Page 236 - The Combined Finite-Discrete Element Method

6

Sensitivity to Initial

Conditions in Combined

Finite-Discrete Element

Simulations

6.1 INTRODUCTION

In the 18th century Pierre Simon de Laplace, in his Philosophical Essayians on Probabil-
ities, concluded that for a vast enough intellect, given initial conditions, the future would
be just like the past, and that nothing would be uncertain. In the context of the combined
finite-discrete element method, this could be interpreted that for a large enough com-
puter, given initial conditions, the motion of each individual particle could be predicted
regardless of the size or nature of the physical problem.
However, in systems such as gas, statistical methods were introduced in 1873 by
J.C. Maxwell; actually scientists were intuitively aware that a deterministic system can
behave in a random way, with randomness occurring in systems with a large number of
particles or degrees of freedom. In 1887, H. Poincare discovered that very complicated
dynamic behaviour can occur even in a simple system such as an idealised three-body
problem. In 1961, Edward Lorenz, while running computer models for weather forecasts,
was the first to notice the heavy dependence of the outputs of the weather model on the
initial conditions supplied. Rounding errors due to the computer program being inter-
rupted and restarted resulted in a completely different weather prediction. He called this
phenomena the ‘butterfly effect’.
The common characteristic of all systems mentioned above is that they never find a
steady state, i.e. they never repeat themselves. They are sensitive to initial conditions, but
they are not periodic. A system that is not periodic and is sensitive to initial conditions
is unpredictable.
An irregular unpredictable behaviour that results from nonlinearity in a dynamic system
is called ‘deterministic chaos’, because the model employed is deterministic, but the results
are not. It can occur in a system with one degree of freedom, as well as in a system with
any number of degrees of freedom.

The Combined Finite-Discrete Element Method A. Munjiza
 2004 John Wiley & Sons, Ltd ISBN: 0-470-84199-0