On auto‐ and cross‐interdependence in interval field finite element analysis

Abstract

Summary This paper discusses the concepts of auto‐ and cross‐interdependence in interval field finite element analysis. In classic interval analysis, independent intervals are used to construct hyper‐rectangular input spaces that correspond to the bounded uncertainty that is present on some model parameters. This is a direct result from the inability of modeling interdependence. Such assumption of complete independence might prove in some cases to be highly over‐conservative. A first example is the modeling of spatial uncertainty, where the interdependence is governed by allowable spatial gradients of field realizations. Secondly, interdependence can also occur in case uncertainty in several structural quantities has the same root cause (eg, the manufacturing process). Recent work by the authors introduced concepts for modeling dependence between intervals in a spatial and multivariate context. However, it is unclear how an analyst has to deal with multiple quantities that have a spatial uncertainty component and are furthermore interdependent. This paper presents an approach to link multiple interval fields using recently introduced convex hull pair constructions and inverse distance weighting interpolation. Two case studies to illustrate the new methodology are included, proving the flexibility of the methodology in the modeling of auto‐ and cross‐interdependence between multiple interval scalars and/or interval fields.

Interval methods for lack-of-knowledge uncertainty in crash analysis

“…. , n e by projecting them onto a non-orthogonal vector space [37]. Further, the input space dimension is reduced if n b < n e , which reduces the computational cost of propagating the interval uncertainty towards bounds on the response quantity of interest.…”

Imprecise random field analysis for non-linear concrete damage analysis

“…which affects the modeling of the dependence between interval scalars [37]. Indeed, the dependence between intervals x I ∈ IR n b (z i ) at locations z i is controlled by the non-orthogonality of the base functions.…”

Inhomogeneous interval fields based on scaled inverse distance weighting interpolation

“…Furthermore, they should behave as unit vectors at the control points to ensure that independent intervals are retained (see [25] for a more thorough discussion). An intuitive definition of basis functions that comply with these requirements is provided by means of Inverse Distance Weighting (IDW) interpolation, as applied in [8].…”