Mechanical Engineering in Uncertainties From Classical Approaches to Some Recent Developments. Группа авторов
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Название: Mechanical Engineering in Uncertainties From Classical Approaches to Some Recent Developments

Автор: Группа авторов

Издательство: John Wiley & Sons Limited

Жанр: Физика

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isbn: 9781119817611

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СКАЧАТЬ the area under the distribution that lies within the failure zone and thus decreases the probability of failure. This example thus illustrates a situation where increasing epistemic uncertainty (that is, increasing the lack of knowledge) decreases the probability of failure. This is somewhat counterintuitive, as one would normally expect that high epistemic uncertainty (that is, a significant lack of knowledge) would lead to unreliable products (namely, having a high probability of failure) and that products can be made more reliable by increasing knowledge (namely, decreasing epistemic uncertainty).

]. In the design phase, the engineer can typically look for ways to modify the structure in order to reduce the probability of failure. Suppose the engineer makes a modification for this purpose (for example, a change in dimensions, in materials), which results in moving the failure area somewhere in the interval [ã, a]. If we consider the real probability distribution over the excitation frequencies f (uniform in [a,b]), this change in the design has the consequence of drastically reducing the probability of failure, since the failure area is outside the interval [a,b]. However, since the real distribution is not known and a wider distribution (uniform in [ã,
]) has been considered to be conservative, the consequence of the change in design is totally imperceptible in this case. The probability of failure remains exactly the same as before the change in design, since the region of failure is still located within the bounds [ã,
]. This third example illustrates situations where the inappropriate modeling of epistemic uncertainties by probability distributions can prevent the proper prediction of tendencies in the variation of the probability of failure due to changes in design.

      We have seen here, through three examples, some of the limitations of an entirely probabilistic modeling of some uncertainties of epistemic nature, especially in the presence of a very small amount of data, which does not allow a specific distribution to be associated with random variables. These limitations have motivated the development of additional approaches, which we shall review in the following sections. A first such approach simply consists of an extension of the probability theory: this is the probability box approach.

and the right bound is a distribution function denoted F. This is illustrated in Figure 1.5.

      Any distribution function comprised between

and F can thus represent the probability distribution of X. The spacing between
and F represents the magnitude of the epistemic uncertainty associated with the lack of knowledge of the distribution of X. If
and F are superimposed, there is no epistemic uncertainty and the aleatory uncertainty is represented by the distribution function FX =
= F.

      A probability box can be interpreted in two different ways: as bounds on the cumulative probability for a given value of x or as bounds on the values of x for a given confidence level. In the example in Figure 1.5, for example, one can read that the probability that x is less than 3 is comprised between 0.05 and 0.3. We can also read that the 95% quantile of X is between x = 8 and 17.

      From a formal point of view, a probability box can be defined, adopting the formalism of Ferson et al. (2015), as follows:

      DEFINITION 1.15.– Let and F be non-decreasing functions from ℝ to [0,1] satisfying the condition

] denote the set of all non-decreasing functions F from ℝ to [0,1] satisfying F (x)≤ F(x)≤
(x),∀ x ∈ ℝ. [F,
] is then called a “probability box”.

      Note that this approach can be seen as an extension of inaccurate probabilities (Walley 1991) to distribution functions and, consequently, to probability distributions. There are different variants existing derived from this concept of a probability box. For example, in addition to bounding the distribution function F with

and F one can impose additional constraints such as:

       – the mean of the probability distributions associated with F must lie within a given interval;

       – the variance of the probability distributions associated with F must lie within a given interval;

       – the probability laws associated with F must belong to a certain class of distributions.

      There are also different types of probability boxes:

       – Distributional probability boxes: the probability distribution is assumed to be known and it is only the parameters of the distribution that are not accurately known, thus generating different possible distribution functions (but all of the same type of distribution). Such an approach will be illustrated in Chapter 7 СКАЧАТЬ