Embed Size (px)
Citation preview
Nonlinear solution methods for infinitesimal perfect plasticity
Christian Wieners
Preprint Nr. 06/11
UNIVERSITÄT KARLSRUHE
Institut für Wissenschaftliches Rechnen
und Mathematische Modellbildung zW RM M
76128 Karlsruhe
Anschrift des Verfassers:
Prof. Dr. Christian WienersInstitut fur Angewandte und Numerische MathematikUniversitat KarlsruheD-76128 Karlsruhe
"!#"$%'&()*+,-&#.+$0/
1325476987:76<;>='?@69AB=.AB48
C3DFE$GIHFJLKFG5MONPQRP$SUTVPXW'YRZ>P)[X\^]_`_`TV[7]a\bQRPXYRcdQRef]a\VgihQRTVYRZdjlkmhQOTVed[XQRP$j P$eYn]a\Iod\^]_pYRTV[$TqYsrtTVeuYRZ>P[$hiewvYRPyxwYuhak3e>hae>\VTVe>P7]QodQRhagaQn]j j Tzedgd]ae>|W%P|wTV_`[$c>_`_fYRZdP]\VgihaQRTqYRZdj Tz[QRP7]\zTV~7]YRTVhiehk3YRZdPd)j PyYRZ>hU|fkmhQ)TVed>e>TqYRP$_`TVj]a\oLPXQ`kmP$[yY)o>\z]_pYRTz[XTVYrNP_`Zdh7WYRZF]YOYRZdPQn]a|dTz]a\QRPXYRcwQRe[$haQ`QRPX_`oLhieF|w_YRh]e3hQ`YRZ>hagihie>]a\dodQRhpP$[XYRTVhaehieYRhYRZ>P%[XhieUSBPXx_`PXY.hkL]a|dj TV_`_`TV>\VP*_pY`QRP$_`_`P$_$.e>_`PXQ`YRTVedgOYRZ>TV_5owQRhsP$[XYRTVhieTVeYRhuYRZdPP7cdTz\VTVdQRTVcdjP7cF]YRTzhaeQRPX_`c>\qYR_Tze"]f_`P$j TV_`j hhYRZ"P7c>]YRTVhieW,Z>TV[nZ[7]eLP_`ha\zSiP7|"r]gaP$e>PyQn]a\VTV~$P7|OPXWYRhie"j PXYRZ>hU|I O\qYRPXQRe>]YRTVSBP$\qr]ae]ao>owQRhiowQRT^]YRP \VTzedP7]QRTV~7]YRTVhiehak%YRZ>PowQRhsP$[XYRTVhieTV_P7c>TVSa]a\VP$eY,YRhYRZ>P>)j PXYRZ>hU|I>W,ZdTz[nZTV__`Zdh7W,eYRhLPj haQRP)QRhidc>_pY]a_,YRZ>P[$\z]a_`_`TV[7]\Qn]i|wT^]\QRPyYRcdQRe0*Z>TV_TV_TV\V\Vc>_pY`Qn]YRP7|Ur]eUc>j PXQRTV[7]\%[$haj oF]QRTz_`haehakLhaYRZj PyYRZ>hU|d_khaQ ]LP$ed[yZdj]QRowQRhid\zPXj
B %¡U¢>£¤)¥¦L¡w§s£% 1¨F=.©ABª5:8¨>«¬F®¯>°7±³²F®²L´^µ>¶a·¶ª5¸<;¹º;>=º6¼»ª*¨F47:;>=b:"4¨F¸9A6¼=³:72.A»;>:72.AB»;>:769©U;>¸4A8AU;>4©2½¨>«6¼=¾.=56m¿:A876¼»;>¸ª5¸<;F8:769©B6¼:n¹IÀ;>=.Á'«sÂ5=.Á.;>»AB=0:;>¸O4A8Â5¸¼:82;ÃIAÄ%AAB=ŨFÄ5:;>6¼=.AÁÇÆp8AALÀALÈ5É.ȼÀ.ÊAB»;>»ÌË ÍÎsÏ ÈÐu=:72.Au¨F:72.AB42;>=.ÁÀ©¨F=.©ABª5:8¨>«3¬F®.¯>°7±ÑÒ7ÓFÒ7²FÔ#Ô#·®0Ó«p¨F4Á698©B4AB:76¼ÕUAÁÖª5¸<;F87:769©B6¼:n¹2;ÃIAu=.¨F:3Ä*AAB=(©¨F=¿8769Á5AB4AÁ×8¹87:AB»;>:769©U;>¸¼¸¼¹ØÆp¨F=5¸¼¹º;«pABÙÚ4A8Â5¸¼:8";>4A(;UÃF;>6¼¸<;>Ä5¸9ALÀALÈ,É.ȼÀÄ0¹Ç1;>487:AB=.8AB=ÇAB:#;>¸RÈË IÎ ;>=.Á1325476987:AB=.8AB=×ËVÛ ÎsÏ ÈÊ2.Aª5Â547ª%¨L8Af¨>«:725698ª;>ª%AB4698:¨ª54¨UÃ69Á5A;"Ä%AB:7:AB4 Â5=.Á5AB48:;>=.Á6¼=5ɨ>«OÜ=.¨UÙ =Ý;>¸¼ÉI¨F476¼:725»8UÀ5:72.AÁ5Ai¿ÃIAB¸9¨Fª5»AB=b:¨>«=.ABÙl;>¸¼ÉI¨F476¼:725»8«p¨F4O6¼=.©B4AB»AB=b:;>¸5ª5¸<;F87:769©B6¼:y¹IÀb;>=.Á:72.A3ª54A8AB=b:;>:769¨F=¨>«.:72.A;>¸¼ÉI¨F476¼:725»86¼=Ý;#Â5=56m¾AÁÞ«p4;>»ABÙ3¨F47ÜÄ0¹(:72.A©¨F=.8AßbÂ.AB=0:u;>ª5ª5¸¼69©U;>:769¨F=Ũ>«©¨F=bÃIAià'ª54¨FÉL4;>»»6¼=5É(:A©25=569ß0Â.A8UÈÊ2.A6¼=.©B4AB»AB=0:;>¸ª54¨FÄ5¸9AB»á6¼=×6¼=¾.=56¼:A876¼»;>¸ª5¸<;F87:769©B6¼:y¹º698©¨F»»¨F=5¸¼¹º8¨F¸¼ÃIAÁ×Äb¹³:72.A(©B¸9¨L8A87:ª54¨>¿â A©B:769¨F=»AB:72.¨ÁãÆs4;FÁ6<;>¸f4AB:7Â547= Ï Àf8AALÀuALÈ3É.ȼÀ"Ë Û Î Èåäy:ÖÙ;F8';>¸¼4AU;FÁ¹æ¨FÄ.8AB47ÃIAÁ6¼=åË bÎ :72;>:Þ:725698;>¸¼ÉI¨F476¼:725»©¨F474A87ª*¨F=.Á58:¨Ç;©¨F=bÃIAiàç»6¼=56¼»6¼Õw;>:769¨F=ت54¨FÄ5¸9AB»è«p¨F4:72.AÝÁ6987ª5¸<;F©AB»AB=b:8UÈé06¼=.©A6¼=:72.AÉIAB=.AB4;>¸©U;F8ALÀ.:72.Aª5476¼»;>¸OÁ6987ª5¸<;F©AB»AB=0:8¨F¸¼Â5:769¨F=Ý698=.¨F:uÂ5=569ßbÂ.ALÀ:725698t»6¼=56¼»6¼Õw;>:769¨F=êª54¨FÄ5¸9AB»©U;>=5=.¨F:3Ä*A87:74769©B:7¸¼¹©¨F=0ÃIAià*ÈëŨF4A¨wÃIAB4UÀ86¼=.©A:72.AtÁ698©B4AB:Atª5476¼»;>¸*8¨F¸¼Â5:769¨F=6¼=(ª%AB4$«`A©B:ª5¸<;F8:769©B6¼:n¹698»A872¿nÁ5ABª*AB=.Á5AB=b: ;>=.Á4¨FÄ5Â.87:A8:76¼»;>:A8;>4At;UÃF;>6¼¸<;>Ä5¸9Af¨F=5¸¼¹#6¼=»AU;F87Â54AtÃF;>¸¼Â.AÁ(87ª;F©A8UÀIÙAt©U;>=5=.¨F:Aiàª*A©B: »A872¿R6¼=.Á5ABª*AB=.Á5AB=b:©¨F=bÃIAB47ÉIAB=.©ALÈì ¹ÅÁÂ;>¸¼6¼:n¹IÀ6¼=.©B4AB»AB=b:;>¸ª5¸<;F8:769©B6¼:n¹©U;>=³Ä%A4Ai«`¨F47»Â5¸<;>:AÁ×;F8;Ö©¨F=.87:74;>6¼=0:©¨F=bÃIAià³»6¼=56¼»6¼Õw;>:769¨F=ª54¨FÄ5¸9AB»í«`¨F4:72.Af87:74A88A8Ù 2569©2Þ©U;>=ÖÄ%Au8¨F¸¼ÃIAÁÀ0ALÈbÉ.ȼÀbÄ0¹Öé5îfï½»AB:72.¨0Á58wÈé06¼=.©Au:72.AfÁÂ;>¸ª54¨FÄ5¸9AB»698 Â5=56m«`¨F47»¸¼¹Þ©¨F=0ÃIAià*À.Ù3A»;U¹ÞAiàª*A©B: 4¨FÄ5Â.87:t©¨F=bÃIAB47ÉIAB=.©ALÈ𠸼ÉI¨F476¼:725»69©U;>¸¼¸¼¹IÀ*:72.A"»;>6¼=ê69Á5AU;©U;>=êÄ%A"6¼¸¼¸¼Â.87:74;>:AÁÅÉIA¨F»AB:74769©U;>¸¼¸¼¹½Æp©i«7Ètñ6¼ÉLÂ54AÞò";>=.Áêñ6¼ÉLÂ54Aó Ï ÈÊ2.At6¼=.©B4AB»AB=0:;>¸ª54¨FÄ5¸9AB»ô6¼=(ª5¸<;F87:769©B6¼:y¹8¨F¸¼ÃIA8:7254AAfAßbÂ;>:769¨F=.8Uõ):72.A3ö F÷øÒaù´9°u«p¨F4:72.Au6¼=.©B4AB»AB=b:¨>«O:72.Afª5¸<;F87:769©87:74;>6¼=,À:72.A¬LÔtÑ%´¼°iÔ°i®úR²FÒa·púpµ(¬L®ûF·púp·pF®.¶u«`¨F4ü¨UÙ«pÂ5=.©B:769¨F=;>=.ÁÞª5¸<;F8:769©u»"Â5¸¼:76¼ª5¸¼69AB4UÀ;>=.Áê:72.AÝ°7ýBù·´·`þBÒa·sù0ÔÌ°7ýBù5²Lúp·pL®×«`¨F4u:72.A8:74A88UÈÊ2.A4;FÁ6<;>¸4AB:7Â547=ØÆp©¨F474A87ª*¨F=.Á6¼=5É:¨Þ;>=çFÒBúsÿÓF®²F´%Ñ%Ò$U°7¬úp·`F® Ï 8¨F¸¼ÃIA8:72.Aü¨wÙÚ47Â5¸9A;>=.Á:72.A©¨F»ª5¸9AB»AB=0:;>476¼:n¹×©¨F=.Á6¼:769¨F=.86¼=.Á5ABª%AB=.Á5AB=0:7¸¼¹«p¨F4ABÃIAB47¹»;>:AB476<;>¸Oª*¨F6¼=b:UÀ*;>=.ÁÅ:725698Æs=.¨F=5¸¼6¼=.AU;>4 Ï »;>:AB476<;>¸O4A8ª%¨F=.8A698t6¼=.8AB47:AÁ'6¼=b:¨(:72.AAß0Â56¼¸¼6¼Ä5476¼Â5»AßbÂ;>:769¨F=,ÈlÊ2.AÅé5îfïÚ»AB:72.¨0Á Æp©¨F474A8ª%¨F=.Á6¼=5ɺ:¨º;>=½Ñ%Ò$U°7¬úp·`F®F®úRº²³ÿ²F´ Ö¶RѲI¬a°Ý©¨F=0:;>6¼=56¼=5É:72.A©¨F=bÃIAiàÝ8AB: ¨>«;FÁ»698876¼Ä5¸9A8:74A88A8 Ï 8¨F¸¼ÃIA8 6¼=ÅABÃIAB47¹Ö87:ABª'¨F=5¸¼¹;¸¼6¼=.AU;>476¼ÕUAÁü¨wÙ-47Â5¸9A;>=.ÁÞ¸¼6¼=¿AU;>476¼ÕUAÁ(©¨F»ª5¸9AB»AB=0:;>476¼:n¹Þ©¨F=.Á6¼:769¨F=.8UÀ;>=.Á:7256983¸¼6¼=.AU;>476¼ÕUAÁ»;>:AB476<;>¸%4A87ª*¨F=.8A6986¼=.8AB47:AÁ6¼=0:¨:72.A
!" $#&%'(( )*( +,.-0/21 11 /2143*65 7 89:+;=<>,?<>@A;"- .\z]a_pYRTV[$TqYsrCB]a|dTz]a\DBOPyYRcdQRe0>d)FEfPXYRZdhU|d_$
G H&I:JLKNM4OLKP*QSRKTUQ:TUJ:MAß0Â56¼¸¼6¼Ä5476¼Â5» AßbÂ;>:769¨F=,È?ºA872.¨wÙ:72;>::725698 876¼»ª5¸¼6m¾©U;>:769¨F=ÇÆs4ABª5¸<;F©B6¼=5É:72.Au«sÂ5¸¼¸*=.¨F=5¸¼6¼=.AU;>4 »;>:AB476<;>¸4A87ª*¨F=.8AÙ 6¼:72':72.A¸¼6¼=.AU;>476¼Õw;>:769¨F= Ï ¸9AU;FÁ58:¨;8Â5Ä.87:;>=b:76<;>¸,Éb;>6¼=Ũ>«)AV©B69AB=.©B¹IÈÊ2.Afª;>ª%AB4698¨F47Éb;>=56¼ÕUAÁ';F83«p¨F¸¼¸9¨wÙ8UÈé06¼=.©Af:72.A©¨F=.87:747Â.©B:769¨F='¨>«)é5îfïl»AB:72.¨Á58698ß0Â56¼:Af:A©25=569©U;>¸RÀÙ3AÁ5A8©B476¼Ä*A:725698u»AB:72.¨0Áݾ.487:u6¼=ê;ÉIAB=.AB4;>¸)8AB:7:76¼=5ɳÆpÊO;>Ä5¸9AÖò Ï À:72.AB=Ý«p¨F4f¨F47:72.¨FÉI¨F=;>¸)ª54¨ â A©B:769¨F=.8ÆpÊO;>Ä5¸9Aó Ï À;>=.Áؾ.=;>¸¼¸¼¹l«p¨F4(6¼=.©B4AB»AB=b:;>¸ª5¸<;F87:769©B6¼:n¹åÆpÊO;>Ä5¸9AXW Ï È?ºAÝ87:;>47:(6¼=éA©B:769¨F=ó³Ù 6¼:72;4ABÃ69ABÙæ¨F==.¨F=.8»¨0¨F:72(»AB:72.¨0Á58UÀ04A©U;>¸¼¸¼6¼=5ÉÉIAB=.AB4;>¸¼6¼ÕUAÁZYfABÙ :¨F=»AB:72.¨Á58«`¨F4\[6¼ª.8©256¼:7Õu«sÂ5=.©B:769¨F=.8;>=.Á½é5îfï »AB:72.¨Á58«`¨F4"©¨F=.87:74;>6¼=0:©¨F=0ÃIAiົ6¼=56¼»6¼Õw;>:769¨F=lª54¨FÄ5¸9AB»86¼=l;>=½;>Ä.87:74;F©B:«s4;>»ABÙ¨F47Ü%ÈYfAià:UÀ3Ù3A4A©U;>¸¼¸t:72.AÞÙ3AB¸¼¸m¿RÜ=.¨UÙ =©B¸9¨L8A87:ª*¨F6¼=b:ª54¨ â A©B:769¨F=æ¨F=0:¨³:72.A'©¨F=bÃIAiàç8AB:¨>«;FÁ»698876¼Ä5¸9A87:74A88A8UÀt;>=.Á@Ù3Aêª54¨Fª*¨L8Aê;lé5îfïô»AB:72.¨ÁØ«`¨F4Ö:72.AêABÃL;>¸¼Â;>:769¨F=¨>«:72.Aêª54¨ â A©B:769¨F=,ÈíÊ2.AB=,ÀÙ3AAiàª5¸<;>6¼=×:72.A»;>6¼=Ç69Á5AU;Ö«p¨F4"87:;>:769©ª5¸<;F87:769©B6¼:y¹³6¼=½éA©B:769¨F=^].ÀOÄ%Ai«`¨F4A#Ù3A(©¨F=.8769Á5AB4:72.A6¼=.©B4AB»AB=b:;>¸ª54¨FÄ5¸9AB» 6¼=6¼=¾.=56¼:A876¼»;>¸,ª5¸<;F8:769©B6¼:n¹Ö6¼='éA©B:769¨F=`_ÈÊ2.AB4ALÀÙ3A©¨F=.8769Á5AB4 6¼:8 ÁÂ;>¸*«`¨F47»Â5¸<;>:769¨F=,À;>=.Á;>=çAià6987:AB=.©AÞ4A87Â5¸¼:698#ª54¨UÃIAÁø«p¨F4#:72.AÞ6¼=.©B4AB»AB=b:;>¸f8¨F¸¼Â5:769¨F= Æsª54¨UÃ69Á5AÁø:72;>:;87Â56¼:;>Ä5¸9AÝé0¸<;>:AB4©¨F=.Á6¼:769¨F=Ý698t8;>:7698$¾AÁ Ï ÈÊ2.A»;>6¼=½4A87Â5¸¼:"698ª54A8AB=0:AÁ×6¼=øéA©B:769¨F=baÙ 2.AB4AÙ3AÖÁ5AB476¼ÃIA:72.A(©B¸<;F88769©U;>¸3ª54¨ â A©B:769¨F=½»AB:72.¨0Á;>=.ÁºÁ5ABÃIAB¸9¨FªÇ:72.A=.ABÙ é5îfï-»AB:72.¨ÁÈ ì ¨F:72Ç;>¸¼ÉI¨F476¼:725»8#;>4A©¨F»ª;>4AÁÇAiàª5¸¼69©B6¼:7¸¼¹Äb¹º;ÞÁ5AB:;>6¼¸9AÁÁ5A8©B476¼ª5:769¨F=ب>«:72.A';>¸¼ÉI¨F476¼:725»69©4AU;>¸¼6¼Õw;>:769¨F=,ÈçÊ2.AB¹½Ù3AB4A'6¼»ª5¸9AB»AB=b:AÁø«`¨F4:72.A87ª*A©B6<;>¸©U;F8A¨>«ÃI¨F=ÞëÝ698A83ª5¸<;F87:769©B6¼:y¹IÀ.;>=.ÁÖ;=bÂ5»AB4769©U;>¸ABÃL;>¸¼Â;>:769¨F=(«`¨F4 ;87:;>=.Á.;>4Á(Ä*AB=.©25»;>47ܪ54¨FÄ5¸9AB»ô698¾.=;>¸¼¸¼¹ÉL6¼ÃIAB=,Èñ.¨F4876¼»ª5¸¼69©B6¼:n¹IÀÙAt4A8:74769©B:¨FÂ548AB¸¼ÃIA83:¨:72.Au©B¸<;F88769©U;>¸%ï4;>=.Á:7¸m¿dctABÂ.88»¨Á5AB¸6¼=(6¼=¾.=56¼:A876¼»;>¸ª*AB4$¿«`A©B:ª5¸<;F87:769©B6¼:n¹Ù 2.AB4Aª5¸<;F87:769©B6¼:n¹698uÁ5AB:AB47»6¼=.AÁ'Ä0¹Ý;'Æsª%¨L8876¼Ä5¸¼¹(»"Â5¸¼:76m¿nÁ6¼»AB=.8769¨F=;>¸ Ï ¹069AB¸9ÁÅ87Â54$«R;F©A;>=.ÁÇ;>=l;F88¨©B6<;>:AÁ³ü¨wÙ47Â5¸9ALÈëê¨F4A¨UÃIAB4UÀ)ÙA4A87:74769©B:¨FÂ548AB¸¼ÃIA8":¨':72.AÖÁ698©B4AB:Aª54¨FÄ5¸9AB»'È'Ê2.AÁ5ABª*AB=.Á5AB=.©A¨F=³Á698©B4AB:76¼Õw;>:769¨F=ºª;>4;>»AB:AB486988:7Â.Á69AÁ³¨F=5¸¼¹ÅAiàª%AB476¼»AB=0:;>¸¼¸¼¹øÆp8AAÊ;>Ä5¸9ASe Ï È"Ê2.A;>=;>¸¼¹87698¨>«:72.A8AÞ»AB:72.¨0Á58Ù 6¼:72ç4A87ª*A©B::¨³»A872¿R6¼=.Á5ABª*AB=.Á5AB=.©A'698Ä*AB¹I¨F=.Ál:72.A'8©¨Fª*A¨>«f:725698Ù3¨F47Ügf:725698Ù3¨FÂ5¸9ÁÖ4AßbÂ56¼4Au:72.Au6¼=bÃIA87:76¼Éb;>:769¨F='¨>«,:72.Auª54¨Fª%¨L8AÁÖ;>¸¼ÉI¨F476¼:725»86¼=«sÂ5=.©B:769¨F=87ª;F©A8;F86¼:698Á5¨F=.A«`¨F4©¨F=b:74¨F¸Oª54¨FÄ5¸9AB»8UÀ.ALÈ5É.ȼÀÄ0¹iht6¼=0:AB47»kj5¸¼¸9AB4AB:UÈ;>¸RÈË'lnmpo Î È
qsr£ utv×££.¡wxv^yI¡wO£¤ut?ºA³872.¨F47:7¸¼¹æ4ABÃ069ABÙ :yÙ3¨ø©B¸<;F88A8¨>«Ù3AB¸¼¸A87:;>Ä5¸¼6982.AÁ»AB:72.¨0Á586¼==.¨F=5¸¼6¼=.AU;>4'ª54¨FÉL4;>»»6¼=5É.õæ;ÉIAB=.AB4;>¸¼6¼ÕUAÁbYfABÙ :¨F=Ç»AB:72.¨Áº«`¨F4=.¨F=.87»¨0¨F:72½«pÂ5=.©B:769¨F=.8UÀ;>=.ÁÇ:72.AÞé5îfï»AB:72.¨Áº«`¨F4#©¨F=.87:74;>6¼=b:ª54¨FÄ5¸9AB»8UÈñ¨F4»¨F4AÁ5AB:;>6¼¸98t¨F=Ý:72.A8A»AB:72.¨0Á58u8AALÀ5ALÈÉ.ȼÀ,Ëz0m |Î È
~ y 0y0¢&p§.y0¤riy.¡U£% vyb¡w£¤$tL [,AB:XÄ%A ;t¾.=56¼:AÁ6¼»AB=.8769¨F=;>¸.87ª;F©A ;>=.Á
f : X −→ XÄ*A ;
[6¼ª.8©256¼:7Õ"©¨F=0:76¼=bÂ.¨FÂ.8t«sÂ5=.©B:769¨F=,ÈÊ2.AB=,Àf698t;>¸¼»¨L8:ABÃIAB47¹0Ù 2.AB4A"Á6 *AB4AB=0:76<;>Ä5¸9ALÀ;>=.ÁÞ«`¨F4
x, v ∈ X:72.AÁ6¼4A©B:769¨F=;>¸Á5AB476¼ÃF;>:76¼ÃIA8
Df(x; v) = lim supt↓0,y→x
1
t
(f(y + tv)− f(y)
)
Aià6987:UÈ"Ê2.A©¨F=0ÃIAiàê20Â5¸¼¸∂f(x) = convDf(x; v) : v ∈ X Ä5Â56¼¸9Á58:72.A#ÉIAB=.AB4;>¸¼6¼ÕUAÁ0;F©¨FÄ56<;>=Ç;F8
6¼:698f6¼=b:74¨ÁÂ.©AÁ³Äb¹³13¸<;>47ÜIA'Æp8AALÀALÈ%É.ȼÀË' Î «p¨F4f:72.AÁ5Ai¾.=56¼:769¨F=×;>=.ÁÅ«p¨F4ª54¨Fª*AB47:769A8¨>«3ÉIAB=.AB4;>¸¼6¼ÕUAÁÁ5AB476¼ÃL;>:76¼ÃIA8 Ï ÈÊ2.Af©¨F474A87ª%¨F=.Á6¼=5É#ÉIAB=.AB4;>¸¼6¼ÕUAÁiYfABÙ :¨F=Þ»AB:72.¨0Á«p¨F4:72.AfAßbÂ;>:769¨F=
f(x) = 04AU;FÁ58
;F8«`¨F¸¼¸9¨UÙ8Uõ87:;>47:76¼=5ÉÙ 6¼:72x0 ∈ X À¾.=.Á ∆xk
8Â.©2':72;>:
0 ∈ f(xk−1) + ∂f(xk−1)∆xk, k = 1, 2, 3, ...;>=.ÁÖ8AB:
xk = xk−1 + ∆xkÈ ð 4AU;>¸¼6¼Õw;>:769¨F=¨>«,:725698»AB:72.¨0Á(4Aß0Â56¼4A8;©2.¨F69©A
Ck ∈ ∂f(xk−1)87Â.©2
:72;>:t:72.A"¸¼6¼=.AU;>4tª54¨FÄ5¸9AB»Ck∆x
k = −f(xk−1)2;F8;8¨F¸¼Â5:769¨F=øÆp8AALÀALÈÉ.ȼÀOË' Î «`¨F4:72.A";>=;>¸¼¹87698f¨>«
ÉIAB=.AB4;>¸¼6¼ÕUAÁ`YfABÙ :¨F='»AB:72.¨Á58 «`¨F4[6¼ª.8©256¼:7Õ"Aß0Â;>:769¨F=.8 Ï È
£ **y.vǧs §=vק.¡w§s£ ñ.¨F4:72.A87ª*A©B6<;>¸©U;F8AÞ:72;>:f = DF
698#:72.AÁ5AB476¼ÃL;>:76¼ÃIA¨>«;©¨F=bÃIAiàª*¨F:AB=b:76<;>¸
F : X −→ RÀ);>=b¹³»6¼=56¼»6¼ÕUAB4
x ∈ X ¨>« F (·) 8¨F¸¼ÃIA8:72.A(AßbÂ;>:769¨F= f(x) = 0À;>=.Á½;
87Â56¼:;>Ä5¸¼¹Á.;>»ª*AÁZYuABÙ :¨F=Þ6¼:AB4;>:769¨F=ÞÉIAB=.AB4;>:A8t;»¨F=.¨F:¨F=569©U;>¸¼¸¼¹(=.¨F=56¼=.©B4AU;F86¼=5É#8AßbÂ.AB=.©A¨>«F (·)876¼=.©A;>¸¼¸
C ∈ ∂f(x);>4Aª%¨L876¼:76¼ÃIA8AB»6m¿nÁ5Ai¾.=56¼:AÅÆp8AALÀOALÈ*É.ȼÀË bÎ «p¨F4ÉL¸9¨FÄ;>¸3©¨F=bÃIAB47ÉIAB=.©A(6¼=³:72.A
Â5=56m«`¨F47»¸¼¹Þ©¨F=0ÃIAià'©U;F8A Ï È
QggQL?KQ*TDPgJM2*LOLKNgQTOLIgggM.gJKQL.KQ:KOLTMKP:SUTJLDTHO?PgMO&KNHKO& £ **y. ¦b£ ut¡U¢.§p %¡vקp O§=vǧD¡w§£% Yf¨UÙÀfÙ3Aº©¨F=.869Á5AB4Å;>= ;FÁ5Á6¼:769¨F=;>¸"©¨F=.87:74;>6¼=0:Uõø¾.=.Á ;»6¼=56¼»6¼ÕUAB4
x ∈ X ¨>« F (·) 87Â5Ä â A©B::¨ g(x) ≤ 0ÀLÙ 2.AB4A
g : X −→ R6983;©¨F=bÃIAià87»¨¨F:72#«sÂ5=.©B:769¨F=,È
äR«5:72.A;FÁ»698876¼Ä5¸9A38AB:698O=.¨F:AB»ª5:y¹IÀI; »6¼=56¼»6¼ÕUAB4¨>«5:72.A3©¨F=.8:74;>6¼=b:)ª54¨FÄ5¸9AB»åAià6987:8UÈOäR«7Àd6¼=;FÁ5Á6¼:769¨F=,À;#87:74769©B:7¸¼¹;FÁ»698876¼Ä5¸9AAB¸9AB»AB=b:
x ∈ X Ù 6¼:72 g(x) < 0Aià698:8UÀ5;[;>ÉL4;>=5ÉIAª;>4;>»AB:AB4
y ≥ 0Aià698:8
87Â.©2Ý:72;>:(x, y)
698u;©B476¼:769©U;>¸,ª*¨F6¼=b:t¨>«:72.A[;>ÉL4;>=5ÉIA«sÂ5=.©B:769¨F=;>¸L(x, y) = F (x) + y g(x)
À5Ù 2569©2698©2;>4;F©B:AB476¼ÕUAÁÅÄb¹Ö:72.AÊ¿n87¹87:AB»
0 = f(x) + yDg(x) ,
g(x) ≤ 0, y ≥ 0, y g(x) = 0 .
Ê2.A©¨F474A87ª*¨F=.Á6¼=5É [;>ÉL4;>=5ÉIAi¿dYfABÙ :¨F=Å»AB:72.¨Á'©¨F=.8769Á5AB48t:72.A[6¼ª.8©256¼:7Õ"©¨F=0:76¼=bÂ.¨FÂ.8uª54¨FÄ5¸9AB»
0 = f(x) + yDg(x) ,
0 = max0, g(x) .ð ÉIAB=.AB4;>¸¼6¼ÕUAÁ¡YfABÙ :¨F= 87:ABª «p¨F4':725698Å87¹87:AB» 4AU;FÁ58Å;F8'«p¨F¸¼¸9¨wÙ8UõØ«p¨F4'ÉL6¼ÃIAB=
(xk−1, yk−1)¾.=.Á
(∆xk,∆yk)87Â.©2':72;>:
0 ∈ f(xk−1) + yk−1Dg(xk−1) + ∂f(xk−1)∆xk + yk−1D2g(xk−1)∆xk + ∆ykDg(xk−1) ,
0 ∈ max0, g(xk−1)+ ∂max0, g(xk−1)Dg(xk−1)∆xk ,
Ù 2.AB4A
∂max0, g(xk−1) =
0 g(xk−1) < 0,conv0, g′(xk−1) g(xk−1) = 0,g′(xk−1) g(xk−1) > 0.
¢ wuy¤£$¥§¦¨vyb¡w£¤3 é06¼=.©A:72.A©[;>ÉL4;>=5ÉIAi¿dYuABÙ :¨F= »AB:72.¨0ÁÁ5¨A8Ö=.¨F:=.A©A88;>476¼¸¼¹æ«sÂ5¸m¾.¸¼¸:72.A©¨F»ª5¸9AB»AB=b:;>47¹ø©¨F=.Á6¼:769¨F=.8
y ≥ 0;>=.Á
y g(x) = 0ÀÙ3AÖÂ.8A(:72.AÞ©¨F474A87ª%¨F=.Á6¼=5ÉÇé5îfï»AB:72.¨ÁÈ
Ê25698698"¨FÄ5:;>6¼=.AÁ×Ä0¹4ABª5¸<;F©B6¼=5ÉÅ:72.AÉIAB=.AB4;>¸¼6¼ÕUAÁªYfABÙ :¨F=×6¼:AB4;>:769¨F=½Äb¹×;Ý8AßbÂ.AB=.©A¨>«tßbÂ;FÁ4;>:769©»6¼=56¼»6¼Õw;>:769¨F=ºª54¨FÄ5¸9AB»8Ù 6¼:72³¸¼6¼=.AU;>476¼ÕUAÁ³©¨F=.87:74;>6¼=0:8UÈÊ2.AB4Ai«p¨F4ALÀ:72.A"ÉIAB=.AB4;>¸¼6¼ÕUAÁ^YuABÙ :¨F=º87:ABª«p¨F4 :72.Ak[;>ÉL4;>=5ÉIA87¹87:AB» 698 »¨Á6m¾AÁ:¨
0 = f(xk−1) + yk−1Dg(xk−1) + Ck∆xk + yk−1D2g(xk−1)∆xk + ∆ykDg(xk−1) ,
Æ$òU; Ï
g(xk−1) +Dg(xk−1)∆xk ≤ 0, yk ≥ 0, yk(g(xk−1) +Dg(xk−1)∆xk
)= 0
Æ$òBÄ Ï
Ù 6¼:72Ý8¨F»ACk ∈ ∂f(xk−1)
Èän=0:74¨0ÁÂ.©B6¼=5É:72.AßbÂ;FÁ4;>:769©;>=.Á'¸¼6¼=.AU;>4 «sÂ5=.©B:769¨F=;>¸98
Fk(∆x) = f(xk−1) ·∆x+1
2∆x ·
(Ck + yk−1D2g(xk−1)
)∆x
gk(∆x) = g(xk−1) +Dg(xk−1)∆x ,
;>=.ÁÂ.876¼=5ÉDg(xk−1) = Dgk(∆x
k)À5:72.A8¹87:AB» Æ$ò Ï 2;F8 :72.A«`¨F47»
0 = DFk(∆xk) + ykDgk(∆x
k) ,
gk(∆xk) ≤ 0, yk ≥ 0, ykgk(∆x
k) = 0 .
Ê25698698:72.AÊ¿n87¹87:AB» ¨>«;Þß0Â;FÁ4;>:769©#ª54¨FÉL4;>»'õf¾.=.Áº;Ö»6¼=56¼»6¼ÕUAB4∆xk
¨>«Fk(·)
87Â5Ä â A©B::¨gk(·) ≤ 0
Èlé0Â.©©A8876¼ÃIAB¸¼¹Ç8¨F¸¼Ã6¼=5ÉæÆ$ò Ï «`¨F4k = 1, 2, 3, ...
4A87Â5¸¼:8#6¼=ø:72.AÝé5îfïã»AB:72.¨Á½Ù 2569©2ç69887Â5»»;>476¼ÕUAÁ³6¼=ºÊ;>Ä5¸9A'òLÈ#ñ.¨F4:72.A;>ª5ª5¸¼69©U;>:769¨F=Ǩ>«:72.A8A#»AB:72.¨0Á58:¨Þª5¸<;F87:769©B6¼:n¹6¼=³:72.A«`¨F¸¼¸9¨UÙ 6¼=5É8A©B:769¨F=.8UÀ5ÙAÙ 6¼¸¼¸,87ª*A©B6m«s¹
XÀFÀf;>=.Á
gÈ
« H&I:JLKNM4OLKP*QSRKTUQ:TUJ:M
é5îfïp¬ Ï é0:;>47:Ù 6¼:72x0 ∈ X ;>=.Á y0 ≥ 0
È3éAB:k = 1
Èé5îfïtò Ï ñ6¼=.Á
∆xk ∈ X ;>=.Á yk ∈ R8¨F¸¼Ã6¼=5É
0 = DFk(∆xk) + ykDgk(x
k−1)gk(∆x
k) ≤ 0, yk ≥ 0, ykgk(∆xk) = 0
ÀÙ 2.AB4A
Fk(∆x) = f(xk−1) ·∆x+ 12∆x ·
(Ck + yk−1D2g(xk−1)
)∆xÀ
Ck ∈ ∂f(xk−1)À;>=.Á
gk(∆x) = g(xk−1) +Dg(xk−1)∆xÈ
éAB:xk = xk−1 + ∆xk
Èé5îfïpW Ï än«
f(xk) + ykDg(xk);>=.Á
max0, g(xk) ;>4A87»;>¸¼¸,AB=.¨FÂ5ÉL2,À87:¨Fª,Èé5îfï>] Ï éAB:
k := k + 1;>=.ÁÉI¨#:¨Öé5îfïò Ï È
ÊO;>Ä5¸9A×òLõÝé5îfïã»AB:72.¨0Áø«p¨F4:72.A';>ª5ª54¨Uà6¼»;>:769¨F=@¨>«;©B476¼:769©U;>¸ª*¨F6¼=b:(x, y) ∈ X × R≥0
¨>«u:72.A[;>ÉL4;>=5ÉIA«sÂ5=.©B:769¨F=;>¸
L(x, y) = f(x) + yg(x)ÈÊ2.A;>¸¼ÉI¨F476¼:725»á©U;>=³Ä*AAià:AB=.Á5AÁ³Ä0¹;Þ8Â56¼:;>Ä5¸9A
Á.;>»ª56¼=5É(8:74;>:ABÉL¹(Ù 2569©2ÝÉLÂ;>4;>=b:AA8uÉL¸9¨FÄ;>¸©¨F=0ÃIAB47ÉIAB=.©A6¼=:72.A©¨F=0ÃIAià'©U;F8AË z0m qÎ È
Û ¢ w0yŦDs£LtAyt¡S£%§s %¡§¢d£&®2y0¦L¡w§s£ ì Ai«p¨F4AuÙ3Af87:7Â.Á¹:72.At«sÂ5¸¼¸*Aß0Â;>:769¨F=.8¨>«,ª5¸<;F8:769©B6¼:n¹IÀÙ3AfÁ698©BÂ.883:72.AfAßbÂ;>:769¨F=.83«`¨F43:72.Af87:74;>6¼=U¯087:74A884AB¸<;>:769¨F=6¼=.Á5ABª%AB=.Á5AB=0:7¸¼¹Ø6¼=@ABÃIAB47¹ç»;>:AB476<;>¸ª%¨F6¼=0:UÈ¡hfAB4ALÀ ABÃIAB47¹Ø87:74;>6¼=@:AB=.8¨F4
ε ∈ Sym(d)698
69Á5AB=b:76m¾AÁêÙ 6¼:72Å6¼:8t¸¼6¼=.AU;>4fAB¸<;F87:769©:7476<;>¸87:74A88θ = C : ε
Èän=Å:725698tª;>ª*AB4UÀ.ÙA4A8:74769©B:u¨FÂ548AB¸¼ÃIA8u:¨»;>:AB476<;>¸»¨Á5AB¸98 Ù 2.AB4A:72.A6¼=.©B4AB»AB=b:;>¸»;>:AB476<;>¸O4A87ª*¨F=.8AÁ5ABª%AB=.Á58u8¨F¸9AB¸¼¹Þ¨F=
θÈ
¦¢Cyp§=vקp u5¢>§yDtI [,AB:Sym(d) = τ ∈ Rd,d : τ T = τ Ä%A:72.A"8AB:f¨>«387¹»»AB:74769©»;>:74769©A8wÈtÊ2.AAB¸<;F87:769©»;>:AB476<;>¸uª54¨Fª%AB47:769A8êÆs6¼=ç:72.A6¼=¾.=56¼:A876¼»;>¸u»¨0Á5AB¸ Ï ;>4A'Á5Ai¾.=.AÁøÄb¹l:72.A°hf¨¨FÜ06<;>=ø:AB=.8¨F4
C : Sym(d) −→ Sym(d)Á5Ai¾.=.AÁÄb¹
C : ε = 2µε + λ trace(ε)IÀfÁ5ABª*AB=.Á6¼=5Éç¨F=@:72.A©[O;>»S±
©¨F=.87:;>=0:8λ, µ > 0
ÈÐu=Sym(d)
ÀF:72.Ahu¨¨FÜ06<;>=":AB=.8¨F4)698ª*¨L876¼:76¼ÃIAÁ5Ai¾.=56¼:ALÀb;>=.Á":72.A3©¨F474A87ª*¨F=.Á6¼=5É6¼=5=.AB4ª54¨ÁÂ.©B:u;>=.Á:72.A;F88¨0©B6<;>:AÁ'AB=.AB47ÉL¹';>4AÁ5AB=.¨F:AÁÝÄb¹
e(σ, τ ) = σ : C−1 : τ , E(σ) =1
2e(σ,σ) .
Ê2.A6¼=.AB¸<;F87:769©»;>:AB476<;>¸,Ä%AB2;UÃ069¨F4u698 »¨Á5AB¸9AÁÄb¹;©¨F=bÃIAiàÞ«pÂ5=.©B:769¨F=
φ : Sym(d) −→ Rr
Á5AB:AB47»6¼=56¼=5É:72.A©¨F=bÃIAià8AB: ¨>«);FÁ»698876¼Ä5¸9A87:74A88A8K = σ ∈ Sym(d) : φ(σ) ≤ 0 È)?ºA;F88Â5»A:72;>:
φ69887»¨¨F:72³«`¨F4
σ 6= 0;>=.Á
φ(0) < 0Èäy=º:72.A©U;F8A¨d«876¼=5ÉL¸9A87Â54$«`;F©A#ª5¸<;F8:769©B6¼:n¹³Ù3A2;ÃIA
r = 1À;>=.Ál«p¨F4
r > 1À:72.AÝ;FÁ»698876¼Ä5¸9A'8AB:
K698:72.A'6¼=b:AB48A©B:769¨F=@¨FÄ5:;>6¼=.AÁl«s4¨F» :72.AÝ©¨F=bÃIAià
©¨F=.87:74;>6¼=0:8φi(σ) ≤ 0
«p¨F4i = 1, ..., r
È
¢ wuyø¦I£% *:y.²O¢>£&®2yb¦I¡w§£% [,AB:PK : Sym(d) −→ K
Ä*A:72.A(¨F47:72.¨FÉI¨F=;>¸ª54¨ â A©B:769¨F=½¨F=0:¨Ý:72.A©¨F=bÃIAià'8AB:¨>«;FÁ»698876¼Ä5¸9A87:74A88A8 Ù 6¼:72Ý4A87ª%A©B: :¨:72.A6¼=5=.AB4ª54¨0ÁÂ.©B:
e(·, ·) ȳ>´:µ¶µ· òD¸º¹,LÒÓF·s¯>°i®
θ ∈ Sym(d)úsÿ°fÑÒ$6°7¬Uúp·pF®
σ = PK(θ) ∈ K·<¶ù0®.·`ýBù.°i´µÝûb°Bún°iÒiÔ#·®%°ûÅþiµ
úsÿ°¶BL´^ùúp·pL®(σ, γ) ∈ Sym(d) ×R
6úsÿ°¶»¼»Z½un¶aµ>¶iún°iÔ
0 = σ − θ + γ · C : Dφ(σ) ,Æ`óL; Ï
φ(σ) ≤ 0, γ · φ(σ) = 0, γ ≥ 0.Æ`ó>Ä Ï
¾º¿:ÀuÀ$Á ¸Ê2.Aª54¨ â A©B:769¨F=Ý698©2;>4;F©B:AB476¼ÕUAÁÝÄ0¹(:72.A©¨F=.87:74;>6¼=b:u»6¼=56¼»6¼Õw;>:769¨F=ݪ54¨FÄ5¸9AB»
σ ∈ Sym(d) : E(σ − θ) = min87Â5Ä â A©B: :¨
φ(σ) ≤ 0 .
QggQL?KQ*TDPgJM2*LOLKNgQTOLIgggM.gJKQL.KQ:KOLTMKP:SUTJLDTHO?PgMO&KNHKO& Âé06¼=.©A
E(·) 698#Â5=56m«`¨F47»¸¼¹½©¨F=0ÃIAiàø;>=.Ál:72.AÞª54¨FÄ5¸9AB» 6988:74769©B:7¸¼¹½;FÁ»698876¼Ä5¸9AºÆsÄ0¹Ç:72.A';F887Â5»ª5:769¨F=φ(0) < 0
Ï À.;"Â5=569ßbÂ.A»6¼=56¼»6¼ÕUAB4σ;>=.Á;Ã[;>ÉL4;>=5ÉIAª;>4;>»AB:AB4
γ ≥ 0Aià6987:87Â.©2:72;>:
(σ, γ)698;
8;FÁ5Á¸9Aª*¨F6¼=b:t¨>«):72.A[;>ÉL4;>=5ÉIA«sÂ5=.©B:769¨F=;>¸
L(σ, γ) = E(σ − θ) + γ · φ(σ)
Ù 2569©2Å698©2;>4;F©B:AB476¼ÕUAÁÝÄb¹Þ:72.A©¨F474A87ª*¨F=.Á6¼=5É Ê¿n87¹87:AB»áÆ`ó Ï È
¢ wuyݦI£% *:yDx£.¡Ay %¡w§&y Ê2.A©¨F474A87ª*¨F=.Á6¼=5É(©¨F=0ÃIAiàª%¨F:AB=0:76<;>¸98f;>4AÁ5AB=.¨F:AÁÝÄb¹
ϕK(θ) = E(θ − PK(θ)
), ψK(θ) = E(θ)−E
(θ − PK(θ)
), θ ∈ Sym(d) .
Ê2.A8Afª%¨F:AB=0:76<;>¸98 Ù 6¼¸¼¸Ä*AuÂ.8AÁ(6¼=i[,AB»»;§_"Ä*AB¸9¨UÙ:¨"«`¨F47»Â5¸<;>:A:72.Aª5476¼»;>¸»6¼=56¼»6¼Õw;>:769¨F='ª54¨FÄ¿¸9AB»'È>Yf¨F:A:72;>:Ù3A2;UÃIAÄ0¹êË Í À&[,AB»'ÈLÄÈ'e Î
DϕK(θ)[η] = e(θ − PK(θ),η
), θ,η ∈ Sym(d) .
Æ"W ϳ>´:µ¶µ· ó&¸§½ÿ°\aù®¬úp·`F®²L´
ψK(·) ·¶¬F®.¯>°±CÅ3®L®.®%°`ÓI²Iúp·¯>°Å²F®û÷3°ÿ²F¯>°
DψK(θ)[η] = e(PK(θ),η) , θ, η ∈ Sym(d) .Æ] Ï
¾º¿:ÀuÀ$Á ¸Ê2.A¨F47:72.¨FÉI¨F=;>¸ª54¨ â A©B:769¨F=PK698 Â5=569ßbÂ.AB¸¼¹Þ©2;>4;F©B:AB476¼ÕUAÁÅÄb¹
e(θ − PK(θ),η − PK(θ)
)≤ 0, θ ∈ Sym(d), η ∈ K .
Æ"_ Ï
än=.8AB47:76¼=5ÉÞÆ"W Ï ÉL6¼ÃIA8DψK(θ)[η] = e(θ,η)−DϕK(θ)[η] = e(PK(θ),η)
À;>=.ÁÙAu¨FÄ5:;>6¼=«s4¨F»Æ"_ Ï«p¨F4θ,η ∈ Sym(d)
ψK(θ)− ψK(η)−DψK(η)[θ − η]
= e(PK(θ),θ)−E(PK(θ))− e(PK(η),η) +E(PK(η))− e(PK(η),θ − η)
= e(PK(θ)− PK(η),θ − PK(θ)
)+E
(PK(θ)− PK(η)
)≥ E
(PK(θ)− PK(η)
).
Ê2bÂ.8UÀDψK(η)[θ−η] ≤ ψK(θ)−ψK(η)
À6RÈbALȼÀDψK
698»¨F=.¨F:¨F=.A;>=.Á:72.AB4Ai«`¨F4AψK698©¨F=bÃIAiàÈ
ñ6¼=;>¸¼¸¼¹IÀ.876¼=.©A0 ∈ K
Ù3A2;UÃIAψK(θ) = E(θ)−E
(θ − PK(θ)
)≥ E(θ)−E(θ − 0) = 0
È
¢ wuyb£$¥¦Æv^yI¡wO£¤Ç. O¤¡p§p 0y5¢>§.y0¤¡¢>£È®Éyb¦L¡w§s£ utL é0:;>47:76¼=5É'Ù 6¼:72σ0 = θ
;>=.Áγ0 = 0
À:72.Aª54¨ â A©B:769¨F=ø©U;>=lÄ*A(©¨F»ª5Â5:AÁlÄ0¹º:72.A'é5îfï»AB:72.¨0ÁÀ6RÈALȼÀ)6¼=øABÃIAB47¹º6¼:AB4;>:769¨F=ç87:ABª½Ù3AÖ2;ÃIAÖ:¨8¨F¸¼ÃIA:72.A©¨F474A87ª*¨F=.Á6¼=5ÉsÊ 87¹8:AB»
0 = σk−1 − θ + γk−1 · C : Dφ(σk−1)
+ ∆σk + γk−1 · C : D2φ(σk−1) : ∆σk + ∆γk · C : Dφ(σk−1) ,
φk(σk) ≤ 0, γk · φk(σk) = 0, γk ≥ 0 ,
Ù 2.AB4Aφk(σ) = φ(σk−1)+Dφ(σk−1) : (σ−σk−1)
698:72.A¸¼6¼=.AU;>476¼ÕUAÁü¨wÙç47Â5¸9ALÈ)Ê2.A=.ABÙç6¼:AB4;>:A8;>4AÉL6¼ÃIAB='Ä0¹
σk = σk−1 + ∆σk;>=.Á
γk = γk−1 + ∆γkÈ
Ê2.A#ÜIAB¹Å¨FÄ.8AB47ÃL;>:769¨F=«`¨F4f:72.A©¨F=.8:747Â.©B:769¨F=ר>«3:72.A#=.ABÙÚ;>¸¼ÉI¨F476¼:725» 6¼=ÇéA©B:769¨F=ªaÈ'WÖÄ*AB¸9¨UÙã698:72;>::72.Aé5îfï@87:ABªÝ©U;>=ÅÄ%A6¼=0:AB47ª54AB:AÁêÉIA¨F»AB:74769©U;>¸¼¸¼¹Å;F8:72.Aª54¨ â A©B:769¨F=ê¨F=b:¨(:72.A2;>¸m«s¿n87ª;F©AÁ5Ai¾.=.AÁÄb¹Þ:72.A¸¼6¼=.AU;>476¼ÕUAÁü¨wÙ-47Â5¸9A
Kk = τ ∈ Sym(d) : φk(τ ) ≤ 0 ⊃ KÙ 6¼:72Ø4A87ª*A©B::¨½;º87Â56¼:;>Ä5¸9Aê;FÁ.;>ª5:AÁØ»AB:74769©LÈ Ê2.AB4Ai«p¨F4ALÀ 4AU;>474;>=5ÉL6¼=5Él:72.A'¾.487:(Aß0Â;>:769¨F=@;>=.ÁÁ5Ai¾.=56¼=5É
Gk = id +γk−1 · C : D2φ(σk−1)ÉL6¼ÃIA8
0 = Gk : σk − θ − γk−1 · C : D2φ(σk−1) : σk−1 + γk · C : Dφ(σk−1) .
Ê H&I:JLKNM4OLKP*QSRKTUQ:TUJ:Mé06¼=.©A
φ698©¨F=0ÃIAià*À
D2φ(σk−1)698 ª*¨L876¼:76¼ÃIA8AB»6m¿nÁ5Ai¾.=56¼:ALÀ%;>=.Á
Gk©U;>='Ä*A6¼=bÃIAB47:AÁÈän=0:74¨0ÁÂ.©B6¼=5É
Ck = G−1k : C
Àθk = G−1
k :(θ−γk−1·C : D2φ(σk−1) : σk−1
) ;>=.ÁÂ.876¼=5ÉDφk(σ
k) = Dφ(σk−1)4ABÙ 476¼:A8:72.Aé5îfïØ87:ABª6¼=':72.Af«`¨F47»
0 = σk − θk + γk · Ck : Dφk(σk) ,
Æ"eL; Ïφk(σ
k) ≤ 0, γk · φk(σk) = 0, γk ≥ 0 .Æ"e>Ä Ï
ð =;>¸9¨FÉI¨FÂ.87¸¼¹º:¨Ë[,AB»»;½òÙA(¨FÄ.8AB47ÃIAÖ:72;>:σk = Pk(θ
k)Ù 2.AB4A
Pk : Sym(d) −→ Kk698:72.A
¨F47:72.¨FÉI¨F=;>¸.ª54¨ â A©B:769¨F=Ù 6¼:724A87ª*A©B:):¨ek(σ, τ ) = σ : C−1
k : τÈÌYf¨F:A3:72;>:
Ck698):72.A;>¸¼ÉI¨F476¼:725»69©
:;>=5ÉIAB=b:u»¨0ÁÂ5¸¼Â.8"Ë ÛÀ132;>ª,ÈLWÈ'WÈVó;>=.ÁÅ132;>ª,ÈL_ÈVóÈ ] Î Èð Éb;>6¼=,Àb87Â.©©A886¼ÃIAB¸¼¹#8¨F¸¼Ã06¼=5ÉÆ"e Ï «`¨F4
k = 1, 2, 3, ...4A87Â5¸¼:86¼=:72.Afé5îfﺻAB:72.¨Á«`¨F4:72.AuABÃF;>¸¼Â;>:769¨F=
¨>«):72.Aª54¨ â A©B:769¨F=,À8AAÊO;>Ä5¸9AóÈ
ï\c¬ Ï éAB:σ0 = θ
;>=.Áγ0 = 0
È3éAB:k = 1
Èï\cò Ï ñ6¼=.Á
σk;>=.Á
γk8¨F¸¼Ã6¼=5É
0 = σk − θk + γk · Ck : Dφk(σk)À
φk(σk) ≤ 0, γk · φk(σk) = 0, γk ≥ 0
ÀÙ 2.AB4A
φk(σ) = φ(σk−1) +Dφ(σk−1) : (σ − σk−1)À
Gk = id +γk−1 · C : D2φ(σk−1);>=.Á
Ck = G−1k : C
À.;>=.Áθk = G−1
k :(θ − γk−1 · C : D2φ(σk−1) : σk−1
) Èï\cW Ï än«
σ − θ + γ · C : Dφ(σ);>=.Á
max0, φ(σk) ;>4A87»;>¸¼¸AB=.¨FÂ5ÉL2,À87:¨Fª,Èï\c] Ï éAB:
k := k + 1;>=.ÁÞÉI¨:¨ï>cò Ï È
ÊO;>Ä5¸9Aóõé5îfï»AB:72.¨Áê«p¨F4:72.A;>ª5ª54¨Uà6¼»;>:769¨F=ר>«:72.Aª54¨ â A©B:769¨F=σ = PK(θ)
«p¨F4;(ÉL6¼ÃIAB=³:7476<;>¸87:74A88
θ ∈ Sym(d)Èäy=Þ86¼»ª5¸9A©U;F8A8Æ
r = 1;>=.ÁÞÃI¨F='ëÅ698A83ü¨UÙ-47Â5¸9A Ï :72.A;>¸¼ÉI¨F476¼:725» 8:¨Fª.8;d«s:AB4
¨F=.A6¼:AB4;>:769¨F=Ý8:ABª,È
Í Î&vÌyLÏ Ð0qÑÌÈtU¡w§¦0§<¡Ò [,AB:r = 1
;>=.Áê¸9AB:φ(σ) = |dev(σ)| −K0
Ä%A":72.AÃI¨F=ëÅ698A8uü¨UÙ47Â5¸9ALÀ5Ù 2.AB4A
dev(σ) = σ − 13 trace(σ)I
ÈÊ25698 ÉL6¼ÃIA8
Dφ(σ) =dev(σ)
|dev(σ)| , D2φ(σ) : η =dev(η)
|dev(σ)| −dev(σ) : dev(η)
|dev(σ)|2dev(σ)
|dev(σ)| ,;>=.Á
C : Dφ(σ) = 2µDφ(σ)ÀC : D2φ(σ) = 2µD2φ(σ)
ÀD2φ(σ) : σ = 0
ÈëŨF4A¨wÃIAB4UÀ)Á6¼4A©B:©¨F»ª5Â5:;>:769¨F=ݹ69AB¸9Á58
(id +2µγD2φ(σ)
)−1= id− 2µγ|dev(σ)|
2µγ + |dev(σ)|D2φ(σ) .
Ê2bÂ.8wÀ«p¨F4 ÉL6¼ÃIAB=σk−1 ;>=.Á γk−1 Ù3A¨FÄ5:;>6¼=
θk = θ − 2µγk−1|dev(σk−1)|2µγk−1 + |dev(σk−1)|D
2φ(σk−1) : θ
;>=.Á
Pk(θk) = θk − 2µmax0, φk(θk) dev(θk)
|dev(θk)|, γk = max0, φk(θk) . Æ(a Ï
äy=:72569887ª*A©B6<;>¸Aià5;>»ª5¸9ALÀ.:72.Aª54¨ â A©B:769¨F=PK698¨FÄ5:;>6¼=.AÁ'6¼=Ý;#86¼=5ÉL¸9A6¼:AB4;>:769¨F=Å87:ABª,À6RÈALȼÀ
PK(θ) = θ − 2µmax0, φ(θ) dev(θ)
|dev(θ)| , γ = max0, φ(θ) , Æ"Ä Ï
8AAË Û,À132;>ª,È:WÈ'Wȼò Î È
QggQL?KQ*TDPgJM2*LOLKNgQTOLIgggM.gJKQL.KQ:KOLTMKP:SUTJLDTHO?PgMO&KNHKO& Óz ¢ wuyv^.§s ÔO¢F§p ¦0§Ìyݧp ç¦I£*v)¥,¡A¡w§s£% 0&ÕÌÈtU¡w§¦0§<¡Ò
ì Ai«p¨F4A ÙAª54¨©AAÁÙ 6¼:72(ï4;>=.Á:7¸m¿dctABÂ.88ª5¸<;F87:769©B6¼:n¹IÀÙ3A6¼¸¼¸¼Â.87:74;>:A:72.A8:;>=.Á.;>4Á(;>ª5ª54¨I;F©2Ö6¼=©¨F»#¿ª5Â5:;>:769¨F=;>¸ª5¸<;F8:769©B6¼:n¹#«`¨F4:72.A876¼»ª5¸9AhuAB=.©Ü0¹#»¨Á5AB¸¨>«,87:;>:769© 6¼=¾.=56¼:A876¼»;>¸%ª5¸<;F87:769©B6¼:n¹IÈÊ2.AB4Ai«`¨F4ALÀ¸9AB:
VÄ%AÞ;'¾.=56¼:AÖAB¸9AB»AB=0:87ª;F©A(¨>«u:72.A(Á6987ª5¸<;F©AB»AB=0:8
uÆpÁ5AB:AB47»6¼=56¼=5ɳ:72.A(¸¼6¼=.AU;>476¼ÕUAÁø87:74;>6¼=.8
ε(u) = sym(Du) ∈ Sym(d)Ï À¸9AB:
ΣÄ*AÖ:72.A'87:74A888ª;F©A';>=.Áø¸9AB:
ΣKÄ%AÞ:72.A'©¨F=0ÃIAiàç8AB:¨>«
87:74A88A8Ù 2569©2ê;>4Aª*¨F6¼=b:yÙ 698A";FÁ»698876¼Ä5¸9A6¼=K ⊂ Sym(d)
Èñ.¨F4;(ÉL6¼ÃIAB=ê¸9¨I;FÁÅ«pÂ5=.©B:769¨F=;>¸
`À%:72.Aª54¨FÄ5¸9AB» ¨>«387:;>:769©6¼=¾.=56¼:A876¼»;>¸ª5¸<;F87:769©B6¼:y¹Å©U;>=Ä%A87:;>:AÁ³;F8
«p¨F¸¼¸9¨wÙ8Uõ)¾.=.ÁÝ;»6¼=56¼»6¼ÕUAB4σ ∈ ΣK
¨>«:72.AÁÂ;>¸AB=.AB47ÉL¹
E(σ) =
∫
ΩE(σ) dx
87Â5Ä â A©B: :¨:72.A¸¼6¼=.AU;>4t©¨F=.87:74;>6¼=0:∫
Ωσ : ε(δu) dx = `(δu), δu ∈ V .
Ê2.A©¨F474A87ª*¨F=.Á6¼=5É [;>ÉL4;>=5ÉIA«sÂ5=.©B:769¨F=;>¸O698 ÉL6¼ÃIAB='Äb¹
L(σ,u, γ) = E(σ) +
∫
Ωγ · φ(σ) dx−
∫
Ωσ : ε(u) dx + `(u) .
;>=.Á'ABÃIAB47¹Ö»6¼=56¼»6¼ÕUAB4u¨>«):72.AÁÂ;>¸,ª54¨FÄ5¸9AB» 698©2;>4;F©B:AB476¼ÕUAÁÝÄ0¹Þ:72.AÊ¿n87¹8:AB»
0 = C−1 : σ − ε(u) + γ ·Dφ(σ),Æ"ÖL; Ï
φ(σ) ≤ 0, γ · φ(σ) = 0, γ ≥ 0,Æ"Ö>Ä Ï
0 =
∫
Ωσ : ε(δu) dx − `(δu), δu ∈ V( × ) .
Æ"ÖF© Ï
ð ª5ª5¸¼¹6¼=5ÉZ[,AB»»;êò:¨Æ"ÖL; Ï ;>=.ÁlÆ"Ö>Ä Ï ÉL6¼ÃIA8σ = PK
(C : ε(u)
) À%;>=.ÁÅ6¼=.8AB47:76¼=5ÉÖ6¼=b:¨³Æ"ÖF© Ï ¹069AB¸9Á58:72.A=.¨F=5¸¼6¼=.AU;>4ÃL;>476<;>:769¨F=;>¸ª54¨FÄ5¸9AB» «`¨F4:72.AÁ6987ª5¸<;F©AB»AB=0:
u ∈ V∫
ΩPK
(C : ε(u)
): ε(δu) dx = `(δu), δu ∈ V .
Ê256986¼¸¼¸¼Â.87:74;>:A8:72.At»;>6¼=Öª5476¼=.©B6¼ª5¸9Af6¼=(=.AU;>47¸¼¹(;>¸¼¸%»AB:72.¨Á5836¼=Ö©¨F»ª5Â5:;>:769¨F=;>¸ª5¸<;F87:769©B6¼:n¹IÈ?ºAf;>6¼»«p¨F4Á5ABÃIAB¸9¨Fª56¼=5ÉÞ;=.ABÙ ©B¸<;F88¨>«8¨F¸¼Â5:769¨F='»AB:72.¨Á58Ù 6¼:72ÝÄ%AB:7:AB4 »;>:72.AB»;>:769©U;>¸ª54¨Fª%AB47:769A8wÈ
Í* 0Ø) §¡AyDtw§=v^&\Ù,¥uÈtw§NÚ4t¡A¡w§s¦`>y0¢Ûyb¦L¡SÌÈt¡w§s¦b§¡4Ò?ºAtÁ5Ai¾.=.Au:72.A «pÂ5¸¼¸Aß0Â;>:769¨F=.83¨>«ß0Â;F876m¿n87:;>:769©tª5¸<;F87:769©B6¼:n¹IÀ©¨F»"Ä56¼=56¼=5É:72.AtÉL¸9¨FÄ;>¸%Aß0Â56¼¸¼6¼Ä5476¼Â5»ôAßbÂ;d¿:769¨F=.86¼=
ΩÀ;>=.ÁÖ:72.Af¸9¨©U;>¸%ª*¨F6¼=b:yÙ 698AAß0Â;>:769¨F=.8«p¨F4:72.A87:74A88$¿n8:74;>6¼=Ö4AB¸<;>:769¨F=Ý;>=.ÁÖ:72.A;F88¨©B6<;>:AÁ
ü¨UÙ-47Â5¸9AÁ5AB:AB47»6¼=.AÁ'Äb¹Þ:72.A¹69AB¸9ÁÞ«pÂ5=.©B:769¨F=φÈ
Ü ¡A* [,AB:Ω ⊂ Rd Æ d = 2, 3
Ï Ä*A#:72.A4Ai«`AB4AB=.©A©¨F=¾.ÉLÂ54;>:769¨F=,À;>=.Á³¸9AB:ΓD ∪ ΓN = ∂Ω
Ä*A;Á5A©¨F»ª%¨L86¼:769¨F=Ũ>«):72.AÄ%¨FÂ5=.Á.;>47¹IÈ?ºAf¾5à;#:76¼»A6¼=b:AB47ÃL;>¸
[0, T ]È
Ê2.Aª54¨FÄ5¸9AB» Á5ABª*AB=.Á58t¨F=:72.A«`¨F¸¼¸9¨UÙ 6¼=5ÉÖÁ.;>:;õ;ª54A8©B476¼Ä*AÁ'Á6987ª5¸<;F©AB»AB=0:uÃIA©B:¨F4
uD : ΓD × [0, T ] −→ Rd
«p¨F4 :72.AA88AB=b:76<;>¸Ä%¨FÂ5=.Á.;>47¹©¨F=.Á6¼:769¨F=.8t¨F=ΓD;>=.Á';#¸9¨I;FÁÞ«pÂ5=.©B:769¨F=;>¸
`(t, δu) =
∫
Ωb(t) · δu dx +
∫
ΓN
tN(t) · δu daÁ5ABª%AB=.Á6¼=5É(¨F='Ä%¨Á¹«`¨F4©AÁ5AB=.876¼:769A8f;>=.ÁÞ:74;F©B:769¨F='«`¨F4©AÁ5AB=.876¼:769A8
b : Ω× [0, T ] −→ Rd, tN : ΓN × [0, T ] −→ Rd .
Ý H&I:JLKNM4OLKP*QSRKTUQ:TUJ:M¢ wuy`yÙ,¥0¡w§s£% 0t£:Ût§p ÎØ §<¡Aytw§=v&p$Èt¡w§s¦b§¡Ò ?ºAÙ;>=0:t:¨Á5AB:AB47»6¼=.AÁ698ª5¸<;F©AB»AB=b:8
u : Ω× [0, T ] −→ Rd ,87:74A88A8
σ : Ω× [0, T ] −→ Sym(d) ,ª5¸<;F87:769©8:74;>6¼=.8
εp : Ω× [0, T ] −→ Sym(d) ,;>=.ÁÝ;ª5¸<;F87:769©»Â5¸¼:76¼ª5¸¼69AB4
λ : Ω× [0, T ] −→ Rr
8;>:7698$«p¹6¼=5É:72.AA88AB=0:76<;>¸,Ä%¨FÂ5=.Á.;>47¹Þ©¨F=.Á6¼:769¨F=.8
u(x, t) = uD(x, t), (x, t) ∈ ΓD × [0, T ] ,:72.A©¨F=.87:76¼:7Â5:76¼ÃIA4AB¸<;>:769¨F=
σ(x, t) = C :(ε(u)(x, t) − εp(x, t)
), (x, t) ∈ Ω× [0, T ]
:72.AAß0Â56¼¸¼6¼Ä5476¼Â5» Aß0Â;>:769¨F=.8
−divσ(x, t) = b(x, t), (x, t) ∈ Ω× [0, T ] ,
σ(x, t)n(x) = tN (x, t), (x, t) ∈ ΓN × [0, T ]ÆsÙ 2.AB4A
n(x)Á5AB=.¨F:A8:72.A¨FÂ5:AB4Â5=56¼:t=.¨F47»;>¸,ÃIA©B:¨F4u¨F=
∂ΩÏ À:72.Aü¨UÙ 47Â5¸9A
d
dtεp(x, t) = λ(x, t) ·Dφ(σ(x, t)) , (x, t) ∈ Ω× [0, T ] ,
Æ$òA¬ Ï;>=.Á:72.A©¨F»ª5¸9AB»AB=0:;>47¹Ý©¨F=.Á6¼:769¨F=.8Æ=;>47Â.82¿dÂ525=¿nÊ,Â.©ÜIAB4 Ï
λ(x, t) · φ(σ(x, t)
)= 0, λ(x, t) ≥ 0, φ(σ(x, t)) ≤ 0, (x, t) ∈ Ω× [0, T ] .
Æ$òLò Ï
l Ü §tU¦I¢CyI¡Ayŧp 0Ø) §¡AyDtw§=v^&pÌÈt¡w§s¦b§¡4ÒÊ2.AÁ698©B4AB:A#»¨Á5AB¸)698©¨F=.87:747Â.©B:AÁ6¼=³:nÙ¨87:ABª.8UÈñ6¼487:UÀ*ÙA#6¼=b:74¨ÁÂ.©A;Ö87ª;>:76<;>¸Á698©B4AB:76¼Õw;>:769¨F=Ä;F8AÁ¨F=¾.=56¼:AAB¸9AB»AB=0:8«p¨F4:72.AÁ6987ª5¸<;F©AB»AB=b:83;>=.ÁÞ;>Â.88Oª*¨F6¼=b:)ÃL;>¸¼Â.A8O«p¨F4)87:74A88A8);>=.Á6¼=b:AB47=;>¸ª;>4;>»AB:AB48UÈÊ2.AB=,À:72.At«sÂ5¸¼¸¼¹Á698©B4AB:Au»¨Á5AB¸%6983¨FÄ5:;>6¼=.AÁÖÄb¹Ö;Ä;F©Ü0Ù;>4ÁißÂ5¸9AB4»AB:72.¨0Á(6¼=(:76¼»ALÈ
Ü §tU¦I¢CyI¡w§.¡w§s£ اs àtu5¦.y [,AB:V ⊂ C0,1(Ω,Rd)
Ä%A;¾.=56¼:AAB¸9AB»AB=b:87ª;F©A87ª;>=5=.AÁÖÄb¹=.¨0Á.;>¸Ä;F87698«pÂ5=.©B:769¨F=.8UÈ>[AB:
V(uD) = v ∈ V : v(x) = uD(x)«p¨F4
x ∈ D,Ù 2.AB4A
D ⊂ ΓD698 :72.A8AB:¨>«;>¸¼¸=.¨Á.;>¸,ª*¨F6¼=b:8u¨F=
ΓDÈ
[,AB:Ξ ⊂ Ω
Ä*Aß0Â;FÁ4;>:7Â54Aª%¨F6¼=0:8u;>=.Á¸9AB:ωξÄ%A©¨F474A87ª*¨F=.Á6¼=5Éß0Â;FÁ4;>:7Â54AÙAB6¼ÉL2b:8f87Â.©2':72;>:
∫
Ωε(v) : ε(w) dx =
∑
ξ∈Ξ
ωξ ε(v)(ξ) : ε(w)(ξ) , v,w ∈ V .
?ºA8AB:Λ = µ : Ξ −→ Rr ;>=.Á Σ = τ : Ξ −→ Sym(d) È[,AB:
ε : V −→ ΣÄ%AÉL6¼ÃIAB=×Äb¹
ε(v)(ξ) = sym(Dv(ξ)
) Èñ.¨F4ÉL6¼ÃIAB=u ∈ V
:725698"Á5Ai¾.=.A8C : ε(u) ∈ Σ
Èäy=º¨FÂ54=.¨F:;>:769¨F=,À:72.A6¼=b:ABÉL4;>¸O698 Â.8AÁ';>¸98¨«p¨F4 :72.Af¾.=56¼:A"87Â5»8
∫
Ωσ : ε dx :=
∑
ξ∈Ξ
ωξ σ(ξ) : ε(ξ) .
QggQL?KQ*TDPgJM2*LOLKNgQTOLIgggM.gJKQL.KQ:KOLTMKP:SUTJLDTHO?PgMO&KNHKO& ᢠwuyâtAyUvקNÚ¤§tU¦b¢yb¡AyâyDÙ,¥u¡w§s£ utø£LÛ'§p ÎØ §<¡Aytw§=v& $Èt¡w§s¦b§¡4Ò Ê2.Aê8AB»6m¿nÁ698©B4AB:AŪ54¨FÄ5¸9AB»Ù 2569©2Å698Á698©B4AB:A6¼='87ª;F©A";>=.Á©¨F=b:76¼=bÂ.¨FÂ.8f6¼=:76¼»A4AU;FÁ58f;F8«`¨F¸¼¸9¨UÙ8UõÁ5AB:AB47»6¼=.A"Á6987ª5¸<;F©AB»AB=b:8
u : [0, T ] −→ V ,87:74A88A8
σ : [0, T ] −→ Σ ,ª5¸<;F87:769©87:74;>6¼=.8
εp : [0, T ] −→ Σ ,;>=.ÁÝ;ª5¸<;F87:769©»"Â5¸¼:76¼ª5¸¼69AB4
λ : [0, T ] −→ Λ8;>:7698$«p¹06¼=5É:72.AA88AB=b:76<;>¸OÄ%¨FÂ5=.Á.;>47¹Þ©¨F=.Á6¼:769¨F=.8
u(t) ∈ V(uD(t)) , t ∈ [0, T ] ,:72.A©¨F=.87:76¼:7Â5:76¼ÃIA4AB¸<;>:769¨F=
σ(ξ, t) = C :(ε(u)(ξ, t)− εp(ξ, t)
), (ξ, t) ∈ Ξ× [0, T ]
ÆsÂ.876¼=5Éσ(ξ, t) := σ(t)(ξ)
«`¨F4σ(t) ∈ Σ
Ï À:72.AAßbÂ56¼¸¼6¼Ä5476¼Â5» AßbÂ;>:769¨F='6¼=ÙAU;>ÜÖ«p¨F47»∫
Ωσ(t) : ε(δu) dx = `(t, δu), t ∈ [0, T ], δu ∈ V( × ) ,
:72.Aü¨UÙ 47Â5¸9Ad
dtεp(ξ, t) = λ(ξ, t) ·Dφ(σ(ξ, t)) , (ξ, t) ∈ Ξ× [0, T ] ,
;>=.Á:72.A©¨F»ª5¸9AB»AB=b:;>47¹Å©¨F=.Á6¼:769¨F=.8Æ=;>47Â.872¿dÂ525=¿nÊÂ.©ÜIAB4 Ï
λ(ξ, t) · φ(σ(ξ, t)
)= 0, λ(ξ, t) ≥ 0, φ(σ(ξ, t)) ≤ 0, (ξ, t) ∈ Ω× [0, T ] .
Ü §tU¦I¢CyI¡w§.¡w§s£ æ§s ø¡w§v^y. Ê2.A»¨Á5AB¸,¨>«O6¼=.©B4AB»AB=b:;>¸6¼=¾.=56¼:A876¼»;>¸,ª5¸<;F87:769©B6¼:y¹(698 ¨FÄ5:;>6¼=.AÁÄ0¹Þ;Á5A©¨F»ª%¨L86¼:769¨F=
0 = t0 < t1 < · · · < tN = T¨>«:72.A:76¼»A6¼=b:AB47ÃL;>¸O;>=.ÁÞ:72.A;>ª5ª5¸¼69©U;>:769¨F=Ũ>«):72.AÄ;F©ÜÙ;>4ÁßÂ5¸9AB4t8©2.AB»ALõ)«p¨F4
n = 1, 2, 3, ...:72.A
=.Aià:6¼=.©B4AB»AB=b:#Á5ABª%AB=.Á58¨F=×:72.A»;>:AB476<;>¸25698:¨F47¹êÁ5A8©B476¼Ä*AÁ³Ä0¹εn−1p
Æp87:;>47:76¼=5ÉÝÙ 6¼:72ε0p = 0
Ï À:72.A#=.ABÙã¸9¨I;FÁ
`n(δu) = `(tn, δu);>=.Á:72.A#=.ABÙ¤ãf6¼4769©25¸9AB:Ä*¨FÂ5=.Á.;>47¹ÅÃL;>¸¼Â.A8
unD = uD(tn)È?ºA
©¨F»ª5Â5:AÝ:72.AêÁ6987ª5¸<;F©AB»AB=0:un ∈ V(unD)
8;>:76987«s¹6¼=5É×:72.AÅA88AB=b:76<;>¸fÄ*¨FÂ5=.Á.;>47¹ç©¨F=.Á6¼:769¨F=.8UÀ :72.A87:74A88
σn ∈ ΣÀ,:72.Aª5¸<;F8:769©87:74;>6¼=
εnp ∈ ΣÀ;>=.Á³:72.Aª5¸<;F87:769©»Â5¸¼:76¼ª5¸¼69AB4
λn ∈ Λ8a;>:7698$«s¹6¼=5É:72.A
©¨F=.87:76¼:7Â5:76¼ÃIA4AB¸<;>:769¨F=
σn(ξ) = C :(ε(un)(ξ)− εnp (ξ)
), ξ ∈ Ξ ,
Æ$òUó Ï:72.A(°7ýBù·s´^·RþiÒi·ùÔ°7ýù²Lúp·`F®
∫
Ωσn : ε(δu) dx = `n(δu), δu ∈ V( × ) ,
Æ$òAW Ï:72.AÁ698©B4AB:76¼ÕUAÁö F÷@Òaù´9°
1
tn − tn−1
(εnp (ξ)− εn−1
p (ξ))
= λn(ξ) ·Dφ(σn(ξ)) , ξ ∈ Ξ ,
;>=.Á:72.A¬FÔuÑ´9°iÔ°i®úR²LÒaµ¬F®ûL·súp·`F®5¶
λn(ξ) · φ(σn(ξ)
)= 0, λn(ξ) ≥ 0, φ(σn(ξ)) ≤ 0, ξ ∈ Ξ .
é06¼=.©A:72.Aª54¨FÄ5¸9AB» 6984;>:Ai¿R6¼=.Á5ABª*AB=.Á5AB=b:UÀ)4A8©U;>¸¼6¼=5ÉŨ>« :72.A:76¼»Aª;>4;>»AB:AB4"Á5¨A8=.¨F:;A©B::72.A»¨0Á5AB¸RÈÊ2bÂ.8UÀ5ÙAÁ5Ai¾.=.A
γn = (tn − tn−1)λn ∈ ΛÀ56RÈ5ALȼÀ.:72.Afü¨UÙ 47Â5¸9A2;F8t:72.Af«p¨F47»
εnp (ξ) = εn−1p (ξ) + γn(ξ) ·Dφ(σn(ξ)) , ξ ∈ Ξ .
Æ$ò2] Ï
ädå H&I:JLKNM4OLKP*QSRKTUQ:TUJ:MʨFÉIAB:72.AB4fÙ 6¼:72½Æ$òUó Ï ;>=.ÁÇÆ$òAW Ï À.Ù3A©U;>=ê87:;>:A:72.A«sÂ5¸¼¸¼¹Á698©B4AB:Aª54¨FÄ5¸9AB» 6¼=Å6¼=¾.=56¼:A876¼»;>¸Oª*AB4$«`A©B:ª5¸<;F87:769©B6¼:y¹%õ½«`¨F4ÞÉL6¼ÃIAB=
εn−1p ∈ Σ
¾.=.Áσn ∈ Σ
Àun ∈ V(unD)
;>=.Áγn ∈ Λ
87Â.©2:72;>::72.AAß0Â56¼¸¼6¼Ä5476¼Â5» Aß0Â;>:769¨F=,À5:72.Aü¨UÙ 47Â5¸9A;>=.Á:72.A©¨F»ª5¸9AB»AB=b:;>47¹Å©¨F=.Á6¼:769¨F=.8u;>4A8;>:76987¾AÁ
(ñÌc
n) σn(ξ) = C :(ε(un)(ξ)− εn−1
p (ξ)− γn(ξ) ·Dφ(σn(ξ))), ξ ∈ Ξ ,
Æ$òA_L; Ï
(1 1
n) φ(σn(ξ)) ≤ 0, γn(ξ) · φ(σn(ξ)
)= 0, γn(ξ) ≥ 0, ξ ∈ Ξ ,
Æ$òA_>Ä Ï
(ß\ß
n)
∫
Ωσn : ε(δu) dx = `n(δu), δu ∈ V( × ) .
Æ$òA_F© Ïñ.¨F4:72.A=.Aià: :76¼»A8:ABª,À
εnp698Á5AB:AB47»6¼=.AÁ'Ä0¹(:72.AÁ698©B4AB:76¼ÕUAÁÞü¨UÙ 47Â5¸9AÆ$ò2] Ï È
¢ wuyÖ¤§tU¦I¢CyI¡Ay(¤¥u&$O¢>£LæÌyv §p ½§p ¦I¢Cyv^y0 %¡A&$ÌÈtU¡w§¦0§<¡Ò. ?ºAt8:7Â.Á¹:72.AuÁÂ;>¸»6¼=56¼»6¼Õw;>:769¨F=ª54¨FÄ5¸9AB»á«p¨F4#;Þ¾5à5AÁº:76¼»AÖ87:ABªº«p4¨F»
tn−1:¨tn;>=.Á׫`¨F4ÉL6¼ÃIAB=
εn−1p ∈ Σ
Èäy=ǨF4Á5AB4:¨'6¼=.©¨F4$¿ª*¨F4;>:A:72.A§ãf6¼4769©25¸9AB:Ä*¨FÂ5=.Á.;>47¹³©¨F=.Á6¼:769¨F=.86¼=0:¨:72.AS[;>ÉL4;>=5ÉIAª;>4;>»AB:AB4UÀÙA4Ai«`¨F47»Â5¸<;>:A:72.AAß0Â56¼¸¼6¼Ä5476¼Â5» Aß0Â;>:769¨F=ÇÆ$òA_F© Ï 6¼=:72.AAß0Â56¼ÃF;>¸9AB=0: «`¨F47»
∫
Ωσn : ε(δu) dx −
∫
ΓD
(σnn) · δu da = `n(δu), δu ∈ V(unD) .Æ$òAe Ï
Ê2.A©¨F474A87ª*¨F=.Á6¼=5É6¼=.©B4AB»AB=0:;>¸ÁÂ;>¸AB=.AB47ÉL¹Þ698Á5Ai¾.=.AÁ'Äb¹
En(σ) =
∫
ΩE(σ − C : εn−1
p
)dx−
∫
ΓD
(σn) · unD da .Æ$òa Ï
Ê2.AB=,À:72.AtÁÂ;>¸*©¨F=bÃIAià»6¼=56¼»6¼Õw;>:769¨F=Þª54¨FÄ5¸9AB» 4AU;FÁ58;F8«`¨F¸¼¸9¨UÙ8UõO¾.=.ÁÞ;»6¼=56¼»6¼ÕUAB4σ ∈ Σ
¨>«:72.AÁÂ;>¸3AB=.AB47ÉL¹ En(·) 8Â5Ä â A©B::¨':72.Aª*¨F6¼=b:nÙ 698A(©¨F=0ÃIAiàש¨F=.87:74;>6¼=b: σ(ξ) ∈ K
«`¨F4ξ ∈ Ξ
;>=.Á³:72.A¸¼6¼=.AU;>4t©¨F=.87:74;>6¼=0:Æ$òAe Ï ÈçkèδLÀu¿n´:µ W&¸é¶¶aùÔ°"úsÿ²Lú3²¶iúpÒi·`¬úp´^µÞ²IûFÔ#·¶¶a·Rþi´9°¶BúpÒ7°i¶¶¶iúR²Iún°
τn ∈ Σ°7±L·<¶Bús¶4Å·(ê°Aê Å
φ(τ n(ξ)
)< 0, ξ ∈ Ξ ,
Æ$òAÄL; Ï∫
Ωτ n : ε(δu) dx = `n(δu), δu ∈ V( × ) .
Æ$òAÄ>Ä Ï½ÿ°i®&ÅÑÒ$0þi´¼°BÔ Æ$òA_ Ï ÿ²>¶²¶F´ùúp·`F®
(σn,un, γn) ∈ Σ×V(unD)×ΛêëêFÒ°7F¯>°BÒ4Å
σn·¶úsÿ°ù®.·pýù5°
Ô#·®·Ô#·ìw°iÒúsÿ°ûFù5²F´O¬F®.¯>°7±ÖÔ#·s®.·sÔ·ìU²Iúp·pF®ÑÒ$0þi´¼°BÔS꾺¿:ÀuÀ$Á ¸é06¼=.©A En(·) 698Â5=56m«p¨F47»¸¼¹º©¨F=0ÃIAiàÇ;>=.Á×876¼=.©A τ k 698ª*¨F6¼=b:nÙ 698A6¼= K
;>=.Á×8a;>:7698$¾A8(Æ$òAe Ï À:72.A;FÁ»698876¼Ä5¸9A#8AB:f698f=.¨F:AB»ª5:y¹ê;>=.Áê:72.AB4Ai«`¨F4A;Â5=569ß0Â.A#8¨F¸¼Â5:769¨F=
σn¨>«:72.A6¼=.©B4AB»AB=b:;>¸ÁÂ;>¸
ª54¨FÄ5¸9AB»Aià6987:8wÈëê¨F4A¨UÃIAB4UÀL876¼=.©Aτ k69887:74769©B:7¸¼¹;FÁ»69886¼Ä5¸9AfÆs6RÈdALȼÀd:72.A é0¸<;>:AB4)©¨F=.Á6¼:769¨F="698«sÂ5¸m¾.¸¼¸9AÁ Ï À
[;>ÉL4;>=5ÉIAÝ»Â5¸¼:76¼ª5¸¼69AB48(un, γn) ∈ V(unD) × Λ
Ù 6¼:72γn ≥ 0
Aià6987:UÀ87Â.©2ç:72;>:(σn,un, γn) ∈
Σ×V(unD)×Λ698u;8;FÁ5Á¸9Aª*¨F6¼=b:t¨>«):72.A[;>ÉL4;>=5ÉIA«sÂ5=.©B:769¨F=;>¸
Ln(σ,u, γ) = En(σ) +
∫
Ωγ · φ(σ) dx−
∫
Ωσ : ε(u) dx +
∫
ΓD
(σn) · u da + `n(u) .
Ê2bÂ.8wÀ(σn,un, γn)
698u;#©B476¼:769©U;>¸,ª%¨F6¼=0:t¨>«):72.A[;>ÉL4;>=5ÉIA«pÂ5=.©B:769¨F=;>¸RõÙA2;ÃIA
DσLn(σ,u, γ)[δσ] =
∫
Ωσ : C−1 : δσ dx−
∫
Ωεn−1p : δσ dx
+
∫
Ωγ ·Dφ(σ) : δσ dx−
∫
Ωδσ : ε(u) dx ,
;>=.ÁDσLn(σn,un, γn)[δσ] = 0
«`¨F4t;>¸¼¸δσ ∈ Σ
698Aß0Â56¼ÃF;>¸9AB=0::¨
C−1 : σn(ξ)− εn−1p (ξ) + γn(ξ) ·Dφ(σn(ξ))− ε(un)(ξ) = 0, ξ ∈ Ξ .
ʨFÉIAB:72.AB4Ù 6¼:72Þ:72.A;FÁ»698876¼Ä56¼¸¼6¼:n¹6¼=K;>=.Á:72.AfAß0Â56¼¸¼6¼Ä5476¼Â5»ôAß0Â;>:769¨F=ºÆ$òAe Ï Àb:725698872.¨wÙ8:72;>:Æ$òA_ Ï
698Aß0Â56¼ÃF;>¸9AB=0::¨:72.AÊ¿n8¹87:AB» ¨>«):72.A6¼=.©B4AB»AB=0:;>¸ÁÂ;>¸,»6¼=56¼»6¼Õw;>:769¨F=Ū54¨FÄ5¸9AB»'È
QggQL?KQ*TDPgJM2*LOLKNgQTOLIgggM.gJKQL.KQ:KOLTMKP:SUTJLDTHO?PgMO&KNHKO& ä6äo Fís>§¡w$vt§p ç¦b£:v)¥¡A¡w§£% 0&ÕÌÈt¡w§s¦b§¡4Ò
?ºAºÁ698©BÂ.88Ý:n٨ةB¸<;F88A8Ũ>«=.¨F=5¸¼6¼=.AU;>48¨F¸¼Â5:769¨F=-»AB:72.¨Á58Ý«p¨F4:72.AÇÁ698©B4AB:Aº6¼=.©B4AB»AB=0:;>¸"ª54¨FÄ¿¸9AB»áÆ$òA_ Ï õ; Ï Ê2.AÖAßbÂ;>:769¨F=.8
(ñÌc
n);>=.Á
(1 1
n)Á5A8©B476¼Ä*A:72.AÖ¨F47:72.¨FÉI¨F=;>¸ª54¨ â A©B:769¨F=ø¨>«u;Ý87Â56¼:;>Ä5¸¼¹×Á5Ai¿
¾.=.AÁ³:7476<;>¸387:74A88¨F=0:¨':72.A;FÁ»698876¼Ä5¸9A(8AB:¨>« 87:74A88A8UÈäy=.8AB47:76¼=5É':725698ª54¨ â A©B:769¨F=Ç6¼=b:¨:72.AAß0Â56¼¸¼6¼Ä5476¼Â5» Aß0Â;>:769¨F=
(ß\ß
n)4A87Â5¸¼:86¼=;=.¨F=5¸¼6¼=.AU;>4fÃF;>476<;>:769¨F=;>¸ª54¨FÄ5¸9AB» «`¨F4t:72.A"Á6987ª5¸<;F©Ai¿
»AB=0:8Ù 2569©26988¨F¸¼ÃIAÁ#Äb¹;tÉIAB=.AB4;>¸¼6¼ÕUAÁ YuABÙ :¨F=»AB:72.¨ÁÈ?æ6¼:7256¼=ABÃIAB47¹6¼:AB4;>:769¨F=87:ABª,ÀL:72.A87:74A88;>=.ÁÅ:72.Aª5¸<;F87:769©"ª;>4;>»AB:AB4;>4A"Á5Ai¾.=.AÁêÄb¹Å:72.A¸9¨©U;>¸)8¨F¸¼Â5:769¨F=³¨>«
(ñ$c
n);>=.Á
(1 1
n)6¼=.Á5ABª*AB=.Á5AB=b:7¸¼¹Þ«`¨F4AU;F©2Å6¼=b:ABÉL4;>:769¨F=ݪ*¨F6¼=b:UÈÊ25698698:72.A8:;>=.Á.;>4Á4AB:7Â547=×;>¸¼ÉI¨F476¼:725» 6¼=º©¨F»ª5Â5:;>:769¨F=;>¸ª5¸<;F87:769©B6¼:y¹IÀ¨>«p:AB=4Ai«pAB474AÁ:¨';F8cf;FÁ6<;>¸0ctAB:7Â547='¨F4f13¸9¨L8A87:ïO¨F6¼=0:uï4¨ â A©B:769¨F=½Æp8AALÀ.ALÈÉ.ȼÀ,Ë Û ÎsÏ È
Ä Ï [6¼=.AU;>476¼Õ6¼=5Éf:72.Aü¨UÙÇ47Â5¸9A4A87Â5¸¼:86¼=:72.A>Ê¿n87¹87:AB» ¨>«.;ßbÂ;FÁ4;>:769©»6¼=56¼»6¼Õw;>:769¨F=ª54¨FÄ5¸9AB»'ÈÊ25698Á5Ai¾.=.A8:72.Até5îfï³»AB:72.¨0ÁÈ)Ê2.A6¼:AB4;>:A88¨F¸¼ÃIA:72.A AßbÂ56¼¸¼6¼Ä5476¼Â5»íAß0Â;>:769¨F=:¨FÉIAB:72.AB4Ù 6¼:72¸¼6¼=.AU;>476¼Õw;>:769¨F=.8
(ñ$c
n,k);>=.Á
(1 1
n,k)¨>«ü¨UÙí:72.A47Â5¸9A;>=.Á³:72.A©¨F»ª5¸9AB»AB=0:;>47¹×©¨F=.Á6¼:769¨F=,È
Ê2.A6¼:AB4;>:769¨F=ê87:¨Fª.8 6m«(ñ$c
n);>=.Á
(1 1
n);>4A8;>:76987¾AÁÞÂ5ª':¨87ÂV©B69AB=b:;F©©BÂ54;F©B¹IÈ
ïk´:µ·u¿nð ]:¸ña®Åúsÿ°¶nÑ5°7¬B·`²F´O¬²F¶°#ò.ó"Ñ´¼²>¶Búp·p¬B·púpµ÷·púsÿÖ¯dL®`ëÝ·¶°i¶)ö F÷çÒiù´¼°\iù0ÒBúsÿ°iÒ¯>²FÒa·`²F®ús¶²FÒ°Ñ>¶a¶a·Rþi´¼°f¶a·s®¬°fúsÿ°ÑÒ7U°7¬úp·`F®Ý¬²F®þ°°i¯d²L´^ù5²Lún°7û(ûF·Ò°7¬úp´µþiµÝÆ"Ä Ï êÕô´·Ô#·s®²Lúp·s®0Ó'Æ$òA_>Ä Ï ·s®ÇÆ$òA_ Ï µ>·R°i´¼û>¶
(ñ$c ′
n) σn(ξ) = C :(ε(un)(ξ)− εn−1
p (ξ)−max0, φ(σn)Dφ(σn(ξ))), ξ ∈ Ξ ,
Æ$òAÖL; Ï
(ß>ß ′
n)
∫
Ωσn : ε(δu) dx = `n(δu), δu ∈ V( × ) .
Æ$òAÖ>Ä Ï
½ÿ°²aÑIÑ%´^·`¬²Lúp·`F® Ų³Ó0°i®%°iÒ7²F´·ìd°7ûªõ"°i÷úRF®Ô°BúsÿUûØûF·Ò°7¬úp´µøúRlúsÿ°Ý¶aµ>¶Bún°iÔ Æ$òAÖ Ï ·<¶¶BúpùûF·R°7ûçþiµöOÿbÒi·<¶Bún°i®5¶°B®êËVÛ Î ê
o p ¢ w0yË¢d£&®2y0¦L¡w§s£ xv^yI¡wO£¤ ãAi¾.=56¼=5É:72.A:7476<;>¸8:74A88
θn(u) = C :(ε(u)− εn−1
p
) Æ`ó¬ Ï4ABÙ 476¼:A8"Æ$òA_L; Ï ;>=.ÁÇÆ$òA_>Ä Ï 6¼=:72.A«`¨F47»
0 = σn − θn(un)− γn · C : Dφ(σn),Æ`óòU; Ï
φ(σn) ≤ 0, γn · φ(σn)
= 0, γn ≥ 0 .Æ`óòBÄ Ï
[,AB»»;ÞòAB=.87Â54A8:72;>:AßbÂ;>:769¨F=ºÆ`óò Ï ©U;>=Ä*Au8¨F¸¼ÃIAÁÖ6¼=.Á5ABª*AB=.Á5AB=b:7¸¼¹(«p¨F4 ABÃIAB47¹(6¼=b:ABÉL4;>:769¨F='ª*¨F6¼=b:ξ ∈ Ξ
Á5Ai¾.=56¼=5É:72.Aª54¨ â A©B:769¨F=
σn = PK(θn(un)) .Æ`óLó Ï
än=.8AB47:76¼=5ÉÇÆ`ó¬ Ï ;>=.ÁçÆ`óLó Ï 6¼=0:¨Þ:72.AAßbÂ56¼¸¼6¼Ä5476¼Â5»áAßbÂ;>:769¨F=æÆ$òA_F© Ï ¹69AB¸9Á58:72.A#=.¨F=5¸¼6¼=.AU;>4ÃL;>476<;>:769¨F=;>¸ª54¨FÄ5¸9AB»ã«`¨F4:72.A6¼=.©B4AB»AB=b:;>¸ª54¨FÄ5¸9AB»6¼=ª5¸<;F87:769©B6¼:n¹*õ«p¨F4ÉL6¼ÃIAB=
εn−1p
ÀL¾.=.Áun ∈ V(unD)
8Â.©2:72;>:∫
ΩPK
(C : (ε(un)− εn−1
p ))
: ε(δu) dx = `n[δu], δu ∈ V( × ) .Æ`óW Ï
³>´:µ¶µ· _&¸§½ÿ°·®%¬BÒ7°BÔ°B®úR²F´°i®*°iÒpÓLµiù0®%¬úp·pL®²F´
I incrn (u) =
∫
ΩψK
(θn(u)
)dx− `n[u], u ∈ V(unD)
Æ`óC] Ï
·<¶3¬F®.¯>°7±êëêLÒ7°7L¯>°iÒÅ· ²¶BúpÒa·`¬úp´µ²bûFÔ#·<¶a¶a·Rþi´¼°¶BúpÒ7°a¶a¶τ n ∈ Σ
¶²Lúp·<¶aµF·s®0ÓÆ$òAÄ Ï °7±L·<¶Bús¶4Åúsÿ°:aù®¬úp·`F®²L´Æ`óC] Ï ÿ²>¶²Ô#·s®.·sÔ·ìw°BÒ
un ∈ V(unD)÷)ÿ0·p¬iÿÞ¶F´^¯F°a¶úsÿ°¯d²FÒi·p²Iúp·pF®%²F´O°7ýBù5²Lúp·`F®@Æ`óW Ï ê
Yu¨F:A:72;>:t6¼='ÉIAB=.AB4;>¸,:72.Aª5476¼»;>¸8¨F¸¼Â5:769¨F='698 =.¨F:Â5=569ß0Â.ALÈ
ädG H&I:JLKNM4OLKP*QSRKTUQ:TUJ:M¾º¿:ÀuÀ$Á ¸Ê2.A©¨F=0ÃIAià6¼:y¹'¨>« I incr
n (·) «p¨F¸¼¸9¨wÙ8 «p4¨F»÷[,AB»»;óÈëê¨F4A¨UÃIAB4UÀOÆ] Ï ÉL6¼ÃIA8
D(ψK
(θn(un)
))[δu] = DψK
(θn(un)
)[Dθn(un)[δu]
]
= e(PK(θn(un)),C : ε(δu)
)= PK(θn(un)) : ε(δu)
;>=.Á:72.AB4Ai«`¨F4A
DI incrn (u)[δu] =
∫
ΩPK
(C : (ε(u)− εn−1
p ))
: ε(δu) dx − `n[δu] .
än«.:72.A;F88Â5»ª5:769¨F=.8¨>«%Ê2.A¨F4AB»øWf;>4A8;>:7698$¾AÁÀI;u8¨F¸¼Â5:769¨F=(σn, γn,un)
¨>« Æ$òA_ Ï Aià6987:8wÈëê¨F4A¨UÃIAB4UÀÄb¹øÆ`óò Ï :72.Aª5476¼»;>¸8¨F¸¼Â5:769¨F=
un8¨F¸¼ÃIA8Aß0Â;>:769¨F=øÆ`óW Ï ÈÊ20Â.8UÀ
un698;(©B476¼:769©U;>¸ª%¨F6¼=0:¨>« I incr
n (·) À;>=.Á'876¼=.©A I incrn (·) 698©¨F=bÃIAiàÀ.6¼: 698u;»6¼=56¼»6¼ÕUAB4UÈ
?ºA3;>ª5ª5¸¼¹; ÉIAB=.AB4;>¸¼6¼ÕUAÁYuABÙ :¨F="6¼:AB4;>:769¨F=#:¨ ¾.=.Á";»6¼=56¼»6¼ÕUAB4¨>« I incrn (·) Ä0¹©¨F»ª5Â5:76¼=5É;u©B476¼:769©U;>¸ª*¨F6¼=b:¨>«
Fn = DI incrn
Æs6RÈALȼÀÄ0¹ê8¨F¸¼Ã6¼=5ÉÞ:72.AAßbÂ;>:769¨F=Fn(un) = 0
Ï Èñ¨F4";(ÉL6¼ÃIAB=×87:;>47:6¼:AB4;>:Aun,0 ∈ V(unD)
À¾.=.ÁÞ6¼=.©B4AB»AB=b:8∆un,k ∈ V( × )
;>=.ÁÁ.;>»ª56¼=5ɪ;>4;>»AB:AB48sn,k ∈ (0, 1]
Ù 6¼:72
0 ∈ Fn(un,k−1) + ∂Fn(un,k−1)∆un,k, un,k = un,k−1 + sn,k∆un,k, k = 1, 2, 3, ...87Â.©2':72;>::72.A8Aß0Â.AB=.©A I incr
n (un,k)698Á5A©B4AU;F86¼=5É.È
132.¨¨L876¼=5É:72.A©¨F=.86987:AB=b:u:;>=5ÉIAB=b:Cn,k ∈ ∂
(PK(θn(un,k−1))
)= ∂PK
(θn(un,k−1)
): C¸9AU;FÁ58:¨
;4AU;>¸¼6¼Õw;>:769¨F=Ũ>«):72.AYfABÙ :¨F=Ý87:ABª6¼=':72.Af«`¨F47»∫
Ωε(∆un,k) : Cn,k : ε(δu) dx = `n[δu] −
∫
ΩPK
(θn(un,k−1)
): ε(δu) dx .
Æ`ó_ Ï
Í Î&vÌyLÏÐ0qX$Èt¡w§s¦b§¡Ò ñ.¨F4φ(σ) = |dev(σ)| −K0
ÙA¨FÄ5:;>6¼=«p4¨F»áÆ"Ä Ï
∂PK(θ)[τ ] = τ − ∂max0,dev(θ)| −K0dev(θ) : dev(τ )
|dev(θ)|dev(θ)
|dev(θ)|
−max0, |dev(θ)| −K0( dev(τ )
|dev(θ)| −dev(θ) : dev(τ )
|dev(θ)|dev(θ)
|dev(θ)|2).
o Nq ¢ wuyÑù\Èî¢C. 0î:yÚriyD¡U£ øvyb¡w£¤3 𠸼:AB47=;>:76¼ÃIAB¸¼¹³¨F=.A©U;>=Ç©¨F=.8769Á5AB4";Ö=.¨F=5¸¼6¼=.AU;>46¼:AB4;d¿:769¨F=Ç«p¨F4":72.A«pÂ5¸¼¸pÊ¿n87¹87:AB» Æ$òA_ Ï ÀOÙ 2.AB4A(:72.A(6¼=.AßbÂ;>¸¼6¼:y¹
φ(σ) ≤ 06984ABª5¸<;F©AÁÇÄb¹º:72.A=.¨F=¿
87»¨¨F:72'AßbÂ;>¸¼6¼:y¹max
0, φ(σ)
= 0õ)¾.=.Á
(σn, γn,un) ∈ Σ×Λ×V(unD)Ù 6¼:72
Fn(σn, γn,un
)= 0 ,
Ù 2.AB4A
Fn(σ, γ,u
)=
σ − C :
(ε(u)− εn−1
p − γ ·Dφ(σ))
max
0, φ(σ)
∫Ω σ : ε[·] dx − `n[·]
.
𠩨F474A87ª%¨F=.Á6¼=5ÉÉIAB=.AB4;>¸¼6¼ÕUAÁYfABÙ :¨F=6¼:AB4;>:769¨F=
0 ∈ Fn(σn,k−1, γn,k−1,un,k−1
)+ ∂Fn
(σn,k−1, γn,k−1,un,k−1
)
∆σn,k
∆γn,k
∆un,k
©U;>=×Ä*AÁ5Ai¾.=.AÁÇÄb¹×;8Â56¼:;>Ä5¸9A(;F©B:76¼ÃIAÖ8AB:87:74;>:ABÉL¹ØÆsÂ.876¼=5ÉÅ:72.A:A©25=569ßbÂ.A8"ª54¨Fª%¨L8AÁº6¼=æË'l ÎsÏ À;>¸m¿:72.¨FÂ5ÉL2'6¼='ÉIAB=.AB4;>¸
∂Fn(σn,k−1, γn,k−1,un,k−1
) 698 =.¨F: 4ABÉLÂ5¸<;>4UÈÊ2.A";>¸¼ÉI¨F476¼:725»69©4AU;>¸¼6¼Õw;>:769¨F=ê2;F8t:72.A"8;>»A8:747Â.©B:7Â54A;F8t:72.Aª54¨ â A©B:769¨F=Ý»AB:72.¨ÁÝÙ 6¼:72Å:72.A8;>»A«`¨F47»Â5¸<;«p¨F4u:72.A©¨F=.876987:AB=0:u¸¼6¼=.AU;>476¼Õw;>:769¨F=º;>=.Áê;»¨Á6m¾AÁÝ476¼ÉL20:$¿R2;>=.Áê869Á5ALÈté06¼=.©A:725698;>ª5ª54¨I;F©26¼=ÉIAB=.AB4;>¸Á5¨A8=.¨F:«sÂ5¸m¾.¸¼¸.:72.A ©¨F»ª5¸9AB»AB=0:;>47¹(©¨F=.Á6¼:769¨F=.8fÆ$òA_>Ä Ï ÀIÙA©¨F=.8769Á5AB4:72.A©¨F474A8ª%¨F=.Á6¼=5Éé5îfïç»AB:72.¨ÁÈ
QggQL?KQ*TDPgJM2*LOLKNgQTOLIgggM.gJKQL.KQ:KOLTMKP:SUTJLDTHO?PgMO&KNHKO& ädo Û ¢ w0y©p§p 0y5¢>§.y0¤O¢>£È®Éyb¦L¡w§s£% úv^yI¡wO£¤ Ê2.Aé5îfï@»AB:72.¨Á'698u¨FÄ5:;>6¼=.AÁÝÄ0¹6¼=.8AB47:76¼=5É:72.A¸¼6¼=.AU;>476¼ÕUAÁ'©¨F»ª5¸9AB»AB=b:;>47¹'©¨F=.Á6¼:769¨F=.8t6¼=b:¨#:72.A[;>ÉL4;>=5ÉIAi¿dYfABÙ :¨F=Ý6¼:AB4;>:769¨F=,õ«p¨F4t;"ÉL6¼ÃIAB='6¼:AB4;>:A(σn−1, γn−1,un−1) ∈ Σ×Λ×V(unD)
Á5Ai¾.=.A:72.A¸¼6¼=.AU;>476¼ÕUAÁü¨wÙ-47Â5¸9A
φn,k(σ) = φ(σn,k−1) +Dφ(σn,k−1) : (σ − σn,k−1)Æ`óe Ï
;>=.Á'©¨F=.8769Á5AB4:72.A«p¨F¸¼¸9¨wÙ 6¼=5É87¹87:AB»'õ¾.=.Á(∆σn,∆γn,∆un) ∈ Σ×Λ×V( × )
Ù 6¼:72
0 = σn,k−1 − C :(ε(un,k−1)− εn−1
p − γn,k−1 ·Dφ(σn,k−1
)
+ ∆σn,k − C :(ε(∆un,k)−∆γn,k ·Dφ
(σn,k−1
)− γn,k−1 ·D2φ
(σn,k−1
): ∆σn,k
),
φn,k(σn,k
)≤ 0, γn,k · φn,k
(σn,k
)= 0, γn,k ≥ 0 ,
0 =
∫
Ωσn,k−1 : ε(δu) dx − `n(δu) +
∫
Ω∆σn : ε(δu) dx, δu ∈ V( × )
Ù 6¼:72(σn, γn,un) = (σn−1, γn−1,un−1) + (∆σn,∆γn,∆un)
ÈÊ25698t©U;>=Ä*A4ABÙ 476¼:7:AB=Å;F8
(ñ$c
n,k) 0 = σn,k − C :(ε(un,k)− εn−1
p − γn,k ·Dφ(σn,k−1
) Æ`óÄL; Ï
− γn,k−1 ·D2φ(σn,k−1
): (σn,k − σn,k−1)
),
(1 1
n,k) φn,k(σn,k
)≤ 0, γn,k · φn,k
(σn,k
)= 0, γn,k ≥ 0 ,
Æ`óÄ>Ä Ï
(ß>ß
n,k)
∫
Ωσn,k : ε(δu) dx = `n(δu), δu ∈ V( × ) .
Æ`óÄF© ÏÊ2.A87:;>=.Á.;>4ÁÝ4AB:7Â547=;>¸¼ÉI¨F476¼:725»©U;>='Ä*A";FÁ.;>ª5:AÁÝ:¨:725698t8¹87:AB» ;F8 «`¨F¸¼¸9¨UÙ8UÈté06¼=.©A
φ698u©¨F=bÃIAiàÀ
D2φ698ª*¨L876¼:76¼ÃIA8AB»6m¿nÁ5Ai¾.=56¼:AÖ;>=.Á³:72.AB4Ai«p¨F4A
Cn,k =(
id +γn,k−1 · C : D2φ(σn,k−1))−1
: C698
Ù3AB¸¼¸m¿nÁ5Ai¾.=.AÁÈän=0:74¨0ÁÂ.©B6¼=5É:72.A»¨0Á6m¾AÁ':7476<;>¸87:74A88
θn,k(u) = Cn,k :(ε(u)− εn−1
p + γn,k−1 ·D2φ(σn,k−1
): σn,k−1
)
;>=.ÁÂ.876¼=5ÉDφ(σn,k−1) = Dφn,k(σ
n,k)4ABÙ 476¼:A8"Æ`óÄL; Ï ;>=.ÁÇÆ`óÄ>Ä Ï 6¼=:72.A«`¨F47»
0 = σn,k − θn,k(un,k)− γn,k · Cn,k : Dφn,k(σn,k),
Æ`óÖL; Ï
φn,k(σn,k
)≤ 0, γn,k · φn,k
(σn,k
)= 0, γn,k ≥ 0 .
Æ`óÖ>Ä Ïð Éb;>6¼=,Àn[AB»»;ò"ÉLÂ;>4;>=0:AA8:72;>:AßbÂ;>:769¨F=ØÆ`óÖ Ï ©U;>=Ä*A"8¨F¸¼ÃIAÁ6¼=.Á5ABª*AB=.Á5AB=b:7¸¼¹Å«`¨F4ABÃIAB47¹Ý6¼=b:Ai¿ÉL4;>:769¨F='ª*¨F6¼=b:
ξ ∈ ΞÈ)?ºA2;ÃIA
σn,k = Pn,k(θn,k(un,k)
),
Æ"W¬ ÏÙ 2.AB4A
Pn,k : Sym(d) −→ Kn,k698 :72.A¨F47:72.¨FÉI¨F=;>¸Oª54¨ â A©B:769¨F=ê¨F=b:¨:72.A2;>¸m«¿n87ª;F©A
Kn,k = τ ∈ Sym(d) : φn,k(τ ) ≤ 0 ⊃ KÙ 6¼:72Ø4A87ª*A©B::¨³:72.AÝ6¼=5=.AB4ª54¨ÁÂ.©B:
en,k(σ, τ ) = σ : (Cn,k)−1 : τÈ@Ê2.AÅ;F88¨0©B6<;>:AÁçAB=.AB47ÉL¹
698Á5AB=.¨F:AÁøÄb¹En,k(σ) = 1
2en,k(σ,σ)À;>=.Ál:72.A©¨F474A87ª*¨F=.Á6¼=5Éש¨F=0ÃIAiàlª*¨F:AB=b:76<;>¸t698#ÉL6¼ÃIAB=øÄb¹
ψn,k(σ) = En,k(σ)−En,k(σ − Pn,k(σ))È
Yu¨wÙØ6¼=.8AB47:76¼=5ÉÆ"W¬ Ï 6¼=0:¨:72.AuAßbÂ56¼¸¼6¼Ä5476¼Â5»ôAßbÂ;>:769¨F=Æ`óÄF© Ï ¹69AB¸9Á583;=.¨F=5¸¼6¼=.AU;>4ÃL;>476<;>:769¨F=;>¸ª54¨FÄ5¸9AB»«p¨F4 :72.Aé5îfïØ87:ABª,õ«`¨F4ÉL6¼ÃIAB='6¼:AB4;>:A8
(σn,k−1, γn,k−1)¾.=.Á
un,k ∈ V(unD)87Â.©2':72;>:
∫
ΩPn,k
(θn,k(u
n,k))
: ε(δu) dx = `n[δu], δu ∈ V( × ) .Æ"Wò Ï
³>´:µ¶µ· e&¸§½ÿ° ÑÒi·Ô²F´O°i®*°iÒpÓLµºiù0®¬Uúp·pF®%²F´.FÒúsÿ°´^·s®%°7²FÒi·ìw°7ûÑÒ7U°7¬úp·`F®ê¶iún°nÑ
I incrn,k (u) =
∫
Ωψn,k
(θn,k(u)
)dx− `n[u], u ∈ V(unD)
Æ"WLó Ï·<¶"¬F®.¯>°7±êkëêFÒ°7F¯>°iÒÅ · ²(¶BúpÒa·`¬úp´µÖ²IûLÔ·¶¶i·`þB´¼°¶BúpÒ7°a¶a¶¶B²Iúp·<¶aµF·s®0ÓºÆ$òAÄ Ï °7±F·¶iús¶Åúsÿ°più0®%¬úp·pL®²F´uÆ"WLó Ïÿ²>¶²(Ô#·®.·sÔ#·ìd°iÒ
un,k ∈ V(unD)÷)ÿ0·p¬iÿÞ¶F´^¯F°a¶úsÿ°¯d²FÒi·p²Iúp·pF®%²F´O°7ýBù5²Lúp·`F®@Æ"Wò Ï ê
ä « H&I:JLKNM4OLKP*QSRKTUQ:TUJ:M¾º¿:ÀuÀ$Á ¸Ê2.Aß0Â;FÁ4;>:769©ÁÂ;>¸,AB=.AB47ÉL¹Ö«pÂ5=.©B:769¨F=;>¸,«`¨F4 :72.A¸¼6¼=.AU;>476¼ÕUAÁ'ª54¨ â A©B:769¨F=Å87:ABª698 ÉL6¼ÃIAB=ÝÄb¹
En,k(σ) =
∫
ΩEn,k
(σ − Cn,k :
(εn−1p − γn,k−1 ·D2φ(σn,k−1) : σn,k−1
))dx−
∫
ΓD
(σn) · unD da ,
;>=.Á#:72.Até5îfïº87:ABª©U;>=Ä*A3«p¨F47»"Â5¸<;>:AÁ;F8«p¨F¸¼¸9¨wÙ8UõO¾.=.Á;f»6¼=56¼»6¼ÕUAB4σn,k
¨>« En,k(·)8Â5Ä â A©B::¨f:72.A
©¨F=.87:74;>6¼=0:8σ(ξ) ∈ Kn,k
«`¨F4ξ ∈ Ξ
;>=.Á':72.A"AßbÂ56¼¸¼6¼Ä5476¼Â5»Aß0Â;>:769¨F=½Æ$òAe Ï Èté06¼=.©AK ⊂ Kn,k
À.:725698»6¼=56¼»6¼Õw;>:769¨F=ݪ54¨FÄ5¸9AB» 698;>¸98¨87:74769©B:7¸¼¹';FÁ»698876¼Ä5¸9ALÈÊ2.AB4Ai«p¨F4ALÀ5:72.AB4AAià6987:8;»6¼=56¼»6¼ÕUAB4
σn,k¨>«
En,k(·);>=.Á©[;>ÉL4;>=5ÉIAª;>4;>»AB:AB48
(γn,k,un,k)8Â.©2º:72;>:
(σn,k, γn,k,un,k)698;Þ8¨F¸¼Â5:769¨F=º¨>«3:72.A
©¨F474A87ª*¨F=.Á6¼=5ÉËÊ¿n8¹87:AB» Æ`óÄ Ï È'Ê20Â.8UÀun,k
8¨F¸¼ÃIA8(Æ"Wò Ï À);>=.Á½;>=;>¸9¨FÉI¨FÂ.87¸¼¹º:¨[,AB»»;X_Ù3A©U;>=Ý872.¨wÙ-:72;>:
un,k698t;#©B476¼:769©U;>¸ª*¨F6¼=b:f;>=.ÁÞ:72.AB4Ai«p¨F4A;»6¼=56¼»6¼ÕUAB4u¨>« I incr
n,k (·) Èëê¨F4A¨UÃIAB4UÀ86¼=.©A En,k(·)698 Â5=56m«`¨F47»¸¼¹©¨F=bÃIAiàÀ.:72.A»6¼=56¼»6¼ÕUAB4
σn,k698 Â5=569ß0Â.ALÈ
Ê2.Afé5îfï½»AB:72.¨Á698=.¨UÙ@4AU;>¸¼6¼ÕUAÁÄb¹:72.At¸¼6¼=.AU;>476¼ÕUAÁª54¨ â A©B:769¨F=;>¸¼ÉI¨F476¼:725»'õ«p¨F487Â56¼:;>Ä5¸9At87:;>47:76¼=5É6¼:AB4;>:A8
(σn,0, γn,0)À.©¨F»ª5Â5:A«`¨F4
k = 1, 2, 3, ...;»6¼=56¼»6¼ÕUAB4
un,k¨>«I incrn,k (·) ÈÊ2.AB=,À.8AB:
σn,k = Pn,k(θ(un,k)), γn,k =max
0, φn,k(θn,k(u
n,k))
Dφ(σn,k−1) : Cn,k : Dφ(σn,k−1);
Æ"WW Ï
:¨FÉIAB:72.AB4(σn,k,un,k, γn,k)
8¨F¸¼ÃIAÆ`óÄ Ï Èé06¼=.©A
un,k698;Ç©B476¼:769©U;>¸ª%¨F6¼=0:Þ¨>«
Fn,k = DI incrn,k
ÀtÙ3AêÂ.8A³;×ÉIAB=.AB4;>¸¼6¼ÕUAÁûYuABÙ :¨F=»AB:72.¨Áæ:¨8¨F¸¼ÃIA:72.A=.¨F=5¸¼6¼=.AU;>4ÃF;>476<;>:769¨F=;>¸3ª54¨FÄ5¸9AB» Æ"Wò Ï õ«p¨F4#;ÞÉL6¼ÃIAB=Ç8:;>47:6¼:AB4;>:A
un,k,0 ∈ V(unD)¾.=.Á
6¼=.©B4AB»AB=b:8∆un,k,m ∈ V( × )
;>=.Á'Á.;>»ª56¼=5ɪ;>4;>»AB:AB48sn,k,m ∈ (0, 1]
Ù 6¼:72
0 ∈ Fn,k(un,k,m−1) + ∂Fn,k(un,k,m−1)∆un,k,m ,
;>=.Áun,k,m = un,k,m−1 +sn,k,m∆un,k,m
«`¨F4m = 1, 2, 3, ...
8Â.©2:72;>::72.At8AßbÂ.AB=.©A I incrn,k (un,k,m)698Á5A©B4AU;F876¼=5É.È
Ê2.Aê4AU;>¸¼6¼Õw;>:769¨F=-¨>«":72.AXYfABÙ :¨F=8:ABª@4Aß0Â56¼4A8Þ:72.A©2.¨F69©A³¨>«;½©¨F=.87698:AB=b:Þ:;>=5ÉIAB=b:Cn,k,m ∈
∂(PK(θ(un,k,m−1))
)= ∂PK
(θ(un,k,m−1)
): Cn,k
;>=.Á4A87Â5¸¼:8 6¼=ê;¸¼6¼=.AU;>4ª54¨FÄ5¸9AB» ¨>«):72.Af«p¨F47»
∫
Ωε(∆un,k,m) : Cn,k,m : ε(δu) dx = `n[δu] −
∫
ΩPn,k
(θn,k(u
n,k,m−1))
: ε(δu) dx .Æ"WC] Ï
Í Î&vÌyLÏÐ0qX$Èt¡w§s¦b§¡Ò ñ.¨F4φ(σ) = |dev(σ)| −K0
ÙA¨FÄ5:;>6¼=«p4¨F»áÆ(a Ï
∂Pn,k(θ)[τ ] = τ − ∂max0,dev(θ)| −K0dev(θ) : dev(τ )
|dev(θ)|dev(θ)
|dev(θ)| .
íütwO£¢d¡'¦I£:v^u5¢F§tU£ 𠸼ÉI¨F476¼:725»69©U;>¸¼¸¼¹IÀ:72.A4AU;>¸¼6¼Õw;>:769¨F=l¨>«:72.Aª54¨ â A©B:769¨F=×»AB:72.¨ÁØÆ`ó_ Ï ;>=.Á:72.A4AU;>¸¼6¼Õw;>:769¨F=l¨>«:72.AÖé5îfï »AB:72.¨ÁºÃ6<;¸¼6¼=.AU;>476¼ÕUAÁǪ54¨ â A©B:769¨F=.8ÞÆ"WC] Ï 2;UÃIA:72.A(8;>»A87:747Â.©B:7Â54AÂ.876¼=5ÉÁ6 AB4AB=b:tª54¨ â A©B:769¨F=.8UÀ.:7476<;>¸87:74A88A8u;>=.Á'©¨F=.876987:AB=0::;>=5ÉIAB=b:8UÀ©i«$ÈÊ;>Ä5¸9AkWÈ1¨F=.©ABª5:7Â;>¸¼¸¼¹IÀ3Ä%¨F:72ç;>¸¼ÉI¨F476¼:725»8#«`¨F¸¼¸9¨UÙ ;êÁ6 *AB4AB=0:87:74;>:ABÉL¹IÈ×Ê2.AÖª54¨ â A©B:769¨F=ø»AB:72.¨0Á½6¼:AB4;>:A8;>¸9¨F=5É;FÁ»698876¼Ä5¸9A
σn,k = PK
(θn(un,k−1)
)∈ K
Â5=0:76¼¸*:72.AfAß0Â56¼¸¼6¼Ä5476¼Â5» 6984AU;F©2.AÁºÆp©i«7È)ñO6¼ÉLÂ54A#ò Ï ÀÙ 2.AB4AU;F8 :72.Aé5îfïø»AB:72.¨ÁÞ6¼:AB4;>:A8u;>¸9¨F=5É
σn,k,m = Pn,k(θn,k(u
n,k,m−1))∈ Kn,k
8;>:7698$«p¹6¼=5É:72.AAß0Â56¼¸¼6¼Ä5476¼Â5» Aß0Â;>:769¨F='Â5=b:76¼¸,:72.A";FÁ»698876¼Ä5¸9A8AB: 6984AU;F©2.AÁÇÆp©i«7Èñ6¼ÉLÂ54Aó Ï È
QggQL?KQ*TDPgJM2*LOLKNgQTOLIgggM.gJKQL.KQ:KOLTMKP:SUTJLDTHO?PgMO&KNHKO& ädÂ
V → Σ
K
u1 u2 u∗
σ1σ2
σ∗
∫Ω σ : ε[·] dx = `n[·]
ñ6¼ÉLÂ54AçòLõlän¸¼¸¼Â.87:74;>:769¨F= ¨>«":72.A³ª54¨ â A©B:769¨F=-»AB:72.¨Á@«p¨F4εp = 0
õǾ.=.Á ;lÁ6987ª5¸<;F©AB»AB=0:ÝÃIA©B:¨F4u ∈ V
8Â.©2':72;>::72.A¨F47:72.¨FÉI¨F=;>¸ª54¨ â A©B:769¨F=ŨF=b:¨K¨>«):72.A:7476<;>¸87:74A88
θ¹69AB¸9Á58 :72.A;FÁ»69886¼Ä5¸9A
87:74A88σ6¼=:72.Au6¼=b:AB48A©B:769¨F=Þ¨>«:72.Af©¨F=bÃIAià(8AB:
K;>=.Á:72.A;V=.Au8ª;F©At¨>«87:74A88387:;>:A88;>:7698$«p¹06¼=5É
:72.A'Aß0Â56¼¸¼6¼Ä5476¼Â5»è©¨F=.Á6¼:769¨F=åÆ«`¨F4:7256986¼¸¼¸¼Â.87:74;>:769¨F=ØÙ3A69Á5AB=0:76m«s¹l:72.A'Á6987ª5¸<;F©AB»AB=0:8uÙ 6¼:72ç:72.AB6¼4
;F88¨©B6<;>:AÁÞ:7476<;>¸87:74A88θ(u)Ï È
V → Σ
K
u1 u2 u∗
σ2
σ∗∂Kn,k
∫Ω σ : ε[·] dx = `n[·]
ñ6¼ÉLÂ54Aóõ)äy¸¼¸¼Â.87:74;>:769¨F=¨>«:72.Aé5îfïl»AB:72.¨0Á(«`¨F4εp = 0
õ)6¼=ÞABÃIAB47¹Öé5îfïø87:ABª,À0¾.=.ÁÖ;Á6987ª5¸<;F©AB»AB=0:un,k
87Â.©2:72;>:f:72.A#¨F47:72.¨FÉI¨F=;>¸ª54¨ â A©B:769¨F=º¨F=b:¨Ö:72.A2;>¸m«¿n8ª;F©AKn,k
Á5Ai¾.=.AÁÅÄ0¹Ý:72.A¸¼6¼=.AU;>476¼ÕUAÁ¸¼6¼=.AU;>476¼ÕUAÁü¨wÙ-47Â5¸9A
φn,k8;>:7698$¾A8 :72.AAßbÂ56¼¸¼6¼Ä5476¼Â5» AßbÂ;>:769¨F=,È
ädÊ H&I:JLKNM4OLKP*QSRKTUQ:TUJ:M
¦¢d£&®2y0¦L¡w§s£ Ôv^yI¡wO£¤ ù§p uyD5¢F§Dyb¤O¢>£&®2yb¦I¡w§£% xvyb¡w£¤þý£$¥¦ÿïp¬ Ï éAB:
ε0p = 0
Àn = 1
È é¬ Ï éAB:ε0p = σ0 = 0
Àγ0 = 0
Àn = 1
Èïtò Ï éAB:
un,0 = un−1 È é%ò Ï éAB:un,0 = un−1 À σn,0 = σn−1 À;>=.Á
γn,0 = γn−1Ê2.AB=,ÀF8AB::72.Apãf6¼4769©25¸9AB:Ä%¨FÂ5=.Á.;>47¹ÃL;>¸¼Â.A8 Ê2.AB=,À08AB:3:72.Aºãf6¼4769©25¸9AB:Ä*¨FÂ5=.Á.;>47¹ÃF;>¸¼Â.A8un,0(x) = unD(x)
Àx ∈ D È un,0(x) = unD(x)
Àx ∈ D ÈéAB:
k = 1È éAB:
k = 1È
éó Ï éAB:un,k,0 = un,k−1 ;>=.Á'8AB: m = 1
Èï3ó Ï 1¨F»ª5Â5:A:72.A:7476<;>¸O87:74A88UÀ éW Ï 1¨F»ª5Â5:A:72.A:7476<;>¸87:74A88UÀ:72.A¨F47:72.¨FÉI¨F=;>¸Oª54¨ â A©B:769¨F= :72.A¸¼6¼=.AU;>476¼ÕUAÁª54¨ â A©B:769¨F=
σn,k = PK(θn(un,k−1)) σn,k,m = Pn,k(θn,k(un,k,m−1));>=.Á
γk,nÀ5ABÃF;>¸¼Â;>:A:72.A4A8769ÁÂ;>¸ ;>=.Á
γk,n,mÀ5ABÃF;>¸¼Â;>:A:72.A4A8769ÁÂ;>¸
Rn,k[δu] = `n(δu)−∫
Ω σn,k : ε(δu) dx
ÈRn,k,m[δu] = `n(δu) −
∫Ω σ
n,k,m : ε(δu) dxÈ
ïpW Ï än« ‖Rn,k‖ 698 =.¨F:87»;>¸¼¸OAB=.¨FÂ5ÉL2,À é] Ï än« ‖Rn,k,m‖ 698 =.¨F:t87»;>¸¼¸OAB=.¨FÂ5ÉL2,ÀÉI¨:¨ï\_ Ï È ÉI¨#:¨(éÄ Ï Èé_ Ï éAB:
un,k = un,k,mÀσn,k = σn,k,m
À;>=.Áγn,k = γn,k,m
Èée Ï äR«:72.A=.¨F47» ¨>«σn,k−C : (ε(un,k)−εn−1
p −γn,k ·Dφ(σn,k))698=.¨F:8»;>¸¼¸,AB=.¨FÂ5ÉL2,À58AB:k := k+ 1
À.;>=.ÁÉI¨#:¨(éó Ï È
ï>] Ï éAB:un = un,k
Àσn = σn,k
Àγn = γn,k
À éÈa Ï éAB:un = un,k
Àσn = σn,k
Àγn = γn,k
À
εnp = εn−1p + γn ·Dφ(σn)
Èεnp = εn−1
p + γn ·Dφ(σn)È
éAB:n := n+ 1
;>=.ÁÉI¨#:¨ïtò Ï È éAB:n := n+ 1
;>=.ÁÉI¨#:¨(é%ò Ï Èïp_ Ï 1¨F»ª5Â5:A éÄ Ï 1¨F»ª5Â5:A
Cn,k ∈ ∂PK(θn(un,k−1)) : C Cn,k,m ∈ ∂Pn,k(θn,k(un,k,m−1)) : Cn,k;>=.Á'©¨F»ª5Â5:A
∆un,k87Â.©2':72;>: ;>=.Á'©¨F»ª5Â5:A
∆un,k,m87Â.©2':72;>:
∫Ω ε(∆un,k) : Cn,k : ε(δu) dx = Rn,k[δu]
È ∫Ωε(∆un,k,m) :Cn,k,m : ε(δu)dx = Rn,k,m[δu]
ÈãAB:AB47»6¼=.Aæ;8Â56¼:;>Ä5¸9AØÁ.;>»ª56¼=5É-«R;F©B:¨F4sn,k ∈ (0, 1]
À8AB:ãfAB:AB47»6¼=.A;Ú8Â56¼:;>Ä5¸9A Á.;>»ª56¼=5É«`;F©B:¨F4sn,k,m ∈ (0, 1]
À8AB:
un,k = un,k−1 + sn,k∆un,kÈ
un,k,m = un,k,m−1 +sn,k,m∆un,k,mÈ
éAB:k := k + 1
;>=.ÁÉI¨:¨ïó Ï È éAB:m := m+ 1
;>=.ÁÉI¨:¨éW Ï È
ÊO;>Ä5¸9AWõ1¨F»ª;>47698¨F=¨>«%:72.A ;>¸¼ÉI¨F476¼:725»69©87:ABª.8«`¨F4:72.A3ª54¨ â A©B:769¨F=»AB:72.¨Á;>=.Á#:72.Até5îfï³»AB:72.¨0ÁÂ.876¼=5É"¸¼6¼=.AU;>476¼ÕUAÁª54¨ â A©B:769¨F=.8UÈñ.¨F43:72.Afé5îfï½»AB:72.¨ÁÀb:72.Af;FÁ5Á6¼:769¨F=;>¸*8:ABªÞée Ï 6984AßbÂ56¼4AÁ(:¨©¨F=¿:74¨F¸b:72.A©¨F=bÃIAB47ÉIAB=.©A¨>«5:72.Aü¨UÙÇ47Â5¸9ALÈ)Ðu=:72.A¨F:72.AB42;>=.ÁÀ>:72.A=.¨F=5¸¼6¼=.AU;>4Oß0Â;FÁ4;>:769©»6¼=56¼»6¼Õw;>:769¨F=ª54¨FÄ5¸9AB»8¨>«:72.Aé5îfïØ87:ABª.8;>4A876¼»ª5¸9AB4 :72;>=:72.Au«sÂ5¸¼¸¼¹(=.¨F=5¸¼6¼=.AU;>4 ª54¨FÄ5¸9AB» 6¼=Þ6¼=.©B4AB»AB=b:;>¸,ª5¸<;F8$¿:769©B6¼:n¹IÈlÊ2.AÞ4AU;>¸¼6¼Õw;>:769¨F=@¨>«;ê87ª*A©B6m¾©Þ»¨0Á5AB¸ 4Aß0Â56¼4A8¨F=5¸¼¹º«`¨F47»Â5¸<;FA«p¨F4:72.AºÆs¸¼6¼=.AU;>476¼ÕUAÁ Ï :7476<;>¸87:74A88wÀª54¨ â A©B:769¨F=Å;>=.ÁÝ©¨F=.876987:AB=b:t:;>=5ÉIAB=b:Æp8AAÊO;>Ä5¸9A]«p¨F4bóª5¸<;F8:769©B6¼:n¹ Ï È
QggQL?KQ*TDPgJM2*LOLKNgQTOLIgggM.gJKQL.KQ:KOLTMKP:SUTJLDTHO?PgMO&KNHKO& äÓ
¦¢>£È®Éyb¦L¡w§s£ úvyb¡w£¤pÏ ©¨F»ª5Â5:Aθn,k,ηn,k, γn,k,σn,k,Cn,k
Ù 6¼:72'¾5àAÁC, PK
θn,k = θn(un,k−1) = C :(ε(un,k−1)− εn−1
p
)
ηn,k =dev(θn,k)
|dev(θn,k)|γn,k = max0, |dev(θn,k)| −K0σn,k = PK(θn,k) = θn,k − 2µγn,kηn,k
Cn,k = C− 2µ sgn(γn,k)ηn,k ⊗ ηn,k − 2µγn,k
|dev(θn,k)|(
dev−ηn,k ⊗ ηn,k)
£u¥§¦÷vyb¡w£¤pÏ ©¨F»ª5Â5:Aθn,k,m, γn,k,m,σn,k,m,Cn,k,m
Ù 6¼:72'»¨0Á6m¾AÁηn,k,Cn,k, Pn,k
ηn,k =dev(σn,k−1)
|dev(σn,k−1)|
Cn,k = C− 2µ2µγn,k−1
2µγn,k−1 + |dev(σn,k−1)|(
dev−ηn,k ⊗ ηn,k)
θn,k,m = θn,k(un,k,m−1) = Cn,k :
(ε(un,k,m−1)− εn−1
p
)
γn,k,m = max0, φ(θn,k,m) + ηn,k : θn,k,mσn,k,m = Pn,k(θ
n,k,m) = θn,k,m − 2µγn,k,mηn,k
Cn,k,m = Cn,k − 2µ sgn(γn,k,m)ηn,k ⊗ ηn,kÊO;>Ä5¸9A\].õ1¨F¸¼¸9A©B:769¨F="¨>«5:72.A«p¨F47»"Â5¸<;FA«`¨F4,:72.AABÃL;>¸¼Â;>:769¨F="¨>«,Æs¸¼6¼=.AU;>476¼ÕUAÁ Ï :7476<;>¸b8:74A88UÀUª54¨ â A©B:769¨F=";>=.Á©¨F=.876987:AB=0:u:;>=5ÉIAB=0:t«`¨F4bóª5¸<;F87:769©B6¼:n¹IÈhuAB4AÙ3A"Â.8A:72.A876¼ÉL=bÂ5» «pÂ5=.©B:769¨F=ÅÁ5Ai¾.=.AÁÅÄb¹
sgn(γ) = 0«p¨F4γ = 0
;>=.Ásgn(γ) = 1
«`¨F4γ > 0
È
srÝ¥uv^yb¢F§¦D&\yDÎ>yb¢F§=vy0 %¡?ºA87:7Â.Á¹ê:72.Aª*AB4$«p¨F47»;>=.©A¨>«:72.A:yÙ3¨Ý;>¸¼ÉI¨F476¼:725»8«p¨F4;ÖÄ*AB=.©25»;>47ܳª54¨FÄ5¸9AB»á6¼=³6¼=¾.=56¼:A876¼»;>¸ª%AB4$«`A©B:ª5¸<;F87:769©B6¼:y¹øÆp8AAË ÎsÏ ÈÊ2.A©¨F»ª5Â5:;>:769¨F=.8;>4A4AU;>¸¼6¼ÕUAÁ³6¼=³:72.A#¾.=56¼:AAB¸9AB»AB=0:"©¨Á5AëË l Î 87Â5ª5ª%¨F47:76¼=5ɪ;>4;>¸¼¸9AB¸»Â5¸¼:76¼ÉL4769ÁÝ»AB:72.¨0Á58wÈñ.¨F4 :72.AÄ*AB=.©25»;>47ÜÞª54¨FÄ5¸9AB» Ù3AÂ.8Au:72.Af«`¨F¸¼¸9¨UÙ 6¼=5É»;>:AB476<;>¸,ª;>4;>»AB:AB48Uõ:72.AAB¸<;F87:769©ª54¨Fª*AB47:769A8;>4AÁ5AB:AB47»6¼=.AÁ³Äb¹ê:72.A#ïO¨F6988¨F=º4;>:769¨
ν = 0.29;>=.Á¨FÂ5=5É'»¨0ÁÂ5¸¼Â.8
E = 206900.00[N/mm2]Á5Ai¾.=56¼=5É:72.A§[;>»S±#ª;>4;>»AB:AB48µ = E/(2(1 + ν))
;>=.Áλ = Eν/((1 + ν)(1 − 2ν))
È"ï¸<;F87:769©B6¼:n¹698tÁ5AB:AB47»6¼=.AÁÅÄb¹Þ:72.A¹69AB¸9ÁÝ87:74A88
K0 = 450.00[N/mm2]ÈÞA¨F»AB:747¹';>=.Á'Ä*¨FÂ5=.Á.;>47¹©¨F=.Á6¼:769¨F=.8
;>4A6¼¸¼¸¼Â.87:74;>:AÁ6¼=ÅñO6¼ÉLÂ54AWõ:72.A4Ai«`AB4AB=.©AÁ5¨F»;>6¼=Ω = (0, 10) × (0, 10) \ B1(10, 0)
Á5A8©B476¼Ä%A8f;ßbÂ;>47:AB4¨>«O;4A©B:;>=5ÉL¸9AuÙ 6¼:72;©B6¼4©BÂ5¸<;>42.¨F¸9ALÈÊ2.A¼ãf6¼4769©25¸9AB: Á.;>:;";>476986¼=5É"Ä0¹87¹»»AB:747¹;>4AtÉL6¼ÃIAB=Äb¹
u1(10, x2) = 0, x2 ∈ (1, 10), u2(x1, 0) = 0, x1 ∈ (0, 9) ,
;>=.Á;¸9¨I;FÁ"«pÂ5=.©B:769¨F=;>¸Á5ABª*AB=.Á6¼=5ɸ¼6¼=.AU;>47¸¼¹"¨F=t ≥ 0
698ÉL6¼ÃIAB=#Äb¹`(t,v) = 100 t
∫ 10
0v(x1, 10) dx1
È?æ6¼:7256¼=Þ;ª5¸<;>=.Au87:74;>6¼=Ö;F88Â5»ª5:769¨F=,À0:72.AuÁ6987ª5¸<;F©AB»AB=b:8 ;>4AuAB»Ä*AÁ5Á5AÁ6¼=0:¨ÃWd¿nÁÄb¹:72.AtAià:AB=.8769¨F=u3 ≡ 0
;>=.ÁÅ;>¸¼¸,»;>:AB476<;>¸©¨F»ª5Â5:;>:769¨F=.8;>4Aª%AB4$«`¨F47»AÁ6¼=`Wd¿nÁÈñ6¼487:UÀÙ3A'87:7Â.Á¹l:72.A=.¨F=5¸¼6¼=.AU;>4Ö©¨F=bÃIAB47ÉIAB=.©AêÄ*AB2;Ã69¨F4«p¨F4Ö;³876¼=5ÉL¸9A:76¼»AÅ87:ABª ÆpÊO;>Ä5¸9AË_ Ï Èç?ºA¨FÄ.8AB47ÃIA#:72;>::72.A87:;>=.Á.;>4Áª54¨ â A©B:769¨F=³»AB:72.¨0Á³©¨F=0ÃIAB47ÉIA8ß0Â;FÁ4;>:769©U;>¸¼¸¼¹Å6¼=:72.A¾.=;>¸6¼:AB4;>:769¨F=.8Ä5Â5:":72.A(6¼:AB4;>:769¨F=ø87:;>47:8#ßbÂ56¼:AÖ87¸9¨wÙ 876¼=.©A(ÉL¸9¨FÄ;>¸ ©¨F=bÃIAB47ÉIAB=.©AÞ4Aß0Â56¼4A8#87:74¨F=5ÉêÁ.;>»ª56¼=5É.ÈÊ2.Aé5îfï³»AB:72.¨Á#;>¸98¨f82.¨UÙ8ß0Â;FÁ4;>:769© ©¨F=bÃIAB47ÉIAB=.©AÄ5Â5:)2.AB4A:72.AÉL¸9¨FÄ;>¸5©¨F=bÃIAB47ÉIAB=.©Atª54¨Fª%AB47:769A8;>4AÄ%AB:7:AB4UÈ ð Éb;>6¼=,À:72.AÉIAB=.AB4;>¸¼6¼ÕUAÁ°YfABÙ :¨F=Þ»AB:72.¨0ÁÖ«`¨F4:72.Au¸¼6¼=.AU;>476¼ÕUAÁ'ª54¨ â A©B:769¨F=»AB:72.¨ÁÞ6¼=ABÃIAB47¹é5îfïÇ8:ABª©¨F=bÃIAB47ÉIA8ß0Â;FÁ4;>:769©U;>¸¼¸¼¹6¼=:72.A ¾.=;>¸.6¼:AB4;>:769¨F=.8wÀ;>=.ÁÖ;>Éb;>6¼=ÉL¸9¨FÄ;>¸©¨F=0ÃIAB47ÉIAB=.©A«`¨F4:72.AßbÂ;FÁ4;>:769©(87Â5Ä¿Rª54¨FÄ5¸9AB»8"4AßbÂ56¼4A887:74¨F=5ÉÝÁ.;>»ª56¼=5ÉêÄ5Â5:¸9A88"Á.;>»ª56¼=5ÉÝ:72;>=×:72.A«sÂ5¸¼¸ª54¨ â A©B:769¨F=»AB:72.¨0ÁÈÐt«O©¨FÂ548ALÀ:72.Af=.¨F=5¸¼6¼=.AU;>4 ª54¨FÄ5¸9AB»8 6¼=Þ:72.A87Â5Ä.87:ABª.8 Á5¨=.¨F:=.AAÁ(:¨Ä*A8¨F¸¼ÃIAÁÞAià5;F©B:7¸¼¹IÈ
ädÝ H&I:JLKNM4OLKP*QSRKTUQ:TUJ:M
ñ6¼ÉLÂ54AWõÌÞA¨F»AB:74769© ©¨F=¾.ÉLÂ54;>:769¨F=Ö;>=.Á"Ä*¨FÂ5=.Á.;>47¹"©¨F=.Á6¼:769¨F=.8uÆs¸9Ai«s: Ï ÀLÁ5Ai«`¨F47»AÁ©¨F=¾.ÉLÂ54;>:769¨F=«`¨F4t = 4.7
Æp©AB=b:AB4 Ï ÀO;>=.ÁÁ6987:7476¼Ä5Â5:769¨F=º¨>«:72.A#AßbÂ56¼ÃL;>¸9AB=b:ª5¸<;F87:769©8:74;>6¼=øÆs476¼ÉL20: Ï «p¨F4:72.AÄ*AB=.©25»;>47ܪ54¨FÄ5¸9AB»'È
äy=;>=Þ6¼=.Aià.;F©B:té5îfïøÃF;>476<;>=0:UÀ:72.A;F©©BÂ54;F©B¹(4Aß0Â56¼4AB»AB=b:8«`¨F4:72.Af¸¼6¼=.AU;>476¼ÕUAÁª54¨ â A©B:769¨F='8:ABª.8©U;>=Ä*A4AÁÂ.©AÁ'Ù 6¼:72.¨FÂ5:u;A©B:76¼=5É:72.A¨wÃIAB4;>¸¼¸)é5îfïØ©¨F=0ÃIAB47ÉIAB=.©ALÈÊ2.A(=.Aià:#ÊO;>Ä5¸9AºÆpÊ;>Ä5¸9AZe Ï 87:7Â.Á69A8:72.AÞÁ5ABª%AB=.Á5AB=.©AÞ¨F=½:72.A(Á698©B4AB:76¼Õw;>:769¨F=ø¸9ABÃIAB¸RÈêäy=ø;>¸¼¸©U;F8A8wÀ:72.AÞÉIAB=.AB4;>¸¼6¼ÕUAÁxYfABÙ :¨F=ø»AB:72.¨0Á58«p¨F4:72.AÞ=.¨F=5¸¼6¼=.AU;>4ÃL;>476<;>:769¨F=;>¸ª54¨FÄ5¸9AB»8ÝÆ`óW Ï ;>=.Á-Æ"Wò Ï ;>4AÉL¸9¨FÄ;>¸¼¸¼¹Ø©¨F=0ÃIAB47ÉIAB=b:ºÆsª54¨UÃ69Á5AÁ;º87Â56¼:;>Ä5¸9AêÁ.;>»ª56¼=5ɽ8:74;>:ABÉL¹ø698Ö;>ª5ª5¸¼69AÁ Ï Ä5Â5:(:72.AÝ=bÂ5»"Ä%AB4¨>«4Aß0Â56¼4AÁ³87:ABª.8:¨¨FÄ5:;>6¼=×;Öª54A8©B476¼Ä*AÁº;F©©BÂ54;F©B¹6986¼=.©B4AU;F876¼=5É'Ù 6¼:72³:72.A#=bÂ5»"Ä%AB4"¨>«3Â5=5Ü=.¨UÙ =.8UÈ[,¨©U;>¸¼¸¼¹IÀLÙ3A¨FÄ.8AB47ÃIAß0Â;FÁ4;>:769©©¨F=bÃIAB47ÉIAB=.©A©B¸9¨L8A3:¨f:72.A38¨F¸¼Â5:769¨F=,ÈÊ25698)698O:72.AAiàª*A©B:AÁÄ%AB2;UÃ069¨F46m«:72.A;F©B:76¼ÃIA8AB:¨>«:72.A©¨F=bÃIAi೪*¨F6¼=b:nÙ 698A©¨F=.87:74;>6¼=0:698Æs»¨F4A¨F4¸9A88 Ï 69Á5AB=b:76m¾AÁÈ ð ¸¼:¨FÉIAB:72.AB4UÀ:725698=bÂ5»AB4769©U;>¸ Aiàª%AB476¼»AB=0:4AiüA©B:8:72.A«`;F©B::72;>::72.Aª5476¼»;>¸ Á6987ª5¸<;F©AB»AB=0:;>ª5ª54¨Uà6¼»;>:769¨F=ø6¼=H1 698"=.¨F:"ÙAB¸¼¸m¿Rª%¨L8AÁõ87:;>Ä5¸9A(;ݪ54769¨F476A8:76¼»;>:A8#;>4AÞ;ÃL;>6¼¸<;>Ä5¸9AÞ¨F=5¸¼¹³6¼=Ç:72.AÖ87ª;F©AÖ¨>«Ä%¨FÂ5=.Á5AÁÁ5Ai«`¨F47»;>:769¨F=.8UÀ.4A87Â5¸¼:76¼=5É6¼=»A872¿nÁ5ABª%AB=.Á5AB=0:fÁ698©B4AB:AA87:76¼»;>:A8Ë ÈqÎ ÈÐu=:72.Aº¨F:72.AB4'2;>=.ÁÀ:72.A×é5îfï »AB:72.¨0ÁÆ`;>=.Á6¼:86¼=.Aià.;F©B:ÃF;>476<;>=0: Ï 872.¨wÙ8»A872¿R6¼=.Á5ABª*AB=.Á5AB=b:ß0Â;FÁ4;>:769©Ö©¨F=0ÃIAB47ÉIAB=.©ALÈ ð ¸¼:72.¨FÂ5ÉL2l=.¨Ý«p¨F47»;>¸ª54¨¨>«t698ÉL6¼ÃIAB=½«`¨F4":725698¨FÄ.8AB47ÃL;>:769¨F=,À:725698698:72.AAiàª%A©B:AÁÝÄ%AB2;UÃ069¨F4876¼=.©A:72.A:72.A"ÁÂ;>¸87:74A88u;>ª5ª54¨Uà6¼»;>:769¨F=6¼=
L2698uÙ3AB¸¼¸m¿Rª*¨L8AÁê;>=.ÁÝ:72.AB4Ai«p¨F4A
»A872¿R6¼=.Á5ABª*AB=.Á5AB=b:;ª54769¨F476,Ä*¨FÂ5=.Á58u;>4A;ÃL;>6¼¸<;>Ä5¸9ALÈ
ï4¨ â A©B:769¨F=Å»AB:72.¨0Á é5îfïæ»AB:72.¨0Á 6¼=.Aià.;F©B:fé5îfïç»AB:72.¨ÁÆpñÌc Ï ÆR1 1 Ï Æ=ß>ß Ï ÆpñÌc Ï ÆR1 1 Ï Æ=ß>ß Ï ÆpñÌc Ï ÆR1 1 Ï Æ=ß>ß Ïε ε
óC].È a?e_eLó òLòA_ȼòUó_¬Ló _È'¬ÖWòA¬Öε
òLòA_ȼòLò2]D¬e _È'¬ÖW¬òAe ¬È¼ò2]]DeÖÄε ε
òaÈ'Ä.a?ÄLóe ]DWÈ'¬W¬_¬ ¬È a?Ö.aFóC]DWε
]DWÈ'¬LóeÖÄ ¬È a?Ö.a?¬__ ¬È'¬C]DĬòLòε ε
òAWȼòa?ÖÖò ¬È a?ÄC]D_C] ¬È'¬.a]DÖÄWε
¬È a?Ä_òA_ ¬È'¬.a]DÖ.a?_ ¬È'¬LóLóLóÖC]ε ε
ÖÈ'eeÄLó_ ¬È'¬¬LóÖC] ¬È'¬¬¬¬.a?Wε
¬È'¬¬LóÖW ¬È'¬¬¬¬.aFó ¬È'¬¬¬¬òUóε ε
eÈ'WĬ_Äε ε ε ε ε ε
ε ε].È'WÄW.aFó
ε εWÈ'WLó_C]DÖ
tn = 4.6À∆tn = 0.3
ε ε¬È'eeÄC]bó
ε < 10−10
ε ε¬È'¬¬òAÄÄ òAeWÄC]ß0Â;FÁ476¼¸<;>:AB4;>¸O©AB¸¼¸98wÀÄ56¼¸¼6¼=.AU;>4¾.=56¼:AAB¸9AB»AB=0:8
ε ε ε
ÊO;>Ä5¸9A_õ)1¨F»ª;>47698¨F=¨>«:72.A©¨F=bÃIAB47ÉIAB=.©A Ä*AB2;Ã69¨F4¨>«:72.Aª54¨ â A©B:769¨F=#»AB:72.¨ÁÀL:72.Aé5îfï»AB:72.¨0ÁÀ;>=.Á";>="6¼=.Aià5;F©B:OÃIAB48769¨F="¨>«5:72.Aé5îfïÝ»AB:72.¨Á«`¨F4¨F=.A:76¼»A387:ABª,Èñ.¨F4:72.A=bÂ5»AB4769©U;>¸:A87:UÀw:72.A3Á5Ai«pA©B:¨>«*:72.AtAß0Â;>:769¨F=.8uÆ$òA_L; Ï ÀÆ$òA_>Ä Ï ÀÆ$òA_F© Ï 6¼=:72.AßÂ5ܸ¼69Á6<;>==.¨F47»Ú6984AÁÂ.©AÁ:¨
ε < 10−10 ÈÌYu¨F:A :72;>::725698 698«R;>4 »¨F4A;F©©BÂ54;>:A:72;>=':72.AÁ698©B4AB:76¼Õw;>:769¨F=êAB474¨F4t¨>«:72.Af¾.=56¼:AAB¸9AB»AB=b:f;>ª5ª54¨Uà6¼»;>:769¨F=,È
QggQL?KQ*TDPgJM2*LOLKNgQTOLIgggM.gJKQL.KQ:KOLTMKP:SUTJLDTHO?PgMO&KNHKO& ädá©AB¸¼¸98 Á5¨>« ï4¨ â A©B:769¨F=Å»AB:72.¨0Á é5îfïæ»AB:72.¨0Á 6¼=.Aià.;F©B:fé5îfïç»AB:72.¨Á
]D¬Öe ÄC]D_¬ Ö§YfABÙ :¨F=Ý87:ABª.8 _é5îfïæ87:ABª.8Æ$òAÄ8Â5Ä.87:ABª.8 Ï _é5îfïæ87:ABª.8Æ$òA¬#87Â5Ä,È ÏòAeWÄC] WWLóÄLó òA¬§YfABÙ :¨F=Ý87:ABª.8 _é5îfïæ87:ABª.8Æ`óò8Â5Ä.87:ABª.8 Ï _é5îfïæ87:ABª.8Æ$ò2]87Â5Ä,È Ïe__We òAWLó¬ÖÄ òaYfABÙ :¨F=Ý87:ABª.8 _é5îfïæ87:ABª.8Æ`óW8Â5Ä.87:ABª.8 Ï _é5îfïæ87:ABª.8Æ$òAe#87Â5Ä,È ÏóeLóò2]] _LóeWWÄ WòYfABÙ :¨F=Ý87:ABª.8 _é5îfïæ87:ABª.8Æ"Wò8Â5Ä.87:ABª.8 Ï _é5îfïæ87:ABª.8Æ`óLó#87Â5Ä,È ÏòA¬C]DÄ_.a?e óòA¬òUó_¬ _e§YfABÙ :¨F=Ý87:ABª.8 _é5îfïæ87:ABª.8Æ]bó8Â5Ä.87:ABª.8 Ï _é5îfïæ87:ABª.8Æ"Wò87Â5Ä,È Ï
ÊO;>Ä5¸9AËeõän:AB4;>:769¨F=Ø©¨FÂ5=0:(Á5ABª*AB=.Á6¼=5ÉרF=ø:72.A'Á698©B4AB:76¼Õw;>:769¨F=æ¸9ABÃIAB¸ «`¨F4:72.Aª54¨ â A©B:769¨F=Ø»AB:72.¨0ÁÀ:72.Aé5îfï»AB:72.¨0ÁÀ;>=.Á³;>=6¼=.Aià.;F©B:fÃIAB4869¨F=¨>«3:72.Aé5îfï-»AB:72.¨0ÁÅ«`¨F4¨F=.A:76¼»A#87:ABªº;>:
tn = 4.6;>=.Á∆tn = 0.3
ÈÇÊ2.AÞ=0Â5»Ä*AB4¨>«f87Â5Ä.87:ABª.8©¨FÂ5=0:8:72.AÞ=bÂ5»"Ä%AB4¨>«fÉIAB=.AB4;>¸¼6¼ÕUAÁxYfABÙ :¨F=ç87:ABª.8«p¨F4f;>¸¼¸)é5îfï87:ABª.8UÈé06¼=.©A":72.A=bÂ5»AB4769©U;>¸)Aiàª%AB=.8A698tÁ5¨F»6¼=;>:AÁÅÄb¹:72.A¸¼6¼=.AU;>4f8¨F¸¼ÃIAB4uÙ 6¼:7256¼=Ý:72.AÉIAB=.AB4;>¸¼6¼ÕUAÁ§YuABÙ :¨F=87:ABª,ÀF:72.A¨wÃIAB4;>¸¼¸©¨F»ª5Â5:76¼=5É:76¼»A698ª54¨Fª*¨F47:769¨F=;>¸5:¨t:72.A=bÂ5»"Ä%AB4¨>«*YuABÙ :¨F=87:ABª.8¨F4 :¨F:;>¸,=0Â5»Ä*AB4u¨>«87Â5Ä.87:ABª.8 6¼=Ý:72.A"é5îfïç6¼:AB4;>:769¨F=,À4A87ª%A©B:76¼ÃIAB¸¼¹IÈ
£ O¦Dp¥0td§£% 𠸼:72.¨FÂ5ÉL2Ç©¨F=bÃIAiສ54¨FÉL4;>»»6¼=5Éêª54¨UÃ69Á5A8";Ä%AB:7:AB4"Â5=.Á5AB487:;>=.Á6¼=5Éê¨>« ©¨F»ª5Â5:;d¿:769¨F=;>¸ª5¸<;F87:769©B6¼:n¹IÀO6¼=³»;>=b¹³;>ª5ª5¸¼69©U;>:769¨F=.8:72.AB4A698=.¨Þ=.AAÁê«p¨F4=.ABÙÚ8¨F¸¼Â5:769¨F=³»AB:72.¨Á58Af:72.A#Ù3AB¸¼¸m¿A87:;>Ä5¸¼69872.AÁª54¨ â A©B:769¨F=º»AB:72.¨ÁÅ698¨>«p:AB=º87Â5ª*AB4769¨F4:¨Þ¨F:72.AB4»AB:72.¨0Á58UÈÐu=³:72.A#¨F:72.AB42;>=.ÁÀ,:725698»AB:72.¨0Ál698#=.¨F:(87:;>Ä5¸9ALÀ3;>=.Áø©B¸9¨L8AÞ:¨:72.AÞ¸¼6¼»6¼:¸9¨I;FÁ-ÆpABÃIAB=ç«p¨F4#4ABÉLÂ5¸<;>476¼ÕUAÁø»¨Á5AB¸98Ù 6¼:72ç87»;>¸¼¸4ABÉLÂ5¸<;>476¼Õw;>:769¨F=Ū;>4;>»AB:AB48 Ï :72.A»AB:72.¨Á»;¹Á5AB:AB4769¨F4;>:ALÈÊ2.AÞ»; â ¨F4;FÁÃL;>=b:;>ÉIAݨ>«é5îfïã»AB:72.¨Á58698#:72.AÞ¸<;>47ÉIAÖüAià6¼Ä56¼¸¼6¼:n¹-ÆpALÈÉ.ȼÀÂ.876¼=5ɺ;>Â5ÉL»AB=0:AÁÔ[;d¿ÉL4;>=5ÉL6<;>=.8 Ï Ù 2569©2º;>¸¼¸9¨UÙ8f:¨AB=52;>=.©A#;>=.Áê87:;>Ä56¼¸¼6¼ÕUA:72.A»AB:72.¨0ÁÅ:¨«R;>4t»¨F4A"ÉIAB=.AB4;>¸ Æ`;>=.ÁÅABÃIAB==.¨F=.©¨F=bÃIAià Ï ª5¸<;F87:769©B6¼:n¹(»¨0Á5AB¸98wÀb6¼=Öª;>47:769©BÂ5¸<;>4 :¨»"Â5¸¼:76m¿R¹069AB¸9ÁÖª5¸<;F87:769©B6¼:y¹(¨F43¾.=56¼:Afª5¸<;F87:769©B6¼:n¹IÀÙ 2.AB4A:72.A4AU;>¸¼6¼Õw;>:769¨F=¨>«:72.Aª54¨ â A©B:769¨F=»AB:72.¨Á698O=.¨F:87:74;>6¼ÉL20:$¿`«p¨F47Ù;>4ÁÈÊ25698Ù 6¼¸¼¸Ä%A87:7Â.Á69AÁ#6¼="«sÂ5:7Â54AÙ3¨F47Ü*È
yDÛdy0¢y ¦DyDt MbC BDwHaGM)JbHLE$GbEIJuM 5JbHLHFJ0D"!9"# <@? %$Ì,CA;d(%.,?% 84d0,Õ@?;d %D%'( N+@%'( 8
9L º;"<É, ,'& (%hij o>cwY7EfPXYRZ0>Oodo>\¼?EfP$[yZb).e>gQRgdV *A,+=.- Lodo0 /*1014532 154 M5C RG6 dA % ,.";(,7@? pN;d,8& "9OTVPXW*P$gw 1455.1 : ; M 4 M=<IH>!E$GIE?># ,+,dÉ+6+ 6 9& N+,k +6<A@+;Õ%'( N+@%'( 0@?;"+2% p B(%hij odcdY7?EfPXYRZ0>Oodo>\¼
EfP$[yZb).e>gQRgdV C+D1455.13- Fodo0 FE/0'1A6 2GH MIuMJ= iJbH>KL 7@? pML6 +,k4,< 6 +,(+6+ # ,% 84( >NTV\VPXr 63E : 154 MIJB"JBNMfMBO5B"."P %'( 8.Qgk C4 (%JRC"+;8,< 6 A;d(% # ,?% 8( >wowQRTVe>giPyQ`vS9PXQR\z]agw
6 //T MUI!VIGwH>WYX . .wH YZM\[`G]^*JB_ZM`Z,a!E7K>< R@?;d %cb<4C%: %$ 0( ;" d&8¼pebN(+6+
6 9& N+, +6< IUpÌEgfwhodYRTVj TV~7]YRTzhae0 : +D155 : - Lodo0 E/108EiEiE *T M`I!jLGwH>WYXB i .dHÞJB T MYk BDFH>!^K><J# =bN,<É@,?<É,dº;"d% @+;Õ%bN(+6+4 6 9& N+,S +6<
E]YRZ>PXj]YRTV[7]a\05QRhigaQn]j j TVe>gd 65AC+D155328- >o>ob 61Al0?EG2 E/ OtM?Z, iJBGLGJBmNM?Z,aWnWYwH06 +,C(+6+ 5o\p +,g ,m7U@? ÕL!4 +, SBha\¼ /5 hk>Ohie>[XhieUSBPXxqhowYRTVj Tz~$]YRTVhie]aeF|tYR_Oodo>\VTV[7]YRTVhae>_$?r\zcW%PXQ7 :
/s MAtJBu^qM 4 !jdHLE.JBwvM 4 !zGLGaW) RC¶d@2@% ( +4,i+x@2<2@? j$@4;"% % %:Õ% M&;d<k +6<º N+@?;"+2% p ,,?+,?% ,.";+% <¶(d,C 5TVey).Q`QRhQ`vx(%haeY`QRhi\V\VP7|,|>]aowYRTVSBP 3 TVe>TqYRPC).\VP$j P$eY\EfPXYRZdhU|d_TVewha\VT^|EfP$[yZ>]aedTz[X_$>)LYRP$TVe0wP7|bVdPXWvdzhaQRL 155.1 >NTV\zPyrdodo0 : 2.*0 : E32
65/ fMtUaIDdH?uwH0 ,.";\,?< 6 +,C% ,."; P ;"+|&;=Õp ,'& F|d|dTV_`hiewv9N#P$_`\VPXr iE32 M=~FJbEaA ,C ;=+6< +,° N+ CR(+4;8 +x@ 6 +,C% ,.";og% % @C oYp +4,C ,SBhi\¼ 1i1 hkP$cd>e>PyQ`vs0PXxUYRPf~$cwQ
E]YRZ>PXj]YRTVL>0P$c>de>PXQ7?0PXTzod~$TVgd 6iE : /1T M ; .aW³JBM 4 !V?wHLEa6 A;d(%È,?%'+(;"@+;=$4;d +,%D ,.p% +D155213- F_`c>dj TqY`YRP7|b : MJM s !jWY]JBM MOM Iau<BwEa #L+4$@?A N +4,% ,.%'( 8 FwodQRTVedgiPXQ`vD9PXQR\z]agd 6iE F2G; M s ? . iabKLK>!¼6 A;d dC;j@!A;",Ã<2;\,dA % ,";(,7@? ÕN;dA,'& >TVQRUZ"ac>_`PyQ7 : !1 OtMWJW C (% P ;"+% ÕÌ , P %'( 8 "%haQn|d]a_$L5]QRTV_$ 63E21 6// M 4 !jdHLEa> d ;dA "<>@+4 ,C *+2 !" - # ,.9Zd+4,d@? >@+;g@;=% % %4)L, p% ,C dTze)hij]TVe)PX[$hij oLhv_`TVYRTVhae¼EfPyYRZ>hU|d_%TzeuU[$TVP$e>[$PO]ae>|,)edgiTVe>PXPXQRTVe>gdB.rOhaQRe>ZUcdLPXQ7ÉB8hao>oLPi'fwU XQRTz]acwxb?h3TqQRhiede>P7]c0?h3BNÞT^|Uv\zcdeF|I]aeF|fwcb.P7|d_$VSBhi\¼ 25 hkA0P$[XYRcwQRPuOhaYRP$_TVe(%hij o>cwYn]YRTVhie>]a\,U[$TVP$e>[XPf]eF|).e>giTVedP$PXQRTVe>gwUodQRTVe>gaPXQ7155G2 >o>o0 *10CE :
H JBKa RGiBG8XIH T JBG<W,JiG!jK^ k!MwHLE!^GiiGZJbH> BE7H"a<B ZJ!^EdHLE$GIHM.?.ZJbH> BE7H?a<BJv,wH>W,JBMo`bN%2<2<;"(3?3G 'i¡¢3£¤?¥i¦3§^¨d©i "'ª¬«¥3¡'8¢G¡3©i§¨S®'
IWRMM-Preprints seit 2004
Nr. 04/01 Andreas Rieder: Inexact Newton Regularization Using Conjugate Gradients as InnerIteractionMichael
Nr. 04/02 Jan Mayer: The ILUCP preconditionerNr. 04/03 Andreas Rieder: Runge-Kutta Integrators Yield Optimal Regularization SchemesNr. 04/04 Vincent Heuveline: Adaptive Finite Elements for the Steady Free Fall of a Body in a
Newtonian FluidNr. 05/01 Gotz Alefeld, Zhengyu Wang: Verification of Solutions for Almost Linear Comple-
mentarity ProblemsNr. 05/02 Vincent Heuveline, Friedhelm Schieweck: Constrained H1-interpolation on quadri-
lateral and hexahedral meshes with hanging nodesNr. 05/03 Michael Plum, Christian Wieners: Enclosures for variational inequalitiesNr. 05/04 Jan Mayer: ILUCDP: A Crout ILU Preconditioner with Pivoting and Row Permuta-
tionNr. 05/05 Reinhard Kirchner, Ulrich Kulisch: Hardware Support for Interval ArithmeticNr. 05/06 Jan Mayer: ILUCDP: A Multilevel Crout ILU Preconditioner with Pivoting and Row
PermutationNr. 06/01 Willy Dorfler, Vincent Heuveline: Convergence of an adaptive hp finite element stra-
tegy in one dimensionNr. 06/02 Vincent Heuveline, Hoang Nam-Dung: On two Numerical Approaches for the Boun-
dary Control Stabilization of Semi-linear Parabolic Systems: A ComparisonNr. 06/03 Andreas Rieder, Armin Lechleiter: Newton Regularizations for Impedance Tomogra-
phy: A Numerical StudyNr. 06/04 Gotz Alefeld, Xiaojun Chen: A Regularized Projection Method for Complementarity
Problems with Non-Lipschitzian FunctionsNr. 06/05 Ulrich Kulisch: Letters to the IEEE Computer Arithmetic Standards Revision GroupNr. 06/06 Frank Strauss, Vincent Heuveline, Ben Schweizer: Existence and approximation re-
sults for shape optimization problems in rotordynamicsNr. 06/07 Kai Sandfort, Joachim Ohser: Labeling of n-dimensional images with choosable ad-
jacency of the pixelsNr. 06/08 Jan Mayer: Symmetric Permutations for I-matrices to Delay and Avoid Small Pivots
During FactorizationNr. 06/09 Andreas Rieder, Arne Schneck: Optimality of the fully discrete filtered Backprojec-
tion Algorithm for Tomographic InversionNr. 06/10 Patrizio Neff, Krzysztof Chelminski, Wolfgang Muller, Christian Wieners: A nume-
rical solution method for an infinitesimal elasto-plastic Cosserat modelNr. 06/11 Christian Wieners: Nonlinear solution methods for infinitesimal perfect plasticity
Eine aktuelle Liste aller IWRMM-Preprints finden Sie auf:
www.mathematik.uni-karlsruhe.de/iwrmm/seite/preprints
