Fully nonlinear FETI method

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(1)Fully nonlinear FETI method Pierre Gosselet, Camille Negrello, Christian Rey. To cite this version: Pierre Gosselet, Camille Negrello, Christian Rey. Fully nonlinear FETI method. Parallel Solution Methods for Systems Arising from PDEs, Sep 2019, CIRM-Luminy, France. 2019. �hal-02424753�. HAL Id: hal-02424753 https://hal.archives-ouvertes.fr/hal-02424753 Submitted on 28 Dec 2019. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) Fully nonlinear FETI method Pierre Gosselet (with Camille Negrello and Christian Rey) LaMcube – Univ. Lille, CNRS, Centrale Lille, FRE 2016 – F-59000, Lille, France DDM for nonlinear problems • (SNK) Solve the nonlinearity at the scale of the subdomains: • (NKS) Newton solver in outer loop, linear DD (Krylov) in inner loop. – Schwarz DD: ASPIN (Cai and Keyes 2002), RASPEN (Dolean et al. 2016), LATIN (Ladevèze, Néron, and – Schwarz DD (Cai, Gropp, et al. 1994); Gosselet 2007), (Badea 2009), global/local non-invasive coupling (Gosselet et al. 2018). – Schur DD (FETI) (Bhardwaj et al. 1999). . . . – Schur DD, “nonlinear relocalization”: primal (Dirichlet) and mixed (Robin) approach (Cresta et al. 2007), dual approach (Pebrel, Rey, and Gosselet 2008) and nonlinear FETIDP and BDDC (Klawonn, Lanser, and – Coupling with inexact Newton technique (Biros and Ghattas 2003). Rheinbach 2014) with large scale assessments in (Klawonn, Lanser, Rheinbach, and Uran 2017a). → Primal/dual/mixed approaches (Negrello, Gosselet, Rey, and Pebrel 2016), Improvement of the impedance of the mixed approach (Negrello, Gosselet, and Rey 2017).. Nonlinear PDE + FE +DD. Fully nonlinear FETI. Decomposed version (block notations): B ub = 0 T gint (u) + gext + T λb = 0 with A λb = 0 • λb : unknown local nodal reaction,        . Nonlinear fixed point T. . . λ̂b = I − Ã A λb ; 0 = R. T. . T. gext + T λb.       . • A / B : primal/dual assembly,. Nonlinear PDE representative of quasi-static structure mechanics:. • Ã / B̃ : scaling AÃT = I, BB̃T B = B AT Ã + B̃T B = I. gint(u) + gext = 0 in Ω. • Small strain hypothesis: ∃ rigib body motions R ∀u, α, gint(u + Rα) = gint(u) RT gint = 0 Let GB = BTR. Assumptions: • gint is differentiable, tangent matrices SPsD • Global problem is well posed • Classical Newton-Raphson would converge.. ub = fnl λ̂b +Rbα gext. gext // Neumann solve optimize α. Find uA ∈ R. T. . . . . Dual approach. Compatibility condition: RT gext + TT λb = 0 Local Neumann-Dirichlet operator ub = fnl (λb; gext) + TRα −1 T PB = I − QB GB GB QB GB GBT λB = λB0 + PB λ̃B with: , QB SPD matrix −1 T T λB0 = −QB GB GB QB GB R gext α can be computed by minimizing the QB -norm of displacement jump.                     . Find λ̃B ∈ R. nB. such that:. . . . . . . . . PBT B fnl BT (λB0 . Scaling. Unbalanced traction λb. Continuous displacement ûb λb = snl(ûb) gext. // Dirichlet solve RBM projector. gext . Aλb 6= 0. T. . ûb = I − B̃ B ub 6= 0. (Rixen and Farhat 1999; Parret-Fréaud et al. 2010). . . . .   . . . . .  .  . • One Newton iteration, starting from λ̂b = BT (λB0 + PB λ̃B ):. such that : A snl A uA; gext = 0 . Scaling. • No contraction ⇒ Krylov solver in linear, Newton in nonlinear. . . Displacement gap. • Identical to FETI with Dirichlet preconditioner in linear. Primal approach. Local Dirichlet–Neumann operator: λb = snl (ub; gext) . Balanced admissible traction λ̂b. Find λ̃B s.t. λ̂b = BT (λB0 + PB λ̃B ) satifies fixed point T T T I − Ã A snl I − B̃ PB B fnl λ̂b − λ̂b = 0. Condensed NL formulations nA. Bub 6= 0 !. • T : trace operator, • subscript b : boundary dof. . . + PB λ̃B ); gext = 0 . → Newton solver, tangent system is usual FETI (Krylov solver + Dirichlet preconditioner). // Solve independant nonlinear Neumann problems ub = fnl(λ̂b) – Assemble displacement gap, compute continuous displacement ûb // Solve independant nonlinear Dirichlet problems λb = B̃snl(ûb) 0 – Assemble reaction lack-of-balance, compute balanced reaction λ̃B – Krylov solver for the (distributed) interface tangent system 0 0 T T T B̃(Dsnl ) I − B̃ PB B (Dfnl) B PB δ = −PB (λ̃B − λ̃B ) – Update reaction λ̂b ← λ̂b + BT PB δ . . . . . . • Equivalent to classical NKS if no inner Newton iterations • Equivalent to FETI-NL if no inner Newton iterations for Dirichlet problem. Illustration: nonlinear thermal problem BDD-NL. Heat equation: ∇ · (K (T ) ∇T ) = 0 Non-classical behavior allowing “rigid body motions”: K (T ) = (1 + ∇T · ∇T )α. #SD α=0 α = 0.2 α = 0.4 α = 0.6. 8 1 6 7 7. 16 1 6 6 6. 32 1 7 8 8. y u = uD2 (x). u = uD1 (x). H x. Fig. 2: Geometry and Dirichlet BCs. BDD-NL. W. #SD α=0 α = 0.2 α = 0.4 α = 0.6. 8 8 20 22 21. 16 8 21 22 21. 32 14 30 31 38. BDD-NL Fig. 3: Temperature map (α = 0.6). #SD α=0 α = 0.2 α = 0.4 α = 0.6. 8 1 11 15 18. 16 1 13 15 17. 32 1 17 29 20. # Global Newton iterations Gains of FETI-precNL (%) FETI-NL FETI-precNL vs. BDD-NL vs. FETI-NL tions, a r u 8 16 32 8 16 32 8 16 32 8 16 32 g fi n o cc i d t e s r o p r e t In m t e b ( ton w e N r e t u o f o e 1 1 1 1 1 1 0 0 0 0 0 0 eracreas t e i d v t o n l a y c r fi K i n r g e i n S • nd in a ) r o i v a h e b r a 5 6 5 3 3 4 50 50 43 40 50 20 tion of nonline ) s n o i t a c i n u m m co 7 8 9 3 3 4 57 50 50 57 63 56 ons) t w e tions (all-to-all N r e n n i ( s tation u p m o c l e l l a 8 7 9 4 4 5 43 33 38 50 43 44 r a p • Limited extra # Krylov iterations Gains of FETI-precNL (%) FETI-NL FETI-precNL vs. BDD-NL vs. FETI-NL 8 16 32 8 16 32 8 16 32 8 16 32 8 8 15 8 8 15 0 0 -6 0 0 0 17 24 32 17 17 29 15 19 3 0 29 9 20 25 40 16 15 27 27 32 13 20 40 33 • Large scale implementation for significant assesse25 23 43 22 20 33 -5 5 13 12 13 23 ment and more general nonlinearities. max # Local Newton iterations • Introducing robustness-enhancing techniques Ratio of FETI-precNL FETI-NL FETI-precNL (adaptive coarse space, multipreconditioning) in vs. BDD-NL vs. FETI-NL 8 16 32 8 16 32 8 16 32 8 16 32 this framework. 1 1 1 2 2 2 2. 2. 2. 2. 2. 2. • Improvement of the load-balancing. 11 14 14 18 14 17 1.6 1.1 1 1.6 1 1.2 18 20 33 22 18 20 1.5 1.2 1.1 1.2 0.9 0.6 24 24 34 29 24 29 1.6 1.4 1.5 1.2 1 0.9. Work in progress. LATEX TikZposter.

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