Page 277 - Process Modelling and Simulation With Finite Element Methods

264 Process Modelling and Simulation with Finite Element Methods

Figures 7.6 and 7.7 are also convergent to the same dielectric constant solution
vector [0.0508, 0.0703, 0.04971 from two substantially different initial guesses.
Nevertheless, neither convergent solution is the one found from the forward
solution. For aesthetic purposes, we could have taken [0.05,0.07, 0.051 as the
choice for the forward problem, and miraculously achieved it by the inversion
problem. Yet the reader would be lured into a false sense of security of the
effectiveness of inverse problems in concretely determining a sufficient answer.
The 40% error in the convergent solution can be reduced with greater resolution
power of smaller electrodes on the boundaries. In reviewing this chapter, I noted
a minor discrepancy: the “set point” for q3=0.16538 was slightly different from
the solution to the forward problem. I had switched platforms, using the PC for
the forward problem, and the linux workstation for the optimization program.
Thus the 40% discrepancy in u2 is largcly due to extreme sensitivity to the
measurement error. But if greater resolution in the composition of the inclusions
in the domain or their positions or sizes are desired, then the better quality
boundary data is diluted across the domain, again possibly obscuring the
“image” of the included data. Image reconstruction is a complicated problem for
capacitance tomography. A good review of applications can be found by
Dyakowski et al. [151. The work of WRB Lionheart and coworkers [16],
especially the EIDORS MATLAB based software package, is the best source of
novel inversion techniques.

Exercise 7.1: Coding efficiency
The modular programming of the “calling” m-file script fmin.m and the
subprograms ect2.m which computes the forward solution, and the objective
function for error minimization errornm.m, is not particularly efficient, though
good for pedagological purposes. To improve the efficiency, the m-file script
should have the FEMLAB model set up defined in global variables, and the
optimization function fminsearch merely changes the dielectric constants. Can
you code this more integrated version?

Exercise 7.2: Unknown diameter rods

An alternative scenario is that the dielectric constant of the inclusions (rods) in
the cylindrical duct is known (say 0.05 of that of the medium) but that the radii
of the rods is unknown. Determine where in the FEMLAB model m-file
function ect2.m the radii of the rods is specified and alter ect2.m appropriately to
compute the forward solution with the radii passed to the m-file function as
arguments. Repeat the search procedure for the inversion from initial guess of
the rod radii. Is this problem any better conditioned than the unknown dielectric
constant problem? How could you improve the error in the estimated radii from
the boundary data?