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Nonlocal Geometric Motion

Nonlocal Geometric Motion. Martin Burger. University of California University Linz, Austria Los Angeles. Geometric Motion. Joint work with: - Marc Droske, Duisburg  INV - Bo Su, MPI Leipzig  GBM - Günther Bauer, Linz Motivated by GBM I Discussions with:

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Nonlocal Geometric Motion

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  1. NonlocalGeometricMotion Martin Burger University of California University Linz, Austria Los Angeles

  2. Geometric Motion • Joint work with: • - Marc Droske, Duisburg  INV • - Bo Su, MPI Leipzig  GBM • - Günther Bauer, Linz • Motivated by GBM I Discussions with: • - Stan Osher, UCLA • - Omar Lakkis, IACM Crete • - Xiaobing Feng, Tennessee Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

  3. Geometric Motion • Brakke 1980 • Evans & Spruck 1991-1994 • Almgren, Taylor, Wang 1993 • Chen, Giga, Goto 1993 • Ley 2001 • Osher & Shu 1989 • Sussman, Smereka & Osher 1993 • Deckelnick & Dziuk 1999-2002 • Chambolle 2002 • 1980s and 1990s: Understanding and computing local geometric motion (in particular mean curvature flow and related flows) • Osher & Sethian 1987 Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

  4. Geometric Motion • But which geometric motion is local? Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

  5. Geometric Motion • In typical applications, part of the velocity is determined by a non-local (physical) effect! Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

  6. Geometric Motion • Nonlocal parts: • Surface Diffusion • Bulk Diffusion • Heat Transfer • Bulk Elasticity • Velocity from Shape Optimization / Inverse Problem • Curvature Dependent Energies Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

  7. Geometric Motion • > 2000: Understanding and computing nonlocal geometric motion? Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

  8. Surface Diffusion • Motion law for normal velocity : • ... Diffusion coefficient • ... Mean curvature • 4-th order PDE, no maximum principle Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

  9. Surface Diffusion • „Challenge: Devise proofs and computations for motion by surface diffusion, including through topological changes and other singularities“ • Jean Taylor, Some Mathematical Challenges in Material Science, Bull AMS, 2002 Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

  10. Surface Diffusion • Analysis for graph-like infinite curve:Baras, Duchon, Robert 1983 • Existence for smooth curves:Elliott, Garcke 1996 Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

  11. Surface Diffusion • Computation for Graphs/Parametrizations:Bänsch, Morin, Nocchetto 2003 • Computation for Level Sets:Chopp, Sethian 2000 (explicit FD)Smereka 2002 (semi-implicit FD) Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

  12. Surface Diffusion • Level Set Formulation • Weak Form Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

  13. Surface Diffusion • Finite Element Simulation Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

  14. Surface Diffusion • That‘s not enough! But ... Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

  15. Surface Diffusion • We really want: • ... Bulk energy term • ... Deposition term Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

  16. Surface Diffusion • Introduce a new variable • „Chemical potential concentration“, also used by Droske, Rumpf (2003) for Willmore flow • Weak form: Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

  17. Surface Diffusion • Semi-implicit approximation, piecewise linear finite elements: • Solution of discretized problem with • multigrid-preconditioned GMRES Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

  18. Surface Diffusion • Still Open: • -Efficient computation of signed distance functions and velocity extension on triangular grids • -Coupling with bulk effects. • -Analysis Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

  19. Surface Diffusion • Future: couple with elasticity • Butterfly shape transition • Colin, Grilhé, • Junqua, 1998 Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

  20. Surface Diffusion • Coupled with bulk diffusion and elasticity: Quantum Dots • PbSe/PbEuTe • Springholz et. Al. 2001 Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

  21. Surface Diffusion • Bulk term given by strain energy density • with misfit strain • Elastic strain and displacement from • with standard relation Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

  22. Surface Diffusion • SiGe Quantum Dots: • height small compared to width (10nm/150nm) • graph representation • Bauer et. Al., 1999 Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

  23. Surface Diffusion • Anisotropic Surface Diffusion for graphs: • 4th-order PDE for height • Bulk is given by Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

  24. Surface Diffusion • Single initial nucleus, no deposition Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

  25. Surface Diffusion • Random initial surface, random deposition Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

  26. Surface Diffusion • Random initial surface, random deposition Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

  27. Surface Diffusion • Random (rough) surface, no deposition Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

  28. Surface Diffusion • Later time Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

  29. Surface Diffusion • 3 D Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

  30. Surface Diffusion • Open Problems: • - No analysis at all for the model (only weak results for the equilibrium state) • - Bulk term is like a third order term. So far treated explicitely  extremely small time steps. Semi-implicit method coupling bulk solver and interface solvers in future ? Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

  31. Solidification • ... temperature, satisfies • ... latent heat • ... indicator function of Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

  32. Solidification • Isotactic Polypropylene • Eder, 1997 Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

  33. Solidification • Analysis: 1D or based on regularity for single object • For • (nonequilibrium Stefan problem) • weak solution concept (varifolds):Soner 1995 Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

  34. Solidification • Polymer Crystals: • , (Eder 1997) • , smooth • Analysis: • Friedman & Velazquez (2002):smooth single crystal, short time existence, 3D • mb (2003): multiple crystals, 1D • Clain (2003): similar chemical attack model, 1D • mb, Su (2004): multiple crystals, 2D, 3D Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

  35. Solidification • Existence proof: use fixed point argument • Velocity positive, but discontinuous. • Need -solutions for level set equation (weaker than viscosity-solution) - Chen, Su 2000 Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

  36. Solidification • Simulation with same technique: Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

  37. Solidification • Simulation with stochastic nucleation: Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

  38. Inverse Problems • Miminize least-square functional by level set method. • Evaluating involves solving PDE dependent on . • Choose from Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

  39. Inverse Problems • Arising problems similar to other nonlocal geometric motions. • Want to have existence and uniqueness of solution regularizing properties! • Similar computational issues Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

  40. Conclusions • Analysis: • Need new solution concepts • Must deal with strong nonlinearities • Lose maximum principles (no global level set methods) • Numerics: • Need efficient coupling with elliptic/parabolic PDE solvers • Need efficient solvers for equations on im-plicit interfaces Bertalmio,Cheng,Osher,Sapiro Nonlocal Geometric MotionGBM / INV Lake Arrowhead 2003

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