A Semi-Explicit Algorithm for the Reconstruction of 3D Planar Cracks
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. A semi-explicit algorithm for the reconstruction of 3D planar cracks Tarek. . , Amel.
(3) . , Mohamed . N ˚ 3063 December 1996 ` THEME 4. ISSN 0249-6399. apport de recherche. .
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(189) V_aX o M 5 V_YhYWXFe\ o Z8\Mce\ K V_ee\fZ]\MONMO`r\O 0^aNldvZ o VW`F?\ K MbY_Mc McYY_VW`FMce¨Æ^ o \ K M ^ o eg\.U*XFmB£ x ¥Zr` i \ K M f}vMceg\.?^a`FM¹£ x ¥æ
(190) M `F^ \gVhkcM%«^ o \ K V_e YWZ8\g\M o Ïe%e\ o ^F`F i Vhegkc^a` \gV[`FXFV_\V_Mce Z]k o ^ ee>\ K Mlk o Z8kc& Zr` i Z8Y_e^ K V_ K o Z i V_MO` \ge`FMOZ o \ K Mlk o Z8k¦\VWdFeO K V_k K i MO`F^r\M\ K MQd o MceMO`FkcM ^8*eVW`FaXFYWZ o Vh\gVhMce i XFM\g^¡`F^a`S¸Z `FV_e K VW`Fe\ o Mcege
(191) V[`r\MO`FeV_\ j%Z8kc\^ o eOp7 MO`\ K V_epZ8egefXvNldF\V_^a` V_eQ`F^r\Q`FMckcMceefZ o j\^¨V[`Fe6X o M\ K M%V i MO`r\V*Z }FV_Y_VW\ j Vh\ 0Z]eQ`FMcM i M i \^d o ^8 M¡Z VWdFek K V_4\ 36eg\fZ }FV_Y_VW\ j o McefXFY_\ Æ^ o ]{ MONSM o VW`Fbk o Z8kc e / Z ` i V_\%0Z]eV[` \ K Md o McegMO` \dvZ dM o MceeMO`r\VWZ8Yº\g^ i M o V_ M¨\ K MM oo ^ o Mceg\VWNlZ8\M©£« 5 ¥^8 Y_MONlN¨Zlxv'Z ` i \ K XFe\g^ i M o V_ M\ K M%M oo ^ o Mce\VWNlZ]\M£« ¥#VW`b\ K Mc^ o MON ~ . . . . . . ED. [ bQVHP WPKL MP"J]W L egcSWLH[ J cdJ eg[ JpPST 'pP"JMegP. . . J K Mlk o Z8kc e%YhVW`FM©}McV[`F.V i MO`r\V'M i
(192) MF`F^8Z o MS o e\r^ VW`F©\^bY_^F^aôZ8\%\ K MM Mckc\e^]è\ K M\ o Xv`FkOZ8\gVh^F`?^8 \ K M©®'^FX o V_M o McmvdvZ `FeV_^a` ^a`:\ K M?Z8kckOX o Z8kcj ^F` \ K M o Mckc^a`Fe\ o XFkc\M i qXvNld { / }M6Æ^ o M}McV[`F VW`r\M o Mce\gM i VW` V_\e>efXvdvdv^ o \O®
(193) VhFX o M z± V_ Mce\ K M9 o MOZ8Y%qXvNld¬£¤^a}F\fZ]V[`FM i }rj\ K Ml*`FV_\MbMcY_MONMO`r\kc^aNldvXF\fZ8\gVh^F`v¥c
(194) Z ` i \ K M o Mckc^a`Fe\ o XFkc\gM i qXvNldFe^a}F\fZ8VW`FM i }rjl\ o Xv`FkOZ]\M i ®&^aX o V_M o eM o V_MceVW`r ^ Y_aVW`Fryvax y 5 yvaZ ` i ; y}vZ8eV_e#Xv`Fkc\V_^a`FeO J è^^a}FegM o ¸Z8\gVh^F`FekOZ `©}Md^ VW` \M i ^aXF\QZ8\\ K V_ee\fZ8rM *. . ~ J K Mk o Z]k¦¨eMcMONe\g^S}M i Mc\Mckc\gM i Mc MO`Z8\\ K M%Y_^¸Mce\ ¸Z8YWXFMce^8
(195) ë´ë Ù .
(196) P. ®
(197) V_aX o M 5 * # #,',( > -.. . . . ? - 6(/ -. - .' 1("-.' ( . . - #% 6(" . JK M o Mckc^a`Fe\ o XFkc\M i q XvNld Q { / @ V_ee^aNMc K Z8\^ ekcV_Y_Y[Z]\VW`F'*Zr` i kOZ `v`F^ \ K MO`FkcMd o ^8FV i MZ `Z]kckcMOdF\fZ }FY_M Z dvd o ^8am VWNlZ]\V_^a`^8
(198) \ K M qXvNldV_\eMcY. . ®
(199) V_aX o Mzg* + " ( -.'%-- "/12 3 6(/ "! .',(*
(200) 6(/ .' 1("-.' ( -? #% (" !e6N^a^r\ K VW`F d o ^FkcM i X o M>è^FXFY i d o ^F}vZ }FY_jbV[N¨d o ^8 M\ K MZ dvd o ^8mFV[N¨Z8\V_^a`b^] \ K M qXvN¨d*}vXF\V_\Q^aXFY i VW`b\ K M e6Z NMp\gV[NSMefd o MOZ i a^ XF\)V_\e
(201) efXvdvd^ o O\ r K V_k K V_ep\ K M M o j%V[`r«^ o N¨Z8\V_^a`MèZ o MeMcMO VW`F>\^QkOZ]\k K ) e Zr`v`F^aXv`FkcM i éêºëºéì.
(202) WV `±eMckc\V_^a`¹£óx ¥cFM%e K Z8Y_Yº8McMOd±\ K MqXvN¨d©Z8ekc^FNldvXF\M i 'Z ` i XFeM«^ o \ K MZ dvd o ^¸mFVWNlZ8\V_^a`±^8ºV_\eefXvdvdv^ o \\ K M «^ Y_Y_^8VW`FeMc\n* q | s
(203) @ Q { / @ £ &¥ @ «^ o Zld o Mcegk o VW}vM i \ K Z]\M%e K Z8Y_Y o M6«M o Z8\QZ]e\ K M !$ Sn" 3 4 . . +V_ MO`lZ%8Z8YWXFM%^8p¹£¤. K hV k K V_eZ%kc^aN¨dvXF\fZ8\V_^a`vZ8Y&kc^ e\#èMZ o MV_Y_Y_VW`Fl\^dvZjv¥c Me K ^aXFY i k K ^a^reM
(204) Z8kckc^ o i VW`F \g^\ K Mc^ o MON ~ %Z ` ^8p^ o i M o Æ^ o Z ` j46 ye^b\ K Z8\QM^a}F\fZ]V[`?Z `M oo ^ o £q
(205) Sq ¥^]è^ o i M o ] < JK M}M K ZOFV_^aX o ^8 \ K MM oo ^ o eOvZ8ee K ^8Q`±VW`'aX o M};vF\fX o `Fe^aXF\ K ^8Mc M o \^l}MSNXFk K }vMc\\gM o \ K Zr` McmvdvMckc\gM i 1000 log-log plot. 20 500 15. log(error). error (%). 0. 10. -500. -1000 5 -1500 0. -2000 30. 40. 50. 60. 70. 80. 0. 100. . number of basis functions. 200. 300. 400. log(nr of functions). ®
(206) VhFX o M ;g* . ? (12 1 .' 1("-.' ( "/ e%ZNlZ]\\M o ^8
(207) Z8kc\O\ K M o McYWZ8\gVh^F`Fe K VWdô}vMc\ McMO`-ôZ ` i XFeM i \^lMce\6Z }FY_Vhe K \ K MM ogo ^ o Mce\gV[N¨Z8\M £ó ¥cvZ8e }vZ]eM i ^a`\ K MXv`FVÆ^ o N"kc^a`r M o MO`FkcM>^8
(208) \ K M®&^aX o V_M o egM o V_MceO' K Vhk K V_eeY_^8'Z ` i \ K V_eQkc^aXFY i }vMZ¨\^F^e\ o ^a`F o M XFV o MONMO`r\*^a}FegM o aVW`Fb\ K M o Mckc^a`Fe\V_\fXF\gM i k o Z8k¦re%^8#'aX o Mzv ^F`FMkOZr`ô`F^ \V_kcM\ K Z8\%\ K Mkc^a`r M o rMO`FkcMl^8 q \+ ^ q?V_e¡Z]k K V_Mc M i óZ o Z K MOZ i \ K MXv`FV«^ o N"kc^a`r M o rMO`FkcM^8 \^+ . . 20 0.07. 2D plot. 3D plot. 0.06. 15. error (%). error (%). 0.05 10. 0.04. 0.03. 5. 0.02 0 2. 3. 4. 5. 6. 7. tolerance number (%). 8. 9. 10. 2. 3. 4. . 5. 6. 7. 8. 9. 10. tolerance number (%). ®
(209) VhFX o M . * . ? -? #% ("+(12 1 kckc^ o i VW`F:\^ô\ K McegMb^a}FeM o 8 Z8\V_^a`FeOZ8Y_Y\ K M©\ o VWZ8Y_e o MOdv^ o \ M i o ^aN`F^8 a^ ` K Z M©}McMO` o Xv`:V_\ K ;ryô}vZ8eV_e Xv`Fkc\V_^a`FeVW`b8{lvZ ` i ; y 3 ; y+Xv`Fkc\V_^a`FeVW`©x]{l K V_k K Z o M \K % M `aXvN}M o e_V \V_eQ`F^ \^ o \ K V_Y_MQ\g^r^S}Mcj ^a` i ë´ë Ù .
(210) z. egV[`FkcM\ K McjZ]k K Y_^8Mce\%MkOZ ` Z]kckc^ o i VW`Fl\^l\ . E^. . . V_Mc M\ K Mkc^a` rM o MO`FkcM^8 \ K MQk o Z8k¦rep Vh\ K Z `¨M oo ^ o £¤Y_Mcee\ K Z `¨yvÏP¨¥
(211) K V_k K eMcMONSeè\g^}vM\ K M o MOZ8k K JK M o McYWZ8\M i \^ Y_M o Z `FkcM`FXvN}vM o e £ 5 VW`ô8{lZ ` i x V[`.x8{¥ K Z Ml}vMcMO`?k K ^ eMO` K M%kOX o Mce^8)'aX o M . . t[ LHR % M cSW c. u vX `v`FV[`F?Z `V i MO` \V'kOZ8\gVh^F`ôZ8Y_ ^ o V_\ K N }vZ8eM i ^a`?NMOZ]efX o MONMO`r\eO&^a`FM K Z8e\^ o MONVW` i \ K Z8\QNSMOZ8efX o M i?i Z8\fZ Z o MefXv} qsMckc\\^b`F^ V_eM \ K MM Mckc\e^8p K V_k K?K ZOrM\^b}vMe\fX i V_M i ) `^aX o kOZ8eM \ K M i Z8\fZZ o M>ejv` \ K Mc\gVhk &V«ÏM ^F}F\fZ8VW`FM i } j Z±*`FV_\M©McY_MONMO`r\lkc^aNldvXF\6Z8\V_^a`:efXv}Fq Mckc\l\^?M oo ^ o eOpZr` i \ K Mcj Z o M K MO`FkcM?Z8Y o MOZ i j `F^ V_ej J ^ \ K M ;31 i " @S
(212)
(213) M K Z MbZ ivi M i Z 63" & ) MO`FM o Z8\gM i } j¹Z®&^ o \ o Zr` o ^FXF\VW`FM èZ ` i o Xv` `FXvNM o V_kOZ8Y&\ o VWZ8Y_e#Æ^ o eMc M o Z8Y o Z]\V_^ e¡£¤ Vh\ Klo McefdMckc\è\g^\ K M%NlZ8mFVWNXvN ¸Z8YWXFM%^8 \ K M fNMOZ8efX o M i i Z8\fZr¥
(214) ^8 \ K V_e Z ivi Vh\gV[`vZ]Y `F^ V_eM ®
(215) V_aX o M V_ Mce%\ K M o McefXFY_\%^8 \ K Mkc^FNldFY_Mc\MV i MO`r\V'kOZ8\V_^a`.Z8Y_ ^ o V_\ K NÆ^ o «^aX oo Z8\V_^ e£ 5 ; ~ y¨Z ` i $~ 5 l¥c R d¨\^ ;!^] Z ivi V_\V_^a`vZ8Y `F^ V_eM v\ K M o Mce6XFYh\ge o MONlZ8VW`le6Z8\V_e Z8kc\^ o j ) \ ~ y ^8ºZ ivi V_\V_^a`vZ8Y)`F^ V_eM vZ g & k o Z8k¦Z dvdMOZ o eVW`\ K M%NV ivi YhM%^] \ K MQk o Z]k¦Y_VW`FM # \ $~ 5 !^8 Z ivi Vh\gVh^F`vZ8Y `F^ V_eM F\ K MQk o Z8k¦SVhekc^aN¨dFYhMc\gMcYhjY_^ e\O. . . ®
(216) V_aX o M * '1 - 0 #%* .'%-.12. '%-.$ .3 ""- -?. -.+( ' 0#% #. kc\fXvZ8Y_Y_j &M Mckc\geè^] \ K M`F^rVhegM K Z M%}vMcMO` o Mckc^ o i M i McrMO`V[`±\ K M%McmvdFYhV_kcV_\dvZ o \ ^8º\ K MZ8Y_ ^ o V_\ K Nb'`vZ NSMcYhjbVW` \ K M i Mc\gM o NVW`vZ8\V_^a`>^8\ K Mk o Z8kc Ïe)Y_VW`FM aegV[`FkcM \ K V_epdvZ o \)V[`r ^ Y_ MceZ8e)èMcY_Yv\ K M`F^ V_ej i Z8\fZ0Æ^ o \ K Mkc^aNldvXF\fZ8\gVh^F` ^]º\ K M}v^FXv` i Z o jV[`r\Mc o Z8Y_eO JK MY_V[`FMV_ekc^ ogo Mckc\Y_jlV i MO` \gVÛ'M i Xvd\^ ~ y ^8 Z ivi V_\V_^a`vZ8Y)`F^ V_eM 'Z ` i V_\V_eQY_^ e\ Z]\0r y b) \ ~ 5 br\ K MY_VW`FMV_ee\V_Y_Yºkc^ ogo Mckc\Y_j o Mckc^¸rM o M i '}vXF\\ K M%k o Z8k¦¨V_\eMcYpV_eQY_^ eg\ ~ yF'¢Z `v`F^aX o / éêºëºéì. . .
(217) ; E . kPSeiW RU[ Y W #QPtegTcdeh~V [Begc"WL [ J. 6 ^Z]eefXvNldF\V_^a`¨^a`¨\ K MYh^FkOZ8\V_^a`^] \ K MQk o Z]k¦reVhe`FMcM i M i F o ^aN"Z+\ K cM ^ o Mc\V_kOZ8YFV_McQdv^ VW`r\O'}rjl\ K MQZ]Yhr^ o V_\ K N \g^ o Mckc^¸rM o \ K MONb ) \V_elVW`r\M o Mce\gV[`F:\ K M o M6«^ o M\^? M o V«j pVW` Z±«Mc cM ma\ o MONMeV_\fXvZ8\V_^a`FeO K ^8" MO`FM o Z8Y\ K M Z]Yhr^ o V_\ K N V_eO> Zr` i Z>NXFYh\gV[dFY_Mk VW` 'aX o M ~ yv. k o 8Z kcbY_^FkOZ8\M i F` MOZ o \ K Mkc^ o `FM o )Z `MONM o Z8kc%M o M\ K M\ Kvo cM MeV_\fXvZ8\V_^a`Fe#k K ^ egMO`>Æ^. o VW`Fbk o Z8kc¹£óZ8\¡Zr`ôXv`va`F^8Q`?}^aXv` i Z o jôdv^rV[`r\f¥c o \ K V_ep\Mce\8\ K M o McefXFY_\ep^]* K V_k K Z o Mèe6XvNlNlZ o V!3cM i. " ( -.'%-- "/ 12. ®
(218) V_aX o M ~ yg* " & + 9 "! ',( ? 2 - 2+.'%- -.' ( " # 12 3 " ( -.'%-- "/$ !. . E
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