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1 Strongly Anisotropic Motion Laws, Curvature Regularization, and Time Discretization Martin Burger Johannes Kepler University Linz SFB Numerical-Symbolic-Geometric.

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Presentation on theme: "1 Strongly Anisotropic Motion Laws, Curvature Regularization, and Time Discretization Martin Burger Johannes Kepler University Linz SFB Numerical-Symbolic-Geometric."— Presentation transcript:

1 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

2 Strongly anisotropic motion laws Oberwolfach, August 20062 Frank Hausser, Christina Stöcker, Axel Voigt (CAESAR Bonn) Collaborations

3 Strongly anisotropic motion laws Oberwolfach, August 20063 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

4 Strongly anisotropic motion laws Oberwolfach, August 20064 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

5 Strongly anisotropic motion laws Oberwolfach, August 20065 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

6 Strongly anisotropic motion laws Oberwolfach, August 20066 Other effects influencing dynamics: - Deposition of atoms on surfaces - Effects induced by electromagnetic forces (Electromigration) Growth Mechanisms

7 Strongly anisotropic motion laws Oberwolfach, August 20067 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)

8 Strongly anisotropic motion laws Oberwolfach, August 20068 Applications: Nanostructures SiGe/Si Quantum Dots Bauer et. al. 99

9 Strongly anisotropic motion laws Oberwolfach, August 20069 Applications: Nanostructures SiGe/Si Quantum Dots

10 Strongly anisotropic motion laws Oberwolfach, August 200610 Applications: Nanostructures InAs/GaAs Quantum Dots

11 Strongly anisotropic motion laws Oberwolfach, August 200611 Applications: Nano / Micro Electromigration of voids in electrical circuits Nix et. Al. 92

12 Strongly anisotropic motion laws Oberwolfach, August 200612 Applications: Nano / Micro Butterfly shape transition in Ni-based superalloys Colin et. Al. 98

13 Strongly anisotropic motion laws Oberwolfach, August 200613 Applications: Macro Formation of Basalt Columns: Giant‘s Causeway Panska Skala (Northern Ireland) (Czech Republic) See: http://physics.peter-kohlert.de/grinfeld.htmld

14 Strongly anisotropic motion laws Oberwolfach, August 200614 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

15 Strongly anisotropic motion laws Oberwolfach, August 200615 Surface energy is given by Standard model for surface free energy Surface Energy

16 Strongly anisotropic motion laws Oberwolfach, August 200616 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

17 Strongly anisotropic motion laws Oberwolfach, August 200617 Surface Attachment Limited Kinetics SALK is a motion along the negative gradient direction, velocity For graphs / level sets

18 Strongly anisotropic motion laws Oberwolfach, August 200618 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

19 Strongly anisotropic motion laws Oberwolfach, August 200619 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)

20 Strongly anisotropic motion laws Oberwolfach, August 200620 Surface diffusion appears in many important applications - in particular in material and nano science Growth of a surface  with velocity Surface Diffusion

21 Strongly anisotropic motion laws Oberwolfach, August 200621 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

22 Strongly anisotropic motion laws Oberwolfach, August 200622 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

23 Strongly anisotropic motion laws Oberwolfach, August 200623 Effective surface free energy of a compressively strained vicinal surface (Shenoy 2004) Surface Energy

24 Strongly anisotropic motion laws Oberwolfach, August 200624 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

25 Strongly anisotropic motion laws Oberwolfach, August 200625 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

26 Strongly anisotropic motion laws Oberwolfach, August 200626 We obtain Energy variation corresponds to fourth-order term (due to curvature variation) Chemical Potential

27 Strongly anisotropic motion laws Oberwolfach, August 200627 Derivative with matrix Curvature Term

28 Strongly anisotropic motion laws Oberwolfach, August 200628 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

29 Strongly anisotropic motion laws Oberwolfach, August 200629 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

30 Strongly anisotropic motion laws Oberwolfach, August 200630 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

31 Strongly anisotropic motion laws Oberwolfach, August 200631 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 !

32 Strongly anisotropic motion laws Oberwolfach, August 200632 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

33 Strongly anisotropic motion laws Oberwolfach, August 200633 MCF – Graph Form Rewrite energy functional in terms of u Local expansion of metric Spatial discretization: finite elements for u

34 Strongly anisotropic motion laws Oberwolfach, August 200634 MCF – Graph Form Time discretization in terms of u Implicit Euler: minimize

35 Strongly anisotropic motion laws Oberwolfach, August 200635 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

36 Strongly anisotropic motion laws Oberwolfach, August 200636 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

37 Strongly anisotropic motion laws Oberwolfach, August 200637 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

38 Strongly anisotropic motion laws Oberwolfach, August 200638 MCF – Graph Form Semi-implicit scheme: with appropriate choice of B we obtain minimization of Equivalent to linear equation

39 Strongly anisotropic motion laws Oberwolfach, August 200639 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 !

40 Strongly anisotropic motion laws Oberwolfach, August 200640 SD can be obtained as the limit (  →0) of minimization subject to Minimizing Movement: SD

41 Strongly anisotropic motion laws Oberwolfach, August 200641 Level set / graph version: subject to Minimizing Movement: SD

42 Strongly anisotropic motion laws Oberwolfach, August 200642 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

43 Strongly anisotropic motion laws Oberwolfach, August 200643 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

44 Strongly anisotropic motion laws Oberwolfach, August 200644 SALK  = 3.5,  = 0.02,  

45 Strongly anisotropic motion laws Oberwolfach, August 200645 SD  = 3.5,  = 0.02,  

46 Strongly anisotropic motion laws Oberwolfach, August 200646 SALK  = 3.5,  = 0.02,  

47 Strongly anisotropic motion laws Oberwolfach, August 200647 SD  = 3.5,  = 0.02,  

48 Strongly anisotropic motion laws Oberwolfach, August 200648 SALK  = 1.5,  = 0.02,  

49 Strongly anisotropic motion laws Oberwolfach, August 200649 SALK  = 1.5,  = 0.02,  

50 Strongly anisotropic motion laws Oberwolfach, August 200650 SALK  = 1.5,  = 0.02,  

51 Strongly anisotropic motion laws Oberwolfach, August 200651 SD  = 1.5,  = 0.02,  

52 Strongly anisotropic motion laws Oberwolfach, August 200652 SD  = 1.5,  = 0.02,  

53 Strongly anisotropic motion laws Oberwolfach, August 200653 Faceting Graph Simulation: mb JCP 04, Level Set Simulation: mb-Hausser-Stöcker-Voigt 06 Adaptive FE grid around zero level set

54 Strongly anisotropic motion laws Oberwolfach, August 200654 Faceting Anisotropic mean curvature flow

55 Strongly anisotropic motion laws Oberwolfach, August 200655 Faceting of Thin Films Anisotropic Mean Curvature Anisotropic Surface Diffusion mb 04, mb-Hausser- Stöcker-Voigt-05

56 Strongly anisotropic motion laws Oberwolfach, August 200656 Faceting of Crystals Anisotropic surface diffusion

57 Strongly anisotropic motion laws Oberwolfach, August 200657 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

58 Strongly anisotropic motion laws Oberwolfach, August 200658 MCM with Obstacles ObstacleEvolution

59 Strongly anisotropic motion laws Oberwolfach, August 200659 MCM with Obstacles ObstacleEvolution

60 Strongly anisotropic motion laws Oberwolfach, August 200660 MCM with Obstacles ObstacleEvolution

61 Strongly anisotropic motion laws Oberwolfach, August 200661 Download and Contact Papers and Talks: www.indmath.uni-linz.ac.at/people/burger from October: wwwmath1.uni-muenster.de/num e-mail: martin.burger@jku.at

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