CN115455833B - Pneumatic uncertainty characterization method considering classification - Google Patents
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Abstract
The invention discloses a pneumatic uncertainty characterization method considering classification. The invention adopts a statistical inference strategy of parameter estimation and goodness-of-fit test, carries out uncertainty characterization modeling on sample data of pneumatic uncertainty variables, classifies the pneumatic uncertainty variables into random variables of single probability distribution, sparse variables of mixed weighted probability distribution formed by multiple distributions, and interval variables of interval forms expressed by upper and lower bounds; and selecting probability distribution with highest coincidence degree with the current sample data as a mathematical model of an uncertainty variable by taking the random variable model and the sparse variable model as candidate models, and adopting an interval model if the probability distribution of all candidate models and the fitting effect of the sample data are not satisfactory. Compared with the traditional subjective assignment method, the method is more obvious in objectivity, more sufficient in theoretical basis, more reference in quantitative evaluation of fitting goodness and applicable to the condition of sufficient or scarcity data.
Description
Technical Field
The invention particularly relates to a pneumatic uncertainty characterization method considering classification, belongs to the technical field of uncertainty characterization modeling methods in the field of pneumatic characteristic uncertainty quantification of aircrafts, and is suitable for aircrafts such as complete machines, full bombs and parts thereof of civil aircrafts, fighter planes, unmanned aerial vehicles, missiles and the like.
Background
Aerodynamic computing is an important task in the aerospace field, and is a problem that aircraft development and use cannot be separated. At present, two calculation approaches of CFD numerical simulation and wind tunnel test mainly exist, and a large number of non-negligible uncertainty factors exist in the two calculation approaches, such as: the method comprises the steps of sealing coefficients of a turbulence model in CFD numerical simulation, incoming flow conditions such as geometrical shapes, attack angles and Mach numbers of an aircraft physical model, measuring instrument errors, bracket and tunnel wall interference of a wind tunnel test and the like. These uncertainty factors lead to non-negligible uncertainties in aerodynamic properties, which are very likely to be very sensitive to certain uncertainties, such as the position of the shock wave on the upper surface of the airfoil and the pressure after the shock wave, and to the variation of the turbulence model closure coefficient, especially for hypersonic, thrust-integrated and other aircraft. These uncertainty factors should therefore be taken into account during the aerodynamic calculation phase, analyzing their propagation law in the aerodynamic analysis system and the impact on the aerodynamic characteristics. Fig. 1 illustrates the propagation of uncertainty to aerodynamic forces/moments in a CFD simulation model, the primary problem of which is uncertainty characterization, namely: and establishing a mathematical model of the uncertainty parameter.
Most of the current research generally models uncertainty directly into random variables of specific distribution according to experience, such as normal distribution where the flight Mach number is assumed to be a nominal value; or modeled by means of "span expansion" as a coarse span range that encompasses all possible values, such as span variables that are considered to be within the maximum and minimum ranges of aircraft surface roughness. The method has the defects that the method depends on experience subjective assignment greatly, the rationality of the method cannot be quantitatively evaluated, the irrational aerodynamic uncertainty characterization model is very likely to cause larger deviation of subsequent aerodynamic uncertainty propagation, for example, the result of the aerodynamic characteristic uncertainty quantification of an aircraft is possibly deviated from reality due to irrational modeling of instrument measurement uncertainty in a wind tunnel test, and an excessively small aerodynamic characteristic uncertainty error band can cause the expected control performance to be not achieved even the flight instability; while an excessive error band would greatly limit control or overall performance improvement.
The Kang provides a sequence statistical modeling method combining a goodness-of-fit test and a model selection scheme for the type identification of probability distribution in Sequential statistical modeling method for distribution, and provides a basic solution idea for determining an optimal distribution model from a plurality of possible probability distributions of random uncertainty; peng studied a method of characterizing variables by comparing the magnitudes of the AD test statistic and the critical value under the condition that only a small number of samples are available in the article Uncertainty analysis of composite laminated plate with data-driven polynomial chaos expansion method under insufficient input data of uncertain parameters, but the AD test method adopted by the method is not suitable for uniformly distributed test, and the calculation formulas of the test statistic and the critical value given by the method are only suitable for normal distribution, and the test of other probability distribution shows limitation. On the basis of previous researches, the invention improves the inspection strategy, effectively solves the two problems, on one hand, adopts the fitting goodness test method of KS inspection to meet the requirement of uniform distribution inspection, and on the other hand, defines the p value as an evaluation index of fitting goodness, and avoids the defect that a critical value formula is only suitable for normal distribution.
Disclosure of Invention
In view of the above, the invention provides a classification-considered aerodynamic uncertainty characterization method, which is based on observation data or high-reliability theoretical analysis data of aerodynamic uncertainty parameters (attack angle, mach number, atmospheric density, turbulence model sealing coefficient, surface roughness, rising resistance coefficient, flutter speed and the like) of an aircraft, provides a statistical inference strategy based on parameter estimation and fitting goodness test, quantitatively evaluates the fitting goodness of candidate probability models, and improves the rationality of aerodynamic uncertainty characterization.
The invention relates to a pneumatic uncertainty characterization method considering classification, which comprises the following steps:
s1, acquiring a sample data set of aerodynamic variables (such as attack angle, mach number, atmospheric density, turbulence model sealing coefficient, surface roughness, rising resistance coefficient, flutter speed and the like) to be determined and characterized; the sample data is generally observation data or high-credibility theoretical analysis data;
s2, assuming the aerodynamic variable to be determined and characterized as a random variable, respectively fitting sample data by adopting a plurality of random variable probability distribution functions to obtain corresponding probability distribution functions;
s3, assuming the aerodynamic variable to be determined and characterized as a sparse variable, wherein the probability distribution function is a weighted sum of continuous random variable probability distribution functions in S2;
s4, calculating the curves and the empirical probability distribution F of the probability distribution functions obtained in S2 and S3 n (x) Is a curve of (2)Distance d between KS The method comprises the steps of carrying out a first treatment on the surface of the According to d KS Solving the fitting goodness p of each probability distribution function under the appointed significance level alpha;
s5, according to the maximum value p in the goodness of fit p max Determining the characterization mode of the aerodynamic variable to be characterized according to the specified significance level alpha:
if p max More than or equal to alpha, the aerodynamic variable to be determined and characterized is characterized as p max Random variables or sparse variables of the corresponding probability distribution functions;
if p max And < alpha, characterizing the aerodynamic variable to be characterized as an interval variable.
Preferably, if the number of samples in the sample dataset of the aerodynamic variable to be characterized is less than the critical number of samples n cv It is directly characterized as an interval variable.
Preferably, 2<n cv <6. More preferably, n cv ＝4。
Preferably, in S2, the random variable probability distribution function includes, but is not limited to, a uniform distribution, a normal distribution, a lognormal distribution, an exponential distribution, a weibull distribution, an extremum distribution, a gamma distribution, a beta distribution, and the like.
Preferably, in the step S4, the distance d is obtained by a KS test method KS 。
Preferably, in the step S3, the weights of the continuous random variable probability distribution functions are distributed by using methods such as a red pool information criterion and a bayesian information criterion.
Preferably, the probability distribution function of the sparse variable is a continuous random variable probability distribution function with the weight less than 0.1.
Preferably, in the step S5, when the characteristic is an interval variable, an interval range is as follows:
wherein ,Iis the lower boundary of the interval,is the upper boundary of the interval; s is(s) min 、s max Respectively minimum and maximum values, σ, in a sample dataset of aerodynamic variables to be characterized s Standard deviation of a sample dataset for a pneumatic variable to be determined for a characterization; and psi is interval margin and reflects the expansion degree of the interval relative to the sample boundary. Preferably, ψ=0.4.
The invention also provides an aircraft pneumatic design method, which adopts the pneumatic uncertainty characterization method considering classification to determine the characterization mode of the pneumatic variable waiting to be determined to be characterized, such as Mach number, attack angle, lift resistance coefficient and turbulence model coefficient, and carries out aircraft pneumatic design based on the characterization mode.
Preferably, the aircraft aerodynamic design includes, but is not limited to: pneumatic uncertainty characterization and propagation of civil aircraft, fighter aircraft, unmanned aerial vehicle, missile complete machine, full missile and parts thereof, CFD model confirmation and verification and the like.
The beneficial effects are that:
1. according to the invention, a statistical inference strategy of parameter estimation and goodness-of-fit test is adopted, uncertainty characterization modeling is carried out on sample data facing to pneumatic uncertainty variables, and the fact that most of the uncertainty variables in the pneumatic characteristic analysis of an aircraft are objectively random uncertainty is considered, wherein the sample data always obey a certain probability distribution (such as air disturbance suffered by the aircraft in the take-off process is completely random deviation); meanwhile, the probability distribution focusing on the single distribution is not necessarily well matched with the data, and a mixed weighted form of a plurality of probability distributions is introduced to serve as a candidate probability model. And selecting the probability distribution with the highest matching degree with the current sample data from the probability distribution as a mathematical model of an uncertainty variable, and if the fitting effect of all candidate probability distributions and the sample data is not satisfactory, considering that the statistical rule of the sample data is weaker, and not modeling the uncertainty as the probability distribution, but adopting an interval model. According to the invention, 7 probability distributions are established as candidate models aiming at the data characteristics of pneumatic uncertainty parameters, and the optimal uncertainty characterization mode is selected.
2. Maximum likelihood estimation is an estimation method for using known sample information to reversely infer probability model distribution parameters most likely to cause the occurrence of sample results (maximum probability), and considering that the effectiveness of maximum likelihood estimation is widely accepted, the method is very mature in parameter estimation, so that the maximum likelihood estimation is adopted to infer the distribution parameters in candidate probability models. On the other hand, KS test is based on an empirical distribution function (Empirical Distribution Function, EDF) to construct test statistics, which can make full use of sample information and is not limited by the number of observed samples, and is a method which is more commonly used in EDF type goodness-of-fit tests, so KS test is selected to evaluate whether the uncertainty variable known sample data and a candidate probability distribution model are goodness-of-fit, the p value is defined as the probability of appearance of a sample observation result or a more extreme result in hypothesis test, and reflects the support degree of the original hypothesis, so that the p value describes the goodness of fit of the sample data with a given distribution, the higher the p value is, the better the goodness of fit of the sample is, and whether the sample obeys the specific distribution of the hypothesis can be measured by using the p value as the goodness-of-fit index of the test.
3. The method can be applied to the whole machine, full bullet and the like of civil aircraft, fighter plane, unmanned aerial vehicle, guided missile and the like, and the pneumatic uncertainty characterization and propagation, CFD model confirmation and verification and the like of the parts thereof.
Drawings
FIG. 1 is an uncertainty propagation.
FIG. 2 is a representation of random, sparse, interval variables.
FIG. 3 is a flow chart of the uncertainty classification characterization modeling method of the present invention.
Fig. 4 is a KS distance illustration.
Fig. 5 is a fitting result of the normal distribution and the lognormal distribution of the drag coefficient.
Detailed Description
The invention will now be described in detail by way of example with reference to the accompanying drawings.
As shown in FIG. 2, the present invention first characterizes the uncertainty as the following 3 types of variables based on the information form of the pneumatic uncertainty parameter.
1) Random variables: the mathematical model is a common single probability distribution;
2) Sparse variables: the mathematical model is a mixed weighted probability distribution formed by a plurality of distributions;
3) Interval variable: the mathematical model is in the form of intervals expressed by upper and lower bounds.
The invention provides an uncertainty classification characterization method, and a specific implementation flow is shown in figure 3. In this embodiment, a uniform distribution, a normal distribution, a lognormal distribution, an exponential distribution, a weibull distribution, and an extremum distribution are selected as the random variable probability distribution function of the candidate model. In practice, the probability distribution may not be limited to these several types, and gamma distribution, beta distribution, and the like may also be used as candidate probability models.
And weighting probability distribution in the random variable candidate model to obtain a sparse distribution candidate model. Wherein the weight of probability distribution of each random variable candidate model is distributed according to methods such as red pool information criterion (Akaike Information Criterion, AIC), bayesian information criterion (Bayesian Information Criterion, BIC) and the like. The AIC method can give quantitative indexes for weighing complexity of the estimated model and superiority of the fitting sample and transverse comparison among various probability models to be selected, and is a classical model selection method.
Calculating distribution parameters of all random variable probability distribution models and sparse variable probability distribution models based on maximum likelihood estimation (Maximum Likelihood Estimation, MLE) according to the observation data of the uncertainty variables, and determining probability density functions of the random variable probability distribution models and the sparse variable probability distribution models; then, quantitatively evaluating the adaptation degree of each candidate model to the observed data and the fitting quality degree by methods such as KS test, AD test, chi-square test and the like; and the characteristic mode of the uncertainty variable is the candidate model corresponding to the optimal fitting degree.
Specifically, the uncertainty classification characterization modeling method provided by the invention comprises the following steps:
step 1: determining the number of data of pneumatic uncertainty variables (such as Mach number of incoming flow, closed coefficient of turbulence model, etc.), if the number of data is less than the critical sample number n cv (comprehensive reference of the invention χ 2 Checksum AD test takes n cv =4) considering that the current available data is too small, and the statistical significance is lacking, and the probability distribution is not fit, and directly representing the variable as an interval variable; otherwise, executing the step 2.
Step 2: assuming that the uncertainty variable is a random variable, the variable sequentially satisfies 6 common probability distributions including uniform distribution, normal distribution, lognormal distribution, exponential distribution, weibull distribution and extremum distribution. And estimating the distributed parameters by adopting a maximum likelihood estimation method.
Step 3: assuming that the uncertainty variable is a sparse variable, the probability distribution is a weighted sum of the 5 distributions except the uniform distribution, and the probability density function is as shown in the formula (1). Weights w of each distribution i And (3) distributing through the red pool information criterion, and neglecting the candidate probability model with the weight less than 0.1 in order to reduce the complexity of the model. The red pool information criterion tends to select a characterization model with few parameters (simple model) and large maximum likelihood function value (high fitting accuracy).
in the formula ,fi (x) Probability density functions respectively representing normal distribution, lognormal distribution, exponential distribution, weibull distribution and extremum distribution.
The reason why the weighting of the uniform distribution is not considered here is that the probability density of the uniform distribution has a large step at the upper and lower boundary points, and the probability density added to result in a mixed weighted distribution also has two large steps, whereas this form of probability distribution hardly exists in practical problems.
Step 4: determining the probability density function PDF: f (x) and the probability distribution function C of the 7 candidate probability distributionsDF: F (x), a complete mathematical model is obtained. Calculation of the empirical probability distribution F of uncertainty parameters Using the KS test method n (x) And the distance d between the two curves of the candidate probability distribution F (x) KS (illustrative schematic is shown in fig. 4), solving for goodness of fit (p-value) at a specified significance level α.
Wherein n is the number of observed sample data; x is x i (1.ltoreq.i.ltoreq.n) is order sample data (arranged from small to large); sup represents the upper bound; when i=n, there is F n (x n+1 )＝F n (∞)＝1；K n Is KS test statistic.
Step 5: according to the maximum value p of the goodness of fit max And the magnitude relation of the significance level alpha, determining the best characterization form of the uncertainty parameter:
(1) if p max Not less than alpha and p max The corresponding distribution type is single probability distribution, and the uncertainty variable is characterized as a random variable;
(2) if p max Not less than alpha and p max The corresponding distribution type is mixed weighted distribution, and the uncertainty variable is characterized as a sparse variable;
(3) if p max And < alpha, characterizing the uncertainty variable as an interval variable, wherein the upper and lower boundaries of the interval satisfy the following formula:
in the formula ,Iis the lower boundary of the interval,is the upper boundary of the interval; s is(s) min 、s max Respectively to be treated ofDetermining minimum and maximum values, σ, in a sample dataset of characterized aerodynamic variables s Standard deviation of a sample dataset for a pneumatic variable to be determined for a characterization; psi is interval margin, reflects the expansion degree of interval relative to sample boundary, and takes 0.4.
In order to verify the effectiveness of the proposed method, it is applied to the uncertainty characterization of the aerodynamic drag coefficient of a certain aircraft wing. Under the condition that the working conditions are that the attack angle alpha=4 and the Mach number Ma=0.9, the drag coefficient c of the model is known x 7 repeated measurements in FL-21 wind tunnel were 0.0210, 0.0206, 0.0204, 0.0210, 0.0208, 0.0205, 0.0203, respectively, literature gave that the drag coefficient obeyed normal distribution, mean value was 0.0207, standard deviation was 0.00029, focusing on whether the characterization modeling conclusion using the proposed method was consistent with literature published results.
At a significance level of α=5%, table 1 shows the goodness of fit (p-value) for the 7 distributions, it can be seen that the p-value is greater than 5% for the 7 distributions, and the goodness of fit for the normal distribution is greatest, so the final characterization result will give the conclusion that the variable x is a random variable, and the best fit distribution should be a normal distribution. It is also noted that the goodness of fit of the normal distribution is almost the same as the lognormal distribution, and it is also possible to characterize the uncertainty of the drag coefficient as a lognormal distribution, as can be seen from fig. 5, the two probability distribution curves given by the characterization result almost coincide.
TABLE 1 goodness of fit (p-value) of candidate probability distribution models
The result of deducing the distribution parameters of each probability distribution by the maximum likelihood estimation method is shown in the table 2, the average value estimated value of the normal distribution is 0.0207, the standard deviation estimated value is 0.0003, the distribution type and the distribution parameters are very consistent with the literature result, and the effectiveness of the method is verified.
TABLE 2 estimation of probability distribution parameters
From this example, the pneumatic uncertainty classification characterization and mathematical modeling methods of the present invention can be demonstrated to be effective.
After the characterization mode of each aerodynamic variable to be characterized is obtained, aerodynamic design works such as aerodynamic uncertainty characterization and propagation, CFD model confirmation and verification and the like can be carried out on the whole machine, the whole bullet and the like of civil aircraft, fighter plane, unmanned aerial vehicle, guided missile and the like and parts thereof based on the characterization mode.
In summary, the above embodiments are only preferred embodiments of the present invention, and are not intended to limit the scope of the present invention. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.
Claims (9)
1. A method of characterizing pneumatic uncertainty in consideration of classification, comprising:
s1, acquiring a sample data set of a pneumatic variable to be determined and characterized;
s2, assuming the aerodynamic variable to be determined and characterized as a random variable, respectively fitting sample data by adopting a plurality of random variable probability distribution functions to obtain corresponding probability distribution functions; wherein the random variable probability distribution function comprises: uniform distribution, normal distribution, lognormal distribution, exponential distribution, weibull distribution, and extremum distribution; estimating parameters of the probability distribution function by adopting a maximum likelihood estimation method;
s3, assuming that the aerodynamic variable to be determined and characterized is a sparse variable, wherein the probability distribution function is a weighted sum of continuous random variable probability distribution functions in S2, and the probability density function is as shown in formula (1):
wherein ,fi (x) Representing the ith continuous random variable probability density function in S2; n is the number of continuous random variables in S2; w (w) i Is the weight;
s4, calculating the curves and the empirical probability distribution F of each probability distribution function obtained by S2 and S3 by using KS test n (x) Distance d between curves of (2) KS Solving the fitting goodness p of each probability distribution function under the appointed significance level alpha;
wherein n is the number of observed sample data; x is x i (1.ltoreq.i.ltoreq.n) is order sample data (arranged from small to large); sup represents the upper bound; when i=n, there is F n (x n+1 )＝F n (∞)＝1；K n Is KS test statistic; f (x) is each probability distribution function obtained by S2 and S3;
s5, according to the maximum value p in the goodness of fit p max Determining the characterization mode of the aerodynamic variable to be characterized according to the specified significance level alpha:
if p max More than or equal to alpha, the aerodynamic variable to be determined and characterized is characterized as p max Random variables or sparse variables of the corresponding probability distribution functions;
if p max <And alpha, characterizing the aerodynamic variable to be characterized as an interval variable.
2. The classification-considered pneumatic uncertainty characterization method of claim 1, wherein if the number of samples in the sample dataset of the pneumatic variable to be characterized is less than a threshold number of samples n cv It is directly characterized as an interval variable.
3.A classification-considered aerodynamic uncertainty characterization method as claimed in claim 2, wherein 2<n cv <6。
4. The classification-considered aerodynamic uncertainty characterization method of claim 1, wherein in S2, the random variable probability distribution function further comprises: gamma distribution and beta distribution.
5. The classification-considered aerodynamic uncertainty characterization method of claim 1, wherein in S3, the weights of each continuous random variable probability distribution function are assigned by a red pool information criterion or a bayesian information criterion.
6. The classification-considered aerodynamic uncertainty characterization method of claim 5, wherein the probability distribution function of the sparse variables is a continuous random variable probability distribution function with a weight less than 0.1.
7. The method for characterizing aerodynamic uncertainty in consideration of classification as claimed in claim 1, wherein in S5, when characterized as interval variable, the interval range is:
wherein ,Iis the lower boundary of the interval,is the upper boundary of the interval; s is(s) min 、s max Respectively minimum and maximum values, σ, in a sample dataset of aerodynamic variables to be characterized s Standard deviation of a sample dataset for a pneumatic variable to be determined for a characterization; and psi is interval margin and reflects the expansion degree of the interval relative to the sample boundary.
8. An aircraft pneumatic design method, characterized in that a characterization mode of each aerodynamic variable to be determined and characterized is determined by adopting the classification-considered pneumatic uncertainty characterization method according to any one of claims 1-7, and the aircraft pneumatic design is performed based on the characterization mode.
9. The method of aerodynamic design of an aircraft of claim 8, wherein the aircraft comprises: civil aircraft, fighter aircraft, unmanned aerial vehicles, and missiles.
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