Dr Jasmin Smajic. ABB Corporate Research Ltd - 1. Introduction to BEM. Baden - Dättwil.

Transcription

1 Dr Jasmi Smajic ABB Corporat Rsarch Bad - Dättwil jasmi.smajic@ch.abb.com Itroductio to BEM ABB Corporat Rsarch Ltd

2 Outli ABB Corporat Rsarch Ltd - Itroductio to th basic ida of BEM Itgral formulatio of D Laplac problm BEM tratmt of D Laplac problm D lctrostatic aalsis ampl Matri comprssio Fast multipl tchiqu Sigular itgrals Itgral formulatio of D dd-currts problm D dd-currts aalsis ampl Coclusios

3 Itroductio to th basic ida of BEM Mathmatical dscriptio Modl paramtrs ABB Corporat Rsarch Ltd - If Ω matrial : frquc : sourcs : E µ ε s f H s J s

4 Itroductio to th basic ida of BEM Modl discrtizatio Itgral formulatio Larg ds liar sstm of quatios ABB Corporat Rsarch Ltd - If Ω BEM schm [ A ]{ } { b}

5 Itroductio to th basic ida of BEM ABB Corporat Rsarch Ltd - 5 Fac picturs as a fial rsult of fild calculatio Magtic lctric fild pottial Elctromagtic rg forc Tmpratur prssur vlocit

6 Itgral formulatio of D Laplac problm Matrials:. Prfct lctric coductors fid pottial lctrod -FIXE. Prfct lctric coductors floatig pottial lctrod - FLOE. Liar homogous dilctrics ABB Corporat Rsarch Ltd - 6 D lctrostatic aalsis R ε ε

7 ABB Corporat Rsarch Ltd - 7 Itgral formulatio of D Laplac problm R R R δ δ [ ] d d δ d δ smooth is if r s fuctio

8 ABB Corporat Rsarch Ltd - 8 Itgral formulatio of D Laplac problm [ ] [ ] { } d thorm s r' [ ] [ ] Itgratio ovr boudar olum itgratio

9 ABB Corporat Rsarch Ltd - 9 Itgral formulatio of D Laplac problm R

10 ABB Corporat Rsarch Ltd - Itgral formulatio of D Laplac problm [ ] [ ] [ ] *

11 ABB Corporat Rsarch Ltd - Itgral formulatio of D Laplac problm BEM is rlatd to th fild sourcs ovr th boudar ' ' ε ε ε pol f f ' ' ε ε ε ε ' ' ε ε ε pol f f Our itgral formulatio dals with a total surfac charg dsit! Total charg Fr charg coductor Polarizd charg dialct. itrfac

12 ABB Corporat Rsarch Ltd - Itgral formulatio of D Laplac problm * * OK!?!! Problm: pottial ad charg dsit ar uows i th sam tim! Solutio:

13 ABB Corporat Rsarch Ltd - Itgral formulatio of D Laplac problm ε ε ε ε Dilctrics itrfac Floatig pottial lctrod Fid pottial lctrod

14 BEM tratmt of D Laplac problm Itgral formulatio I h Ω BEM discrtizatio Larg ds liar sstm of quatios [ A ]{ } { b} BEM discrtizatio Wightd rsidual mthod wi [ I h ] i... N Ω ABB Corporat Rsarch Ltd - Poit matchig collocatio alri w δ i w N i i i

15 ABB Corporat Rsarch Ltd - 5 BEM tratmt of D Laplac problm Poit matchig collocatio w i i δ Ω i i i N i h I ;... ; j j j N Liar odal triagular lmt Cocpt of lmtal cotributio i i i h I F [ ]{ } { } N N b K F F Ω Ω d N C d N K j i j i j i othrwis b i i j i

16 BEM tratmt of D Laplac problm Larg ds liar sstm of quatios [ A ]{ } { b} Itrativ solvr has to b usd BiC MRES Asmmtric matri has b producd Prcoditioig of th matri is vr importat Matri comprssio is cssar ABB Corporat Rsarch Ltd - 6 Fast multipol algorithm improvmt for matri comprssio ad solutio spd

17 D lctrostatic ampl ABB Corporat Rsarch Ltd - 7

18 D lctrostatic ampl ABB Corporat Rsarch Ltd - 8

19 D lctrostatic ampl ABB Corporat Rsarch Ltd - 9

20 D lctrostatic ampl ABB Corporat Rsarch Ltd -

21 D lctrostatic ampl ABB Corporat Rsarch Ltd -

22 D lctrostatic ampl ABB Corporat Rsarch Ltd -

23 D lctrostatic ampl Basic solutio data N5; N Full matri siz 5 5 Full matri mmor 7MB CPU tim 6 sc matri assmblag solutio fild calc. CPU tim matri gratio sc Diagoal prcoditioig MRES covrgc i itrat. ABB Corporat Rsarch Ltd - CPU tim sstm solutio sc CPU tim fild calculatio 7 sc

24 ABB Corporat Rsarch Ltd - Matri comprssio fast multipol tchiqu Krl pasio >> I m m Y X K K K farfild ; µ ν ν µ µ ν η Farfild coditio Talor- Multipol- Cbsv- pasio MRES with clustrig F T u Y F X u N u A v ~ τ τ τ Matri-vctor multiplicatio

25 Matri comprssio fast multipol tchiqu ABB Corporat Rsarch Ltd - 5 MRES with clustrig ~ v A u N u X τ F T F Y u τ τ Matri-vctor multiplicatio

26 Matri comprssio fast multipol tchiqu Basic solutio data with comprssio N5; N Full matri siz 5 5 Nar-fild matri mmor MB CPU tim sc matri assmblag solutio fild calc. CPU tim matri gratio sc Diagoal prcoditioig MRES covrgc i itrat. ABB Corporat Rsarch Ltd - 6 CPU tim sstm solutio sc CPU tim fild calculatio sc

27 Sigular itgrals Poit collocatio K i N j dω i CP itgrals K wa sigularit strog sigularit hpr sigularit Flat lmts: aaltical itgratio ABB Corporat Rsarch Ltd - 7 Curvd lmts: - Numrical itgratio auss quadratur - Coordiat trasformatios Duff trasformatio

28 Itgral formulatio of D dd-currts problm D dd-currt problm at idustrial frqucis H J E jωb B µ µ rh J E B J H H E E K curl Mawll quatios Costitutiv rlatios div Mawll quatios Itrfac coditios ABB Corporat Rsarch Ltd - 8

29 Itgral formulatio of D dd-currts problm H formulatio H H δ H S Ω H c imprssd fild fild du to th coil scodar mag. fild H H δ H S ABB Corporat Rsarch Ltd - 9 H Ω jωµh H Ω Ω H Hδ Ω µ H µ µ H Ω H δ H H S Itrfac coditios

30 Itgral formulatio of D dd-currts problm H formulatio ABB Corporat Rsarch Ltd - J ξ [ J η K η ξ ] [ ] [ ] t t m η ξ ξ η ξ η H ξ Hδ ξ Ω m ξ Ω µ ξ µ Ω J virtual currt m ξ [ m η ξ ξ η ξ ] Ω ξ [ η η ξ ] [ ξ ξ ] J K H H virtual magtic charg ξ η K η ξ η η - j r r η ξ η ξ δ η ξ η ξ Ω ωµ µ r r η ξ

31 D dd-currts aalsis ampl f 5Hz ABB Corporat Rsarch Ltd -

32 D dd-currts aalsis ampl ABB Corporat Rsarch Ltd -

33 D dd-currts aalsis ampl ABB Corporat Rsarch Ltd -

34 D dd-currts aalsis ampl ABB Corporat Rsarch Ltd -

35 D dd-currts aalsis ampl Basic solutio data with comprssio N78; N555 Full matri siz Nar-fild matri mmor MB CPU tim sc matri assmblag solutio fild calc. CPU tim matri gratio 7 sc Diagoal prcoditioig MRES covrgc i 5 itrat. ABB Corporat Rsarch Ltd - 5 CPU tim sstm solutio 7 sc CPU tim fild calculatio 6 sc

36 Coclusios ABB Corporat Rsarch Ltd - 6 W hav illustratd BEM applicatio to D lctrostatic ad D dd-currts problms I th cas of homogous ad liar matrials BEM is domiat i compariso b FEM Obviousl BEM is vr powrful simulatio tool for crtai class of lctromagtic problms Ds matri ad sigular itgrals ar th basic drawbac of BEM Matri comprssio tchiqus fast multipol mthod or ACA ar promisig tchiqus for th futur of BEM.

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