Keywords: non-deterministic analysis, epistemic uncertainty, aleatory uncertainty, interval number, fuzzy number, design criteria, finite element analysis.

Research into non-probabilistic finite elements started in the mid nineteen nineties. The basic idea of extending the conventional steps of the deterministic finite element to their interval or fuzzy equivalent was soon abandoned as repetitive operations on dependent numbers could not take into account dependency between these numbers. As a result, conservatism on the interval or fuzzy finite element result was unacceptably high, making these results useless. The conventional finite element procedure of element matrix assembly, system matrix assembly and matrix inversion is only valid with a fully deterministic data set.

The finite element (FE) problem with interval or fuzzy data can only be solved after some kind of conversion to a deterministic problem or to a set of deterministic problems. The transformation method that has been developed at the Universität Stuttgart is generally applicable to all kinds of numerical processes, including FE analysis. A significant disadvantage is that the number of deterministic FE runs increases exponentially with the number of uncertain parameters. Another general concept is optimisation, which is directly applicable if a single output quantity is required. Response surface models allow for a considerable reduction of the total analysis time. Both the transformation method and the optimisation concept have been used successfully for a wide range of applications, in linear and non-linear static analysis, and in linear dynamic analysis.

The paper will give an overview of the achievements that have been realised so far. Examples will be presented, featuring different analysis types and different application areas. Attention will also be paid to the performance of the analysis procedure, including accuracy and computation times.

The paper will conclude with an outlook into the challenges that are still ahead. Data representation and visualisation will have to be improved to allow for a wider utilisation of interval and fuzzy FE, and an appropriate concept will be required to describe input and output parameter fields that take into account the relation between parameter values at different spatial positions.

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