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Geotechnical News • December 2012
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GROUNDWATER
Finally, it is worth underlining that all
numerical solutions have converged
in the numerical sense for this 1D
example. This illustrates again that the
code numerical convergence does not
mean convergence towards the physi-
cally correct solution.
Conclusion to this paper
This paper has examined a simple
1D case of a vertical column under
transient conditions, using uniform
meshes, the variable being the time–
step size. The code took several steps
to converge numerically but it always
converged. The numerical conver-
gence criterion was a relative error
on the modulus of the pore pressure
vector below 10
–6
. The numerical con-
vergence was slower than for saturated
problems, due to the highly non–linear
equations for unsaturated conditions.
Different numerical solutions were
obtained, one for each element size
and time step. In short, the finer the
mesh and the smaller the time step, the
more correct the solution. However,
this does not mean that we should
finely discretize any problem in space
and time. A few basic principles
should be observed. They are provided
hereafter.
First, we must have a preliminary
idea of how the hydraulic head var-
ies within our study domain. For a
first appraisal we can use a coarse
mesh, which will give us a first rough
solution. We must examine this first
solution 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 condi-
tion. 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 dimen-
sion of what was thought to be the last
mesh. The verification mesh should
give the same results (heads, gradi-
ents, velocities, 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 must use the
verification mesh firstly 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. In some cases however, the mesh
must be locally finer for transient
conditions than for steady–state condi-
tions, a situation which is not exam-
ined in this short paper.
General concluding remarks for
the four papers
The four short papers on numerical
convergence have not made refer-
ence to, and not used semi–automatic
and automatic meshing refinement
methods, which are very useful
utilities available in several ground-
water codes (e.g., SoilVision 2012,
Comsol 2011). The short papers’ goal
was to introduce, as simply as pos-
sible, the h–convergence tests (e.g.,
Roache 1994, 2009) to finite element
users who, most often, have not had
advanced mathematical courses on
finite elements.
The four papers have proposed simple
convergence tests that can be done by
deactivating the automatic meshing
system if there is one, in order to find
the optimal element size for a given
problem (optimal for an accepted level
of relative error). Once the optimal
size is found, then the automatic
meshing method can be reactivated,
using the optimal uniform grid as the
starting reference: the meshing system
will reduce the total number of nodes
and elements (mesh refining and
coarsening), thus reducing the calcula-
tion time for next calculations with the
same grid.
The mesh coarsening is essential for
huge parametric studies using large
grids, such as steady–state saturated
problems with 10
5
–10
6
elements, and
transient unsaturated problems with
10
4
–10
5
elements, the latter being
much more complicated (mathemati-
cally and numerically) to solve, due to
the highly non–linear partial differen-
tial equations (PDEs). However, this
was not important for the examples in
the four short papers, because there
was no parametric study (except for
the uniform element size) and the
calculations times did not exceed a
few seconds for saturated steady–
state problems and a few minutes for
unsaturated transient problems.
References
Chapuis, R.P. 2010. Influence of ele-
ment size in numerical studies of
seepage: Large–scale or regional
studies. Geotechnical News, 28(4):
31–34.
Chapuis, R.P. 2012a. Influence of
element size in numerical studies
of seepage: Small–scale details.
Geotechnical News, 30(1): 32–35.
Chapuis, R.P. 2012b. Influence of ele-
ment size in numerical studies of
seepage: Unsaturated zones, steady
state. Geotechnical News, 30(3):
27–30.
Chapuis, R.P., and Chenaf, D. 2010.
Driven field permeameters: Rein-
venting the wheel? Geotechnical
News, 28(1): 37–42.
Chapuis, R.P., and Dénes, A. 2008.
Écoulement saturé et non saturé de
l’eau souterraine vers des drains
en aquifère à nappe libre. Cana-
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1210–1223.