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Strong and Weak Formulations of Electromagnetic Problems

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Title: Strong and Weak Formulations of Electromagnetic Problems


1
Strong and Weak Formulations of Electromagnetic
Problems
Patrick Dular, University 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
4
Electromagnetic models
All phenomena are described by Maxwell equations
  • 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

5
Maxwell equations
Maxwell equations
curl h j t d curl e t b div b 0 div d
rv
Ampère equation
Faraday equation
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) rv charge
density (C/m3)
6
Material constitutive relations
Constitutive relations
b m h ( bs) d e e ( ds) j s e ( js)
Magnetic relation
Dielectric relation
Ohm law
Constants (linear relations) Functions of the
fields(nonlinear materials) Tensors (anisotropic
materials)
Characteristics of materials
m magnetic permeability (H/m) e dielectric
permittivity (F/m) s electric conductivity
(W1m1)
Possible sources
bs remnant induction, ... ds ... js source
current in stranded inductor, ...
7
Electrostatics
Basis equations
Type of electrostatic structure
boundary conditions
curl e 0 div d r d e e
n e G0e 0 n d G0d 0
e electric field (V/m) d electric flux density
(C/m2) r electric charge density
(C/m3) e 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
Formulation for the exterior region W0 the
dielectric regions Wd,j In each conducting
region Wc,i v vi v vi on Gc,i
8
Electrostatic
9
Electrokinetics
Basis equations
Type of electrokinetic structure
boundary conditions
curl e 0 div j 0 j s e
G0j
n e G0e 0 n j G0j 0
G0e,1
Wc
e electric field (V/m) j electric current density
(C/m2) s electric conductivity (W1m1)
G0e,0
e?, j?
V v1 v0
Electric scalar potential formulation
div s grad v 0 with e grad v
Wc Conducting region
Formulation for the conducting region Wc On
each electrode G0e,i v vi v vi on
G0e,i
10
Magnetostatics
Type of studied configuration
Equations
curl h j div b 0
Ampère equation
Magnetic conservationequation
Constitutive relations
W Studied domain Wm Magnetic domain Ws Inductor
b m h bs j js
Magnetic relation
Ohm law source current
11
Magnetodynamics
Type of studied configuration
Equations
curl h j curl e t b div b 0
Ampère equation
Faraday equation
Magnetic conservationequation
Constitutive relations
W Studied domain Wp Passive conductor and/or
magnetic domain Wa Active conductor Ws Inductor
b m h bs j s e js
Magnetic relation
Ohm law source current
12
Magnetodynamics


Inductor (portion 1/8th)
Stranded inductor -uniform current density (js)
Massive inductor -non-uniform current density (j)
13
Magnetodynamics - Joule losses
Foil winding inductance - current density (in a
cross-section)
With air gaps, Frequency f 50 Hz
All foils
14
Magnetodynamics - Joule losses
Transverse induction heating (nonlinear physical
characteristics,moving plate, global quantities)
Eddy current density
Search for OPTIMIZATION of temperature profile
Temperature distribution
15
Magnetodynamics - Forces
16
Magnetodynamics - Forces
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!
17
Inductive and capacitive effects
Magnetic flux density
Electric field
  • Frequency and time domain analyses
  • Any conformity level

Resistance, inductance and capacitance versus
frequency
18
Continuous mathematical structure
Domain W, Boundary W Gh U Ge
Basis structure
Function spaces Fh0 Ì L2, Fh1 Ì L2, Fh2 Ì L2,
Fh3 Ì L2
dom (gradh) Fh0   f Î L2(W) grad f Î L2(W)
, f½Gh 0 dom (curlh) Fh1   h Î L2(W)
curl h Î L2(W) , n Ù h½Gh 0 dom (divh)
Fh2   j Î L2(W) div j Î L2(W) , n . j½Gh 0
gradh Fh0 Ì Fh1 , curlh Fh1 Ì Fh2 , divh Fh2
Ì Fh3
Boundary conditions on Gh
Sequence
Basis structure
Function spaces Fe0 Ì L2, Fe1 Ì L2, Fe2 Ì L2,
Fe3 Ì L2
dom (grade) Fe0   v Î L2(W) grad v Î L2(W)
, v½Ge 0 dom (curle) Fe1   a Î L2(W)
curl a Î L2(W) , n Ù a½Ge 0 dom (dive)
Fe2   b Î L2(W) div b Î L2(W) , n . b½Ge 0
gradh Fe0 Ì Fe1 , curle Fe1 Ì Fe2 , dive Fe2
Ì Fe3
Boundary conditions on Ge
Sequence
19
Electrostatic problem
Basis equations
curl e 0
div d r
d e e
Ì
É
grad
div
e
d
e

e

d
e
curl
curl
e
d
div
grad
e
d
e grad v
d curl u
"e" side
"d" side
20
Electrokinetic problem
Basis equations
curl e 0
div j 0
j s e
Ì
É
grad
div
e
j
s

e

j
e
curl
curl
e
j
div
grad
e
j
e grad v
j curl t
"e" side
"j" side
21
Magnetostatic problem
Basis equations
curl h j
div b 0
b m h
Ì
É
h grad f
b curl a
"h" side
"b" side
22
Magnetodynamic problem
Basis equations
curl e t b div b 0
curl h j
b m h
j s e
Ì
É
b curl a
h t grad f
"h" side
"b" side
e t a grad v
23
Discretization of Electromagnetic Problems
Nodal, edge, face and volume finite elements
24
Discrete mathematical structure
Continuous problem
Continuous function spaces domain Classical and
weak formulations
Discrete problem
Discrete function spaces piecewise definedin a
discrete domain (mesh)
Finite element method
Objective
Questions
To build a discrete structureas similar as
possibleas the continuous structure
Classical weak formulations ? Properties of
the fields ?
25
Discrete mathematical structure
Finite element Interpolation in a geometric
element of simple shape
f
Finite element space Function space Mesh
Sequence of finite element spaces Sequence of
function spaces Mesh

26
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

27
Finite elements
  • Unisolvance
  • " u Î PK , u is uniquely defined by the degrees
    of freedom
  • Interpolation
  • Finite element space
  • Union of finite elements (Kj, PKj, SKj) such as
  • the union of the Kj fill the studied domain (º
    mesh)
  • some continuity conditions are satisfied across
    the element interfaces

Degrees of freedom
Basis functions
28
Sequence of finite element spaces
Geometric elements
Mesh
Geometric entities
Sequence of function spaces
29
Sequence of finite element spaces
si , i Î N
Point evaluation
Nodal value
Nodalelement
si , i Î E
Curve integral
Circulation along edge
Edge element
si , i Î F
Surface integral
Flux across face
Face element
si , i Î V
Volume integral
Volume integral
Volume element
Finite elements
Bases
30
Sequence of finite element spaces
si , i Î N
value
si , i Î E
tangential component
grad S0 Ì S1
si , i Î F
normal component
curl S1 Ì S2
si , i Î V
discontinuity
div S2 Ì S3
Sequence
Conformity
31
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

32
Mesh of electromagnetic devices
  • Electromagnetic fields enter the materials up to
    a distance depending of physical characteristics
    and constraints
  • Skin depth d (dltlt if w, s, m gtgt)
  • mesh fine enough near surfaces (material
    boundaries)
  • use of surface elements when d 0

33
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,
    ...)

34
Classical and weak formulations
Partial differential problem
Classical formulation
Notations
L u f in W B u g on G W
u º classical solution
Weak formulation
Continuous level system Discrete level
n n system Þ numerical solution
v º test function
u º weak 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-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
37
Strong and weak formulations
Equations
in W
Scalar potential f
curl h js div b rs
h hs ? grad f , with curl hs js
Strongly satisfies
Constitutive relation
Strongly satisfies
b m h
Vector potential a
b bs curl a , with div bs rs
Boundary conditions (BCs)
38
Strong formulations
39
Grad-div weak formulation
grad-div Green formula
integration in W and divergence theorem
grad-div scalar potential f weak formulation
40
Curl-curl weak formulation
curl-curl Green formula
integration in W and divergence theorem
curl-curl vector potential a weak formulation
41
Grad-div weak formulation
1
Use of hierarchal TF fp' 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
1
or
Local FE problems
42
A posteriori error estimation (1/2)
V
Electrokinetic / electrostatic problem
Electric scalar potential v(1st order)
Electric field
Higher order hierarchal correction vp(2nd order,
BFs and TFs on edges)
Field discontinuity directly
Large local correction? Large error
43
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
2
Local FE problems
44
A posteriori error estimation (2/2)
V
Magnetostatic problem
Magnetodynamic problem
Magnetic vector potential a(1st order)
Fine mesh
skin depth
Conductive core
Magnetic core
Coarse mesh
Higher order hierarchal correction ap (2nd
order, BFs and TFs on faces)
Large local correction? Large error
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