Skin Effect Description in Electromagnetism with a Multiscaled Asymptotic Expansion

(1)

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Skin Effect Description in Electromagnetism with a Multiscaled Asymptotic Expansion

Gabriel Caloz, Monique Dauge, Erwan Faou, Victor Péron

To cite this version:

Gabriel Caloz, Monique Dauge, Erwan Faou, Victor Péron. Skin Effect Description in Electromag-

netism with a Multiscaled Asymptotic Expansion. Asymptotic methods, mechanics and other appli-

cations, Aug 2009, Rennes, France. �inria-00528527�

(2)

Skin-Effect Description in Electromagnetism with a Scaled Asymptotic Expansion

Gabriel CALOZ Monique DAUGE Erwan FAOU Victor P ´ERON

IRMAR Universit ´e de Rennes 1

Macadam Workshop, RENNES 31.08.2009

(3)

References

GABRIELCALOZ, MONIQUEDAUGE, VICTORP ´ERON(2009)

Uniform Estimates for Transmission Problems with High Contrast in Heat Conduction and Electromagnetism

MONIQUEDAUGE, ERWANFAOU, VICTORP ´ERON(2009) Asymptotic Behavior at High Conductivity of Skin Depth in Electromagnetism

(4)

Transmission Problems with High Contrast Media

ELECTROMAGNETISM

Ω

₋

Ω

+

Σ

Ω

₋Highly conductingmaterial

⊂⊂ Ω

:Conductivity

σ

₋

≡ σ

¹

Σ = ∂Ω

₋: Interface

Ω

₊Insulating or dielectricbody:Conductivity

σ

₊

=

0 Aim: Understand the electromagnetic phenomenon as

σ → ∞

(5)

Transmission Problems with High Contrast Media

HEAT CONDUCTION

Ω

−

Ω

+

Σ

Ω

₋

⊂⊂ Ω

:heat conductivity a₋

Σ = ∂Ω

₋: Interface

Ω

+:heat conductivity a₊

Aim: Describe the transmission phenomenon as

|

^a−

a₊

|

¹

(6)

Outline

1 Uniform Estimates for Transmission Problems

2 3D Multiscaled Asymptotic Expansion

3 Numerical Simulations

(7)

Outline

1 Uniform Estimates for Transmission Problems

2 3D Multiscaled Asymptotic Expansion

3 Numerical Simulations

(8)

Transmission Problems in High Contrast Media

Ω

₋

Ω

₊

Σ Ω

Heat transfer equation

(

P_a

)

:

div

a grad

ϕ =

f

a

= (

^a+

,

^a₋

)

^s.t.

|

^a−

| / |

^a⁺

| → ∞

Maxwell equations

(

P_σ

)

:

curl

E

−

ⁱ

ωµ

0H

=

0 and

curl

H

+ (

i

ωε

0

− σ)

E

=

j

σ = (σ

₊

, σ

₋

)

s.t.

σ

₊

=

0 and

σ

₋

≡ σ → ∞

(9)

Issues

1 Uniform piecewise estimates and regularity in Sobolev norms for solutions

ϕ

of

(

P_a

)

2 Uniform

L

²estimates for solutions

(

E

,

H

)

of

(

P_σ

)

Hypothesis

Σ

is a bounded Lipschitz surface in

R

³

Scalar case:

H.P. HUY, E. SANCHEZ-PALENCIA(1974)

Limit problem and Strong convergence results as a₋

a₊

→ ∞

S. HASSANI, S. NICAISE, A. MAGHNOUJI(2009) Maxwell case:

H. HADDAR, P. JOLY, H.N. NGUYEN(2008)

Uniform estimates under a stronger regularity assumption on

Σ

(10)

Scalar Problem

Variationnal Problem

V_D

= H

¹₀

(Ω)

or V_N

= { ϕ ∈ H

¹

(Ω) | Z

Ω

ϕ d

x

=

0

} (

VP_a

)

: Find

ϕ ∈

^V

=

V_Dor V

=

V_Nsuch that

∀ ψ ∈

^V

, Z

Ω+

a₊

∇ ϕ

⁺

· ∇ ψ

⁺

d

x

+ Z

Ω₋

a₋

∇ ϕ

⁻

· ∇ ψ

⁻

d

x

=

− Z

Ω

f

ψ d

^x

+ (

^a+

−

^a−

) Z

Σ g

ψ d

^s

Hypothesis (H_a)

1 f

∈ L

²

(Ω)

and g

∈ L

²

(Σ)

2

R

Ωf

d

x

= R

Σg

d

s

=

0 if V

=

V_N and

R

Σg

d

s

=

0 if V

=

V_D

(11)

Uniform Estimates

Theorem

Let us assume that a₊

6 =

0. There exist

ρ

0

>

0 independent of a₊such that for all a₋

∈ {

^z

∈ C | |

^z

| > ρ

0

|

^a+

|}

, and for all data

(

f

,

g

)

satisfying

(

H_a

)

,

(

VP_a

)

has a unique solution

ϕ ∈

V , which is piecewise

H

³^/²and

k ϕ

⁺

k

³₂,Ω+

+ k ϕ

⁻

k

³₂,Ω₋

6

C_ρ₀

|

^a⁺

|

⁻¹

k

^f

k

⁰,Ω

+ k

^g

k

⁰,Σ

with C_ρ₀

>

0, independent of a₊, a₋, f , and g.

GABRIELCALOZ, MONIQUEDAUGE, V. P ´ERON(2009)

Key of the proof: asymptotic expansion for

ϕ

with respect to the powers of

ρ

⁻¹

:=

^a+

(

a₋

)

⁻¹and convergence of these series in the piecewise

H

³^/²-norm

(12)

Remarks

1 Similar estimate when the roles of a₊and a₋are exchanged. More precise estimate where a₊and a₋play symmetric roles

2 The assumption that

Σ

is Lipschitz is necessary

3 For polyhedral Lipschitz interface

Σ

, uniform piecewise

H

^sestimates:

k ϕ

⁺

k

^s,Ω+

+ k ϕ

⁻

k

^s,Ω₋

6

C_ρ₀

|

^a+

|

⁻¹

k

^f

k

^s−²,Ω

+ k

^g

k

s−³₂,Σ

for s

6

s_Σwith some³₂

<

s_Σ

6

2

4 For smooth interface

Σ

, uniform piecewise

H

^sestimates for any s

>

2

(13)

Maxwell Problem

Framework and Boundary Conditions

(

P_σ

) curl

E

−

ⁱ

ωµ

0H

=

0 and

curl

H

+(

i

ωε

0

− σ)

E

=

j

σ = (

0

, σ)

1 E

×

ⁿ

=

0 and H

·

ⁿ

=

0 on

∂Ω

2 E

·

ⁿ

=

0 and H

×

ⁿ

=

0 on

∂Ω

1 B. C. 1: j

∈ H(div, Ω) = {

^u

∈

^L²

(Ω) | div

u

∈ L

²

(Ω) }

2 B. C. 2:

j

∈ H

0

(div, Ω) = {

^u

∈ H(div, Ω) |

^u

·

ⁿ

=

0 on

∂Ω }

Hypothesis (SH)

The angular frequency

ω

is not an eigenfrequency of the problem







curl

E

−

ⁱ

ωµ

0H

=

0 and

curl

H

+

i

ωε

0E

=

0 in

Ω

₊

E

×

ⁿ

=

0 and H

·

ⁿ

=

0 on

Σ

B

.

C

.

1 or B

.

C

.

2 on

∂Ω

(14)

Maxwell Problem

Uniform L²(Ω)Estimate

Theorem

Under Hypothesis

(

^SH

)

, there are

σ

0and C

>

0, such that for all

σ > σ

0, the Maxwell problem

(

P_σ

)

with B.C.2 and j

∈ H

0

(div, Ω)

has a unique solution

(

E

,

H

)

in L²

(Ω)

², which satisfies:

k

^E

k

⁰,Ω

+ k

^H

k

⁰,Ω

+ √

σ k

^E

k

⁰,Ω₋

6

C

k

^j

k

H(div,Ω)

Similar result for B.C.1 and j

∈ H(div, Ω)

GABRIELCALOZ, MONIQUEDAUGE, V. P ´ERON(2009)

Application: Convergence of asymptotic expansion at high conductivity

(15)

Maxwell Problem

Proof of Maxwell Uniform Estimate

Lemma

Under Hypothesis (SH), there are

σ

0and C₀

>

0 such that if

σ > σ

0any solution

(

E

,

H

) ∈

^L²

(Ω)

²of problem

(

P_σ

)

with B.C.2 and data

j

∈ H

0

(div, Ω)

satisfies the estimate

k

^E

k

⁰,Ω

6

C₀

k

^j

k

H(div,Ω)

Similar statement for B.C.1 and j

∈ H(div, Ω)

. Keys of the proof:

Electrical Field Decomposition into a regular field and a gradient field C. AMROUCHE, C. BERNARDI, M. DAUGE, V. GIRAULT(1998) Scalar Estimates used for the gradient field

∇ ϕ

(16)

Outline

1 Uniform Estimates for Transmission Problems

2 3D Multiscaled Asymptotic Expansion

3 Numerical Simulations

(17)

References and Notations

Asymptotic expansion as

σ → ∞

of solutions of

(

^Pσ

)

^when

Σ

^{is smooth:}

E. STEPHAN, R.C. MCCAMY(1983-84-85) Plane Interface and Eddy Current Approximation H. HADDAR, P. JOLY, H.N. NGUYEN(2008)

Smooth Interface and Impedance boundary conditions V. P ´ERON(2009)

Smooth Interface and Perfectly Conducting Electric or Magnetic Wall B.C.

Hypothesis

1

Σ

is an oriented smooth compact surface

2 Perfectly Conducting Magnetic Wall b.c.

3 j

∈ H

0

(div, Ω)

and j

=

0 in

Ω

₋

4 Hypothesis (SH) on

ω

(18)

Asymptotic Expansion

δ :=

r ωε

0

σ →

⁰ ^as

σ → ∞

By Theorem there exists

δ

0s.t. for all

δ 6 δ

0, there exists a unique solution

(

E_(δ)

,

H_(δ)

)

to

(

P_σ

)

Then it is possible to construct series expansions in powers of

δ

:

E⁺

(δ)

(

x

) ≈ X

j>0

δ

^jE⁺_j

(

x

) ,

E⁻

(δ)

(

x

) ≈ χ(

y₃

) X

j>0

δ

^jW_j

(

y_β

,

^y³

δ )

E⁺_j

∈ H(curl, Ω

₊

)

, W_j

∈ H(curl, Σ × R

₊

)

profiles

(

^y_β

,

^y3

)

: “normal coordinates” to

Σ

in a tubular neighborhood

U

−of

Σ

ⁱⁿ

Ω

₋

Validation of the asymptotic expansion

Remainders

Aim: proving estimates for remainders R_m_;_δ

R_m_;_δ

:=

^E_(δ)

−

m

X

j=0

δ

^jE_j in

Ω

Evaluation of the right hand side when the Maxwell operator is applied to R_m_;_δ:



 



 



curl curl

R⁺_m_;_δ

− κ

²

α

₊R⁺_m_;_δ

=

0 in

Ω

₊

curl curl

R⁻_m_;_δ

− κ

²

α

₋R⁻_m_;_δ

=

j⁻_m_;_δ in

Ω

₋

R_m_;_δ

×

ⁿ

Σ

=

0 on

Σ

curl

R_m_;_δ

×

ⁿ

Σ

=

g_m_;_δ on

Σ

curl

R⁺_m_;δ

×

ⁿ

=

0 on

∂Ω

with

α

+

=

^{1 and}

α

₋

=

¹

+

ⁱ

/δ

²

(20)

Validation of the asymptotic expansion

Estimates for Remainders

k

^j⁻m;δ

k

²,Ω₋

+ k

^gm;δ

k

¹₂,Σ

+ k curl

_Σg_m_;_δ

k

³₂,Σ

6

C_m

δ

^m⁻¹

with C_m

>

0 independent of

δ

. Theorem

Under Hypothesis in the framework above, for all m

∈ N

_and

δ ∈ (

0

, δ

0

]

, the remainders R_m_;_δsatisfy the optimal estimates

k

^R⁺m;δ

k

⁰,Ω+

+ k curl

R⁺_m_;δ

k

⁰,Ω+

+ δ

⁻¹²

k

^R⁻m;δ

k

⁰,Ω₋

+ δ

¹²

k curl

^R⁻_m_;_δ

k

⁰,Ω₋

6

C_m⁰

δ

^m⁺¹

(21)

Electrical Profiles

W_j

=: ( W

α,j

;

w_j

)

in coordinates

(

y_β

,

Y₃

)

with Y₃

=

^y_δ³

W₀

=

0

W

α,1

= −

^jα,0

(

y_β

) e

^−λ^Y³ and w₁

=

0

W

α,2

= h λ

⁻¹

b_α^σj_σ,₀

− H

^jα,0

−

^jα,1

+

Y₃

b^σ_αj_σ,₀

− H

^jα,0

i (

y_β

) e

^−λ^Y³

w₂

= − λ

⁻¹D_αj₀^α

(

y_β

) e

^−λ^Y³

with

H

mean curvature of

Σ

j_α,_k

(

y_β

) := λ

⁻¹

(curl

E⁺_k

×

ⁿ

)

α

(

y_β

,

0

)

and

λ = κ e

⁻ⁱ^π/⁴

(22)

Curl Operator and Levi-Civita Tensor

Ω

₋

h

Σ Σ

h

U

− n

Levi-Civita Tensor

:

^ijk

=

0

(

ⁱ

,

^j

,

^k

) .q

det a_αβ

(

^h

)

a_αβ

(

h

)

: metric on

Σ

hand

0

(

i

,

j

,

k

) =

sign of

(

i

,

j

,

k

)

Curl Operator in Normal Parameterization:

( ( ∇ ×

^E

)

^α

=

³^βα

(∂

₃^hE_β

− ∂

_βE₃

)

on

Σ

h

( ∇ ×

^E

)

³

=

³^αβD^h_αE_β on

Σ

h

D_α^h :Covariant Derivativeon

Σ

h

(23)

Magnetical Profiles

V_j

=: ( V

j^α

;

v_j

)

in

(

y_β

,

Y₃

)

coordinates

V₀

(

y_β

,

Y₃

) =

H₀

(

y_β

) e

^−λ^Y³

,

V

1^α

(

y_β

,

Y₃

) = h

H^α₁

(

y_β

) +

Y₃

H

^H^α0

+

b_σ^αH^σ₀

(

y_β

) i

e

^−λ^Y³

,

v₁

(

y_β

,

Y₃

) = λ

⁻¹D_αH^α₀

(

y_β

) e

⁻^λ^Y³

with H^α_j

(

y_β

) := (

n

×

^H⁺j

) ×

ⁿ^α

(

y_β

,

0

)

Comparison with

H. HADDAR, P. JOLY, H.N. NGUYEN(2008)

(24)

Skin Effect and Skin Depth

1D model ofSkin Effect :

{

^z

>

⁰

} =

half-space conductor body

k

^Eσ

(

z

) k =

E₀

e

⁻

z

`(σ)

, `(σ) :=

r

2

ωµ

0

σ

^{Skin Depth}

`(σ) = δ/

Re

λ .

3D model of Skin Effect, using profiles V₀, V₁, V

(

y_α

,

y₃

) :=

V₀

(

y_α

,

^y³

δ ) + δ

V₁

(

y_α

,

^y³

δ )

Definition

Suppose that for all y_α

, k

^V

(

y_α

,

0

) k 6 =

0. The Skin Depth in a conductor body at conductivity

σ

and y_αon a Regular Surface

Σ

is the positive real

L (σ,

y_α

)

defined on

Σ

taking the smallest value s.t.

k

^{V y}α

, L (σ,

y_α

) k =

¹

e k

^V

(

y_α

,

0

) k

(25)

Asymptotic Behavior for Skin Depth

Theorem

Let

Σ

be a Regular Surface with mean curvature

H

. Suppose that H₀

(

y_α

) 6 =

0. The Skin Depth

L (σ,

y_α

)

has the asymptotic expansion

L (σ,

y_α

) = `(σ)

1

+ H (

y_α

) `(σ) + O (σ

⁻¹

)

MONIQUEDAUGE, ERWANFAOU, V. P ´ERON(2009) Key of the proof:

k

^V

(

y_α

,

y₃

) k

²

= h

k

^H⁰

k

²

+

2

δ

Re

h

^H⁰

,

H₁

i +

2y₃

H k

^H⁰

k

²

+ O (δ,

y₃

) i

e

⁻^2y³^/`(σ)

(26)

Outline

1 Uniform Estimates for Transmission Problems

2 3D Multiscaled Asymptotic Expansion

3 Numerical Simulations

(27)

Numerical Simulations

FEM Discretization

Magnetic Field

Framework:

Ω

₋and

Ω

Axisymmetric Domains

F Axisymmetric & Orthoradial

→

^HσAxisymmetric & Orthoradial 2D Problem

→

^accuracy

Variational Form : Unknown H_σ

=

H_σ,θ

(

r

,

z

) ∈

^V

O r

z

Ω

^m₋

Ω

^m₊

Finite Element Library Melina

(28)

Numerical simulations

Mesh Refinement

Ω

₋&

Ω

coaxial ellipsoidal domains

→ Ω

^m₋&

Ω

^mcoaxial half-ellipsis

(29)

Numerical Simulations

Skin Effect in ellipsoidal geometry

|

^{Im H}σ

|

^for

σ =

5S

.

m⁻¹

|

^{Im H}σ

|

^for

σ =

80S

.

m⁻¹

(30)

Asymptotic Behavior for Skin Depth in Ellipsoidal Geometry

|

^{Im H}σ

|

^for

σ =

5S

.

m⁻¹

Conductor

= Ω

^m₋(

H >

0) Conductor

= Ω

^m₊(

H <

0)

(31)

Asymptotic Behavior for Skin Depth in Ellipsoidal Geometry

|

^{Im H}σ

|

^for

σ =

5S

.

m⁻¹

Conductor

= Ω

^m₋ Conductor

= Ω

^m₊

(32)

Numerical Post-Treatment

Linear Regression

Extract

|

^H^σ

(

^h

) |

along edges of the mesh of

Ω

^m₋^{when z}

=

^0:^h

=

²

−

^r^.

Linear Regression in skin

(σ) →

ⁿ

(σ)

points

→

”numerical” slope s

(σ)

log₁₀

|

^Hσ

(

h

) | =

s

(σ)

h

+

b

0 0.5 1 1.5 2

−6

−5

−4

−3

−2

−1 0 1 2

2 − r

0 0.5 1 1.5 2

−14

−12

−10

−8

−6

−4

−2 0 2

2 − r

◦ :

log₁₀

|

^Hσ

(

2

−

^r

) |

^when

σ =

5

◦ :

log₁₀

|

^Hσ

(

2

−

^r

) |

^when

σ =

80 n

(σ) =

7 , s

(σ) = −

³

,

65542 n

(σ) =

3 , s

(σ) = −

¹⁶

,

279162

(33)

Post-Treatment

Accurary of Asymptotics

Troncated Asymptotic Expansion :

H

⁻_θ,₁

:= H

⁻_θ,0

+ δ H

⁻_θ,1

log₁₀

| H

⁻_θ,₁

(.,

2

−

^r

) | =

c

+ β(σ, Σ)(

2

−

^r

) + O (

^√¹_σ

;

2

−

^r

)

”Theoretical” slope

β(σ, Σ) :=

_{ln 10}¹

H −

_`(σ)¹

Relative Error between slopes: error

(σ) :=

^β(σ,Σ)−

s(σ) β(σ,Σ)

σ

5 20 80

skin

(σ)

(cm) 10

.

3 5

.

15 2

.

58

n

(σ)

7 5 3

s

(σ) −

³

,

65542

−

⁷

,

883903

−

¹⁶

,

279162

β(σ, Σ) −

³

,

673319

−

⁷

,

889506

−

¹⁶

,

32188 error

(σ)

(

%

) 0

,

48 0

,

07 0

,

26