calculating, adopt the partial derivative of method of difference computation structure parameter at revised interval intermediate value place, to obtain the approximate expression of formula (11), then in this approximate expression enterprising row iteration optimization in basis, the interval radius of correcting principle parameter:
According to the interval radius of structure actual measurement response, the interval radius by following formula correcting principle parameter:
(13)
In formula
with
be respectively the theoretical interval radius responded with actual measurement in structure jth rank; M is the number of structural response; S.t. constraint condition is represented; UB, LB represent the Lower and upper bounds of structural parameters; R
2for interval radius residual error, be used as the objective function of interval radius here; Min [] represents the function minimized in bracket.
To the invention has the beneficial effects as follows structural response approximate representation based on taylor series expansion to be the linear function of interval parameter, set up the interval intermediate value of response with this, the one-to-one function relation of intermediate value, radius between radius and parameter region, and be two deterministic models correction solution procedurees by interval model correction problem reduction.Compared to prior art, the invention has the advantages that: 1) be two deterministic models correction solution procedurees by interval model correction problem reduction, to carry out substep correction to intermediate value between parameter region and radius, make interval model makeover process more simple; 2) the incidental interval extension problem of interval arithmetic process is avoided; 3) for the situation that distribution range between parameter region is less, there is higher computational accuracy; 4) not high to the size requirements of actual measurement response data amount; 5) Modifying model process decreases the interference of subjective factor, and acquired results is more true and reliable.
Accompanying drawing explanation
Fig. 1 is the realization flow figure of the embodiment of the present invention.
Embodiment
The present invention is based on the interval model modification method of Taylor series expansion, first based on Taylor series expansion, set up the interval intermediate value of the interval intermediate value of structural response, radius and structural parameters, the function corresponding relation of radius, as the theoretical foundation of interval model correction, then carry out interval model correction as follows:
Step 1: according to the interval intermediate value of structure actual measurement response, adopt deterministic models correction technique, the interval intermediate value of correcting principle parameter;
Step 2: in order to simplify calculating, adopts the partial derivative of method of difference computation structure parameter at revised interval intermediate value place, with the approximate expression of the function corresponding relation of the interval radius of the interval radius and structural parameters that obtain structural response;
Step 3: according to the interval radius of structure actual measurement response, adopt deterministic models correction technique, in the enterprising row iteration optimization in basis of the approximate expression that step 2 obtains, the interval radius of correcting principle parameter.
Further, the above-mentioned interval model modification method based on Taylor series expansion, specifically carries out as follows:
One, based on the theoretical foundation of the interval model modification method of Taylor series expansion:
The range format of note structural parameters x is
,
be called structure interval parameter, wherein,
, n represents the dimension of structure interval parameter, and subscript I represents interval number.Interval of definition number
, then
by the interval upper bound
, lower bound
or interval intermediate value
, radius
can be expressed as:
(1)
1) by the interval intermediate value x of structural response y at structural parameters
cnear carry out Taylor series expansion (now do not consider interval radius, be reduced to deterministic parameter by parameter x), obtain:
(2)
When the degree of uncertainty of parameter x is less, formula (2) is blocked, only retention constant item and once item, can obtain:
(3)
Now calculating for simplifying, the partial derivative of finite difference method structural parameters can be utilized, that is:
(4)
In formula
structural parameters
small perturbation;
2) in order to further consider interval radius, x is adopted complete range format x
irepresent, then the interval type of structural response is:
(5)
For symmetric interval
perseverance has:
(6)
Therefore formula (5) is expressed as further:
(7)
Obtain the interval upper bound of structural response thus
, lower bound
and interval intermediate value
, radius
be respectively:
(8)
(9)
(10)
(11)
Formula (10) establishes the interval intermediate value of structural response and the function corresponding relation of the interval intermediate value of structural parameters, and formula (11) establishes the function corresponding relation of the interval radius of structural response and the interval radius of structural parameters.Therefore, follow-uply just deterministic models correction technique can be adopted, iteration optimization on the basis of formula (10) and (11), with this interval intermediate value of correcting principle parameter and radius respectively.Wherein during the intermediate value of corrected parameter interval, structural parameters can be reduced to deterministic parameter correction; And during the radius of corrected parameter interval, in order to simplify computation process, adopt the partial derivative of method of difference computation structure parameter at revised interval intermediate value place, to obtain the approximate expression of formula (11), then in this approximate expression enterprising row iteration optimization in basis, the interval radius of correcting principle parameter.
Two, based on the specific implementation process of the interval model modification method of Taylor series expansion
According to interval intermediate value, radius and the interval intermediate value of parameter, the one-to-one function relation of radius of said structure response, therefore interval model correction problem reduction can be two deterministic optimization inverse problems by i.e. formula (10) and formula (11):
Problem one: the interval intermediate value of structural parameters is revised
According to the interval intermediate value of structure actual measurement response, adopt deterministic models correction technique, the interval intermediate value of correcting principle parameter.
(12)
In formula
with
be respectively the theoretical interval intermediate value responded with actual measurement in structure jth rank; M is the number of structural response; S.t. constraint condition is represented; UB, LB represent the Lower and upper bounds of structural parameters; R
1for interval intermediate value residual error, be used as the objective function of interval intermediate value here; Min [] represents the function minimized in bracket;
Problem two: the interval radius of structural parameters is revised
Calculate for simplifying, adopt the partial derivative of method of difference computation structure parameter at revised interval intermediate value place, to obtain the approximate expression of formula (11), then in this approximate expression enterprising row iteration optimization in basis, and according to the interval radius that structure actual measurement responds, adopt deterministic models correction technique to carry out the interval radius of correcting principle parameter.
(13)
In formula
with
be respectively the theoretical interval radius responded with actual measurement in structure jth rank; M is the number of structural response; S.t. constraint condition is represented; UB, LB represent the Lower and upper bounds of structural parameters; R
2for interval radius residual error, be used as the objective function of interval radius here; Min [] represents the function minimized in bracket.
Inevitably there is uncertainty in the material of engineering structure, geometric parameter, boundary condition and condition of contact, in Modifying model process, therefore consider that the uncertainty of above-mentioned parameter is very necessary, model modification method can be made 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.Now adopt the stochastic response based on probability statistics may have the larger error of calculation, or even infeasible.Therefore, when measurement data is less, the stochastic response based on interval analysis has good advantage.
The present invention is based on taylor series expansion by structural response approximate representation is the linear function of interval parameter, set up the interval intermediate value of response with this, the one-to-one function relation of intermediate value, radius between radius and parameter region, and be two deterministic models correction solution procedurees by interval model correction problem reduction.Present invention, avoiding the incidental interval extension problem of interval arithmetic process, simultaneously less to distribution range between parameter region situation, there is higher accuracy of identification and optimizing process is simple, fast.The present invention is under the insufficient prerequisite of structural information, " truly " interval distribution of identification parameter is carried out according to the interval of structure actual measurement response, interval model correction problem is sought to solve in deterministic framework, enormously simplify the complexity of interval probabilistic model correction problem, be convenient to practical implementation.
Be more than preferred embodiment of the present invention, all changes done according to technical solution of the present invention, when the function produced does not exceed the scope of technical solution of the present invention, all belong to protection scope of the present invention.
Claims (3)
1. the interval model modification method based on Taylor series expansion, it is characterized in that: first based on Taylor series expansion, set up the interval intermediate value of the interval intermediate value of structural response, radius and structural parameters, the function corresponding relation of radius, as the theoretical foundation of interval model correction, then carry out interval model correction as follows:
Step 1: according to the interval intermediate value of structure actual measurement response, adopt deterministic models correction technique, the interval intermediate value of correcting principle parameter;
Step 2: in order to simplify calculating, adopts the partial derivative of method of difference computation structure parameter at revised interval intermediate value place, with the approximate expression of the function corresponding relation of the interval radius of the interval radius and structural parameters that obtain structural response;
Step 3: according to the interval radius of structure actual measurement response, adopt deterministic models correction technique, in the enterprising row iteration optimization in basis of the approximate expression that step 2 obtains, the interval radius of correcting principle parameter;
Wherein, set up the interval intermediate value of the interval intermediate value of structural response, radius and structural parameters, the function corresponding relation of radius, carry out as follows:
The range format of note structural parameters x is
, x
ibe called structure interval parameter, wherein,
, n represents the dimension of structure interval parameter, and subscript I represents interval number; Interval of definition
, then x
iby the interval upper bound
, lower bound
or interval intermediate value x
c, radius x
rcan be expressed as:
(1)
1) by the interval intermediate value x of structural response y at structural parameters
cnear carry out Taylor series expansion, now do not consider interval radius, be reduced to deterministic parameter by parameter x, obtain:
(2)
When the degree of uncertainty of parameter x is less, formula (2) is blocked, only retention constant item and once item, can obtain:
(3)
In order to simplify calculating, utilize the partial derivative of finite difference method structural parameters, that is:
(4)
In formula
structural parameters x
ismall perturbation;
2) in order to further consider interval radius, x is adopted complete range format x
irepresent, then the interval type of structural response is:
(5)
For symmetric interval e
iperseverance has:
(6)
Therefore formula (5) is expressed as further:
(7)
Obtain the interval upper bound of structural response thus
, lower bound
and interval intermediate value y
c, radius y
rbe respectively:
(8)
(9)
(10)
(11)
Formula (10) establishes the interval intermediate value of structural response and the function corresponding relation of the interval intermediate value of structural parameters, and formula (11) establishes the function corresponding relation of the interval radius of structural response and the interval radius of structural parameters; Therefore, follow-uply just deterministic models correction technique can be adopted, iteration optimization on the basis of formula (10) and (11), with this interval intermediate value of correcting principle parameter and radius respectively; During the interval intermediate value of wherein correcting principle parameter, structural parameters can be reduced to deterministic parameter correction; And during the radius of corrected parameter interval, in order to simplify calculating, adopt the partial derivative of method of difference computation structure parameter at revised interval intermediate value place, to obtain the approximate expression of formula (11), then in this approximate expression enterprising row iteration optimization in basis, the interval radius of correcting principle parameter.
2. the interval model modification method based on Taylor series expansion according to claim 1, it is characterized in that: first according to the interval intermediate value of said structure response and the function corresponding relation of interval intermediate value of structural parameters, be deterministic optimization inverse problem by interval intermediate value correction problem reduction:
According to the interval intermediate value of structure actual measurement response, the interval intermediate value by following formula correcting principle parameter:
(12)
In formula
with
be respectively the theoretical interval intermediate value responded with actual measurement in structure jth rank; M is the number of structural response; S.t. constraint condition is represented; UB, LB represent the Lower and upper bounds of structural parameters; R
1for interval intermediate value residual error, be used as the objective function of interval intermediate value here; Min [] represents the function minimized in bracket.
3. the interval model modification method based on Taylor series expansion according to claim 1, it is characterized in that: in order to simplify calculating, adopt the partial derivative of method of difference computation structure parameter at revised interval intermediate value place, to obtain the approximate expression of formula (11), then in this approximate expression enterprising row iteration optimization in basis, the interval radius of correcting principle parameter:
According to the interval radius of structure actual measurement response, the interval radius by following formula correcting principle parameter:
(13)
In formula
with
be respectively the theoretical interval radius responded with actual measurement in structure jth rank; M is the number of structural response; S.t. constraint condition is represented; UB, LB represent the Lower and upper bounds of structural parameters; R
2for interval radius residual error, be used as the objective function of interval radius here; Min [] represents the function minimized in bracket.
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| US5883818A (en) * | 1996-08-29 | 1999-03-16 | International Business Machines Corporation | Method for generating an improved model for evaluating the operation of an integrated circuit design |
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| US5883818A (en) * | 1996-08-29 | 1999-03-16 | International Business Machines Corporation | Method for generating an improved model for evaluating the operation of an integrated circuit design |
| CN102999678A (en) * | 2012-12-26 | 2013-03-27 | 北京航空航天大学 | Nonlinear multi-target range robust optimization based automobile noise reduction method |
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| 基于随机响应面模型的随机模型修正方法;张秋虎等;《第22届全国结构工程学术会议论文集第3册》;20130810;第466-469页 * |
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