Numerical methods in electromagnetism and applications
Transcript of 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
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
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
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
Industrial applications of electromagnetism
Electrodes for electric arc furnaces
Induction furnaces
Electric motors
Microwave heating
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave 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
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
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.
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
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)
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
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.
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
Metallurgy of silicon II
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
The ELSA electrode I
Graphite coreMotion system
Casing
Clamps
Pre-baked paste
Liquid paste
Solid paste
Nipple
Supportsystem
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
The ELSA electrode II
Modulus of the current density, |Jh|, in conductors.
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
The ELSA electrode III
Magnetic potential Φh in dielectric.
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
The ELSA electrode IV
|Jh|: Horizontal section of one of the electrodes.
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
The ELSA electrode V
|Jh|: Vertical section of one of the electrodes.
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
Induction furnaces
Photographs taken from http://www.ameritherm.com
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
Industrial induction furnace
Silicon formelting andpurification
Graphitecrucible
Refractorylayers
Water-cooledcoil
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
Mathematical modelling
Multi-physics problem
Three coupled models corresponding to three different areas ofphysics.
Thermal modelElectromagnetic model
Hydrodynamic Model
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
Numerical simulation I
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
Numerical simulation II
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
Numerical simulation III
VELOCITY
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
Numerical simulation IV
VELOCITY
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
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
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
Usage of electric motors I
Electric motors are a reliable way of transforming electrical energyinto movement.
Photo by Zurecks.
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
Usage of electric motors II
From classical uses. . . .
Photo by Harrihealey02.
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
Usage of electric motors III
. . . to present or future applications.
Photo by Tony Hisgett.
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
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.
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
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
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
Electromagnetic analysis II
Figure: Stator: coils and laminated core
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
Electromagnetic analysis III
Figure: Magnetic laminated core
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
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
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
Motor sketch
A motor is composed of many pieces:
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
Electromagnetic simulation: 2D mesh
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
Electromagnetic simulation: magnetic flux density
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
Electromagnetic simulation: losses
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
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.
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
Temperature I
FEM GLPM
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
Temperature II
FEM GLPM
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
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
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
Microwave oven
Figure: Domain.
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
Microwave oven
Figure: Domain.
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
Numerical simulation: electric field
Figure: Norm of the electric field.
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
Numerical simulation: temperature I
Figure: Temperature.
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Electrodes for electric arc furnacesInduction furnacesElectric motorsMicrowave heating
Numerical simulation: temperature II
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
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.
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Maxwell’s equationsConstitutive lawsHarmonic eddy currents model
Electromagnetic modelling: constitutive laws
J = σE + v ×B, σ: electric conductivity
D = εE , ε: electric permittivity,
B = µH, µ: magnetic permeability.
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Maxwell’s equationsConstitutive lawsHarmonic eddy currents model
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
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Maxwell’s equationsConstitutive lawsHarmonic eddy currents model
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
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Maxwell’s equationsConstitutive lawsHarmonic eddy currents model
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.
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
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
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Strong problemWeak formulation
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.
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Strong problemWeak formulation
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)
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Strong problemWeak formulation
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.
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Strong problemWeak formulation
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.
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Strong problemWeak formulation
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
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Strong problemWeak formulation
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).
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Strong problemWeak formulation
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.
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Strong problemWeak formulation
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)
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Strong problemWeak formulation
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
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
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.
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Scalar magnetic potentialWeak formulation
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.
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Scalar magnetic potentialWeak formulation
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.
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Scalar magnetic potentialWeak formulation
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.
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
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.
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Finite element approximationDiscretized problems. Error analysisComputer implementationNumerical experiments
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
.
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Finite element approximationDiscretized problems. Error analysisComputer implementationNumerical experiments
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
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Finite element approximationDiscretized problems. Error analysisComputer implementationNumerical experiments
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 .
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Finite element approximationDiscretized problems. Error analysisComputer implementationNumerical experiments
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
].
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Finite element approximationDiscretized problems. Error analysisComputer implementationNumerical experiments
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.
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Finite element approximationDiscretized problems. Error analysisComputer implementationNumerical experiments
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
.
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Finite element approximationDiscretized problems. Error analysisComputer implementationNumerical experiments
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
].
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Finite element approximationDiscretized problems. Error analysisComputer implementationNumerical experiments
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.
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Finite element approximationDiscretized problems. Error analysisComputer implementationNumerical experiments
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.
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Finite element approximationDiscretized problems. Error analysisComputer implementationNumerical experiments
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.
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Finite element approximationDiscretized problems. Error analysisComputer implementationNumerical experiments
Numerical results
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Finite element approximationDiscretized problems. Error analysisComputer implementationNumerical experiments
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
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Finite element approximationDiscretized problems. Error analysisComputer implementationNumerical experiments
Numerical results
Magnetic potential in the dielectric domain.
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Finite element approximationDiscretized problems. Error analysisComputer implementationNumerical experiments
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.
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications
Introduction. Industrial applicationsElectromagnetic modellingMagnetic field formulation
Magnetic field/magnetic potential formulationNumerical solution
Finite element approximationDiscretized problems. Error analysisComputer implementationNumerical experiments
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.
Alfredo Bermudez de Castro Numerical methods in electromagnetism and applications