CN107633117B - Global sensitivity analysis method based on Gaussian process model - Google Patents
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Abstract
The disclosure provides a global sensitivity analysis method based on a Gaussian process model, and relates to the technical field of system robustness analysis and structure optimization design. The analysis method comprises the following steps: randomly acquiring a group of sample points based on the distribution parameters of the input variables, and obtaining corresponding output response quantity according to the input variables by utilizing a preset relation; establishing a multi-output Gaussian process model according to the input variables and the corresponding output response quantities; deducing a calculation formula of a global sensitivity index according to the distribution type of the input variable and the expression of the multi-output Gaussian process model; and carrying out global sensitivity analysis on a mechanism part based on the calculation formula of the global sensitivity index. According to the method, the global sensitivity index can be accurately and efficiently obtained through a small number of sample points, so that the calculation efficiency of the structure optimization design can be improved by performing global sensitivity analysis.
Description
Technical Field
The disclosure relates to the technical field of system robustness analysis and structure optimization design, in particular to a global sensitivity analysis method based on a Gaussian process model.
Background
In engineering design applications, global sensitivity analysis is an important theoretical tool for researching the influence degree of input parameters of a mechanical system on output response. In recent years, various global sensitivity analysis methods have been rapidly developed. Variance of assumed variables such as Sobol and Iman can fully describe uncertainty indexes of model output, and a global sensitivity analysis method based on variance is provided. Borgnonovo suggests that moment-independent global sensitivity indicators reflect differences in the importance of the underlying variables. Cui researches global sensitivity indexes based on independent variance and moment under the action of random excitation, and is applied to sensitivity analysis research of a track of a shaper. The girth smart provides a global sensitivity index based on a dynamic response parameter, and researches the influence of a random uncertainty input parameter of a structural system under random excitation on the structural dynamic response. The Suzhong super researches the influence of the change of the hinge gap distribution parameters on the overall sensitivity of the motion precision of the cabin door connecting rod mechanism. The method is high and the overall sensitivity index for measuring the influence of the basic random variable on the power reliability is provided by analyzing and transforming the conditional probability density function in the composite random vibration system. The Lvhui swallow effectively reduces the complexity of the problem of aviation gear vibration optimization containing high-dimensional parameters by using a global sensitivity index based on variance. The research of various scholars at home and abroad promotes the application of the global sensitivity analysis theory in the research of the input-output relationship of a mechanical system.
Taking the civil aircraft field as an example, the horizontal tail of the aircraft is an important mechanism for ensuring the longitudinal stability and the maneuverability of the aircraft and improving the efficiency of control surfaces, and the rotating shaft of the horizontal tail of the aircraft can be operated by a rocker arm to drive the horizontal tail to deflect so as to control the deflection direction and the movement attitude of the aircraft, so that the horizontal tail mechanism system of the aircraft plays an important role in the safe operation of the aircraft. Based on the method, in order to prolong the service life of the civil aircraft, improve the flight safety and the robustness of the mechanism system, the method has very important significance in carrying out global sensitivity analysis on typical mechanism components. However, the existing global sensitivity analysis method usually requires a large sample size and is time-consuming to calculate, thereby restricting the application of the method in engineering practice.
It is to be noted that the information disclosed in the above background section is only for enhancement of understanding of the background of the present disclosure, and thus may include information that does not constitute prior art known to those of ordinary skill in the art.
Disclosure of Invention
It is an object of the present disclosure to provide a global sensitivity analysis method based on a gaussian process model, thereby overcoming, at least to some extent, one or more of the problems due to the limitations and disadvantages of the related art.
Additional features and advantages of the disclosure will be set forth in the detailed description which follows, or in part will be obvious from the description, or may be learned by practice of the disclosure.
According to an aspect of the present disclosure, there is provided a global sensitivity analysis method based on a gaussian process model, including:
randomly acquiring a group of sample points based on the distribution parameters of the input variables, and obtaining corresponding output response quantity according to the input variables by utilizing a preset relation;
establishing a multi-output Gaussian process model according to the input variables and the corresponding output response quantities;
deducing a calculation formula of a global sensitivity index according to the distribution type of the input variable and the expression of the multi-output Gaussian process model;
and carrying out global sensitivity analysis on a mechanism part based on the calculation formula of the global sensitivity index.
In an exemplary embodiment of the present disclosure, the global sensitivity analysis method further includes: and acquiring key parameters of the mechanism component according to the result of the global sensitivity analysis.
In an exemplary embodiment of the present disclosure, the obtaining the key parameter of the mechanism component according to the result of the global sensitivity analysis includes:
acquiring a target sensitivity index of which the index value is greater than a preset value in the global sensitivity index according to the result of the global sensitivity analysis;
and acquiring the corresponding input variable as a key parameter of the mechanism component according to the target sensitivity index.
In an exemplary embodiment of the disclosure, the obtaining the corresponding output response according to the input variable by using a preset relationship includes:
obtaining corresponding output response quantity according to the input variable by using the following formula;
g0,l＝E[g(x,l)]；
gi(xi,l)＝E[g(x,l)|xi]-g0,l；
……
wherein y ═ y1,y2,...,ym) For multi-output function, y is the output response, x ═ x1,x2,...,xn) Is a random input variable, g (x, l) is yl。
In an exemplary embodiment of the present disclosure, the global sensitivity index includes a global sensitivity main index and a global sensitivity total index;
the expression of the global sensitivity main index is as follows:
the expression of the global sensitivity total index is as follows:
wherein C is a function ylVariance matrix of CiAs a function gi(xiL) variance matrix of C)12...n(y1,...,ym) As a function g1...n(x1,x2,...,xnL) variance matrix, Tr [ C ]]As a function of ylIs the variance matrix of (1) to trace Tr [ C ]i]Tracing the variance matrix of the function g (x, l), Tr [ C ]12...n(y1,...,ym)]Is a pair function g1...n(x1,x2,...,xn,l) The variance matrix of (2) is traced.
In an exemplary embodiment of the present disclosure, the expression of the multiple output gaussian process model is:
μy(x)＝h(x)+r(x)R-1(Y-BH)；
wherein, muy(x) The output of the Gaussian process model is H (x), a specific regression function sequence is H, (x), a regression coefficient matrix is B, a regression function matrix is H, and a functional function output value of a sample point when the Gaussian process model is established is Y;
r (x) is the spatial relationship between x and N sample points, i.e. an Nx 1-dimensional vector, and the ith element is ri(x)＝R(xi,x)；
R is a spatial function matrix, and the (i, j) th element is:
and omega is a roughness coefficient.
In an exemplary embodiment of the present disclosure, the input variables follow a standard normal distribution.
In an exemplary embodiment of the present disclosure, in the expression of the global sensitivity index:
wherein,
is an N x 1 dimensional vector, t (x)i)＝∫r(x)φ(xi)dxiIs an N-dimensional vector, phi (x)k) Is a probability density function of a standard normal distribution.
In an exemplary embodiment of the present disclosure, the output response amount includes: stress or strain of the mechanism component when subjected to an external force; the input variables include: one or more of an inner diameter and an outer diameter of a failure portion of the mechanism component, a yield strength of a material selected for the mechanism component, and an external force to which a predetermined portion of the mechanism component is subjected.
In an exemplary embodiment of the present disclosure, the mechanism component comprises a horizontal tail rotor shaft mechanism.
The global sensitivity analysis method provided by the exemplary embodiment of the disclosure completes the establishment of a gaussian process model based on a small number of sample points, and can efficiently and accurately obtain the global sensitivity index of each input variable by means of a derived global sensitivity calculation formula, so as to perform global sensitivity analysis on the mechanism component, and improve the calculation efficiency of structural optimization design.
It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory only and are not restrictive of the disclosure.
Drawings
The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate embodiments consistent with the present disclosure and together with the description, serve to explain the principles of the disclosure. It is to be understood that the drawings in the following description are merely exemplary of the disclosure, and that other drawings may be derived from those drawings by one of ordinary skill in the art without the exercise of inventive faculty.
FIG. 1 schematically illustrates a first flow chart of a global sensitivity analysis method in an exemplary embodiment of the present disclosure;
FIG. 2 schematically illustrates a second flow chart of a global sensitivity analysis method in an exemplary embodiment of the present disclosure;
fig. 3 schematically illustrates a simplified model diagram of an aircraft horizontal tail mechanism in an exemplary embodiment of the disclosure.
Detailed Description
Example embodiments will now be described more fully with reference to the accompanying drawings. Example embodiments may, however, be embodied in many different forms and should not be construed as limited to the examples set forth herein; rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the concept of example embodiments to those skilled in the art. The described features, structures, or characteristics may be combined in any suitable manner in one or more embodiments. In the following description, numerous specific details are provided to give a thorough understanding of embodiments of the disclosure. One skilled in the relevant art will recognize, however, that the subject matter of the present disclosure can be practiced without one or more of the specific details, or with other methods, components, devices, steps, and the like. In other instances, well-known technical solutions have not been shown or described in detail to avoid obscuring aspects of the present disclosure.
For ease of description, spatial relationship terms such as "below …," "below …," "lower," "above …," "upper," and the like may be used herein to describe one element or feature's relationship to another element or feature (or other elements or features) as illustrated. It will be understood that the spatially relative terms are intended to encompass different orientations of the device in use or operation in addition to the orientation depicted in the figures. For example, if the device in the figures is turned over, elements described as "below" or "beneath" other elements or features would then be oriented "above" the other elements or features. Thus, the exemplary term "below …" can include orientations of both "above …" and "below …". The device may be otherwise oriented (rotated 90 degrees or at other orientations) and the spatial relationship descriptors used herein interpreted accordingly.
Furthermore, the drawings are merely schematic illustrations of the present disclosure and are not necessarily drawn to scale. The thicknesses and shapes of the layers in the drawings are not to be construed as true scale, but merely as a matter of convenience for illustrating the disclosure. The same reference numerals in the drawings denote the same or similar parts, and thus their repetitive description will be omitted.
The exemplary embodiment provides a global sensitivity analysis method based on a gaussian process model, as shown in fig. 1, including:
s1, randomly acquiring a group of sample points based on the distribution parameters of the input variables, and obtaining corresponding output response quantity according to the input variables by utilizing a preset relation;
s2, establishing a multi-output Gaussian process model according to the input variables and the corresponding output response quantities;
s3, deriving a calculation formula of a global sensitivity index according to the distribution type of the input variables and the expression of the multi-output Gaussian process model;
and S4, carrying out global sensitivity analysis on a mechanism component based on the calculation formula of the global sensitivity index.
It should be noted that: the global sensitivity analysis method provided by the present example embodiment may be applied to any mechanical system to obtain a global sensitivity index of the influence of an input variable on the output response of the mechanical system.
The global sensitivity analysis method provided by the exemplary embodiment of the disclosure completes the establishment of a gaussian process model based on a small number of sample points, and can efficiently and accurately obtain the global sensitivity index of each input variable by means of a derived global sensitivity calculation formula, so as to perform global sensitivity analysis on the mechanism component, and improve the calculation efficiency of structural optimization design.
Based on the above steps, as shown in fig. 2, after performing the global sensitivity analysis on a mechanism component, the global sensitivity analysis method may further include:
and S5, acquiring key parameters of the mechanism component according to the result of the global sensitivity analysis.
Therefore, the global sensitivity analysis method achieves the purpose of dimension reduction by neglecting factors which have small influence on output results, so that the calculation efficiency of structural optimization design can be improved, and theoretical guidance is provided for improving the robustness design of mechanical systems such as typical mechanism systems of aviation aircrafts.
The global sensitivity analysis method in the present exemplary embodiment is described in detail below with reference to the drawings.
In step S1, a set of sample points is randomly obtained based on the distribution parameters of the input variables, and a corresponding output response is obtained according to the input variables corresponding to the sample points by using a preset relationship.
In this exemplary embodiment, the input variables are preferably a set of random input variables, which may correspond to a set of efficient and uniform sample points drawn at random, and the random input variables may follow any one of a plurality of distribution manners, such as a normal distribution, a binomial distribution, a poisson distribution, etc., and specific distribution parameters may include expectation and variance, etc. The predetermined relationship may be a functional relationship or a corresponding relationship established by finite element software simulation analysis. The output response quantity may be an output response obtained based on a specific preset relationship according to the input variable.
For example, the input variables may be the inside diameter D or outside diameter D of a failure site of an aircraft horizontal tail rotor mechanism, the yield strength σ of the selected materialbAnd external force applied to the preset part of the horizontal tail rotating shaft mechanism; the output response quantity can be stress or strain of the horizontal tail rotating shaft mechanism when the horizontal tail rotating shaft mechanism is subjected to external force; the preset relationship may be a multi-output function of the horizontal tail rotation shaft mechanism, that is, a function for describing a system state of the horizontal tail rotation shaft mechanism. Thus, when the horizontal tail rotor shaft mechanism of the airplane is subjected to external force, the multi-output function can describe the internal change of the horizontal tail rotor shaft mechanism, such as stress or strain, and the horizontal tail rotor shaft mechanism fails once the stress or strain exceeds the range which the horizontal tail rotor shaft mechanism can bear.
In step S2, a multi-output gaussian process model is established according to the sample points and the output response quantities.
In this exemplary embodiment, the nature of the multiple-output gaussian process model is a functional correspondence relationship, which can obtain an output response according to an input variable. That is, the multi-output gaussian process model has the same function as the multi-output function in the above steps, but after modeling through the gaussian process, the global sensitivity analysis can be performed more efficiently and accurately by using some parameters of the model. It should be noted that: the establishment of the multi-output gaussian process model can be realized by a special program package.
For example, the multi-output gaussian process model can be established by inputting the input variables and the corresponding output response quantities, which are randomly acquired in the above steps, into a specific package for establishing the gaussian process model.
In step S3, a calculation formula of a global sensitivity index is derived from the distribution type of the input variables and the expression of the multiple output gaussian process model.
In this exemplary embodiment, the global sensitivity index may include a global sensitivity main index and a global sensitivity total index, and these two sensitivity indexes may be defined based on the sorting deformation of the multi-output function in the above steps. On the basis, the calculation formula of the global sensitivity index, such as the calculation formula of each factor contained in the calculation formula, can be deduced by combining the expression of the multi-output Gaussian process model.
It should be noted that: since the expression of the gaussian process model is determined, the specific parameter value in the expression needs to be determined for establishing the gaussian process model, and the derivation of the calculation formula of the global sensitivity index only needs the expression of the gaussian process model, the derivation of the calculation formula of the global sensitivity index can be performed before the gaussian process model is established or after the gaussian process model is established, which is not specifically limited in this embodiment.
For example, according to the Sobol theory, the multiple output function of the horizontal tail rotating shaft mechanism of the airplane is decomposed into the sum of component functions, the output response variance is decomposed into the sum of the variances of all decomposition items, and then the trace is solved at the same time to obtain the deformation form of the multiple output function, so that the global sensitivity index defined on the basis can quantify the influence of the input variable on the output response variance, and further influences the robustness of the system. After the global sensitivity index is defined, the calculation formula of the global sensitivity index needs to be derived by means of the expression of the gaussian process model and by combining the distribution characteristics of the input variables, such as the standard normal distribution, and specifically, the calculation formula can be derived for each factor, so that the calculation complexity is simplified, and the calculation efficiency is improved.
In step S4, a global sensitivity analysis is performed on a mechanism component based on the calculation formula of the global sensitivity index.
In this exemplary embodiment, the analysis process of the global sensitivity index is a process of calculating according to the calculation formula of the global sensitivity index obtained in the above step, and may obtain an index value of the global sensitivity index according to a plurality of random sample points.
In step S5, key parameters of the mechanism component are acquired from the result of the global sensitivity analysis.
In the present exemplary embodiment, the key parameter refers to an input variable that has a large influence on the output response of the mechanism component, and an input variable that has a small or even almost no influence on the output response of the mechanism component is negligible.
Based on this, the step S5 may specifically include:
s501, acquiring a target sensitivity index of which the index value is larger than a preset value in the global sensitivity index according to the result of the global sensitivity analysis;
and S502, acquiring a corresponding input variable as a key parameter of the mechanism component according to the target sensitivity index.
The preset value is a reference value used for measuring the influence degree of the input variable on the output response quantity, and the size of the reference value can be reasonably set according to the robustness design index of the mechanical system.
The global sensitivity analysis method of the present disclosure is exemplarily described below by taking an aircraft horizontal tail rotor shaft mechanism as an example. Referring to fig. 3, the input variables may include: inner diameter D and outer diameter D of failure part of mechanism component, yield strength sigma of selected materialbAnd external force applied to a preset part of the mechanism component; the output response may include: stress or strain of the mechanism component when subjected to an external force; wherein the failure positions are the first to fifth failure interfaces shown in the figure.
In the field of aviation aircrafts, an airplane horizontal tail rotating shaft mechanism plays an important role in safe operation of airplanes, so that the safety condition of an airplane horizontal tail mechanism system is very important in the service stage of airplanes. The modern civil airliner management department puts higher requirements on the safety of civil aircrafts, and the technical design requirements of a plurality of airplane horizontal tail rotating shaft mechanisms are added in the civil aircraft pilot standard. Based on the above, in order to prolong the service life of the civil aircraft and improve the flight safety and the robustness of the mechanism system, the global sensitivity analysis of the typical components of the civil aircraft mechanism, such as the horizontal tail rotating shaft mechanism, is necessary.
At present, most scholars research global sensitivity analysis indexes in a theoretical stage, but the research of applying the proposed theoretical method to large-scale complex engineering in engineering practice is rare, and especially the application of the proposed theoretical method to global sensitivity analysis of aviation aircraft mechanism systems is relatively lacked. In the service process of a civil aircraft, the horizontal tail rotating shaft mechanism controls the flight attitude of the aircraft through multiple actions, and the process enables the components of the horizontal tail rotating shaft mechanism to bear repeated load action, causes performance degradation represented by abrasion or fatigue, and influences the normal operation of the horizontal tail rotating shaft mechanism, thereby causing the horizontal tail rotating shaft mechanism to fail and even causing a series of major flight accidents.
The implementation mode aims to improve the system robustness and safety of the civil aircraft mechanism, and selects a horizontal tail rotating shaft mechanism of a certain type of civil aircraft to carry out global sensitivity analysis, so that a global sensitivity index of the mechanism is obtained. Referring to fig. 3, five failure interfaces are sequentially selected from left to right on the horizontal tail rotating shaft mechanism, and the multi-output function of the horizontal tail rotating shaft mechanism is recorded as:
y＝(y1,y2,...,ym) (1)；
sobol proposes that the functional function is decomposed into the sum of component functions, i.e.:
wherein:
g0,l＝E[g(x,l)] (3)；
gi(xi,l)＝E[g(x,l)|xi]-g0,l (4)；
……
in equations (1) to (6), y is an output response amount, and x is (x)1,x2,...,xn) For random input variables, g (x, l) is yl。
In this embodiment, the global sensitivity index can quantify the influence of the input variable x on the variance of the output response y, thereby affecting the robustness of the mechanical system.
Based on this, Sobol proposes that the variance of the output response y can be decomposed into the sum of the variances of the decomposition terms on the basis of equation (2), that is:
wherein, C (y)1,...,ym) As a function ylVariance matrix of Ci(y1,...,ym) As a function gi(xiL), and so on.
On the basis, traces are simultaneously found on two sides of the formula (7) to obtain:
based on this, in order to quantify the contribution of the input variables, the global sensitivity index may include a global sensitivity main index and a global sensitivity total index.
The expression of the global sensitivity main index is as follows:
the expression of the global sensitivity total index is as follows:
wherein C is a function ylVariance matrix of CiAs a function gi(xiL) variance matrix of C)12...n(y1,...,ym) As a function g1...n(x1,x2,...,xnL) variance matrix, Tr [ C ]]As a function of ylIs the variance matrix of (1) to trace Tr [ C ]i]Tracing the variance matrix of the function g (x, l), Tr [ C ]12...n(y1,...,ym)]Is a pair function g1...n(x1,x2,...,xn,l) The variance matrix of (2) is traced.
For a multiple output gaussian process model, its predicted values at other input sample points can be expressed as:
μy(x)＝h(x)+r(x)R-1(Y-BH) (11)；
wherein, muy(x) The output of the Gaussian process model is H (x), a specific regression function sequence is H, (x), a regression coefficient matrix is B, a regression function matrix is H, and a functional function output value of a sample point when the Gaussian process model is established is Y;
r (x) is the spatial relationship between x and N sample points, i.e. an Nx 1-dimensional vector, and the ith element is ri(x)＝R(xi,x)；
R is a spatial function matrix, and the (i, j) th element is:
and omega is a roughness coefficient.
Based on the process, the establishment of the multi-output Gaussian process model can be completed according to the input variable and the output response quantity of the horizontal tail rotating shaft mechanism.
On the basis, a calculation formula of the global sensitivity index can be deduced according to the distribution type of the input variables and the expression of the multi-output Gaussian process model.
In this embodiment, taking the example that the input variable obeys the standard normal distribution, the global sensitivity main index is measured
And the global sensitivity total index
Is further derived. It should be noted that: the input variables may also be subject to other distribution types, in which case a similar derivation is only necessary.
For C (y)1,...,ym) The derivation is as follows:
wherein:
for C (y)1,...,ym) The derivation is as follows:
in the formula (16), the first and second groups,
is a vector of dimension N × 1, s (x)i) The m-th element of (a):
wherein:
total indicator for global sensitivity
Equation (10) can be further derived as follows:
wherein, C～iFunction g without i for any subscript～i(xiL) of a variance matrix, which is further derivable as:
in the formula (22), t (x)i)＝∫r(x)φ(xi)dxiIs an N-dimensional vector, phi (x)k) Probability density function of standard normal distribution, t (x)i) The m-th element of (a):
wherein:
based on the above theoretical analysis process, the global main sensitivity index can be obtained according to the above formula (12) and formula (16)
According to the above-mentioned formula (12) and formula (22), the global sensitivity total index can be obtained
In order to prove the beneficial effect of the technical scheme of the disclosure, a group of global sensitivity indexes obtained by random input variables based on the horizontal tail rotating shaft mechanism according to the formula are provided below. The distribution type of input variables of the horizontal tail rotating shaft mechanism is shown in a table 1; wherein X ═ X (X)1,X2,X3,X4,X5,X6,X7,X8,X9)，X1Is the outer diameter D of the first failure interface1，X2Is the outer diameter D of the second failure interface2，X3Is the outer diameter D of the third failure interface3，X4Is the outer diameter D of the fourth and fifth failure interfaces4＝D5，X5Is the inner diameter d of the second to fifth failure interfaces2＝d3＝d4＝d5，X6Yield strength sigma of material for flat tail rotor shaft mechanismb，X7Is the outward force applied to point M, X8Downward force on point T, X9The upward force applied to the point M.
TABLE 1
The global sensitivity index obtained based on the above calculation formula is shown in table 2; wherein, the bracket in the table 2 is the corresponding calculation times of the corresponding method. Compared with the Monte Carlo method, the calculation times required by the global sensitivity analysis method in the technical scheme are obviously reduced and the calculation efficiency is obviously improved under the condition of achieving the same calculation precision.
TABLE 2
Based on the above-mentioned global sensitivity analysis results, the input variable X is divided6And X7Besides, other input variables have little influence on the output response of the horizontal tail rotating shaft mechanism, so that the key parameter of the horizontal tail rotating shaft mechanism is the input variable X6And X7The parameters represented may be used to optimize the design of the plant with an emphasis on the input variable X6And X7。
Other embodiments of the disclosure will be apparent to those skilled in the art from consideration of the specification and practice of the disclosure disclosed herein. This application is intended to cover any variations, uses, or adaptations of the disclosure following, in general, the principles of the disclosure and including such departures from the present disclosure as come within known or customary practice within the art to which the disclosure pertains. It is intended that the specification and examples be considered as exemplary only, with a true scope and spirit of the disclosure being indicated by the following claims.
It will be understood that the present disclosure is not limited to the precise arrangements described above and shown in the drawings and that various modifications and changes may be made without departing from the scope thereof. The scope of the present disclosure is limited only by the appended claims.
Claims (6)
1. A global sensitivity analysis method based on a Gaussian process model is characterized by comprising the following steps:
randomly acquiring a group of sample points based on the distribution parameters of the input variables, and obtaining corresponding output response quantity according to the input variables by utilizing a preset relation;
establishing a multi-output Gaussian process model according to the input variables and the corresponding output response quantities;
deducing a calculation formula of a global sensitivity index according to the distribution type of the input variable and the expression of the multi-output Gaussian process model;
performing global sensitivity analysis on a mechanism component based on a calculation formula of the global sensitivity index;
the obtaining of the corresponding output response according to the input variable by using a preset relationship includes:
obtaining corresponding output response quantity according to the input variable by using the following formula;
g0,l＝E[g(x,l)]；
gi(xi,l)＝E[g(x,l)|xi]-g0,l；
……
wherein y ═ y1,y2,...,ym) For multi-output function, y is the output response, x ═ x1,x2,...,xn) Is a random input variable, g (x, l) is yl；
The global sensitivity index comprises a global sensitivity main index and a global sensitivity total index;
the expression of the global sensitivity main index is as follows:
the expression of the global sensitivity total index is as follows:
wherein C is a function ylVariance matrix of CiAs a function gi(xiL) variance matrix of C)12...n(y1,...,ym) As a function g1...n(x1,x2,...,xn,l) Of the variance matrix Tr [ C ]]As a function of ylIs the variance matrix of (1) to trace Tr [ C ]i]Tracing the variance matrix of the function g (x, l), Tr [ C ]12...n(y1,...,ym)]Is a pair function g1...n(x1,x2,...,xn,l) The trace is solved by the variance matrix;
the expression of the multiple output Gaussian process model is as follows:
μy(x)＝h(x)+r(x)R-1(Y-BH)；
wherein, muy(x) The output of the Gaussian process model is H (x), a specific regression function sequence is H, (x), a regression coefficient matrix is B, a regression function matrix is H, and a functional function output value of a sample point when the Gaussian process model is established is Y;
r (x) is the spatial relationship between x and N sample points, i.e. an Nx 1-dimensional vector, and the ith element is ri(x)＝R(xi,x)；
R is a spatial function matrix, and the (i, j) th element is:
omega is a roughness coefficient;
in the expression of the global sensitivity index:
wherein,
is an N x 1 dimensional vector, t (x)i)＝∫r(x)φ(xi)dxiIs an N-dimensional vector, phi (x)k) Is a probability density function of a standard normal distribution.
2. The global sensitivity analysis method according to claim 1, further comprising:
and acquiring key parameters of the mechanism component according to the result of the global sensitivity analysis.
3. The global sensitivity analysis method according to claim 2, wherein the acquiring key parameters of the mechanism component from the result of the global sensitivity analysis includes:
acquiring a target sensitivity index of which the index value is greater than a preset value in the global sensitivity index according to the result of the global sensitivity analysis;
and acquiring the corresponding input variable as a key parameter of the mechanism component according to the target sensitivity index.
4. The global sensitivity analysis method of claim 1, wherein the input variables follow a standard normal distribution.
5. The global sensitivity analysis method of any one of claims 1-4, wherein the outputting the response comprises: stress or strain of the mechanism component when subjected to an external force; the input variables include: one or more of an inner diameter and an outer diameter of a failure portion of the mechanism component, a yield strength of a material selected for the mechanism component, and an external force to which a predetermined portion of the mechanism component is subjected.
6. The global sensitivity analysis method of any one of claims 1-4, wherein the mechanical component comprises a horizontal tail rotor shaft mechanism.
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