CN108920786A - A kind of bounded-but-unknown uncertainty analysis method based on chebyshev approximating polynomial - Google Patents
A kind of bounded-but-unknown uncertainty analysis method based on chebyshev approximating polynomial Download PDFInfo
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Abstract
The bounded-but-unknown uncertainty analysis method based on chebyshev approximating polynomial that the invention discloses a kind of, this method are applied to engineering structure analysis of uncertainty.Firstly, selecting and utilizing sectionRationally characterization bounded-but-unknown uncertainty variable；Secondly, precision as needed and the order for being fitted function estimated determine number and position with point；Then, to the corresponding response for solving uncertain system with point；Finally, being calculated using the obtained response for respectively matching point and obtaining fitting function.The numerical example shows that the bounded-but-unknown uncertainty based on chebyshev approximating polynomial method propagates analysis method and can obtain the response section of accurate uncertain system in the biggish situation of offset radii, provides a kind of new method for bounded-but-unknown uncertainty analysis.
Description
Technical field
The present invention relates to the technical fields of the bounded-but-unknown uncertainty of engineering structure analysis, in particular to more based on Chebyshev
The bounded-but-unknown uncertainty analysis method of item formula fitting.
Background technique
In practical projects since material property is horizontal, the limitation of process water equality objective condition, random material characteristic, if
Meter and manufacturing defect, different loading environments etc. cause the various uncertain safety and reliabilities to engineering and generate
It influences.In order to effectively measure the above uncertain influence generated to engineering, meet the reliability and safety of engineering system
It is required that quickly becoming an important branch of Multidisciplinary Optimization based on probabilistic optimization design.
Uncertainty propagation analysis is the pith of uncertainty optimization design, can further be used to make reliable
Decision.The field of structure safety evaluation and structural design optimization is often served in analysis of uncertainty, with system to uncertain
Property parameter response be solve object.By taking structural design optimization as an example, be substantially different schemes according to given criterion into
Row trap queuing, but the crucial natural ordering relationship for being uncertain response and being not present between similar real number, and this sequence is closed
System is inevitable structure safety evaluation or design optimization, solves the needs of multidisciplinary uncertain problem for not true
Method for qualitative analysis proposes requirement.
Existing Uncertainty Analysis Method is broadly divided into two classifications of probabilistic method and non-probabilistic method, probabilistic method warp
The development for having crossed many decades has reached its maturity, and in the field, widely used method includes Monte Carlo method, sound at present
Answer face method etc..A large amount of and comprehensive unascertained information is to construct the exact probability distribution of uncertain parameters to use probability side
Method solves the prerequisite of uncertain problem, and these information access processes are excessively cumbersome, practical for certain engineerings to ask
Topic can not even obtain.In order to cope with the Solve problems of the infull uncertain system of this type of information, non-probability theory method is met the tendency of
And it gives birth to.
In non-probabilistic method, bounded-but-unknown uncertainty analysis method is important method, bounded-but-unknown uncertainty analysis method
Without going into seriously the specific structure of system and the intension of uncertain parameters, only by obtain uncertain parameters median with
The upper and lower bounds of responses that offset radii can carry out system output solves, so that the judge of structural safety performance is completed, for future
The update of uncertainty structure analysis and design concept, has important facilitation.Therefore, how to develop bounded-but-unknown uncertainty
Analysis method and realize in sample information deficiency situation to large complicated engineering system carry out non-probabilistic uncertainty analysis and
Optimization design is following one of the main contents for expanding uncertainty optimization design theoretical system.
In addition to traditional interval algorithm, most of non-probability interval Uncertainty Analysis Methods are unfolded based on first order Taylor
Formula and Global sensitivity analysis.However, the error of these methods is larger when mission nonlinear degree is higher.And traditional illiteracy
Special calot's method calculating is costly, and the present invention can obtain the response of system output variables using chebyshev approximating polynomial method
Section, compared to existing bounded-but-unknown uncertainty analysis method,
Summary of the invention
The technical problem to be solved by the present invention is to：It overcomes the shortcomings of existing methods, provides a kind of multinomial based on Chebyshev
The bounded-but-unknown uncertainty analysis method of formula fitting.Chebyshev approximating polynomial method computational efficiency in ensuring engineering design
More preferably computational accuracy is obtained simultaneously, is the good supplement of existing method.
The technical solution adopted by the present invention is：Bounded-but-unknown uncertainty analysis method based on chebyshev approximating polynomial,
This method is applied to engineering structure analysis of uncertainty, and implementation step is as follows：
Step 1：According to the uncertain situation for needing to consider in design, such as material properties, geometric attribute, load
Attribute selects material property, geometric dimension and loading environment etc. as uncertain parameters and obtains its bound, utilizes area
BetweenThe rationally set of characterization uncertain parameters, whereinXIndicate the lower bound of input parameter,Indicate input parameter
The upper bound, the set X of uncertain parameters includes uncertain variable xi, (i=1,2 ... n)；
Step 2：Number and position of the uncertain variable with point are set；
Step 3：It obtains engineering structure and is respectively matching response at point；
Step 4：Structural response at respectively being matched based on gained, solves the coefficient of each basic function；
Step 5：By the coefficient of gained basic function, fitting function is obtained；
Step 6：Based on gained fitting function, the bound of Solution of Optimization system output response is used.
Further, bounded-but-unknown uncertainty parameter depends on the uncertainty for needing to consider in design in the step 1
Situation considers the collective effect selection of material properties, geometric attribute, load attribute, selects material property, geometric dimension and load
Condition etc. is used as uncertain parameters, and the set for inputting uncertain parameters can be expressed as
Further, in the step 2 number with point of uncertain variable by being fitted the order of function and required
Precision determine.
Further, it is as follows in the process for respectively matching the response at point that system is obtained in the step 3：
When needing to solve a certain response with point, adjust each uncertain parameters to this with each representated by point
Uncertain parameters value obtains system response in this case using engineering method.
Further, basic function is Chebyshev polynomials in the step 4.
Further, in the step 5 fitting function be each basic function and its coefficient product summation, i.e.,：
Wherein f=f (x1,x2,…,xn) it is fitting function,For corresponding basic function,For corresponding basic function
Coefficient.
Further, the bound of system output response is obtained by solving the most value of fitting function in the step 6
's.
The advantages of the present invention over the prior art are that：The present invention provides the new think ofs of one kind of bounded-but-unknown uncertainty analysis
Road, makes up and perfect traditional Uncertainty Analysis Method based on probability theory, used to be based on Chebyshev polynomials
On the one hand the bounded-but-unknown uncertainty analysis method of fitting process can substantially reduce the dependence to sample information, on the other hand can be with
Its high-precision characteristic is made full use of, bounded-but-unknown uncertainty problem biggish for offset radii obtains accurate structural response
Bound.
Detailed description of the invention
Fig. 1 is the overview flow chart of the bounded-but-unknown uncertainty analysis method the present invention is based on chebyshev approximating polynomial；
Specific embodiment
With reference to the accompanying drawing and specific embodiment further illustrates the present invention.
The invention proposes a kind of the bounded-but-unknown uncertainty analysis method based on chebyshev approximating polynomial, specific steps
It is as follows：
Step 1：According to the uncertain situation for needing to consider in design, such as material properties, geometric attribute, load
Attribute selects material property, geometric dimension and loading environment etc. as uncertain parameters and obtains its bound, utilizes area
BetweenThe rationally set of characterization uncertain parameters, whereinXIndicate the lower bound of input parameter,Indicate input parameter
The upper bound, the set X of uncertain parameters includes uncertain variable xi, (i=1,2 ... n)；
Step 2：Number and position of the uncertain variable with point are set, wherein with point of uncertain variable
Several and position determines that specific derivation process is seen below by the order and required precision for being fitted function：
Uncertain parameters after unitization are expressed as：
Enabling X is the set of uncertain parameters, and Z is the set of structural response, is had：
Use Chebyshev polynomials (Chebyshev polynomials) as basic function in this method.Work as weight functionWhen section is [- 1,1], by sequence { 1, x ..., xn... } and the obtained orthogonal polynomial of orthogonalization is exactly to cut
Than avenging husband's multinomial, expression-form is：
Tn(x)=cos (n arccos x), x ∈ [- 1,1]
Wherein n is constant.
By trigonometric identity：
Cos (n+1) θ=2cos θ cos n θ-cos (n-1) θ
Enable x=cos θ to obtain the final product：
Step 3：It obtains structure and is respectively matching response at point；
Step 4：Structural response at respectively being matched based on gained, solves the coefficient of each basic function.Basic function coefficient solved
Journey is as follows：
Problem is tieed up for 1,
zi(Xj) indicate ziIn jth time uncertain parameters XjThe value at place.
By Gauss-Chebyshev's integral：
Wherein point upIt is
Simultaneously：
Then have：
It is generalized to multi-dimensional form, is had：
Wherein k indicates the number of uncertain parameters, i1,…,ikIt indicates for x1,…,xkBasic function order,
i1,…,ik=0,1,2 ..., n；
Step 5：By the coefficient of gained basic function, fitting function is obtained, fitting function calculation formula is as follows：
By Gauss-Chebyshev's integral it is found that the truncated error of each coefficient of digital simulation function is：
Wherein f (x) is real function expression formula.
It can be concluded that obtained fitting function is more accurate when the acquirement of q value is bigger.However from the above equation, we can see that, each coefficient
The calculation times of solution be all qk, influence of the q value to calculation amount be huge.Assuming that f (x) is a t rank multinomial, as q >=(r+
1) when/2, the truncated error that above formula is calculated is approximately 0, can guarantee fitting precision very well at this time.
Step 6：Based on gained fitting function, the bound of Solution of Optimization system output response is used.
Embodiment 1
This example compared traditional bounded-but-unknown uncertainty analysis method and chebyshev approximating polynomial method, function table
It is as follows up to formula：
The central value and offset radii of each uncertain parameters are as shown in table 1.
Table 1
Four kinds of methods obtain that the results are shown in Table 2, wherein representing Taylor expansion (Taylor expansion with TEM
Method), MCM represents Monte Carlo method (Monte Carlo method), and CIM represents chebyshev approximating polynomial method
(Chebyshev interval method).And as a comparison with analytic solutions, demonstrate chebyshev approximating polynomial method for
The precision of high-order nonlinear problem.
Table 2
It is not difficult to find out from upper table, for being only to solve for y there are the system of cubic term1Upper and lower bounds of responses, Tai Lezhan
Opening method result obtained obviously confirms that error is excessive, far beyond acceptable requirement in engineering, does not just have yet
Continue to probe into necessary.The cycle-index of the random process of Monte Carlo method setting is 107It is secondary, however precision is still low
In chebyshev approximating polynomial method；When cycle-index raising can also obtain more accurate result.But firstly there are difficulty
Point is how to obtain the sample point of sufficiently large quantity；In addition to this, it is on the one hand newly increased with the raising of cycle-index
The boundary benefit of circulation reduces, and on the other hand the duration that is averaged of the operation of Monte Carlo method program at this time has had reached 102Second
Rank allows to obtain enough sample points, in engineering compared to the operation time of the Millisecond with two methods of point collocation
It is middle time-consuming and laborious using Monte Carlo method progress bounded-but-unknown uncertainty analysis, it is still very unadvisable behavior.
Embodiment 2
In order to more fully show that, to the actual applicability of engineering, the square plate that the present invention is 1 meter for side length quivers
Vibration problem has carried out bounded-but-unknown uncertainty analysis.Plate thickness is selected as uncertain parameters, median 0.01m, offset radii is
0.0005m；Elasticity modulus and density are selected as material uncertain parameters, elasticity modulus median is 70GPa, and offset radii is
1GPa；Density median is 2700kg/m3Offset radii is 100kg/m3；Material modulus of shearing be 27GPa, Poisson's ratio 0.3, one
Side is fixed.System output is critical divergence speed, solves value and offset radii among it.It is the key step of analysis below：
(1) for three uncertain factors, the selection for matching point is carried out first：
Wherein ti,Ei,deniRespectively indicate plate thickness, the elasticity modulus, density respectively matched at point；
(2) the critical divergence speed of plate respectively matched at point is solved using business software Nastran, totally 125 groups of data；
(3) chebyshev approximating polynomial bounded-but-unknown uncertainty respectively will be substituted into point data and analyze program, solve plate and face
The median and offset radii of boundary's divergence speed.
Solving result is as shown in table 5.It can be seen that the bounded-but-unknown uncertainty analysis method based on chebyshev approximating polynomial
With preferable engineering adaptability.
Table 5
In conclusion the invention discloses a kind of bounded-but-unknown uncertainty analysis side based on chebyshev approximating polynomial
Method, this method are applied to single, multidisciplinary analysis of uncertainty field.Firstly, selecting and utilizing sectionRationally characterization
Bounded-but-unknown uncertainty variable；Secondly, precision as needed and the order for being fitted function estimated determine the number with point
With position；Then, to the corresponding response for solving uncertain system with point；Finally, utilizing the obtained response for respectively matching point
Value calculates and obtains fitting function.
The above is only specific steps of the invention, are not limited in any way to protection scope of the present invention；All use is equal
Transformation or equivalence replacement and the technical solution that is formed, all fall within rights protection scope of the present invention.
Part of that present invention that are not described in detail belong to the well-known technology of those skilled in the art.
Claims (7)
1. the bounded-but-unknown uncertainty analysis method based on chebyshev approximating polynomial, this method is uncertain applied to engineering structure
Property analysis, which is characterized in that realize steps are as follows：
Step 1：According to the uncertain situation for needing to consider in engineering design, including material properties, geometric attribute, load
Attribute selects material property, geometric dimension and loading environment as uncertain parameters and obtains its bound, utilizes sectionThe rationally set of characterization uncertain parameters, whereinXIndicate the lower bound of input parameter,Indicate input parameter
The upper bound, the set X of uncertain parameters include uncertain variable xi, (i=1,2 ... n)；
Step 2：Number and position of the uncertain variable with point are set；
Step 3：It obtains engineering structure and is respectively matching response at point；
Step 4：Structural response at respectively being matched based on gained, solves the coefficient of each basic function；
Step 5：By the coefficient of gained basic function, fitting function is obtained；
Step 6：Based on gained fitting function, the bound of Solution of Optimization system output response is used.
2. the bounded-but-unknown uncertainty analysis method according to claim 1 based on chebyshev approximating polynomial, feature
It is：Bounded-but-unknown uncertainty parameter depends on the uncertain situation for needing to consider in design in the step 1, considers material
Expect attribute, geometric attribute, load attribute collective effect selection, select material property, geometric dimension and loading environment etc. as
Uncertain parameters, the set for inputting uncertain parameters can be expressed as
3. the bounded-but-unknown uncertainty analysis method according to claim 1 based on chebyshev approximating polynomial, feature
It is：The number with point of uncertain variable is determined by the order and required precision for being fitted function in the step 2.
4. the bounded-but-unknown uncertainty analysis method according to claim 1 based on chebyshev approximating polynomial, feature
It is：It is as follows in the process for respectively matching the response at point that system is obtained in the step 3：
When needing to solve a certain response with point, make each uncertain parameters adjust to this with point it is representative it is each not really
Qualitative parameter value obtains system response in this case using engineering method.
5. the bounded-but-unknown uncertainty analysis method according to claim 1 based on chebyshev approximating polynomial, feature
It is：Basic function is Chebyshev polynomials in the step 4.
6. the bounded-but-unknown uncertainty analysis method according to claim 1 based on chebyshev approximating polynomial, feature
It is：Fitting function is the summation of each basic function and the product of its coefficient in the step 5, i.e.,：
Wherein f=f (x1,x2,…,xn) it is fitting function,For corresponding basic function,It is for correspond to basic function
Number.
7. the bounded-but-unknown uncertainty analysis method according to claim 1 based on chebyshev approximating polynomial, feature
It is：The bound of system output response is obtained by solving the most value of fitting function in the step 6.
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| CN110457768A (en) * | 2019-07-18 | 2019-11-15 | 东南大学 | Consider the configuration method of the MEMS device parameter based on reliability under fabrication error |
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