An interval finite element method based on the Neumann series expansion

Uncertainty is common in the practical engineering. The interval finite element method is an interval method which introduces the numerical computational method of finite element to structural uncertainty analysis. The aim of the interval finite element analysis is to obtain the upper and lower response bounds of the structure with interval uncertain parameters, where solving the interval finite element equilibrium equations is the key issue. But the solution of interval linear equations belongs to a class of NP-hard problems which are often difficult to solve. This paper classifies and defines a type of linearly decomposable interval finite element problems, which exist commonly in practical engineering. To solve this type of problems, an interval finite element method based on Neumann series is proposed. It is named as the linearly decomposable interval finite element problem if the stiffness matrix in the interval finite element analysis formulation can be expressed as a linear superposition of a set of independent interval variables when the interval uncertain parameter is expressed as a linear superposition form of the independent interval variables. For this kind of problems, the inverse of the stiffness matrix can be represented by its Neumann series expansion. Thus the explicit expressions of structural responses with interval variables can be then obtained, with which the upper and lower bounds of the structural response can be solved efficiently. Finally, two numerical examples show the effectiveness and accuracy of the proposed method. © 2020, Chinese Journal of Theoretical and Applied Mechanics Press. All right reserved.