Méthode de décomposition de domaine avec maillages non conformes pour l électromagnétisme

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2 A. Buffa Méthode de décomposition de domaine avec maillages non conformes pour l électromagnétisme Annalisa Buffa Istituto di Analisi Numerica del C.N.R. Pavia (Italy) Laboratoire Jacques Louis Lions (Paris) En collaboration avec: Y. Maday, F. Rapetti, F. Bouillault, F. Ben Belgacem 1

3 A. Buffa Outline Domain decomposition method with non matching grids: A Model problem and its variational formulation; Definition of the mortar method; Convergence results; Applications magnetostatics; 3D electric engines. 2

4 curl E = t B div (εe) = ρ curl µ 1 B = ε t E + J s + σe div (B) = 0 E n Ω = 0 B n Ω = 0 E(, 0) = E 0 B(, 0) = B 0. Good parallel/domain decomposition methods are of key importance in electromagnetic computation. huge calculation: 3D vectors in the space, sometimes 6D vectors in the space. Often the coefficients involved (ε, µ, σ) are discontinuous at the interface between different materials. 3

5 Maxwell s equations, model problem curl E = t B curl µ 1 B = ε t E + J s + σe E n Ω = 0 B n Ω = 0 E(, 0) = E 0 B(, 0) = B 0. u + curl curl u = f u n = 0 at Ω Ω Γ 12 Ω 2 Γ 23 Ω 1 Ω 3 Γ 13 4

6 3D continuous/discrete problem H 0 (curl, Ω) = {u L 2 (Ω) 3 : curl u L 2 (Ω) 3, u n = 0 at Ω} Variational formulation: = {(u i ) i L 2 (Ω i ) 3, curl u i L 2 (Ω i ) 3, u n = 0 at Ω u i n = u j n at Γ ij } u H 0 (curl, Ω) v H 0 (curl, Ω) Ω u v + Ω curl u curl v = Ω f v Discretization: Independent grids on each subdomain, T i,h ; Edge Elements in each subdomain; Easy and optimal coupling among subdomains; ([ uh n ], ϕ h ) Γ ij = 0 ϕ h?? Mortar method: the constraint is in the discrete space 5

7 Σ = Skeleton, Σ = L l=1 Γ l, non-overlapping partition (T i,h, X i,h ), conforming H(curl, Ω i ) finite elements, i = 1,.., N X h = i X i,h + boundary condition at Ω (T Γ l,h, M l,h) finite elements on Γ l, l = 1,.., L M h = l M l,h We set u h = (u 1 h,.., un h ) a(u, v) = M ( i=1 Ω i u i h vh i + curl ui h curl h) vi dω u, v Xh b(u, λ) = L [ l=1 uh Γ l n ] λ h dγ u X h, λ h M h Discrete space: X h = {v h X h s.t. b(v h, ϕ h ) = 0 ϕ h M h } Discrete problem: Find u h X h such that a(u h, v h ) = Ω fv h v h X h 6

8 Edge Elements Families of H(curl, Ω) conforming finite elements, Tetrahedra Nédélec ( 80): (EE 0 1(Ω i )) ˆv : ˆT R 3, ˆv = a + b x ; a, b R 3 e ˆv ˆτ ˆT ϕ ds ϕ P 0 (e) e = e i, i = 1,., 6 (6 D.o.F.) Nédélec ( 86): (EE 1 (Ω i )) ˆv : ˆT R 3, ˆv P 1 ( ˆT ) 3 e ˆv ˆτ ˆT ϕ ds ϕ P 1 (e) e = e i, i = 1,., 6 (12 D.o.F.) Tangetial Traces: λ = u n (n outward normal) EE 0 1(Ω i ) n i RT 1 ( Ω i ) EE 1 (Ω i ) n i BDM 1 ( Ω i ) Ω i non-smooth surfaces H(div)-conforming, normal components. 7

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10 Choice of discrete spaces X i,h EE 1, T l,h := {v i(l),h n, v i(l),h X i(l),h } Tl,h 0 = T l,h H 0 (div, Γ l ), M l,h T l,h such that dim ( ) ( ) M h = dim T 0 l,h. M l,h = T l,h no existence of solutions. slave master P 0 P 0 P 0 P 1 P 1 P 0 Γ Γ 9

11 Constraint space formulation X h = {v h X h s.t. b(v h, ϕ) = 0 ϕ M h } Find u h X h such that a(u h, v h ) = Ω fv h v h X h We have a non conforming approximation on Ω: Lemma: (Berger-Scott-Strang) If curl u H 1 (Ω), then: u u h,curl inf vh X h u v h,curl + L sup vh X h l=1 1/2,Γ l < [ u h n l ] l, (curl u) T > 1/2,Γl v h,curl (curl u) T = n (curl u n) 2,curl = M i=1 2 curl,ω i (broken norms) 10

12 Error Estimates When max h i 1 i M min 1 i M h i (mesh sizes) u u h,curl h ln h u,2,curl,ω Ingredients Π h : L 2 (Γ) T 1 h,0 R Γ ( Π h)ϕ h = 0 ϕ h M h Π h L 2 (Γ). L 2 (Γ) I i h : H 1 (curl, Ω i ) EE 1 (Ω i ) interpolation operator u i Ihu i i curl + h 1 i u i Ihu i i L 2. h s i u i s+1,ωi v i,h n 0,Γ. h 1/2 i v i,h n 1/2 (H inverse inequality (Γ)) (curl u) T s,bγ. h 1/2 1 q 00 ln h 1 (curl u) T s+1/2,γ Sobolev inequality The regularity of u is requested only in a neighborhood of Γ. 11

13 same mesh size different mesh size 0.3 Error on the energy value (h_1 = h_2) 0.6 Error on the energy value (h_1 different from h_2) matching non matching h_1 > h_2 h_1 < h_2 1.2 ln ( ( W_a W_h ) / W_a ) ln ( ( W_a W_h ) / W_a ) ln ( h ) ln ( h ) 12

14 Magnetostatics curl(ν curl u) = J in Ω u n = 0 on Ω. here u : curl u = B and u solution implies u + φ is solution. Variational formulation T : curl T = J, and Given T L 2 (Ω) 3, find u H 0 (curl, Ω) such that Ω ν curl u curl v dω = Ω T curl v dω, v H 0(curl, Ω) Discrete Variational formulation Given T L 2 (Ω) 3, find u h X h such that Ω ν curl u h curl v h dω = Ω T curl v h dω, v h X h 13

15 Features of the discrete problem: We need to construct a basis for X h... their are supported on 3/4 tetrahedra; Ti T T interface mesh integration mesh interface mesh The final system has the form: Mu h = T h where the matrix M has a non-trivial kernel. The system has a solution since T h range{m}. Conjugate gradient method! 14

16 Table 1: Conforming discretizations meshes dof CG residual/ 10 7 CG iterations energy/10 6 I - I II - II III - III IV - IV V - V VI - VI VII - VII Analytic value of the energy: 2208 MJ 15

17 Table 2: Non-conforming discretizations meshes dof CG residual/10 7 CG iterations energy/10 6 I - I rotated III - I I - III I - III rotated II - II rotated II - IV II - IV rotated III - III rotated IV - IV rotated V - V rotated VI - VI rotated VI - VII (*) VII - VI (*) VII - VII rotated Analytic value of the energy: 2208 MJ 16

18 σ u 1 t + curl (ν curl u 1) = J s Ω 1 (0) curl (ν curl u 2 ) = J s Ω 2 div(r t u) = 0 I ]0, T [ R T m = Ω 1 r σ u 1 t + J s curl u1 dω J dω dt + k ω = T m, ω = dθ dt (R t (u 1 )) τ,γ (x, t) = (u 2 ) τ,γ (x, t) (R t (ν curl u 1 )) τ,γ (x, t) = (ν curl u 2 ) τ,γ (x, t) where R t u = R t (u(r t x, t)). Theorems: Existence, uniqueness for J suff. large, discretization by mortar edge element + Implicit/explicit Euler in time. 17

19 Time discretisation and coupling initial configuration rotor θ 0 computation of the magnetic potential u temporal increment calculation of the magnetic torque C solution of the mechanical equations and ω θ Coupling flowchart θ 0 +θ rotor application of the computed angle θ Time discretisation by a simple implicit/explicit Euler. Download at annalisa 18