Page 268 - Process Modelling and Simulation With Finite Element Methods

Coupling Variables Revisited 255

have spatially uniform, but unknown values, since the diffusive terms (r) do not
change the imposed neutral boundary conditions. All of the dynamics come
from the requirement that the charges are fixed at steady state, e.g. Fl=ql-
0.77067=0. Now to the solver parameters selection. Select Weak solution form.
Depending on the mood of your platform, you should get a variety of error
messages upon selecting the stationary nonlinear solver. The common error is
“Inf or NaN repeatedly found in solution. Returned solution has not converged.”
The exact Jacobian does this; the numeric Jacobian takes longer to arrive at the
same spot.

Error Message
A companion message to “Inf or NaN repeatedly found in solution. Returned
solution has not converged.” is the error message “Stepsize too small. Returned
solution has not converged.” The latter must be the most commonly encountered
error message, as it is the symptom of many different ills: A short list includes:
1. Inconsistent model leading to a singular system. For instance, a badly
posed boundary condition that can never be satisfied would never
converge to a solution. The damping factor (i.e. step size) will be cut
down until it reaches machine precision, but Newton’s method will
never provide a direction of decreasing error.
2. Unresolved physics. This pretty much means that you need more grid
somewhere. Try the adaption option for the solver.
3. Your problem could simply be poorly posed or ill-conditioned. This is
frequently due to large disparity in length scales or time scales at which
complexity is generated in your problem. Try cutting down
dimensionless complexity parameters like Reynolds, Rayleigh, or Peclet
numbers to a size appropriate to your grid resolution or pack elements
into supposed locations of boundary layers.
In the case of the ECT inversion problem, both explanation 1 and 3 fit the
problem, as we explore further below.
The iterative solver should give a variation on the error theme - the
preconditioning matrix has three rows that are all zero. If you try the linear
solver, however, the story is different. It finds the solution for phi and quite
readily determines values of the qi near the imposed values. The dielectric
constants ui, however, are all extremely large magnitude, O(1014). As we
discussed in chapter 1, this behavior is consistent with a singular linearized
operator, specifically with three zero rows.
How can this happen? Easily. The coupling variables ql, q2, and q3 are
not differentiated correctly to form the contributions necessary for the Jacobian
to be non-singular. They are treated as pseudo-constants that are not updated
during the Newton solver operation. Consequently, the three equation model in