Dual Finite Element Formulations and Associated Global Quantities for Field-Circuit Coupling

Transcription

1 Dual Finite Element Formulations and Associated Global Quantities for Field-Circuit Coupling Edge and nodal finite elements allowing natural coupling of fields and global quantities Patrick Dular, Dr. Ir., Research associate F.N.R.S. Dept. of Electrical Engineering - University of Liège - Belgium Patrick.Dular@ulg.ac.be May

2 Constraints in partial differential problems Local constraints (on local fields) Boundary conditions i.e., conditions on local fields on the boundary of the studied domain Interface conditions e.g., coupling of fields between sub-domains Global constraints (functional on fields) Flux or circulations of fields to be fixed e.g., current, voltage, m.m.f., charge, etc. Flux or circulations of fields to be connected e.g., circuit coupling Weak formulations for finite element models Essential and natural constraints, i.e., strongly and weakly satisfied 2

3 Constraints in electromagnetic systems Coupling of scalar potentials with vector fields e.g., in h-φ and a-v formulations Gauge condition on vector potentials e.g., magnetic vector potential a, source magnetic field h s Coupling between source and reaction fields e.g., source magnetic field h s in the h-φ formulation, source electric scalar potential v s in the a-v formulation Coupling of local and global quantities e.g., currents and voltages in h-φ and a-v formulations (massive, stranded and foil inductors) Interface conditions on thin regions i.e., discontinuities of either tangential or normal components Interest for a correct discrete form of these constraints Sequence of finite element spaces 3

4 Sequence of finite element spaces Tetrahedral (4 nodes) Geometric elements Hexahedra (8 nodes) Prisms (6 nodes) Mesh Nodes i N Edges i E Faces i F Geometric entities Volumes i V S 0 S 1 S 2 S 3 Sequence of function spaces 4

5 Sequence of finite element spaces Functions Properties Functionals Degrees of freedom S 0 S 1 S 2 S 3 {s i, i N} {s i, i E} {s i, i F} {s i, i V} j j j s i (x j ) = δ ij i, j N s i.dl s i. n ds s i dv =δ ij i, j E =δ ij i, j F =δ ij i, j V Point evaluation Curve integral Surface integral Volume integral Nodal value Circulation along edge Flux across face Volume integral Nodal element Edge element Face element Volume element Bases u K = i φ i (u)s i Finite elements 5

6 Sequence of finite element spaces Base functions Continuity across element interfaces Codomains of the operators S 0 S 1 S 2 S 3 {s i, i N} value {s i, i E} tangential component grad S 0 S 1 {s i, i F} normal component curl S 1 S 2 {s i, i V} discontinuity div S 2 S 3 S 0 S 1 grad S 0 S 2 curl S 1 S 3 div S 2 Conformity Sequence S 0 grad S 1 curl S 2 div S 3 6

7 Magnetodynamic problem with global constraints Equations curl h = j curl e = t b div b = 0 Boundary conditions n h Γh = 0, n b Γe = 0 e dl = V γ i Voltage i Global conditions for circuit coupling, n jds = I Γ j i i Current Constitutive relations b = µ h j = σ e Inductor 7

8 Weak formulations Green formulae grad - div formula involved in weak formulations Notations ( ab, ) ab, Ω Γ def = ab dv def = Ω Γ abds ( u, grad v ) Ω + ( div u, v ) Ω = < n u, v > Γ Γ = Ω n Weak global quantity of flux type Domain Ω curl - curl formula ( curl u, v ) Ω ( u, curl v ) Ω = < n u, v > Γ u, v H 1 (Ω), v H 1 (Ω) Weak global quantity of circulation type 8

9 h-φ and t-ω weak formulations Magnetic scalar potential in nonconducting regions Ω C c φ h r = grad φ in Ω C c with h = h s + h r Reaction magnetic field h-φ magnetodynamic formulation Source magnetic field ( µ hh, ') + ( σ 1curlh, curlh') +< n e, h' > = 0 h' F hφ ( Ω) t Ω Ω s Γ c e (1) t-ω magnetodynamic formulation (similar) How to couple local and global quantities? h, φ V i, I i 9

10 Current as a strong global quantity Characterization of curl-conform vector fields : h or t h = he se, v S 1 ( Ω) e E h = h s + v + I c k E k k n N n n i C i i c c C φ E c : edges in Ω c Basis functions N cc : nodes in Ω cc and on Ω C c C : cuts Coupling of edge end nodal finite elements Explicit constraints for circulations and zero curl i.e. currents I i Circulation basis function, associated with a group of edges from a cut its circulation is equal to 1 along a closed path around Ω c c i = grad q i Elementary geometrical entities (nodes, edges) and global ones (groups of edges) 10

11 Voltage as a weak global quantity Discrete weak formulation ( µ hh, ') + ( σ 1curlh, curlh') +< n e, h' > = 0 h' F hφ ( Ω) t Ω Ω s Γ c e h = h s + v + I c k E k k n N n n i C i i c c C φ system of equations (symmetrical matrix) Test function h' = s k, v n classical treatment, no contribution for < > Γe Test function h' = c i contribution for < > Γe n e, h' = n e, c = n e, grad q = e dl = V s Γ s i Γ s i Γ γ s i h h h i Electromotive force Weak global quantity 11

12 Voltage as a weak global quantity and circuit relations n e, c = e dl = V s i Γ s i h γ Source of e.m.f. Electromotive force in (1) 1 t i Ω i Ω i ( µ hc, ) + ( σ curlh, curlc ) = V c Weak circuit relation between V i and I i for inductor i t (Magnetic Flux) + Resistance Current = Voltage Natural way to compute a weak voltage! Better than an explicit nonunique line integration 12

13 Massive and stranded inductors Massive inductor Direct application Stranded inductor Additional treatment Tree technique... r j Ω s, j s, j h = h + I h s h F h φ hs ( Ω) Number of turns Reaction field ( µ hh, ') + ( σ 1curlh, curlh') + ( σ 1 j, curlh') +< n e,' h > Γ = 0 t Ω Ω s Ω h'=h s,j 1 ( µ hh, ) + I ( σ j, curlh ) = V t s, j Ω s, j s, j s, j Ω j c Source field due to a magnetomotive force N j (one basis function for each stranded inductor) s w s e h' ( Ω) F h φ hs Weak circuit relation between V j and I j for stranded inductor j Natural way to compute the magnetic flux through all the wires! 13

14 Stranded inductors - Source field h = h s + v + I c k E k k n N n n i C i i c c C φ Simplified source field Projection method Source ( curl h, curl h') = ( j, curl h') sj, Ω sj, Electrokinetic problem 1 Ω sj, sj, h' ( Ω ) F h, sj, s, j ( σ curl h, curl ') = 0 Source = N j sj h Ω h' ( Ω ) F h s, j With gauge condition (tree) & boundary conditions 14

15 Stranded inductors - Magnetic flux Physical and geometrical interpretation of the circuit relation ( µ hh, s, j ) Ω Natural way to compute the magnetic flux through all the wires! 15

16 a-v weak formulation Magnetic vector potential - Electric scalar potential a v e = t a grad v in Ω c, with b = curl a in Ω a-v magnetodynamic formulation ( µ 1 curla, curla') + ( σ a, a') + ( σgrad v, a') Ω Ω Ω t c c ( j, a') = 0, a' F ( Ω) s Ω s a (1) How to couple local and global quantities? a, v V i, I i 16

17 Voltage as a strong global quantity With a' = grad v' in (1) ( σ ta, grad v') Ω + ( σ grad v, grad v') Ω = < n j, v' > Γ v' F v ( Ω c ) c c j (2) Weak form of div j = 0 At the discrete level : implication only true when grad F v (Ω c ) F a (Ω) OK with nodal and edge finite elements Otherwise : consideration of the 2 formulations (1) and (2) with a penalty term for gauge condition 17

18 Voltage as a strong global quantity Unit source electric scalar potential v 0 (basis function for the voltage) v = Vi v0 i Γ j i Needs a finite element resolution! Electrokinetic problem (physical field) ( σ grad v, grad v') =, v' F ( Ω ) 0 Ω 0 c v c Generalized potential (nonphysical field) i n Γ v0 = s = s j i n Direct expression Reduced support 18

19 Current as a weak global quantity and circuit relations i < n j, s > =< n j, 1> = n jds= I Γ Γ Γ j i j i j i i for massive inductor i in (2) I = ( σ, grads ) + ( σ grad v, grads ) i i i ta Ω Ω c c I = ( σ, grads ) + V ( σgrad v, grads ) i i ta i i Ω i 0 Ω c Weak circuit relation between V i and I i for massive inductor i c Natural way to compute a weak current! Better than an explicit nonunique surface integration 19

20 Circuit relation for stranded inductors From the a-formulation I = ( σ, grad v ) + V ( σ grad v, grad v ) j ta 0, j Ω j 0, j 0, j From the h-formulation sj, sj, 1 t sj, Ω j sj, sj, Ωsj, j ( µ hh, ) + I ( σ j, curlh ) = V Ω cannot be used ( µ hh, ) = ( bh, ) = ( curlah, ) t s, j Ω t s, j Ω t s, j Ω t( µ hh, s, j) Ω = t( a, curlhs, j) Ω + t < n ah, s, j > Ω ( µ hh, ) = ( aj, ) = ( a, j ) t s, j Ω t s, j Ω t s, j Ωsj, 1 t( aj, sj, ) Ω + Ij( σ jsj,, jsj, ) Ω = Vj sj, sj, Weak circuit relation between V j and I j for massive inductor j 20

21 Equivalent current density (1) Explicit distribution of the current density N def j js = t = w = w S j I unit t aw dω s + R I j = V Ω j s 21

22 Equivalent current density (2) Source electric scalar potential Source electric vector potential Projection method Source Projection method Source ( σ grad v, grad v') = ( j, grad v') 0 Ω s0 Ω sj, sj, ( curlh, curlh') = ( j, curlh') sj, Ω sj, Ω sj, sj, v' F v ( Ω s, j ) h' F h ( Ω s, j ) Electrokinetic problem Source Electrokinetic problem ( σ grad v, grad v') = < n j, v' > 0 Ω s Γ sj, Jj, 1 ( σ curlh, curl ') = sj h Ω, sj, 0 v' F v ( Ω s, j ) Source = N j h' ( Ω ) F h s, j Tensorial conductivity With gauge condition (tree) & boundary conditions 22

23 Application - Massive inductor Inductor-Core system in air a-form., 200 Hz 11 Resistance R (µω/m) h-form., 200 Hz µ r,core = a-form., 50 Hz 5 4 h-form., 50 Hz Number of elements (1/4th) µ r,core = 1, 10, 100, σ = m S/m Frequency f = 50, 200 Hz 3 Computation of resistance and inducance Inductance L (µh/m) h-form., 50 Hz µ r,core = 100 a-form., 50 Hz h-form., 200 Hz Complementarity between a-v and h-φ formulations validation at global level a-form., 200 Hz Number of elements 23

24 Application - Stranded inductor Inductor-Core system in air µ r,core = 10 Computation of a source field h s h µ r,core = 10 φ h-form., µ r =100 a-form., µ r = h-form., µ r =10 Computation of reaction field, total field and inducance Inductance L (H/m) a-form., µ r = Complementarity between h-φ and a-v formulations validation at global level h-form., µ r =1 a-form., µ r = Number of elements 24

25 Application Inductor-Core system in air bz(x), h-form. bz(x), a-form. bz(z), h-form. bz(z), a-form. µ r,core = 100 σ = m S/m bz (T) Mesh quality factor = x (y=0, z=0) and z (x=0, y=0) (m) (1/16th) bz(x), h-form. bz(x), a-form. bz(z), h-form. bz(z), a-form Mesh quality factor = 7 Enforcement of the current I j bz (T) A 0.02 Complementarity between h-φ and a-v formulations validation at local level x (y=0, z=0) and z (x=0, y=0) (m) 25

26 Application Inductor-Core system in air Computation of the inductance 3D coil 1 Inductance L (H/m) D Axi 2D Axi 2D Axi h-form., µ r =10 a-form., µ r =10 h-form., µ r =100 a-form., µ r =100 h-form., µ r =10000 a-form., µ r =10000 Inductance L (H/m) h-form., µ r =100 a-form., µ r =100 h-form., µ r =10 a-form., µ r = h-form., µ r =1 2D Axi a-form., µ r = Number of elements h-form., µ r =1 a-form., µ r = Mesh quality factor Axisymmetrical coil Complementarity between a-v and h-φ formulations validation at global level 26

27 Foil winding circuit relations # turns Circuit relation for one foil Nf Ii= ( σ ta,gradvs,i) + V ( σgradv,gradv ) L α Total thickness of all the foils Foil winding and its continuous representation Ω βγ i s,i s,i Ω βγ Voltage N L f α I i Ω α Domain of the foil V'( α) dα = V ( α) Continuum for V i (α) and weak form of circuit relation V i + σ,v'( α)gradv ) ( t a s, i Ω αβγ + Vi ( α)( σgradvs,i,v'( α)gradvs, i) = 0 V'( α) IR([0,L i α ]) Ω αβγ coordinate α Limited support of v s,i! Whole foil region Need of basis functions for V i (α) (polynomial or 1D finite element approximations)! Anisotropic conductivity! 27

28 Spatially dependent global quantities Foil Inductor-Core system in air Voltage (mv) Global voltage - 0 order poly. Global voltage - 1st order poly Global voltage - 2nd order poly. Global voltage - 3rd order poly. Global voltage - 6 piecewise constants 0.27 Massive foils (6) Massive foils (12) Massive foils (18) Position α (mm) Voltage of the foils in an n-foil 3D winding (n = 6, 12, 18) and its continuum in the associated foil region approximated by complete and piecewise polynomials 28

29 Conclusions - Global quantities General method for the definition of global quantities natural coupling between local quantities (scalar and vector fields) and global quantities (flux and circulation) for various formulations of various physical problems for all kinds of geometrical models (2D, 3D) for linear or nonlinear material characteristics for various finite elements (geometry and degree) For efficient treatment of coupled problems within a finite element problem through external lumped circuits 29

30 Conclusions - h-φ formulation h-φ magnetodynamic finite element formulations with massive and stranded inductors Use of edge and nodal finite elements for h and φ Natural coupling between h and φ Definition of current in a strong sense with basis functions either for massive or stranded inductors Definition of voltage in a weak sense Natural coupling between fields, currents and voltages 30

31 Conclusions - a-v formulation a-v 0 Magnetodynamic finite element formulation with massive and stranded inductors Use of edge and nodal finite elements for a and v 0 Definition of a source electric scalar potential v 0 in massive inductors in an efficient way (limited support) Natural coupling between a and v 0 for massive inductors Adaptation for stranded inductors: several methods Natural coupling between local and global quantities, i.e. fields and currents and voltages 31