Analysis, modeling and simulation of multiscale problems

703 Pages·2006·9.228 MB·English

by  Alexander Mielke

Mielke(Ed.)
Analysis,ModelingandSimulationofMultiscaleProblems
Alexander Mielke
Editor
Analysis,
Modeling and
Simulation of
Multiscale
Problems
With167Figures,32ColoredPlates,and11Tables
123
Editor
AlexanderMielke
Weierstraß-InstitutfürAngewandte
AnalysisundStochastik
Mohrenstraße39
10117Berlin,Germany
and
InstitutfürMathematik
Humboldt-UniversitätzuBerlin
RudowerChaussee25
12489Berlin,Germany
E-mail:[email protected]
LibraryofCongressControlNumber:2006932292
MathematicsSubjectClassification(2000):35-XX,49-XX,70-XX,74-XX,81-XX,92-XX
ISBN-10 3-540-35656-8 SpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-35656-1 SpringerBerlinHeidelbergNewYork
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Preface
L’´etude approfondie de la nature
est la source la plus f´econde
des d´ecouvertes math´ematiques.
J.B.J. Fourier (1768–1830)
Recent technological advances allow us to study and manipulate matter
ontheatomicscale.Thus,thetraditionalbordersbetweenmechanics,physics
and chemistry seem to disappear and new applications in biology emanate.
However,modeling matter on the atomistic scale ab initio, i.e., starting from
the quantum level, is only possible for very small, isolated molecules. More-
over, the study of mesoscopic properties of an elastic solid modeled by 1020
atoms treated as point particles is still out of reach for modern computers.
Hence, the derivation of coarse grained models from well accepted fine-scale
models is one of the most challenging fields. A proper understanding of the
interactionofeffectsondifferentspatialandtemporalscalesisoffundamental
importance for the effective description of such structures. The central ques-
tionarisesastowhichinformationfromthe smallscalesisneededtodescribe
the large-scale effects correctly.
BasedonexistingresearcheffortsintheGermanmathematicalcommunity
we proposed to the Deutsche Forschungsgemeinschaft (DFG) to strengthen
the mathematical basis for attacking such problems. In May 1999 the DFG
decided to establish the
DFG Priority Program (SPP 1095)
Analysis, Modeling and Simulation of Multiscale Problems.
After another reviewing process the official start in September 2000 involved
about 25 research groups all over Germany, see
http://www.mathematik.uni-stuttgart.de/∼mehrskalen
VI Preface
for information about these groups and their activities. In the sequel, this
programhas inspired a number of multiscale initiatives worldwide,for exam-
ple, the Center for Integrative Multiscale Modeling and Simulation (CIMMS)
at CalTech, the Bath Institute for Complex Systems and the European Net-
work Multi-scale modeling and characterization of phase transformations in
advanced materials (MULTIMAT).
Theaimofourpriorityprogramhasbeentocombinedifferentexpertisein
the mathematical community and to enhance the development of mathemat-
ical methods and concepts for the study of multiscale problems. On the one
hand, these problems are driven by needs in applications in physics, chem-
istry,orbiology.Onthe otherhand,the challengeto bridgetheoriesatdiffer-
entscales leadsto deepandfundamental questionswithin mathematics.This
book tries to span this variety of the subject by surveying and highlighting
the work done in the groups of the priority program. In the vast mathemati-
cal area of multiscale problems the programhas focused its efforts on several
specific fields that can be characterized by the following subjects:
• weak-convergencemethods, relaxations, and Gamma convergence
• Young measures, gradient Young measures, Wigner measures
• homogenization, averaging,and adiabatic evolution
• singular perturbations, boundary layers,interfaces, and point defects
• formation, stability, and dynamics of sharp interfaces
• microstructuresinelasticsolids,thinfilmsorrods,andinmicro-magnetism
• derivation of continuous models from spatially discrete models
• quantum-chemical and quantum-mechanical models
• quantumandsemi-classicalmodelsforsemi-conductorsandtheircoupling
• numerical algorithms for microstructures in solids
• exponential integrators for Hamiltonian systems with fast oscillations
• almost invariant sets and metastability
Mostoftheworkreportedinthisbookhasbeensupportedsubstantiallyby
the DFG through the priority program. The major part of the funding went
into positions for PhD students and post-doctoral researchers. In fact, this
programhasbeenanidealwaytoeducateanewgenerationofyoungscientists
to cope with the future challenges in mathematical modeling, analysis and
numerics. We are glad to see that about half of the contributors to this book
are such young scientists. The additional support for travel and the guest
program has proved to be crucial for the interaction and mutual stimulation
of the participating groups, in particular the annual colloquia as wells as a
large series of specialized workshops.
Bernhard Nunner and Frank Kiefer from the DFG made our life as easy
as possible concerning the administrative duties while pushing our scientific
Preface VII
achievements to new frontiers. In this goal, they could rely on the steady
support and expertise of the referees Wolfgang Dahmen, Gero Friesecke,
Thomas Y. Hou, Bernhard Kawohl, Volker Mehrmann, Umberto Mosco,
MarekNiezg´odka,TomaˇsRoub´ıˇcek,StefanSauter,ValeryP.Smyshlyaev,and
Gabriel Wittum. These referees have been valuable advisors for the projects
in this programandplayedanessentialrole in shaping our viewof the math-
ematics for multiscale problems.
Finally, Alexander Mielke would like to thank Stefanie Siegert. Her ded-
ication and precise management was often hidden in the background, but
constituted an essential part of the smooth coordination of the program.
Of course, such a priority programon multiscale problems cannot be suc-
cessfulwithoutthecreativeworkoftheinvolvedresearchersandwithouttheir
openness for collaboration. We, as the initiators of this program, are grate-
ful to all these companions in this endeavor and hope that the achievements
within the program will prove to be rewarding for all of them.
Mu¨nchen Folkmar Bornemann
Leipzig Stephan Luckhaus
Berlin Alexander Mielke (coordinator)
Leipzig Stefan Mu¨ller
May 2006
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Contents
Exponential Estimates in Averaging and Homogenisation
Karsten Matthies ................................................ 1
Multiscale Problems in Solidification Processes
Christof Eck, Harald Garcke, Bjo¨rn Stinner ........................ 21
Multiscale Modeling of Epitaxial Growth:
From Discrete-Continuum to Continuum Equations
Lev Balykov, Vladimir Chalupecky, Christof Eck, Heike Emmerich,
Ganeshram Krishnamoorthy, Andreas Ra¨tz, Axel Voigt ............... 65
On Asymptotic Limits of Cahn-Hilliard Systems
with Elastic Misfit
Harald Garcke, David Jung Chul Kwak ............................. 87
Simulations of Complex Microstructure Formations
Britta Nestler, Frank Wendler .....................................113
Multiple Scales in Phase Separating Systems
with Elastic Misfit
Harald Garcke, Martin Lenz, Barbara Niethammer, Martin Rumpf,
Ulrich Weikard ..................................................153
A General Theory for Elastic Phase Transitions in Crystals
Steffen Arnrich, Thomas Blesgen, Stephan Luckhaus .................179
Relaxation and the Computation of Effective Energies
and Microstructures in Solid Mechanics
S¨oren Bartels, Carsten Carstensen, Sergio Conti, Klaus Hackl, Ulrich
Hoppe, Antonio Orlando .........................................197
X Contents
Derivation of Elastic Theories for Thin Sheets
and the Constraint of Incompressibility
Sergio Conti, Georg Dolzmann.....................................225
Domain Walls and Vortices in Thin Ferromagnetic Films
Matthias Kurzke, Christof Melcher, Roger Moser.....................249
Wavelet-Based Multiscale Methods for
Electronic Structure Calculations
Heinz-Ju¨rgen Flad, Wolfgang Hackbusch, Hongjun Luo, Dietmar Kolb ..299
Multi-Scale Modeling of Quantum Semiconductor Devices
Anton Arnold, Ansgar Ju¨ngel......................................331
Electronic States in Semiconductor Nanostructures and
Upscaling to Semi-Classical Models
Thomas Koprucki, Hans-Christoph Kaiser, Ju¨rgen Fuhrmann..........365
Shape Optimization of Biomorphic Ceramics with
Microstructures by Homogenization Modeling
Ronald H.W. Hoppe, Svetozara I. Petrova ...........................395
Discrete Free Energy Functionals for Elastic Materials
with Phase Change
Thomas Blesgen, Stephan Luckhaus, Luca Mugnai ...................425
Continuum Descriptions for the Dynamics in Discrete
Lattices:
Derivation and Justification
Johannes Giannoulis, Michael Herrmann, Alexander Mielke ...........435
Wave Trains, Solitons and Modulation Theory in FPU Chains
Wolfgang Dreyer, Michael Herrmann, Jens D.M. Rademacher .........467
On the Effect of Correlations, Fluctuations and Collisions
in Ostwald Ripening
Barbara Niethammer, Felix Otto, Juan J.L. Vela´zquez ................501
Radiative Friction for Charges Interacting with the
Radiation Field: Classical Many-Particle Systems
Sebastian Bauer, Markus Kunze ...................................531
Numerical Integrators for Highly Oscillatory Hamiltonian
Systems: a Review
David Cohen, Tobias Jahnke, Katina Lorenz, Christian Lubich.........553
Energy Level Crossings in Molecular Dynamics
Folkmar Bornemann, Caroline Lasser, Torben Swart .................577
Contents XI
Motion of Electrons in Adiabatically Perturbed
Periodic Structures
Gianluca Panati, Herbert Spohn, Stefan Teufel.......................595
Graph Algorithms for Dynamical Systems
Michael Dellnitz, Mirko Hessel-von Molo, Philipp Metzner, Robert
Preis, Christof Schu¨tte ...........................................619
Conditional Averaging for Diffusive Fast-Slow Systems:
A Sketch for Derivation
Jessika Walter, Christof Schu¨tte ...................................647
Colored Plates ................................................683
List of Contributors
Anton Arnold Thomas Blesgen
Institut fu¨r Analysis und Scientific Max-Planck-Institut fu¨r Mathematik
Computing in den Naturwissenschaften
Technische Universita¨t Wien Inselstraße 22–26
Wiedner Hauptstraße 8 04103 Leipzig
1040 Wien, Austria
Folkmar Bornemann
Steffen Arnrich Zentrum fu¨r Mathematik
Fakulta¨t fu¨r Mathematik und Technische Universita¨t Mu¨nchen
Informatik Boltzmannstraße 3
Universita¨t Leipzig 85747 Garching bei Mu¨nchen
Augustusplatz 10/11
04103 Leipzig Carsten Carstensen
Institut fu¨r Mathematik
Lev Balykov Humboldt-Universita¨t zu Berlin
Crystal Growth group Rudower Chaussee 25
Research center caesar 12489 Berlin
Ludwig-Erhard-Allee 2
53175 Bonn Vladimir Chalupecky
Department of Mathematics
So¨ren Bartels Faculty of Nuclear Sciences and
Institut fu¨r Mathematik Physical Engineering
Humboldt-Universita¨t zu Berlin Czech Technical University
Rudower Chaussee 25 Trojanova 13
12489 Berlin 120 00 Prague 2, Czech Republic
Sebastian Bauer David Cohen
Fachbereich Mathematik Mathematisches Institut
Universita¨t Duisburg-Essen Universita¨t Tu¨bingen
45117 Essen 72076 Tu¨bingen