Accuracy of PML
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|
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:
- Long-time stability of this PML has been verified numerically.
- The critical time-step of the PML for explicit analysis is the same as that for the corresponding elastic element.