CN111931394A - De-nesting analysis method for non-probability mixed reliability index - Google Patents

De-nesting analysis method for non-probability mixed reliability index Download PDF

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CN111931394A
CN111931394A CN202010428579.7A CN202010428579A CN111931394A CN 111931394 A CN111931394 A CN 111931394A CN 202010428579 A CN202010428579 A CN 202010428579A CN 111931394 A CN111931394 A CN 111931394A
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CN111931394B (en
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李国发
陈泽权
陈传海
杨兆军
何佳龙
田海龙
于立娟
王继利
罗巍
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Jilin University
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Abstract

The invention discloses a de-nesting analysis method of a non-probability mixed reliability index, which comprises the following steps: determining a function and a variable of a structure, converting a variable space of the function of the structure, approximating the function by using a linear approximation model and a two-point self-adaptive model, performing inner-layer iteration and outer-layer iteration parameter updating by using an HL-RF method, and performing inner-layer iteration and outer-layer iteration parameter updating by using the HL-RF method. Compared with the traditional nested method, the method provided by the invention has the following remarkable advantages: the iterative convergence speed is greatly improved compared with the traditional first-order reliability method; a two-point self-adaptive nonlinear approximation model is adopted to construct an approximation function, and the calculated convergence rate is not limited to the linear approximation precision of the traditional first-order reliability method any more; the resources and time required by analysis and calculation are effectively reduced: the response extreme value of the extreme state function is approximately obtained by a function approximation method, the extreme value analysis process of the nested function response value is not needed, the main workload of each iteration step is realized by calculating a few real functions, and the resources and time required by calculation are effectively reduced; the calculation precision is ensured while the function calling times are reduced.

Description

De-nesting analysis method for non-probability mixed reliability index
Technical Field
The invention belongs to the field of structural engineering reliability, and particularly relates to a structural reliability analysis technology under the condition of mixing a non-probability variable and a traditional probability variable, and a de-nesting analysis method of a non-probability mixed reliability index.
Technical Field
A structural reliability model is established based on a traditional probability model, and a method for analyzing and optimizing reliability on the basis is mature. The large sample data is a precondition for obtaining the accurate distribution of uncertain variables, but in practical engineering, whether from the aspect of cost or time, a large amount of sample data of all uncertain variables cannot be obtained. In view of this situation, researchers have proposed a variety of non-probabilistic models in conjunction with traditional probability theory to describe the cognitive uncertainty that affects the reliability of a structure. After a non-probability mixed reliability model of a structure is established by using a traditional probability model and a non-probability model, structural reliability analysis of mixing of the non-probability model and the probability model needs to be carried out.
The p-box theory is used as a novel non-probability model theory, and cognitive determination variables are described through a probability box of upper and lower limit envelopes of a cumulative distribution function. The p-box can describe the cognitive uncertain variables, whether in a distributed form of unknown uncertain variables or in a distributed form of known variables. In addition, the p-box theory has good compatibility with other common non-probability theories, namely other non-probability models can be converted to the p-box model to a certain extent, so that the p-box theory can be adopted to establish a more extensive non-probability model for structural reliability analysis containing cognitive uncertain variables.
At present, most non-probability mixed reliability index analysis methods need to establish a double-layer nested optimization model, wherein the outer layer is a traditional structural reliability optimization model, and the inner layer is an optimization model of a structural function limit value taking a cognitive uncertain variable as an optimization vector. Due to the existence of the inner layer structure, the nested optimization structure needs a large number of function functions of the structure, and the calculation efficiency is low.
Disclosure of Invention
The invention aims to provide a non-probability mixed reliability index de-nesting analysis method based on a function approximation model, which comprises the following steps:
step one, determining a function and a variable of a structure:
1.1 determining variables affecting the reliability of the structure, including n random variables X ═ X (X), according to design and analysis requirements1,x2,…,xn) And m p-box variables Y ═ (Y)1,y2,…,ym)；
1.2 acquiring a functional function g (X, Y) of a structure to be analyzed by utilizing a mechanical theory or finite element simulation method;
step two, converting the variable space of the structural function:
2.1 structural function g (X, Y) obtained in step one, by probability transformation, UX＝Φ-1(FX(X)) and UY＝Φ-1(FY(Y)), a function G (U) of the structure in the normal space can be obtainedX,UY) I.e. G (U)X,UY)＝g(X,Y)；
Wherein: fX(X) is a cumulative distribution function of a random variable X, FY(Y) is the cumulative distribution function of the p-box variable Y,. phi-1Expressed as the inverse of the standard normal distribution;
2.2 initializing parameters (1), s 1,
Figure RE-GDA0002613954060000011
r 11, namely performing linear expansion on the mean value point of each variable;
wherein s is used for recording the outer layer iteration number,
Figure RE-GDA0002613954060000012
for the design flare point for the s-th iteration,
Figure RE-GDA0002613954060000013
solving for the s-th outer iteration the resulting maximum possible failure point at the upper limit of the non-probabilistic reliability index, and
Figure RE-GDA0002613954060000014
expressed as the maximum possible failure point, r, obtained when solving the lower bound of the non-probabilistic reliability indexsIs the non-linear exponent for the s-th iteration,
Figure RE-GDA0002613954060000015
are respectively as
Figure RE-GDA0002613954060000016
The random variable component and the p-box variable component of (1);
2.3 variable transformation, standard Normal space variable U ═ U (U)X,UY) By probability transformation
Figure RE-GDA0002613954060000017
Figure RE-GDA0002613954060000021
Conversion back to the original space of the function to obtain (X, Y); y isL,YRUpper and lower limits for the p-box variables of the structure, respectively; while
Figure RE-GDA0002613954060000022
And YF -1inverse functions representing the cumulative probability distribution function upper and lower limits of the p-box variable, respectively;
step three, approximating the function by using a linear approximation model and a two-point self-adaptive model:
3.1 at an arbitrary point (X, Y), the structural function g (X, Y) is approximated as an approximate expression for the p-box variable Y using a linear approximation method, i.e.
Figure RE-GDA0002613954060000023
Where j is 1,2,3, …, m,
Figure RE-GDA0002613954060000024
3.2 according to the non-Linear index rsAt point of point
Figure RE-GDA0002613954060000025
Establishing information about g (X, Y)c) Is a two-point adaptive non-linear approximation model, i.e.
Figure RE-GDA0002613954060000026
Wherein:
Figure RE-GDA0002613954060000027
Figure RE-GDA0002613954060000028
and i is 1,2,3, …, n, xi∈X，xi,s∈Xs-1，
Figure RE-GDA0002613954060000029
3.3 model combining the linear approximation of 3.1 and the two-point adaptive non-linear approximation of 3.2, the approximation representing a functional function of the structure, has
Figure RE-GDA00026139540600000210
Obtaining an extremum of the response of the function at an arbitrary point (X, Y) in approximation;
3.4 the interval theory approximately obtains the upper and lower limits of the response of the function at any point (X, Y), specifically:
lower limit:
Figure RE-GDA00026139540600000211
upper limit:
Figure RE-GDA00026139540600000212
fourthly, performing inner layer iteration by using an HL-RF method:
4.1 initializing parameter (2), parameter (2) being k,
Figure RE-GDA00026139540600000213
the initialization operation is such that k is 1,
Figure RE-GDA00026139540600000214
wherein k is used for recording the number of inner layer iterations,
Figure RE-GDA00026139540600000215
to solve for
Figure RE-GDA00026139540600000216
Designing expansion points of the kth iteration of the corresponding inner layer;
4.2 obtaining the current nonlinear index r by iterative computationsUpper (lower) limit of the structural reliability index of (2)
Figure RE-GDA0002613954060000031
It includes:
1) obtaining the upper (lower) limit of the reliability index of the kth iteration structure of the inner layer in the s-th iteration of the outer layer by combining an HL-RF method
Figure RE-GDA0002613954060000032
The specific formula is as follows:
Figure RE-GDA0002613954060000033
Figure RE-GDA0002613954060000034
wherein the content of the first and second substances,
Figure RE-GDA0002613954060000035
expressed as a function in the standard normal space
Figure RE-GDA0002613954060000036
The upper (lower) limit of (c),
Figure RE-GDA0002613954060000037
is shown as
Figure RE-GDA0002613954060000038
A first partial derivative vector of;
2) the convergence condition (1) is set as
Figure RE-GDA0002613954060000039
Wherein, the constant is a relatively small constant and is generally set to 0.001;
3) judging whether the convergence condition (1) is met; if the convergence condition (1) is met, acquiring the current nonlinear index rsIs most preferred
Figure RE-GDA00026139540600000310
And corresponding thereto
Figure RE-GDA00026139540600000311
Carrying out the next step; if not, updating k to 1 in k +1 and returning to 3.2) and carrying out iterative calculation again;
4.3 setting a convergence condition (2), when s is more than or equal to 2,
Figure RE-GDA00026139540600000312
judging whether the convergence condition (2) is met; if the convergence condition (2) is met, the final optimal non-probability mixed reliability index is obtained
Figure RE-GDA00026139540600000313
And corresponding maximum possible failure point
Figure RE-GDA00026139540600000314
And go to step five;
if the s is not matched or is 1, carrying out the next step;
updating outer iteration parameters:
4.4 calculated by 4.3 of step four
Figure RE-GDA00026139540600000315
And the expansion point of the last iteration
Figure RE-GDA00026139540600000316
Acquisition using probability transformation
Figure RE-GDA00026139540600000317
And (X)s-1,Ys-1)；
4.5, updating the iteration times s to be s + 1;
4.6 pairs of non-linear indexes rsUpdating to obtain relevant parameters of the approximate model when the s-th iteration is performed, wherein the updated optimization formula is as follows:
Figure RE-GDA00026139540600000318
returning to the third step to start the next round of iterative computation;
step five, judging the symbol and obtaining the final reliability index beta E [ beta ]L,βR]And betaR(L)Corresponding maximum possible failure point
Figure RE-GDA00026139540600000319
5.1 Using the formula βR(L)＝sgn(GR(L)(0,0))βR(L)＝sgn(gR(L)(μX,μY))βR(L)For the obtained non-probability mixed reliability indexβR(L)The signs are distinguished to obtain the final betaR(L)(ii) a Wherein sgn (. cndot.) is expressed as a sign function, and is 1 when the input value is positive, and-1 when the input value is negative, (mu.)X,μY) The mean point of the variable expressed as a function in the original space;
5.2 integrating the results obtained in the third step and the 5.1 to finally obtain the non-probability mixed reliability index beta E [ beta ] of the analyzed structureL,βR]Upper (lower) limit beta of mixed reliability index with non-probabilityR(L)Corresponding maximum possible failure point
Figure RE-GDA0002613954060000041
The invention provides a de-nesting analysis method of a non-probability mixed reliability index, which comprises the following steps: 1. determining a function and a variable of a structure, 2, converting a variable space of the function of the structure, 3, approximating the function by using a linear approximation model and a two-point self-adaptive model, 4, carrying out inner layer iteration and updating outer layer iteration parameters by using an HL-RF method, and 5, carrying out inner layer iteration and updating outer layer iteration parameters by using the HL-RF method.
Compared with the traditional nested method, the method provided by the invention has the following remarkable advantages:
1. compared with the traditional first-order reliability method, the iterative convergence speed of the method provided by the invention is greatly improved: a two-point self-adaptive nonlinear approximation model is adopted to construct an approximation function, and the calculated convergence rate is not limited to the linear approximation precision of the traditional first-order reliability method any more;
2. the method provided by the invention efficiently reduces resources and time required by analysis and calculation: the response extreme value of the extreme state function is approximately obtained by a function approximation method, the extreme value analysis process of the nested function response value is not needed, the main workload of each iteration step is realized by calculating a few real functions, and the resources and time required by calculation are effectively reduced;
3. the method provided by the invention reduces the calling times of the function and ensures the calculation precision.
Drawings
FIG. 1 is a simplified flow diagram of a method for de-nesting analysis of a non-probabilistic mixed reliability index in accordance with the present invention;
FIG. 2 is a flow chart of a method for de-nesting analysis of a non-probabilistic mixed reliability index of the present invention;
FIG. 3 is a schematic diagram of a cumulative probability distribution function for a p-box variable;
FIG. 4 is a schematic diagram of a non-probabilistic mixed extreme state function surface;
FIG. 5 is a schematic diagram of a structure to be analyzed in the embodiment;
FIG. 6 is a diagram of iterative convergence of an embodiment using the method of the present invention and a conventional nesting method.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
As shown in fig. 1 to fig. 3, the method for de-nesting analysis of a non-probability mixed reliability index based on a function approximation model of the present invention mainly includes the following steps:
step one, determining a function and a variable of a structure:
1.1 determining variables affecting the reliability of the structure, including n random variables X ═ X (X), according to design and analysis requirements1,x2,…,xn) And m p-box variables Y ═ (Y)1,y2,…,ym)；
1.2 acquiring a functional function g (X, Y) of a structure to be analyzed by utilizing a mechanical theory or finite element simulation method;
step two, converting the variable space of the structural function:
2.1 structural function g (X, Y) obtained in step one, by probability transformation, UX＝Φ-1(FX(X)) and UY＝Φ-1(FY(Y)), a function of the structure in the standard normal space can be obtainedG(UX,UY) I.e. G (U)X,UY)＝g(X,Y)；
Wherein: fX(X) is a cumulative distribution function of a random variable X, FY(Y) is the cumulative distribution function of the p-box variable Y,. phi-1Expressed as the inverse of the standard normal distribution;
2.2 initializing parameters (1), s 1,
Figure RE-GDA0002613954060000051
r 11, namely performing linear expansion on the mean value point of each variable;
wherein s is used for recording the outer layer iteration number,
Figure RE-GDA0002613954060000052
for the design flare point for the s-th iteration,
Figure RE-GDA0002613954060000053
solving for the s-th outer iteration the resulting maximum possible failure point at the upper limit of the non-probabilistic reliability index, and
Figure RE-GDA0002613954060000054
expressed as the maximum possible failure point, r, obtained when solving the lower bound of the non-probabilistic reliability indexsIs the non-linear exponent for the s-th iteration,
Figure RE-GDA0002613954060000055
are respectively as
Figure RE-GDA0002613954060000056
The random variable component and the p-box variable component of (1);
2.3 variable transformation, standard Normal space variable U ═ U (U)X,UY) By probability transformation
Figure RE-GDA0002613954060000057
Figure RE-GDA0002613954060000058
Conversion back to functionObtaining (X, Y) in an original space; y isL,YRUpper and lower limits for the p-box variables of the structure, respectively; while
Figure RE-GDA0002613954060000059
And YF -1inverse functions representing the cumulative probability distribution function upper and lower limits of the p-box variable, respectively;
step three, approximating the function by using a linear approximation model and a two-point self-adaptive model:
3.1 at an arbitrary point (X, Y), the structural function g (X, Y) is approximated as an approximate expression for the p-box variable Y using a linear approximation method, i.e.
Figure RE-GDA00026139540600000510
Where j is 1,2,3, …, m,
Figure RE-GDA00026139540600000511
3.2 according to the non-Linear index rsAt point of point
Figure RE-GDA00026139540600000512
Establishing information about g (X, Y)c) Is a two-point adaptive non-linear approximation model, i.e.
Figure RE-GDA00026139540600000513
Wherein:
Figure RE-GDA00026139540600000514
Figure RE-GDA00026139540600000515
and i is 1,2,3, …, n, xi∈X，xi,s∈Xs-1，
Figure RE-GDA00026139540600000516
3.3 model combining the linear approximation of 3.1 and the two-point adaptive non-linear approximation of 3.2, the approximation representing a functional function of the structure, has
Figure RE-GDA00026139540600000517
Obtaining an extremum of the response of the function at an arbitrary point (X, Y) in approximation;
3.4 the interval theory approximately obtains the upper and lower limits of the response of the function at any point (X, Y), specifically:
lower limit:
Figure RE-GDA0002613954060000061
upper limit:
Figure RE-GDA0002613954060000062
fourthly, performing inner layer iteration by using an HL-RF method:
4.1 initializing parameter (2), parameter (2) being k,
Figure RE-GDA0002613954060000063
the initialization operation is such that k is 1,
Figure RE-GDA0002613954060000064
wherein k is used for recording the number of inner layer iterations,
Figure RE-GDA0002613954060000065
to solve for
Figure RE-GDA0002613954060000066
Designing expansion points of the kth iteration of the corresponding inner layer;
4.2 obtaining the current nonlinear index r by iterative computationsUpper (lower) limit of the structural reliability index of (2)
Figure RE-GDA0002613954060000067
It includes:
1) obtaining the s-th iteration on the outer layer by combining HL-RF methodIn generation, the upper (lower) limit of the reliability index of the kth iteration structure of the inner layer
Figure RE-GDA0002613954060000068
The specific formula is as follows:
Figure RE-GDA0002613954060000069
Figure RE-GDA00026139540600000610
wherein the content of the first and second substances,
Figure RE-GDA00026139540600000611
expressed as a function in the standard normal space
Figure RE-GDA00026139540600000612
The upper (lower) limit of (c),
Figure RE-GDA00026139540600000613
is shown as
Figure RE-GDA00026139540600000614
A first partial derivative vector of;
2) the convergence condition (1) is set as
Figure RE-GDA00026139540600000615
Wherein, the constant is a relatively small constant and is generally set to 0.001;
3) judging whether the convergence condition (1) is met; if the convergence condition (1) is met, acquiring the current nonlinear index rsIs most preferred
Figure RE-GDA00026139540600000616
And corresponding thereto
Figure RE-GDA00026139540600000617
Is carried out byOne step; if not, updating k to 1 in k +1 and returning to 3.2) and carrying out iterative calculation again;
4.3 setting a convergence condition (2), when s is more than or equal to 2,
Figure RE-GDA00026139540600000618
judging whether the convergence condition (2) is met; if the convergence condition (2) is met, the final optimal non-probability mixed reliability index is obtained
Figure RE-GDA00026139540600000619
And corresponding maximum possible failure point
Figure RE-GDA00026139540600000620
And go to step five;
if the s is not matched or is 1, carrying out the next step;
updating outer iteration parameters:
4.4 calculated by 4.3 of step four
Figure RE-GDA0002613954060000071
And the expansion point of the last iteration
Figure RE-GDA0002613954060000072
Obtaining (X) using a probability transformations,Ys c) And (X)s-1,Ys-1)；
4.5, updating the iteration times s to be s + 1;
4.6 pairs of non-linear indexes rsUpdating to obtain relevant parameters of the approximate model when the s-th iteration is performed, wherein the updated optimization formula is as follows:
Figure RE-GDA0002613954060000073
returning to the third step to start the next round of iterative computation;
step five, judging the symbol and obtaining the final reliability index beta E [ beta ]L,βR]And betaR(L)Corresponding maximum possible failure point
Figure RE-GDA0002613954060000074
5.1 Using the formula βR(L)＝sgn(GR(L)(0,0))βR(L)＝sgn(gR(L)(μX,μY))βR(L)For the obtained non-probability mixed reliability index betaR(L)The signs are distinguished to obtain the final betaR(L)(ii) a Wherein sgn (. cndot.) is expressed as a sign function, and is 1 when the input value is positive, and-1 when the input value is negative, (mu.)X,μY) The mean point of the variable expressed as a function in the original space;
5.2 integrating the results obtained in the third step and the 5.1 to finally obtain the non-probability mixed reliability index beta E [ beta ] of the analyzed structureL,βR]Upper (lower) limit beta of mixed reliability index with non-probabilityR(L)Corresponding maximum possible failure point
Figure RE-GDA0002613954060000075
The following are specific examples of the present invention, which are not intended to limit the present invention. The embodiment will be described with reference to fig. 4 and 5.
Step one, determining a function and a variable of a structure:
1.1 determining variables affecting the reliability of the structure, including n random variables X ═ X (X), according to design and analysis requirements1,x2,…,xn) And m p-box variables Y ═ (Y)1,y2,…,ym)。
In this embodiment, since the number of random variables is 4 and the number of p-box variables is 2, n is 4 and m is 2, the present embodiment is a one-vibrator system.
In the present embodiment, the random variable is represented by X ═ X (X)1,x2,x3,x4) And the p-box variable is represented by Y ═ (Y)1,y2). Wherein x1Representing the mass, x, of the slider of the vibrator system2Denotes the resistance of the vibrator system, x3Representing the magnitude of the force, x, applied to the vibrator system4Representing the time of force, y, of the vibrator system1And y2Expressed as the spring rate of the spring of the vibrator system.
In this example, the specific information of all variables is as follows in table 1:
TABLE 1 distribution of variables in vibrator system of this example
Figure RE-GDA0002613954060000076
Figure RE-GDA0002613954060000081
1.2, acquiring a functional function of a structure to be analyzed by using a mechanical theory or a finite element simulation method and the like.
In this embodiment, the specific formula of the functional function of the structure is as follows:
Figure RE-GDA0002613954060000082
wherein the content of the first and second substances,
Figure RE-GDA0002613954060000083
step two, converting the variable space of the structural function:
2.1 structural function g (X, Y) obtained in step one, by probability transformation, UX＝Φ-1(FX(X)) and UY＝Φ-1(FY(Y)), a function G (U) of the structure in the normal space can be obtainedX,UY) I.e. G (U)X,UY)＝g(X,Y)；
Wherein: fX(X) is a cumulative distribution function of a random variable X, FY(Y) is the cumulative distribution function of the p-box variable Y,. phi-1Expressed as the inverse of the standard normal distribution;
2.2 initializing parametersThe number (1), s ═ 1,
Figure RE-GDA0002613954060000084
r 11, namely performing linear expansion on the mean value point of each variable;
wherein s is used for recording the outer layer iteration number,
Figure RE-GDA0002613954060000085
for the design flare point for the s-th iteration,
Figure RE-GDA0002613954060000086
solving for the s-th outer iteration the resulting maximum possible failure point at the upper limit of the non-probabilistic reliability index, and
Figure RE-GDA0002613954060000087
expressed as the maximum possible failure point, r, obtained when solving the lower bound of the non-probabilistic reliability indexsIs the non-linear exponent for the s-th iteration,
Figure RE-GDA0002613954060000088
are respectively as
Figure RE-GDA0002613954060000089
The random variable component and the p-box variable component of (1);
2.3 variable transformation, standard Normal space variable U ═ U (U)X,UY) By probability transformation
Figure RE-GDA00026139540600000810
Figure RE-GDA00026139540600000811
Conversion back to the original space of the function to obtain (X, Y); y isL,YRUpper and lower limits for the p-box variables of the structure, respectively; while
Figure RE-GDA00026139540600000812
And YF -1upper limit of cumulative probability distribution function representing p-box variables, respectivelyAn inverse function of the lower limit;
step three, approximating the function by using a linear approximation model and a two-point self-adaptive model:
3.1 at an arbitrary point (X, Y), the structural function g (X, Y) is approximated as an approximate expression for the p-box variable Y using a linear approximation method, i.e.
Figure RE-GDA00026139540600000813
Where j is 1,2,3, …, m,
Figure RE-GDA00026139540600000814
3.2 according to the non-Linear index rsAt point of point
Figure RE-GDA00026139540600000815
Establishing information about g (X, Y)c) Is a two-point adaptive non-linear approximation model, i.e.
Figure RE-GDA00026139540600000816
Wherein:
Figure RE-GDA0002613954060000091
Figure RE-GDA0002613954060000092
and i is 1,2,3, …, n, xi∈X，xi,s∈Xs-1，
Figure RE-GDA0002613954060000093
3.3 model combining the linear approximation of 3.1 and the two-point adaptive non-linear approximation of 3.2, the approximation representing a functional function of the structure, has
Figure RE-GDA0002613954060000094
Obtaining an extremum of the response of the function at an arbitrary point (X, Y) in an approximation；
3.4 the interval theory approximately obtains the upper and lower limits of the response of the function at any point (X, Y), specifically:
lower limit:
Figure RE-GDA0002613954060000095
upper limit:
Figure RE-GDA0002613954060000096
fourthly, performing inner layer iteration by using an HL-RF method:
4.1 initializing parameter (2), parameter (2) being k,
Figure RE-GDA0002613954060000097
the initialization operation is such that k is 1,
Figure RE-GDA0002613954060000098
wherein k is used for recording the number of inner layer iterations,
Figure RE-GDA0002613954060000099
to solve for
Figure RE-GDA00026139540600000910
Designing expansion points of the kth iteration of the corresponding inner layer;
4.2 obtaining the current nonlinear index r by iterative computationsUpper (lower) limit of the structural reliability index of (2)
Figure RE-GDA00026139540600000911
It includes:
1) obtaining the upper (lower) limit of the reliability index of the kth iteration structure of the inner layer in the s-th iteration of the outer layer by combining an HL-RF method
Figure RE-GDA00026139540600000912
The specific formula is as follows:
Figure RE-GDA00026139540600000913
Figure RE-GDA00026139540600000914
wherein the content of the first and second substances,
Figure RE-GDA00026139540600000915
expressed as a function in the standard normal space
Figure RE-GDA00026139540600000916
The upper (lower) limit of (c),
Figure RE-GDA0002613954060000101
is shown as
Figure RE-GDA0002613954060000102
A first partial derivative vector of;
2) the convergence condition (1) is set as
Figure RE-GDA0002613954060000103
Wherein, the constant is a relatively small constant and is generally set to 0.001;
3) judging whether the convergence condition (1) is met; if the convergence condition (1) is met, acquiring the current nonlinear index rsIs most preferred
Figure RE-GDA0002613954060000104
And corresponding thereto
Figure RE-GDA0002613954060000105
Carrying out the next step; if not, updating k to 1 in k +1 and returning to 3.2) and carrying out iterative calculation again;
4.3 setting a convergence condition (2), when s is more than or equal to 2,
Figure RE-GDA0002613954060000106
judging whether the convergence condition (2) is met; if it is coincident withThe final optimal non-probability mixed reliability index is obtained according to the convergence condition (2)
Figure RE-GDA0002613954060000107
And corresponding maximum possible failure point
Figure RE-GDA0002613954060000108
And go to step five;
if the s is not matched or is 1, carrying out the next step;
updating outer iteration parameters:
4.4 calculated by 4.3 of step four
Figure RE-GDA0002613954060000109
And the expansion point of the last iteration
Figure RE-GDA00026139540600001010
Acquisition using probability transformation
Figure RE-GDA00026139540600001011
And (X)s-1,Ys-1)；
4.5, updating the iteration times s to be s + 1;
4.6 pairs of non-linear indexes rsUpdating to obtain relevant parameters of the approximate model when the s-th iteration is performed, wherein the updated optimization formula is as follows:
Figure RE-GDA00026139540600001012
returning to the third step to start the next round of iterative computation;
step five, judging the symbol and obtaining the final reliability index beta E [ beta ]L,βR]And betaR(L)Corresponding maximum possible failure point
Figure RE-GDA00026139540600001013
5.1 Using the formula βR(L)＝sgn(GR(L)(0,0))βR(L)＝sgn(gR(L)(μX,μY))βR(L)For the obtained non-probability mixed reliability index betaR(L)The signs are distinguished to obtain the final betaR(L)(ii) a Wherein sgn (. cndot.) is expressed as a sign function, and is 1 when the input value is positive, and-1 when the input value is negative, (mu.)X,μY) The mean point of the variable expressed as a function in the original space;
5.2 integrating the results obtained in the third step and the 5.1 to finally obtain the non-probability mixed reliability index beta E [ beta ] of the analyzed structureL,βR]Upper (lower) limit beta of mixed reliability index with non-probabilityR(L)Corresponding maximum possible failure point
Figure RE-GDA00026139540600001014
In this embodiment, the method is used for solving the non-probability mixed reliability index upper limit betaRThen, through three iterations, the nonlinear index rsIn the three iteration processes, the values are respectively [1,2.115 and 0.081%]To finally obtain betaR＝1.745，
Figure RE-GDA0002613954060000111
In this embodiment, the invention solves the non-probability mixed reliability index lower limit betaLTime, non-linearity index rsAre respectively [1,2.043,4.99 ]]To finally obtain betaL＝1.634，
Figure RE-GDA0002613954060000112
As a comparison of this example, a conventional nested optimization method was used for analysis. The method obtains beta epsilon [ beta ] through 4 times of iterative computation no matter solving the upper limit or the lower limit of the mixed reliability index of the non-probabilityL,βR]＝[1.634,1.745]. The specific information of the method of the present invention and the conventional method in this embodiment is shown in tables 2 and 3.
TABLE 2 comparison of the method of the present invention to the traditional nested method for solving the lower bound of the reliability index
Figure RE-GDA0002613954060000113
TABLE 3 comparison of the method of the present invention with the conventional nested method when solving the upper limit of the reliability index
Figure RE-GDA0002613954060000114
In this embodiment, as can be seen from tables 2 and 3, compared with the prior art, the present invention can effectively reduce the number of times of function calls of the structure on the premise of ensuring the precision, and has an obvious efficiency advantage.
It will be understood by those skilled in the art that the foregoing is only a preferred embodiment of the present invention, and is not intended to limit the invention, and that any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the scope of the present invention.

Claims (9)

1. The non-probability mixed reliability index de-nesting analysis method comprises the following steps:
step one, determining a function and a variable of a structure:
1.1 determining variables influencing the reliability of the structure according to design and analysis requirements;
1.2 obtaining the function of the structure to be analyzed by using the mechanics theory or finite element simulation method
Figure RE-42851DEST_PATH_IMAGE001
Wherein
Figure RE-422011DEST_PATH_IMAGE002
It is meant that the random variable is,
Figure RE-785996DEST_PATH_IMAGE003
represents a p-box variable;
step two, converting the variable space of the structural function:
2.1 structural function from step one
Figure RE-461084DEST_PATH_IMAGE004
Obtaining a function of the structure in the standard normal space by probability transformation
Figure RE-910520DEST_PATH_IMAGE005
(ii) a Wherein
Figure RE-358950DEST_PATH_IMAGE006
Expressed as a random variable in a standard normal space,
Figure RE-561130DEST_PATH_IMAGE007
is a p-box variable under a standard normal space;
2.2 initializing a parameter (1) of outer layer iteration, and performing linear expansion on the mean value point of each variable;
2.3 variable transformation, variables whose structure is in the Standard Normal space
Figure RE-420502DEST_PATH_IMAGE008
Conversion back to raw space of functional function by probability conversion
Figure RE-107966DEST_PATH_IMAGE009
；
Figure RE-343776DEST_PATH_IMAGE010
，
Figure RE-668971DEST_PATH_IMAGE011
Lower and upper limits for the p-box variables of the structure, respectively;
step three, approximating the function by using a linear approximation model and a two-point self-adaptive model:
3.1 structuring the function in its original space by means of linear approximation
Figure RE-699244DEST_PATH_IMAGE012
Approximated as a function of p-box
Figure RE-608425DEST_PATH_IMAGE013
An approximate expression of (c);
3.2 establishing a correlation between the first and second iterations based on the non-linearity index at the s-th iteration
Figure RE-913505DEST_PATH_IMAGE014
The two-point adaptive nonlinear approximation model of (1);
wherein
Figure RE-90277DEST_PATH_IMAGE015
The midpoint of the p-box variable is indicated,
Figure RE-291451DEST_PATH_IMAGE016
；
3.3 combining the models of linear approximation of (3.1) and two-point adaptive non-linear approximation of (3.2), approximating a functional function representing a structure;
3.4 approximate obtaining of function at any point by using interval theory
Figure RE-687929DEST_PATH_IMAGE017
The upper and lower limits of the response;
fourthly, carrying out inner layer iteration and updating outer layer iteration parameters by using HL-RF method
Inner layer iteration is carried out by using an HL-RF method:
4.1 initializing parameters (2) of the inner-layer iteration;
4.2 calculate the upper or lower limit of the reliability index of the structure of the s-th iteration
Figure RE-734382DEST_PATH_IMAGE018
Wherein
Figure RE-288030DEST_PATH_IMAGE019
Expressed as an upper limit
Figure RE-394527DEST_PATH_IMAGE020
For the lower limit, it includes:
1) obtaining the upper limit or the lower limit of the reliability index of the kth iteration structure of the inner layer in the s-th iteration of the outer layer by combining an HL-RF method
Figure RE-543879DEST_PATH_IMAGE021
；
2) Setting a convergence condition (1), and judging whether the convergence condition (1) is met; if the convergence condition (1) is met, acquiring the upper limit or the lower limit of the optimal non-probability reliability index of the current non-linear index
Figure RE-174450DEST_PATH_IMAGE022
And corresponding maximum possible failure point
Figure RE-810967DEST_PATH_IMAGE023
And carrying out the next step; if not, returning to 1) in (3.2) to repeat iterative calculation;
4.3, setting a convergence condition (2) and judging whether the convergence condition (2) is met; if the convergence condition (2) is met, acquiring the final optimal non-probability mixed reliability index upper limit or lower limit
Figure RE-370256DEST_PATH_IMAGE024
And corresponding maximum possible failure point
Figure RE-725014DEST_PATH_IMAGE025
And go to step five;
if the convergence condition (2) is not met or
Figure RE-378849DEST_PATH_IMAGE026
If so, carrying out the next step;
updating outer iteration parameters:
4.4 calculated by step four
Figure RE-387650DEST_PATH_IMAGE027
And of the last iteration
Figure RE-101528DEST_PATH_IMAGE028
Obtaining by using probability transformation
Figure RE-225473DEST_PATH_IMAGE029
And
Figure RE-932266DEST_PATH_IMAGE030
；
wherein the content of the first and second substances,
Figure RE-DEST_PATH_IMAGE031
to represent
Figure RE-622005DEST_PATH_IMAGE032
The medium random variable is converted back to the coordinates of the original space of the functional function,
Figure RE-DEST_PATH_IMAGE033
is shown as
Figure RE-86878DEST_PATH_IMAGE034
The medium p-box variable is converted back to the median of the coordinates of the original space of the function;
4.5 updating outer iteration times
Figure RE-416228DEST_PATH_IMAGE035
；
4.6, updating the parameters so as to obtain relevant parameters of the approximate model when the iteration is performed for the s time, and returning to the third step to start the next iteration calculation;
step five, judging the symbol and obtaining the final reliability index
Figure RE-428177DEST_PATH_IMAGE036
And
Figure RE-893794DEST_PATH_IMAGE037
corresponding maximum possible failure point
Figure RE-464321DEST_PATH_IMAGE038
；
5.1 the obtained non-probabilistic mixed reliability index is subjected to
Figure RE-280968DEST_PATH_IMAGE039
The signs are distinguished to obtain the final
Figure RE-831029DEST_PATH_IMAGE040
；
5.2 integrating the results obtained in the third step and the (5.1) to finally obtain the non-probability mixed reliability index of the analyzed structure
Figure RE-416731DEST_PATH_IMAGE041
Upper or lower bound of mixed reliability index with non-probability
Figure RE-161089DEST_PATH_IMAGE042
The corresponding maximum possible failure point.
2. The method for de-nesting analysis of non-probabilistic mixed reliability indices of claim 1, wherein:
the linear approximation method in 3.1 in the third step is specifically expressed as
Figure RE-199452DEST_PATH_IMAGE043
；
Wherein
Figure RE-818784DEST_PATH_IMAGE044
，
Figure RE-524572DEST_PATH_IMAGE045
Expressed as the number of p-box variables,
Figure DEST_PATH_672307DEST_PATH_IMAGE041
。
3. the method for de-nesting analysis of non-probabilistic mixed reliability indices of claim 1, wherein:
the two-point adaptive non-linear approximate model in 3.2 in step three is specifically expressed as
Figure RE-228140DEST_PATH_IMAGE047
；
Wherein:
Figure RE-120004DEST_PATH_IMAGE048
Figure RE-680298DEST_PATH_IMAGE049
；
and also
Figure RE-512599DEST_PATH_IMAGE050
，
Figure RE-791133DEST_PATH_IMAGE051
。
4. The method for de-nesting analysis of non-probabilistic mixed reliability indices of claim 1, wherein:
the function of the approximate expression structure in step three 3.3 is specifically expressed as:
Figure RE-486688DEST_PATH_IMAGE052
。
5. the method for de-nesting analysis of non-probabilistic mixed reliability indices of claim 1, wherein:
3.4 function in step three at any point
Figure RE-416336DEST_PATH_IMAGE053
Upper or lower limit of response
Figure RE-421201DEST_PATH_IMAGE054
The concrete expression is as follows: lower limit:
Figure RE-672185DEST_PATH_IMAGE055
：
upper limit:
Figure RE-155119DEST_PATH_IMAGE056
。
6. the method for de-nesting analysis of non-probabilistic mixed reliability indices of claim 1, wherein:
the parameter (2) of 4.1 in the fourth step is
Figure RE-207782DEST_PATH_IMAGE057
The initialization operation is
Figure RE-117969DEST_PATH_IMAGE058
,
Figure RE-856249DEST_PATH_IMAGE059
；
Wherein k is used for recording the number of inner layer iterations,
Figure RE-408453DEST_PATH_IMAGE060
to solve for
Figure RE-47114DEST_PATH_IMAGE061
Design of the k-th iteration of the time-corresponding inner layerAnd (4) expanding points.
7. The method for de-nesting analysis of non-probabilistic mixed reliability indices of claim 1, wherein:
acquisition of HL-RF method of 1) in step four, 4.2
Figure RE-393781DEST_PATH_IMAGE062
The specific formula of (A) is as follows:
Figure RE-619357DEST_PATH_IMAGE063
wherein the content of the first and second substances,
Figure RE-444094DEST_PATH_IMAGE064
，
Figure RE-205770DEST_PATH_IMAGE065
expressed as a function in the standard normal space
Figure RE-457760DEST_PATH_IMAGE066
The upper limit or the lower limit of (c),
Figure RE-436211DEST_PATH_IMAGE067
，
Figure RE-64639DEST_PATH_IMAGE068
is shown as
Figure RE-677891DEST_PATH_IMAGE069
The first partial derivative vector of (a).
8. The method for de-nesting analysis of non-probabilistic mixed reliability indices of claim 1, wherein:
the updated optimization formula of 4.6 in step four is:
Figure RE-117094DEST_PATH_IMAGE070
。
9. the method for de-nesting analysis of non-probabilistic mixed reliability indices of claim 1, wherein:
step five, 5.1
Figure RE-566530DEST_PATH_IMAGE071
The specific formula for distinguishing the signs is as follows:
Figure RE-250846DEST_PATH_IMAGE072
wherein, therein
Figure RE-469337DEST_PATH_IMAGE073
Expressed as a sign function, 1 when the input value is positive, 1 when the input value is negative,
Figure RE-79441DEST_PATH_IMAGE074
expressed as the mean point of the variable in the original space of the function.
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