Page 224 - Process Modelling and Simulation With Finite Element Methods

Simulation and Nonlinear Dynamics 21 1

same problem from initially noisy conditions. The most dangerous mode is
expected to be observed asymptotically as long as it is smaller than nonlinear
interactions. If the operator is non-self-adjoint, however, this is not necessarily
the case (see [lo]). So interestingly, eigensystem analysis informs about the
results of simulations, even with stationary solutions. In the case of the Benard
problem, the stationary solution returns the no motion base state, even in the
situation that the eigensystem analysis identifies critical or growing modes. The
second example, viscous fingering, is a paradigm for simulation of evolving
instabilities - the base state is moving and changing with time, and the
instabilities formed have complex nonlinear interactions. The eigensystem
analysis in the uniform outflow Darcy’s Law model did not show instability -
neutral stability was enforced by the choice of outflow boundary conditions.
This uniformity was relaxed in a model based on the streamfunction-vorticity
generation equation and periodic boundary conditions that permitted unstable
growth. In this example, noisy initial conditions were introduced directly in the
simulation by a random number (normally distributed) modulating the
concentration base state. As we claimed in the introduction, by simulation we
normally expect some element of randomness is modelled. This case is the least
controversial use of randomness in a simulation - noisy or uncertain initial
conditions. Thereafter, the simulation is a completely deterministic model. In
general, FEMLAB can be used for simulating more complicated stochastic
processes by alternating random processes and deterministic ones. In this case,
there is exactly one such cycle.
It should be noted that noisy initial conditions may not be necessary in such
simulations simply due to the approximation error in FEM analysis and roundoff
errors in truncation of fixed precision arithmetic. Since the user has control over
error tolerances, stochasticity can be simulated by using unconverged or
unresolved analysis, but this is a dangerous practice as the statistics of the noise
so introduced may be unquantifiable, and the ‘simulation’ may just be numerical
instability. A more controlled simulation with quantifiable levels of noise is
preferable.
As averse to classical linear stability theory, the application of FEM analysis
and subsequent interrogation of the eigensystem analysis of the FEM operator is
not limited to a specific type of basis functions - typically “normal modes.” The
advantage of normal modes is that the transform space that is dual to the physical
space has useful measures as coordinates - wavenumber, for instance, specifies
the lengthscale characterizing the associated eigenmode. With FEM
eigensystem analysis, the growth rates are elucidated for whatever the natural
growing mode(s) turns out to be, but the eigenmode does not have an
unequivocal length scale, for instance. Where the normal modes are
eigenmodes, the FEM methodology usually shows this qualitatively with regard
to the patterns in the eigenmode. Figure 5.18, however, shows that normal