Page 186 - Process Modelling and Simulation With Finite Element Methods

Chapter 5

SIMULATION AND NONLINEAR DYNAMICS

W.B.J. ZIMMERMAN
Department of Chemical and Process Engineering, University of Shefield,
Newcastle Street, Sheffield SI 3JD United Kingdom

E-mail: w.zimmerman @she$ac.uk

Eigensystem analysis of the linearized operator derived by FEM analysis (the stiffness
matrix) is a powerful tool for characterizing the local stability of transient evolution of
nonlinear dynamical systems governed by pdes and for parametric stability of stationary,
nonlinear problems. Here we discuss how to perform such an analysis in the context of
two complex systems - Benard convection and viscous fingering instabilities. The later
are simulated from “white noise” initial conditions added to a base flow. The linear
stability theory in both cases assumes that the noisy initial conditions include all
frequencies, and thus whichever eigenvalue has the largest real part corresponds to the
eigenmode that grows most rapidly. FEM eigenanalysis is shown to reproduce the
predictions of linear stability theory with good agreement, but is more general in regimes
of applicability.

5.1 Introduction

Modelling versus Simulation
So far, we have been concerned with the use of FEM for computational
modeling. The model could be expressed as a well posed mathematical system,
typically PDEs with boundary and initial conditions, possibly algebraic
constraints. Such systems are theoretically deterministic, i.e. the state of the
system can be known up to any arbitrary accuracy at any given time. By
simulation, something different is usually understood - the physics of the system
includes some element of randomness in its temporal development. So we don’t
expect a simulation to be perfectly accurate in all details. Simulations are
expected to mimic the microscopic behaviour of complex systems, typically by
posing interaction rules for subsystems from which the global, coordinated
behaviour of the whole system emerges. Where the low level interaction rules of
the system are particularly poorly physically based, the simulation predictions
about global emergent properties must be validated by experiment, perhaps even
semi-empirically fitted.

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