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 Cottbus-Senftenberg, Germany Research Center M ATHEON Mathematics for key technologies RICAM SpecSem – Workshop on Analysis and Numerics of Acoustic and Electromagnetic Problems 2016, Linz, Oct 18th 2016

Thin conducting shielding sheets Maxwell equations in eddy current approximation curl curl E + i µσω E = − i ωµ 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, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 2 / 29

Thin conducting shielding sheets Maxwell equations in eddy current approximation curl curl E + i µσω E = − i ωµ 0 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

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, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 2 / 29

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, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 2 / 29

Thin conducting shielding sheets Eddy current model ε b curl curl E + i µσω E = − i ωµ 0 J a f = − iωµ 0 j 0 Ω ε int Ω ε ext K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 3 / 29

Thin conducting shielding sheets Eddy current model ε b curl curl E + i µσω E = − i ωµ 0 J a f = − iωµ 0 j 0 Two important effects of the thin sheet (of thickness ε ) Ω ε int ⊲ Shielding effect – in conductors induced currents Ω ε ext 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 ≈ 8 mm K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 3 / 29

Thin conducting shielding sheets Eddy current model curl curl E + i µσω E = − i ωµ 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 ≈ 8 mm ε ∼ δ ε ≪ δ ε ≫ δ K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 3 / 29

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

Optimal basis inside the sheet Eddy current model in 2D (TM polarisation) − ∆ u ε ( x ) + i ωµ 0 σ ( x ) u ε ( x ) = − i ωµ 0 j 0 ( x ) Approximation of higher order without reduction to an interface Ansatz for the solution inside the sheet � N − 1 u ε int ( t , s ) ≈ u ε i =0 φ ε i ( s , t ) u ε � ε int , N ( t , s ) = int , i ( t ) . ext inspired by: Vogelius, M. and Babuˇ ska, I., Math. Comp. 37, 1981. ε t 0 � ε � ε s with N ≥ 2 linear independent basis functions φ ε int ext i n spanning V ε N , and u ε int , i ∈ H 1 ( � Γ). ∂� � κ 2 Basis functions φ ε 0 , φ ε 1 in the kernel of − ∂ 2 κ s − 1+ s κ ∂ s + i ωµ 0 σ + 4(1+ s κ ) 2 , cosh( √ i ωµ 0 σ s ) 1 φ ε { φ ε , 0 } κ = 1 , [ φ ε 0 ( s , κ ) = √ 1 + s κ cosh( √ i ωµ 0 σ ε 2 ) , , 0 ] κ = 0 , sinh( √ i ωµ 0 σ s ) 1 φ ε { φ ε , 1 } κ = 0 , [ φ ε 1 ( s , κ ) = √ 1 + s κ 2 sinh( √ i ωµ 0 σ ε 2 ) , , 1 ] κ = 1 , K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 5 / 29

Optimal basis inside the sheet Eddy current model in 2D (TM polarisation) − ∆ u ε ( x ) + i ωµ 0 σ ( x ) u ε ( x ) = − i ωµ 0 j 0 ( x ) Approximation of higher order without reduction to an interface Ansatz for the solution inside the sheet � N − 1 u ε int ( t , s ) ≈ u ε i =0 φ ε i ( s , t ) u ε � ε int , N ( t , s ) = int , i ( t ) . ext inspired by: Vogelius, M. and Babuˇ ska, I., Math. Comp. 37, 1981. ε t 0 � ε � ε s int ext with N ≥ 2 linear independent basis functions φ ε i n int , i ∈ H 1 ( � spanning V ε N , and u ε Γ). ∂� � κ 2 φ ε 2 j , φ ε κ Basis functions 2 j +1 , j ∈ N 0 in the kernel of ( − ∂ 2 4(1+ s κ ) 2 ) j +1 , s − 1+ s κ ∂ s + i ωµ 0 σ + � P j ( s ) φ ε { φ ε , 2 j } κ = δ j , 0 , [ φ ε 2 j ( s , κ ) = √ 1 + s κ cosh( i ωµ 0 σ s ) , , 2 j ] κ = 0 � P j ( s ) φ ε { φ ε [ φ ε √ 1 + s κ sinh( 2 j +1 ( s , κ ) = i ωµ 0 σ s ) , , 2 j +1 } κ = 0 , , 2 j +1 ] κ = δ j , 0 K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 5 / 29

Optimal basis inside the sheet κ 2 Basis functions φ ε i , i ∈ N 0 such that ( − ∂ 2 4(1+ s κ ) 2 ) φ ε i = ε − 2 φ ε κ s − 1+ s κ ∂ s + i ωµ 0 σ + i − 2 � P j ( s ) φ ε { φ ε , 2 j } κ = δ j , 0 , [ φ ε 2 j ( s , κ ) = √ 1 + s κ cosh( i ωµ 0 σ s ) , , 2 j ] κ = 0 � P j ( s ) φ ε { φ ε [ φ ε 2 j +1 ( s , κ ) = √ 1 + s κ sinh( i ωµ 0 σ s ) , , 2 j +1 } κ = 0 , , 2 j +1 ] κ = δ j , 0 1 1 1 φ ε int , 0 φ ε φ ε 0.5 0.5 0.5 int , 4 int , 2 κ = +8 0 0 0 κ = − 8 φ ε φ ε int , 3 φ ε int , 5 int , 1 -0.5 -0.5 -0.5 - ε ε - ε ε - ε ε 0 0 0 2 s 2 2 s 2 2 s 2 i ωµ 0 σ = 1000, ε = 0 . 1 K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 6 / 29

Optimal basis inside the sheet κ 2 Basis functions φ ε i , i ∈ N 0 such that ( − ∂ 2 4(1+ s κ ) 2 ) φ ε i = ε − 2 φ ε κ s − 1+ s κ ∂ s + i ωµ 0 σ + i − 2 � P j ( s ) φ ε { φ ε , 2 j } κ = δ j , 0 , [ φ ε 2 j ( s , κ ) = √ 1 + s κ cosh( i ωµ 0 σ s ) , , 2 j ] κ = 0 � P j ( s ) φ ε { φ ε [ φ ε 2 j +1 ( s , κ ) = √ 1 + s κ sinh( i ωµ 0 σ s ) , , 2 j +1 } κ = 0 , , 2 j +1 ] κ = δ j , 0 � � κ 2 κ − ∂ 2 Decomposition − ∆ + i ωµ 0 σ = s − 1 + s κ∂ s + i ωµ 0 σ + + A ( s , κ ) 4(1 + s κ ) 2 � �� � scales with ε , depends on σ with regular pertubation, independent of σ � � κ 2 1 1 A ( s , κ ) = − 1 + s κ∂ t 1 + s κ∂ t − 4(1 + s κ ) 2 K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 6 / 29

Optimal basis inside the sheet κ 2 Basis functions φ ε i , i ∈ N 0 such that ( − ∂ 2 4(1+ s κ ) 2 ) φ ε i = ε − 2 φ ε κ s − 1+ s κ ∂ s + i ωµ 0 σ + i − 2 � P j ( s ) φ ε { φ ε , 2 j } κ = δ j , 0 , [ φ ε 2 j ( s , κ ) = √ 1 + s κ cosh( i ωµ 0 σ s ) , , 2 j ] κ = 0 � P j ( s ) φ ε { φ ε [ φ ε 2 j +1 ( s , κ ) = √ 1 + s κ sinh( i ωµ 0 σ s ) , , 2 j +1 } κ = 0 , , 2 j +1 ] κ = δ j , 0 � � κ 2 κ − ∂ 2 Decomposition − ∆ + i ωµ 0 σ = s − 1 + s κ∂ s + i ωµ 0 σ + + A ( s , κ ) 4(1 + s κ ) 2 � �� � scales with ε , depends on σ with regular pertubation, independent of σ � � κ 2 1 1 A ( s , κ ) = − 1 + s κ∂ t 1 + s κ∂ t − 4(1 + s κ ) 2 N u ε for u ε smooth enough Interpolation I ε ⌊ N − 1 ⌊ N 2 ⌋ ⌋ � � 2 I ε N u ε ( s , t ) = ε 2 j φ ε 2 j ( s , κ ) A N , j ( s , κ ) { u ε } κ + ε 2 j φ ε 2 j +1 ( s , κ ) A N , j ( s , κ )[ u ε ] κ j =0 j =0 K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 6 / 29

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 u ε smooth enough there exists a constant C independent of σ such that int ) ≤ C ε N − 1 int ) ≤ C ε N + 1 | w ε N − u ε | H 1 (Ω ε 2 , � w ε N − u ε � L 2 (Ω ε 2 . inf inf w ε N ∈ V ε N ⊗ H 1 ( � w ε N ∈ V ε N ⊗ H 1 ( � Γ) Γ) N u ε for u ε smooth enough Interpolation I ε � ⌊ N � ⌊ N − 1 2 ⌋ ⌋ I ε N u ε ( s , t ) = j =0 ε 2 j φ ε 2 j ( s ) A N , j ( s , κ ) { u ε } κ + ε 2 j φ ε 2 j +1 ( s ) A N , j ( s , κ )[ u ε ] κ 2 j =0 K.Schmidt RICAM SpecSem W1, 18.10.2016 FEM + impedance conditions for thin electromagnetic shielding sheets 7 / 29