Research On Algorithms Of Reliability Analysis And Global Sensitivity Analysis Of The Structures

There are various uncertainties of model input variables in aircraft structures.These uncertainties result in the uncertainty of the output performance through the transformation of the inputs-output relationship.The uncertainty of performance directly decides the safety and robustness of aircraft structures.Based on the structural uncertainty analysis theory,this paper mainly focuses on the structural reliability analysis and the global sensitivity analysis,and establishes various efficient methods from different perspectives to calculate the reliability and the global sensitivity indices.At the same time,a lifetime model to assess the safety of the time-variant structures is studied.The detailed contents are summarized as follows:(1)For the reliability analysis of static structures and systems,firstly,this paper improves the importance sampling method,and weight functions of importance samples are defined to reveal the contribution degree of importance samples to the failure probability.By constructing the maximum relative error of the failure probability less than a pre-set threshold to screen the samples with less contribution to the estimation of failure probability,and by ignoring the influence of these samples on failure probability and directly setting their indicator function values of failure domain as zeros,the actual number of model evaluations can be greatly reduced under the satisfied accuracy of estimating the failure probability.Subsequently,the adaptive Kriging meta-model is nested into the proposed improved importance sampling method,which further enhances the efficiency of static reliability analysis from the perspective of efficiently sampling,reducing the size of candidate sampling pool and meta-modelling.For the cases that the importance sampling probability density function(PDF)is difficult to be constructed,a stratified Kriging combined with the adaptive radial-based importance sampling method is constructed to efficiently analyze the static reliability especially for the small failure probability.For the multiple failure modes,a new learning function is proposed.Since recognizing the extreme failure mode is replaced by recognizing an easily identified failure mode in the proposed new learning function,it makes the results obtained by adaptive Kriging method more robust.The advantages of all proposed methods are validated by the numerical and engineering examples.(2)For the reliability analysis of time-variant cases,the first time to failure(FTTF)based time-variant reliability analysis model is constructed,In the proposed method,solving the differential performance equation is transformed to solve the performance equation,then the complexity of solving equation is greatly reduced.An efficient PDF estimation technique combined with the meta-model method is proposed to estimate the solution of the FTTF based model,and it involves a double-loop process.In the inner loop,the corresponding FTTF is searched by the one-dimensional Kriging model,while in the outer loop,the PDF of the difference function between the FTTF and the upper boundary of the time interval of interest is estimated by the fractional moment based maximum entropy theory.Then,the time-variant failure probability is estimated by integral of the obtained PDF within the failure domain.Besides,the paper also constructs the reliability lifetime model under a give reliability constraint.Based on the single-loop Kriging meta-model and the dichotomy method,an efficient algorithm of estimating the reliability lifetime of time-variant structures is proposed.Results of numerical and engineering cases show the good adaptability of the proposed methods to engineering problems.(3)For solving the variance-based global sensitivity indices,the following three contents are studied in this paper.Firstly,a moment estimation method by combining space-partition idea and unscented transformation(UT)is proposed.By partitioning space and employing the decreased nonlinearity degree of the performance function in the segmentation subspace,the UT can be well utilized for solution of the model due to the strong capability of exploring the probability space and efficiency of UT.Based on the proposed moment estimation method,the computational formulas of four sensitivity indices including the variance-based global sensitivity index,the variance-based regional sensitivity index,the W index and its improvement are derived,on which the four kinds of variance-based sensitivity indices can be simultaneously obtained by repeatedly using the integration points generated in the process of estimating of the moment by the proposed space-partition with UT method.Therefore,more abundant sensitivity information from different perspectives can be efficiently provided to designers by the proposed method simultaneously.Secondly,by considering the importance of total variance-based sensitivity index,the multiplication dimensionality reduction approximation of the performance function combined with the simulation is constructed.This method unifies the computational formulas of all order variance-based sensitivity indices by the expectation of the squared differences function between the performance function and the expected performance function conditional on the input variables.By this method,every order variance-based sensitivity index can be obtained simultaneously.Finally,the law of total expectation and the law of total variance in the successive intervals without overlapping are proved,on which,an improved version of space-partition method is constructed.The improved space-partition method enhances the robustness of the space-partition method.The accuracy and efficiency of the proposed method are verified by some numerical examples,and the proposed method are also applied to engineering applications.(4)For solving the failure probability-based global sensitivity indices,the relevant efficient algorithms are constructed from different perspectives.Firstly,an efficient algorithm is established based on combination of the fractional moment,the maximum entropy and the Nataf transformation.The unconditional PDF of the performance function in the proposed algorithm is estimated by the maximum entropy process constrained by the fractional moment,and the joint PDF between performance function and input variable is estimated by the Nataf transformation.The computational cost of this proposed method only is spent in the process of estimating the fractional moment by the multiplication reduction integration formula.Due to the use of the multiplication dimensionality reduction strategy,the computational cost of the proposed method increases linearly with the dimensionality of model input variables,thus the problem of 'curse of dimensionality'is extremely alleviated.Secondly,two kinds of algorithms are proposed from the numerical simulation perspective,and their computational cost is independent of the dimensionality of model inputs.The first category is the space-partition based algorithm.Through partitioning the sample points of output into different subsets according to different inputs,the proposed method can efficiently evaluate all the sensitivity indices concurrently by one group of sample points.Aiming at the problems with easily-obtained design point and difficultly-obtained design point,the importance sampling method and the weight density sampling method are inducted in the proposed method,respectively.The second category is based on the combination of the Bayes formula and the subset simulation method,which convents the original expression of the failure probability-based global sensitivity index into a new equivalent form,in which only the unconditional failure probability and the failure-conditional PDF of each model input variable are required.And they can be obtained simultaneously by the same group of the output samples,which improves the efficiency of the proposed method.Finally,the concept of space partition and the meta-model are inducted in the Bayes formula based method,where the estimation of failure-conditional PDF into the estimation of failure-conditional probability,and this transformation enhances the accuracy and efficiency of estimating the failure probability-based global sensitivity indices especially for the problems with disconnected multiple failure domains.This proposed algorithm reduces the difficulty of the algorithm,expands the scope of application of the algorithm and greatly improves the efficiency.The results of numerical and engineering examples verify the adaptability and high efficiency of the proposed methods for different kinds of problems.(5)For solving the Borgonovo moment-independent global sensitivity indices,two kinds of algorithms are constructed in this paper.Firstly,based on using the same samples in the integration grid repeatedly,Borgonovo moment-independent global sensitivity indices can be estimated by the maximum entropy technique.This algorithm can repeatedly use the same samples for estimate the unconditional fractional moments,the conditional fractional moments of model output and the one-dimensional expectation of conditional variable.Then,the unconditional and conditional PDFs of model output are estimated by using the fractional moments based maximum entropy technique.The computational cost of this proposed method only generates in the process of estimating the fractional moment by the multiplication dimensionality reduction integration formula.It is worth noting that the first proposed algorithm also can be extended to estimate the variance-based global sensitivity indices and the failure probability-based global sensitivity indices.Secondly,an efficient radial basis function based algorithm is established to estimate the Borgonovo moment-independent global sensitivity indices from the perspective of meta-model.Based on the radial basis function,the analytical computational formulas of the first four-order moments of model output are derived.Then,the unconditional and conditional PDFs of model output are obtained by the Edgeworth expansion.Based on them,Borgonovo moment-independent sensitivity indices are finally estimated by a series of one-dimensional integrals.The computational cost of this method to obtain all inputs' sensitivity indices is independent of the dimensionality of model inputs.Results of numerical and engineering examples are verified the accuracy and efficiency of the proposed methods.