Page 187 - Process Modelling and Simulation With Finite Element Methods

174 Process Modelling and Simulation with Finite Element Methods

Equivalence ?
With the above classification scheme, computational modeling and simulations
would appear to be wholly distinct - models are deterministic and physically
based; simulations are stochastic and semi-empirically based. This dichotomy
blurs, however, with modem understanding of complex systems. SA Billings
and coworkers [l], for instance, have developed a data analysis technique for
patterns in spatio-temporal systems that can identify the best PDE system within
a candidate class that captures the nonlinear dynamics in experimental systems.
The technique finds a rule based description for cellular automata that is
consistent with the complex system pattern development. By limiting the types
of PDE terms available to the model, an inverse mapping from interaction rules
to PDE description can be elucidated. So, the common usage of muddling the
terms ‘modelling’ and ‘simulation’ is justified by this functional equivalence.
No doubt this is the “new kind of science” that Wolfram [2] is espousing;
dynamics can be equated to simulation schemes (new science) which are
equivalent to (nonlinear) pde systems derivable from physical laws (old science).
Where the new science wins is that the applicable physical laws may be two
complicated to describe in full a priori, but those that are being expressed in the
complex system may be easily identifiable by finding the interaction rules that
are consistent with the global emergent behaviour. Koza and coworkers at
Stanford [3] have long been proponents of the view that the trick is to find the
computer program that meets the physical requirements. Genetic programming
is an approach to letting the program consistent with the observations to
assemble itself.

Bridging the Gap: Nonlinear Dynamics
The gap between modeling deterministic systems and simulating stochastic ones
is bridged by the nature of nonlinear dynamics and complex systems. The
principle feature of a class of nonlinear dynamics - chaotic systems - is that of
extreme sensitivity to initial conditions. States that are not particularly far
initially in some sense become very far apart eventually. Before chaos theory
became better understood in the late 1970s, it was conventional wisdom that in
dissipative systems, equilibrium states or periodic oscillations would be the time
asymptotic attractors for all initial states of the system. When a dynamical
system is extremely sensitive to initial noise or uncertainty, such a system is
termed complex. The paradigm in fluid dynamics is turbulence. Shear
instability of flows “at high Reynolds number” lead to complexity of the motion
at millimeter scales all the way up to thousands of kilometers in the atmosphere,
for instance. Even though the temporal state of the system is theoretically
deterministic, our uncertainty in the initial state is such that the system is
indistinguishable from a stochastic one for practical purposes.