Environmental Engineering Reference
In-Depth Information
periods, the ensuing numerical model can become very large. In such situations, the
analyst is tempted to limit the lateral extension of the mesh as much as possible.
The definition of the fictitious lateral boundary (position and values of variables)
requires some care, because the solution of a coupled problem depends on the
response characteristics of interacting fields. It is very easy to introduce errors and
one or more of the following situations may occur [SIM 89b]:
- errors do not always involve all of the field variables and do not necessarily
affect the field variables in which the forcing functions are applied, hence we may
control the wrong variable;
- errors may appear after a certain time lag, depending on the specific rate of
each field. Sometimes the analysis in not sufficiently long to emphasize the errors;
- comparison with known normalized solutions can guarantee the solution is in a
correct form, but not the size of the values.
The control of a unique variable or for a limited time span at the onset of the
transient period or the comparison with normalized solutions may hence entail an
unjustified confidence in the results obtained, which may not correspond to the
actual facts.
To obtain reliable and economic solutions, the use of one of the following
approaches is very convenient: reducing the domain by adopting boundary element
formulations [BEE 92] or analytical solutions for the boundary areas and
introducing them into the numerical model [AUS 93]. Unfortunately, very few
analytical solutions can be found in the literature for the coupled problems with
which we are concerned. Another possibility is to reduce the dimensions of the
numerical model at each time step by using partitioned solutions [PAR 83, SCH 85].
In the finite element method, the use of infinite elements can be very
advantageous [BET 92, SIM 87]. We refer here to the approach in which the
unbounded real domain is transformed into a limited one by using suitable singular
coordinate transformations (mapped infinite elements).
Within the transformed limited domain, the usual polynomial approximations are
introduced. The resulting elements do not require particular quadrature rules; hence
the use of this type of elements is very simple and allows for a realistic analysis of
the boundary conditions. The solutions obtained by this approach are very good and,
at the same time, the number of unknowns required is limited.
