A 2D Finite Element Simulation of Liquid Coupled Ultrasonic NDT System

Transcription

1 A 2D Finit lmnt Simulation of Liquid Coupld Ultrasonic NDT Systm Prathamsh N. Bilgund and Lonard J. Bond Iowa Stat Univrsity, Cntr for Nondstructiv valuation, Ams, IA Corrsponding author: Abstract. Th aim of this work is to improv modlling capabilitis and rliability of wav propagation modls using a commrcial finit lmnt packag (COMSOL). Th currnt modl focusss on invstigating th rror and accuracy with th chang in spatial and tmporal discrtization. To incras th rliability and inclusivnss of th finit lmnt mthod, wav propagation has bn modlld in solid mdium with a cylindrical dfct (sid drilld hol), in a fluid mdium and in a fluidsolid immrsion modl. Th numrical prdictions ar validatd through comparisons with availabl analytical solutions and xprimntal data. Th modl is bing dvlopd to incorporat additional complxity and rangs of proprtis, including opration at lvatd tmpratur. INTRODUCTION A puls-cho ultrasonic non-dstructiv tsting (NDT) systm using longitudinal wavs in high tmpratur liquid mdium (watr, liquid mtal or a moltn salt) has bn proposd in th past for inspction of advancd small modular ractors [1, 2]. Prforming xprimnts in a moltn salt or liquid mtal at tmpratur (~250C) is both xprimntally challnging and xpnsiv. In th past, many authors hav studid pizo-lctric matrials and thir prformanc paramtrs at high tmpratur [2, 3, and 4]. To provid gratr insight into transducr prformanc and to rduc th cost of xprimntal vrification, a modling approach can b adoptd. This papr focuss on validation and vrification of bas modls for th wav propagation in solids and fluids. Th purpos of such basic validation is to b abl to build up complx computational modls with tmpratur dpndncy, including prformanc phnomnon which ar closr to th NDT inspction ndd for small modular ractors. Th currnt work discusss th finit lmnt modlling of pizolctric transducrs for application in liquid coupld NDT systm. Th modl will initially b validatd with xprimnts at room tmpratur using coolant surrogats which is in this cas is simply, watr. For simulating th wav propagation phnomnon, it is ncssary to rsolv th shortst wavlngth and hnc th highst frquncy in th spctrum. This rquirs gratr mphasis on th maximum lmnt siz to b usd in th mshing and tim stp for convrgnc of th solution. Th prsnt work contributs towards dtrmining th optimal lmnt siz and tim stp through an itrativ approach with considration towards th accuracy and solution tim for th computational modl. Th rquird data gnratd by ths itrations can b usd to mak th dsign cycl mor fficint by applying diffrnt combinations of lmnt siz and tim stp. Th modl uss a two dimnsional finit lmnt mthod and mploys commrcially availabl cod (COMSOL) which solvs th quations of dynamic quilibrium in th tim domain. Thr simpl cass ar invstigatd: wav propagation in an aluminum block with a sid drilld hol (SDH), wav propagation in watr and in watr with an immrsd aluminum plat. In th past, many authors hav studid modlling of absorbing boundary to rduc th computational grid siz. A task to invstigat th potntial for us of absorbing boundaris in COMSOL is discussd. Finally, th modl rsults from thr distinct cass ar validatd with th xprimntal data.

2 THORTICAL BACKGROUND Th accuracy of th finit lmnt modl dpnds upon th constitutiv quations of th modls, matrial proprtis and discrtization of th modl. Constitutiv quations of Pizolctric Finit lmnts Svral authors hav implantd a finit lmnt mthod to modl th pizolctric transducr and masur th prformanc paramtrs [5, 6]. In th pizolctric dvic, th govrning matrix quation rlating to mchanical and lctrical quantitats ar givn by: whr T is mchanical strss, T c S s D S c is th lastic stiffnss matrix undr constant lctric fild, S is th mchanical strain, is pizolctric strss constant, is th lctric fild, D is th lctric displacmnt and s is th is th lctrical prmittivity undr constant strain S. This form of quation is calld th strss-charg quation which has bn implmntd in this modl. Th pizolctric dvic can also b modlld using th strain-charg quations givn by: S S T d (3) T D dt (4) Whr S th mchanical strain undr constant lctric fild, d is th pizo lctric charg constant, and T is th lctrical prmittivity undr constant mchanical strss. During analysis, th mchanical displacmnt and lctric potntial ar found at ach nodal point in th pizolctric dvic. Morovr, a polynomial intrpolation basis nds to b usd to dscrib th continuity for displacmnt and lctric fild [6]. Hnc, by applying th variation principl to quation (1), (2), a discrt finit lmnt quation systm can b obtaind as follows [7]: 2 u u M C K u K uu 2 uu uu u F t t (1) (2) (5) K u K Q u (6) whr u is th mchanical displacmnt vctor, is th lctric potntial vctor, F is th mchanical forc vctor and Q is th lctric charg vctor for ach lmnt dfind in th pizolctric dvic, M is th mass matrix, C is th mchanical damping matrix, K is th mchanical stiffnss matrix, K is th di-lctric matrix, K is th uu u pizo-lctric coupling matrix. Ths trms ar compltly dfind by Lrch [6].Th solution for th pizolctric lmnts can b obtaind by solving a st of linar algbraic quations with a symmtric band structur. Th valus of u,, F, and Q ar th globally assmbld fild quantitis. Mchanical damping plays an important rol in th dynamic rspons, attnuation of vibration and hnc radiatd acoustic wavform [7]. Rayligh damping can b applid to th tim domain and it assums that th damping matrix C is th linar combination of mass matrix and mchanical stiffnss matrix as dfind in quations 5 and 6. This uu can b rprsntd as: [ C ] [ M ] [ K ] (7) uu uu uu Whr and ar th Rayligh constants for th mass and stiffnss matrics rspctivly In th currnt modl, viscous damping is considrd in which = 0 and > 0. Morovr, th backing matrial, matching layr and insulating cas and stl outr body casing hav bn assignd nods with linar lastic matrial proprtis and Rayligh damping cofficints. For ths linar lastic matrials, th pizolctric coupling matrix K bcoms a null matrix [5] and hnc thr is no coupling btwn th mchanical forc vctor and lctric charg vctor as dfind in quations (5) and (6). As a rsult, computational modl distinguishs th pizo-lmnt from th othr st of matrials and th pizolctric ffct is only modld as a ral tim phnomnon. uu uu u

3 Prssur Acoustics Modlling Th prssur acoustics modul consists of th wav propagating mdium and a rflctor. Th attnuation in th acoustic mdium can b modlld by using complx valud spd of sound and dnsity [8].Prssur acoustic problms modlld using COMSOL involv solving for th small acoustic prssur variations p on top of th stationary background prssur P0. Mathmatically, this rprsnts a linarization (small paramtr xpansion) of th dpndnt variabls around th stationary quiscnt valus [8]. Th govrning quations for ths problms ar th momntum consrvation quation (ulr s quation) and th mass consrvation quation (continuity quation). In prssur acoustics all procsss ar assumd to b rvrsibl adiabatic (isntropic). Thus th wav propagating mdia for th currnt modl is a losslss fluid mdium. Th govrning quation for th prssur acoustics transint analysis problm is givn by: 2 1 p 1 P q Q (8) 2 2 t d m c t P P P t b 2 u. T F (10) 2 v t whr is th dnsity of th wav propagating mdium, c is th spd of sound in th mdium, P is th gaug b prssur, q is dipol sourc, Q is th monopol acoustic sourc [8] and f d m v is th body forc pr unit volum. Th trm c 2 rprsnts bulk modulus of th fluid. MATRIALS AND GOMTRY Lad Zirconat Titanat (PZT)-5A is usd as th activ lmnt in th pizolctric transducr. Th matrial proprtis for PZT-5A can b found in th COMSOL usr guid and th matrial library [8]. Th backing matrial for th pizolctric transducr is assumd to b poxy loadd tungstn powdr. Th matching layr consists of poxy. Prformanc of th matching layr and backing matrial is affctd by th oprating tmpratur of th ultrasonic masurmnt systm. Th insulating cas is mad of nylon whil th transducr outr body casing is mad of stainlss stl. Proprtis for ths matrials ar givn in svral rfrncs including Mdina t al. [7]. TABL 1. Matrial proprtis for transducr modling. Matrial Dnsity (Kg/m 3 ) Poison Ratio lasticity β damping modul(n/m 2 ) Araldit Araldit/Tungstn Nylon Stl (9) Sid drilld hol (c) FIGUR 1. Cas 1 - Aluminum block; Cas 2 - only watr; (c) Cas 3 - watr-aluminum

4 Th modl consists of th pizolctric and prssur acoustics moduls as shown blow: Pizolctric dvic Watr Rflctor: Aluminum Block - FIGUR 2. Structur of modl. Backing matrial Insulating cas Outr Body Casing Pizo-lmnt Matching layr FIGUR 3. Pizo-lctric dvic and Prssur acoustics modul Th pizo lmnt PZT-5A is on half wavlngth (~λ/2) thick. Th matching layr is a quartr wavlngth (~λ/4). To provid a ralistic dynamic rspons for th transducr, it is ncssary to modl a particularly backing layr, th insulating cas, and outr body. Th thicknss of backing matrial in th currnt modl is 10mm.Th thicknss of insulating cas is 0.5mm whil thicknss of outr body stl casing is 1mm in th prsnt modl. Ths lmnts all contribut towards damping of th vibration of th activ lmnt. This damping rducs th mchanical quality factor Q and thus incrass th bandwidth [9]. Wid bandwidth is particularly ncssary for NDT applications. BOUNDARY CONDITIONS It is ncssary to coupl th pizolctric dvic modul with th acoustics modul to obtain th dsird pattrn of radiatd wavs. This is mad possibl by applying a boundary load in th pizolctric dvic intrfac at a boundary common to both mdia. Th sam boundary is assignd a normal acclration in th acoustics modl. Th acoustic analysis provids th acoustic load to th structural analysis whil th structural analysis provids acclration to th acoustic analysis. This coupls th physics in th two moduls. This is also achivd by applying continuity principl in th displacmnt fild. A floating potntial nod is dfind to giv a puls to th pizolctric lmnt and to rciv cho rspons. This rquirs th xcitation to hav a potntial dfind in trms of charg. A zro charg nod is dfind for th boundary with null charg dnsity. A rigid boundary nod has bn assignd to boundaris at which th normal acclration is rquird to b zro [8]. Wavs rflcting from th sid walls of th modl incras th dgrs of frdom for th modl [10]. Hnc th modl bcoms computationally xpnsiv and solution tim incrass. Absorbing boundaris rduc such rflctions by allowing th outgoing wav to lav th computational domain with minimum rflctions. This is ncssary for th wll posd solution of th partial diffrntial quations.

5 FIGUR 4. xampls of tim domain wav forms: without absorbing boundaris and with absorbing boundaris. In a COMSOL tim domain modl, this can b achivd by using low rflcting boundaris, and cylindrical/sphrical wav radiation nods. Fig. 4a and Fig. 4b clarly show th diffrncs sn in wavform at th boundary with and without an absorbing layr. DISCRTIZATION For th wav propagation modl it is important to avoid aliasing and rsolv th shortst wavlngth and hnc th highst frquncy in th spctrum. For a structurd msh th avrag rsolution diffrs significantly btwn th dirction paralll to grid lins and dirctions rotatd 45 0 to on of th axs. Mor importantly, th dirction of wav propagation is not known in advanc. Hnc, an unstructurd typ of msh is prfrrd ovr a structurd msh. Th maximum lmnt siz in th msh can b givn by: c h (11) F N N Whr h=maximum lmnt siz (m) c=spd of sound in wav propagating mdium F 0=Highst frquncy in th spctrum (Hz) λ= wavlngth of mdium, N= numbr of lmnts pr wavlngth 0 A scond ordr triangular Lagrang lmnt is usd for mshing in COMSOL. Th finit lmnt msh rprsnts solution fild of th problm. This solution fild is computd at th nodal points and thn intrpolatd using a polynomial basis function. In such a cas, th scond ordr lmnt givs bttr accuracy for th solution, whn compard to a first ordr lmnt, du to us of a gratr numbr of nodal points. Apart from spatial discrtization, tmporal discrtization is also important for th stability of numrical mthod and hnc th convrgnc of th solution. COMSOL tim dpndnt simulation by dfault us an implicit tim mthod to solv th partial diffrntial quations. Gnralizd-α [11, 12] is th implicit mthod usd in which th α paramtr controls th numrical dissipation at highr frquncis. It provids rlativly lss dissipation as compard to a Backward Diffrntiation formula (BDF) which svrly dissipats nrgy at highr frquncis [8]. Th dgr of dissipation can invrsly affct th accuracy of solution at th highr frquncy. Th absolut and rlativ tolrancs dfind in COMSOL control th rror at ach intgration stp. Morovr, for th stability of th algorithm using th gnralizd-α mthod, it is ncssary for th rror growth rat to b constant. This can b achivd b dfining th Courant-Fridrichs-Lwy (CFL) numbr [13] for th tim stp as shown blow: c t CFL (12) h Whr t th tim stp (sc), c and h ar as dfind in quation (11). Hnc th tim stp can b dfind as h* CFL t c (13) Whr 0<CFL 1 (14)

6 T(SC) %D T(SC) %D For th stability of th algorithm, th distanc travlld by a wav in on tim stp should not xcd th lngth of on spatial stp h as dfind in quations (11) and (13). For applications whr all shap functions ar quadratic, th CFL is takn as approximatly 0.2 [6]. But to calculat an optimum CFL for spcific cass that ar bing simulatd, itrations hav bn prformd by varying lmnt siz and tim stp. Using th data book valu of spd of sound in th mdium [14], th prcntag diffrnc (%D) has bn calculatd for th data book spd and computd spd of wav in th mdium. Th solution tim (T) is rcordd against N and CFL to stimat computation xpns. Ths itrations can also hlp in dvloping a combination of diffrnt tim stp and lmnt siz which can giv prliminary rsults within accptabl tolranc and mor importantly in shortr solution tims. For th cas-1, wavs in an Aluminum block with a 3mm sid drilld hol, th following is data obtaind from a sris of modl: CFL=1 CFL=0.6 CFL=0.2 CFL=0.8 CFL=0.4 CFL= CFL=1 CFL=0.4 CFL=0.1 CFL=0.6 CFL= N N 9 10 FIGUR 5. Invstigation of computational fficincy. Diffrnc vs. numbr of lmnt pr wavlngth (N) and solution tim (T) vs. N. From Fig. 5a, it can b sn that with an incrasd numbr of lmnts pr wavlngth and rduction in CFL, th prcntag diffrnc btwn thortical wav spd and computd wav spd rducs. Lss than 5% diffrnc is considrd to b th accptabl rsult for th all th cass that ar simulatd. Hnc CFL<0.4 is obsrvd to giv accptabl valus for rsult. From Fig. 5b for CFL<0.4, th solution tim sms to follow an xponntial form with incras in th numbr of lmnts. Hnc, a tradoff is ndd btwn numbr of lmnts and tim stp which is th basic purpos of prforming ths itrations. To dtrmin this trad off, w also plot th prcntag diffrnc (%D) btwn th data book wav spd and computd wav spd, with chang in CFL. This is rprsntd as: N=6 N=8 N= CFL N=6 N=8 N= CFL FIGUR 6. Invstigation of computational fficincy- Prcntag diffrnc vs CFL and Solution tim vs CFL From Fig. 6a, it is sn that at CFL=0.2, th prcntag diffrnc (%D) sms to convrg irrspctiv of th numbr of lmnts. Hnc, by considring th data from Fig. 5a through (6b), N=8 and CFL=0.2 is usd for running th simulation for validation with th xprimntal data in all thr cass discussd in th currnt work. Th data in Tabl 2 can b usd to plot Figs. 5a through 6b:

7 TABL 2. Prcntag diffrnc and solution for varying lmnt siz and tim stp. CFL C thory* C computd (m/s) %Diffrnc Computation tim(sc) m/s N= N= N= VALIDATION PROCDUR To validat th computd data, xprimntal data is rquird for all thr cass. Hnc, simpl puls-cho masurmnts wr mad using a 2.2 MHz transducr with nominal diamtr of 13mm for th activ lmnt (Olympus V306). Th data is normalizd for th as of comparison with th computd data. For cas-1 with an aluminum block, Sonotch gl is usd as couplant. Cas 2 consists of th puls-cho rspons in watr, whil cas 3 is th rspons in watr and with a 6.5 mm thick aluminum plat, st as a rflctor. Th st up for xprimnts can b sn in Fig. 7: (c) FIGUR 7. xprimntal configurations for validation. Cas 1 Aluminum block; Cas 2 watr; Cas 3 Aluminum plat immrsd in watr RSULTS Th normalizd puls cho xprimntal and computd data for all thr cass can b compard. Cas 1: Wav propagation in a 50mm thick aluminum block with 3mm diamtr sid drilld hol 8.5us 16.9us 24.5us 8.3us 16.5us 24us FIGUR 8. xprimntal data and Computd data Th puls-cho rspons is masurd at pak amplitud of th wav for th xprimntal and computd data. As shown in Fig.8a, th first cho is rcivd from wavs rflctd from th cylindrical dfct (sid drilld hol - SDH).

8 Th scond cho is du to th wavs rflctd back from bottom surfac of th aluminum. Th third cho is also from th SDH. Th amplitud of th wavs rflctd from th SDH rduc significantly btwn th first and last puls. This can b sn in Fig. 8a) and b. Figur 9a shows plan wav and dg wav which ar du to th dgs of th transducr. Fig. 9b through 10b shows th wav filds du to th cylindrical rflctor. FIGUR 9. Transmittd puls from transducr and Wav passing through sid drilld hol FIGUR 10. Wavs rflcting from sid drilld hol and Wavs rflcting from bottom surfac of aluminum block Cas 2: Wav propagation in watr 24.9us 25.3us FIGUR 11. xprimntal data and Computd data Figur 11 a) through 12b) shows, puls-cho rspons for masurmnts in watr. Th wav arrival tim masurd at th pak amplitud for th xprimnt and th simulation shows a diffrnc of lss than 2%. For th cas 3, a 6.5mm thick aluminum plat is immrsd in watr. Th wav arrival tim and pak amplitud tim ar considrd to calculat th diffrnc btwn th xprimntal and computd data.

9 FIGUR 12. Transmittd puls into watr and Rflctd puls from th bottom surfac Cas 3: Wav propagation in watr and aluminum plat 15.2us 15.6us 15.1us 16.1us FIGUR 13. xprimntal data and Computd data Sinc thr is th possibility of variation in pak amplitud tim, th wav arrival tim is also obsrvd. Th wav arrival tims diffr by lss than 1% for cas 3 btwn th xprimntal and simulation data. For cass 2 and 3, in th modls, unlik th xprimntal data, a puls is rflctd off th on of th boundaris aftr th incidnc puls, and this is shown in th computd data wavform in Fig.11b and 13b. This anomaly is currntly bing invstigatd. DISCUSSIONS Th prcntag diffrnc for th puls-cho rspons btwn xprimntal and computd data is tabulatd for all thr cass: TABL 3. Diffrnc in puls cho rspons btwn xprimntal and computd data. Cas % maximum Diffrnc Cas 1-Aluminum block 2.4 Cas 2-watr 1.6 Cas 3-Aluminum and watr 3.2 As sn from th Tabl 3, th computational modl is validatd by th xprimntal data to within 5% of diffrnc in th wav arrival and wav pak amplitud tims. lmnt siz and tim stp ar varid for spcific modl to dtrmin optimum lmnt siz in th msh and tim stp. Ths itrations can also hlp in dvloping a combination of diffrnt tim stp and lmnt siz which can giv prliminary rsults within accptabl tolranc and mor

10 importantly in lss solution tim. It can mak th dsign cycl mor fficint. Th prsnt work uss msh siz of 8 lmnts pr wavlngth and tim stp at CFL =0.2. Absorbing boundaris can furthr b xplord for ffctiv damping of outgoing wavs at th sid walls of computational gomtry. By adding xtrnal lctric circuit to th transducr modl, voltag sourc can b accuratly modlld CONCLUSIONS Thus, th nw computational modl can now b dvlopd in nar futur for addd complxity such as modlling tmpratur dpndncy on prformanc paramtrs of activ lmnt of transducr. This will hlp to addrss th issus of dgradation of transducr prformanc paramtrs which is has bn critical for inspction capabilitis in nuclar ractors. ACKNOWLDGMNT Th work is fundd by th U.S. Dpartmnt of nrgy s offic of Nuclar nrgy undr Nuclar nrgy Univrsity programs (NUP). Th authors would lik to acknowldg gnrous support of th U.S. Dpartmnt of nrgy. RFRNCS 1. G. J. Posakony, R. V. Harris, D. L. Baldwin, A. M. Jons, and L. J. Bond, (2012) High tmpratur ultrasonic transducrs for in-srvic inspction of liquid mtal fast ractors, Proc., I Intrnational Ultrasonics Symposium, Orlando, FL, Octobr 2011, pp Lonard J. Bond, Jffry W. Griffin, Grald J. Posakony, Robrt V. Harris, and David L. Baldwin. "Matrials issus in high tmpratur ultrasonic transducrs for undr-sodium viwing," in Rviw of Progrss in Quantitativ Nondstructiv valuation, ds. D. O. Thompson and D.. Chimnti, (Amrican Institut of Physics, 1430, Mlvill, NY), 31, (2012). 3. Rymantas Kazys, Algirdas Volisis, Rimondas Slitris, Liudas Mazika, Rudi Van Niuwnhov, Ptr Kupschus, and Hamid Ait Abdrrahim, "High tmpratur ultrasonic transducrs for imaging and masurmnts in a liquid Pb/Bi utctic alloy," Ultrasonics, Frrolctrics and Frquncy Control, I Transactions, 52 (4), (2005). 4. R. Kažys, Algirdas Volišis, and Birutė Volišinė, "High tmpratur ultrasonic transducrs: rviw," Ultragarsas (Ultrasound) 63 (2), 7-17 (2008). 5. Najib N. Abboud, Grgory L. Wojcik, David K. Vaughan, John Mould Jr, David J. Powll, and Lisa Nikodym, "Finit lmnt modling for ultrasonic transducrs," in Mdical Imaging'98, pp Intrnational Socity for Optics and Photonics (1998). 6. Rinhard Lrch, "Simulation of pizolctric dvics by two-and thr-dimnsional finit lmnts," Ultrasonics, Frrolctrics and Frquncy Control, I Transactions, 37 (3), (1990). 7. J San Migul Mdina, Flávio Buiochi, and Júlio C. Adamowski, "Numrical modling of a circular pizolctric ultrasonic transducr radiating watr," in ABCM symposium Sris in Mchatronics, 2, (2006). 8. COMSOL Usr s guid 4.4-COMSOL Multiphysics rfrnc Manual 9. J. A. Gallgo-Juarz, "Pizolctric cramics and ultrasonic transducrs," Journal of Physics : Scintific Instrumnts 22 (10), 804 (1989). 10. M. Drozdz, fficint Finit lmnt modlling of ultrasound wavs in lastic mdia, PhD thss, Mchanical nginring Dpartmnt, Imprial Collg London, J. Chung and G. M. Hulbrt, "A tim intgration algorithm for structural dynamics with improvd numrical dissipation: th gnralizd-α mthod," Journal of applid mchanics 60 (2), (1993). 12. Knnth. Jansn, Christian H. Whiting, and Grgory M. Hulbrt, "A gnralizd-< i> α</i> mthod for intgrating th filtrd Navir Stoks quations with a stabilizd finit lmnt mthod," Computr Mthods in Applid Mchanics and nginring, 190 (3), (2000). 13. Richard Courant, Kurt Fridrichs, and Hans Lwy, "Übr di partilln Diffrnznglichungn dr mathmatischn Physik," Mathmatisch Annaln, 100 (1), (1928). 14. L. W. Schmrr, Fundamntals of ultrasonic nondstructiv valuation: A modling approach, Nw York: Plnum (1998).