An interval finite element method for the analysis of structures with spatially varying uncertainties

Abstract

Finite element analysis of linear-elastic structures with spatially varying uncertain properties is addressed within the framework of the interval model of uncertainty. Resorting to a recently proposed interval field model, the uncertain properties are expressed as the superposition of deterministic basis functions weighted by particular unitary intervals. An Interval Finite Element Method (IFEM) incorporating the interval field representation of uncertainties is formulated by applying an interval extension in conjunction with the standard energy approach. Uncertainty propagation analysis is performed by adopting a response surface approach which provides approximate explicit expressions of response bounds requiring only a few deterministic analyses. Then, the whole procedure is implemented in ABAQUS' environment by coding User Subroutines and Python scripts. 2D plane stress and bending problems involving uncertain Young's modulus of the material are analyzed. The accuracy of the proposed IFEM as well as response variability under spatially dependent uncertainty are investigated.

Robust uncertainty quantification in structural dynamics under scarse experimental modal data: A Bayesian-interval approach

“…More specifically, the interval finite element method has been well established [1,2,3,4] and implemented to determine intervals on output quantities of Finite Element (FE) models subject to interval uncertainties. Also, recent advances in surrogate modelling [5,6], (anti-)optimization [7,8,9] and interval field theory [10,11,12,13,14] further extends the applicability of interval analysis in numerical dynamical analysis. A next step towards practical application of this method is the capturing of uncertain input data in a suitable format by defining and possibly estimating intervals from available data [15,16,17].…”

Operator norm theory as an efficient tool to propagate hybrid uncertainties and calculate imprecise probabilities

“…As a drawback, intervals provide only the worst and best case structural response to the analyst [7,8]. Furthermore, the modelling of dependencies between uncertain parameters requires dedicated methods based on the projection of θ I to a non-orthogonal basis [9], the admissible set decomposition method [10] or using affine arithmetic [11]. Alternatively, also convex set approaches can be applied to represent Type-II uncertain properties [12].…”

Inhomogeneous interval fields based on scaled inverse distance weighting interpolation

“…Following this framework of interval fields, locally defined intervals are expanded through the model domain based on a set of a priori defined basis functions. Multiple definitions of basis functions can be found in literature, which are based on inverse distance weighting [8], affine arithmetic [9,10,11], radial basis functions [12], a spatial averaging method [13],…”