High Order Nystrom Method for Acoustic Scattering

Transcription

1 High Ordr Nystro Mthod for Acoustic Scattring Kun Chn, a), Siing Yang, Jiing Song, b) and Ron Robrts, Dpt. Elctrical and Coputr Enginring, Iowa Stat Univrsity, As, IA, USA Dpt. Arospac Enginring, Iowa Stat Univrsity, As, IA, USA Cntr for Nondstructiv Evaluation, Iowa Stat Univrsity, As, IA, USA a) b) Corrsponding author: Abstract. Whil high frquncy approxiation thods ar widly usd to solv flaw scattring in ultrasonic nondstructiv valuation, full wav approachs basd on intgral quations hav grat potntials du to thir high accuracy. In this work, boundary intgral quations for acoustic wav scattring ar solvd using high ordr Nystro thod. Copard with boundary lnts thod, it faturs th coincidnc of th sapls for intrpolation basis and quadratur, which aks th far-fild intraction fr fro nurical intgration. Th singular intgral is dalt with using th Duffy transforation, whil fficint singularity subtraction tchniqus ar ployd to valuat th nar singular intgrals. This approach has th as to go high ordr so highly accurat rsults can b obtaind with fwr unknowns and fastr convrgnc, and it is also anabl to incorporat fast algoriths lik th ulti-lvl fast ulti-pol algorith. Th convrgnc of th approach for diffrnt ordrs of lnts and intrpolation basis functions is invstigatd. Nurical rsults ar shown to validat this approach. INTRODUCTION In ultrasonic nondstructiv valuation (NDE), flaw scattring plays a vry iportant rol. For xapl, in th rciprocity basd ultrasonic asurnt odl, th rcivd voltag of th syst is proportional to an intgral on th flaw surfac rlatd to surfac tractions and vlocitis, and th Thopson-Gray odl gos on stp furthr, stating that th rcivd voltag is proportional to th scattring aplitud []. In this papr, w confin our intrst to acoustic wav scattring by flaws, for which nurous solving approachs xist. solutions ar accurat, yt only availabl for sipl gotris lik sphr, cylindr, tc.[]; approxiat thods such as Kirchhoff and Born approxiations [] ar vry popular aong practitionrs du to th capability to dlivr rsults in ral ti, but suffr fro inaccuracis sinc it is hard to track th rror; full wav solutions, aong which th finit diffrnc thod (FDM) [4], finit lnts thod (FEM) [5] and boundary lnts thod (BEM) [6] hav ost popularity, giv rlativly or accurat rsults than approxiat thods, but ar usually coputational xpnsiv and ti consuing, which rquirs incorporation of fast algoriths [7-8] if on intnds to solv scattring by larg and coplx flaws. In this work, th acoustic scattring by a hoognous flaw is forulatd as a boundary intgral quation (BIE), which is thn discrtizd into a linar syst using high ordr Nystro thod [8]. In trs of local corrction for Nystro thod, an fficint sch is proposd to addrss th singular and nar-singular intgrals for high ordr lnts. Copard with boundary lnts thod, our iplntation of Nystro thod has th spcial fatur that th sapls for intrpolation and quadratur coincid, as a rsult of which thr is no nd to prfor nurical intgral for far fild intraction. Furthror, this approach njoys as to go high ordr and anability for incorporating fast algoriths lik th fast ulti-pol algorith, which aks it appropriat to solv practical larg and coplx probls. Nurical rsults for sphrs and sphroids ar copard with analytic solution to validat th approach, and scattring by othr shaps of flaws ar also prsntd. FORMULATION Th boundary intgral quation for acoustic scattring by a flaw nclosd by th surfac S is writtn as g( xx, ) pinc ( x) p( ) g(, ) q( ) ds x x x x p( ), S S n x x ()

2 whr x and x ar th fild and sourc points, pinc is th incidnt prssur (assud to b a plan wav), p is th total prssur on th flaw surfac, q p n is th noral drivativ of th prssur, g( xx, ) xp( ikr) (4 R), R R x- x, k is th wavnubr, and n nˆ, with nˆ bing th outward unit surfac noral. To solv th abov probl, w nd on or condition sinc both p and q ar unknowns. Hr w considr th soft boundary condition p 0 and hard boundary condition q 0. (a) (b) (c) FIGURE. Triangular sh for a sphr (a) and flat (b) and quadratic (c) lnts. To discrtiz th abov intgral quation into a atrix quation, w nd a dscription of th flaw surfac, as wll as a st of basis functions with which w can xpand th surfac prssur filds. In our work, w choos to approxiat th flaw surfac with a nubr of triangular lnts, as illustratd in Fig. for a sphr, sinc triangular sh is asy to gnrat and rfin. But it should b ntiond that quadrilatral lnts ar also quit popular, and in so advancd applications, thy ar usd in conjunction with triangular lnts [9]. Mathatically, th points on an lnt is dtrind by x ij () ξ x B ij () ξ () ij whr xij is th sapld points (rd dots in Fig. ) for th triangl, and B () ξ is th Lagrang intrpolation polynoial dfind via products of Silvstr-Frrari polynoials [0]. Th sapld nods corrspond to a st of siplx coordinats in an isoscls right triangl givn by ξ ij i, j, M i j M () with i, j 0,,, M, i j M ( M is th ordr of th lnt). Th first and scond ordr lnts ar shown in Fig. (b) and (c), whr and 6 nods ar usd rspctivly. Nxt, w xprss th prssur and its noral drivativ on an lnt p ( x ) and q ( x ) using intrpolation fro k k its valus on so sapld points p and q (grn dots in Fig. ) k p ( x) p L k ( ξ ) (4) k q ( x) q L k ( ξ) (5) i j whr th intrpolation polynoial is dfind as L () ξ c ( )( ) and th cofficints ar sought such as to satisfy k kij l th l Lk( ξ l) k, whr ξl is th l sapld nod for prssur, k is th Kronckr dlta function, and L k() ξ Lk() ξ J( ξk) J() ξ is noralizd fro L () k ξ for convninc. Hr, ξl ar not chosn arbitrarily, but xactly th sapls for th Gaussian quadratur rul on a triangl. Th nubr of sapls for th first 4 quadratur ruls, nubrd 0-, is,, 6,, rspctivly. This choic givs bst prforanc in th nurical intgration, and has othr advantags to b rvald latr. Also, w notic that w can asily go high ordr by incrasing th ordr of th intrpolation function Lk () ξ, which is indpndnt of th gotric dscription. To illustrat th procdur of Nystro discrtization, w writ () in a or copact for as K(, ) ( xx x) ds pinc ( x ) (6) S whr p, K g n for hard boundary condition, and q, K g for soft boundary condition. For convninc, w drop th p in () now sinc th tr is asily found and will b addd latr. By valuating (6) at x, whr (,) l is th global indx for th nod l in lnt, aftr intrpolating th unknown in ach lnt fro th nod valus k, w hav

3 k K (, ) L k( x x x) ds p inc( x) (7) S th whr K ( xx, ) K( xx, ) ( x ), ( x) is th support of th lnt, Lk Lk if p, and Lk L k if q. In a or concis annr, w hav n Kn pinc (8) th whr n n(, k) is th global indx for th k nod in lnt, and K K (, ) L ( ) ds x x x (9) n k S whr S is lnt. Whn x is far fro S, th krnl is quit sooth, and w us Gaussian quadratur rul to yild Kn wk l ( x, ξl) Lk( ξl) (0) wk k ( x, ξk), no su on k l whr w hav usd th fact that Lk( ξl) k. W can s that this is vry sipl, just th valu of th K( x, xn) ultiplid by a wight, and thr is no nd to prfor intgral any or. This siplicity aks it asir for us to incorporat fast algoriths lik fast ulti-pol thod. Whn x S, th intgral is singular; but on can show that K( xx, ) O( R), R 0, thrfor, th intgral can b prford using th Duffy transforation dirctly. Howvr, whn x S but ar in th vry vicinity of S, w hav th nar-singularity issu, naly th intgrand has hug variations, rndring nurical intgration, vn if it b high ordr Gaussian quadratur, hard to convrg. Sinc w ar using high ordr Nystro thod, this issu has to b addrssd if w ar to fulfil th advantags of high ordr thod high accuracy and high ordr convrgnc. To this nd, w propos an fficint singularity subtraction sch. In dtail, for soft boundary condition, w nd to valuat th following intgral ikr J ( ξk ) I Lk ( ξ) ds () S 4 RJ( ξ) Th basic ida of th singularity subtraction is that w subtract th asyptotic tr fro th singular krnl so th rsultd krnl bcos sooth, whil th tr xtractd is valuatd analytically. To b spcific, w hav ikr J( ξk) J( ξk) Lk( ξ) I Lk ( ξ) ds d S 4R 4 R f J( ) 4 ξ () ξ Rf whr Rf xx f, xf x( ξ0) h ( ξ0), h = ξ- ξ 0, is th tangnt vctor, and ξ0 is th siplx coordinat for th xpansion point. It is th point on th lnt which is narst to th obsrvation point, and can b found by Nwton thod. Th scond intgral can b carrid out on a flat triangl spannd by x for f ξ in th rfrnc isoscls right triangl dnotd as. Sinc Lk ( ξ ) is polynoials of, this intgral can b valuatd analytically in a rcursiv annr []. Th first intgral has now a sooth intgrand, and can b valuatd using high ordr Gaussian quadratur. For hard boundary condition, w will ncountr th intgral g( xx, ) I L ( ) ds 4 ξ S n () ikr ikr ˆ ( ) L ds 4 n R ξ S R which can b valuatd as I I I, whr and ikr ikr k I ˆ L( ) ds 4 n R S R Rf R ξ (4) f k 4 R f 4 R f ( )( ) ( )( ) I L ξ R dξ L ξ R dξ (5)

4 Sinc L( ξ)( ) Ris rly a polynoial of, I can b valuatd analytically. And I can b handld by Gaussian quadratur. It should b pointd out that, though th abov singularity subtraction is intndd for nar singular intgrals, thy ar qually applicabl to th singular intgrals, naly whn x S. It can b sn that th abov approach can dal with not just flat lnts, but curvd ons. It is also capabl of handling krnls of diffrnt singularitis, not confind to wakly singular krnls. Now w ar abl to calculat all th lnts of K n. Thn w can ov on to solv th linar syst (8) to gt th filds on th surfac, fro which w can calculat th scattring aplitud as A( aˆ ; ˆ ) [ ( ˆ ˆ ) ]xp( ˆ i as ikp s q ik s) ds 4 n a x a (6) S whr a ˆ i and a ˆ s ar unit dirction vctor for th incidnt fild and scattring dirction, and p p p0, q q p0 with p 0 th aplitud of p inc. Scattring Aplitud.5.5 Nystro Scattring Aplitud Nystro /dgr /dgr FIGURE. Scattring aplituds for soft sphr calculatd using iso-paratric quadratic lnts. (a) ka, N 80 ; (b) ka 5, N 0. NUMERICAL RESULTS Figur shows th scattring aplituds for soft sphrs of radii ka and 5. Th incidnt wav is a plan ikz wav. Iso-paratric quadratic triangular lnts, naly with th intrpolation ordr bing two for both pinc (a) th lnt and th basis functions, ar usd in both cass in th figur, and th nubr of lnts N 80 for (a), and N 0 for (b). Th rsults ar copard with analytic solutions. W can s that good agrnt is obsrvd btwn our rsults and th analytic solution. Th axiu rlativ rror for (a) is 0.5%, whil that for (b) is 5%. Th rror for (b) is largr sinc th siz of th scattrr ovr wavlngth is 5 tis largr than that in (a), but w only incras th nubr of unknowns by a factor of 4. Considring that ka 5 givs a vry larg sphr, our rsults ar still quit satisfactory. Of cours, if w incras th nubr of lnts, th accuracy will iprov. In both (a) and (b), it is notd that th backscattring is uch sallr than th forward scattring. (b)

5 Scattring Aplitud Nystro /dgr /dgr (a) (b) FIGURE. Scattring aplituds for hard oblat sphroid (ajor axis = k ) calculatd using 80 iso-paratric quadratic lnts. Aspct ratio: for (a) and 0 for (b). In Fig., rsults ar shown for scattring fro hard oblat sphroids of ajor axis k orintd in a way that th sytry axis is in z dirction. Th aspct ratio is for (a) and 0 for (b). W ar intrstd in sphroids bcaus high aspct ratio sphroids potntially can b usd to odl so cracks. Th incidnt fild is th sa as Fig. and 80 iso-paratric quadratic lnts ar usd for both (a) and (b). Th scattring aplituds of sphroids can b solvd analytically in th sphroidal coordinat syst. Fig. xhibits vry good agrnt btwn th Nystro thod and analytic solution. In fact, th axiu rlativ rror is lss than %. A coparison of Fig. (a) and (b) infors us that th forward scattring dcrass and backscattring incrass whn w incras th aspct ratio for th oblat sphroid. Whn th aspct ratio is as high as 0, w s ost scattrd nrgy gos in th broadsid of th sphroid, and a iniu is sn in th plan containing th ajor axis ( ). This can b intrprtd as th consqunc of th sphroid approaching th planar circular disc. Scattring Aplitud 5 4 Nystro Scattring Aplitud.5 Rl. RMS Error st nd rd /dgr 0 0 (a) (b) FIGURE 4. Scattring fro a soft cubic flaw. (a) Scattring aplitud; (b) convrgnc for diffrnt basis ordrs. W also calculatd th scattring fro a cubic void with kd (d is sid). Sinc th gotry is itslf flat, thr is no gotric rror in our odling. Th incidnt fild is still in th z dirction, and th obsrvation plan is prpndicular to 4 paralll dgs and contains th z axis. Fig. 4 (a) givs th scattring aplitud, and (b) shows th convrgnc for diffrnt ordr of basis. It is up to th xpctation that highr ordr basis yild or accurat rsults and bttr convrgnc. 0-4 No. Unknowns/

6 CONCLUSIONS This papr applis th highr ordr Nystro thod to solv acoustic scattring probls. Nystro thod is asy to go high ordr, and has advantags of high accuracy and convrgnc rat. It is also vry anabl for incorporating fast algoriths du to th siplicity of th far fild intraction. An fficint singularity subtraction sch is proposd, which can handl both singular and nar-singular intgrals of diffrnt singular ordrs for curvd triangular lnts. Nurical rsults for scattring by sphrs, sphroids, and cubs hav donstratd th validity of th proposd approach. ACKNOWLEDGEMENTS This work was supportd by th NSF Industry/Univrsity Cooprativ Rsarch Progra of th Cntr for Nondstructiv Evaluation at Iowa Stat Univrsity. REFERENCES. L. W. Schrr and S. Song, Ultrasonic Nondstructiv Evaluation Systs: Modls and Masurnts (Springr, Nw York, 007), pp P. L. Uslnghi, T. B Snior and J. J. Bowan, Elctroagntic and Acoustic Scattring Sipl Shaps (HPC, Nw York,988).. L. W. Schrr, Fundantals of Ultrasonic Nondstructiv Evaluation: A Modling Approach (Springr, Nw York, 998), pp S. Wang, J. Acoust. Soc. A, 99 (4), F. Ihlnburg, Finit Elnt Analysis of Acoustic Scattring (Springr, Nw York, 998), pp Y. Liu and S. Chn, Coput. Mthods Appl. Mch. Engrg. 7, (999). 7. L. Shn and Y. J. Liu, Coput. Mch. 40, (007). 8. L. F. Canino, J. J. Ottusch, M. A. Stalzr, J. L. Vishr and S. M. Wandzura, J. Cop. Phys., 46, (998). 9. R. Chang and V. Loakin, IEEE Trans. Antnnas Propagat., 6 (), (0). 0. P. P. Silvstr and R. L. Frrari, Finit Elnts for Elctrical Enginrs (Cabridg Univrsity Prss, Cabridg), S. Jarvnpaa, M. Taskinn and P. Yla-Oijala, Int. J. Nur. Mth. Engng. 58, (00).