A perfectly matched layer (PML) is an absorbing layer model that
— when placed next to an elastic bounded domain —
absorbs nearly perfectly all waves traveling outward from the
bounded domain, without any reflection from the interface
between the bounded domain and the PML.
The outgoing wave is absorbed and attenuated in the PML. There
is some reflection of the wave from the fixed outer boundary of
the PML, but that reflected wave can be made as small as
desired. Therefore, an accurate model of an unbounded domain may
be obtained even if the PML is placed very close to the
excitation.
PML was originally developed for electromagnetic waves in
seminal works
by Bérenger
and Chew
in 1994, and followed up by extensive investigation of
electromagnetic PMLs by numerous researchers, as well as
extensions to other fields such as elastic waves for seismic
applications.
Most of these formulations and implementations used
finite-difference split-field methods to implement the PML,
which had two disadvantages: (i) the finite-difference
methods could not be used easily with finite-element models
for structures, and (ii) the split-field formulation often
led to long-time instability.
These shortcomings were rectified for elastic PMLs by Basu
and Chopra
[2003
![[pdf]](images/pdficon_large.gif)
,
2004
,
2009
] by developing a
displacement-based finite-element implementation that
allowed explicit analysis, thus enabling realistic analysis
of three-dimensional soil-structure systems.
Consider a semi-infinite rod — a simple model of an
unbounded half-space — where only rightward waves are
allowed:
The equations for the elastic medium of this rod can be
converted into equations for a perfectly matched medium
(PMM), which is mathematically designed to damp out waves
using a damping function 
that increases in the unbounded direction:
This PMM may be placed next to a bounded elastic rod to
absorb and damp out all waves traveling outward from the
bounded medium:
The medium is mathematically designed not to reflect any
portion of the waves at its interface to the elastic rod,
this being the perfect matching property of the
medium.
This PMM may be truncated where the wave is sufficiently
damped, to give the perfectly matched layer:
There will be some reflection from the truncated end of the
PML, but the amplitude of the reflected wave, given by
![|R| = exp[-2F(L_P)], F = int f dx](equations/reflc.png)
is controlled by 
and 
, and can be made as small as desired.
The attenuation function is typically chosen as
Typically, 
works
best for finite-element analysis, and
may
be chosen from simplified discrete analysis. LS-DYNA
automatically chooses an optimal value of
according to the depth of the layer.
The depth of the layer may be chosen so that the
layer is about 5–8 elements deep, with the mesh
density in the PML chosen to be similar to that in the
elastic medium.
PML has been implemented in LS-DYNA for elastic, elastic
fluid and acoustic media, and may be used through one of the
following cards:
Some requirements of the PML materials are:
-
The PML material should form a parallelepiped box around
the bounded domain.
-
This box must be aligned with the coordinate axes.
-
The outer boundary of the PML should be fixed.
-
The material in the bounded domain near the PML should
be linear.
-
The PML material constants should match the linear
bounded material.
-
The PML layer should have 5–8 elements through the
depth, with a smaller depth if the PML is distant from
the excitation.
-
The PML layer cannot withstand any static load.
Two examples are presented here: 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.
