A posteriori estimates and mesh adaptation for the thermistor - PowerPoint PPT Presentation

A posteriori estimates and mesh adaptation for the thermistor problem Claire CHAUVIN ∗ , Christophe TROPHIME ∗ , Pierre SARAMITO † claire.chauvin@grenoble.cnrs.fr ∗ Grenoble High Magnetic Field Laboratory † Laboratoire Jean Kuntzmann CNRS, 25 avenue des Martyrs, Grenoble Campus Universitaire, Grenoble claire.chauvin@grenoble.cnrs.fr (claire.chauvin@grenoble.cnrs.fr) 1 / 14

Outline The Grenoble High Magnetic Field Lab 1 The thermistor problem: study and numerical resolution 2 A posteriori estimate 3 Application to adaptation 4 (claire.chauvin@grenoble.cnrs.fr) 2 / 14

1. The Grenoble High Magnetic Field Lab Resistive magnets (34 Tesla , 30000 A , 20 MW ) Water cooling, high flow (20 l / s ) Polyhelice Geometrical optimization [1]. Heating effect and mechanical stress: Joule, Lorentz. (claire.chauvin@grenoble.cnrs.fr) 3 / 14

1. The Grenoble High Magnetic Field Lab Potential φ in a cut of a 3D Magnet... ... on an helix under some simplifications. Magnetic field b and current density j : ∀ x ∈ Ω , j ( x ) = σ ( u ) ∇ φ ( x ) , Z ∀ x ∈ ω , b ( x ) = j ( y ) ∧ ∇ G ( x , y ) dy , µ Ω Estimate on the numerical error on b ? (claire.chauvin@grenoble.cnrs.fr) 4 / 14

2. The thermistor problem Model Find ( φ , u ) : Ω → R s.t.:  − div ( σ ( u ) ∇φ ) = f in Ω ,   σ ( u ) | ∇φ | 2 − div ( κ ( u ) ∇ u ) = in Ω ,     φ = φ 0 on Γ 1 ,  ( C ) − σ ( u ) ∇φ . n = on Γ 2 , 0   = on Γ 2 , u u w     − κ ( u ) ∇ u . n = on Γ 1 . 0  Ω , ω ⊂ R 2 , ω ∩ Ω = ∅ . σ et κ bounded, Lipschitz-continuous on R + ∗ . Difficulties Geometry: highly non convex, fissures, holes. Numeric: mesh, method? A posteriori estimate on b ? (claire.chauvin@grenoble.cnrs.fr) 5 / 14

2. The thermistor problem Model Let Ω be a polygonal domain, and ω i the interior angle between two consecutive edges of Ω . Let ω ∗ i s.t.: ω ∗ i = ω i if the two edges have the same BC, ω ∗ i = 2 ω i else. Let ω ∗ = max i ( ω ∗ i ) If κ , σ : Ω �→ R , ∈ C m (Ω) and f ∈ L s (Ω) , s > 1 . ⇒ u , φ ∈ H 1 + 2 / q (Ω) , q > max ( q ∗ , 2 ) , q ∗ = 2 πω ∗ = i ( C h ) : Find u h ∈ V h and φ h ∈ W h s.t. Z Z Ω σ ( u h ) | ∇φ h | 2 v h d x , ∀ v h ∈ V h Ω κ ( u h ) ∇ u h . ∇ v h d x = Z Z Ω σ ( u h ) ∇φ h . ∇ψ h d x = f ψ h d x , ∀ ψ h ∈ W h Ω (claire.chauvin@grenoble.cnrs.fr) 6 / 14

2. The thermistor problem Model If more general conditions on Ω , well-posed problem, associated to the limit-problem [2,3]: Let τ ( θ ) = σ o κ − 1 ( u ) , ( C ′ ) : Find θ ∈ L 2 (Ω) and φ ∈ H 1 0 (Ω) s.t. Z Z Ω τ ( θ ) | ∇φ | 2 (∆ − 1 ξ ) d x , ∀ ξ ∈ L 2 (Ω) Ω θ . ξ d x = Z Z f ψ d x , ∀ ψ ∈ H 1 (Ω) Ω τ ( θ ) ∇φ . ∇ψ d x = Ω 0 (Ω) , s < 1 / 2 solutions of ( C ′ ) . ⇒ ∃ ( θ , φ ) ∈ H s (Ω) × H 1 = (claire.chauvin@grenoble.cnrs.fr) 7 / 14

2. The thermistor problem Numerical simulation Relaxed Fixed-Point Algorithm: Z Z Z u n + 1 v d x +∆ t Ω σ ( u n ) | ∇φ n + 1 | 2 v d x Ω κ ( u n ) ∇ u n + 1 ∇ v d x = ∆ t Ω Z u n v d x + ∀ v ∈ V h , Ω Z Z Ω σ ( u n ) ∇φ n + 1 ∇ w d x = ∀ w ∈ W h , f w d x Ω u n + 1 = ∑ u n + 1 v i , v i finite element of order k on quads. i i (gmsh, deal.ii) Stop when � u n + 1 − u n � L 2 < ε . (claire.chauvin@grenoble.cnrs.fr) 8 / 14

3. A posteriori estimates Looking forward an estimate η s.t. � u − u h � H 1 + � φ − φ h � H 1 � η � h p . Theoretical estimate [5] with residual and edge terms. Kelly estimate of u defined on an element K [6]: � 2 � κ ( u h ) ∂ u h K ( u ) = h Z η 2 ⇒ convergence order γ ds = ∂ n 24 ∂ K Comparison of γ + 1 with λ solution of [3]: � u hn − 1 − u hn − 2 � L 2 (Ω) � u hn − u hn − 1 � L 2 (Ω) = ( h n − 2 / h n − 1 ) λ − 1 ( 1 ) 1 − ( h n / h n − 1 ) λ = ⇒ Validation of Kelly estimate. = ⇒ Numerical indicators for the error: γ et λ . (claire.chauvin@grenoble.cnrs.fr) 9 / 14

3. A posteriori estimates Comparison with an analytical solution | φ − φ h | H 1 (Ω) η φ Error | φ h | H 1 (Ω) . Estimate | φ h | H 1 (Ω) . Q 1 Q 2 Q 3 λ γ λ γ λ γ rel. err. rel. err. rel. err. φ 1.99 1.97 0.84 2.97 2.95 2.73 3.99 3.51 2.85 u 2.00 2.00 0.98 2.98 2.54 2.49 3.96 2.41 2.96 Convergence orders for φ and u and several Q k . rel. err. = � u − u h � L 2 . � u � L 2 (claire.chauvin@grenoble.cnrs.fr) 10 / 14

4. Application to adaptation η φ Non convex geometry η φ + η u Refinment criteria: η u u (claire.chauvin@grenoble.cnrs.fr) 11 / 14

4. Application to adaptation Non convex geometry Q 1 Q 2 Q 3 γ λ γ λ γ λ φ 1.42 0.70 1.43 0.67 1.45 0.67 u 1.52 0.55 1.82 0.53 1.76 0.54 Convergence orders for φ and u and several Q k . ⇒ Ok with the theory, u and φ ∈ H s , with s > 1 . 66. = (claire.chauvin@grenoble.cnrs.fr) 12 / 14

Conclusion and perspectives Numerical study of an efficient A posteriori estimate for the thermistor problem [3]. Efficiency of the adaptive scheme. Limitation by hanging nodes on quads? Future developpements: A posteriori error estimate for the magnetic field, by means of a optimal control approach [7]. (claire.chauvin@grenoble.cnrs.fr) 13 / 14

References [1] C. Trophime et al., Magnetic Field Homogeneity Optimization of the Giga-NMR Resistive Insert , IEEE Trans. Appl. Supercond., 16 (2), 1509–1512, 2005. [2] C. Bernardi et al., A model for two coupled turbulent fluids. Part I and III . [3] C. Chauvin et al., A posteriori estimates and adaptive FEM for the thermistor problem, en préparation. [4] W. Allegretto et al., A posteriori error analysis for FEM of thermistor problems , Int. J. Numer. Anal. Model., 3 (4), 413–436, 2006. [5] D.W. Kelly et al., A posteriori error analysis and adaptive processes in the finite element method - Part I: Error analysis , Int. J. Num. Meth. Engrg., 19 , 1593–1619, 1983. [6] W. Bangerth, R. Hartmann et G. Kanschat, deal.II — a General Purpose Object Oriented Finite Element Library , ACM Trans. Math. Software, 33 (4), 2007. [7] R. Becker et R. Rannacher, A feed-back approach to error control in finite element methods: basic analysis and examples , East-west J. Num. Math., 4 , 237–264, 1996. (claire.chauvin@grenoble.cnrs.fr) 14 / 14