Coupled FETI/BETI for Nonlinear Potential Problems

Transcription

1 Coupled FETI/BETI for Nonlinear Potential Problems U. Langer 1 C. Pechstein 1 A. Pohoaţǎ 1 1 Institute of Computational Mathematics Johannes Kepler University Linz {ulanger,pechstein,pohoata}@numa.uni-linz.ac.at This work has been supported by the Austrian Science Fund Fonds zur Förderung der wissenschaftlichen Forschung (FWF) under the grant Project P

2 Outline Motivation Nonlinear Magnetostatic Maxwell Coupled FETI/BETI for Linear Potential Problems Weak Formulation Tearing and Interconnecting Schur Complement Regularization Dual Problem Preconditioning Nonlinear Potential Problems Approximation of Nonlinear Parameter Newton + Coupled FETI-BETI First Numerical Results (Linear) Concluding Remarks and Outlook 2

3 Motivation for Coupled FETI/BETI Solving Nonlinear Field Problems Magnetostatic 2D Maxwell Why FETI/BETI? Parallel Computing Why Coupling? Benefit from both FEM and BEM techniques References FETI: [Farhat, Roux, Klawonn, Widlund, Brenner] BETI/Coupling: [Langer, Steinbach] coil core coil air 3

4 Non-overlapping Domain Decomposition Ω R n n = 2, 3 bounded Ω = i I Ω i Γ = Ω Γ i = Ω i Γ ij = Γ i Γ j n i... outward unit normal vector to Ω i Γ Γ i Ω i Ω j Γjk Ω k 4

5 Linear Potential Problem Potential Problem α i = α i (x)! [α i u ] = f u = 0 in Ω i on Γ (α i u) n i + (α j u) n j = 0 on Γ ij Weak formulation Find u H 1 0(Ω): α i u v dx = Ω i i I Ω f v dx v H 1 0(Ω) 5

6 BEM Subdomains I = I BEM I FEM such that α i const for i I BEM Local representation: α i u v dx = Ω i Ω i α i u }{{} =f v dx + Γ i u α i v ds x }{{ n } i =S i u N i f for i I BEM Global formulation: i I FEM Ω i α i u v dx + = i I FEM Ω i f v dx + i I BEM i I BEM (S i u) v ds x = Γ i (N i f) v ds x Γ i 1 v H0 (Ω) 6

7 Discretization Triangulation of Ω i, i I FEM V i,h H 1 (Ω i ) H 1 0(Ω) Triangulation of Γ i, i I BEM V i,h {v H 1/2 (Γ i ) : v Γi Γ = 0}) conforming (K i,h u i, v i ) = α i u i,h v i,h dx (f i, v i ) = f v i,h dx Ω i Ω i (Si,h BEM u i, v i ) (S i u i,h )v i,h ds x (fi,h BEM, v i ) = (N i f) v i,h dsx Γ i Γ i [ 1 ] Si,h BEM := D i,h + 2 M i,h + Ki,h V 1 i,h [ 1 2 M i,h + K i,h ] 7

8 Tearing and Interconnecting Tearing: Unknowns are now (u i ) i I Interconnecting: Continuity enforced by B i u i = 0 i I I FEM = {1,..., q} Linear System: I BEM = {q + 1,..., p} K 1,h B K q,h B q S BEM q+1,h B q S BEM p,h B p B 1 B q B q+1 B p 0 u 1. u q u q+1. u p λ = f 1. f q f BEM q+1. f BEM p 0 8

9 FEM Schur Complement Optionally: Elimination of inner FEM-unknowns via Schur Complement S FEM i,h f FEM i,h = K CC i,h Ki,h IC [Ki,h] II 1 Ki,h CI = f C i,h KIC i,h [K II i,h] 1 f I i,h Linear System: S FEM 1,h B S FEM q,h B q S BEM q+1,h B q S BEM p,h B p B 1 B q B q+1 B p 0 u 1. u q u q+1. u p λ = f FEM 1. f FEM q f BEM q+1. f BEM p 0 9

10 FETI/BETI Regularization, Dual Problem In floating domains, i.e. Γ I Γ = : S FEM/BEM i,h = S FEM/BEM i,h +β i e i e i Elimination of u i : u i = [ ] 1 [ SFEM/BEM FEM/BEM i,h f i B i ] + γi e i Dual problem: F := i I B [ ] 1B SFEM/BEM i i,h i, ( ) ( ) F G λ = γ G G := (B ie i ) i I floating ( d e ) Projected dual problem: P := I G(G G) 1 G P F λ = P d 10

11 FETI/BETI Preconditioners Projected dual problem solved with CG-iteration. Preconditioners: [Langer, Steinbach, Klawonn, Widlund, Brenner] C 1 α FETI = (BC 1 C 1 BETI = (BC 1 Spectral Equivalence: α B ) 1 BC 1 α B ) 1 BC 1 α [ i I B i S FEM/BEM i,h B i [ B i D i,h B i i I ] ] C 1 α C 1 α B )(BC 1 α B ) 1 B )(BC 1 α B ) 1 S BEM i,h S FEM i,h S i,h D i,h Condition Estimate: κ(p C 1 FETI/BETI P P F P ) (1 + log(h/h)) 2 11

12 Nonlinear Potential Problems Nonlinear Coefficient ν i : R + 0 R+, ν i ( ) const for i I BEM s ν(s)s strongly monotone and Lipschitz-continuous [ν i ( u ) u ] = f u = 0 ν i ( u ) u n i + ν j ( u ) u n j = 0 in Ω i on Γ on Γ ij Problem: In magnetostatics, ν i not available in analytical form Approximation 12

13 Approximation of B-H-Curves Magnetostatics / Maxwell: B-H-curve 1.2 [ν i ( u ) u] 2D 3D [ν i ( A ) }{{ A } ] } {{ =B } =H B in T H in A/m reluctivity nu H = ν( B ) B ν(s):= f 1 (s) s B = f( H ) nu(b) Physical Properties: B B-H-Curve f reluctivity ν f C 1 (R + 0 R+ 0 ) ν C1 (R + R + ) f(0) = 0 ν(s)s strongly monotne: f ( ) (s) µ 0 > 0 = ν(s)s ν(t)t (s t) 1/L s t 2 lim f (s) = µ 0 ν(s)s Lipschitz-continuous: s = L := max s 0 f (s) < ν(s)s ν(t)t 1/µ 0 s t 13

14 Approximation of B-H-Curves [Pechstein, Jüttler, Gfrerer] Given Data Pairs B-H-curve data B 1 (H k, B k ) k=1,...,n with f(h k ) B k δ k H Approximation of f in the class of strongly monotone cubic C 1 splines, such that f(h k ) B k c δ k Minimization of H N 0 [ f (s) ] 2 ds ω(s) Quadratic Optimization Problem Spline Representation of (Data Dependent Functional) f Newton+Trick Fast Evaluation of ν, ν, etc. 14

15 Approximation of B-H-Curves Results nu - interproximated B-H-curve - Su7a2.dat - monotone cubic C^1 B-spline (rel. tolerance 0.005) 1e+06 data function nu0 2 interproximated B-H-curve - SiFe.dat - monotone cubic C^1 B-spline (rel. tolerance 0.005) data function asymptotics nu(b) B B H interproximated B-H-curve - QMS3L.dat - monotone cubic C^1 B-spline (rel. tolerance 0.005) interproximated B-H-curve - QMS3L.dat - monotone cubic C^1 B-spline (rel. tolerance 0.005) 1.6 data function asymptotics 1.6 data function asymptotics B 0.8 B H H 15

16 Nonlinear Potential Problem Nonlinear Coefficient ν i C 1 (R + 0 R+ ), ν i ( ) const for i I BEM ν i (t)t strongly monotone, Lipschitz-continuous reluctivity nu [ν i ( u ) u ] = f u = 0 in Ω i on Γ nu(b) ν i ( u ) u n i + ν j ( u ) u n j = 0 on Γ ij 100 Weak formulation: Find u H 1 0 (Ω): ν i ( u ) u v dx + i I FEM Ω i = f v dx + i I FEM Ω i (ν (S i ) i I BEM Γ i i I BEM B i u)v ds x = Γ i (N i f)v ds x 1 v H0 (Ω) 16

17 Newton Iteration Initial u (0), e.g. u (0) = 0 u (k+1) = u (k) + ρ k w (k) i I FEM Ω i = where [ζ i ( u (k) ) w (k)] v dx + i I FEM Ω i i I BEM Γ i f v ν i ( u (k) ) u (k) v dx + ( S (ν i ) i w (k)) v ds x = i I BEM Γ i [ (Ni f) ( S i u (k))] v ds x ζ i (p)q := ν i ( p )q + ν i ( p ) (p q)p p R n \ {0} q R n p ζ i (0)q := ν i ( 0 )q q R n 17

18 FETI/BETI for the k-th Newton Equation Define (K i,h (u)w, v) = Ω i [ ζi (u h ) w h ] vh dx (r (k), v) = (f i i K i (u (k), v) for i I FEM (r (k),bem, v) = (f BEM i i Si,h BEM u (k) ) for i I BEM Linear System K 1,h (u i ) B K q,h (u i ) B q S q+1,h BEM B q S p,h BEM Bp B 1 Bq B q+1 Bp 0 w (k) 1. w (k) q w (k) q+1. w (k) p λ = r (k) 1. r q r (k),bem q+1. r (k),bem p 0 18

19 Inner vs. Outer Iteration Outer Iteration: Inner Iteration: Newton Projected Preconditioned CG Stopping Criterion (PPCG): P F λ (i) < ε (k) CG λ 0 Stopping Criterion (Newton): ( r (k),inner ) i < ε i I FEM Newton r 0 additionally measure flux jump on the interfaces Γ ij 19

20 Numerical Results (Linear) model problem: coil-core configuration core µ r = 10 3 coil / coil f = ±10 3 elsewhere µ r = 1, f = 0 Dirichlet: u Ω = 0 ν = 1 µ r µ 0 µ 0 = 4.π.10 7 OSTBEM 20

21 Numerical Results (Linear) 260 coupling nodes, 911 inner nodes, 276 Lagrange parameters preconditioner iterations residual total (sec) one step (sec) FETI (C α = 1) e BETI e FETI (Kla-Wid) e BETI e coupling nodes, 3271 inner nodes, 548 Lagrange parameters preconditioner iterations residual total (sec) one step (sec) FETI (C α = 1) e BETI e FETI (Kla-Wid) e BETI e

22 Numerical Results (Linear) PPCG residual - magnet.inp PPCG residual - magnet.inp 1 Scaled Schur Complement Preconditione type Klawonn-Widlund (Euclid) Scaled Hypersingular Preconditione type Klawonn-Widlund (Euclid) Scaled Schur Complement Preconditione type C_diag = 1 (Euclid) Scaled Hypersingular Preconditione type C_diag = 1 (Euclid) 1 Scaled Schur Complement Preconditione type Klawonn-Widlund (Euclid) Scaled Hypersingular Preconditione type Klawonn-Widlund (Euclid) Scaled Schur Complement Preconditione type C_diag = 1 (Euclid) Scaled Hypersingular Preconditione type C_diag = 1 (Euclid) relative residual e-06 1e-08 relative residual e-06 1e-08 1e-10 1e-10 1e-12 1e-12 1e iteration 1e iteration 22

23 Numerical Results (Linear) potential u B -field 23

24 Concluding Remarks and Outlook Family of FEM/BEM Domain Decomposition Techniques Coupled FETI/BETI with two efficient and robust Preconditioners Efficient and fast handling of B-H-Curves Balancing inner and outer iteration (ε (k) CG, ε Newton) Meshrefinement (adaptive) Use multilevel structure for good initials u (0) and Multigrid Preconditioning Exploit various levels of elimination w.r.t. parallel computing (K i, S i, BETI saddle-point-problem, etc. Nonlinear Tearing and Interconnecting 24

25 Nonlinear Tearing and Interconnecting Solution u of the local nonlinear boundary value problem [ν i ( u ) u ] = f u = g u = v i in Ω i on Γ i Γ on Γ i \ Γ defines the Nonlinear Dirichlet-to-Neumann-map T i [f, g] : H 1/2 (Γ i \ Γ) H 1/2 (Γ i \ Γ) : v i ν i ( u ) u ν i ( ) const = T i [f, g](v i ) = S i g + S i v i N i f Ω i floating: T i [f, g](v i ) = T i [f](v i ), kernel = constant functions Possible Realization of T i,h [f, g](v i ): Solve nonlinear Dirichlet problem (g, v i, f) with (damped) Newton Last iterate u (k) h determines Neumann data via Schur Complement. n i 25

26 Nonlinear Tearing and Interconnecting Nonlinear Neumann-to-Dirichlet-Map (nonfloating) Solve Dirichlet problem [ν i ( u ) u ] = f, in Ω i, u = g, on Γ i Γ, ν i ( u ) u/ n i = t i, on Γ i \ Γ, with a (damped) Newton, take Dirichlet data T i [f, g] 1 (t i ) Nonlinear Neumann-to-Dirichlet-Map (floating) Solve a regularized version of the local Neumann problem [ν i ( u ) u ] = f ν i ( u ) u/ n i = t i in Ω i on Γ i with (damped) Newton, take Dirichlet data T i [f, g] 1 (t i ) 26

27 Nonlinear Tearing and Interconnecting Global Problem t i = T i [f, g](u i ) u i = u j t i + t j = 0 on Γ i on Γ ij on Γ ij Eliminating (t i ) Nonlinear Variational Skeleton Problem T i [f, g](u) v ds x = 0, v H 1/2 ( i I Γ i \ Γ). i I Γ i \Γ Discrete approximation of the nonlinear operators: T FEM/BEM i,h [f, g], T FEM/BEM i,h [f, g] 27

28 Nonlinear Tearing and Interconnecting Nonlinear FETI/BETI System T FEM/BEM i,h [f, g](u i ) + B T i λ = 0, i I, B i u i = 0. Eliminate (u i ) via solution of local (nonlinear) problems Define i I u i = T FEM/BEM i,h [f, g] ( B T i λ) + γ i e i }{{} + constant F (λ) := i I B i T FEM/BEM i,h [f, g] ( B T i λ), G := (B i e i ) i I, Ωi floating, 28

29 Nonlinear Tearing and Interconnecting Nonlinear Dual Equation F (λ) + Gγ = 0 G T λ = 0 Linear Projection P := I G(G T G) 1 G T Projected Nonlinear Dual Equation P F (λ) = 0 Solution λ γ (u i ) global field u h 29

30 Nonlinear Tearing and Interconnecting - To Do Existence of a unique solution λ (Regularity of F ( )) Fixpoint or Newton Iteration + Analysis Preconditioning 30