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Strong and Weak Formulations of

Electromagnetic Problems

Patrick Dular, Patrick Dular, University of Liège - FNRS, BelgiumUniversity of Liège - FNRS, Belgium

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Content

Formulations of electromagnetic problems♦ Maxwell equations, material relations

♦ Electrostatics, electrokinetics, magnetostatics, magnetodynamics

♦ Strong and weak formulations

Discretization of electromagnetic problems♦ Finite elements, mesh, constraints

♦ Weak finite element formulations

3

Formulations

of Electromagnetic Problems

Maxwell equations

Electrostatics

Electrokinetics

Magnetostatics

Magnetodynamics

4

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 = v

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) v 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 = h (+ bs)

d = e (+ ds)

j = e (+ js)

Constitutive relations

Magnetic relation

Dielectric relation

Ohm law

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

Possible sources

magnetic permeability (H/m) dielectric permittivity (F/m) electric conductivity (–1m–1)

Characteristics of materials

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• Formulation for • the exterior region 0

• the dielectric regions d,j

• In each conducting region c,i : v = vi v = vi on c,i

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

Electrostatics

Type of electrostatic structure

0 Exterior region

c,i Conductorsd,j Dielectric

div grad v = –

with e = – grad v

Electric scalar potential formulation

curl e = 0div d = d = e

Basis equations

n e | 0e = 0

n d | 0d = 0

& boundary conditions

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Electrostatic

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• Formulation for • the conducting region c

• On each electrode 0e,i : v = vi v = vi on 0e,i

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

Electrokinetics

Type of electrokinetic structure

c Conducting regiondiv grad v = 0

with e = – grad v

Electric scalar potential formulation

curl e = 0div j = 0j = e

Basis equations

n e | 0e = 0

n j | 0j = 0

& boundary conditions

0e,0

0e,1

0j

e=?, j=?

c

V = v1 – v0

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curl h = j

div b = 0

Magnetostatics

Equations

b = h + bs

j = js

Constitutive relations

Type of studied configuration

Ampère equation

Magnetic conservationequation

Magnetic relation

Ohm law & source current

Studied domain

m Magnetic domain

s Inductor

j

m

s

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curl h = j

curl e = – t b

div b = 0

Magnetodynamics

Equations

b = h + bs

j = e + js

Constitutive relations

p

a

Va

I a

s

js

Type of studied configuration

Ampère equation

Faraday equation

Magnetic conservationequation

Magnetic relation

Ohm law & source current

Studied domain

p Passive conductorand/or magnetic domain

a Active conductors Inductor

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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 lossesFoil 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 parallelnon-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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dom (grade) = Fe0 = { v L2() ; grad v L2() , ve = 0 }

dom (curle) = Fe1 = { a L2() ; curl a L2() , n ae = 0 }

dom (dive) = Fe2 = { b L2() ; div b L2() , n . be = 0 }

Boundary conditions on e

dom (gradh) = Fh0 = { L2() ; grad L2() , h = 0 }

dom (curlh) = Fh1 = { h L2() ; curl h L2() , n hh = 0 }

dom (divh) = Fh2 = { j L2() ; div j L2() , n . jh = 0 }

Boundary conditions on h

Function spaces Fe0 L2, Fe

1 L2, Fe2 L2, Fe

3 L2

Function spaces Fh0 L2, Fh

1 L2, Fh2 L2, Fh

3 L2Basis structure

Basis structure

Continuous mathematical structure

gradh Fh0 Fh

1 , curlh Fh1 Fh

2 , divh Fh2 Fh

3

Fh0 grad h Fh

1 curl h Fh2 div h Fh

3Sequence

gradh Fe0 Fe

1 , curle Fe1 Fe

2 , dive Fe2 Fe

3

Fe3 div e Fe

2 curl e Fe1 grad e Fe

0Sequence

Domain , Boundary = h U e

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"e" side "d" side

Electrostatic problem

Basis equations

curl e = 0 div d = d = e

e = – grad v d = curl u

e = dd

(u)

grad

curl

div

e

e

e

e

0

0

(–v)Fe0

Fe1

Fe2

Fe3

div

curl

grad

d

d

d

Fd3

Fd2

Fd1

Fd0

eS 0

Se1

Se2

Se3

Sd3

Sd2

Sd1

Sd0

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"e" side "j" side

Electrokinetic problem

Basis equations

curl e = 0 div j = 0j = e

e = – grad v j = curl t

e = jd

(t)

grad

curl

div

e

e

e

e

0

0

(–v)Fe0

Fe1

Fe2

Fe3

div

curl

grad

j

j

j

F j3

F j2

F j1

F j0

eS 0

Se1

Se2

Se3

S j3

S j2

S j1

S j0

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"h" side "b" side

Magnetostatic problem

Basis equations

curl h = j div b = 0b = h

hS 0

Sh1

Sh2

Sh3

S e3

S e2

S e1

S e0

h = – grad b = curl a

h = bb

0

(a)

grad

curl

div

h

h

h

h

j

0

(–)Fh0

Fh1

Fh2

Fh3

div

curl

grad

e

e

e

Fe3

Fe2

Fe1

Fe0

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"h" side "b" side

Magnetodynamic problem

hS 0

Sh1

Sh2

Sh3

S e3

S e2

S e1

S e0

h = t – grad b = curl a

curl h = jcurl e = – t b

div b = 0b = h

Basis equations

j = e

h

j

0

h = bb

0

e

grad

curl

div

h

h

h

je

(–)

(–v)

Fh0

Fh1

Fh2

Fh3

div

curl

grad

e

e

e

(t)

(a, a )*

Fe3

Fe2

Fe1

Fe0

e = – t a – grad v

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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 & domainClassical and weak formulations

Continuous problem

Finite element method

Discrete function spaces piecewise definedin a discrete domain (mesh)

Discrete problem

Discretization Approximation

Classical & weak formulations ?Properties of the fields ?

QuestionsTo build a discrete structure

as similar as possibleas the continuous structure

Objective

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Discrete mathematical structure

Sequence of finite element spacesSequence of function spaces & Mesh

Finite element spaceFunction space & Mesh

fi

i

fi

i

Finite elementInterpolation in a geometric element of simple shape

+ f

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Finite elements

Finite element (K, PK, K)♦ K = domain of space (tetrahedron, hexahedron, prism)

♦ PK = function space of finite dimension nK, defined in K

♦ K = set of nK degrees of freedom represented by nK linear functionals i, 1 i nK, defined in PK and whose values belong to IR

IR

cod(f)K = dom(f) f P

xf(x)

(f)i

K

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Finite elements

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

Interpolation

Finite element spaceUnion of finite elements (Kj, PKj, Kj) 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

uK i (u) pi

i1

nK

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Sequence of finite element spaces

Geometric elementsTetrahedral

(4 nodes)Hexahedra

(8 nodes)Prisms(6 nodes)

Mesh

Geometric entities

Nodesi N

Edgesi E

Facesi F

Volumesi V

Sequence of function spaces

S0 S1 S2 S3

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Sequence of finite element spaces

{si , i N}

{si , i E}

{si , i F}

{si , i V}

Bases Finite elements

S0

S1

S2

S3 Volumeelement

Point evaluation

Curve integral

Surface integral

Volume integral

Nodal value

Circulation along edge

Flux across face

Volume integral

Functions Functionals Degrees of freedomProperties

si (x j) ij i, j N

si . n dsj ij

i, j F

si dvj ij

i, j V

uK i (u) si

i

Faceelement

Edgeelement

Nodalelement

si . dlj ij

i, j E

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Sequence of finite element spaces

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

S0 grad S1 curl S2 div S3

Sequence

S0

S1 grad S0

S2 curl S1

S3 div S2

S0

S1

S2

S3

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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

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Electromagnetic fields enter the materials up to a

distance depending of physical characteristics and

constraints

♦ Skin depth (<< if , , >>)

♦ mesh fine enough near surfaces (material boundaries)

♦ use of surface elements when 0

2

Mesh of electromagnetic devices

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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, ...)

Mesh of electromagnetic devices

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u weak solution

u v v v v( , L* ) ( f , ) Qg ( ) ds

0 , V()

Classical and weak formulations

u classical solution

L u = f in B u = g on =

Classical formulation

Partial differential problem

Weak formulation

( u , v ) u(x) v(x) dx

, u, vL2 ()

( u , v ) u(x) . v(x) dx

, u, v L2 ()

Notations

v test function Continuous level : systemDiscrete level : n n system numerical solution

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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 forfinite 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- 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- formulation,

source electric scalar potential vs in the a-v formulation

Coupling of local and global quantities♦ e.g., currents and voltages in h- 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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Strong and weak formulations

curl h = js

div b = s

b = h

Constitutive relation

Equations

Boundary conditions (BCs)

,h b

s s n h n h n b n b

in

h = hs grad , with curl hs = js

b = bs + curl a , with div bs = s

Scalar potential

Vector potential a

gradb b

u n a n

heach non-connected portion of constant

Strongly satisfies

Strongly satisfies

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Strong formulations

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

Electrokinetics

curl e = 0 , div d = s , d = e , e = grad v or d = ds + curl u

Electrostatics

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

Magnetostatics

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

Magnetodynamics

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Grad-div weak formulation

1 1grad div div( ) , ( ), ( )v v v v H u u u u H

, , ',(( ,g 'rad , 0') , '')bs bss R

n b n bb

(div , ') , 'bs s n n bbb

, 'Electrokinetics: ( , grad ') , ' , ' (div , ') , 'j v js s j vv v v v v R j n j n j j n j n j

, 'Electrostatics: ( , grad ') ( , ') , ' , ' (div , ') , 's d v ds s d vv v v v v v R d n d n d d n d n d

( ,grad ) (div , ) ,v v v u u n u

grad-div Green formula

,add on both sides: ( , ') , '

s bss n b

integration in and divergence theorem

grad-div scalar potential weak formulation

( grad )s b h

sMagnetostatics: 0

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Curl-curl weak formulation

curl-curl Green formula

,add on both sides ( , '): , '

hs js s j n h aa

integration in and divergence theorem

curl-curl vector potential a weak formulation

( ,curl ) (cu l , ) ,r u v u n u vv

1curl curl div( ) , , ( ) u v v u v u u v H

(curl , ') , 'hss n ha aj n hh

, , ',(( , 0',curl ') ,') 's j bh as hs R n h n h aaj ah a

1( curl )s h b a

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Grad-div weak formulation

1

, ',( ,grad ) ( , )' ' ', 0s b pp p ps s bR n bb

,( ,grad ') ' '( , ) , 0

s bp p p s ps bb nb

Use of hierarchal TF p' 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 :

, ''( ,grad )pp p bR bor Local FE problems1

42

A posteriori error estimation (1/2)V

Hig

he

r o

rder

hie

rarc

hal

co

rrec

tio

n v

p

(2n

d o

rder

, B

Fs

an

d T

Fs

on

ed

ge

s)

Ele

ctri

c sc

alar

po

ten

tia

l v

(1st

ord

er)

Large local correction Large error

Coarse mesh Fine mesh

Electric field

Field discontinuity directly

Electrokinetic / electrostatic problem

43

Curl-curl weak formulation

2

, , '( ,curl ) ( , ) ,' 0' 's j h pp p ps h as R j n hah a a

, ''( ,curl )pp p ahR h a

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

44

A posteriori error estimation (2/2)

Magnetostatic problem Magnetodynamic problem

Mag

net

ic v

ecto

r p

ote

nti

al

a(1

st o

rder

)

Hig

he

r o

rder

hie

rarc

hal

co

rrec

tio

n a

p (

2nd

ord

er,

BF

s an

d T

Fs

on

fac

es)

Coarse mesh

Fine mesh

skin depth

Large local correction Large error

Conductive coreMagnetic core

V