Page 190 - Process Modelling and Simulation With Finite Element Methods

Simulation and Nonlinear Dynamics 177

The error norm of (5.5) is just one of many weighted errors that can be
defined, e.g.

also defines a measure for any set of weights wp0. The choice of all weights
equal in (5.5) only makes sense for a convergence criteria if all degrees of
freedom are expected to range over the same scale. One of the rationales for
dimensional analysis of physical models is to condition all degrees of freedom to
range over a unit scale. In any unconditioned model, the range of scales
expected a priori for different degrees of freedom would not be expected to be
identical. FEMLAB 2.3 introduced the “automatic scaling of variables” feature,
that estimates the appropriate weights wi automatically or permits user pre-
defined scales. The release notes point out that in a structural mechanics
application, displacements might be submillimeter, yet stresses could be
megapascals. Without scaling of variables, numerically small quantities would
have degrees of freedom contributing little to convergence criteria, and
numerically large quantities would be unduly restricted by convergence criteria
using formula (5.5).
In summary, excepting the case of chaotic states, linear theory can identify
whether a stationary or periodic state is unstable. Regardless, it also identifies
the mode(s) that are asymptotically attractive for the perturbation. For instance,
if an eigenvalue is complex, then the frequency of oscillation of the perturbation
can be predicted, along with the decay or growth rate. Furthermore, the
eigenvector associated with the eigenvalue with greatest real part should be the
pattern of degrees of freedom that a disturbance evolves into. Using FEM
models, representing these eigenmodes is straightforward. They are elements of
the space of all possible solutions, so any postprocessing that can be done on a
solution can be done to an eigenmode as well. FEMLAB, for instance, can be
“tricked” into displaying and analyzing eigenmodes as though they were
solutions.

Chapter Organization
This chapter can only be a survey of the range of models that can be used in
simulations. The theme of the chapter is to illustrate how features of the
MATLABEEMLAB computational engines can be used for simulation. A
strong undercurrent, however, is awareness of how nonlinear dynamics is
important in computational modeling. Our first case study, Rayleigh-Benard
convection, is simply a stationary nonlinear system for which convergence is
difficult to achieve because of the inseparability of parasitic time dependent