Structural Interval Finite Element Analysis Based On Dimension-reduction Algorithm

Structural response analysis is of great importance in the design and maintenance of structures.However,traditional numerical analysis is based on deterministic system parameters and ignores the uncertainties of materials,geometry and loads that exist in real engineering.These uncertainties lead to uncertainty of structural response and impose new requirements on structural analysis and design.Interval analysis requires only the determination of the bounds of the parameters,not their probability distribution functions and fuzzy membership functions.It has gradually become an important complement to probability methods and fuzzy methods.Based on the dimension-reduction algorithm,this dissertation develops the interval finite element method by combining Neumann series,improved interval algorithm and Epsilon algorithm to solve the problem of uncertain structure analysis.Provision of theoretical support for structural analysis and design in the presence of interval uncertainty.The specific research of the dissertation is as follows:(1)To solve the bounds of the static response of the structure,the interval finite element method based on univariate dimension-reduction is proposed.First,the univariate dimensionreduction algorithm is constructed by generalized Taylor series expansion.The global stiffness matrix,the load vector and the element elasticity matrix of the structure are represented as a linear superposition of the median and the univariate perturbation radius.Modified Neumann series are then used to approximate the inverse of the interval stiffness matrix.Finally,the upper and lower bounds of displacement response and element stresses of the structure are solved using the improved interval algorithm.(2)For structures with large uncertainties,an adaptive subinterval finite element method is proposed to solve the upper and lower bounds of the structural response.First,the interval variables are divided into several subintervals,and then the extreme values of structure response in each subinterval combination are solved using the univariate dimension-reduction-based interval finite element method.The response bounds of the structure in the whole uncertainty region are obtained by the interval union operation.In addition,the adaptive strategy can reasonably divide the subintervals according to the fitness value to improve the efficiency of the algorithm.(3)To solve structural response problems with high-dimensional and large uncertainty parameters,an interval finite element method based on bivariate dimension-reduction algorithm is proposed.First,a bivariate dimension-reduction algorithm is derived to decompose the interval stiffness matrix and the interval load vector of the structure into median,univariate and bivariate terms.Then,to resolve the dependence between the interval parameters,the adjoint sensitivity analysis of the structural response is performed.The adjoint sensitivity method greatly improves the computational efficiency compared to the traditional difference method,since it only needs to solve the inverse of the median of the interval stiffness matrix.Finally,the structural response bounds are solved using Neumann series and an improved interval algorithm.(4)In order to solve the problem that the computational accuracy and efficiency depend on the order of truncation,a bivariate dimension-reduction-based interval finite element method accelerated by the Epsilon algorithm is proposed.By extending the Epsilon algorithm to vector operations and accelerating the displacement response sequence,the accuracy and efficiency of bivariate dimension-reduction-based interval finite element method are effectively improved.An integrated framework for interval finite element analysis is built to programmatically solve the structural uncertainty response through collaboration between the data manager,interval analysis module,and finite element software.