Page 79 - Process Modelling and Simulation With Finite Element Methods

66 Process Modelling and Simulation with Finite Element Methods

demarcation between regions that have already received the information, regions
that will receive the information, and possibly regions that will never receive the
information. If the system is linear or quasi-linear (i.e. some coefficient depends
on the dependent variable or a lower order partial derivative than that it
multiplies), this classification system and the intuition about how information is
transported serves as a robust guide to second order systems. For nonlinear
systems, however, nonlinearity can destroy the information transport structure.
In nonlinear systems, information may be “bound”, i.e. never transferred, beyond
given attractors, or it may be created from noise (one view) or lost (a different
view) by forgetting initial conditions in a given window in time.

2.1.1 Poisson’s equation: An elliptic PDE
A modest variant on Laplace’s equation is the Poisson equation:

V2u = f (x)

We saw this equation in 1 -D form in (1.19) which described heat transfer in a
nonuniform medium with a distributed source. Here, the thermal conductivity is
uniform. In order to give a different spin on (2.6), one should note that it is the
equation satisfied by the streamfunction with an imposed vorticity profile:
V2y/ = --u) (x )

There are two common types of vortices that are easy to characterize - the
Rankine vortex, where vorticity is constant within a region, and the point-source
vortex, where vorticity falls off rapidly and thus is idealized as point vortex.
One might be curious about the streamlines generated by these two vortex types.

Start up FEMLAB and enter the Model Navigator:

Model Navigator
Select 2-D dimension
Select Classical PDEs + Poisson’s Equation
Element: Lagrange - quadratic
More>>

This application mode gives us one dependent variable u, but in a 1-D space
with coordinate x. Now we are in a position to set up our domain. Pull down the
Draw menu and select Circle/Ellipse.