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Molecular and multiscale methods for the numerical simulation of materials

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(1)Molecular and multiscale methods for the numerical simulation of materials Frédéric Legoll. To cite this version: Frédéric Legoll. Molecular and multiscale methods for the numerical simulation of materials. Mathematics [math]. Université Paris 6 (UPMC), 2004. English. �tel-01986889�. HAL Id: tel-01986889 https://tel.archives-ouvertes.fr/tel-01986889 Submitted on 19 Jan 2019. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

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(35)   .  $ .       .                  .  +   ­!ˆ ‡™˜¦”{ƒf›> ‡¢‹„ˆ>                             „Œ +   ± Ž|¢‹ˆ žd†> ˜ ‡¢‹ˆ{°,Ž|ˆ{”K™d†Mž˜ƒ{Š‹ -                         )" + Œ §hƒfœ†>™d¢–›MŽ|ŠT†3¸{Ž’œ{Š‹†Mž                   +  0Œ   ¨A„Š–Š‹†M›> d¢–„ˆ$, 46²_Ž’™dœ,„ˆ{¢–›S„žd›>¢–Š–Š–Ž’ ‡|™dž        +  0Œ   *²{† ¼ †>{Š–†>™{™dfŠ–†Mœ               +  0Œ  Œ ¡ Ž|™j ‡¢–›>Š–†>ž¢–ˆKŽ4 ‡™dƒfˆ{›xŽ’ d†M” ¬†>ˆ{ˆ_Ž|™˜”¦«  |ˆ{†Mžr| ‡†>ˆ‰ d¢ Ž|Š +  0Œ  “ ¥S#Š N¹Ž|ˆf†=›³²_Ž|¢–ˆfG ž                  +  0Œ #" ´ ˆf†#{Ž|™˜ d¢–›MŠ‹†[¢–ˆKŽ ”{„0ƒ {Š–B† 1-†MŠ–ŠXt| d†Mˆ ‡¢–Ž|Š      + “ *²{†#›xŽ|ž˜†s 46žjš‚ž˜ ‡†>œž 1¢‹ d² œ4ƒ{Š‹ d¢–{Š‹†Sœ,†> ‡Ž|ž˜ ‡Ž {Š–†=žj ³Ž’ d†Mž   +  /“   ´ ˆf†#{Ž|™˜ d¢–›MŠ‹†[¢–ˆKŽ ”{„0ƒ {Š–B† 1-†MŠ–ŠXt| d†Mˆ ‡¢–Ž|Š      +  /“   ¥S#Š N¹Ž|ˆf†=›³²_Ž|¢–ˆfG ž                  +#" ¨A„ˆ{›>Š–ƒ{ž˜¢–„ˆfž                       +  ¥Sft†>ˆ{”{¢ ¸                        +  0  ¥Sždž˜ƒ{œ,f ‡¢‹„ˆ{ž„ˆ$ ‡²{†#©hŽ|œ¢‹Š‹ d„ˆ{¢ Ž’Jˆ 4 ƒ{ˆ{›& ‡¢‹„ˆ H    +  0  ¾ ¢–„f²_Ž|ˆ ‡¢‹ˆ{†sŽ|ždž˜ƒ{œ,f ‡¢‹„ˆ             . . . .         .               

(36)         . Œ0  Œ0  0Œ  Œ 0Œ  “ 0Œ #" 0Œ   0Œ  .                          . . ­!ˆ ‡™˜¦”{ƒf›> ‡¢‹„ˆ>                          *²{†#›M„œ,{Š‹†> ‡†[Ž|ˆ_Ž|Š š¦ž˜¢–ž4E ‡²f† d«·”{¢–œ,†Mˆfžd¢–|ˆ_Ž|Šr²_Ž|™dœ,„ˆ{¢‹›[„ž˜›M¢–Š‹Š Ž’ d„™ ¥Sf{™dx¸¦¢–œŽ’ ‡¢‹„ˆ$4E ‡²{†#Žx•„†>™‡Ž|°|†,  *²f†#›>„ˆ ‡¢‹ˆ‚ƒf„ƒ{ž›xŽ|ž˜†       €¦†>œ¢ «!”{¢‹žd›M™˜†> d†sŽ}•|†M™dŽ|°„†Mž                     º_ƒ{Š–Š š ”f¢–žd›>™d†& ‡†#Žx•„†M™dŽ|°„†>ž                     ® †MœŽ|D™ N‰ž„ˆ$ ‡²{†s¢‹œ{Š‹†Mœ,†Mˆ ³Ž¹ ‡¢–|ˆŽ|ˆ{”Kˆ‰ƒ{œ,†M™d¢‹›xŽ|ŠX†&¸¦t†>™d¢‹œ†>ˆ‰ dž  ¥Sft†>ˆ{”{¢ ¸ _žd„œ,†[ ‡†>›³²{ˆ{¢–›MŽ|Š™d†>ždƒ{Š ‡ž                      .       . 

(37)         .   |‘   |‘   |Œ   *"      „‘     |“   |“   )".                         .    !          & 

(38)                   

(39)    .             . . . .

(40) .        l }‘„‘  l }‘   l }‘„Œ  l }‘)".    .  /“   ­!ˆ ‡™˜¦”{ƒf›> ‡¢‹„ˆ>                            l„}Œ /“   ¥œ†& ‡²{‚”$ ‡°„†>ˆ{†M™dŽ’ ‡†sœ,†xŽ|ž˜ƒ{™d†s¢–ˆ •’Ž|™˜¢ Ž|ˆ Ž|Š‹°„„™˜¢‹ ‡²fœž         l„" . “/ Œ ¥ žd¢‹œ{Š‹†=Ž|{fŠ–¢–›MŽ’ ‡¢‹„ˆ                        l„ “/ “ =¯ †Mˆ{†>™‡Ž|Š‹¢#6M†>”C¯#Ž|ƒfždžd¢–Ž|ˆ ± „œ,†Mˆ *²{†>™dœ,„ž˜ ‡Ž’ d d¢–ˆ{°            l„ /“  “/  §h„™˜œ Ž|ŠC4 „™dœ 4 „™5 ‡²{†'¯#¯ ± * ”fš‚ˆ_Ž’œ¢‹›Mž            l„ /“  “/  §hƒ{œ,†M™˜¢–›xŽ’ŠT™d†>ždƒ{Š ‡žK                     l ¦ “/#" ¥Sft†>ˆ{”{¢ ¸                              l   /“ #"+  §h„ž˜Ÿ&«0©w¦}•|†M™w›³²_Ž|¢–ˆfž 4 „™8§ ?(*8Ž|ˆ{”K§ ¡ *a†>ˆ{žd†>œ{Š‹†Mž      l   /“ #"+  ¯'¯ ± * ”¦š¦ˆ{Ž|œ¢‹›M!ž  ¨5Ž|žd† M = 2, N = 1, d = 1          l  /“ #"+ Œ ¡ ™˜¦, 4X 4r ‡²f†w›>„ˆ{ž˜†M™˜•’Ž’ d¢–„ˆ, 4r ‡²{††>ˆ{†M™˜°|š 4 „™¤ d²{†h¯'¯ ± * ”¦š¦ˆ{Ž|œ¢‹›Mž’}Œ„‘ /“ #"+ “ §h„ˆ¦«·†&¸fŽ|›& s{™d†>žd†M™j•’Ž’ ‡¢‹„ˆ  4l ‡²{† œ,†xŽ|ž˜ƒ{™d† 4 |™S ‡²{†Ž’Š–°„„™˜¢‹ d²{œ.¬¢‹ƒ  ( *  )"¦f™d„r„žd†>” 4 „™6¯'¯ ± * ”fš‚ˆ_Ž|œ,¢–›>$ž  ›xŽ’žd†5 4f ‡²{† 4 ™˜†M†-_Ž|™˜ d¢–›>Š–†fxŒ .

(41)   .  . “/#"+#" h§  „ˆ¦( «·* †&¸f Ž|›& s{™d†>žd†M™j•’Ž’ ‡¢‹„ˆ 4l ‡²{† ,œ †xŽ|ž˜ƒ{™d†4 |™S ‡²{†Ž’Š–°„„™˜¢‹ d²{œ.¬¢‹ƒ *" {™d|t„ž˜†M” 4 „™w¯#¯ ± * f” š‚ˆ_Ž|œ,¢–›>ž$_Žœ,„™d†[°|†Mˆ{†>™‡Ž|Š›MŽ|žd† }Œ .  .      

(42)                  .              .  . . 

(43)  .     $

(44) . . "+  ­!ˆ ‡™˜¦”{ƒf›> ‡¢‹„ˆ>                            lx“  "+  ¥Sˆ{Ž|Š‹š‚žd¢‹ž4 žd†>•|†M™dŽ|Š6Ž’Š–°„„™˜¢‹ d²{œž                    lx“ +"  +  ® † 4 †M™d†>ˆ{›M†#{™˜{Š–†>œIŽ|ˆf”CœŽ|¢–ˆ$ž˜†> ˜ ‡¢–ˆf°5            lx“ +"  +  ± †> d²{¦”fž _Ž’žd†M”K„ˆ ‡²{†=›MŽ|ˆ{›>†MŠ–Š–Ž’ ‡¢‹„ˆ$4E ‡²{†s¢‹ˆ‰ d†M™ 4 Ž|›M† 4 „™˜›M†>ž   l"|‘ +"    +  ¥ <{™dž˜ hŽ|Š‹°„„™˜¢‹ ‡²fœ                  l"|‘ +"    +  ¥Sˆ{| d²{†M™wŽ|Š‹°„„™˜¢‹ ‡²fœ                 l"*" +"    + Œ ¥ §w†!1 d„ˆ Ž’Š–°„„™˜¢‹ d²{œ                l"! +"  + Œ µ 6xŽM1AŽ Ž|Š‹°„„™˜¢‹ ‡²fœž-                     l"! +"    Œ0  ¥SˆCŽ|Š–°„|™d¢‹ d²{œ _Ž|ž˜†M” |ˆ d²{†sž˜ d™d†Mž˜žSŽ¹  ‡²{†s¢–ˆ d†M™24 Ž’›M†  l"! +"    Œ0  ¥SˆCŽ|Š–°„|™d¢‹ d²{œ _Ž|ž˜†M” |ˆ d²{†st|žd¢‹ d¢–„ˆ Ž¹  ‡²{†s¢–ˆ d†M™24 Ž’›M† lf +"  + “ ± †> d²{¦”fžL_Ž|ž˜†M” |ˆ ‡²f†›>„ˆ ‡¢–ˆ‰ƒ{¢ !š 45 d²{†”f¢–ždfŠ Ž|›>†Mœ,†Mˆ 4Ž¹ ' d²{† ¢–ˆ ‡†>™24 Ž|›>†                           l  +"    “/  ¥SˆK¢‹ ‡†>™‡Ž’ d¢‹•|†sŽ|Š–°„|™d¢‹ d²{œ               l  +"    “/  ¥œ,†> d²{‚”K¢‹ˆ{ždf¢–™d†>” 46¢‹ˆ ‡†M°|™‡Ž|Št™d†Mf™d†Mž˜†Mˆ ³Ž¹ ‡¢–|ˆ    l)" +"  +#" ¨A„ˆ{›MŠ‹ƒ{žd¢‹„ˆ{ž                         l "+ Œ §hƒfœ†>™d¢–›MŽ|ŠT†3¸{Ž’œ{Š‹†Mž                         l +"  Œ0  ¨A„œ{Ž|™d¢‹žd„ˆ 4E ‡²{†sœ,†> d²{¦”fž„ˆ ždœŽ|Š–ŠTžjš‚ž˜ ‡†>œž        l +"  Œ+‹  j% £lŽ|ž˜š ‡¢–œ,{Š–†>œ†>ˆ S' Ž|Š–°|„™d¢ ‡²{œ,ž;          l +"  Œ+‹  µ 6xŽM1AŽ Ž|Š‹°„„™˜¢‹ ‡²fœ                  l

(45) +"  Œ+‹ Œ ¨A„ˆ{›MŠ‹ƒ{žd¢‹„ˆ                     l„ +"  Œ0  ¥S{{Š‹¢–›MŽ’ ‡¢‹„ˆ ‡,Š Ž’™d°„†>™5ž˜š‚ž˜ d†Mœ,ž                l’“. "+ “ A¨ „ˆ{›>Š–ƒ{ž˜¢–„ˆ                              l! 

(46)      

(47)    *      %                 &       *                  &      . 0  ­!ˆ ‡™˜¦”{ƒf›> ‡¢‹„ˆ>                            l f 0    * ²{†Ž¹ ‡„œ,¢–žj ‡¢–›SŽ|ˆ{” ›M|ˆ‰ d¢–ˆ‰ƒ{ƒ{œf™d{Š‹†Mœ,ž           l  0   ¥›>„ƒ{{Š‹†M” {™˜{Š–†>œ                      l |“ 0  Œ ´ ƒ¦ ‡Š–¢‹ˆ{†[E 4 ‡²f†#™˜†Mž˜ƒ{Š‹ dž                     l  0   * ²{†#›xŽ|ž˜†s¤4 Ž ›>„ˆ •„†&¸C†>Š Ž|žj ‡¢–›=†>ˆ{†M™˜°|š$”{†>ˆ{žd¢ !š W            l  0 +  ¡ ™˜„r†M™˜ d¢–†>žw, 4 d²{†=•|Ž’™d¢ Ž¹ ‡¢–|ˆ_Ž|Šr{™df Š–†Mœ,ž             l  0 +  ¾ †!_< ˆf¢‹ ‡¢‹„ˆ E 4 ‡²{†s_Ž|™j ‡¢ ‡¢–|ˆ                  l „‘ 0 + Œ ¨A„œ{Ž|™d¢‹žd„ˆ E 4 ‡²{†#Ž’ d„œ,¢–ž˜ d¢–›h{™df Š–†Mœ;Ž|ˆ{” ‡²{†s›>„ƒ{{Š‹†M” {™d{Š–†>œ f 0 Œ  * ²{†'¬†Mˆ{ˆ_Ž’™d”¦«  „ˆ{†Mž›MŽ|žd†                      l  0 Œ0   * ²{†'›>„ˆ ‡¢‹ˆ‚ƒfƒ{œI{™˜{ Š–†>œ                   l  * 0 Œ0   * ²{†Ž¹ ‡„œ,¢–žj ‡¢–›h{™d{ Š–†>œ                     |‘f 0 Œ+ +   * ²{†'›MŽ|žd†s 4 ˆ{)r ¦”fš54 „™d›>†              |‘  0 Œ+ +   * ²{†'°|†Mˆ{†>™‡Ž|ŠT›MŽ|žd†                   |‘|“ . .

(48)  .  . 0 Œ0 Œ * ²{†'ˆ{Ž’ ‡ƒ{™dŽ|ŠT›>„ƒ{{Š‹†M”CŽ|{{™˜ Ž|›³²H                  ¦}‘ 0 Œ0 “  ¥ œ,‚”{¢<_†>”$›M„ƒf{Š–†>” Ž|f{™d Ž’›³²                   ¦  0 Œ0#" ¾ †!<_ˆf¢‹ ‡¢‹„ˆ 4E ‡²{†s_Ž|™j ‡¢ ‡¢–|ˆ                      „„ . . . . .   

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(51)  

(52)              . +  !­ ˆ ‡™˜¦”{ƒf›> ‡¢‹„ˆ>                           +   * ²{†#œ¢‹›M™˜„žd›>„{¢–›Sœ‚”{†MŠ                      + Œ  * ²{†#²{„œ,„°„†>ˆ{¢#6MŽ’ ‡¢‹„ˆ{™˜¦›>†M”{ƒ{™˜†B                  + Œ0  ¨AŠ Ž|ž˜žd¢–›MŽ|ŠT²{|œ„°|†Mˆ{¢x6 Ž’ d¢–„ˆf™d‚›M†M”fƒ{™d†K           + Œ0  ©h„œ,„°„†>ˆ{¢#M6 Ž’ ‡¢‹„ˆE 4 ‡²{†sr„Š‹š‚›>™˜š‚ž˜ ‡Ž|ŠŠ ŽM1           + Œ+ +   * ²{†#•‚¢‹žd›M|{Š Ž|žj ‡¢‹›S ‡†>™dœ              + Œ+ +  ¡ „žj ‡ƒ{Š–Ž’ ‡†>”CœŽ|›M™˜„žd›>„{¢‹›=œ‚”{†MŠ 4 „™5 ‡²f†#r„Š š¦›>™˜š‚ž˜ ‡Ž|Š + “ §hƒfœ†>™d¢–›MŽ|ŠT™˜†MždƒfŠ‹ ‡žG                        +# " ¨A„ˆ{›>Š–ƒ{ž˜¢–„ˆfž                            .  .     

(53) 

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‡™d†A_Ž|™˜$ |r„ƒ{™ Ÿ>•‚¢‹ d†M™¤”{†>ž~† t†> dž~”{† r„™d ” ’¢–Š‚†Mžj lˆ{ŸM›>†MždždŽ|¢–™˜†”{†5›>„ˆ{žd¢‹”{ŸM™˜†M™ ”{†>žhœŽ’ dŸM™d¢–Ž|ƒ¦¸ ždƒ ,ždŽ|œœ,†Mˆ w°„™dŽ|ˆ{”{!ž  ¡ „ƒf™S ‡™dŽ|¢‹ d†M™w›M†°|†Mˆ{™˜†4”{†4{™˜ {Š‹¿Mœ,†M$ž rƒ{ˆ{†4Ž|{{™˜¦›³²{†›M„ˆ{ž˜¢–žj ‡† %_Ž|™˜ d¢‹ d¢–„ˆ{ˆf†M™wŠ–†#”{’« œŽ|¢–ˆf†5”{†›xŽ’Š–›MƒfŠ ¹†>@ %[ƒf d¢–Š–¢‹žd†>™ ƒ{ˆœ,¦”f¿MŠ–†AŽ’ ‡|œ¢‹ž˜ ‡¢ ‰ƒ{†A”_Ž’ˆ{ž~Š–†>&ž 6M„ˆf†Mž ‰ƒ{¢{Š–†Aˆ{ŸM›>†Mždž˜¢‹ d†Mˆ M  ‡Ž|ˆ{”{¢‹ž ‰ ƒ  ƒfˆoœ,‚”{¿MŠ‹†'”{† ›>„ˆ ‡¢‹ˆ‚ƒfƒ{œ †>ž˜ sƒf d¢–Š‹¢–ždŸ#”_Ž|ˆfž[Š–†4™˜†Mžj ‡†4”{ƒ œŽ’ ‡Ÿ>™d¢–Ž| ƒ  µ ˆ°„™dŽ|ˆ{” ˆ{|œ {™d†”{†œ,Ÿ> d²{¦”f†Mžsœ4ƒ{Š ‡¢ «·ŸM›³²{†>Š–Š‹†MžSf™d‚›MŸM”{Ž|ˆ '”f†,›>†{™d¢‹ˆ{›M¢‹t† „ˆ sŸ& ‡Ÿ{™d|t„ž˜ŸM†>ž'™˜Ÿ&« ›>†Mœ,œ†>ˆ‰$ ¤¨A†>žœ,Ÿ> d²{¦”f†Mž™˜†Mr„žd†>ˆ 4žd|ƒf•„†>ˆ‰ 4ždƒ{™'ƒ{ˆ ›M™˜¢‹ d¿M™d†,†Mœ,{¢‹™d#¢ ‰ƒ{†r†M™˜œ†& d ³Ž’ˆ‰ '”{† ›³²{|¢–žd¢‹™=†Mˆo›³²_Ž ‰ƒ{†,t|¢–ˆ s”{ƒoœ Ž’ dŸM™˜¢ Ž|ƒŠ‹† œ‚”{¿MŠ‹† F ”f¢–žd›>™d†& '|ƒo›>„ˆ ‡¢‹ˆ‚ƒ I ‰ƒ{¢~†>ž˜ 'ƒf ‡¢‹Š–¢‹ždŸ  §w„ƒ{ž5ˆ{|ƒ{žA¢‹ˆ‰ dŸM™˜†Mždž˜„ˆ{žŽ’ƒ›³²_Ž|{¢ ‡™˜† %#Š  Ž|ˆ_Ž|Š š‚žd†wˆ‚ƒfœŸ>™d#¢ ‰ƒ{†h ”  ƒ{ˆ†&¸¦†>œ{Š‹†h d™d¿Mž-žd¢‹œ{Š‹†. ”  ƒ{ˆ{†s ‡†>Š–Š‹†[œŸ& ‡²{‚”{,† _†MˆKœ†& d ³Ž’ˆ‰ "Š  Ž’›M›M†>ˆ hž˜ƒ{™Š–†=›>™d¢‹ d¿M™˜†= ”  Ž’”_Ž|f d¢‹•‚¢‹ dŸ  ¨A„ˆ{›>†M™dˆ{Ž|ˆ AŠ‹†Mž¤ ‡²f¿Mœ,†Mžlœ†>ˆ ‡¢–|ˆ{ˆ{ŸMž~›>¢ «·”{†Mž˜ždƒ{!ž ‚Š‹†Mž~›>„ˆ ‡™d¢ {ƒf d¢–„ˆ{ž~|™d¢–°|¢–ˆ_Ž|Š‹†Mž ”{†›M†> ˜ ‡† d²{¿Mž˜†'ž˜„ˆ '  "Š  Ž’ˆ_Ž|Š‹š‚ž˜†4ˆ‰ƒ{œŸ>™d¢ ‚ƒf†,”f†Mž=œ,Ÿ> ‡²f¦”{†>ž=›MŠ–Ž|ždž˜#¢ ‰ƒ{†Mžs”{† ”¦š¦ˆ{Ž|œ¢ ‰ƒ{† œ|Š–ŸM›>ƒ{Š Ž’¢–™d†#r„ƒ{™ Š‹†h›MŽ|Š–›>ƒ{Š ”{†wœ}š„†>ˆ{ˆ{†>žA ‡†>œr„™d†>Š–Š‹†Mžl†Mˆ, ‡†Mœ,{ž-Š–„ˆ{0°  †> -Š Žs{™d„r„ž˜¢‹ ‡¢‹„ˆ†> AŠ  Ž|ˆ_Ž|Š š¦ž˜† ”f†hž˜›³²{ŸMœŽ|žct†>™dœ,†> d ‡Ž|ˆ cƒ{ˆ{†›M„ˆ •|†M™d°|†Mˆ{›>†[{Š–ƒfžc™dŽ|{¢‹”{†”{†Mžcœ, š|†Mˆ{ˆf†Mžl ‡†Mœ,r„™d†>Š–Š–†>ž •|†M™˜žhŠ‹†Mƒ{™Š‹¢–œ,¢‹ ‡†h†Mˆ ‡†>œ{ž¢‹0ˆ <_ˆ{¢ Š–Ž[›M„ˆfž˜ ‡™˜ƒ{›> d¢–„'ˆ  t„ƒf™~›M†M™j ³Ž|¢‹ˆ{ž~ž˜š‚žj ‡¿Mœ,†Mžc”fš‚ˆ_Ž|œ,#¢ ‰ƒ{†>žlƒf ‡¢‹Š–¢–ž˜ŸMž †Mˆ,”fš‚ˆ_Ž|œ,#¢ ‰ƒ{†œ,’« Š‹ŸM›>ƒ{Š Ž|¢‹™d,† †> ‚ƒf¢c›>„ˆ{ž˜†M™˜•|†Mˆ ƒ{ˆ{†œ†>ždƒ{™˜†  ”  Ž|Š–°|„™d¢ ‡²{œ,†Mž= ”  ¢–ˆ dŸM°„™dŽ’ ‡¢‹„ˆ ‚ƒf¢-›>„ˆ¦« ž˜†M™j•„†Mˆ S›M†> ˜ ‡†#œ†>ždƒ{™˜† . . . . .

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