CN116305593B - Global sensitivity analysis method with strong portability - Google Patents
Global sensitivity analysis method with strong portability Download PDFInfo
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- CN116305593B CN116305593B CN202310580529.4A CN202310580529A CN116305593B CN 116305593 B CN116305593 B CN 116305593B CN 202310580529 A CN202310580529 A CN 202310580529A CN 116305593 B CN116305593 B CN 116305593B
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- G06F30/00—Computer-aided design [CAD]
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- G06F30/15—Vehicle, aircraft or watercraft design
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- Y02T—CLIMATE CHANGE MITIGATION TECHNOLOGIES RELATED TO TRANSPORTATION
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Abstract
The invention discloses a global sensitivity analysis method with strong portability, which belongs to the technical field of aircraft design, and aims to accurately determine the influence of uncertainty variables on aircraft design in the existing aircraft design process, and comprises the following steps: s1: establishing an aircraft design model; s2: performing uncertainty variable analysis on the aircraft design model to obtain an analysis result; s3: establishing a mapping model of the analysis result and the global sensitivity analysis result; s4: according to the mapping model, establishing a sensitivity measurement Sobol index based on direct variance decomposition; s5: and obtaining a global sensitivity analysis result according to the analysis result and a sensitivity measurement Sobol index value corresponding to the analysis result.
Description
Technical Field
The invention relates to the technical field of aircraft design, in particular to a global sensitivity analysis method with strong portability.
Background
The uncertainty propagation process examines the effect of the uncertainty of the model input on the output, but cannot calculate the degree of contribution of each uncertainty variable to the output. In uncertainty analysis, the forward process is the propagation of uncertainty from input to output, and the reverse process is sensitivity analysis, i.e., the degree of sensitivity of an output variable to input uncertainty is analyzed. Sensitivity analysis can be divided into two major categories, namely local sensitivity analysis and global sensitivity analysis, and each has advantages and disadvantages.
The local sensitivity analysis focuses on the local influence of parameters on the model, and mainly takes the derivative of the output of the model at a nominal value and the input as a sensitivity index. The local sensitivity analysis has the advantages of convenient calculation and higher calculation efficiency. The solving method mainly comprises a finite difference method, a direct derivation method, a green function method and the like. When the variation range of the model parameters is small and the output and input of the model are in a linear relationship, namely the model is a linear model, the local sensitivity analysis is practical and has higher precision. However, local sensitivity analysis also has its limitations. First, local sensitivity analysis is difficult to evaluate parameters with uncertainty, because it can only calculate the nominal sensitivity, and cannot examine the entire parameter variation range. In most cases, local sensitivity analysis sometimes fails to provide information about the effect of a change in the range of a parameter on the model when the parameter is changing within an uncertainty range, which can only provide information about a point within the parameter definition domain. Second, the input parameters of the model cannot vary too much and the model nonlinearity cannot be too strong. In addition, local sensitivity does not allow the calculation of the mutual coupling between the individual parameters.
The global sensitivity analysis can overcome the limitation of local sensitivity, not only considers the influence of the distribution and the form of probability density functions of each factor, but also can change all factors simultaneously during analysis, and can be free from the limitation of a model and examine the contribution degree of input parameters to output in the whole parameter change range.
Disclosure of Invention
The invention aims to provide a global sensitivity analysis method with strong portability, so as to accurately determine the influence of uncertainty variables in the existing aircraft design process on the aircraft design.
The technical scheme for solving the technical problems is as follows:
the invention provides a global sensitivity analysis method with strong portability, which comprises the following steps:
s1: establishing an aircraft design model;
s2: performing uncertainty variable analysis on the aircraft design model to obtain an analysis result;
s3: establishing a mapping model of the analysis result and the global sensitivity analysis result;
s4: according to the mapping model, establishing a sensitivity measurement Sobol index based on direct variance decomposition;
s5: and obtaining a global sensitivity analysis result according to the analysis result and a sensitivity measurement Sobol index value corresponding to the analysis result.
Optionally, in the step S3, the mapping model is:
wherein, the liquid crystal display device comprises a liquid crystal display device,for global sensitivity analysis results,/->Is the analysis result and isdVector of uncertainty variable inputs.
Optionally, the S4 includes:
s41: assuming that the inputs are independently and uniformly distributed in the unit hypercube, decomposing the mapping model to obtain a decomposed mapping model;
s42: defining all decomposition terms of the decomposed mapping model according to the condition expected values;
s43: squaring all the decomposition terms and integrating to obtain a new function model;
s44: obtaining a decomposed variance expression model according to the new function model;
s45: and establishing a sensitivity measurement Sobol index based on direct variance decomposition by using the decomposed variance expression model.
Optionally, in S41, the decomposed mapping model is:
wherein, the liquid crystal display device comprises a liquid crystal display device,is constant (I)>Is->Function of->Is->And->Is a function of (a) and (b),das the number of uncertainty variables,is the firstiUncertainty variable and->∈[0，1]，i=1,2, · ·j · ,d。
Optionally, in S42, all decomposition terms are:
wherein, the liquid crystal display device comprises a liquid crystal display device,is constant (I)>Is->Function of->Is->And->Is a function of (a) and (b),das the number of uncertainty variables,is the firstiUncertainty variable and->∈[0，1]，i=1,2, · ·j · ,d，dAs the number of uncertainty variables,for mathematical expectations +.>Mathematical expectations for output->For the presence of uncertainty variable only +.>Is used as a condition for the (c) of the (c),for presence of uncertainty variable->、/>Is desirable for the conditions of (2).
Optionally, in S43, the new function model is:
wherein, the liquid crystal display device comprises a liquid crystal display device,for the variance of the global sensitivity analysis results, +.>Is the analysis result and isdVector of individual uncertainty variable inputs, +.>Is constant (I)>Is->Function of->Is->And->Is a function of (a) and (b),dis the number of uncertainty variables, +.>Is the firstiUncertainty variable and->∈[0，1]，i=1,2, · ·j · ,d，dIs the number of uncertainty variables, +.>Indicating the number->Representing a first input uncertainty variable, +.>Representing the s-th input uncertainty variable.
Optionally, in S44, the decomposed variance expression model is:
wherein, the liquid crystal display device comprises a liquid crystal display device,the total variance is represented as such,dis the number of uncertainty variables, +.>Representing uncertainty variable +.>Independent contribution to the output variance, and +.>，/>Representing uncertainty variable +.>、/>Coupling contribution to output variance, and +.>，/>Indicating output, ->Indicating the presence of uncertainty variable +.>Total contribution to output variance, +.>Representing uncertainty variable +.>、/>Total contribution to output variance, +.>Indicating that only uncertainty variable +.>Mathematical expectations of time, < >>Indicating the presence or absence ofDeterministic variable->Condition expectations at time->Indicating the presence of uncertainty variable +.>、/>Mathematical expectations of time, < >>Indicating the presence of uncertainty variable +.>、/>Condition expectations at time->Representing uncertainty variable +.>Independent contribution to total variance, +.>Is the firstiUncertainty variable and->∈[0，1]，i=1,2, · ·j · ,d。
Alternatively, the sensitivity metric Sobol index based on direct variance decompositionThe method comprises the following steps:
wherein, the liquid crystal display device comprises a liquid crystal display device,representing the total variance>Representing uncertainty variable +.>Independent contributions to output variance, and，/>indicating output, ->Indicating the presence of uncertainty variable +.>Total contribution to output variance, +.>Indicating the presence of only uncertainty variablesX i Mathematical expectations of time, < >>Indicating the presence of uncertainty variable +.>Condition expectations at time->Is the firstiUncertainty variable and->∈[0，1]，i=1,2, · ·j · ,d。
The invention has the following beneficial effects:
the global sensitivity analysis method provided by the invention can be applied to the design of an aircraft on one hand, and has strong portability no matter whether the layer of wings are full turbulence or laminar flow; on the other hand, the limitation of local sensitivity can be overcome, the influence of the distribution and the form of probability density functions of all factors is considered, all factors can be changed simultaneously during analysis, the influence of uncertainty variables in the existing aircraft design process on the aircraft design can be accurately determined without being limited by a model and by examining the contribution degree of input parameters to output in the whole parameter change range.
Drawings
FIG. 1 is a flow chart of a global sensitivity analysis method with strong portability of the present invention.
Detailed Description
The principles and features of the present invention are described below with reference to the drawings, the examples are illustrated for the purpose of illustrating the invention and are not to be construed as limiting the scope of the invention.
The invention provides a global sensitivity analysis method with strong portability, which is shown in a reference figure 1 and comprises the following steps:
s1: establishing an aircraft design model;
s2: performing uncertainty variable analysis on the aircraft design model to obtain an analysis result;
s3: establishing a mapping model of the analysis result and the global sensitivity analysis result;
the mapping model is as follows:
wherein, the liquid crystal display device comprises a liquid crystal display device,for global sensitivity analysis results,/->Is the analysis result and isdVector of uncertainty variable inputs.
S4: according to the mapping model, establishing a sensitivity measurement Sobol index based on direct variance decomposition;
optionally, the S4 includes:
s41: assuming that the inputs are independently and uniformly distributed in the unit hypercube, decomposing the mapping model to obtain a decomposed mapping model;
the decomposed mapping model is as follows:
wherein, the liquid crystal display device comprises a liquid crystal display device,is constant (I)>Is->Function of->Is->And->Is a function of (a) and (b),das the number of uncertainty variables,is the firstiUncertainty variable and->∈[0，1]，i=1,2, · ·j · ,d。
S42: defining all decomposition terms of the decomposed mapping model according to the condition expected values;
all decomposition terms are:
wherein, the liquid crystal display device comprises a liquid crystal display device,is constant (I)>Is->Function of->Is->And->Is a function of (a) and (b),dis the number of uncertainty variables, +.>Is the firstiUncertainty variable and->∈[0，1]，i=1,2, · ·j · ,d，dAs the number of uncertainty variables,for mathematical expectations +.>Mathematical expectations for output->For the presence of uncertainty variable only +.>Is used as a condition for the (c) of the (c),for presence of uncertainty variable->、/>Is desirable for the conditions of (2).
S43: squaring all the decomposition terms and integrating to obtain a new function model;
by means of all the decomposition terms, it can be determined that,only follow->Changes (called->Major effect), ->Is->And->The effect of the simultaneous variation, which is called second order interaction, other higher order terms have similar definitions. Based on this, a new functional model obtained by squaring and integrating all the decomposition terms is:
wherein, the liquid crystal display device comprises a liquid crystal display device,for the variance of the global sensitivity analysis results, +.>Is the analysis result and isdVector of individual uncertainty variable inputs, +.>Is normalCount (n)/(l)>Is->Function of->Is->And->Is a function of (a) and (b),das the number of uncertainty variables,is the firstiUncertainty variable and->∈[0，1]，i=1,2, · ·j · ,d，dIs the number of uncertainty variables, +.>Indicating the number->Representing a first input uncertainty variable, +.>Representing the s-th input uncertainty variable.
S44: obtaining a decomposed variance expression model according to the new function model;
it can be seen that the left side of the equation for the new functional model is equal to the global sensitivity analysis resultThe term on the right is the variance term, decomposing with respect to the set, resulting in a decomposed variance expression model of:
wherein, the liquid crystal display device comprises a liquid crystal display device,the total variance is represented as such,dis the number of uncertainty variables, +.>Representing uncertainty variable +.>Independent contribution to the output variance, and +.>，/>Representing uncertainty variable +.>、/>Coupling contribution to output variance, and +.>，/>Indicating output, ->Indicating the presence of uncertainty variable +.>Total contribution to output variance, +.>Representing uncertainty variable +.>、/>Total contribution to output variance, +.>Indicating that only uncertainty variable +.>Mathematical expectations of time, < >>Indicating the presence of uncertainty variable +.>Condition expectations at time->Indicating the presence of uncertainty variable +.>、/>Mathematical expectations of time, < >>Indicating the presence of uncertainty variable +.>、/>Condition expectations at time->Representing uncertainty variable +.>Independent contribution to total variance, +.>Is the firstiUncertainty variable and->∈[0，1]，i=1,2, · ·j · ,d。
S45: and establishing a sensitivity measurement Sobol index based on direct variance decomposition by using the decomposed variance expression model.
The decomposed variance expression model shows how the variance of the model output is decomposed into terms attributable to each input, and the interactions between them, all taken together is the total variance of the model output. Based on the above, the present invention establishes a sensitivity metric Sobol index based on a direct variance decomposition, whichThe method comprises the following steps:
wherein, the liquid crystal display device comprises a liquid crystal display device,representing the total variance>Representing uncertainty variable +.>Independent contributions to output variance, and，/>indicating output, ->Indicating the presence of uncertainty variable +.>For output varianceTotal contribution(s)>Indicating the presence of only uncertainty variablesX i Mathematical expectations of time, < >>Indicating the presence of uncertainty variablesCondition expectations at time->Is the firstiUncertainty variable and->∈[0，1]，i=1,2, · ·j · ,d。
Also referred to as a "first order sensitivity index" or "main effect index". It reflects the contribution to the output variance normalized by the total variance.
S5: and obtaining a global sensitivity analysis result according to the analysis result and a sensitivity measurement Sobol index value corresponding to the analysis result.
The input of the global sensitivity analysis is the analysis result of an uncertainty variable, the output is the Sobol index of different variables, and the larger the value of the Sobol index is, the larger the influence of the variable in the range is.
The foregoing description of the preferred embodiments of the invention is not intended to limit the invention to the precise form disclosed, and any such modifications, equivalents, and alternatives falling within the spirit and scope of the invention are intended to be included within the scope of the invention.
Claims (6)
1. A global sensitivity analysis method with strong portability, characterized in that the global sensitivity analysis method with strong portability comprises:
s1: establishing an aircraft design model;
s2: performing uncertainty variable analysis on the aircraft design model to obtain an analysis result;
s3: establishing a mapping model of the analysis result and the global sensitivity analysis result;
s4: according to the mapping model, establishing a sensitivity measurement Sobol index based on direct variance decomposition;
s5: obtaining a global sensitivity analysis result according to the analysis result and a sensitivity measurement Sobol index value corresponding to the analysis result;
in the step S3, the mapping model is:
Q＝F(X)
wherein Q is a global sensitivity analysis result, X is an analysis result and is a vector input by d uncertainty variables;
the step S4 comprises the following steps:
s41: assuming that the inputs are independently and uniformly distributed in the unit hypercube, decomposing the mapping model to obtain a decomposed mapping model;
s42: defining all decomposition terms of the decomposed mapping model according to the condition expected values;
s43: squaring all the decomposition terms and integrating to obtain a new function model;
s44: obtaining a decomposed variance expression model according to the new function model;
s45: and establishing a sensitivity measurement Sobol index based on direct variance decomposition by using the decomposed variance expression model.
2. The global sensitivity analysis method with strong portability according to claim 1, wherein the decomposed mapping model is:
wherein f 0 Is constant, f i Is X i Letter of (1)Number f ij Is X i And X j D is the number of uncertainty variables, X i Is the ith uncertainty variable and X i ∈[0，1]，
3. The global sensitivity analysis method with strong portability according to claim 1, wherein in S42, all decomposition terms are:
f 0 ＝E(Q)
f i (X i )E(Q|X i )-f 0
f ij (X i ，X j )＝E(Q|X i ，X j )-f 0 -f i -f j
wherein f 0 Is constant, f i Is X i F is a function of (f) ij Is X i And X j D is the number of uncertainty variables, X i Is the ith uncertainty variable and X i ∈[0，1]，d is the number of uncertainty variables, E () is the mathematical expectation, E (Q) is the mathematical expectation of the output, E (Q|X) i ) To only have uncertainty variable X i Is the condition of E (Q|X i ，X j ) To be provided with uncertainty variable X i 、X j Is desirable for the conditions of (2).
4. The global sensitivity analysis method with strong portability according to claim 1, wherein in S43, the new function model is:
wherein, the liquid crystal display device comprises a liquid crystal display device,for the variance of the global sensitivity analysis result, X is the analysis result and is the vector of d uncertainty variable inputs, f 0 Is constant, f i Is X i F is a function of (f) ij Is X i And X j D is the number of uncertainty variables, X i Is the ith uncertainty variable and X i ∈[0，1]I=1, 2, j, d, d is the number of uncertainty variables, s represents the number, i 1 Representing a first input uncertainty variable, i s Representing the s-th input uncertainty variable.
5. The global sensitivity analysis method with strong portability according to claim 1, wherein in S44, the decomposed variance expression model is:
where Var (Y) represents the total variance, d is the number of uncertainty variables, V i Representing uncertainty variable X i Independent contributions to output variance, andV ij representing uncertainty variable X i 、X j Coupled contribution to output variance, andy represents output, ->Representing the presence of uncertainty variable X i Total contribution to output variance, +.>Representation ofUncertainty variable X i 、X j Total contribution to output variance E X～i () Indicating the presence of only uncertainty variable X i Mathematical expectation of time, E X～i (Y|X i ) Representing the presence of uncertainty variable X i Conditions at that time expect, E X～ij () Representing the presence of uncertainty variable X i 、X j Mathematical expectation of time, E X～ij (Y|X i ，X j ) Representing the presence of uncertainty variable X i 、X j Conditions at that time expect, V j Representing uncertainty variable X j Independent contribution to total variance, X i Is the ith uncertainty variable and X i ∈[0，1]，/>
6. The global sensitivity analysis method with strong portability according to any one of claims 1-5, wherein the sensitivity metric Sobol index S based on direct variance decomposition i The method comprises the following steps:
wherein Var (Y) represents the total variance, V i Representing uncertainty variable X i Independent contributions to output variance, andy represents output, ->Representing the presence of uncertainty variable X i Total contribution to output variance E X～i () Indicating the presence of only uncertainty variable X i Mathematical expectation of time, E X～i (Y|X i ) Representing the presence of uncertainty variable X i Conditions in that case expect X i For the ith uncertaintyVariable and X i ∈[0，1]，/>
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| CN112906251A (en) * | 2021-04-16 | 2021-06-04 | 云南电网有限责任公司 | Analysis method and system for reliability influence factors of power distribution network |
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| US20160055125A1 (en) * | 2014-06-27 | 2016-02-25 | The Arizona Board Of Regents On Behalf Of The University Of Arizona | Methods and systems for determining global sensitivity of a process |
| CN111898245A (en) * | 2020-06-29 | 2020-11-06 | 西北工业大学 | Wear reliability and sensitivity analysis method for aircraft cabin door pressure release valve mechanism |
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| CN110634534A (en) * | 2018-06-06 | 2019-12-31 | 中国石油化工股份有限公司 | Chemical process parameter sensitivity determination method based on extended Fourier amplitude analysis |
| CN112906251A (en) * | 2021-04-16 | 2021-06-04 | 云南电网有限责任公司 | Analysis method and system for reliability influence factors of power distribution network |
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