Shape sensing of aerospace structures by coupling of isogeometric analysis and inverse finite element method

Transcription

1 Shap snsng of arospac structurs by couplng of sogomtrc analyss and nvrs fnt lmnt mthod dnan Kfal and Erkan Otrkus Unvrsty of Strathclyd, 00 Montros Strt, Glasgow G4 0LZ, UK Ths papr prsnts a novl sogomtrc nvrs Fnt Elmnt Mthod (FEM) formulaton, whch coupls th NURBS-basd sogomtrc analyss (IG) togthr wth th FEM mthodology for shap snsng of complx/curvd thn shll structurs. Th prmary goal s to b gomtrcally xact rgardlss of th dscrtzaton sz and to obtan a smoothr shap snsng vn wth lss numbr of stran snsors. For ths purpos, an sogomtrc Krchhoff- Lov nvrs-shll lmnt (KLS) s dvlopd on th bass of a wghtd-last-squars functonal that uss mmbran and bndng stran masurs consstnt wth th Krchhoff- Lov shll thory. Th novl KLS lmnt mploys NURBS not only as a gomtry dscrtzaton tchnology, but also as a dscrtzaton tool for dsplacmnt doman. Thrfor, ths dvlopmnt srvs th followng bnfcal aspcts of th IG for th shap snsng analyss basd on FEM mthodology: () xact rprsntaton of computatonal gomtry, () smplfd msh rfnmnt, () smooth (hgh-ordr contnuty) bass functons, and fnally (4) ntgraton of dsgn and analyss n only on computatonal doman. Th supror capablts of KLS lmnt for shap snsng of curvd shlls ar dmonstratd by varous cas studs ncludng a pnchd hmsphr and a partly clampd hyprbolc parabolod. Fnally, th ffct of snsor locatons, numbr of snsors, and th dscrtzaton of th gomtry on soluton accuracy s xamnd. I. Introducton Ovr th last fw dcads, Structural Halth Montorng (SHM) has bcom a ky tchnology of th futur arospac structurs to achv sgnfcant mprovmnt n safty, rlablty, and affordablty. typcal SHM systm provds ral-tm vhcl halth nformaton va snsors and softwar to mtgat accdnts du to falur and achv saf landng. Th nvrs Fnt Elmnt Mthod (FEM) s a stat-of-th-art mthodology orgnally ntroducd by Tsslr and Spanglr [, ] for ral-tm rconstructon of full-fld structural dsplacmnts n plat and shll structurs that ar nstrumntd by stran snsors. Ths nvrs problm, commonly known as shap snsng, s th most crucal componnt of a typcal SHM systm. Snc th frst publcaton of FEM algorthm [, ], many dffrnt numrcal and xprmntal studs hav bn dvotd to xpand th horzon of th FEM mthodology n th ltratur. For nstanc, Tsslr and Spanglr [] dvlopd a thr-nod nvrs shll lmnt (MIN) utlzng lowst-ordr ansoparamtrc C 0 -contnuous shap functons and adoptng th knmatc assumptons of th Mndln s plat thory. Mor rcntly, Kfal and co-workrs [4] hav rcntly formulatd a four-nod quadrlatral nvrs-shll lmnt, QS4, basd upon a wghtd-last-squars functonal. Ths nw lmnt ncluds hrarchcal drllng rotaton dgrs-of-frdom (DOF) and furthr xtnds th practcal usfulnss of FEM for shap-snsng analyss of larg-scal structurs. Morovr, Ghrlon and co-workrs [5] formulatd a robust nvrs fram lmnt that uss knmatc assumptons of Tmoshnko bam thory ncludng strtchng, bndng, transvrs-shar and torson dformaton mods. Furthrmor, Crraccho t al. [6] has rcntly mprovd th orgnal FEM formulaton Tsslr and Spanglr [, ] by addng th knmatc assumptons of rcntly dvlopd Rfnd Zgzag Thory [7] n ordr to prform SHM of mult-layrd compost and sandwch structurs. part from thortcal dvlopmnts, Kfal and Otrkus [8] prformd shap-snsng of a longtudnally and transvrsly stffnd plat as a fundamntal applcaton of th FEM framwork to SHM of marn structurs. Smlarly, Kfal and Otrkus [9] prsntd a mor sophstcatd applcaton of FEM to marn structurs namly dsplacmnt and strss montorng of a chmcal tankr basd on FEM algorthm. Mor rcntly, Kfal and Otrkus [0] hav dmonstratd th applcaton of th FEM mthodology PhD Rsarchr, Dpartmnt of Naval rchtctur, Ocan, and Marn Engnrng. ssocat Profssor, Dpartmnt of Naval rchtctur, Ocan, and Marn Engnrng, I Mmbr.

2 for montorng mult-axal dformatons and strsss of a Panamax contanrshp advancng n bam sa wavs. ll ths nvstgatons hav so far provd that th FEM framwork s an accurat, robust, and fast shap- and strsssnsng algorthm Morovr, sogomtrc analyss (IG) orgnally ntroducd by Hughs t al. [], prsnts smooth functon spacs such as Non-Unform Ratonal B-spln (NURBS) to numrcally solv a numbr of ngnrng problms. Rcntly, IG rcvd a grat dal of attnton from both acadmy and ndustry (.g., rfr to []) du to th hghordr contnuty ( C, p > 0) provdd by NURBS n both gomtrc doman and physcal smulaton. In fact, a p smoothr shap snsng can b obtand basd on th FEM framwork, f th shap functons nsur a hgh-ordr contnuty throughout th lmnt ntror and dg ntrfac. Thrfor, th man and novl am of ths study s to carry out a bttr shap snsng analyss of arospac structurs by couplng th NURBS-basd IG togthr wth th FEM mthodology. For ths purpos, a rotaton-fr sogomtrc nvrs-shll lmnt s dvlopd basd on th knmatc assumptons of th Krchhoff-Lov plat/shll thory and ths nvrs-shll lmnt lablld as KLS. Th shap snsng capablty of th KLS lmnt s valdatd by solvng two numrcal xampls, namly pnchd hmsphr and partly clampd hyprbolc parabolod. Fnally, th ffct of snsor locatons, numbr of snsors and th dscrtzaton of th gomtry on soluton accuracy s xamnd. II. Rotaton-fr sogomtrc nvrs-shll lmnt formulaton novl sogomtrc Krchhoff-Lov nvrs-shll lmnt, labld KLS, s dvlopd on th bass of FEM last-squars formulaton. Ths dvlopmnt coupls th NURBS-basd IG togthr wth th FEM mthodology for shap snsng analyss of thn shll structurs such as arospac structurs. In th followng formulaton, Grk ndcs tak th valus of and whl th Latn ndcs rang from to. Th convctv coordnats ar β = [ β β β ] T whr β dnots n-plan coordnats and β dfns thcknss drcton of th shll. Th KLS lmnt has a unform shll thcknss β [ h, + h] and th matral ponts locatd at th md-surfac of th shll ar dscrbd as β = 0. Th paramtrc coordnats ξ = [ ξ η ζ] T usd to dfn th NURBS bass functons ar assocatd wth th convctv coordnats β as physcal spac. Th ncssary poston vctors to dvlop th KLS formulaton, translaton dgr-of-frdoms (DOFs) of componnts ( v, v, v ) ar dmonstratd n Fgur. th control pont ( d, d, d ), and dsplacmnt vctor Fgur. (a) KLS lmnt. (b) Translatonal DOFs of th control pont. Th poston vctor to a matral pont on th md-surfac of th KLS lmnt can b dscrbd by th fnt sum of NURBS lmnt bass functons as x x = B B () ( β, β ) R( ξ, η) R

3 whr R s th NURBS bass functons and gomtry of th KLS lmnt. Th covarant bas vctors of md-surfac B ar th coordnats of th control ponts that dfns th physcal confguraton by takng th partal drvatvs of x wth rspct to convctv coordnats ( β, β ) = = R, β a can b obtand n rfrnc β as x a a B () Usng th bass vctors a, a unt-magntud vctor fld (th drctor vctor) that s prpndcular to th tangnt plan of any pont blongs to md-surfac can b dfnd as F F( β, β ) = a a a a Thn, th poston vctor x, th drctor vctor F, and th thcknss drcton convctv coordnat () β can b utlzd to dfn a poston vctor to an arbtrary matral pont of th shll body n th rfrnc confguraton as X X( β, β, β ) = x( β, β ) + β F ( β, β ) (4) Th dsplacmnt vctor v rprsntng th dsplacmnt fld btwn rfrnc and currnt confguratons of th shll body can b dfnd as v v ( β, β ) = [ v ( β, β ) v ( β, β ) v ( β, β )] T (5a) and th sam NURBS bass functons usd for th physcal gomtry dscrtzaton can b utlzd to xprss th dsplacmnt fld as v v = d d (5b) ( β, β ) R(, ) ξ η R whr d = [ d d d] T s th translatonal DOFs of ach control pont that ar usd to dfn th local bass functons th KLS lmnt. ccordng to th Krchhoff-Lov hypothss as th drctor vctor F rmans normal to th md-surfac, dformaton of any arbtrary pont n th shll body can b dfnd from a lnarzd rotaton θ of th drctor vctor F as [] ( ) u u( β, β, β ) = v+ β θ F (6a) whr th rotaton vctor θ rprsnts an lmnt of tangnt spac spannd by n-plan bas vctors a and can b dfnd as θ θ = φ + φ v F φ v F φ (6b) ( β, β ) φ φ (, ) (, ) = a φ a a (6c) whr φ s th drctor of rotaton angls φ. It s mportant to not that, th dffrnc vctor θ F rprsnts th dffrnc btwn th drctor of th rfrnc and currnt confguraton of th shll body. For small dformatons, th lnarzd thr-dmnsonal Grn-Lagrang stran n curvlnar coordnats can b dfnd as j = j G G (7a) wth j = ( u, G j + u, j G ) (7b) whr G s th contravarant bas vctors of th shll and j s th Grn-Lagrang stran tnsor coffcnts. Th ndvdual componnts of Grn-Lagrang stran tnsor coffcnts j can b computd as

4 ( ( ) ) = v a + β v F + F a θ (8a),,,, ( ( ) ) = v a + β v F + F a θ (8b),,,, ( ( ) ( ) ) = γ = v a + v a + β v F + v F + F a θ + F a θ (8c),,,,,,,, = 0 (8d) For th sak of brvty, th non-zro Grn-Lagrang stran tnsor coffcnts of KLS lmnt can b wrttn n a compact vctor form as + β + ζ γ m b d ( ) kd ( ) B d B d (9a) N [ cp T d = d d d ] (9b) whr d ( ) s th mmbran strans assocatd wth strtchng of th md-surfac, kd ( ) s th bndng curvaturs, d s translatonal DOF of all th control ponts, and th suprscrpt m B and N s total numbr of control ponts. Th matrcs cp b B, gvn n Eq. (9a), contan th drvatvs of th NURBS shap functons corrspondng to mmbran and bndng actons of th KLS lmnt, rspctvly. Dscrt n-stu stran masurmnts obtand from on-board snsors ar th fundamntal nput of th FEM formulaton. To calculat mmbran and bndng scton strans xprmntally, th n-stu stran rostts should b locatd on top and bottom surfacs of th KLS lmnt, as shown n Fgur. Fgur. Dscrt surfac strans masurd by stran rostts wthn KLS lmnt at X X ( β, β, ± h) locatons. Th xprmntally masurd (n-stu) mmbran scton strans and curvaturs analytc countrparts, ( ) d and ( ) k that corrspond to thr kd gvn by Eq. (9a), can b dtrmnd from th masurd surfac strans at N s dscrt locatons X X ( β, β, ± h) ( =,..., N s ) locatd wthn th lmnt. Ths n-stu scton strans ar computd as follows [, ] + = ( + ) (0a) and + k = ( ) (0b) h whr 4

5 = [ γ ] T (0c) and = [ γ ] T (0d) ar th masurd surfac strans, wth th suprscrpts + and dnotng th quantts that corrspond to th top and bottom surfac locatons, rspctvly. Th FEM formulaton rconstructs th dformd shap of a dscrtzd structur by mnmzng a last-squars functonal wth rspct to th nodal DOF of th ntr dscrtzaton. For an ndvdual KLS lmnt, ths functonal, Φ( d ), accounts for th mmbran and bndng dformatons and can b xprssd as Φ ( d ) = ( d ) + k( d ) k (a) Th squard norms xprssd n Eq. (a) can b wrttn n th form of th normalzd Eucldan norms as d ( ) d ( ) d (b) and N s N s = ( h) kd ( ) k kd ( ) k d (c) N s N s = whr rprsnts th md-plan ara of th lmnt. Mnmzng th FEM last-squars functonal, gvn by Eq. (a), wth rspct to th nodal dsplacmnt DOF gvs rs to Φ( d ) δ Φ( d ) = Γd Ε = 0 d (a) or smply Γd = Ε (b) whr Γ s th local lft-hand-sd matrx, Ε s th local rght-hand-sd vctor that s a functon of th masurd stran valus, and d s translatonal DOF of all th control ponts that ar th unknowns n th analyss. Th lmnt m b Γ matrx can b xplctly wrttn n trms of th matrcs B and B as Γ = ( B m ) T B m + ( h) ( B b ) T B b d (c) Th Ε vctor s a functon of th numbr of stran snsors wthn th lmnt as wll as th masurd scton-stran valus, and s gvn as Ε B B k d (d) N s m T b T = ( ) + ( h) ( ) Ns = Onc th lmnt matrx quatons ar stablshd, th lmnt contrbutons to th global lnar quaton systm of th dscrtzd structur can b valuatd. ftr that, th rsultng systm of quatons can b rducd to wllcondtond systm of quatons by prscrbng problm-spcfc dsplacmnt boundary condtons. Th soluton of ths systm of quatons provds th dformd structural shap at any ral-tm. Th pnchd hmsphr III. Numrcal rsults Th pnchd hmsphr problm proposd and studd by many rsarchrs [4, 5] s solvd hr as a bnchmark problm. Th hmsphrc shll s subjctd to th four dffrnt concntratd loads as shown n Fgur. 5

6 Th prscrbd boundary condtons ar th mnmum rqurd to prvnt rgd body motons. In othr words, th apx of th hmsphr along Z-drcton nds to b fxd n ordr to lmnat th rgd body moton. Lnar, lastc and sotropc matral proprts E and v, magntud of th concntratd forc F, radus of th hmsphr R, and thcknss of th hmsphr h ar gvn n Fgur. Ths problm s challngng n trms of dmonstratng bndng capablty of th KLS lmnt bcaus t xhbts almost non of mmbran strans. Morovr, doubly curvd gomtry and pont loadng mak ths problm hghly snstv to lockng phnomna. By utlzng th advantag of symmtry, only on quartr of th structur s modlld and symmtry constrant and loadng boundary condtons ar appld. Th quantty of ntrst s U,.., th dsplacmnt along th drcton of th loadng F at pont (rfr to Fgur ). Th assocatd rfrnc soluton s gvn as U = m n [4, 5]. Fgur. Pnchd hmsphr. n accurat rfrnc soluton s stablshd by prformng a convrgnc study usng drct FEM analyss. Th most rfnd msh consstd of 7500 unformly dstrbutd rctangular lmnts, possssng DOF. Th valu of th dsplacmnt at pont s found as U = 0.09 m from th hgh-fdlty FEM analyss, whch s n a farly wll agrmnt wth th rfrnc soluton. Thrfor, th FEM dflctons and rotatons ar usd to calculat th smulatd stran-snsor strans. In th currnt FEM analyss, fv dffrnt KLS dscrtzaton ar gnratd by unformly dvdng dgs of on quartr of th hmsphr nto, 4, 6, 8, and 0 sgmnts, rspctvly. In ach KLS msh, two stran rostts ar placd at th cntrod of ach KLS lmnt, on on th top and th othr on th bottom surfac. ccordng to ths arrangmnt of stran snsors, all of th KLS lmnts ar nstalld wth stran snsors for ach FEM analyss. n xampl of stran rostt confguraton for 4 sgmnts pr dg s llustratd for th KLS dscrtzaton n Fgur 4. Fgur 4. Dscrtzaton (4 sgmnts pr dg) of on-fourth of th hmsphr usng KLS lmnts wth topand bottom-surfac stran rostts pr ach lmnt. 6

7 Onc th FEM analyss ar prformd usng KLS modl, th sam shap snsng analyss ar also conductd usng th QS4 lmnt [4] n ordr to mak a clar comparson btwn KLS and QS4 lmnts. Th U dsplacmnt rsults ar plottd wth rspct to th numbr of snsors on th ach dg, as dpctd n Fgur 5. Ths rsult clarly dmonstrats th gratr shap snsng accuracy of th KLS lmnt n comparson to QS4 lmnt. In fact, usng KLS lmnt allows us to mprov th accuracy of shap snsng analyss vn f a vry coars msh (wth a low numbr of stran-snsors) s usd for th analyss. Ths s du to th fact that th polynomal dgr of th NURBS can b lvatd wthout changng th locaton of knots, hnc th numbr of lmnts wll rman unchangd. Ths fatur of th NURBS bass functon can b xplotd for obtanng mor accurat rsults. Fgur 5. U dsplacmnt rsult comparson btwn KLS and QS4 lmnts. Th contour plots of th U dsplacmnt ar dpctd n Fgur 6 whr contour plots for FEM analyss ar graphcally almost dntcal to thos of FEM analyss. Morovr, th prcnt dffrnc btwn FEM and FEM prdctons for maxmum valu of th U dsplacmnt s approxmatly 5.8% (rfr to Fgur 6). Ths obsrvatons confrm th supror bndng capablty of th KLS lmnt, vn f a low-fdlty dscrtzaton (4 sgmnts pr dg) wth fw numbr of snsors (.., 6 = stran rostts n total) s usd n th shap snsng analyss. Rmarkably, ths prdctons dmonstrat th hgh qualty prcson of sogomtrc FEM solutons for shap snsng analyss of a complx and doubly curvd gomtry. Fgur 6. Contour plots of U dsplacmnt for KLS modl dpctd n Fgur 4; comparson btwn hghfdlty FEM and FEM analyss. partly clampd hyprbolc parabolod ccordng to Bath and co-workrs [6], partly clampd hyprbolc parabolod problm s an xcllnt tst for lockng n bndng-domnatd stuatons. Thrfor, shap snsng of th hyprbolc parabolod s prformd usng 7

8 th KLS lmnt n ordr to bttr assss th capablty of th KLS lmnt aganst th lockng phnomnon. Th md-surfac of th hyprbolc parabolod s dfnd as =, ( XY, ) [ L/; L/] Z X Y () It s vry straghtforward to construct ths surfac usng scond ordr B-splns. s dpctd n Fgur 7, th shll s clampd from th dg locatd at X = L and a gravtatonal load of g s appld. Lnar, lastc and sotropc matral proprts and xact gomtry of th shll s also gvn n Fgur 7. Utlzng th symmtry of th problm, only on half of th shll s modlld for th computaton and rlvant symmtry constrant boundary condtons ar appld. Th quantty of ntrst s W,.., th vrtcal dsplacmnt along Z-drcton at pont whch s th mdpont of th dg locatd at X =+ L, as shown n Fgur 7. Th assocatd rfrnc soluton s found as W = 6.94 mm n [6]. Fgur 7. Hyprbolc parabolod. convrgnc study was prformd usng drct FEM analyss and an accurat rfrnc soluton s stablshd for smulatng snsor strans. Th hghst fdlty msh has 500 unformly dstrbutd rctangular lmnts and 6806 DOF. Th hgh-fdlty FEM analyss prdcts ths vrtcal dsplacmnt as W = mm whch agrs wll wth ts assocatd rfrnc soluton [6]. In th followng FEM analyss, ths problm s analyzd usng fv dffrnt KLS dscrtzaton whr dgs of th gomtry ar unformly dvdd nto, 4, 6, 8, and 0 sgmnts, rspctvly. Smlar to th prvous cas study (hmsphr problm), ach KLS lmnt has two stran rostts n ach KLS modl. Thrfor, fv dffrnt FEM analyss of th parabolod ar prformd consdrng all th KLS lmnts nstalld wth stran snsors. To gv an xampl of stran rostt confguratons, an KLS modl wth 6 sgmnts pr dg s prsntd n Fgur 8. Fgur 8. Dscrtzaton (6 sgmnts pr dg) of half of th hyprbolc parabolod usng KLS lmnts wth top- and bottom-surfac stran rostts pr ach lmnt. 8

9 part from th KLS/FEM analyss, th sam shap snsng analyss of th hyprbolc parabolod ar also conductd usng th QS4 modls wth sam numbr of stran snsors pr dg. s dpctd n Fgur 9, th W dsplacmnt rsults obtand usng KLS and QS4 lmnts ar compard for varyng numbr of snsors. Rmarkably, ths rsult valdats th supror shap snsng capablty of th KLS lmnt. Fgur 9. W dsplacmnt rsult comparson btwn KLS and QS4 lmnts. Furthrmor, n Fgurs 0, contour plots of W dsplacmnts ar compard btwn th FEM and hgh-fdlty FEM analyss. In ths fgurs, th FEM contours corrspond to th FEM analyss that uss th KLS modl (6 sgmnts as prsntd n Fgur 8) wth 6 stran rostts n total. Th prcnt dffrnc btwn th FEM and FEM for th maxmum valus of th W dsplacmnt s only 4.% (rfr to Fgur 0). s can b sn from ths plots, both th FEM and drct FEM contours ar graphcally ndstngushabl. Ths rsults dmonstrat th supror bndng prdctons of KLS lmnt, spcally consdrng th low-fdlty msh usd n th FEM analyss. Fgur 0. Contour plots of W dsplacmnt for KLS modl dpctd n Fgur 8; comparson btwn hghfdlty FEM and FEM analyss. IV. Conclusons Ths papr focuss on th shap snsng of complx shll structurs by couplng th NURBS-basd IG togthr wth th FEM mthodology. n sogomtrc Krchhoff-Lov nvrs-shll lmnt (KLS) s dvlopd for th numrcal mplmntaton of th proposd couplng btwn IG and FEM (sogomtrc FEM mthodology). Th prsntd KLS formulaton s applcabl to prform shap-snsng analyss of complx thn plat and shll structurs by usng th stran data obtand from randomly dstrbutd snsors on th structur. Th bndng capablty of th KLS lmnt s xamnd by carryng out two numrcal smulatons ncludng a pnchd hmsphr and a partly clampd hyprbolc parabolod. s a rsult, th bndng ffcncy of KLS lmnt has bn justfd vn usng th low-fdlty KLS dscrtzaton. It has bn dmonstratd that th KLS lmnt has th advantag for shap snsng 9

10 of complx shll structurs n comparson to QS4 lmnt. Morovr, t has bn confrmd that vn wth a vry coars msh (wth a low numbr of stran-snsors), th KLS lmnt provds mprovd rsults by ncrasng th polynomal dgr of th NURBS shap functon wthout changng th locaton of knots. Ths fatur can b xplotd n th futur to obtan mor accurat shap snsng rsults for dffrnt typ of lmnts that wll b dvlopd basd on th proposd couplng. Rfrncs [] Tsslr,., and Spanglr, J. L., varatonal prncpal for rconstructon of lastc dformaton of shar dformabl plats and shlls, NS TM-445, 00. [] Tsslr,., and Spanglr, J. L., last-squars varatonal mthod for full-fld rconstructon of lastc dformatons n shar-dformabl plats and shlls, Computr Mthods n ppld Mchancs and Engnrng, Vol. 94, No., 005, pp. 7, 9. [] Tsslr,., and Spanglr, J. L., Invrs FEM for full-fld rconstructon of lastc dformatons n shar dformabl plats and shlls, Procdngs of nd Europan Workshop on Structural Halth Montorng, Munch, Grmany, 004. [4] Kfal,., Otrkus, E., Tsslr,., and Spanglr, J. L., quadrlatral nvrs-shll lmnt wth drllng dgrs of frdom for shap snsng and structural halth montorng, Engnrng Scnc and Tchnology, an Intrnatonal Journal, Vol. 9, 06, pp. 99,. [5] Ghrlon, M., Crraccho, P., Matton, M., D Scuva, M., and Tsslr,., Shap snsng of D fram structurs usng an nvrs Fnt Elmnt Mthod, Intrnatonal Journal of Solds and Structurs, Vol. 49, No., 0, pp. 00,. [6] Crraccho, P., Ghrlon, M., D Scuva, M., and Tsslr,., novl approach for dsplacmnt and strss montorng of sandwch structurs basd on th nvrs Fnt Elmnt Mthod, Compost Structurs, Vol. 7, 05, pp. 69, 76. [7] Tsslr,., D Scuva, M., and Ghrlon, M., consstnt rfnmnt of frst-ordr shar dformaton thory for lamnatd compost and sandwch plats usng mprovd zgzag knmatcs, Journal of Mchancs of Matrals and Structurs, Vol. 5, No., 00, pp. 4, 67. [8] Kfal,., and Otrkus, E., Structural Halth Montorng of marn structurs by usng nvrs Fnt Elmnt Mthod, nalyss and Dsgn of Marn Structurs V, 05, pp [9] Kfal,., and Otrkus, E., Dsplacmnt and strss montorng of a chmcal tankr basd on nvrs fnt lmnt mthod, Ocan Engnrng, Vol., 06, pp., 46. [0] Kfal,., and Otrkus, E., Dsplacmnt and strss montorng of a Panamax contanrshp usng nvrs fnt lmnt mthod, Ocan Engnrng, Vol. 9, 06, pp. 6, 9. [] Hughs, T. J., Cottrll, J.., and Bazlvs, Y., Isogomtrc analyss: CD, fnt lmnts, NURBS, xact gomtry and msh rfnmnt, Computr Mthods n ppld Mchancs and Engnrng, Vol. 94, No. 9, 005, pp. 45, 495. [] Cottrll, J.., Hughs, T. J., and Bazlvs, Y., Isogomtrc nalyss: Toward Intgraton of CD and FE, 009, John Wly and Sons. [] Echtr, R., Ostrl, B., and Bschoff, M, hrarchc famly of sogomtrc shll fnt lmnts, Computr Mthods n ppld Mchancs and Engnrng, Vol. 54, 0, pp. 70, 80. [4] MacNal, R. and Hardr, R., proposd standard st of problms to tst fnt lmnt accuracy, Fnt Elmnts n nalyss and Dsgn, Vol., 985, pp., 0. [5] Blytschko, T., Stolarsk, H., Lu, W., Carpntr, N., and Ong, J. S., Strss projcton for mmbran and shar lockng n shll fnt lmnts, Computr Mthods n ppld Mchancs and Engnrng, Vol. 5, No., 985, pp., 58. [6] Bath, K. J., Ioslvch,., and Chapll, D., n valuaton of th MITC shll lmnts, Computrs and Structurs, Vol. 75, No., 000, pp., 0. 0