Page 246 - The Combined Finite-Discrete Element Method

COMBINED FINITE-DISCRETE ELEMENT SYSTEMS 229

Figure 6.21 Sensitivity to initial conditions in experimental investigations; results of three iden-
tical experiments (courtesy of J.P. Latham).

Sensitivity to initial conditions that is observable in the combined finite-discrete element
simulations is not purely an academic exercise. It is a physical property of the combined
finite-discrete element system, in the same way that energy dissipation is a property of such
a system. It has practical implications and needs to be taken seriously. For instance, it can
be demonstrated that some ‘results’, such as the position of an individual discrete element
in a big pile of particles, is probably due to chance, i.e. is probably a random variable. A
result such as the overall density of the pile at state of rest has no probability associated
with it, and is a completely deterministic variable. In short, results of the combined finite-
discrete element simulation can be grouped into two categories: probabilistic variables
and deterministic variables.
The combined finite-discrete element simulation is based on rational mechanics, and is
deterministic in its formulation without inclusion of any theory of probability. However,
due to the sensitivity to initial conditions, all results are not deterministic, and some of
the results are clearly random variables.
This has important consequences in both verification and validation of combined finite-
discrete element models. Verification is in essence a process of checking that the model is
implemented properly. One of standard verification tests is to run a problem that contains
symmetry. The results should also show the same symmetry. In the combined finite-
discrete element method, some loss of symmetry may not indicate that the implementation
was wrong, as clearly demonstrated by the 2D example shown in Figure 6.7.
Validation is a process in which the underlying assumptions of any model are tested,
usually by comparison to either theoretical or experimental results. In the light of the
sensitivity to initial conditions, as explained above, it is important to carefully identify
variables to be measured and compared. Probabilistic results may not be of much use
in the validation process, and an effort should be made to ensure that the results used
for comparison are deterministic results (i.e. results not sensitive to initial conditions).
Using, for instance, the positron tracking motion of a single particle may not yield much
in terms of results, except confirming that such a result belongs to a group of probabilistic
results. Using X-ray imaging to record velocity profiles over the whole domain would
probably exhibit no sensitivity to initial conditions, i.e. obtained velocity maps would be
very similar, regardless of small perturbations in initial conditions.