PML Theory

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Con­sid­er a se­mi-in­fi­nite rod — a sim­ple mod­el of an un­bound­ed half-space — where on­ly right­ward waves are al­lowed:

Wave in semi-infinite rod

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 f(x) that in­creas­es in the un­bound­ed di­rec­tion:

Wave in PMM

This PMM may be placed next to a bound­ed elas­tic rod to ab­sorb and damp out all waves trav­el­ing out­ward from the bound­ed medi­um:

PMM absorbs and attenuates waves

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:

Truncating the PMM gives the PML

There will be some re­flec­tion from the trun­cat­ed end of the PML, but the am­pli­tude of the re­flect­ed wave, giv­en by


is con­trolled by f and Lp, and can be made as small as de­sired.

The at­ten­u­a­tion func­tion is typ­i­cal­ly cho­sen as

attenuation function

Typ­i­cal­ly, m=2 works best for fi­nite-el­e­ment analy­sis, and f0 may be cho­sen from sim­pli­fied dis­crete analy­sis. LS-DY­NA au­to­mat­i­cal­ly choos­es an op­ti­mal val­ue of f0 ac­cord­ing to the depth of the lay­er.

The depth Lp 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.