The convolution computation for Perfectly Matched Boundary Layer algorithm in finite differences

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1 The convoluton computaton for Perfectly Matched Boundary Layer algorthm n fnte dfferences Herman Jaramllo May 10, Introducton Ths s an exercse to help on the understandng on some mportant ssues on CPML) Convolutonal Perfect Matchng Layer algorthms to attenuate edge effects n fnte dfferences. From the many references, I cte Roden and Gedney s [4], and Pasalc and McGarry [2]. Also, some good notes 1 by Steven Johnson. 2 The Stretchng Factor The dea well explaned by Steven Johnson s as follows. There are two zones. The nsde zone where we want to preserve the numercs and a buffer zone the boundares and ts surroundngs) where we want to attenuate reflectons so that the numercal experments looks lke smulated n a free space medum wth boundares at nfnty. At the buffer zone some dstance away from the center), the wave equatons are analytcally contnued nto the upper or lower, dependng on the Fourer transform beng studed. For causal functons the sgns of the k vector and ω are opposte. That s, a plane wave s wrte n ts smplest form as e k x ωt.) half complex plane so that the oscllaton character changes to an 1 math.mt.edu/ stevenj/18.369/pml.pdf 1

2 exponental dampng character. The trck s done by the change of varable assumng 1D for smplcty, but the dea s general) x x + fx) 2.1) where 1. The functon f acts as the dampng factor. When f 0 then the problem s not changed. Ths occurs n the zone of nterest. Based on physcal ntuton f s chosen so that df dx σ xx), 2.2) ω and ths s explaned next. The reason for the ω n the denomnator s that ths electon wll make the problem ndependent of frequency for a gven reflecton angle. That s, t wll avod dspersve frequency dependent) behavor. To see ths, we observe from equatons 2.1 and 2.2 that xx) x + ω x σ x x )dx, wth the coordnate where the dampng layer starts. See for example that for f 0 n the new coordnate system a plane wave along the x wth k k x k 1 ) drecton, n the new coordnate x, could be represented as e k x e kx e k ω x σ xx )dx and snce c p k/ω s the phase velocty whch s constant n non dspersve meda) we see that the attenuaton represented by the factor e k ω x σ xx )dx. s ndependent of frequency for a gven propagaton drecton). To be sure that the attenuaton factor works correctly, we need to guarantee that x σ x x )dx s postve for postve k, assumng only postve frequences ω. Now, f the plane wave s defned as e k x+ωt the sgns for and k should be opposte). Note that f the wave s travelng along the negatve drecton k < 0 and the ntegral s negatve snce the upper ndex s smaller than the lower ndex and then stll the ntegral contrbuton s negatve, so stll the exponent s negatve) then also here the factor s a dampng factor. 2

3 The smplest electon of σ x s a constant n whch case we would get the mappng x 1 + σ ) x s x x ω where s x s called a stretchng factor. Roden and Gedney s [4] use a more general stretchng factor stll wth constant), s +, wth α > 0 and 1. Whle the factor can be seen as an overall scalng, the factor α s justfed as shft to the pole away from the real axs, whch wll mprove accuracy for grazng ncdence due to source proxmty to the boundary or large offsets. Martn et. al., [3] reaffrm Roden and Gedney s [4] statements. Here and n what follows the subndex means x, f 1, y f 2 and z f 3; for the three dmensonal space. Note that n ther work the sgn conventon s opposte, snce the magnary unt s n the denomnator. They descrbe the plane wave wth a mnus - sgn n front of the phase φ k x ωt. Martn et. al., [3] use the stretchng factor s x 1 + σ x ω where, as n Roden and Gedney, the sgn conventon s opposte to the one used here. Komattsch and Martn [1] also clam that wth the help of the new parameter α ntroduced by Roden and Gedney, better accuracy s obtaned at grazng ncdence angles. Snce a product n the frequency/wavenumber doman s convoluton n tme/space doman, and the FDTD Fnte Dfference Tme Doman) wave equaton s mplemented n tme doman, we need to fnd the nverse Fourer Laplace) transform of the stretchng factor. After fndng the stretchng factor we should apply convoluton. Next secton deals wth fndng the nverse Laplace transform of the stretchng factor, and the fnal secton shows a recurson formula that speeds up the convolutonal mplementaton. 3

4 3 Laplace Transform of the Stretchng factor The stretchng parameter s gven by so s + α + ωɛ ) 1 s α + ωɛ α + sɛ 0 α + sɛ 0 + ) 1 κ 1 α + sɛ 0 + So the nverse Laplace transform of 1/s s gven by ) L 1 1 δt) σ L 1 1 s κ 2 α + sɛ 0 + κ ) δt) σ κ 2 ɛ L s + α ɛ 0 + ɛ 0 δt) σ ) κ 2 ɛ e α ɛ + ɛ t 0 ut) 0 δt) + ζ t). wth γ ζ t) σ κ 2 ɛ e γ t ut) 0 α + σ ) 1ɛ0 α + σ ) ɛ 0 ɛ 0 Ths agrees wth Roden and Gedney s [4] result. 4 Implementaton by Convoluton At the end, we should convolve the functon ζ t) wth a dfferental operator. Let us refer to the tme varable t as n for the n th grd coordnate and 4

5 use a super ndex for t. Let us call the tme convoluton at some gven x, as ψ n x and at the grd pont x, n t). Then by the defnton of convoluton ψ n ζ x ) n n t 0 ) n t τ ζ x τ)dτ We now make use of the fact that we have a the data n a grd, and f a staggered grd method s used, the partal dervatve, s defned half a tme step between and m + 1) t, so ψ n n 1 m0 n 1 m+1) t ) n t τ ζ τ)dτ m+1) t ) n m+1/2) ζ τ)dτ m0 n 1 m0 Z m) ) n m+1/2), where Z m) m+1) t ɛ 0 κ 2 ɛ 0 κ 2 ζ τ)dτ m+1) t e γ t m+1) t e γ t ut) γ σ ɛ 0 κ 2 γ e mγ t e γ t 1 ) κ 2 α + e ) a e α + ɛ 0 α + ) [ ) ɛ 0 e α + t ɛ 0 1 ] wth a κ 2 α + [e α + ) t ɛ 0 1 ] 5 κ 2 α + b 1)

6 as n Roden and Gedney s [4]. Here as n Komattsch and Martn [1]. ) b e α + t ɛ The recurson formula We found ψ n wth so n 1 m0 Z m) ) n m+1/2) Z 0) ) n+1/2 + n 2 m0 Z m + 1) ) n m+1+1/2), Z m + 1) a e α + ) m+1) t ɛ 0 Z m)e α + ) t ɛ 0 Z m)b. ψ n Z 0) ) n+1/2 + n 2 m0 Z m + 1) ) n m+1+1/2) n 2 Z 0) ) n+1/2 + b Z m) ) n 1 m+1/2) m0 Z 0) ) n+1/2 + b ψ n 1 a ) n+1/2 + b ψ n 1 snce Z 0) a. Ths s equaton 26) n Komattsch and Martn [1]. Ths equaton, thanks to the recursve propertes of the exponental, provdes an effcent mplementaton of the boundary layer method. 4.2 A partcular mplementaton Komattsch and Martn [1] plemented the CPMBL, as follows: For the functon they used the symbol d and defned as d x ) d 0 x /L) N 6

7 where: x 1 x, x 2 y, x 3 z N 2 d 0 N + 1)v log R c 2L, v velocty R c L n d 100 thckness of absorbng layer Also they chose 1.0 The value of α s pcked lnear from the maxmum at the PML entrance wth a value of α max πf 0 wth f 0 the domnant frequency of the source wavelet) and 0 at the boundary. At the boundary they mposed Drechlet boundary condtons of v x v z 0. The varables: d, a, b, and α should be computed both at any nteger and nteger+half grd locaton accordng to the staggered grd recep. References [1] Komattsch M and R. Martn. An unsplt convolutonal perfectly matched layer mproved at grazng ncdence for sesmc wave equaton. Geophyscs, 725):SM155 SM167, [2] D. Pasalc and McGarry R. Convolutonal perfectly matched layer for sotropc and ansotropc acoustc wave equatons. SEG Techncal Program Expanded Abstracts, pages , [3] Martn R, D. Komattsch, and A. Ezzan. An unsplt convolutonal perfectly matched layer mproved at grazng ncdence for sesmc wave propagaton n poroelastc meda. Geophyscs, 734):T51 T61, [4] J. A. Roden and S. D. Gedney. Convoluton pml cpml): an effcent fdtd mplementaton of cfs-pml for arbtrary meda. Mcrowave and Optcal Technology Letters, 275): ,