Dual Finite Element Formulations and Associated Global Quantities - - PowerPoint PPT Presentation

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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

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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

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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 hs Coupling between source and reaction fields – e.g., source magnetic field hs in the h-φ formulation, source electric scalar potential vs 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

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Sequence of finite element spaces

Geometric elements

Tetrahedral

(4 nodes)

Hexahedra

(8 nodes)

Prisms

(6 nodes)

Mesh Geometric entities

Nodes

i ∈ N

Edges

i ∈ E

Faces

i ∈ F

Volumes

i ∈ V

S0 S1 S2 S3

Sequence of function spaces

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Sequence of finite element spaces

Functions

Functionals

Degrees of freedom Properties {si , i ∈ N} {si , i ∈ E} {si , i ∈ F} {si , i ∈ V}

Bases Finite elements

S0 S1 S2 S3 Volume element Point evaluation Curve integral Surface integral

si(x j) = δij

∀ i, j ∈N

si . n ds

j

∫

= δij

∀ i, j ∈F

si dv

j

∫

= δij

∀ i, j ∈V

si . dl

j

∫

= δij

∀ i,j ∈E

Nodal element Nodal value Circulation along edge Edge element Flux across face Face element Volume integral Volume integral uK = φi(u) si

i

∑

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Sequence of finite element spaces

Base functions Continuity across element interfaces Codomains of the

• perators

S0 S1 S2 S3

S0 S1 grad S0 S2 curl S1 S3 div S2

value {si , i ∈ N} tangential component {si , i ∈ E} grad S0 ⊂ S1 normal component {si , i ∈ F} curl S1 ⊂ S2 discontinuity {si , i ∈ V} div S2 ⊂ S3

Conformity

S0

grad

 →   S1

curl

 →   S2

div

 →   S3

Sequence

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Magnetodynamic problem with global constraints

Constitutive relations Boundary conditions Global conditions for circuit coupling n × hΓh = 0 , n ⋅ bΓe = 0

e l ⋅ =

∫

d V

i

i γ

,

n j ⋅ =

∫

ds I

j i

i Γ

Inductor Voltage Current Equations curl h = j curl e = – ∂t b div b = 0 b = µ h j = σ e

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Weak formulations

( )

a b a b , ,

Ω Ω Γ Γ

= ⋅ =

∫ ∫

def

dv a b a b ds

def

Notations Green formulae

involved in weak formulations

( u , grad v )Ω + ( div u , v )Ω = < n · u , v >Γ grad - div formula Γ = ∂Ω Domain Ω n

Weak global quantity of flux type

( curl u , v )Ω – ( u , curl v )Ω = < n × u , v >Γ curl - curl formula u, v ∈ H1(Ω), v ∈ H1(Ω)

Weak global quantity of circulation type

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h-φ and t-ω weak formulations

Magnetic scalar potential in nonconducting regions Ωc

C

(1)

hr = – grad φ in ΩcC ∂ µ σ

t s

curl curl

c e

( , ' ) ( , ' ) , ' h h h h n e h

Ω Ω Γ

+ +< × > =

−1

∀ ∈ h' ( ) Fhφ Ω

t-ω magnetodynamic formulation (similar) with h = hs + hr Source magnetic field Reaction magnetic field φ h-φ magnetodynamic formulation How to couple local and global quantities ? Vi, Ii h, φ

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Current as a strong global quantity

Characterization of curl-conform vector fields : h or t

Ec : edges in Ωc Nc

C : nodes in Ωc C and on ∂Ωc C

C : cuts

Elementary geometrical entities (nodes, edges) and global ones (groups of edges)

Basis functions

‘Circulation’ basis function, associated with a group of edges from a cut → its circulation is equal to 1 along a closed path around Ωc

ci

i

grad q = − h s v = ∈

∈

∑

h S

e e e E

, ( )

1 Ω

h s v c = + +

∈ ∈ ∈

∑ ∑ ∑

h I

k k k E n n n N i i i C

c c C φ

Coupling of edge end nodal finite elements Explicit constraints for circulations and zero curl

i.e. currents Ii

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Voltage as a weak global quantity

Discrete weak formulation

h s v c = + +

∈ ∈ ∈

∑ ∑ ∑

h I

k k k E n n n N i i i C

c c C φ

∂ µ σ

t s

curl curl

c e

( , ' ) ( , ' ) , ' h h h h n e h

Ω Ω Γ

+ +< × > =

−1

∀ ∈ h' ( ) Fhφ Ω

system of equations (symmetrical matrix)

n e h n e c n e e × = × = × − = ⋅ =

∫

s s i s i s i

h h h i

grad q dl V , ' , ,

Γ Γ Γ γ

Test function h' = sk, vn → classical treatment, no contribution for < · >Γe Test function h' = ci → contribution for < · >Γe Electromotive force Weak global quantity

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Electromotive force

n e c e × = ⋅ =

∫

s i s i

h

dl V ,

Γ γ

∂ µ σ

t i i i

curl curl V

c

( , ) ( , ) h c h c

Ω Ω

+ = −

−1

Voltage as a weak global quantity and circuit relations

Source of e.m.f. in (1) Weak circuit relation between Vi and Ii for inductor i “ ∂t (Magnetic Flux) + Resistance × Current = Voltage ” Natural way to compute a weak voltage ! Better than an explicit nonunique line integration

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Massive and stranded inductors

Stranded inductor

h h h = +

∈

∑

r s j s j j

I

s

, , Ω

h ∈Fh hs

φ

( ) Ω

∂ µ σ σ

t s

curl curl curl

c w

( , ' ) ( , ' ) ( , ' ) h h h h j h

Ω Ω Ω

+ +

− − 1 1

+< × > = n e h

s

e

, '

Γ

Source field due to a magnetomotive force Nj (one basis function for each stranded inductor) Number of turns Reaction field Massive inductor Direct application Additional treatment Tree technique ...

∀ ∈ h' ( ) Fh hs

φ

Ω

h'=hs,j

∂ µ σ

t s j s j s j s j j

I curl V

s

( , ) ( , )

, , , ,

h h j h

Ω Ω

+ =−

−1

Weak circuit relation between Vj and Ij for stranded inductor j Natural way to compute the magnetic flux through all the wires !

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Stranded inductors - Source field

h s v c = + +

∈ ∈ ∈

∑ ∑ ∑

h I

k k k E n n n N i i i C

c c C φ

Simplified source field Source Projection method

( , ') ( , ')

, ,

, ,

curl curl curl

s j s j

s j s j

h h j h

Ω Ω

=

∀ ∈ h' ( )

,

Fh

s j

Ω

Electrokinetic problem

( , ')

,

,

σ− =

1

curl curl

s j

s j

h h Ω

∀ ∈ h' ( )

,

Fh

s j

Ω

Source = Nj With gauge condition (tree) & boundary conditions

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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 !

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a-v weak formulation

e = – ∂t a – grad v in Ωc , with b = curl a in Ω

a Magnetic vector potential - Electric scalar potential v a-v magnetodynamic formulation (1) ( , ') ( , ') ( , ') µ σ∂ σ

−

+ +

1curl

curl grad v

t

c c

a a a a a

Ω Ω Ω

− = ∀ ∈ ( , ') , ' ( ) j a a

s a

s

F

Ω

Ω

How to couple local and global quantities ? a, v Vi, Ii

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Voltage as a strong global quantity

With a' = grad v' in (1) ( , ') ( , ') , ' σ∂ σ

t

grad v grad v grad v v

c c j

a n j

Ω Ω Γ

+ = < ⋅ >

∀ ∈ v Fv

c

' ( ) Ω

(2) Weak form of div j = 0 At the discrete level : implication only true when grad Fv(Ωc) ⊂ Fa(Ω) OK with nodal and edge finite elements Otherwise : consideration of the 2 formulations (1) and (2) with a penalty term for gauge condition

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Voltage as a strong global quantity

v V v

i i i

j

=

∈

∑

Γ

( , ') , ' ( ) σ grad v grad v v F

c

v c Ω

Ω = ∀ ∈ v s s

i n n

j i

0 =

=

∈

∑

Γ

Generalized potential (nonphysical field) Needs a finite element resolution ! Direct expression

Reduced support Unit source electric scalar potential v0 (basis function for the voltage)

Electrokinetic problem (physical field)

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Current as a weak global quantity and circuit relations

< ⋅ > =< ⋅ > = ⋅ =

∫

n j n j n j , , s ds I

i i

j i j i j i

Γ Γ Γ

1 for massive inductor i I grads grad v grads

i t i i

c c

= + ( , ) ( , ) σ∂ σ a

Ω Ω

in (2) I grads V grad v grads

i t i i i i

c c

= + ( , ) ( , ) σ∂ σ a

Ω Ω

Weak circuit relation between Vi and Ii for massive inductor i Natural way to compute a weak current ! Better than an explicit nonunique surface integration

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Circuit relation for stranded inductors

I grad v V grad v grad v

j t j j j j

s j s j

= + ( , ) ( , )

, , ,

, ,

σ∂ σ a

Ω Ω

From the a-formulation

cannot be used

From the h-formulation ∂ µ σ

t s j j s j s j j

I curl V

s j

( , ) ( , )

, , ,

,

h h j h

Ω Ω

+ = −

−1

∂ µ ∂ ∂

t s j t s j t s j

curl ( , ) ( , ) ( , )

, , ,

h h b h a h

Ω Ω Ω

= = ∂ µ ∂ ∂

∂Ω t s j t s j t s j

curl ( , ) ( , ) ,

, , ,

h h a h n a h

Ω Ω

= + < × > ∂ µ ∂ ∂

t s j t s j t s j s j

( , ) ( , ) ( , )

, , , ,

h h a j a j

Ω Ω Ω

= = ∂ σ

t s j j s j s j j

s j s j

I V ( , ) ( , )

, , ,

, ,

a j j j

Ω Ω

+ = −

−1

Weak circuit relation between Vj and Ij for massive inductor j

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Equivalent current density (1)

Explicit distribution of the current density j t w w

s j j def unit

N S I = = = ∂t

s j j

d R I V

s

a w ⋅ + = −

∫

Ω

Ω

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Equivalent current density (2)

Source electric vector potential Source electric scalar potential Source Source Projection method Projection method

( , ') ( , ')

, ,

, ,

curl curl curl

s j s j

s j s j

h h j h

Ω Ω

= ( , ') ( , ')

, ,

σgrad v grad v grad v

s j s j

s Ω Ω

= −j ∀ ∈ v Fv

s j

' ( )

,

Ω

∀ ∈ h' ( )

,

Fh

s j

Ω

( , ')

,

,

σ− =

1

curl curl

s j

s j

h h Ω

∀ ∈ h' ( )

,

Fh

s j

Ω

With gauge condition (tree) & boundary conditions Electrokinetic problem ( , ') , '

, ,

σgrad v grad v v

s j J j

s Ω Γ

= < ⋅ > n j Source Electrokinetic problem ∀ ∈ v Fv

s j

' ( )

,

Ω

Source = Nj Tensorial conductivity

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Application - Massive inductor

Inductor-Core system in air

2.4 2.5 2.6 2.7 2.8 2.9 3 2000 4000 6000 8000 10000 12000 Inductance L (µH/m) Number of elements h-form., 50 Hz a-form., 50 Hz h-form., 200 Hz a-form., 200 Hz 4 5 6 7 8 9 10 11 12 13 2000 4000 6000 8000 10000 12000 Resistance R (µΩ/m) Number of elements h-form., 50 Hz a-form., 50 Hz h-form., 200 Hz a-form., 200 Hz

µr,core = 100 µr,core = 100 (1/4th) µr,core = 1, 10, 100 , σ = 5.9 107m S/m Frequency f = 50, 200 Hz

Computation of resistance and inducance

Complementarity between a-v and h-φ formulations → validation at global level

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Application - Stranded inductor

Inductor-Core system in air

µr,core = 10

Computation of reaction field, total field and inducance

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 1000 2000 3000 4000 5000 6000 Inductance L (H/m) Number of elements h-form., µr=1 a-form., µr=1 h-form., µr=10 a-form., µr=10 h-form., µr=100 a-form., µr=100

φ hs

µr,core = 10

Computation of a source field h

Complementarity between h-φ and a-v formulations → validation at global level

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Application

Inductor-Core system in air

• 0.02

0.02 0.04 0.06 0.08 0.1 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 bz (T) x (y=0, z=0) and z (x=0, y=0) (m) bz(x), h-form. bz(x), a-form. bz(z), h-form. bz(z), a-form.

Mesh quality factor = 3 (1/16th) µr,core = 100 σ = 5.9 107m S/m 1 A

• 0.02

0.02 0.04 0.06 0.08 0.1 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 bz (T) x (y=0, z=0) and z (x=0, y=0) (m) bz(x), h-form. bz(x), a-form. bz(z), h-form. bz(z), a-form.

Mesh quality factor = 7

Enforcement of the current Ij

Complementarity between h-φ and a-v formulations → validation at local level

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Application

Inductor-Core system in air Computation of the inductance

3D coil

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2000 4000 6000 8000 10000 12000 14000 Inductance L (H/m) Number of elements h-form., µr=1 a-form., µr=1 h-form., µr=10 a-form., µr=10 h-form., µr=100 a-form., µr=100 h-form., µr=10000 a-form., µr=10000 2D Axi 2D Axi 2D Axi 2D Axi 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2 3 4 5 6 7 8 9 Inductance L (H/m) Mesh quality factor h-form., µr=1 a-form., µr=1 h-form., µr=10 a-form., µr=10 h-form., µr=100 a-form., µr=100

Axisymmetrical coil

Complementarity between a-v and h-φ formulations → validation at global level

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Foil winding circuit relations

Foil winding and its continuous representation

coordinate α

βγ βγ

Ω Ω α

σ + ∂ σ = ) v grad , v grad ( V ) v grad , ( I L N

i , s i , s i i , s t i f

a

Circuit relation for one foil

∫

α

Ω α

α α − d ) ( ' V I L N

i f

αβγ

Ω

α ∂ σ + ) v grad ) ( ' V , (

i , s ta

) v grad ) ( ' V , v grad ( ) ( V

i , s i , s i

= α σ α +

αβγ

Ω

]) L , ([ IR ) ( ' V

α

∈ α ∀

# turns

) ( V V

i i

α =

Continuum for Vi(α) and weak form of circuit relation

Limited support of vs,i ! Domain of the foil Voltage Total thickness of all the foils Whole foil region

Need of basis functions for Vi(α) (polynomial or 1D finite element approximations) ! Anisotropic conductivity !

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Spatially dependent global quantities

Foil Inductor-Core system in air

0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 5 10 15 20 Voltage (mV) Position α (mm) 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 Massive foils (6) Massive foils (12) Massive foils (18)

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

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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

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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

• r stranded inductors

– Definition of voltage in a weak sense – Natural coupling between fields, currents and voltages

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Conclusions - a-v formulation

a-v0 Magnetodynamic finite element formulation with

massive and stranded inductors

Use of edge and nodal finite elements for a and v0 – Definition of a source electric scalar potential v0 in massive inductors in an efficient way (limited support) – Natural coupling between a and v0 for massive inductors – Adaptation for stranded inductors: several methods – Natural coupling between local and global quantities, i.e. fields and currents and voltages