Institut de Recherche MAthématique de Rennes

Transcript

1 Oberwolfach meeting on Computational Electromagnetism Feb 22 - Feb 28 Singularities of electromagnetic fields in the Maxwell and eddy current formulations Martin COSTABEL, Monique DAUGE, Serge NICAISE Institut de Recherche MAthématique de Rennes

2 Situation Ω C Conductor body Ω C : polyhedron with boundary B. Conductivity σ C, permittivity ε C and permeability µ C. Exterior region Ω E := B(0,R) \ Ω C. Conductivity σ E =0, permittivity ε E and permeability µ E. Region of interest Ω=B(0,R)=Ω C Ω E Ω E with σ, ε and µ defined on Ω. Perfect conductor boundary conditions on Ω = Ω E \ B

3 Outline Maxwell problem for an imperfect conductor. Gauge conditions Regularized variational form. Singularities of Type 1 and 2. Sobolev and weighted Sobolev regularity. Eddy current problem. Gauge conditions Regularized variational form. Eddy current pb as a limit of Maxwell (as δ ωεσ 1 0 ). Singularities of Type 1 and 2 (are also limits as δ 0 ). Regularity: The field inside a convex conductor is more regular than outside. Singularities of electromagnetic fields in the eddy current limit 2

4 Maxwell problem for a conductor Conductor body Ω C : polyhedron with boundary B (connected components B i ). Conductivity σ C, permittivity ε C and permeability µ C. Exterior region Ω E := B(0,R) \ Ω C for R large enough. Conductivity σ E =0, permittivity ε E and permeability µ E. Region of interest Ω=B(0,R)=Ω C Ω E. Conductivity σ, permittivity ε and permeability µ defined on Ω. Frequency ω, fixed. Perfect conductor boundary conditions on Ω = Ω E \ B. Source current density j 0 in L 2 (R 3 ) 3, with support in Ω C and divergence free, i.e. div j 0 =0in Ω C and j 0 n =0on B. Find the electromagnetic field (E, H) (i) curl E = iω µh in Ω, (Maxwell) (ii) curl H = (σ + iω ε)e + j 0 in Ω, (iii) E n = 0 & H n =0 on Ω. Plan Regularity and Singularities for a polyhedral conductor Eddy 3

5 Conditions on the divergence of the electric field Taking the divergence of equation (ii) Therefore div αe =0, with α C = σ C + iω ε C and α E = iω ε E. (1) div E C =0 and div E E =0 (2) α C E C n = α E E E n on B. Equation (ii) also yields that E C = curl ψ with ψ =(iωε C + σ C ) 1 (H J 0 ) where J 0 is a vector potential for j 0. By localization around B i : E C n ds = B i div{curl(µ i ψ)} dx =0. Ω C Combining with (2) B i E E n ds =0 Plan Regularity and Singularities for a polyhedral conductor Eddy 4

6 Regularized variational formulation We propose a variational space independent of σ, ε and ω : { Y(Ω) = u L 2 (Ω) 3 : curl u L 2 (Ω) 3, div u C L 2 (Ω C ), div u E L 2 (Ω E ) } u n =0 on Ω, B i u E n ds =0. The associate variational forms are for u, v Y(Ω) ( a(u, v) = µ 1 curl u curl v ω 2 εu v ) dx + iω σ C u C v C dx Ω Ω C a reg (u, v) =a(u, v) + div u C div v C dx + Ω C div u E div v E dx. Ω E The electric field E solution of (Maxwell) is the only solution of E Y(Ω), v Y(Ω), a reg (E, v) = iω(j 0,v) ΩC. Plan Regularity and Singularities for a polyhedral conductor Eddy 5

7 Corner singularities, general recipe Fix a corner O (here O B ). Let (ρ, ϑ) be polar coordinates centered in O. Around O, the 3D space is shared into two complementary cones Γ C and Γ E reproducing the sharing by Ω C and Ω E : R 3 = Γ C Γ E. The general recipe consists in looking for solutions u of the form ρ λ U(ϑ) of the principal part of the associated elliptic bvp, with zero RHS: curl(µ 1 C curl u C) div u C =0 in Γ C, curl(µ 1 E curl u E ) div u E =0 in Γ E, [u n] =0, [αu n] =0 on I:= Γ C = Γ E, [µ 1 curl u n] =0, [curl u n] =0 on I, [div αu] =0, [α 1 n div αu] =0 on I. Plan Regularity and Singularities for a polyhedral conductor 6

8 Corner singularities, Maxwell recipe Set q C = div α C u C, q E = div α E u E and ψ C = µ 1 C q C =0 in Γ C, q E =0 in Γ E, [q] =0, [α 1 n q]=0 on I. curl ψ C = q C, div(µ C ψ C )=0 in Γ C, curl ψ E = q E, div(µ E ψ E )=0 in Γ E, [ψ n] =0, [µψ n] =0 on I. curl u C = µ C ψ C, div α C u C = q C in Γ C, curl u E = µ E ψ E, div α E u E = q E in Γ E, [u n] =0, [αu n] =0 on I. curl u C, ψ E = µ 1 E curl u E. Type 3: Type 2: Type 1: general q particular ψ, particular u q =0, general ψ, particular u q =0, ψ =0, general u Plan Regularity and Singularities for a polyhedral conductor 7

9 Maxwell Corner Singularities of Type 1 u C = Φ C and u E = Φ E with div Φ C =0 in Γ C, div Φ E =0 in Γ E, [Φ] = 0, [α n Φ] = 0 on I. The singular density Φ is in H 1 loc (R3 ) and is a singularity of the Laplace transmission problem with α : α = α C = σ C + iω ε C in Γ C and α = α E = iω ε E in Γ E. The singularities Φ=ρ λ ϕ(ϑ) with Re λ>0 and λ(λ +1)=ν eigenvalue and ϕ eigenvector of the problem (with G C =Γ C S 2 and G E =Γ E S 2 ) ϕ H 1 (S 2 ), ψ H 1 (S 2 ) α C ϕ ψ + G C α E ϕ ψ = ν G E { G C α C ϕψ + } α E ϕψ G E For wedges, this reduces to a 1D angular version: θ and λ 2 = ν. Plan Regularity and Singularities for a polyhedral conductor Eddy 8

10 Maxwell Corner Singularities of Type 2 ψ C = Ψ C and ψ E = Ψ E with div Ψ C =0 in Γ C, div Ψ E =0 in Γ E, [Ψ] = 0, [µ n Ψ] = 0 on I. The density Ψ is a singularity ρ λ ψ(ϑ) of the Laplace transmission problem with µ : µ = µ C in Γ C and µ = µ E in Γ E. Then u =(λ +1) 1 (µ Ψ x r) with r solution of r C =0 in Γ C, r E =0 in Γ E, [r] =0, [α n r]=[αµ]( Ψ x) n on I. Plan Regularity and Singularities for a polyhedral conductor Eddy 9

11 Sobolev regularity Let β α and β µ be the limiting regularity Sobolev exponents for the transmission Laplace operators div α and div µ respectively. Note that 3 2 <β µ < 2 and 1 <β α. Then E C H s (Ω C ) and E E H s (Ω E ), s <min{β α 1,β µ }. Moreover with E = Φ +E reg E reg C Hs (Ω C ) and E reg E H s (Ω E ), s <min{β α,β µ }. Plan Regularity and Singularities for a polyhedral conductor Eddy 10

12 Weighted Sobolev spaces Isotropic spaces (Kondrat ev) and m N, β R Wβ {u m(ω) = L 2 loc (Ω) : α, α m, α u L 2 (V 0 ), } c corner, rc β+ α α u L 2 (Vc). Isotropic spaces (Nazarov-Plamenevskii) D = C or E, and m N, β R { K m β (Ω D)= u L 2 loc (Ω D) : α, α m, α u L 2 (V 0 ), c corner, e edge, r β+ α c α u L 2 (V 0 c ) r β+ α e α u L 2 (V e 0 ) and rβ+ α e α u L 2 (V e }. c) Anisotropic spaces (with α =(α,α 3 ) transverse - longitudinal to the edge) { M m β (Ω D)= u L 2 loc (Ω D) : α, α m, α u L 2 (V 0 ), c corner, rc β+ α α u L 2 (V c 0) e edge, r β+ α e α u L 2 (V e 0) and rα 3 c r β+ α e α u L 2 (V c e ) } Plan Regularity and Singularities for a polyhedral conductor 11

13 Weighted (anisotropic) Sobolev regularity Suppose j 0 smooth. Let β α and β µ be the limiting regularity Sobolev exponents for the transmission Laplace operators div α and div µ respectively. Splitting of the electric field E = Φ +E reg with E reg C M β (Ω C) and E reg E M β (Ω E), β <min{β α,β µ } and the potential Φ=Φ 0 +Φ 1 +Φ reg, Φ reg H (Ω) and Φ 0 C M β (Ω C), Φ 0 E M β (Ω E), Φ 1 W β (Ω). Plan Regularity and Singularities for a polyhedral conductor Conclusion 12

14 Eddy current equations Recall Maxwell equations. Find the electromagnetic field (E, H) curl E = iω µh in Ω, (Maxwell) curl H = (σ + iω ε)e + j 0 in Ω, E n = 0 & H n =0 on Ω. The Eddy Current equations are defined for σ>>εby setting ε to 0 in (Maxwell). Find the electromagnetic field (E, H) curl E = iω µh in Ω, (Eddy c.) curl H = σe + j 0 in Ω, E n = 0 & H n =0 on Ω. Plan Regularity and Singularities for a polyhedral conductor Maxwell 13

15 Conditions on the divergence of the electric field Taking the divergence of the 2d equation div σe =0, with σ = σ C in Ω C and σ E =0. Therefore div E C =0 and E C n =0 on B We have nothing on div E E. We impose as gauge conditions, the conditions we obtained from (Maxwell) before passing to the limit: div E E =0 and B i E E n ds =0 Plan Regularity and Singularities for a polyhedral conductor Maxwell 14

16 The eddy current equations as limiting equations Set ωε = δ ε where δ>0 is a small parameter ε has the same order of magnitude as σ C. The assumption that δ is small reflects the fact that the product ωε is small with respect to σ C. We write (Maxwell) and (Eddy c.) in a unified way, for δ>0 and δ =0, resp.: curl E δ = iω µh δ in Ω, (P δ ) curl H δ = (σ + iδ ε)e δ + j 0 in Ω, E δ n = 0 & H δ n =0 on Ω. Plan Regularity and Singularities for a polyhedral conductor Maxwell 15

17 Regularized variational formulations We propose a variational space independent of σ, ε and ω : { Y(Ω) = u L 2 (Ω) 3 : curl u L 2 (Ω) 3, div u C L 2 (Ω C ), div u E L 2 (Ω E ) } u n =0 on Ω, B i u E n ds =0. The associate variational forms are for u, v Y(Ω) a δ ( (u, v) = µ 1 curl u curl v ωδ εu v ) dx + iω σ C u C v C dx Ω Ω C a δ reg (u, v) =aδ (u, v) + div u C div v C dx + Ω C div u E div v E dx. Ω E The electric field E solution of (P δ ) solves E δ Y(Ω), v Y(Ω), a δ reg (E, v) = iω(j 0,v) ΩC. Plan Regularity and Singularities for a polyhedral conductor Maxwell 16

18 The eddy current limit Theorem (i) δ 0 > 0 s. t. the sesquilinear forms a δ reg are uniformly coercive for δ [0,δ 0]. (ii) We have the uniform bound: C >0, δ [0,δ 0 ], E δ Y(Ω) C. (iii) We have the convergence as δ 0 : C >0, δ [0,δ 0 ], E δ E 0 Y(Ω) Cδ. The singularities are of Type 1 and 2, also in the limit δ =0, and they depend continuously on δ.../... Plan Regularity and Singularities for a polyhedral conductor Maxwell 17

19 Eddy current Corner Singularities of Type 1 u C = Φ C and u E = Φ E with div Φ C =0 in Γ C, div Φ E =0 in Γ E, [Φ] = 0, n Φ C =0 on I. We have either (i) or (ii) (i) Φ C is a singularity of the Laplace Neumann problem, Φ E has the same degree λ. (ii) Φ C =0and Φ E is a singularity of the Laplace Dirichlet problem. Skip to Type 2 Plan Regularity and Singularities for a polyhedral conductor Maxwell 18

20 Using the harmonic extension P E from G C := Γ C S 2 into G E := Γ E S 2, we can write the singularities eigenvalue pb in the unified way for δ =0and δ>0 : Find (ϕ C, ϕ 0 ) H 1 (G C ) H0 1(G E), (ψ C, ψ 0 ) H 1 (G C ) H0 1(G E) : a δ (ϕ C, ϕ 0 ; ψ C, ψ 0 ) = νb δ (ϕ C, ϕ 0 ; ψ C, ψ 0 ) The singularity of Type 1 is u = Φ with Φ C = r λ ϕ C and Φ E = r λ (P E ϕ C + ϕ 0 ). Let η = iδ ε E. σ C + iδ ε C The forms a δ and b δ depend continuously on δ [0,δ 0 ] : a δ = ϕ C ψ C + η G C P E ϕ C P E ψ C + G E G E ϕ 0 ψ 0 and b δ = G C ϕ C ψ C + η G E ( PE ϕ C P E ψ C + ϕ 0 P E ψ C ) + G E ( PE ϕ C ψ 0 + ϕ 0 ψ 0 ). Plan Regularity and Singularities for a polyhedral conductor Maxwell 19

21 Eddy current Corner Singularities of Type 2 ψ C = Ψ C and ψ E = Ψ E with div Ψ C =0 in Γ C, div Ψ E =0 in Γ E, [Ψ] = 0, [µ n Ψ] = 0 on I. The density Ψ is a singularity ρ λ ψ(ϑ) of the Laplace transmission problem with µ : µ = µ C in Γ C and µ = µ E in Γ E. Then u =(λ +1) 1 (µ Ψ x r) with r solution of r C =0 in Γ C, r E =0 in Γ E, [r] =0, n r C = µ C ( Ψ C x) n on I. Plan Regularity and Singularities for a polyhedral conductor Maxwell 20

22 Sobolev regularity Let β Dir E, βneu C and β µ be the limiting regularity Sobolev exponents for the Dirichlet pb on Ω E, Neumann pb on Ω C and the transmission pb div µ respectively: Then 3 2 <β µ < 2 and 3 2 < min{βneu C, βdir E } < 2 E C H s (Ω C ) Moreover s <min{βc Neu 1,β µ }. and { E E H s (Ω E ), s <min E = Φ +E reg } min{βc Neu, βdir E } 1,β µ. with E reg C Hs (Ω C ) s <min{β Neu C, β µ} and E reg E H s (Ω E ), s <min{min{βc Neu, βdir E },β µ}. Plan Regularity and Singularities for a polyhedral conductor Maxwell 21

23 Conclusion: How to approximate these solutions? Combining with FEM techniques already investigated for Maxwell equations in polyhedral bodies, we may hope that the following will provide good approximations (considered the nasty singularities): Curl-Conforming Elements (first NÉDÉLEC family of edge elements) used with a Lagrange multiplier in Ω E (KIKUSHI formulation) Weighted Regularization Method with nodal FEM in Ω C and Ω E. Singular Complement Method (for axisymmetric domains only). The performances of all methods may hopefully be dramatically improved by the use of anisotropic refinement along edges and at corners, and higher degree polynomials. Plan Regularity and Singularities for a polyhedral conductor Maxwell 22