Two examples are presented hereto demonstrate the accuracy of a PML model: the first gives a visual demonstration of the absorption of waves by the PML, and the second shows the efficacy of the PML model even with small bounded domains.
Consider a half-space, with a uniform vertical force applied over a square area on its surface:
We first choose the following PML model — with 5 elements through the PML — to demonstrate the wave absorption:
The wave propagation may be seen in the following movie: (note the dark band in the PML in the edges)
However, the PML is most effective when it is close to the excitation:
The following figure shows the above PML in cross-section, with 8 elements through the PML, along with a dashpot model of the same size used for comparison.
An extended mesh model is used as a benchmark:
We apply a vertical force:
and calculate the vertical displacements at the center and at the corner of the area:
Clearly, the PML model produces accurate results, borne out by the computed error in the results:
Model | Center displacement | Corner displacement |
PML | 5% | 6% |
Dashpots | 46% | 85% |
But more striking is the cost of the PML model, which is found to be similar to the dashpot model, but a tiny fraction of the cost of the extended mesh model:
Model | Elements | Time steps | Wall-clock time |
---|---|---|---|
PML | 4 thousand | 600 | 30 secs |
Dashpots | 4 thousand | 900 | 15 secs |
Extd. mesh | 10 million | 900 | 35 proc-hrs |
The PML and dashpot results were obtained from LS-DYNA running on a desktop workstation, whereas the extd. mesh results required a specially parallelised and optimised code running on a supercomputer.
Clearly, PML guarantees accurate results at low cost. A slightly shallower PML, e.g. one 5-elements deep, would still have produced close to accurate results.
We may also mention here that: