Método de Elementos Finitos em Eletromagnetismo

Prévia do material em texto

Finite Element Modeling 
of Electromagnetic Systemsof Electromagnetic Systems
Mathematical and numerical tools
Prof. Mauricio Valencia Ferreira da Luz, Dr.Prof. Mauricio Valencia Ferreira da Luz, Dr.
Departamento de Engenharia Elétrica – EEL
Universidade Federal de Santa Catarina – UFSC
Email: mauricio.luz@ufsc.br
1
GRUCADGRUCAD
Formulations 
of Electromagnetic Problemsof Electromagnetic Problems
Electrostatics
Maxwell equations
Electrokinetics
Magnetostaticsg
Magnetodynamics
2
Scalar and vector potentials, gauge, cuts, etc.
Maxwell equationsMaxwell equations
curl h = j + t d
Maxwell equations
Ampère equation
curl e = – t b
div b = 0
Faraday equation
div b 0
div d = v
Conservation equations
Principles of electromagnetism
Physical fields and sources
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)  charge density (C/m3)
3
j current density (A/m ) v charge density (C/m )
Material constitutive relationsMaterial constitutive relations
b =  h
Constitutive relations
Magnetic relation
d =  e
j =  e
Dielectric relation
Oh lj  e Ohm law
Characteristics of materials
Constants (linear relations)
 magnetic permeability (H/m)
 dielectric permittivity (F/m)
l t i d ti it ( 1 1)
Characteristics of materials
( )
Functions of the fields
(nonlinear materials)
 electric conductivity (–1m–1)
4
Tensors (anisotropic materials)
Electrostatic formulationElectrostatic formulation
T f l t t ti t tBasis equations Type of electrostatic structure
curl e = 0
div d = 
d
Basis equations
n  e | 0e = 0
d |  0
& boundary conditions
e electric field (V/m)
d electric flux density (C/m2)
d =  e n  d | 0d = 0
 electric charge density (C/m3)
 dielectric permittivity (F/m)
0 Exterior region
 C d t
div  grad v = – 
Electric scalar potential formulation
• Formulation for • the exterior region 0
• the dielectric regions d j
c,i Conductorsd,j Dielectric
with e = – grad v
5
 the dielectric regions d,j
• In each conducting region c,i : v = vi  v = vi on c,i
Electrokinetic formulationElectrokinetic formulation
T f l t ki ti t tBasis equations Type of electrokinetic structure
curl e = 0
div j = 0
j
Basis equations
n  e | 0e = 0
j |  0
& boundary conditions
0e 1
0j
e electric field (V/m)
j electric current density (C/m2)
j =  e n  j | 0j = 0
0e,0
0e,1
e=?, j=?
c
 electric conductivity (–1m–1)
V 1 0
c Conducting regiondiv  grad v = 0
Electric scalar potential formulation V = v1 – v0
• Formulation for • the conducting region c
• On each electrode 0e i : v = vi  v = vi on 0e i
with e = – grad v
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 On each electrode 0e,i : v v  v v on 0e,i
Quasi-stationary approximationQuasi-stationary approximation
curl h = j + t d
Di i l th
C d i
Dimensions << wavelength
Conduction current
density
Displacement current
density> > >
curl h = j
Electrotechnic apparatus (motors, transformers, ...)
Frequencies from Hz to a few 100 kHz
Applications
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Frequencies from Hz to a few 100 kHz
Magnetodynamics - eddy currentsMagnetodynamics - eddy currents
T f t di d fi ti
curl h = j
Equations


Type of studied configuration
Ampère equation
curl e = – t b
div b = 0
 p
 s
js
Ampère equation
Faraday equation
div b = 0
Va
I a
Magnetic conservation
equation
b =  h
Constitutive relations
 a
 Studied domainb  h
j =  e
Magnetic relation
Ohm law
p Passive conductor
and/or magnetic domain
a Active conductor
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s Inductor
Magnetic constitutive relationMagnetic constitutive relation
 Diamagnetic and paramagnetic materials
b =  h r relative magnetic permeability = r 0
 Diamagnetic and paramagnetic materials
Linear material r  1 (silver, copper, aluminium)
 Ferromagnetic materialsg
Nonlinear material r >> 1 , r = r(h) (steel, iron)
b-h characteristic of steel r-h characteristic of steel
9
Magnetostatic formulationsMagnetostatic formulations

m
j
m
s
Maxwell equations
(magnetic - static)( g )
curl h = j
div b = 0
b =  h
a Formulation Formulation
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"h" side "b" side
Magnetostatic formulationsMagnetostatic formulations
Basis equations
curl h = j div b = 0b =  h
Basis equations
(h) (b)(m)
a Formulation Formulation
Magnetic scalar potential  Magnetic vector potential a
b = curl ah = hs – grad   (h) OK (b) OK 
c rl ( –1 c rl a ) j
hs given such as curl hs = j
(non-unique)
di ( ( h grad  ) ) 0 ( ) & ( ) ( ) & ( )
Multivalued potential Non-unique potential
curl ( –1 curl a ) = jdiv (  ( hs – grad  ) ) = 0  (b) & (m) (h) & (m) 
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Cuts Gauge condition
Multivalued scalar potentialMultivalued scalar potential
K l f th l (i d i )Kernel of the curl (in a domain )
ker ( curl ) = { v : curl v = 0 }
grad
dom(grad)
cod(grad) curl
dom(curl)
ker(curl)
cod(curl)
. 0
curl
cod ( grad )  ker ( curl )
curl
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cod ( grad )  ker ( curl )
Multivalued scalar potential - CutMultivalued scalar potential - Cut
OK
curl h = 0 in 
h = – grad  in 
Scalar potential 

OK

g 
?
Circulation of h along path AB in 
AB AB
h . dl

   grad  . dl

  A  B
I


()
Closed path AB (AB)
surrounding a conductor (with current I)
() A – B = 0  I ! ! !
 must be discontinuous ... through a cut
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  = I
 g
Vector potential - gauge conditionVector potential - gauge condition
OK
div b = 0 in 
b = curl a in 
Vector potential a

OK

?
Non-uniqueness of vector potential a
b = curl a = curl ( a + grad  ) Qex.:w(r)=r
Gauge condition
Coulomb gauge div a = 0 O
Gauge condition
P
 vector field with non-closed lines
Gauge a .  = 0
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linking any 2 points in 
Magnetodynamic formulationsMagnetodynamic formulations

h- Formulation a* Formulation p

 s
js
h  Formulation a Formulation
 a
Va
I a
Maxwell equations
(quasi-stationary)(quasi stationary)
curl h = j
curl e = – t b
di b 0
b =  h
j =  e
a-v Formulationt- Formulation div b = 0
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"h" side "b" side
Magnetodynamic formulationsMagnetodynamic formulations
Basis equations
curl h = j
curl e = – t b
div b = 0b =  h
Basis equations
(h) (b)j =  e
t- Formulationh- Formulation
Magnetic field h
Magnetic scalar potential 
Electric vector potential t
Magnetic scalar potential 
curl hs = js
h ds c
h = hs – grad  ds cC
 (h) OK j = curl t
(h) OK 
h = t – grad 
curl (–1 curl t) + t ( (t – grad )) = 0
div ( (t grad )) 0
s js
curl (–1 curl h) + t ( h) = 0
di ( (h grad )) 0
g
 in c 
i  C
 (b) 
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div ( (t – grad )) = 0div ( (hs – grad )) = 0  in cC  + Gauge
Magnetodynamic formulationsMagnetodynamic formulations
Basis equations
curl h = j
curl e = – t b
div b = 0b =  h
Basis equations
(h) (b)j =  e
a-v Formulationa* Formulation
Magnetic vector potential a*
Magnetic vector potential a
Electric scalar potential v
b = curl a
(b) OK 
e = – t a – grad v
b = curl a*
 (b) OK
e = – t a*
curl (–1 curl a) + grad v + t a = jscurl (–1 curl a*) +  t a* = js
t gt
 (h) 
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+ Gauge in + Gauge in cC
Continuous mathematical structureContinuous mathematical structure
Domain , Boundary  = h U e
dom (gradh) = Fh0 = {   L2() ; grad   L2() , h = 0 }
Function spaces Fh0  L2, Fh1  L2, Fh2  L2, Fh3  L2Basis structure
(g h) h {  ( ) g  ( )  h }
dom (curlh) = Fh1 = { h  L2() ; curl h  L2() ,n  hh = 0}
dom (divh) = Fh2 = { j  L2() ; div j  L2() , n . jh = 0 }
Boundary conditions on gradh Fh0  Fh1 curlh Fh1  Fh2 divh Fh2  Fh3 Boundary conditions on hgradh Fh  Fh , curlh Fh  Fh , divh Fh  Fh
Fh 0
grad h  Fh1 curl h  Fh 2 div h  Fh3Sequence
dom (grade) = Fe0 = { v  L2() ; grad v  L2() , ve = 0 }
dom (curl ) = F 1 = { a  L2() ; curl a  L2() n  a = 0 }
Function spaces Fe0  L2, Fe1  L2, Fe2  L2, Fe3  L2Basis structure
dom (curle) = Fe1 = { a  L2() ; curl a  L2() , n  ae = 0 }
dom (dive) = Fe2 = { b  L2() ; div b  L2() , n . be = 0 }
Boundary conditions on egradh Fe0  Fe1 , curle Fe1  Fe2 , dive Fe2  Fe3
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Fe3
div e  Fe2 curl e  Fe1 grad e  Fe 0Sequence
Magnetostatic problemMagnetostatic problem
Basis equationsBasis equations
curl h = j div b = 0b =  h
F 0 F 3
hS
0 Se
3
 
 h = b
b
0
grad h
h
(–)Fh0
Fh
1
div e
Fe
3
Fe
2
Sh
1
2
Se
2
1
(a)
curl
div
h
h
jFh2
3
curl
grad
e
e
Fe
1
0Sh
2
Sh
3
Se
1
Se
0
0Fh
3 Fe
0
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"h" side "b" side
h e
h = – grad  b = curl a
Magnetodynamic problemMagnetodynamic problem
Basis equations
curl h = j
curl e = – t b
div b = 0b =  h
Basis equations
j =  e
0F 0 F 3
hS
0 Se
3
 
h
 h = b
b
0
grad h
(–)Fh0
Fh
1
div e
(t)
Fe
3
Fe
2
Sh
1
2
Se
2
1
j e
curl
div
h
h
je
Fh
2
3
curl
grad
e
e
(a, a )* Fe
1
0Sh
2
Sh
3
Se
1
Se
0
0 (–v)Fh
3 Fe
0
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"h" side "b" side
h e
h = – grad  b = curl a
Classical and weak formulationsClassical and weak formulations
Partial differential problem
L u = f in 
B  
Classical formulation
( u , v )  u(x) v(x) dx , u, vL2 ()
Notations
u  classical solution
B u = g on  =  
( u , v )  u(x) . v(x) dx

 , u, v L2 ()
u v v v v( L* ) ( f )  Q ( ) ds  0  V()
Weak formulation

u  weak solution
u v v v v( , L )  ( f , )  Qg ( ) ds

  0 ,  V()
v  test function Continuous level :    system
Discrete level : n  n system
21
y
 numerical solution
Classical and weak formulationsClassical and weak formulations
Application to the magnetostatic problempp g p
curl h = j
di b 0
Magnetostatic
div b = 0
b =  h
n  h h = 0 n . b e = 0
h e
g
classical formulation
h e
Weak form lation
( b , grad ' ) = 0 ,  '  () with () = {   H0() ; h = 0 }
( div b ' ) + < n b ' > = 0  '  () Weak formulation
of div b = 0
(+ boundary condition)
( div b ,  ) + < n . b ,  > = 0 ,    ()
div b = 0

n . b e = 0


b =  h & h = hs – grad  (with curl hs = j)  curl h = j
(  (h grad ) grad ' ) = 0  '  () Magnetostatic
22
(  (hs – grad ) , grad  ) = 0 ,    () gweak formulation with 
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
 e.g., circuit coupling
Weak formulations for
finite element models
Essential and natural constraints
23
Essential and natural constraints, 
i.e., strongly and weakly satisfied
Constraints in electromagnetic systemsConstraints in electromagnetic systems
C li f l t ti l ith t fi ld Coupling of scalar potentials with vector fields
– e.g., in h- and a-v formulations
 Gauge condition on vector potentials 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
Interest for a 
“correct” discrete Coupling of local and global quantities
– e.g., currents and voltages in h- and a-v formulations 
(massive, stranded and foil inductors)
“correct” discrete 
form of these 
constraints
 Interface conditions on thin regions
– i.e., discontinuities of either tangential or normal components
Sequence of 
finite element 
spaces
24
spaces
Discretization 
of Electromagnetic Problemsof Electromagnetic Problems
Nodal, edge, face and volume finite elementsNodal, edge, face and volume finite elements
25
Discrete mathematical structureDiscrete mathematical structure
Continuous function spaces & domain
Classical and weak formulations
Continuous problem
Classical and weak formulations 
Discretization Approximation
Discrete function spaces piecewise defined
Discrete problem
Finite element method
Discrete function spaces piecewise defined
in a discrete domain (mesh) 
Classical & weak formulations  ?
P ti f th fi ld ?
Questions
To build a discrete structure
as similar as possible
Objective
26
Properties of the fields  ? as similar as possibleas the continuous structure
Discrete mathematical structureDiscrete mathematical structure
Finite element
Interpolation in a geometric + f
Finite element space
p g
element of simple shape
Function space 
& Mesh  fi
i

Sequence of finite element spaces
Sequence of function spaces 
& Mesh  fi
i



27
Finite elementsFinite elements
Fi it l t (K P  ) Finite element (K, PK, K)
– K = domain of space (tetrahedron, hexahedron, prism)
– PK = function space of finite dimension nK, defined in KPK 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 IRK g
cod(f)
K = dom(f) f  PK
x
f(x)

 (f)i
28
IR
i
Finite elementsFinite elements
U i l Unisolvance
 u  PK , u is uniquely defined by the degrees of freedom
 Interpolation Interpolation
Degrees of freedom
 ( )
nK
 Finite element space
Basis functions
uK  i (u) pi
i1

 Finite element space
Union of finite elements (Kj, PKj, Kj) such as :
 the union of the Kj fill the studied domain ( mesh)j ( )
 some continuity conditions are satisfied across the element 
interfaces
29
Sequence of finite element spacesSequence of finite element spaces
Geometric elements
Tetrahedral
(4 nodes)
Hexahedra
(8 nodes)
Prisms
(6 nodes)(4 nodes) (8 nodes) (6 nodes)
MeshMesh
Nodes Edges Faces Volumes
Geometric entities
i  N i  E i  F i  V
Sequence of function spaces
S0 S1 S2 S3
30
Sequence of function spaces
Sequence of finite element spacesSequence of finite element spaces
P i
Functions Functionals Degrees of freedomProperties
N d l{si , i  N}
{si , i  E}
S0
S1
Point 
evaluation
Curve 
i t l
Nodal value
Circulation 
l d
si (x j)   ij
 i, j N
Edge
element
Nodal
element
si . dl
j   ij{ i , }
{si , i  F}S2
integral
Surface 
integral
along edge
Flux across 
face
si . n ds
j   ij i, j F

Face
element
elementj  i, j E
{si , i  V}S3 Volumeelement
Volume 
integral
Volume 
integral
si dv
j  ij i, j V
Bases Finite elements
uK  i (u) si
i

31
i
Sequence of finite element spacesSequence of finite element spaces
Base 
functions
Continuity across 
element interfaces
Codomains of the 
operators
{ i } lS0 {si , i  N} value
{si, i  E} tangential component grad S0  S1
S0
S1 d S0
S0
S1
{si , i  F} normal component curl S1  S2
S1 grad S0
S2 curl S1
S2
3 {si , i  V} discontinuity div S2  S3
S3 div S2
S3
Conformity Sequence
32
S0 grad  S1 curl  S2 div  S3
Function spaces S0 et S3Function spaces S et S
F h d i N l fi ldFor each node i  N  scalar field
0si (x) = pi (x)  S0
0 0
0
si (x) pi (x)  S
1 at node i
0 0
0pi = 1node i
pi  1 at node i0 at all other nodes

00
0pi continuous in 
For each For each VolumeVolume v v  V V  scalar fieldscalar field
33
sv = 1 / vol (v)  S3
Edge function space S1Edge function space S
F h d {i j} E t fi ldFor each edge eij = {i, j}  E  vector field
se  S1  ese ij  pj grad pr
rN F,ji 
  pi grad pr
rN F,ij 

NF mn  {i N ; i  fmop (q ) , o, p,q  n}
Illustration of the vector field seDefinition of the set of nodes NF,mn-
e ij j
m
p
m
q
p
NF,mn {i N ; i  fmop (q ) , o, p,q  n} e ij
s e
i
j
m
o
n n
m
o
NF, ij- NF, ji
-
34
N.B.: In an element : 3 edges/node
, j
Edge function space S1Edge function space S
je iji
j Geometric interpretation
of the vector field se
NF, ji-p j grad pr
j
p j g pr
rN F j, i 

 NF, ji se ij  pj grad pr
rN F,ji 
  pi grad pr
rN F,ij 

e ij
i
j e ij
s e
i
j
NF, ij-
 pi grad pr NF, ij- NF, ji-
35
, j
rN F,ij  NF,ij 
F, ij
Function space S2Function space S
F h f t f F t fi ldFor each facet f  F  vector field 
f = fijk(l) = {i, j, k (, l) } = {q1, q2, q3 (, q4) }
sf  af pq c
c1
# N f grad prrN F,qc q c1   grad prrN F,qc q c1 
2
Illustration of the vector field sf
sf  S2
3  af = 2
i
k
NF, ij-
N
NF, kj-
f
f
#Nf =
4  af = 1
j
N ji-
NF, ki-
NF, ik-
f ijk
36
NF, ji
NF, jk-
Particular subspaces of S1Particular subspaces of S
ApplicationsKernel of the curl operator
Ampere equation
in a domain cC
without current 
h  S1() ; curl h = 0 in cC    h  ?
w t out cu e t
(c)
Gauged subspace
Gauge condition
on a vector potential
a  S1() ; b = curl a  S2()  a  ?
Gauge a  = 0
Definition of a
generalized source field hs
such that curl h = j
Gauge a .  = 0
to fix a
37
such that curl hs js
Kernel of the curl operatorKernel of the curl operator
cCCase of simply connected domains

H { h S1 () ; curl h  0 in cC }
c
c
Ec
C C
h l
lE cC
h  h a s a
eE
  h k s k
kE c
  sl
h = – grad  in cC
Nc , EcC C
Nc , EcC C
(interface)
cc
h l  h . dl
lab
  grad  . dl
lab
  a l   b l
hkn'
h 
h  hk s k
kE c
   s l
lE cC
 a l  b l
n Base of H  basis functions of
• inner edges of c
• nodes of cC,with
h  hk sk
kE c
  n vn
nN cC

vn  snj
38
c ,
with those of c
n nj
njE cC

Kernel of the curl operatorKernel of the curl operator
Case of multiply connected domains
H { h S1 () ; curl h  0 in cC }
h d  i  C
+ – – = +eci – –eci = Ii
 = cont + disc
di i i
(cuts)
h = – grad  in cC
qi • defined in cC
discontinuity
of disc disc  I i q i
iC

• unit discontinuity across eci
• continuous in a transition layer
• zero out of this layer
i h
h  hk sk
kA c
  cont n vn
nN cC
  Ii ci
iC

with
edges of cC
starting from a node of the cut
ci  snj
njA c C
nN eci
j N C
 Basis of H  basis functions of
• inner edges of c
• nodes of cC
39
and located on side '+'
but not on the cut
jN cC 
jN eci
c
• cuts of C
Gauged subspace of S1Gauged subspace of S
Gauged space in 
b = curl a a  a e se
eE
  S1 () , b  bf sf
fF
  S2 ()with
b f  i(e,f ) ae
eE
 , f F matrix form: bf CFE ae=
Face-edge
Tree  set of edges connecting
(in ) all the nodes of  without 
forming any loop (E)

Face edge
incidence matrix
treeo g a y oop ( )
Co-tree  complementary set of the tree (E)
Gauged space of S1()

E 
co-tree

E a  a i si  S 1 ()
S1() = {a  S1() ; aj = 0 ,  j  E}

g p ( )
40
i E 
 ( )
Basis of S1()  co-tree edge basis functions
(explicit gauge definition)

Global QuantitiesGlobal Quantities
and
Circuit Relations
Through essential and natural conditions
Natural coupling between 
local quantities (scalar and vector fields) 
and global quantities (flux and circulation)...
... for efficient treatment of coupled problems, 
ithi fi it l t bl th h t l l d i it
41
within a finite element problems or through external lumped circuits
Weak formulationsWeak formulations
Notations
involved in weak formulations  a b a b,  

 



def
dv
a b a b ds
def
Green formulae
,  a b a b ds
 =  n( u , grad v ) + ( div u , v ) = < n · u , v >
grad - div formula
Domain Weak global quantity of flux type
( u , grad v ) ( div u , v ) n u , v 
curl - curl formula
u, v  H1(), v  H1()( curl u , v ) – ( u , curl v ) = < n  u , v >
42
Weak global quantity of circulation type
Global quantities of flux typeGlobal quantities of flux type
Nodal finite element space S0 defined on a mesh 
v v s v Sn nN  , ( )0 
Nodal finite element space S0 defined on a mesh 
Boundary conditions  v constant et known
Nodes of 
n nn N , ( )
Floating potential  v constant and unknown
Constraints
Basis functions ?
 relations between coefficients vn ...
Degree of freedom ?
El t t ti f l ti ith l t i l t ti l
       grad v grad v v v Ss ed, ' , ' , ' ( )  n d 0 0
Electrostatic formulation with electric scalar potential v
0
Magnetostatic formulation with magnetic scalar potential 
43
          grad grad Ss hb, ' , ' , ' ( )  n b 0 0
Floating scalar potential constraintsFloating scalar potential constraints
Characterization of the potential in the whole domain
v v s v sn nn N
f nn Nf Cv ff
   
Characterization of the potential in the whole domain
Explicit constraints for floating potential
s s f Cf nn N ff   ,with
Explicit constraints for floating potential
v v s v s v Sn nn N
f f
f Cv f
     , 0
Basis functions
Floating potential basis function,
associated with a group of nodes Nf
Elementary geometrical
entities (nodes) and
44
global ones (groups of nodes)
Discrete global quantities of flux typeDiscrete global quantities of flux type
Discrete weak formulationDiscrete weak formulation
       grad v grad v v v Ss ed, ' , ' , ' ( )  n d 0 0
system of equations
(symmetrical matrix)
v v s v s v Sn nn N
f f
f C ev f
     , 0
with
(symmetrical matrix)
Test function v' = sn  classical treatment, no contribution for n d s v, ' 
n d n d n d n d        f fv s d Q' 1
n
Test function v' = sf  contribution for n d s v f, ' 
s f
n d n d n d n d   s s s sv s d Qf f f f, , ,    1
Total electric charge in conductor fSimilarly
W k l b l titi
45
n b n b    s f s fs df f,     Total magnetic flux through f
Weak global quantities
Matrix of the system of equationsMatrix of the system of equations
( ' ) ' ( ' ) ' ( )d d S 0 0( , ' ) , ' ( , ' ) , ' ( )         grad v grad v v v v Ss rs   n s 0 0
X
sn sf
X
0
0
0 vnsn =
X
0 XX
X
(n  Nv)
0 X X X
X
X 1 vf
Qf
sf
(f  Cf)
46
Qf
+ Relations : vf fixed, Qf fixed or vf = f(Qf) Symmetrical matrix
Discrete global quantities of flux typeDiscrete global quantities of flux type
El t i h k l b l titElectriccharge as a weak global quantity
    Q grad v grad s f Cf f f , ,
Computation with natural average , i.e. volume integral in a transition layer
(which is the support of sf)
in perfect accordance with the discretized 
weak formulation of the problem
 differentiation  d
No computational intermediary, like e.g.
^ differentiation  d
 distribution of n  d on a surface
 surface integration
^
Not natural with a weak Q df   n d 
47
conservation of flux
Q d
f
  n d 
Application to
electromagnetic problems
Electrostatics Electrokinetics Magnetostatics Magneto-thermal(thermal part)
Generalized
problem
Equations
curl e = 0 curl e = 0 curl h = js curl p = 0
div d =  div j = 0 div b = 0 div q = Q
d =  e j =  e b =  h q = k p
curl r = ks
div s = 
s =  r
Potential def.
e = – grad v e = – grad v h = hs – grad 
with curl hs = js
p = – grad T
e = electric field e = electric field h = magnetic field p = gradient of temperature
r = rs – grad v
Fields
field r g p g p
d = electric flux density j = current density b = magnetic flux density q = heat flux density
 = electric charge density — — Q = thermal source density
— — js = source current density —
v = electric scalar potential v = electric scalar potential  = magnetic scalar potent. T = temperature
field s
field 
field ks
scalar potential v p p  g p p
 = electric permittivity  = electric conductivity  = magnetic permeability k = thermal conductivity
electric flux,
i.e. electric charge
electric current magnetic flux heat flux
floating floating floating floating
p
characteristic 
Global flux
(weak sense)
Global floating
48
g
electric potential
g
electric potential
g
magnetic potential
g
temperature
g
potential
Global quantities of circulation typeGlobal quantities of circulation type
Edge finite element space S1 defined on a mesh 
v s v  v Se eE , ( )1 
Edge finite element space S1 defined on a mesh 
Boundary conditions  v constant et known
Edges of 
 e ee E , ( )
Constraints for circulation of v
Constraints
Basis functions ?
 relations between coefficients ve ...
Degree of freedom ?
M t t ti f l ti ith ti t t ti l
( , ' ) , ' ( , ' ), ' ( )      1 1curl curl Ss eha a n h a j a a  
Magnetostatic formulation with magnetic vector potential a
Magnetodynamic formulation with magnetic field and scalar potential h-
49
  t s hcurl curl Sc e( , ' ) ( , ' ) , ' , ' ( )h h h h n e h h         1 10
Constraints of circulation typeConstraints of circulation type
Characterization of a curl-conform vector field in the whole domainCharacterization of a curl conform vector field in the whole domain
Explicit constraints for circulation and zero curl
v s v  v Se ee E , ( )1 
Explicit constraints for circulation and zero curl
v s v c      v Ik kk E n nn N i ii Cc cC 
Ec : edges in c
NcC : nodes in cC and on cC
C : cuts
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 
Elementary geometrical
entities (nodes edges) and
along a closed path around c
c i igrad q 
50
entities (nodes, edges) and
global ones (groups of edges)
Global quantities of circulation typeGlobal quantities of circulation type
Discrete weak formulationDiscrete weak formulation
( , ' ) , ' ( , ' ), ' ( )      1 1curl curl Ss eha a n h a j a a  
system of equations
(symmetrical matrix)
with (v  a)
v s v c      v Ik kk E n nn N i ii Cc cC  (symmetrical matrix)
Test function a' = sk, vn  classical treatment, no contribution for < · >h
n h a n h c n h h       i igrad q dl'
k n h
Test function a' = ci  contribution for < · >h
n h a n h c n h h      s s i s i sCh h h igrad q dl, , ,   
Magnetomotive force
Similarly
W k l b l titi
51Electromotive force
Weak global quantities
n e c e   s i s ih dl V,  
Global quantities of circulation typeGlobal quantities of circulation type
n h a n h c n h h       s s i s i sgrad q dl, ' , ,  n h a n h c n h hs s i s i sCh h h igrad q dl, , ,   
Magnetomotive force
Surface h is made simply connected
Source of e.m.f.
Electromotive force
n e c e   s i s ih dl V,  
52
Magnetodynamic model 
with enforced magnetic fluxes
curl h = j
Equations Boundary conditions
n  h  i = 0 , n b  ds i  (i=1, 2, …) ,
enforced fluxes
curl e = – t b
div b = 0
 i , 
i
i  ( , , ) ,
n · b  0 = 0 , n · j  0 = 0div b 0
Constitutive relations
b =  h
j =  ej  e
53
Eddy currents & Joule losses = ?
Complementary 3D formulationsComplementary 3D formulations
Magnetodynamic h-formulation
  t scurl curl e( , ' ) ( , ' ) , 'h h h h n e h       1 0  h' ( )Fh1 
Magnetodynamic a-formulation
( , ' ) ( , ' ) , '        1 0curl curl t s ha a a a n h a    a' ( )Fe1 
How to enforce global fluxes ?
54
How to enforce global fluxes ?
Magnetodynamic h-formulationMagnetodynamic h-formulation
Magnetodynamic h-formulation
  t scurl curl e( , ' ) ( , ' ) , 'h h h h n e h       1 0  h' ( )Fh1 
n  j | 0 = 0  a surface scalar potential  can be defined on 0
n  h = – n  grad  on 0
n  h | i = 0   can also be defined on each flux gate i as a floating potential
No cut for  !
55
How to discretize h and , and enforce fluxes ?
No cut for  !
Magnetodynamic h-formulationMagnetodynamic h-formulation
Discrete weak formulationDiscrete weak formulation
  t scurl curl e( , ' ) ( , ' ) , 'h h h h n e h       1 0  h' ( )Sh1 
system of equations
(symmetrical matrix)
with (v  h)
v s v c      v Ik kk E n nn N i ii Cc cC  (symmetrical matrix)
Test function h' = sk, vn  classical treatment, no contribution for < · >e
n e h n e c n e e l         s s i s i s t igrad q d'     
k n e
Test function h' = ci  contribution for < · >e
n e h n e c n e e l   s s i s i s t ie igrad q d, , ,  0 0   
Time derivative of the flux
W k l b l tit
56
Weak global quantity
Magnetodynamic a-formulationMagnetodynamic a-formulation
Magnetodynamic a-formulation
( , ' ) ( , ' ) , '        1 0curl curl t s ha a a a n h a    a' ( )Fe1 
g y
n  b | 0 = 0  a surface scalar potential v can be defined on 0
d
n  curl a | 0 = 0 n  a = – n  grad v on 0
v is multivalued  cuts are necessary !
57How to discretize a and v, and enforce fluxes ?
Magnetodynamic a-formulationMagnetodynamic a-formulation
Discrete weak formulationDiscrete weak formulation
( , ' ) ( , ' ) , '        1 0curl curl t s ha a a a n h a    a' ( )Se1 
system of equations
with (v  a)
v s v c      v Ik kk E n nn N i ii Cc cC 
Test function a' = sk, vn  classical treatment, no contribution for < · >h
directly related to fluxes
n h a n h c n h h l       s s i s i i h sgrad q d, ' , ,  
k n h
Test function a' = ci  contribution for < · >h
n h a n h c n h h l   s s i s i with cuts sCuth igrad q d, , ,  0 0
Magnetomotive force (m.m.f.)
W k l b l tit
58
Weak global quantity
Magnetodynamic problem 
with global constraints
curl h = j
Equations Boundary conditions
Global conditions
for circuit couplingn  hh = 0 , n  be = 0
curl e = – t b
div b = 0
for circuit coupling
e l  d V
i
i , n j  ds Iji i
Voltage Currentdiv b 0
Constitutive relations
Voltage Current
b =  h
j =  ej  e
59Inductor
h- formulationh- formulation
 h- magnetodynamic finite element formulations with 
massiveand stranded inductors
 Use of edge and nodal finite elements for h and 
– Natural coupling between h and 
– Definition of current in a strong sense with basis functions either for massive 
or stranded inductors
– Definition of voltage in a weak senseDefinition of voltage in a weak sense
– Natural coupling between fields, currents and voltages
60
h- and t- weak formulationsh- and t- weak formulations
Magnetic scalar potential in nonconducting regions cC

hr = – grad  in cC with h = hs + hr
Reaction magnetic field
h- magnetodynamic formulation
  t scurl curl( , ' ) ( , ' ) , 'h h h h n e h      1 0  h' ( )Fh 
Source magnetic field
(1)
t sc e( , ) ( , ) ,   ( )h
t- magnetodynamic formulation (similar)
How to couple local and global quantities ?
g y ( )
61
How to couple local and global quantities ?
h,  Vi, Ii
Current as a strong global quantityCurrent as a strong global quantity
Characterization of curl-conform vector fields : h or tCharacterization of curl conform vector fields : h or t
Coupling of edge end nodal finite elements
Explicit constraints for circulations and zero curl
h s v  h Se ee E , ( )1 
i.e. currents Ii
p
h s v c      h Ik kk E n nn N i ii Cc cC 
Ec : edges in c
NcC : nodes in cC and on cC
C : cuts
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 
Elementary geometrical
entities (nodes edges) and
along a closed path around c
c i igrad q 
62
entities (nodes, edges) and
global ones (groups of edges)
Voltage as a weak global quantityVoltage as a weak global quantity
Discrete weak formulationDiscrete weak formulation
  t scurl curl c e( , ' ) ( , ' ) , 'h h h h n e h      1 0  h' ( )Fh 
system of equations
(symmetrical matrix)
h s v c      h Ik kk E n nn N i ii Cc cC 
(symmetrical matrix)
Test function h' = sk, vn  classical treatment, no contribution for < · >e
n e h n e c n e e        i i igrad q dl V'
k n e
Test function h' = ci  contribution for < · >e
n e h n e c n e e   s s i s i s ih h h igrad q dl V, , ,   
Electromotive force
W k l b l tit
63
Weak global quantity
Voltage as a weak global quantity
and circuit relations
Source of e.m.f.
Electromotive force
Source of e.m.f.
n e c e   s i s ih dl V,  
in (1)   t i i icurl curl Vc( , ) ( , )h c h c   1
Weak circuit relation between V and I for inductor i
N t l t t k lt !
Weak circuit relation between Vi and Ii for inductor i
“ t (Magnetic Flux) + Resistance  Current = Voltage ”
64
Natural way to compute a weak voltage !
Better than an explicit nonunique line integration
Massive and stranded inductorsMassive and stranded inductors
Massive inductor
Direct application
Massive inductor
Stranded inductor
Additional treatment
Stranded inductor
h h h  r s j s jI h Fh h ( ) Number of turns
Tree technique ...
h h h r s j s jj Is , , h Fh hs ( )
Source field due to a magnetomotive force Nj
(one basis function for each stranded inductor)
Reaction field
   t scurl curl curlc w( , ' ) ( , ' ) ( , ' )h h h h j h    1 1    n e hs e, '  0
 h' ( )Fh hs h‘ = h j
Weak circuit relation between Vj and Ij for stranded inductor j
  t s j s j s j s j jI curl Vs( , ) ( , ), , , ,h h j h  1
h hs,j
65
Weak circuit relation between Vj and Ij for stranded inductor j
Natural way to compute the magnetic flux through all the wires !
a-v formulationa-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 methodsAdaptation for stranded inductors: several methods
– Natural coupling between local and global quantities, i.e. fields and currents 
and voltages
66
a-v weak formulationa-v weak formulation
Magnetic vector potential - Electric scalar potential
a v
e = – t a – grad v in c , with b = curl a in 
a-v magnetodynamic formulation
(1)
( , ') ( , ') ( , ')    1 curl curl grad vt c ca a a a a  
   ( , ') , ' ( )j a as aF 0
How to couple local and global quantities ?
( , ) , ( )js as
67
How to couple local and global quantities ?
a, v Vi, Ii
Voltage as a strong global quantityVoltage as a strong global quantity
( ' ) ( ' ) ' grad v grad v grad v va n j     F' ( )
With a' = grad v' in (1)
(2)
Weak form of div j = 0
( , ) ( , ) , t grad v grad v grad v vc c ja n j        v Fv c' ( )
At the discrete level : implication only true when grad Fv(c)  Fa()
OK with nodal and edge finite elements
Otherwise : consideration of the 2 formulations (1) and (2) 
with a penalty term for gauge condition
68
Current as a weak global quantity
and circuit relations
        n j n j n j, ,s ds Ii iji ji ji  1
for massive inductor i
j
i iin (2) I grads grad v gradsi t i ic c ( , ) ( , ) a  in (2)
I grads V grad v gradsi t i i i ic c ( , ) ( , ) a  0
Weak circuit relation between Vi and Ii
for massive inductor i
N t l t t k t !
69
Natural way to compute a weak current !
Better than an explicit nonunique surface integration
Circuit relation for stranded inductorsCircuit relation for stranded inductors
From the a-formulation
 I l V( ) ( )h h j h1
I grad v V grad v grad vj t j j j js j s j ( , ) ( , ), , ,, , a 0 0 0 
From the h-formulation
cannot be used directly...
grad v0,j = ?
  t s j j s j s j jI curl Vs j( , ) ( , ), , , ,h h j h   1
   t s j t s j t s jcurl( , ) ( , ) ( , ), , ,h h b h a h   t s j t s j t s j, , ,  
    t s j t s j t s jcurl( , ) ( , ) ,, , ,h h a h n a h     
   t s j t s j t s j s j( , ) ( , ) ( , ), , , ,h h a j a j   
Weak circuit relation between Vj and Ij
 t s j j s j s j js j s jI V( , ) ( , ), , ,, ,a j j j   1
70
Weak circuit relation between Vj and Ij
for massive inductor j
Equivalent current density (1)Equivalent current density (1)
Explicit distribution of the current density
j t w ws
j def
unit
N
S
I  js
j
unitS
t s j jd R I V
s
a w    
71
Equivalent current density (2)Equivalent current density (2)
SS
Projection method
Source electric vector potential
Source
Source electric scalar potential
Projection method
Source
( , ') ( , '), ,, ,curl curl curls j s js j s jh h j h 
h ' ( )F 
jProjection method
( , ') ( , ')
, ,
grad v grad v grad v
s j s js0 0  j
 F' ( )  h ' ( ),Fh s j
Electrokinetic problemElectrokinetic problem
 v Fv s j' ( ),
Source
( , '), , 1 0curl curls j s jh h 
 h ' ( )Fh s j
( , ') , '
, ,
grad v grad v v
s j J js0     n j
 v Fv s j' ( )  h ( ),Fh s j
With gauge condition (tree) &
Source = Nj
 v Fv s j( ),
Tensorial conductivity
72
With gauge condition (tree) & 
boundary conditions
Foil Winding Inductors
Adaptation of massive inductor circuit relation
To avoid the explicit definition of multiple foils, the voltage of the foils is 
defined as a continuous unidirectional spatially dependent global quantity, in 
the direction perpendicular to the foils, which enables to express the 
elementary foil circuit relations, relating currents and voltages. ...
... A convenient form of these circuit relations is developed and its continuous 
extension is shown to weaklydefine the constraints on currents peculiar to foil 
windings, i.e. allowing skin effect in the direction of the height of the foils
hil i i i i h l f il
73
while maintaining a constant current in each elementary foil.
Foil winding circuit relationsFoil winding circuit relations
Circuit relation for one foil Domain of the foil
 
 )vgrad,vgrad(V)vgrad,(I
L
N
i,si,sii,sti
f a
# turns
)(VV ii 
Foil winding and
   d)('VIL
N
i
f
 )vgrad)('V,( i,sta
Total thickness of all the foils
Voltage Continuum for Vi() and weak form of circuit relation
Foil winding and 
its continuous representation
L 
])L0([IR)('V
0)vgrad)('V,vgrad)(V( i,si,si  
])L,0([IR)('V 
Wh l f il i
coordinate  Whole foil region
Need of basis functions for Vi()
(polynomial or 1D finite element
74
approximations)
! Anisotropic conductivity !
Limited 
support of vs,i !
Foil winding reactorFoil winding reactor
75
ApplicationApplication
Foil winding reactor
Lamination stack
r = 1000lam = 2.5 107 Sm-1
dlam = 0.3 mm
1/8th of the 
structure
10 foils, Al
foil = 3.7 107 Sm-1
d 1 1
Foil winding
76
dfoil = 1.1 mm
ApplicationApplication
Current density (in a cross section of the foil winding)
No airgaps, Frequency f = 50 Hz
All foils Foil continuum
77
ApplicationApplication
Current density (in a cross section of the foil winding)
No airgaps, Frequency f = 500 Hz
All foils Foil continuum
78
ApplicationApplication
Current density (in a cross section of the foil winding)
With airgaps, Frequency f = 50 Hz
All foils Foil continuum
79
ApplicationApplication
Current density (in a cross section of the foil winding)
With airgaps, Frequency f = 500 Hz
All foils Foil continuum
80
Circuit relation for stranded inductorsCircuit relation for stranded inductors
From the a-formulation
 I l V( ) ( )h h j h1
I grad v V grad v grad vj t j j j js j s j ( , ) ( , ), , ,, , a 0 0 0 
From the h-formulation
cannot be used directly...
grad v0,j = ?
  t s j j s j s j jI curl Vs j( , ) ( , ), , , ,h h j h   1
   t s j t s j t s jcurl( , ) ( , ) ( , ), , ,h h b h a h   
OK 
if explicit approximation of
v0,j in a cross-section
t s j t s j t s j, , ,  
    t s j t s j t s jcurl( , ) ( , ) ,, , ,h h a h n a h     
...
   t s j t s j t s j s j( , ) ( , ) ( , ), , , ,h h a j a j   
Weak circuit relation between Vj and Ij
 t s j j s j s j js j s jI V( , ) ( , ), , ,, ,a j j j   1
81
Weak circuit relation between Vj and Ij
for massive inductor j
Circuit relation with turn voltagesCircuit relation with turn voltages
Circuit relation for one turn Domain of the foil
# turns
),(VV ii    )vgrad,vgrad(V)vgrad,(IS
N
i,si,sii,sti
i
i a
Stranded inductor and
Surface area of the cross-section
Voltage Continuum for Vi() and weak form of circuit relation
  d),('VISN ii  )vgrad),('V,( i,staStranded inductor and 
its continuous representation
  Si ,
0)vgrad),('V,vgrad),(V( i,si,si  
IR),('V 
Whole inductor region
IR),(V 

Whole inductor region
Need of basis functions for Vi()
(polynomial or 2D finite element

82
approximations)
! Anisotropic conductivity !
Limited 
support of vs,i !
ApplicationApplication
Inductor (1/8th of structure) - Current density
83
With varying turn voltage,
i.e., stranded inductor
With constant turn voltage,
i.e., massive inductor
With varying turn voltage
and isotropic conductivity
ApplicationApplication
Turn voltage continuum of a stranded inductor (1/8th of structure)
84
References
1. P. Dular, “Modélisation du champ magnétique et des courants induits dans des
systèmes tridimensionnels non linéaires” Tese de Doutorado Universidade desystèmes tridimensionnels non linéaires , Tese de Doutorado, Universidade de
Liège, Bélgica, 1996.
2. J. P. A. Bastos, N. Sadowski, "Electromagnetic Modeling by Finite Elements",
M l D kk I 2003 N Y k USA (490 ) (ISBN 0824742699)Marcel Dekker, Inc, 2003, New York, USA (490pp) (ISBN 0824742699)
3. N. Ida, J. P. A. Bastos, "Eletromagnetism and calculation of fields", Springer-Verlag,
1992, New York, USA (458 pp) (ISBN 0-387-97852-6).
4. M. V. Ferreira da Luz, “Desenvolvimento de um Software para Cálculo de Campos
Eletromagnéticos 3D Utilizando Elementos de Aresta, Levando em Conta o
Movimento e o Circuito de Alimentação”, Tese de Doutorado, Universidade Federal
de Santa Catarina, 2003.
85