finite element modeling of electromagnetic systems · 2 introduction formulations of...

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Finite Element Modeling of Electromagnetic Systems

Mathematical and numerical tools

Unit of Applied and Computational Electromagnetics (ACE) Dept. of Electrical Engineering - University of Liège - Belgium

Patrick Dular, Christophe Geuzaine

October 2009

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Introduction

 Formulations of electromagnetic problems ♦ Maxwell equations, material relations ♦ Electrostatics, electrokinetics, magnetostatics, magnetodynamics ♦ Strong and weak formulations

 Discretization of electromagnetic problems ♦ Finite elements, mesh, constraints ♦ Very rich content of weak finite element formulations

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Formulations of Electromagnetic Problems

Maxwell equations

Electrostatics

Electrokinetics

Magnetostatics

Magnetodynamics

Waves

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Electromagnetic models   Electrostatics

♦  Distribution of electric field due to static charges and/or levels of electric potential

  Electrokinetics ♦  Distribution of static electric current in conductors

  Electrodynamics ♦  Distribution of electric field and electric current in materials (insulating

and conducting)

  Magnetostatics ♦  Distribution of static magnetic field due to magnets and continuous

currents

  Magnetodynamics ♦  Distribution of magnetic field and eddy current due to moving magnets

and time variable currents

  Wave propagation ♦  Propagation of electromagnetic fields

All phenomena are described by Maxwell equations

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Maxwell equations

curl h = j + ∂t d

curl e = – ∂t b

div b = 0

div d = ρv

Maxwell equations

Ampère equation

Faraday equation

Conservation equations

Principles of electromagnetism

h magnetic field (A/m) e electric field (V/m) b magnetic flux density (T) d electric flux density (C/m2) j current density (A/m2) ρv charge density (C/m3)

Physical fields and sources

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Constants (linear relations)

Functions of the fields (nonlinear materials)

Tensors (anisotropic materials)

Material constitutive relations

b = µ h (+ bs)

d = ε e (+ ds)

j = σ e (+ js)

Constitutive relations

Magnetic relation

Dielectric relation

Ohm law

bs remanent induction, ... ds ... js source current in stranded inductor, ...

Possible sources

µ magnetic permeability (H/m) ε dielectric permittivity (F/m) σ electric conductivity (Ω–1m–1)

Characteristics of materials

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• Formulation for • the exterior region Ω0 • the dielectric regions Ωd,j

• In each conducting region Ωc,i : v = vi → v = vi on Γc,i

e electric field (V/m) d electric flux density (C/m2) ρ electric charge density (C/m3) ε dielectric permittivity (F/m)

Electrostatics

Type of electrostatic structure

Ω0 Exterior region Ωc,i Conductors Ωd,j Dielectric

div ε grad v = – ρ

with e = – grad v

Electric scalar potential formulation

curl e = 0 div d = ρ d = ε e

Basis equations

n × e | Γ0e = 0 n ⋅ d | Γ0d = 0

& boundary conditions

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Electrostatics

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• Formulation for • the conducting region Ωc • On each electrode Γ0e,i : v = vi → v = vi on Γ0e,i

e electric field (V/m) j electric current density (C/m2) σ electric conductivity (Ω–1m–1)

Electrokinetics

Type of electrokinetic structure

Ωc Conducting region div σ grad v = 0

with e = – grad v

Electric scalar potential formulation

curl e = 0 div j = 0 j = σ e

Basis equations

n × e | Γ0e = 0 n ⋅ j | Γ0j = 0

& boundary conditions

Γ0e,0

Γ0e,1 Γ0j

e=?, j=?

Ωc

V = v1 – v0

10 "e" side "d" side

Electrostatic problem

Basis equations curl e = 0 div d = ρ d = ε e

e = – grad v d = curl u

⊂ ⊃

ε e = d d

ρ

( u )

grad

curl

div

e

e

e

e

0

0

(– v ) F e 0

F e 1

F e 2

F e 3

div

curl

grad

d

d

d

F d 3

F d 2

F d 1

F d 0

e S 0

S e 1

S e 2

S e 3

S d 3

S d 2

S d 1

S d 0

11 "e" side "j" side

Electrokinetic problem

Basis equations curl e = 0 div j = 0 j = σ e

e = – grad v j = curl t

⊂ ⊃

σ e = j d

ρ

( t )

grad

curl

div

e

e

e

e

0

0

(– v ) F e 0

F e 1

F e 2

F e 3

div

curl

grad

j

j

j

F j 3

F j 2

F j 1

F j 0

e S 0

S e 1

S e 2

S e 3

S j 3

S j 2

S j 1

S j 0

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u ≡ weak solution

u v v v v ( , L * ) - ( f , ) + Q g ( ) ds Γ ∫ = 0 , ∀ ∈ V ( Ω )

Classical and weak formulations

u ≡ classical solution

L u = f in Ω B u = g on Γ = ∂Ω

Classical formulation

Partial differential problem

Weak formulation

( u , v ) = u ( x ) v ( x ) d x Ω ∫ , u , v ∈ L 2 ( Ω )

( u , v ) = u ( x ) . v ( x ) d x Ω ∫ , u , v ∈ L 2 ( Ω )

Notations

v ≡ test function Continuous level : ∞ × ∞ system Discrete level : n × n system ⇒ numerical solution

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Classical and weak formulations Application to the magnetostatic problem

curl e = 0 div d = 0 d = ε e

n × e ⏐Γe = 0 n . d ⏐Γd = 0 Γe Γd

Electrostatic classical formulation

Weak formulation of div d = 0

(+ boundary condition)

( d , grad v' ) = 0 , ∀ v' ∈ V(Ω)

d = ε e & e = – grad v ⇔ curl e = 0

( – ε grad v , grad v' ) = 0 , ∀ v' ∈ V(Ω) Electrostatic weak formulation with v

with V(Ω) = { v ∈ H0(Ω) ; v⏐Γe = 0 }

( div d , v' ) + < n . d , v' >Γ = 0 , ∀ v' ∈ V(Ω)

div d = 0 ⇓

n . d ⏐Γd = 0 ⇓

⇒

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Classical and weak formulations Application to the magnetostatic problem

curl h = j div b = 0 b = µ h

n × h ⏐Γh = 0 n . b ⏐Γe = 0 Γh Γe

Magnetostatic classical formulation

Weak formulation of div b = 0

(+ boundary condition)

( b , grad φ' ) = 0 , ∀ φ' ∈ Φ(Ω)

b = µ h & h = hs – grad φ (with curl hs = j) ⇔ curl h = j

( µ (hs – grad φ) , grad φ' ) = 0 , ∀ φ' ∈ Φ(Ω) Magnetostatic weak formulation with φ

with Φ(Ω) = { φ ∈ H0(Ω) ; φ⏐Γh = 0 }

( div b , φ' ) + < n . b , φ' >Γ = 0 , ∀ φ' ∈ Φ(Ω)

div b = 0 ⇓

n . b ⏐Γe = 0 ⇓

⇒

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Quasi-stationary approximation

Conduction current density

Displacement current density > > >

curl h = j + ∂t d

curl h = j

Electrotechnic apparatus (motors, transformers, ...) Frequencies from Hz to a few 100 kHz

Applications

Dimensions << wavelength

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curl h = j

div b = 0

Magnetostatics

Equations

b = µ h + bs

j = js


Type of studied configuration

Ampère equation

Magnetic conservation equation

Magnetic relation

Ohm law & source current

Ω Studied domain Ωm Magnetic domain

Ωs Inductor

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curl h = j

curl e = – ∂t b

div b = 0

Magnetodynamics

Equations

b = µ h + bs

j = σ e + js


Type of studied configuration

Ampère equation

Faraday equation

Magnetic conservation equation

Magnetic relation

Ohm law & source current

Ω Studied domain Ωp Passive conductor and/or magnetic domain Ωa Active conductor Ωs Inductor

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Magnetic constitutive relation

  Diamagnetic and paramagnetic materials Linear material µr ≈ 1 (silver, copper,

aluminium)

  Ferromagnetic materials Nonlinear material µr >> 1 , µr = µr(h) (steel, iron)

b = µ h µr relative magnetic permeability µ = µr µ0

b-h characteristic of steel µr-h characteristic of steel

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Magnetodynamics Inductor (portion : 1/8th)

Stranded inductor - uniform current density (js)

Massive inductor - non-uniform current density (j)

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Magnetodynamics - Joule losses Foil winding inductance - current density (in a cross-section)

All foils

With air gaps, Frequency f = 50 Hz

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Transverse induction heating (nonlinear physical characteristics,

moving plate, global quantities)

Eddy current density

Temperature distribution

Search for OPTIMIZATION of temperature profile

Magnetodynamics - Joule losses

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Magnetodynamics - Forces

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Magnetic field lines and electromagnetic force (N/m) (8 groups, total current 3200 A)

Currents in each of the 8 groups in parallel non-uniformly distributed!

Magnetodynamics - Forces

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Inductive and capacitive effects

  Frequency and time domain analyses

  Any conformity level

Magnetic flux density Electric field

Resistance, inductance and capacitance versus frequency

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Maxwell equations (magnetic - static)

Magnetostatic formulations

a Formulation φ Formulation

curl h = j div b = 0

b = µ h

"h" side "b" side

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Magnetostatic formulations

a Formulation φ Formulation

Multivalued potential Cuts

Non-unique potential Gauge condition

Magnetic scalar potential φ Magnetic vector potential a

curl ( µ–1 curl a ) = j

b = curl a hs given such as curl hs = j

(non-unique) div ( µ ( hs – grad φ ) ) = 0

h = hs – grad φ

curl h = j div b = 0 b = µ h

Basis equations

(h) (b) (m)

⇒ (h) OK (b) OK ⇐

← (b) & (m) (h) & (m) →

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Multivalued scalar potential

Kernel of the curl (in a domain Ω) ker ( curl ) = { v : curl v = 0 }

dom(grad)

dom(curl)

ker(curl)

cod(grad)

cod ( grad ) ⊂ ker ( curl )

cod(curl)

. 0

grad

curl

curl

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AB AB

h . d l γ ∫ = - grad φ . d l

γ ∫ = φ A - φ B

Multivalued scalar potential - Cut

curl h = 0 in Ω h = – grad φ in Ω

Scalar potential φ ⇒ ?

OK ⇐

Circulation of h along path γAB in Ω

⇒ φA – φB = 0 ≠ I ! ! !

Closed path γAB (A≡B) surrounding a conductor (with current I)

Δ φ = I

φ must be discontinuous ... through a cut

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Vector potential - gauge condition

div b = 0 in Ω b = curl a in Ω

Vector potential a ⇒ ?

OK ⇐

b = curl a = curl ( a + grad η ) Non-uniqueness of vector potential a

Coulomb gauge div a = 0

ex.: w(r)=r

ω vector field with non-closed lines linking any 2 points in Ω

Gauge a . ω = 0

Gauge condition

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Magnetodynamic formulations

"h" side "b" side

Maxwell equations (quasi-stationary)

h-φ Formulation

a-v Formulation t-ω Formulation

a* Formulation

curl h = j curl e = – ∂t b

div b = 0

b = µ h j = σ e

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t-ω Formulation h-φ Formulation

Magnetic field h Magnetic scalar potential φ

Electric vector potential t Magnetic scalar potential ω

curl (σ–1 curl t) + ∂t (µ (t – grad ω)) = 0

div (µ (t – grad ω)) = 0


div b = 0 b = µ h

Basis equations

(h) (b)

curl hs = js

h ds Ωc h = hs – grad φ ds Ωc

C ⇒ (h) OK

j = σ e

curl (σ–1 curl h) + ∂t (µ h) = 0

div (µ (hs – grad φ)) = 0

j = curl t (h) OK ⇐

h = t – grad ω

← in Ωc →

← in ΩcC →

← (b) →

+ Gauge

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a-v Formulation a* Formulation

Magnetic vector potential a* Magnetic vector potential a Electric scalar potential v

curl (µ–1 curl a) + σ (∂t a + grad v)) = js


div b = 0 b = µ h

Basis equations

(h) (b) j = σ e

curl (µ–1 curl a*) + σ ∂t a* = js

b = curl a (b) OK ⇐

e = – ∂t a – grad v

b = curl a*

⇒ (b) OK e = – ∂t a*

+ Gauge in Ω

← (h) →

+ Gauge in ΩcC

33 "h" side "b" side

Magnetostatic problem

Basis equations curl h = j div b = 0 b = µ h

h = – grad φ b = curl a

⊂ ⊃

34 "h" side "b" side

Magnetodynamic problem

h = t – grad φ b = curl a

⊂ ⊃


div b = 0 b = µ h

Basis equations

j = σ e

e = – ∂t a – grad v

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dom (grade) = Fe0 = { v ∈ L2(Ω) ; grad v ∈ L2(Ω) , v⏐Γe = 0 }

dom (curle) = Fe1 = { a ∈ L2(Ω) ; curl a ∈ L2(Ω) , n ∧ a⏐Γe = 0 }

dom (dive) = Fe2 = { b ∈ L2(Ω) ; div b ∈ L2(Ω) , n . b⏐Γe = 0 }

Boundary conditions on Γe

dom (gradh) = Fh0 = { φ ∈ L2(Ω) ; grad φ ∈ L2(Ω) , φ⏐Γh = 0 }

dom (curlh) = Fh1 = { h ∈ L2(Ω) ; curl h ∈ L2(Ω) , n ∧ h⏐Γh = 0 }

dom (divh) = Fh2 = { j ∈ L2(Ω) ; div j ∈ L2(Ω) , n . j⏐Γh = 0 }

Boundary conditions on Γh

Function spaces Fe0 ⊂ L2, Fe

1 ⊂ L2, Fe2 ⊂ L2, Fe

3 ⊂ L2

Function spaces Fh0 ⊂ L2, Fh

1 ⊂ L2, Fh2 ⊂ L2, Fh

3 ⊂ L2 Basis structure

Basis structure

Continuous mathematical structure

gradh Fh0 ⊂ Fh

1 , curlh Fh1 ⊂ Fh

2 , divh Fh2 ⊂ Fh

3

F h 0 grad h ⎯ → ⎯ ⎯ ⎯ F h

1 curl h ⎯ → ⎯ ⎯ F h 2 div h ⎯ → ⎯ ⎯ F h

3 Sequence

gradh Fe0 ⊂ Fe

1 , curle Fe1 ⊂ Fe

2 , dive Fe2 ⊂ Fe

3

F e 3 div e ← ⎯ ⎯ ⎯ F e

2 curl e ← ⎯ ⎯ ⎯ F e 1 grad e ← ⎯ ⎯ ⎯ ⎯ F e

0 Sequence

Domain Ω, Boundary ∂Ω = Γh U Γe

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Discretization of Electromagnetic Problems

Nodal, edge, face and volume finite elements

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Discrete mathematical structure

Continuous function spaces & domain Classical and weak formulations

Continuous problem

Finite element method

Discrete function spaces piecewise defined in a discrete domain (mesh)

Discrete problem

Discretization Approximation

Classical & weak formulations → ? Properties of the fields → ?

Questions To build a discrete structure

as similar as possible as the continuous structure

Objective

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Discrete mathematical structure

Sequence of finite element spaces Sequence of function spaces & Mesh

Finite element space Function space & Mesh + f i

i "

+ f i i "

⎧ ⎨ ⎩ ⎪

⎫ ⎬ ⎭ ⎪

Finite element Interpolation in a geometric element of simple shape

+ f

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Finite elements

  Finite element (K, PK, ΣK) ♦  K = domain of space (tetrahedron, hexahedron, prism) ♦  PK = function space of finite dimension nK, defined in K ♦  ΣK = set of nK degrees of freedom

represented by nK linear functionals φi, 1 ≤ i ≤ nK, defined in PK and whose values belong to IR

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Finite elements

  Unisolvance ∀ u ∈ PK , u is uniquely defined by the degrees of freedom

  Interpolation

  Finite element space Union of finite elements (Kj, PKj, ΣKj) such as :

  the union of the Kj fill the studied domain (≡ mesh)   some continuity conditions are satisfied across the element

interfaces

Basis functions

Degrees of freedom

u K = φ i ( u ) p i i = 1

n K

∑

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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

Sequence of function spaces S0 S1 S2 S3

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{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 Volume integral

Nodal value

Circulation along edge Flux across

face Volume integral

Functions Functionals Degrees of freedom Properties

Face element

Edge element

Nodal element

∀ i , j ∈ E

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Base functions

Continuity across element interfaces

Codomains of the operators

{si , i ∈ N} value

{si , i ∈ E} tangential component grad S0 ⊂ S1

{si , i ∈ F} normal component curl S1 ⊂ S2

{si , i ∈ V} discontinuity div S2 ⊂ S3

Conformity S 0 grad

⎯ → ⎯ ⎯ S 1 curl ⎯ → ⎯ ⎯ S 2 div

⎯ → ⎯ ⎯ S 3

Sequence

S0

S1 grad S0

S2 curl S1

S3 div S2

S0

S1

S2

S3

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Function spaces S0 et S3

For each node i ∈ N → scalar field

0

0 0

0 0

0 0

0

0

0 pi = 1 node i

si (x) = pi (x) ∈ S0

p i = 1 at node i 0 at all other nodes

⎧ ⎨ ⎩

p i continuous in Ω

sv = 1 / vol (v) ∈ S3

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Edge function space S1

For each edge eij = {i, j} ∈ E → vector field

se ∈ S1

N.B.: In an element : 3 edges/node

Illustration of the vector field se Definition of the set of nodes NF,mn -

s e ij = p j grad p r r ∈ N F , j i

∑ - p i grad p r r ∈ N F , i j

∑

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Edge function space S1

Geometric interpretation of the vector field se

p j grad p r r ∈ N F j , i

∑ ⊥ N F , j i

s e ij = p j grad p r r ∈ N F , j i

∑ - p i grad p r r ∈ N F , i j

∑

- p i grad p r r ∈ N F , i j

∑ ⊥ N F , i j

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Function space S2

For each facet f ∈ F → vector field f = fijk(l) = {i, j, k (, l) } = {q1, q2, q3 (, q4) }

Illustration of the vector field sf

s f = a f p q c c = 1

# N f

∑ grad p r r ∈ N F , q c q c + 1 ∑ ⎛

⎝ ⎜

⎞

⎠ ⎟ ∧ grad p r r ∈ N F , q c q c - 1

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟

sf ∈ S2

3 → af = 2 #Nf =

4 → af = 1

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Particular subspaces of S1

Ampere equation in a domain Ωc

C without current

(↔ Ωc)

Applications

Gauge condition on a vector potential

Definition of a generalized source field hs

such that curl hs = js

h ∈ S1(Ω) ; curl h = 0 in ΩcC ⊂ Ω → h ≡ ?

a ∈ S1(Ω) ; b = curl a ∈ S2(Ω) → a ≡ ? Gauge a . ω = 0 to fix a

Kernel of the curl operator

Gauged subspace

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Ω c ∂ Ω c

E c

h k

φ n

φ n'

Ω c C

N c , E c C C N c , E c C C

(interface)

Base of H ≡ basis functions of • inner edges of Ωc • nodes of Ωc

C, with those of ∂Ωc

H = { h ∈ S 1 ( Ω ) ; curl h = 0 in Ω c C }

with

h = h k s k k ∈ E c

∑ + φ n v n n ∈ N c

C ∑

v n = s nj nj ∈ E c

C ∑

h l l ∈ E c

C h = h a s a

e ∈ E ∑ = h k s k

k ∈ E c

∑ + s l ∑

h l = h . dl l ab

∫ = - grad φ . dl l ab

∫ = φ a l - φ b l

( h = h k s k k ∈ E c

∑ + ) s l l ∈ E c

C ∑ a l - φ b l φ

h = – grad φ in ΩcC

Case of simply connected domains

50


H = { h ∈ S 1 ( Ω ) ; curl h = 0 in Ω c C }

φ+ – φ– = φ⏐Γ+eci – φ⏐Γ–

eci = Ii

qi • defined in ΩcC

• unit discontinuity across Γeci • continuous in a transition layer • zero out of this layer

φ = φcont + φdisc

discontinuity of φdisc

with

edges of ΩcC

starting from a node of the cut and located on side '+'

but not on the cut

h = h k s k k ∈ A c

∑ + φ cont n v n

n ∈ N c C

∑ + I i c i i ∈ C ∑

c i = s nj nj ∈ A c

C

n ∈ N ec i j ∈ N c

C +

j ∉ N ec i

∑

φ disc = I i q i i ∈ C ∑

Basis of H ≡ basis functions of • inner edges of Ωc • nodes of Ωc

C • cuts of C

(cuts) h = – grad φ in Ωc

C

Case of multiply connected domains

51

Gauged subspace of S1

b = curl a

co-tree "E

Tree ≡ set of edges connecting (in Ω) all the nodes of Ω without

forming any loop (E) Co-tree ≡ complementary set of the tree (E)

" "

a = a e s e e ∈ E ∑ ∈ S 1 ( Ω ) , b = b f s f

f ∈ F ∑ ∈ S 2 ( Ω )

a = a i s i i ∈ "E ∑ ∈

"S 1 ( Ω )

S1(Ω) = {a ∈ S1(Ω) ; aj = 0 , ∀ j ∈ E} " "

Gauged space of S1(Ω)

with

b f = i ( e , f ) a e e ∈ E ∑ , f ∈ F matrix form: bf CFE

ae =

Face-edge incidence matrix

Gauged space in Ω

tree "E

Basis of S1(Ω) ≡ co-tree edge basis functions (explicit gauge definition)

"

52

  Electromagnetic fields extend to infinity (unbounded domain)

♦  Approximate boundary conditions:

  zero fields at finite distance

♦  Rigorous boundary conditions:

  "infinite" finite elements (geometrical transformations)

  boundary elements (FEM-BEM coupling)

  Electromagnetic fields are confined (bounded domain) ♦  Rigorous boundary conditions

Mesh of electromagnetic devices

53

  Electromagnetic fields enter the materials up to a distance depending of physical characteristics and constraints

♦  Skin depth δ (δ<< if ω, σ, µ >>)

♦  mesh fine enough near surfaces (material boundaries)

♦  use of surface elements when δ → 0


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  Types of elements

♦  2D : triangles, quadrangles

♦  3D : tetrahedra, hexahedra, prisms, pyramids

♦  Coupling of volume and surface elements

  boundary conditions

  thin plates

  interfaces between regions

  cuts (for making domains simply connected)

♦  Special elements (air gaps between moving pieces, ...)


55

u ≡ weak solution

u v v v v ( , L * ) - ( f , ) + Q g ( ) ds Γ ∫ = 0 , ∀ ∈ V ( Ω )

Classical and weak formulations

u ≡ classical solution

L u = f in Ω B u = g on Γ = ∂Ω

Classical formulation

Partial differential problem

Weak formulation

( u , v ) = u ( x ) v ( x ) d x Ω ∫ , u , v ∈ L 2 ( Ω )

( u , v ) = u ( x ) . v ( x ) d x Ω ∫ , u , v ∈ L 2 ( Ω )

Notations

v ≡ test function Continuous level : ∞ × ∞ system Discrete level : n × n system ⇒ numerical solution

56

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

57

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

58

Complementary 3D formulations

Magnetodynamic h-formulation

Magnetodynamic a-formulation

How to enforce global fluxes ?

59

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 ♦  etc.

60

a-v formulation

  a-vs Magnetodynamic finite element formulation with massive and stranded inductors

  Use of edge and nodal finite elements for a and vs ♦  Definition of a source electric scalar potential vs in massive inductors in an

efficient way (limited support) ♦  Natural coupling between a and vs for massive inductors ♦  Adaptation for stranded inductors: several methods ♦  Natural coupling between local and global quantities, i.e. fields and currents

and voltages ♦  etc.

Dual Procedures to take Inductors into account in

Magnetodymic Formulations

Edge and nodal finite elements allowing natural coupling of fields and global quantities

Magnetodynamics with voltages and currents

Numerical modeling with circuit coupling

Magnetodynamic problem with global constraints

curl h = j

curl e = – ∂t b

div b = 0

b = µ h

j = σ e


Equations Boundary conditions Global conditions

for circuit coupling

Inductor

Voltage Current

h-φ and t-ω weak formulations

Magnetic scalar potential in nonconducting regions ΩcC

h-φ magnetodynamic formulation

How to couple local and global quantities ? h, φ Vi, Ii

φ

(1)

t-ω magnetodynamic formulation (similar)

with h = hs + hr

Source magnetic field

Reaction magnetic field

i.e. currents Ii Ec : edges in Ωc

NcC : nodes in Ωc

C and on ∂ΩcC

C : cuts

Current as a strong global quantity Characterization of curl-conform vector fields : h or t

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

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

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

Voltage as a weak global quantity

Discrete weak formulation

system of equations (symmetrical matrix)

Test function h' = sk, vn → classical treatment, no contribution for < · >Γe

Test function h' = ci → contribution for < · >Γe

Electromotive force

Weak global quantity

Voltage as a weak global quantity and circuit relations

Electromotive force

Source of e.m.f.

Natural way to compute a weak voltage ! Better than an explicit nonunique line integration

in (1)

Weak circuit relation between Vi and Ii for inductor i

Direct application

Massive and stranded inductors

Massive inductor

Additional treatment Stranded inductor

Weak circuit relation between Vj and Ij for stranded inductor j

Source field due to a magnetomotive force Nj (one basis function for each stranded inductor)

Number of turns

Reaction field

h'=hs,j

Natural way to compute the magnetic flux through all the wires !

Tree technique ...

Stranded inductors - Source field

With gauge condition (tree) & boundary conditions

Projection method


Source

Source = Nj

Simplified source field

Stranded inductors - Magnetic flux Physical and geometrical interpretation of the circuit relation

Natural way to compute the magnetic flux

through all the wires !

a-v weak formulation

Magnetic vector potential - Electric scalar potential

a-v magnetodynamic formulation

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

a v

(1)

Voltage as a strong global quantity

Weak form of div j = 0

With a' = grad v' in (1)

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

(2)

72

Voltage as a strong global quantity Unit source electric scalar potential v0 (basis function for the voltage)

Electrokinetic problem (physical field)

Generalized potential (nonphysical field)

Needs a finite element resolution !

Direct expression

Reduced support

Current as a weak global quantity and circuit relations

Weak circuit relation between Vi and Ii for massive inductor i

in (2)

for massive inductor i

Natural way to compute a weak current ! Better than an explicit nonunique surface integration

Circuit relation for stranded inductors

Weak circuit relation between Vj and Ij for massive inductor j

cannot be used

From the h-formulation

From the a-formulation

Equivalent current density (1)

Explicit distribution of the current density

Equivalent current density (2)

With gauge condition (tree) & boundary conditions

Projection method


Source = Nj

Source electric vector potential

Source

Source electric scalar potential

Projection method


Source

Source

Tensorial conductivity

Conclusions

  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

Conclusions

  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

79

Application - Massive inductor Inductor-Core system in air

Computation of resistance and inducance

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

(1/4th)

2.4

2.5

2.6

2.7

2.8

2.9

3

2000 4000 6000 8000 10000 12000

Indu

ctan

ce L

(µH

/m)

Number of elements

h-form., 50 Hz

a-form., 50 Hzh-form., 200 Hz

a-form., 200 Hz

4

5

6

7

8

9

10

11

12

13

2000 4000 6000 8000 10000 12000

Res

ista

nce

R (!

/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

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

80

Application - Stranded inductor Inductor-Core system in air

Computation of reaction field, total field and inducance

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

Indu

ctan

ce 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

µr,core = 10

Computation of a source field

φ

hs

h

µr,core = 10

81

-0.02

0

0.02

0.04

0.06

0.08

0.1

0 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.

-0.02

0

0.02

0.04

0.06

0.08

0.1

0 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.

Application Inductor-Core system in air

Enforcement of the current Ij

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

(1/16th)

Mesh quality factor = 3 µr,core = 100 σ = 5.9 107m S/m

1 A

Mesh quality factor = 7

82

Application

Computation of the inductance

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

Axisymmetrical coil

3D coil

Inductor-Core system in air

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2000 4000 6000 8000 10000 12000 14000

Indu

ctan

ce 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=100002D Axi2D Axi

2D Axi

2D Axi0.6

0.8

1

1.2

1.4

1.6

1.8

2

2 3 4 5 6 7 8 9

Indu

ctan

ce 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

83

Strong and weak formulations

curl h = js

div b = ρs

b = µ h

Constitutive relation

Equations

Boundary conditions (BCs)

in Ω

h = hs - grad φ , with curl hs = js

b = bs + curl a , with div bs = ρs

Scalar potential φ

Vector potential a

Strongly satisfies

84

Strong formulations

curl e = 0 , div j = 0 , j = σ e , e = - grad v or j = curl u

Electrokinetics

curl e = 0 , div d = ρs , d = ε e , e = - grad v or d = ds + curl u

Electrostatics

curl h = js , div b = 0 , b = µ h , h = hs - grad φ or b = curl a

Magnetostatics

curl h = j , curl e = – ∂t b , div b = 0 , b = µ h , j = σ e + js , ...

Magnetodynamics

85

Grad-div weak formulation

grad-div Green formula integration in Ω and divergence theorem

grad-div scalar potential φ weak formulation

86

Curl-curl weak formulation

curl-curl Green formula integration in Ω and divergence theorem

curl-curl vector potential a weak formulation

87

Grad-div weak formulation

1 Use of hierarchal TF φp' in the weak formulation

Error indicator: lack of fulfillment of WF

... can be used as a source for a local FE problem (naturally limited to the FE support of each TF) to calculate the higher order correction bp to be given to the actual solution b for satisfying the WF... solution of :

or Local FE problems 1

88

YX

Z

A posteriori error estimation (1/2) V

Y

XZ

Hig

her o

rder

hie

rarc

hal c

orre

ctio

n v p

(2

nd o

rder

, BFs

and

TFs

on

edge

s)

Elec

tric

sca

lar

pot

entia

l v

(1st

ord

er)

Y

XZ

Large local correction ⇒ Large error

Coarse mesh Fine mesh

Electric field Y

XZ

1.2

1.4

1.6

1.8

2

0 5 10 15 20 25 30 35 40 45

Elec

tric

field

(V/m

m)

Solution eRefined solution e + ep

-0.2

-0.1

0

0.1

0.2

0 5 10 15 20 25 30 35 40 45

Elec

tric

field

(V/m

m)

Position along top electrode (mm)

Correction ep

Field discontinuity directly

Electrokinetic / electrostatic problem

89

Curl-curl weak formulation

2 Use of hierarchal TF ap' in the weak formulation

Error indicator: lack of fulfillment of WF

... also used as a source to calculate the higher order correction hp of h... solution of :

Local FE problems 2

90

A posteriori error estimation (2/2) Magnetostatic problem Magnetodynamic problem

Y

XZ

Y

XZ

Y

XZ

Mag

netic

vec

tor

pot

entia

l a

(1st

ord

er)

Hig

her o

rder

hie

rarc

hal

corr

ectio

n a p

(2nd

ord

er,

BFs

and

TFs

on

face

s)

Y

XZ

Coarse mesh

Fine mesh

skin depth

Z

Y

X

Large local correction ⇒ Large error

Conductive core Magnetic core

V

finite element modeling of electromagnetic systems · 2 introduction formulations of...

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