SIMULATION TOOLS FOR OPTIMAL DESIGN AND INTERPRETATION OF GUIDED WAVE INSPECTIONS

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1 SIMULATION TOOLS FOR OPTIMAL DESIGN AND INTERPRETATION OF GUIDED WAVE INSPECTIONS A. Lhémery, K. Jezzie, V. Baroia ad S. Mahaut CEA, LIST, F Gif-sur-Yvette, Frace Abstract. The paper reviews theoretical ad umerical tools developed at CEA for the simulatio of odestructive testig ivolvig guided propagatio of elastic waves i the compoets uder test. I waveguides, flaws or variatios of guide shape (discotiuities) cause scatterig. Icidet modes reflect or are coverted ito ew modes; their trasmissio through discotiuities is similarly affected. Guided waves (GW) NDT methods rely o the aalysis of reflected or trasmitted modes relatively to what happes i a soud guide. Simulatio tools are essetial to hadle the iheret complexity of GW propagatio ad scatterig; they help to optimize cofiguratios, to iterpret measuremets. As GW aalysis ofte refers to mode amplitudes, it is advatageous to exploit the modal ature of GW i the simulatio itself. This simplifies the iterpretatio of overall or partial results (cosiderig the various pheomea ivolved); this avoids developig post-processig techiques ad ruig their time-cosumig computatios as required if results are calculated with o referece to modes. Two modal formulatios to simulate NDT-GW measuremets (pitch-catch, pulse-echo) are used to lik a semi-aalytic fiite elemet code for the modal solutio i arbitrary guides, models of radiatio ad receptio by trasducers ad a specific fiite elemet model for GW scatterig by arbitrary discotiuities. The paper first reviews these tools. The, examples of iterest for NDT illustrate their capabilities to address complex cofiguratios ad to help iterpretatio. Itroductio Elastic guided waves (GW) propagate at log rage i the thickess of parts of regular shape which thickess is of the same order of magitude as wavelegths. This property is very attractive for the odestructive evaluatio (NDE) of large structures sice it limits or eve avoids trasducer scaig; this reduces the overall duratio ad cost of the examiatio ad makes its implemetatio easier [1,2]; GW are also measured i NDE by Acoustic Emissio (AE) of pressure vessels ad ca be passively ad actively used i Structural Health Moitorig (SHM). Other itrisic properties of the physical behavior of GW ted to lesse their iterest. i) most GW are dispersive their speed is frequecy depedet, ii) they are multi-modal at a give frequecy, several modes coexist, their umber growig with frequecy; iii) modes couple whe iteractig with a discotiuity of the guide; iv) sice their wavelegth compares with structure thickess, spatial resolutio is limited. All these characteristics make difficult the iterpretatio of results as well as the desig of optimal testig cofiguratios. Simulatio tools ca costitute the appropriate mea to overcome these difficulties ad are expected by idustrial coceivers of GW ispectios. It is our objective to address these idustrial eeds. I this paper, we review the modelig approach adopted for simulatig NDE methods ivolvig GW propagatio ad the tools used or developed at CEA to implemet it. The, advatages of the simulatio approach are illustrated by examples of NDE examiatios ivolvig GW propagatio. 1. Theory 1.1 Modelig approach I geeral, sice most guided waves are dispersive, GW testig is operated i a limited frequecy badwidth. Typically, excitatio sigals are i the form of a sigle frequecy (CW) sigal modulated i amplitude (Gaussia wave packets, toe bursts etc.); therefore, it

2 is atural to model GW i the frequecy domai. Typical waveforms measured are the sythesized by Fourier trasform over a limited spectrum. I what follows, models are described uder CW assumptio where ω deotes the agular frequecy. A first property of GW propagatio is that, at a give frequecy, GW ca be decomposed as (complex-valued) liear combiatio of eigemodes peculiar to the sectio of the structure (perpedicular to its guidig axis deoted by z) ad to its stiffess. The kowledge of the set of modes ad their behavior is sufficiet to depict the wave behavior of ay elastodyamic quatity (particle displacemet or velocity, stress) of a arbitrary field. The th mode of this set is described at a give frequecy by: i) its waveumber β, real for the fiite umber of propagative modes, imagiary for the fiite umber of evaescet modes or icludes a imagiary part for the ifiite umber of ihomogeeous modes, ii) the correspodig particle displacemet vector i the ivariat sectio of the guide u% ( x, y). The CW displacemet u associated to a wavefield writes = ( ) ( x, y, z; t( )) A (, ) j z t x y e β u ω u% ω, (1) where A deotes the th amplitude i the decompositio. The kowledge of mode behavior ad dispersio characteristics is a essetial step for uderstadig complex pheomea arisig i a guidig structure: measured or simulated sigals are the iterpreted i referece to modes. Typical questios cocer the ability of modes to be trasmitted through or reflected o a guide discotiuity, to be coverted ito other modes i the iteractio etc. If simulated results are computed regardless of the modal ature of GW, they are very ofte post-processed (at a cosiderable computatio cost) to be evetually iterpreted as variatios of mode amplitudes. Therefore, we made the choice to develop simulatio tools fudametally o the basis of the modal descriptio of waves i each portio of the structure that propagates GW. A secod property of GW propagatio as applied to NDT, is that it ca be described as a global pheomeo i the homogeeous portios of the structure but the way GW iteract is otherwise domiated by local pheomea (e.g., trasducer diffractio both i radiatio ad i receptio, scatterig by a defect, by a variatio of geometrical or material properties of the structure ad by ay ihomogeeity of the structure). GW are especially attractive for their ability to propagate over large distaces; NDT is operated by trasducer(s) ad aims primarily at detectig defects which are localized. Thus, the GW/NDT simulatio tools must deal with differet scales correspodig to various pheomea. Moreover, idustrial eeds for simulatio suppose that tools ca be used itesively. Clearly, a sigle method caot be effective for both local ad global computatios. We believe that various pheomea at differet scales require various models. A further igrediet is ecessary to give these models the possibility to work all together at the same time ad i syergy. Next paragraph recalls two overall formulatios [3] that costitute this cetral igrediet, followed by the descriptio of models used or developed for computig modes, trasducer diffractio effect o them ad their scatterig by ihomogeeities. 1.2 Overall Modal Formulatios Two cofiguratios (see Fig. 1) are cosidered for which two formulatios were derived [3]. The first (resp. secod) is a pulse-echo (resp. pitch-catch) cofiguratio. The two frequecy-depedet expressios give by Eqs. (2) of the sigal received s i (ω), i=1, 2, were obtaied usig some mathematical properties of guided modes (bi-orthogoality) ad the electro-mechaical theorem of reciprocity proposed by Auld [4].

3 1) defect 2) trasducer T/R icidet reflected z - emitter icidet ihomogeeity z - z + trasmitted (differet guide) receiver Figure 1: 1) A sigle trasducer is used i pulse-echo cofiguratio. 2) Two separated trasducers (oe emitter, oe receiver) are positioed o two differet waveguides i a pitch-catch cofiguratio. s iω = A A R e ( ) r e i( β + βm ) z, 1 ω m m P m iω s ω A A T e e = ( ) r e iβmz iβ ( L z ' ), 2 m m P m where P is the electrical power provided to the emitter. + e A m ad (2a) (2b) r A are respectively the amplitude of mode m radiated by the trasducer ad the amplitude of sesitivity to mode of the trasducer i receptio; they stad for trasducer diffractio effects. R m (resp. T m ) is the reflectio (resp. trasmissio) coefficiet for the icidet mth mode ad the reflected (resp. trasmitted) th mode; they stad for the scatterig by a ihomogeeity of the guide(s). z - deotes the distace betwee the emitter ad the scatterig zoe; L z + deotes the distace betwee the scatterig zoe ad the receiver i cofiguratio 2 (alog aother axis z i this case); these distaces appear together multiplied by β, the wave umber of the th mode of a give guide, i expoetial terms which are propagators of GW i the various guides ivolved. I practice, as soo as trasducers are sufficietly distat from the scatterig zoe, the two discrete sums i equatio (2) ca be restricted to the sole propagative modes. I these formulas, the various local pheomea (causig variatio of mode amplitude) ad the global propagatio i homogeeous guides are mathematically separated. These formulas ca admit differet methods for computig the various terms of the double discrete sums. Aother crucial poit about them is that they make it possible to combie existig results for some of the terms with ew results for other terms, opeig oto vast post-processig capabilities. It is possible to use a scatterig matrix with several amplitudes relative to differet trasducers without re-computig the whole simulatio; this costitutes a ecoomical way of optimizig testig cofiguratios. Examples of such capabilities will be discussed i the results give i this paper. 1.3 Mode computatio by the Semi-Aalytical Fiite Elemet () method There are may methods i the literature for computig modal solutios; some are more appropriate tha others for a give applicatio. Our aim beig to offer geeric tools, the semi-aalytical fiite elemet method ( method, see [5] for example) appeared to be very well suited to our eeds. This method ivolves a fiite elemet computatio i the guide sectio, allowig the computatio of both wave vectors ad modal displacemets i the sectio as beig the eigevalues ad eigevectors (resp.) of a quadratic system of equatios; this system is the discrete form of a variatioal problem i the guide sectio. Sice this is a fiite elemet computatio, it allows oe to deal with all sorts of characteristics of the guides (sectio shape, costitutive materials). As it is restricted to the sectio, it is computatioally very efficiet. The propagatio is otherwise accouted for by

4 meas of aalytic propagators i the guidig directio ormal to the sectio cosidered, at o computatioal cost (same fuctio whatever the rage). I Eqs. (2), this model gives the solutio for the expoetial propagator terms. 1.4 Local models of trasducer diffractio There are basically two cases to distiguish: the trasducer(s) ca be positioed either, i) o the guidig surface; ii) o the guide sectio. Each case requires a specific model for computig the amplitude of the modes. The, it is ecessary to derive models adapted to the trasductio that takes place; this depeds o the type of trasducer used (piezoelectric, EMAT, magetostrictive). Agai, this problem is addressed i the literature for various cases of idustrial iterest. I their preset implemetatios, our models deal with piezotrasducers assumed to be sources of ormal stresses all over their active surface. For trasducers actig from the sectio, a specific variatioal formulatio has bee derived which is discretized o the same elemets as those used for computig modes by [6]. The case of ouiformly excited trasducers the applied stress is made variable alog the active surface has bee treated allowig us to propose two methods for selectig oe sigle mode chose amog possibly may modes [7]. For those actig from the guidig surface, a surface itegratio over the trasducer area must be computed. I the case of a agled probe a very commo way of selectig a mode at a give workig frequecy by phase coicidece, it ca eve be computed aalytically [8]. I all cases, the results are expected to be give i the form of a liear combiatio of modes, as i both refereces cited here [6, 8]. I Eqs. (2), these models give the solutio for the terms A ad 1.5 Local models of scatterig by defects ad by guide discotiuities Computig the scatterig by a guide ihomogeeity is a difficult task. Cotrary to bulk waves typically used i NDT, guided waves have i essece a wavelegth comparable with the dimesios of the guide sectio ad with the size of the ihomogeeity that scatters GW. I the former case of bulk waves, scatterig ca be accurately computed by meas of approximatios (high frequecy); i the latter case of GW, derivig suitable approximatios is almost impossible. Our aim is to write the solutio of the scatterig problem i the form of a matrix of complex coefficiets the reflectio R m ad trasmissio T m coefficiets i Eqs. (2), assumig that modal solutios i all the guidig structures coected to the local zoe of scatterig are kow. This matrix liks a iput vector costituted by the coefficiets of decompositio of the icomig wave i its guidig structure, to output vectors costituted by the coefficiets of decompositio of the outgoig waves i their guidig structure. A efficiet method was proposed [3] for plaar cracks of arbitrary shape i a otherwise homogeeous guide, assumig that the crack surface belogs to the guide cross-sectio. By takig advatage of the symmetry, a variatioal formulatio was derived, discretized o the elemets of the calculatio. To deal with arbitrary flaw shapes or guide ihomogeeities (sectio or material variatios, juctio etc.), a origial fiite elemet (FE) scheme has bee developed with the further goal to limit the computatio zoe to a miimal size, for computer efficiecy. A full demostratio of the mathematical derivatio is give i [9] ad a exteded review ca be foud i [10]. The computatio zoe beig ecessarily of fiite size, its boudaries with all the guidig structures coected to it must be trasparet for elastic waves: they must ot reflect the icomig waves or reflect outgoig waves. Thus, the mai task i this developmet was the obtaiig of artificial boudary coditios edowig trasparecy. Radiatio coditios at ifiity are brought back to the artificial boudaries by buildig a operator couplig the fiite elemets iside the FE zoe to the modal solutios outside the e m A. r

5 FE zoe i the waveguides. The operator combies the displacemet compoets with axial stresses (axes of the various guides); mathematically this is called a Dirichlet-to-Neuma operator. By doig so, a origial mixed variatioal formulatio was derived combiig the displacemet ad a multiplier associated to the axial stresses. The scattered field is projected o modal solutios i guides through the use of bi-orthogoality relatios expressed for all guides, this beig doe while solvig the FE system. Note that it is straightforward to itroduce iteral sources iside the FE zoe; this allows us to simulate Acoustic Emissio testig for example. 2. Examples of applicatio to NDT Oe applicatio to NDT is treated here to illustrate possible uses of the tools described. The scatterig of guided waves by a juctio of three idetical guides is cosidered. The juctio is cosidered as flaw-free i a first series of computatios ivolvig differet kid of trasducers actig from the guidig surface of plate #1 ad operatig pulse-echo cofiguratio. The, results are obtaied i the case of the same juctio which ow cotais a surface-breakig crack. Both kids of results are compared. Post-processig capabilities of our simulatio tools are used to study the effects of mode selectio o measured waveforms. 2.1 Scatterig of GW by a flaw-free juctio The juctio cosidered (fig. 2) is that of three plates (40-mm-thick) made of steel. The radiatio is operated from plate #1 ad reflectio ad trasmissio from ad through the juctio i plates (#1-3) are studied. plate # 3 (meters) trasmitted 40 mm T / R icidet reflected 40 mm plate # 1 (meters) Juctio 40 mm trasmitted plate # 2 (meters) Figure 2: 1) The juctio of three idetical guides (plates). I plate #1, a trasducer works i pulse-echo mode o the guidig surface. The scatterig matrix is computed over the badwidth [ ] khz. Figure 3 shows how the various computatioal methods work together: i each waveguide, the modal solutio is computed usig the method. I the juctio, the FE scheme is used. At the boudaries, as part of the FE computatio, specific trasparet boudary coditios are applied o which the total field is decomposed oto the modal solutios i the various uiform waveguides.

FE zoe + trasparet boudaries Figure 3: Detail of the computatio methods used: method for the modal solutios i guides; FE method icludig special trasparet boudary coditios i the juctio.

6 FE zoe + trasparet boudaries Figure 3: Detail of the computatio methods used: method for the modal solutios i guides; FE method icludig special trasparet boudary coditios i the juctio. Figure 4 shows the total field iside the juctio for a icidet S0 mode at the ceter frequecy of 56.5 khz. u x u y Figure 4: Total displacemet field (left: u x compoet, right: u y compoet) i the juctio cosiderig a S0 icidet mode arisig from left guide, at the ceter frequecy of 56.5 khz. The FE computatio results i three scatterig matrices (reflectio, trasmissio) for the various possible icidet modes (three propagative modes i the frequecy rage) scattered as propagative modes i the guides. Results are show i Figure 5 as variatios of reflectio ad trasmissio coefficiets i the frequecy rage cosidered. R A0 A1 R A0 A0 R A0 S0 R S0 S0 T (2) R S0 A0 R S0 A1 R A1 A1 R A1 A0 R A1 S T (2) S0 S0 S0 A0 T (2) S0 A1 T (2) A0 A0 T (2) A0 S0 T (2) A0 A1 T (2) A1 A1 T (2) A1 S0 T (2) A1 A0 T (3) A0 A0 T (3) A0 A1 T (3) A0 S0 T (3) A1 S0 T (3) A1 A1 T (3) S0 S0 T (3) S0 A0 T (3) S0 A1 T (3) A1 A Figure 5: For A0 (top), S0 (middle) or A1 (bottom) icidet mode i plate #1, scatterig coefficiets i plates #1-3 as fuctios of frequecy (khz). Left: reflectio i plate #1 Ceter: trasmissio i plate # 2 Right: trasmissio i plate #3.

7 2.2 Scatterig of GW by a juctio cotaiig a surface-breakig crack The same juctio is cosidered agai but this time, the presece of a flaw iside the juctio is simulated, as show by Fig. 7. The flaw is a surface-breakig crack which height is half the thickess of the guidig plates liked together by the juctio. FE zoe + trasparet boudaries crack Figure 6: Same as Fig. 3, but the juctio cotais ow a surface-breakig crack. Figure 7 shows the total field i the juctio cotaiig the flaw, for the same coditios as those take for the flaw-free case (show i Fig. 4). The presece of the crack creates a strog discotiuity of the displacemet which is clearly visible i the results. u x u y Figure 7: Same as Fig. 4, but the juctio cotais ow a surface-breakig crack. Similarly, the various coefficiets are show i Figure 8. This time, reflectio coefficiets are of far higher amplitude, as compared to results for the flaw-free juctio: the crack reflects much more eergy i the guide where the icomig waves propagate (plate #1). Trasmissio i plate #2 decreases more tha that i plate #3. Globally, the behaviour of the various coefficiets as fuctios of the frequecy becomes more complex: resoaces due to the crack are likely to be at the origi of such a behaviour. Further umerical studies would be required to fully uderstad this behaviour as a fuctio of the crack-legth. 2.3 Sythesis of waveforms for the three trasducers i pulse-echo cofiguratio Results for the scatterig matrices are ow processed by meas of the overall formulas [Eqs. (2)], for various trasducers. We cosider three differet agled-probes actig o plate #1 i pulse-echo mode. The trasducer is positioed 2 meters away from the juctio. Shoe agles are take so that phase coicidece occurs betwee a bulk logitudial CW i the shoe ad a chose mode at the ceter frequecy (56.5 khz): the agles are 7, 25.2, 19.5 for A0, S0 ad A1 modes, respectively. Their active legth equals 50 mm. The excitatio pulse is Gaussia (10% badwidth at -6dB).

8 R A0 A1 R A0 S0 R A0 A0 R S0 S0 T (2) R S0 A0 R S0 A1 R A1 A0 R A1 A1 R A1 S T (2) S0 A1 T (2) A1 A0 T (2) A0 S0 T (2) A0 A1 T (2) A0 A0 T (2) S0 S0 S0 A0 T (2) A1 A1 T (2) A1 S0 T (3) A0 A1 T (3) A0 A0 T (3) S0 A0 T (3) A1 S0 T (3) A1 A1 T (3) S0 S0 T (3) A0 S0 T (3) S0 A1 T (3) A1 A Figure 8: Same as Fig. 5, but for the juctio cotaiig a surface breakig crack. Waveforms simulated are show o Fig. 9. This figure compares the waveforms arisig from the ucracked ad the cracked juctio, cosiderig the various trasducers used. The waveforms are displayed at the same amplitude scale to make the quatitative compariso easier. Note that i Eqs. (2), each term of the double sum ca be see as a sigal itself. Therefore, oe ca easily iterpret sigals by decomposig them ito elemetary sigals. A0 wedge i T/R mode S0 wedge i T/R mode A1 wedge i T/R mode flaw abset flaw preset Figure 9: Waveforms received by the trasducers (A0 (left colum), S0 (middle colum) or A1 (right colum) trasducers) o plate #1 used i pulse-echo. Top lie: juctio free of flaw. Bottom lie: cracked juctio. Time-scale for the various waveforms is [ ] ms. Same amplitude scale for the various waveforms. Eve if iterpretig coefficiets gives a deep isight i the uderlyig pheomea, waveforms are what a actual measure provides; oe of the iterests of simulatio studies is that iterpretig the first kid of results (scatterig coefficiets) greatly helps the

9 iterpretatio of waveforms. I the preset results, it is obvious that reflected sigals are of higher amplitude whe arisig from the cracked juctio. But the detailed study of idividual coefficiets allows a better uderstadig of the complex pheomea that take place iside the scatterig zoe. Coclusio A modelig approach for GW / NDT simulatio has bee described; wavefields i guides are decomposed o modes ad two formulas lik local ad o-local models of pheomea typical of GW measuremets to deal with either pulse-echo or pitch-catch cofiguratios. The method is used for computig log rage propagatio i uiform guides. GW scatterig ca be computed i some simple cases (crack ormal to the propagatio axis) by a scheme derived from. A specific FE method has bee derived for computig the scatterig by arbitrary ihomogeeities; it icludes exact trasparet artificial boudaries for reducig the size of the FE zoe, thus the computatio costs. Overall modal formulas offer may post-processig capabilities that help data iterpretatio as illustrated by some examples give herei. These tools will be implemeted i future versios of CIVA software platform [11]. Experimetal validatio i complex cases, icludig applicatios to o-destructive testig methods such as SHM or AE are i progress. Refereces [1] P. Cawley, M. J. S. Lowe, D. N. Alleye, B. Pavlakovic ad P. D. Wilcox, Mat. Eval. 61, pp , [2] J. L. Rose, J. Pressure Vessel Techol. 124, pp , [3] K. Jezzie ad A. Lhémery, Review of Progress i QNDE 26, pp , [4] B. A. Auld, Wave Motio 1, pp. 3-10, [5] T. Hayashi, W. J. Sog ad J. L. Rose, Ultrasoics 41, pp , [6] K. Jezzie ad A. Lhémery, Review of Progress i QNDE 25, pp , [7] A. Lhémery ad K. Jezzie, Review of Progress i QNDE 27, pp , [8] J. Li ad J. L. Rose, J. Acoust. Soc. Am. 109, pp , [9] V. Baroia, A.-S. Boet-BeDhia ad É. Luéville, accepted for publicatio i J. Comput. Appl. Math., 2010), o-lie at [10] V. Baroia, A.-S. Boet-BeDhia, A. Lhémery ad É. Luéville, Spriger Proc. Phys. 193, p , [11] details o preset capabilities of CIVA software platform available i wwwciva.cea.fr

APPLICATION NOTE UNDERSTANDING EFFECTIVE BITS

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