PML Absorbing Boundary

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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.

PML absorbs outgoing waves

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.

Background

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.

Theory

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

Large

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.

Implementation

PML has been im­ple­ment­ed in LS-DY­NA for elas­tic, flu­id and acoustic me­dia, and may be used through one of the fol­low­ing cards:

  • MAT_­PML_­ELAS­TIC (MAT_­230), cor­re­spond­ing to MAT_­ELAS­TIC (MAT_­001)
  • MAT_­PML_­ELAS­TIC_­FLU­ID (MAT_­230_­FLU­ID), cor­re­spond­ing to MAT_­ELAS­TIC_­FLU­ID (MAT_­001_­FLU­ID)
  • MAT_­PML_­ACOUSTIC (MAT_­231), cor­re­spond­ing to MAT_­ACOUSTIC (MAT_­090)
  • MAT_­PML_­OR­THO/­ANISOTROP­IC (MAT_­245), cor­re­spond­ing to MAT_­OR­THO/­ANISOTROP­IC_­ELAS­TIC (MAT_­002 and MAT_­002_­ANIS)
  • MAT_­PML_­NULL (MAT_­246), cor­re­spond­ing to MAT_­NULL (MAT_­009); to be used on­ly with EOS_­LIN­EAR_­POLY­NO­MI­AL or EOS_­GRUNEISEN.

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:

  • The PML ma­te­r­i­al should form a par­al­lelepiped box around the bound­ed do­main, and the box should be aligned with the co­or­di­nate ax­es.
  • The out­er bound­ary of the PML should be fixed.
  • The PML lay­er may typ­i­cal­ly have 5–8 el­e­ments through the depth.
  • The PML ma­te­r­i­al should not be sub­ject­ed to any sta­t­ic load.

Results

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:

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
label

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.

Examples

PML model of half space

 
 

References