An h-adaptive unfitted finite element method for interface elliptic - PowerPoint PPT Presentation

An h-adaptive unfitted finite element method for interface elliptic boundary value problems Eric Neiva 1 , 3 Santiago Badia 2 , 3 Monash Workshop on Numerical Differential Equations and Applications 2020, MWNDEA 2020 , Feb. 2020. 1 Universitat Politècnica de Catalunya (UPC), BCN, Spain. 2 Centre Internacional de Mètodes Numèrics en Enginyeria (CIMNE), BCN, Spain. 3 Monash University, Clayton, Victoria, Australia. Supported by: Grant-id.: 2017FIB00219

Towards multiphysics and multiscale applications with h-AgFEM Previous talk In this talk Unfitted interfaces : in Ω in Y Ω out ´ ∇ ¨ p α ∇ u q “ f Unfitted boundaries : u “ 0 on B Ω ´ ∆ u “ f in Ω � u � “ 0 on Γ u “ u D n out ¨ � α ∇ u � “ 0 on B Ω on Γ e.g. Powder-bed 3D printing POWDER SOLID Building platform E. Neiva ¨ UPC–Monash ¨ 2020 2/16

An overview of unfitted FE methods for interface problems Interface Poisson problem Structure of approximation in Ω in Y Ω out ´ ∇ ¨ p α ∇ u q “ f u “ 0 on B Ω � u � “ 0 on Γ n out ¨ � α ∇ u � “ 0 on Γ Approximation method Known results A h p u h , v h q . “ p α ∇ u h , ∇ v h q Ω in Y Ω out • Naive SIPM+FEM: τ unbounded for arbitrarily small cut size [HH02] u ¨ n ` , � v h � @ D ´ t t α ∇ u h u Γ • Robustness to cut location via, e.g., u ¨ n ` D ´ @ � u h � , t t α ∇ v h u stabilization (CutFEM) [Bur+15] Γ ` x τ � u h � , � v h � y Γ • Robustness to material contrast via harmonic t t¨u u and diag. prec. [Bur+15] SIPM or other Nitsche formulations E. Neiva ¨ UPC–Monash ¨ 2020 3/16

Is AgFEM suitable for interface elliptic BVPs? Contents: • Construction of aggregated FE spaces • Approximation of the interface problem • Numerical analysis • Numerical experiments • Verification in uniform meshes • Robustness w.r.t. cut location • Robustness w.r.t. material contrast • Robustness and optimality in tree-based meshes E. Neiva ¨ UPC–Monash ¨ 2020 4/16

Construction of AgFE spaces in interface problems Recalling the rationale: improve conditioning by removing problematic DOFs $ , & . ÿ V agg : “ % u P V h : u ˆ “ C ˆ‚ u ‚ @ˆ P P h ‚P masters pˆq - touched ‚ well-posed dofs untouched ˆ problematic dofs ( P ) aggregated E. Neiva ¨ UPC–Monash ¨ 2020 5/16

Construction of AgFE spaces in interface problems Recalling the rationale: improve conditioning by removing problematic DOFs $ , & . ÿ V agg : “ % u P V h : u ˆ “ C ˆ‚ u ‚ @ˆ P P h ‚P masters pˆq - touched ‚ well-posed dofs untouched ˆ problematic dofs ( P ) aggregated E. Neiva ¨ UPC–Monash ¨ 2020 5/16

Construction of AgFE spaces in interface problems Recalling the rationale: improve conditioning by removing problematic DOFs $ , & . ÿ V agg : “ % u P V h : u ˆ “ C ˆ‚ u ‚ @ˆ P P h ‚P masters pˆq - touched ‚ well-posed dofs untouched ˆ problematic dofs ( P ) aggregated E. Neiva ¨ UPC–Monash ¨ 2020 5/16

Construction of AgFE spaces in interface problems Recalling the rationale: improve conditioning by removing problematic DOFs $ , & . ÿ V agg : “ % u P V h : u ˆ “ C ˆ‚ u ‚ @ˆ P P h ‚P masters pˆq - touched ‚ well-posed dofs untouched ˆ problematic dofs ( P ) aggregated E. Neiva ¨ UPC–Monash ¨ 2020 5/16

Construction of AgFE spaces in interface problems Recalling the rationale: improve conditioning by removing problematic DOFs $ , & . ÿ V agg : “ % u P V h : u ˆ “ C ˆ‚ u ‚ @ˆ P P h ‚P masters pˆq - touched ‚ well-posed dofs untouched ˆ problematic dofs ( P ) aggregated E. Neiva ¨ UPC–Monash ¨ 2020 5/16

Construction of AgFE spaces in interface problems Recalling the rationale: improve conditioning by removing problematic DOFs $ , & . ÿ V agg : “ % u P V h : u ˆ “ C ˆ‚ u ‚ @ˆ P P h ‚P masters pˆq - touched ‚ well-posed dofs untouched ˆ problematic dofs ( P ) aggregated E. Neiva ¨ UPC–Monash ¨ 2020 5/16

Construction of AgFE spaces in interface problems In our context: constrain by cell aggregation at both subregions Ñ V in , agg Ñ V out , agg h h V agg “ V in , agg ˆ V out , agg h h h Analogously, given n different materials, V agg “ V 1 , agg ˆ . . . ˆ V n , agg . h h h E. Neiva ¨ UPC–Monash ¨ 2020 6/16

Approximation of interface problem with Nitsche’s method Local FE operators: For any K P T h , we define A K p u h , v h q . . “ a bulk p u h , v h q ` a Γ “ l bulk K p u h , v h q , l K p v h q p v h q , K K with ´ ¯ a bulk α in ∇ u in h , ∇ v in α out ∇ u out h , ∇ v out p u h , v h q “ Ω in X K ` ` ˘ Ω out X K , K h h A E l bulk f in , v in f out , v out @ D p v h q “ Ω in X K ` Ω out X K , K h h and n out ¨ t � u h � , n out ¨ t a Γ @ D @ D K p u h , v h q “ ´ t α ∇ u h u u , � v h � Γ X K ´ t α ∇ v h u u Γ X K ` x τ K � u h � , � v h � y Γ X K , where τ K “ β K h ´ 1 K , with β K ą 0 large enough, is a stabilization parameter and weighted average t t¨u u given by α out α in α in ` α out p¨q in ` α in ` α out p¨q out . E. Neiva ¨ UPC–Monash ¨ 2020 7/16

Approximation of interface problem with Nitsche’s method Local FE operators: For any K P T h , we define A K p u h , v h q . . “ a bulk p u h , v h q ` a Γ “ l bulk K p u h , v h q , l K p v h q p v h q , K K with ´ ¯ a bulk α in ∇ u in h , ∇ v in α out ∇ u out h , ∇ v out p u h , v h q “ Ω in X K ` ` ˘ Ω out X K , K h h A E l bulk f in , v in f out , v out @ D p v h q “ Ω in X K ` Ω out X K , K h h and n out ¨ t � u h � , n out ¨ t a Γ @ D @ D K p u h , v h q “ ´ t α ∇ u h u u , � v h � Γ X K ´ t α ∇ v h u u Γ X K ` x τ K � u h � , � v h � y Γ X K , where τ K “ β K h ´ 1 K , with β K ą 0 large enough, is a stabilization parameter and weighted average t t¨u u given by α out α in α in ` α out p¨q in ` α in ` α out p¨q out . E. Neiva ¨ UPC–Monash ¨ 2020 7/16

interface-AgFEM inherits key boundary-AgFEM inverse inequalities Trace inverse inequality for unfitted boundary [BVM18] For any v h P V agg and K P T Γ h , h } n ¨ ∇ v h } 2 0 , Γ D X K ď C B h ´ 1 K } ∇ v h } 2 0 , Ω K , where Ω K is the domain of the aggregate where K belongs and C B independent of mesh size and cut location . Trace inverse inequality for unfitted interface For any v h P V agg and K P T Γ h , h α in α out › n out ¨ t ´ ¯ › 2 0 , Γ X K ď C B h ´ 1 › › } ∇ v h } 2 0 , Ω K in ` } ∇ v h } 2 t α ∇ v h u u , α in ` α out K 0 , Ω K out is the aggregate domain at Ω in { out and C B independent of mesh size and cut where Ω in { out K location . E. Neiva ¨ UPC–Monash ¨ 2020 8/16

interface-AgFEM inherits key boundary-AgFEM inverse inequalities Trace inverse inequality for unfitted boundary [BVM18] For any v h P V agg and K P T Γ h , h } n ¨ ∇ v h } 2 0 , Γ D X K ď C B h ´ 1 K } ∇ v h } 2 0 , Ω K , where Ω K is the domain of the aggregate where K belongs and C B independent of mesh size and cut location . Trace inverse inequality for unfitted interface For any v h P V agg and K P T Γ h , h α in α out › n out ¨ t ´ ¯ › 2 0 , Γ X K ď C B h ´ 1 › › } ∇ v h } 2 0 , Ω K in ` } ∇ v h } 2 t α ∇ v h u u , α in ` α out K 0 , Ω K out is the aggregate domain at Ω in { out and C B independent of mesh size and cut where Ω in { out K location . E. Neiva ¨ UPC–Monash ¨ 2020 8/16

Summary of numerical analysis Well-posedness and a priori error estimates Let V p h q . 0 p Ω q X H 2 p Ω in Y Ω out q and define for any v P V agg “ V agg ` H 1 the norms h h . V p h q . | v | 2 ÿ β K h ´ 1 K } � v � } 2 } v } 2 “ | v | 2 1 , Ω in Y Ω out ` | v | 2 “ 0 , Γ X K , ˚ , ˚ K P T Γ ˆˇ ˙ 2 . ˇ v in ˇ ˇ 2 ÿ and ||| v ||| 2 “ } v } 2 h 2 ˇ v out ˇ ˇ V p h q ` 2 , Ω ` X K ` . ˇ ˇ V p h q K 2 , Ω ´ X K ˇ K P T cut The following results hold: (stability if β K Á 2 α in α out for all u h P V agg A p u h , u h q Á } u h } 2 α in ` α out ) V p h q h A p u, v q À ||| u ||| 2 V p h q ||| v ||| 2 for all u, v P V p h q (continuity) V p h q for all u h P V agg } u ´ u h } 1 , Ω in Y Ω out À h p (optimal convergence in H 1 ) and u P V p h q h } u ´ u h } 0 , Ω in Y Ω out À h p ` 1 for all u P V p h q and u h P V agg (optimal convergence in L 2 ) h where the constants are independent of cut location . E. Neiva ¨ UPC–Monash ¨ 2020 9/16