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Strongly Anisotropic Motion Laws, Curvature Regularization, and Time Discretization

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##### Strongly Anisotropic Motion Laws, Curvature Regularization, and Time Discretization

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**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**Collaborations**• Frank Hausser, Christina Stöcker, Axel Voigt (CAESAR Bonn) Strongly anisotropic motion laws Oberwolfach, August 2006**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**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**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**Growth Mechanisms**• Other effects influencing dynamics: • Deposition of atoms on surfaces • Effects induced by electromagnetic forces(Electromigration) Strongly anisotropic motion laws Oberwolfach, August 2006**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**Applications: Nanostructures**• SiGe/Si Quantum Dots • Bauer et. al. 99 Strongly anisotropic motion laws Oberwolfach, August 2006**Applications: Nanostructures**• SiGe/Si Quantum Dots Strongly anisotropic motion laws Oberwolfach, August 2006**Applications: Nanostructures**• InAs/GaAs Quantum Dots Strongly anisotropic motion laws Oberwolfach, August 2006**Applications: Nano / Micro**• Electromigration of voids in electrical circuits Nix et. Al. 92 Strongly anisotropic motion laws Oberwolfach, August 2006**Applications: Nano / Micro**• Butterfly shape transition in Ni-based superalloys Colin et. Al. 98 Strongly anisotropic motion laws Oberwolfach, August 2006**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**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**Surface Energy**• Surface energy is given by • Standard model for surface free energy Strongly anisotropic motion laws Oberwolfach, August 2006**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**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**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**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**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**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**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**Surface Energy**• Effective surface free energy of a compressively strained vicinal surface (Shenoy 2004) Strongly anisotropic motion laws Oberwolfach, August 2006**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**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**Chemical Potential**• We obtain • Energy variation corresponds to fourth-order term (due to curvature variation) Strongly anisotropic motion laws Oberwolfach, August 2006**Curvature Term**• Derivative • with matrix Strongly anisotropic motion laws Oberwolfach, August 2006**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**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**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**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**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**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**MCF – Graph Form**• Time discretization in terms of u • Implicit Euler: minimize Strongly anisotropic motion laws Oberwolfach, August 2006**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**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**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**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**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**Minimizing Movement: SD**• SD can be obtained as the limit (t→0) of minimization • subject to Strongly anisotropic motion laws Oberwolfach, August 2006**Minimizing Movement: SD**• Level set / graph version: subject to Strongly anisotropic motion laws Oberwolfach, August 2006**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**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**SALK e = 3.5, a = 0.02, 10t = 5 10-4**Strongly anisotropic motion laws Oberwolfach, August 2006**SD e = 3.5, a = 0.02, 10t = 5 10-5**Strongly anisotropic motion laws Oberwolfach, August 2006**SALK e = 3.5, a = 0.02, 10t = 2.8 10-3**Strongly anisotropic motion laws Oberwolfach, August 2006**SD e = 3.5, a = 0.02, 10t = 2.8 10-5**Strongly anisotropic motion laws Oberwolfach, August 2006**SALK e = 1.5, a = 0.02, 10t = 6.66 10-3**Strongly anisotropic motion laws Oberwolfach, August 2006**SALK e = 1.5, a = 0.02, 10t = 6.66 10-3**Strongly anisotropic motion laws Oberwolfach, August 2006**SALK e = 1.5, a = 0.02, 10t = 6.66 10-3**Strongly anisotropic motion laws Oberwolfach, August 2006