ME 78 FIIE ELEME MEHOD Chper. Ioprerc Forlon Se fncon h ed o defne he eleen geoer ed o defne he dplceen whn he eleen ode r Eleen Lner geoer Lner dplceen ode Be Eleen Qdrc geoer Qdrc dplceen We gn he e locl coordne e o ech eleen. h coordne e clled he nrl coordne e. he dvnge of choong h coordne e eer o defne he hpe fncon nd he negron over he rfce of he eleen eer (we wll e nercl negron whch ch pler n he nrl coordne e nd cn e cled o he cl re he ep n dervng he eleenl ffne rce re he e: Sep Selec eleen pe Sep Selec dplceen fncon Sep Defne rn/dplceen re/rn relon Sep Derve eleen ffne r nd eqon.
-D r Eleen For -D lner r eleen he nrl coordne e for n eleen : - he nrl coordne re reled o he glol coordne hrogh whchwecnolveforhe : [( ( ] or n r for : [ ] where ow followng he render of he ep ecoe ch pler. Sep Selec dplceen fncon [ ] Sep Defne / nd /σ relon Recll h we hd he followng relon: ( d d hen pplng he chn rle of dfferenon we hve.
( d d d d h B L L he re/rn relon epreed : h: σ D where D E σ EB Sep. Derve he eleen ffne r nd eqon he ffne r e K ( L AE B B d whch h n negrl over whch we hve o conver o n negrl over. h done hrogh he rnforon: L f ( d f ( J d where J he Jcon nd for he ple r eleen : And Vol!! K ( e d J d AE L L L L / L d L AE L.
. CS Eleen we chooe nrl coordne e hown nd defne he geoer n er of he nrl coordne e : whchwecnwrenrfor: Whch cn e olved : A β α β α β α..
. In h ce hee re he hpe fncon he of he hpe fncon nwhere on he eleen dd o oe h n h ce he re pl nd. Sep : Chooe he dplceen fncon we cn pl wre he eleen dplceen fncon of nodl dof n he e for ed o decre he geoer:
. ( ( or Ψ Sep : Srn/dplceen nd re/rn relon In -D he rn dplceen relon re: nd or n r for : B In -D he re/rn relon re: DB D σ Where D depend on wheher plne re or plne rn condon prevl(ee Chper for del
.7 Snce he hpe fncon re fncon of he nrl coordne nd no nd we ppl he chn rle : Le conder he followng B o hen for he dervve of he hpe fncon wh repec o he glol coordne e we pl hve: β β β And he rn re wren : A β β β β β β Sep. Derve he eleen ffne r nd eqon Ll we e he PMPE o on he ffne eqon : V S rc od V ds dv dv X P B DB Snce ll he er n B re conn nd ng he hckne nd erl propere re conn over he eleen we hve: f K where DB B K A
.8 Lner Srn rngle (LS Eleen Agn we chooe he e nrl coordne e for he CS we defne he geoer n er of he nrl coordne e hrogh he hpe fncon : Whch we cn olve for he hpe fncon n er of he nrl coordne follow: Le e qdrc fncon of nd ( we cn epre fncon of nd : - - o whch en h here re nknown coeffcen o e deerned for ech hpe fncon...
.9 Ung he followng nforon h node we wn nd ll oher hen we ge eqon for ech hpe fncon nd we cn olve for he coeffcen nd we hve: ( ( ( or recognzng h hen we hve ( ( (( ( ( ( In h ce hee look lke he of he hpe fncon nwhere on he eleen dd o
. Incdenll he hpe fncon n he glol coordne e for nce eleen wh de lgned wh he nd e wold look oehng lke h: / /( / / /( / /( / / / / / /( / / / h h h h h h h h h h h h wh we e he Ioprerc forlon!!! Sep : Chooe he dplceen fncon We cn pl wre he eleen dplceen fncon of nodl dof n he e for ed o decre he geoer: ( ( or Ψ
. Sep : Srn/dplceen nd re/rn relon In -D he rn/dplceen relon re: nd or n r for : In -D he re/rn relon re: DB D σ Where D depend on wheher plne re or plne rn condon prevl(ee Chper for del So how do we conrc he B r? Le defne he followng r 8 8 B o nd le he Jcon r e (noe h
. J hen he er n he B r re pl erced fro he prodc o B J Sep. Derve he eleen ffne r nd eqon Ll we e he PMPE o on he ffne eqon : V S rc od V ds dv dv X P B DB We e Gn qdrre o perfor he negron over he eleen (oe h B nd n he ove re fncon of he nrl coordne nd
Gn Qdrre (ercl Inegron A we w he dervon of he ffne reqre h we perfor n negron over he eleen (h coe fro he defnon of he nernl rn energ nd when we ele he force vecor. Ofen h dffcl o do eplcl nle o re ng Mhec o we rn o nercl negron echnqe. oe h n he eleen forlon we re choong he fncon for of he dplceen (hence ndrecl he for of he rn nd re whch pper n he nernl rn energ he prncple ehnd Gn Qdrre h f we know he fnconl for of wh we re rng o negre hen here cern ner of pon where we need o evle he fncon whch wll gve n ec repreenon of he negrl. G Forl: I f ( di n n W f ( We evle n negrl evlng he fncon we wn o negre dcree pon n n nd lpl h n ppropre wegh Rle: n n negron pon rle n n order ccrc Eple: negron pon wll negre order polnol ecl - n n W.
negron pon wll negre rd order polnol ecl - n n / ± W W G Forl n -Denon: I / / W f ( dd W f ( W f ( d W W f ( Eple: CS (conn rn rngle Here we ed lner dplceen fncon whch en h he rn feld (nd he re feld conn over he eleen. he fnd he negrl of conn.e. he re nder he crve we need onl evle h fncon one pon. For CS h pon loced n he cener of he rngle n he nrl coordne e h pon loced. nd he correpondng wegh. -node ver -node rnglr eleen.
. Rernng o he node LS eleen we hd B nd whch were epreed n er of he nrl coordne. For hee eleen we hve G pon wh locon nd wegh :......... W gp W gp W gp h gve degree of precon of (negre nd order polnol ecl o we now hve for he ffne r ( ( ( (... gp gp gp n plne plne e W da da DB J B DB J B DB J B J DB B J DB B DB B k where he Jcon h lred een conrced (when we fored he B r : de J Slrl we cn perfor he negrl pperng n he force vecor oe: he fcor of ½ coe fro he re of he rngle n pce
-D Brck Eleen rl coordne e For ech eleen we gn locl coordne e repreened nd whch oh pn he rnge fro o over he re of he eleen Lner dplceen fncon Qdrc dplceen fncon ( noded (8 noded ode Brck BLner Qd - we chooe nrl coordne e hown nd defne he geoer n er of he nrl coordne e : -.
.7 or rher n er of he hpe fncon nd he nodl coordne : whchwecnwrenrfor: Here he hpe fncon re ( ( ( ( ( ( ( ( h we hve he of he hpe fncon nwhere on he eleen dd o
.8 Sep : Chooe he dplceen fncon we cn pl wre he eleen dplceen fncon of nodl dof n he e for ed o decre he geoer: ( ( or Ψ Sep : Srn/dplceen nd re/rn relon Agn he -D rn dplceen relon re: nd or n r for :
.9 A B In -D he re/rn relon re: DB D σ Where D depend on wheher plne re or plne rn condon prevl (gn ee Chper for del B re fncon of nd nd no nd o we hve o ppl he chn rle of dfferenon gn. h e we hve: d Or n r for : nd re dffcl o evle nd re no o we cn wre:
. J nd J where he deernn of he Jcon J : J de So we ge new B whch eql B now fncon of nd. Sep. Derve he eleen ffne r nd eqon Ll we e he PMPE o on he ffne eqon : V S rc od A ds dv da X P DB B n whch we rnfor he negrl n he - plne o negrl over he - plne fro o hrogh he rnforon nd e Gn Qdrre o perfor he negron ( n A W dd J f dd f J DB B ( (
. 8-ode Brck Eleen rl coordne e For ech eleen we gn locl coordne e repreened nd whch oh pn he rnge fro o over he re of he eleen Qdrc dplceen fncon he hpe fncon re: ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 8 7 lernvel we chooe nrl coordne e hown nd defne he geoer n er of he nrl coordne e : 8 7 8 7 oe h h n ncoplee qdrc polnol. 7 8
G pon node nd 8 node rck eleen Recll h we hd ed he for of he dplceen : h lner pproon no lner n ever drecon nd o perhp pon no ffcen Qd end o eh nle. he followng ode hve no rn herefore we e ore pon. - - - -.
PACH ES Vld of Ioprerc Eleen Crcl e for vld he pch e Serve necer nd ffcen condon for he correc convergence of fne eleen forlon Bc de o ele ll ner of eleen o h le one node whn he pch hred ore hn wo eleen. he ondr node of he odel re loded e of conenl derved nodl lod correpondng o e of conn re. Eple A pch e for σ for -node eleen F F F F A pch e e perfored for ll conn re e dended of he eleen A ccefl pch e revel h he eleen - wll dpl e of conn rn - wll no rn when eced o rgd od oon - cople wh dcen eleen when eced o e of conn rn.
ercl eple of LS Specf he nodl coordne ccord 88. < 8 < 8 < 8.< 8.7 < 8.7.<<; Merl propere nd pln re D r plnre; E.; ν.; D E ν ν :8 ν < 8ν < : >>; Specf he hpe fncon ; n H L; n H L; n H L; n ; n ; n ; For he Bno r nd deerne he Jcon MrFor@ Bno 88D@n D D@n D D@n D D@n D D@n D D@n D< 8D@n D D@n D D@n D D@n D D@n D D@n D<<D H L H L J H L H L MrFor@J Splf@Bno.ccordDêêChopD J.. MrFor@Bno Splf@Invere@JD.BnoDD..888889..77778.7.888889.888889.77778.77778.7.888889..888889.77778.888889 J..77778...7.....7.77778..77778..77778 For he B r MrFor@ B 88Bno@@ DD Bno@@ DD Bno@@ DD Bno@@ DD Bno@@ DD Bno@@ DD < 8 Bno@@ DD Bno@@ DD Bno@@ DD Bno@@ DD Bno@@ DD Bno@@ DD< 8Bno@@ DD Bno@@ DD Bno@@ DD Bno@@ DD Bno@@ DD Bno@@ DD Bno@@ DD Bno@@ DD Bno@@ DD Bno@@ DD Bno@@ DD Bno@@ DD<<D..888889..77778.7.888889.888889.77778.77778.7.888889..888889.77778.888889..77778...7.....7.77778..77778..77778..77778..888889...77778.7...7.888889.888889...7.77778.77778.7.77778..77778.888889..888889..77778.77778.888889 k Sr forng he ffne r MrFor@K@ D De@JD Splf@rnpoe@BD.D.BDD Perfor nercl negron(evle he g pon MrFor@ Kloc ê.ê HK@..D K@..D K@..DLD.8.7..7..7.8.9..8.7.98.7..8.8.8.78.8.9.89..78.9787..8 9.9788.88.989.8 7.97..88.898.77.9.7.8.88.8.87.8.8.7.7..87.97..8.989.87 7.788.98 7.98.88..899.9.9.7.78.8.8.98....8.9.98.79.8.8 7.97.8 7.98..979.7.77 7.79.88.89.9.9..7.88..7.8 7.7.997.9.7..89.88.7..8.77 7.7..787.87..8..898..899.9 7.79.997.787...9.7.78.77.87.9.98.88.9.87..99.7 k.98.9787.9.97.9.79.89.7..9.7. { {.
Appl he ondr condon R ; R ; R ; R ; R.; R ; R ; R.; R ; ; ; ; ; MrFor@ fvec 88R< 8R< 8R< 8R< 8R< 8R< 8R< 8R< 8R< 8R< 8R< 8R<<D MrFor@ vec 88< 8< 8< 8< 8< 8< 8< 8< 8< 8< 8< 8<<D Solve@Kloc.vec fvecd 88.7.79.788.8.898.89.978.98 R. R. R.9 7 R.99<< MrFor@ fvec 88R< 8R< 8R< 8R< 8R< 8R< 8R< 8R< 8R< 8R< 8R< 8R<<D MrFor@ vec 88< 8< 8< 8< 8< 8< 8< 8< 8< 8< 8< 8<<D..7.8..898..79.89.99.978..788 k { k.98 { MrFor@revec@ D Splf@D.B.vecDD.8.. k.997.98.99.789.77.988 Deerne he ree he G pon MrFor@revec@..DD MrFor@revec@..DD MrFor@revec@..DD.997.78.9.98..78 k.99 { k.79 { nd k.887 { {.