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, 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
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, fluid and acoustic media, and may be used through one of the following cards:
Please see the latest draft manual for the input format of each card.
Each PML material is meant to be placed next to — and absorbs waves from — a bounded domain composed of the material to which it corresponds. The material in the bounded domain near the PML should be linear, and the material constants of the PML should match those of the linear bounded material. To facilitate this, the input format of each PML material is largely similar to the corresponding linear material.
Some further requirements for the PML materials are:
Two examples are presented here to 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: