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Nonlocal Geometric Motion: Understanding and Applications

Explore the latest advancements in nonlocal geometric motion, including surface and bulk diffusion, heat transfer, elasticity, and shape optimization. Discover the theoretical foundations and computational methods in this cutting-edge field.

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Nonlocal Geometric Motion: Understanding and Applications

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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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