Geotechnical News • September 2012
33
GROUNDWATER
flow condition – yielded elements no
more than 10-cm high for our example
(Fig. 1). Therefore, the proposed rule-
of-thumb worked fairly well for the
example.
Finally, we observe that the
θ
and
K
functions, those which were
input functions for the code, are
not respected by the final numerical
solutions (once the code has met the
criteria for numerical convergence)
for elements between 100 and 10 cm
(Figs. 5 and 6). This illustrates that the
code numerical convergence does not
mean convergence towards the physi-
cally correct solution.
Conclusion
This paper has examined a simple 1D
case of a vertical column in steady
state, using uniform meshes, the only
variable being the element size or
height. The code took a few hundred
steps to converge numerically. The
numerical convergence criterion was
a relative error on the modulus of the
pore pressure vector below 10
-6
. The
numerical convergence is much slower
than for saturated problems, due to the
highly non-linear equations for unsatu-
rated conditions.
Different numerical solutions were
obtained, one for each element size.
In short, the finer the mesh, the more
correct the solution. However, this
does not mean that we should finely
discretize any problem. A few basic
principles should be observed. They
are provided hereafter.
First, we must have a preliminary
idea of how the hydraulic head varies
within our study domain. For a first
appraisal we can use a coarse mesh,
which will give us a first rough solu-
tion. We must examine this first solu-
tion and find out the zones with high
variations of
h
,
θ
, and
K
. These zones
are those where our mesh must be
refined. For a second appraisal, we can
use finer meshes in the zones of high
variations. For unsaturated zones, a
rule-of-thumb is to restrict the element
height to the value giving a maximum
change of one order of magnitude
for
K
in a no-flow condition. When
examining the second solution we
may find that some local refinements
are still needed. Once we are satisfied
with the last refinement and believe
that further refinement would add
nothing, we should not be satisfied
with our belief, but prove it. We can
prepare a verification mesh in which
all elements are half the dimension of
what was thought to be the last mesh.
The verification mesh should give the
same results (heads, gradients, veloci-
ties, flow rates, etc.) as our last mesh.
If this is the case, then we have proved
that we had designed and retained the
correct mesh. Note that the computing
time for the verification mesh will be
about four to nine times longer than
the time for our final and correct mesh.
Thus, we will avoid using the verifica-
tion mesh for transient problems (the
verification could last many hours) but
use it first for faster-to-solve steady-
state problems.
Once this choice of mesh has been
proven to be adequate for steady-
state condition, it can then be used
in transient conditions for which the
time increments must then be selected
to ensure proper convergence at each
time.
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