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On optimal FEM and impedance conditions for thin electromagnetic - - PowerPoint PPT Presentation
On optimal FEM and impedance conditions for thin electromagnetic - - PowerPoint PPT Presentation
On optimal FEM and impedance conditions for thin electromagnetic shielding sheets Kersten Schmidt Research Center Matheon, Berlin, Germany, Institut f ur Mathematik, Technische Universit at Berlin, Germany Institut f ur Mathematik, BTU
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Thin conducting shielding sheets
Maxwell equations in eddy current approximation curl curl E +iµσω E = −iωµ0 J Remedies ⊲ thin sheet basis ⊲ approximate transmission conditions ⊲ boundary integral formulation
K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 2 / 29
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Thin conducting shielding sheets
Maxwell equations in eddy current approximation curl curl E = J [E × n]Γ = 0 [curl E × n]Γ = Z(ω, σ, d) ET Remedies ⊲ thin sheet basis ⊲ approximate transmission conditions ⊲ boundary integral formulation
K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 2 / 29
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Thin conducting shielding sheets
Maxwell equations in eddy current approximation curl curl E = J [E × n]Γ = 0 [curl E × n]Γ = Z(ω, σ, d) ET Remedies ⊲ thin sheet basis ⊲ approximate transmission conditions ⊲ boundary integral formulation
K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 2 / 29
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Thin conducting shielding sheets
b a f = −iωµ0j0 ε Ωε
ext
Ωε
int
Eddy current model curl curl E +iµσω E = −iωµ0 J
K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 3 / 29
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Thin conducting shielding sheets
b a f = −iωµ0j0 ε Ωε
ext
Ωε
int
Eddy current model curl curl E +iµσω E = −iωµ0 J Two important effects of the thin sheet (of thickness ε) ⊲ Shielding effect – in conductors induced currents diminish electromagnetic fields (behind the conductor) ⊲ Skin effect – major current flow in a boundary layer (skins of the conductor)
◮ Skin depth in solid body δ =
- 2
µ0σω
◮ Copper at 50 Hz → δ ≈ 8mm
K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 3 / 29
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Thin conducting shielding sheets
ε ∼ δ ε ≪ δ ε ≫ δ Eddy current model curl curl E +iµσω E = −iωµ0 J Two important effects of the thin sheet (of thickness ε) ⊲ Shielding effect – in conductors induced currents diminish electromagnetic fields (behind the conductor) ⊲ Skin effect – major current flow in a boundary layer (skins of the conductor)
◮ Skin depth in solid body δ =
- 2
µ0σω
◮ Copper at 50 Hz → δ ≈ 8mm
K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 3 / 29
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Outline
1 Optimal basis inside the sheet 2 Impedance transmission conditions (ITCs) 3 Boundary integral equations for impedance transmission conditions
K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 4 / 29
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Optimal basis inside the sheet
Eddy current model in 2D (TM polarisation) −∆uε(x) + iωµ0σ(x) uε(x) = −iωµ0j0(x) Approximation of higher order without reduction to an interface Ansatz for the solution inside the sheet uε
int(t, s) ≈ uε int,N(t, s) =
N−1
i=0 φε i (s, t) uε int,i(t).
inspired by: Vogelius, M. and Babuˇ ska, I., Math. Comp. 37, 1981.
with N ≥ 2 linear independent basis functions φε
i
spanning V ε
N, and uε int,i ∈ H1(
Γ).
ε n s t ε
ext
ε
ext
ε
int
∂
Basis functions φε
0, φε 1 in the kernel of −∂2 s − κ 1+sκ ∂s + iωµ0σ + κ2 4(1+sκ)2 ,
φε
0(s, κ) =
1 √1 + sκ cosh(√iωµ0σs) cosh(√iωµ0σ ε
2 ),
{φε
,0}κ = 1, [φε ,0]κ = 0,
φε
1(s, κ) =
1 √1 + sκ sinh(√iωµ0σs) 2 sinh(√iωµ0σ ε
2 ),
{φε
,1}κ = 0, [φε ,1]κ = 1, K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 5 / 29
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Optimal basis inside the sheet
Eddy current model in 2D (TM polarisation) −∆uε(x) + iωµ0σ(x) uε(x) = −iωµ0j0(x) Approximation of higher order without reduction to an interface Ansatz for the solution inside the sheet uε
int(t, s) ≈ uε int,N(t, s) =
N−1
i=0 φε i (s, t) uε int,i(t).
inspired by: Vogelius, M. and Babuˇ ska, I., Math. Comp. 37, 1981.
with N ≥ 2 linear independent basis functions φε
i
spanning V ε
N, and uε int,i ∈ H1(
Γ).
ε n s t ε
ext
ε
ext
ε
int
∂
Basis functions φε
2j, φε 2j+1,j ∈ N0 in the kernel of (−∂2 s − κ 1+sκ ∂s + iωµ0σ + κ2 4(1+sκ)2 )j+1,
φε
2j(s, κ) =
Pj(s) √1 + sκ cosh(
- iωµ0σs),
{φε
,2j}κ = δj,0, [φε ,2j]κ = 0
φε
2j+1(s, κ) =
Pj(s) √1 + sκ sinh(
- iωµ0σs),
{φε
,2j+1}κ = 0,
[φε
,2j+1]κ = δj,0 K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 5 / 29
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Optimal basis inside the sheet
Basis functions φε
i , i ∈ N0 such that (−∂2 s − κ 1+sκ∂s + iωµ0σ + κ2 4(1+sκ)2 )φε i = ε−2φε i−2
φε
2j(s, κ) =
Pj(s) √1 + sκ cosh(
- iωµ0σs),
{φε
,2j}κ = δj,0, [φε ,2j]κ = 0
φε
2j+1(s, κ) =
Pj(s) √1 + sκ sinh(
- iωµ0σs),
{φε
,2j+1}κ = 0,
[φε
,2j+1]κ = δj,0
s φε
int,0
φε
int,1
κ = +8 κ = −8
- ε
2 ε 2
- 0.5
0.5 1 s φε
int,2
φε
int,3
- ε
2 ε 2
- 0.5
0.5 1 s φε
int,4
φε
int,5
- ε
2 ε 2
- 0.5
0.5 1
iωµ0σ = 1000, ε = 0.1
K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 6 / 29
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Optimal basis inside the sheet
Basis functions φε
i , i ∈ N0 such that (−∂2 s − κ 1+sκ∂s + iωµ0σ + κ2 4(1+sκ)2 )φε i = ε−2φε i−2
φε
2j(s, κ) =
Pj(s) √1 + sκ cosh(
- iωµ0σs),
{φε
,2j}κ = δj,0, [φε ,2j]κ = 0
φε
2j+1(s, κ) =
Pj(s) √1 + sκ sinh(
- iωµ0σs),
{φε
,2j+1}κ = 0,
[φε
,2j+1]κ = δj,0
Decomposition −∆ + iωµ0σ =
- − ∂2
s −
κ 1 + sκ∂s + iωµ0σ + κ2 4(1 + sκ)2
- scales with ε, depends on σ
+ A(s, κ) with regular pertubation, independent of σ A(s, κ) = − 1 1 + sκ∂t
- 1
1 + sκ∂t
- −
κ2 4(1 + sκ)2
K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 6 / 29
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Optimal basis inside the sheet
Basis functions φε
i , i ∈ N0 such that (−∂2 s − κ 1+sκ∂s + iωµ0σ + κ2 4(1+sκ)2 )φε i = ε−2φε i−2
φε
2j(s, κ) =
Pj(s) √1 + sκ cosh(
- iωµ0σs),
{φε
,2j}κ = δj,0, [φε ,2j]κ = 0
φε
2j+1(s, κ) =
Pj(s) √1 + sκ sinh(
- iωµ0σs),
{φε
,2j+1}κ = 0,
[φε
,2j+1]κ = δj,0
Decomposition −∆ + iωµ0σ =
- − ∂2
s −
κ 1 + sκ∂s + iωµ0σ + κ2 4(1 + sκ)2
- scales with ε, depends on σ
+ A(s, κ) with regular pertubation, independent of σ A(s, κ) = − 1 1 + sκ∂t
- 1
1 + sκ∂t
- −
κ2 4(1 + sκ)2 Interpolation Iε
Nuε for uε smooth enough
Iε
Nuε(s, t) = ⌊ N
2 ⌋
- j=0
ε2jφε
2j(s, κ)AN,j(s, κ){uε}κ + ⌊ N−1
2
⌋
- j=0
ε2jφε
2j+1(s, κ)AN,j(s, κ)[uε]κ K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 6 / 29
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Optimal basis inside the sheet
Lemma (Best-approximation error)
For any even N (curved sheet) any N (straight sheet or curved sheet N ≤ 4) and uε smooth enough there exists a constant C independent of σ such that inf
wε
N∈V ε N ⊗H1(
Γ)
|w ε
N − uε|H1(Ωε
int) ≤ CεN− 1 2 ,
inf
wε
N∈V ε N ⊗H1(
Γ)
w ε
N − uεL2(Ωε
int) ≤ CεN+ 1 2 .
Interpolation Iε
Nuε for uε smooth enough
Iε
Nuε(s, t) =
⌊ N
2 ⌋
j=0 ε2jφε 2j(s)AN,j(s, κ){uε}κ +
⌊ N−1
2
⌋ j=0
ε2jφε
2j+1(s)AN,j(s, κ)[uε]κ K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 7 / 29
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Optimal basis inside the sheet
Eddy current model in 2D (TM polarisation) −∆uε(x) + iωµ0σ(x) uε(x) = −iωµ0j0(x) Semi-discretization W ε
N :=
- u ∈ H1(Ω) : u|Ωε
ext ∈ H1(Ωε
ext), u|Ωε
int ∈ V ε
N ⊗ H1(
Γ)
- Seek uε
N ∈ W ε N such that
- Ω
∇uε
N · ∇vN dx +
- Ωε
int
iωµ0σuε
NvN dx = −
- Ω
iωµ0j0vN dx ∀vN ∈ W ε
N
Lemma (Semi-discretization error)
For any even N (curved sheet) any N (straight sheet or curved sheet N ≤ 4) and uε smooth enough it holds for uε
N ∈ W ε N
|uε
N − uε|H1(Ωε
int) ≤ CεN− 1 2 ,
uε
N − uεL2(Ωε
int) ≤ CεN+ 1 2 ,
uε
N − uεH1(Ωε
ext) ≤ Cε2N−1.
K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 8 / 29
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Optimal basis inside the sheet
Semianalytical study for circular arc with κ = 1
2, iωµ0σ = 1 ε.
|uε
N − uε|H1(Ωε
int) ≤ CεN− 1 2 ,
uε
N − uεH1(Ωε
ext) ≤ Cε2N−1.
⊲ e.g., four functions ⇒ sixth-order scheme O(ε7) (outside the sheet) ⊲ easily increasing order by enrichment with higher optimal basis functions ⊲ pre-computation of integrals in s ⇒ surface variables Error in the H1-seminorm inside the sheet. Error in H1-seminorm outside the sheet.
K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 9 / 29
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Content
1 Optimal basis inside the sheet 2 Impedance transmission conditions (ITCs) 3 Boundary integral equations for impedance transmission conditions
K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 10 / 29
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Impedance transmission conditions (ITCs)
− ε 2 ε 2
Ωε
ext
Ωε
ext
Ωε
int Γ x y
Original problem curl curl E = −iωµ0 J in Ωε
ext
curl curl E +iωµ0σ E = 0 in Ωε
int
(1)
− ε 2 ε 2
Ω0
ext
Ω0
ext
Γ x y
Reduced problem with ITC-1-0 (Levi-Civita’1902) curl curl E0 = −iωµ0 J in Ω0
ext
[E0 × n] = 0
- n Γ
[curl E0 × n] − iωµ0σε{E0,T} = 0
- n Γ
(2) ⊲ E0 defined Ω0
ext approximates E in Ωε ext
⊲ layer correction inside Ωε
int can be computed a-posteriori
⊲ limit for ε → 0 for σ = σ(ε) ∼ ε−1
K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 11 / 29
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Impedance transmission conditions (ITCs)
− ε 2 ε 2
Ωε
ext
Ωε
ext
Ωε
int Γ x y
Original problem (TM mode) −∆u = f in Ωε
ext
−∆u + α δ2 u = 0 in Ωε
int
(1) ⊲ Skin depth δ serves as a parameter
− ε 2 ε 2
Ω0
ext
Ω0
ext
Γ x y
Reduced problem with ITC-1-0 (Levi-Civita’1902) −∆u0 = f in Ω0
ext
[u0] = 0
- n Γ
[∂nu0] − αε δ2 {u0} = 0
- n Γ
(2) ⊲ u0 defined in Ω0
ext approximates u in Ωε ext
⊲ layer correction inside Ωε
int can be computed a-posteriori
⊲ limit for ε → 0 for δ(ε) ∼ √ε
K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 11 / 29
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Impedance transmission conditions (ITCs)
− ε 2 ε 2
Ωε
ext
Ωε
ext
Ωε
int Γ x y
Asymptotic problem (TM mode) −∆uε = f in Ωε
ext
−∆uε + α δ2(ε)uε = 0 in Ωε
int
(1) ⊲ Skin depth δ serves as a parameter
− ε 2 ε 2
Ω0
ext
Ω0
ext
Γ x y
Reduced problem with ITC-1-0 (Levi-Civita’1902) −∆u0 = f in Ω0
ext
[u0] = 0
- n Γ
[∂nu0] − αε δ2 {u0} = 0
- n Γ
(2) ⊲ u0 defined in Ω0
ext approximates u in Ωε ext
⊲ layer correction inside Ωε
int can be computed a-posteriori
⊲ family ITC-1-N of transmission conditions derived by asymptotic expansion with δ → δ(ε) ∼ √ε, and
K.S. and S. Tordeux, ESAIM: M2AN, 45(6): 1115–1140, 2011.
K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 11 / 29
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Impedance transmission conditions (ITCs)
Reduced problem with transmission conditions ITC-1-0 (Levi-Cevita’1902) −∆u0 = f in Ω0
ext
[u0] = 0
- n Γ
[∂nu0] − αε δ2 {u0} = 0
- n Γ
⊲ O(ε) : error in exterior decreases linearly with ε along δ(ε) ∼ √ε (proven) ⊲ surprise : even if ε ≫ δ → 0 ⊲ extra accuracy for ε ≪ δ → ∞ u − u0H1(Ωε
ext)
0.1 mm 1 mm 10 mm 0.1 mm 1 mm 10 mm Sheet thickness ε Skin depth δ
K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 12 / 29
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Impedance transmission conditions (ITCs)
Reduced problem with transmission conditions ITC-1-1 −∆uε,1 = f in Ω0
ext
[uε,1] = 0
- n Γ
[∂nuε,1] − αε δ2
- 1 − αε2
6δ2
- {uε,1}
= 0
- n Γ
⊲ O(ε2) : error in exterior decreases like ε2 along δ(ε) ∼ √ε (proven) ⊲ but only O(ε) in case of ε ≫ δ → 0, no improvement when increasing order N from 0 to 1 ITC-1-0 ITC-1-1
0.1 mm 1 mm 10 mm 0.1 mm 1 mm 10 mm Sheet thickness ε Skin depth δ 0.1 mm 1 mm 10 mm 0.1 mm 1 mm 10 mm Sheet thickness ε Skin depth δ
K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 13 / 29
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Impedance transmission conditions (ITCs)
Reduced problem with transmission conditions ITC-1-2 −∆uε,2 = f in Ω0
ext
[uε,2] + αε3
12δ2 {∂nuε,2} = 0
- n Γ
[∂nuε,2] − αε δ2
- 1 − αε2
6δ2 + ε2 12
7α2ε2
20δ4 + ∂2 Γ
- {uε,2} = 0
- n Γ
⊲ O(ε3) : error in exterior decreases like ε3 along δ(ε) ∼ √ε (proven) ⊲ but convergence to wrong solution in case of ε ≫ δ → 0, not robust in δ anymore, worse than for low orders N = 0, 1 ITC-1-0 ITC-1-1 ITC-1-2
K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 14 / 29
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Impedance transmission conditions (ITCs)
Reduced problem with transmission conditions ITC-1-2 −∆uε,2 = f in Ω0
ext
[uε,2] + αε3
12δ2 {∂nuε,2} = 0
- n Γ
[∂nuε,2] − αε δ2
- 1 − αε2
6δ2 + ε2 12
7α2ε2
20δ4 + ∂2 Γ
- {uε,2} = 0
- n Γ
⊲ O(ε3) : error in exterior decreases like ε3 along δ(ε) ∼ √ε (proven) ⊲ but convergence to wrong solution in case of ε ≫ δ → 0, not robust in δ anymore, worse than for low orders N = 0, 1 Let ε fixed and ⊲ δ → ∞ : [uε,2] → 0, [∂nuε,2] → 0 on Γ → no shielding ⊲ δ → 0 : {∂nuε,2} → 0, {uε,2} → 0 on Γ → perfect electric b.c. (PEC) ⇒ only valid results for large enough skin depth δ
0.1 mm 1 mm 10 mm 0.1 mm 1 mm 10 mm Sheet thickness ε Skin depth δ
K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 14 / 29
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Impedance transmission conditions (ITCs)
Reduced problem with transmission conditions ITC-2-1 (derived for δ(ε) ∼ ε)
K.S. and A. Chernov, SIAM J. Appl. Math., 73(6): 1980–2003, 2013.
−∆uε,1 = f in Ω0
ext
[uε,1] + ε
- 1 −
2δ √αε tanh( √αε 2δ
- {∂nuε,1} = 0
- n Γ
[∂nuε,1] − 2√α tanh(
√αε 2δ )
δ − 1
2
√αε tanh(
√αε 2δ )
{uε,1} = 0
- n Γ
⊲ O(ε2) : error in exterior decreases like ε2 along δ(ε) ∼ ε (proven) ⊲ we observe (numerically) O(ε2) independent of δ(ε) Let ε fixed and ⊲ δ → ∞ : [uε,1] → 0, [∂nuε,1] → 0 on Γ → no shielding ⊲ δ → 0 : [uε,1] + ε{∂nuε,1} → 0 {uε,1} + ε
4 [∂nuε,1] → 0 on Γ
→ perfect electric b.c. (PEC) at Γε ⇒ robust results w.r.t. skin depth δ / conductivity σ
0.1 mm 1 mm 10 mm 0.1 mm 1 mm 10 mm Sheet thickness ε Skin depth δ
K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 15 / 29
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Impedance transmission conditions (ITCs)
Electromagnetic scattering by thin shielding sheet of thickness ε curl curl Eε −(kε)2 Eε = 0 + Silver-M¨ uller b.c. ⊲ with complex wave-number kε = kext = ω2µext
- ǫext + i σext(ε)
ω
- ,
in Ωε
ext,
kε
int = ω2µint
- ǫint + i σint(ε)
ω
- ,
in Ωε
int.
Reduced problem with transmission conditions ITC-2-1 (derived for σint(ε) ∼ ε−2)
- V. P´
eron, K.S. and M. Durufle, SIAM J. Appl. Math., 76(3): 1031–1052, 2016.
curl curl Eε,1 −k2
ext Eε,1 = 0
in Ω0
ext
Eε,1 × n
- Γ
- Eε,1 × n
- Γ
- = ε
L1 L3 L3 L2
- 1
µext (curl Eε,1)T
- Γ
- (
1 µext curl Eε,1)T
- Γ
+ Silver-M¨ uller b.c. ⊲ with the operators Li = Ai curlΓ curlΓ −BiId and constants Ai, Bi ⊲ decoupling of ITCs if material parameters are the same on both sides of Γ ⇒ L3 = 0
K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 16 / 29
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Impedance transmission conditions (ITCs)
Electromagnetic scattering by thin shielding sheet of thickness ε Variational formulation for reduced problem with transmission conditions ITC-2-1 where V =
- v ∈ H(curl, Ω0
ext), v × n ∈ L2 t (∂Ω)
- ,
W = {v ∈ L2
t (Γ), curlΓ v ∈ L2(Γ)} .
Find (Eε,1, λε, µε) ∈ V × W × W such that for all (E′, λ′, µ′) ∈ V × W × W
- Ω+∪Ω−
1 µext curl Eε,1 · curl E′ − κ2
ext
µext Eε,1 ·E′ dx−
- ∂Ω
iκext µext Eε,1 × n ·E′ × n dS −
- Γ
n ×λε n ×µε
- ·
- [E′
T]
{E′
T}
- dS = r.h.s. ,
- Γ
- [n × Eε,1]
{n × Eε,1}
- ·
- λ′
µ′
- + ε A
- curlΓ λε
curlΓ µε
- ·
- curlΓ λ′
curlΓ µ′
- − ε B
- λε
µε
- ·
- λ′
µ′
- dS = 0 .
⊲ with A = A1 A3 A3 A2
- , B =
B1 B3 B3 B2
- K.Schmidt
RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 17 / 29
SLIDE 29
Impedance transmission conditions (ITCs)
Electromagnetic scattering by thin shielding sheet of thickness ε Impedance transmission conditions ITC-2-1 for spherical sheet ⊲ Discretization with N´ ed´ elec’s elements of the first kind on hexahedral curved elements and its tangential traces
x1 = 0 x2 = 0 x3 = 0
10−6 10−4 10−2 100 102 10−4 10−3 10−2 10−1 100
÷ 2.02 ÷ 4.07 ÷ 1.05 ÷ 2.01
σε2 Relative L2 error
PEC, ε = 0.02 PEC, ε = 0.01 ITC-2-1, ε = 0.02 ITC-2-1, ε = 0.01
⊲ Impedance transmission conditions are robust w.r.t. skin depth δ / conductivity σ
K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 18 / 29
SLIDE 30
Content
1 Optimal basis inside the sheet 2 Impedance transmission conditions (ITCs) 3 Boundary integral equations for impedance transmission conditions
K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 19 / 29
SLIDE 31
Boundary integral equations for impedance transmission conditions
Reduced problem with transmission conditions on a closed Lipschitz curve/surface Γ −∆U = F in Rd\Γ [γ1U] − β {γ0U} = 0
- n Γ
[γ0U] = 0
- n Γ
(3) γ0, γ1 ... Dirichlet, Neumann traces on Γ, β ... impedance parameter BVP is singularly perturbed for large β (homogeneous Dirichlet b.c. in the limit |β| → ∞) Γ supp(F)
Mathematical model for thin conducting sheets in electromagnetics (d = 2)
→ K.S. and S. Tordeux, ESAIM: M2AN, 2011 → K.S. and A. Chernov, SIAM J. Appl. Math., 2013 and references therein.
K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 20 / 29
SLIDE 32
Boundary integral equations for impedance transmission conditions
Reduced problem with transmission conditions on a closed Lipschitz curve/surface Γ −∆U = F in Rd\Γ [γ1U] − β {γ0U} = 0
- n Γ
[γ0U] = 0
- n Γ
(3) γ0, γ1 ... Dirichlet, Neumann traces on Γ, β ... impedance parameter BVP is singularly perturbed for large β (homogeneous Dirichlet b.c. in the limit |β| → ∞)
Mathematical model for thin conducting sheets in electromagnetics (d = 2)
→ K.S. and S. Tordeux, ESAIM: M2AN, 2011 → K.S. and A. Chernov, SIAM J. Appl. Math., 2013 and references therein.
Boundary integral equations and BEM for impedance boundary conditions
→ A. Bendali and L. Vernhet, CRAS, 1995, L. Vernhet, M2AS, 1999, A. Bendali, 2000.
Boundary integral equations and BEM for several kind of transmission conditions
→ K.S. and R. Hiptmair, Discrete Contin. Dyn. Syst. Ser. S, 2015 and references therein. Aim: Numerical analysis of BEM on uniform meshes in dependence of large parameter β,
- r small parameter ε := β−1, and the smoothness of Γ
When is BEM on uniform meshes ε-robust ?
K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 20 / 29
SLIDE 33
Boundary integral equations for impedance transmission conditions
2nd order elliptic BVP with transmission conditions on a closed Lipschitz curve/surface Γ −∆U = F in Rd\Γ (3a) [γ0U] = 0
- n Γ
(3c) Representation formula U = −S [γ1U] + D [γ0U]
=0
+N F with single layer potential S and Newton potential NF (S φ)(x) :=
- Γ
G(x − y)φ(y)dy (N F)(x) :=
- R2 G(x − y)F(y)dy
G(x − y) =
- − 1
2π log(|x − y|),
d = 2,
1 4π|x−y|,
d = 3. Mean of Dirichlet traces gives the single layer operator V := {γ0S ·} : H−1/2+s(Γ) → H
1/2+s(Γ),
Taking mean traces on Γ {γ0U} = −V [γ1U] + γ0NF (5)
K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 21 / 29
SLIDE 34
Boundary integral equations for impedance transmission conditions
2nd order elliptic BVP with transmission conditions on a closed Lipschitz curve/surface Γ −∆U = F in Rd\Γ (3a) [γ0U] = 0
- n Γ
(3c) Representation formula U = −S [γ1U] + D [γ0U]
=0
+N F with single layer potential S and Newton potential NF (S φ)(x) :=
- Γ
G(x − y)φ(y)dy (N F)(x) :=
- R2 G(x − y)F(y)dy
G(x − y) =
- − 1
2π log(|x − y|),
d = 2,
1 4π|x−y|,
d = 3. Mean of Dirichlet traces gives the single layer operator V := {γ0S ·} : H−1/2+s(Γ) → H
1/2+s(Γ),
Taking mean traces on Γ {γ0U} = −V [γ1U] + γ0NF (5) First transmission condition {γ0U} = ε [γ1U] (3b)
K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 21 / 29
SLIDE 35
Boundary integral equations for impedance transmission conditions
2nd order elliptic BVP with transmission conditions on a closed Lipschitz curve/surface Γ −∆U = F in Rd\Γ (3a) ε [γ1U] − {γ0U} = 0
- n Γ
(3b) [γ0U] = 0
- n Γ
(3c) Single layer operator V := {γ0S ·} : H−1/2+s(Γ) → H
1/2+s(Γ).
Mean Dirichlet trace of representation formulation {γ0U} = −V [γ1U] + γ0NF (5) Boundary integral equations for φ = [γ1U] (insert (3b) in (5)) (εId + V )φ = γ0NF ⊲ Singularly perturbed for ε → 0 (|β| → ∞) ⊲ expect internal layers at corners of Γ
(or points of lower smoothness)
K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 22 / 29
SLIDE 36
Boundary integral equations for impedance transmission conditions
Singularly perturbed boundary integral equations for φ = [γ1U] (ε Id + V )φ = γ0NF Variational formulation: Seek φ ∈ L2(Γ) such that for all φ′ ∈ L2(Γ) bε(φ, φ′) := ε
- φ, φ′
+
- V φ, φ′
=
- γ0NF, φ′
Bilinear form bε is L2(Γ)-elliptic bε(φ, φ) ≥ εφ2
L2(Γ)
⇒ φL2(Γ) ≤ ε−1γ0NFL2(Γ) and H−1/2(Γ)-elliptic since
- V φ, φ′
φ2
H−1
/2(Γ)
(with a constant indep. of ε)
⇒ φ2
H−1
/2(Γ) bε(φ, φ) = γ0NF, φ ≤ γ0NFH1 /2(Γ)φH−1 /2(Γ)
⇒ φH−1
/2(Γ) γ0NFH1 /2(Γ).
K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 23 / 29
SLIDE 37
Boundary integral equations for impedance transmission conditions
Variational formulation: Seek φ ∈ L2(Γ) such that for all φ′ ∈ L2(Γ) bε(φ, φ′) := ε
- φ, φ′
+
- V φ, φ′
=
- γ0NF, φ′
(??) BEM discretization: Seek φ ∈ Vh such that for all φ′ ∈ Vh bε(φh, φ′
h) := ε
- φh, φ′
h
- +
- V φh, φ′
h
- =
- γ0NF, φ′
h
- (7)
where Vh is defined on mesh Th of (curved) panels K as
S−1 (Γh) :=
- vh ∈ L2(Γ) : vh ∈ P0(K) ∀K ∈ Th
- ,
ℓ = 0 S0
1 (Γh) :=
- vh ∈ L2(Γ) ∩ C(Γ) : vh ∈ P1(K) ∀K ∈ Th
- ,
ℓ = 1
Γh n K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 24 / 29
SLIDE 38
Boundary integral equations for impedance transmission conditions
Variational formulation: Seek φ ∈ L2(Γ) such that for all φ′ ∈ L2(Γ) bε(φ, φ′) := ε
- φ, φ′
+
- V φ, φ′
=
- γ0NF, φ′
(??) BEM discretization: Seek φ ∈ Vh such that for all φ′ ∈ Vh bε(φh, φ′
h) := ε
- φh, φ′
h
- +
- V φh, φ′
h
- =
- γ0NF, φ′
h
- (7)
where Vh is defined on mesh Th of (curved) panels K as
S−1 (Γh) :=
- vh ∈ L2(Γ) : vh ∈ P0(K) ∀K ∈ Th
- ,
ℓ = 0 S0
1 (Γh) :=
- vh ∈ L2(Γ) ∩ C(Γ) : vh ∈ P1(K) ∀K ∈ Th
- ,
ℓ = 1
Theorem (Stability and a-priori error estimates)
Let Th be a mesh of Γ with mesh width h. Then, φh ∈ Vh ⊂ L2(Γ) solution of (7) satisfies φhL2(Γ) ≤ ε−1γ0NFL2(Γ) φhH−1
/2(Γ) γ0NFH1 /2(Γ).
For Vh = S−1
0 (Γh) (ℓ = 0) or Vh = S0 1(Γh) (ℓ = 1) and Γ ∈ C ℓ+1,1 it holds
φ − φhL2(Γ) ε−ℓ−5/2hℓ+1γ0NFHℓ+1
/2(Γ).
K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 24 / 29
SLIDE 39
Boundary integral equations for impedance transmission conditions
Variational formulation: Seek φ ∈ L2(Γ) such that for all φ′ ∈ L2(Γ) bε(φ, φ′) := ε
- φ, φ′
+
- V φ, φ′
=
- γ0NF, φ′
(??) BEM discretization: Seek φ ∈ Vh such that for all φ′ ∈ Vh bε(φh, φ′
h) := ε
- φh, φ′
h
- +
- V φh, φ′
h
- =
- γ0NF, φ′
h
- (7)
Theorem (Stability and a-priori error estimates)
[...] For Vh = S−1
0 (Γh) (ℓ = 0) or Vh = S0 1(Γh) (ℓ = 1) and Γ ∈ C ℓ+1,1 it holds
φ − φhL2(Γ) ε−ℓ−5/2hℓ+1γ0NFHℓ+1(Γ).
Theorem (Higher order regularity estimates)
For Γ ∈ C s+j+1,1 with 0 ≤ j ≤ s it holds φHs+1
/2(Γ) εj−sγ0NFHs+j+3 /2(Γ).
K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 25 / 29
SLIDE 40
Boundary integral equations for impedance transmission conditions
Variational formulation: Seek φ ∈ L2(Γ) such that for all φ′ ∈ L2(Γ) bε(φ, φ′) := ε
- φ, φ′
+
- V φ, φ′
=
- γ0NF, φ′
(??) BEM discretization: Seek φ ∈ Vh such that for all φ′ ∈ Vh bε(φh, φ′
h) := ε
- φh, φ′
h
- +
- V φh, φ′
h
- =
- γ0NF, φ′
h
- (7)
Theorem (Higher order regularity estimates)
For Γ ∈ C s+j+1,1 with 0 ≤ j ≤ s it holds φHs+1
/2(Γ) εj−sγ0NFHs+j+3 /2(Γ).
Theorem (Improved a-priori error estimates)
For Vh = S−1
0 (Γh) (ℓ = 0) or Vh = S0 1(Γh) (ℓ = 1) and Γ ∈ C 2ℓ+3,1 it holds
φ − φhL2(Γ) hℓ+1γ0NFH2ℓ+7
/2(Γ).
Proof: Asymptotic expansion of BEM solution φh = φ0,h + δφ0,h.
Theorem (ε-robust stability estimates)
For Γ ∈ C 2,1 it holds φhL2(Γ) γ0NFH5
/2(Γ).
K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 25 / 29
SLIDE 41
Boundary integral equations for impedance transmission conditions
“Stadium” interface Γ ∈ C 1,1
× ×
R R
Γh
1.0 10−5 10−4 10−3 10−2 10−1 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 mesh-width h φh − φL2(Γ) ℓ = 0 β = 70i β = 5600i β = 448000i 2.0 1.5 10−5 10−4 10−3 10−2 10−1 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 mesh-width h φh − φL2(Γ) ℓ = 1 β = 70i β = 5600i β = 448000i
Solution φ computed w. hp-adaptive FEM using the C++ library Concepts (www.tu-berlin.de/?concepts)
K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 26 / 29
SLIDE 42
Boundary integral equations for impedance transmission conditions
Rectangular interface Γ ∈ C 0,1
×
m Rm b a
Γh
1.0 10−5 10−4 10−3 10−2 10−1 10−6 10−5 10−4 10−3 10−2 10−1 mesh-width h φh − φL2(Γ) ℓ = 0 β = 70i β = 5600i β = 448000i 1.52 1.47 0.5 10−5 10−4 10−3 10−2 10−1 10−6 10−5 10−4 10−3 10−2 10−1 mesh-width h φh − φL2(Γ) ℓ = 1 β = 70i β = 5600i β = 448000i
Solution φ computed w. hp-adaptive FEM using the C++ library Concepts (www.tu-berlin.de/?concepts)
K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 27 / 29
SLIDE 43
Boundary integral equations for impedance transmission conditions
⊲ Transmission conditions of Type II have form (e. g., shielding element by Nakata et.al.) [γ1U] − (β1 − β2∂2
Γ) {γ0U} = 0
- n Γ,
[γ0U] = 0
- n Γ
Boundary integral equation as mixed formulation (1st kind) for φ := [γ1U] ∈ H−1/2(Γ), u := {γ0U} ∈ H1(Γ) V Id −Id β1Id − β2∂2
Γ
φ u
- =
γ0N f
- Variational formulation
- V φ, φ′
Γ
+
- u, φ′
Γ
=
- γ0N f , φ′
Γ
−
- φ, u′
Γ + β1
- u, u′
Γ + β2
- ∂Γu, ∂Γu′
Γ = 0
⊲ singularly pertubed BIE for β1 ≫ 1 (high frequency) or β2 ≪ 1 (always)
K.S. and R. Hiptmair, Discrete Contin. Dyn. Syst. Ser. S, 2015
K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 28 / 29
SLIDE 44