A BRIEF SUMMARY OF SOME PML FORMULATIONS AND DISCRETIZATIONS FOR THE VELOCITY-STRESS EQUATION OF SEISMIC MOTION ABSTRACT

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1 A BRIEF SUMMARY OF SOME PML FORMULATIONS AND DISCRETIZATIONS FOR THE VELOCITY-STRESS EQUATION OF SEISMIC MOTION JOZEF KRISTEK PETER MOCZO * AND MARTIN GALIS Faculty of Mathematcs Physcs and Informatcs Comenus Unversty Mlynská dolna F Bratslava Slovak Reublc (moczo@fmh.unba.sk) Geohyscal Insttute Slovak Academy of Scences Dúbravská cesta Bratslava Slovak Reublc Receved: March 9 009; Revsed: July 7 009; Acceted: July ABSTRACT The erfectly matched layer s an effcent tool to smulate nonreflectng boundary condton at boundares of a grd n the fnte-dfference modelng of sesmc wave roagaton. We show relatons between dfferent formulatons of the erfectly matched layer wth resect to ther three key asects - slt/unslt classcal/convolutonal wth the general/secal form of the stretchng factor. Frst we derve two varants of the slt formulatons for the general form of the stretchng factor. Both varants naturally lead to the convolutonal formulatons n case of the general form of the stretchng factor. One of them L-slt reduces to the well-known classcal slt formulaton n case of the secal form of the stretchng factor. The other R-slt remans convolutonal even for the secal form of the stretchng factor. The R-slt formulaton eventually leads to the equatons dentcal wth those obtaned straghtforwardly n the unslt formulaton. We also resent an alternatve tme dscretzaton of the unslt formulaton that s slghtly algorthmcally smler than the dscretzaton resented recently. We mlement the dscretzaton n the 3D velocty-stress staggered-grd fnte-dfference scheme - 4thorder n the nteror grd nd-order n the erfectly matched layer. Ke y word s : erfectly matched layer nonreflectng boundary condton fntedfference modelng sesmc waves. INTRODUCTION The erfectly matched layer (PML) s robably the most effcent method to revent reflectons of sesmc waves at artfcal boundares of the comutatonal regon that s at boundares of the dscrete satal grd. The mortance of PML n the numercal modelng of sesmc wave roagaton has been recently very well recognzed. Ths s clear from 459

2 numerous artcles e.g. Chen et al. (000) Collno and Tsogka (00) Komattsch and Trom (003) Marcnkovch and Olsen (003) Festa and Nelsen (003) Wang and Tang (003) Basu and Chora (004) Festa and Vlotte (005) Festa et al. (005) Martn et al. (005) Martn and Komattsch (006) Ma and Lu (006) Komattsch and Martn (007) Drossaert and Gannooulos (007ab) Moczo et al. (007) Martn et al. (008ab) Basu (009). Komattsch and Martn (007) rovded a very good comrehensve revew of the develoment of the PML theory and ts alcatons to the numercal modelng of sesmc wave roagaton ncludng the oneerng artcles (Bérenger ; Chew and Weedon 994; Chew and Lu 996) as well as the most recent ones. Therefore we refer to the revew by Komattsch and Martn (007) nstead of develong another revew here. At the same tme our short artcle was nsred by the fact that we have not found n the lterature exlanaton of relatons between dfferent PML formulatons or ther classfcaton. For examle Komattsch and Martn (007) frst exlaned the classcal slt (usng drectonal decomostons of the stress tensor and dvergence of the stress tensor) PML formulaton n velocty and stress and made a ont about the lmtaton of the formulaton n case of the grazng ncdence. Then they contnued wth the convolutonal PML (C-PML) technque that mroves the accuracy of the dscrete PML at the grazng ncdence. They onted out the advantage of the unslt formulaton whch does not requre the slt arts of the artcle velocty vector comonents and stress-tensor comonents and consequently does not ncrease the number of deendng varables. Whle t was easy to follow both nce exostons and understand dffcultes of the frst and advantages of the second formulaton we realzed that the artcle does not exlctly address the relatons of three key asects of the PML formulatons - slt/unslt wavefeld vs classcal/convolutonal algorthm vs general/secal form of the stretchng factor. Our concern was suorted by the mresson one may have from the two mentoned chaters n the artcle by Komattsch and Martn (007): slt classcal and secal make one formulaton unslt convolutonal and general make the other - no menton of the relaton between the two formulatons. We have not found exlanaton of the relatons and classfcaton of the formulatons n other artcles ether. In ths artcle we am to clarfy the relatons. We start wth the decomoston n the slt formulaton. Then we derve slt and unslt formulatons for the general and secal forms of the stretchng factors and show the relatons between them. Fnally as a comlementary result we show two alternatve tme dscretzatons of the C-PML formulaton the corresondng algorthms and results of numercal tests.. DIRECTIONAL DECOMPOSITION IN THE SPLIT FORMULATION Drectonal decomostons of the dvergence of the stress tensor and the stress tensor tself at a ont are the key asects of the slt PML formulaton. Here we brefly recall and exlan the decomoston before we formulate equatons for the slt formulatons. The equaton of moton wthout the body-force term and Hooke s law are () u j j 460

3 and j uk k j u j uj. () In order to nterret acton/meanng of only one of the stress-tensor dervatves on the r.h.s. of Eq.() e.g. xy x assume that xy x s the only non-zero satal stress-tensor dervatve. Then u x 0 u y xy x u z 0. Assumng u x ux 0 and u u 0 at some reference tme we have z z u y xy x xy u y x. (3) Ths means that the above secfcaton defnes a D roblem wth dslacement and stress olarzed n the y-drecton and roagatng n the x-drecton. Consder now the meanng of term xx x. Assume that xx x s the only non-zero satal stress-tensor dervatve. Then u x xx x u y 0 u z 0. Assumng u y uy 0 and u u 0 at some reference tme we have z z u x xx x xx x ux x. (4) Ths means that the above secfcaton defnes a D roblem wth dslacement and stress olarzed n the x-drecton and roagatng n the x-drecton. The meanng of the other satal stress-tensor dervatves n the equaton of moton and terms on the r.h.s. of Hooke s law can be shown analogously. Corresondngly the equaton of moton can be wrtten as x y z x xx x yx y zx z x x x u x y z y xy x yy y zy z y y y u x y z z xz x yz y zz z z z z u. (5) j Here means a body force actng at a ont and causng at that ont a moton olarzed n the -th drecton and havng tendency to roagate from that ont n the j-th drecton. Hooke s law can be arranged n the form u u u u u u u u u xy uy x ux y u u u u xx x x y y z z yy x x y y z z zz x x y y z z yz z y y z zx z x x z (6) 46

4 Here Mu j (M beng an arorate modulus) means that art of the stress-tensor comonent at a ont whch has tendency to roagate from that ont n the j-th drecton. Eqs.(5) and (6) thus show drectonal decomostons of the dvergence of the stress tensor and the stress tensor tself at a ont resectvely: the decomostons are determned by the drectons of the satal dervatves. 3. THE PML FORMULATIONS Instead of Hooke s law () for the sotroc medum we can consder the more concse general form c c u (7) j jkl kl jkl k l where c jkl s tensor of the elastc coeffcents. Because our goal s a PML for the velocty-stress FD scheme we wll further consder equatons for the artcle velocty v v (8) Consderng qr xyz j j j cjkl k l v. (9) 3.. The Slt Formulaton let denote a coordnate drecton erendcular to a lanar nterface between the nteror regon and the PML and q r drectons erendcular to drecton. Decomoston of the artcle velocty and stress tensor qr v v v (0) qr j j j () yelds the equaton of moton and Hooke s law n the forms v j j j () qr v j j j (3) j c jkl k l l qr j c jkl k l l v (4) v. (5) An alcaton of the Fourer transform to Eqs.() and (4) gves v (6) j j j j c jkl k l l v. (7) 46

5 We use the same symbols for the quanttes n the frequency and tme domans. A relacement of the satal dfferentaton wth resect to x n Eqs.(6) and (7) by the satal dfferentaton wth resect to x x s x wth the so-called stretchng factor s and and beng n general functons of x gves ether (8) (9) s v (0) j j j j s c v () jkl k l l or v j j j () s j c jkl v k l l. (3) s We wll recognze the L-slt formulaton based on manulatons wth Eqs.(0) and () whereas Eqs.() and (3) wll be the bass for the R-slt formulaton L-Slt A substtuton of s n Eqs.(0) and () accordng to Eq.(9) leads to j j j v (4) Defne j c jkl v k l l. (5) ( ) = v + (6) j ( ) = j. (7) + 463

6 Then Eqs.(4) and (5) become æ ö ç + = j j j + ( ) çè ø v (8) æ ö ç + j = cjklv k l l + j ( ). (9) çè ø In order to remove the magnary unt from the denomnator we rewrte Eqs.(6) and (7): ( + ) ( ) = v (30) ( + ) j ( ) = j. (3) An alcaton of the nverse Fourer transform to Eqs.(8)(3) yelds v + v = j j j + (3) j + j = cjklv k l l + j (33) + = v (34) Here and j + j = j. (35) j are functons of tme. It s clear that and j are addtonal varables (so-called memory varables) obeyng ordnary dfferental equatons (34) and (35). They are ntroduced n order to avod a drect calculaton of the convolutons that would otherwse aear n Eqs.(3) and (33) and consequently avod memory requrements for the hstory of the artcle velocty and stress. Eqs.(0) () (3) (5) and (3)(35) make the fnal system to be solved. If we consder the secal case wth and 0.e. s = + (36) then Eqs.(3)(35) consderng defntons (6) and (7) reduce to the well known slt PML formulaton v + v = (37) j j j j j c jkl k l l + = v. (38) 464

7 Note that n ths secal case we could n fact drectly aly the damng terms roortonal to v and j n Eqs.() and (4). An alcaton of the Fourer transform to the modfed equatons would then reveal that the addton of the damng terms s equvalent to relacement of the dfferentaton wth resect to x by the dfferentaton wth resect to x R-Slt Substtutng defnton of the stretchng factor s Eq.(9) nto Eqs.() and (3) we obtan b v æ ö = - j j ç j è a + ø (39) where æ b ö j = - cjklv k l l (40) ç è a + ø a b. (4) Defne Then b ( ) =- j j j a + (4) b j ( ) =- cjklv k l l. a + (43) v j j j (44) j cjklv k l l j. (45) In order to remove the magnary unt from the denomnator we rewrte Eqs.(4) and (43): a b (46) j a j j bc v. (47) j j jkl k l l An alcaton of the nverse Fourer transform to Eqs.(44)(47) yelds 465

8 Here and v j j j (48) j cjklv k l l j (49) a b (50) j j j a bc v. (5) j j jkl k l l j are functons of tme. Smlarly to the L-slt case and j are addtonal (memory) varables. Lookng at Eqs.(3) (5) and (48)(5) we can realze that t s ossble to sum u Eqs.(3) and (48) as well as Eqs.(5) and (49). We obtan v j j j j j j (5) j cjklvk l l j cjklv k l l. (53) Thus Eqs.(50)(53) make the fnal system of equatons to be solved. Note that n the secal case wth and 0 both arameters a and b are equal to. Eqs.(48)(53) however do not change whch means that the R-slt formulaton remans convolutonal even n the case of the secal form of the stretchng factor. 3.. The Unslt Formulaton In the so-called unslt formulaton we manulate drectly wth the entre equaton of moton and Hooke s law that s wth Eqs.(8) and (9). An alcaton of the Fourer transform to these equatons yelds v (54) j j c v. (55) j jkl k l A relacement of the satal dfferentatons wth resect to resect to x yelds x by dfferentatons wth b v a j j j j j j (56) b j cjklvk l l cjkl k l l a v. (57) 466

9 Substtuton of varables and j defned by Eqs.(4) and (43) n Eqs.(56) and (57) resectvely and a subsequent alcaton of the nverse Fourer transform to the equatons yeld v j j j j j j (58) j cjklvk l l j cjklv k l l. (59) Clearly the addtonal varables and j satsfy dfferental equatons (50) and (5) and Eqs.(58) and (59) are the same as Eqs.(5) and (53). In other words we see that the unslt formulaton leads to the same fnal equatons as the R-slt case of the slt formulaton. Let us recall that the resented unslt formulaton s equvalent to that ntroduced by Komattsch and Martn (007) The Summary of the Formulatons All treated formulatons are summarzed n Table whch clearly mas dfferent formulatons and ther relatons. For concseness the table lsts only equatons for the artcle velocty. It s clear that the general form of the stretchng factor s gven by Eq.(9) naturally mles the addtonal functons (memory varables) n both the slt and unslt formulatons f a drect calculaton of the convolutonal terms s to be avoded. In the case of the L-slt formulaton and secal choce of the stretchng factor gven by Eq.(36) the resultng equatons reduce to the well known equatons wth a smle damng term - Eqs.(37) and (38). The case of the R-slt formulaton eventually leads to the equatons dentcal wth those obtaned straghtforwardly n the unslt formulaton. 4. TIME DISCRETIZATION OF THE UNSPLIT FORMULATION Consder the followng aroxmatons at the tme level m: m m m (60) m m m. (6) t Alcaton of aroxmatons (60) and (6) to Eq.(50) yelds m m m at b t a t at j j j. (6) 467

10 Table. Summary of all formulatons for velocty n the PML regon erendcular to drecton. Formulaton for velocty n the PML regon erendcular to drecton : Slt: Unslt L-Slt R-Slt General Case: Ω Ω Ω leads to: Ω Ω Secal Case: Ω Ω Ω Ω 468

11 m If we relate Eq.(58) to the tme level m we need. Usng Eqs.(60) and (6) we obtan Usng Eq.(63) we can rewrte Eq.(58): m m m b t a t at j j j. (63) m bt m m m j j j j j j a t a t v. (64) Then the fnal form of the equaton of moton wth the memory varables and corresondng addtonal equatons s obtaned after substtutng a and b from Eq.(4) n m Eqs.(6) and (64) and aroxmatng v by the central-dfference formula: v t t t m m j j j t m m m v j j j t t t (65) m m m t j j j. (66) The tme dscretzaton of the consttutve relaton (59) and addtonal equatons for the memory varables s analogous. The fnal system s m m t m j j t cjkl k l l t v m m j cjklvk l l t t m m j j t t c t m jklvk l l. (67) (68) The dscretzaton gven by Eqs.(65)(68) s an alternatve to the dscretzaton resented by Komattsch and Martn (007). Ther dscretzaton can be wrtten n the form smlar to that n Eqs.(65)(68): 469

12 . v m m m v j j j t ex t ex t m m j j j m m ex t m ex t j j j ex t m m m j j t c jklvk l l ex t m m j cjklvk l l (69) (70) (7) m m j ex tj m ex t c jkl k l l. v The dfference between the two dscretzatons s n coeffcents. It s due to dfferent tme ntegratons of Eqs.(50) and (5). The ntegraton chosen by Komattsch and Martn (007) would be the exact ntegraton n case of homogeneous Eqs.(50) and (5) that s f b 0. Both dscretzatons are nd-order accurate n tme f b 0 whch s the case. Numercal tests are necessary to comare the two aroaches. 5. NUMERICAL COMPARISON OF THE TWO DISCRETIZATIONS We resent numercal results for a wavefeld generated by a ont sngle force n a homogeneous sotroc elastc medum. The confguraton s close to that consdered by Komattsch and Martn (007). The 3D model of the relatvely thn slce has shae of the rectangular aralleleed wth dmensons m 3. P-wave seed S-wave seed and densty are resectvely 3300 m/s m/s and 800 kg/m 3. The ont sngle-force source s located at x = 790 m y = 470 m and z = 30 m assumng that one corner of the rectangular aralleleed s located at the orgn of the Cartesan coordnate system and ts short edge s arallel wth the x-axs. The force s orented at 35 n the (xy) lane. The source-tme functon used n the numercal smulatons s Gabor sgnal - a harmonc carrer wth a Gaussan enveloe s s s s() t ex tt cos tt. Here f t 0t s (7) 470

13 f = 5 Hz s redomnant frequency s = 4 controls the wdth of the sgnal = / s a hase shft and ts 0.45 s f. The comutatonal doman was covered wth a unform staggered grd. Wth the grd sacng of 0 m n the three Cartesan drectons the numbers of grd sacngs n the three drectons were and 640. The smulatons for the nteror grd were erformed usng the 4th-order n sace nd-order n tme velocty-stress staggered-grd fntedfference scheme. A 00 m (0 grd sacngs) thck PML was aled on the sx sdes of the aralleleed. The same damng rofle was aled n all coordnate drectons: 0 L xyz 0 = 34.9 L = 00 m. The other arameters are = = f. For exlanaton of ths choce see Komattsch and Martn (007). At the external boundares of the PML (and thus the grd) the artcle velocty s set equal to zero (rgd boundary). The fnte-dfference aroxmaton aled n the PML s nd-order n sace and tme. The tme ste was.4 ms. The smulated tme wndow s 40 s the number of erformed tme stes s Fg.. Total energy n the nteror art of the grd (PML not ncluded) as a functon of tme levels. The 4th-order n sace and nd-order n tme velocty-stress staggered-grd fnte-dfference scheme was aled to the nteror art of the grd. The nd-order n tme and sace scheme was aled n the PML. One curve corresonds to the tme dscretzaton resented n ths artcle the other to the dscretzaton suggested by Komattsch and Martn (007). 47

14 Followng Komattsch and Martn (007) we evaluate the total energy n the nteror art of the grd (the PML excluded) as a functon of tme. The total energy defned as E v jj (73) should decay wth tme after the source stos radatng energy. Fg. shows the total energy n the grd as a functon of tme level obtaned by two numercal smulatons. One smulaton was erformed wth the use of our dscretzaton Eqs.(65)(68) the other wth the use of the dscretzaton resented by Komattsch and Martn (007) Eqs.(69)(7). The two curves ractcally concde for all tme levels. We can see ths agreement as a confrmaton of accuracy of both dscretzatons. We note that the energy-decay curve dffers slghtly from that resented by Komattsch and Martn (007): The oscllatons are due to the 4th-order scheme aled n the nteror grd and not n the PML. We saw ths clearly on the snashots (not shown here). The resultng resdual hgh-frequency numercal nose s resonsble for the fact that the amount of energy stablzes n the grd - the grd s not caable to rocess the hgh-frequency nose. However the nose level s about 8 orders of magntude weaker than the useful sgnal. Therefore t does not ose a ractcal roblem. 6. CONCLUSIONS The PML may be slt or unslt classcal or convolutonal wth the general or secal form of the stretchng factor. We showed relatons between dfferent formulatons of the PML wth resect to ther three key asects. We derved two varants of the slt formulatons for the general form of the stretchng factor. The L-slt varant has the stretchng factor on the left-hand sde of the equaton of moton and consttutve law the R-slt varant on the rght-hand sde. Both varants naturally lead to convolutonal formulatons n case of the general form of the stretchng factor. The L-slt varant reduces to the well-known classcal slt formulaton n case of the secal form of the stretchng factor. The R-slt formulaton remans convolutonal even for the secal form of the stretchng factor. The R-slt formulaton eventually leads to the equatons dentcal wth those obtaned straghtforwardly n the unslt formulaton. We resented a tme dscretzaton of the unslt formulaton whch s a slghtly algorthmcally smler alternatve to the tme dscretzaton resented by Komattsch and Martn (007). The latter s shown n the form consstent wth our dscretzaton. We mlemented both dscretzatons n the 3D velocty-stress staggered-grd fnte-dfference scheme. The nteror grd was solved wth the 4th-order whereas the PML wth the ndorder scheme n sace both beng the nd-order n tme. Numercal tests showed a very good level of agreement of the two dscretzatons. Acknowledgements: Ths work was suorted n art by the Slovak Research and Develoment Agency under the contract No. APVV (roject OPTIMODE) VEGA Project /403/07 and ITSAK-GR MTKD-CT Project. 47

15 References Basu U Exlct fnte element erfectly matched layer for transent three-dmensonal elastc waves. Int. J. Numer. Meth. Eng Basu U. and Chora A.K Perfectly matched layers for transent elastodynamcs of unbounded domans. Int. J. Numer. Meth. Eng Bérenger J.P A erfectly matched layer for the absorton of electromagnetc waves. J. Comut. Phys Bérenger J.P Three-dmensonal erfectly matched layer for the absorton of electromagnetc waves. J. Comut. Phys Chen Y.H. Coates R.T. and Robertsson J.O.A Extenson of PML ABC to Elastc Wave Problems n General Ansotroc and Vscoelastc Meda. Schlumberger OFSR Research Note Schlumberger Cambrdge U.K. Chew W.C. and Lu Q Perfectly Matched Layers for elastodynamcs: a new absorbng boundary condton. J. Comut. Acoust Chew W.C. and Weedon W.H A 3-D erfectly matched medum from modfed Maxwell s equatons wth stretched coordnates. Mcrow. Ot. Technol. Lett Collno F. and Tsogka C. 00. Alcatons of the PML absorbng layer model to the lnear elastodynamc roblem n ansotroc heterogeneous meda. Geohyscs Drossaert F.H. and Gannooulos A. 007a. Comlex frequency shfted convoluton PML for FDTD modellng of elastc waves. Wave Moton Drossaert F.H. and Gannooulos A. 007b. A nonslt comlex frequency-shfted PML based on recursve ntegraton for FDTD modelng of elastc waves. Geohyscs 7 T9T7. Festa G. Delavaud E. and Vlotte J.-P Interacton between surface waves and absorbng boundares for wave roagaton n geologcal basns: D numercal smulatons. Geohys. Res. Lett. 3 L0306. Festa G. and Nelsen S PML Absorbng Boundares. Bull. Sesmol. Soc. Amer Festa G. and Vlotte J.P The Newmark scheme as velocty-stress tme-staggerng: An effcent PML mlementaton for sectral-element smulatons of elastodynamcs. Geohys. J. Int Komattsch D. and Martn R An unslt convolutonal Perfectly Matched Layer mroved at grazng ncdence for the sesmc wave equaton. Geohyscs 7 SM55SM67. Komattsch D. and Trom J A erfectly matched layer absorbng boundary condton for the second-order sesmc wave equaton. Geohys. J. Int Marcnkovch C. and Olsen K On the mlementaton of erfectly matched layers n a threedmensonal fourth-order velocty-stress fnte dfference scheme. J. Geohys. Res Ma S. and Lu P Modelng of the erfectly matched layer absorbng boundares and ntrnsc attenuaton n exlct fnte-element methods. Bull. Sesmol. Soc. Amer Martn R. and Komattsch D An otmzed convoluton-erfectly matched layer (C-PML) absorbng technque for 3D sesmc wave smulaton based on a fnte-dfference method. Geohys. Res. Abstracts

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