The Perfectly Matched Layer (PML)

Transcription

1 9/13/216 EE 533 Electomagnetic Analsis Using Finite Diffeence Time Domain Lectue #13 The Pefectl Matched Lae (PML) Lectue 13 These notes ma contain copighted mateial obtained unde fai use ules. Distibution of these mateials is stictl pohibited Slide 1 Lectue Outline Review of Lectue 12 Backgound Infomation The Uniaial Pefectl Matched Lae (UPML) Convolutional PML (CPML) Calculating the PML paametes Incopoating a UPML into Mawell s equations Lectue 13 Slide 2 1

2 9/13/216 Review of Lectue 12 Lectue 13 Slide 3 Effect of Windowing on FDTD Spectum wt W h t H Long simulation time Wide window Naow window spectum Less bluing High fequenc esolution H W wt W h t H Modeate simulation time Modeate window Modeate window spectum Moe bluing Reduced fequenc esolution H W wt ht W H Shot simulation time Naow window Wide window spectum Much bluing Poo fequenc esolution H W Lectue 13 Slide 4 2

3 9/13/216 Ensuing Sufficient Iteations to Resolve f Given the time step t, the Nquist sampling theoem quantifies the highest fequenc that can be esolved. 1 1 fma Note: In pactice, ou should set t N 2 t Nf t ma Given the FDTD simulation uns fo STEPS numbe of iteations, the fequenc esolution is 2 fma 1 f STEPS t STEPS Theefoe, to esolve the fequenc esponse down to a esolution of f, the numbe of iteations equied is t 1 1 STEPS t f Notes: 1. This is elated to the uncetaint pinciple. 2. You can calculate kenels fo ve finel spaced fequenc points, but the actual fequenc esolution is still limited b the windowing effect. You data will be blued. Lectue 13 Slide 5 Dielectic Smoothing Conventional Repesentation on Gid Repesentation on Gid with Dielectic Smoothing Phsical Device Lectue 13 Slide 6 3

4 9/13/216 Anisotopic Dielectic Smoothing Given the simulation poblem defined b D 1 1 smooth E We can impove the convegence ate b smoothing the dielectic function accoding to ˆN 2 n nn nn ˆ 2 N nn n nn 2 nn nn n Lectue 13 Slide 7 Dielectic Smoothing of a Sphee Given a sphee with dielectic constant =5. in ai and in a gid with N =N =N =25 cells, the dielectic tenso afte smoothing is coss section coss section Lectue 13 Slide 8 4

5 9/13/216 2 Gid Technique The 2 gid is onl used fo building devices on a Yee gid. It is not used anwhee else! Lectue 13 Slide 9 Concept of the 2 Gid The Conventional 1 Gid Due to the staggeed natue of the Yee gid, we ae effectivel getting twice the esolution. It now makes sense to talk about a gid that is at twice the esolution, the 2 gid. j j j i i i The 2 gid concept is useful because we can ceate devices (o PMLs) on the 2 gid without having to think about whee the diffeent field components ae located. In a second step, we can easil pull off the values fom the 2 gid whee the eist fo a paticula field component. H H E Lectue 13 Slide 1 5

6 9/13/216 Pasing the Mateials Fom 2 to 1 Gid ER UR UR H H E Lectue 13 Slide 11 Backgound Lectue 13 Slide 12 6

7 9/13/216 Tensos Tensos ae a genealiation of a scaling facto whee the diection of a vecto can be alteed in addition to its magnitude. Scala Relation V av V av Tenso Relation a a a a V a a a V V a a a V Lectue 13 Slide 13 Reflectance fom a Suface with Loss Comple efactive inde n n j n odina efactive inde etinction coefficient Reflectance fom a loss suface ai R 1n 1n ** Loss contibutes to eflections n n j Lectue 13 Slide 14 7

8 9/13/216 Reflection, Tansmission and Refaction at an Inteface: Isotopic Case Angles inc ef 1 n sin n sin Snell s Law n, 1 1 n, 2 2 TE Polaiation 2cos11cos2 TE cos cos t TE cos1 cos cos n efactive inde in egion i i impedance in egion i TM Polaiation Lectue 13 i Slide 15 t TM TM 2cos2 1cos1 cos cos cos1 cos cos Mawell s Equations in Anisotopic Media Mawell s cul equations in anisotopic media ae: H j E E j H These can also be witten in a mati fom that makes the tenso aspect of and moe obvious. H E H j E H E E H E j H E H Lectue 13 Slide 16 8

9 9/13/216 Tpes of Anisotopic Media Thee ae thee basic tpes of anisotopic media: iso iso iso isotopic o o e uniaial a b c biaial Note: tems onl aise in the offdiagonal positions when the tenso is otated elative to the coodinate sstem. Lectue 13 Slide 17 (, ) Vs. (, ) Thee ae two was to incopoate loss into Mawell s equations. At ve low fequencies and/o fo time domain analsis, the (, ) sstem is usuall pefeed. H J jd E j E j E At high fequencies and in the fequenc domain, (, ) is usuall pefeed. H jd j E The paametes ae elated though j We use this fo FDTD Note: It does not make sense to have a comple and a conductivit. Lectue 13 Slide 18 9

10 9/13/216 Mawell s Equations in Doubl Diagonall Anisotopic Media Mawell s equations fo diagonall anisotopic media can be witten as H E E H H j E E j H H E E H We can genealie futhe b incopoating loss. E H j E E H j j E E H je H E j H H E j j H H E j H Lectue 13 Slide 19 Scatteing at a Doubl Diagonal Anisotopic Inteface Refaction into a diagonall anisotopic mateials is descibed b Reflection fom a diagonall anisotopic mateial is sin TE TM bc sin 1 2 acos acos acos acos bcos bcos bcos bcos a b c Sacks, Zacha S., et al. "A pefectl matched anisotopic absobe fo use as an absobing bounda condition." IEEE Tans. Antennas and Popagation, Vol. 43, No. 12, pp , Lectue 13 Slide 2 1

11 9/13/216 Notes on a Single Inteface It is a change in impedance that causes eflections Snell s Law quantifies the angle of tansmission Angle of tansmission and eflection does not depend on polaiation The Fesnel equations quantif the amount of eflection and tansmission Amount of eflection and tansmission depends on the polaiation Lectue 13 Slide 21 The Uniaial Pefectl Matched Lae (UPML) S. Zacha, D. Kingsland, R. Lee, J. Lee, A Pefectl Matched Anisotopic Absobe fo Use as an Absobing Bounda Condition, IEEE Tans. on Ant. and Pop., vol. 43, no. 12, pp , Lectue 13 Slide 22 11

12 9/13/216 Bounda Condition Poblem If we model a wave hitting some device o object, it will scatte the applied wave into potentiall man diections. We do NOT want these scatteed waves to eflect fom the boundaies of the gid. We also don t want them to eente fom the othe side of the gid (peiodic boundaies).?? How do we pevent this? Lectue 13 Slide 23 No PAB in Two Dimensions If we model a wave hitting some device o object, it will scatte the applied wave into potentiall man diections. Waves at diffeent angles tavel at diffeent speeds though a bounda. Theefoe, the PAB condition can onl be satisfied fo one diection, not all diections. fastest slowest Lectue 13 Slide 24 12

13 9/13/216 How We Pevent Reflections in Lab In the lab, we use anechoic foam to absob outgoing waves. Lectue 13 Slide 25 Concept of a PML -lo PML -lo PML -hi PML -hi PML Lectue 13 Slide 26 13

14 9/13/216 Absobing Bounda Conditions We can intoduce loss at the boundaies of the gid! Absobing Bounda Lectue 13 Slide 27 Oops!! But if we intoduce loss, we also intoduce eflections fom the loss egions!! R 1n 1n Lectue 13 Slide 28 14

15 9/13/216 Match the Impedance We need to intoduce loss to absob outgoing waves, but we also need to match the impedance of the poblem space to pevent eflections. intoduce loss hee j adjust this to contol impedance Lectue 13 Slide 29 Moe Touble? B eamining the Fesnel equations, we see that we can onl pevent eflections fom the inteface at one fequenc, one angle of incident, and one polaiation. cos cos cos TE 2 1 2cos11cos2 cos1 cos cos cos TM 2 1 1cos12cos2 cos2 Lectue 13 Slide 3 15

16 9/13/216 Anisotop to the Rescue!! It tuns out we can pevent eflections at all angles and fo all polaiations if we allow ou absobing mateial to be doubldiagonall anisotopic. and and Lectue 13 Slide 31 Poblem Statement fo the PML Fee Space, % 1 2 1, 1% Lectue 13 Slide 32 16

17 9/13/216 Designing Anisotop fo Zeo Reflection (1 of 3) We need to pefectl match the impedance of the gid to the impedance of the absobing egion. Fo an absobing bounda in ai, this condition can be thought of as evewhee One eas wa to ensue impedance is pefectl matched to ai is: a b c Lectue 13 Slide 33 Designing Anisotop fo Zeo Reflection (2 of 3) If we choose bc 1, then the efaction equation educes to sin bc sin sin No efaction! With this choice, the eflection coefficients educe to TE TM acos1 bcos2 a b acos bcos a b 1 2 acos1 bcos2 a b acos bcos a b 1 2 These ae no longe a function of angle!! Lectue 13 Slide 34 17

18 9/13/216 Designing Anisotop fo Zeo Reflection (3 of 3) If we futhe choose a b, the eflection equations educe to TE TM a b a b a b a b Reflection will alwas be eo egadless of fequenc, angle of incidence, o polaiation!! Recall the necessa conditions: bc 1 and a b Lectue 13 Slide 35 The PML Paametes (1 of 3) So fa, we have a b c a b 1 c Thus, we can wite ou PML in tems of just one paamete s. s S s s j 1 s This is fo a wave tavelling in the + diection incident on one bounda. This fom of tenso is wh we call this a uniaial PML. Lectue 13 Slide 36 18

19 9/13/216 The PML Paametes (2 of 3) We potentiall want a PML along all the bodes. 1 s s s S s S s S s 1 s s s 1 These can be combined into a single tenso quantit. ss s ss s ss s S S S S Lectue 13 Slide 37 The PML Paametes (3 of 3) The 3D PML can be visualied this wa S ss s ss s ss s s 1 s 1 s 1 s 1 s 1 s 1 Lectue 13 Slide 38 19

20 9/13/216 The PML is Not a Bounda Condition A numeical bounda condition is the ule ou follow when a finite diffeence equation efeences a field fom outside the gid. The PML does not addess this issue. It is simpl a wa of incopoating loss while peventing eflections so as to absob outgoing waves. Sometimes it is called an absobing bounda condition, but this is still a misleading title because the PML is not a bounda condition. Lectue 13 Slide 39 Convolutional Pefectl Matched Lae (CPML) Lectue 13 Slide 4 2

21 9/13/216 The Uniaial PML Mawell s equations with uniaial PML ae: E k S S H H k S E ss s ss s ss s Lectue 13 Slide 41 Reaange the Tems We can bing the PML tenso to the left side of the equations and associate it with the cul opeato. 1 E k H S H k 1 S The cul opeato is now S 1 1 s s s s ss 1 1 ss s s 1 s 1 s s s s 1 s 1 1 s 1 s s s s s s s s s s Lectue 13 Slide 42 E 21

22 9/13/216 Stetched Coodinates Ou new cul opeato is S 1 1 s 1 1 s 1 1 s 1 s s s s s s s s s s s s s s s The factos s, s, and s ae effectivel stetching the coodinates, but the ae stetching into a comple space. Lectue 13 Slide 43 Dop the Othe Tems We dop the non stetching tems. s 1 s 1 s s s s 1 1 s s s 1 s s s s s s s s 1 1 s 1 s 1 s s s s s s Justification s 1 1 s s s Inside the PML, s = s = 1. This is valid evewhee ecept at the eteme cones of the gid whee the PMLs ovelap. This also implies that the UPML and SC PML have neal identical pefomance in tems of eflections, sensitivit to angle of incidence, polaiation, etc. Lectue 13 Slide 44 22

23 9/13/216 Mawell s Equations with a SC PML Mawell s equations befoe the PML is added ae E k H H k E The SC PML is incopoated as follows. s E j H H j E s s 1 1 s s 1 1 s s 1 1 s s Lectue 13 Slide 45 Vecto Epansion Mawell s equations with a SC PML epand to Full Anisotopic 1 H 1 H k E E E s s 1 H 1 H k E E E s s 1 H 1 H k E E E s s 1 E 1 E k H H H s s 1 E 1 E k H H H s s 1 E 1 E k H H H s s Diagonall Anisotopic 1 H 1 H k E s s 1 H 1 H k E s s 1 H 1 H k E s s 1 E 1 E s s 1 E 1 E k H s s 1 E 1 E kh s s k H Lectue 13 Slide 46 23

24 9/13/216 Convolutional PML A convolutional PML is a SC PML that is solved efficientl in the time domain. We have a multiplication opeation in the fequenc domain. This becomes convolution in the time domain. 1 H 1 H k E s s 1 H 1 H k E s s 1 H 1 H k E s s 1 E 1 E s s 1 E 1 E k H s s 1 E 1 E kh s s Lectue 13 Slide 47 k H UPML Vs. SC PML Benefits Uniaial PML Has a phsical intepetation Models can be fomulated and implemented without consideing the PML in the fequenc domain Stetched Coodinate PML Benefits Less computationall intensive in time domain Moe efficient implementation in the time domain Matices ae bette conditioned. Dawbacks Can be moe computationall intensive to implement in timedomain Resulting matices ae less well conditioned in the fequendomain Dawbacks Must be accounted fo in the fomulation and implementation of the numeical method. Not intuitive to undestand Lectue 13 Slide 48 24

25 9/13/216 Incopoating a UPML into Mawell s Equations Lectue 13 Slide 49 Wh a UPML A CPML (o SC PML) is cuentl the state of theat in FDTD. A UPML offes benefits to fequenc domain models that ae useful fo beginnes. The models can be fomulated and implemented without consideing the PML. The PML is incopoated simpl b adjusting the values of pemittivit and pemeabilit at the edges of the gid. Implementing a UPML in this couse offes bette continuit going into the following couse, Computational Electomagnetics. Lectue 13 Slide 5 25

26 9/13/216 Incopoating a UPML into Mawell s Equations Befoe incopoating a PML, Mawell s equations in the fequencdomain ae E j H D E H E j D We can incopoate a PML independent of the actual mateials on the gid as follows: E j SH D E H E j S D Lectue 13 Slide 51 Nomalie Mawell s Equations with UPML We nomalie the electic field quantities accoding to 1 1 E E E Mawell s equations with the UPML and nomalied fields ae E j S H c j D E H E SD c D D c D Lectue 13 Slide 52 26

27 9/13/216 Mati Fom of Mawell s Equations Mawell s equations can be witten in mati fom as 1 E s ss H 1 1 E j ss s H c 1 E sss H 1 H E s ss D j 1 H E ss s D c 1 H E sss D D E D E D E Lectue 13 Slide 53 Assume Onl Diagonal Tensos Hee we assume that [], [], and [] contain onl diagonal tems. 1 E s ss H 1 1 E j ss s H c 1 E sss H 1 H E s ss D j 1 H E ss s D c 1 H E sss D D E D E D E Lectue 13 Slide 54 27

28 9/13/216 Vecto Epansion of Mawell s Equations E j s H c j H E s D c D E E E ss j H c s E E ss j H c s E E ss j H c s D D D H E E E H s s c s j E D j E D H H ss H c s H j s s E D c s Lectue 13 Slide 55 Final Fom of Mawell s Equations with UPML E j s H c E 1 E c j1 1 1 H j j j 1 c E E H j1 1 1 j j j 1 c E E j H j j j j H E s D c 1 H H j D c E j j j 1 H H j D c E j j j 1 H H j D c E j j j D E D D D E E E Lectue 13 Slide 56 28

29 9/13/216 Calculating the PML Paametes Lectue 13 Slide 57 The Pefectl Matched Lae (PML) The pefectl matched lae (PML) is an absobing bounda condition (ABC) whee the impedance is pefectl matched to the poblem space. Reflections enteing the loss egions ae pevented because impedance is matched. Reflections fom the gid boundaies ae pevented because the outgoing waves ae absobed. PML PML Poblem Space PML PML Lectue 13 Slide 58 29

30 9/13/216 Calculating the PML Loss Tems Fo best pefomance, the loss tems should incease gaduall into the PMLs. s s s 1 j 1 j 1 j 3 2t L 3 2 t L 3 2 t L Lectue 13 Slide 59 L? length of the PML in the? diection Visualiing the PML Loss Tems 2D Fo best pefomance, the loss tems should incease gaduall into the PMLs. Lectue 13 Slide 6 3

31 9/13/216 Note About /L, /L, and /L The following atios povide a single quantit that goes fom to 1 as ou move though a PML egion. and and L L L,, position within PML L, L, L sie of PML We can calculate the same atio using intege indices fom ou gid. n n o L NXLO NXHI n n o L NYLO NYHI n o L NZLO n NZHI n = 1, 2,..., NXLO n = 1, 2,..., NXHI n = 1, 2,..., NYLO n = 1, 2,..., NYHI n = 1, 2,..., NZLO n = 1, 2,..., NZHI 1 L 1 L Lectue 13 Slide 61 Visualiing the Calculation of in 2D % ADD XLO PML fo n = 1 : NXLO o(nxlo-n+1,:) =... end % ADD XHI PML fo n = 1 : NXHI o(n-nxhi+n,:) =... end () = ma (/L ) 3 () = () = ma (/L ) 3 1 NXLO Lectue 13 Slide 62 N-NXHI+1 N 31

32 9/13/216 Visualiing the Calculation of in 2D 1 NYLO () = ma (/L ) 3 % ADD YLO PML fo n = 1 : NYLO o(:,nylo-n+1) =... end () = 1 N NYHI + 1 N () = ma (/L ) 3 % ADD YHI PML fo n = 1 : NYHI o(:,n-nyhi+n) = end Lectue 13 Slide 63 Pocedue fo Calculating and on a 2D Gid 1. Initialie and to all eos.,, 2. Fill in ais PML egions using two fo loops. NXLO NXHI 3. Fill in ais PML egions using two fo loops. NYLO NYHI Lectue 13 Slide 64 32