FEM-FEM and FEM-BEM Coupling within the Dune Computational Software Environment

Transcription

1 FEM-FEM and FEM-BEM Coupling within the Dune Computational Software Environment Alastair J. Radcliffe Andreas Dedner Timo Betcke Warwick University, Coventry University College of London (UCL) U.K. Radcliffe (Warwick + UCL UK) FEM-FEM / FEM-BEM Coupling Heidelberg, July 14-16, / 20

2 Outline - Ultimate Objective: FEM-BEM Coupling; - Model Elliptic Problem; - Iterative versus Non-iterative; - Overlapping v. Non-overlapping; - Dirichlet-Neumann Coupling (D-N) - - Example: side by side boxes (D-N); - Robin Coupling (R-R) - - Example: overlapping nested boxes (R-R); - Dirichlet-Dirichlet Coupling (D-D) - - Example: bulk-surface coupling (D-D). Radcliffe (Warwick + UCL UK) FEM-FEM / FEM-BEM Coupling Heidelberg, July 14-16, / 20

3 Target Application: Sonic Brain Tumour Heating Radcliffe (Warwick + UCL UK) FEM-FEM / FEM-BEM Coupling Heidelberg, July 14-16, / 20

4 Target Application: Coulomb Explosions Figure : Deformation pathways of highly electrically charged fluid droplets. Radcliffe (Warwick + UCL UK) FEM-FEM / FEM-BEM Coupling Heidelberg, July 14-16, / 20

5 Ultimate Objective: FEM-BEM Coupling Figure : Computational domains and boundaries for the general problem. Radcliffe (Warwick + UCL UK) FEM-FEM / FEM-BEM Coupling Heidelberg, July 14-16, / 20

6 Model Elliptic Problem Consider the following general elliptic problem defined on Ω: L[u Ω ] = f (1) and a linear elliptic problem defined on Ω L [u ] = g = 0 (2) satisfying a compatibility condition on the interface Ω Ω of the form L[u Ω ] = L [u ], f = 0 on Ω Ω (3) We will focus on the Laplace cases: L = L = or = + I (4) Consider the following two coupling conditions: γu Ω + δ u Ω n = γu + δ u n on Γ (5) γ u + δ u n = γ u Ω + δ u Ω n on Γ (6) Non-overlapping implies Γ = Γ & γ = γ & δ = δ. Radcliffe (Warwick + UCL UK) FEM-FEM / FEM-BEM Coupling Heidelberg, July 14-16, / 20

7 Iterative versus Non-Iterative Coupling - FEM and BEM very different approaches; - Sparse matrices for FEM; - Dense matrices for BEM; - Different preconditioning/solution strategies needed for each: eg fast multipole for BEM; - A single mixed sparse and dense system matrix difficult to precondition and solve optimally; - Each problem better solved seperately with custom preconditioners and solvers; - Iterative exchange of information between schemes until convergence; - Both Dirichlet (D) and Neumann (N) data can be exchanged, or Robin (R) combinations of both. Radcliffe (Warwick + UCL UK) FEM-FEM / FEM-BEM Coupling Heidelberg, July 14-16, / 20

8 Overlapping versus Non-Overlapping - Typical trade-off between cost and solution: - Faster convergence (fewer iterations) with greater overlapping but greater solution duplication on overlap ( waste of computational effort); - Difficulties defining a conforming smooth surface within an existing volume mesh for over-lapping?; - Non-conforming BEM surface boundary within volume mesh would require interpolation and thus loss of accuracy? Radcliffe (Warwick + UCL UK) FEM-FEM / FEM-BEM Coupling Heidelberg, July 14-16, / 20

9 Implementation within Dune Software Environment - Dune-Fem module using Dune core modules for FEM part; - First-order linear FE shape functions η and ξ used; - Surface-Extraction from volume mesh to form surface mesh for mixed dimension coupling; - Filtering of initial volume mesh before surface extraction if overlap required; - Dune-Grid-Glue used to match elements from the two different meshes up for updates; - Bem++ used to solve problem on extracted surface mesh; - Iteration between Dune-Volume and Bempp-Surface solutions until convergence. Radcliffe (Warwick + UCL UK) FEM-FEM / FEM-BEM Coupling Heidelberg, July 14-16, / 20

10 Some schemes Some iterative schemes for FEM-FEM coupling... (... as a precursor to FEM-BEM coupling...) Radcliffe (Warwick + UCL UK) FEM-FEM / FEM-BEM Coupling Heidelberg, July 14-16, / 20

11 Non-overlapping (D-N) Schwarz scheme (Daoqi Yang) Dirichlet step (γ = 1 & δ = 0), Weighting parameter α L[u 2 n+1 ] = f on Ω L [v 2 n+1 ] = g on Ω u 2 n+1 = α u 2 n + (1 α) v 2 n on Γ v 2 n+1 = α u 2 n + (1 α) v 2 n on Γ Neumann step (γ = 0 & δ = 1), Weighting β, φ = u n ψ = v n L[u 2 n+2 ] = f on Ω L [v 2 n+2 ] = g on Ω φ 2 n+2 = βφ 2 n+1 + (1 β) ψ 2 n+1 on Γ ψ 2 n+2 = βφ 2 n+1 + (1 β) ψ 2 n+1 on Γ Radcliffe (Warwick + UCL UK) FEM-FEM / FEM-BEM Coupling Heidelberg, July 14-16, / 20

12 Side by Side Boxes Example (α = β = 1/2) This is Codim1Extractor on a <3,3> grid! added 8 subfaces Error at iteration 11 = = This is Codim1Extractor on a <3,3> grid! added 32 subfaces Error at iteration 11 = = EOC( 1 ) = This is Codim1Extractor on a <3,3> grid! added 128 subfaces Error at iteration 11 = = EOC( 2 ) = Radcliffe (Warwick + UCL UK) FEM-FEM / FEM-BEM Coupling Heidelberg, July 14-16, / 20

13 Side by Side Boxes Parameter Choices α + β choice do not seem to influence number of iterations required; β significantly removed from 0.5 causes greater oscillations (between Dirichlet and Neumann steps) about final solution value. Note: Neumann step with L = L = (as above) only solvable if an actual (non-coupling) boundary is always present (as above). In the absence of such a boundary, require at least L = L = + I (as next) to solve with purely Neumann data on the boundary. Radcliffe (Warwick + UCL UK) FEM-FEM / FEM-BEM Coupling Heidelberg, July 14-16, / 20

14 Overlapping Scheme (Chen, Xu & Zhang) Variant Maintain TWO collections of Robin data g m 1 = γ 1 u m + φ m = γ 1 v m + ψ m g m 2 = γ 2 u m φ m = γ 2 v m ψ m To be used in the RHS s of two problems (now let L = L = + I) ( u η + uη) + γ 1 uη = f η + g 1 η Ω Γ Ω Γ ( v ξ + v ξ) + γ 2 vξ = gξ + g 2 ξ Ω Dirichlet updates to Robin data (no need to calculate derivatives) g m+1 2 = g m 1 + (γ 2 + γ 1 ) u m g m+1 1 = g m 2 + (γ 1 + γ 2 ) v m g m+1 1 = θ g m 1 + (1 θ) gm 1 Will use following variant (γ 2 = 1 γ 1 ) requiring derivatives g m+1 1 = γ 1 [α v m + (1 α) u m ] + β ψ m + (1 β) φ m g m+1 2 = γ 2 [α u m + (1 α) v m ] + β φ m + (1 β) ψ m Γ Radcliffe (Warwick + UCL UK) FEM-FEM / FEM-BEM Coupling Heidelberg, July 14-16, / 20 Γ Γ

15 Nested Boxes Example (α = β = 1/2, γ = 0.7) This is Codim1Extractor on a <3,3> grid! added 48 subfaces This is Codim1Extractor on a <3,3> grid! added 192 subfaces Error at iteration 8 = = This is Codim1Extractor on a <3,3> grid! added 192 subfaces This is Codim1Extractor on a <3,3> grid! added 768 subfaces Error at iteration 9 = = EOC( 1 ) = This is Codim1Extractor on a <3,3> grid! added 768 subfaces This is Codim1Extractor on a <3,3> grid! added 3072 subfaces Error at iteration 9 = = EOC( 2 ) = junk Radcliffe (Warwick + UCL UK) FEM-FEM / FEM-BEM Coupling Heidelberg, July 14-16, / 20

16 Nested Boxes Parameter Choices γ 1 α β iterations error γ 1 α β iterations error γ 1 α β iterations error Reminder: g m+1 1 = γ 1 [α v m + (1 α) u m ] + β ψ m + (1 β) φ m g m+1 2 = γ 2 [α u m + (1 α) v m ] + β φ m + (1 β) ψ m Radcliffe (Warwick + UCL UK) FEM-FEM / FEM-BEM Coupling Heidelberg, July 14-16, / 20

17 Coupled Bulk-Surface FEM-FEM (Elliott and Ranner) Governing equations and b.c. s u + u = f in Ω u v + u n = 0 on Γ Γ v + v + u n = g on Γ Force with f and g found from a chosen analytic solution u(x, y, z) = exp [ x (x 1) y (y 1)] v(x, y, z) = [1 + x (1 2 x) + y (1 2 y)] exp [ x (x 1) y (y 1)] Weak formulation of above suggests a D-D iterative coupling... ( u η + uη) + uη = f η + vη Ω Γ Ω Γ ( Γ v Γ ξ + v ξ) + vξ = gξ + uξ Γ Γ Radcliffe (Warwick + UCL UK) FEM-FEM / FEM-BEM Coupling Heidelberg, July 14-16, / 20 Γ Γ

18 Bulk-Surface Example This is Codim1Extractor on a <3,3> grid! added 48 subfaces Error at iteration 10 = = EOC( 1 ) = This is Codim1Extractor on a <3,3> grid! added 192 subfaces Error at iteration 10 = = EOC( 2 ) = This is Codim1Extractor on a <3,3> grid! added 768 subfaces Error at iteration 10 = = EOC( 3 ) = This is Codim1Extractor on a <3,3> grid! added 3072 subfaces Error at iteration 10 = = EOC( 4 ) = junk Radcliffe (Warwick + UCL UK) FEM-FEM / FEM-BEM Coupling Heidelberg, July 14-16, / 20

19 Conclusions - Iterative the best approach for FEM-BEM coupling to allow the use of specialist preconditioners for each solver type; - Robin data exchange the most useful and flexible; - Effect of differing overlaps to be investigated; - Work to replace surface FEM solver (Dune) in a suitable new bulk-surface problem above with a BEM solver (Bem++) now underway... Radcliffe (Warwick + UCL UK) FEM-FEM / FEM-BEM Coupling Heidelberg, July 14-16, / 20

20 The End Thank you for your attention!!! * Daoqi Yang A Parallel Nonoverlapping Schwarz Domain Decomposition Method for Elliptic Interface Problems, IMA preprint #1508, University of Minnesota, August 1997; * Wenbin Chen, Xuejun Xu, Shangyou Zhang On the optimal convergence rate of a Robin-Robin domain decomposition method Journal of Computational Mathematics; * Charles M. Elliott and Thomas Ranner Finite element analysis for a coupled bulk surface partial differential equation IMA Journal of Numerical Analysis Radcliffe (Warwick + UCL UK) FEM-FEM / FEM-BEM Coupling Heidelberg, July 14-16, / 20