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Homogenization of the
Eddy Current Problem in 2D
Karl Hollaus and Joachim Schoberl
Vienna University of Technology
Institute for Analysis and Scientific Computing
September 20, 2010
Laplace Problem
Motivation:
Fig.: FE-Model with 100laminations.
Fig.: FE-Model with 100 laminations(Detail, lower right corner).
Finite element models of laminated media lead to large equation systems.
Homogenization overcomes this problem!
Contents
Motivation
Electrostatic problemI Boundary value problem
I Weak form
I Multiscal ansatz
I Homogenization
I Numerical example
Eddy current problemI Boundary value problem
I Weak form
I Multiscal ansatz
I Homogenization
I Numerical example
Contents
Motivation
Electrostatic problemI Boundary value problem
I Weak form
I Multiscal ansatz
I Homogenization
I Numerical example
Eddy current problemI Boundary value problem
I Weak form
I Multiscal ansatz
I Homogenization
I Numerical example
Contents
Motivation
Electrostatic problemI Boundary value problem
I Weak form
I Multiscal ansatz
I Homogenization
I Numerical example
Eddy current problemI Boundary value problem
I Weak form
I Multiscal ansatz
I Homogenization
I Numerical example
Contents
Motivation
Electrostatic problemI Boundary value problem
I Weak form
I Multiscal ansatz
I Homogenization
I Numerical example
Eddy current problemI Boundary value problem
I Weak form
I Multiscal ansatz
I Homogenization
I Numerical example
Electrostatic ProblemLaminated medium:
Fig.: Problem model.
λi ... Material parametern ... No. laminationsd = d1 + d2
ff ... Fill factorff = d1
d1+d2
For example:Electrostatics
λ=ε ... Electric permittivityu=V ... Electric scalar potential
Electrostatic Problem
Boundary value problem:
Fig.: Boundary value problem.
Ω ... Domain
Ω = Ω0 ∪ Ωm
Γ ... Boundary
Γ = ΓD ∪ ΓN
−∇(λ(x , y)∇u(x , y)) = 0 in Ω (1)
u = uD on ΓD (2)
λ∂u
∂n· n = α on ΓN (3)
Electrostatic Problem
Weak Form:
Multiplication of (1) by a test function v and integration over Ω yields
−∫
Ω
∇(λ∇(u))v dΩ = 0
and integration by parts leads to∫Ω
λ∇u · ∇v dΩ−∫S
λ∂u
∂nv · ds = 0.
Considering boundary conditions (2, 3) and v = 0 on ΓD yields the weakform ∫
Ω
λ∇u · ∇v dΩ = 0. (4)
Electrostatic Problem
Reference solution:
Fig.: Reference solution u of one halfe of the domain.
Reference solution: Laminations are modeled individually!
Electrostatic Problem
Reference solution:
Fig.: Reference solution.
Fig.: Solution u, mean value u0 andenvelope u1 along x at y = 0.
Fig.: Periodic micro shapefunction φ(x), one periode isshown.
Electrostatic Problem
Two scale ansatz:
Thus, the following ansatz can be made:
u(x , y) = u0(x , y) + φ
(x
p
)u1(x , y) (5)
u0 ... mean valueu1 ... envelope of the staggered partφ ... periodic micro shape function
Inserting (5) into the bilinear form (4) and caring out simplemanipulations leads to∫
Ω
λ(∇u0 +∇φu1 + φ∇u1) · (∇v0 +∇φv1 + φ∇v1) dΩ = 0. (6)
Electrostatic Problem
Homogenization of the weak form:
The finite element matrix would be calculated by
∫ΩFE
∂u0
∂x∂u0
∂y
u1∂u1
∂x∂u1
∂y
T
λ 0 λφx λφ 00 λ 0 0 λφλφx 0 λφ2
x λφφx 0λφ 0 λφφx λφ2 00 λφ 0 0 λφ2
∂v0
∂x∂v0
∂y
v1∂v1
∂x∂v1
∂y
dΩ.
With φx = ∇φ.
The micro shape function φ is a highly oscillating function.
To homogenize the weak form, the coefficients λ, λ∇φ, λφ, etc. areaveraged over the periode d :
λ =1
d
∫ d
0
λ
(x
p
)dx =
λ1d1 + λ2d2
d
Electrostatic Problem
Homogenization of the weak form:
λφx =1
d
∫ d
0
λ
(x
p
)φx
(x
p
)dx = 2
λ1 − λ2
d
λφ =1
d
∫ d
0
λ
(x
p
)φ
(x
p
)dx = 0
λφ2x =
1
d
∫ d
0
λ
(x
p
)φx
(x
p
)φx
(x
p
)dx =
4
d(λ1
d1+λ2
d2)
λφxφ =1
d
∫ d
0
λ
(x
p
)φx
(x
p
)φ
(x
p
)dx = 0
λφ2 =1
d
∫ d
0
λ
(x
p
)φ
(x
p
)φ
(x
p
)dx =
λ1d1 + λ2d2
3d
Electrostatic Problem
Homogenization of the weak form:
Homogenized finite element matrix:
∫ΩFE
∂u0
∂x∂u0
∂y
u1∂u1
∂x∂u1
∂y
T
λ 0 λφx 0 00 λ 0 0 0
λφx 0 λφ2x 0 0
0 0 0 λφ2 0
0 0 0 0 λφ2
∂v0
∂x∂v0
∂y
v1∂v1
∂x∂v1
∂y
dΩ.
The homogenized medium can be modeled like a bulk by a coarse finiteelement mesh!
Choice for the ansatz: u0 ∈ H1(Ω), u1 ∈ L2(Ωm)
Electrostatic Problem in 2D
Numerical Example:
Fig.: Numerical model.
ff ... Fill factorff = d1
d1+d2
ff = 0.9
λ0 = 1
λ1 = 1000
λ2 = 1
ub − ua = 2
No.laminations = 10
Electrostatic Problem
Comparison of the results:
Fig.: Reference solution. Fig.: Homogenized solution.
Electrostatic Problem
Comparison of the results: flux density, x-component
Fig.: Reference solution. Fig.: Homogenized solution.
Electrostatic Problem
Comparison of the results: flux density, y-component
Fig.: Reference solution. Fig.: Homogenized solution.
”Energy” stored in the media:
Wlam = 0.586 and Whom = 0.608
Eddy Current Problem in 2D
Boundary value problem:
Fig.: Boundary value problem.
Ω ... DomainΩ = Ω0 ∪ Ωm
Γ ... Boundary
Γ = ΓH ∪ ΓB
µ ... Magnetic permeability
σ ... Electric conductivity
J 0 ... Impressed current density in a coil
B ... Magnetic flux density
Assumptions:- Linear material properties
- Time harmonic case
- Steady state
Eddy Current Problem
Maxwell’s equations for the eddy current problem:
curlH = J in Ωm (7)
curlE = −jωB (8)
divB = 0 (9)
J = σE (10)
B = µH (11)
curlH = J 0 in Ω0 (12)
divB = 0 (13)
B = µH (14)
H × n = K on ΓH (15)
B · n = b on ΓB (16)
Eddy Current Problem
Boundary value problem:
Magnetic vector potential:
B = curlA (17)
Faraday’s law:
E = −jωA. (18)
Ampere’s law:
curl(µ−1curlA)− jωσA = J 0 in Ω (19)
Boundary conditions:
µ−1curlA× n = K on ΓH (20)
A× n = α on ΓB . (21)
Eddy Current Problem
Weak form:
∫Ω
(curl(µ−1curlA) + jωσA)v dΩ =
∫Ω0
J 0v dΩ. (22)
(23)
Integration by parts yields
∫Ω
µ−1curlAcurlv dΩ +
∫Γ
µ−1curlA× v dΓ +
∫Ω
jωσAv dΩ =
∫Ω0
J 0v dΩ
Weak form with homogeneous boundary condition on ΓH :∫Ω
µ−1curlAcurlv dΩ + jω
∫Ω
σAv dΩ =
∫Ω0
J 0v dΩ. (24)
Eddy Current Problem
Multiscale ansatz:
Fig.: Eddy currents in laminations, 2D problem.
A = A0 + φ
(0
A1
)+
∇(φw)φ∇w
(25)
A0 represents the mean value of the solution
A1 considers J y and
w models J x at the end of the laminates
Eddy Current Problem
Modified weak form:
Inserting the approaches (27) into the weak form (26) yields:∫Ω
µ−1curl(A0 + φ(0,A1)T +∇(φw)
)curl
(v0 + φ(0, v1)T +∇(φq)
)dΩ
+jω
∫Ω
σ(A0 + φ(0,A1)T +∇(φw)
) (v0 + φ(0, v1)T +∇(φq)
)dΩ =
∫Ω0
J 0v dΩ
(26)∫Ω
µ−1curl(A0 + φ(0,A1)T + φ∇w
)curl
(v0 + φ(0, v1)T + φ∇q
)dΩ
+jω
∫Ω
σ(A0 + φ(0,A1)T + φ∇w
) (v0 + φ(0, v1)T + φ∇q
)dΩ =
∫Ω0
J 0v dΩ
(27)
Eddy Current Problem
Finite element matrix: ”Stiffness”
Multiscale ansatz:
A = A0 + φ(0,A1)T +∇(φw)
Bilinearform: ∫Ω
µ−1curlAcurlv dΩ
Homogenized finite element matrix:∫ΩFE
(curlA0
A1
)T (ν νφxνφx νφ2
x
)(curlv0
v1
)dΩ,
with ν = µ−1.
Eddy Current Problem
Finite element matrix: ”Mass”
Multiscale ansatz:
A = A0 + φ(0,A1)T +∇(φw)
Bilinearform:
jω
∫Ω
σAv dΩ
Homogenized finite element matrix:
jω
∫ΩFE
(A0)x(A0)y
A1
w∂w∂x∂w∂y
T
σ 0 0 σφx σφ 00 σ σφ 0 0 σφ
0 σφ σφ2 0 0 σφ2
σφx 0 0 σφ2x σφxφ 0
σφ 0 0 σφxφ σφ2 0
0 σφ σφ2 0 0 σφ2
(v0)x(v0)y
v1
q∂q∂x∂q∂y
dΩ
Eddy Current Problem
Finite element matrix: ”Stiffness”
Multiscale ansatz:
A = A0 + φ(0,A1)T + φ∇w
Bilinearform: ∫Ω
µ−1curlAcurlv dΩ
Homogenized finite element matrix:
∫ΩFE
curlA0
A1∂w∂y
T ν νφx νφxνφx νφ2
x νφ2x
νφx νφ2x νφ2
x
curlv0
v1∂q∂y
dΩ,
with ν = µ−1.
Eddy Current Problem
Finite element matrix: ”Mass”
Multiscale ansatz:
A = A0 + φ(0,A1)T + φ∇w
Bilinearform:
jω
∫Ω
σAv dΩ
Homogenized finite element matrix:
jω
∫ΩFE
(A0)x(A0)y
A1∂w∂x∂w∂y
T
σ 0 0 σφ 00 σ σφ 0 σφ
0 σφ σφ2 0 σφ2
σφ 0 0 σφ2 0
0 σφ σφ2 0 σφ2
(v0)x(v0)y
v1∂q∂x∂q∂y
dΩ
Eddy Current Problem
Multiscale ansatz:
A = A0 + φ
(0
A1
)+
∇(φw)φ∇w
We propose the following choice:
A0, v0 ∈ H(curl ,Ω)
A1, v1 ∈ L2(Ωm)
w , q ∈ H1(Ωm)
φ ∈ Hper (Ωm)
Again, a coarse finite element mesh suffices for the homogenized medium!
Eddy Current Problem in 2D
Numerical example:
Fig.: Numerical example,dimensions in mm.
α = 1.0Vs/mff = 0.9
µ0 = 4π10−7Vs/Am
µ = µrµ0
µr1 = 1000
σ1 = 2 · 106S/m
f = 50Hz
δ ... Penetration depth
δ = 1.6mm
Eddy Current Problem
Comparison of the results: Eddy current density J
Fig.: Reference solution. Fig.: Homogenized solution.
... solution in the upper left corner.
d1 = 1.8mm
10 laminations
Eddy Current Problem
Comparison of the results: Losses and computational costs
Table: Eddy current losses
Losses in W/md in mm Laminations Ansatz ∇(φw) Ansatz φ∇(w)
0.5 11.59 11.58 11.481.0 45.70 45.46 44.742.0 177.2 174.16 169.4
Table: Computational costs for d=0.5mm and 40 laminations
Model FE NDOFLaminated 104 452 783 555
Homogenized 1 286 11 117
FE ... No. finite elementsNDOF ... No. degrees of freedom