The Study On The Generalized Pareto Distribution Fitting Method For Strong Nonlinear And High Reliability Problems

There exist lots of uncertainties in the structural analysis of engineering,such as loads,material properties,geometry,initial conditions,manufacturing tolerances and boundary conditions,etc.The reliability method evaluates the failure probability of structure by the probability and statistics theory.Compared with the traditional safety coefficient method,the reliability method can evaluate the failure behavior of the structure under the influence of uncertain factors,and obtain the reasonable structural design or safety assessment.As the engineering structure becomes more complex,larger and lighter,especially with the application of new materials and technologies,the reliability requirements for structures are increaseed greatly,and more uncertainty factors are involved.How to evaluate the high reliability of highly nonlinear function with high efficiency and accuracy is an urgent need in the current engineering field.Traditional reliability analysis methods,such as first order second moment method and maximum entropy method based on moment information,may not be applicable.The generalized Pareto diustribution(GPD)method is developed based on the extreme value theory,which is widely used in the prediction of extreme events.The determinations of threshold and unknown parameters are the key problems for GPD method,which is mainly based on a large amount of existing data.However,the computational cost for obtaining samples is an important factor for in the reliability evaluation of engineering structures,which also restrict the application of GPD in the field of engineering.The surrogate model is an effective way to improve the efficiency of reliability evaluation,and the efficiency of GPD method is improved by combining the surrogate model in this paper.A least square fitting method for GPD function based on quantiles is proposed for the evaluation of high reliability problems with highly nonlinear functions.The main contents of this paper are as follows.(1)It is found that only the tail samples are significant for GPD fitting,and a large number of non-tail samples are useless.For avoiding the low efficiency of GPD method caused by non-tail sample calculation,a radial basis function network(RBF)assisted sampling method is proposed to improve the computational efficiency.Moreover,the tail sample validation method is also proposed to ensure the accuracy of GPD fitting.The performance of GPD is investigated with the root mean square error considering different numbers of tail samples and total samples.The accuracy and efficiency of the proposed method are verified by numerical examples with nonliner functions.(2)For highly nonlinear problems,the RBF may have large error with Latin hypercube sampling method,leading to failure of RBF assisted sampling method.Therefore,this paper proposes an RBF updating method.It is found that the accuracy of the RBF predictions is much affected by the distance between test samples and training samples.Therefore,in the tail sample selected by the current RBF network,the sample with the largest distance from the training sample is selected to update the RBF network model.Furthermore,the tail samples are divided into several segments according to the predictions of the RBF network,selecting an updat:ing sample in each segment to avoid the early convergence of the RBF updating.The prediction accuracy of the RBF network is improved rapidly by the proposed updating method,and the RBF predictions are used for GPD fitting.(3)In order to improve the stability of GPD estimation,this paper proposes a quantile based least squares fitting method for GPD function fitting.The method uses several quantiles instead of a large number of tail sample points to fit the GPD function,and achieves a more accurate estimation.With regard to the quantile determination,the improved multiple-single sample updating method based on U-criterion for Kriging model is proposed to determine the quantiles efficiently.The proposed method is verified by 3 numerical examples,with more accurate and efficient results of quantiles,compared with the results obtained by the traditional maximum likelihood method and MC S,.(4)Considering the randomess of the size and distribution of voids in the matrix of the composite conponent in airplane,the random model of the matrix with the given porosity is estabilished using the represent volume element method(RVE).The affect of the matrix porosity on the material property are discussed.RBF is introduced to accelerate the sampling process.Considering the uncerta:in number of random variables caused by the uncertain number of random voids under the given porosity,the Kevin-Luff transform(K-L)method is introduced to solve this problem in RBF model.Meanwhile,a crowded distance based sampling method is also proposed to achieve uniform sampling to avoid the overlearning problem in RBF surrogate model.Finally,reliability analysis was carried out for the composite wing box with voids and delaminations,in which the randomness of the material parameters is calculated utiliz:ing the RBF based random RVE method.Then quantile based GPD fitting method is used to evaluate the bearing capacity for the specified reliability index.