Transcript Strong and Weak Formulations of Electromagnetic Problems Patrick Dular, 1

Strong and Weak Formulations of
Electromagnetic Problems
Patrick Dular, University of Liège - FNRS, Belgium
1
Content
 Formulations of electromagnetic problems
♦ Maxwell equations, material relations
♦ Electrostatics, electrokinetics, magnetostatics, magnetodynamics
♦ Strong and weak formulations
 Discretization of electromagnetic
♦ Finite elements, mesh, constraints
♦ Weak finite element formulations
problems
2
Formulations
of Electromagnetic Problems
Electrostatics
Electrokinetics
Maxwell equations
Magnetostatics
Magnetodynamics
3
Electromagnetic models

Electrostatics
All phenomena are described by Maxwell equations
♦ 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
4
Maxwell equations
Maxwell equations
curl h = j + t d
Ampère equation
curl e = – t b
Faraday equation
div b = 0
div d = rv
Conservation equations
Principles of electromagnetism
Physical fields and sources
h
b
j
magnetic field (A/m)
magnetic flux density (T)
current density (A/m2)
e electric field (V/m)
d electric flux density (C/m2)
rv charge density (C/m3)
5
Material constitutive relations
Constitutive relations
b = m h (+ bs)
Magnetic relation
d = e e (+ ds)
Dielectric relation
j = s e (+ js)
Ohm law
Characteristics of materials
m
e
s
magnetic permeability (H/m)
dielectric permittivity (F/m)
electric conductivity (W–1m–1)
Constants (linear relations)
Functions of the fields
(nonlinear materials)
Tensors (anisotropic materials)
Possible sources
bs remnant induction, ...
ds ...
js source current in stranded inductor, ...
6
Electrostatics
Basis equations
curl e = 0
div d = r
d=ee
e
d
r
e
Type of electrostatic structure
& boundary conditions
n  e | G0e = 0
n  d | G0d = 0
electric field (V/m)
electric flux density (C/m2)
electric charge density (C/m3)
dielectric permittivity (F/m)
Electric scalar potential formulation
div e grad v = – r
with e = – grad v
W0 Exterior region
Wc,i Conductors
Wd,j Dielectric
• the exterior region W0
• the dielectric regions Wd,j
• In each conducting region Wc,i : v = vi  v = vi on Gc,i
• Formulation for
7
Electrostatic
8
Electrokinetics
Basis equations
curl e = 0
div j = 0
j=se
e
j
s
Type of electrokinetic structure
& boundary conditions
n  e | G0e = 0
n  j | G0j = 0
electric field (V/m)
electric current density (C/m2)
electric conductivity (W–1m–1)
G0j
G0e,0
Wc
e=?, j=?
V = v1 – v0
Electric scalar potential formulation
div s grad v = 0
G0e,1
Wc
Conducting region
with e = – grad v
• Formulation for • the conducting region Wc
• On each electrode G0e,i : v = vi  v = vi on G0e,i
9
Magnetostatics
Type of studied configuration
Equations
curl h = j
W
Ampère equation
Wm
div b = 0
Magnetic conservation
equation
Ws
j
Constitutive relations
b = m h + bs
j=
js
W
Magnetic relation
Ohm law
& source current
Studied domain
Wm Magnetic domain
Ws Inductor
10
Magnetodynamics
Type of studied configuration
Equations
curl h = j
curl e = – t b
div b = 0
W
Ampère equation
Ws
js
Faraday equation
Wp
Magnetic conservation
equation
Ia
Wa
Va
Constitutive relations
b = m h + bs
j = s e + js
W
Magnetic relation
Ohm law
& source current
Studied domain
Wp Passive conductor
and/or magnetic domain
Wa Active conductor
Ws Inductor
11
Magnetodynamics
Inductor (portion : 1/8th)
Stranded inductor uniform current density (js)
Massive inductor non-uniform current density (j)
12
Magnetodynamics - Joule losses
Foil winding inductance - current density (in a cross-section)
With air gaps, Frequency f = 50 Hz
All foils
13
Magnetodynamics - Joule losses
Transverse induction heating
(nonlinear physical characteristics,
moving plate, global quantities)
Eddy current density
Search for OPTIMIZATION
of temperature profile
Temperature distribution
14
Magnetodynamics - Forces
15
Magnetodynamics - Forces
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!
16
Inductive and capacitive effects
Magnetic flux density

Frequency and
time domain
analyses

Any conformity
level
Electric field
Resistance, inductance and capacitance versus frequency
17
Continuous mathematical structure
Domain W, Boundary W = Gh U Ge
Basis structure
Function spaces Fh0  L2, Fh1  L2, Fh2  L2, Fh3  L2
dom (gradh) = Fh0 = { f  L2(W) ; grad f  L2(W) ,
fGh = 0 }
dom (curlh) = Fh1 = { h  L2(W) ; curl h  L2(W) ,n  hGh = 0 }
dom (divh) = Fh2 = { j  L2(W) ; div j  L2(W) , n . jGh = 0 }
gradh Fh0  Fh1 , curlh Fh1  Fh2 , divh Fh2  Fh3
Sequence
grad
Boundary conditions on Gh
curl
div
h
h
h
Fh 0 

Fh1 

Fh 2 
Fh 3
Basis structure Function spaces Fe0  L2, Fe1  L2, Fe2  L2, Fe3  L2
dom (grade) = Fe0 = { v  L2(W) ; grad v  L2(W) ,
vGe = 0 }
dom (curle) = Fe1 = { a  L2(W) ; curl a  L2(W) , n  aGe = 0 }
dom (dive) = Fe2 = { b  L2(W) ; div b  L2(W) , n . bGe = 0 }
gradh Fe0  Fe1 , curle Fe1  Fe2 , dive Fe2  Fe3
Sequence
div
Boundary conditions on Ge
curl
grad
e
e
e
Fe 3 

Fe 2 

Fe1 

Fe 0
18
Electrostatic problem
Basis equations
curl e = 0

F e0
S e0
F e1
S e1
F e2
S e2
F e3
S e3
"e" side
grad e
Fd3
r
(–v)
e
curl e
div d = r
d=ee
e e =d
0
d
(u)
div e
div d
curl d
grad d
0
e = – grad v

Fd2
S d3
Fd1
S d2
Fd0
S d1
S d0
d = curl u
"d" side
19
Electrokinetic problem
Basis equations
curl e = 0

F e0
S e0
F e1
S e1
F e2
S e2
F e3
S e3
"e" side
grad e
F j3
r
(–v)
e
curl e
div j = 0
j=se
se =j
d
0
(t)
div e
div j
curl j
grad j
0
e = – grad v

F j2
S j3
F j1
S j2
F j0
S j1
S j0
j = curl t
"j" side
20
Magnetostatic problem
Basis equations
curl h = j

F h0
S h0
F h1
S h1
F h2
S h2
3
Fh
grad h
div b = 0
b=mh
h
F e3
0
(–f)
div e
m h =b
b
curl h
j
(a)
div h
curl e
grad e
0

F e2
S e3
F e1
S e2
0
S e1
Fe
0
3
Sh
"h" side
h = – grad f
Se
b = curl a
"b" side
21
Magnetodynamic problem
Basis equations
curl h = j

F h0
S h0
F h1
S h1
F h2
S h2
Fh
3
b=mh
j=se
(t) h
curl h
j
3
Fe
0
(–f)
grad h
curl e = – t b
div b = 0
div e
m h =b
b

F e2
curl e
1
e (a, a*) F e
grad e
(–v)
F e0
jse
div h
0
S e3
S e2
S e1
0
3
Sh
"h" side
h = t – grad f
b = curl a
e = – t a – grad v
Se
"b" side
22
Discretization
of Electromagnetic Problems
Nodal, edge, face and volume finite elements
23
Discrete mathematical structure
Continuous problem
Continuous function spaces & domain
Classical and weak formulations
Discretization
Approximation
Discrete problem
Discrete function spaces piecewise defined
in a discrete domain (mesh)
Finite element method
Questions
Classical & weak formulations  ?
Properties of the fields  ?
Objective
To build a discrete structure
as similar as possible
as the continuous structure
24
Discrete mathematical structure
Finite element
Interpolation in a geometric
element of simple shape
+ f
Finite element space
Function space
& Mesh
+
Uf
i
i
Sequence of finite element spaces
Sequence of function spaces
& Mesh
+




U
i

fi 


25
Finite elements

Finite element (K, PK, SK)
♦ K = domain of space (tetrahedron, hexahedron, prism)
♦ PK = function space of finite dimension nK, defined in K
♦ SK = set of nK degrees of freedom
represented by nK linear functionals fi, 1  i  nK,
defined in PK and whose values belong to IR
f  PK
K = dom(f)

cod(f)
f(x)
x
IR
f i(f)
26
Finite elements

Unisolvance
" u  PK , u is uniquely defined by the degrees of freedom

Interpolation
Degrees of freedom
nK
uK 
 f (u) p
i
i
i 1

Basis functions
Finite element space
Union of finite elements (Kj, PKj, SKj) such as :
 the union of the Kj fill the studied domain ( mesh)
 some continuity conditions are satisfied across the element
interfaces
27
Sequence of finite element spaces
Geometric elements
Tetrahedral
Hexahedra
Prisms
(4 nodes)
(8 nodes)
(6 nodes)
Mesh
Nodes
iN
Edges
iE
Faces
iF
Volumes
iV
Geometric entities
S0
S1
S2
S3
Sequence of function spaces
28
Sequence of finite element spaces
S0
S1
S2
S3
Functions
Properties
Functionals
{si , i  N}
si (x j )   ij
Point
evaluation
Curve
integral
Surface
integral
Volume
integral
{si , i  E}
{si , i  F}
{si , i  V}
" i, j N
 s . dl  
 s . n ds  
 s dv  
i
j
ij
" i, j E
i
j
ij
" i, j F
j
Bases
i
ij
" i, j V
uK 
 f (u) s
i
i
Degrees of
freedom
Nodal value
Circulation
along edge
Flux across
face
Volume
integral
i
Nodal
element
Edge
element
Face
element
Volume
element
Finite elements
29
Sequence of finite element spaces
S0
Base
functions
Continuity across
element interfaces
{si , i  N}
value
Codomains of the
operators
S0
S1
S2
S3
{si , i  E}
{si , i  F}
{si , i  V}
tangential component
normal component
discontinuity
Conformity
grad S0  S1
S1
grad S0
S2
curl S1
S3
div S2
curl S1  S2
div S2  S3
Sequence
grad
curl
div
S0 
S1 
S 2 
S3
30
Mesh of electromagnetic devices

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
31
Mesh of electromagnetic devices

Electromagnetic fields enter the materials up to a
distance depending of physical characteristics and
constraints
♦ Skin depth  (<< if w, s, m >>)
2

wsm
♦ mesh fine enough near surfaces (material boundaries)
♦ use of surface elements when   0
32
Mesh of electromagnetic devices

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, ...)
33
Classical and weak formulations
Partial differential problem
Classical formulation
Notations
in W
on G = W
( u, v ) 
u  classical solution
( u,v ) 
Lu=f
Bu=g

u( x) v( x) dx

u( x) . v( x) dx , u, v L2 (W)
, u, v L2 (W)
W
W
Weak formulation

( u , L* v ) - ( f , v ) + Q g ( v ) ds  0 , " v V(W)
G
u  weak solution
v  test function
Continuous level :    system
Discrete level :
n  n system
 numerical solution
34
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
35
Constraints in
electromagnetic systems

Coupling of scalar potentials with vector fields
♦ e.g., in h-f 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-f formulation,
source electric scalar potential vs in the a-v formulation

Coupling of local and global quantities
♦ e.g., currents and voltages in h-f 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
36
Strong and weak formulations
in W
Equations
curl h = js
div b = rs
Scalar potential f
Strongly satisfies
h = hs - grad f , with curl hs = js
f each non-connected portion of G  constant
h
Constitutive relation
b=mh
Vector potential a
b = bs + curl a , with div bs = rs
n  h G  n  hs , n  b G  n  bs
h
n  a G  n  grad u G
b
b
b
Boundary conditions (BCs)
37
Strong formulations
Electrokinetics
curl e = 0 , div j = 0 , j = s e , e = - grad v or j = curl u
Electrostatics
curl e = 0 , div d = rs , d = e e , e = - grad v or d = ds + curl u
Magnetostatics
curl h = js , div b = 0 , b = m h , h = hs - grad f or b = curl a
Magnetodynamics
curl h = j , curl e = – t b , div b = 0 , b = m h , j = s e + js , ...
38
Grad-div weak formulation
u  grad v + v div u  div(v u) , u  H 1(W), v  H 1(W)
grad-div Green formula
(u,grad v)W + (div u, v)W   n  u, v W
integration in W and divergence theorem
add on both sides: - (rs , f ')Ws ,r +  n  bs , f ' Gb
- (b,grad f ')W - (rs , f ')Ws ,r +  n  bs , f ' Gb +  n  b, f '  Gf  Rb,f'  0
b m( hs - grad f)
 (div b - rs , f ')W -  n  b - n  bs , f ' Gb
grad-div scalar potential f weak formulation
Magnetostatics: rs  0
Electrokinetics: - ( j , grad v ')W +  n  js , v '  G j +  n  j, v '  Gv  (div j, v ')W -  n  j - n  js , v '  G j  R j , v '
Electrostatics: - (d , grad v ')W - (r, v ')Ws +  n  ds , v ' Gd +  n  d , v ' Gv  (div d - r, v ')W -  n  d - n  ds , v ' Gd  Rd , v '
39
Curl-curl weak formulation
u  curl v - v  curl u  div(v  u) , u, v  H 1(W)
curl-curl Green formula
(u,curl v)W - (curl u, v)W  -  n  u, v W
integration in W and
divergence theorem
add on both sides: - ( js , a ')Ws , j +  n  hs , a ' Gh
( h, curl a ')W - ( js , a ')Ws , j +  n  hs , a '  Gh +  n  h, a ' Gb  Rh,a '  0
h m-1 (bs + curl a)
 (curl h - js , a ')W -  n  h - n  hs , a ' Gh
curl-curl vector potential a weak formulation
40
Grad-div weak formulation
Use of hierarchal TF fp' in the weak formulation
1
- (b,grad f p ')W - (rs , f p ')Ws ,r +  n  bs , f p '  Gb  Rb,f p '  0
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 :
- (b + bp ,grad f p ')W - (rs , f p ')Ws ,r +  n  bs , f p '  Gb  0
or
(bp ,grad f p ')W  - Rb,f p '
1
Local FE problems
41
A posteriori error estimation (1/2)
Electric scalar
potential v
(1st order)
Electrokinetic / electrostatic problem
Electric field
Coarse mesh
Higher order hierarchal correction vp
(2nd order, BFs and TFs on edges)
V
Fine mesh
Field discontinuity directly
Large local correction
 Large error
42
Curl-curl weak formulation
Use of hierarchal TF ap' in the weak formulation
2
( h,curl a p ')W - ( js , a p ')Ws , j +  n  hs , a p '  Gh  Rh,a p '  0
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
( hp ,curl a p ')W  - Rh,a p '
43
A posteriori error estimation (2/2)
V
Higher order hierarchal
correction ap (2nd order,
BFs and TFs on faces)
Magnetic vector
potential a
(1st order)
Magnetostatic problem
Magnetodynamic problem
Fine mesh
skin depth
Magnetic core
Coarse mesh
Conductive core
Large
local correction
 Large error
44