CN113343433B - KKT condition and differential evolution algorithm-based first-order reliability analysis method - Google Patents
KKT condition and differential evolution algorithm-based first-order reliability analysis method Download PDFInfo
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Abstract
The invention discloses a first-order reliability analysis method based on KKT conditions and a differential evolution algorithm, which is characterized in that an equivalent reliability analysis model with penalty function parameters capable of changing in a self-adaptive manner is established based on the KKT conditions so as to solve the problems of poor analysis precision and low calculation efficiency of the equivalent reliability analysis model with the penalty function parameters increasing exponentially in strong nonlinearity problems; the improved differential evolution optimization algorithm with the self-adaptive cross operation mechanism is used for solving an equivalent structural element reliability analysis model, and further calculating the failure probability. The method has good universality and adaptability when a first-order reliability method is used for evaluating the reliability and the safety degree of the complex engineering structure in the fields of civil engineering, mechanical engineering, aerospace and the like, the convergence is fast, the precision is high, and the problem of poor reliability analysis result precision and the like caused by the fact that the most probable failure point search algorithm is early trapped in local optimization can be avoided by a new penalty function coefficient determining mode in an equivalent reliability analysis model.
Description
Technical Field
The invention relates to the technical field of structural reliability analysis, in particular to an aspect of analyzing structural reliability by adopting a primary second moment method based on a group intelligent optimization algorithm, and specifically relates to an equivalent reliability analysis model based on a KKT condition and a reliability analysis method of a differential evolution group intelligent optimization algorithm.
Background
The structural reliability design and analysis method aims at evaluating the reliability of the normal working capacity of structures or products (such as building structures, bridges, power systems, electronic products and the like) in the fields of civil engineering, mechano-electronics, aerospace and the like, establishes a structural function taking geometric dimension, load and the like as influencing factors, and quantifies the reliability of the structures on the basis of the functional function. Compared with the traditional deterministic analysis method, the method has the advantages that a plurality of influence factors of the structure are fully considered, the safety of structure operation is improved, and the sensitivity analysis is carried out on the basis, so that the influence factors are more favorably improved, and the reliability of the structure is effectively improved.
The primary second moment reliability method is based on the first-order linear approximation of a functional function which embodies the working capacity of a structure or reflects the critical state of the structure capable of working safely, is widely used for evaluating the reliability of the structure or a product due to simple and convenient calculation, but has the problem of unconvergence such as periodic oscillation, bifurcation and the like in the calculation process of the strong nonlinear functional function. Although some methods, such as a stability transformation method based on chaotic feedback control, a step-variable method, and the like, overcome the problem of unstable calculation to a certain extent, and improve the robustness of the algorithm, the requirements of a first-order second-order moment method, which is a fast calculation method, on the accuracy and stability of reliability analysis in practical engineering, especially when a strong nonlinear function exists, cannot be met.
The primary second moment method is essentially a constrained optimization problem for evaluating the reliability of structures or products in the fields of civil engineering, mechano-electronics, aerospace and the like. At present, swarm intelligent optimization algorithms such as particle swarm and the like are used for evaluating the reliability of a structure or a product, the methods are used as a global optimization algorithm, the condition that convergence cannot be achieved is avoided, but the penalty function coefficient of an equivalent unconstrained optimization problem grows exponentially along with the number of iteration steps, and for the condition that some function functions change at a slow speed in the iteration process and the strong nonlinear problem, the constraint of a limit state function is possibly lost due to the excessively fast growth, so that the equivalent model and an original model have great deviation, and the problems of poor accuracy of a reliability analysis result, slow convergence and the like are brought.
According to an optimization theory, an equivalent unconstrained optimization model is adjusted in a self-adaptive mode to maintain a good equivalent relation with an original model, so that a group intelligent optimization algorithm can be more effectively applied to a primary second moment method of structural reliability, a more accurate reliability analysis result is obtained, and the convergence of an iteration process is improved.
The differential evolution optimization algorithm is an efficient group intelligent optimization algorithm, and provides a potential possibility for efficient solution of an equivalent unconstrained optimization problem of structural reliability analysis.
Therefore, a more reasonable unconstrained equivalent optimization model for structural reliability analysis is established according to an optimization theory, and meanwhile, the solution is carried out by combining an efficient differential evolution group intelligent optimization algorithm, so that the method has important significance in the aspect of carrying out structural or product reliability evaluation by adopting an efficient primary second moment method in the fields of civil engineering, mechano-electronics, aerospace and the like.
Disclosure of Invention
The invention aims to solve the defects in the prior art and provides a first-order reliability analysis method based on a KKT condition and a differential evolution algorithm.
The purpose of the invention can be achieved by adopting the following technical scheme:
a reliability analysis method based on a KKT condition equivalent reliability analysis model and a differential evolution group intelligent optimization algorithm comprises the following steps:
s1, specifying a function g (x) reflecting the normal working capacity or the safe working critical state of the structure or the product in the field to be analyzed, and determining a random variable x [ x ] of the function according to the statistical data of the actual operation of the structure or the product 1 ,…,x i ,…]And probability distribution information thereof, wherein x i The ith random variable is used as the analysis field of civil engineering, mechanical electronics and aerospace;
s2, and enabling the random variable x to be equivalent to a standard normal space u ═ u 1 ,u 2 ,…,u i ,…]Generating population by initializing in standard normal space, where u i Is x i An equivalent standard normal random variable;
s3, solving for the condition that g (u) is 0
Minimum reliability analysis constrained equivalence optimization problem, which in turn equates to unconstrained minimization
Where ξ is a penalty function and λ is a penalty coefficient;
s4, setting the initial value of the penalty coefficient lambda to 0.1, using an equivalent unconstrained optimization problem as a solving target by a differential evolution optimization algorithm, converting the population in the standard normal space into an original space, substituting the original space into an objective function to evaluate the fitness of the population, and searching a new most probable failure point;
s5, updating the penalty coefficient according to a mode obtained by a KKT condition until meeting a control condition ξ (g (u)) < kappa, and stopping updating, wherein kappa is a small value for controlling whether the penalty coefficient lambda needs to be updated or not;
s6, checking the reliability index beta calculated in the k-1 and k steps (k-1) And beta (k) Whether or not to satisfy
If so, outputting the reliability index beta at the moment (k) And gives the probability of failure phi (-beta) (k) ) Otherwise, returning to step S4 to continue searching for a new most likely failure point, where e is a small positive number of control loop endings and Φ (-) is a normal distribution.
Further, in step S5, if the penalty function is an absolute value function, the penalty coefficient is updated according to the KKT condition as follows, and the current λ is calculated 1 ,λ 2 … … and then take their minimum value, i.e.:
then comparing the obtained penalty coefficient lambda with the penalty coefficient of the previous k-1 step, and finally taking the larger value as a new penalty coefficient of the k step, wherein the expression is as follows: lambda [ alpha ] (k) ＝max(min(λ 1 ,λ 2 ,...,λ n ),λ (k-1) ) Wherein the content of the first and second substances,
is the partial derivative of the function.
Further, the intelligent optimization algorithm of the differential evolution group in step S4 achieves the purposes of reducing the calculation amount and improving the calculation efficiency by using a mode in which an individual has an adaptive mutation operation mechanism, wherein an operator in the adaptive mutation operation mechanism is dynamically adaptively adjusted according to the number of continuous successes or the fitness value of the individual in an iteration process in a form not limited to triangular distribution.
Further, the initialization population generated in step S2 may be generated by any random or quasi-random monte carlo method.
Further, the penalty function ξ in step S3 takes the absolute value function or the square function.
Compared with the prior art, the invention has the following advantages and effects:
(1) the penalty function equivalent unconstrained structure reliability analysis model established according to the KKT condition has better equivalence with the original structure reliability analysis model, so that the accuracy of the reliability analysis result is ensured.
(2) The invention applies the high-efficiency global optimization algorithm of the differential evolution group intelligent optimization algorithm to the solution of the reliability problem, ensures the convergence and stability of the first-order second-order moment method, and can not fall into unstable states of solution oscillation, chaos and the like.
(3) The method combines a penalty function equivalent unconstrained structure reliability analysis model established based on the KKT condition with a differential evolution group intelligent optimization algorithm, greatly expands the effectiveness and the universality of a first-order second-order moment method in the strong nonlinear reliability analysis problem, and has important significance in the field of reliability analysis.
Drawings
FIG. 1 is a flowchart of a reliability analysis method based on a KKT condition equivalent reliability analysis model and a differential evolutionary group intelligent optimization algorithm provided by the present invention;
FIG. 2 is a flow chart of the differential evolution optimization algorithm of the present invention;
FIG. 3 is a graph showing the variation trend of the reliability index obtained by the methods in example 1 of the present invention;
FIG. 4 is a schematic diagram of a master-slave two-degree-of-freedom power system in embodiment 2 of the present invention;
fig. 5 is a graph showing the variation trend of the reliability index obtained by the methods in embodiment 2 of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are some, but not all, embodiments of the present invention. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
Examples
The embodiment discloses a reliability analysis method based on a KKT condition equivalent reliability analysis model and a differential evolution group intelligent optimization algorithm, which comprises the following steps of:
s1, specifying the functional function g (x) reflecting the normal working capacity or safe working critical state of the structure or product in the fields of civil engineering, mechano-electronics, aerospace and the like to be analyzed, and determining the random variable x ═ x of the functional function according to the statistical data of the actual operation of the structure or product 1 ,…,x i ,…]And probability distribution information thereof, wherein x i Is the ith random variable;
s2, and enabling the random variable x to be equivalent to a standard normal space u ═ u 1 ,…,u i ,…]Generating population by initializing in standard normal space, where u i Is x i An equivalent standard normal random variable;
the initialization population generated in step S2 may be generated using any random or quasi-random monte carlo method.
S3, if g (u) is 0, solving
Minimum reliability analysis constrained equivalence optimization problem, which in turn equates to unconstrained minimization
Where ζ is a penalty function and λ is a penalty coefficient;
in this step S3, the penalty function ξ is an absolute value function or a square function.
S4, setting the initial value of the penalty coefficient lambda to 0.1, using an equivalent unconstrained optimization problem as a solving target by a differential evolution optimization algorithm, converting the population in the standard normal space into an original space, substituting the original space into an objective function to evaluate the fitness of the population, and searching a new most probable failure point;
in the step S4, the differential evolution group intelligent optimization algorithm adopts a mode that an individual has a self-adaptive cross operation mechanism, so as to achieve the purposes of reducing the calculation amount and improving the calculation efficiency. And an operator in the self-adaptive mutation operation mechanism is dynamically and self-adaptively adjusted according to the continuous success times or the adaptability value of the individual in the iteration process in a form not limited to triangular distribution.
S5, updating the penalty coefficient according to a mode obtained by a KKT condition until meeting a control condition ξ (g (u)) < kappa, and stopping updating, wherein kappa is a small value for controlling whether the penalty coefficient lambda needs to be updated or not;
in step S5, if the penalty function is an absolute function, the penalty coefficient λ is updated according to the KKT condition as follows, and the current λ is calculated 1 ,λ 2 … … and then take their minimum value, i.e.:
then comparing the obtained penalty coefficient lambda with the penalty coefficient of the previous k-1 step, and finally taking the larger value as a new penalty coefficient of the k step, wherein the expression is as follows: lambda [ alpha ] (k) ＝max(min(λ 1 ,λ 2 ,...,λ n ),λ (k-1) ) Wherein, g u1 (u),g u2 (u), … is a functionPartial derivatives of
S6, checking the reliability index beta calculated in the k-1 and k steps (k-1) And beta (k) Whether or not to satisfy
If so, outputting the reliability index beta at the moment (k) And gives the probability of failure phi (-beta) (k) ) Otherwise, returning to step S4 to continue searching for a new most likely failure point, where e is a small positive number of control loop endings and Φ (-) is a normal distribution.
Example 1
Fig. 1 is a flowchart of a reliability analysis method based on a KKT condition equivalent reliability analysis model and a differential evolutionary group intelligent optimization algorithm provided in this embodiment, which includes 7 steps in total. FIG. 2 is a flow chart of an intelligent optimization algorithm for a differential evolution cluster, wherein operators in an adaptive mutation operation mechanism achieve 15 times of dynamic adaptive adjustment according to individual continuous success times in an iteration process in a triangular distribution mode. This example 1 further illustrates the present invention in a 6-dimensional application example.
Step S1, specifying the function g (x) of the structure to be analyzed, as follows:
random variable x ═ x of function 1 ,…,x i ,…]And the probability distribution information thereof are as follows:
TABLE 1 EXAMPLE 1 random variable characteristics
| Variables of | Type of distribution | Mean value of | Variance (variance) |
| x 1 | Lognormal method | 120 | 12 |
| x 2 | Lognormal method | 120 | 12 |
| x 3 | Lognormal method | 120 | 12 |
| x 4 | Lognormal method | 120 | 12 |
| x 5 | |
50 | 15 |
| x 6 | |
40 | 12 |
Step S2, the random variable x is equivalent to a standard normal space u ═ u 1 ,…,u i ,…]Generating an initialization population in a standard normal space by a random Monte Carlo sampling method, wherein the population size N is 80;
step S3, solving so that the constraint of g (u) ═ 0 is satisfied
Minimum reliability analysis constrained equivalence optimization problem, which in turn is equivalent to unconstrained minimization
In the optimization problem of (1), xi is taken as an absolute value function, and lambda is a penalty coefficient;
s4, setting the initial value of a penalty coefficient lambda to be 0.1, converting the population in the standard normal space into the original space by using an equivalent unconstrained optimization problem as a solving target through a differential evolution optimization algorithm, substituting the original space into an objective function to evaluate the fitness of the population, and searching a new most probable failure point;
step S5, respectively updating the penalty coefficients according to the mode obtained by the KKT condition and according to the exponential mode with the base 2, until meeting the control condition ξ (g (u)) < κ, and the updating is stopped, wherein κ is 10;
in step S6, the reliability index beta calculated in the k-1 and k-steps is checked (k-1) And beta (k) Whether or not to satisfy
If so, outputting the reliability index beta at the moment (k) And gives the probability of failure phi (-beta) (k) ) Otherwise, returning to step S4 to update the population size, and continuing to search for a new most probable failure point, where epsilon is 10 in this embodiment -6 。
A comparison of the reliability analysis method disclosed in example 1 with the reliability indicator convergence history of other methods is shown in FIG. 3(λ @) 1 For a penalty factor, λ, which varies exponentially 2 In order to obtain a penalty coefficient according to a KKT condition, DE is a differential evolution algorithm, HLRF is a classic first second moment method, FSML and FAL are variable step length algorithms and improved algorithms thereof,STM is a chaotic control algorithm), as can be seen in fig. 3, λ 2 DE converges to a higher accuracy reliable indicator 2.34882, ratio λ, in generation 42 1 Faster DE convergence. For the problems of complex strong nonlinearity and oscillation, the reliability analysis method based on the KKT condition equivalent reliability analysis model and the differential evolution group intelligent optimization algorithm has good universality and rapid convergence, achieves good first-order analysis precision, and meets the actual requirements of engineering.
Example 2
The embodiment continuously discloses a reliability analysis method based on a KKT condition equivalent reliability analysis model and a differential evolution group intelligent optimization algorithm, wherein an operator in a self-adaptive mutation operation mechanism achieves dynamic self-adaptive adjustment for 15 times according to individual continuous success times in an iteration process in a triangular distribution mode, and the reliability analysis method comprises the following steps:
step S1, specifying a function g (x) of a master-slave two-degree-of-freedom power system structure to be analyzed (as shown in the schematic diagram of fig. 4), as follows:
g＝F s -k s p[E(x s 2 )] 1/2
wherein, ω is p ＝(k p /m p ) 0.5 ，ω s ＝(k s /m s ) 0.5 ，ω a ＝(ω p +ω s )/2，ξ a ＝(ξ p +ζ s )/2， v＝m s /m p ，η＝(ω p -ω s )/ω a And p is 3. Random variable x ═ m for functional functions p ，m s ，…]And the probability distribution information thereof are as follows:
TABLE 1 EXAMPLE 2 random variable characteristics
| Variables of | Type of distribution | Mean value of | Variance (variance) |
| m p | Logarithm of | 1 | 0.1 |
| m s | Logarithm of | 0.01 | 0.001 |
| k p | Logarithm of | 1 | 0.2 |
| k s | Logarithm of | 0.01 | 0.002 |
| ζ p | Logarithm of | 0.05 | 0.02 |
| ζ s | Logarithm of | 0.02 | 0.01 |
| F s | Logarithm of | 15 | 1.5 |
| S 0 | Logarithm of | 100 | 10 |
Step S2, the random variable x is equivalent to a standard normal space u ═ u 1 ，...，u i ，...]Generating an initialization population in a standard normal space by a random Monte Carlo sampling method, wherein the population size N is 80;
step S3, solving so that the constraint of g (u) ═ 0 is satisfied
Minimum reliability analysis constrained equivalence optimization problem, which in turn equates to unconstrained minimization
In the optimization problem, xi is taken as an absolute value function, and lambda is a penalty coefficient;
s4, setting the initial value of a penalty coefficient lambda to be 0.1, converting the population in the standard normal space into the original space by using an equivalent unconstrained optimization problem as a solving target through a differential evolution optimization algorithm, substituting the original space into an objective function to evaluate the fitness of the population, and searching a new most probable failure point;
step S5, respectively updating the penalty coefficients according to the mode obtained by the KKT condition and according to the exponential mode with 2 as the base number until meeting the control condition ξ (g (u)) < κ, and the updating is stopped, wherein κ is 0.1 in the embodiment;
step S6, checking the reliability index beta calculated in the k-1 step and the k step (k-1) And beta (k) Whether or not to satisfy
If so, outputting the reliability index beta at the moment (k) And gives the probability of failure phi (-beta) (k) ) Otherwise, returning to step S4 to update the population size, and continuing to search for a new most probable failure point, where epsilon is 10 in this embodiment -4 。
A comparison of the reliable indicator convergence history of the method of example 2 with other methods is shown in FIG. 5, where it can be seen from FIG. 5 that λ 2 -DE convergence ratio λ 1 DE is fast and can converge to a higher accuracy reliable indicator 2.12453 in generation 26, while FAL, HLRF cannot. For complex strong nonlinear problems, compared with other methods, the reliability analysis method based on the KKT condition equivalent reliability analysis model and the differential evolution group intelligent optimization algorithm can always converge quickly, and a reliability analysis result reaching first-order precision is obtained.
The above embodiments are preferred embodiments of the present invention, but the present invention is not limited to the above embodiments, and any other changes, modifications, substitutions, combinations, and simplifications which do not depart from the spirit and principle of the present invention should be construed as equivalents thereof, and all such changes, modifications, substitutions, combinations, and simplifications are intended to be included in the scope of the present invention.
Claims (4)
1. A reliability analysis method based on a KKT condition equivalent reliability analysis model and a differential evolution group intelligent optimization algorithm is characterized by comprising the following steps:
s1, specifying a function g (x) reflecting the normal working capacity or the safe working critical state of the structure or the product in the field to be analyzed, and determining a random variable x [ x ] of the function according to the statistical data of the actual operation of the structure or the product 1 ,…,x i ,…]And probability distribution information thereof, wherein x i For the ith random variable, the field to be analyzed includes earthworkStroke, mechatronics, and aerospace;
s2, equivalent random variable x to standard normal space u ═ u 1 ,u 2 ,…,u i ,…]Generating population by initializing in standard normal space, where u i Is x i An equivalent standard normal random variable;
s3, solving for the condition that g (u) is 0
Minimum reliability analysis constrained equivalence optimization problem, which in turn equates to unconstrained minimization
Where ξ is a penalty function and λ is a penalty coefficient;
s4, setting the initial value of the penalty coefficient lambda to 0.1, using an equivalent unconstrained optimization problem as a solving target by a differential evolution optimization algorithm, converting the population in the standard normal space into an original space, substituting the original space into an objective function to evaluate the fitness of the population, and searching a new most probable failure point;
s5, updating the penalty coefficient according to a mode obtained by a KKT condition until meeting a control condition ξ (g (u)) < kappa, and stopping updating, wherein kappa is a value for controlling whether the penalty coefficient lambda needs to be updated or not;
in step S5, if the penalty function is an absolute function, the penalty coefficient λ is updated according to the KKT condition as follows, and the current λ is calculated 1 ,λ 2 … … and then take their minimum value, i.e.:
then comparing the obtained penalty coefficient lambda with the penalty coefficient of the previous k-1 step, and finally taking the larger value as a new penalty coefficient of the k step, wherein the expression is as follows: lambda [ alpha ] (k) ＝max(min(λ 1 ,λ 2 ,...,λ n ),λ (k-1) )
Wherein the content of the first and second substances,
is the partial derivative of the function;
s6, checking the reliability index beta calculated in the k-1 and k steps (k-1) And beta (k) Whether or not to satisfy
If so, outputting the reliability index beta at the moment (k) And gives the probability of failure phi (-beta) (k) ) Otherwise, return to step S4 to continue searching for a new most likely failure point, where e is a positive number of control loop endings and Φ (·) is a normal distribution.
2. The reliability analysis method according to claim 1, wherein the intelligent differential evolution optimization algorithm in step S4 adopts a mode that each of the algorithms has an adaptive mutation mechanism, and an operator in the adaptive mutation mechanism is dynamically adaptively adjusted according to the number of continuous successes or an adaptive value of each of the operators in an iterative process in a triangular distribution manner.
3. The reliability analysis method based on the KKT condition equivalent reliability analysis model and the differential evolution group intelligent optimization algorithm of claim 2, wherein the initialization population generated in the step S2 can be generated by any random or quasi-random monte carlo method.
4. The reliability analysis method based on the KKT condition equivalent reliability analysis model and the differential evolution group intelligent optimization algorithm as claimed in claim 3, wherein the penalty function ξ in step S3 takes the absolute value function or the square function.
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