Page 284 - Process Modelling and Simulation With Finite Element Methods

Coupling Variables Revisited 27 1

Extrusion Coupling Variables

An extrusion coupling variable was named after one of its most common uses; it
maps information from a domain of dimension n to one of higher dimension
n+l. Yet extrusion is only one of its potential uses, which are generalized as
interpolation, projection, or mapping, depending on the information passed. The
other two coupling variable types - scalar and projection - perform integrations
over their source domains (or subdomains) and are thus able to be incorporated
in integral equations. Extrusion coupling variables map detailed or distributed
information from one domain to another, with the destination position selected
by the local mesh transformation. So extrusion variables are useful
intermediaries in models with multi-domain coupling. Yet they need not be
defined on domains of different geometries. In FEMLAB seminars, the common
example given of extrusion coupling variables is for aesthetic reasons.
Frequently, given the symmetry in a physical configuration, the model can be
solved over only part of the domain or even a lower dimension, yet the real
physical configuration is required to visualize the solution. So, for instance, in a
cylindrical duct, axi-symmetry may only require solution in the r-z plane, yet
visualization on the cylinder may be desirable. Extrusion over the 0-coordinate
of the r-z solution will permit the desired visualization. Suppose placement of
baffles with hexagonal symmetry in the domain permitted solution over a wedge
of 0 E [O,n/3] with r and z bounded. Yet, if visualization is required over the
whole duct, extrusion of the wedge to the other fiver wedges would permit this.
So extrusion coupling variables may merely extend information for
postprocessing into other domains.

Integral Equations
Integral equations are distinguished by containing an unknown function within
an integral. As with differential equations, linear systems are the best
characterized and therefore most commonly occurring. The classification system
is straightforward.
If the integration limits are fixed, the equation is termed of Fredholm type.
If one limit is a variable, it is termed of Volterra type.
If the unknown function appears only under the integral sign, it is labeled as
of the “first kind.” If it appears both inside and outside the integral, it is labeled
of the “second kind.”
Here are the four combinations symbolically:
Fredholm integral equations of the first kind:

(7.16)