Non Deterministic Approaches

Engineering design is the activity of design and development of technical products. A technical product is built to fulfil a well specified function under more or less well prescribed conditions of utilisation. The complexity of modern technical products tends to increase systematically, increasing the need for thorough design analysis. This process consists of a number of analysis verifications on a virtual product. A common procedure for design verification is finite element analysis, a numerical method for the simulation of the effect of mechanical or thermal loads on a product. As most product parameters are undetermined in the initial phases of design, a range of non-deterministic properties have to be taken into account. This paper discusses the effects of non-determinism on engineering analysis using the finite element method (FE).

Over the past decade, the probabilistic FE analysis has gained a large popularity in the area of engineering analysis with non-deterministic parameters. Recently, a number of non-probabilistic approaches for non-deterministic analysis are emerging. The Interval FE (IFE) analysis is based on the interval concept for the description of non-deterministic model properties, and so far has been studied only on an academic level [1-3]. The Fuzzy FE (FFE) analysis is basically an extension of the IFE analysis, and has been studied in a number of specific research domains: static structural analysis [4,5], dynamic analysis [6-8], geotechnical engineering [9,10], multi-body kinematics [11], steady-state analysis of rolling [12], analysis of smart structures [13] and analysis of fibre-reinforced composite materials [14]. The numerical procedures developed for the non-probabilistic approaches are all strongly influenced by the specific properties of the analysed physical phenomenon, and only academic examples with very limited size and complexity are considered.

The non-probabilistic approaches broaden the possibilities, but simultaneously they complicate the choice for the analyst. Therefore, it is important to study to what extent they can be an alternative in the areas where probabilistic analysis has become the standard. The growing interest for non-probabilistic methods for non-deterministic numerical analysis mainly originates from criticism on the credibility of probabilistic analysis when it is based on limited information. Especially when extremely high reliabilities are analysed based on numerical models, design engineers often remain very sceptic regarding the trustworthiness of the numerical predictions. The recent development of the non-probabilistic approaches stems from the argumentation that this lack of credibility is always present in probabilistic analysis results, but generally remains unaccounted for. It is argued that the non-probabilistic concepts could be more appropriate to model certain types of non-deterministic information, resulting in a better representation of the simulated non-deterministic physical behaviour. Also, it is believed that a full probabilistic description of a non-deterministic event is not always required. Especially in early design stages, when objective probabilistic information often is not available, non-probabilistic concepts are considered to be of great value. It is the aim of this paper to critically review this argumentation, and to study to what extent the non-probabilistic methods can be considered as useful alternatives to the existing probabilistic approach.