CN107247831A - It is a kind of based on iteration by dimension method multidisciplinary bounded-but-unknown uncertainty analysis method - Google Patents
It is a kind of based on iteration by dimension method multidisciplinary bounded-but-unknown uncertainty analysis method Download PDFInfo
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- CN107247831A CN107247831A CN201710385227.6A CN201710385227A CN107247831A CN 107247831 A CN107247831 A CN 107247831A CN 201710385227 A CN201710385227 A CN 201710385227A CN 107247831 A CN107247831 A CN 107247831A
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Abstract
The invention discloses a kind of based on multidisciplinary bounded-but-unknown uncertainty analysis method of the iteration by dimension method, belong to multidisciplinary analysis of uncertainty field.First, interval is utilizedRationally characterize poor information, a small number of uncertain parameters under the conditions of；Secondly, the initial nominal value of uncertain variables is set；Then, using the maximum and minimum value point that uncertain variables are obtained by dimension method；Finally, calculate remaining parameter and judge whether convergence, if convergence, the response of output system output variable is interval, otherwise, carry out circulation next time until convergence.The numerical example shows, the response interval that analysis method results in accurate system output variables is propagated based on the multidisciplinary bounded-but-unknown uncertainty by dimension iterative method, compared with Monte Carlo method, its amount of calculation is also acceptable, and a kind of new method is provided for the analysis of multidisciplinary bounded-but-unknown uncertainty.
Description
Technical field
The present invention relates to the technical field of multidisciplinary analysis of uncertainty, more particularly to based on iteration by the multidisciplinary of dimension method
Bounded-but-unknown uncertainty analysis method.
Background technology
Multidisciplinary design optimization design shows huge potentiality on processing complex engineering problems, and designer can excavate
Coupling effect and cooperative effect between subject is so as to improve design.However, due to lacking knowledge, random material in Practical Project
Expect characteristic, design and manufacturing defect, different loading environments etc. cause various uncertainties.It is fast with modern technologies
The requirement that speed development and robustness and security to engineering system are increasingly improved, based on probabilistic Multidisciplinary Optimization
Quickly become an important branch of Multidisciplinary Optimization.
Multidisciplinary uncertainty propagation analysis is the pith of uncertainty optimization design, can further be used to do
Go out reliable decision-making.Multidisciplinary Uncertainty Analysis Method can be divided into probabilistic method and non-probabilistic method, due to grinding for probabilistic method
Study carefully that history is longer, possessed more complete mathematical theory basis and be also more continually employed in practice in engineering.
However, probabilistic method needs substantial amounts of sample number strong point to describe the probability distribution of uncertain parameters, this is to a certain extent
Limit the application of probabilistic method.By contrast, non-probabilistic method needs further research still in the exploratory stage.
In non-probabilistic method, interval model is an important method, and Novel Interval Methods only need to know uncertain
Property parameter border, without know uncertain parameters specifically distribution or membership function form, so as to greatly reduce
To the demand of initial data.In interval model, variable is represented by two scalars, i.e. floor value and upper dividing value, interval
Algorithm is to carry out arithmetical operation with the interval of description uncertain parameters, and it is applied to various fields, including structure
Analysis and dynamic analysis, however, these applications are confined to the scope of a very little, relate generally to some simple questions.Because
Interval Computation produces potentially conservative result, moreover, the amount of calculation of interval arithmetic is also big, because its processing is interval, and
It is not exclusively digital.
Except traditional interval algorithm, most of non-multidisciplinary bounded-but-unknown uncertainty analysis methods of probability are safe based on single order
Strangle expansion and Global sensitivity analysis.However, when mission nonlinear degree is higher, the error of these methods is larger.And pass
The Monte Carlo method computational costs of system is higher, and the present invention can obtain the sound of multidisciplinary system output variable using iteration by dimension method
Should be interval, computational accuracy is very high, and compared to Monte Carlo method, computational efficiency is also greatly improved.
The content of the invention
The technical problem to be solved in the present invention is：Overcome the shortcomings of existing method there is provided it is a kind of based on iteration by many of dimension
Subject bounded-but-unknown uncertainty analysis method.Iteration is very high by dimension method computational accuracy, compared to Monte Carlo method, and computational efficiency is also obtained
It is the good supplement of existing method to increasing substantially.
The technical solution adopted by the present invention is：Based on multidisciplinary bounded-but-unknown uncertainty analysis method of the iteration by dimension method, use
To make reliable decision-making, implementation step is as follows：
Step one：According to real abstract, the hypothesis situation made, the shortage situation of knowledge, physical dimension and loading bar
Part, material property determines the bound of input parameter, utilizes intervalPoor information, minority are rationally characterized under the conditions of
Uncertain parameters set, wherein,XThe lower bound of input parameter is represented,Represent the upper bound of input parameter.Uncertain variables
Set X include subject uncertain variables xi, (i=1,2 ... n) with system uncertain variables xs, wherein xiOnly go out in subject i
It is existing, xsIt is the uncertain input of multiple subjects.
Step 2：The nominal value X of uncertain input variable when the l times circulation is setl n, l initial value is 1；
Step 3：Perform the most value point X that the l times circulation is obtained by Wei Falail m；
Step 4：The nominal value for making the uncertain variables of the l+1 times circulation is Xl m, i.e. Xl+1 n=Xl m；
Step 5：Calculate residual error coefficient and judge whether iterative process restrains, if convergence, performs step 6, otherwise,
Return to step one, residual error coefficient CsysCalculation formula be：
Csys=| Xl+1 n-Xl m|
Step 6：System output is obtained in X by deterministic parsingl mThe value Z at place, this be system output maximum or most
Small value.
Further, bounded-but-unknown uncertainty parameter depends on abstract, the hypothesis situation of reality in the step one, knowledge
Shortage situation, physical dimension and loading environment, the collective effect of material property, the set of input uncertain parameters can be stated
For
Further, in the step 2 uncertain input variable initial nominal value X1 nIts interval intermediate value is taken as, i.e.,
Further, using by dimension method, to find, to be most worth process a little as follows in the step 3：
As a certain uncertain variables x of searchingiMost value point when, other uncertain variables are fixed as its nominal value, with cutting
Than snow husband's fitting of a polynomial system output variables and xiFunctional relation, and find the most value point of the fitting function.
Further, the most value point X of the uncertain variables obtained in the step 4 with the l timesl mInstead of the l+1 times iteration
Uncertain variables nominal value Xl+1 n。
Further, residual error coefficient C in step 5 described in step 5sysCalculation formula be：
Csys=| Xl+1 n-Xl m|
Further, the interval response that system is exported in step 6 is by the most value point X in uncertain inputl mPlace passes through
What deterministic parsing was obtained.
The advantage of the present invention compared with prior art is：The invention provides the one of multidisciplinary bounded-but-unknown uncertainty analysis
Kind of new approaches, make up and perfect traditional multidisciplinary Uncertainty Analysis Method based on probability theory, used based on repeatedly
In generation, by the multidisciplinary bounded-but-unknown uncertainty analysis method of dimension method, on the one hand can significantly reduce the dependence to sample information, another
Aspect can make full use of its high-precision characteristic, accurate system output interval response be obtained, with traditional Monte Carlo method
Compare, iteration is also greatly lowered by the amount of calculation of dimension method, saves the calculating time.
Brief description of the drawings
Fig. 1 is that the present invention is directed to the overall procedure by the multidisciplinary bounded-but-unknown uncertainty analysis method of dimension method based on iteration
Figure；
Fig. 2 is the dimensional airfoil simplified model figure with chain of command in the present invention；
Fig. 3 is implementing procedure figure of the iteration by dimension method in embodiment in the present invention；
Fig. 4 is by the interval iteration course figure of the response of the angle of attack of dimension method in the present invention based on iteration；
Fig. 5 is the pressure cloud atlas of dimensional airfoil in the present invention；
Fig. 6 be in the present invention dimensional airfoil centre-of-pressure position with chain of command drift angle variation diagram.
Embodiment
Below in conjunction with the accompanying drawings and specific embodiment further illustrates the present invention.
The present invention propose it is a kind of based on iteration by dimension method multidisciplinary bounded-but-unknown uncertainty analysis method, specific steps are such as
Under：
Step one：According to real abstract, the hypothesis situation made, the shortage situation of knowledge, physical dimension and loading bar
Part, material property determines the bound of input parameter, utilizes intervalPoor information, minority are rationally characterized under the conditions of
The set of uncertain parameters, wherein,XThe lower bound of input parameter is represented,Represent the upper bound of input parameter.Uncertain variables
Set X includes subject uncertain variables xi, (i=1,2 ... n) with system uncertain variables xs, wherein xiOnly go out in subject i
It is existing, xsIt is the uncertain input of multiple subjects.
Step 2：The nominal value X of uncertain input variable when the l times circulation is setl n, l initial value is 1, is set not true
The initial nominal value for determining input variable is its interval intermediate value, i.e.,
Step 3：Perform the most value point X that the l times circulation is obtained by Wei Falail m, obtain the l times circulation by dimension method uncertain
The process being most worth a little of variable is as follows：
First, uncertain variables are converted into its canonical form, i.e.,Wherein
ei∈[-1,1],es∈ [- 1,1], it is assumed that X is the set of all uncertain input variables, Z is the set of system output variables, i.e.,
X=[x1,…,xn,xs], Z=[z1,…,zn], it can then obtain：
Wherein e=[e1,…,en,es]T, operator ο represents corresponding element multiplication.Assuming that j-th of element is u in e, (-
1≤u≤1), other elements are 0, i.e.,：
ej=0 ..., u ... 0 }
1,…,j,…
So jth time input variable XjIt can be write as：
With the functional relation of chebyshev approximating polynomial system output variables and uncertain input variable.Assuming that Hr=
Span{L0(u),L1(u),…,Lr(u) it is } sub-spaces of r rank Chebyshev polynomials, then target is just changed into seeking one
The Least squares approximation multinomial T of seriesr ij(u)∈HrTo be fitted the functional relation of system output and input, i.e.,：
Whereinzi(Xj) represent ziIn jth time uncertain input
Variable XjThe value at place.According to Gauss-Chebyshev's integration,
Wherein point up, (p=1,2 ..., q) it is Lq(u) the conventional q of chebyshev approximating polynomial in root, engineering
Value be 10, therefore q=10 is taken in the present invention,
Then it can obtain：
Then it can be obtained by the fitting function relation of system output variables and uncertain input variable：
According to fitting function, it is easy to can be obtained by the most value point of uncertain input variable.
Step 4：The most value point X of the uncertain variables obtained for the l timesl mInstead of the uncertain variables name of the l+1 times iteration
Value Xl+1 n。
Step 5：Calculate residual error coefficient and judge whether iterative process restrains, if convergence, performs step 5, otherwise,
Return to step one, residual error coefficient CsysCalculation formula be：
Csys=| Xl+1 n-Xl m|
Step 6：System output is obtained in X by deterministic parsingl mThe value Z at place, this be system output maximum or most
Small value.
Embodiment 1：
The characteristics of in order to more fully understand the invention and its applicability actual to engineering, the present invention is for shown in Fig. 2
Two-dimensional wing with chain of command has carried out multidisciplinary bounded-but-unknown uncertainty analysis.The chord length of two-dimensional wing is b, e in embodiment1With
e2The position of elastic shaft and chain of command is determined, α is the wing angle of attack, if it is just when inclined under wing；β is chain of command drift angle and set
For just when under chain of command partially.
In pneumatic subject, the size of the angle of attack influences the size of aerodynamic moment；And in structure subject, aerodynamic moment it is big
The size of small and influence the angle of attack.Therefore, angle of attack α and torque M are chosen as system output variables.In pneumatic subject, torque is
Calculated by highly-accurate nephelometric titrimetry instrument, i.e. computational fluid dynamics (CFD).In structure subject, it is assumed that in the angle of attack be zero
In the case of, torsion spring no elastic deformation, then the angle of attack can be using following formula calculating：
Wherein KαIt is the torsional rigidity of wing, it is contemplated that flight unstability and the dispersiveness of material etc., by flight speed
Spend v, chain of command drift angle β, torsional rigidity KαIt is taken as uncertain variables.Table 1 shows these uncertain variables central values and deviation
Coefficient.
Table 1
Fig. 3 shows that iteration finds the process of extreme point by dimension method in this example, and module DOE passes through certainty pneumostatic bullet
Response of the acquisition system output in set point is analyzed, CATIA is used for parameter model, carrying out automatic mesh using ICEM draws
Point, aerodynamic analysis is carried out in Fluent, and MATLAB is used to perform by dimension method, all these to be collected by iSIGHT
Into.Clearly as amount of calculation needed for employing CFD Monte Carlo method is too big, so being not suitable for solving this problem.Iteration is by dimension
Method, by Wei Fa and the first order Taylor method of development be used, respectively, to solve this problem.Differential in the first order Taylor method of development is limited
Difference is replaced, i.e.
Table 2 shows iteration by the iterative process being most worth in dimension method a little.By dimension method when finding it and being most worth, first by β and
KαIt is taken as its nominal value -3.3deg and 160Nm/deg-1, calculated in v interval range at corresponding 10 Gauss integrations point
Torque and angle of attack response, then with the functional relation of chebyshev approximating polynomial torque or the angle of attack and speed v, then can
To obtain speed v most value point by fitting function.β and K can similarly be obtainedαMost value point.Iteration is exactly with for the first time by dimension method
The most value point calculated replaces the nominal value of second uncertain variables, then with the most value that uncertain variables are calculated by dimension method, so
Reciprocation cycle, until convergence.
Table 2
As shown in figure 4, iteration just restrains by the iterative process of dimension method within 3 steps, and therefore, calculating of the iteration by dimension method
Amount is acceptable.In this embodiment, in given interval, when β is more than a certain critical value, aerodynamic moment is with speed
Monotone decreasing, the contrast when β is less than the critical value.As shown in figure 5, in given interval, β change divides pressure
Cloth influences very little.But β change causes Center of Pressure to move, it is that Center of Pressure even has been moved to bullet to exceed a certain critical value to β
After property axle, as shown in Figure 6.
Table 3 shows the result by being obtained by dimension method, iteration by Wei Fa and the first order Taylor method of development.Because iteration is by dimension method
The most value of the system output of acquisition is to be obtained in obtained most value point by deterministic parsing, therefore iteration is by dimension method acquisition
As a result it is necessarily available.But, by the angle of attack obtained by dimension method and the upper bound of torque than iteration by the much smaller of dimension method.
So there is error by dimension method.In addition, the phase between the result that the result that first order Taylor expansion is obtained is obtained with iteration by dimension method
It is also quite big to error because this problem is nonmonotonic, first order Taylor Expansion Solution by no means dull problem when have sizable mistake
Difference, in this embodiment, the error of the first order Taylor method of development also include the calculation error that differential is replaced with finite difference.So
Iteration is a kind of good method for being used for solving multidisciplinary bounded-but-unknown uncertainty analysis by dimension method, because its precision is high and counts
Calculation amount is also little.
Table 3
In summary, the present invention proposes a kind of multidisciplinary bounded-but-unknown uncertainty based on iteration by dimension method and propagates analysis side
Method.First, interval is utilizedRationally characterize poor information, a small number of uncertain parameters under the conditions of；Secondly, set
The initial nominal value of uncertain variables；Then, using the maximum and minimum value point that uncertain variables are obtained by dimension method；Finally, count
Calculate remaining parameter and judge whether convergence, if convergence, the response of output system output variable is interval, otherwise, carries out next
Secondary circulation is until convergence.
It the above is only the specific steps of the present invention, protection scope of the present invention be not limited in any way；It is all to use equivalent
Technical scheme formed by conversion or equivalence replacement, all falls within rights protection scope of the present invention.
Non-elaborated part of the present invention belongs to the known technology of those skilled in the art.
Claims (7)
1. based on multidisciplinary bounded-but-unknown uncertainty analysis method of the iteration by dimension method, for making reliable decision-making, its feature exists
In realizing that step is as follows：
Step one：According to real abstract, the hypothesis situation made, the shortage situation of knowledge, physical dimension and loading environment, material
Material characteristic determines the bound of input parameter, utilizes intervalRationally characterize poor information, minority not true under the conditions of
The set of qualitative parameter, wherein, X represents the lower bound of input parameter,Represent the upper bound of input parameter, the set of uncertain variables
X includes subject uncertain variables xi, (i=1,2 ... n) with system uncertain variables xs, wherein xiOnly occur in subject i, xs
It is the uncertain input of multiple subjects；
Step 2：The nominal value X of uncertain input variable when the l times circulation is setl n, l initial value is 1；
Step 3：Perform the most value point X that the l times circulation is obtained by Wei Falail m；
Step 4：The nominal value for making the uncertain variables of the l+1 times circulation is Xl m, i.e. Xl+1 n=Xl m；
Step 5：Calculate residual error coefficient and judge whether iterative process restrains, if convergence, perform step (2), otherwise, return
Return step (1), residual error coefficient CsysCalculation formula be：
Csys=| Xl+1 n-Xl m|
Step 6：System output is obtained in X by deterministic parsingl mThe value Z at place, this is the maximum or minimum of system output
Value.
2. according to claim 1 based on multidisciplinary bounded-but-unknown uncertainty analysis method of the iteration by dimension method, its feature exists
In：Bounded-but-unknown uncertainty parameter depends on abstract, the hypothesis situation, the shortage situation of knowledge, geometry of reality in the step one
Size and loading environment, the collective effect of material property, the set of input uncertain parameters can be expressed as
3. according to claim 1 based on multidisciplinary bounded-but-unknown uncertainty analysis method of the iteration by dimension method, its feature exists
In：The initial nominal value X of uncertain input variable in the step 21 nIts interval intermediate value is taken as, i.e.,
4. according to claim 1 based on multidisciplinary bounded-but-unknown uncertainty analysis method of the iteration by dimension method, its feature exists
In：Using by dimension method, to find, to be most worth process a little as follows in the step 3：
As a certain uncertain variables x of searchingiMost value point when, other uncertain variables are fixed as its nominal value, use Chebyshev
Fitting of a polynomial system output variables and xiFunctional relation, and find the most value point of the fitting function.
5. according to claim 1 based on multidisciplinary bounded-but-unknown uncertainty analysis method of the iteration by dimension method, its feature exists
In：The most value point X of the uncertain variables obtained in the step 4 with the l timesl mInstead of the uncertain variables name of the l+1 times iteration
Adopted value Xl+1 n。
6. according to claim 1 based on multidisciplinary bounded-but-unknown uncertainty analysis method of the iteration by dimension method, its feature exists
In：Residual error coefficient C in the step 5sysCalculation formula be：
Csys=| Xl+1 n-Xl m|。
7. according to claim 1 based on multidisciplinary bounded-but-unknown uncertainty analysis method of the iteration by dimension method, its feature exists
In：The interval response that system is exported in the step 6 is by the most value point X in uncertain inputl mPlace passes through deterministic parsing
Obtain.
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| CN108920786A (en) * | 2018-06-20 | 2018-11-30 | 北京航空航天大学 | A kind of bounded-but-unknown uncertainty analysis method based on chebyshev approximating polynomial |
| CN110334425A (en) * | 2019-06-26 | 2019-10-15 | 上海工程技术大学 | One kind being used for the probabilistic improvement interval algorithm of engineering structure |
| CN113886947A (en) * | 2021-09-13 | 2022-01-04 | 北京航空航天大学 | Aircraft static aeroelastic system output state quantity interval determination method based on iteration strategy |
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Cited By (6)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN108920786A (en) * | 2018-06-20 | 2018-11-30 | 北京航空航天大学 | A kind of bounded-but-unknown uncertainty analysis method based on chebyshev approximating polynomial |
| CN108920786B (en) * | 2018-06-20 | 2022-10-25 | 北京航空航天大学 | Section uncertainty analysis method based on Chebyshev polynomial fitting |
| CN110334425A (en) * | 2019-06-26 | 2019-10-15 | 上海工程技术大学 | One kind being used for the probabilistic improvement interval algorithm of engineering structure |
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| CN113886947A (en) * | 2021-09-13 | 2022-01-04 | 北京航空航天大学 | Aircraft static aeroelastic system output state quantity interval determination method based on iteration strategy |
| CN113886947B (en) * | 2021-09-13 | 2023-04-14 | 北京航空航天大学 | Aircraft static aeroelastic system output state quantity interval determination method based on iteration strategy |
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