Interval fields to represent uncertainty on the output side of a static FE analysis

Intervals have been used extensively for the representation of parametric uncertainty in Finite Element Models when the available information is insufficient to build representative probabilistic models. Although the complexity of the interval concept is rather limited, the numerical problem arising at the core of the Interval Finite Element Method was proven to be non-trivial. Many research activities have focussed on this problem over the last decade. Still, one of the principal shortcomings of the current interval finite element procedures is that they are intrinsically not capable of representing dependencies among uncertain input and/or output quantities. This poses important limitations on the applicability of the interval concept in non-deterministic finite element analysis. These limits arise in the model definition phase as well as in the post-processing phase. In order to cope with this problem, this paper introduces the concept of interval fields. The focus in this work is on the static analysis of mechanical structures with interval uncertainties in the context of the Finite Element Method. However, the presented methodology is generic, and can be readily extended to any kind of finite element problem including interval uncertainties. In the first part of the paper, the theoretical background of the concept is presented, and it is shown how explicit and implicit interval fields can be used to represent dependent uncertain parameters in the model definition phase and in the post-processing phase. Furthermore, the applicability and accuracy of both types of interval fields are studied by means of a numerical validation example.