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Transcript of 1 Strong and Weak Formulations of Electromagnetic Problems Patrick Dular, University of Liège -...
1
Strong and Weak Formulations of
Electromagnetic Problems
Patrick Dular, Patrick Dular, University of Liège - FNRS, BelgiumUniversity of Liège - FNRS, Belgium
2
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
5
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
6
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
7
• 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
8
Electrostatic
9
• 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
10
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
11
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
12
Magnetodynamics Inductor (portion : 1/8th)
Stranded inductor -uniform current density (js)
Massive inductor -non-uniform current density (j)
13
Magnetodynamics - Joule lossesFoil winding inductance - current density (in a cross-section)
All foils
With air gaps, Frequency f = 50 Hz
14
Transverse induction heating
(nonlinear physical characteristics,moving plate, global quantities)
Eddy current density
Temperature distribution
Search for OPTIMIZATION of temperature profile
Magnetodynamics - Joule losses
15
Magnetodynamics - Forces
16
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
17
Inductive and capacitive effects
Frequency and time domain analyses
Any conformity level
Magnetic flux density Electric field
Resistance, inductance and capacitance versus frequency
18
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
19
"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
20
"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
21
"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
22
"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
23
Discretization
of Electromagnetic Problems
Nodal, edge, face and volume finite elements
24
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
25
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
26
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
27
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
28
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
29
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
30
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
31
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
32
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
33
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
34
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
35
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
36
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
37
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
38
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
39
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
40
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
41
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