Transcript Folie 1 - uni

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 1
Collaborations

Frank Hausser, Christina Stöcker, Axel Voigt
(CAESAR Bonn)
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Introduction
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.
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Growth Mechanisms
Various fundamental surface growth
mechanisms can determine the dynamics, most
important:
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Attachment / Detachment of atoms to / from
surfaces
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Diffusion of adatoms on surfaces
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Growth Mechanisms
Other effects influencing dynamics:
- Anisotropy
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-
Bulk diffusion of atoms (phase separation)
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Exchange of atoms between surface and bulk
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Elastic Relaxation in the bulk
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Surface Stresses
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Growth Mechanisms
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-
Other effects influencing dynamics:
Deposition of atoms on surfaces
Effects induced by electromagnetic forces
(Electromigration)
-
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Isotropic Surface Diffusion
Simple model for surface diffusion in the isotropic
case:
Normal motion of the surface by minus surface
Laplacian of mean curvature
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Can be derived as limit of Cahn-Hilliard model
with degenerate diffusivity (ask Harald Garcke)
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Applications: Nanostructures
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SiGe/Si Quantum Dots
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Bauer
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et. al. 99
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Applications: Nanostructures
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SiGe/Si Quantum Dots
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Applications: Nanostructures
InAs/GaAs
Quantum Dots
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Applications: Nano / Micro
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Electromigration of voids in electrical circuits
Nix et. Al. 92
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Applications: Nano / Micro
Butterfly shape transition in Ni-based
superalloys
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Colin et. Al. 98
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Applications: Macro
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Formation of Basalt Columns:
Panska Skala
(Czech Republic)
Giant‘s Causeway
(Northern Ireland)
See: http://physics.peter-kohlert.de/grinfeld.htmld
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Energy
The energy of the system is composed of
various terms:
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Total Energy =
(Anisotropic) Surface Energy +
(Anisotropic) Elastic Energy +
Compositional Energy +
.....

We start with first term only
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Surface Energy
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Surface energy is given by
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Standard model for surface free energy
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Chemical Potential
Chemical potential m 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)
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For surfaces represented by a graph, the
chemical potential is the functional derivative of
the energy
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Surface Attachment Limited Kinetics
SALK
is a motion along the negative gradient
direction, velocity
For
graphs / level sets
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Surface Attachment Limited Kinetics
Surface attachment limited kinetics appears in
phase transition, grain boundary motion, …
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Isotropic case: motion by mean curvature
 Additional
curvature term like Willmore flow
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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
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Numerical simulation of regularized model
-Hausser-Voigt
2004 (parametric)
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Surface Diffusion
Surface diffusion appears in many important
applications - in particular in material and nano
science

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Growth of a surface G with velocity
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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

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Surface Energy
In several situations, the surface free energy
(respectively its one-homogeneous extension)
is not convex. Nonconvex energies can result
from different reasons:
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Special materials with strong anisotropy:
Gjostein 1963, Cahn-Hoffmann1974
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Strained Vicinal Surfaces: Shenoy-Freund 2003
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Surface Energy
Effective surface free energy of a compressively
strained vicinal surface (Shenoy 2004)
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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)
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Anisotropic Surface energy
Cubic anisotropy, surface energy becomes nonconvex for
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e > 1/3
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Faceting of the surface
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Microstructure possible without curvature term
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Equilibria are local energy minimizers only
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Chemical Potential
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We obtain
Energy variation corresponds to fourth-order
term (due to curvature variation)
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Curvature Term
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Derivative
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with matrix
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Analysis and Numerics
Existing results:
- Studies of equilibrium structures, Gurtin 1993,
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Spencer 2003, Cecil-Osher 2004
Numerical simulation of asymptotic model
(obtained from long-wave expansion), Golovin-
Davies-Nepomnyaschy 2002 / 2003
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Discretization: Gradient Flows
SD and SALK can be obtained as the limit of
minimizing movement formulation (De Giorgi)
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with different metrics d between surfaces, but
same surface energies
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Discretization: Gradient Flows
Natural first order time discretization.
Additional spatial discretization by constraining
manifold and possibly approximating metric and
energy
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Discrete manifold determined by
representation (parametric, graph, level set, ..)
+ discretization (FEM, DG, FV, ..)
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Gradient Flow Structure
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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
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Shape metric translates to norm (scalar
product) for normal velocities !
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Gradient Flow Structure
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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
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MCF – Graph Form
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Rewrite energy functional in terms of u
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Local expansion of metric
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Spatial discretization: finite elements for u
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MCF – Graph Form
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Time discretization in terms of u
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Implicit Euler: minimize
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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
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Implicit Euler 2: minimize
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MCF – Graph Form
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Explicit Euler: minimize
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Time step restriction: minimizer exists only if
quadratic term (metric) dominates linear term
This yields standard parabolic condition
by interpolation inequalities
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MCF – Graph Form
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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
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MCF – Graph Form
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Semi-implicit scheme: with appropriate choice
of B we obtain minimization of
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Equivalent to linear equation
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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 !
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Minimizing Movement: SD
SD can be obtained as the limit (t →0) of
minimization


subject to
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Minimizing Movement: SD
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Level set / graph version:
subject to
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Numerical Solution
Basic idea: Semi-implicit time discretization +
Splitting into two / three second-order equations +
Finite element discretization in space
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Natural variables for splitting:
Height u, Mean Curvature k, Chemical potential m
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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
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SALK e = 3.5, a = 0.02, 10t = 5 10-4
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SD e = 3.5, a = 0.02, 10t = 5 10-5
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SALK e = 3.5, a = 0.02, 10t = 2.8 10-3
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SD e = 3.5, a = 0.02, 10t = 2.8 10-5
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SALK e = 1.5, a = 0.02, 10t = 6.66 10-3
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SALK e = 1.5, a = 0.02, 10t = 6.66 10-3
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SALK e = 1.5, a = 0.02, 10t = 6.66 10-3
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SD e = 1.5, a = 0.02, 10t = 3.33 10-3
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SD e = 1.5, a = 0.02, 10t = 1.66 10-3
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Faceting
Graph Simulation: mb JCP 04,
 Level Set Simulation: mb-Hausser-Stöcker-Voigt 06

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Adaptive FE grid around zero level set
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Faceting
 Anisotropic
mean curvature flow
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Faceting of Thin Films
Anisotropic
Mean Curvature
Anisotropic
Surface Diffusion
mb 04,
mb-HausserStöcker-Voigt-05
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Faceting of Crystals
 Anisotropic
surface diffusion
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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

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MCM with Obstacles

Obstacle
Evolution
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MCM with Obstacles

Obstacle
Evolution
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MCM with Obstacles

Obstacle
Evolution
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Download and Contact

Papers and Talks:
www.indmath.uni-linz.ac.at/people/burger
from October:
wwwmath1.uni-muenster.de/num

e-mail: [email protected]
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