CN108710713B - Bayesian point matching method for solving statics response containing interval parameter structure - Google Patents
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Abstract
The invention discloses a Bayesian point matching method for solving statics response containing interval parameter structures. Different from the traditional point matching method that a fixed point is selected in a structural parameter interval to construct an orthogonal polynomial to approximate the response of the structure to an interval variable, the method adaptively selects the point in the interval parameter domain. Firstly, according to the selected sample points and the corresponding structural response, the posterior information (including the mean and variance of the structural response function) of the structural response function is obtained based on the Bayesian theory; then constructing two index functions according to posterior information; and then carrying out next round of matching points according to the index functions, and calculating the structural response value of a new sample point until a convergence criterion is reached. The maximum and minimum values of the structural response corresponding to the previously selected sample points can be considered as the approximate upper and lower bounds, respectively, of the hydrostatic response of the interval-parameter-containing structural system.
Description
Technical Field
The invention relates to the technical field of structural statics response analysis containing interval uncertain parameters, in particular to a Bayesian point matching method for solving statics response containing interval parameter structures.
Background
Structural analysis has taken an important place in aerospace, mechanical engineering, vehicle engineering and civil engineering. Before the structure is in service, various indexes of displacement, stress, strain and the like of the structure need to be checked so as to ensure that the structure can work safely within the economic life of the structure. Structural analysis is also the most basic and most important link in structural design. And statics checking of structures is the most core and basic part in structural analysis. In conventional structure analysis processes, parameters of a structure are often treated as a definite value. After the whole analysis process is finished, a certain result is obtained.
However, the service environment of the engineering structure is relatively complex, and in addition to the uncontrollable process of the structure during the machining process, the dispersibility of the materials constituting the structure and other factors, various parameters of the structure are uncertain. These uncertainties may have a relatively large impact on the results of the structural analysis. The traditional method of describing uncertainty is probabilistic, but in order to obtain a probability density curve for a parameter, a large number of experiments are required to obtain sufficient data. However, in reality, not so much data is available, which greatly limits the application of the probabilistic approach. To solve this problem, scholars propose fuzzy sets, evidence theories, and the like. However, when a fuzzy method is used, an expert is required to distribute membership functions according to the experience of the expert; when applying evidence theory, it is also necessary to assign probability densities to individual focal elements by "experts".
When the uncertainty is represented by the interval, only the upper and lower boundaries of the parameter need to be known, which can play a great role under the conditions of poor data and less information. The traditional methods for solving the propagation of the uncertainty of the interval comprise a Monte Carlo method, a vertex method, a perturbation method, a traditional matching method and the like. The Monte Carlo method needs to call finite elements for many times, and the calculated amount is very large; the vertex method is only applicable to the problem of monotony; the perturbation method can only solve approximate solutions of small uncertainty intervals and weak non-linear problems; the traditional spotting method selects sample points at fixed positions, does not consider the form of a function, and therefore has insufficient precision and efficiency.
Because the structural statics analysis problem of the structure containing the interval uncertainty parameters plays a very important role in the fields of machinery, ships, vehicles and aerospace, the establishment of the statics analysis method containing the interval uncertainty parameters, which is small in calculation amount and high in precision, has a remarkable practical significance.
Disclosure of Invention
The technical problem to be solved by the invention is as follows: the invention provides a Bayesian point matching method for solving statics response of a structure containing interval parameters. The method obtains posterior distribution of a structural response function based on Bayesian theory according to the selected sample points; constructing an index function according to the posterior information; guiding a new round of matching points by an index function until a global convergence index is met; and finally, selecting the maximum and minimum corresponding structural response functions in the selected sample points as the upper and lower bounds of the structural response.
The technical scheme adopted by the invention is as follows: a Bayesian point matching method for solving statics response containing interval parameter structure mainly comprises the following steps:
the first step is as follows: determining an interval uncertainty variable alphaIAnd its interval field Θ. Determining a Gaussian kernel function
Where θ is a parameter of the Gaussian kernel function; alpha is alpha(i)，α(j)∈αIIs any two points in the uncertainty domain. A convergence tolerance epsilon is determined. The end points in the interval are selected, and then a point is selected randomly in the interval.
The second step is that: calculating the structural response values u (alpha) of the selected points by using a finite element method, and adding the points and the corresponding structural response values into the set omega; selecting the maximum value u (alpha) of the structural response function values in the set omega(+)) And a minimum value u (. alpha.)(-))。
The third step: the covariance matrix of the points that have been selected is calculated. Let it be assumed that M points, respectively α, have been selected in the uncertainty region at this time(1),...,α(M)The structural response values are respectively u (alpha)(1)),...,u(α(M)) Then the covariance matrix can be expressed as:
the fourth step: the inner layer optimization using genetic algorithms yields points that are likely to maximize the index. First, a population is initialized (the population corresponds to being in an uncertain domain)Randomly selected points) and then calculates the covariance between each individual in the population and the point that has been selected (assuming a)(*)Is any individual in the population):
k＝[k(α(1),α(*))…k(α(M),α(*))]T
and then according to the previously selected sample set omega, calculating the posterior mean value mu (alpha) of each individual by using the Bayesian theory(*)Omega) and the posterior variance sigma2(α(*)|Ω)：
μ(α(*))＝kTK-1u1:M,σ2(α(*))＝1-kTK-1k
Wherein u is1:M＝[u(α(1)),...,u(α(M))]TIs a column vector consisting of the structural response values of the previously selected M points. And then calculating the fitness value (namely an index function value) of each individual of the quasisome according to the obtained posterior mean value and variance:
and respectively selecting an individual with larger EI _ UP or EI _ LO, performing intersection and variation by using the rule of the genetic algorithm, and generating a new population until the convergence criterion of the genetic algorithm is met. Then, the current individual value and the corresponding fitness value are output.
The fifth step: if the fitness value is smaller than the convergence tolerance epsilon of the algorithm, the next step is carried out, and if the fitness value is larger than the convergence tolerance epsilon of the algorithm, the second step is returned.
And a sixth step: selecting the minimum and maximum values of Ω as the lower bound of the statics response of the interval-containing parametric structureuAnd (alpha) and upper bound
The principle of the invention is as follows: different from the traditional point matching method that a fixed point is selected in a structural parameter interval to construct an orthogonal polynomial to approximate the response of the structure to an interval variable, the method adaptively selects the point in the interval parameter domain. Firstly, according to the selected sample points and the corresponding structural response, the posterior information (including the mean and variance of the structural response function) of the structural response function is obtained based on the Bayesian theory; then constructing two index functions according to posterior information; and then carrying out next round of matching points according to the index functions, and calculating the structural response value of a new sample point until a convergence criterion is reached. The maximum and minimum values of the structural response corresponding to the previously selected sample points can be considered as the approximate upper and lower bounds, respectively, of the hydrostatic response of the interval-parameter-containing structural system.
Compared with the prior art, the invention has the advantages that: the invention solves the problem of statics response analysis of a structure containing interval uncertainty in a mode of adaptively selecting sample points. Compared with the traditional matching method, the method has the advantages that the calculated points are few, and the precision is high; compared with a vertex method and a perturbation method, the method can process more complicated problems and has higher precision; compared with the Monte Carlo method, the method has similar precision, but the calculation amount is far less than that of the Monte Carlo method. Therefore, the invention provides a method for analyzing the structural statics response containing interval uncertain parameters with high efficiency and high precision.
Drawings
FIG. 1 is a flow chart of the present invention for solving a statics response solution for a structure containing interval parameters;
FIG. 2 is a schematic view of a square articulated steel frame structure to which the present invention is directed;
FIG. 3 is a graphical representation of the steel frame structure displacement response obtained with various uncertainties according to the present invention and other prior art methods, wherein FIG. 3(a) is a graphical representation of the lateral displacement of the steel frame structure obtained by a different method; FIG. 3(b) is a schematic diagram of longitudinal displacement of a steel frame structure obtained by a different method;
FIG. 4 is a schematic representation of the deformation of the steel frame structure obtained with different parameter values at an uncertainty level of 0.1, wherein FIG. 4(a) is a schematic representation of the deformation of the steel frame structure when the uncertainty parameter assumes a nominal value; FIG. 4(b) is a schematic illustration of the steel frame structure at an upper deformation boundary; FIG. 4(c) is a schematic view of the steel frame structure at the lower limit of deformation.
Detailed Description
The invention is further described with reference to the following figures and detailed description.
As shown in fig. 1, the present invention provides a bayesian collocation method for solving a statics response including an interval parameter structure, including the following steps:
(1) determining an interval uncertainty variable alphaIAnd its interval field Θ. The determination of the gaussian kernel function is carried out,
where θ is a parameter of the Gaussian kernel function; alpha is alpha(i)，α(j)∈αIIs any two points in the uncertainty domain. A convergence tolerance epsilon is determined. The end points in the interval are selected, and then a point is selected randomly in the interval.
(2) Calculating the structural response values u (alpha) of the selected points by using a finite element method, and adding the points and the corresponding structural response values into the set omega; selecting the maximum value u (alpha) of the structural response function values in the set omega(+)) And a minimum value u (. alpha.)(-))。
(3) The covariance matrix of the points that have been selected is calculated. Let it be assumed that M points, respectively α, have been selected in the uncertainty region at this time(1),...,α(M)The structural response values are respectively u (alpha)(1)),...,u(α(M)) Then the covariance matrix can be expressed as:
(4) the inner layer optimization using genetic algorithms yields points that are likely to maximize the index. First, a population is initialized (population corresponds to being indeterminateRandomly selected points within the domain) and then calculate the covariance between each individual in the population and the point that has been selected (assuming a)(*)Is any individual in the population),
k＝[k(α(1),α(*))…k(α(M),α(*))]T
and then according to the previously selected sample set omega, calculating the posterior mean value mu (alpha) of each individual by using the Bayesian theory(*)Omega) and the posterior variance sigma2(α(*)|Ω)：
μ(α(*))＝kTK-1u1:M,σ2(α(*))＝1-kTK-1k
Wherein u is1:M＝[u(α(1)),...,u(α(M))]TIs a column vector consisting of the structural response values of the previously selected M points. And then calculating the fitness value (namely an index function value) of each individual of the quasisome according to the obtained posterior mean value and variance:
and respectively selecting an individual with larger EI _ UP or EI _ LO, performing intersection and variation by using the rule of the genetic algorithm, and generating a new population until the convergence criterion of the genetic algorithm is met. Then, the current individual value and the corresponding fitness value are output.
(5) If the fitness value is smaller than the convergence tolerance epsilon of the algorithm, the next step is carried out, and if the fitness value is larger than the convergence tolerance epsilon of the algorithm, the second step is returned.
(6) Selecting the minimum and maximum values of Ω as the lower and upper bounds u (α) and u (α) of the hydrostatic response of the interval-containing parametric structure
Example (b):
in order to more fully understand the characteristics of the invention and the practical applicability of the invention to engineering, the invention carries out statics response analysis and solution of parameter uncertainty aiming at the square steel frame structure shown in figure 2. The square steel frame is formed by hinging two L-shaped steel frames. The side length of the steel frame is L1000 mm; the cross section of the steel frame is also square, and the width of the cross section of the steel frame is 10 mm. Two forces with equal magnitude and opposite directions are respectively acted on the upper and lower hinge points of the steel frame. Because the load is large, the steel frame is seriously deformed, and a large deformation theory needs to be considered when the steel frame statics response calculation is carried out. Considering now the load applied to the steel frame and the modulus of elasticity of the steel frame as interval uncertainty parameters, the load and modulus of elasticity can be expressed as follows
P＝[Pc-βPc,Pc+βPc],E＝[Ec-βEc,Pc+βEc]
Wherein P isc＝1350N，Ec＝2.15×105Respectively, are nominal values for the load and the modulus of elasticity, beta is the uncertainty level, and the vertical displacement W and the horizontal displacement U of the steel frame vertex obtained by different methods are shown as the graph when beta is changed from 0 to 0.1. Wherein, the Monte Carlo simulation uses 100000 sample points to calculate, and the obtained value is regarded as the true value; bayesian collocation methods use 22 sample points for the calculation. Fig. 3 shows that the bayesian collocation method can obtain higher calculation precision with smaller calculation cost. Fig. 4 shows that even under small uncertainty disturbance, the calculated steel frame deformation results are quite different, so that it is necessary to consider the influence of uncertainty in actual engineering.
In conclusion, the invention provides a method for determining interval parameter-containing structure statics response based on Bayesian point matching theory. Different from the traditional point matching method that a fixed point is selected in a structural parameter interval to construct an orthogonal polynomial to approximate the response of the structure to an interval variable, the method adaptively selects the point in the interval parameter domain. Firstly, according to the selected sample points and the corresponding structural response, the posterior information (including the mean and variance of the structural response function) of the structural response function is obtained based on the Bayesian theory; then constructing two index functions according to posterior information; and then carrying out next round of matching points according to the index functions, and calculating the structural response value of a new sample point until a convergence criterion is reached. The maximum and minimum values of the structural response corresponding to the previously selected sample points can be considered as the approximate upper and lower bounds, respectively, of the hydrostatic response of the interval-parameter-containing structural system.
The above are only the specific steps of the present invention, and the protection scope of the present invention is not limited in any way; the method can be expanded and applied to the field of statics response solving of other structural forms containing interval parameters, and all technical schemes formed by adopting equivalent transformation or equivalent replacement fall within the protection scope of the invention.
The invention has not been described in detail and is part of the common general knowledge of a person skilled in the art.
Claims (1)
1. A Bayesian point matching method for solving statics response containing interval parameter structure is characterized in that: the method obtains posterior distribution of a structural response function based on Bayesian theory according to the selected sample points; constructing an index function according to the posterior information; guiding a new round of matching points by an index function until a global convergence index is met; and finally, selecting the maximum and minimum corresponding structural response functions in the selected sample points as the upper and lower bounds of the structural response, wherein the implementation steps of the method are as follows:
the first step is as follows: determining an interval uncertainty variable alphaIAnd its interval field Θ, determine the gaussian kernel function:
where θ is a parameter of the Gaussian kernel function; alpha is alpha(i)，α(j)∈αIDetermining convergence tolerance epsilon for any two points in an uncertain domain, selecting an end point in an interval domain, and then selecting a point in the interval at will;
the second step is that: calculating the structural response values u (alpha) of the selected points by using a finite element method, and adding the points and the corresponding structural response values into the set omega; selecting the maximum value u (alpha) of the structural response function values in the set omega(+)) And a minimum value u (. alpha.)(-))；
The third step: calculating covariance matrix of selected points, selecting M points in uncertain domain, respectively alpha(1),...,α(M)The structural response values are respectively u (alpha)(1)),...,u(α(M)) Then the covariance matrix can be expressed as:
the fourth step: performing inner layer optimization by using a genetic algorithm to obtain a point with the maximum index, initializing a population, wherein the population corresponds to the randomly selected point in an uncertain domain, and calculating the covariance, alpha, between each individual in the population and the selected point(*)Is any one individual in the population of individuals,
k＝[k(α(1),α(*)) … k(α(M),α(*))]T
and then according to the previously selected sample set omega, calculating the posterior mean value mu (alpha) of each individual by using the Bayesian theory(*)Omega) and the posterior variance sigma2(α(*)|Ω)：
μ(α(*)|Ω)＝kTK-1u1:M,σ2(α(*)|Ω)＝1-kTK-1k
Wherein u is1:M＝[u(α(1)),…,u(α(M))]TThe method is characterized in that the method is a column vector consisting of the structural response values of M points selected previously, and then the fitness value, namely an index function value, of each individual of a quasisome is calculated according to the obtained posterior mean value and variance:
respectively selecting an individual with a larger EI _ UP or EI _ LO, performing intersection and variation by using the rule of the genetic algorithm, generating a new population until the convergence criterion of the genetic algorithm is met, and then outputting the current individual value and the corresponding fitness value;
the fifth step: if the fitness value is smaller than the algorithm convergence tolerance epsilon, entering the next step, and if the fitness value is larger than the algorithm convergence tolerance epsilon, returning to the second step;
and a sixth step: selecting the minimum and maximum values of Ω as the lower bound of the statics response of the interval-containing parametric structure
And upper bound
Firstly, selecting an uncertain variable, a value interval and tolerance of the uncertain variable according to the actual engineering condition, and selecting a Gaussian kernel function and a parameter value thereof; then, the end points in the interval are selected, and one point in the interval is selected as a parameter sample point.
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