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The objective of this paper is to critically review the emerging non-probabilistic approaches for uncertainty treatment in finite element analysis. The paper discusses general theoretical and practical aspects of both the interval and fuzzy finite element analysis. First, the applicability of the non-probabilistic concepts for numerical uncertainty analysis is discussed from a theoretical viewpoint. The necessary conditions for a useful application of the non-probabilistic concepts are determined, and are proven to be complementary rather than competitive to the classical probabilistic approach. The second part of the paper focuses on numerical aspects of the interval finite element method. It describes two principal strategies for the implementation, i.e. the anti-optimisation and the interval arithmetic approach, and gives a state-of-the-art of the interval finite element algorithms available from literature. It is shown how the application of the interval arithmetic approach to the classical finite element procedure can result in a severe overestimation of the uncertainty on the output, and the main sources of this conservatism are identified. A numerical example in the final part of the paper illustrates the capabilities of the different strategies on an eigenfrequency analysis of a built-up benchmark structure.

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This paper gives an overview of recent advances in the field of nonprobabilistic uncertainty quantification. Both techniques for the forward propagation and inverse quantification of interval and fuzzy uncertainty are discussed. Also the modeling of spatial uncertainty in an interval and fuzzy context is discussed. An in depth discussion of a recently introduced method for the inverse quantification of spatial interval uncertainty is provided and its performance is illustrated using a case studies taken from literature. It is shown that the method enables an accurate quantification of spatial uncertainty under very low data availability and with a very limited amount of assumptions on the underlying uncertainty. Finally, also a conceptual comparison with the class of Bayesian methods for uncertainty quantification is provided.