On optimal FEM and impedance conditions for thin electromagnetic shielding sheets

Transcript

1 On optimal FEM and impedance conditions for thin electromagnetic shielding sheets Kersten Schmidt Research Center Matheon, Berlin, Germany, Institut für Mathematik, Technische Universität Berlin, Germany Institut für Mathematik, BTU Cottbus-Senftenberg, Germany Research Center MATHEON Mathematics for key technologies RICAM SpecSem Workshop on Analysis and Numerics of Acoustic and Electromagnetic Problems 2016, Linz, Oct 18th 2016

2 Thin conducting shielding sheets Maxwell equations in eddy current approximation curl curl E +iµσω E = iωµ 0 J Thin conducting sheets shields electric and magnetic fields Challenges: high gradients in thickness directions high aspect ratio of the sheets K.Schmidt RICAM SpecSem W1, FEM + impedance conditions for thin electromagnetic shielding sheets 2 / 29

3 Thin conducting shielding sheets Maxwell equations in eddy current approximation curl curl E +iµσω E = iωµ 0 J Remedies thin sheet basis approximate transmission conditions boundary integral formulation K.Schmidt RICAM SpecSem W1, FEM + impedance conditions for thin electromagnetic shielding sheets 2 / 29

4 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) E T Remedies thin sheet basis approximate transmission conditions boundary integral formulation K.Schmidt RICAM SpecSem W1, FEM + impedance conditions for thin electromagnetic shielding sheets 2 / 29

5 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) E T Remedies thin sheet basis approximate transmission conditions boundary integral formulation K.Schmidt RICAM SpecSem W1, FEM + impedance conditions for thin electromagnetic shielding sheets 2 / 29

6 Thin conducting shielding sheets Eddy current model curl curl E +iµσω E = iωµ 0 J ε b a f = iωµ0j0 Ω ε int Ω ε ext K.Schmidt RICAM SpecSem W1, FEM + impedance conditions for thin electromagnetic shielding sheets 3 / 29

7 Thin conducting shielding sheets Eddy current model ε curl curl E +iµσω E = iωµ 0 J b a 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 µ 0 σω Copper at 50 Hz δ 8mm f = iωµ0j0 Ω ε int Ω ε ext K.Schmidt RICAM SpecSem W1, FEM + impedance conditions for thin electromagnetic shielding sheets 3 / 29

8 Thin conducting shielding sheets Eddy current model curl curl E +iµσω E = iωµ 0 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 µ 0 σω Copper at 50 Hz δ 8mm ε δ ε δ ε δ K.Schmidt RICAM SpecSem W1, FEM + impedance conditions for thin electromagnetic shielding sheets 3 / 29

9 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, FEM + impedance conditions for thin electromagnetic shielding sheets 4 / 29

10 Optimal basis inside the sheet Eddy current model in 2D (TM polarisation) u ε (x) + iωµ 0σ(x) u ε (x) = iωµ 0j 0(x) Approximation of higher order without reduction to an interface Ansatz for the solution inside the sheet uint(t, ε s) uint,n(t, ε s) = N 1 i=0 φε i (s, t) uint,i(t). ε inspired by: Vogelius, M. and Babuška, I., Math. Comp. 37, with N 2 linear independent basis functions φ ε i spanning V ε N, and u ε int,i H 1 ( Γ). ε ext ε int n ε ext ε s t 0 Basis functions φ ε 0, φ ε 1 in the kernel of 2 s κ 1+sκ s + iωµ 0σ + κ2 4(1+sκ) 2, φ ε 0(s, κ) = φ ε 1(s, κ) = 1 cosh( iωµ 0σs) 1 + sκ cosh( iωµ 0σ ε ), {φε,0} κ = 1, [φ ε,0] κ = 0, 2 1 sinh( iωµ 0σs) 1 + sκ 2 sinh( iωµ 0σ ε ), {φε,1} κ = 0, [φ ε,1] κ = 1, 2 K.Schmidt RICAM SpecSem W1, FEM + impedance conditions for thin electromagnetic shielding sheets 5 / 29

11 Optimal basis inside the sheet Eddy current model in 2D (TM polarisation) u ε (x) + iωµ 0σ(x) u ε (x) = iωµ 0j 0(x) Approximation of higher order without reduction to an interface Ansatz for the solution inside the sheet uint(t, ε s) uint,n(t, ε s) = N 1 i=0 φε i (s, t) uint,i(t). ε inspired by: Vogelius, M. and Babuška, I., Math. Comp. 37, with N 2 linear independent basis functions φ ε i spanning V ε N, and u ε int,i H 1 ( Γ). ε ext ε int n ε ext ε s t 0 Basis functions φ ε 2j, φ ε 2j+1,j N 0 in the kernel of ( s 2 κ 1+sκ s + iωµ 0σ + κ2 4(1+sκ) 2 ) j+1, φ ε 2j(s, κ) = φ ε 2j+1(s, κ) = P j(s) 1 + sκ cosh( iωµ 0σs), {φ ε,2j} κ = δ j,0, [φ ε,2j] κ = 0 P j(s) 1 + sκ sinh( iωµ 0σs), {φ ε,2j+1} κ = 0, [φ ε,2j+1] κ = δ j,0 K.Schmidt RICAM SpecSem W1, FEM + impedance conditions for thin electromagnetic shielding sheets 5 / 29

12 Optimal basis inside the sheet Basis functions φ ε i, i N 0 such that ( 2 s φ ε 2j(s, κ) = φ ε 2j+1(s, κ) = κ s + iωµ0σ + κ2 1+sκ 4(1+sκ) 2 )φ ε i = ε 2 φ ε i 2 P j(s) 1 + sκ cosh( iωµ 0σs), {φ ε,2j} κ = δ j,0, [φ ε,2j] κ = 0 P j(s) 1 + sκ sinh( iωµ 0σs), {φ ε,2j+1} κ = 0, [φ ε,2j+1] κ = δ j, φ ε int,0 κ = +8 κ = 8 φ ε int, ε 0 ε 2 s φ ε int,2 φ ε int, ε 0 ε 2 s 2 iωµ 0σ = 1000, ε = φ ε int,4 φ ε int, ε 0 ε 2 s 2 K.Schmidt RICAM SpecSem W1, FEM + impedance conditions for thin electromagnetic shielding sheets 6 / 29

13 Optimal basis inside the sheet Basis functions φ ε i, i N 0 such that ( 2 s φ ε 2j(s, κ) = φ ε 2j+1(s, κ) = κ s + iωµ0σ + κ2 1+sκ 4(1+sκ) 2 )φ ε i = ε 2 φ ε i 2 P j(s) 1 + sκ cosh( iωµ 0σs), {φ ε,2j} κ = δ j,0, [φ ε,2j] κ = 0 P j(s) 1 + sκ sinh( iωµ 0σs), {φ ε,2j+1} κ = 0, [φ ε,2j+1] κ = δ j,0 ( Decomposition + iωµ 0σ = s 2 κ 1 + sκ s + iωµ0σ + κ 2 ) + A(s, κ) 4(1 + sκ) }{{ 2 } scales with ε, depends on σ with regular pertubation, independent of σ A(s, κ) = 1 ( ) 1 κ sκ t 1 + sκ t 4(1 + sκ) 2 K.Schmidt RICAM SpecSem W1, FEM + impedance conditions for thin electromagnetic shielding sheets 6 / 29

14 Optimal basis inside the sheet Basis functions φ ε i, i N 0 such that ( 2 s φ ε 2j(s, κ) = φ ε 2j+1(s, κ) = κ s + iωµ0σ + κ2 1+sκ 4(1+sκ) 2 )φ ε i = ε 2 φ ε i 2 P j(s) 1 + sκ cosh( iωµ 0σs), {φ ε,2j} κ = δ j,0, [φ ε,2j] κ = 0 P j(s) 1 + sκ sinh( iωµ 0σs), {φ ε,2j+1} κ = 0, [φ ε,2j+1] κ = δ j,0 ( Decomposition + iωµ 0σ = s 2 κ 1 + sκ s + iωµ0σ + κ 2 ) + A(s, κ) 4(1 + sκ) }{{ 2 } scales with ε, depends on σ with regular pertubation, independent of σ A(s, κ) = 1 ( ) 1 κ sκ t 1 + sκ t 4(1 + sκ) 2 Interpolation I ε Nu ε for u ε smooth enough N 2 j=0 N 1 2 INu ε ε (s, t) = ε 2j φ ε 2j(s, κ)a N,j (s, κ){u ε } κ + ε 2j φ ε 2j+1(s, κ)a N,j (s, κ)[u ε ] κ j=0 K.Schmidt RICAM SpecSem W1, FEM + impedance conditions for thin electromagnetic shielding sheets 6 / 29

15 Lemma (Best-approximation error) Optimal basis inside the sheet For any even N (curved sheet) any N (straight sheet or curved sheet N 4) and u ε smooth enough there exists a constant C independent of σ such that inf w w N ε V N ε N ε u ε H 1 (Ω ε H1 int ( Γ) ) Cε N 2 1, inf w w N ε V N ε N ε u ε L 2 (Ω ε H1 int ( Γ) ) Cε N Interpolation I ε Nu ε for u ε smooth enough I ε Nu ε (s, t) = N 2 j=0 ε2j φ ε 2j(s)A N,j (s, κ){u ε } κ + N 1 2 j=0 ε 2j φ ε 2j+1(s)A N,j (s, κ)[u ε ] κ K.Schmidt RICAM SpecSem W1, FEM + impedance conditions for thin electromagnetic shielding sheets 7 / 29

16 Optimal basis inside the sheet Eddy current model in 2D (TM polarisation) u ε (x) + iωµ 0σ(x) u ε (x) = iωµ 0j 0(x) } Semi-discretization WN {u ε := H 1 (Ω) : u Ω ε ext H 1 (Ω ε ext ), u Ω ε V ε int N H1 ( Γ) Seek un ε WN ε such that un ε v N dx + Ω Ω ε int iωµ 0σuNv ε N dx = iωµ 0j 0v N dx Ω v N W ε N Lemma (Semi-discretization error) For any even N (curved sheet) any N (straight sheet or curved sheet N 4) and u ε smooth enough it holds for u ε N W ε N u ε N u ε H 1 (Ω ε int ) Cε N 1 2, u ε N u ε L 2 (Ω ε int ) Cε N+ 1 2, u ε N u ε H 1 (Ω ε ext ) Cε 2N 1. K.Schmidt RICAM SpecSem W1, FEM + impedance conditions for thin electromagnetic shielding sheets 8 / 29

17 Semianalytical study for circular arc with κ = 1 2, iωµ0σ = 1 ε. Optimal basis inside the sheet Error in the H 1 -seminorm inside the sheet. u ε N u ε H 1 (Ω ε int ) Cε N 1 2, Error in H 1 -seminorm outside the sheet. u ε N u ε H 1 (Ω ε ext ) Cε 2N 1. e.g., four functions sixth-order scheme O(ε 7 ) (outside the sheet) easily increasing order by enrichment with higher optimal basis functions pre-computation of integrals in s surface variables K.Schmidt RICAM SpecSem W1, FEM + impedance conditions for thin electromagnetic shielding sheets 9 / 29

18 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, FEM + impedance conditions for thin electromagnetic shielding sheets 10 / 29

19 y Original problem Impedance transmission conditions (ITCs) Ω ε ext ε 2 0 ε 2 Ω ε int Γ curl curl E = iωµ 0 J in Ω ε ext curl curl E +iωµ 0σ E = 0 in Ω ε int (1) Ω ε ext x Reduced problem with ITC-1-0 (Levi-Civita 1902) y ε 2 0 Ω 0 ext Γ curl curl E 0 = iωµ 0 J [curl E 0 n] iωµ 0σε{E 0,T } = 0 in Ω 0 ext [E 0 n] = 0 on Γ on Γ (2) ε 2 Ω 0 ext E 0 defined Ω 0 ext approximates E in Ω ε ext layer correction inside Ω ε int can be computed a-posteriori x limit for ε 0 for σ = σ(ε) ε 1 K.Schmidt RICAM SpecSem W1, FEM + impedance conditions for thin electromagnetic shielding sheets 11 / 29

20 y Impedance transmission conditions (ITCs) Original problem (TM mode) Ω ε ext ε 2 0 ε 2 Ω ε int Γ u = f in Ω ε ext u + α δ 2 u = 0 in Ωε int (1) Ω ε ext Skin depth δ serves as a parameter x Reduced problem with ITC-1-0 (Levi-Civita 1902) y ε 2 0 ε 2 Ω 0 ext Γ u 0 = f in Ω 0 ext [u 0] = 0 on Γ [ nu 0] αε {u0} δ2 = 0 on Γ (2) Ω 0 ext x u 0 defined in Ω 0 ext approximates u in Ω ε ext layer correction inside Ω ε int can be computed a-posteriori limit for ε 0 for δ(ε) ε K.Schmidt RICAM SpecSem W1, FEM + impedance conditions for thin electromagnetic shielding sheets 11 / 29

21 y Impedance transmission conditions (ITCs) Asymptotic problem (TM mode) Ω ε ext ε 2 0 ε 2 Ω ε int Ω ε ext Γ x u ε = f in Ω ε ext u ε + α δ 2 (ε) uε = 0 in Ω ε int Skin depth δ serves as a parameter Reduced problem with ITC-1-0 (Levi-Civita 1902) (1) y ε 2 0 ε 2 Ω 0 ext Γ u 0 = f in Ω 0 ext [u 0] = 0 on Γ [ nu 0] αε {u0} δ2 = 0 on Γ (2) Ω 0 ext x u 0 defined in Ω 0 ext approximates u in Ω ε ext layer correction inside Ω ε 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): , K.Schmidt RICAM SpecSem W1, FEM + impedance conditions for thin electromagnetic shielding sheets 11 / 29

22 Impedance transmission conditions (ITCs) Reduced problem with transmission conditions ITC-1-0 (Levi-Cevita 1902) u 0 = f in Ω 0 ext [u 0] = 0 on Γ [ nu 0] αε {u0} δ2 = 0 on 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 10 mm u u 0 H 1 (Ω ε ext ) Skin depth δ 1 mm 0.1 mm 0.1 mm 1 mm 10 mm Sheet thickness ε K.Schmidt RICAM SpecSem W1, FEM + impedance conditions for thin electromagnetic shielding sheets 12 / 29

23 Reduced problem with transmission conditions ITC-1-1 Impedance transmission conditions (ITCs) u ε,1 = f in Ω 0 ext [u ε,1 ] = 0 on Γ [ nu ε,1 ] αε ( 2 ) 1 αε δ 2 6δ {u ε,1 } 2 = 0 on Γ O(ε 2 ) : error in exterior decreases like ε 2 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 10 mm ITC mm ITC-1-1 Skin depth δ 1 mm Skin depth δ 1 mm 0.1 mm 0.1 mm 0.1 mm 1 mm 10 mm Sheet thickness ε 0.1 mm 1 mm 10 mm Sheet thickness ε K.Schmidt RICAM SpecSem W1, FEM + impedance conditions for thin electromagnetic shielding sheets 13 / 29

24 Reduced problem with transmission conditions ITC-1-2 Impedance transmission conditions (ITCs) [ nu ε,2 ] αε ( 2 1 αε δ 2 6δ 2 + ε2 12 u ε,2 = f in Ω 0 ext [u ε,2 ] + αε3 { 12δ nu ε,2 } = 0 on Γ ( 2 7α 2 ε δ Γ)) 2 {u ε,2 } = 0 on Γ 4 O(ε 3 ) : error in exterior decreases like ε 3 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, FEM + impedance conditions for thin electromagnetic shielding sheets 14 / 29

25 Reduced problem with transmission conditions ITC-1-2 Impedance transmission conditions (ITCs) [ nu ε,2 ] αε ( 2 1 αε δ 2 6δ 2 + ε2 12 u ε,2 = f in Ω 0 ext [u ε,2 ] + αε3 { 12δ nu ε,2 } = 0 on Γ ( 2 7α 2 ε δ Γ)) 2 {u ε,2 } = 0 on Γ 4 O(ε 3 ) : error in exterior decreases like ε 3 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 10 mm Let ε fixed and δ : [u ε,2 ] 0, [ nu ε,2 ] 0 on Γ no shielding δ 0 : { nu ε,2 } 0, {u ε,2 } 0 on Γ perfect electric b.c. (PEC) only valid results for large enough skin depth δ Skin depth δ 1 mm 0.1 mm 0.1 mm 1 mm 10 mm Sheet thickness ε K.Schmidt RICAM SpecSem W1, FEM + impedance conditions for thin electromagnetic shielding sheets 14 / 29

26 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): , [u ε,1 ] + ε ( 1 2δ u ε,1 = f in Ω 0 ext αε tanh( αε 2δ ) { nu ε,1 } = 0 on Γ [ nu ε,1 ] 2 α tanh( αε ) 2δ δ 1 2 αε tanh( αε ) {uε,1 } = 0 on Γ 2δ O(ε 2 ) : error in exterior decreases like ε 2 along δ(ε) ε (proven) we observe (numerically) O(ε 2 ) independent of δ(ε) Let ε fixed and δ : [u ε,1 ] 0, [ nu ε,1 ] 0 on Γ no shielding δ 0 : [u ε,1 ] + ε{ nu ε,1 } 0 {u ε,1 } + ε 4 [ nuε,1 ] 0 on Γ perfect electric b.c. (PEC) at Γ ε robust results w.r.t. skin depth δ / conductivity σ 10 mm 1 mm 0.1 mm 0.1 mm 1 mm 10 mm Sheet thickness ε K.Schmidt RICAM SpecSem W1, FEM + impedance conditions for thin electromagnetic shielding sheets 15 / 29 Skin depth δ

27 Electromagnetic scattering by thin shielding sheet of thickness ε Impedance transmission conditions (ITCs) curl curl E ε (k ε ) 2 E ε = 0 + Silver-Müller b.c. ( ) k ext = ω 2 µ ext ɛ ext + i σ ext(ε), in Ω ε with complex wave-number k ε ω ext, = ( ) kint ε = ω 2 µ int ɛ int + i σ int(ε), in Ω ε int. Reduced problem with transmission conditions ITC-2-1 (derived for σ int(ε) ε 2 ) V. Péron, K.S. and M. Durufle, SIAM J. Appl. Math., 76(3): , ω curl curl E ε,1 kext 2 E ε,1 = 0 in Ω 0 ext ( [ E ε,1 n ] ) ( ) Γ { E ε,1 n } L1 L 3 = ε L 3 L 2 Γ + Silver-Müller b.c. { 1 [ µ ext (curl E ε,1 ) T }Γ ( 1 µ ext curl E ε,1 ) T ]Γ with the operators L i = A i curl Γ curl Γ B i Id and constants A i, B i decoupling of ITCs if material parameters are the same on both sides of Γ L 3 = 0 K.Schmidt RICAM SpecSem W1, FEM + impedance conditions for thin electromagnetic shielding sheets 16 / 29

28 Electromagnetic scattering by thin shielding sheet of thickness ε Impedance transmission conditions (ITCs) Variational formulation for reduced problem with transmission conditions ITC-2-1 where { } V = v H(curl, Ω 0 ext), v n L 2 t ( Ω), W = {v L 2 t (Γ), curl Γ v L 2 (Γ)}. Find (E ε,1, λ ε, µ ε ) V W W such that for all (E, λ, µ ) V W W Γ 1 curl E ε,1 curl E κ2 ext E ε,1 E iκ ext dx E ε,1 n E n ds Ω + Ω µ ext µ ext Ω µ ext ( ) ( ) n λ ε [E n µ ε T ] {E ds = r.h.s., T } ( ) ( [n E ε,1 ] {n E ε,1 } λ µ ( ) ( ) A1 A 3 B1 B 3 with A =, B = A 3 A 2 B 3 B 2 Γ ) ( ) ( ) ( ) ( curlγ λ ε curl Γ λ λ ε + ε A curl Γ µ ε ε B curl Γ µ µ ε λ µ ) ds = 0. K.Schmidt RICAM SpecSem W1, FEM + impedance conditions for thin electromagnetic shielding sheets 17 / 29

29 Electromagnetic scattering by thin shielding sheet of thickness ε Impedance transmission conditions ITC-2-1 for spherical sheet Impedance transmission conditions (ITCs) Discretization with Nédélec s elements of the first kind on hexahedral curved elements and its tangential traces x3 = 0 x2 = 0 x1 = 0 Relative L 2 error PEC, ε = 0.02 PEC, ε = ITC-2-1, ε = 0.02 ITC-2-1, ε = σε 2 Impedance transmission conditions are robust w.r.t. skin depth δ / conductivity σ K.Schmidt RICAM SpecSem W1, FEM + impedance conditions for thin electromagnetic shielding sheets 18 / 29

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, FEM + impedance conditions for thin electromagnetic shielding sheets 19 / 29

31 Boundary integral equations for impedance transmission conditions Reduced problem with transmission conditions on a closed Lipschitz curve/surface Γ U = F in R d \Γ [γ 1U] β {γ 0U} = 0 on Γ [γ 0U] = 0 on Γ (3) γ 0, γ 1... 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, FEM + impedance conditions for thin electromagnetic shielding sheets 20 / 29

32 Boundary integral equations for impedance transmission conditions Reduced problem with transmission conditions on a closed Lipschitz curve/surface Γ U = F in R d \Γ [γ 1U] β {γ 0U} = 0 on Γ [γ 0U] = 0 on Γ (3) γ 0, γ 1... 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, 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, or small parameter ε := β 1, and the smoothness of Γ When is BEM on uniform meshes ε-robust? K.Schmidt RICAM SpecSem W1, FEM + impedance conditions for thin electromagnetic shielding sheets 20 / 29

33 Boundary integral equations for impedance transmission conditions 2nd order elliptic BVP with transmission conditions on a closed Lipschitz curve/surface Γ Representation formula U = F in R d \Γ (3a) [γ 0U] = 0 on Γ (3c) U = S [γ 1U] + D [γ 0U] +N F }{{} =0 with single layer potential S and Newton potential NF (S φ)(x) := G(x y)φ(y)dy (N F )(x) := G(x y)f (y)dy Γ R { 2 1 log( x y ), d = 2, 2π G(x y) = 1, d = 3. 4π x y Mean of Dirichlet traces gives the single layer operator Taking mean traces on Γ V := {γ 0S } : H 1/2+s (Γ) H 1/2+s (Γ), {γ 0U} = V [γ 1U] + γ 0NF (5) K.Schmidt RICAM SpecSem W1, FEM + impedance conditions for thin electromagnetic shielding sheets 21 / 29

34 Boundary integral equations for impedance transmission conditions 2nd order elliptic BVP with transmission conditions on a closed Lipschitz curve/surface Γ Representation formula U = F in R d \Γ (3a) [γ 0U] = 0 on Γ (3c) U = S [γ 1U] + D [γ 0U] +N F }{{} =0 with single layer potential S and Newton potential NF (S φ)(x) := G(x y)φ(y)dy (N F )(x) := G(x y)f (y)dy Γ R { 2 1 log( x y ), d = 2, 2π G(x y) = 1, d = 3. 4π x y Mean of Dirichlet traces gives the single layer operator Taking mean traces on Γ V := {γ 0S } : H 1/2+s (Γ) H 1/2+s (Γ), {γ 0U} = V [γ 1U] + γ 0NF (5) First transmission condition {γ 0U} = ε [γ 1U] (3b) K.Schmidt RICAM SpecSem W1, FEM + impedance conditions for thin electromagnetic shielding sheets 21 / 29

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 R d \Γ (3a) ε [γ 1U] {γ 0U} = 0 on Γ (3b) [γ 0U] = 0 on Γ (3c) Single layer operator V := {γ 0S } : H 1/2+s (Γ) H 1/2+s (Γ). Mean Dirichlet trace of representation formulation {γ 0U} = V [γ 1U] + γ 0NF (5) Boundary integral equations for φ = [γ 1U] (insert (3b) in (5)) (εid + V )φ = γ 0NF Singularly perturbed for ε 0 ( β ) expect internal layers at corners of Γ (or points of lower smoothness) K.Schmidt RICAM SpecSem W1, FEM + impedance conditions for thin electromagnetic shielding sheets 22 / 29

36 Boundary integral equations for impedance transmission conditions Singularly perturbed boundary integral equations for φ = [γ 1U] (ε Id + V )φ = γ 0NF Variational formulation: Seek φ L 2 (Γ) such that for all φ L 2 (Γ) b ε(φ, φ ) := ε φ, φ + V φ, φ = γ 0NF, φ Bilinear form b ε is L 2 (Γ)-elliptic b ε(φ, φ) ε φ 2 L 2 (Γ) and H 1/2 (Γ)-elliptic since φ L 2 (Γ) ε 1 γ 0NF L 2 (Γ) V φ, φ φ 2 H 1/2 (Γ) (with a constant indep. of ε) φ 2 H 1/2 (Γ) bε(φ, φ) = γ0nf, φ γ0nf H 1/2 (Γ) φ H 1/2 (Γ) φ H 1/2 (Γ) γ0nf H 1/2 (Γ). K.Schmidt RICAM SpecSem W1, FEM + impedance conditions for thin electromagnetic shielding sheets 23 / 29

37 Boundary integral equations for impedance transmission conditions Variational formulation: Seek φ L 2 (Γ) such that for all φ L 2 (Γ) b ε(φ, φ ) := ε φ, φ + V φ, φ = γ 0NF, φ (??) BEM discretization: Seek φ V h such that for all φ V h b ε(φ h, φ h) := ε φ h, φ h + V φh, φ h = γ0nf, φ h (7) where V h is defined on mesh T h of (curved) panels K as S 1 0 (Γ h ) := S 0 1 (Γ h) := {v h L 2 (Γ) : v h P 0 (K) K T h }, l = 0 {v h L 2 (Γ) C(Γ) : v h P 1 (K) K T h }, l = 1 Γ h n K.Schmidt RICAM SpecSem W1, FEM + impedance conditions for thin electromagnetic shielding sheets 24 / 29

38 Boundary integral equations for impedance transmission conditions Variational formulation: Seek φ L 2 (Γ) such that for all φ L 2 (Γ) b ε(φ, φ ) := ε φ, φ + V φ, φ = γ 0NF, φ (??) BEM discretization: Seek φ V h such that for all φ V h b ε(φ h, φ h) := ε φ h, φ h + V φh, φ h = γ0nf, φ h where V h is defined on mesh T h of (curved) panels K as S 1 0 (Γ h ) := S 0 1 (Γ h) := {v h L 2 (Γ) : v h P 0 (K) K T h }, l = 0 {v h L 2 (Γ) C(Γ) : v h P 1 (K) K T h }, l = 1 (7) Theorem (Stability and a-priori error estimates) Let T h be a mesh of Γ with mesh width h. Then, φ h V h L 2 (Γ) solution of (7) satisfies φ h L 2 (Γ) ε 1 γ 0NF L 2 (Γ) φ h H 1/2 (Γ) γ0nf H 1/2 (Γ). For V h = S 1 0 (Γ h ) (l = 0) or V h = S 0 1 (Γ h ) (l = 1) and Γ C l+1,1 it holds φ φ h L 2 (Γ) ε l 5/2 h l+1 γ 0NF H l+1/2 (Γ). K.Schmidt RICAM SpecSem W1, FEM + impedance conditions for thin electromagnetic shielding sheets 24 / 29

39 Boundary integral equations for impedance transmission conditions Variational formulation: Seek φ L 2 (Γ) such that for all φ L 2 (Γ) b ε(φ, φ ) := ε φ, φ + V φ, φ = γ 0NF, φ (??) BEM discretization: Seek φ V h such that for all φ V h b ε(φ h, φ h) := ε φ h, φ h + V φh, φ h = γ0nf, φ h (7) Theorem (Stability and a-priori error estimates) [...] For V h = S 1 0 (Γ h ) (l = 0) or V h = S 0 1 (Γ h ) (l = 1) and Γ C l+1,1 it holds φ φ h L 2 (Γ) ε l 5/2 h l+1 γ 0NF H l+1 (Γ). Theorem (Higher order regularity estimates) For Γ C s+j+1,1 with 0 j s it holds φ H s+1/2 (Γ) εj s γ 0NF H s+j+3/2 (Γ). K.Schmidt RICAM SpecSem W1, FEM + impedance conditions for thin electromagnetic shielding sheets 25 / 29

40 Boundary integral equations for impedance transmission conditions Variational formulation: Seek φ L 2 (Γ) such that for all φ L 2 (Γ) b ε(φ, φ ) := ε φ, φ + V φ, φ = γ 0NF, φ (??) BEM discretization: Seek φ V h such that for all φ V h b ε(φ h, φ h) := ε φ h, 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 φ H s+1/2 (Γ) εj s γ 0NF H s+j+3/2 (Γ). Theorem (Improved a-priori error estimates) For V h = S 1 0 (Γ h ) (l = 0) or V h = S 0 1 (Γ h ) (l = 1) and Γ C 2l+3,1 it holds φ φ h L 2 (Γ) h l+1 γ 0NF H 2l+7/2 (Γ). Proof: Asymptotic expansion of BEM solution φ h = φ 0,h + δφ 0,h. Theorem (ε-robust stability estimates) For Γ C 2,1 it holds φ h L 2 (Γ) γ 0NF H 5/2 (Γ). K.Schmidt RICAM SpecSem W1, FEM + impedance conditions for thin electromagnetic shielding sheets 25 / 29

41 Boundary integral equations for impedance transmission conditions Stadium interface Γ C 1,1 R R Γ h 10 1 l = l = φh φ L 2 (Γ) φh φ L 2 (Γ) mesh-width h β = 70i β = 5600i β = i mesh-width h β = 70i β = 5600i β = i Solution φ computed w. hp-adaptive FEM using the C++ library Concepts ( K.Schmidt RICAM SpecSem W1, FEM + impedance conditions for thin electromagnetic shielding sheets 26 / 29

42 Boundary integral equations for impedance transmission conditions b R m Rectangular interface Γ C 0,1 m a Γ h 10 1 l = l = φh φ L 2 (Γ) φh φ L 2 (Γ) β = 70i β = 70i 10 6 β = 5600i β = i 10 6 β = 5600i β = i mesh-width h mesh-width h Solution φ computed w. hp-adaptive FEM using the C++ library Concepts ( K.Schmidt RICAM SpecSem W1, FEM + impedance conditions for thin electromagnetic shielding sheets 27 / 29

43 Boundary integral equations for impedance transmission conditions Transmission conditions of Type II have form (e. g., shielding element by Nakata et.al.) [γ 1U] (β 1 β 2 2 Γ) {γ 0U} = 0 on Γ, [γ 0U] = 0 on Γ Boundary integral equation as mixed formulation (1st kind) for φ := [γ 1U] H 1/2 (Γ), u := {γ 0U} H 1 (Γ) Variational formulation ( V Id Id β 1Id β 2 2 Γ ) ( ) φ = u ( ) γ0n f 0 V φ, φ + u, φ = γ Γ Γ 0N f, φ Γ φ, u + Γ β1 u, u + Γ β2 Γ u, Γ u = 0 Γ singularly pertubed BIE for β 1 1 (high frequency) or β 2 1 (always) K.S. and R. Hiptmair, Discrete Contin. Dyn. Syst. Ser. S, 2015 K.Schmidt RICAM SpecSem W1, FEM + impedance conditions for thin electromagnetic shielding sheets 28 / 29

44 Summary Impedance conditions for thin electromagnetic shielding sheets FEM with optimal basis inside the sheet σ-robust convergence of high order in thickness ε Impedance transmission conditions families ITC-1-N and ITC-2-N of transmission conditions derived by asymptotic expansion with δ(ε) ε or δ(ε) ε σ-robust convergence for ITC-1-0 (Levi-Civita), ITC-1-1, ITC-2-1 (also in 3D for Maxwell scattering, NtD operators mixed formulation) Singularly perturbed boundary integral equation (second kind) (β 1 Id + V )U = γ 0NF β-robust stability and a-priori error estimates for Γ smooth enough Outlook convolution quadrature for impedance transmission conditions in time-domain Impedance transmission conditions for eddy current model in 3D thin electromagnetic sheets with corners (boundary layer + singularities) K.Schmidt RICAM SpecSem W1, FEM + impedance conditions for thin electromagnetic shielding sheets 29 / 29