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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
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Strongly anisotropic motion laws
Oberwolfach, August 2006 2
Frank Hausser, Christina Stöcker, Axel Voigt (CAESAR Bonn)
Collaborations
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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
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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
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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
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Other effects influencing dynamics:
- Deposition of atoms on surfaces
- Effects induced by electromagnetic forces (Electromigration)
Growth Mechanisms
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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)
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Applications: Nanostructures SiGe/Si Quantum Dots
Bauer et. al. 99
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Applications: Nanostructures SiGe/Si Quantum Dots
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Applications: NanostructuresInAs/GaAs Quantum Dots
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Applications: Nano / Micro Electromigration of voids in electrical circuits
Nix et. Al. 92
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Applications: Nano / Micro Butterfly shape transition in Ni-based superalloys
Colin et. Al. 98
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Applications: Macro Formation of Basalt Columns:
Giant‘s Causeway
Panska Skala (Northern Ireland) (Czech Republic)
See: http://physics.peter-kohlert.de/grinfeld.htmld
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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
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Surface energy is given by
Standard model for surface free energy
Surface Energy
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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
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Surface Attachment Limited KineticsSALK 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, …
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
- Numerical simulation of regularized model-Hausser-Voigt 2004 (parametric)
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Surface diffusion appears in many important applications - in particular in material and nano science
Growth of a surface with velocity
Surface Diffusion
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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
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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
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Effective surface free energy of a compressively strained vicinal surface (Shenoy 2004)
Surface Energy
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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
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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
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We obtain
Energy variation corresponds to fourth-order term (due to curvature variation)
Chemical Potential
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Derivative
with matrix
Curvature Term
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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
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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
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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
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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 !
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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
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MCF – Graph Form Rewrite energy functional in terms of u
Local expansion of metric
Spatial discretization: finite elements for u
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MCF – Graph Form Time discretization in terms of u
Implicit Euler: minimize
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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
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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
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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
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MCF – Graph Form Semi-implicit scheme: with appropriate choice
of B we obtain minimization of
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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SD can be obtained as the limit ( →0) of minimization
subject to
Minimizing Movement: SD
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Level set / graph version:
subject to
Minimizing Movement: SD
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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
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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
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SALK = 3.5, = 0.02,
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SD = 3.5, = 0.02,
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SALK = 3.5, = 0.02,
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SD = 3.5, = 0.02,
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SALK = 1.5, = 0.02,
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SALK = 1.5, = 0.02,
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SALK = 1.5, = 0.02,
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SD = 1.5, = 0.02,
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SD = 1.5, = 0.02,
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Faceting Graph Simulation: mb JCP 04, Level Set Simulation: mb-Hausser-Stöcker-Voigt 06
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-Hausser-
Stöcker-Voigt-05
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Strongly anisotropic motion laws
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Faceting of Crystals Anisotropic surface diffusion
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Strongly anisotropic motion laws
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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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Strongly anisotropic motion laws
Oberwolfach, August 2006 58
MCM with Obstacles Obstacle Evolution
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Strongly anisotropic motion laws
Oberwolfach, August 2006 59
MCM with Obstacles Obstacle Evolution
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Strongly anisotropic motion laws
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MCM with Obstacles Obstacle Evolution
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