BASIC PRINCIPLES OF ACOUSTIC EMISSION TOMOGRAPHY

Transcription

1 bstract BSIC PRINCIPLES OF COUSTIC EMISSION TOMOGRPHY FRNK SCHUBERT Fraunhofer-Insttute for Nondestructve Evaluaton (IZFP), Dresden, Germany The present paper descrbes the basc prncples of acoustc emsson (E) tomography Ths method uses acoustc emsson events as pont sources and combnes the usual teratve localzaton algorthm of E testng wth algorthms for travel tme tomography lke RT (algebrac reconstructon technque) The procedure s equvalent to the soluton of the generalzed nverse localzaton problem n locally sotropc heterogeneous meda and leads to a new magng technque where n addton to the source postons the volume of the specmen s vsualzed n terms of a locally varyng wave speed dstrbuton It s shown by numercally obtaned data sets that the algorthm leads to a more accurate localzaton of acoustc emsson events and offers totally new perspectves for acoustc emsson magng and for acoustc tomography n general 1 Introducton and Outlne Localzaton algorthms n acoustc emsson (E) testng mostly use the assumpton of a homogeneous background medum wth constant wave speed n order to determne the locaton of E events However n practce, the structures under nvestgaton are nhomogeneous n many cases, e wave speeds are changng n space and tme due to heterogenetes of the mcrostructure (eg grans and pores), the effect of structural components (eg tendon ducts n concrete), and materal changes caused by the damage mechansm tself (eg crack growth) These heterogenetes lmt the accuracy of source localzaton algorthms In order to overcome these drawbacks the usual localzaton algorthm of E testng can be combned wth travel tme tomography by usng the E events as acoustc pont sources In ths context a re-localzaton, e an update of the current source postons, has to be performed after each tomographc nverson resultng n an teratve procedure wth alternatng steps of source localzaton and tomography Ths method s n prncple known from geophyscs where earthquakes are located and used for tomographc magng of the earth s nteror Secton 2 frst summarzes the fundamentals of teratve E localzaton and descrbes how the underlyng equatons can smply be generalsed to heterogeneous meda Secton 3 brefly descrbes the dfferent approaches of tomographc magng wth dffractng and non-dffractng sources payng partcular attenton to algebrac reconstructon technques (RT) Secton 4 shows how the two concepts of localzaton and travel-tme tomography can be combned resultng n an teratve algorthm for acoustc emsson tomography called E-TOMO In Secton 5 numercal E data obtaned by the elastodynamc fnte ntegraton technque (EFIT) are used to demonstrate the physcal soundness of the proposed method It s further shown that the E- TOMO approach offers totally new perspectves, not only for E magng but also for tradtonal acoustc tomography snce E events can also be produced artfcally at the outer surfaces of the specmen under nvestgaton (eg by pencl-lead breaks or hammer mpacts) Fnally an outlook s gven how the present algorthms could be mproved by usng further advanced tomographc magng technques and how E tomography can be verfed expermentally J coustc Emsson, 22 (2004) coustc Emsson Group

2 2 Source Localzaton Let us suppose we have = 1,,N sensors and the P-wave arrval tmes of a sngle E event have been determned by usng an approprate pckng algorthm In a frst approxmaton the arrval tme at sensor s gven by S S 2 S 2 S ( x x) + ( y y) + ( z z) 2 r r t = + t = + t c c, (1) where r S = (x S, y S, z S ) s the known poston of sensor and r = (x, y, z) s the unknown locaton of the E event to be determned by the localzaton algorthm t represents the source tme of the E event and s usually measured relatve to a trgger level Eq (1) s based on a smple straght ray model regardng the source-sensor travel path In ths context c s the mean effectve wave speed along ray In most cases, E localzaton s done under the assumpton of a homogeneous background medum and thus, c = c = constant for all rays, = 1,,N On the other hand, n a heterogeneous medum the effectve wave speed can be dfferent from one ray to another, e c c j for j, n general In Eq (1) we have four unknowns, namely the three source coordnates x, y, z and the source tme t Thus, at least four dfferent sensor arrval tmes are needed to solve the underlyng system of equatons Snce Eq (1) represents a nonlnear system of equatons, a closed analytcal soluton s not avalable n general and thus, t has to be solved by an teratve method For that purpose we start wth some ntal values x 0, y 0, z 0, t 0 and suppose that the effectve wave speeds c are all known By usng the same straght ray model as explaned above we can now calculate the theoretcal arrval tmes usng Eq (1) In general, these theoretcal tmes t, 0 wll be dfferent from the measured arrval tmes, T From the dfferences t = T t correcton values for the next teraton x, y, z, t can be obtaned Snce the measured arrval tmes can be wrtten as a functon of x, y, z, and t, T f f f f f ( x, y, z, t) = t,0 + t = t, + x + y + z + t, (2) x y z t = 0 the arrval tme dfference t s expressed as total dfferental of f In matrx form we have or n short, t t 1 N f1 x = f N x f1 y f N y f1 z f N z f1 t x y z f t N t, 0, (3) = F s, (4) t where t and F are known whle s s unknown If the number of sensors s N = 4, the soluton of Eq (4) s well-defned, namely 1 s = F t (5), the 148

3 If N > 4, the problem s over-determned In ths case a least-square approxmaton wth regard to arrval tme dfferences s gven by the normal equaton, s = T 1 T ( F F ) F t (6) Usng the result of Eq (6), the update of source coordnates and source tme from teraton 0 to teraton 1, or more general from teraton k to k+1 can now be expressed as ( k + 1) ( k ) x = x + R x y z t ( k + 1) ( k + 1) ( k + 1) = y = z = t ( k ) ( k ) ( k ) + R y + R z + R t t x z y, (7) where the R j s wth 0 R j 1 and j = x, y, z, t are relaxaton parameters In order to ensure convergence of the teratve method, values of R 01 are typcally used 3 coustc Tomography In acoustc tomography, data collected from sendng acoustc waves through an object at many dfferent angles are used to compute an mage of changes of a physcal quantty wthn the object under nvestgaton (eg attenuaton or wave speed n transmsson tomography or acoustc mpedance msmatch n reflecton tomography) In tradtonal acoustc tomography the sound waves are passed from a seres of sources to recevers at varous locatons around the specmen (Fg 1) Fg 1 Source/recever geometry for tradtonal acoustc tomography Usually each transducer s used as both, source and recever, resultng n at most N 2 ray paths (exact number depends on transducer confguraton and operaton mode (reflecton or transmsson)) f the number of transducers s N In order to ensure a hgh depth of ray coverage, a large number of evenly dstrbuted transducers are necessary For example, the travel tmes between each source and recever are measured In ths case the resultng tomographc mages represent the locally varyng wave speed dstrbuton nsde the specmen lternatvely, the ampltudes of the transmtted tme-doman sgnals at the recevers can be measured and the varatons n attenuaton may be reconstructed The latter method, however, makes hgh demands on the trans- 149

4 ducer couplng condtons nother possblty s the measurement of reflected sgnals and the reconstructon of the acoustc mpedance msmatch between matrx and scatterers ( reflecton tomography ) Mathematcally, there are two major classes of tomographc reconstructon technques [1] One class, the transform based methods are usng the Fourer-slce theorem for non-dffractng sources and the Fourer-dffracton theorem for dffractng sources, respectvely These methods are fast but have the restrcton that the data must be acqured on evenly spaced sets of straght rays ( projectons ) second type of methods employs teratve procedures to reconstruct an mage The most wdely used of these methods s called algebrac reconstructon technque (RT) Other related teratve technques are SIRT (smultaneous teratve reconstructon technque) and SRT (smultaneous algebrac reconstructon technque) These technques are less effcent than transform-based methods but they have several advantages They can be used wth rregular samplng geometres, ncomplete data sets, and may ncorporate curved ray paths In recent years powerful ray-tracng algorthms have been developed correctng for both, dffracton and refracton of the wave feld In the present paper, a transmsson RT algorthm wth straght ray paths has been used n a frst step due to ts smplcty and ts potental for further ray-tracng enhancements as mentoned above Ths mples by no means that RT represents the best method for the problem descrbed n ths paper The RT algorthm adjusts the estmated slowness values, s j = 1/c j, of the dscrete tomography cells (,j) n a systematc fashon untl the computed arrval tmes, t p, for ray p (p = 1,,N p ) match the measured arrval tmes, T p For that purpose, the path lengths, l p j, of each ray p n tomography cell (,j) and the computed arrval tmes for teraton k are used to obtan a new estmate of the slowness for teraton k+1: ( k + 1) ( k ) s = s + R s, (8) wth and j s t p j j j p l j = ( Tp t p ), (9) p 2 ( l ) = s, j l ( k ) p j j, j j (10) The summaton n Eqs (9) and (10) s performed over all tomography cells (,j) passed by ray p R s a relaxaton parameter used to mprove stablty and convergence of the method Small values of R between 001 und 01 are typcal and provde some flterng behavor RT makes the slowness adjustment on a ray-by-ray bass [2] In contrast to that, the SIRT method uses an average correcton for all rays appled to each tomography cell [3], but was not employed here 4 coustc Emsson Tomography Wth the basc prncples descrbed n the prevous two sectons t s now easy to depct the concept of E tomography usng E events nsde the specmen as tomographc pont sources (Fg 2) Only sensors are needed at the outer surfaces of the specmen Naturally, a hgh depth of 150

5 ray coverage s reached snce the number of E events typcally les n the range of some hundreds or thousands Fg 2 Source/recever geometry for E tomography We start wth an ntal guess of the slowness dstrbuton, typcally a homogeneous medum wth constant wave speed (step 0) In step 1 we perform an ntal localzaton of the E sources usng the method descrbed n Secton 2 The ntal source locatons are then used to perform an RT tomography as descrbed n Secton 3 (step 2) The result s an updated heterogeneous model wth locally varyng slowness and wave speed, respectvely Ths heterogeneous model can now be used to perform a re-localzaton of the E sources usng dfferent effectve wave speeds for each ray (step 1 agan) In a further step the modfed and mproved source postons permt an mproved tomography result (step 2 agan) whch n turn leads to better source postons (step 1) and so on (step 2 step 1) Ths teratve procedure s repeated untl the squares of arrval-tme dfferences between measured and computed tmes (resdues) reach a mnmum or a predefned accuracy level The procedure s equvalent to the soluton of the generalzed nverse localzaton problem n locally sotropc heterogeneous meda and leads to a new magng technque where n addton to the source postons the volume of the specmen s vsualzed n terms of a locally varyng wave speed dstrbuton Whle n tradtonal acoustc tomography the number of avalable ray paths and thus, the depth of coverage s lmted by the number of actuators and recevers, n E tomography hundreds or thousands of E events are typcally avalable Therefore, a relatvely small number of sensors, eg N = 8 or less, can be compensated by pckng a correspondngly larger number of E events Of course, the locatons of E events are usually restrcted to some actve areas lke damage zones leadng to a more or less non-unform coverage of rays However, the large number of rays should also compensate for ths nhomogenety to some extent Moreover, magng of the mmedate vcnty of damage zones s of hgh practcal nterest and n these zones, an extremely hgh depth of ray coverage and thus, a hgh spatal resoluton of the tomographc mage should be achevable 151

6 nother mportant advantage of E tomography compared to tradtonal E testng s connected wth the use of unwanted E events For example n an arcraft usually the majorty of E events s produced by sources of nterference lke jonts and great effort s requred to separate these unwanted sgnals from wanted sgnals generated by cracks or other defects In E tomography, these unwanted sources can also be used for tomographc magng provded that they can also be treated as transent pont sources Whle tradtonal E analyss provdes nformaton about the locaton of actve E regons, E tomography vsualzes both, actve and non-actve regons of the specmen Therefore, acoustcally passve defects n the structure can also be dentfed n prncple The lst of advantages of E tomography as descrbed above s by no means complete Snce E events can also be produced artfcally, eg by pencl-lead breaks or hammer mpacts at the surface of a specmen, new paradgms n acoustc NDT tomography n general are magnable as explaned n somewhat more detal n the outlook t the end of ths secton t s mportant to pont out that E tomography represents a purely algorthmc extenson of tradtonal E analyss usng exactly the same raw data, e no addtonal expenses concernng data acquston are necessary n general 5 E Tomography wth Numercal Data In order to demonstrate the physcal soundness of E tomography, numercal E data sets calculated by the elastodynamc fnte ntegraton technque (EFIT, [4]) were used For that purpose, the cross-secton of an exstng concrete specmen ( mm 2 ) wth steel renforcement and tendon duct was chosen (see Fg 2 and [5]) The specmen was used for fatgue tests and was assgned for E measurements as well The cross secton n Fg 2 shows four steel renforcement bars wth dameters of 22 mm n the corners (c P = 5900 m/s, c S = 3200 m/s, ρ = 7820 kg/m 3 ) and an ungrouted polyethylene tendon duct wth nner dameter of 100 mm and a wall thckness of 3 mm n the mddle of the model (c P = 2300 m/s, c S = 1200 m/s, ρ = 950 kg/m 3 ) 16 sensors were placed at the outer surfaces of the specmen usng a dstance of 11 cm to each other In ths frst step the concrete matrx was approxmated as homogeneous background medum (c P = 3950 m/s, c S = 2250 m/s, ρ = 2050 kg/m 3 ), the sensors as pont detectors Moreover dealzed sotropc E sources were realzed, generatng P-waves only s shown n [5] these smplfyng assumptons can easly be dropped and more realstc models ncludng aggregates, pores, grouted tendon ducts flled wth mortar and steel strands as well as E sources wth more realstc crack mechansms can be smulated n further nvestgatons but are omtted here total of 40 E events wth random locatons nsde the model (but restrcted to the concrete matrx) were calculated by the numercal EFIT code Fgure 3 exemplary shows the wave front snapshots of sx E events taken at a tme of 253 µs after source exctaton n each case The pctures clearly show the nteracton of the pressure waves wth tendon duct, steel renforcement, and outer surfaces of the specmen, respectvely The normal components of partcle velocty were calculated at the ndvdual sensor postons shown n Fg 2 fter that, the frst-arrval tmes of the tme-doman sgnals were determned by usng an automated pckng algorthm based on the Hnkley crteron [6] These arrval tmes then served as nput for the E tomography algorthm called E-TOMO 152

7 The ray coverage due to the 40 E sources s shown n Fg 4 total of = 640 rays was used for E tomography Due to the random locatons of the E events a more or less unform coverage of the model was reached Only the corners are not covered It can also be seen that the steel cable n the bottom left corner s covered less than the remanng three cables due to a lower densty of E events n ths regon Snce the locatons of all E events n the smulaton are exactly known, the accuracy of the tradtonal localzaton algorthm as descrbed n Secton 2 can be evaluated and compared wth the E-TOMO algorthm Snce the tradtonal algorthm uses a homogeneous background medum, the results strongly depend on the chosen P-wave velocty of the matrx as shown n Fg 5 In ths case, a mnmum source locaton error of ± 2 mm s reached at c P 3930 m/s Fg 3 Wave front snapshots of sx dfferent E events Fg 4 Ray coverage of the model accordng to 40 E events and 16 sensors 153

8 If the mean effectve wave speed vares n the range of a few percent whch s the typcal naccuracy n expermental measurements the mean error of source locaton ncreases by a few hundred percent as can be seen n Fg 5 Devatons of only 3-4% n acoustc P-wave speed ( 3930 ± 150 m/s) lead to an ncrease of source locaton error by % (from 2 mm at the mnmum up to 6 and 10 mm, respectvely) In order to avod ths problem, the wave speed could be ncorporated as a further unknown nto the tradtonal localzaton algorthm However, our nvestgatons revealed that n ths case the mnmum error of source locaton amounts to about ± 45 mm whch s sgnfcantly larger than that obtaned by the normal localzaton algorthm wth constant wave speed (±2 mm) Moreover, the enhanced algorthm turned out to be numercally less stable The results of both localzaton algorthms are summarzed n Table 1 and compared Fg 5 Mean error of source locaton as a functon of matrx wave speed as obtaned by the tradtonal (normal) localzaton algorthm Table 1 Results for normal localzaton, enhanced localzaton, and E tomography Normal localsaton (homogeneous matrx); Unknowns: x,y,(z),t Enhanced localsaton (homogeneous matrx); UnknownsÊ: x,y,(z),t,c P coustc emsson tomography (heterogeneous matrx); Unknowns: x,y,(z),t, c P (x,y,z) Mean resdual 071 µs 072 µs 012 µs Mnmum resdual 042 µs 042 µs 004 µs Maxmum resdual 117 µs 114 µs 035 µs Mean source locaton error Mnmum source locaton error Maxmum source locaton error Mean source tme error Mnmum source tme error Maxmum source tme error 220 mm 454 mm 090 mm 031 mm 017 mm 010 mm 498 mm 100 mm 382 mm 025 µs 141 µs 011 µs 004 µs 001 µs 0001 µs 053 µs 425 µs 043 µs 154

9 wth the results of E tomography The spatal coordnate z s gven n parentheses snce the underlyng problem s effectvely 2-D For the normal localzaton the values at c P = 3950 m/s are lsted (compare Fg 5) The results for E tomography, obtaned by the model n Fg 4 (40 E events, 16 sensors), are sgnfcantly better than for the tradtonal algorthms snce heterogenety of the medum s taken nto account Moreover, each new E event leads to a better approxmaton of the spatal velocty feld and thus, also to a better localzaton of the precedng E events Therefore, ncreasng the number of E events n E tomography mproves the overall accuracy of localzaton whle tradtonal localzaton of E events s performed ndependently from each other, e the mean error s more or less ndependent from the number of E events 0 teratons 5 teratons 10 teratons 15 teratons 20 teratons Fg 6 Tomographc mages of the model after 0, 5, 10, 15, and 20 teratons of the E-TOMO algorthm (28 28 pxels wth ant-alasng) Fgure 6 shows tomographc mages after 0 (homogeneous start model), 5, 10, 15, and 20 teratons of the E-TOMO algorthm obtaned at the model from Fg 4 (40 E events, 16 sensors) One can see that after about teratons the wave speed model becomes more and more stable The steel renforcement cables wth locally ncreased wave speed (red color) n the corners as well as the ungrouted tendon duct wth effectvely decreased wave speed (blue color) n the mddle of the model are clearly vsble It s obvous that the reconstructon of the steel cable n the bottom left corner s worse than for the remanng three cables, whch s most lkely caused by the worse ray coverage as shown n Fg 4 155

10 Fgure 7 shows the nfluence of the number of E events on the tomography result (after 20 teratons n each case) The correspondng pctures represent the wave speed model usng 5, 10, 20, 30, and 40 E events for tomographc reconstructon If the number of avalable sources and thus, the number of rays s ncreased the model becomes better and better However, smlar to Fg 6 a saturaton behavor can be observed f the ray coverage of the tomography cells reaches a certan level further ncrease of the number of rays would not automatcally mprove the qualty of the mage f we keep the number of tomography cells constant Instead we could ncrease the number of cells and thus, the spatal resoluton of the tomographc mage Wth typcal numbers of E events n the range of hundreds or thousands mage resolutons that can never be reached wth tradtonal acoustc tomography seem to be possble 5 E events 10 E events 20 E events 30 E events 40 E events Fg 7 Tomographc mages of the model usng 5, 10, 20, 30, and 40 E events for reconstructon (28 28 pxels wth ant-alasng) However, as can be seen n Fgs 6 and 7 there are stll some artfacts n the tomographc mages most lkely caused by the smple straght-lne approxmaton of the ray model Due to the large dfferences n acoustc mpedance between the matrx and the scatterers (steel renforcement and ungrouted tendon duct, respectvely) ths straght-lne approxmaton s not suffcently fulflled n the present case Thus, takng dffracton and refracton effects wth curved ray paths nto account should lead to sgnfcantly mproved mages n the future nother reason for some of the artfacts s the non-unform ray coverage of the model leadng to large dfferences n the number of rays passng through the equally szed tomography cells s a consequence, workng wth adaptvely szed tomography cells should lead to a more even ray densty per cell and thus, to a better tomographc mage 156

11 6 Summary and Outlook In the present paper t has been shown that E tomography represents an mportant mprovement of tradtonal E analyss leadng to sgnfcantly better localzaton of E events due to consderaton of heterogeneous meda The method leads to a new magng technque where n addton to the source postons the volume of the specmen s vsualzed n terms of a locally varyng wave speed dstrbuton part from that, E tomography has self-contaned relevance snce n the tradtonal sense unwanted E events can be used for tomographc magng, too Moreover, structural elements and acoustcally passve defects can also be vsualzed Fnally, also E events artfcally generated, eg by pencl-lead breaks or hammer mpacts at outer surfaces of the specmen can be used for tomographc magng Thus, t seems that n many cases tradtonal acoustc tomography wth fxed and nflexble ray coverage could be replaced by adaptve E tomography where tomographc nverson takes place on a ray-by-ray or rather an event-by-event bass Based on the prevous teraton the nvestgator could nteractvely decde f and where a hgher resoluton of the tomographc mage s necessary and consequently, where the next E event should be placed manually s has been shown n the dscusson of the tomographc mages n Fgs 6 and 7, there s stll much room for mprovements of the underlyng algorthms Varetes of algebrac reconstructon technques usng curved ray paths should be tested for applcablty n E tomography n the future lso the development of transform based tomography algorthms smlar to the fltered back-projecton algorthm seems to be worthwhle Moreover, snce n nondestructve evaluaton strong scatterers are typcal, the development of algorthms for E reflecton tomography (eg fltered back-propagaton or synthetc aperture focusng technque, SFT [7]) should be consdered So far, the physcal soundness of E tomography has been shown by dealzed numercal data only In the future, more realstc models ncludng heterogenety of the concrete matrx and more complcated crack mechansms wll be nvestgated Real measurements on concrete beams (usng pencl-lead breaks as E sources) and alumnum plates (usng Lamb waves for tomography) are under way n order to verfy E tomography expermentally Based on the results obtaned so far t can be expected that E tomography has great potental for future applcatons offerng totally new perspectves for E magng and acoustc tomography n general References [1] Kak, C, and Slaney, M, Prncples of Computerzed Tomographc Imagng, IEEE Press, New York, 1988 [2] Schechter, RS, Mgnogna, RB, and Delsanto, PP, Ultrasonc tomography usng curved ray paths obtaned by wave propagaton smulatons on a massvely parallel computer, J coust Soc m 100 (4), , 1996 [3] Mgnogna, RB, and Delsanto, PP, parallel approach to acoustc tomography, J coust Soc m 99 (4), , 1996 [4] Fellnger, P, Marklen, R, Langenberg, K J, and Klaholz, S, Numercal modellng of elastc wave propagaton and scatterng wth EFIT Elastodynamc fnte ntegraton technque, Wave Moton 21, 47-66,

12 [5] Schubert, F, and Schechnger, B, Numercal modellng of acoustc emsson sources and wave propagaton n concrete, NDTnet, the e-journal of Nondestructve Testng, 7 (9), 2002 ( [6] Renhardt, HW, and Grosse, CU, Contnuous montorng of settng and hardenng of mortar and concrete, Constructon and Buldng Materals, 18, , 2004 [7] Langenberg, KJ, Berger, M, Kreutter, T, Mayer, K, and Schmtz, V, Synthetc aperture focusng technque sgnal processng, NDT Internatonal, 19 (3), ,