Boussinesq's analytical solution for the problem of a point load on a
half-space gives the vertical displacement as
where r and z are the radial and
vertical distance from the point load, respectively. This equation clearly
shows the
singularity at the point of application of the load (0).
Here we compare the displacement variation along a vertical line beneath the
point load where, for the given elastic properties,
This analytical result is plotted in
Figure 3,
together with results obtained with the finite and infinite element models.
It is clear that the results obtained with the infinite element meshes show
a significant improvement over the finite element meshes with the same number
of elements, and that the infinite elements provide reasonable accuracy even
with such relatively coarse modeling. In this case the load is a point load, so
that the infinite elements can be focused on the pole of the solution.
Infinite elements: circular load on half-space
considers a distributed load, for which the infinite element mesh design is not
as obvious.
Flamant's analytical solution for the problem of a line load on a half-space
gives the vertical displacement along a vertical line beneath the line load as
where d is an arbitrary large distance at which the
displacement is assumed to be zero (see the discussion in
Infinite Elements).
Here, we have chosen to fix the far-field nodes on the infinite elements so
that
8.0. This analytical result is plotted in
Figure 4.
The results obtained with the finite and infinite element models are also shown
in this figure. Even though the infinite elements contain displacement
interpolations in the infinite direction with terms of order
while the analytical solution is of a
nature, they provide a significant improvement over the solutions obtained with
finite elements only.
