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Strong and Weak Formulations of Electromagnetic

Problems

Patrick Dular, University of Liège - FNRS, Belgium

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

Formulations of Electromagnetic Problems

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

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)

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, ...

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

Electrostatic

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

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

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

Magnetodynamics

Inductor (portion 1/8th)

Stranded inductor -uniform current density (js)

Massive inductor -non-uniform current density (j)

Magnetodynamics - Joule losses

Foil winding inductance - current density (in a

cross-section)

With air gaps, Frequency f 50 Hz

All foils

Magnetodynamics - Joule losses

Transverse induction heating (nonlinear physical

characteristics,moving plate, global quantities)

Eddy current density

Search for OPTIMIZATION of temperature profile

Temperature distribution

Magnetodynamics - Forces

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!

Inductive and capacitive effects

Magnetic flux density

Electric field

- Frequency and time domain analyses
- Any conformity level

Resistance, inductance and capacitance versus

frequency

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

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

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

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

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

Discretization of Electromagnetic Problems

Nodal, edge, face and volume finite elements

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 ?

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

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

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

Sequence of finite element spaces

Geometric elements

Mesh

Geometric entities

Sequence of function spaces

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

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

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

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

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,

...)

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

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

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

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)

Strong formulations

Grad-div weak formulation

grad-div Green formula

integration in W and divergence theorem

grad-div scalar potential f weak formulation

Curl-curl weak formulation

curl-curl Green formula

integration in W and divergence theorem

curl-curl vector potential a weak formulation

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

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

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

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