A Transmission Problem in Electromagnetism with a Singular Interface

(1)

HAL Id: inria-00528523

https://hal.inria.fr/inria-00528523

Submitted on 22 Oct 2010

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

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A Transmission Problem in Electromagnetism with a Singular Interface

Gabriel Caloz, Monique Dauge, Erwan Faou, Victor Péron, Clair Poignard

To cite this version:

Gabriel Caloz, Monique Dauge, Erwan Faou, Victor Péron, Clair Poignard. A Transmission Problem

in Electromagnetism with a Singular Interface. 6th Singular Days on Asymptotic Methods for PDEs,

Apr 2010, Berlin, Germany. �inria-00528523�

(2)

A Transmission Problem in Electromagnetism with a Singular Interface

GABRIELCALOZ² MONIQUEDAUGE² ERWANFAOU² V. P ´ERON¹

CLAIRPOIGNARD¹

1Projet MC2, INRIA Bordeaux Sud-Ouest 2IRMAR, Universit ´e de Rennes 1

6th Singular Days on Asymptotic Methods for PDEs WIAS Berlin, April 29 – May 1, 2010

V. P ´eron A Transmission Problem in Electromagnetism with a Singular Interface 1 / 33

(3)

The Skin Effect : A Model Problem

σ

+

= ⁰ σ ¹ Ω

₋

Ω

₊

Σ

Ω

₋Highly Conductingbody

⊂⊂ Ω

:Conductivity

σ

₋

≡ σ

¹

Σ = ∂Ω

₋: Interface

Ω

₊Insulating or Dielectricbody:Conductivity

σ

₊

=

0

TheSkin Effect: rapid decay of electromagnetic fields with depth inside the conductor.

TheSkin Depth:

`(σ) = p

2

/ωµ

0

σ

(4)

References

V. P ´ERON(PhD 09)

G. CALOZ, M. DAUGE, V. P ´ERON(JMAA 10)

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

M. DAUGE, E. FAOU, V. P ´ERON(CRAS 10)

Asymptotic Behavior for High Conductivity of the Skin Depth Electromagnetism

Aim : Understanding the influence of the geometry of a conducting body on the skin effect in electromagnetism.

(5)

Outline

1 Framework

2 3D Multiscaled Asymptotic Expansion

3 Axisymmetric Problems

4 Finite Element Computations

5 Numerical simulations of skin effect

6 Postprocessing

(6)

Outline

1 Framework

2 3D Multiscaled Asymptotic Expansion

3 Axisymmetric Problems

4 Finite Element Computations

5 Numerical simulations of skin effect

6 Postprocessing

(7)

Framework

Maxwell Problem

(

P_σ

) curl

E

−

ⁱ

ωµ

0H

=

0 and

curl

H

+ (

i

ωε

0

− σ)

E

=

j

σ = (

0

, σ

¹

)

j

∈ H

0

(div, Ω) = {

^u

∈ L

²

(Ω) | div

u

∈ L

²

(Ω),

u

·

ⁿ

=

0 on

∂Ω }

Ω

₋

Ω

+

Σ

Ω

Perfectly Conducting Magnetic Wall B. C.:

E

·

ⁿ

=

0 and H

×

ⁿ

=

0 on

∂Ω

(8)

Existence of solutions

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 on

Σ

B

.

C

.

on

∂Ω

Theorem (CALOZ, DAUGE, P., 09)

If the surface

Σ

is Lipschitz, under Hypothesis

(

^SH

)

, there exist

σ

0and C

>

0, such that for all

σ > σ

0,

(

P_σ

)

with B.C. and j

∈ H(div, Ω)

has a unique solution

(

E

,

H

)

in

L

²

(Ω)

², and

k

^E

k

⁰,Ω

+ k

^H

k

⁰,Ω

+ √

σ k

^E

k

⁰,Ω₋

6

C

k

^j

k

H(div,Ω)

Application: Convergence of asymptotic expansion for large conductivity

(9)

Outline

1 Framework

2 3D Multiscaled Asymptotic Expansion

6 Postprocessing

(10)

References and Notations

Asymptotic Expansion as

σ → ∞

of solutions of

(

^Pσ

)

^when

Σ

is smooth : E.P. STEPHAN, R.C. MCCAMY(83-84-85)

Plane Interface and Eddy Current Problems

H. HADDAR, P. JOLY, H.N. NGUYEN(08) Generalized Impedance Boundary Conditions

G. CALOZ, M. DAUGE, V. P ÉRON(JMAA 10) M. DAUGE, E. FAOU, V. P ÉRON(CRAS 10) P ÉRON(09)

Hypothesis

1

Σ

is a Smooth Surface

2

ω

satisfies the Spectral Hypothesis (SH)

3 j is smooth and j

=

0 in

Ω

₋

(11)

Asymptotic Expansion

δ := p

ωε

0

/σ −→

⁰ ^as

σ → ∞

By Theorem there exists

δ

0s.t. for all

δ 6 δ

0, the solution H_(δ)to

(

P_σ

)

:

H⁺_(δ)

(

x

) ≈

^H⁺0

(

x

) + δ

H⁺₁

(

x

) + O (δ

²

)

H⁻

(δ)

(

x

) ≈

^H⁻0

(

x

; δ) + δ

H⁻₁

(

x

; δ) + O (δ

²

)

with H⁻_j

(

x

; δ) = χ(

y₃

)

V_j

(

y_β

,

^y³

δ ) .

(

^y_β

,

^y3

)

: “normal coordinates” to

Σ

in a tubular region

U

−of

Σ

ⁱⁿ

Ω

₋

H⁺_j

∈ H(curl, Ω

₊

)

and V_j

∈ H(curl, Σ × R

₊

)

profiles

. k

^H⁻j

(

^x

; δ) k

⁰,Ω₋

6

C_j

√

δ

^{for all} ^j

∈ N

(12)

Profiles of the Magnetic Field

V_j

=: ( V

j^α

;

v_j

)

in coordinates

(

y_β

,

Y₃

)

withY₃

=

^y_δ³

V₀

(

y_β

,

Y₃

) =

h₀

(

y_β

) e

^−λ^Y³

V

1^α

(

^y_β

,

^Y3

) = h

h^α₁

+

^Y3

H

^h^α0

+

^b^α_σ ^h^σ₀

i

(

^y_β

) e

^−λ^Y³

Here,

H

mean curvature of

Σ

h₀

(

y_β

) = (

n

×

^H⁺0

) ×

ⁿ

(

y_β

,

0

)

and h^α_j

(

y_β

) := (

H⁺_j

)

^α

(

y_β

,

0

) λ = ω √ ε

0

µ

0

e

⁻ⁱ^π/⁴

(13)

Application

A New Definition of the Skin Depth

V_(δ)

(

^yα

,

^y3

) :=

^H⁻_(δ)

(

^x

),

^yα

∈ Σ,

⁰

≤

^y³

<

^h0

Definition

Let

Σ

be a smooth surface, and j s.t. V_(δ)

(

y_α

,

0

) 6 =

0. The skin depth is the smallest length

L (σ,

y_α

)

defined on

Σ

s.t.

k

^V(δ) y_α

, L (σ,

y_α

)

k = k

^V(δ)

(

y_α

,

0

) k e

⁻¹

Theorem (DAUGE, FAOU, P., 10)

Let

Σ

be a regular surface with mean curvature

H

, and assume h₀

(

y_α

) 6 =

0.

L (σ,

y_α

) = `(σ)

1

+ H (

y_α

) `(σ) + O (σ

⁻¹

)

, σ → ∞

Key of the proof:

k

^V(δ)

k

²

= h

k

^h⁰

k

²

+

2y₃

H k

^h⁰

k

²

+

2

δ

Re

h

^h⁰

,

h₁

i + O (δ+

y₃

)

²

i

e

⁻^2y³^/`(σ)

(14)

Outline

1 Framework

3 Axisymmetric Problems

6 Postprocessing

(15)

Axisymmetric domains

The meridian domain

O r

z

Ω

^m₋

Ω

^m₊

Γ

^m

Σ

^m

Figure:The meridian domainΩ^m= Ω^m₋∪Ω^m₊∪Σ^m

(16)

Reduction problem



 

 

(curl

H

)

r

=

¹_r

∂

_θH_z

− ∂

zH_θ

, (curl

H

)

_θ

= ∂

zH_r

− ∂

rH_z

, (curl

H

)

z

=

¹_r

∂

r

(

rH_θ

) − ∂

_θH_r

.

The Maxwell problem is axisymmetric : the coefficients do not depend on the angular variable

θ

.

(17)

Reduction problem



 

 

(curl

H

)

r

=

¹_r

∂

_θH_z

− ∂

zH_θ

, (curl

H

)

_θ

= ∂

zH_r

− ∂

rH_z

, (curl

H

)

z

=

¹_r

∂

r

(

rH_θ

) − ∂

_θH_r

.

The Maxwell problem is axisymmetric : the coefficients do not depend on the angular variable

θ

.

H is axisymmetric iff

˘

H

:=

H_r

,

H_θ

,

H_z

does not depend on

θ

.

Assume that the right-hand side is axisymmetric and orthoradial. Then, H_(δ)is axisymmetric and orthoradial :

H

˘

_(δ)

(

r

, θ,

z

) = (

0

,

h_(δ)

(

r

,

z

),

0

).

(18)

Configurations chosen for computations

Configuration A

O r₀ r₁ r

z

Ω

^m₋

Ω

^m₊

Σ

^m

a

Figure:The meridian domainΩ^min configuration A

(19)

Configurations chosen for computations

Configuration B

O c d

r z

a b

Ω

^m₋

Ω

^m₊

Σ

^m

Figure:The meridian domainΩ^min configuration B

(20)

Outline

1 Framework

4 Finite Element Computations

6 Postprocessing

(21)

Finite Element Method

High order elements available in the finite element library M ´elina h^p_(δ)^,^M: the computed solution of the discretized problem with an interpolation degree p and a mesh

M

A^p_σ^,^M

:= k

^h^p_(δ)^,^M

k

L²₁(Ω^m

−) with

σ = ωε

0

δ

⁻²

k

^v

k

²L²₁(Ω^m

−)

= Z

Ω^m

−

|

^v

|

²^rdrdz

.

In the computations, the angular frequency

ω =

3

.

10⁷.

(22)

Interpolation degree

Configuration B

We first check the convergence when the interpolation degree p of the finite elements increases.

We consider the discretized problem with different degrees: Q_p, p

=

1

, ...,

20, and with 2 different meshes:

Figure:The meshesM₃_andM₆

(23)

We represent the absolute value of the difference between the weighted norms A^p_σ^,^M³and A²⁰_σ^,^M⁶, versus p in semilogarithmic coordinates

SCHWAB,SURI(96)

theoretical results of convergence for the p-version of problems with boundary layers

(24)

We plot in log-log coordinates the weighted norm A¹⁶_σ^,^M³with respect to

σ

with red circles.

The figure shows that A¹⁶_σ^,^M³behaves like

σ

⁻¹^/⁴(solid line) when

σ → ∞

^.

This behavior is consistent with the asymptotic expansion.

(25)

Configuration A

We consider a family of meshes with square elements

M

_k, with size h

=

1

/

k

Figure:MeshesM₂_{, and}M₃in configuration A

(26)

Configuration A

We represent the absolute value of the difference between A^p_σ^,^M²and A¹⁶_σ^,^M³, versus p in semilogarithmic coordinates

The figure shows that A^p_σ^,^M²approximates A¹⁶_σ^,^M³better than 10⁻⁴when p

>

12.

(27)

Outline

1 Framework

5 Numerical simulations of skin effect

6 Postprocessing

(28)

Skin effect in configuration B

|

^{Im h}⁺_(δ)

| = O (δ) .

Thus, the imaginary part of the computed field is located in the conductor.

Figure:Configuration B.|^ImHσ|^whenσ=5 andσ=80

(29)

Skin effect in configuration A

Figure:Configuration A.|^ImHσ|^whenσ=^{5 and}σ=⁸⁰

(30)

Influence of the Geometry on the Skin effect

Configuration B

H >

0 on the left, and

H <

0 on the right

Figure:|^ImHσ|^,σ=⁵

The skin depth is larger when the mean curvature of the conducting body surface is larger.

(31)

Outline

1 Framework

6 Postprocessing

(32)

Configuration B

We perform numerical treatments from computations in configuration B.

We extract values of log₁₀

| H

_σ

|

ⁱⁿ

Ω

^m₋along the axis z

=

0 : y₃

=

2

−

^{r .}

Figure:On the leftσ=20. On the right,σ=80.

The curves exactly behave like lines: the exponential decay is obvious.

(33)

Configuration A

We extract values of log₁₀

| H

_σ

|

ⁱⁿ

Ω

^m₋along the diagonal axis r

=

z

Here,

ρ = p

(

1

−

^r

)

²

+ (

1

−

^z

)

²is the distance to the corner point a

(

r

=

1

,

z

=

1

)

.

(34)

The exponential decay is not obvious in configuration A.

To measure a possible exponential decay, we define the slopes

˜

s_i

(σ) :=

^log¹⁰

| H

_σ

(

r_i

,

z_i

) | −

^log10

| H

_σ

(

z_i₊₁

,

r_i₊₁

) | ρ

i+1

− ρ

i

.

Here,

ρ

i

:= p

(

1

−

^rⁱ

)

²

+ (

1

−

^zⁱ

)

²is the distance from the extraction points

(

r_i

,

z_i

)

to a .

Figure:The graphs of the slopes˜^si(σ)

(35)

Asymptotics in the conducting part

In configuration A, the slopes tend to 0, which means that there is no exponential convergence near the corner.

Nevertheless, a sort of exponential convergence is restored in a region further away from the corner.

The principal asymptotic contribution inside the conductor is a profile globally defined on a sector

S

(of opening ^π

2) solving the model Dirichlet problem

( −

ⁱ

(∂

_X²

+ ∂

_Y²

)

v₀

+ κ

²v₀

=

0 in

S ,

v₀

=

h⁺₀

(

a

)

on

∂ S ,

instead the 1D problem in configuration B

( −

ⁱ

∂

_Y²v₀

+ κ

²v₀

=

0 for 0

<

Y

< + ∞ ,

v₀

=

h⁺₀ for Y

=

0

.

A Transmission Problem in Electromagnetism with a Singular Interface

HAL Id: inria-00528523

https://hal.inria.fr/inria-00528523

Submitted on 22 Oct 2010

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.