Page 286 - Process Modelling and Simulation With Finite Element Methods

Coupling Variables Revisited 213
explore some aspects of FEMLAB’s ability to compute solutions to integral and
integro-differential equations.

Solving a Fredholm Integral Equation of the Second Kind
Zimmerman [23] gives the derivation of a Fredholm integral equation of the
second kind as an intermediate in the solution for the drag on a thin disk in
broadside motion in a cylindrical duct. The variation on (7.17) is slight:
1
g (n) = 1 + Ej K (n, t)g (t)dt (7.20)
0
where &<I is a small parameter. The kernel K was bounded, so a theorem in
integral equation theory [21] ensures that a solution for g(x) can be found by
iteration, with each iterate improving in accuracy by at least one order of
correction in E. Zimmerman [23] demonstrated a solution by series expansion in
powers of x and E, albeit relying on numerical computation of the series
coefficients. As the kernel of that problem is not particularly tractable (it too
was expanded in powers of x and t), a simpler kernel will be selected here for
demonstration. The FEMLAB implementation is a tour de force in projection
and extrusion coupling variables.
As alluded to in section 7.4, projection variables are the variable of choice
for a line integration that returns a function. Although we wish to achieve a line
integration of the form

(7.21)
0
it is easier to achieve

where g2(x1,x2)=gl(x), with the mapping (x2 t x) and extruded along the x2
coordinate. Figure 7.10 shows this graphically for the kernel K(x,t)=sin(2nx t).
The left figure is extruded along the horizontal coordinate after being mapped to
the vertical. Alternatively, think of the left figure as the projection of the right 2-
D domain onto its vertical axis. The computation (7.22) has the intermediate
swelling by one dimension of the domain in order to preserve the functionality of
the line integral though the projection coupling variable. If this concept is clear,
then the FEMLAB implementation is merely “turning the crank.”