Evaluation of Defect Shape Based on Inverse Analysis Considering the Resolution of Magnetic Sensor

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1 E-Journal of Advancd Maintnanc Vol.3, No. (20) -0 Japan Socit of Maintnolog Evaluation of Dfct Shap Basd on Invrs Analsis Considring th Rsolution of Magntic Snsor Shogo NAKASUMI,* and akauki SUZUKI National Institut of Advancd Industrial Scinc and chnolog (AIS), East -2- Namiki, sukuba, Ibaraki , Japan ABSRAC his papr prsnts a tchniqu for improving th accurac of non-dstructiv valuation using magntic flux lakag. Rspons function that associats magntic dipols and magntic flux dnsitis in trms of gomtrical coordinat is modifid so as to captur th avrag magntic flux valu along th snsor lngth, that is consistnt with xprimntall masurd magntic flux. Invrs analsis of a smi-lliptical surfac dfct is conductd through numrical simulation. B using th modifid rspons function, th rstord dfct shap bcoms mor propr and rmarkabl improvmnts ar indicatd, spciall in low lift-off cass. KEYWORDS magntic flux lakag, flux gat snsor, invrs analsis, non-dstructiv valuation, magntic dipol, ikhonov rgularization ARICLE INFORMAION Articl histor: Rcivd 28 March 20 Accptd 24 Ma 20. Introduction Nondstructiv valuation that idntifis structural dfcts such as cracks and pitting is important for saft and rliabl structur valuation. Structural matrials ar oftn magntic substancs, making non-dstructiv valuation using lctromagntism ffctiv. h magntic flux lakag dnsit mthod is an inspction mthod b which dfcts ar dtctd b masuring th lakag flux dnsit causd b th magntic charg on crack surfacs. Howvr, quantitativ valuation of dfct siz, shap, and othr faturs cannot b prformd soll b masuring lakag flux dnsit. o valuat ths dfcts quantitativl, it is ncssar to rstor th distribution of th magntic chargs b formulating a rspons function that indicats a gomtrical rlation btwn magntic chargs on th dfct and th masurd magntic flux lakag dnsit, and b conducting invrs analsis [-4]. h flux gat snsor (FG snsor, blow) is a highl snsitiv magntic snsor of about 0-7 that has attractd rcnt attntion bcaus can b usd from a low lift-off ara (th distanc btwn th sampl and th obsrvation plan) of about mm to high lift-off ara of about 0 mm [5-7]. FG snsor masurmnts giv high accurac from a low lift-off ara. Gnrall, howvr, th spatial chang rat of th magntic flux dnsit bcoms vr larg in low lift-off aras, and such stp changs within a small ara cannot b accuratl capturd if masurd using an FG snsor of about 3 mm in lngth. As a rsult, thr is a possibilit that th high snsitivit of th FG snsor will not lad to a highl accurat valuation of th dfct shap. o th authors knowldg, thr is no prvious rsarch that conducts invrs analsis taking into considration th influnc of snsor lngth. With a high lift-off ara, on th othr hand, attnuation of th magntic flux dnsit bcoms significant, and masurmnt nois gratl influncs th invrs analsis. his papr assums masurmnt of magntic lakag flux dnsit using an FG snsor. h rspons function is improvd in ordr to considr th influnc of snsor lngth, and th rsults of vrifing th magntic flux dnsit distribution through forward analsis and th dfct shap * Corrsponding author, nakasumi.shogo@aist.go.p

3mm mm Flux gat snsor Lift off Masurmnt lin Dfct z x Magntic dipol 30mm momnt vctor Fig.. Masurmnt of magntic flux lakag rstoration through invrs analsis ar dscribd. 2.

2 3mm mm Flux gat snsor Lift off Masurmnt lin Dfct z x Magntic dipol 30mm momnt vctor Fig.. Masurmnt of magntic flux lakag rstoration through invrs analsis ar dscribd. 2. Driving a rspons function that considrs snsor rsolution 2.. Formulation of magntic charg-flux dnsit Figur shows th apparanc of a spcimn of lctromagntic matrial with a surfac slit crack. h dirction of th normal of th structur surfac is takn as th z-axis, and a vrtical dirction to th crack surfac is takn as th -axis. Positiv and ngativ magntic chargs xisting on th crack surfacs ar assumd to b magntic dipols. A magntic dipol has a non-zro lmnt m onl in th dirction of th -axis. As th flux dnsit has non-zro -axis valu lmnts B, w will us it in this stud and flux dnsit blow should b takn to man B. Whn th coordinats of th magntic dipol ar dscribd as x = ( x,, z ), th flux dnsit obsrvd at th x = x, z, is givn b Equation ()[8]. point ( ) ( ) = ( ) B x F x x m dx () Ω Hr, Ω is th total ara from which th magntic dipols xit, and F ( ) x is a function whos input variabls ar th rlativ coordinats of th obsrvation point to th magntic dipol, as dscribd b Equation (2) F ( x,, z) μ0 = 4π ( x 2 + z ) ( x + + z ) Hr, μ 0 is th spac prmabilit. Not that Equation (2) is applicabl in cass whr magntic dipols xist at th origin. h sum of th discrt magntic dipols is substitutd for th intgration in Equation (): N ( ) = ( ) = 52 B x F x x m (3) Hr, th suprscript in trms x and m is th indx of th magntic dipol, and N is th numbr of magntic dipols. Equation (3) holds at vr obsrvation point on a plan with a givn constant (2) 2

3 Normalizd magntic flux dnsit [a.u.] lift-off valu. Aftr som manipulation, Equation (4) is obtaind Liftoff = mm Liftoff = 5mm Liftoff =0mm Y position [mm] Fig. 2. Distribution of normalizd magntic flux (x = 0 masurmnt lin) 2 N B F F F m B F m = M M MN N B F F m (4) i Hr, th suprscript i in th trm B is th indx, and M is th numbr of obsrvation points. is th rspons function of Equation (2), which rlats an obsrvation point i to a magntic dipol. Equation (4) can b rwrittn as Equation (5). In th following, w call F of Equation (5) as a rspons function matrix. B= Fm (5) i F 2.2. Influnc of lift-off on flux dnsit distribution Equation (4) givs th magntic flux dnsit of th configurd magntic dipols as obsrvd at an arbitrar point, and is considrd a forward analsis. In th following, w vrif how th lift-off valu influncs th flux dnsit distribution obtaind b this forward analsis. For th tst pic in Fig., Fig. 2 shows th -axis distributions of th flux dnsit obtaind on th lin x = 0 with lift-off st to mm, 5 mm, and 0 mm. h graph lin corrsponding to ach lift-off is normalizd b dividing b ach maximum valu so that th maximum valu bcoms. Not that along th z-axis, th flux dnsit dcrass in proportion to approximatl th -3 powr of th z coordinat. A basic tndnc of th distribution whr lift-off is mm is confirmd blow. h flux dnsit is 0 at infinit distanc. Howvr, it dcrass whn approaching th crack walls ( = 0 mm), and thn bcoms a ngativ valu. Aftr raching a minimum valu in th vicinit of = 2 mm, it incrass stpl, attaining a pak-lik distribution of maximum valu on th crack surfac. h FG snsor assumd in this stud is of clindrical shap, 3 mm in lngth and about mm in diamtr (Fig. ). h masurd valu of th magntic flux is an avrag valu of th magntic flux pntrating through th full intrnal lngth of th snsor. hr is a possibilit that flux dnsitis showing rapid incrass and dcrass cannot b masurd accuratl in th vicinit of < 2 mm, which is almost th sam as th lngth of th snsor. On th othr hand, in th cas whr lift-off is 0 mm, spatial changs of th magntic flux ar dulld, making th influnc of th lngth of th snsor rlativl small as compard with th cas whr lift-off is mm. 3

4 B FG snsor B L L 2L Mthod of corrcting th rspons function As mntiond abov, thr is a possibilit that th FG snsor will not accuratl captur stp changs around th crack surfac for a low lift-off ara of about mm. In considration of such, in this sction w modif th rspons function matrix of Equation (5). It is ralistic to assum that distribution of th flux dnsit in th snsor is wightd towards th vicinit of th snsor s cntr. In this stud, howvr, th simpl man valu is usd to modif th obsrvd flux dnsit as a masur simplification. Flux dnsit B ( x 0 ) masurd b th snsor at th point x 0 = ( x0, 0, z0) is givn as follows: masur + L x = (,, ) 2L (6) ( ) B 0 0 B x 0 t z 0 dt 0 L Hr, 2L is th lngth of th snsor. h avraging oprator acts on th rspons function. As a rsult, aftr manipulation similar to Equations () (3), Equation (6) lads to Equation (7) and Equation (8). N ( 0) = ( 0 ) masur = B x F x x m (7) + L = (,, ) 2L (8) F 0 F x 0 t z 0 dt 0 L L L Fig. 3. Concptual imag of trapzoidal intgration in th FG snsor lngth intrval h right-sid trm F in Equation (8) is givn b Equation (2). Sinc it is difficult to prform th intgration in Equation (8) analticall within th rang of ral numbrs, it is modifid according to th composit trapzoid rul, thus giving Equation (9). m 2L F { F ( x0,, z0) + F ( x0, +, z0) } (9) i i 2L i= 2 m Hr, m is th trapzoid numbr (th numbr of partitions), and i ( i=,2,, m+ ) is th -coordinat of th division point that divids th -dirction sction 0 L< < 0 + L into m sctions. Using F from Equation (9), w obtain th modifid rspons function matrix F in Equation (0). 4

5 Magntic flux dnsit [a.u.] Y position [mm] (a) z=mm S. Nakasumi, t al. / Evaluation of Dfct Shap Basd on Invrs Analsis B B' (m=2) B' (m=5) Magntic flux dnsit [a.u.] Y position [mm] (b) z=0mm Fig. 4. Effct of rspons function rvision for magntic flux dnsit distribution B B' (m=2) B' (m=5) 2 N F F F 2 F F = M MN F F (0) 2.4. Flux dnsit distribution b modifid rspons function his sction vrifis th ffct of th modifid rspons function matrix. Blow, th flux dnsit obtaind using F from Equation (0) in th forward analsis is dscribd as B. hat is to sa, Equation () holds. B = Fm () Figur 4 shows th magntic flux dnsit distributions obtaind b forward analsis in th cas whr lift-off is mm and 0 mm in th spcimn shown in Fig.. Hr, B and B ar th flux dnsitis obtaind b th forward analsis shown in Equation (5) and Equation (), rspctivl, and m of B is th numbr of trapzoidal divisions. In Fig. 4(a), th distribution of B (m=2) is diffrnt from that of B bcaus th approximation shown in Equation (9) and Fig.3 is not nough. As m in incrass, th distribution of B convrgs graduall on a crtain curv, and our trials rsultd in that m = 5 is suitabl valu in trms of accurac and computational cost. Compard with th distribution of B, that of B (m=5) loss th stpnss at = 0 mm and it rounds smoothl. Morovr, th position whr th sign rvrss movs about mm outward. On th othr hand, B and B (m=5) showd almost th sam bhavior in Fig. 4(b). In summar, th distribution of th flux dnsit obsrvd in a low lift-off ara is gratl influncd b modifing th rspons function, and th influnc is smallr in a high lift-off ara. 3. Evaluation of dfct shap b invrs analsis 3.. Driving a numrical analsis modl and simulatd flux dnsit In th prcding sction w dvlopd a rspons function that considrs th influnc of th lngth of th snsor, and forward analsis confirmd th ffct. In this sction, w conduct invrs analsis to dvlop a practicabl modl and vrif th ffct. h modl uss a smi-lliptical surfac crack on a tst pic. Figur 5 shows an xtrnal viw. h radii of th long and short axs ar 5 mm and 2.5 mm, rspctivl. Flux dnsit, which bcoms input data for th invrs analsis, is originall obtaind through xprimntal masurmnt. In this papr, howvr, mathmatical calculation of th forward analsis 5

Masurmnt plan F.G. Snsor Lift off z x 2.5mm =0 cross 0mm sction 30mm Fig. 5. Magntic dipol configuration for a smi-lliptical surfac dfct z Magntic dipol 0.25mm 0.25mm x 5.0mm Magntic flux dnsit [a.u.] Magntic flux dnsit [a.

5 in a grid. h configuration intrval is st to 0.25 mm in both th x and z dirction, and th valu of th magntic dipol locatd within th smi-lliptical ara was st to b -.

6 Masurmnt plan F.G. Snsor Lift off z x 2.5mm =0 cross 0mm sction 30mm Fig. 5. Magntic dipol configuration for a smi-lliptical surfac dfct z Magntic dipol 0.25mm 0.25mm x 5.0mm Magntic flux dnsit [a.u.] Magntic flux dnsit [a.u.] [mm] is usd as a substitut for xprimntal masurmnt. W will thrfor call this th simulatd flux dnsit. h drivation procss is xplaind blow. h magntic dipols ar configurd on th sction = 0 of Fig. 5 in a grid. h configuration intrval is st to 0.25 mm in both th x and z dirction, and th valu of th magntic dipol locatd within th smi-lliptical ara was st to b -. Morovr, obsrvational points for th flux dnsit ar configurd in a grid along th x and axis at 0.5 mm intrvals in th obsrvation plan (th x plan whr z quals th lift-off) within th ara -5 < x <5 (mm), -20 < < 20 (mm). As a rsult, th numbr of obsrvation points M = 494 and th numbr of magntic dipols N = 70. From th discussion of sctions 2.2 and 2.3, th magntic flux dnsit that bcoms th input data for th invrs analsis should not b B of Equation (5), but rathr B of Equation (). In addition, masurmnt nois must also b considrd. h nois lvl is st to % of th maximum valu of B, and th distribution is assumd to b a uniform random numbr with a man valu of 0. h simulatd magntic flux dnsit is thrfor givn b B in Equation (2). [mm] (a) Lift-off = mm (b) Lift-off = 0 mm Fig. 6. Magntic flux dnsit distribution of a smi-lliptical surfac dfct b forward analsis including % rror B = B + R = Fm + R (2) Hr, R is th nois vctor mntiond abov, and m is a vctor indicating th distribution of magntic dipols shown in Fig. 5. Figur 6 shows th distribution of B obtaind b Equation (2) whn lift-off is mm and 0 mm. his magntic flux dnsit is calld th simulatd flux dnsit, and th licitation procss is xplaind blow Application of ikhonov rgularization Using F of Equation (0) and B of Equation (2), th distribution of th magntic dipols that 6

7 Rank F F' Forward analsis Magntic dipol configuration m ( ) Magntic flux dnsit B = F m Invrs analsis Magntic dipol rstration m = F F + αi F B ikhonov Rgularization min{ Fm B + αm Im } Lift off [mm] Fig. 7. Rspons function matrix ranks Add Error (%) B = B + R = F m+ R Fig. 8. Invrs analsis procdur should b obtaind b th invrs analsis bcoms th m in Equation (3) that minimizs th valuation function J. 2 J = Fm B (3) Not that m in Equation (3) is an output valu of th invrs analsis, whil m of Equation (2) is an input valu, which is obtaind from th forward analsis. h proprtis of matrix F dtrmin whthr th appropriat m is obtaind b minimizing J. In th following, F and F of Equation (4) ar vrifid from a viwpoint of linar indpndnc. Figur 7 shows th rank (th numbr of indpndnt linar quations) of F and F with lift-off varing from mm to 0 mm. h numbr of obsrvation points M = 494, thus it would sm that M quations xist. h numbr of indpndnt quations found in natur, howvr, ar 600 or lss, fwr than th numbr of dipols N that should b rqustd. Morovr, it can b sn that th rank dcrass as lift-off grows, and th rank of F bcoms lowr than that of F for all lift-off valus. wo factors contribut to incrasing th numbr of simultanous quations, on bing th dullnss of th magntic flux dnsit for high lift-off valus as shown in Fig. 2, th othr bing th avraging of th magntic flux dnsit. If th solution is obtaind from simultanous quations in which linar indpndnc dcrass, th influnc of nois bcoms xcssivl prdominant, and as a rsult so-calld vibration solutions ar oftn obtaind. o obtain an appropriat solution, thrfor, w will modif Equation (3) using th ikhonov rgularization mthod [9]. Naml, w minimiz J 2 in Equation (4) instad of J in Equation (3). J 2 2 = Fm B + αm Im (4) Hr,α is a rgularization paramtr and I is a unit matrix. h first and scond trms on th right-hand sid ar calld th rsidual trm and th rgularization trm, rspctivl. Whn J 2 is minimizd, and if M > N, m is obtaind b Equation (5). ( α ) m = F F + I F B (5) Equation (5) is a solution in th cas whr th modifid rspons function matrix F is usd for th invrs analsis. As mntiond latr in th numrical xampls in nxt sction, m in Equation (6) bcoms th solution, whn th xisting rspons function F in Equation (5) is usd for th invrs analsis. ( ) m= F F+αI F B (6) 7

Rgularizd trm 0 3 0 2 0 0 0 α= 0-2 α= 0 - +++++++++++++++++ Fm B + αm Im 2 +++++++++++++++++ Fm B + αm Im 2 0-0 0 2 Rsiduals trm Fig. 9.

8 Rgularizd trm α= 0-2 α= Fm B + αm Im Fm B + αm Im Rsiduals trm Fig. 9. L-curv (lift-off = mm) (a) Distribution of m (α = 0 - ) (b) Distribution of m (α = 0-2 ) Fig. 0. Distribution of rstord magntic dipols (lift-off = mm) A flowchart of th forward analsis and th invrs analsis dscribd abov is shown in Fig Evaluation of dfct shap Figur 9 shows th curv dscribd b th plot points whn th rgularization paramtr α varis 7 from 0 3 to 0 and lift-off is mm. h horizontal and vrtical axs ar th rsidual trm norm and th rgularization trm norm, rspctivl. his curv is calld th L-curv, and it is thought th α that corrsponds to th vicinit of th bnd location of th L-curv givs th most appropriat distribution of magntic dipols [0,]. Examining th appropriat rgularization paramtr locatd in this vicinit basd on th rsult imag of th invrs analsis, α = 0 was obtaind for m and 2 α = 0 was obtaind for m. h α giving th bst magntic distribution for m tnds to b somwhat largr than that of th bnd point of th L-curv. Figur 0 shows th distribution of magntic dipols obtaind b th invrs analsis. Figurs 0(a) and 0(b) show th distribution of m in Equation (6) and m in Equation (5), rspctivl. his notation is th sam for Fig. 2, which is mntiond latr. Black solid lins in ths figurs show th boundar (crack front lin) of th assumd crack surfac. h magntic dipols of m in Equation (2) ar ngativ valu. Howvr, m in Fig. 0(a) has positiv valu in th vicinit of th surfac (z = 0 mm), which obviousl diffrs from m. his tndnc was sn vn if α was somwhat changd. It sms that this phnomnon occurs bcaus th position at which th sign of th magntic flux dnsit rvrss around th crack is diffrnt according to diffrncs in th rspons function, as shown in Fig. 2. On th othr hand, in m of Fig. 0(b), th sign of th magntic dipol dos not rvrs lik m in Fig. 0(a), and th magntic dipol is rproducd accuratl in th surfac vicinit. As a rsult, a profil almost corrsponding to th assumd crack front was obtaind. Figurs and 2 show th L-curv and magntic dipol distribution, rspctivl, whr lift-off is 0 mm. In Fig., m and m show almost th sam bhavior. his mans that th diffrnc btwn m and m in Fig. is rducd from th viwpoint of th rgularization as compard with Fig. 9. Similarl, th diffrnc btwn Figs. 2(a) and 2(b) is also rducd as compard with Fig. 0. If both ar compard in dtail, howvr, th crack shap of Fig. 2(b) is mor accurat than that of Fig. 2(a), which signifis that th ffct of th modifid rspons function is still ffctiv. 8

9 Rgularizd trm α= 0-6 α= Fm B + αm Im Fm B + αm Im Rsiduals trm Fig.. L-curv (lift-off = 0 mm) (a) Distribution of m (α = 0-5 ) (b) Distribution of m (α = 0-6 ) Fig. 2. Distribution of rstord magntic dipols (lift-off = 0 mm) From low lift-off ara to high lift-off ara, thrfor, th ffctivnss of modifing th rspons function matrix is confirmd for rproducing th crack shap mor accuratl. Furthrmor, this ffctivnss bcoms spciall rmarkabl with low lift-off aras. 4. Conclusions h following conclusions wr obtaind about th magntic flux lakag dnsit mthod and valuation of dfct shaps using invrs analsis.. h influnc of th lngth of th flux gat snsor was considrd, and a nw rspons function that outputs th avrag valu of th magntic flux dnsit that pntrats th snsor was proposd. 2. Invrs analsis of a smi-lliptical surfac crack using th ikhonov rgularization mthod was conductd using th proposd rspons function. As a rsult, a rmarkabl improvmnt ffct was obtaind at th vicinit of th surfac of crack for low lift-off aras. h improvmnt was also sn for high lift-off aras, though not so rmarkabl. Rfrncs [] D. Minkov and. Shoi, Mthod for sizing of 3-D surfac braking flaws b lakag flux, ND&E intrnational, Vol.3-5, 998, pp [2] D. Minkov, J. L and. Shoi, Stud of crack invrsions utilizing dipol modl of a crack and hall lmnt masurmnts, Journal of Magntism and Magntic Matrials, Vol.27, 2000, pp [3] R. Baskaran and M. P. Janawadkar, Imaging dfcts with rducd spac invrsion of magntic flux lakag filds, ND&E intrnational, Vol.40, 2007, pp [4] S. akaa, G. Prda, K. Dmachi,. Uchimoto and K. Mia, Rconstruction of magntization from magntic flux lakag for valuation of matrial dgradation, Elctromagntic nondstructiv valuation (V), IOS Prss, 200, pp [5] Magntics tchnical committ of th institut of Elctrical Enginrs of Japan, Magntics - Fundamntals 9

10 and Applications - p.7, CORONA Publishing Co., LD 999 [6]. Suzuki, A. rasaki, A. Sasamoto, Y. Nishimura and. ramoto, Nondstructiv valuation of frromagntic structural matrials using FG snsor, Elctromagntic Nondstructiv Evaluation (XII), IOS Prss, 2009, pp [7] S. akaa,. Suzuki, Y. Matsumoto, K. Dmachi and M. Usaka, Magntic microstructur of th snsitizd SUS304 stainlss stl, Elctromagntic Nondstructiv Evaluation (VII), IOS Prss, 2006, pp [8]. Suzuki, A. Sasamoto, Y. Nishimura and. ramoto, Matrial charactrization using magntic forc microscop, Intrnational ournal of applid lctromagntics and mchanics, Vol.28, No.,2, pp63-69 [9] A. N. ikhonov and V. Y. Arsin, Solutions of ill-posd problms, Halstd Prss, 977 [0] C. Hansn, Analsis of discrt ill-posd problms b mans of th L-Curv, Socit for Industrial and Applid Mathmatics, Vol.34, No.4, 992, pp [] F. Koima, N. Kasai, Y. Nagashima, Idntification of matrial flaws using Hc-SQUID b rgularizd invrs analsis, Journal of th Japan Socit of Applid Elctromagntics and Mchanics, Vol.8, No., 2000, pp9-5 0

EXST Regression Techniques Page 1

EXST Regression Techniques Page 1 EXST704 - Rgrssion Tchniqus Pag 1 Masurmnt rrors in X W hav assumd that all variation is in Y. Masurmnt rror in this variabl will not ffct th rsults, as long as thy ar uncorrlatd and unbiasd, sinc thy