Strongly Anisotropic Motion Laws, Curvature Regularization, and Time Discretization

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Strongly Anisotropic Motion Laws, Curvature Regularization, and Time DiscretizationMartin Burger

Johannes Kepler University LinzSFB Numerical-Symbolic-Geometric Scientific ComputingRadon Institute for Computational & Applied Mathematics

Westfälische Wilhelms Universität Münster

Strongly anisotropic motion laws

Oberwolfach, August 2006 2

Frank Hausser, Christina Stöcker, Axel Voigt (CAESAR Bonn)

Collaborations



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



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



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



Other effects influencing dynamics:

- Deposition of atoms on surfaces

- Effects induced by electromagnetic forces (Electromigration)

Growth Mechanisms



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)



Applications: Nanostructures SiGe/Si Quantum Dots

Bauer et. al. 99



Applications: Nanostructures SiGe/Si Quantum Dots



Applications: NanostructuresInAs/GaAs Quantum Dots



Applications: Nano / Micro Electromigration of voids in electrical circuits

Nix et. Al. 92



Applications: Nano / Micro Butterfly shape transition in Ni-based superalloys

Colin et. Al. 98



Applications: Macro Formation of Basalt Columns:

Giant‘s Causeway

Panska Skala (Northern Ireland) (Czech Republic)

See: http://physics.peter-kohlert.de/grinfeld.htmld



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



Surface energy is given by

Standard model for surface free energy

Surface Energy



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



Surface Attachment Limited KineticsSALK is a motion along the negative gradient direction, velocity

For graphs / level sets



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



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)



Surface diffusion appears in many important applications - in particular in material and nano science

Growth of a surface with velocity

Surface Diffusion



F ... Deposition flux Ds .. Diffusion coefficient... Atomic volume ... Surface density k ... Boltzmann constant T ... Temperature n ... Unit outer normal ... Chemical potential =

energy variation

Surface Diffusion



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

Surface Energy



Effective surface free energy of a compressively strained vicinal surface (Shenoy 2004)

Surface Energy



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



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



We obtain

Energy variation corresponds to fourth-order term (due to curvature variation)

Chemical Potential



Derivative

with matrix

Curvature Term



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

Analysis and Numerics



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



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



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 !



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



MCF – Graph Form Rewrite energy functional in terms of u

Local expansion of metric

Spatial discretization: finite elements for u



MCF – Graph Form Time discretization in terms of u

Implicit Euler: minimize



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



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



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



MCF – Graph Form Semi-implicit scheme: with appropriate choice

of B we obtain minimization of

Equivalent to linear equation



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 !



SD can be obtained as the limit ( →0) of minimization

subject to

Minimizing Movement: SD



Level set / graph version:

subject to

Minimizing Movement: SD



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



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



SALK = 3.5, = 0.02,



SD = 3.5, = 0.02,



SALK = 3.5, = 0.02,



SD = 3.5, = 0.02,



SALK = 1.5, = 0.02,



SD = 1.5, = 0.02,



Faceting Graph Simulation: mb JCP 04, Level Set Simulation: mb-Hausser-Stöcker-Voigt 06

Adaptive FE grid around zero level set



Faceting Anisotropic mean curvature flow



Faceting of Thin Films Anisotropic

Mean Curvature

Anisotropic Surface Diffusion

mb 04, mb-Hausser-

Stöcker-Voigt-05



Faceting of Crystals Anisotropic surface diffusion



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



MCM with Obstacles Obstacle Evolution



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Strongly Anisotropic Motion Laws, Curvature Regularization, and Time Discretization

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