Accuracy of PML

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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:

Force on a half-space

We first choose the fol­low­ing PML mod­el — with 5 el­e­ments through the PML — to demon­strate the wave ab­sorp­tion:

Large PML model of half-space

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:

Small PML model of half-space

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.

Cross-section of PML model
Cross-section of dashpot model

An ex­tend­ed mesh mod­el is used as a bench­mark:

Extended mesh model of half-space

We ap­ply a ver­ti­cal force:

Vertical force

and cal­cu­late the ver­ti­cal dis­place­ments at the cen­ter and at the cor­ner of the area:

Center displacement
Corner displacement

Clear­ly, the PML mod­el pro­duces ac­cu­rate re­sults, borne out by the com­put­ed er­ror in the re­sults:

PML error
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.