Uncertainty Analysis For Structures With Aleatory And Epistemic Hybrid Uncertainties And Its Application In Topology Optimization

In engineering structures,there are various types of unavoidable uncertainties related to material properties,boundary conditions,loads,etc.Generally,traditional uncertainty analysis methods are based on random theory,and we need to use probability model to describe the parametric uncertainty.However,a large number of experimental samples are required to establish accurate probability distributions of the random uncertain parameters.In engineering problems,due to testing difficulties or cost constraints,it is very difficult to gain complete information to accurately describe the probability distributions,while it is relatively easy to get their interval bounds for limited information.In recent years,with the continuous increase in complexity of practical engineering structures and our high requirements for structural safety,engineers have started encountering a type of more complex and also very important aleatory and epistemic hybrid uncertainty problem,in which there exist both aleatory uncertainty described by random variables and epistemic uncertainty described by interval variables.Hence,it is important to incorporate the hybrid uncertainties in the structure analysis and design.At present,a series of problems are required to be solved in the random-interval hybrid uncertainty analysis.In this work,these problems will be investigated for the hybrid uncertainty analysis and applications for structures.Firstly,a reliability analysis method for structures with random-interval uncertainties considering correlations is proposed.Secondly,two hybrid uncertain propagation methods are developed based on orthogonal polynomials and dimension reduction schemes to achieve both good efficiency and accuracy.Moreover,robust topology optimization methods are developed for structures with hybrid uncertainties.In this case,the research works in this thesis are as follows:(1)The random-interval uncertainty model considering correlations and the structural reliability analysis method are proposed to solve the reliability analysis problem of correlated variables.In order to more conveniently express the correlations of random variables,random-interval variables and interval variables,sample correlation coefficients are introduce to build a unified uncertain model.Then through affine coordinates relevant variables are transformed into independent variables.An equivalent reliability analysis model is put forward in the independent space,and the corresponding failure probability interval are calculated.(2)Hybrid uncertainty propagation analysis methods based on orthogonal polynomials and dimension reduction schemes are proposed.In the orthogonal polynomial based method,the general polynomial chaos are used for the random uncertainties and the Chebyshev inclusion functions are used for the interval uncertainties.A hyperbolic index is employed to reduce the terms in the expansion and save the computational cost as the increasing of the number of the uncertain variables.In the dimensional reduction based method,the univariate dimensional reduction integration is employed for the random analysis,and the dimensional reduction model is used for the interval analysis.In this case,the original problem is transformed into a series of one dimensional problems.The proposed methods can keep good numerical precision while improve the efficiency of the uncertainty analysis.(3)A robust topology optimization(RTO)method based on level sets is developed for structures with hybrid uncertainties.The space parameters with sufficient information are regarded as random fields,while the bounded parameters without sufficient information are treated as intervals.The Karhunen-Loève(KL)expansion is applied to discretize random fields into a finite number of random variables,and the original hybrid uncertainty analysis is transformed into a problem with only random and interval parameters,to which the orthogonal Polynomial based analysis method is employed for the uncertainty analysis.RTO is formulated to minimize a weighted sum of the mean and standard variance of the objective function.The sensitivity information with respect to the design variables can be obtained after the uncertainty analysis,and the gradient based topology optimization method is applied for the analysis.(4)A RTO method is proposed for the concurrent design of cellular composites with an array of identical microstructures considering random interval hybrid uncertainties.Considering random-interval hybrid uncertainties,a robust concurrent topology optimization framework is formulated to optimize both the composite macrostructure and the material microstructure based on a parametric level set method.The robust objective function is defined based on the interval mean and interval variance of the response.The uncertain propagation approach based on the univariate dimension reduction method is employed to estimate the interval mean and standard variance.The sensitivity information of the robust objective function can be obtained after the uncertainty analysis.A gradient based topology optimization method is employed for the optimization analysis.