# 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 |

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:

- 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.