Chapter 4 NUMERICAL METHODS FOR SOLVING BOUNDARY-VALUE PROBLEMS

Transcription

1 Chaptr 4 NUMERICL METHODS FOR SOLVING BOUNDRY-VLUE PROBLEMS

2 00 4. Varatoal formulato two-msoal magtostatcs Lt th followg magtostatc bouar-valu problm b cosr ( ) J (4..) 0 alog ΓD (4..) 0 alog ΓN (4..) whr ΓD ΓN Γ s th bouar of th two-msoal smplcoct oma, whl a ar vctor pottal a magtc prmablt, rspctvl. ssumg rctagular coorats, o has z a J Jz, whras clrcal coorats ϕ a J J ϕ. It s assum that J a th sco rvatv of ar cotuous so that th tgral of both ss of (4..) sts. wa to appromat th soluto to (4..) (4..) s to rla th ffrtal formulato of th bouar-valu problm; to ths, th avrag of both ss of (4..), wght b a sutabl tst fucto, s cosr. ccorgl, f u s a tst fucto whch s cotuous up to ts sco rvatv, from (4..) o has ( ( ) + J) 0 u (4..4) a, b tgratg ovr th oma, o gts u ( ) + Ju 0 Du to vctor tt (.4), tag ϕ u a V, t follows (4..5) ( u ) u + Ju 0 B applg Gauss s thorm o obtas Γ u Γ u + Ju (4..6)

3 0 u Γ + u Γ u + Ju 0 ΓD ΓN (4..7) Du to bouar cotos (4..) a (4..) th frst two trms ar zro; thrfor, t rsults u + Ju 0 (4..8) whch s th varatoal quato assocat to th ffrtal quato (4..). Now, tag u δ whr δ s th lmtar varato of, (4..8) bcoms or, quvaltl δ + Jδ 0 δ J 0 (4..9) (4..0) whch stats th cssar coto for to b a sta pot of th fuctoal χ( ) J (4..) I othr wors, f s a soluto of ffrtal quato (4..) subct to (4..) a (4..), th s a soluto also of varatoal quato (4..0) a s such to gv org to a sta pot of fuctoal (4..). Covrsl, havg f th rg fuctoal of th smpl-coct oma as χ or, thas to (..5) whr χ J B B J (4..) (4..)

4 0 ( ) B,,0 (4..4) (s Scto..), quato (4..) follows. I fact, lt th frst-orr varato δχ of fuctoal χ, whch rprsts mus th co-rg of th oma for gv currt st (s Scto..5), b cosr. It rsults δχ δ Du to (.4) wth δχ δ ( ) ( ) + ( δ) ( ) δ ϕ δ a V Jδ Jδ o has ( δ ) δ ( ) Th, applg Gauss s thorm, t follows δχ δ Γ δ ( + J) Γ δ Γ + δ Γ δ ΓD ΓN If fulfls (4..) a (4..), o has δχ δ ( + J) Jδ J δ (4..5) (4..6) ( + J) (4..7) (4..8) Sc th co-rg has a sta pot wh δχ 0, th Posso s quato + J 0 (4..9) s vrf; (4..9) s call th Eulr s quato assocat to fuctoal (4..) s a rsult, th quvalc btw th sarch of a soluto to Posso s quato a th sarch of a sta pot of a rg fuctoal has b

5 0 prov. Th two approachs ar ow as Rtz s mtho a Galr s mtho, rspctvl. ccorg to th lattr, a umrcal procur appromatg th mmzato of th rg fuctoal s vlop. Th followg rmars ar applcabl. ) Drchlt s coto (4..) s a sstal bouar coto, bcaus th valu of must b forc at last a pot of th bouar. ) Homogous Numa s coto (4..) s a atural bouar coto, bcaus t s alra ta to accout both (4..7) a (4..7). 4. Ft lmts for two-msoal magtostatcs 4.. Dscrtzato of rg fuctoal Lt th followg cotuous problm b cosr: f th sta pot of χ( ), cost alog ΓD, 0 alog Γ N, whr χ ( ) + J (4..) s th rg fuctoal assocat to a smpl-coct oma whch rctagular coorats ar assum. I (4..) t s suppos that s cotuous up to ts frst rvatv, whl J s assum to b a cotuous fucto. Lt b scrtz b mas of a gr of tragular lmts subct to th followg costrats (Fg. 4.): - two aact lmts o ot ovrlap; - o vrt of a tragl blogs to th g of a aact tragl.

6 04 a b c Fg. 4. Eampls of corrctl (a,b) a corrctl (c) shap tragls. Th followg scrtzato of problm (4..) s trouc: f th sta pot of χ( ) for th t of tragls of th gr, upo th coto that th rstrcto of pottal to a lmt of th gv gr s rprst b a lar polomal a cost alog ΓD, 0 alog ΓN. s a cosquc, th whol oma th pottal s appromat b a pcws-lar fucto. Gv a umbrg of gr os (,,...,), th pcws-lar fuctos (, ) at o, (, ) 0 at all th othr os,, ar call global shap fuctos (Fg. 4.). ca b wrtt as (, ) (, ) (4..) (4..) whr s th uow valu of (,) at -th o a (,) vars larl.

7 05 Fg Dtal of a gr: rprstato of th global shap fucto assocat to -th o. Sha tragls show th lar varato of th fucto. ftr substtutg (4..) (4..) o obtas ( ) + + χ,...,, J (4..4) that rprsts th scrt vrso of (4..). I (4..4) th trms pt of ust ca b sparat from thos pt of. Eplctl, o gts ( ) χ J J,...,, (4..5) I orr (,,..., ) to ma fuctoal (4..4) sta, t must b

8 06 χ 0,,,..., From (4..5) a (4..6) o obtas (4..6) J 0,,,..., + (4..7) If shoul b ot that (4..7) rprsts a lar sstm of quatos uows. If th sstm s prss matr form, th trs of th coffct matr H(,) bcom h +,,,,..., whl th trs of th sourc vctor (,) ar J,,,..., Th, sstm (4..7) ca b wrtt as (4..8) (4..9) H (4..0) whr H(,) s th rluctac (H - m) matr, whl (,) a (,) ar oal pottal (Wbm - ) a oal currt () vctors, rspctvl. It s as to ralz that (4..8) fuctos, ca b trchag,.. matr H s smmtrc. Th problm of fg a sta pot of fuctoal (4..4) s th ruc to th soluto of a lar sstm govr b matr H a sourc trm. It shoul b ot that sstm (4..0) s sgular; orr ts soluto to b uqu, t s cssar to f th valu of pottal of all D os whr (4..) hols; at last o o locat alog bouar Γ must b costra.

9 Local shap fuctos rctagular coorats Rfrrg to a tragl of th gr, th followg local shap fuctos (, ) at o,, (, ) 0 at th othr two os atclocws (4..) wth lar varato wth rspct to (,) ca b trouc; th rprst th rstrcto of (4..) to o of th tragl. Rfrrg to a tragl of vrtcs V (, ), V (, ), V (, ), th followg fuctos whr (, ) [( ) + ( ) + ( )] (, ) (, ) S S S [( ) + ( ) + ( )] [( ) + ( ) + ( )] (4..) S (4..) s th ara of th tragl cosr, ar lar both a. Th fulfl cotos (4..) a ar th local shap fuctos. gomtrc trprtato of (4..) s gv Fg. 4.. V S S V S V V s h-s Fg. 4. Gomtrc trprtato of local shap fuctos.

10 08 Cosrg a r pot V(,), th rato s S S h S + S + S S (4..4) s call ara coorat ξ rfrr to vrt ; gral, ara coorats ar f as Th followg proprts hol It ca b prov that S ξ,,, (4..5) S 0 ξ, ξ,,, (4..6) ξ,,, (4..7) I fact, S s obta b substtutg vctor [ ] th -th row of ara matr trmat (4..); cosrg (4..5) a (4..), (4..7) mmatl follows. Cosqutl, th rstrcto of pottal (,) to th gv tragl s (, ) ( ) [ ( ) ( ) ( )],,,, (4..8) whr (,, ) ar th oal valus of pottal th tragl tslf. 4.. Coffct matr a sourc vctor If (4..8) s appl to all th tragular lmts composg th gr, th tr h s th sum of cotrbutos of ach lmt h +,,,,..., (4..9) whr, ar global shap fuctos. Howvr, t s asl s that th cotrbuto of a tragl to th tgral (4..9) s zro f thr th -th or th -th o os ot blog to

11 09 tragl tslf. s a cosquc, th maort of trms formg th tgral s zro a th matr s spars. I ths rspct, t s covt to f th local coffct matr H (,) assocat to a sgl tragl havg ara S wth trs h l + l l,, l,, (4..0) whr, l ar local shap fuctos. From (4..), b a cclc prmutato of cs, t rsults a + + S S b + + S S (4..) wth,, a 4, 5, 4, 5. Th trs of th local coffct matr (4..0) ar th gv b a a b b a a b b h l + l l + l l,, l,, (4..) 4 S 4S I practc, th trs of th global matr ar assmbl startg from trs of local matrcs, for th corrspoc btw local a global umbrg of os s uqu. Th assmblg ruls wll b clarf latr o b mas of a ampl. I a smlar wa, t s possbl to costruct (4..9) b assmblg local cotrbutos; fact J J,,,...,,,, Th local sourc vctor assocat to a sgl tragl has tr (4..) J,,, (4..4) I partcular, f currt st J s assum to b costat th lmt JS cosr, ach local sourc trm s qual to.

12 From pottal to fl ssumg that th magtc pottal ( ), s appromat b a lar polomal o ach lmt of th gr, f (,, ) ar th valus of pottal vrtcs V (, ), V (, ), V (, ) of tragl, rspctvl, th ucto fl ( ) z B, B, B B s appromat o ach lmt b 0,, B (4..5) From (4..) a (4..8) t rsults a a a b b b S S B B (4..6) ppartl, matr a a a b b b S (4..7) appromats th curl oprator. Th followg rmar ca b put forwar. t th trfac btw two tragls, havg a g commo, both ormal compot B of ucto fl a tagtal compot of magtc fl ar ot cotuous. I othr wors, fl compots ar appromat b mas of pcws costat fuctos all ovr th oma. t H t B