Fast Boundary Element Methods Fast Boundary Element Methods ur - PowerPoint PPT Presentation

Fast Boundary Element Methods Fast Boundary Element Methods ur Angewandte Analysis und Numerische Simulation SFB 404 Mehrfeldprobleme in der Kontinuumsmechanik • surface triangulation of complex structures • exterior boundary value problems • moving boundaries • direct computation of the complete Cauchy data at Stuttgart Universit¨ Institut f¨ Special Radon Semester, Linz, November 16, 2005 – p.1/3

Fast Boundary Element Methods Fast Boundary Element Methods ur Angewandte Analysis und Numerische Simulation SFB 404 Mehrfeldprobleme in der Kontinuumsmechanik Indirect ansatz for the Dirichlet boundary value problem of the Laplacian: 1 1 � u ( x ) = ( V w )( x ) := | x − y | w ( y ) ds y = g ( x ) for x ∈ Γ = ∂ Ω . 4 π Γ Fast Multipole Method : (related to Hierarchical Matrices) • separation of variables for the kernel by spherical harmonics • hierarchy provides fast application of matrix vector multiplication ( O ( N log 2 N )) Preconditioning : BPX, AMG, boundary integral operator of opposite order at Stuttgart Applications : • Laplace • Collision detection Universit¨ Institut f¨ • Linear Elastostatic • Electromagnetics (Breuer) • Stokes system • Acoutics (Fischer, Gaul) Special Radon Semester, Linz, November 16, 2005 – p.2/3

Domain Decomposition Methods Domain Decomposition Methods ur Angewandte Analysis und Numerische Simulation Boundary Element Tearing and Interconnecting method (BETI) [Langer, Steinbach 2003] (FETI [Farhat, Roux 1991; Klawonn,Widlund 2001; . . . ]): SFB 404 Mehrfeldprobleme in der Kontinuumsmechanik u i − u j = 0 local vectors u i = A i u Ω 1 Ω 2 Ω 1 Ω 2 Ω 1 Ω 2 Ω 3 Ω 4 Ω 3 Ω 4 Ω 3 Ω 4 at Stuttgart Cooperations: Outlook: • saddle point formulations • real life applications Universit¨ Institut f¨ • AMG • coupling with finite • coupling elements • nearly incompressible • inverse problems, materials optimization Special Radon Semester, Linz, November 16, 2005 – p.3/3