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 2003 1
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 Weak formulations for finite element models � e.g., circuit coupling Essential and natural constraints, i.e., strongly and weakly satisfied 2
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, Interest for a source electric scalar potential v s in the a-v formulation “correct” discrete form of these � Coupling of local and global quantities constraints – e.g., currents and voltages in h- φ and a-v formulations (massive, stranded and foil inductors) Sequence of finite element � Interface conditions on thin regions spaces – i.e., discontinuities of either tangential or normal components 3
Sequence of finite element spaces Geometric elements Tetrahedral Hexahedra Prisms (4 nodes) (8 nodes) (6 nodes) Mesh Nodes Edges Faces Volumes i ∈ N i ∈ E i ∈ F i ∈ V Geometric entities S 0 S 1 S 2 S 3 Sequence of function spaces 4
Sequence of finite element spaces Degrees of Functions Properties Functionals freedom Point Nodal s i (x j ) = δ ij {s i , i ∈ N} S 0 Nodal value element evaluation ∀ i, j ∈ N ∫ Curve Circulation = δ ij Edge s i . d l { s i , i ∈ E} S 1 element integral along edge j ∀ i,j ∈ E ∫ Surface Flux across Face = δ ij s i . n ds { s i , i ∈ F} S 2 element integral face j ∀ i, j ∈ F ∫ Volume Volume Volume = δ ij {s i , i ∈ V} s i dv S 3 element integral integral j ∀ i, j ∈ V ∑ u K = φ i (u) s i Finite elements Bases i 5
Sequence of finite element spaces Base Continuity across Codomains of the functions element interfaces operators {s i , i ∈ N} S 0 value S 0 grad S 0 ⊂ S 1 { s i , i ∈ E} S 1 tangential component S 1 grad S 0 curl S 1 ⊂ S 2 { s i , i ∈ F} S 2 normal component S 2 curl S 1 div S 2 ⊂ S 3 {s i , i ∈ V} S 3 discontinuity S 3 div S 2 Conformity Sequence grad curl div S 0 S 1 S 2 S 3 → → → 6
Magnetodynamic problem with global constraints Equations Boundary conditions Global conditions n × h Γ h = 0 , n ⋅ b Γ e = 0 curl h = j for circuit coupling curl e = – ∂ t b , ∫ ∫ ⋅ = ⋅ = e d l V n j ds I i i γ Γ i i j div b = 0 Voltage Current Constitutive relations b = µ h j = σ e Inductor 7
Weak formulations Notations def Green formulae ( ) ∫ = ⋅ a b , a b dv Ω involved in weak formulations Ω def ∫ = a b , a b ds Γ Γ grad - div formula n Γ = ∂Ω ( u , grad v ) Ω + ( div u , v ) Ω = < n · u , v > Γ Domain Ω Weak global quantity of flux type curl - curl formula u , v ∈ H 1 ( Ω ), v ∈ H 1 ( Ω ) ( curl u , v ) Ω – ( u , curl v ) Ω = < n × u , v > Γ Weak global quantity of circulation type 8
h- φ and t- ω weak formulations Magnetic scalar potential in nonconducting regions Ω c C φ h r = – grad φ in Ω cC with h = h s + h r Reaction magnetic field Source magnetic field h- φ magnetodynamic formulation ∂ µ + σ − 1 +< × > = ∀ ∈ F h φ Ω ( h h , ' ) ( curl h , curl h ' ) n e h , ' 0 h ' ( ) Ω Ω Γ t s e c (1) t- ω magnetodynamic formulation (similar) How to couple local and global quantities ? h, φ V i , I i 9
Current as a strong global quantity Characterization of curl-conform vector fields : h or t ∑ 1 Ω Coupling of edge end nodal finite elements = ∈ h h s , v S ( ) e e ∈ e E Explicit constraints for circulations and zero curl ∑ ∑ ∑ = + C φ + h h s v I c k k n n i i i.e. currents I i ∈ ∈ ∈ k E n N i C c c E c : edges in Ω c Basis functions ‘Circulation’ basis function, C and on ∂ Ω c C : nodes in Ω c N c C associated with a group of edges C : cuts 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
Voltage as a weak global quantity Discrete weak formulation − 1 ∂ µ + σ +< × > = ∀ ∈ F h φ Ω ( h h , ' ) ( curl h , curl h ' ) n e h , ' 0 h ' ( ) Ω Ω Γ t s c e ∑ ∑ ∑ = + C φ + h h s v I c system of equations k k n n i i ∈ ∈ ∈ k E n N i C c c (symmetrical matrix) Test function h ' = s k , v n → classical treatment, no contribution for < · > Γ e Test function h ' = c i → contribution for < · > Γ e ∫ × = × = × − = ⋅ = n e , ' h n e , c n e , grad q e dl V s s i s i s i Γ Γ Γ γ h h h i Electromotive force Weak global quantity 11
Voltage as a weak global quantity and circuit relations Source of e.m.f. ∫ × = ⋅ = n e , c e dl V s i s i Γ γ h Electromotive force in (1) − 1 ∂ µ + σ = − ( h c , ) ( curl h , curl c ) V Ω Ω t i i i 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
Massive and stranded inductors Massive inductor Direct application Stranded inductor Additional treatment Tree technique ... ∑ Number of turns = + h ∈ F h h s Ω h h I h ( ) φ r s j , s j , ∈ Ω j s Source field due to a magnetomotive force N j Reaction field (one basis function for each stranded inductor) − − ∂ µ + σ + σ +< × > = 1 1 ( h h , ' ) ( curl h , curl h ' ) ( j , curl h ' ) n e h , ' 0 Ω Ω Ω Γ t s s c w e ∀ ∈ Ω h ' F h h s ( ) φ h '= h s,j − 1 ∂ µ + σ =− ( h h , ) I ( j , curl h ) V Ω Ω t s j , s j , s j , s j , j s 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
Stranded inductors - Source field ∑ ∑ ∑ = + C φ + h h s v I c Simplified source field k k n n i i ∈ ∈ ∈ k E n N i C c c Source Projection method = ( curl h , curl h ') ( j , curl h ') Ω Ω s j , s j , s j , s j , ∀ ∈ Ω h ' F h ( ) s j , Electrokinetic problem σ − 1 = ( curl h , curl h Ω ') 0 s j , s j , ∀ ∈ Ω h ' F h ( ) s j , Source = N j With gauge condition (tree) & boundary conditions 14
Stranded inductors - Magnetic flux Physical and geometrical interpretation of the circuit relation µ h h s j ( , ) Ω , Natural way to compute the magnetic flux through all the wires ! 15
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 − 1 curl µ + σ∂ + σ ( a , curl a ') ( a a , ') ( grad v , ') a Ω Ω Ω t c c (1) − = ∀ ∈ Ω ( j , ') a 0 , a ' F ( ) Ω s a s How to couple local and global quantities ? a, v V i , I i 16
Voltage as a strong global quantity With a' = grad v' in (1) (2) σ∂ + σ = < ⋅ > ( a , grad v ') ( grad v grad v , ') n j , ' v ∀ ∈ Ω v ' F v ( ) Ω Ω Γ t c c c j 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
Voltage as a strong global quantity Needs a finite element Unit source electric scalar potential v 0 resolution ! (basis function for the voltage) ∑ i = v V v i 0 ∈ Γ i j Electrokinetic problem (physical field) σ grad v = ∀ ∈ Ω ( , grad v ') 0 , v ' F ( ) Ω 0 v c c ∑ Generalized potential 0 = i = v s s n i ∈ Γ Direct expression n (nonphysical field) j Reduced support 18
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