CN114169192B - Uncertainty analysis method of thermodynamic coupling system based on rough set theory - Google Patents
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Abstract
The invention discloses a rough set theory-based uncertainty analysis method for a thermodynamic coupling system, relates to the field of mechanical engineering, and is used for solving the problems that the thermodynamic coupling system relies on priori knowledge and is low in efficiency when performing uncertainty analysis. The method comprises the following steps: determining a domain of uncertain parameters of the coupling system, and constructing a knowledge base of the domain according to experimental measurement points; constructing an upper approximation set and a lower approximation set and calculating approximation precision aiming at any uncertain region in the domain; establishing an initial Kriging model through Latin hypercube sampling and system finite element calculation, and realizing the parameter correction of the Kriging model by utilizing a maximum likelihood estimation and mode search method; based on the maximum mean square error point adding criterion, a self-adaptive kriging model is established, and then the upper and lower response bounds of the upper and lower approximation sets are calculated, so that the response approximation accuracy is obtained. By applying the rough set theory, the uncertainty analysis of the thermodynamic coupling system is performed without relying on priori knowledge, and the efficiency is high.
Description
Technical Field
The invention belongs to the field of mechanical engineering, and particularly relates to a thermodynamic coupling system uncertainty analysis method based on a rough set theory.
Background
Uncertainty analysis is a precondition for reliability analysis and robustness optimization, and refers to a process of quantitatively describing and calculating system output response analysis containing uncertainty parameters aiming at uncertainty in a system. Along with the development of numerical simulation technology, the uncertainty widely exists in engineering practice due to factors such as model simplification errors, load and material property changes, processing errors and measurement errors. For a simple structure, the influence of uncertainty factors is small, but for a complex system, accumulation and coupling among uncertainty factors can seriously influence the stability and reliability of the system, and how to effectively realize quantification of uncertainty parameters and uncertainty propagation analysis becomes an urgent need for the actual problems of the complex system, particularly the engineering of a thermodynamic coupling system.
The uncertainty quantification model commonly used at present mainly comprises a probability model, a fuzzy model and a convex set model. In probability models, the uncertain parameters are usually described as random variables, and the establishment of the probability model requires a large amount of sample information to determine accurate probability features, such as probability distribution functions, moments, and the like, so in the case of small samples, the probability model has a certain limitation. In the fuzzy model, the concept or parameter values of certain things have ambiguity, and a membership function is often constructed based on engineering experience and test data to determine a rough range. Compared with a probability model, the convex set model only needs a small amount of sample information, the uncertainty of the parameters is represented by using the upper limit and the lower limit, the convex set model is more convenient to build, but when uncertainty analysis is carried out on a complex system, the sample distribution characteristics of the uncertain parameters are more complex, and the convex set model has the limitation of poor quantification effect and the like.
Developing uncertainty propagation analysis based on an uncertainty quantification model is another important content of uncertainty analysis methods. With the increasing complexity of engineering problems, traditional uncertainty propagation analysis methods such as sampling methods, perturbation methods, numerical integration methods and the like face contradictions that the calculation accuracy and the calculation efficiency are not adjustable. The complex thermodynamic coupling system presents the characteristics of high nonlinearity, uncertainty discipline intersection and the like, and is suitable for the uncertainty propagation analysis method and needs to be studied in depth.
Thus, in view of the foregoing, there is a need for a highly efficient uncertainty analysis method for a thermally coupled system that does not rely on any prior knowledge.
Disclosure of Invention
In view of the above, the invention provides a thermodynamic coupling system uncertainty analysis method based on rough set theory, which is used for solving the problems that the thermodynamic coupling system uncertainty analysis method depends on any priori knowledge and has lower efficiency.
The invention provides a thermodynamic coupling system uncertainty analysis method based on rough set theory, which comprises the following steps:
step one: determining a domain of a plurality of uncertain parameters of the thermodynamic coupling system, and constructing a knowledge base of the domain according to experimental measuring points in the domain;
step two: constructing an upper approximation set and a lower approximation set of the uncertain region based on the knowledge base aiming at the uncertain region in the discussion domain, taking the upper approximation set and the lower approximation set as uncertainty quantification models, and calculating the approximation precision of the uncertain region;
Step three: establishing a finite element model of the thermal coupling system, extracting sample points in the discussion domain by adopting a random Latin hypercube sampling method, and obtaining the response of the thermal coupling system at the sample points by adopting a finite element calculation method;
step four: establishing an initial kriging proxy model according to the sample points and the response thereof, and correcting the kriging proxy model by a maximum likelihood estimation and mode searching method;
Step five: in the theory domain, searching a maximum value point of the mean square error of the corrected Kerling proxy model by utilizing a maximum mean square error criterion, calculating a system response of the maximum value point by finite elements, and reconstructing the Kerling proxy model by taking the maximum value point and the corresponding response as newly added data until a convergence condition of the maximum mean square error is met;
Step six: and taking the reconstructed agent model as an uncertainty propagation analysis model, calculating the upper response bound and the lower response bound of the upper approximation set and the upper response bound and the lower response bound of the lower approximation set by using the agent model, and defining and calculating response approximation accuracy.
The uncertainty analysis method of the thermodynamic coupling system based on the rough set theory provided by the invention has the following advantages:
(1) Compared with the traditional uncertain parameter quantification method, the method for representing the upper and lower approximate sets based on the rough set theory can be used for improving the accuracy of parameter uncertainty characterization by using an uncertain region with any complex shape as an application object.
(2) In the process of constructing the knowledge base, the domain is divided into dividing blocks with regular shapes based on the known experimental measuring point set, so that the subsequent response interval calculation is more convenient.
(3) The kriging agent model in the invention can carry out model correction and update sample data according to the dotted criterion, and has self-adaptability. The self-adaptive kriging agent model is used as a black box to operate, so that the calculation times of the finite element model can be reduced, the global accuracy of calculation is ensured, and the calculation efficiency is effectively improved.
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FIG. 1 is a flow chart of a rough set theory-based thermodynamic coupling system uncertainty propagation analysis method in an embodiment of the invention;
FIG. 2 is a flow diagram of a rough set theory-based thermodynamic coupling system uncertainty propagation analysis method in an embodiment of the invention;
FIG. 3 is a schematic structural view of a three-dimensional airfoil in an embodiment of the invention.
Detailed Description
In order to make the above objects, features and advantages of the embodiments of the present invention more comprehensible, the technical solutions of the embodiments of the present invention will be described clearly and completely with reference to the accompanying drawings. It will be apparent that the described embodiments are only some, but not all, embodiments of the invention. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
The rough set theory is used as a mathematical tool for processing the uncertainty problem, any priori knowledge outside the data set required by the uncertainty problem is not required to be provided, the potential rule is revealed only by using the information of the data set, and the method has good combination with other uncertainty analysis methods. The uncertainty analysis is carried out on the thermodynamic coupling system under the condition of lacking priori knowledge by utilizing the rough set theory, and the method has important academic and engineering application values.
The invention is further described below with reference to the drawings and examples.
Referring to fig. 1 and 2, an embodiment of the present invention provides a thermodynamic coupling system uncertainty analysis method based on rough set theory, including the following steps:
Step one: and determining the domains of a plurality of uncertain parameters of the thermodynamic coupling system, and constructing a knowledge base of the domains according to experimental measuring points in the domains.
In some possible embodiments, the uncertainty parameter generally includes a plurality, and step one may include the following:
First, the range of variation of each uncertain parameter is represented in the form of intervals, namely each uncertain parameter has a variation interval, and the universe is the Cartesian product of a plurality of uncertain parameter intervals. Then, a plurality of experimental measurement points are obtained through an experimental method, wherein each experimental measurement point comprises a set of values of each uncertain parameter, for example, when the uncertain parameter comprises a thermal expansion coefficient and a thermal conduction coefficient, each experimental measurement point comprises a thermal expansion coefficient value and a thermal conduction coefficient value. And sequencing and de-duplicating the values of the same uncertain parameter of a plurality of experimental measuring points, and forming a breakpoint set of the uncertain parameter with the upper and lower boundaries of the variation interval of the uncertain parameter.
For example, n experimental measurement points include n thermal expansion coefficient values, n is a positive integer greater than 1, and the n thermal expansion coefficient values are sorted and de-duplicated, i.e. the same thermal expansion coefficient value only remains one, and the upper and lower bounds of the variation interval of the thermal expansion coefficient are increased to form a breakpoint set of the thermal expansion coefficient. And obtaining a plurality of breakpoint sets of a plurality of uncertain parameters, and constructing a knowledge base of the discourse domain based on the breakpoint sets.
Specifically, the uncertainty parameters generally include a plurality, each of which is bounded, and the range of variation of the uncertainty parameters is characterized by an interval, and the uncertainty parameter X i can be described by referring to the following formula:
wherein m represents the number of uncertain parameters, X i represents the ith uncertain parameter, Representing the i-th uncertainty parameter variation interval,/>And/>The lower and upper bounds of the variation interval, respectively.
All uncertainty parameters within the thermodynamic coupling system can be represented as X, x= (X 1,X2,...,Xm)T, whereThe domain U of the thermodynamic coupling system can be expressed as:
Wherein "x" represents a cartesian product.
The experimental measurement points are data points for measuring a plurality of uncertain parameters in an experimental mode, and the experimental measurement points form an experimental measurement point set w=x 1,x2,...,xl. Wherein l represents the number of experimental measurement points, x i＝(xi1,xi2,...,xim)T represents the ith experimental measurement point, x ij represents the jth uncertain parameter value of the ith experimental measurement point, the jth uncertain parameter value of each experimental measurement point in the experimental measurement point set W is sequenced, the repeated value is removed, and the lower bound of the variation interval of the jth uncertain parameter is increasedAnd upper bound of variation interval/>Obtaining a group of break point sets from small to large corresponding to the j-th uncertain parameters, and obtaining m groups of break point sets corresponding to m uncertain parameters by adopting the mode, wherein the m groups of break point sets are expressed as follows:
where x i,j represents the i-th breakpoint of the order from small to large corresponding to the j-th uncertainty parameter, and the subscript k j represents the number of breakpoints corresponding to the j-th uncertainty parameter, which satisfies the inequality k j +.ltoreq.l+2 (j=1, 2.., m).
And constructing a knowledge base of the domain through the m groups of breakpoint sets. Specifically, each breakpoint set disperses its corresponding uncertain parameter variation interval into a union of multiple subintervals:
Because the subintervals are non-empty and the intersection sets are empty sets, the division relation is satisfied, if the uncertain parameters are mutually independent, one subinterval is taken from among the multiple subintervals corresponding to each uncertain parameter, the division blocks in the theoretical domain are defined as Cartesian products of the taken multiple subintervals, and the Cartesian products are uniformly represented by the symbol pi as follows:
Wherein, R j break points,/>, representing the j-th set of break pointsRepresenting the corresponding order/>, of the j-th uncertain parameterDiscrete interval as left boundary, if/>1 St breakpoint of j-th group breakpoint set, interval/>Conversion to amenorrhea/>The partitioned blocks appear as regular hyper-cubes in the geometric space.
Using the partitioning blocks, the knowledge base V of the domain is expressed as:
wherein N represents a positive integer and wherein, The union operation of all the above-mentioned dividing blocks in the domain is represented, and the knowledge base of the domain is represented as a set of a plurality of rule supercubes in the geometric space.
Based on the partitioned blocks, an equivalence relation is defined, and for any element a, b and c on one partitioned block, they are considered to satisfy the following equivalence relation R, namelyAnd/>And then establishing equivalence classes of uncertain parameters by using equivalence relations:
[a]R＝{b|b∈π,(a,b)∈R}；
wherein a is a representative element of an equivalence class, and R is an equivalence relation in a dividing block.
Illustratively, referring to FIG. 3, an example of an on-wing thermal coupling system is shown, in which the thermal expansion coefficient α, the elastic modulus E, and the thermal conductivity λ of the material are used as uncertainty parameters, α ε [1.88×10 -5,2.2×10-5]℃-1 ], E ε [64,71] GPa, λ ε [200,237] W/(m.K) are the variation intervals of uncertainty parameters, and the argument is the Cartesian product of the variation intervals of the 3 uncertainty parameters:
U＝[1.88×10-5,2.2×10-5]×[64,71]×[200,237]；
Table 1 initial 5 test point samples
Obtaining an experimental measurement point set W= { x 1,x2,x3,x4,x5 } about 3 uncertain parameters in an experimental mode, as shown in table 1, sorting the j uncertain parameter values of each experimental measurement point in the experimental measurement point set W, removing the repeated values, and increasing the lower boundary of the variation interval of the j uncertain parametersAnd upper bound of variation interval/>Obtaining a group of break point sets from small to large corresponding to the j-th uncertain parameters, taking elastic modulus as an example, and obtaining the break point sets as follows:
W1＝{64,64.0684,65.6074,67.7244,69.0749,69.7306,71}；
the 3 groups of breakpoint sets corresponding to the 3 uncertain parameters are obtained in the mode, and are expressed as follows:
Where x i,j represents the i-th breakpoint of the order from small to large corresponding to the j-th uncertainty parameter, and the subscript k j represents the number of breakpoints corresponding to the j-th uncertainty parameter, which satisfies the inequality k j.ltoreq.7 (j=1, 2, 3).
Constructing a knowledge base of a domain through the m groups of breakpoint sets, wherein the breakpoint sets are used for dispersing a plurality of uncertain parameter change intervals into a union set of a plurality of subintervals:
Taking the elastic modulus as an example, the dispersion is as follows:
Since each subinterval is non-empty and the intersection is an empty set, the partitioning relationship is satisfied, any one subinterval is taken out of the multiple subintervals corresponding to each uncertain parameter, and the partitioning block in the theoretical domain is defined as the Cartesian product of the taken multiple subintervals, and the partitioning block is collectively represented by the symbol pi as follows:
wherein "×" represents the cartesian product, R j break points,/>, representing the j-th set of break pointsRepresenting the corresponding order/>, of the j-th uncertain parameterDiscrete interval as left boundary, if/>1 St breakpoint of j-th group breakpoint set, interval/>Conversion to amenorrhea/>The partitioned blocks appear as regular hyper-cubes in the geometric space.
Taking a first subinterval corresponding to the thermal expansion coefficient, the elastic modulus and the thermal conductivity coefficient as an example, a dividing block is established as follows:
π＝[1.88×10-5,1.8818×10-5]×[64,64.0684]×[200,206.8470]；
using the partitioning blocks, the knowledge base V of the domain is expressed as:
wherein N represents a positive integer and wherein, The union operation of all the above-mentioned dividing blocks in the domain is represented, and the knowledge base of the domain is represented as a set of a plurality of rule supercubes in the geometric space.
Based on the partitioned blocks, an equivalence relation is defined, and for any element a, b and c on one partitioned block, they are considered to satisfy the following equivalence relation R, namelyAnd/>And then establishing equivalence classes of uncertain parameters by using equivalence relations:
[a]R＝{b|b∈π,(a,b)∈R}；
wherein a is a representative element of an equivalence class, and R is an equivalence relation in a dividing block.
Step two: for an uncertain region in a theoretical domain, constructing an upper approximation set and a lower approximation set of the uncertain region based on a knowledge base, taking the upper approximation set and the lower approximation set as uncertainty quantification models, and calculating the approximation precision of the uncertain region.
Specifically, for any given uncertain region A in the thermodynamic coupling system domain U, the equivalence class and the knowledge base of the step one are utilized to establish the following approximate set up and down:
Wherein, Representing empty set,/>An upper approximation set of the uncertainty region a is represented, which is a set made up of elements within the argument U that certainly belong to or are likely to belong to the uncertainty region a, and R (a) is a lower approximation set of the uncertainty region a, which is a set made up of elements within the argument U that certainly belong to the uncertainty region a, by which a complex boundary of the uncertainty region a can be approximated from an inside and outside angle.
The approximation accuracy characterizes the approximation of the approximation set to the uncertainty region, calculated as follows:
Where |·| is the cardinality of the collection. When the number of uncertain parameters is 2, |·| represents area; when the number of uncertain parameters is 3, |·| represents volume.
On the basis of the above example, i.e. the example of the thermal coupling system on the wing in step one, for any given uncertainty area a within the universe U, an ellipsoid is used as uncertainty area, and the ellipsoid equation for these 3 uncertainty parameters (coefficient of thermal expansion α, modulus of elasticity E and coefficient of thermal conductivity λ) is:
[X-Xc]TMΛ-1MT[X-Xc]＝1；
wherein, x= (X 1,X2,X3)T is a vector containing three uncertain parameters of thermal conductivity, elastic modulus and thermal expansion coefficient, X c represents an ellipsoid center, M is an orthogonal transformation matrix, Λ -1 is a diagonal matrix, and the uncertain region a is an ellipsoid inner region.
Based on the constructed equivalence class and knowledge base of the discourse domain, constructing an upper and lower approximate set of the uncertain region A:
Wherein, Representing empty set,/>An upper approximation set of the uncertainty region a is represented, which is a set made up of elements within the argument U that certainly belong to or are likely to belong to the uncertainty region a, and R (a) is a lower approximation set of the uncertainty region a, which is a set made up of elements within the argument U that certainly belong to the uncertainty region a, by which a complex boundary of the uncertainty region a can be approximated from an inside and outside angle.
The approximation accuracy characterizes the approximation of the approximation set to the uncertainty region, calculated as follows:
Where |·| is the cardinality of the collection. Since the number of uncertain parameters is 3, |·| represents the volume operation, and the approximation accuracy obtained by the upper and lower approximation sets is 0.4632.
Step three: and establishing a finite element model of the thermodynamic coupling system, extracting sample points in a domain by adopting a random Latin hypercube sampling method, and obtaining the response of the thermodynamic coupling system at the sample points by adopting a finite element calculation method.
Firstly, establishing a finite element model of a thermodynamic coupling system through finite element simulation analysis software, randomly extracting n sample points in a domain by using a Latin hypercube sampling method to form a sample point set S= { S 1,s2...,sn }, whereinRepresenting the i-th sample point,/>The j-th uncertain parameter value representing the i-th sample point. Next, the system response F at the sample point s i is calculated by the finite element model (s i), and the calculated F (s i) is formed into a system response vector f= (F (s 1),f(s2),...,f(sn))T) corresponding to the sample point set.
The agent model method approximately characterizes the response process of the complex system by establishing a mathematical model and gradually becomes an important means for uncertainty propagation analysis in the engineering field. Meanwhile, the accuracy and efficiency of the proxy model are closely related to the number of the constructed model sample points, and the development of the self-adaptive sampling proxy model has important significance for saving cost and improving efficiency. Therefore, based on the rough set theory and the self-adaptive agent model, uncertainty quantification and propagation analysis of the thermodynamic coupling system are carried out, the defect of the existing uncertainty theory can be overcome, and the actual engineering problem can be solved more reasonably and efficiently.
For example, based on the example in the second step, a finite element model of the thermodynamic coupling system is built through finite element simulation analysis software, 30 sample points are randomly extracted in the domain by using Latin hypercube sampling method, and a sample point set S= { S 1,s2...,s30 }, whereinRepresenting the i-th sample point,/>The j-th uncertain parameter value representing the i-th sample point is calculated by a finite element model, the system response F (s i) at the s i sample point is calculated, and the calculated F (s i) is formed into a system response vector F= (F (s 1),f(s2),...,f(s30))T, the response data is the maximum displacement of a certain monitoring point of the wing tip) corresponding to the sample point set.
Step four: and establishing an initial kriging proxy model according to the sample points and the response thereof, and correcting the kriging proxy model by a maximum likelihood estimation and mode searching method.
Specifically, the system response f(s) caused by any input vector s e U containing m-dimensional uncertainty parameters is expressed as an approximate combination of the following regression model and the stochastic model:
Where g(s) = (g 1(s),g2(s),...,gp(s))T represents a vector of p regression functions for s, e.g., a one-term regression function, where p=m+1, (g 1(s),g2(s),...,gp(s))＝(1,s1,...sm) m is the number of uncertain parameters, s j represents the j-th uncertain parameter value of s, β= (β 1,β2,...,βp)T represents the parameter corresponding to the regression function. Z(s) is considered to be a random process with mean 0 and variance σ 2. Assuming ω and ζ are any two input vectors within the universe U, the covariance between z (ω) and z (ζ) is:
Wherein, The method is characterized by representing a correlation function with a correlation coefficient vector theta, selecting a Gaussian model as the correlation function, and the expression is as follows:
Where ω j and ζ j are the j-th uncertainty parameter values of ω and ζ, respectively, and θ j is the component of the correlation coefficient θ on the j-th uncertainty parameter.
According to the sample point set S= { S 1,s2...,sn }, establishing a regression function matrix of the sample points as follows:
Meanwhile, the following correlation coefficient matrix can be calculated:
Wherein, And r (S) represents a correlation coefficient matrix of the random process between the sample points, and r (S) represents a correlation coefficient matrix of the random process between the sample point set S and any input vector S in the domain U.
Consider the system response caused by the input vector sIs a linear weighting of the system response F caused by the sample point set S, and the following model is built:
Where η= (η 1,η2,...,ηn)T is a weighting coefficient while the system response f(s) due to any input vector s e U can be represented as an approximate combination of a regression model and a random model, the prediction error at any input vector s is calculated as follows:
Where z= (Z (s 1),z(s2),...,z(sn))T represents the error matrix at the sample points, z=z(s) represents the error at the input variable s, G T η is the basis function at the input vector s resulting from the weighted summation of the sample points:
GTη＝g(s)；
The mean square error at the predicted input vector is thus calculated as:
the problem is converted into an optimization problem with eta as an independent variable and the minimum mean square error under the constraint of an equation is solved. The lagrangian multiplier is constructed and the η is graded as follows:
Wherein ζ is Lagrangian multiplier, by definition The following set of equations is obtained:
The solution of the equation set is:
the solution eta of the equation set is taken into a linear weighting model of the system response at the predicted input vector to obtain:
and meanwhile, the predicted mean square error at the input vector s can be obtained:
Wherein the method comprises the steps of And for the variance sigma 2 and the correlation coefficient vector theta in the mean square error, adopting a maximum likelihood estimation method and a mode search optimization algorithm to realize parameter correction. The model can predict the system response and the mean square error at any input vector S in the domain after the initial sample point set S and the corresponding system response F are constructed.
Illustratively, on the basis of the example of step four, a linear function is selected as the regression function, i.e. p=4, whose regression function vector is, for any input vector s in the domain: (g 1(s),g2(s),g3(s),g4(s))＝(1,s1,s2,s3), wherein (s 1,s2,s3) corresponds to the coefficient of thermal expansion, the modulus of elasticity and the coefficient of thermal conductivity at s, respectively. An initial kriging proxy model is established with 30 sample points and corresponding system responses, wherein the variance σ 2 and the correlation coefficient vector θ in the mean square error are corrected to σ 2 = 0.1402 and θ= (7.0711,1.1663,0.1184) by using a maximum likelihood estimation and pattern search optimization algorithm.
Step five: in the domain, searching a maximum value point of the mean square error of the corrected Kerling proxy model by using a maximum mean square error criterion, and reconstructing the Kerling proxy model by using the maximum value point as a newly added sample point until a convergence condition is met.
Specifically, any input vector s in the domain U is taken as an independent variable, the maximum predicted mean square error at the s position is obtained by using the established kriging proxy model, and the following optimization model is established:
max
s.t.s∈U；
And obtaining a maximum point s * of a predicted mean square error value of the initial kriging proxy model in the theoretical domain by adopting a genetic algorithm, and calculating a corresponding response value f by using a finite element model (s *). And taking the maximum value point of the mean square error and the response of the corresponding system as newly added data, reconstructing a Kerling proxy model, and when the calculated maximum mean square error curve meets the following convergence condition, establishing the proxy model to meet the global precision in the theoretical domain.
(1) The maximum mean square error value at all input vectors s in the universe is smaller than a given threshold delta 1.
(2) The average value of the phase difference of the maximum mean square error values obtained by five times in succession at all input vectors s in the domain is smaller than a given threshold delta 2.
The thresholds delta 1 and delta 2 are reasonably given according to engineering experience and precision requirements.
By way of example, based on the example of the fifth step, the optimization model is built based on the initial kriging proxy model, taking the maximum mean square error at any input vector s in the domain as a prediction target, and taking the domain boundary as a constraint condition:
max
s.t.s∈U；
And obtaining a maximum point s * of a predicted mean square error value of the initial kriging proxy model in the theoretical domain by adopting a genetic algorithm, and obtaining a corresponding response value f (s *) of the thermodynamic coupling system by finite element simulation calculation. And taking the maximum value point of the mean square error and the response of the corresponding system as newly added data, reconstructing a Kerling proxy model, and when the calculated maximum mean square error curve meets the following convergence condition, establishing the proxy model to meet the global precision in the theoretical domain.
(1) The maximum mean square error value at all input vectors s within the universe is less than a given threshold δ 1 =0.05.
(2) The average value of the phase difference of the maximum mean square error values obtained by five times in succession at all input vectors s in the domain is smaller than a given threshold δ 2 =0.01.
Based on the maximum mean square error point adding criterion and the convergence condition, 48 sample points are newly added, and a kriging proxy model meeting the global precision is established with the initial 30 sample points.
Step six: and taking the reconstructed agent model as an uncertainty propagation analysis model, calculating an upper response bound and a lower response bound of the upper approximation set and an upper response bound and a lower response bound of the lower approximation set by using the agent model, and calculating response approximation accuracy.
Specifically, based on the kriging proxy model satisfying global accuracy in the argument domain, for the upper approximation set of the uncertainty region a in the second stepAnd a lower approximation set R (A), obtaining upper and lower response limits of the thermodynamic coupling system:
Wherein f represents the mapping relation between the input vector obtained by the global precision kriging agent model and the system response; For the upper approximation set/> The lower bound of the response interval obtained in the step (a); /(I)For the upper approximation set/>The upper bound of the response interval obtained in the step (a); f L is the lower bound of the response interval found in the lower approximation set R (A); f U is the upper bound of the response interval found in the lower approximation set R (A);
the response approximation precision of the system under the current equivalence relation R and the uncertain region A is calculated as follows:
Where || denotes the cardinality, here the absolute value, i.e. the length of the interval. The response approximation accuracy may approximate how close the system response caused by the set of upper and lower approximations approximates the system response caused by the uncertainty region.
Illustratively, based on the example in the fifth step, the set of the upper approximations is established based on the kriging proxy model satisfying the global accuracy in the domainAnd obtaining upper and lower response limits of the thermodynamic coupling system in the lower approximation set R (A):
wherein f represents the mapping relation between the input vector obtained by the global precision kriging agent model and the system response; For the upper approximation set/> The lower bound of the response interval obtained in the step (a); /(I)For the upper approximation set/>The upper bound of the response interval obtained in the step (a); f L is the lower bound of the response interval found in the lower approximation set R (A); f U is the upper bound of the response interval found in the lower approximation set R (A);
the response approximation precision of the system under the current equivalence relation R and the uncertain region A is calculated as follows:
where || represents the cardinality, here the absolute value.
And solving the upper and lower bounds of the system response and the response approximate accuracy under the current knowledge base as follows:
TABLE 2 upper and lower approximation set system response upper and lower bounds and response approximation accuracy
In the embodiment of the invention, based on the up-and-down approximate set representation method adopted by the rough set theory, the application object can be an uncertain region with any complex shape, thereby improving the accuracy of parameter uncertainty characterization; on the other hand, the kriging proxy model can be modified by the model, and sample data is updated according to the dotted criterion, so that the kriging proxy model has self-adaption. The self-adaptive kriging agent model is used as a black box to operate, so that the calculation times of the finite element model can be reduced, the global accuracy of calculation is ensured, and the calculation efficiency is effectively improved. In addition, in the process of constructing the knowledge base, the domain is divided into dividing blocks with regular shapes based on the known experimental measuring point set, so that the subsequent response interval calculation is more convenient.
In this specification, each embodiment or implementation is described in a progressive manner, and each embodiment focuses on a difference from other embodiments, and identical and similar parts between the embodiments are all enough to refer to each other.
In the description of the present specification, reference is made to "one embodiment," "some embodiments," "an exemplary embodiment," "an example," "a particular instance," or "some examples," etc., means that a particular feature, structure, material, or characteristic described in connection with the embodiment or example is included in at least one embodiment or example of the invention. In this specification, schematic representations of the above terms do not necessarily refer to the same embodiments or examples. Furthermore, the particular features, structures, materials, or characteristics described may be combined in any suitable manner in any one or more embodiments or examples.
Finally, it should be noted that: the above embodiments are only for illustrating the technical solution of the present invention, and not for limiting the same; although the invention has been described in detail with reference to the foregoing embodiments, it will be understood by those of ordinary skill in the art that: the technical scheme described in the foregoing embodiments can be modified or some or all of the technical features thereof can be replaced by equivalents; such modifications and substitutions do not depart from the spirit of the invention.
Claims (1)
1. The uncertainty analysis method of the thermodynamic coupling system based on the rough set theory is characterized by comprising the following steps of:
step one: determining a domain of a plurality of uncertain parameters of the thermodynamic coupling system, and constructing a knowledge base of the domain according to experimental measuring points in the domain;
step two: constructing an upper approximation set and a lower approximation set of the uncertain region based on the knowledge base aiming at the uncertain region in the discussion domain, taking the upper approximation set and the lower approximation set as uncertainty quantification models, and calculating the approximation precision of the uncertain region;
Step three: establishing a finite element model of the thermodynamic coupling system, extracting sample points in the discussion domain by adopting a random Latin hypercube sampling method, and obtaining the response of the thermodynamic coupling system at the sample points by adopting a finite element calculation method;
step four: establishing an initial kriging proxy model according to the sample points and the response thereof, and correcting the kriging proxy model by a maximum likelihood estimation and mode searching method;
Step five: in the theory domain, searching a maximum value point of the mean square error of the corrected Kerling proxy model by utilizing a maximum mean square error criterion, calculating a system response of the maximum value point by finite elements, and reconstructing the Kerling proxy model by taking the maximum value point and the corresponding response as newly added data until a convergence condition of the maximum mean square error is met;
step six: using the reconstructed agent model as an uncertainty propagation analysis model, calculating a response upper bound and a response lower bound of the upper approximation set and a response upper bound and a response lower bound of the lower approximation set by using the agent model, and defining and calculating response approximation accuracy;
Determining a domain of a plurality of uncertain parameters of the thermodynamic coupling system, and constructing a knowledge base of the domain according to experimental measuring points in the domain, wherein the method comprises the following steps:
Representing each of the uncertain parameters of the thermodynamic coupling system in the form of intervals, wherein the universe is a Cartesian product of a plurality of uncertain parameter intervals;
Obtaining a plurality of experimental measurement points in the discussion domain through an experimental method, wherein each experimental measurement point comprises a group of values of each uncertain parameter;
The values of the same uncertain parameter in the plurality of experimental measuring points are de-duplicated, and a breakpoint set of the uncertain parameter is formed by the values of the same uncertain parameter in the plurality of experimental measuring points and the upper bound and the lower bound of the interval of the uncertain parameter;
Constructing a knowledge base of the discourse domain based on a plurality of breakpoint sets;
Based on a plurality of breakpoint sets, constructing a knowledge base of the discourse domain, including:
dispersing the interval of the uncertain parameters corresponding to each breakpoint set into a union set of a plurality of subintervals;
If a plurality of uncertain parameters are mutually independent, any one of a plurality of subintervals corresponding to each uncertain parameter is taken, the dividing blocks in the domain are Cartesian products of the taken subintervals, and a knowledge base of the domain is a union set of the dividing blocks;
Using the reconstructed proxy model as an uncertainty propagation analysis model, calculating an upper response bound and a lower response bound of the upper approximation set and an upper response bound and a lower response bound of the lower approximation set by using the proxy model, and defining and calculating response approximation accuracy, wherein the method comprises the following steps:
for the uncertainty region Upper approximation set/>Solving the upper bound/>, of the response by using the reconstructed proxy modelAnd response lower bound/>；
For the uncertainty regionThe following approximate set/>Solving the upper bound of the response by using the reconstructed kriging modelAnd response lower bound/>；
The response approximation accuracy is:。
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