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Geotechnical News • December 2012
37
GROUNDWATER
Chapuis, R.P., Chenaf, D.,
Acevedo, N., Marcotte, D., Chou-
teau, M. 2005. Unusual drawdown
curves for a pumping test in an
unconfined aquifer at Lachenaie,
Quebec: Field data and numerical
modeling. Canadian Geotechnical
Journal, 42: 1133–1144.
Comsol, 2011. COMSOL Multi-
physics user’s guide – version
4.2, COMSOLAB, Stockholm,
Sweden.
Richards, L.A. 1931. Capillary con-
duction of liquids through porous
mediums. Physics, 1: 318–333.
Roache, P.J. 2009. Fundamentals of
verification and validation. Her-
mosa Publishers, Socorro, NM
Roache, P.J. 1994. Perspective: A
method for uniform reporting of
grid refinement studies. ASME
Journal of Fluids Engineering,
116: 405–413.
SoilVision, 2012. SVFlux User’s
Manual. SoilVision Systems Ltd,
Saskatoon, Canada.
Benefits of adaptive automatic mesh refinement
Igor Petrovic, Murray Fredlund
Semi-automatic mesh generation is
a time-consuming and error-prone
process. This is particularly true for
engineering computations where
the mesh requires varying levels of
complexity. This paper studies two
numerical models that produce con-
verged solutions with the assistance of
automatic adaptive mesh refinement,
AMR. The studies illustrate how auto-
mated adaptive mesh refinement can
reduce modeling time as well as errors
during the modeling process. The
AMR solutions were performed using
the SVFlux / FlexPDE software. The
results are discussed in the contexts
of the solutions published by Chapuis
(2012). Chapuis (2012) analyzed the
same example problems while using
user-controlled mesh design when
performing the numerical solutions.
Types of errors that occur in
finite element analysis
The mathematical type of errors intro-
duced into the finite element solution
of a given differential equation can
be attributed to three basic sources
(Reddy, 2006):
1. domain approximation errors –due
to approximation of domain,
2. quadrature and finite arithmetic er-
rors – these are errors due to the
numerical evaluation of integrals
and the numerical computation on
a computer,
3. approximation errors – these are er-
rors due to the approximation of
the solution through interpolation
functions.
This list does not consider errors
in programming, and differences
between the numerical model and
the real physics. For more complete
list see for example Oberkampf et al.
(1995) and Roache (2009).
Convergence
The main problem in any numerical
model which needs to be addressed
consist of the questions of
how good
the approximation is and how it can be
systematically improved to approach
the exact answer
. The answer to the
first question presumes knowledge of
the exact solution.
The second question can be answered
from studies in interpolation theory.
The finite element approximation is
known to converge in the energy norm
when ||e||
<
Ch
p
, for
p
> 0, where
h
is
the distance between nodes on a uni-
form mesh (the characteristic element
length),
p
is called the rate of conver-
gence. The rate depends on the degree
of the polynomial used to approximate
true solution
u
the order of the highest
derivative of
u
in the weak form, and
whether there are local singularities in
the domain. The constant, C, is inde-
pendent of
u
and will be influenced by
the shape of the domain and whether
Dirichlet or Neumann boundary condi-
tions are employed. Typically
p
=
k
+
1-
m
> 0 where
k
is the degree of the
highest complete polynomial used in
the interpolation and
m
is the order
of the highest derivative of
u
in the
weak form. The above equation for the
error would be a straight line plot for a
log-log plot of error versus mesh size.
In that case the slope of the line is the
rate of convergence,
p
(Akin, 2005).
Finite element adaptive mesh
refinement, AMR
An adaptive mesh refinement proce-
dure measures the adequacy of the
mesh and refines the mesh wherever
the estimated error is large. The
system iterates the mesh refinement
and solution until a user-defined error
tolerance is achieved. The most com-
mon criterion in general engineering
use is that of prescribing a total limit
of the estimated error computed in the
energy norm as described in previous
chapter. Often this estimated error is
specified to not exceed a specified
percentage of the total norm of the
solution. An adaptive mesh refinement
procedure is used to reduce estimated