Ellipsoid multi-subdomain analysis method for complex equipment uncertainty under small sample

Abstract

The present invention provides an ellipsoid multi-subdomain analysis method for the uncertainty of complex equipment under small samples, which belongs to the technical field of uncertainty analysis of complex equipment, including: when the sample size is small, constructing an uncertainty ellipsoid domain according to the type of uncertainty parameter. Then, the ellipsoid domain with higher uncertainty is converted into a standard domain, and then subdivided into a plurality of simplexes with smaller uncertainties. By calculating the structural response boundary of the simplex, and dividing each structural response boundary into subintervals and assigning different weights, the possibility of the complex equipment in the uncertainty ellipsoid domain is calculated, and the reliability of the complex equipment under uncertain conditions can be analyzed based on this. The present invention can realize the accurate measurement of multi-source uncertain parameters under small sample conditions, and realize accurate uncertainty propagation analysis of complex equipment structural response in a non-probabilistic framework, and has a good application prospect in the uncertainty analysis of complex equipment structural response.

Description

Technical Field

[0001] The present invention generally relates to the technical field of uncertainty analysis of complex equipment, and specifically to an ellipsoid multi-subdomain analysis method for uncertainty of complex equipment under small samples. Background Art

[0002] In actual engineering, especially in the design, manufacturing and service process of complex equipment structures and products in the fields of mechanical manufacturing, bridge construction and aerospace, there are inevitably deviations and uncertainties related to material properties, geometric characteristics, measurement errors, service conditions, etc. When using deterministic theories and methods to design and develop complex equipment, small fluctuations and differences will cause huge deviations in the structural response of complex equipment, significantly affecting structural performance and equipment safety. Therefore, it is urgent to develop uncertainty analysis methods to improve structural reliability and system robustness in engineering practice.

[0003] The current uncertainty analysis methods mainly include two categories: probability method and non-probabilistic method. However, for complex equipment, it is difficult to obtain a large amount of sample information through physical experiments and numerical simulations, which makes the traditional probability method impractical in solving the uncertainty of complex mechanical equipment. Therefore, it is very necessary to develop non-probabilistic uncertainty analysis methods suitable for complex engineering problems under small sample conditions.

[0004] The variable distribution characteristics of complex equipment are diverse, and there is a coupling relationship between some variables. This makes the traditional uncertainty propagation method based on non-probabilistic models suitable for small uncertainty problems and only the upper and lower boundaries of the response function can be obtained during uncertainty propagation. Compared with the traditional probability method that can obtain the probability distribution of all values, the response information that can be obtained is less.

[0005] Therefore, how to carry out uncertainty analysis under finite sample conditions and obtain the reliability of complex equipment more accurately is a common problem for all kinds of complex equipment. Developing advanced uncertainty analysis methods to obtain more response information has important theoretical significance and engineering practical value for improving the reliability and safety of complex equipment.

[0006] In the prior art, when evaluating the reliability of a complex device, it is necessary to first use a large number of specific values ​​of uncertainty parameters as samples to obtain the degree of probability, and then evaluate the reliability based on the degree of probability. In practice, it is difficult to know the specific values ​​of uncertainty parameters, and it is even more difficult to obtain a large number of uncertainty parameters as samples; the existing methods often significantly reduce the accuracy of reliability evaluation when there are fewer samples. Summary of the invention

[0007] In view of the above-mentioned defects or deficiencies in the prior art, it is desired to provide an ellipsoid multi-subdomain analysis method for complex equipment uncertainty under small samples.

[0008] The present invention provides an ellipsoid multi-subdomain analysis method for complex equipment uncertainty under small sample conditions, including:

[0009] S1: Obtaining quantitative indicators of various types of uncertainty parameters of complex equipment; the quantitative indicators include the mean, interval range and uncertainty level of the uncertainty parameters;

[0010] S2: constructing an uncertainty ellipsoid domain according to the quantitative indicators of the uncertainty parameters of various types; the uncertainty ellipsoid domain is used to quantify the uncertainty parameters and the coupling relationship between the uncertainty parameters of various types;

[0011] S3: converting the uncertainty ellipsoid domain into a standard domain;

[0012] S4: dividing the standard domain into a plurality of simplices; the simplex is a geometric body formed by connecting a plurality of points, and the plurality of simplices are used to approximately represent the standard domain;

[0013] S5: Calculate the structural response boundaries of multiple simplexes;

[0014] S6: Obtaining the possibility of the complex equipment in the uncertainty ellipsoid domain according to the structural response boundaries of the plurality of simplexes;

[0015] S7: Evaluate the reliability of the complex equipment based on the probability.

[0016] According to the technical solution provided by the present invention, S4: dividing the standard domain into a plurality of simplexes, including:

[0017] S4-1: decomposing the standard domain into a preset number of ellipsoidal subdomains on average;

[0018] S4-2: Divide the plurality of ellipsoid subdomains into the preset number of d simplices respectively; the d simplices are geometric bodies formed by connecting a plurality of points, and the degree of approximation between the preset number of d simplices and the standard domain is lower than the degree of approximation between the plurality of simplices and the standard domain; wherein d is the dimension of the d simplices, which is 1 less than the number of types of the uncertainty parameter, and is used to decompose the ellipsoid subdomains;

[0019] S4-3: Map the multiple d simplices to the corresponding ellipsoid subdomains respectively to obtain multiple simplices.

[0020] According to the technical solution provided by the present invention, S4-1: decomposing the standard domain into a preset number of ellipsoidal subdomains on average, including:

[0021] S4-1-1: Obtaining a scaling factor of the standard domain; the scaling factor is equal to the product of a preset number and a small variable; the small variable is a positive number infinitely close to 0;

[0022] S4-1-2: Decompose the standard domain into a preset number of ellipsoid sub-domains on an even basis according to the scaling factor.

[0023] According to the technical solution provided by the present invention, S4-3: mapping the plurality of d simplices to the corresponding ellipsoid subdomains respectively to obtain a plurality of simplices, including:

[0024] S4-3-1: Obtain the number of subdivisions of each ellipsoidal subdomain and the structural response of the complex equipment;

[0025] S4-3-2: Generate a color matching matrix according to the number of subdivisions, the dimension of the uncertainty ellipsoid domain and the structural response of the complex equipment; the dimension of the uncertainty ellipsoid domain is equal to the number of types of uncertainty parameters;

[0026] S4-3-3: dividing the d simplex into a plurality of initial simplexes according to the corresponding element definition in the color matching matrix; the initial simplex has a plurality of initial vertices; the plurality of initial vertices include a plurality of geometric vertices of the initial simplex and additional vertices required for subdividing each side of the initial simplex by the number of subdivisions;

[0027] S4-3-4: normalize the coordinates of all the initial vertices, map the initial vertices to the corresponding ellipsoid subdomain, and obtain the coordinates of all vertices of the simplex;

[0028] S4-3-5: Based on the scaling factor and the coordinates of all the vertices of the simplex, calculate the function values ​​of the structural response of all the vertices of the simplex in the standard domain and the function values ​​of the structural response in the ellipsoid domain.

[0029] According to the technical solution provided by the present invention, S5: calculating the structural response boundaries of multiple simplexes includes:

[0030] S5-1: Generate a direction cosine matrix and a difference vector matrix for each simplex;

[0031] S5-2: Calculating the extreme points of the simplex structural response according to the direction cosine matrix and the difference vector matrix;

[0032] S5-3: Calculate the upper boundary and lower boundary of the simplex structural response according to the extreme value points of the simplex structural response; the upper boundary and the lower boundary constitute the structural response boundary of the simplex.

[0033] According to the technical solution provided by the present invention, S6: obtaining the possibility of the complex equipment in the uncertainty ellipsoid domain according to the structural response boundaries of the plurality of simplexes, including:

[0034] S6-1: Calculating the upper and lower boundaries of the structural response of the complex equipment according to the structural response boundaries of the plurality of simplexes;

[0035] S6-2: According to the upper and lower boundaries of the structural response of the complex equipment, the possibility of the complex equipment in the uncertainty ellipsoid domain is calculated.

[0036] According to the technical solution provided by the present invention, S6-1: calculating the upper boundary and the lower boundary of the structural response of the complex equipment according to the structural response boundaries of the plurality of simplexes, comprising:

[0037] S6-1-1: Calculating the structural response boundaries of the plurality of d simplexes according to the structural response boundaries of the plurality of simplexes;

[0038] S6-1-2: Calculate the upper and lower boundaries of the ellipsoid subdomain structural response according to the structural response boundaries of the plurality of d simplexes;

[0039] S6-1-3: Based on the upper and lower boundaries of the structural response of the ellipsoid subdomain, the upper and lower boundaries of the structural response of the complex equipment are calculated.

[0040] According to the technical solution provided by the present invention, S6-2: calculating the possibility of the complex equipment in the uncertainty ellipsoid domain according to the upper boundary and the lower boundary of the structural response of the complex equipment, including:

[0041] S6-2-1: Divide the upper boundary and the lower boundary of the structural response of the complex equipment into a plurality of boundary sub-intervals;

[0042] S6-2-2: assigning weights to the plurality of boundary subintervals; the weights being positively correlated with the boundary subintervals;

[0043] S6-2-3: Calculate the possibility of the complex equipment in the uncertainty ellipsoid domain based on the upper and lower boundaries of the responses of multiple ellipsoid sub-domain structures, multiple boundary sub-intervals and weights corresponding to the multiple boundary sub-intervals.

[0044] The beneficial effects of the present invention are:

[0045] When the sample size of uncertainty parameters is small, an uncertainty ellipsoid domain is constructed according to the type of uncertainty parameters to quantify the uncertainty parameters and the influence of the coupling relationship between them on the reliability and robustness of complex equipment. In order to reduce the difficulty of calculation uncertainty on the calculation process, the ellipsoid domain with higher uncertainty is converted into a standard domain, and then the standard domain is subdivided into multiple simplexes with smaller uncertainty. By calculating the structural response boundary of each simplex with smaller uncertainty, dividing each structural response boundary into subintervals and assigning different weights, the possibility of the complex equipment in the uncertainty ellipsoid domain is calculated, based on which the reliability of the complex equipment under uncertain conditions can be analyzed. The present invention can realize the accurate measurement of multi-source uncertain parameters under small sample conditions, and realize accurate uncertainty propagation analysis of complex equipment structural response in a non-probabilistic framework, and has a good application prospect in the uncertainty analysis of complex equipment structural response. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] Other features, objects and advantages of the present invention will become more apparent from the detailed description of non-limiting embodiments made with reference to the following drawings:

[0047] Figure 1 It is a flowchart of the ellipsoid multi-subdomain analysis method for complex equipment uncertainty under small samples;

[0048] Figure 2 It is a schematic diagram of the ellipsoid domain when the uncertainty parameter dimension is equal to 2;

[0049] Figure 3 Schematic diagram of the standard domain d simplex and simplex when the uncertainty parameter dimension is equal to 2;

[0050] Among them, 1. Ellipsoid domain; 2. Standard domain; 3. d simplex; 4. Simplex; 5. Vertex; 6. Initial vertex. DETAILED DESCRIPTION

[0051] The present invention will be further described in detail below in conjunction with the accompanying drawings and embodiments. It is to be understood that the specific embodiments described herein are only used to explain the relevant invention, rather than to limit the invention. It should also be noted that, for ease of description, only the parts related to the invention are shown in the accompanying drawings.

[0052] It should be noted that, in the absence of conflict, the embodiments of the present invention and the features in the embodiments can be combined with each other. The present invention will be described in detail below with reference to the accompanying drawings and in combination with the embodiments.

[0053] Please refer to Figure 1 The present invention provides an ellipsoid multi-subdomain analysis method for complex equipment uncertainty under small sample conditions, including:

[0054] Determine in advance the various parameters that affect complex equipment and select the types of uncertain parameters that need to be studied and analyzed;

[0055] S1: Obtaining quantitative indicators of various types of uncertainty parameters of complex equipment; the quantitative indicators include the mean, interval range and uncertainty level of the uncertainty parameters;

[0056] Since the specific values ​​of various parameters of complex equipment are difficult to measure or calculate, and the coupling relationship between them is also unknown; therefore, we can only roughly estimate the quantitative indicators of uncertainty parameters based on the actual data that can be obtained.

[0057] In this embodiment, the thermal-mechanical coupling system of an aerial camera is taken as an example; Table 1 is a quantitative index table of uncertainty parameters of this complex equipment:

[0058] Table 1 Uncertainty parameter quantitative index table

[0059]

[0060] Among them, the uncertainty level of a type of uncertainty parameter is equal to the difference between the maximum and minimum values ​​of the interval range divided by 2 times the mean. The uncertainty level, mean, maximum and minimum values ​​of the interval range can be converted to each other through the above formula.

[0061] According to Table 1, in this embodiment, there are four types of uncertainty parameters, so the number of types or dimensions of uncertainty parameters is four.

[0062] S2: constructing an uncertainty ellipsoid domain according to the quantitative indicators of the uncertainty parameters of various types; the uncertainty ellipsoid domain is used to quantify the uncertainty parameters and the coupling relationship between the uncertainty parameters of various types; referring to Figure 2 ;

[0063] In this embodiment, the interval range of the uncertainty parameter and the correlation coefficient matrix are used to calculate the non-probabilistic covariance matrix C of the complex equipment, wherein the correlation coefficient is represented by the correlation coefficient matrix R.

[0064] The process of calculating the non-probability covariance matrix C based on the correlation coefficient matrix R and the interval range of the uncertainty parameter includes:

[0065] The value of the i-th row and j-th column in the non-probability covariance matrix C The calculation method is shown in Formula 1:

[0066] Formula 1;

[0067] in, represents the interval width of the i-th parameter, represents the interval width of the jth parameter, represents the value of the i-th row and j-th column in the correlation coefficient matrix R, that is, the uncertainty parameter and uncertainty parameters The correlation coefficient of . i and j are both positive integers.

[0068] Then, the non-probability covariance matrix C is inverted to obtain the characteristic matrix According to Formula 2, the ellipsoid domain 1 of the uncertainty parameters of the thermal-mechanical coupling system of the aerial camera is calculated, and the final calculation structure is shown in Formula 3:

[0069] Formula 2:

[0070]

[0071] Formula 3;

[0072] in, represents the uncertainty ellipsoid domain, X represents the uncertainty parameter matrix, X C The midpoint matrix of the ellipsoid domain 1 representing the uncertainty parameters is specifically expressed as: , n represents the number of columns of the midpoint matrix;

[0073] G represents the feature matrix, represents the transpose of the matrix and E represents the exponent 10.

[0074] S3: Convert the uncertainty ellipsoid domain into the standard domain 2, reference Figure 3 ; The standard domain 2 is a circle when the dimension is 2 and a sphere when the dimension is 3.

[0075] The process of converting the ellipsoid domain to the standard domain 2 includes:

[0076] Obtain the dimension of the uncertainty parameter, which is 4 in this embodiment;

[0077] Obtaining structural responses of complex rigs , correlation coefficient matrix R;

[0078] Perform Cholesky decomposition on the correlation coefficient matrix R and obtain a The matrix ; n represents the number of columns of the midpoint matrix; as shown in Formula 4:

[0079] Formula 4;

[0080] The structural response of complex equipment in the uncertainty ellipsoid domain Transformation to the structural response of a complex device in the standard domain 2 ;

[0081] The definition of standard domain 2 is shown in formula 5:

[0082] Formula 5;

[0083] in, Indicates standard domain 2, represents a vector in the standard domain.

[0084] S4: Divide the standard domain into a plurality of simplexes; the simplex is a geometric body formed by connecting a plurality of points, and the plurality of simplexes are used to approximately represent the standard domain, including:

[0085] S4-1: decomposing the standard domain 2 into a preset number of ellipsoid subdomains on average; the uncertainty of the ellipsoid subdomains is less than that of the standard domain 2;

[0086] The specific steps include:

[0087] S4-1-1: Obtaining a scaling factor of the standard domain 2; the scaling factor is equal to the product of a preset number and a small variable; the small variable is a positive number infinitely close to 0;

[0088] S4-1-2: Decompose the standard domain 2 into a preset number of ellipsoid subdomains according to the scaling factor; in this embodiment, the number is set to m.

[0089] The decomposition process is uniform decomposition. For example, when the dimension of the uncertainty parameter is 2, it is uniformly decomposed into multiple sectors. When the dimension is 3, it is uniformly decomposed into multiple spherical cones.

[0090] Specifically, the scaling factor The choice of is set manually according to the actual data distribution. Indicates that the radius of the standard domain is share, That is, the preset number; equivalent to obtaining concentric circles (2D) or concentric spheres (3D), represents a tiny variable, and the tiny variable is a positive number infinitely close to 0. Therefore, the scaling factor is equal to the product of the preset number and the tiny variable. Decomposing the standard domain 2 into a plurality of ellipsoidal subdomains can be understood as: dividing the standard domain 2 into a plurality of subdomains along the radius.

[0091] The following are examples:

[0092] In other cases, when the dimension of the uncertainty parameter is 2, an ellipsoid subdomain is a quarter-circle arc; different ellipsoid subdomains have the same center.

[0093] In other cases, when the dimension of the uncertainty parameter is 3, an ellipsoid subdomain is a one-eighth spherical shell; different ellipsoid subdomains have the same sphere center.

[0094] S4-2: Divide the plurality of ellipsoid subdomains into the preset number of d simplices respectively; the d simplices are geometric bodies formed by connecting a plurality of points, and the degree of approximation between the preset number of d simplices and the standard domain is lower than the degree of approximation between the plurality of simplices and the standard domain; wherein d is the dimension of the d simplices, which is 1 less than the number of types of the uncertainty parameter, and is used to decompose the ellipsoid subdomains; in this embodiment, d is equal to 3;

[0095] Specifically, in order to make the uncertainty parameter more realistic, in this embodiment, a simplex 4 is used to approximate an ellipsoid subdomain. All simplices can approximate all ellipsoid subdomains, i.e., the standard domain. The higher the subdivision degree of the simplex 4, the closer it is to the shape of the real ellipsoid subdomain.

[0096] d Simplex 3 is the initial state of the ellipsoid subdomain; when the dimension of the uncertainty parameter is 2, it is Figure 3 When the dimension is 3, d-simplex 3 is a tetrahedron. There is still a clear difference between this and the actual ellipsoid subdomain.

[0097] Therefore, in order to more closely approximate the ellipsoid subdomain in the shape of a quarter arc, it needs to be further subdivided into simplex 4.

[0098] S4-3: Mapping the plurality of d simplices to the corresponding ellipsoid subdomains respectively to obtain a plurality of simplices, including:

[0099] S4-3-1: Obtain the subdivision number k of each ellipsoid subdomain and the structural response of the complex equipment ;

[0100] S4-3-2: Based on the subdivision number k, the dimension of the uncertainty ellipsoid domain and the structural response of the complex equipment Generate a color matching matrix M; the dimension of the uncertainty ellipsoid domain is equal to the number of types of uncertainty parameters;

[0101] After segmentation, we get Initial simplexes 4, and initial vertices 6. The larger the number of subdivisions, the closer the simplex is to the ellipsoid subdomain. Thus, a complex ellipsoid domain can be approximated by multiple simpler simplexes.

[0102] The process of generating the color matching matrix is ​​a prior art and will not be described in detail here.

[0103] S4-3-3: dividing the d simplex into a plurality of initial simplexes according to the corresponding element definition in the color matching matrix; the initial simplex has a plurality of initial vertices; the plurality of initial vertices include a plurality of geometric vertices of the initial simplex and additional vertices required for subdividing each side of the initial simplex by the number of subdivisions;

[0104] The additional vertices required for subdividing the edges of the initial simplex by the number of subdivisions are not located on the ellipsoid subdomain.

[0105] Specifically, Figure 3 For the case where the dimension of the uncertainty parameter is 2, we obtain 4 d-simplexes 3; Figure 3 The number of subdivisions is chosen to be 2, so one d simplex 3 gives 4 initial simplices.

[0106] The coordinates of the initial vertex 6 are defined according to the u-th column of the color matching matrix M, where u represents a positive integer.

[0107] Before mapping the coordinates of vertex 5, vertex 5 of the simplex is on the hypotenuse of the triangle corresponding to d-simplex 3.

[0108] In order to make the simplex more approximate the ellipsoid subdomain, the coordinates of vertex 5 need to be mapped. The process includes:

[0109] S4-3-4: normalizing the coordinates of all the initial vertices 6 to map all the initial vertices 6 to corresponding ellipsoid subdomains to obtain the coordinates of all the vertices 5 of a plurality of simplexes;

[0110] The coordinate unitization process includes:

[0111] Dividing the vector representing the coordinates of the initial vertex 6 by the second norm of the vector can obtain the coordinate vector mapped to the point on the ellipsoid subdomain, that is, the coordinates of the vertex 5.

[0112] Therefore, all vertices 5 are in the shape of the ellipsoid subdomain, which can make the obtained polygon closer to the shape of the standard domain 2 and ensure that the analysis structure is closer to the true value.

[0113] S4-3-5: According to the scaling factor and the coordinates of all the vertices of the simplex, calculate the function value of the structural response of all the vertices of the simplex under the standard domain and the function value of the structural response under the ellipsoidal domain ,The specific process includes:

[0114] Calculate the function value of the structural response at the center position under the standard domain 2 and the function value of the structural response of all the vertices 5 under the standard domain 2; when the dimension is 2, the center is the center of the circle, when the dimension is 3, the center of the sphere, and so on;

[0115] Substitute the center coordinates of the ellipsoid subdomain obtained after mapping into the function value of the structural response under standard domain 2 ;

[0116] Substitute the vertex coordinates on the ellipsoid subdomain obtained after mapping into the function value of the structural response under the ellipsoid domain 2 .

[0117] S4-3-6: Using the coordinates of the vertex 5, the function value of the structural response at the center position under the standard domain 2 and the function value of the structural response at the vertex 5 under the ellipsoid domain 1 Storing the characteristics of the simplex; equivalent to obtaining the simplex.

[0118] Since the features of all simplexes are stored through vertex coordinates, coordinate position function values ​​and circle center coordinate position function values, the features of multiple simplexes can be stored using this step method.

[0119] S5: Calculates the structural response boundaries of multiple simplexes, including:

[0120] S5-1: Generate a direction cosine matrix for each simplex and the difference vector matrix ;

[0121] S5-2: According to the direction cosine matrix and the difference vector matrix , calculate the extreme points of the simplex structural response ;

[0122] The calculation method is: Substitute the value of a point into ,when is non-negative linear and , it indicates that the point is an extreme point.

[0123] S5-3: Calculate the upper boundary of the simplex structural response according to the extreme point of the simplex structural response and lower bound ; The upper limit of the boundary and the lower boundary The structural response boundary that constitutes the simplex .

[0124] Wherein, the upper bound of the simplex structural response is is equal to the maximum value of the function values ​​of a plurality of simplex vertex positions;

[0125] The lower bound of the structural response of the simplex is equal to the minimum value of the function values ​​of a plurality of simplex vertex positions;

[0126] S6: According to the structural response boundaries of multiple simplexes, the possibility of obtaining the complex equipment in the uncertainty ellipsoid domain ,include:

[0127] S6-1: Calculate the upper boundary of the structural response of the complex equipment based on the structural response boundaries of the plurality of simplexes and the lower boundary ,include:

[0128] S6-1-1: Calculating the structural response boundaries of the plurality of d simplexes 3 according to the structural response boundaries of the plurality of simplexes;

[0129] The structural response bound upper bound of the d simplex 3 is equal to the maximum value among the upper bounds of the structural responses of the multiple simplices decomposed by the d simplex;

[0130] The structural response lower bound of the d simplex 3 is equal to the minimum value among the lower bounds of the structural responses of the multiple simplices decomposed by the d simplex;

[0131] The upper and lower bounds of the structural response of the d simplex 3 constitute the structural response bounds of the d simplex 3. .

[0132] S6-1-2: Calculate the upper boundary of the ellipsoid subdomain structural response based on the structural response boundaries of the plurality of d simplex 3 and the lower boundary ;

[0133] The upper boundary of the ellipsoid subdomain structure response is equal to the maximum value among the upper bounds of the structural responses of the multiple d-simplexes contained in the ellipsoid subdomain;

[0134] The lower boundary of the ellipsoid subdomain structure response is equal to the minimum value among the lower bounds of the structural responses of the multiple d-simplexes contained in the ellipsoid subdomain;

[0135] S6-1-3: According to the upper boundary of the ellipsoid subdomain structure response and the lower boundary , calculate the upper boundary of the structural response of the complex equipment and the lower boundary .

[0136] The upper boundary of the structural response of the complex equipment is equal to the maximum value among the upper bounds of all ellipsoidal subdomain structural responses;

[0137] The lower bound of the structural response of the complex equipment It is equal to the minimum value of the lower limit of all ellipsoid subdomain structure responses;

[0138] S6-2: Calculating the probability of the complex equipment in the uncertainty ellipsoid domain according to the upper boundary and the lower boundary of the structural response of the complex equipment, including:

[0139] S6-2-1: Divide the upper boundary and the lower boundary of the structural response of the complex equipment into a plurality of boundary sub-intervals; in this embodiment, there are H sub-intervals;

[0140] The specific division method is to evenly divide the boundary of the structural response of the complex equipment into H boundary sub-intervals, and each boundary sub-interval is expressed as , where h=1, 2,...H.

[0141] S6-2-2: When the dimension is 2, the area of ​​each ellipsoid subdomain is used; when the dimension is 3, the volume of each ellipsoid subdomain is used; and so on. In high dimensions, weights are assigned to multiple boundary subintervals according to the law that the weights are positively correlated with the high-dimensional volume of the boundary subinterval. ;

[0142] Taking dimension = 2 as an example, the ratio of the areas between the two ellipsoid subdomains is equal to the ratio of the weights.

[0143] S6-2-3: Obtain the probability of the complex equipment in the uncertainty ellipsoid domain according to the upper and lower boundaries of the multiple ellipsoid subdomain structural responses, the multiple boundary subintervals and the weights corresponding to the multiple boundary subintervals. .

[0144] The calculation process is as follows:

[0145] The structural response boundary of the simplex The interval and each boundary subinterval The size of the interval of the overlap part is the structural response boundary of the d simplex 3 The ratio of the interval sizes is defined as the overlap rate ;

[0146] For example, the interval sizes of [1,3] and [2,4] are both 2; and the interval size of the overlapping part of the two is 1;

[0147] The weights corresponding to the multiple boundary subintervals are The corresponding multiple coincidence rates Multiply and add to get the cumulative sum ;

[0148] The weight corresponding to the boundary subinterval The corresponding coincidence rate The product and cumulative sum of The ratio of is the probability of complex equipment in each boundary subinterval .

[0149] The above steps are used to calculate the probability of the structural response (in this embodiment, the deformation of the motherboard and the temperature of the CPU) in the thermal-mechanical coupling system of the aerial camera. .

[0150] S7: Evaluate the reliability of the complex equipment according to the possibility, and obtain a conclusion on whether the reliability of the current complex equipment meets the actual needs. The detailed process belongs to the prior art and will not be described here.

[0151] In order to reduce the difficulty of the calculation process caused by the calculation uncertainty, this embodiment converts the ellipsoid domain 1 with higher uncertainty into the standard domain 2, and then subdivides the standard domain 2 into multiple simplexes with smaller uncertainty. By calculating the structural response boundary of each simplex with smaller uncertainty, and dividing each structural response boundary into subintervals and assigning different weights, the possibility of the complex equipment in the uncertainty ellipsoid domain is calculated, based on which the possibility of the structural response of the complex equipment under uncertain conditions can be analyzed. The present invention can realize the accurate measurement of multi-source uncertain parameters under small sample conditions, and realize accurate uncertainty propagation analysis of the structural response of complex equipment in a non-probabilistic framework, and has a good application prospect in the uncertainty analysis of the structural response of complex equipment.

[0152] The more samples the conventional Monte Carlo method has, the more accurate it is. However, when faced with fewer samples, the accuracy of the Monte Carlo method drops significantly.

[0153] The above scheme of the present embodiment does not require a large number of samples, and can achieve the same accuracy of complex equipment analysis as the Monte Carlo method when the number of samples is large, thus filling the technical gap of the existing methods.

[0154] The above description is only a preferred embodiment of the present invention and an explanation of the technical principles used. Those skilled in the art should understand that the scope of the invention involved in the present invention is not limited to the technical solution formed by a specific combination of the above technical features, but should also cover other technical solutions formed by any combination of the above technical features or their equivalent features without departing from the inventive concept. For example, the above features are replaced with the technical features with similar functions disclosed in the present invention (but not limited to) to form a technical solution.

Claims

1. Ellipsoid multi-subdomain analysis method for complex equipment uncertainty under small sample conditions, characterized by: include: S1: Obtaining quantitative indicators of various types of uncertainty parameters of complex equipment; the quantitative indicators include the mean, interval range and uncertainty level of the uncertainty parameters; S2: constructing an uncertainty ellipsoid domain according to the quantitative indicators of the uncertainty parameters of various types; the uncertainty ellipsoid domain is used to quantify the uncertainty parameters and the coupling relationship between the uncertainty parameters of various types; S3: converting the uncertainty ellipsoid domain into a standard domain; S4: dividing the standard domain into a plurality of simplices; the simplex is a geometric body formed by connecting a plurality of points, and the plurality of simplices are used to approximately represent the standard domain; S5: Calculate the structural response boundaries of multiple simplexes; S6: Obtaining the possibility of the complex equipment in the uncertainty ellipsoid domain according to the structural response boundaries of the plurality of simplexes; S7: Evaluate the reliability of the complex equipment based on the probability.

2. The ellipsoid multi-subdomain analysis method for complex equipment uncertainty under small sample conditions according to claim 1 is characterized in that: S4: Divide the standard domain into a plurality of simplexes, including: S4-1: decomposing the standard domain into a preset number of ellipsoidal subdomains on average; S4-2: Divide the plurality of ellipsoid subdomains into the preset number of d simplices respectively; the d simplices are geometric bodies formed by connecting a plurality of points, and the degree of approximation between the preset number of d simplices and the standard domain is lower than the degree of approximation between the plurality of simplices and the standard domain; wherein d is the dimension of the d simplices, which is 1 less than the number of types of the uncertainty parameter, and is used to decompose the ellipsoid subdomains; S4-3: Map the multiple d simplices to the corresponding ellipsoid subdomains respectively to obtain multiple simplices.

3. The ellipsoid multi-subdomain analysis method for complex equipment uncertainty under small sample conditions according to claim 2 is characterized in that: S4-1: Decomposing the standard domain into a preset number of ellipsoidal subdomains on average, including: S4-1-1: Obtaining a scaling factor of the standard domain; the scaling factor is equal to the product of a preset number and a small variable; the small variable is a positive number infinitely close to 0; S4-1-2: Decompose the standard domain into a preset number of ellipsoid sub-domains on an even basis according to the scaling factor.

4. The ellipsoid multi-subdomain analysis method for complex equipment uncertainty under small sample conditions according to claim 3 is characterized in that: S4-3: Mapping the plurality of d simplices to the corresponding ellipsoid subdomains respectively to obtain a plurality of simplices, including: S4-3-1: Obtain the number of subdivisions of each ellipsoid subdomain and the structural response of the complex equipment; S4-3-2: Generate a color matching matrix according to the number of subdivisions, the dimension of the uncertainty ellipsoid domain and the structural response of the complex equipment; the dimension of the uncertainty ellipsoid domain is equal to the number of types of uncertainty parameters; S4-3-3: dividing the d simplex into a plurality of initial simplexes according to the corresponding element definition in the color matching matrix; the initial simplex has a plurality of initial vertices; the plurality of initial vertices include a plurality of geometric vertices of the initial simplex and additional vertices required for subdividing each side of the initial simplex by the number of subdivisions; S4-3-4: normalize the coordinates of all the initial vertices, map the initial vertices to the corresponding ellipsoid subdomain, and obtain the coordinates of all vertices of the simplex; S4-3-5: Based on the scaling factor and the coordinates of all the vertices of the simplex, calculate the function values ​​of the structural response of all the vertices of the simplex in the standard domain and the function values ​​of the structural response in the ellipsoid domain.

5. The ellipsoid multi-subdomain analysis method for complex equipment uncertainty under small sample conditions according to claim 2 is characterized in that: S5: Computes structural response boundaries for multiple simplexes, including: S5-1: Generate a direction cosine matrix and a difference vector matrix for each simplex; S5-2: Calculating the extreme points of the simplex structural response according to the direction cosine matrix and the difference vector matrix; S5-3: Calculate the upper boundary and lower boundary of the simplex structural response according to the extreme value points of the simplex structural response; the upper boundary and the lower boundary constitute the structural response boundary of the simplex.

6. The ellipsoid multi-subdomain analysis method for complex equipment uncertainty under small sample conditions according to claim 5 is characterized in that: S6: Obtaining the possibility of the complex equipment in the uncertainty ellipsoid domain according to the structural response boundaries of the plurality of simplexes, including: S6-1: Calculating the upper and lower boundaries of the structural response of the complex equipment according to the structural response boundaries of the plurality of simplexes; S6-2: According to the upper and lower boundaries of the structural response of the complex equipment, the possibility of the complex equipment in the uncertainty ellipsoid domain is calculated.

7. The ellipsoid multi-subdomain analysis method for complex equipment uncertainty under small sample conditions according to claim 6 is characterized in that: S6-1: Calculating the upper and lower boundaries of the structural response of the complex equipment according to the structural response boundaries of the plurality of simplexes, including: S6-1-1: Calculating the structural response boundaries of the plurality of d simplexes according to the structural response boundaries of the plurality of simplexes; S6-1-2: Calculate the upper and lower boundaries of the ellipsoid subdomain structural response according to the structural response boundaries of the plurality of d simplexes; S6-1-3: Based on the upper and lower boundaries of the structural response of the ellipsoid subdomain, the upper and lower boundaries of the structural response of the complex equipment are calculated.

8. The ellipsoid multi-subdomain analysis method for complex equipment uncertainty under small sample conditions according to claim 7 is characterized in that: S6-2: Calculating the probability of the complex equipment in the uncertainty ellipsoid domain according to the upper boundary and the lower boundary of the structural response of the complex equipment, including: S6-2-1: Divide the upper boundary and the lower boundary of the structural response of the complex equipment into a plurality of boundary sub-intervals; S6-2-2: assigning weights to the plurality of boundary subintervals; the weights being positively correlated with the boundary subintervals; S6-2-3: Calculate the possibility of the complex equipment in the uncertainty ellipsoid domain based on the upper and lower boundaries of the responses of multiple ellipsoid sub-domain structures, multiple boundary sub-intervals and weights corresponding to the multiple boundary sub-intervals.

Images