Asymptotic Description of the Magnetic Potential near a Corner Singularity in the Bidimensional Eddy-Current Problem

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Asymptotic Description of the Magnetic Potential near a Corner Singularity in the Bidimensional Eddy-Current

Problem

Monique Dauge, Patrick Dular, Laurent Krähenbühl, Victor Péron, Ronan Perrussel, Clair Poignard

To cite this version:

Monique Dauge, Patrick Dular, Laurent Krähenbühl, Victor Péron, Ronan Perrussel, et al.. Asymp-

totic Description of the Magnetic Potential near a Corner Singularity in the Bidimensional Eddy-

Current Problem. WCCM 2012, Jul 2012, São Paulo, Brazil. �hal-00768026v2�

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Asymptotic Description of the Magnetic Potential near a Corner Singularity in the Bidimensional

Eddy-Current Problem

Monique Dauge^? Patrick Dular^♥ Laurent Kr ähenb ühl^♦ Victor P éron^♠ Ronan Perrussel^† Clair Poignard^♣

?IRMAR (Rennes),^♥ACE (Li `ege),^♦Amp `ere (Lyon),^♠EPI Magique3D & LMAP, UPPA (Pau),

†LAPLACE (Toulouse),^♣EPI MC2 (Bordeaux)

WCCM 2012, Sao Paulo, July 8-13.

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Motivations

A usual model for electrical engineering applications

Dielectric relaxation vs magnetic diffusion:

τ_e=

σ τ_m=µσL²

L,

,

σ

,

µ

: characteristic length, permittivity, conductivity and permeability.

= ⇒

magneto-quasistatic(eddy-current) model.

curl E

= − ∂(µ

H

)

∂

t div

(µ

H

) =

0

,

curl H

= σ

E

+

∂(E)

∂t div

(

E

) =

0

.

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Motivations

Usage of the magneto-quasistatic model

Electrothermics

(a) Position of the inductor. (b) Induced current density.

Figure :Surface steel hardening of a shift fork. PhD thesis Sven Wanser (’93 - Renault+EDF/ECL).

Others: non-destructive testing, most of electromechanical systems.

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A computational simplification

Skin Effect

Skin effect: “exponential decrease” of the electromagnetic field in a volume conductor,i.e.

δ =

r 1

µπ

f

σ

L

. δ

: penetration (skin) depth.

Instead of strongly meshing close to the surface of the volume conductors:surface impedance conditions. Most usual:

Z_δ

(

H

×

n

) =

E_t

,

withZ_δ

=

¹

+

i

σδ .

n: unitary normal to the conductor,E_t: tangential component of the electric field.

For smooth enough surface, the impedance can even be “more accurate”.

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A computational simplification

Difficulties in the corner

Example in a domain decomposition process.

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A computational simplification

Difficulties in the corner

Few authors have considered impedance modifications :

“Surface impedance boundary conditions near corners and edges:

rigorous consideration”. [IEEE Trans. on Mag., 2002]

Application of the same asymptotic expansion as in the smooth surface case. Impedance obtained:

Z_x

= −

^tan

(π/

6

)

ρσ

,

Z_y

=

^tan

(π/

6

)

ρσ

.

= ⇒

blows up in the cornerfor any

σ < +∞

.

It is not validfor non-magnetic materials with

σ < +∞

[Buretet al, IEEE Trans. on Mag., 2012]

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Statement of the Problem

The magnetic vector potentialAsatisfies







−∆A

⁺

= µ

0Jin

Ω

₊

,

−∆A

⁻

+

4iζ²

A

⁻

=

0 in

Ω

−

, A

⁺

=

0 on

Γ,

[A]

_Σ

=

0

,

on

Σ, [∂

n

A]

_Σ

=

0

,

on

Σ.

⁽¹⁾

with

ζ

²

= κµ

0

σ/

4, and a smooth dataJ.

Γ Ω

+

Σ

c

ω Ω

−

Aim : Description of the magnetic potentialAnear the cornercin 2D

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Statement of the Problem

Weak solutions

Variational Formulation

Find

A ∈

H₀¹

(Ω)

such that

∀A

_∗

∈

H₀¹

(Ω)

Z

Ω+

∇A

⁺

·∇A

⁺_∗ dx

+

Z

Ω−

∇A

⁻

·∇A

⁻_∗

+

4iζ²

A

⁻

A

⁻_∗ dx

= µ

0

Z

Ω J

A

∗

Proposition

There exists a unique solution

A

in H₀¹

(Ω)

to problem(1). Moreover,

A

belongs to H⁵²^−ε

(Ω)

for any

ε >

0.

In particular, bySobolev embedding, we obtain that

A

belongs to

C

¹

(Ω)

.

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Corner asymptotics

In general

A

does not belong to

C

²

(Ω)

but

A

possesses a corner asymptotic expansion as the distancer tocgoes to zero

A(

r

, θ) ∼

r→0 Λ⁰^,⁰S⁰^,⁰(r, θ)

+

X

k>¹

X

p∈{0,1}

Λ^k^,^pS^k^,^p(r, θ)

Key points for the behavior of

A

in the vicinity ofc:

•

Derivation of thesingular functionsS^k^,^p

•

Determination of thesingular coefficientsΛ^k^,^p

We can provide a constructive procedure to determineS^k^,^p

S^k^,^p(r, θ)

=

r^kcos

(

k

θ−

p

π/

2

)

| {z }

kernel functionsof

∆

+

r^kX

j>1

(iζ

²r²

)

^j

j

X

n=0

logⁿr

Φ

^k_j_,^,_n^p

(θ)

| {z }

shadow terms

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Outline

1 The case

ζ =

0 : For the determination ofcoefficientsΛ^k^,^pwe introduce the method ofdual singular functions

2 The case

ζ 6=

0 : we introduce the method ofquasi-dual singular functions. This method is an approximate method. We prove estimates for its convergence.

3 Numerical simulations illustrate the theoretical results.

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Laplace Operator ( ζ = 0)

We consider the solution

A

to

(

−∆A = µ

0Jin

Ω, A =

0 on

Γ,

with a smooth right hand side with support outside the interior pointc.

A

admits the Taylor expansion atc:

A(

r

, θ) ∼

r→0 Λ⁰^,⁰

+

X

k>¹

X

p∈{0,1}

Λ^k^,^pr^kcos(kθ−pπ/2)

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Methods to extract the coefficients Λ

^k^,^p

The method of moments :

It uses the orthogonality of the angular partscos(kθ−pπ/2)

The dual function method:

It is the application to the present smooth case of a general method [Mazya-Plamenevskii, ’78] valid forcorner coefficientextraction

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The Dual Function Method

Dual harmonic functions singular atr

=

0 :

k^k₀^,^p

(

r

, θ) =







−

¹

2

π

^log^r

,

if k

=

0

,

p

=

0

,

1 2k

π

^r

−k

cos

(

k

θ−

p

π/

2

),

if k >¹

,

p

=

0

,

1

.

ForR

>

0, let us introduce the bilinear form

J

_R

(

K

,

A

) =

Z

r=R

(

K

∂

rA

− ∂

rK A

)

Rdθ.

Proposition

Let

A

be the solution to the Laplace equation with a smooth right hand side with support outside the ball

B(

c

,

R

)

. Then

J

_R

(k

⁰₀^,⁰

, A) =

Λ⁰^,⁰ and

J

_R

(k

^k₀^,^p

, A) =

Λ^k^,^p

,

k>¹

,

p

=

0

,

1

.

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The Quasi-Dual Function Method ( ζ 6= 0)

[Costabelet al, ’04; ’12]: The QDFM for straight edges, circular edges and homogeneous operators with constant coefficients.

We use quasi-dual functionsK^k_m^,^p:

K^k_m^,^p(r, θ)

=

k^k₀^,^p

(

r

, θ) +

m

X

j=1

(iζ

²

)

^j k^k_j^,^p

(

r

, θ)

| {z }

shadow term

wherek^k_j^,^pare solution to

(

∆k

^k_j^,^p⁺

=

0

,

in

S

₊

,

∆k

^k_j^,^p⁻

=

4k^k_j₋^,^p₁⁻

,

in

S

₋

in the space :

r⁻^k⁺^2jSpan

log^qr

Φ(θ),

q 6^j

+

1

, Φ ∈ C

¹

(

T

), Φ

^±

∈ C

^∞

(

T±

)

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Extraction of coefficients

The extraction ofΛ^k^,^pis performed through the evaluation of quantities

J

_R

(K

^k_m^,^p

, A),

k

=

0

,

1

,

2

, . . .

Theorem

Let

A

be the solution to problem(1), under the assumptions of the introduction.

Let k

∈

N^{and p}

∈ {

0

,

1

}

(p

=

0if k

=

0). Letmsuch that2m+2>k.

There exist coefficients

J

^k^,^p^;^k⁰^,^p⁰independent of R and

A

such that

J

R

(K

^k_m^,^p

, A) =

R→0 Λ^k^,^p

+

[k/2]

X

`=1

J

^k^,^p^;^k⁻²^`,^pΛ^k⁻²^`,^p

+ O(

R⁻^kR₀^2m⁺²logR

) ,

where R0 =ζR(1+p

|logR|) .

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Extraction of coefficients

Example

•

Fork

=

0

Λ⁰^,⁰

=

R→0

J

_R

(K

⁰_m^,⁰

, A) + O(

R²₀⁺^2mlogR

)

•

Fork

=

1

Λ¹^,⁰

=

R→0

J

_R

(K

¹_m^,⁰

, A) + O(

R⁻¹R₀^2m⁺²

)

•

Fork

=

2, we needm>¹ Λ²^,⁰

=

R→0

J

_R

(K

²₁^,⁰

, A) − J

²^,⁰^;⁰^,⁰Λ⁰^,⁰

+ O(

R⁻²R₀⁴logR

)

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The Problem











−∆A

⁺

=

0 in

Ω

₊

,

−∆A

⁻

+

4iζ²

A

⁻

=

0 in

Ω

−

, A

⁺

= |θ|

π

^on

Γ,

[A]

_Σ

=

0

,

on

Σ, [∂

n

A]

_Σ

=

0

,

on

Σ.

Ω

: disk of radius 50 mm.

Conducting sector :

ω = π/

4

ζ =

1

/(

5

√

2

)

mm⁻¹corresponds to a physical skin depth of 5 mm.

Γ

Ω

₊

Σ

Ω

−

c

ω

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Finite Element Solution

We useP₂finite elementss

(b) Real part of the solutionA. (c) Imaginary part of the solutionA.

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Accuracy for the computation of Λ

⁰^,⁰

as a function of R

RecallΛ⁰^,⁰

= J

_R

(K

⁰m^,⁰

, A) + O(

R²₀⁺^2mlogR

)

.

We plot

|J

_R

(K

⁰m^,⁰

, A) −

A|_c

|

w.r.t. the radiusR, form

=

0

,

1

10⁻⁴ 10⁻³ 10⁻²

10⁻¹⁰ 10⁻⁸ 10⁻⁶ 10⁻⁴ 10⁻²

RadiusR.

m

=

0 R²₀log

(

R

)

m

=

1 R⁴₀log

(

R

)

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Computation of the coefficient Λ

¹^,⁰

RecallΛ¹^,⁰

= J

_R

(K

¹m^,⁰

, A) + O(

R⁻¹R^2m₀ ⁺²

)

. We plot

|J

_R

(K

¹m^,⁰

, A) −

J_R_min(K¹m^,⁰,A)|.

10⁻⁴ 10⁻³ 10⁻²

10⁻⁷ 10⁻⁵ 10⁻³ 10⁻¹ 10¹

RadiusR

m

=

0 R⁻¹R²₀ m

=

1 R⁻¹R⁴₀

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Computation of the coefficient Λ

²^,⁰

RecallΛ²^,⁰

= J

_R

(K

²₁^,⁰

, A) − J

²^,⁰^;⁰^,⁰Λ⁰^,⁰

+ O(

R⁻²R⁴₀logR

)

. We plot

|J

_R

(K

²₁^,⁰

, A) −

J_R_min(K²₁^,⁰,A)|.

10⁻⁴ 10⁻³ 10⁻²

10⁻⁵ 10⁻³ 10⁻¹ 10¹ 10³

RadiusR.

m

=

1 Real part Imaginary part R⁻²R₀⁴log

(

R

)

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Comparison of the FE solution and of the local expansion

Expansion restricted to order 1:

JR_min(K⁰₁^,⁰,A)(1+iζ²s⁰₁^,⁰)

+

JR_min(K¹₁^,⁰,A)s¹₀^,⁰

,

Expansion restricted to order 2: adding to the above expression the term

(J_R_min(K²₁^,⁰,A)− J²^,⁰^;⁰^,⁰J_R_min(K⁰₁^,⁰,A))s²₀^,⁰

.

(d) Expansion restricted to order 1. Real part. (e) Expansion restricted to order 2. Real part.

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Thank you for your attention!

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