Sources of numerical non-determinism in general FE modelling

Definitions

In literature, the use of the terminology error, uncertainty and variability is not unambiguous. Different researchers apply the same terminology but the meaning attached to these is rather inconsistent. This necessitates a profound clarification of the terminology for each publication which treats uncertainties. This work does not propose a new terminology, but applies the terminology proposed by Oberkampf [15]. Some additional nuances are, however, necessary in order to enable clear distinction between probabilistic and non-probabilistic quantities in the remainder of this chapter.

The term variability covers the variation which is inherent to the modelled physical system or the environment under consideration. Generally, this is described by a distributed quantity defined over a range of possible values. The exact value is known to be within this range, but it will vary from unit to unit or from time to time. Ideally, objective information on both the range and the likelihood of the quantity within this range is available. Some literature refers to this variability as aleatory uncertainty or irreducible uncertainty, referring to the fact that even when all information on the particular property is available, the quantity cannot be deterministically determined.

An uncertainty is a potential deficiency in any phase or activity of the modelling process that is due to lack of knowledge. The word potential stresses that the deficiency may or may not occur. This definition basically states that uncertainty is caused by incomplete information resulting from either vagueness, nonspecificity or dissonance [16]. Vagueness characterises information which is imprecisely defined, unclear or indistinct. It is typically the result of human opinion on unknown quantities ("the density of this material is around $x$ "). Nonspecificity refers to the availability of a number of different models that describe the same phenomenon. The larger the number of alternatives, the larger the nonspecificity. Dissonance refers to the existence of conflicting evidence of the described phenomenon, for instance when there is evidence that a quantity belongs to disjoint sets. Possibly, limited objective information is available, for instance when a range of possible values is known. In most cases, however, information on uncertainties is subjective and based on some expert opinion. Others in literature refer to this uncertainty as reducible, epistemic or subjective uncertainty.

An error is defined as a recognisable deficiency in any phase of modelling or simulation that is not due to lack of knowledge. The fact that the error is recognisable states that it should be identifiable through examination, and as such is not caused by lack of knowledge. This means that the error could be avoided by an alternative approach which is known to be more accurate, but which is possibly limited in practical applicability by computational cost or other practical considerations. A further distinction between acknowledged and unacknowledged errors is possible. Errors will not be considered further in this paper.

Figure 1 summarises the definitions in this section with their main characteristics in the context of the FE methodology.

Figure 1: Occurrence of variabilities, uncertainties and errors in the FE procedure

Discussion and extension of the definitions

The above definitions of uncertainty and variability are fairly straightforward and comprehensible. However, they are not mutually exclusive, since a variability could be subject to lack of knowledge when information on its range or likelihood within the range is missing. This is for instance the case for every design dimension subject to tolerances, but without further specification of manufacturing process or supplier. The tolerances represent the bounds on the feasible domain, but there is no information on the likelihood of the possible values within these bounds. Consequently, because there is a lack of knowledge, such a variability is also an uncertainty. It is referred to here as an uncertain variability. Some vague knowledge may be available ("the mean value is approximately $x$ ") but also nonspecificity may play an important role in the uncertainty, for instance in choosing an appropriate model to describe a random quantity. Opposed to the uncertain variability, a certain variability refers to a variability the range and likelihood of which are exactly known.

On the other hand, it appears logical to state that every property in a numerical model corresponding to a physical quantity is a variability, since it will eventually have a range of possible values and a likelihood inside this range in the physical model. This argumentation implies that all uncertainties are also variabilities. In practice however, the majority of model properties are implemented as constant deterministic values in the numerical model. Though they are subject to variation, the influence of their variability on the analysis result is considered to be negligible. Often, uncertainties refer to a possible lack of knowledge in these deterministic properties. This type of uncertainty is referred to as invariable uncertainty. Note that invariable in this case does not mean that the property cannot change over different analyses. According to the definition of uncertainty, it will change when additional information is gathered that decreases the amount of uncertainty. The invariable uncertainties typically occur in model properties for model parts that are difficult to describe numerically, but considered constant in the final physical product (connections, damping, ...). Other examples are design properties which have negligible variability but which are not defined exactly in an early design stage. Figure 2 gives a graphical illustration of the proposed subdivision of the definitions for uncertainty and variability.

The group of variabilities may be further subdivided into two categories. Inter-sample variability is the property of a population of nominally identical realisations of a particular product, with each individual element of the population possibly exhibiting scatter. Intra-sample variability is a property of one particular realisation --- of which other realisations possibly exist --- that exhibits one or more properties that may change over time, due to temperature differences, ageing, ...