Page 301 - Process Modelling and Simulation With Finite Element Methods

288 Process Modelling and Simulation with Finite Element Methods

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03,
02
025L D 18
0 16
02-1 0 14
0 16 0 12
01
01 -
om
005- 0 06
004
0 D
0 02
005 ' 0
Nm BsBe4l
Figure 7.18 Solution nl(v) to (7.27) with graduated mesh of Figure 7.17 for tau=lO. Left: 1-D
solution. Right: 2-D solution for extrusion variable N2=nl(v2,vl-v2).
However, given that it was non-convergent after ~=1.175 the last time we tried a
stationary nonlinear solution, the graduated mesh is an improvement. The
difficulty with the standard uniform mesh that leads to ill-posedness could be
that the 2-D abstract domain was coarser than the I-D solution space. Thus, the
full contribution of the 2-D convolution integrals was lumped into too few bins
to permit the inversion. Or possibly it is that the graduated mesh resolves the
solution tremendously better. But how good is this solution strategy?
Nicmanis and Hounslow [27] compute their solution for 2=200 and compare
with the analytic solution of Hounslow 1301. Can we do the same? I coded a
MATLAB m-file script to use parametric continuation out to 2=200 by steps of
A~=0.5. It crashed at T =lS with the ubiquitous step-size too small error after
slowly converging for nearby T. Eigenanalysis again shows that the stiffness
matrix is nearly singular. So this promising solution strategy is still not fully
effective. How can we alter it to achieve better performance?

Last Chance Saloon: ActivatingIDeactivating Variables With
Solve for Variables
Perhaps you noticed that we are solving in the abstract domain for n2, which at
steady state should be the trivial solution of F=O, i.e. n2=Nl*N2. n2 is pretty
useless to us, but as it is a diagonal system at steady state, it should not be hard
to solve, right? And we do have to solve for something in our fictitious domain,
don't we? Wrong! Even the trivial diagonal solution for F=O uses sparse matrix
solvers with somewhere around 4600 back substitutions. Eventually, this work
will lead to an ill-conditioned numerical solution due to round-off error alone.
Furthermore, we do not need to solve for anything in our fictitious 2-D domain.
We can disable the solution for n2.
Multiphysics: Solve for Variables
Select and highlight only Geoml: 1 variable gen form (gl): nl