Improved Perfectly Matched Layers for Acoustic Radiation and Scattering Problems

Transcription

1 Excerpt from the Proceedigs of the COMSOL Coferece 2008 Haover Improved Perfectly Matched Layers for Acoustic Radiatio ad Scatterig Problems Mario Zampolli *,1, Nils Malm 2, Alessadra Tesei 1 1 NURC NATO Research Cetre, La Spezia (Italy), 2 COMSOL AB, Stockholm (Swede), *Correspodig author: Viale Sa Bartolomeo 400, La Spezia, Italy, zampolli@urc.ato.it Abstract: Perfectly matched layers (PML) are a efficiet alterative for emulatig the Sommerfeld radiatio coditio i the umerical solutio of wave radiatio ad scatterig problems. The key igrediet of the PML formulatio is the complex scalig fuctio, which cotrols the aisotropic dampig of the PML. The objective of this study is to propose a modified complex scalig fuctio capable of providig the user with two advatages: (i) miimizatio of the spurious reflectios at the physical domai-pml iterface for propagatig ad evaescet fields at all agles of icidece ad (ii) stability with respect to the frequecy parameter (which reduces the meshig effort i broadbad applicatios). Numerical results are preseted for radiatio from a circular pisto ad for scatterig from a rigid sphere. Overall, the modified formulatio is more stable at lower frequecies, while some potetial difficulties arisig i high-frequecy radiatio problems remai to be addressed. Keywords: acoustics, radiatio, scatterig, perfectly matched layers, waves. 1. Itroductio The capability of emulatig the Sommerfeld radiatio coditio, which requires that outgoig waves propagate out towards ifiity i the absece of reflectig boudaries, is a critical compoet of ay umerical code cocered with the solutio of wave problems. For tools based o bouded computatioal domais, such as fiite-elemet tools, the fiite-sized computatioal regio is trucated by a outer boudary. This boudary represets the ideal iterface betwee the fiite regio, modeled by the computatioal domai, ad the surroudig ifiite medium. I order to satisfy the radiatio boudary coditio, outgoig waves must traverse such a ideal boudary without beig reflected. Oe method for defiig o-reflectig boudaries cosists of surroudig the fiite computatioal regio with a perfectly matched layer (PML). The PML is a o-physical layer, iside which the wave equatio has bee modified with a aisotropic dampig, which icreases with distace i the directio perpedicular to the iterface with the physical domai. The result is that waves eterig the PML are absorbed oly i the outgoig directio, while the wave compoets tagetial to the iterface betwee the physical domai ad the PML remai uaffected. This approach was itroduced origially by Béreger [1] i 1994 for electromagetic waves. Reviews of PML research, with a particular emphasis o acoustics, ca be foud for example i Refereces [2] ad [3]. The mai advatages of the PML formulatio, compared to other umerical radiatio boudary coditios, are the relatively straightforward implemetatio via a complex coordiate scalig, ad the adaptability to geeric covex boudaries [3, 4]. This makes it possible to miimize the size of the computatioal domai, particularly i those cases where the physical domai caot be circumscribed easily by spheroids or similar shapes, which are usually required by other umerical radiatio boudary coditios like Dirichlet-to-Neuma maps or ifiite elemets. Oe major difficulty associated with PML formulatios is the choice of the scalig fuctio, which defies the absorptio of the outgoig waves i the PML. I Sectio 2 it is show how the scalig fuctio proposed i Ref. [3], ad the correspodig versios implemeted i Comsol [5], suffer from iaccuracies at very low frequecies (ka << 1, where k is the acoustic wave umber ad a is a characteristic size of the computatioal domai). The source of the iaccuracies lies i the discretizatio of the evaescet wave field compoets, which decay steeply iside the PML. Examples are provided for two differet test problems: radiatio from a baffled rigid circular pisto, ad plae wave scatterig from a rigid sphere. A modificatio to the scalig strategies of Refs. [3, 5] is proposed i Sec. 3. The ew PML formulatio is more accurate at the low to mid

2 frequecies, compared to the scaligs of Refs. [3, 5], while the high-frequecy performace for radiatio problems requires some further work. 2. Iaccuracies at low frequecies, caused by the evaescet wave-field A aalysis of the plae-wave dampig i a fiite-thickess PML shows that a oewavelegth thick PML, defied as i Refs. [3, 5], reduces spurious reflectios to about -100dB relative to the outgoig wave. The cotiuous aalysis also shows that evaescet waves, which are aturally decayig, are absorbed eve more strogly, depedig o the expoet of the evaescet wave (Fig. 4 i Ref. 3). O the other had, discretizatio problems arise with the strogly evaescet wave-field at very low frequecies. 2.1 Radiatio from a baffled pisto The first test problem to illustrate the difficulties of discretizig the evaescet wave field at low frequecies is that of the radiatio from a baffled rigid circular pisto (Fig. 1). Figure 2. Soud pressure level o the axis at a distace of 0.1a. Note the slight deviatio of the COMSOL solutio from the referece at ka<0.1. I the figure, results for the PML with quadratic scalig [5] ad 12 layers of quadratic elemets are compared to a aalytical referece solutio (Eq , Ref. 6) i the frequecy bad ka = The agreemet betwee the umerical result ad the referece solutio is good at high frequecies, while errors are visible below ka=0.1. The problems at low ka are more visible i Fig. 3, showig the relative error i the pressure at a/10 for a umber of differet meshes. Figure 1. Pisto model geometry ad mid-resolutio mesh with the physical domai i the lower left corer, surrouded by PML domais. The pisto radius is a, ad the costat acceleratio o the pisto face is set to a omial value of c 2 /a, with c beig the speed of soud. The acoustic pressure, sampled at a/10 distace from the ceter of the pisto alog the axis of symmetry, is plotted as a fuctio of the odimesioal wave umber ka i Fig. 2. Figure 3. Logarithmic plot of the relative error as fuctio of ka for PMLs with 3, 6, ad 12 layers of quadratic elemets.

3 2.2 Scatterig from a rigid sphere Similar discretizatio errors as those ecoutered i the pisto radiatio problem occur also i scatterig problems. As a example, oe ca study the axisymmetric scatterig of a uit-amplitude plae wave from a rigid sphere (Fig. 4). Figure 6. Relative error as a fuctio of ka, for PMLs with 3, 6, ad 12 layers of quadratic elemets (logarithmic scale). The error is >10% at ka<0.1, also for the fier meshes. Figure 4. The rigid sphere is modeled with a PML i direct cotact with the surface, here show with a lowresolutio mesh. I Fig. 5, the backscattered pressure, sampled o the surface of the sphere, is plotted as a fuctio of ka, with a represetig the radius of the sphere. Also i this case, the agreemet with a aalytical referece solutio, obtaied from a spherical harmoic series represetatio [7], is very good at the higher frequecies, while discretizatio errors similar to those show i Fig. 3 appear at the lower frequecies (Fig. 6). 3. Modified PML scalig Oe way i which PML s ca be defied is via a complex-valued chage of coordiates [4]. I this paper, the stadard approach is modified by defiig a complex PML coordiate, i which the real ad the imagiary part are scaled separately. The correspodig coordiate trasformatio, illustrated here for a PML which absorbs waves i the x-directio, is defied by i s x = + + d pml ( s) x0 xr ( s) σ ( s ) s ω x0 where s=(x-x 0 )/(x 1 -x 0 ) is the dimesioless distace across the PML i the directio ormal to the iterface with the physical domai. This iterface is located at x 0, while x 1 deotes the outer boudary of the PML. Desigig a efficiet PML requires a careful choice of the dampig fuctio σ ad of the real part scalig fuctio, x r. 3.1 Ubouded scalig fuctios Figure 5. Backscatter soud pressure level sampled o the surface of the sphere. A straightforward plae-wave aalysis shows that for propagatig waves, the imagiary part of the scaled coordiate is resposible for the dampig of the outward travellig wave, while the real part of the scaled coordiate affects the oscillatio of the solutio iside the PML. I the cotiuous aalysis, it ca therefore be show [2] that a rapidly growig imagiary part of the

4 scaled coordiate leads to a efficiet PML. The scalig of the real part does ot affect the dampig of the propagatig wave compoets. Based o this observatio, Bermudez et al. suggest a dampig fuctio σ proportioal to 1/(x 1 -x), which correspods to a imagiary part of the scalig fuctio x pml proportioal to the logarithm of (1-s). This leads to a PML formulatio which is perfectly oreflectig for propagatig waves, provided that the mesh resolutio ca be icreased arbitrarily. 3.2 Low-ka correctios A similar aalysis applied to evaescet waves shows that the decay of the wave iside the PML is affected by the real part of the scalig fuctio, while the imagiary part of the scalig gives rise to spurious ati-causal waves which must be absorbed to avoid errors i the solutio. Hece, it is to be expected that the real part of the scalig fuctio, x r, becomes importat whe the features of the particular problem iduce substatial evaescet compoets. I particular, this teds to happe at low ka, where the PML eeds to resolve details o the geometry scale, a, ad the wave scale, 1/k, simultaeously. Ay fiite elemet implemetatio of a PML by defiitio has a limited resolutio, which is quatified i this paper by the umber of elemet layers through the PML thickess. Choosig a efficiet scalig fuctio for low ka requires distributig the available resolutio betwee the differet scales i the pressure field. A scalig whose real ad imagiary parts grow slowly as fuctios of the odimesioal PML coordiate, s, effectively moves the ier elemet layers closer to the physical domai, thereby icreasig the resolutio of the short scales. The errors at low ka, illustrated i Sectio 2, are caused by the scalig described i [3,5] ad built ito COMSOL Multiphysics: c x = x + (1 i) pml 0 s f with =2. This scalig does ot supply sufficiet resolutio to the rapidly decayig compoets of the wave field. Icreasig the expoet improves the result for low ka, but teds to decrease the accuracy for medium to high frequecies, sice the resolutio power at the propagatig wave scale decreases i this case. 3.3 A combied approach The above observatios suggest a scalig fuctio which combies the beefits of the polyomial, wavelegth-idepedet scalig of [3,5], but ehaced with a ka-depedet expoet, with the ubouded imagiary part suggested by Bermudez et al. [2]. Numerical experimets suggest the form c r xpml = x0 + A ( s + i log 2 ( 1 s ) i f where 1 log10 ka, ka < 1 r = 1, ka 1 ka 1 log < = 10, ka 10 i 10 1, ka 10 ad A=0.25 is a weakly problem-depedet parameter. 4. Numerical experimets The suggested scalig fuctio was implemeted as a User defied PML scalig i the COMSOL Multiphysics Acoustics Module. A structured mesh like the oes show i Fig. 1 ad 3 was used i all cases, together with secod order Lagrage shape fuctios. For the baffled pisto, a combiatio of a cylidrical PML i the radial directio ad a Cartesia PML i the axial directio was chose. As Fig. 7 shows, the suggested scalig fuctio ca produce a cosistetly accurate solutio for low to medium frequecies. Comparig to the scalig suggested by [3,5], the proposed method gives almost a order of magitude smaller error at the same computatioal cost for ka<10.

5 Figure 7. For the pisto problem, the suggested scalig fuctio reduces the error at low to mid frequecies cosiderably. Note, however, the rapid growth of the relative error above ka=10 for the ew scalig ad 6 elemet layers. However, for ka>10, the relative error grows rapidly with icreasig frequecy ad i the regio 10<ka<20 it exceeds the error produced by the stadard scalig by a order of magitude. A more detailed aalysis of the frequecy bad 10<ka<200 shows that the error usig COMSOL s stadard PMLs starts to icrease rapidly aroud ka=30 ad from there follows the same tred as the error produced usig the modified scalig suggested here. The authors believe that these errors are caused by evaescet wave field compoets with wavelegths larger tha c/f. The dampig of such waves is limited by the maximum value of the real part of the scalig fuctio, which is equal to A i the proposed scalig, but equal to 1 for the polyomial scalig. Iitial experimets support this reasoig, showig that icreasig A moves the poit where the error starts to rise towards higher ka. The results from the scatterig case, show i Fig. 8, are uambiguous. The suggested scalig performs better tha the stadard fixedexpoet polyomial scalig over the etire frequecy rage. I particular, the error obtaied with oly 3 elemet layers is well below 1% for ka>1. Figure 8. For the scatterig problem, the suggested scalig ad 3 elemet layers produces results comparable to the =2 polyomial scalig with 6 layers. 5. Coclusios The suggested combied scalig approach promises higher accuracy at lower computatioal cost for geeral simulatios with low to moderate ka, as well as for higher-frequecy scatterig problems. Further research is eeded to detect a priori which problems require a larger real part of the scalig fuctio. The possibility of a frequecy-adaptive or ubouded scalig of the real part is also beig cosidered as a possibility for future work. 6. Refereces 1. J.P. Béreger, A perfectly matched layer for the absorptio of electromagetic waves, J. Comput. Phys., 114, (1994). 2. A. Bermudez, L. Hervella-Nieto, A. Prieto, R. Rodriguez, A optimal perfectly matched layer with ubouded absorbig fuctio for timeharmoic acoustic scatterig problems, J. Comput.Phys., 223, , (2007). 3. M. Zampolli, A. Tesei, F.B. Jese, N. Malm, J.B. Blottma, A computatioally efficiet fiite elemet model with perfectly matched layers applied to scatterig from axially symmetric objects, J. Acoust. Soc. Am., 122, (2007).

6 4. F. Collio ad P. Mok, The perfectly matched layer i curviliear coordiates, SIAM J. Sci. Comput., 19, (1998). 5.COMSOL AB, Comsol Multiphysics Acoustics Module, User s Maual, (2007). 6. A.D. Pierce, Acoustics A Itroductio to Its Physical Priciples ad Applicatios, Acoustical Society of America, Melville NY (1991). 7. J.J. Bowma, T.B.A. Seior, P.L.E. Usleghi, Electromagetic ad Acoustic Scatterig by Simple Shapes, North-Hollad Publ. Co. (1969).