Stochastic Finite Element Method For Reliability Analysis Of Complex Mechanical Structures

Spatial variabilities are crucial factors for safety analysis and design of engineering structures.Therefore,the finite element method in conjunction with the stochastic methods is a significant research topic in structural reliability engineering.The thesis starts from numerical methods for spatial variabilities discretization and simulation,and presents an efficient framework for structural reliability analysis based on the stochastic finite element methods.The research primary includes:(1)Meshless and high-order Galerkin based numerical methods for discretization and simulation of random fieldsThe utility of Karhunen-Loeve expansion allows to represent the random field as the summation of a series of deterministic functions.By this way,statistical approximation of the spatially varying random property can berealized by means of a finite number of random variables.However,the K-L expansion depends crucially upon analytical eigen-solutions of the Fredholm integral equation of the second kind.Especially with an emerging requirement for a multivariate random field simulation along with complex geometries,the exact K-L expansion is seldom applicable for realistic engineering problems.To this end,a polynomial element based Galerkin approach is introduced to deal with the integral eigenvalue problem(IEVP)for complex geometries.To implement,various types of high-order elements are derived based on orthogonal polynomials in numerical analysis.Besides,to deal with highdimensional integrals produced by multivariate Galerkin projections,the numerical Gausstype interpolation schemes are introduced to approximate the true but computationally demanding high-dimensional integrals.To access numerical accuracy of the polynomial element based Galerkin approach,the convergence rate is assessed by means of global variance and covariance errors.Several examples have demonstrated high accuracy and efficiency of the proposed Galerkin approach(2)Uncertainty analysis of structural systems by means of the stochastic finite element approachTo introduce spatial random field of structural material properties and load into the finite element model of structure mechanics analysis,various random factors can be remained in the structural response analysis.By combining with the box-girder structure widely applied to bridge engineering and a turbine disc structure,the paper provides the numerical method of stochastic finite element analysis for complex engineering structures,which provide a more accurate analytical model for engineering applications.The results fully exhibit the advantages of stochastic finite element method(SFEM)in a variety of structural strength analysis.(3)An adaptive sparse regression method for structural reliability analysis under the spatial variabilityThe reliability analysis of a structural system is typically evaluated based on a multivariate model that describes underlying mechanistic relationship between the system's input and output random variables.In this regard,the thesis presents a sparse regression method for structural reliability analysis based on the generalized polynomial chaos(gPC)expansion.However,results from the global sensitivity analysis have justified that it is unnecessary to contain all polynomial terms in the gPC surrogate model,instead of comprising a rather small number of principle components only.Therefore,by utilizing the standard gPC basis functions to constitute an explanatory dictionary,an adaptive sparse regression approach characterized by introducing the most significant explanatory variable in each iteration is presented.A statistical approach for detecting and excluding spuriously explanatory polynomials is also introduced to maintain the high sparsity of the non-intrusive meta-modelling result.Combined with a variety of low-discrepancy schemes in generating training samples,structural reliability and global sensitivity analysis of originally true but computationally demanding models are alternatively realized based on the sparse gPC surrogate method in conjunction with the brutal MCS method.