Least-Squares Finite Element Methods with Ap- plications ...Least-Squares Finite Element Methods...
30
PREPRINT 2002-10 Least-Squares Finite Element Methods with Ap- plications in Electromagnetics Rickard Bergstr ¨ om Chalmers Finite Element Center CHALMERS UNIVERSITY OF TECHNOLOGY G¨ oteborg Sweden 2002
Transcript of Least-Squares Finite Element Methods with Ap- plications ...Least-Squares Finite Element Methods...
PREPRINT 2002-10
Least-Squares Finite Element Methods with Ap-plications in Electromagnetics
Rickard Bergstrom
Chalmers Finite Element CenterCHALMERS UNIVERSITY OF TECHNOLOGYGoteborg Sweden 2002
Preprint 2002-10
Least-Squares Finite Element Methods withApplications in Electromagnetics
Rickard Bergstrom
Chalmers Finite Element CenterChalmers University of Technology
SE–412 96 Goteborg SwedenGoteborg, December 2002
Least-Squares Finite Element Methods with Applications in ElectromagneticsRickard BergstromNO 2002-10ISSN 1404–4382
Chalmers Finite Element CenterChalmers University of TechnologySE–412 96 GoteborgSwedenTelephone: +46 (0)31 772 1000Fax: +46 (0)31 772 3595
Printed in SwedenChalmers University of TechnologyGoteborg, Sweden 2002
!"#$ %%'&( *)+) ,-"#!&* /.02143 5276'*8&9:;9;"#)=<?>A@B4.
CEDGFHI)J (7 <9:7" 529;)(KFLMONP&&RQST1=3 *)-"#57%U &V<9:*)UW( X) ,I<U352 &EY6'( (Z [\]6^<W &*_=( &&_L <`439;52( &&*_L" aNT*"3* b'&*]52<3 &GNP*"4# <c9:2"+ <U %*O*0_ aG@EBOCEDGFd : ,A (Z) 5_`523&*,e) 5P%5) bU&9;ENT*";<U(Z)443)3f9; #(T%g529: bU&**V) ,h& 5 &*_7<Yig54f) 52<#52G) ,9; *3U<# a#bU3#jNT*")3#I5)#;(Z) 5k#5T3& 52*[
l
! "$#&%(') * $#+ ,, -/.0$1% ) 12.34%5 76 8"$/.9
:<;>=@?BADCFE&GIH/CFJLKNMFCPOQLRS"H=1H RUT H/C0VBWYXZVDWLW[V
\ ][^<_Y`ba9cdef_Yghai^
jlknmoqprm@sutwvlswx1yPm@zp|s~Pktpp|op|p|k1t~pt/9Y2~mswy)sytm@oqpzwphsqyPm@okzwss~ik-vihp|n@phzZmp| hp|¡fqsp¢D@oq£/ki¤m@o/sm@ hp¥b¦§p¨zwphswyPmosbmN£$ hktm@kiq©Dphzwphktm@op|x/ymtqkshª h@kswtqtytq@pphx1yPmtks|ªPktp|zwm@ |pm@k0D@ykPm@zu£< h@kqtk-sh¥«-Y s$m¬phk-phzmoip|t-®¡ft®tp¯@ooN¡fqk¬0¯phmty-zphs|ªbsp|p@ª°p@¥±¬¥ªY²/ p|®m@k´³ykv
hBphzw¬phzµ±¶F·¸ªB¹@¤m@k¬0µ lº ·¸ªm@k7µ l ·¸»
¼ m@-oq rm@qotl£Itn¬phk-phzmo>ªBsswo£3Np|zswDp| hqphLªPzsut¢zp|zfsw£/swtphishª
¼ sutm@qotl£¯oqoN¡fsqzp| |to£½¯zw¾¡Zphoqo¿Dswphk-phss¨@«tp h@k1tk1yy-s¢-zoqph0ª
¼ p|ssp|kt¤mo¿Dy-kPm@zu£< hk-qtks§mr£3BpiDswph<¡Zprm@À1oq£@ªm@k
¼ tpzphswyoqtk¬I-s |zptp4sw£/swtphÁ@phx1yPmtk-s¢qs§sw£/iptzq D@sqtq@pp|Pk-qtp¥
jlknPm@zwt |yo¤m@z|ª@«-~ZqsYm@-o hm@oqpt4mFÂ/¡p|oo¸ÃÄsYp|x1yPmtqksYqkÅPzsutbzwphz~¯@z0¥«ÆÇtÅt-pphzw¬phk- hpIphx1yPmFt@ksqk hoqyphȪL«-Y/phsk-@tswy-©Èp|z¯z@Étp$swyzqysswoyt@ks¢¡f- mN£/ h |yz$qk® |phzwtm@kU³4m@oqphzwÀ/qk7ip|t/shªsphpªbp@¥±¬¥ª°¹@¤m@k-¬ª«Æ7yLªbmkUÊN/k-phooq¢µ lrË ·mk®t-pBÀ£"¹@¤m@k¬0µ lº ·¸¥¦§p½sutzwk¬k-zÌzwphsqyPm@oikqqrmtk"@§«-Y qk7ts4swtm@kPm@zw7¯zw"ª[m@À@p|s$ |nv
y-tmtqk@fswk¬@yo¤m@zsoqy-tqksqÍi hy-oqtr¥Îk@tphz$qÍi hy-oqtl£7 hk |phzwksItp½¡Zphq¬tqk¬2@ft-p©Èp|zphktZzp|s-yPm@os|¥~jlkitqsZPmDp|zhª¡pm@zwphswsZt-phsp¢zwop|s|ª¡fqtn¯/ hysk½sutmt ¢m@kntqpPm@zw@k -zoqphish¥¦§p$m@Dp|z¨qsfz¬mk|ph"m@s¢¯@ooN¡fs|¥jlk/ph |tk º ¡Zp$-zphswphkttpm@¬k-p|t@swtmFt -zoqph
m@k7¯zÅy-o¤mtpitp½oqprm@sutwvlswx1yPm@zp|sÅip|t-Lª~kU/ph |tk Ë ¡p½zNp9m0zqzZm@kmD@swtp|zqzphsutq½mtphs|ªPkph t@k"Ïn¡pm@-oq£3tpip|t/0tim@¬k-p|t@swtmFt -zoqphishªm@k-k"p| |tk¶¡ZppÐÂtphk0tpip|t<tntipvÑPm@zikq ¨-zoqphish¥
º
_ <dÅ` ^g_ ^3_ _2a9c ! "$#&%(')#&#+*-,.%0/"21435+%!617"2ÎswsyiptPmt98;:=<?>@BADC&8 @ sZm$m@qkqkFEHGNª1¡fphzwp8Çm@k3phm@ I8 @ mzpByk-ph9m@k-3phtp|z@ hom@ssJ CLKMC ªP-y-t¢½mr£<BpkkvÑ hkpÐÂDªDz§ |kpÐÂȪÈm@k0phk-@tpt-pqk1tphzum@ hpBp|tl¡Zphp|k2zp|¬k-s8 @ m@kN8O£QP @ ONª¡fqtIRTSVUPªswphp~¬y-zp l ¥¿p|t
WX: >Y @BADC W@[Z º ¥ l
¡fqtHW @ : µ]\ C ^8 @ l· G ¥YΨsswyp¨tPmtZprm 3syBm@qk½Pmrp¢tp¨½m¬kp|t fBphziprmoqqtl£?_a` bdce:_ @f _hg$¡fphzwpF_ @f4ikj s4 hk-swtmk1tÅm@kl_hgÅstp½m¬kp|t iDp|ziprm@-otl£2qk7¯zp|p½swPm@ hpª_egm:Ïonqp l jdrtsvuxw "¥~¦§pm@¬k-p|t@swtmFt sw£/swtphÁtp|k0tm@À@p|s¢tp¯z0»~Pk-zyW sy 0tPmt
|p_ r+C y.:~ kQ8 @LZ º ¥ º m|.y.: j qkQ8 @Z º ¥ º Pyz: j @kNP Z º ¥ º r
m@k9t-p4qktphzum@ hp |ktqksµM_ r+C ypD·D: j kzP @ O Z º ¥ Ë m
µyD·D: j kzP @ O Z º ¥ Ë PoqL¥ u p|zp´qstpnpÂ/tphzqzyktkzm@obkt-piByk-Pm@zw£P m@k®m<Â-p|yktkzw½m@o~kprm qk1tphzwzYqktphzum@ hpP @ ONª/m@k"µM~D ·D: oq? g- 9NL&FvI¿«¡ftmP @ ONª1p|k@tp|st-pwy-9kIm@ hzwss§t-p4qktphzum@ hpP @ O ¥EF)Ne ^¡¦§-pm@swsyi-tqktPmt«t-pm@qks«m@zp hk@pÂI@z@- hom@ss¢J CLKMC qsk-@t~zwprm@oqsut Y¯@zphk-¬kp|phzwk¬m@oq hmtqks¡fp|zp¡Zp$tl£/q rm@oqoq£3@koq£9Pmr@pÅ¿sw qt-½mks|¥Æ®p¡foqo¿sw hyswsm@k0zp|o¤mF½tqs§zp|¬yo¤mzqtl£ h@kswtzm@qk1tf¡fp|k"¡ZpPmrp4zNp|p|zz@z¢p|swtmtp|s§k"ph tqk Ë ¥
Ω1
Ω 2
Γ12
Γ
~q¬yzp l »b¦§pk@tmFt@k0y-sph9¡fp|k2mIzp|¬k9qs§soqqt§qk1tnsyzwph¬qks|¥
Ë
L I* *#&" "21-" "2 a# '3,$"¢'¿p|t @ Z R: l Z Z Z Bpnph |Bswqtqks@t-pswyD-½mks 8 @ kt-ª«p¥Ä¬-¥qª[tptzmphzm@o>ª[p|opviphktsªÈm@k-2opt : >@BADC @ -phk@tptp$zwphsy-oqtqk¬iph |Bswqtqk0@T8¥¢¿p|t-phk@tp$t-piphs3¯y-k |tk<pPkp|"£` : : ¤m@0ª¸¥Äp¥qª-mÅphm@sy-zp@[tpo/ rmoDswhp@[tpp|opviphktsfk3tpp|sLªmk9op|tH: mF [phktptp¬oqPm@oDp|s0sqhp@¥¢kvѽmt k-¬iphs-phsm@zwpIm@oqoN¡p|LªBy-t¡ZpIm@swsyipÅo/ rmo«x1yPmsvÑyk¯ziqtl£9mkm½ikq½m@oLm@k¬oqp |kt@kk<t-ptzm@k¬yomtqkLª/sp|p$²zwphkk-phz¢m@k htt$µ·>¥¿p|t
Wx: >Y @BADC W@ Z º ¥ÄÏ
¡fqtIW @ p|kph"1£W @ :! 4´µ" g 8 @ · G »# $` %$ f Z ¯zfmoo& ' @)( Z
¡fp|zp*$ f qs§tp4spt@Ym@ooDph tz¨D@oq£/ki¤m@oqs@~ph¬@zphp4op|ssftPmk"zfp|x/ym@o«t,+/¥¦§/y-sxW$qst-p¨swp|t¿m@oqoPp| hp|¡fqsp¢ph tzZD@oq£/ki¤m@o/¯yk t@ksZLp|¬zp|p+¡fq <m@zp¢ h@k1tk1yy-sZqkt-psy-D@½m@qks08 @ sy 0tPmFtrªPk<phm@ p|op|p|k1thª- $`. /$ f 0Ð¥Pzt-pip|zzwzm@kPmoq£/sqs¯ooqN¡fk¬3BphoqN¡ªÈ¡pnkp|phtpů@ooN¡fqk¬<m@-zrÂmtqkzwBphzwtl£
@xW1ªZsp|p@ªp¥Ä¬¥ªµ l Ï·f¯znmzw@ ¥´¦§p|zpqsim@k qktphzwD@o¤mtkUBphzmt@z n :32 >@BADC)n @ ¡fqtn @ »Bµ]\ C 8 @ · G546 Wswy "tPmt§¯z7 F82 >@ ADC|µ\ ^8 @ · G ªPtfo-s9 ln& 9;: K =<>"?-@ r : ` $` @ K A#B DC Z E : j Z l Z º ¥ ¶@
¡fp|zpFV:&ik¿G+ l Z ªIHf0fsftpPmt @bphoqphiphkts¨kphq¬1D@zk-¬JªDm@ktp$ |kswtm@kt"qs¢qkp|Dp|kphkt¢@«tpiphsw"Pmzm@ip|tphzKL¥)L MON"l1-"¢')#QP 'RKS5+"¢'UT *#&""217"2"2 #z"e#N%0/VX*#NVl"¢DW6%S /5-X
T / * a#&"¢5Y ,$" ,$%! / *L#*7%! '¦§p4soqy-tqkIy tnzwop| º ¥ º [ZÈ º ¥ Ë qki|phsZtpophm@swtuvlsx1yPmzphs§¯yk- |tqkPm@o
\È^yI: >] @BADC*^ 9 _°| p ^_ r+C yIX~[ 9`_b c 9 |.y 9`_b cGa º ¥cb ]Cedd@gf O d > ^ 9
r+C0h _ji_°µM_ r+C ypB· 9 _k c l 9 r+C0h _ µ]yD· 9 _k c l a 9 r+C0h _ µ]yD· 9 _k u p|zp¡Zp7Pmr@p qktz/y |ph m¡p|¬tQ_8k5tp2@oy-pqk1tph¬zm@otphzis9 hktm@kqk¬t-p hyzwo
phx1yPmFt@kL¥ ¦§snsimkPmtyzmo§s rmok-¬@tp9phx1yPmtksim@k-´tp"oqprm@sutwvÑsx1yPm@zwphsI¯yk t@kPm@oqst1ys$Dpttp|zÅPm@om@k |phUBp|tl¡ZphphkUtp3 hy-zomk®t-p3q@phzw¬phk |p< hk-qtks|¥¦§p½Pm@zmptphz i_ h@zzp|sBkst½t-pÅÅyot-o hmtqk3¡fqt_k"tp$@oy-pktp|¬zm@oqshªBy-tqsm@kmr@phzm¬pÅm@oyp
Ï
!#"$&%(')* +%
~q¬yzp º » YÂ-m@ioqphs<@½mFt qk¬´m@kÇkkvѽmFt qk¬tzw¤m@k¬@yo¤mtk-s½@4tl¡ syBm@qksh¥jlk tp0zswt½ rm@swp@ª¡Zp2Pmrp0tp" q hpDt sutzwk¬mk ¡Zprm@À´phk¯z |phiphkt3@ktp|zwm@ |p h@kqtk-sh¥°Pztpkkvѽmt k-¬I¬zqLªN¡Zp|p|zhª¡p4Åysutfysp¡Zprm@À<p|k-¯z |phiphktr¥
sqk hp!_3sk@tp|kphnk$tpqktphzum@ hp¥Æ7p¡foqozptyzwkÅt¨tppÐÂm |tbpPkt@k$@ts~mrphzm@¬p¥"zwphNp|zhªBk9tpBykPmzw£0m@k0ktphzwm hphs|ªPtp-, _ P«lvÑkzws hm@op|"£% r+C0h _ Pms¨Bphp|kyswphk-swtphm@i@Ètpf hiy-tmtqkPm@oqoq£4 hy$Dp|zs@p\ C0h _ ^PÑvlk@z0ªsp|p²/ pim@k<³yk|Dp|z¬p|z¨µÄ¶F·m@kÅtp§zp¯phzwphk |phsbtp|zp|kL¥bPzkkvѽmFt qk¬iphs-phsbqtY½mr£Dpk-ph hp|ssm@zw£Ityspfm@knmr@phzm¬pm@oqsn¯z¿¥Z¦§p4oprmswtwvÑsx1yPm@zwphs¨pt/my-k1ts§tnPkqk¬its¢ikqqhphz|»~Pkzy W sy tmt
\È^yI: k-. / \D º ¥10@
EF)Ne ?¨@tptPmt«tpBykPmzw£ h@kqtk º ¥ º N[m@ktpktp|zwm@ |pZ |ktqksZ º ¥ Ë ¿m@zwpiBsp|n¡ZprmÀ/o£¥/tzk-¬oq£IqBswphnD@ykPm@zu£im@kktp|zwm@ |p¢ |ktqks°Pmrp¢Dp|phksy¬¬@phswtphkÇp¥Ä¬¥ ¹@¤mk¬ª¨Æ7yLªmkÊ«Nqkphoqonµ lNË ·¸ªytmzpkkv>tzwq/¤m@o¢t´iop|p|kt<kÇBoq£/p|zm@om@qks$swk |p3tp<k@zm@oZqsÅk@tn-p|Pkp| pp|zw£1¡fphzwp@¥ Pyzutp|zhª°phk-¯@z hp|p|ktItzwy¬Ut-pophm@swtuvlswx/ym@zp|s¯yk tqkPm@obm@À@p|st4DswsqopIt"yspq©Dphzwphkt4Pkqtpnphop|p|kt$swPm@ hp|s$k7phtp|zsqp½tpsyzum@ hp|shª°m@k-®tp hk@phz¬@phk |p<zBphzutqphs$tppt/®qsBp|twtp|z¡fp|kUysqk¬¡Zprm@À1oq£3iDswph9 h@kqtk-shªswphp-2swtm@Bpho«m@k43m@y¬@p½µ lOl ·«m@k"²m@im@k65Zm@k¬µ Ë ·¸¥
Î8kph |phssm@zw£0 |kt@k3¯zfmůyk |tkQyWÇtnsmtsu £<phx1yPmtk7 º ¥10@ªqs
oq7 g
88:9\D^y 9 iyn<; º (=D^y Z iyI?>w iyn: j Z º ¥c
¶
¯zfm@oqo iy Wª¡fp|zp=By Z iyÅ(: >] @BADC [|py Z |p iyI b c |.y Z |. iyI b c º ¥
]Cedd@ f O d > ^ [r+C i_°µM_ r+C ypB· Z i_bµ]_ r+C iykpB· k c l
[ r+C µ]yD· Z µ iyB· k c l a[ r+C y Z iyL k Z
m@k>u iyn: >] @BADC _~ Z _¢| p _ r+C iyIw7bdc º ¥ l j
¦§1ysswphphÀ1qk¬Im$ikqqhp|z~t*\IkHW oprmst$t-p¨m@zmtqkPm@o-z-oph0»[Pk yW swy 3tPmFt=By Z iyÅ: >u iyI Z º ¥ lOl
¯zfm@oqo iy W¥¦§p$oqprm@sutwvÑsx1yPm@zwphsfPkqtpp|op|p|k1t¢ip|t0sfpPkp|2£"swphp|À/qk¬3mk2m@-zrÂmtqkIyÅqkWswy "tPmt \DyN(: k-
. / \D Z º ¥ lº ¡fqt9 hzwzphswD@kk-¬im@zw¤mtkPmoB¯zw"»PkNyW4sy 0tPmFt
=By Z iyÅ: >u iyI Z º ¥ lNË ¯zfm@oqo iy W¥ &`°`bai` _Yg _L9 35 *-%(5 *"¢5+5+%(5 "2'd#+*7#"¢'Æ7p4Bph¬@k9£<ktzw-y hqk¬Itpphk-phz¬£<kzw
` `` $`` ` _ : =B Z ¯z7 FW Z Ë ¥ l ¨k-phz$@yz4m@sswyt@kshª` `` h`` `Ds4m9kzwÌk W)¡fp|k®tp|zppÂsutsm"ykqx1ypswoyt@kt
zwop|Ì º ¥ º )ZÈ º ¥ Ë ¥Pyzwtphz|ª¡p4Pmrptp¯oqoN¡fk-¬$ktphzBomtqk3phzwzzfp|swtq½mFtp@¥
Tt$ ^¡ 2 >@ ADChµ\ ^8f · G Z l !" #%$&(')!"*+&(,&-$.!"#&( nÌ»2 >@BADC|µ]\ 8fl· G 46 W /0-12!"
`` ` ln& $`` `Q< " -@ r+C ` $` @ Z Ë ¥ º 3 F : qkLG+0 l Z !"54 3 " &('76849:$54(;&( _ !"54 8=<
b
§¿p|tm: n& ¥bÆ®ps-¡ tPmFt` `` ¢` `` _ < " ] r _ 9 9 _ ` ¢` _ CLK Ë ¥ Ë
¦§p4phswzwph0zphswyoqt Ë ¥ º Ztp|k0¯@ooN¡fs¯zwÌ º ¥ ¶@¥°Pzt-p4qktphzwzZtphzws|ª-¡Zpqzp| |to£3Pmr@p9 |p 9 _ < " ` ¢` _ CLK Z Ë ¥ÄÏm@k 9 |. 9 _ < " ` ¢` _ CLK Ë ¥ ¶@Pzt-p4Byk-Pm@zw£½tphzw ¡Zp4¬@p|t
9 r+C0h _ 9 _k : ] kA 9
r+C0h _ 9 _ k Ë ¥cb< ] kA "?
r+C 9 9 0 r+C 9 9 ` ¢` CLK < " ]
kA r _ 9 9 _ ` ¢` _ CLK Z
¡fp|zp3¡Zp3ysp|tptzm hp3qkphx1yPm@oqqtl£ 9 9 _ < " 9 9 0 r+C 9 9 ` $` CLK ¥~kPmooq£@ª«¯@zt-pktphzwm hpawyi9tphzws¡Zpy-sptptz¤mk¬opkp|x/ym@otl£9 r+C0h _ µ D· 9 _k c l < 9 r+C0h _ @ 9 _k c 9 r+C0h _ O 9 _k l Z Ë ¥10@m@k 9 r+C0h _ i_ µ]_ r+C pD· 9 _k c l < 9 r+C0h _ i_ ^_ @ r+C @ pz 9 _k c Ë ¥c
9 r+C0h _ i_ ^_ O r+C O p 9 _k l Zm@k9t-phk9tzphmt§t-ph¡fqt<tpsmptph kqx1yp$msf¯@ztpBykPmzw£3tphz0¥
EF)e ^¡ÅjÑLtp¢iphs-phs @ ½mt ½knt-p¨qk1tphzum@ hp|sm@k-4n& IsmtqswPp|stp¨qktphzum@ hp¢ h@kvtqks4 º ¥ Ë ª-tphk9tp |kswtm@kt ")sfm@oqsnk-phBphkp|kt@ _sqk hpt-p4qktphzum@ hptphzism@kqsL¥
¨N¡¡Zpm@zpzphm@-£3tnswtmFtptp¯oqoN¡fk-¬Åzp|syotr» ) e y 2 >@BADC|µ]\ 8fl· G ! &('7/9 +&( #&! <#"$"&% !"54 yW !$ $5&(' #) !": &('7/9 +&( 49+* 4 6! <#"-,(%"</. ! 0-&(2+!" " 549:$549 & / 0- 2!"
` ``MyVy`` ` < " @ r+C `MyN` @ Z Ë ¥3 F : qkLG+0 l Z <
0
§¿p|t: y Vy$p|k@tptpphzzwzh¥°¦§p|k` ``&``` _ : =B Z yVyF Ë ¥ l j
: =B Z yn¢y n$ylyF: =B Z yn¢yÅ Z
¡fp|zp9¡p9yswp9tp"³4mophzwÀ1kUzut¬@kPm@oqqtl£ =B Z iyI: j ª°¯z iy Wª°kUt-p0om@swtÅphx1yPm@oqqtl£¥¨swk¬Itp-2m@y £vu ¡m@zÅqkphx1yPmoqtl£3m@k0q/qk¬$1£V```&`` `¡Zp$mzzpmt
`` ` &` ``Q< ` ``Myn¢yQ` ``<>" -@ r+C ` yN` @ Z Ë ¥ lOl ¡fp|zp¡pyswph2¿p|m Ë ¥ l qk3tpo¤mswtfk-phx1yPm@oqqtl£¥
L9L 3 %!'d#"¢5 *-%!5 *"25+5+%(5 "¢')#*#&"2'Î sqoqpÅ rmo hy-o¤mtk¬p|stPmttpp|kphzw¬@£k@z @Ztpnp|zz@z¡f-phk®y-sk-¬"m9qok-prm@z¢¯zp|zq@phU¯z@ oqprm@sutwvlswx1yPm@zp|sIzqk hqop|s$swioq£p|x/ym@osÅtp3zp|sqyPm@o¸¥¦§/y-sn¡Zp< rmkUswtmFtp3t-p¯oqoN¡fk¬ÅmnBsutphzwzwLphzwzz§p|swtmtpqk9tpphkp|z¬@£9k-z0¥ ) e y W 8! &('7/9 +&( #& ! <#"$"&% !"54 yI W 8!$ $5&(' #) !"# &('7/ *+&(49+* 4 6 ! <#"-,(%"</.
` ``My y``` _ : ] yF ^yF Ë ¥ lº
3 = ' &) -*4"/2!"' ^yFE 49+* 4 6
^yFT:
9 _b[|pz_ r+C yvX~¿ 9 9 |.y 9 C _ 9 r+C0h _ i_bµM_ r+C y pzD· 9 k9 C @ r+C0h _ µyvB· 9 Z Ë ¥ lNË
3 F ! / 0*+&(2 / 0- 2!" F : l &( !90-- 3 P: !"54 F : º &( 3 <
EF) e ¨kp|zyzmssy--tk-shªDtphswpIzp|syotsm@zwpÅiprm@kqk¬@¯yo[swk |pÅ¡pÅmN@pnphky-¬zp|¬yom@ztl£7@k®tp½soqy-tqkly½¥jlkUizwp½zphm@oqswt isqtyPmtks4¡fphzwpI8 s$@koq£7¿sw qt@ªY¡ZpPmrpy 2.>@BADCµ\ ^8f · G ¯zF É l@º Z l ·¸ªswphpΨzwy pªf²phzwkPm@zw>ª 3my¬p"m@k¢mrm@zwtµ º ·>ªLm@ktpIoqprm@sutwvÑsx1yPm@zwphsPktpÅp|op|p|k1tpt/¬phk-phzmooq£0m@qosft< h@k1@phzw¬pIt<tp$ hzwzph ts@oy-tkLª¿swphpp@¥±¬¥3²zm@Åoqp@ª«[m@rmzNm@k®Ê~m@sw hm@À´µ0N·ztpnzp|¬yom@ztl£zp|syots@ 2@swtmDp|oµ l j ·¸¥ u N¡Zp|@phzhªLtpÅyswpÅ@b¡Zprm@À0BykPmzw£m@k2qk1tphzum@ hpÅ |ktqks¢swtqoo[¬phsmt-phzwp|tq rm@o
h@k1@phzw¬phk |p@ªBsp|p 2sutm@Bpho[mk 3my¬piµ lOl ·¸¥PzI½mk1£m@oq hmtqkshª~kp3¡m@ktsntiprm@swyzp<t-p<phzzwzIkUtp|zix1yPmk1tqtphstPm@k t-p
phk-phz¬£7k-z0¥½jlk tpzphm@qkphz@Ztqssph tqkLª¡Zp½y-sp½-yPm@otl£m@zw¬yiphktst0phzwq@p½m@k®mBswtphzqzÈphswtmtp¯@ztpt-p4p|zzwz§k9phk-phz¬£ª
ff f : lºb _
r+C `MyN` _ lºb _r+C ` y` _ Ë ¥ l Ï1
"zwphNp|zhªLts¨phswtmtp4s-¡fstPmt¡ZpÅ hm@kPmrpÅ |kp|z¬p|k hpnp|@phk¡fphk y lWªÈm@sqs¨t-p rmsp¯z¢mI¬phkp|zm@o[¿qs t@½m@qk3¡fqt0 |zk-phzs|¥
e $ ^¡ I + ¢2^)¦«i-zNp4phsutmtphs@«t-p4p|zzwz§k3tpphkp|z¬@£@ª¡pswtm@zwtf£9suty-£/qk¬Itp¯oqoN¡fk¬z@op|"»Pk® Z a\ |qp Z 8f9p\ Cg ^8f§smtqsw £/k-¬
|pz_¢|pH | : kQ8 @ Z Ë ¥ l ¶@m|. :"! kQ8 @Z Ë ¥ l ¶ : j Z Npz: j @kNP Z Ë ¥ l ¶ r
¡fqt9ktp|zwm@ |p4 |ktqkµM_ r+C zpB·D: j kQP @ O Z Ë ¥ l bm
µ#HB·D: j kQP @ O Z Ë ¥ l b@µM_¢|pzpB·D: j kQP @ O Z Ë ¥ l b@ r
µ ·D: j kQP @ O Ë ¥ l b@u p|zp3¡Zp9ktzwy- hphtp<sm@ hpQ\ [| p Z 8f4: ! X µ]\ 8f · G »(| p ; µ]\ 8fl· G ( ¥ ¦§qszwop| sZm@kPm@o£/sph½k 2-phkLª 3yLª-mk%$Èy2µ F·>ª1¡fphzwpqtqsZswN¡fk3tPmtt-phzp¨pÂsutsmykqx1yps@oy-tkL¥Yy-zwt-phzizwp@ª ;
j ¯zfm@oo&8¡fqtN|. : j ¥e $ ')(*^+ , N) -.;/2d01¨swk¬7tp u phoq-oqt"p| hiBst@kLªswphpp@¥±¬¥²ssmrt µ b·>ª¡Zp rmk ¡fztp"tp2p|zz@z V:y yN: y g ^y g y32 Ъ~¡fp|zp3¡Zp3PmrpQ| g : j ªbm@k 2 : |54 ¯z$swps hm@o¤m@z¯yk tqk64¥YÆ®p /swp =:=y g y g Z Ë ¥ l 0m@À1k¬m|7 : j m@k-3t1ys : j ªmk%! swy 0tPmt§tsmtqswPp|stpm@y/Â-qom@zw£I¡Zprm@Àz@op|"»Pk8! \ C 8fsy 0tPmFt
|9! Z | : >] @BADC [| 1_ r+C y 2 Z b c ]Cedd@ f O d > µM_r+C y 2 B· Z k c l Z Ë ¥ l
¯z$m@oo =\ C ^8f¥¦§s$z-oph qs¡ZphooZD@sph´m@k-¡p<Pmr@p3tp3p|x1yPm@otl£ Å|9! Z y32 :^_ r+C y32 Z y 2 sqk hpy32 sfmn¬zmp|k1tfYmIs hm@o¤mz¯yk |tkÈ¥¦~mÀ/qk¬9tpikk-phzz/y t$ Ë ¥ l ¶@m¡fqtq_ r+C g :_ r+C y g ;y g m@k7@ Ë ¥ l ¶¨¡fqt|. 2 : x| y 2 ª£/phoqs¯ztpz¬@1tfm@k0s-p
^_ r+C y g Z y g ^_ r+C y g Z y g Ç ! Z |.y 2 Ë ¥ l :_ r+C y g Z y g ^_ r+C y g Z y g $ [| ! Z y 2 5.! Z µy 2 B· k ]Cedd@ f O d > .!
Z µ]y 2 D·¤ k c l: ff f [|. Z µy 2 D·¤ k ]Cedd@ f O d > [|. Z µ]y 2 D·¤ k c l
Æ®pt/y-s¨¬pt ff f : >] @BADC ^ ^_ r+C y g Z y g b)c ^_ r+C y g Z y g bdc _ r+C y 2 Z y 2 bdc a Ë ¥ º j
]Cedd@ f O d > [|. Z µy 2 B· k c l |. Z µ]y 2 D·¤ k: >] @BADC?^ [| pz_$|p Z _ r+C g b c [| eZ _ r+C g b c |. Z | 2 b c a ]Cedd@ f O d > [|. Z µ 2 B· k c l [|. Z µ 2 B· k
: >] @BADC*^ [|p Z |p g bdc [|. Z | 2 bdc _ r+C $Z |. g bdc a ]Cedd@ f O d >^ _¢|;p
Z µM_ r+C g pD·¤ k c l 0^_ r+C $Z µ 2 B· k c l 0[| Z µ 2 B· k c l a|. Z µM 2 ·¤ k
: >] @BADC*^ [|p Z |p 7bdc&[| Z |.7bdc a ]Cedd@ f O d > ^ ^_¢| p
Z µ]_ r+C ¢pvD·¤ k c l [|;p Z µ + D·¤ k c l a [|;p Z µ 2 B· k: >] @BADC*^ [|pU HlnÈ Z | p bdc [|./ 4lnD Z |. bdc a ]Cedd@ f O d >^ 0
C0h _ji_ r+C _¢|;p r+C0h _ µ]_ r+C !nÈ&pD· Z i_& r+C0h _ µM_ r+C ¢pvB· k c l [ C0h _ |. ? r+C0h _ µHn ÈD· Z r+C0h _ µ B· k c l a
0 C0h _ |. ? r+C0h _ µq HlnÈ2D· Z r+C0h _ µ D·¤ k l j
¦§p¯ooq¡fqk¬Åp|swtq½mFtp¯ztpphzwzzfk<tpphkp|z¬@£<qstp|zp|1£0zwNphÈ¥
) e &('7/ *+&( :& , <#"&% 3 4 !"%!8!90-0-&(.4" :& , <#" % !"54 , <#"(% y &('7/ *+&(& ! <#"$"&% !"54 y !$ $5&(' #) !" +&( ! <#"-,(% &( "6 &(/9549 4 !( yI ^y5< ] yN= È Z Ë ¥ º'l
3 ' &) 3 -*4"/2!"' yF 49+* 4 , <#"-,(% !"54 ' &) 3 (9 = D
= È:
9 |p 4lnÈ 9 9 |./ HnÈ 9 C _ 9 C0h _ i_ r+C _$|p? r+C0h _ µM_ r+C Hln ÈapB· 9 k9 C @ C0h _ |. ? r+C0h _ µq 4lnDD· 9 Z Ë ¥ ºº
3 F !( , <#"-,(%"<
EF) e "Îswsyik-¬9 2 >@ ADC µ\ ^8f · G m@k-zy 2 >@BADC µ]\ ^8f · G ª-tpisutfiDzutm@kt h@ksp|x/y-phk |p½@Z¦§ph@zph Ë ¥ÄÏ9stPmt¯zm<¬p|ip|tzu£2swy 7tPmt i º ªL¡ZpnPmrpi hk/vp|z¬p|k hpkt-pip|kphzw¬@£p|@phk hkp|z¬p|k hp½ktpnswoy-tktsp|oq rmk7ktBpnzN@phL¥¦§pm@kPmoq£/sqsqk5µ ·Zm@kǵ º ·Z½m@Àphsqt4zphm@skm@opit"Dp|oppitPmt º Yª~¡ftl ) lº Z l ·>ªm@À1k¬Åtstzypk9m@k£< rmsphs|¥ZÎ(m |tf¯yzwtphz§qkq rmtph9£9yz§k1yiphzq rm@oÈpÂDp|zqp|k1tsh¥
a i^_~a4«__Ygre"!`ba$# % &l%!k3 Sv##*7%! T1H't"$# S3Îot@y¬ik@tpÂoq hto£Åiphktqkphiknt-p¨zwp|/@ysZswph t@kshª/kpfmN£imos¡Zphq¬tZtp¨-q©Dphzuvphkt¨tphzws¨k9tpophm@swtuvlsx1yPmzphs¨¯yk |tkm@o[1£0 |kswtm@kts¡ft@y-t m@k¬qk¬tpÅm@km@oq£/sqsh¥§Æ7pPmrpysp|3tqsZBsswqotl£$t½mÀ@p¨tpp|k-¯zw hphiphkt@Lt-pktphzwm hp¨tphzwsswtzk¬@phzh¥°Pz°t-pPzwswtÅ-zoqph ¡Zp¡Zphq¬tph7tpntp|zis |ktm@qkk¬9t-p½kzw½mob hiD@kphkts4£7m0m@ tz l j G ª¡fqop§tpftmk¬phkt¤m@oP |kt@kn¡m@sop tZy-k1¡Zphq¬tp|L¥@zYtp¢sp| hk½z-oph |kswp|zphȪ/¡Zp¡Zph¬@1tphÅtptmk¬phkt¤m@o/ hk-qtk4£Im¨m@ tz l j G mkÅop ttpk@zm@o1 hk-qtqk$y-k1¡Zphq¬tp|L¥¦§qs¢ - |p4~ |pÍ |p|k1tsfPm@sfDp|phk"Pm@sp|"@k"k1yiphzw rmoLpÂBphzqp|ktsh¥ÎPm-ttl£½¡m@sfPm@swph9k<tpmIBswtphzwzwÈzp|syotfk<¦§p|zp| Ë ¥±ÏªPm@k-"mk"mssy-p|0zwph¬y/v
o¤mzqtl£I :8Ï @Ë @ [ªBtpÅswoy-tk9titp$yPm@o«z@op|"¥§Æ7p$Pmrpt1yskts@oqp|t-pÅyPm@ozwop| k1yiphzw rmooq£@¥Y¦§pphoqphiphkt¢k- rmFtzy-sph9¡m@s¢tphk
\ : C0h G yF Z ¯Ï¥ l ¡fqt yFm@sfk9phx1yPmtk7 Ë ¥ lrË ¥
lOl
[qkprmz yPmzmt ¢p¯phzp|k hp¨i@~p|op|p|k1ts ¶ j ¶ 0 l j º Ï 0 joj v¨n@k/phs l ¶ l j ËË Ë b v @ f >¹ b 0 p l j rts j j lNË j l p l j rts j j j l j ?p l j rts3>¹ Ë Ë@ËË p l j r j j 0 Ë ¶ l p l j r j j j Ë b l Ïmp l j r 4>¹ Ï #¶ p l j r+C g j j Ë@Ë Ï j º p l j r+C g j j l ¶ Ï 0 Ë^l p l j r+C g¦m@op l »0¦§p |iy-tp| m@¬kptq 0p|kphzw¬p|si¯@zÊbzwop| l ªy-sk-¬«-Y mk qph |p|¡fswpoqkprmznm@k x1yPmzmt 0D@oq£/ki¤m@oZphoqphiphkts|ª§ hiPm@zwph ¡fqt zp¯phzp|k hp"@moyp|s§tp"zp|o¤mtqpphzwzzs¬p|k<k<Pm@zwphkt-phsqsh¥°¦§pzwp|¯phzwphk |p@moyp|s§m@zp¨¯z tl¡nqp|ksqkPm@oB h@ytmtksk-pmFtfβ²)µ Ï·>¥
% L q5+%!61-" $ ^¡ I/( 2^)¦§pi¬ph@ptzw£@tqs4-zoqph sp|s hzwBph7k ~¬@yzpÅÏP¸mÐ¥I¦§p-zoqph smFÂsu£/ip|tz kizwphzt4½m@Àp¢tl¡ZÅqp|ksqkPm@oP |iy-tmFt@ks°Dswsqop¢m@sZzp|¯p|zp|k hp¥bÎ t-zphp¢qp|ksqkPm@o/p¡¾ hm@k Dp½sphp|k k ~q¬yzp3¶¥¦§p<i-pho |kswswtsÅ@¨m@kUqzk £/ok-phzÅ |zp½phk |z |op|U1£m7 |Bphzi¡fkqk¬¥Ç¦§p" hk-¬yzmFt@k siphk- ho@sph5qk mzm@k swyzzwykp|5£ m Dr´¡fqtBphzw¯p| |toq£< |ky tqk¬nsyzum@ hp|sh¥¦§p4¡fqkqk¬Is§i-phooqph9m@s¢mIi¬p|kph@ysf hBphzf |o¸¥
3mtm¯zÅtsnzwop| m@zp9zwpho¤mFtq@p<½m¬kp|t 9Bphziprmoqqtqphs _ f K 4: l j m@k-;_ f K q:_ f K @ f : l m@k- _hg: Ï nVp l j rts uw m@ktpI |yzzwphkt4phk-sqtl£~ s h@kswtm@ktN@phztpI |zswssp| |tk0@«tp h@o¿m@k-9tpt@tm@oL |yzzwphktqs l Î$¥¢p¯phzwphk |p$ h@ytmtksk9tl¡iiphk-s@ks§kp4£<β² m@k-"zwphBzwtphqk7µ Ï@·¸ªP¬mrpt-pm@oy-phs@tpn½m@¬@kp|t nphkp|z¬qphs4ktpi©Èp|zphkt4mtp|z¤mosm@soqswtph k ¦~m@-op l ª[¡fphzwpt-pm@¬kpt phkp|z¬@£9qs§p|Pk-ph£
bdce: lºbdc y\ ¯Ï¥ º
$ &++2( (7 (¦§qs$zwop|Ì¡m@s$soph®sy | hphswsw¯yoqoq£7t¬@/m@ | hyzm@ |£@ªbsp|p3¦~mop l ¥"Î Pphoqoqkpio@t4qss-¡fk k¬yzwpI¶¥jlk ~¬y-zp Ë ¸m¢¡Zpnott-pIzphomtpÅp|zzwzktpIm@¬kpt $phkp|z¬@£2m@sm¯yk tqk½@Ètp¨ph¬zwphp|s@L¯zwphp|0ª ª¯z°x1yPm@zmtq mk3oqkprmzBo£k-m@oPmss°¯yk tqkshªm@k0k9¬yzwp Ë PZtpphzwzzfqk hmt@zÏ¥ l m@k-9tpophm@swtuvlsx1yPmzphsf¯yk tqkPm@o¿mzpoq@ttphL¥Æ®p rm@kÅk-@tptPmttpZzmtp@ |kphzw¬phk- hpfs[tpsm@ipZ¯zDtÅoqkphm@z«m@kÅx1yPm@-zmt Bo£1v
ki¤mos§ypttp$oqN¡&zp|¬yo¤mzqtl£3@btpz@op|"ªko£<tp$m@ h |yzm |£"s¢q©Dphzwphktr¥fÎk-@tp|zk- hmtz[@-tpZoN¡ zp|¬yo¤mzqtl£s[tPmttpZophm@swtuvlsx1yPmzphs¯yk |tkm@o1p|s~kt~p| hzwprm@swp¡fqtzpÐvPkp|p|ktrª-qkim |tZqtk |zphm@sp|sh¥ u N¡Zp|p|zhª-tpqk hzwprm@swp¨qsZsw½moop|zY¡fqt½x1yPm@zmtq ¨Bo£k-m@os|¥
lº
0 1 2 3 4 5 6 7 8 9
x 105
10−4
10−3
10−2
10−1
100
AirFeCuTotal
- !&
0 1 2 3 4 5 6 7 8 9
x 105
10−5
10−4
10−3
10−2
10−1
Error indicatorLeast−squares functional
') )% & ) '
~q¬yzp Ë » 2iy-tmtqks[¯zYÊbz@op| l ¡fqtÅx1yPm@zmtq ¯soq$oqkp|s[m@kÅoqkphm@zZPmsp|Iok-phsPm@sws§¯yk tqksh¥
0.20
0.20 2.00
0.05
0.301.00
1.00
Iron
Copper
Air
! #"
2.001.00
1.00
0.05
0.100.300.20
0.01
0.20
0.200.30
Copper
AirIron
! ! $ &%
~q¬yzpbÏ-»«³p|ptzu£@/tp°tl¡ZmFÂqsw£/ip|tzq Yz@op|s|¥¦§p°qp|ksqks[m@zwp°¬q@phk4kptp|zsh¥
lNË
~q¬yzp4¶/»b¦§p½m@¬@kp|t Pp|o9oqkphs§qk"mIswo |ptzwy¬9t-ptzp|piphkswkm@oÈswoy-tk<~t-pmFÂsu£/ip|tz Ê°zwoqph l ¥
~q¬yzp*b-» 3ptm@qo[@Yt-p$iphs2m tp|zm@m@-tpzp|kphiphktr¥¨¦§pÅPmzwt¨sN¡fks¢tp4t"@bt-pzwk9 hzwp@bÊbzwop| l ¥
l Ï
¿qkprm@z yPmzmt ¢p¯phzp|k hp¨i@phoqphiphkts lNº b 0 j º l ## b Ë v¢n@~k-p|s º ¶ ¶ Ë 0 º b lNË b º v @ f >¹ l ?p l j rts j º j j 0 ¶ 0 Ë p l j rts j º b l l j º ¶ p l j r <¸¹ Ë º 0Ïmp l j r j j #b Ë Ë Ï1¶?p l j r j j b#b Ë ¶"#b?p l j r 4>¹ º l=lNË p l j r j Ï j 0 l ¶ l bmp l j r j ¶ 0¶@ Ë ¶ b@Ïmp l j r¦m@op º »0¦§p |iy-tp| m@¬kptq 0p|kphzw¬p|si¯@zÊbzwop| º ªy-sk-¬«-Y mk qph |p|¡fswpoqkprmzfm@kx1yPmzmt 4D@oq£/ki¤m@oDphoqphiphkts|ªD |m@zp|"¡fqt"zwp|¯p|zphk- hp$m@oqyphs|¥¦§p$zwp|¯p|zphk- hpm@oy-phsfm@zwp¯zw¾tl¡Z½-iphkswkPmoÈ h@ytmtkskp4mtfβ²µ Ï@·¸¥
% )L q5+%!61-" $ ^¡ I/( 2^)¦§pÅswph |kzwop|Ás¨m@osw3m@kmFÂsu£iip|tzw 4½m¬kp|tswtmtq z-oph0ªy-t¨tp¬phip|tzw£9qsizpI |-o hmtp|tPm@k k7Êbz-oph l ¥jÑt4qs4¬@qp|k7qk ~¬y-zpIϯ¥¦§p |Bphz¡fk-k¬9qst-p4sm@ip@ªPy-tftpz@k<Pm@zwt¢Pms¢Bphp|k"pÐÂ/tphk-phmkmoiswt§p|k hoqsp|s§tp h@o>¥b¦§-p4-zoqphPm@s¨Dp|phktphsutph"¡fqtN_ f K !: l j ms¨¡Zphoo«m@s¢¡fqtN_ f K !: l j _ ªPtp$@tphz¢Pmtm½m@zwp4tp$sm@ipm@sfk0Êbz@op| l ¥
$ EF/( (¦§qszwoqph PmsktBphp|k®smtqswm@ tzwoq£"s@oqp|7¡ft7Y7¥«@ztpi rm@swpi¡ft _ f K :l j _ ¡ZpswtoqoPmrp" hk@phz¬@phk |p@ªfy-tn¡fqt B/zim@ h |yzm@ £ªswphp"~¬@yzp40m@k ¦~m@-op º ¥´¦§p h@ytmtks¡ftI_ f K : l j s§k-@tfsy | hp|ssw¯y-oYm@k9kt¨zwphBzwtph9¯yzwtphz|¥Æ®p hm@k0m@oswIsp|ptPmt§tpyswp@«x/ym@zmFt Do£/ki¤m@oqsZ/phs§k-@t§¬pBp|twtphz§zwphswyoqtshªt-p
sqk¬yom@ztqphsmzpt/IikPmk1tfqk3tqsf¬phip|tzw£@¥
9_^Yghai^ _Ya _ _Yg v$`( ai^gre e0 ="¢'&,$5 *73v#*7%! %YF#-N".35+%!617"2Îs¢mItqz9zwop|¾tpm@sw£/iptzw rmoL hk-y |tz§¡fqtmIoqpz@op|0µ l ¶F·¸ªm@sfoqoysutzmFtphk3~¬y-zp l j ª-PmsDp|phk9yswphL¥°jÑt§ |ksqswts@m@k9m@oqyqky- o¤mFtp¨¡ft<mÅ-opª-om@ hp|<ykp|z§m h@Dp|zb¡fkqk¬ªi-phop|nm@smi¬phk-phys° |o¸¥¦§p|zp¨m@zp§k-su£iip|tzwp|sbkItqsbz@op|"¥¦§p½m@oyik-yÉo¤mFtpnPm@sm" |ky ttl£@;: Ë ¶ º b p l j s w m@k tp½m¬kp|t IBphz viprm@qotl£½qs_ f K : l ªDms¢qk0tp4m@zfmk"tp hBphz|¥§qk hp4ki½m¬kp|t mtp|zm@oÈqs¢-zphswphktt-phzp½m@zpkswk¬@yo¤m@zwqtphsms$kt-p½zwp|/y-stl¡Z2z@op|s|¥0jlk-swtphm@Lª~¡Zp½¡fqoob¬pt$qky |phph--£9 hyzwzp|k1ts¨qk<tp h@ky tk-¬moyikqy o¤mFtp@¥¦§pn hqo~qs rm@zwzw£/k¬9m<swk1ys@Pm@ot@tm@o~ |yzzwphkt4@ º 0Ï º Î$¥È¦§pI¯zphx1yp|k |£qs¶ j u mk
t-p4 |yzzwphkt¨phkswqtl£<qs§ hk-swtmk1t¨Np|zftp hzwss¢sp| |tqkL¥
l ¶
0 1 2 3 4 5 6
x 105
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1AirFeCuTotal
- !&
0 1 2 3 4 5 6
x 105
10−3
10−2
10−1
100
Error indicatorLeast−squares functional
') )% & ) '
~q¬yzp0» 2iy-tmtqks[¯zYÊbz@op| º ¡fqtÅx1yPm@zmtq ¯soq$oqkp|s[m@kÅoqkphm@zZPmsp|Iok-phsPm@sws§¯yk tqksh¥
~q¬yzp»b¦§p½m@¬@kp|t Pp|o9oqkphs§qk"mIswo |ptzwy¬9t-ptzp|piphkswkm@oÈswoy-tk<~t-pmFÂsu£/ip|tz Ê°zwoqph º ¥
l b
~q¬yzp -» 3ptm@qotpÅiphsw7mF tphzm@Pm@-tq@pÅzpPkphiphkth¥$¦§pIPmzwtswN¡fk7qsm½t2/qp|¡@t-p4qktphzwz@tpzwk9 hzwp4YÊbz@op| º ¥
L MON""RS#*-%a 'qk hptp hy-zzp|ktqs"swk1ys@Pm@omk tp¯zphx1yp|k |£ s9oq¡ª¨¡p7ysp tp x1yPm@sqswtmtq tipvPm@zw@k p|x1yPmtqks
|p : !U!y kQ8 @LZ >¶¥ l m|p\: ~ kQ8 @LZ >¶/¥ l P| y.: j kQ8 @LZ >¶¥ l r
¡fp|zpy : _$\ Z >¶¥ º m~N: ~ h >¶/¥ º P¨tp§tPmFtZp|x1yPmtqk>¶¥ l YioqphstPmta|X ~z: j ¥Y¦§-p¨qk1tphzum@ hpf h@kqtk-s°¯@zYtqs-zoqph
t-phk0Dp| hiphsµ pB·D: j kQP @ O Z >¶¥ Ë mµ]\ pB·D: j kQP @ O Z >¶/¥ Ë Pµ ~IB·D: j kQP @ O Z >¶¥ Ë rµ]yB·D: j kQP @ O Z >¶/¥ Ë P
¡fp|zpH´s4m0ykqt4k@zm@o~t0tpiktphzwm hpm@k5µ Ä·°-phk@tphs4tpwyim@ |zsws4tpswyzwm hp3msBp|¯zwp@¥½k |pm_ f : l qktpn¡fopIzwph¬qkLª¿t-pim@¬k-p|tq ÅPphoq7qs hktk1yys4Np|zswyzwm@ |phs|¥¦§psm@ip¨qstzyp¢¯zZtp¨tm@k¬p|k1t¤m@o |Bkp|kt@¿tpp|op| |tzw fPphoqLª ª1¡foqpŸ¶¥ Ë Nbioqphstmt oV: j ªL¡f-phzp phktphst-pÅPphoq qksqp$tpim@oyik-yÁomtpªLsqk hp ~V: j qkt-pmzh¥ ¨n¯yzwtphzfzp|swtz tqksfm@-oq£3@k9tpphoqph |tzq PphoqL¥
l 0
¦§p4Dy-kPm@zu£< hk-qtksm@oqph0m@zwp pN: j kQP Z >¶¥±ÏmyN: j kQP >¶/¥ÄÏP
¦§pIom@ hp|p|kt@°tp$y-tp|zD@ykPm@zu£qsk@tsBph hPphqk2phsw hzw-tk@°tpÅtp|swt hm@sp¥Æ7pPmrpphk |oswph½tp |oBm@ktpm@oyik-yo¤mFtp¢k3m4 hyBp[m@zwrÂ-q½mFtpho£Itzwphp¢tiphs°t-psqhp@~mIs-p4qk<tpomtp¥
)L MON" 17"¢'d# P 'jRS5+"¢'=Y %(5 S1-#+*-%! Yp|kity¬Itpm@¬kpt ZPphoqIsY |ktk1y@ysbkÅtp¡foqp§zp|¬kȪ@¡p§sutqoo/Pmrp§tqk1tz/y |pt-p$qs h@k1tk1yy-sphoqphiphktsm@o@k¬itp$swyzwm hphs|ªÈy-p$titp9wy"k9tpkzw½m@o[ |Bkp|kt@ i¥p|twtqk¬$ytp¨oprmswtwvÑsx1yPm@zwphs¯yk |tkm@oP¯ztssw£/sutph¾@¿p|x/ymtqksZtphk½oprmst$t-p¯oqoN¡fk¬$pÂzp|sswkLª
\D Z \: G] @BADC ^ 9 |p qU(_$\ 9 _bdc ¸¶¥ ¶@
9 |p\ ;~ 9 _b)c 9 |./_$\ 9 _b)c a ]Cedd@ f O d G ^ 9
r+C0h _ µ pD· 9 _k c l 9 r+C0h _ I 9 _k a ]Cedd@ f O d G ^ 9
r+C0h _ µ]\ pzD· 9 _k c l 9 r+C0h _ µ]_$\B· 9 _k c l a 9 r+C0h _ µ pD· 9 _k 9 r+C0h _ µM_$\D· 9 _k Z
¡fp|zp$tpswph |k@Ytp$qktphzum@ hptp|zissw¬k £3tmt¨tp$k-zm@o[ hiBkphkt¨@Ytp Pphoqk-s-ptp4m@oqyqky- swyoq9Dphp|zik<tpqk1tphzum@ hp¥¦§p |ktqksb¯zZm4ikqÅy)@ \D Z \Y¬pftp¢¯ooq¡fqk¬m@zmtqkPm@o/¯z$yo¤mtků@z
: Z \2»~Pk-8swy "tPmt= b Z i¢ = C Z i$ = _ Z i$ = G Z i(: >u i Z ¸¶¥cb
¯zfm@oqo iª¡fphzwp= b Z iT: G] @BADC [|p Ud_ !\ Z | p i U(_ i\ bdc >¶¥ 0@m
[|p\ Z |p i\ i$7bdc [|./_$\ Z | /^_ i\-b)c Z
>w iT: G] @BADC -~ Z |p i\ i$ bdc >¶/¥10P
l
m@k9¯ztpktp|zwm@ |ptphzws|ª= C Z iT: ]Cedd@ f O d G [
r+C µ pzD· Z µ ipzD·¤ k c l >¶¥ m0 r+C Z i ¿ k Z
= _ Z iT: ]Cedd@ f O d G ^ [r+C µ]\ pD· Z µ i\ pzB· k c l >¶/¥c@P
0 r+C µ]_e\ D· Z µ]_ i\ B· k c l a Z= G Z iT: 0 r+C µ pB· Z µ i pD·¤ k >¶¥ @ r
0 r+C µ]_e\D· Z µ]_ i\ B· k ¨@tpÅtPmFtk2tphswpIpÂzwphsswk-s¡Zpnm@zwpÅprmok-¬½¡fqt2 hioqpÂ"ph tzpho-sh¥jlkzm@ tqsp@ªkpsp|Pm@zmtp|s«tpZzwprm@o/m@k-q½m@¬@kPm@zu£m@zwts~m@k-4t/y-s~Pm@s[tf¡ZzÀ¡ft lº y-kÀ1kN¡fk$m@zm@op|sh¥
% &l%!k3 Sv##*7%! T1H5+"¢'-S1#&'Îs9pÐÂ-Bph tp|Lªk5tpm@swphk |p7@4swk¬y-o¤m@zwqtqphs|ªfts½z-oph hy-oÇDpswoq@ph sy | hp|ssw¯y-ooq£¡fqt9«-Y7¥2iPm@zwswk½¡fqt<pÂBphzqp|ktm@oDPmtmÅqss-¡fk0k9¬yzwp lOl ¸mÐ¥Y¦§p hy-zzp|ktsk-y hp|k2t-pmoyikqyÁomtpImzpIs-¡fk k¬yzwp lº ¥ u N¡Zp|p|zhª«ypÅt9tpIsqhpI@t-pzwop|"ªN¡Zp°Pmrpkoq£Dp|phkm@opYtf hiy-tpy-sk-¬¨oqkphm@zLBoq£/ki¤moPmss¿¯yk- |tqks[m@k-ktzphm@ qk¬itpphswzwph2m h hy-zm@ £¥¦§-p |kphzw¬phk- hpty¬@"qs¢¬@/0m@sfsN¡fkqk0~¬y-zp lOl ¯ªsqk hptp , _ kzw@«tpzwphsqyPm@o¿qs§phx1y@mophkt§tntpphzwzzfk9p|kphzw¬@£<kzw"¥
R25
R50150
150
294
294
18
25
10018
30
19
108
108
A1 B1A1
B1
Aluminium
Coil
Coil
Hole HoleAluminium
~q¬yzp l j »¦§pi¬p|ptzu£@§Êbz@op| Ë ªm¯zwktqp|¡ªt qp|¡¥3¦§-pqp|ksqks4m@zwp¬phk0k9ioqoip|tphzwsh¥
jlk"tsfpÂ-m@ioqp¡Zp4k-@tmN@p$mkk- hzphm@sqk¬nzp|s-yPm@o>ªPswphp~¬@yzp lOl ¯ª¡f- hm@yswphzwop|sqk tp½zwp|/@yspÐÂ-m@-ophs|¥ u p|k hpªYqt4Pms$Bphphk7D@ssqopItyswpt-poqprm@sutwvlswx1yPm@zp|s
l
0 0.05 0.1 0.15 0.2 0.25−4
−2
0
2
4
6
8
10x 10
−3
x [m]
Re(
Bz )
calculated datameasurements
& ! "% "$ % "
5 6 7 8 9 10 11 12
x 104
120
140
160
180
200
220
240
260
280
|| R
(uh)
|| 0
Number of elements
)% & ) ' &
¬yzwp lOl » 2@ytmtks¯@zfÊ°zwoqph Ë ¡fqt9ok-prm@zm@sqs¯yk |tk-sh¥
~q¬yzwp lº »°p| |t@z¢-o@tf@«t-p4qky |ph0 hyzwzp|k1t¢qk9tp4m@oy-qky omtpYÊbz@op| Ë ¥
º j
zp|sqyPm@o[m@s§zwp|Pk-phiphkt |ztphzwkÈ»\ : 9 |p qU(_$\ 9 _ ¸¶¥
9 |p\ X~ 9 _ 9 |./_$\ 9 _ Z
º'l
`T^eµ l ·~zwswtuvlzwphzsu£/swtp|Áophm@swt¢sx1yPmzphs¢qosw£¥
¥µ º · 2¥PÎizy- p@ª 2¥²p|zkPmz¸ª7¥3m@y¬@p@ªPm@k- $¥B³qzm@yotr¥ °ph |tz§B@tp|k1t¤m@oqs§ktzwphpÐvlqp|ksqkPm@oDkksw/@t9@½m@qksh¥ !" < & 4( $ $5' < 0 <ª º'l » º@Ë Zj#b@Ϫ l #¥
µ Ë ·$³n¥-²m@im@k u ¥ 5Zm@k¬¥bÎ&oqprm@sutwvÑsx1yPm@zwphs§Pktpp|ophiphkt¢m@km@oq£/sqs¯z©Èzm@ |tk<-zoqphish¥ 2< /) < 5!"' <ª Ë 0 º » b#b¶ Zjb º ª º j j j ¥
µ Ï@· 4¥²phzw¬swtz! 0ªÎ$¥²Zkp|skLª 2¥B¹@-kskȪB7¥ ³I¥[m@zswkLª 5Å¥¿yLªPmk#"I¥m@Åyp|osswkL¥ÎPm-tpPktpphoqphiphktfip|t-sfqk<phop| |tzm@¬kptq hsh¥b¦«ph -k hm@o ¢p|D@zwt º ª/¡Zph-sjlkswtqty-tp@Î-oqph"mtphmtq hs4¯jl¦¢2ª l "-¥
µ±¶F·Ê¥Ä²¥²/ p|<m@k7¥ 3Å¥B³yk|yzw¬phz|¥Zktpp|op|p|k1t¢ip|t-s§@~oqprm@sutwvÑsx1yPm@zwphsftl£Bp@¥ %$ '& <ªÏ j Ï1л10 !Zj Ë 0ª l #-¥
µ b·4Î$¥P²ZssmN/tr¥)( &) $5/ +!"*+&(5!"' ' -0 &) !" * &) ¥YÎ hm@phi Êbzwphss|ª l #-¥µ 0F·$¹-¥ u ¥P²zm@Åoqp@ª ¥ 3Å¥«mrm@zwBªm@k-2¹-¥Ä§¥Ê~m@sw h¤mÀB¥°Î8ophm@swtuvlswx/ym@zp|s¨mzm@ "m@sp|"k"msw hzwp|tpik1ys§k-p4qkkp|z§z/y t¢¯@zPzsut¨@zp|z¢su£/swtp|s|¥ !" <*( &) $ <ªb#bP º'l л Ë ¶ Z ¶¶/ª l 0¥
µ ·$D¥ 2¥²zp|kkphz¢m@k-b¥ 4¥B |@tth¥ . !"&) !"*+0 !"' . -&( 6 & # ' &) & 4( ¥zwk¬p|zuv °phzom@¬ª l @Ï-¥
µ · $~¥ ¥2phkȪ Å¥3yLªm@k¹-¥$ÈyL¥°~qkqtpphoqphiphktfip|t-s§¡ft9½mFt qk¬nm@kkkmt k¬Iiphswphsf¯z¢m¡Zphoqo¿phx1yPmtk-sf¡ft9qs h@k1tk1yy-s¢ |pÍi hp|ktsh¥ % < /) < 5!"' <ª Ë 0» l ¶Ï º Z l ¶ 0 j ª º j j j ¥
µ l j ·$7¥ 2@swtmDp|o>¥«Î( h/phzw hq@poqkphm@z°¯z ¯zf"mFÂ/¡p|oo>ñsp|x/ymtqksh¥+2< !" < 5!"' < $ $5' <ªl ¶ 0 º » ¶ º 0Z-¶Ï l ª l l ¥
µ lOl ·$7¥ 2sutm@Bpho[m@k- ¥3m@y¬p¥ ¨k0z,phswyoqtmtfp4phk-sqt,pDy-z§ophs-,p|x/ymtqks§pmFÂ/¡Zphoqoz,p|¬yo¤mzs,php|s§Pm@ksfyk0m@k-poqs t|phkÈ¥.( </$ < 0 ! 4 < 0 <10 !" 32 < !" <ªËº 0*» @Ï Zj¶Ïª l -¥
µ lº ·²¥ v ¥P¹@¤m@k¬-¥ !(54+76 /2!"- * #;' &) ) & 498 . -&( 6 !"54!$ $5'7+0 !" +&(2= 0-&) $5/ +!"*+&(5!"';: /9*4 4"6(5! ) +0- !"54 ' -0*-&) !" *+0- ¥°-zk-¬phz v °phzwo¤m@¬-ª l -¥
µ lNË ·²¥ v ¥P¹@¤m@k¬-ª¹-¥-Æ7yLªm@k"b¥ÄÎ¥ÊN/kp|oo¸¥~¦§p4zw¬qk3@syzwy-s¢swoyt@ks§k hiy-tmtqkPm@oBphoqph tz@½m@¬@kp|t hs|¥2<( &) $ <0 96 <ª lNº ¶» l j Ï Z lNº@Ë ª l #b-¥
µ l Ï@·b¥ ¥ |@tt¨m@k"D¥$Èm@k¬¥°~qkqtpphoqphiphktfktphzBomtqk3@kksw/@t9¯y-k |tkssmtsu £qk¬Iyk-Pm@zw£9 |ktqksh¥ !" <( &) $ <ª¶Ï»±Ï Ë Z/Ï Ë ª l j ¥
ºº
µ l ¶F·b¥¦y-zkp|zh¥²phk- ½mzÀ<zwop|s¯@ztpm@oqPmtk@~p|-£9 |yzzwphkt |iy-tp|z§ h/phs|¥( 0 ª-» lº@Ë Z º'l b-ª l j ¥
º@Ë
º Ï
q) ( . 8')^)/ / ) 2 . (
) $ ' * $ & ' !
! #"$"% ! ! $ < < ' &')(+* , - .0/21435 # 6387:91 ; 1=< 9).
> ! % ' $< ' $ & '$ ? @& 11 28AB 2DC*2 @
E ) ) ' ' ! GF & AHI2 &2KJ ')* 9 2
!L < ') M ! ')!!& & &
N O $ & ' QP J @ ' D@?;R@& 'TS 4J ' 9)VUXW .'T*ZY[' -
!\ ] & $') ' ( * ' ') C ' _^ &
` & & ') (ba0c * ' & & $ C ' _^ &
d e %( gfhc ] F ) i A# K - -
QD e %( gfhc ]V] F ')A# K - -
Q ) < ')&&& =j & 91, (+* 3 K 3&8 #k V2
Qk ( $ ' &&& 4j & ' F :') AHI2 &2KJ ')* 9 2
QD> ( $ ' ')!!& ?j & ' ! %$ F AHI2 &2KJ ')* 9 2
QRE & < & .' , ' 9 ' &k3k7:91 ; 1=< 9). 7 ' . lA#@k9
QkL & ! ')& &% ' & 91, (+* 3 K & 9 S ; .-
QDN ') e %mc c ' ! n @ 1 &Ko 91 4A# K,p
Qk\ ' & ') $ &q $r_h ' <63$A=s,p 63 & h, - .0/2143ptBhI 2@k 63$ush 63 v7w; 1=< 9x
º ¶
QD` & & & < ' pP P I2 @! !1 ?t$91 pI2 @V2
QDd c $' & ') ' ' & ) ' I2 @! !1
! ' < ' & & t$91 KI 2@k 63DI2 @& !1 A K6 - -
6 & & '! ' ! $ A# 'T
')&&& ')&&& ! & - e &% $
!5 - ) ' & ' '0F') & q <& ' q
!5! - ) ' & ' '0F! $ q <& ' q
!5> O ! ' % ' % ' e &% n @ 1 &Ko 91 4A# K,p
!5E ' ' )sP I2 @! !1 ?t$91 pI2 @V2
!5!L e mc e & < ') .' , ' 9 ' &
!5N c f ] " %Qc $'8f ' ! : ] "< $ ! ') I2 @! !1 4A# K6 - -
!5!\ ' % ! ) fhc 7 k @ W .'T* .V263kt$9 KI2 @k 8A# K6 - -
!5` & & & < $ !P I2 @! !1
!5d ')$ ') < ') ') 8P P I2 @! !1 ?t$91 pI2 @V2
!5QD )% & ! ! - ') )' * ' &')(+* , - .0/21
¦§p|sp4zp|zqktsrm@k0Bp4@-tm@qkph<¯zw !#"$&%('*),+.-,/0#/1+ $(-(+2&-354 /
