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.