numerical methods in electromagnetism and applications

Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation

Magnetic field/magnetic potential formulationNumerical solution

Numerical methods in electromagnetism and applications

Alfredo Bermudez de Castro

Departamento de Matematica Aplicada, Universidade de Santiago de Compostela. Spain

Colloquium del Departamento de Matematicas de laUniversidad Carlos IIIMadrid April 17, 2012

Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications



Outline

Introduction. Industrial applicationsElectrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating

Electromagnetic modellingMaxwell’s equationsConstitutive lawsHarmonic eddy currents model

Magnetic field formulationStrong problemWeak formulation

Magnetic field/magnetic potential formulationScalar magnetic potentialWeak formulation

Numerical solutionFinite element approximationDiscretized problems. Error analysisComputer implementationNumerical experiments




Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating

Industrial applications of electromagnetism

Electrodes for electric arc furnaces

Induction furnaces

Electric motors

Microwave heating





Electrodes for electric arc furnaces

Contract with Ferroatlantica I+D company. Also financed bythe the Spanish government), FEDER and Xunta de Galicia

FA is interested in silicon production

FA invented a new compound electrode called ELSA in the1990. It is the world leader in the sector of silicon ofmetallurgical quality

Numerical simulation has helped Ferroatlantica for ELSAdesign and operation, and also for marketing





Silicon

Silicon (Si) is the second most abundant element in the earth’scrust after oxygen.

In natural form, it can be found mainly as silicon dioxide(Silica, SiO2) and silicates.

In particular, quartz and sand are two of the most commonforms.





Applications of silicon

Depending on its purity, silicon has a wide variety of applications:

Ferrosilicon (silicon steels, it can contain more than 2% ofother materials)

Metallurgical silicon (e.g. silicon-aluminum alloys, it containsabout 1% of other elements)

Chemical silicon (silicones)

Solar silicon (solar cells)

Electronic silicon (semiconductors, the purest silicon, “9N” =99.9999999 of purity)





Metallurgy of silicon I

Silicon is produced industrially by reduction of silicon dioxidewith carbon by a reaction which can be written in a simple wayas follows:

Si O2 + 2C −→ Si+ 2CO.





Metallurgy of silicon II





The ELSA electrode I

Graphite coreMotion system

Casing

Clamps

Pre-baked paste

Liquid paste

Solid paste

Nipple

Supportsystem





The ELSA electrode II

Modulus of the current density, |Jh|, in conductors.





The ELSA electrode III

Magnetic potential Φh in dielectric.





The ELSA electrode IV

|Jh|: Horizontal section of one of the electrodes.





The ELSA electrode V

|Jh|: Vertical section of one of the electrodes.





Induction furnaces

Photographs taken from http://www.ameritherm.com





Industrial induction furnace

Silicon formelting andpurification

Graphitecrucible

Refractorylayers

Water-cooledcoil





Mathematical modelling

Multi-physics problem

Three coupled models corresponding to three different areas ofphysics.

Thermal modelElectromagnetic model

Hydrodynamic Model





Numerical simulation I





Numerical simulation II





Numerical simulation III

VELOCITY





Numerical simulation IV

VELOCITY





Electric motors

Contract with ORONA Company in the framework of theproject NET0LIFT to design new lift technologies

This project was co-financed by the Spanish researchprogramme CENIT

Our tasks were related to the numerical simulation of electricmotors





Usage of electric motors I

Electric motors are a reliable way of transforming electrical energyinto movement.

Photo by Zurecks.





Usage of electric motors II

From classical uses. . . .

Photo by Harrihealey02.





Usage of electric motors III

. . . to present or future applications.

Photo by Tony Hisgett.





Numerical simulation of electric motors

Electromagnetic energy is dissipated into heat through twodifferent mechanisms: Joule effect and hysteresis

The released heat causes the temperature rise of the motor

This is a very important limiting factor in designing electricmotors.

Numerical simulation is nowadays an essential tool foroptimum design.

It is done in two steps:

1 electromagnetic analysis and

2 thermal analysis.





Electromagnetic analysis I

Electromagnetic analysis aims at determining the eddy currentswhich are responsible for Joule heating.

Electromagnetic models are obtained from Maxwell equations

Three-dimensional electromagnetic simulation of the wholemotor is still a challenge





Electromagnetic analysis II

Figure: Stator: coils and laminated core





Electromagnetic analysis III

Figure: Magnetic laminated core





Electromagnetic analysis IV

The presence of a laminated core composed of isolated thinplates makes this problem difficult because very fine meshesneed to be used (several geometric scales)

The material of the laminate has nonlinear magnetic behaviourwith hysteresis

For these reasons, motor designers usually employ simplified2D transient magnetic models.

These 2D distributed parameter models are coupled withlumped parameter (or circuit) models





Motor sketch

A motor is composed of many pieces:





Electromagnetic simulation: 2D mesh





Electromagnetic simulation: magnetic flux density





Electromagnetic simulation: losses





Heat transfer simulation. 3D FE mesh

Having many pieces of different size and local phenomena can leadto large meshes . . .

A coarse mesh of the motor.





Temperature I

FEM GLPM





Temperature II

FEM GLPM





Microwave heating and food technology

Household microwave ovens work by passing non-ionizingmicrowave radiation, at a frequency of 2.45 GHz.

Some substances in the food, like water and fat, absorb energyfrom the microwaves in a process called dielectric heating(lossy dielectrics).

Raytheon Company sold the first microwave oven in 1947,derived from radar technology developed during the World WarII.

The compact versions become popular from 1967.

Microwave heating is very important for food technology, inparticular, for defrosting food





Microwave oven

Figure: Domain.





Numerical simulation: electric field

Figure: Norm of the electric field.





Numerical simulation: temperature I

Figure: Temperature.





Numerical simulation: temperature II




Maxwell’s equationsConstitutive lawsHarmonic eddy currents model

Electromagnetic modelling: Maxwell’s equations

Maxwell’sequations

∂D∂t − curlH = −J in R

3,∂B∂t + curlE = 0 in R

3,

divB = 0 in R3

divD = in R3

H: Magnetic field, E : Electric field,D: Electric displacement, J : Current density,B: Magnetic induction, : Charge density,t: Time.





Electromagnetic modelling: constitutive laws

J = σE + v ×B, σ: electric conductivity

D = εE , ε: electric permittivity,

B = µH, µ: magnetic permeability.





Harmonic regime. Eddy currents model I

Assumptions

∂D∂t can be neglected (low frequency)

F(x, t) = Re [eiωt F(x)] (alternating current)

ω: angular frequency, i: imaginary unit





Harmonic regime. Eddy currents model II

Time harmonic eddy current model

curlH = J

curlE = −iωB

divB = 0

B = µH

J = σE

E(x) = O(|x|−1) uniformly for |x| → ∞H(x) = O(|x|−1) uniformly for |x| → ∞

Fields are complex valued





Harmonic regime. Eddy currents model III

Different formulations are possible

Magnetic field/scalar magnetic potential (H/ϕ)

Magnetic vector potential/scalar electric potential (A/V )

Electric field (E)

Primitive of the electric field with respect to time (A∗ or u)

...

For the harmonic regime see, for instance, the book

Eddy Current Approximation of Maxwell Equations.

Theory, Algorithms and Applications

A. Alonso Rodrıguez and A. ValliSpringer, 2010.




Strong problemWeak formulation

The magnetic field formulation

Ω is a bounded domain such that

Ω = ΩC ∪ΩD

ΩC : conductors (σ > 0)ΩD: dielectrics (air) (σ = 0)

Conductors are not assumed tobe totally included in Ω





Boundary conditions

The boundary of the domain splits as follows:

∂Ω = ΓC∪ Γ

D

where

ΓC

:= ∂ΩC ∩ ∂Ω,

ΓD

:= ∂ΩD ∩ ∂Ω.

ΓI:= ∂ΩC ∩ ∂ΩD.

Boundary conditions:

E× n = g on ΓC,

H× n = f on ΓD.





Strong problem

SP.- To find the magnetic field H and the electric field E in Ω,satisfying

curlH = 0 in ΩD, (1)

curlH = J = σE in ΩC , (2)

iωµH+ curlE = 0 in Ω, (3)

div(µH) = 0 in Ω, (4)

E× n = g on ΓC, (5)

H× n = f on ΓD. (6)





Weak formulation I

We eliminate the electric field E in the previous system:Let us make the scalar product of (3) by a test field G such that

curlG = 0 in ΩD and G× n = 0 on ΓD.

Then, let us integrate in Ω. We get

∫

ΩiωµH · G+

∫

ΩcurlE · G = 0 in Ω,

Now we use a Green’s formula to transform the second integral:

∫

ΩcurlE · G =

∫

ΩE · curl G+

∫

ΓE · G× ndΣ.





Weak formulation II

We have

0 =

∫

ΩiωµH · G+

∫

ΩcurlE · G

=

∫

ΩiωµH · G+

∫

ΩC

E · curl G+

∫

ΓC

E · G× ndΣ

=

∫

ΩiωµH · G+

∫

ΩC

E · curl G+

∫

ΓC

n× (E× n) · G× n,dΣ

=

∫

ΩiωµH · G+

∫

ΩC

E · curl G+

∫

ΓC

n× g · G× ndΣ

=

∫

ΩiωµH · G+

∫

ΩC

E · curl G−∫

ΓC

g × n · G× ndΣ.





Weak formulation III

Now we use (2) to deduce

E =1

σcurlH in ΩC .

By replacing this equality we finally get

∫

ΩiωµH · G+

∫

ΩC

1

σcurlH · curl G =

∫

ΓC

g × n · G× ndΣ





Function spaces

H(div,Ω) :=G ∈ L2(Ω)3 : divG ∈ L2(Ω)

,

H(curl,Ω) :=G ∈ L2(Ω)3 : curlG ∈ L2(Ω)3

.

Hr(curl,Ω) :=G ∈ Hr(Ω)3 : curlG ∈ Hr(Ω)3

, r > 0.

Each of these spaces is endowed with its natural norm, i.e.,

‖G‖2Hr(curl,Ω) = ‖G‖2Hr(Ω)3 + ‖ curlG‖2Hr(Ω)3 .

H1/200 (Γ): space of functions defined on Γ that extended by 0

to ∂Ω \ Γ belong to H1/2(∂Ω).

H−1/200 (Γ): dual space of H

1/200 (Γ).





Weak formulation in terms of H

WP.- To find H ∈ V such that

H× n = f in H−1/200 (Γ

D)3,

iω

∫

ΩµH · G+

∫

ΩC

1

σcurlH · curl G =

⟨g × n, G× n

⟩ΓC

, ∀G ∈ V0.

where

V = G ∈ H(curl,Ω) : curlG = 0 in ΩD ,V0 =

G ∈ V : G× n = 0 in H

−1/200 (Γ

D)3.





Existence of weak solution

Theorem.-If there exists Hf ∈ V such that Hf × n = f in

H−1/200 (Γ

D)3, then the weak problem WP has a unique solution.

The proof is standard (Lax-Milgram Lemma)





Properties of the weak solution

Theorem.- Let H ∈ V be the solution of problem WP. LetB = µH ∈ L2(Ω)3, J = curlH ∈ L2(Ω)3, andE = ( 1σJ)|ΩC

∈ L2(ΩC)3. Then:

divB = 0 in Ω,

iωµH+ curlE = 0 in ΩC ,

E× n = g in H−1/200 (Γ

C)3,

H× n = f in H−1/200 (Γ

D)3,

J = 0 in ΩD.

Remark: The electric field is not uniquely determined in thedielectric domain




Scalar magnetic potentialWeak formulation

Cutting surfaces

Bossavit and Verite (1982): magnetic field in ΩC / scalarmagnetic potential in ΩD.

Σj “cut”surface:

Σj ⊂ ΩD, j = 1, . . . , J .

∂Σj ⊂ ∂ΩD, j = 1, . . . , J .

Σj ∩ Σk = ∅ for j 6= k.

ΩD:= ΩD \⋃j=J

j=0 Σj

is simply connected.





The kernel of the curl operator

Let T be the linear space of H1(ΩD) defined by

T =Ψ ∈ H1(Ω

D) : [[Ψ]]Σj

= constant, j = 1, . . . , J.

For all G ∈ V , there exists a unique scalar field Ψ ∈ T /C,

such that G|ΩD= grad Ψ.





Magnetic field/magnetic potential formulation

Problem WPP: To find (H, Φ) ∈ W such that

grad Φ× n = f in H−1/200 (Γ

D)3,

iω

∫

ΩC

µH · G+

∫

ΩC

1

σcurlH · curl G

+iω

∫

ΩD

µgrad Φ · grad ¯Ψ =

⟨g × n, G× n

⟩ΓC

, ∀(G, Ψ) ∈ W0.

where

W :=(G, Ψ) ∈ H(curl,Ω

C)× (T /C) : (G|grad Ψ) ∈ H(curl,Ω)

,

W0 :=(G, Ψ) ∈ W : grad Ψ× n = 0 in H

−1/200 (Γ

D)3.





Magnetic field/magnetic potential formulation

Theorem.- If there exists Hf ∈ V such that Hf × n = f in

H−1/200 (Γ

D)3, then problem WPP has a unique solution (H, Φ),

with (H|grad Φ) being the unique solution of problem WP.




Finite element approximationDiscretized problems. Error analysisComputer implementationNumerical experiments

Numerical solution: Finite Element Method

Ω, ΩC and ΩD are Lipschitz polyhedra.

Th: family of regular thetraedral meshes of Ω.

For every mesh Th, each element K ∈ Th is contained either inΩ

Cor in Ω

D.





Nedelec edge elements (Nedelec, 1980)

H(curl,Ω) is approximated by:

N h(Ω) := Gh ∈ H(curl,Ω) : Gh|K ∈ N (K) ∀K ∈ Th .where

N (K) :=Gh ∈ P1(K)3 : Gh(x) = a× x+ b, a,b ∈ C

3, x ∈ K.

Degrees of freedom of a function Gh ∈ N (K):∫

eGh · te for the six edges e of K

.





Nedelec edge elements. An important property

Fields of the form

Gh(x) = a× x+ b, a,b ∈ C3,

have constant tangential component along any straight line inthe space

In particular, this is true along the 6 edges of any tetrahedron

This function vector space has dimension 6

The values of the tangential components are taken asinterpolation conditions to determine a and b in eachtetrahedron of the mesh





Discretizing problem WP I

Problem DWP: To find Hh ∈ Vh such that

Hh × n = fI

on ΓD,

iω

∫

ΩµHh · Gh +

∫

ΩC

1

σcurlHh · curl Gh

=

∫

ΓC

g × n · Gh × n ∀Gh ∈ V0h,

where

fI:= two-dimensional Nedelec interpolant of n× f ,

Vh := Gh ∈ N h(Ω) : curlGh = 0 on ΩD ,V0

h := Gh ∈ Vh : Gh × n = 0 on ΓD .





Discretizing problem WP II

Theorem.- Let us assume that the solution H of problem WP

satisfies H|ΩC∈ Hr(curl,Ω

C) and H|ΩD

∈ Hr(ΩD)3, with

r ∈ (12 , 1].Then, f

Iis well defined by the 2D Nedelec interpolant of n× f ,

problem DMP has a unique solution Hh, and

‖H−Hh‖H(curl,Ω) ≤ Chr[‖H‖Hr(curl,Ω

C) + ‖H‖Hr(Ω

D)3

].





Discretizing problem WPP I

Finite dimensional space to approximate T :

Th := Ψh ∈ Lh(ΩD) : [[Ψh]]Σj

= constant, j = 1, . . . , J,

being

Lh(ΩD) :=

Ψh ∈ H1(Ω

D) : Ψh|K ∈ P1(K) ∀K ∈ T Ω

D

h

.

The curl-free vector fields in N h(ΩD) admit a multivaluedpotential in Th.





Discretizing problem WPP II

Problem DWPP: To find (Hh, Φh) ∈ Wh such that

grad Φh × n = fI

on ΓD,

iω

∫

ΩC

µHh · Gh +

∫

ΩC

1

σcurlHh · curl Gh

+iω

∫

ΩD

µgrad Φh · grad ¯Ψh =

∫

ΓC

g × n · Gh × n ∀(Gh, Ψh) ∈ W0h,

Wh :=(Gh, Ψh) ∈ N h(ΩC)× (Th/C) : (Gh|grad Ψh) ∈ H(curl,Ω)

,

W0h :=

(Gh, Ψh) ∈ Wh : grad Ψh × n = 0 on Γ

D

.





Discretizing problem WPP III

Theorem.- Let us assume that the solution (H, Φ) of problem HP

satisfies H ∈ Hr(curl,ΩC) and grad Φ ∈ Hr(Ω

D)3, with

r ∈ (12 , 1].Then, problem DWPP is well posed, it has a unique solution(Hh, Φh), and

‖H−Hh‖H(curl,ΩC) + ‖grad Φ− grad Φh‖L2(Ω

D)3

≤ Chr[‖H‖Hr(curl,Ω

C) + ‖grad Φ‖Hr(Ω

D)3

].





Computer implementation of problem DWPP

It is necessary to impose the following constraints:

(Gh|grad Ψh) ∈ H(curl,Ω):Elimination of the degrees of freedom of Gh on the interface by static

condensation.

[[Ψh]]Σj= constant, which arise from the definition of Th:

Ψh|Σ−

j= Ψh|Σ+

j+ [[Ψh]]Σj

= Ψh|Σ+j+ chj , j = 1, . . . , J.

The boundary condition grad Φh × n = fIon Γ

Dis imposed

by means of a Lagrange multipler defined on ΓD.





Numerical experiments

Coaxial cable: Ω is a cylindrical domain which consists of twodifferent conductors Ω

C1and Ω

C2separated by a dielectric ΩD.

An alternating current J goes through the innermostconductor along its axis.





Analytical solution

By using a cylindrical coordinate system:

In the innermost conductor: H(r) = c I1(γ1r)eθ.In the dielectric domain: H(r) = I

2πreθ.

In the outer conductor: H(r) = (d I1(γ3r) + e K1(γ3r))eθ,

being

γ1 =√iωµσ1, γ3 =

√iωµσ3

I1 and K1 modified Bessel functions.

c, d, e constants obtained by using the boundary and theinterface conditions.





Numerical results





Numerical results

Error curve for the magnetic field H (log-log scale).

104

105

106

101

102

Rel

ativ

e er

ror

(%)

Number of d.o.f.

Percentual errors

y=Ch





Numerical results

Magnetic potential in the dielectric domain.





References

A. Bermudez, D. Gomez, M.C. Muniz, P. Salgado, R. Vazquez, Numericalmodelling of industrial induction, in Advances of Induction & Microwave

Heating of Mineral and Organic Materials p. 75–100, INTECH Open AccessPublisher. Rijeka. 2011.

A. Bermudez, C. Reales, R. Rodrıguez, P. Salgado, Numerical analysis of afinite-element method for the axisymmetric eddy current model of an inductionfurnace. IMA J. Numer. Anal., 30 p. 654–676, 2010.

A. Bermudez, D. Gomez, P. Salgado, Eddy-current losses in laminated coresand the computation of an equivalent conductivity. IEEE Transactions on

Magnetics, 44, p. 4730–4738, 2008.





References

A. Bermudez, R. Rodrıguez, P. Salgado, A finite element method for themagnetostatic problems in terms of scalar potentials. SIAM Journal on

Numerical Analysis, 46, p. 1338–1363, 2008.

A. Bermudez, D. Gomez, M. C. Muniz, P. Salgado, Transient numericalsimulation of a thermoelectrical problem in cylindrical induction heatingfurnaces. Advances in Computational Mathematics, 26, p. 39–62, 2007.

A. Bermudez, R. Rodrıguez, P. Salgado, FEM for 3D eddy current problems inbounded domains subject to realistic boundary conditions. An application tometallurgical electrodes. Archives of Computational Methods in Engineering,

12 (1), p. 67–114, 2005.

A. Bermudez, R. Rodrıguez, P. Salgado, A finite element method withLagrange multipliers for low frequency harmonic Maxwell equations. SIAMJournal on Numerical Analysis, 40 (5), p. 1823–1849, 2002.


numerical methods in electromagnetism and applications

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