Non Deterministic Approaches

The reliability of a product is defined as the likelihood that it will successfully fulfil its intended task over a predefined period in time under specific environmental conditions. Numerical reliability analysis based on probabilistic analysis is very popular because when realistic data is used, it can give a clear indication of the likelihood of failure of the analysed structure. As such, it can be usefully applied in an economical product analysis taking into account the cost associated with failure. This probabilistic reliability analysis is broadly applied and already incorporated in generally accepted design specifications in civil engineering. However, its application in mechanical engineering is far less standardised. This is mainly due to the plenitude of different mechanical products, which all require a different amount of reliability. Hence, there are very few standards for reliability in mechanical design. Each product designer applies rules which are based on experience rather than on general engineering standards.

According to its definition, reliability belongs clearly to the probabilistic framework in the frequentist context. On the one hand, this complicates probabilistic analysis of designs intended for limited production, since the fact that the product is only produced in limited quantity strongly complicates a decent a posteriori verification of the non-deterministic numerical predictions. Furthermore, for most designs intended for limited production, an unverifiably high reliability is requested (e.g. spacecraft). But an even bigger problem lies in probabilistic reliability analysis in the absence of trustworthy objective information. As discussed in section "Numerical concepts for non-deterministic numerical modelling", while applying the probabilistic concept for the representation of subjective information is possible, results from such an analysis should definitely not be interpreted as indication for an absolute frequency of occurrence. The subjectiveness devaluates the use of the probabilistic results in a reliability context. This subjectiveness of (parts of) the information is not always detected. For instance, neglecting unknown correlation between properties by assuming them as independent is a common simplification that is sometimes implicitly made, but that can have important consequences. This implicit assumption of independence between probabilistic quantities was one of the important errors that were the source of the Challenger space shuttle disaster [44]. In this case, the impact of different extreme weather conditions on the launch was analysed for each condition individually beforehand. The impact of a combination of more than one of these events, however, was never checked. Although each of the events had a very low probability of occurrence, the probability of their combination proved to be not simply a multiplication of the probability of the single events. The correlation between the conditions was clearly misjudged, leading to a plausible but unaccounted for weather situation with disastrous consequences.

The lack of credibility of numerical predictions of reliability is generally compensated by safety factors. However, one could argue that using these safety factors after applying sophisticated and computationally expensive numerical procedures is not a really economical situation. Much effort is spent on a numerical prediction, which, in the end, still has to be corrected based on practical experience. In this context, the non-probabilistic approaches could prove their value.

The application of the interval concept in numerical reliability studies is often referred to as anti-optimisation. This name stems from the fact that from all numerical models within the interval input boundaries, the one with the least favourable analysis result is the most interesting from reliability point of view. Finding this least favourable result is mathematically equivalent to performing a numerical optimisation aimed at the worst case result with respect to the input intervals.

The concept of anti-optimisation has been introduced as the basis for a non-probabilistic reliability framework [45]. This requires an evolution from a reliability concept as probability of failure towards range of acceptable behaviour. This means that the design must assure that the performance remains within an acceptable domain, without specifying a likelihood of failure. Reliability then becomes a crisp criterion distinguishing between either acceptable or unacceptable designs. The most important benefit of the anti-optimisation concept is that it broadens the objectivity of reliability studies to uncertain variabilities with known range, because the interval model perfectly represents these uncertainties without the need for subjective input. For instance, this enables a fast assessment of dimension tolerances on a design, without knowing the actual distribution of the dimension within the bounds of the prescribed tolerance. For some cases, it can be shown that the anti-optimisation procedure results in the same choice of design parameters as a probabilistic analysis if the required reliability tends to one [46]. The anti-optimisation in this case proves to be far less expensive in computation time.

The numerical implementation of the anti-optimisation approach is subject to an important requirement. Since the result of the analysis is the source of a crisp decision between acceptable and unacceptable designs, approximate results should always be kept on the safe side of the exact result. This means that if approximate solution procedures are used in the numerical implementation, they should guarantee conservatism in their result. On the other hand, this conservatism should not be excessively high in order for the result to be of any practical value.

Also the fuzzy concept can be usefully applied in a reliability framework to perform a possibilistic reliability analysis. In the interpretation of the membership function as a degree of possibility, the fuzzy outcome of an analysis could be used to define a possibility of failure. This possibility is clearly influenced by the subjectiveness that is implicitly incorporated in the fuzzy input of the analysis. This means that for the same problem, different analysts can and generally will end up with different possibilities of failure. This could be compensated by defining a personal threshold value for the allowed possibility of failure in the final decision on acceptable or unacceptable designs. However, due to the necessary amount of personal interpretation of the analyst, possibility of failure only has a relative value. Therefore, this approach is extremely difficult to standardise in a general reliability framework.

A different application of the fuzzy concept in reliability analysis is based on the use of the membership function as limit CDFs as explained in section "Numerical concepts for non-deterministic numerical modelling". It was shown by Ferrari et al. [47] that if the input membership functions represent boundaries on the CDFs of the input parameters, the membership function resulting from fuzzy analysis on this input forms reliable boundaries on the actual CDF* of the result. Therefore, the fuzzy result of a FFE analysis can be used to derive bounds on the probability of failure. A simple example illustrates this. Suppose that a FFE analysis results in a membership function $\mu_{bar \lambda}(\lambda)$ representing a crucial eigenfrequency of a design as illustrated in figure 6. Suppose furthermore that a crisp criterion states that the design is acceptable if this eigenfrequency is kept below the value $\lambda^{**}$. The fuzzy result envelopes the exact CDF* of the eigenfrequency. This means that the bounds on the probability that the eigenfrequency of the design lies below $\lambda^{**}$ can be derived from the fuzzy result. The probability interval is obtained from taking the value of the envelope curves at $\lambda^{**}$ as indicated in the figure by $ul P'_f$ and $bar P'_f$. The most conservative statement resulting from the analysis is that the probability of failure equals $(1 - ul P'_f)$ in the worst case.

Figure 6: Example of the application of the fuzzy outcome of a FFE analysis to predict bounds on the probability of failure

It is clear that also the above non-probabilistic reliability methods are subject to the limitation that whenever there is subjective information involved in the problem definition, the results can not be interpreted as absolute measures of design quality. In an absolute reliability context, the amount of expert knowledge required in the distinction between a good or bad design is proportional to the amount of subjectiveness incorporated in the description of the non-determinism. Still, subjective analysis can be of great value when used in a relative framework, as for instance a design optimisation procedure.