PREPRINT 2001–08

Adaptive finite element computation of 3Dmagnetostatic problems in potential formulation

Marzia Fontana

Chalmers Finite Element CenterCHALMERS UNIVERSITY OF TECHNOLOGYGoteborg Sweden 2001

Preprint 2001–08

Adaptive finite element computation of 3Dmagnetostatic problems in potential formulation

Marzia Fontana

Chalmers Finite Element CenterChalmers University of Technology

SE–412 96 Goteborg SwedenGoteborg, April 2001

Adaptive finite element computation of 3Dmagnetostatic problems in potential formulationMarzia FontanaNO 2001–08ISSN 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 2001

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ÄY¦eZYfEcaWY¦njV¼L ½ WXZYblhZUa_Ypcax^3fEZfZ^%WY`%jzeZ^©-YGoYfnW^!UaZ`%Y~x>|ZfZ_W^%cqfZlM¦^%WX«_cq[dhUa_W±l|ZhZhcqo£W^3fx¤w^ 6 Y 6 Uall|Z[d^3fEgQfZcqfÓÉYocÂ1Ua`%|ZYl´cafZ`%jcqf²lcq[dYO_cq[dhUa_W«l|ZZlYGWcax 6³Ï fs^3[dh cqo£WUafnWl|EZlhU1_Y¤^%f¨IG¼L ½ ^3lBWXEYf I ¼L ½ wn¦XZ^%_X\^%leZYfZYGerUalBWXZYp_`%cql|EoYpcaxV¼9 ½ ^%f I¼L ½6yu W~_UafY¡lXZc¦fVWXU1W

I ¼L ½ D u= _ I ¼9 ½ °)=@? R D F aI ¼ )6 ½

I ¼9 ½ W|ZofZl¨cq|EW¨Wc Y´WXZY«fU1W|EoUa`MlhUa_GY\Uallc _^3U1WYGeÂWcOZfZ^%WYdYG`3YG[YGf¢W¡lcq`3|EW^3cqfEl¤caxlY_GcqfZecqoeZYoBcq|ZfZeZUaoj¯1Ua`%|ZYhZocqZ`%Y[dly¦^%WXdcq|ZfEeUaoj_cqfEeZ^%W^3cqfZlMUa`3`cax ^%o^%_XZ`3YWCWj hY 6 Ë~cqXZYGoYGf¢W`%jawWXEY¡lhUa_GY

I ¼L ½ D u= _3 I ¼L ½ °)=@? D F ¼ )6hn½

_kU1fY^3fnWoc eZ|Z_YGerx>cqo -¤wUal¦U¯fU1W|ZoU1`)lYWW^%fZg¯¦XZYGfVcaWX ^%o^3_XE`3YGWUafZe Y|Z[UafZf¼caoÊ cqZ^3f ½ cq|ZfZeU1oji_GcqfZeZ^W^%cqfZl¤U1oY¯hZoYlYfnWkwWXEY¯x>cqo[dYogq^%aYfcqf 6 caWY¯WXU1W¤^3f I ¼L ½ WXEYlYG[^%fZcqo[ M = M , 5SR I D M = M NGc 5eR I wEcqEWUa^3fEYe\x>ocq[·¼ )6½ x>cao D wE^%lbUfZcao[ÐY¥¢|Z^%1Ua`%YfnW~WcWXEY µ cqcq`3YfZcao[ ¼ )6Fn½ ¼UafZeUafU1`3cqgqca|Zl`jx>cao I ¼L ½½6 cqo[l M M , 5SR I UafZe M M NGc UaoYoYG`!U1WYeiWcYUa_XcaWXEYo¤WXEocq|ZgaXWXZY¯x>cq`%`3c¦^3fZg ] À ¾ ¿ À ¹ZaÁ °¤^%x ^3lcq|ZfEeZYe)w)WXZYoYY^%lWlpU«_cqfElWU1f¢W JF eZYGh YGfZeZ^%fZg«cqfEÒl|Z_XWXU]W

M = MONGc 5SR I n M = M Ndc 5eR I \ ^ =_ I ¼9 ½ I ¼ )6An½

u [d[YGeZ^!U1WY`jqwMU1f^3fZYG¥¢|Ua`3^WjÂUafUa`%cqgqcq|ElzWc³¼ )6An½ XZcq`3eEl¯UagnU1^3f¦XEYf²_cqfEl^3eEYo^%fZgQU1WzWXZY\`3YGxWXUafEeVl^3eZY M = M , 5SR I ^3fZl£WYkU1eVcax M = M NGc 5SR I 6u frcq|ZfEeUaoj«1Ua`%|ZYhZocqZ`%Y[dlx>|ZfE_GW^%cqfZlXZUaYWc YUall^3gqfZYGeOUal~eZU1WUcaYohU1oWl~cax$eZcq[Ua^3f

cq|ZfZeU1o^3YGl 6B» X¢|ZlGw ^Wb[\U1¶aYl¦lYfZlYzWc«^%f¢Woc eZ|Z_GY¤WXZY¨lhZUa_Y

IL[ H ¼ ½ D b ° ¼cqo ½ °2=_# I ¼L ½ °)= ? D b \ ¼ )6 Fn½

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"! #"$&%

M = M , 5SR I \ ^ b _ IL[ H ¼ ½ I ¼ )6 ½

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\ HbUaoYoYUa` f¢|Z[ YGoll|Z_XÍWXU]W

I JHkwZWXZYfOWXEYoY¨Y+E^%lWlU«_cafZlWUafnW JF l|Z_XVWXU1W

M = M , 5SR I n M = M I , 5SR I M = M , c 5eR I \ ^ b _ c ¼9 ½ ¼ )6 q½

¦XZYGoY D ¼ a½ I v HkwZx>cqo¦Uafnj F n n ,6

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V

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v _ D F ¼ V$6½

D \ ¼ V$6 q½

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D ¼ V$6gVn½ _ D F

\¼ V$6i^¢½

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C D V! _ C#" D F ¼ V$6 wa½ C D V !%$ v '& C#" D F ¼ V$6 fq½

R C#" D V R C ¼ V$6 ½ R _ C#" D F ¼ V$6 hq½

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v D F\

¼ V$6 Aq½

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D ¼ V$6½_ D \ ¼ V$6 q½

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For each element: compute elemental stiffness matrix; compute elemental RHS due to source field; compute (possible) elemental RHS due to interface

Compute source fieldby Urankar’s method

Read toleranceRead refinement factor

Global FE assembly

Do Neumann nodes exist?

Compute source fieldby Gaussian integration

Compute line integrals on Dirichlet nodesSubstitute Dirichlet values at RHS

List all boundary Dirichlet nodesList all boundary Neumann nodesList all interface nodes

Is conductor a coil withpolygonal cross section?

Compute line integrals on interface nodes

Do Dirichlet nodes exist?

Write nodal potential of reduced domain on fileWrite nodal potential of total domain on file

Solve FE linear system

number of refinements?Is iteration j less than max

For each element: compute local error indicator Adaptive method?

Write nodal field of reduced domain on file Write nodal field of total domain on file

Compute normal source field on Neumann nodesAdd Neumann condition to RHS

START

Is union of total materialssimply connected?

Choose uniform/adaptiverefinement method

For each boundary surface: set Dirichlet/Neumann type

For each material domain: set total/reduced type

Adaptive method?

Does union of reduced materialscontain all conductor regions?

j = j+1 j = 0 Read max number of mesh refinements Read order of Gaussian quadrature

Compute global error estimator

For each element: mark refinement level

STOP

Error less thantolerance?

Refine marked elements according to levelDefine new mesh

Refine all elements uniformlyDefine new mesh

Compute magnetic field on total/reduced domainCompute magnetic energies on materials

Compute gradients of total/reduced potential

Read geometryRead physical dataRead mesh

YES

YES

NO

NO

YES

NO

NO

YES

YES

YES

NO

YES YES

NO

NO

NO

NO

YES

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% "D) 4UH96G:&6B>B2CA9F a42C5,?6"-|= : 7 A 7 $= :@iS-.-J6;cA4&= >m-.2-F96",M 7 :Y= A~cH96ka4&6;d-=2%R9:Y> 7 :&6%M5R<c 7 -.234E6

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Copper

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; AcEH9= :L> 7 :&6%MNFR96pc2cH<6pA2CA t >"2CAod6y>1C6B23-J6;cA4@iP2%7DcH96g7h6B44D23- 7 1CA96;cE= >JF23- 7 = AzMW6"QR9=?a9ab6BF8D=?cEH>"234&A96"4&: 7 A9F6BF16B:"M 7 a 7 4nc=?>"R, 7 4z> 7 4E6t=?:+4E6B^R<=4E6;F=?A cH96t>B23-JaR<c 7 c=2AW2%78cH96 ,?= A96Q=?Aoc6 t134 7 , :qF96B*9A9= A1fcH96qHDR-Ja9:5b6;cx8s6B6;AYa?2cE6BAoc= 7 , : 7 cqcH<6= AocE6"4D7 7 >B6%M%234qd 7 ,R96B:|2%7 7 a?2:E:E=?5, 6 ¬ =4&= >(H,?6;c5!23R9A9F 7 4Di>"2CA9F<=?c=2CAKMG a!2C:&:E= 5, 6E8 7 ic2 = A<>"4E6 7 :E6ScH96 7 >B>BR4 7 >ih27D:R9>(H>, =?A96J= Aoc6B134 7 ,g>B23- taR<c 7 c=2CA9:m=?:mcA2^2 7 d32C=?FJencA2^2C,2CA13fPcA4&6B6;:"M 7 :@=?cN8I23R, FF2->B>"R4=?A cH9=?:m> 7 :E6Mdz=-J= A1 7 c@cEH9= :BM 7

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Number of elements

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0 1 2 3 4 5 6

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Number of elements

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