Applied Mathematical Modelling

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1 Appled Mathematcal Modellng 35 (2011) Contents lsts avalable at ScenceDrect Appled Mathematcal Modellng journal homepage: Applcaton of tme reverse modelng on ultrasonc non-destructve testng of concrete Erk H. Saenger a,b, *, Georg Karl Kocur c, Roman Jud d, Manuel Torrlhon d a Geologcal Insttute, ETH Zurch, Zurch, Swtzerland b Spectrases AG, Gesserestrasse 5, 8005 Zurch, Swtzerland c Insttute of Structural Engneerng, ETH Zurch, Zurch, Swtzerland d Semnar for Appled Mathematcs, ETH Zurch, Zurch, Swtzerland artcle nfo abstract Artcle hstory: Receved 10 July 2009 Receved n revsed form 16 July 2010 Accepted 16 July 2010 Avalable onlne 27 July 2010 Keywords: Tme reverse modelng Imagng Fnte-dfferences Wave propagaton Source localzaton Non-destructve testng Tme reverse modelng (TRM) s appled to localze and characterze acoustc emsson usng a numercal concrete model. Am s to transform a method wthn exploraton geophyscs to non-destructve testng. In contrast to prevous tme reverse applcatons, no sngle event or frst onset tme dentfcaton s appled. The method s descrbed from a mathematcal pont of vew. So-called source TRM wth lmted knowledge of boundary values s compared wth so-called full TRM where a complete set of boundary condtons s used. The resultng localzaton accuracy of both approaches s smlar. Wth a known three-dmensonal analytcal soluton we demonstrate the applcablty and the lmtatons of the two-dmensonal wave propagaton method solvng the elastodynamc wave equaton. Wth the help of CT mages we are able to dgtalze a concrete specmen and to verfy a used numercal concrete model. TRM localzaton usng ths hghly scatterng materal s feasble usng the rotated staggered fnte-dfference method. We demonstrate the localzaton of acoustc emsson wth a lmted number of sensors and usng effectve elastc propertes. Source characterstcs can also be recovered. Goal s to apply our method to acoustc emssons measured durng experments carred out on concrete and renforced concrete specmen. Ó 2010 Elsever Inc. All rghts reserved. 1. Introducton Tme reverse modelng (TRM) s appled n several felds of scence such as medcal and earth scences [1]. Pror tme reverse studes focus on flterng sngle events out of recordngs of low sgnal to nose (S/N)-rato [2] or on spatal and temporal accuracy of sngle event localzaton [3]. We want to transfer a TRM approach wthn exploraton geophyscs to applcatons wthn the feld of non-destructve testng (NDT). The fundamental problem s to dentfy a source locaton by solvng the elastodynamc wave equaton wth lmted data recorded at the boundares of the modelng doman. However, those nverson problems are often ll-posed. We dscuss applcaton lmtatons and soluton strateges. Stener et al. [4] appled TRM of the recorded surface wave feld to better understand the passve low-frequency sesmc wave feld around hydrocarbon reservors and to determne whether some of the low-frequency sgnals orgnate from hydrocarbon reservors. The TRM methodology uses numercal algorthms for two-dmensonal elastc wave propagaton n combnaton wth an magng condton. Ths magng condton s mplemented because the measured low-frequency * Correspondng author at: Geologcal Insttute, ETH Zurch, Zurch, Swtzerland. E-mal addresses: erk.saenger@erdw.ethz.ch (E.H. Saenger), kocur@bk.baug.ethz.ch (G.K. Kocur), roman.jud@math.ethz.ch (R. Jud), matorrl@- math.ethz.ch (M. Torrlhon) X/$ - see front matter Ó 2010 Elsever Inc. All rghts reserved. do: /j.apm

2 808 E.H. Saenger et al. / Appled Mathematcal Modellng 35 (2011) tremor sgnals were more or less contnuous n tme and no ndvdual events or frst arrval tmes could be detected. Durng TRM emtted sesmc energy s propagated back to ts orgn. The sesmograms recorded wth synchronzed three-component sesmometers are reversed n tme and mplemented as sources for numercal wave extrapolaton. The sgnals are propagated backwards through the velocty model. The backward propagated energy wll focus at the source locaton, f the velocty model s accurate [3,4]. Concrete as a strongly heterogeneous and densely packed composte materal represents a very mportant but also very dffcult object for ultrasonc NDT methods. Due to the hgh densty of scatterng consttuents and nclusons, ultrasonc wave propagaton n ths materal conssts of a complex mxture of multple scatterng, mode converson and dffusve energy transport [5]. For a better understandng of the effect of aggregates, porosty and of crack dstrbuton on elastc wave propagaton n concrete and to optmze nverse reconstructon technques, e.g. Impact echo methods [6], t s useful to smulate the wave propagaton and scatterng process explctly n the tme doman [7 9]. Acoustc emssons (AE) are caused by stran energy release due to rreversble processes such as crackng or nternal frcton n the materal consdered. Acoustc emsson analyss (AEA) has become a promsng method to evaluate the condton of concrete structures [10,11]. Qualtatve procedures make use of basc parameters of recorded sgnals and try to dentfy the stage of degradaton and to estmate the load hstory. Quanttatve procedures attempt to dentfy characterstcs of an AE source and therefore have to consder the wave propagaton between source and sensors. Proftng from the methods of geophyscs, consderable results have been acheved wth respect to determnaton of onset tmes (pckng), source localzaton and moment tensor analyss. Further progress depends on the handlng of dfferent crack dstrbutons and cracked concrete tself as the medum for elastc wave propagaton, consderng also other elements of structural concrete such as renforcement bars and prestressng tendons. The paper ams at establshng a tool for localzaton and further evaluaton of acoustc emssons n structural concrete and shall contrbute to better understandng on the relaton between crack growth and AE actvty. Concrete pervaded by cracks serves as a medum for elastc wave propagaton and shall be observed, descrbed, nterpreted and modeled. The procedure ncludes physcal tests, to record and vsualze crack formaton and to obtan real AE data. Numercal forward modelng s used to evaluate and understand the physcal tests. In ths paper a feasblty study on a numercal concrete specmen s performed for ths planned approach. The challengng aspect for us s to transfer the tme reverse modelng localzaton procedure by Stener et al. [4] (descrbed above) from geophyscal exploraton to the feld of non-destructve testng n order to mage acoustc emssons occurrng durng crack nucleaton and growth n concrete. Wth TRM, recevers located lke those n the physcal model are consdered as sources n the numercal smulaton. AE are traced back n tme, and the locaton of the generatng fracture process wll become vsble as a concentraton of elastc energy. Advantages of ths method are the ndependency of detecton/pckng algorthms, the capablty to handle low sgnal to nose ratos, and a possble dentfcaton of the moment tensor of the AE sources. In the frst part of the paper we brefly revew the TRM approach by Stener et al. [4]. In addton we perform a numercal accuracy test based on a 3D analytcal soluton. For the second part we generate a 2D numercal concrete model and determne the correspondng effectve elastc propertes. Ths allows a 2D plan stran feasblty study on the applcablty of TRM n non-destructve testng. We conclude the paper wth an outlook to 3D applcatons usng real AE data. 2. Tme reverse modelng of waves In practse TRM s based on sgnals obtaned from experments whch are send back nto a specmen as sources n a tme reverse manner. In ths paper we nvestgate the potental of the TRM approach usng a computatonal framework. Ths means that a forward smulaton s used to produce sgnals wth full control of ther orgn. The forward computaton s typcally based on an exctaton localzed n space and tme. The waves emtted from ths exctaton are smulated up to a certan end tme, whle dsplacement values are recorded at specfc boundary-ponts over tme. These tme-dependent values serve as sgnals n subsequent TRM smulatons. Some mathematcal detals are descrbed n ths secton. More detals can be found n the work of Fnk [12]. The computatons of ths paper are based on the lnear ansotropc elastc wave equaton for the dsplacement feld u ðx; tþ 2R d (d = 2 n ths paper) whch reads: q g o tt u ¼ o j c jkl o l u k þ f n X ½0; TŠ: ð1þ The spatal doman of nterest s denoted by X R d wth postons x 2 X and we consder tme t 2 [0,T] wth some end tme T. We use ndex notaton for vectors and tensors wth summaton conventon and o ^¼ o=ox. In general, the equaton features an nhomogenety f (x,t), whch represents a space and tme-dependent body force. The gravtatonal densty s gven by q g (x) and c jkl (x) denotes the stffness tensor, whch gves the momentum tensor by m j = c jkl o l u k. The equaton contans no dampng term, such that t s form-nvarant wth respect to tme-nverson transformaton t t. For more nformaton about the modelng, see for example [13]. Intal condtons are requred for u and the velocty o t u, whle boundary condtons can be of Drchlet or Neumann type for u. The am of TRM s to solve an nverse problem for the wave equaton approxmately. Note, that t s not the task to fnd a spatal and/or temporal dstrbuton of a body force from boundary recordngs, whch s an ll-posed problem n general.

3 E.H. Saenger et al. / Appled Mathematcal Modellng 35 (2011) Instead the bass s an essentally homogeneous wave equaton and the am s to reconstruct a localzed ntal condton for the dsplacement feld from recorded boundary values. We dstngush between two dfferent TRM procedures: source tme reverse modelng (source TRM) and full tme reverse modelng (full TRM). An overvew about these procedures s dsplayed n Fg Forward computaton A standard forward computaton based on (1) s utlzed to produce boundary sgnals whch enter TRM smulatons. In ths paper we always apply the so-called rotated staggered fnte-dfference scheme to dscretze the wave equaton. For a descrpton of the numercal method see [14,15]. In our forward computatons the ntal dsplacement and velocty are not drectly ntalzed but generated n the frst tme steps by a body force. Hence, the ntal condtons are set to zero both for u and ts velocty, whle the body force s chosen to be: f ðx; tþ ¼ R ðx; tþ t 2 ½0; t s Š; ð2þ 0 t > t s ; whch vanshes for tme t > t s wth a start-up tme t s T. Typcally, R s chosen localzed n space around a poston x s wth a specfc exctaton pattern for example a second dervatve of a Gaussan. After tme t s a localzed non-vanshng dsplacement feld s generated whch can be consdered as actual ntal condton emttng waves towards the boundares. To mplement a free surface on the boundary of X Neumann condtons for u are used. The am of the TRM smulaton below s to fnd an approxmaton to the orgnal source poston x s Source tme reverse modelng Durng a forward computaton values of dsplacement are recorded by recevers on the boundary ox of the specmen. The locatons of the recevers are denoted by S ¼ x ð1þ ; x ð2þ ;...; x ðnþ ox; ð3þ where N s the total number of source postons. In the followng we wll typcally use N = 12. The tme seres of the dsplacement at poston x (k) s wrtten: u ðkþ ðtþ ¼u x ðkþ ; t ; ð4þ wth tme t 2 [0,T]. These tme seres serve as nput data for a TRM smulaton. The poston arrangement can be vared to evaluate the reproducton ablty of the TRM smulaton. The TRM smulaton s agan based on the wave Eq. (1) usng the same coeffcents from the forward computaton as well as x 2 X and t 2 [0,T]. No body force s present throughout the computaton, f = 0. Intal condtons for u and o t u are vanshng, such that the equaton s drven by boundary condtons. On ox the recorded sgnals u ðkþ are fed as sources nto the doman. Formally, we wrte: u ðx; tþ ¼u ðkþ ðt tþ for x 2 S ox ð5þ u ðx; tþ ¼0 for x 2 ox n S; ð6þ such that nhomogeneous Drchlet data s gven exclusvely n the source locatons S. Note, that the tme seres s fed nto the computaton backwards n tme. Hence, the TRM smulaton reverses the forward computaton. The term source TRM emphaszes the way the tme sgnals are mplemented n the algorthm,.e. as sources of wave exctatons. source TRM s not complete by defnton n the sense that the equaton s provded wth tme-reversed rece- Fg. 1. A forward smulaton of elastc wave propagaton s used to generate synthetc nput data for full and source TRM.

4 810 E.H. Saenger et al. / Appled Mathematcal Modellng 35 (2011) ver-sgnals at every boundary-pont. Only a few selected ponts are used. In bref, we do not provde the equaton wth the full set of nformaton. It turns out that only a few boundary-ponts have to be provded wth tme-reversed sgnals to acheve very good results. Note that the source TRM has also been appled successfully to real models,.e. usng recever data (representng the forward smulaton) to carry out a numercal tme reverse smulaton n order locate a real physcal wave exctng source. Such a real data example wthn exploraton geophyscs can be found n [4]. In the numercal method the actual doman X s supplemented by a layer of almost vacuum wth zero Drchlet condtons at the outer computatonal boundary as descrbed above. Hence, the tme sgnals u ðkþ are nserted nsde the numercal grd on the boundary grd ponts ox of the medum specmen. To avod scatterng they are supermposed to any exstng values at these grd ponts that are the results of nteror and surface waves. By ths technque the sgnals are nterpreted as tme seres of localzed ntal condtons whose evolutons are supermposed n a tme-delayed manner. The waves emtted from the boundary sources durng a source TRM smulaton wll nterfere constructvely n the dsplacement feld and the locaton of strongest nterference s taken as an approxmaton of the source locaton x s of the orgnal ntal condton. In order to easly dsplay the result of ths nterference n a TRM smulaton we ntroduce the so-called TRM-feld defned by TRMðxÞ :¼ max ku ðx; tþk; ð7þ t2½0;tš for every pont x 2 X. Ths means, n order to mage the convergent wave focusng on the ntal source, we store the maxmum partcle dsplacement for each grd pont throughout the entre tme of modelng. The hghest value of the TRM-feld fnally makes t possble to locate the orgnal source locaton,.e. the source locaton of the forward smulaton Full tme reverse modelng Due to tme nvarance an ntal condton for a homogeneous wave equaton can be exactly recovered n a tme-reversed computaton from the tme seres of all boundary values for tmes t 2 [0,T] and the dsplacement feld and ts velocty at tme T. An approxmaton to the ntal condtons can be found when only the dsplacement feld u ðfwdþ ðx; TÞ and ts velocty o t u ðfwdþ ðx; TÞ at tme T are gven from a forward smulaton but no knowledge of the boundary values s avalable. For such a tme reverse smulaton we feed n the fnal felds as ntal condtons: u ðx; t ¼ 0Þ ¼u ðfwdþ ðx; TÞ n X; ð8þ o t u ðx; t ¼ 0Þ ¼ o t u ðfwdþ ðx; TÞ n X; ð9þ where the velocty s taken negatve n correspondence to the tme reversal. Snce the numercal method s based on a twostep tme ntegrator, see [14], such a reversed computaton s easy to realze from the last two dsplacement felds at tme T and T Dt. Both felds are smply used as ntal felds for the two-step ntegrator n a reversed order. Boundary condtons are the same as n the forward computaton. The result of such a full TRM smulaton can be vewed as benchmark for a source TRM snce much more nformaton s used and typcally the localzaton of the ntal exctaton s much better, see the example n the next secton. However, full TRM s typcally not usable n practce due to the lack of knowledge n realstc stuatons Example for comparson To llustrate the dfferent procedures we wll compare full TRM to source TRM for a generc example. We created a homogeneous 2D model surrounded by a thn vacuum layer. The model (grd spacng h = m) conssts of a cm area n whch the compressonal and shear wave velocty s set to v p = 3987 and v s = 2328 m/s; the densty s q g = 2376 kg/m 3.An ntal body force source (second dervatve of a Gaussan wth f fund = 100 khz) n horzontal drecton s placed at (400,300) and marked wth a whte crcle (Fg. 2(a)). The modelng s done wth second order tme update and a second order spatal dfferentaton operator as descrbed n [14] usng a tme step Dt = s. The full wave feld (.e. the vertcal and horzontal dsplacement feld) s recorded at 12 sensor postons (marked wth crosses n Fg. 2(a)) durng the full length of the smulaton. In addton, the complete wave feld s stored at two consecutve tme steps at the end. Ths s the necessary nput data for the source TRM and full TRM, respectvely. Fg. 2(b)) shows the TRM-feld whch results from the full TRM smulaton. It accurately shows largest values around the locaton of the ntal source. The TRM-feld of the source TRM smulaton s dsplayed n Fg. 2(c)). Remarkably, the ntal source locaton s also accurately recovered TRM source patterns In NDT t s relevant not only to detect the ntal source localzaton but also the exctaton pattern n the momentum tensor. To smulate dfferent exctatons we model the external force f of the forward computaton (2) by f ðx; tþ ¼o j ðx; tþ ; ð10þ m ðexþ j

5 E.H. Saenger et al. / Appled Mathematcal Modellng 35 (2011) Fg. 2. A snapshot of the vertcal dsplacement wave feld at one tmestep durng the forward smulaton usng a homogeneous velocty wthn a block (a). The orgnal source poston s marked wth a whte crcle. The result of the full TRM calculaton (b) gves the best possble nverse source localzaton (.e. the maxmum of the dsplayed feld). The source TRM approach uses as nput data only the recorded dsplacement at 12 recever postons marked wth 12 black crosses. The orgnal source poston can be dentfed wth ths method. that s, derved from a gven momentum tensor m ðexþ j. We want to brefly demonstrate that t s possble to extract the specfc form of m ðexþ j from a tme reverse computaton wth only few source nputs on the boundary. In the example of (2) above we used a body force source n horzontal drecton. Ths force has been modeled by a moment tensor wth vanshng entres except for the m 11 component whch was gven by a localzed wavelet w(x,t) localzed around the source locaton x s. We nvestgated two more dfferent source types namely an exploson and an arbtrary moment tensor. The defntons are gven by (a) Horzontal force: m 11 = w(x,t), m 22 = m 12 = m 21 =0 (b) Exploson: m 11 = m 22 = w(x,t), m 12 = m 21 =0 (c) Arbtrary: m 12 = m 12 = m 22 = w(x,t), m 11 =0 The TRM-feld for the horzontal force has been already shown n Fg. 2(b) and s enlarged n Fg. 3(a). However, dfferent ntal exctaton types wll gve dfferent source patterns. In Fg. 3(b) and (c) we show the pattern for an exploson and the arbtrary source wth the same source wavelet w(x, t), respectvely. It can be observed that the TRM-feld s able to dstngush between dfferent exctaton characterstcs. A catalogue of typcal TRM-felds from generc exctatons and a tmedependent analyss of the stress-feld wll be usefull to dentfy the exctaton n realstc TRM smulatons D analytcal forward soluton Two-dmensonal (2D) models have the advantage that they requre relatvely modest computng power n comparson to three-dmensonal (3D) ones. Most stuatons n nature, however, are three-dmensonal, so that two-dmensonal smulatons represent a somewhat artfcal model of the real stuaton. More precsely, our 2D setup mples a 2D plan stran boundary condton. Ths secton dscusses a specfc effect: vertcal offset of source localzaton. The smulatons we carred Fg. 3. Examples of governng source exctatons n geophyscs and NDT wth characterstc radaton patterns detected by the TRM-feld.

6 812 E.H. Saenger et al. / Appled Mathematcal Modellng 35 (2011) out were motvated by observatons made for 2D tme reverse smulatons wth a recever lne placed drectly above the known source and wth other recever lnes postoned n a certan horzontal dstance away from the frst lne. It was observed that the TRM-feld of maxmum partcle dsplacement localzed the source n dfferent depths dependng on the recever lne chosen for the 2D tme reverse smulaton. In order to understand those effects, we use a 3D analytcal soluton of the elastodynamc wave equaton for a pont force n a homogeneous, sotropc and unbounded medum as descrbed n Ak and Rchards [13]. If a pont force f n drecton e j wth general tme-varyng ampltude X 0 (t) acts at a partcular fxed pont O whch we choose to be the orgn of a fxed Cartesan coordnate system and f the medum s elastc, sotropc, homogeneous and unbounded, the dsplacement u (x,t) at pont x and tme t n drecton e s gven by the followng explct formula: u ðx; tþ ¼ 1 4pq 3c Z 1 r=b c j d j sx r 3 0 ðt sþds þ 1 r=a 4pqa c 1 c 2 j r X 0 t r 1 1 c a fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 4pqb 2 c j d j r X 0 t r ; ð11þ b fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ðnfþ where the followng notatons have been used: r :¼jxj denotng the dstance of pont x from the pont, where the pont force f s actng. c :¼ x /r ( = 1, 2, 3) denotng cosne of drecton. a, b and q denotng P-wave, S-wave veloctes and densty of the medum. d j s Kronecker s delta. NF stands for the near feld part of the soluton. FFP stands for the far feld of the compressonal (P-) wave part of the soluton. FFS stands for the far feld of the shear (S-) wave part of the soluton. A vertcal-force source wth the same characterstc as used for the forward smulaton descrbed above s used (f fund = 100 khz, Dt = s, second dervatve of a Gaussan). The homogeneous medum parameter of ths example was used as well (v p = 3987 m/s and v s = 2328 m/s; the densty s q = 2376 kg/m 3 ). There are three mportant dfferences to the numercal forward smulaton descrbed n Secton 2.5. Frst, the analytcal soluton s a full 3D soluton of the elastodynamc wave equaton. Therefore we also obtan analytcal forward recever data wth an offset to the plane ncludng the source (Lne 1 n Fg. 4(a)). Second, the analytcal soluton s avalable for an unbounded homogeneous elastc medum only. Thrd, the recevers are placed n lnes only n one drecton away from the source poston. Ths non-symmetrc confguraton wll nfluence the localzaton accuracy (shown below). The used model sze s typcal for engneerng applcatons but the results can be easly transferred to geophyscal problems. The 2D source TRM seres of smulatons are based on the setup shown n Fg. 4(a), whch conssts of 7 lnes wth 12 recevers placed n a vertcal dstance of 337 grd unts from the source. The length of the ndvdual recever lnes s fve tmes as large as the dstance from the source to the frst recever lne. Recever lne 1 s placed drectly vertcal above the source, recever lne 2 s located 50 grd unts n y-drecton from recever lne 1 and so on. In order to comply wth the condton unbounded medum gven by the analytcal soluton, we chose the dmenson of the numercal grd n the tme reverse smulaton as large that the P-wave would not reach the edge of the model durng the smulaton. Frst, we calculated the recever data of recever lne 1 usng the 3D analytcal soluton. We then run the two-dmensonal source TRM smulaton based on ths set of data. Ths procedure was successvely repeated for the remanng lnes. Fg. 5 ðffpþ ðffsþ (a) Setup (b) localzaton depths vs offset depth [grd unts] offset [grd unts] localzaton of 2D Source TRM Fg. 4. The setup for the 3D numercal forward smulaton s shown on the left hand sde. The determned source depth versus the offset s dsplayed on the rght hand sde.

7 E.H. Saenger et al. / Appled Mathematcal Modellng 35 (2011) Fg. 5. TRM-feld for Lne 1 where the recevers were located drectly above the pont source (grd spacng h = m). The tme-reversed sgnals, determned by an analytcal soluton, are nserted at 12 ponts marked wth a whte crcle. The cross s the maxmum value of the TRM-feld. There s a vertcal localzaton error wth respect to the orgnal source locaton (whte quadrat). shows the 2D-TRM-feld resultng on the plane drectly under the correspondng recever lne. The whte square marks the locaton of the orgnal source and the black cross the maxmum value. It s vsble that the locaton of the maxmum value shfts vertcally downwards wth ncreasng offset of the recever lne relatve to the source locaton (Fg. 4(b)). We dsplay the offset of the recever lnes on the x-axs and the depth ndcated by the correspondng TRM-felds on the y-axs. We observe a clear trend to deeper depths the farther a recever lne s located from the orgnal source. Ths can be explaned by the non-symmetrc dstrbuton of the recevers around the source n combnaton wth the non-zero wavelength of the propagatng elastc waves. However, for most practcal applcatons the observed localzaton accuracy s acceptable. More generally, we want to pont out that the analytcal forward soluton presented here can also used for an analytcal TRM approach n an unbounded homogeneous medum [16]. 3. NDT-applcaton: localzaton of acoustc emssons n concrete 3.1. Numercal concrete model A numercal concrete model n 2D wth randomly dstrbuted concrete consttuents smlar to [17] s presented (Fg. 6(a)). The model dsplays an arbtrary cross secton of the numercal concrete specmen. Plan stran s assumed for each cross secton. The concrete specmen s smplfed by spatal randomly dstrbuted ellpses (grans) and crcles (ar vods) and flled between wth homogeneous cement paste. The aggregates and ar vods are effectvely modeled as nfnte cylnders. The gran-sze dstrbuton s transferred from a real concrete mx and s n agreement wth Fuller s curve [18]. Ar nclusons are estmated wth 2%, a common lmt percentage n practce. The elastc materal propertes for the grans (v p = 4180 m/ s, v s = 2475 m/s, q = 2610 kg/m 3 ), the ar vods (v p = 0 m/s, v s = 0 m/s, q = kg/m 3 ) and for the cement paste (v p = 3950 m/s, v s = 2250 m/s, q = 2050 kg/m 3 ) are allocated to grd ponts for the fnte-dfference algorthm (grd spacng h = m). For reasons of verfcaton computer tomography (CT) screens of a concrete cube (Fg. 6(b)) are provded. Com- Fg. 6. Numercal concrete model (10 10 cm) versus CT screen of concrete specmen (12 12 cm). Dsplayed are normalzed densty values. Dark colors correspond to ar nclusons, grey to the cement paste and lgth color defnes the grans.

8 814 E.H. Saenger et al. / Appled Mathematcal Modellng 35 (2011) parson of propertes of the ntroduced numercal concrete model versus the CT model (real concrete) shows a good agreement and suggests ts effectveness and applcablty to non-destructve testng of concrete. Moreover, the numercal concrete model can be straghtforward vared wth dfferent gran sze and ar vod dstrbutons for further parameter studes. However, t s also possble to use the CT screens as nput for our numercal smulatons. Ths s llustrated n the secton outlook. We choose here the numercal concrete model because, (1) t can be generated wth a full control of the gran-sze dstrbuton, and (2) we want to avod artefacts due to segmentaton errors. Segmentaton of raw CT data s one dffcult and necessary processng step (e.g. [19] and references theren) Forward smulaton wth an arbtrary chosen momentum tensor source In order to create a synthetc but realstc data set we use the followng model setup. An arbtrary chosen momentum tensor source wth source tme-functon as descrbed n Secton 2.5 s used. The dscretzaton detals are adopted from the prevous secton. In the snapshot shown (Fg. 7(a)) a wave feld excted at (400, 300) s llustrated. Due to the heterogeneous concrete model a lot of scatterng can be observed compared to the wave feld n Fg. 2(a). Two-components (horzontal and vertcal dsplacement) of the emtted waves are recorded at 12 sensor postons of sensors marked by black crosses on the boundares Reverse smulaton wth 12 sensors usng exact velocty model The tme reverse computaton s executed for two sensor modfcatons, sensors wth two components (dsplacement normal and parallel to surface) and sensors wth one component (dsplacement normal to surface) recorded. Stener et al. [4] appled TRM successfully on data measured wth three-components sesmometers. For most NDT applcatons (.e. for pezoelectrc sensors) the case where one-component data s recorded on sensor postons s relevant. The performance s analyzed consderng two-component sensor modfcaton and dscussed. As descrbed before by usng both components the tme-reversed propagatng waves focus on the source coordnates they orgnated from. An excellent result s obtaned (Fg. 7(b)) n that the radaton pattern of the nduced source can be vsual dentfed (marked by the whte crcle) by means of the characterstc accordng to Fg. 3(c). Usng only the dsplacement component normal to concrete surface a good focus can be observed (Fg. 7(c)). The radaton pattern seems blurred, but represents a satsfyng achevement f only one-component data used for reverse computaton s consdered Effectve elastc propertes of the used numercal concrete sample To obtan effectve veloctes of the numercal concrete sample (Fg. 6(a)) we use a technque descrbed n detal n [20]. A revew of ths and related methods s gven n [21]. We apply a body force plane source at the top of the model. The plane wave generated n ths way propagates through the numercal concrete model. Wth two horzontal planes of recevers at the top and at the bottom, t s possble to measure the tme-delay of the peak ampltude of the mean plane wave caused by the nhomogeneous regon. Wth the tme-delay (compared to a homogeneous reference model) one can estmate the effectve velocty of the compressonal and shear wave. The source wavelet n our experments s always the frst dervatve of a Gaussan wth a domnant frequency of Hz and wth a tme ncrement of Dt = s. As a result, we have determned the effectve compressonal wave velocty to v p,eff = 3987 m/s and the effectve shear wave velocty to v s,eff = 2328 m/s. Fg. 7. Forward smulaton and reverse smulaton n a heterogeneous medum for two sensor modfcatons.

9 E.H. Saenger et al. / Appled Mathematcal Modellng 35 (2011) Fg. 8. Performance of tme reverse modelng usng effectve elastc propertes for two dfferent sensor modfcatons. Fg. 9. CT screens of uncracked concrete cubod ( cm) wth post processed numercal model for wave propagaton smulaton Reverse smulaton wth 12 sensors usng effectve elastc propertes for concrete Gven the effectve elastc propertes (EEP) wth v p,eff and v s,eff a reverse computaton s executed smlar to Secton 3.3. The forward smulaton s performed based on the heterogeneous medum as descrbed n Secton 3.2, but for tme reverse modelng the prevous determned EEPs are used nstead. Consderng both sensor components a very good result can be acheved (Fg. 8(a)). In the backward propagaton drecton no scatterng wll occur and compared to reverse smulaton wth the exact velocty model (Fg. 7(a)) a convergence of elastc energy can be observed. The TRM source pattern s clearly vsble. Usng only the dsplacement component normal to the concrete surface a problematc result can be obtaned (Fg. 8(b)). Artfacts due to surface waves are sgnfcantly vsble around the boundares and the source pattern s dsplayed rudmentary. Possble solutons to mprove the localzaton are to place more sensors on the surface and to magnfy the energy nduced nto the model and to nterpolate the zero values on the boundary between sensor postons wth respect to the ncomng waves httng the boundary. An rregular sensor arrangement may be nvestgated to mprove the performance of the method and to clarfy ther nfluence on the wave focus. 4. Outlook The forthcomng approach ams at extendng the ntroduced two-dmensonal TRM method to three dmensons. We show the frst steps of ths approach. A concrete specmen screened n thn slces (Fg. 9(a)) s consdered and after post processng (e.g. threshold-segmentaton) vsualzed as a complete three-dmensonal model (Fg. 9(b)). The dgtal format s benefcal for further consderatons such as smulatons of elastc wave propagaton.

10 816 E.H. Saenger et al. / Appled Mathematcal Modellng 35 (2011) The segmented CT data can be read and translated nto a feasble format for wave propagaton computatons on a FD grd. Elastc materal propertes such as p-wave velocty c p and S-wave velocty c s are allocated to the segmented aggregates. Numercal smulatons startng a double-couple source exctaton (Fg. 9(c)) are performed on the hgh performance parallel computng cluster at the central computng facltes of ETH Zurch. The nfluence of densty dstrbuton of aggregates, ar vods percentage and crack dstrbuton on elastc wave propagaton wll be nvestgated separately. The numercal results are compared to data obtaned from physcal tests and are to be dscussed. The TRM method presented n ths paper was transferred from exploraton geophyscs (km-scale) to acoustc emsson analyss of a sngle concrete sample (cm-scale). Therefore we assume that ths technque can also be transferred to other NDT applcatons such as tubes nspecton n nuclear power plants or gas-ppe nspecton bured under ground. Most mportant s that the dscussed method s able to localze a (secondary) source of acoustc waves. Ths can be for example a crack whch can scatter an elastc wave n a thn plate of steel (plan stran). However, for such an applcaton we recommend to perform numercal feasblty studes as presented n ths work. 5. Concluson Tme reverse modelng usng the elastodynamc wave equaton s, due to the ncreasng computatonal possbltes, nowadays fast and accurate. We used the rotated staggered FD grd to calculate effectve elastc propertes of concrete. Our numercal modelng can be consdered as an effcent and well-controlled computer experment. The numercal smulatons show that source areas and characterstcs of acoustc emssons can be located usng TRM. Wth our feasblty study we demonstrate that our approach s ready to be appled n the laboratory for a deeper understandng of experments n the area of non-destructve testng. We have demonstrated that wth a lmted number of sensors and an effectve homogeneous elastc model the accuracy of localzaton s acceptable. Acknowledgments The authors are grateful because the CT screens used (Fgs. 6(a) and 9(a)) were kndly provded by Dr. Thomas Frauenfelder from the Insttute of Dagnostc Radology of the Unversty Hosptal Zurch. Constructve anonymous revewers have mproved ths work sgnfcantly. Prof. Thomas Vogel from the Insttute of Structural Engneerng of ETH Zurch s acknowledged for supportng the methods used. E.H. Saenger thanks the DFG (Deutsche Forschungsgemenschaft) for ther support through the Hesenberg Programme (SA 996/1-1). M. Torrlhon s funded through an EURYI-Award of the European Scence Foundaton. References [1] M. Fnk, Tme-reversed acoustcs, Sc. Am. (1999) [2] H. Kao, S.-J. 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