Research On Highly Efficient And Precise Methods For Reliability Analysis With Epistemic Uncertainties

Uncertainty can be categorized into two distinct types: aleatory and epistemic.Aleatory uncertainty,also known as the random uncertainty,requires precise probabilistic model.When experimental data are not enough to obtain the probabilistic model with prescribed confidence,it is required to tackle the uncertain variable with epistemic uncertainty theory.The proposition of epistemic uncertainty theories makes it possible that representing uncertainties,propagating uncertainties and making decisions strictlyconform to the amount of information obtained in hand.Reliability analysis is a major way foruncertaintiespropagation.Probabilistic reliability analysis(PRA)with only aleartory uncertainties has been deeply researched so far while the research on reliability analysis with epistemic uncertainties(RAEU)is still in its preliminary stage.This dissertation focuses on the investigation of RAEU.To accurately estimatethe bounds of failure probability with calling the performance function(PF)as few as possible,an integrated set of methodologies based on active learning Kriging(ALK)model are proposed.The major contribution of this dissertation can be summarized into three points.(I)A so-called extreme signtheorem(EST)which is adaptable to different kinds of epistemic theories is originally proposed.EST indicates that a surrogate model only precisely predicting the sign of PF can meet the requirement of RAEU in accuracy.(II)To construct a Kriging model which can precisely predict the sign of performance function,a so-called “Expected Risk Function”(ERF)is elaborated according to the uncertain information provided by a Kriging model.(III)According to the characteristics of different representative models of epistemic uncertainties,ALK models are adjusted and improved so that the proposed method can behavethe best in each situation.The contents of this dissertation are listed as follows.(1)Hybrid reliability analysis with both random and interval variables is researched.Aleatory variables are described with probabilistic models and epistemic variables are represented by interval model.EST is proposed and the deduction that only rightly predicting the sign of performance function can meet the accuracy requirement of HRA is obtained.To construct such a Kriging model,ERF is elaborated.Based on ERF,the method termed as ALK-HRA is proposed.ALK-HRA is oriented to precisely predicting the sign of PF and focuses much attention on the approximation of PF in the narrow region around limit state.Therefore,ALK-HRA can obtain accurate results at the minimal computational cost.Furthermore,for rare event estimation,the importance sampling method for HRA is proposed and is integrated into ALK-HRA method.The new method is termed as ALK-HRA-IS.Several cases are studied to demonstrate the accuracy and efficiency of ALK-HRA and ALK-HRA-IS.(2)Probabilistic and convex set hybrid reliability analysis(termed as HRA-Convex)is investigated.Aleatory variables are described with probabilistic models and epistemic variables are represented by multi-ellipsoid convex model.Multi-ellipsoid convex model can indicate the correlation between epistemic variables.To overcome the defects of existing literatures,the maximum failure probability is proposed to measure the reliability of HRA-Convex and the corresponding Monte Carlo simulation method(MCS)is offered.The sampling method to generate points constrained in the multi-ellipsoid region is elaborated.These points are treated as candidate points to build the ALK model and thus the ALK-HRA-Convex method is developed.ALK-HRA-Convex only predicts the sign of PF in the multi-ellipsoid.Therefore,it is economic in the number of function evaluations.(3)Hybrid reliability analysis with Probabilistic and probability–box variables(termed as HRA-P-box)is researched.Aleatory variables are described with probabilistic models and epistemic variables are modeled by P-box model.The so-called interval Monte Carlo simulation method(IMCS)from existing literatures is introduced and the optimization method is fused into IMCS.This method is termed as OIMCS.By integrating ALK model and OIMCS,the method ALK-OIMCS for HRA-P-box is proposed.However,it is figured out that ALK-OIMCS approximates the PF in a region larger than the necessity of OIMCS and wasted efforts arise in ALK-OIMCS.To further reduce the computational expense,ALK-OIMCS is improved.A new sampling method is originally developed to generate candidate points for P-box variables.With this method,the candidate points can cover the region neither more or less than the necessity of OIMCS.ALK model constructedbased on those candidate points can assure the accuracy of HRA-P-box with function evaluations as few as possible.The improved method is termed as Adv-ALK-OIMCS.(4)Evidence-theory-based reliability analysis(ETRA)is researched.Only epistemic uncertainties exist in ETRA and they are represented by evidence theory.The adaption of EST is proved under the situation only comprising epistemic variables.Then the method based on ALK model for ETRA is developed.The new method is termed as ALK-ETRA.In ALK-ETRA,IMCS rather than Cartesian product method is adopted to estimate the bounds of failure probability,so that the curse of dimensionality is alleviated.Furthermore,we propose the Karush-Kuhn-Tuckerconditions based optimization(KKTO)method to calculate the extrema of the Kriging model.KKTO makes the bounds estimation of failure probability more efficient.(5)United uncertainty analysis(UUA)is researched.In UUA,aleatory variables are tackled with probabilistic models and epistemic variables are modeled by evidence theory.To overcome the curse of dimensionality of existing methods,the random-set-based Monte Carlo simulation(RS-MCS)is introduced.Although the former mentioned ALK-ETRA method is available to perform UUA,ALK-ETRA is not the optimal method.ALK-ETRA approximates the sign of PF throughout the evidence space which results in wasted computational expense.Therefore,a new sampling method is proposed to generate the candidate points that can just envelop the region of RS-MCS.With those candidate points,ALK model is constructed and thus the so-called ALK-RS-MCS is proposed.The efficiency and accuracy of ALK-RS-MCS are compared with the existing methods through four examples.