1 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 Westfälische Wilhelms Universität Münster

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

Strongly anisotropic motion laws Oberwolfach, August Surface diffusion processes 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. Introduction

Strongly anisotropic motion laws Oberwolfach, August Various fundamental surface growth mechanisms can determine the dynamics, most important: - Attachment / Detachment of atoms to / from surfaces - Diffusion of adatoms on surfaces Growth Mechanisms

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

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

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

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

Strongly anisotropic motion laws Oberwolfach, August Applications: Nanostructures SiGe/Si Quantum Dots

Strongly anisotropic motion laws Oberwolfach, August Applications: Nanostructures InAs/GaAs Quantum Dots

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

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

Strongly anisotropic motion laws Oberwolfach, August Applications: Macro Formation of Basalt Columns: Giant‘s Causeway Panska Skala (Northern Ireland) (Czech Republic) See:

Strongly anisotropic motion laws Oberwolfach, August 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 Energy

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

Strongly anisotropic motion laws Oberwolfach, August Chemical potential  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 Chemical Potential

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

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

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

Strongly anisotropic motion laws Oberwolfach, August F... Deposition flux D s.. Diffusion coefficient ... Atomic volume ... Surface density k... Boltzmann constant T... Temperature n... Unit outer normal ... Chemical potential = energy variation Surface Diffusion

Strongly anisotropic motion laws Oberwolfach, August 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-Hoffmann Strained Vicinal Surfaces: Shenoy-Freund 2003 Surface Energy

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

Strongly anisotropic motion laws Oberwolfach, August 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) Curvature Regularization

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

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

Strongly anisotropic motion laws Oberwolfach, August Derivative with matrix Curvature Term

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

Strongly anisotropic motion laws Oberwolfach, August 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 Discretization: Gradient Flows

Strongly anisotropic motion laws Oberwolfach, August 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,..) Discretization: Gradient Flows

Strongly anisotropic motion laws Oberwolfach, August 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 V n Shape metric translates to norm (scalar product) for normal velocities !

Strongly anisotropic motion laws Oberwolfach, August 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 V n

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

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

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

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

Strongly anisotropic motion laws Oberwolfach, August Basic idea: Semi-implicit time discretization + Splitting into two / three second-order equations + Finite element discretization in space Natural variables for splitting: Height u, Mean Curvature , Chemical potential  Numerical Solution

Strongly anisotropic motion laws Oberwolfach, August 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 Spatial Discretization

Strongly anisotropic motion laws Oberwolfach, August SALK  = 3.5,  = 0.02,  

Strongly anisotropic motion laws Oberwolfach, August SD  = 3.5,  = 0.02,  

Strongly anisotropic motion laws Oberwolfach, August SALK  = 3.5,  = 0.02,  

Strongly anisotropic motion laws Oberwolfach, August SD  = 3.5,  = 0.02,  

Strongly anisotropic motion laws Oberwolfach, August SALK  = 1.5,  = 0.02,  

Strongly anisotropic motion laws Oberwolfach, August SALK  = 1.5,  = 0.02,  

Strongly anisotropic motion laws Oberwolfach, August SALK  = 1.5,  = 0.02,  

Strongly anisotropic motion laws Oberwolfach, August SD  = 1.5,  = 0.02,  

Strongly anisotropic motion laws Oberwolfach, August SD  = 1.5,  = 0.02,  

Strongly anisotropic motion laws Oberwolfach, August Faceting Graph Simulation: mb JCP 04, Level Set Simulation: mb-Hausser-Stöcker-Voigt 06 Adaptive FE grid around zero level set

Strongly anisotropic motion laws Oberwolfach, August Faceting Anisotropic mean curvature flow

Strongly anisotropic motion laws Oberwolfach, August Faceting of Thin Films Anisotropic Mean Curvature Anisotropic Surface Diffusion mb 04, mb-Hausser- Stöcker-Voigt-05

Strongly anisotropic motion laws Oberwolfach, August Faceting of Crystals Anisotropic surface diffusion

Strongly anisotropic motion laws Oberwolfach, August Obstacle Problems Numerical schemes obtained again by approximation of the energy and metric for time discretization, finite element spatial discretization Local optimization problem with bound constraint (general inequality constraints for other obstacles) Explicit scheme: additional projection step Semi-implicit scheme: quadratic problem with bound constraint, solved with modified CG

Strongly anisotropic motion laws Oberwolfach, August MCM with Obstacles ObstacleEvolution

Strongly anisotropic motion laws Oberwolfach, August MCM with Obstacles ObstacleEvolution

Strongly anisotropic motion laws Oberwolfach, August MCM with Obstacles ObstacleEvolution

Strongly anisotropic motion laws Oberwolfach, August Download and Contact Papers and Talks: from October: wwwmath1.uni-muenster.de/num