Uncertainty Analysis Methods Considering Epistemic Parameters Or Multimodal Aleatory Parameters

Uncertainties associated with material properties,geometric sizes,external loads and computational models widely exist in practical engineering problems.Generally,these uncertainties are small in value,but they may cause large diviations of structural performance and even structural failures when they coupled.Therefore,quantifying and controlling uncertainties are becoming critical for ensuring the quality and safety of engineering structrues.According to the different generation mechanisms and physical meanings,uncertainties can be categorized into two distinct types: aleatory uncertainty and epistemic uncertainty.Aleatory uncertainty analysis methods are supported by probability theory;their theoretical study and practical application are quite mature.Epistemic uncertainty analysis methods are non-probabilistic theories,including possibility theory,interval analysis theory,fuzzy theory,evidence theory etc.Among these theories,evidence theory gets great concern and becomes a main tool of epistemic uncertainty analysis for employing a more flexible framework to represent uncertainty and can deal with all kinds of uncertainty effectively.Athough great progresses have been made,there are still many problems left unsolved in probability-theory-based aleatory uncertainty analysis and evidence-theory-based epistemic uncertainty analysis.For examples,probability-theory-based aleatory uncertainty analysis is difficult to solve problems like the strongly nonlinear problem,the ultra-high dimensional problem,the problem involving multimodal aleatory input variables and the efficiency and accuracy problem,and evidence-theory-based epistemic uncertainty analysis has problems like the large-scale computational problem,the correlational problem,the hybrid uncertainty problem and the system reliability problem.With the increasing attention of multimodal aleatory parameters in engineering,the difficulty to solve the problem involving multimodal aleatory input variables has become an important issue in probability-theory-based aleatory uncertainty analysis.And the large-scale computational problem has always been a critical issue that limits the application of evidence-theory-based epistemic uncertainty analysis in engineering practice.As a result,the following studies are carried out in this dissertation:(1)For epistemic uncertainty problems with all inputs are evidence variables and monotonous performance functions,an efficient epistemic uncertainty analysis method using evidence theory is proposed.Firstly,a transformation form evidence variable to Johnson p-box is performed by Moment-metching method,so as to get the continuous expression of the evidence variable.Subsequently,the probability bound analysis is conducted for the Johnson p-box and the response probability distribution based on monotonicity analysis.Through the probability bound analysis,the Johnson p-box propagation problem becomes two aleatory uncertainty problems.Finally,uncertainties are propagated efficiently by univariate dimension reduction method and maximum entropy method,and the epistemic uncertainty analysis using evidence theory is completed at the same time.(2)For aleatory uncertainty problems involving multimodal distribution variable with performance function has no interaction term or weak interaction terms,a multimodal aleatory uncertainty analysis method based on dimension reduction integration is proposed.Firstly,a generalized maximum entropy method(GMEM)is developed by extending the four-moment constaints of maximum entropy method to n-moment constaints.Subsequently,a maximum entropy iteration loop is proposed based on GMEM.Finally,the order of GMEM constraints and response proabability distribution are calculated by univariate dimension reduction method and maximum entropy iteration loop.(3)For aleatory uncertainty problems involving multimodal distribution variable with performance function has strong interaction terms,a multimodal aleatory uncertainty analysis method based on sparse grid numerical integration is proposed.In the method,Normalized Moment Based Quadrature Rule(NMBQR)is first introduced into sparse grid numerical integration menthod to calculate one-dimensional Gaussian quadrature nodes.Then,the order of GMEM constraints and response probability distribution are calculated by sparse grid numerical integration method and maximum entropy iteration loop,and the multimodal aleatory uncertainty analysis is also completed.