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Strongly Anisotropic Motion Laws, Curvature Regularization, and Time Discretization
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  1. Westfälische Wilhelms Universität Münster Strongly Anisotropic Motion Laws, Curvature Regularization, and Time Discretization Martin Burger Johannes Kepler University Linz SFB Numerical-Symbolic-Geometric Scientific Computing Radon Institute for Computational & Applied Mathematics

  2. Collaborations • Frank Hausser, Christina Stöcker, Axel Voigt (CAESAR Bonn) Strongly anisotropic motion laws Oberwolfach, August 2006

  3. Introduction • Surface diffusionprocesses appear in various materials science applications, in particular in the (self-assembled) growth of nanostructures • Schematic description: particles are deposited on a surface and become adsorbed (adatoms). They diffuse around the surface and can be bound to the surface. Vice versa, unbinding and desorption happens. Strongly anisotropic motion laws Oberwolfach, August 2006

  4. Growth Mechanisms • Various fundamentalsurface growth mechanismscan determine the dynamics, most important: • Attachment / Detachment of atoms to / from surfaces • Diffusionof adatoms on surfaces Strongly anisotropic motion laws Oberwolfach, August 2006

  5. Growth Mechanisms • Other effects influencing dynamics: • Anisotropy • Bulk diffusion of atoms (phase separation) • Exchange of atoms between surface and bulk • Elastic Relaxationin the bulk • Surface Stresses Strongly anisotropic motion laws Oberwolfach, August 2006

  6. Growth Mechanisms • Other effects influencing dynamics: • Deposition of atoms on surfaces • Effects induced by electromagnetic forces(Electromigration) Strongly anisotropic motion laws Oberwolfach, August 2006

  7. Isotropic Surface Diffusion • Simple model for surface diffusion in the isotropic case: Normal motion of the surface by minus surfaceLaplacian of mean curvature • Can be derived as limit of Cahn-Hilliard model with degenerate diffusivity (ask Harald Garcke) Strongly anisotropic motion laws Oberwolfach, August 2006

  8. Applications: Nanostructures • SiGe/Si Quantum Dots • Bauer et. al. 99 Strongly anisotropic motion laws Oberwolfach, August 2006

  9. Applications: Nanostructures • SiGe/Si Quantum Dots Strongly anisotropic motion laws Oberwolfach, August 2006

  10. Applications: Nanostructures • InAs/GaAs Quantum Dots Strongly anisotropic motion laws Oberwolfach, August 2006

  11. Applications: Nano / Micro • Electromigration of voids in electrical circuits Nix et. Al. 92 Strongly anisotropic motion laws Oberwolfach, August 2006

  12. Applications: Nano / Micro • Butterfly shape transition in Ni-based superalloys Colin et. Al. 98 Strongly anisotropic motion laws Oberwolfach, August 2006

  13. Applications: Macro • Formation of Basalt Columns: Giant‘s Causeway Panska Skala (Northern Ireland) (Czech Republic) See: http://physics.peter-kohlert.de/grinfeld.htmld Strongly anisotropic motion laws Oberwolfach, August 2006

  14. Energy • The energy of the system is composed of various terms: Total Energy = (Anisotropic) Surface Energy + (Anisotropic) Elastic Energy + Compositional Energy + ..... • We start with first term only Strongly anisotropic motion laws Oberwolfach, August 2006

  15. Surface Energy • Surface energy is given by • Standard model for surface free energy Strongly anisotropic motion laws Oberwolfach, August 2006

  16. Chemical Potential • Chemical potentialm is the change of energy when adding / removing single atoms • In a continuum model, the chemical potential can be represented as a surface gradient of the energy (obtained as the variation of total energy with respect to the surface) • For surfaces represented by a graph, the chemical potential is the functional derivative of the energy Strongly anisotropic motion laws Oberwolfach, August 2006

  17. Surface Attachment Limited Kinetics • SALK is a motion along the negative gradient direction, velocity • For graphs / level sets Strongly anisotropic motion laws Oberwolfach, August 2006

  18. Surface Attachment Limited Kinetics • Surface attachment limited kinetics appears in phase transition, grain boundary motion, … • Isotropic case: motion by mean curvature • Additional curvature term like Willmore flow Strongly anisotropic motion laws Oberwolfach, August 2006

  19. Analysis and Numerics • Existing results: • Numerical simulation without curvature regularization, Fierro-Goglione-Paolini 1998 • Numerical simulation of Willmore flow, Dziuk-Kuwert-Schätzle 2002, Droske-Rumpf 2004 • Numerical simulation of regularized model • Hausser-Voigt 2004 (parametric) Strongly anisotropic motion laws Oberwolfach, August 2006

  20. Surface Diffusion • Surface diffusion appears in many important applications - in particular in material and nano science • Growth of a surface G with velocity Strongly anisotropic motion laws Oberwolfach, August 2006

  21. Surface Diffusion • F ... Deposition flux • Ds .. Diffusion coefficient • W ... Atomic volume • s ... Surface density • k ... Boltzmann constant • T ... Temperature • n ... Unit outer normal • m ... Chemical potential = energy variation Strongly anisotropic motion laws Oberwolfach, August 2006

  22. Surface Energy • In several situations, the surface free energy (respectively its one-homogeneous extension) is not convex. Nonconvex energies can result from different reasons: • Special materials with strong anisotropy:Gjostein 1963, Cahn-Hoffmann1974 • Strained Vicinal Surfaces: Shenoy-Freund 2003 Strongly anisotropic motion laws Oberwolfach, August 2006

  23. Surface Energy • Effective surface free energy of a compressively strained vicinal surface (Shenoy 2004) Strongly anisotropic motion laws Oberwolfach, August 2006

  24. Curvature Regularization • In order to regularize problem (and possibly since higher order terms become important in atomistic homogenization), curvature regularization has beeen proposed by several authors (DiCarlo-Gurtin-Podio-Guidugli 1993, Gurtin-Jabbour 2002, Tersoff, Spencer, Rastelli, Von Kähnel 2003) Strongly anisotropic motion laws Oberwolfach, August 2006

  25. Anisotropic Surface energy • Cubic anisotropy, surface energy becomes non-convex for e > 1/3 • Faceting of the surface • Microstructure possible without curvature term • Equilibria are local energy minimizers only Strongly anisotropic motion laws Oberwolfach, August 2006

  26. Chemical Potential • We obtain • Energy variation corresponds to fourth-order term (due to curvature variation) Strongly anisotropic motion laws Oberwolfach, August 2006

  27. Curvature Term • Derivative • with matrix Strongly anisotropic motion laws Oberwolfach, August 2006

  28. Analysis and Numerics • Existing results: • Studies of equilibrium structures, Gurtin 1993, Spencer 2003, Cecil-Osher 2004 • Numerical simulation of asymptotic model (obtained from long-wave expansion), Golovin-Davies-Nepomnyaschy 2002 / 2003 Strongly anisotropic motion laws Oberwolfach, August 2006

  29. Discretization: Gradient Flows • SD and SALK can be obtained as the limit of minimizing movement formulation (De Giorgi)with different metrics d between surfaces, but same surface energies Strongly anisotropic motion laws Oberwolfach, August 2006

  30. Discretization: Gradient Flows • Natural first order time discretization. Additional spatial discretization by constraining manifold and possibly approximating metric and energy • Discrete manifold determined by representation (parametric, graph, level set, ..) + discretization (FEM, DG, FV, ..) Strongly anisotropic motion laws Oberwolfach, August 2006

  31. Gradient Flow Structure • Expansion of the shape metric (SALK / SD) where denotes the surface obtained from a motion of all points in normal direction with (given) normal velocity Vn • Shape metric translates to norm (scalar product) for normal velocities ! Strongly anisotropic motion laws Oberwolfach, August 2006

  32. Gradient Flow Structure • Expansion of the energy (Hadamard-Zolesio structure theorem) where denotes the surface obtained from a motion of all points in normal direction with (given) normal velocity Vn Strongly anisotropic motion laws Oberwolfach, August 2006

  33. MCF – Graph Form • Rewrite energy functional in terms of u • Local expansion of metric • Spatial discretization: finite elements for u Strongly anisotropic motion laws Oberwolfach, August 2006

  34. MCF – Graph Form • Time discretization in terms of u • Implicit Euler: minimize Strongly anisotropic motion laws Oberwolfach, August 2006

  35. MCF – Graph Form • Time discretization yields same order in time if we approximate to first order in t • Variety of schemes by different approximations of shape and metric • Implicit Euler 2: minimize Strongly anisotropic motion laws Oberwolfach, August 2006

  36. MCF – Graph Form • Explicit Euler: minimize • Time step restriction: minimizer exists only if quadratic term (metric) dominates linear term This yields standard parabolic condition by interpolation inequalities Strongly anisotropic motion laws Oberwolfach, August 2006

  37. MCF – Graph Form • Semi-implicit scheme: minimize with quadratic functional B • Consistency and correct energy dissipation if B is chosen such that B(0)=0 and quadratic expansion lies above E Strongly anisotropic motion laws Oberwolfach, August 2006

  38. MCF – Graph Form • Semi-implicit scheme: with appropriate choice of B we obtain minimization of • Equivalent to linear equation Strongly anisotropic motion laws Oberwolfach, August 2006

  39. MCF – Graph Form • Semi-implicit scheme is unconditionally stable, only requires solution of linear system in each time step • Well-known scheme (different derivation) Deckelnick-Dziuk 01, 02 • Analogous for level set representation • Approach can be extended automatically to more complicated energies and metrics ! Strongly anisotropic motion laws Oberwolfach, August 2006

  40. Minimizing Movement: SD • SD can be obtained as the limit (t→0) of minimization • subject to Strongly anisotropic motion laws Oberwolfach, August 2006

  41. Minimizing Movement: SD • Level set / graph version: subject to Strongly anisotropic motion laws Oberwolfach, August 2006

  42. Numerical Solution • Basic idea: Semi-implicit time discretization + Splitting into two / three second-order equations + Finite element discretization in space • Natural variables for splitting: Heightu, Mean Curvaturek, Chemical potentialm Strongly anisotropic motion laws Oberwolfach, August 2006

  43. Spatial Discretization • Discretization of the variational problem in space by piecewise linear finite elements • and P(u) are piecewise constant on the triangularization, all integrals needed for stiffness matrix and right-hand side can be computed exactly Strongly anisotropic motion laws Oberwolfach, August 2006

  44. SALK e = 3.5, a = 0.02, 10t = 5 10-4 Strongly anisotropic motion laws Oberwolfach, August 2006

  45. SD e = 3.5, a = 0.02, 10t = 5 10-5 Strongly anisotropic motion laws Oberwolfach, August 2006

  46. SALK e = 3.5, a = 0.02, 10t = 2.8 10-3 Strongly anisotropic motion laws Oberwolfach, August 2006

  47. SD e = 3.5, a = 0.02, 10t = 2.8 10-5 Strongly anisotropic motion laws Oberwolfach, August 2006

  48. SALK e = 1.5, a = 0.02, 10t = 6.66 10-3 Strongly anisotropic motion laws Oberwolfach, August 2006

  49. SALK e = 1.5, a = 0.02, 10t = 6.66 10-3 Strongly anisotropic motion laws Oberwolfach, August 2006

  50. SALK e = 1.5, a = 0.02, 10t = 6.66 10-3 Strongly anisotropic motion laws Oberwolfach, August 2006