Page 270 - Process Modelling and Simulation With Finite Element Methods

Coupling Variables Revisited 257

5. Solve problem by simply highlighting all the variables in Solve for
variables menu.
6. In order to run this recipe, you need FEMLAB 2.3 (2.2 won't do).
We would recommend this treatment for all nonlinear boundary couplings, not
just those in time-dependent models. Why? Because it signals to FEMLAB that
a non-standard boundary condition with nonlinear coupling should be treated
with symbolic contributions to the assembly of the FEM system of matrices. See
the Reference Manual for the command
[K , N , L , MI =as semble ( f em)
and the description in Chapter 2 (52.3.1) for the FEMLAB implementation of
boundary conditions by Lagrange multipliers for a better understanding of why
nonlinearity should be treated this way.
Even though this aside has been focused on the difficulties of nonlinear
boundary constraints, nonlinear coupling constraints suffer a related problem.
The FEMLAB symbolic engine does not contribute the full dependency (or even
any in our case) of the coupling variables on the degrees of freedom. So the
Jacobian matrix formed can be incomplete or inaccurate. The numeric Jacobian
also lacks this dependency, as we found in the last section. The time-dependent
solver lacks it as well, but the nonlinear coupling does manifest itself, with a
delay of one time step as the coupling variable is updated at each time step.
Since modern time stepping algorithms have quality control built-in, even stiff
nonlinearity can be ferreted out by the time dependent solver.
Figure 7.2 shows six frames of the potential contours for times t=O.O1
through t=l. Quantitatively, very little change occurs in the potential lines out to
t=4. Wait a moment. The ECT problem, (7.1) and BCs defined in Figure 7.1, is
time independent. So what is the time scale for t? Answer: completely fictitious.
For stationary problems, a pseudo-time is another way of iterative solving. If a
time-asymptotic, steady-state emerges, then the time-dependent solver hase done
its job.
Another way of thinking of the time-dependent form of (7.1) is as an
analogous problem in heat or mass transfer in a heterogeneous medium. The
forward problem determines the boundary flux at fixed temperature
(concentration) boundary segments interspersed among insulated segments. The
inverse problem is to determine the distribution of transport coefficients
internally in the heterogeneous medium consistent with the measured boundary
fluxes.
In this context, the time scale is that for conduction and is physically
meaningful. The potential (temperature or concentration) diffuses into the
domain from the source boundary segment. Initially, the domain has uniform
potential different from the source, Field lines are warped by the inclusions of
non-uniform diffusivity. The field lines largely lead to flux out of the domain