CN113051851B - Sensitivity analysis method under mixed uncertainty - Google Patents
Sensitivity analysis method under mixed uncertainty Download PDFInfo
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Abstract
The invention discloses a sensitivity analysis method under mixed uncertainty. Firstly, quantifying mixed uncertainty by using a probability box, and degrading the probability box into a distribution function after separating cognitive uncertainty components in the probability box; after the random uncertain components in the probability box are separated, the probability box is degenerated into an interval number, and the boundary is the average value of the distribution of the upper boundary and the lower boundary of the probability box. Therefore, cognitive and random uncertain components in the input parameters of the flow field can be respectively and independently propagated, and the output response of the flow field is obtained. The method can analyze the interactive influence of cognition and random uncertainty in the flow field input parameters on cognition and random uncertainty in the output response, thereby pertinently reducing one or two uncertainties in the flow field input parameters, saving the time cost and the economic cost of uncertainty design and avoiding the waste of manpower and material resources.
Description
Technical Field
The invention belongs to the technical field of reliability, and particularly relates to a sensitivity analysis method under mixed uncertainty.
Technical Field
Working media of modern large engineering equipment are mostly fluid, hydrodynamic experiments such as wind tunnel experiments of aircrafts and vehicles, water tunnel experiments of naval vessels, wind wave resistance experiments of offshore drilling platforms and the like cannot be carried out in the design process, and the hydrodynamic experiments have very obvious effects in the fields of aerospace, energy engineering, vehicles, ships, power electronics and the like. The fluid mechanics experiment of large-scale engineering equipment is high in cost and time-consuming in the experiment process, and the simple experiment simulation is not suitable for the design of modern large-scale engineering equipment. With the maturing of the computing power of modern computers, the function of the numerical simulation method of the fluid mechanics experiment is increasingly prominent: for example, the Boeing company in the united states uses numerical simulation techniques in the whole machine design of Boeing 787, which saves 30% and 55% of time compared with Boeing 777 and Boeing 767, and the cost is greatly reduced. However, the reliability of the numerical simulation result depends on whether the selected numerical simulation model can accurately reflect the real physical phenomenon and the experimental result. Due to the complexity of the actual physical process of a flow field (such as turbulence, transition, boundary layer separation and gust load) and the deviation of subjective cognition of people, a large amount of random uncertainty and cognitive uncertainty exist in the numerical simulation of the fluid mechanics experiment, and the two uncertainties are often coupled together to form mixing uncertainty, such as incoming flow parameters (incidence angle and Mach number) and turbulence model parameters (Karman constant) with the mixing uncertainty. When these parameters are used as the input of the numerical simulation of the fluid mechanics experiment and the output response is calculated, the output response will fluctuate due to the propagation and accumulation of the mixing uncertainty, and the subsequent structural design using the simulation result will be unreliable. Therefore, a reliable and robust numerical simulation method for a fluid mechanics experiment is urgently needed, the influence of input mixing uncertainty parameters on output response is considered, and an uncertainty margin is designed in advance to guarantee high-reliability structural design.
The core of the numerical simulation of the fluid mechanics experiment is numerical dispersion, however, the numerical dispersion process is relatively complex, the input parameters of the flow field are various (inlet and outlet pressure, turbulence model parameters and the like), and the uncertainty in different input parameters can generate different influences on the output response of the flow field simulation. In most cases, however, only a small number of parameters will have a significant effect on the output response. Therefore, there is an urgent need in engineering for a method of screening important input variables to simplify the computational model and reduce the computational burden imposed by complex fluid mechanics models. Sensitivity analysis is a method for calculating the contribution degree of input variables to the uncertainty of output response, and can realize the importance ranking of the input variables. On the one hand, the unimportant variables can be fixed to a certain constant by the sensitivity analysis to reduce the dimensionality of the input variables; on the other hand, the uncertainty of the output response can be greatly reduced by reducing the uncertainty of the important input variable, so that an important basis is provided for the robust design of a complex system.
The traditional sensitivity analysis method is based on single uncertainty, and the method is rarely researched for the coexistence or mixing uncertainty of multiple uncertainties. At present, the complexity of a large-scale engineering system is gradually increased, and people find that the input variables of the complex system can be represented by the condition of coexistence of multiple uncertainties, and even the random uncertainties and the cognitive uncertainties of some input variables are coupled with each other to form mixed uncertainties. In such a case, continuing to use conventional sensitivity analysis methods will not analyze how important the system input variables are in the presence of multiple uncertainties or mixed uncertainties. Therefore, how to perform sensitivity analysis under mixed uncertainty becomes a research hotspot in academia and industry, and a sensitivity analysis method under mixed uncertainty is urgently needed.
The sensitivity analysis method mainly comprises the following two types: local Sensitivity Analysis (LSA), which is defined as the partial derivative of the system response function at a nominal value to the input variables, and Global Sensitivity Analysis (GSA), which reflects only the Sensitivity information of the response function at the nominal value. The global sensitivity analysis researches the influence degree of the uncertainty of the input variable in the whole value-taking domain on the uncertainty of the output response, is more accurate than the result of the local sensitivity analysis, and is suitable for a relatively complex nonlinear system. Conventional global sensitivity analysis methods include a scanning method, a differential method, variance-based sensitivity analysis, moment-independent sensitivity analysis, information-quantity-based sensitivity analysis, and the like. However, the sensitivity analysis method and the development thereof are both based on the assumption that the input variables only have random uncertainty or cognitive uncertainty, and can perform more accurate sensitivity analysis on the input variables only containing a single type of uncertainty in the complex engineering system. Due to the unclear identification of physical degradation mechanisms of large complex engineering systems and the inherent randomness of the systems, random and cognitive uncertainties are often coupled together, called mixing uncertainty, and none of the above conventional sensitivity analysis methods can be directly applied to mixing uncertainty.
The problem of how to quantify random uncertain components contained when input has mixed uncertainty is rarely studied at home and abroad at present. The mixing uncertainty can be quantified generally using a probability box, and the cognitive uncertainty component it contains can be generally measured in terms of the area of the probability box envelope. The traditional way of quantifying random uncertainty is to use variance, however, the probability box envelops numerous cumulative distribution function curves, and it is difficult to find the variance of one curve to quantify the random uncertainty component contained in the probability box. In addition, when mixed uncertainty exists in input variables, input cognitive and random uncertainty influences output response cognitive and random uncertainty to different degrees, and uncertain interaction influence is generated, and the analysis result of the interaction influence directly influences the robustness optimization design of a system. Therefore, it is necessary to specifically quantify the random uncertainty component and the cognitive uncertainty component in the mixture uncertainty and analyze the interactive effect of uncertainty in the input variables on uncertainty in the output response.
Compared with the common simple system simulation, the fluid mechanics experiment numerical simulation is more complex, and the input parameters of the flow field are more various, so that the method is a great challenge to the sensitivity analysis. Up to now, sensitivity analysis under the mixed uncertainty aiming at the numerical simulation of the hydrodynamics experiment is still blank at home and abroad.
Disclosure of Invention
In order to solve the problems in the prior art, the invention provides a sensitivity analysis method under mixing uncertainty.
The technical scheme adopted by the invention is as follows: a sensitivity analysis method under mixing uncertainty specifically comprises the following steps:
s1: uncertain components of numerical simulation input mixed uncertain parameters of the fluid mechanics experiment are not removed, uncertainty propagation is carried out through a two-layer nesting algorithm, output response distribution is established, and the area and the maximum variance of the output response are obtained;
s2: removing uncertain components of numerical simulation input mixed uncertain parameters of a fluid mechanics experiment, carrying out uncertain propagation through a two-layer nesting algorithm, establishing flow field output response distribution and obtaining the area and the maximum variance of the output response;
s3: and performing interactive sensitivity analysis on each input parameter of the flow field by using the flow field output response probability box area and the maximum variance obtained in the steps S1 and S2 to obtain the importance ranking of each input parameter.
Further, step S1 specifically includes the following sub-steps:
s101: outer layer of two-layer nested algorithm for parameterization of probability box
Quantifying mixed uncertainty in input parameters, performing sampling propagation random uncertainty in a high-dimensional probability space by using an LHS method, and obtaining N through mapping s Interval value sequence composed of group flow field input parameters
S102: establishing a numerical simulation model g (X) of a fluid mechanics experiment, wherein X is an input parameter of the flow field in the step S101, g (X) is taken as a target function in the inner layer of the two-layer nested algorithm, each input parameter is taken as a decision variable, and the interval value sequence obtained in the step S101 is taken as a decision variable
Establishing an interval value constrained optimization model for a feasible domain, maximizing and minimizing the established objective function, and obtaining N through optimization s Interval of group flow field output response
As shown in fig. 2, the two-layer nested algorithm includes an outer layer and an inner layer, where the outer layer obtains N by using Latin Hypercube Sampling (LHS) s Grouping intervals of flow field input parameters, and spreading random uncertainty; inner layer utilization flowEstablishing an optimization model by using a numerical simulation model g (X) of a body mechanics experiment, and obtaining N through optimization s And (4) forming a response interval of flow field output, and propagating cognitive uncertainty. The solution method of the inner layer optimization includes, but is not limited to, a simplex method, an inner point method and other algorithms.
S103: establishing an empirical probability box of the output response according to the output response interval obtained in the step S102
Here, an Empirical Probability box (Empirical P-box) refers to a Probability box whose boundary Distribution is determined by an Empirical Cumulative Distribution Function (Empirical circumferential Distribution Function, Empirical CDF).
S104: empirical probability box obtained according to step S103
Establishing the area of the flow field output response probability box
And maximum variance
The boundary distribution of the empirical probability box is formed by empirical cumulative distribution, and when the two-layer nested algorithm is adopted, the outer layer sampling quantity N is s Approaching infinity, the empirical probability box will approximate the true output response probability box.
The "probability box maximum variance" is found by the maximum variance CDF. As shown in FIG. 3, the probability box parameters are exemplified by the normal probability box
The flow field output response probability box can be constructed by the interval number obtained by optimizing the inner layer of a two-layer nesting algorithm, and the cumulative probability corresponding to the intervals meets the requirement
The maximum variance CDF includes the following three CDF curves:
section I: output response probability box upper bound
Where y is an element of [ y ∈ [ 0 ,y m ]A segment in which y 0 Is the upper boundary of the probability box
The leftmost endpoint of (a); y is m Is the upper boundary of the probability box
A breakpoint is to be found, satisfy
Wherein
Distributing functions for upper boundaries
Inverse function of p m Is y m A corresponding cumulative probability;
and a section II: uniformly distributed y in CDF belongs to [ y ∈ [) m ,y m+1 ]A segment in which, in the case of a segment,
is the lower boundary of the probability box
And cumulative probability p m+1 The right end of the corresponding response interval. It is noted that y m And y m+1 The corresponding cumulative probability values are adjacent, so only y needs to be determined m The value of (a).
Section III: output response probability box lower boundF Y (y) in
Wherein, the first and the second end of the pipe are connected with each other,
is the lower boundary of the probability boxF Y The rightmost endpoint of (y).
The specific establishment process of the maximum variance CDF provided by the invention is as follows:
sampling the maximum variance CDF to obtain points and corresponding cumulative probabilities on the section i and the section iii:
i.e. y 0:m Represents section I from y 0 To y m Each output response of the flow field, p 0:m Is y 0:m The corresponding cumulative probability.
Represents a segment III from y m+1 To
Each of the output responses of the flow field of (c),
is composed of
The corresponding cumulative probability.
The sampling point on the section II is special compared with the sections I and III, and the response value of the section II is positioned at the breakpoint y to be solved m And response values y adjacent to their corresponding cumulative probabilities m+1 In the meantime. y is m And y m+1 Determined by the inner optimization of the two-layer nesting algorithm, but the section ii between the two is not determined by the inner optimization. When the outer layer sampling quantity N s Approaching infinity, y m And y m+1 Distance y between m+1 -y m I will approach 0, so the invention ignores y m And y m+1 The curvature of the CDF curve changes, the CDF is uniformly distributed as a distribution function of the section II, and the number of sampling points on the section II is proportional to the cumulative probability of the corresponding section:
namely, it is
Representing an ceiling function.
Since the segment ii is a segment of the CDF with uniform distribution, a linear interpolation method can be used to obtain a sampling point and a corresponding cumulative probability of the segment ii:
the goal is to find the breakpoint y m The variance of the sample set of sample points is maximized:
namely, it is
Represents the difference between the i +1 th cumulative probability and the i-th cumulative probability of the j (j ═ 1,2,3) th segment;
means j (j is 1,2,3) th section and cumulative probability
Outputting a response value by the corresponding ith flow field; μ is the mean of the samples:
the following constraints are established:
and then the maximum variance of the numerical simulation output response probability box of the fluid mechanics experiment is obtained.
Further, step S2 includes the following steps:
s201: respectively and independently eliminating uncertain components (cognitive uncertain components and random uncertain components) of numerical simulation input mixed uncertain parameters of the fluid mechanics experiment, performing sampling propagation random uncertainty in a high-dimensional probability space by using an LHS (left-right hand learning) method at the outer layer of a two-layer nested algorithm, and obtaining N through mapping s Interval value sequence composed of group flow field input parameters (incoming flow incidence angle, Mach number and the like)
The variation form after removing the cognitive uncertainty of the flow field input parameters is shown in fig. 4, and when the cognitive uncertainty of the flow field input parameters is removed, the probability box degenerates into an enveloped cumulative distribution function; when the random uncertainty of the flow field input parameters is eliminated, the probability box is degenerated into an interval number, and the boundary is the mean value of the boundary distribution of the probability box.
S202: establishing a fluid mechanics experiment numerical simulation model g (X '), wherein X' is a flow field input parameter after uncertainty (cognition/randomness) is eliminated in the fluid mechanics experiment numerical simulation. The inner layer of the two-layer nested algorithm takes the model as a target function, takes each input parameter as a decision variable, and takes the flow field input parameter interval value sequence obtained in S201
Establishing an interval value constrained optimization model for a feasible region, maximizing and minimizing the established objective function, and obtaining N through optimization s Interval of group flow field output response (resistance raising coefficient, transition position, etc.)
S203: respectively establishing an empirical probability box for flow field output response after removing cognitive uncertainty components and random uncertainty components in the ith input parameter
And
s204: according to the obtained experience probability box
And
deriving the area of the output response probability box
And maximum variance
Wherein the content of the first and second substances,
and the j-th condition of output response after the cognitive uncertain components of the ith flow field input parameters are eliminated is shown.
And (4) representing the output response after the random uncertain components of the ith flow field input parameter are removed.
Further, the specific process of step S3 is as follows:
s301: input parameters by eliminating flow fieldArea of output response probability box after cognition uncertainty
And maximum variance
Respectively establishing interactive sensitivity indexes of input cognition to output cognition and input cognition to output randomness:
namely, it is
And
the area and the maximum variance of the response probability box are respectively output when the uncertainty components of the mixed uncertainty parameters are input in the numerical simulation of the fluid mechanics experiment without being removed in the step S104.
S302: area of output response probability box after eliminating random uncertainty of flow field input parameter
And maximum variance
Respectively establishing interactive sensitivity indexes of input random to output random and input random to output cognition:
s303: and sorting the importance degrees of the influence of each input parameter of the flow field on the output response according to the interactive sensitivity indexes obtained in S301 and S302.
S304: the method of're-averaging Bootstrap self-service sampling' is utilized, namely input cognition pair output cognition pair obtained by selecting different sampling sample number pairs from a sampling sample pool BS
Interactive sensitivity index, input-cognitive versus output randomness
Performing Bootstrap self-service sampling on the interactive sensitivity indexes, calculating the mean value and standard deviation of the two sensitivity indexes, extracting the sample set, averaging the mean value set and the standard deviation set, and taking the average value set and the standard deviation set as corresponding final sensitivity indexes
And
the invention has the beneficial effects that: the method can separate cognitive and random uncertain components in the complex flow field input mixed uncertain parameters, and provides support for simplifying the complexity of the flow field input parameters. The flow field environment under the real working condition is very complex, and the flow field parameters (inflow parameters, turbulence parameters and the like) have mixing uncertainty due to the mutual coupling of environmental disturbance (random uncertainty) and cognitive deviation (cognitive uncertainty). The method comprises the steps of firstly, quantifying mixed uncertainty by using a probability box, and degrading the probability box into a distribution function after cognitive uncertainty components in the probability box are separated; after the random uncertain components in the probability box are separated, the probability box is degenerated into an interval number, and the boundary is the average value of the distribution of the upper boundary and the lower boundary of the probability box. Therefore, cognitive and random uncertain components in the input parameters of the flow field can be respectively and independently propagated, and the output response of the flow field is obtained.
The method can analyze one or more input parameters which have large influence on the output response of the complex flow field, and provides guarantee for pertinently reducing the uncertainty of the input parameters of the flow field and realizing the high-efficiency, reliable and steady structural design. Considering the complex and changeable flow field environment, the previous deterministic or single-type uncertain flow field input parameter setting cannot meet the structural design of modern large-scale equipment. The method can analyze the interactive influence of cognition and random uncertainty in the flow field input parameters on cognition and random uncertainty in the output response, thereby pertinently reducing one or two uncertain components in the flow field input mixed uncertainty parameters, saving the time cost and the economic cost of uncertainty design and avoiding the waste of manpower and material resources.
Drawings
FIG. 1 is a flow chart of the method of the present invention;
FIG. 2 is a schematic diagram of a two-layer nested propagation method of the present invention;
FIG. 3 is a schematic diagram of the maximum variance of the numerical simulation output probability box of the fluid mechanics experiment of the present invention;
FIG. 4 is a schematic diagram of a hybrid uncertainty decoupling method of the present invention;
FIG. 5 is a diagram showing the relationship among the numerical simulation, sensitivity analysis and uncertainty analysis of the fluid mechanics experiment;
FIG. 6 is a schematic diagram of a far-field grid structure for numerical simulation of wind tunnel tests according to an embodiment of the present invention.
FIG. 7 is a schematic diagram of an output response when cognitive uncertainty in wind tunnel test incoming flow parameters is removed according to an embodiment of the present invention;
FIG. 8 is a schematic diagram of an output response when cognitive uncertainty in wind tunnel test turbulence model parameters is rejected in an embodiment of the present invention;
FIG. 9 is a schematic diagram of an output response when random uncertainties in wind tunnel test incoming flow parameters are eliminated according to an embodiment of the present invention;
FIG. 10 is a schematic diagram of an output response when random uncertainties in wind tunnel test turbulence model parameters are eliminated according to an embodiment of the present invention.
Detailed Description
As shown in fig. 1, the wind tunnel test is used as an embodiment of the fluid mechanics experiment, and the method comprises the following steps:
s1: establishing output response distribution under the condition that the uncertainty of the input parameters of the wind tunnel test is not eliminated, and obtaining the area and the maximum variance of a flow field output response probability box;
s2: establishing output response distribution under the condition of eliminating uncertainty of input parameters of a wind tunnel test, and obtaining the area and the maximum variance of a flow field output response probability box;
s3: and (4) carrying out sensitivity sequencing on the flow field input parameters by adopting the obtained area and maximum variance of the flow field output response, and obtaining importance sequencing of the input parameters on the flow field output response.
The following takes the analysis of the influence of uncertainty of 7 input parameters, such as main parameters of A spalar-almiras (S-A) turbulence model and incoming flow parameters (incidence angle and mach number), on the simulation of the nacA0012 airfoil profile as an example, and the three steps are described in detail:
first, the geometric parameters of the far field where the airfoil profile is located and the input parameters of the model according to the embodiment of the present invention are described, and the present invention is applicable to the parameter setting that satisfies the corresponding objective facts and the physical principles.
As shown in fig. 6, a far-field C-type grid is drawn for the NACA0012 airfoil, with the left boundary of the grid being 15 chord lengths away from the leading edge of the airfoil and the right boundary being 20 chord lengths away from the leading edge of the airfoil. The specific grid parameters are shown in table 1:
TABLE 1
The input parameters of the wind tunnel test are shown in table 2 and table 3, table 3 is the S-A turbulence model parameter, table 4 is the incoming flow parameter, and the mixing uncertainty input is considered and quantified by A parameterized probability box.
TABLE 2
TABLE 3
| Mean value | Standard deviation of | |
| Mach number | [0.65,0.7] | [0.02,0.04] |
| Angle of attack | [4°,5.5°] | [0.3,0.5] |
The following describes the steps of an embodiment of the present invention:
(1) the uncertain components of each input parameter of the wind tunnel test are not removed, and the output response (lift-drag ratio C) is established L /C D Etc.) and obtains the area of the output response probability box and the maximum variance index.
a. Sampling probability spaces in which 7 input parameters such as attack angle, Mach number and the like in 5 turbulence model parameters and incoming flow parameters in the SA turbulence model are located by using an LHS (left hand right) sampling method to obtain 10 5 A set of 7-dimensional probability samples P.
b. Probability box according to 7 input parameters will 10 5 Mapping the group probability samples to the value space of each input parameter of the flow field to obtain 10 5 7-dimensional interval value sample of group flow field input parameter valuesSet [ X ]]。
c. According to a two-layer nesting algorithm, the random uncertainty in the input parameters of the wind tunnel test is propagated at the outer layer, the cognitive uncertainty in the input parameters of the wind tunnel test is propagated at the inner layer, and the output response (lift-drag ratio C) is obtained L /C D Etc.) 10 of 5 Group interval.
d. According to the obtained 10 5 And (4) group interval, constructing the distribution of wind tunnel test output response (lift-drag ratio and the like) through an empirical probability box.
e. Establishing wind tunnel test output response (lift-drag ratio C) L /C D Etc.), maximum variance.
(2) Eliminating the cognitive and random uncertain components of each input parameter of the wind tunnel test and establishing the output response (lift-drag ratio C) of the wind tunnel test L /C D Etc.) and obtains the area, maximum variance, of the output response probability box.
a. Respectively and independently eliminating the cognitive uncertainty components in 7 wind tunnel test input parameters, obtaining 100 groups of CDF curves obtained by eliminating the cognitive uncertainty, and sampling the probability space of each newly obtained input parameter by using an LHS (left-hand Log) sampling method to obtain 10 5 A set of 7-dimensional probability samples.
As shown in table 4, the mach number μ ∈ [0.65,0.7] is input in the wind tunnel test; σ ∈ [0.02,0.04] for example: after the cognitive uncertainty is eliminated, the parameters of the CDF are respectively selected randomly from the intervals [0.65 and 0.7] and the intervals [0.02 and 0.04], and a plurality of CDF curves belonging to the same distribution function family are obtained.
b. 10 to be obtained 5 The 7-dimensional probability samples are mapped to the value space of the wind tunnel test input parameters, and the wind tunnel test output response (lift-drag ratio C) is constructed by an empirical probability box through a two-layer nested algorithm L /C D Etc.).
c. And respectively obtaining the area and the maximum variance of the output response probability box when 7 wind tunnel test input parameter cognitive uncertainty components are eliminated.
d. Random uncertain components in 7 wind tunnel test input parameters are respectively and independently removed, and the area and the maximum variance of an output response probability box when the random uncertain components of the 7 input parameters are removed are finally obtained by the method.
As shown in table 4, the mach number of the input parameter of the wind tunnel test is taken as an example: after the random uncertainties are eliminated, the original probability box degenerates to the interval [0.65,0.7], only one case.
(3) And establishing an interactive sensitivity index under mixed uncertainty of analyzing the input parameters of the wind tunnel test to the output response importance, and obtaining the importance sequence of the input parameters to the output response influence in the wind tunnel test.
a. Establishing 100 groups of input cognition pairs to output cognition by utilizing the area and the maximum variance of the output response probability box obtained by removing the cognition uncertainty components in the 7 wind tunnel test input parameters
Input cognition versus output randomization
The interactive sensitivity index of (1).
b. Performing're-averaging Bootstrap self-sampling', namely {2 multiplied by 10 } sampling sample pool BS 5 :2×10 4 :4×10 5 Selecting different sampling sample quantities to perform Bootstrap self-service sampling, calculating the mean value and standard deviation of the two sensitivity indexes, extracting the sample set, averaging the mean value set and the standard deviation set, and taking the average value set and the standard deviation set as corresponding final sensitivity indexes
And
c. the area and the maximum variance of an output response probability box obtained by removing random uncertain components in 7 wind tunnel test input parameters are utilized to establish input random pair output random
Input random pair output awareness
The interactive sensitivity index of (1).
d. 4 interactive sensitivity indexes obtained by the utilization
The importance of 7 wind tunnel test input parameters consisting of 5 turbulence model parameters and 2 incoming flow parameters to the output response is ranked, the results are respectively shown in tables 4-7, and the corresponding output response probability boxes are shown in fig. 7-10.
FIGS. 7-8 show output response lift-to-drag ratio C in removing cognitive uncertainty in wind tunnel test input parameters L /C D In which: FIG. 7(a) and FIG. 7(b) show the output response lift-drag ratio C in eliminating the cognitive uncertainty in the input incoming flow parameter Mach number and angle of attack of the wind tunnel test L /C D 8(A) -8 (e) show the output response lift-to-drag ratio C in the rejection of cognitive uncertainty in the input S-A turbulence model parameters, respectively L /C D A change in (c). As can be seen from fig. 7 to 8, when the cognitive uncertainty of the wind tunnel test input parameter is eliminated, compared with the area of the output response lift-drag ratio probability box without the cognitive uncertainty, the maximum variance is reduced but the degree of reduction is different, the incoming flow parameters (mach number and attack angle) represented in fig. 7 have a larger influence on the output response, while the turbulence model parameters represented in fig. 8 have a relatively smaller influence on the output response, and the specific index quantization is shown in tables 4 to 5.
FIGS. 9-10 show the output response lift-to-drag ratio C in the rejection of random uncertainties in wind tunnel test input parameters L /C D In which: FIG. 9(a) and FIG. 9(b) show the output response lift-drag ratio C in eliminating the random uncertainties in the wind tunnel test input incoming flow parameter Mach number and angle of attack L /C D 10(A) -10 (e) respectively show the output response lift-to-drag ratio C in the rejection of random uncertainties in the input S-A turbulence model parameters L /C D A change in (c). As can be seen from FIGS. 9-10, when the random uncertain components of the input parameters of the wind tunnel test are eliminated, the random uncertain components are not eliminated compared with those without eliminationThe areas and the maximum variances of the deterministic result output response lift-to-drag ratio probability boxes are reduced, but the reduction degrees are different, which shows that the random uncertainty components of different flow field input parameters have different influences on the output response, and the specific index quantization is shown in tables 6-7.
TABLE 4
| Sorting | Inputting | Sensitivity index | |
| 1 | Mach number | 0.5321 (0.0015) | |
| 2 | Angle of attack | 0.3684 (0.0015) | |
| 3 | c v1 | 0.0136 (6.91e-4) | |
| 4 | κ | 0.0084 (1.52e-4) | |
| 5 | c b1 | 0.0078 (1.96e-4) | |
| 6 | c w2 | 0.0061 (1.22e-4) | |
| 7 | σ | 0.0024 (3.40e-5) |
TABLE 5
| Sorting | Inputting | Sensitivity index | |
| 1 | Mach number | 0.2743 (0.0025) | |
| 2 | Angle of attack | 0.1780 (0.0012) | |
| 3 | c v1 | 0.0109 (6.62e-4) | |
| 4 | κ | 0.0061 (1.19e-4) | |
| 5 | c b1 | 0.0056 (1.56e-4) | |
| 6 | c w2 | 0.0055 (1.11e-4) | |
| 7 | σ | 0.0021 (1.98e-5) |
TABLE 6
| Sorting | Inputting parameters | Sensitivity index |
| 1 | c v1 | 0.6405 |
| 2 | c b1 | 0.6388 |
| 3 | κ | 0.2444 |
| 4 | Mach number | 0.1653 |
| 5 | σ | 0.0668 |
| 6 | Angle of attack | 0.0181 |
| 7 | c w2 | 3.12e-4 |
TABLE 7
| Sorting | Inputting parameters | Sensitivity index |
| 1 | c v1 | 0.6983 |
| 2 | c b1 | 0.6958 |
| 3 | κ | 0.2481 |
| 4 | σ | 0.0764 |
| 5 | Mach number | 0.0150 |
| 6 | Angle of attack | 0.0142 |
| 7 | c w2 | 5.67e-4 |
Claims (4)
1. A sensitivity analysis method under mixing uncertainty specifically comprises the following steps:
s1: the uncertainty components of the fluid mechanics experiment numerical simulation input mixed uncertainty parameters are not removed, uncertainty propagation is carried out through a two-layer nesting algorithm, output response distribution is established, and the area and the maximum variance of an output response probability box are obtained;
the method specifically comprises the following steps:
s101: two-layer nested algorithm outer-layer to parameterize probability boxes
Quantifying mixed uncertainty in flow field input parameters, using Latin Hypercube Sampling (LHS) method to sample and spread random uncertainty in high-dimensional probability space, and obtaining N through mapping s Interval value sequence composed of group flow field input parameters
S102: establishing a numerical simulation model g (X) of a fluid mechanics experiment, wherein X is an input parameter of the flow field in the step S101, g (X) is used as a target function in the inner layer of the two-layer nested algorithm, each input parameter is used as a decision variable, and the interval value sequence obtained in the step S101 is used as a decision variable
Establishing an interval value constrained optimization model for a feasible domain, maximizing and minimizing the established objective function to obtain N s Interval of group flow field output response
S103: establishing an empirical probability box of the output response according to the output response interval obtained in the step S102
S104: empirical probability box obtained according to step S103
Obtaining the area of the flow field output response probability box
And maximum variance
S2: removing uncertain components of numerical simulation input mixed uncertain parameters of a fluid mechanics experiment, performing uncertain propagation through a two-layer nesting algorithm, establishing flow field output response distribution and obtaining the area and the maximum variance of an output response probability box;
the method specifically comprises the following steps:
s201: respectively and independently eliminating in the numerical simulation of the hydrodynamics experimentInputting uncertain components of mixed uncertain parameters into a flow field, performing sampling propagation random uncertainty in a high-dimensional probability space by using an LHS (length of sales) method at the outer layer of a two-layer nested algorithm, and obtaining N through mapping s Interval value sequence composed of group flow field input parameters
S202: establishing a fluid mechanics experiment numerical simulation model g (X '), wherein X' is a flow field input parameter after uncertainty is eliminated in fluid mechanics experiment numerical simulation; the inner layer of the two-layer nested algorithm takes the model as a target function, takes each input parameter as a decision variable, and takes the flow field input parameter interval value sequence obtained in S201
Establishing an interval value constrained optimization model for a feasible domain, maximizing and minimizing the established objective function to obtain N s Interval of group flow field output response
S203: respectively establishing an empirical probability box for flow field output response after removing cognitive uncertainty components and random uncertainty components in the ith input parameter
And
s204: according to the obtained experience probability box
And
obtaining the area of the output response probability box
And maximum variance
Wherein the content of the first and second substances,
represents the jth situation of output response after the identification uncertainty component of the ith flow field input parameter is removed,
the output response after the random uncertain components of the ith flow field input parameters are removed is shown;
s3: carrying out interactive sensitivity analysis on each input parameter of the flow field by using the area and the maximum variance of the flow field output response probability box obtained in the steps S1 and S2 to obtain importance ranking of each input parameter;
the specific process is as follows:
s301: area of output response probability box after eliminating flow field input parameter cognitive uncertainty
And maximum variance
Respectively establishing interactive sensitivity indexes of input cognition to output cognition and input cognition to output randomness:
s302: after random uncertainty of input parameters of flow field is eliminatedArea of output response probability box
And maximum variance
Respectively establishing interactive sensitivity indexes of input random to output random and input random to output cognition:
s303: according to the interactive sensitivity indexes obtained in S301 and S302, ranking the importance of the influence of each input parameter of the flow field on the output response;
s304: by using a re-averaging Bootstrap self-service sampling method, namely selecting different sampling sample numbers from a sampling sample pool BS to obtain an input cognition interactive sensitivity index to output cognition
Input-cognitive versus output random interactive sensitivity index
Performing Bootstrap self-service sampling, calculating the mean value and standard deviation of the re-extracted sample set of the two sensitivity indexes, and averaging the mean value set and the standard deviation set to obtain the corresponding final sensitivity index
And
2. the sensitivity analysis method under the mixing uncertainty according to claim 1, characterized in that the uncertainty components are specifically: cognitive uncertainty components and random uncertainty components.
3. The method according to claim 2, wherein the empirical probability box specifically refers to: the boundary distribution is a probability box of an empirical cumulative distribution.
4. The method of claim 3, wherein the maximum variance of the probability box of step S104 is determined by the maximum variance of the probability box
The maximum variance CDF is obtained by a maximum variance Cumulative Distribution Function (CDF), and includes the following three CDF curves:
section I: output response probability box upper bound
Where y is an element of [ y ∈ [ 0 ,y m ]A segment in which y 0 Is the upper boundary of the probability box
The leftmost end point of; y is m Is the upper boundary of the probability box
A breakpoint to be solved is obtained by the inverse function of the upper boundary distribution function, and satisfies
Wherein
Distributing functions for upper boundaries
The inverse function of (d);
and a section II: uniformly distributed y in CDF belongs to [ y ∈ [) m ,y m+1 ]A segment in which, in the case of a segment,
is the lower boundary of the probability boxF Y (y) and cumulative probability p m+1 The right end point of the corresponding response quantity interval;
section III: output response probability box lower boundF Y (y) in
Wherein the content of the first and second substances,
is the lower boundary of the probability boxF Y (y) rightmost endpoint;
the specific process of establishing the maximum variance CDF is as follows:
sampling the maximum variance CDF to obtain each point on the section I and the section III and the corresponding cumulative probability:
y (1) ＝y 0:m
p (1) ＝p 0:m
wherein, y 0:m Indicates section I from y 0 To y m Each output response of the flow field, p 0:m Is y 0:m The corresponding cumulative probability of the event is,
represents a segment III from y m+1 To
Each of the output responses of the flow field of (c),
is composed of
A corresponding cumulative probability;
the response quantity value on the section II is positioned at the breakpoint y to be solved m And response values y adjacent to their corresponding cumulative probabilities m+1 C, y m And y m+1 Determined by the inner optimization of a two-layer nesting algorithm, y m And y m+1 The CDF is uniformly distributed as the distribution function of the section ii, and the number of sampling points on the section ii is proportional to the cumulative probability of the corresponding section:
namely, it is
Represents an upward rounding function;
since the segment ii is a segment of the CDF with uniform distribution, a linear interpolation method can be used to obtain a sampling point and a corresponding cumulative probability of the segment ii:
the goal is to find the breakpoint y m The variance of the sample set of sample points is maximized:
i.e., μ is the mean of the sample points, expressed as:
the following constraints are established:
and then the maximum variance of the numerical simulation output response probability box of the fluid mechanics experiment is obtained.
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