Corner Asymptotics of the Magnetic Potential in the Eddy-Current Model

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Corner Asymptotics of the Magnetic Potential in the Eddy-Current Model

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.. Corner Asymptotics of the Magnetic Potential in the Eddy-Current Model. JSA 2013 - Journées Singulières Augmentées, Aug 2013, Rennes, France. �hal-00931735�

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Corner Asymptotics of the Magnetic Potential in the Eddy-Current Model

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)

JSA 2013,

Conf ´erence en l’honneur de Martin Costabel pour ses 65 ans, Rennes, August 26-30.

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The Configuration

Ω−:a conducting body(σ >0)

Ω₊: a dielectric medium c: a corner

ω∈(0,2π): the angle of the corner

Γ Ω₊

Σ

c ω Ω−

We describe the Magnetic potential in the vicinity of the cornerc.

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The Magnetic Potential

The eddy-current problem

The magnetic vector potentialAsatisfies







−∆A⁻+4iζ²A⁻ =0 inΩ−,

−∆A⁺ =µ0JinΩ+, A⁺ =0 onΓ,

[A]_Σ=0, onΣ, [∂nA]_Σ=0, onΣ. ⁽¹⁾

Hereζ² =κµ0σ/4>0,

J: a smooth data, vanishing near the cornerc.

Proposition

There exists a unique solutionAin H₀¹(Ω)to problem(1). Moreover,A

belongs to H⁵²^−ε(Ω)for anyε >0. In particular,Abelongs toC¹(Ω).

Apossesses acorner asymptoticexpansion nearc.

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





−∆A⁻+4iζ²A⁻ =0 inΩ−,

[A]_Σ=0, onΣ, [∂nA]_Σ=0, onΣ. ⁽¹⁾

Proposition

belongs to H⁵²^−ε(Ω)for anyε >0. In particular,Abelongs toC¹(Ω).

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





−∆A⁻+4iζ²A⁻ =0 inΩ−,

[A]_Σ=0, onΣ, [∂nA]_Σ=0, onΣ. ⁽¹⁾

Proposition

belongs to H

5

2−ε(Ω)for anyε >0. In particular,Abelongs toC¹(Ω).

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Corner Asymptotics

To generalize the Taylor Expansion Corner Asymptotics involve

1 Thesingular functions(primalanddual) :

belong to the kernel of the considered operator inR²^.

2 Thesingular coefficients:

its calculation requires the knowledge ofdual singular functions.

Aim: To explicit the Corner Asymptotics ofAnear the cornerc.

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Motivation : The eddy-current phenomenon

The limit problem inlarge frequency/high conductivity:

(−∆A⁺₀ =µ0JinΩ+,

A⁺₀ =0 on∂Ω+. ⁽²⁾

Ω+

c ω

IfΩ−has a convex corner, i.e.ω∈(0, π), Problem (2) hasnonC¹

singularities

Problem (1) hasC¹singularities

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(−∆A⁺₀ =µ0JinΩ+,

A⁺₀ =0 on∂Ω+. ⁽²⁾

Ω+

c ω

singularities

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(−∆A⁺₀ =µ0JinΩ+,

A⁺₀ =0 on∂Ω+. ⁽²⁾

Ω+

c ω

singularities

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Motivation : Multi-scale analysis for eddy-currents

BURETet al(IEEE Trans. on Mag.⁰12) Eddy currents and corner singularities

Whenδ =q

2

κµ0σ 1, there holds

A_δ ≈ A₀+aδ^αV(·

δ) +R_δ.

HereVis aprofile defined inR²^{such that}

−∆V+2iV1^Ω−=0 nearc,

andα= ₂_π−ω^π .

=⇒ The knowledge of thesingularitiesof−∆ +4iζ²1^Ω−is ”crucial”.

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Whenδ =q

2

δ) +R_δ.

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Whenδ =q

2

δ) +R_δ.

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Whenδ =q

2

δ) +R_δ.

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Main difficulty

The considered operator is

−∆ +iκµ0σ1Ω−=

(−∆ +iκµ0σ inΩ−,

−∆ inΩ₊.

Thesingularitiesare generated by the termiκµ0σ1Ω−.

=⇒ the derivation of the Corner Asymptotics is not obvious.

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Main difficulty

The considered operator is

−∆ +iκµ0σ1Ω−=

(−∆ +iκµ0σ inΩ−,

−∆ inΩ₊.

Thesingularitiesare generated by the termiκµ0σ1Ω−.

=⇒ the derivation of the Corner Asymptotics is not obvious.

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Our Reference

M. DAUGEet al(To appear in MMAS - INRIA RR 8204)

Corner asymptotics of the magnetic potential in the eddy-current model

Apossesses a corner asymptotic expansion

A(r, θ) ∼

r→0 Λ⁰^,⁰S⁰^,⁰(r, θ)+X

k>1

X

p∈{0,1}

Λ^k^,^pS^k^,^p(r, θ),

(r, θ): polar coordinates centered atc

S^k^,^p: primal singular functions

Λ^k^,^p: singular coefficients

We provide a constructive procedure to determine theprimalanddual singularities

We generalize the method of moments and we introduce the method of quasi-dual functionsto determineΛ^k^,^p

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Our Reference

A(r, θ) ∼

r→0 Λ⁰^,⁰S⁰^,⁰(r, θ)+X

k>1

X

p∈{0,1}

Λ^k^,^pS^k^,^p(r, θ),

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Our Reference

A(r, θ) ∼

r→0 Λ⁰^,⁰S⁰^,⁰(r, θ)+X

k>1

X

p∈{0,1}

Λ^k^,^pS^k^,^p(r, θ),

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Our Reference

A(r, θ) ∼

r→0 Λ⁰^,⁰S⁰^,⁰(r, θ)+X

k>1

X

p∈{0,1}

Λ^k^,^pS^k^,^p(r, θ),

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Outline

1 The caseζ =0 :

We introduce the method ofmomentsand the method ofdual functions.

2 The caseζ 6=0 :

We provide a constructive procedure to determine thesingularities.

We introduce the method ofquasi-dual singular functions.

3 Numerical simulations

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Laplace Operator (ζ = 0)

We consider the solutionAto

(−∆A=µ0JinΩ A=0 onΓ Aadmits 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)

| {z }

=s^k^,^p(r,θ)

s^k^,^p: harmonic polynomials

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Methods to extract the coefficientsΛ^k^,^p

We use dual harmonic functions :

k^k^,^p(r, θ) =







− ¹

2π ^log^r, if k=0, p=0,

1 2kπ ^r

−k

cos(kθ−pπ/2), if k>¹, p=0,1.

The method of moments The dual function method

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The Method of Moments

ForR >0, we introduce the formM_R:

M_R(K,A) = ¹

R

Z

r=R

K A Rdθ.

AssumeJhas a support outside the ballB(c,R).

Proposition

LetAbe the solution to the Laplace equation. Then

M_R(1,A) =2πΛ⁰^,⁰ and M_R(k^k^,^p,A) = ¹

2kΛ^k^,^p, k >¹, p=0,1.

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The Dual Function Method

V.G. MAZ’YA, B.A. PLAMENEVSKII(Amer. Math. Soc. Trans. (2)⁰84) On the coefficients in the asymptotic of solutions of the elliptic boundary problem in domains with conical points

ForR >0, let us introduce the bilinear form

J_R(K,A) = Z

r=R

(K∂rA−∂rK A) Rdθ.

Proposition

LetAbe the solution to the Laplace equation. Then

J_R(k⁰^,⁰,A) =Λ⁰^,⁰ and J_R(k^k^,^p,A) =Λ^k^,^p, k>¹,p =0,1.

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Thesingularities

ThesingularitiesUof−∆ +4iζ²1^S−:

(−∆U+4iζ²U=0 inS₋,

−∆U=0 inS₊.

r S−

S₊

θ=0

θ=ω/2 Ω+

c Ω−

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Description of theSingularities

V.A. KONDRATEV(Trudy Moskov. Mat. Obˇsˇc.⁰67)

Boundary value problems for elliptic equations in domains with conical or angular points

U= u₀

|{z}

leading part

+X

j>1

(iζ²)^j u_j

|{z}

shadow

,

We solve

∆u0=0, ∆u1 =4u₀1^S−, . . . , ∆uj =4u_j₋₁1^S−, u_j ∈S^λ =Span

n

r^λlog^qrΦ(θ), q∈N, Φ∈ C¹(T), Φ^±∈ C^∞(T±)o

Hereλ∈Z^,

T=R/(2πZ),T−= (−ω/2, ω/2)andT+ =T\T−.

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U= u₀

|{z}

leading part

+X

j>1

(iζ²)^j u_j

|{z}

shadow

,

We solve

∆u0=0, ∆u1 =4u₀1^S−, . . . , ∆uj =4u_j₋₁1^S−, u_j ∈S^λ =Span

n

r^λlog^qrΦ(θ), q∈N, Φ∈ C¹(T), Φ^±∈ C^∞(T±)o

Hereλ∈Z^,

T=R/(2πZ),T−= (−ω/2, ω/2)andT+ =T\T−.

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U= u₀

|{z}

leading part

+X

j>1

(iζ²)^j u_j

|{z}

shadow

,

We solve

∆u0=0, ∆u1 =4u01^S−, . . . , ∆uj =4uj−11^S−, u_j ∈S^λ =Span

n

r^λlog^qrΦ(θ), q∈N, Φ∈ C¹(T), Φ^±∈ C^∞(T±) o

Hereλ∈Z^,

T=R/(2πZ),T−= (−ω/2, ω/2)andT+ =T\T−.

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Existence of the shadows

Lemma

Letλ∈Z^andf∈T^λ−². Then, there existsu∈S^λsuch that∆u=f. Moreover

(i) Ifλ∈Z\ {0},degu6^degf+1, (ii) Ifλ=0,degu6^degf+2.

Here,

T^λ=Span

n

r^λlog^qrΨ(θ), q∈N, Ψ∈L²(T), Ψ^±∈ C^∞(T±) o

.

The degree ofgis its degree as polynomial of logr.

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Proof of Lemma

M. DaugeElliptic Boundary Value Problems in Corner Domains –

Smoothness and Asymptotics of Solutions, Lecture Notes in Mathematics, Vol. 1341, Berlin 1988

We introduce theMellin symbolM associated with∆ M(µ) = (µ²+∂_θ²), θ∈T.

Setting

f=

degf

X

q=0

r^λ−²log^qrΨq ∈T^λ−²,

and

u=Res µ=λ

n

r^µM(µ)⁻¹

degf

X

q=0

q! Ψq

(µ−λ)^q⁺¹ o

∈S^λ,

we check that∆u=f.

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Primal and Dual Singularities

Primal singular functionsSbelong toH¹in any bounded neighborhood

Bofc.

Dual singular functionsKdo not belong toH¹in such a neighborhoodB. The functionsKare needed for the determination of the coefficientsΛinvolved in the asymptotics.

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1. A basis for Primal Singularities

For(k,p) = (0,0)and for(k,p)∈N^∗× {0,1},

S^k^,^p = s^k^,^p

|{z}

leading part

+X

j>1

(iζ²)^j s^k_j^,^p

|{z}

shadow

.

Lemma

For any j >1, there existss^k_j^,^p ∈S^k⁺^2j, withdegs^k_j^,^p 6j, satisfying

∆s^k_j^,^p =4s^k_j₋^,^p₁1^S−.

Application : A(r, θ) ∼

r→0 Λ⁰^,⁰S⁰^,⁰(r, θ)+X

k>¹

X

p∈{0,1}

Λ^k^,^pS^k^,^p(r, θ)

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2. A basis for Dual Singularities

For(k,p) = (0,0)and for(k,p)∈N^∗× {0,1},

K^k^,^p = k^k^,^p

|{z}

leading part

+X

j>¹

(iζ²)^j k^k_j^,^p

|{z}

shadow

.

Lemma

For any j >1, there existsk^k_j^,^p ∈S⁻^k⁺^2j, withdegk^k_j^,^p 6^j+1, satisfying

∆k^k_j^,^p =4k^k_j₋^,^p₁1^S−.

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3. The Quasi-Dual Function Method

M. COSTABELet al(SIAM⁰04)

A quasidual function method for extracting edge stress intensity functions

S. SHANNONet al(Preprint⁰12)

Extracting generalized edge flux intensity functions by the quasidual function method along circular 3-D edges

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

K^k_m^,^p =k^k^,^p+

m

X

j=1

(iζ²)^j k^k_j^,^p

|{z}

shadow

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The Quasi-Dual Function Method

Extraction of coefficients

Theorem

LetAbe the solution to problem(1). Let k ∈N^{and p}∈ {0,1}(p=0if k=0). Then for allmsuch that2m+2>k,

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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Examples

• Fork=0

Λ⁰^,⁰ =

R→0

J_R(K⁰_m^,⁰,A) +O(R²₀⁺^2mlogR)

• Fork=1

Λ¹^,⁰ =

R→0 J_R(K¹_m^,⁰,A) +O(R⁻¹R₀^2m⁺²)

• Fork=2, we needm>¹

Λ²^,⁰ =

R→0 J_R(K²₁^,⁰,A)− J²^,⁰^;⁰^,⁰Λ⁰^,⁰+O(R⁻²R₀⁴logR)

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Key for the Proof of Theorem

We use quasi-primal singularitiesS^k_m^,^p:

S^k_m^,^p =s^k^,^p+

m

X

j=1

(iζ²)^j s^k_j^,^p

|{z}

shadow

Forε∈(0,R), we evaluate

J_R(K^k_m,S^k_m⁰0)− J_ε(K^k_m,S^k_m⁰0) =

R→0O(R⁻^k⁺^k⁰R₀^2m⁺²logR)

with any chosenk⁰, andm⁰ >^m.

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4. Calculation of singularities

M. COSTABEL, M. DAUGE(Math. Nachr.⁰93)

Construction of corner singularities for Agmon-Douglis-Nirenberg elliptic systems.

We use appropriate complex variablesz_±instead of the polar coordinates :

z₋=z(whenz∈ S₋) andz₊=−z(whenz∈ S₊) We use an Ansatz involving only integer powers of

z_±,¯z_±,logz_±,and log¯z_±.

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The first shadow of primal singularities

Example : expression ofs⁰₁^,⁰

s⁰₁^,⁰(r, θ) = ^sinω π ^r

2

logr cos 2θ−θsin 2θ +r²

1− ^cosω cos 2θ

2

inS₋

s⁰₁^,⁰(r, θ) = ^sinω π ^r

2

logr cos 2θ−θ+sin 2θ

+r² cosω cos 2θ

2 in S₊

withθ₊=θ−πsgnθ

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The first shadow of primal singularities

Example : Expression ofs¹₁^,⁰

s¹₁^,⁰(r, θ) = ^{2 sin}ω+sin 2ω

6π ^r

3

logr cos 3θ−θsin 3θ

+r³

_cosθ

2 −^cosω cos 3θ

3

, in S₋

s¹₁^,⁰(r, θ) = ^{2 sin}ω+sin 2ω

6π ^r

3

logr cos 3θ−θ₊sin 3θ

+r³ cos 2ω cos 3θ

6 , in S₊

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Framework

Ω: disk of radius 50 mm Conducting sector :ω=π/4

ζ =1/(5

√

2)mm⁻¹ Source :A⁺= ^|θ|₂_π on∂Ω

Γ

Ω+

Σ Ω−

c ω

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Finite Element Solution

We plot the real part and the imaginary part of the FE solution.

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Accuracy for the computation ofΛ⁰^,⁰as a function ofR

There holds

Λ⁰^,⁰ =

R→0

J_R(K⁰_m^,⁰,A) +O(R₀²⁺^2mlogR).

We plot|J_R(K⁰m^,⁰,A)−A|_c|/|A|_c|as a function ofR(m=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)