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METODA KONAČNIH ELEMENATA
(MKE) U ELEKTROMAGNETIZMUDoc.dr.sc. Bojan Trkulja
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PREGLED PREDAVANJA
� Numeričke metode u elektromagnetizmu� Osnovne osobine metode konačnih elemenata� Primjer razvoja metode konačnih elemenata na
elektrostatskom 1D modeluPrimjer primjene MKE u programskim paketima� Primjer primjene MKE u programskim paketima
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NUMERIČKE METODE U
ELEKTROMAGNETIZMU
� Proračun elektromagnetskog polja važan je dio proračuna električnih uređaja (radne karakteristike uređaja, standardi, EMC)
� Metode proračuna� Najčešće metoda konačnih elemenata (MKE, eng. � Najčešće metoda konačnih elemenata (MKE, eng.
Finite element methods, FEM)� Programski paketi Ansys, Infolytica MagNet, ElecNet,
VectorFields Opera, CEDRAT Flux
� Metoda rubnih elemenata (Boundary element Methods, BEM)� Programski paketi IntegratedSoft
� Metoda konačnih razlika u vremenskoj domeni (FDTD, Finite difference Time Domain methods)
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METODA KONAČNIH ELEMENATA
� Danas je praktički standard u industrijskim proračunima u elektromagnetizmu
� Višedisciplinarno područje � Modeliranje geometrije problema� Matematičko modeliranje� Matematičko modeliranje� Matrična algebra� Numeričke metode u proračunu� Programiranje
� Dodatno – problemi različitih proračuna u međudjelovanju� Elektromagnetski, termički, dinamika fluida
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METODA KONAČNIH ELEMENATA – PRISTUP
RJEŠAVANJU PRIMJENOM PROGRAMSKOG
PAKETA
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MAXWELL – DEFINICIJA MODELA
PRORAČUNA
� Proračun električnog polja� Elektrostatsko polje – uvjetovano statičkim naponom
na vodičima i razdiobom statičkih naboja na vodiču� DC vođenje – analiza prilika u vodičima i
dielektricima uslijed statičkog polja� AC vođenje – analiza prilika u vodičima i
dielektricima uslijed vremenski promjenjivog polja
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MAXWELL – DEFINICIJA GEOMETRIJE U 2D
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MAXWELL – MODELIRANJE GEOMETRIJE
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MAXWELL – KREIRANJE OBJEKATA 1/5
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MAXWELL – KREIRANJE OBJEKATA 2/5
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MAXWELL – KREIRANJE OBJEKATA 3/5
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MAXWELL – KREIRANJE OBJEKATA 4/5
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MAXWELL – KREIRANJE OBJEKATA 5/5
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MAXWELL – DEFINICIJA MATERIJALA1/3
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MAXWELL – DEFINICIJA MATERIJALA 2/3
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MAXWELL – DEFINICIJA MATERIJALA 3/3
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MAXWELL – ZADAVANJE RUBNIH UVJETA I
IZVORA
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MAXWELL – ZADAVANJE RUBNIH UVJETA I
IZVORA
� Rubu elementa zadaje se rubni uvjet/izvor
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RUBNI UVJETI 1/2
� Balloon – prazni prostor u okolini prostire se u beskonačnost
� Value – propisivanje potencijala na granici� Symmetry
� Even (isti predznak na obje strane granice)� Even (isti predznak na obje strane granice)� Odd (suprotan predznak na obje strane granice)
� Resistance – za modeliranje tankih slojeva (problem s generiranjem mreže)
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RUBNI UVJETI 2/2
� Impedance – inducirane struje u vodičima s malom dubinom prodiranja
� Master/Slave – polje na slave granici podudara se s poljem na master granici� Korisno za geometrijski periodičke strukture – npr. � Korisno za geometrijski periodičke strukture – npr.
dizajn motora
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ZADAVANJE IZVORA
� Solid – zadavanje izvora (struje, napona, naboja) označenom objektu
� Sheet – zadavanje izvora (struje, napona, naboja) označenom rubu objekta
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MAXWELL – ZADAVANJE IZVORA I RUBNIH
UVJETA
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POSTAVKE RJEŠAVANJA PROBLEMA
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ADAPTIVNA ANALIZA
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GENERIRANJE MREŽE
� Iterativni postupak u rješavanju problema� Current – korištenje zadnje generirane mreže u
postupku iteracije� Manual – ručno postavljanje mreže u željenoj
regijiregiji
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IZBOR NUMERIČKE METODE U PRORAČUNU
� Auto – sustav određuje pristup� Direct – uvijek konvergentan, ali spor za velike
sustave � ICCG – incomplete conjugate gradient solver,
� Ponekad nije konvergentan� Ponekad nije konvergentan� Brži za velike matrice sustava
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ITERATIVNI POSTUPAK U RJEŠAVANJU
PROBLEMA
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PRIKAZ REZULTATA 1/4
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PRIKAZ REZULTATA 2/4
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PRIKAZ REZULTATA 3/4
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PRIKAZ REZULTATA 4/4
� Vidna nedostatna točnost→prilagodba proračuna
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PRILAGODBA ŽELJENOG POSTOTNOG
ODSTUPANJA
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RUČNO GENERIRANJE MREŽE
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TOČNIJE RJEŠENJE
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KREIRANJE LINIJE I 2D PLOT PO LINIJI
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� Diskretizacija domene proračuna konačnim elementima
� 1D
2D
POGLED U STRUKTURU MKE PROGRAMA
� 2D
� 3D
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IZBOR BAZNIH FUNKCIJA NA ELEMENTU
1D
� Linearne funkcije na 1D elementu
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IZBOR BAZNIH FUNKCIJA NA ELEMENTU
2D
� Linearne funkcije na 2D elementu
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PROMATRAMO POISSONOVU JEDNADŽBU U
1D
� Poissonova jednadžba
� Rubni uvjeti
� Analitičko rješenje postoji
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GALERKINOV PRISTUP RJEŠAVANJU
SUSTAVA
� Postavlja se forma jednadžbe
� Za težinske funkcije biraju se bazne funkcijeParcijalna integracija � Parcijalna integracija
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GALERKINOV PRISTUP RJEŠAVANJU
SUSTAVA 2/3
� Konačno
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GALERKINOV PRISTUP RJEŠAVANJU
SUSTAVA 3/3
� Formiranje matričnog zapisa
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ZAPIS ZA JEDAN LINEARNI ELEMENT ZA
POISSONOVU JEDNADŽBU
� Izvedeni izrazi prema prethodnim formulama
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SASTAVLJANJE SUSTAVA JEDNADŽBI 1/4
� Općenito
� 1
� 2
� 3
� .....
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SASTAVLJANJE SUSTAVA JEDNADŽBI 2/4
� Vrijedi
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SASTAVLJANJE SUSTAVA JEDNADŽBI 3/4
� Matrični zapis – rijetko popunjena matrica MKE
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SASTAVLJANJE RUBNIH UVJETA –DIRICHLETOVI UVJETI PREMA TESTNOM
PRIMJERU
� Propisan potencijal na granici – redukcija dimenzije
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PRIMJER PRORAČUNA 1
� Zadani elementi duljine 2cm
� Treba izračunati raspodjelu potencijala po elementima
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PRIMJER PRORAČUNA 2
� Izračun elemenata za formulaciju sustava
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PRIMJER PRORAČUNA 3
� Postavljanje rubnih uvjeta za napon na početku i kraju V
1=1, V
5=0
V1
=1, V2
=0,07, V3
= -0,4, V4
= -0,43, V5=0
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PRIMJER PRORAČUNA 4
� Usporedba s analitičkim izrazom
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ZAKLJUČAK
� Pokazan primjer proračuna metodom konačnih elemenata sa svim koracima� profesionalnim programom Maxwell� na jednostavnom modelu s potpunom matematičkom
formulacijomformulacijom
� MKE � robusna metoda � široko implementirana u programskim paketima� razvoj u zreloj fazi – implementacije na spregnutim
proračunima, optimizacije � Solveri se izvode za paralelne arhitekture