Modelling of interfaces and free boundaries
|
|
- Brett Anderson
- 6 years ago
- Views:
Transcription
1 University of Regensburg Regensburg, March 2009
2 Outline 1 Introduction 2 Obstacle problems 3 Stefan problem 4 Shape optimization
3 Introduction What is a free boundary problem? Solve a partial differential equation (PDE) in a domain whose boundary is unknown in advance.
4 Introduction What is a free boundary problem? Solve a partial differential equation (PDE) in a domain whose boundary is unknown in advance. Task: Determine both the free boundary and the solution of the PDE
5 boundary conditions: standard boundary conditions for PDEs additional conditions to specify free boundary
6 boundary conditions: standard boundary conditions for PDEs additional conditions to specify free boundary Example: Dirichlet problem: u = f in Ω u = 0 on Γ = Ω. If Γ is free: Specify u n = 0 on Γ
7 One main strategy for free boundary problems: Reformulate the problem in such a way that the free boundary disappears Methods to achieve this: Use variational inequalities Use discontinuous variables and weak formulations
8 Where free boundaries appear in applications? obstacle problems for a membrane contact problems for an elastic body finance - Black-Scholes models porous media flow tumor growth free surface flow coating flows epitaxial growth shape optimization problems (reduce drag by changing the shape shape is given with the help of a free boundary. PDE to be solved Navier-Stokes equation)
9 Obstacle problems Obstacle problems and variational inequalities Consider: elastic membrane as graph u : Ω R, Ω R d obstacle ϕ : Ω R boundary data u 0 : Ω R force f : Ω R Minimize such that E(u) := Ω ( ) 1 2 u 2 fu dx min u ϕ in Ω u = u 0 on Ω
10 boundary of {u > ϕ} is the free boundary on {u > ϕ} the PDE u = f holds
11 Let u be a minimum and v is assumed to fulfill the constraints. Then it holds 0 d dε E(u + ε(v u)) ( ) ε=0 = u (v u) f (v u) dx. It holds N := {u > ϕ} is open Choose v = u ± εζ with ζ C0 (N) [ ( u) f ] ζ dx = 0 for all ζ C0 (N) Ω Ω
12 Hence u f = 0 in N Furthermore: [ ( u) f ] ζ dx 0 for all ζ C0 (Ω) with ζ 0 Ω and hence u f 0 for almost all x Ω.
13 It can be shown (if data are smooth enough): u C 1 (Ω). Hence u = ϕ and u n = ϕ n (BE CAREFUL) on free boundary Γ := N. Define active set A = Ω \ N. obstacle ϕ 0
14 Reformulation as complementarity system u f 0 u ϕ ( u f )(u ϕ) = 0 in Ω, u = u 0 on Ω, u = ϕ on Γ, u n = ϕ n on Γ.
15 Reformulation with Lagrange multipliers Define Lagrange multiplicator { 0 if x N, µ(x) = u(x) f (x) if x A, We then obtain u f = µ u ϕ µ 0 µ(u ϕ) = 0 in Ω.
16 Reformulation as semi-smooth equation Rewrite u ϕ µ 0 µ(u ϕ) = 0 in Ω as µ = max(0, µ c(u ϕ)) in Ω. where c > 0
17 Formulation as free boundary problem Find N Ω, a free boundary Γ = N and u : N R, such that u f = 0 in N u = u 0 on Ω u = ϕ, u n = ϕ n on Γ u ϕ in N ϕ f 0 in Ω \ N
18 The free boundary problem for an elastic membrane can be rewritten as minimization problem variational inequality complementarity problem with the help of Lagrange multipliers with the help of a semi-smooth equation
19 Situations in which free boundary problems can be reformulated as a variational inequality: contact of an elastic body with an obstacle (vector valued variational inequality) dam problem. Filtration of water through a dam. Free boundary separates wet and dry parts. Black-Scholes variational inequalities (pricing of options) parabolic variational inequality solidification problems (Stefan problem) thin obstacle problem (Signorini problem) obstacle problems for plates
20 Stefan problem Stefan problem models melting and solidification Derivation from energy conservation Variables: T : temperature u: internal energy density L: latent heat q: heat flux For simplicity We set specific heat c v = 1 mass density ρ = 1 q = T u(t ) = { T solid phase T + L liquid phase
21 Energy conservation: For all V R d we have d udx = q n ds x, n outer normal dt V If V lies in pure phases V V ( t u + q)dx = 0 t u + q = 0, q = T t T T = 0
22 Let Γ be the free boundary between liquid and solid. If Γ V we obtain using a transport theorem: d udx = t u [u] l dt svds x V V Γ V [u] l s : jump of u across free boundary V : normal velocity of the free boundary. x(t) Γ(t) then we define V (x(t), t) := ẋ(t) n. Remark: [u] l s = L
23 V q n = V q + Γ V [q] l s n ds x Combining both equations and using the fact that V is arbitrary gives Stefan condition LV = [ T ] n on Γ This is only one condition on Γ. How many conditions do we expect?
24 Possibilities for two more conditions I II T = 0 on Γ (two boundary conditions) βv = γκ T on Γ (Gibbs-Thomson law) where β: kinetic undercooling coefficient γ: capillarity coefficient κ = Γ n : mean curvature How to treat the free boundary Case I : We can hide the free boundary Case II : We discuss serveral possibilites to treat the free boundary
25 Classical Stefan problem is equivalent to t T T = 0 in liquid + solid T = 0 on Γ LV = [ T ] n on Γ t (T + Lχ {T >0} ) = T in distributional sense }{{} u(t )= where χ {T >0} is the characteristic function of the set {T > 0}.
26 invert Solve t u = β(u) in weak sense degenerate parabolic PDE analysis difficult but simple numerical methods available
27 Now βv = γκ T instead of T = 0 (avoids mushy regions, takes interface energy into account) How to represent the free boundary At least five possibilities: as a graph use distance function to a reference manifold using a parameterization use level set approach use phase field approach
28 Graph approach Write free boundary Γ(t) as Γ(t) = {(x, h(x, t)) x Ω } with height function h : Ω [0, T ) R 1+ h 2 V = th ( κ = ( β t h =γ 1+ h 2 h 1+ h 2 h 1+ h 2 highly nonlinear parabolic PDE in variable h ) ) T
29 Parametric approach I Choose (d 1)-dimensional reference surface M R d (e.g. free boundary at time t = 0). Search for maps X (., t) : M R d, t [0, T ) normal velocity: X t n, n has to be computed from X mean curvature: trace of second fundamental form important relation Γ X = κn Γ : Laplace Beltrami operator on Γ Γ X = 1 d 1 ( g θ i g ij ) g X θ j i,j=1
30 Parametric approach II Two possibilities to write the equation βv = γκ T a) βx t = γ Γ X Tn = βx t = (γκ T )n b) Velocity always in normal direction (numerical approaches based on this formulation: Dziuk, Schmidt) βx t n = γκ T Γ X = κn Allows also for solutions X which have a velocity in tangential direction (numerical approach based on this formulation: Barrett, Garcke, Nürnberg)
31 Often additional aspects have to be considered Examples: several phases triple junctions anisotropy replace κ by anistropy curvature κ γ Sharp interface computations for anisotropic Stefan problem (Barrett, G., Nürnberg)
32 For anisotropic energies the Wulff shape is the energy minimizers (Numerical computation: Barrett, G., Nürnberg) ) Coupling at triple junctions Young s law has to be satisfied (Numerical computation:barrett, G., Nürnberg) )
33 Level set approach I Look for Γ(t) as zero level set of φ : R d [0, ) R i.e. Γ(t) = {x R d φ(x, t) = 0}. Require: φ 0 on Γ. We obtain: n = φ φ, V = φ t φ, κ = φ φ βv = γκ T can be written as β φ t φ = γ φ φ T
34 Level set approach II highly nonlinear degenerate not defined if φ = 0 β φ t φ = γ φ φ T φ has to be computed away from the interface and φ R d \Γ has no interpretation Mathematical theory: use viscosity solutions (Crandall, Ishii, Lions; Evans, Spruck; Chen, Giga, Goto). Numerical approaches: Osher, Fedkiv and Sethian very efficient, many applications
35 Phase field approach I Replace V = κ + T β = γ = 1 by ε t ϕ = (ε ϕ 1 ε ψ (ϕ)) T 1 2 V = 1 2 κ T ψ(ϕ) = 9 32 (ϕ2 1) 2, ε > 0 small asymptotic limit ε 0
36 Phase field approach II In a phase field approach the free boundary (a hypersurface) is replaced by a diffuse interface }{{} interface size proportional to ε phase field simulation for solidification (M. Krä)
37 Phase field approach III Solve ε t ϕ = ε ϕ 1 ε ψ (ϕ) T t (T + ϕ) = T, L = 1 in order to approximate the Stefan problem with Gibbs-Thomson law. Phase field methods have been used for many interface problems: Examples: solidification interfaces in binary and multicomponent alloys grain growth (Allen-Cahn equation) coarsening phenomena (Cahn-Hilliard equation) epitaxial growth in thin films (quantum dot formation) fluid flow problems with interfaces electromigration topology optimization image processing
38 Advantages / Disadvantages of the approaches graph and parametric approach do not allow for topology changes level set and phase field approach handle topology changes implicitely graph and parametric approach need only (d 1)-dimensional variables computational advantage phase field and level set approaches need d-dimensional fields. Involve regularization parameters which might not have a physical significance
39 Phase field simulations I several scales and phenomena in material modelling experimentally observed dendrites
40 Phase field simulations II eutectic growth with multiple scales (phase field simulation, G. + Nestler) dendritic growth (phase field simulation, G. + Nestler)
41 Phase field simulations III eutectic growth in 3d with isotropic energies (phase field simulation, G. + Nestler) eutetic growth in 3d with anisotropic energies (phase-field simulation, G. + Nestler)
42 Introduction Obstacle problems Stefan problem Shape optimization Phase field simulations IV alignement of interfaces driven by anisotropic elasticity (G., Weikard) monotectic growth (Nestler, Wheeler)
43 Phase field simulations V surface diffusion in heteroepitaxial thin film growth Island formation in thin film epitaxial growth
44 Introduction Obstacle problems Stefan problem Shape optimization Phase field simulations VI t = t = t = t = Vector-valued Allen-Cahn equation (Blank, G., Sarbu, Styles) t = t = t = t = topology optimization (Blank, G., Sarbu, Styles)
45 Shape optimization THIS PART WILL BE ON THE BLACKBOARD!!!!!!!!!!!!!!!!!!!
46 Literatur Deckelnick, K., Dziuk, G., and Elliott, C.M., Computation of geometric partial differential equations and mean curvature flow. Acta Numer. 14 (2005), Eck, C., Garcke, H., and Knabner, P., Mathematische Modellierung. Springer-Lehrbuch. Springer-Verlag 2008 Elliott, C.M., Ockendon, J.R., Weak and variational methods for moving boundary problems. Research Notes in Mathematics 59. Pitman Boston, Mass.-London, 1982, iii+213 pp. Friedman, A., Variational principles and free-boundary problems, Wiley 1982 Journal Interfaces and Free Boundaries, EMS Publishing House
47 Literatur Kinderlehrer, D. and Stampacchia, G., An introduction to variational inequalities and their applications, Pure and Applied Mathematics 88. Academic Press, Inc., New York-London (1980), xx+313 pp. Osher, S. and Fedkiw, R., Level Set Methods and Dynamic Implicit Surfaces. Springer-Verlag New York, Applied Mathematical Sciences, 153, 2002, BOOK. Sethian, J.A., Level Set Methods. Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision and Materials Sciences. Cambridge University Press, (First edition) Sokolowski, J and Zolesio, J.-P., Introduction to Shape Optimization, Springer, 1991.
A sharp diffuse interface tracking method for approximating evolving interfaces
A sharp diffuse interface tracking method for approximating evolving interfaces Vanessa Styles and Charlie Elliott University of Sussex Overview Introduction Phase field models Double well and double obstacle
More informationA finite element level set method for anisotropic mean curvature flow with space dependent weight
A finite element level set method for anisotropic mean curvature flow with space dependent weight Klaus Deckelnick and Gerhard Dziuk Centre for Mathematical Analysis and Its Applications, School of Mathematical
More informationNumerical Approximation of Phase Field Models
Numerical Approximation of Phase Field Models Lecture 2: Allen Cahn and Cahn Hilliard Equations with Smooth Potentials Robert Nürnberg Department of Mathematics Imperial College London TUM Summer School
More informationUniversität Regensburg Mathematik
Universität Regensburg Mathematik On the stable discretization of strongly anisotropic phase field models with applications to crystal growth John W. Barrett, Harald Garcke and Robert Nürnberg Preprint
More informationNumerical Methods of Applied Mathematics -- II Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.336 Numerical Methods of Applied Mathematics -- II Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
More informationLECTURE 4: GEOMETRIC PROBLEMS
LECTURE 4: GEOMETRIC PROBLEMS Department of Mathematics and Institute for Physical Science and Technology University of Maryland, USA Tutorial: Numerical Methods for FBPs Free Boundary Problems and Related
More informationLECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES)
LECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES) RAYTCHO LAZAROV 1 Notations and Basic Functional Spaces Scalar function in R d, d 1 will be denoted by u,
More informationClassical solutions for the quasi-stationary Stefan problem with surface tension
Classical solutions for the quasi-stationary Stefan problem with surface tension Joachim Escher, Gieri Simonett We show that the quasi-stationary two-phase Stefan problem with surface tension has a unique
More informationOn Multigrid for Phase Field
On Multigrid for Phase Field Carsten Gräser (FU Berlin), Ralf Kornhuber (FU Berlin), Rolf Krause (Uni Bonn), and Vanessa Styles (University of Sussex) Interphase 04 Rome, September, 13-16, 2004 Synopsis
More informationGeneralized Phase Field Models with Anisotropy and Non-Local Potentials
Generalized Phase Field Models with Anisotropy and Non-Local Potentials Gunduz Caginalp University of Pittsburgh caginalp@pitt.edu December 6, 2011 Gunduz Caginalp (Institute) Generalized Phase Field December
More informationEXISTENCE OF SOLUTIONS TO THE CAHN-HILLIARD/ALLEN-CAHN EQUATION WITH DEGENERATE MOBILITY
Electronic Journal of Differential Equations, Vol. 216 216), No. 329, pp. 1 22. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS TO THE CAHN-HILLIARD/ALLEN-CAHN
More informationOn the Parametric Finite Element Approximation of Evolving Hypersurfaces in R 3
On the Parametric Finite Element Approximation of Evolving Hypersurfaces in R 3 John W. Barrett Harald Garcke Robert Nürnberg Abstract We present a variational formulation of motion by minus the Laplacian
More informationMotivation Power curvature flow Large exponent limit Analogues & applications. Qing Liu. Fukuoka University. Joint work with Prof.
On Large Exponent Behavior of Power Curvature Flow Arising in Image Processing Qing Liu Fukuoka University Joint work with Prof. Naoki Yamada Mathematics and Phenomena in Miyazaki 2017 University of Miyazaki
More informationwith deterministic and noise terms for a general non-homogeneous Cahn-Hilliard equation Modeling and Asymptotics
12-3-2009 Modeling and Asymptotics for a general non-homogeneous Cahn-Hilliard equation with deterministic and noise terms D.C. Antonopoulou (Joint with G. Karali and G. Kossioris) Department of Applied
More informationA phase field model for the coupling between Navier-Stokes and e
A phase field model for the coupling between Navier-Stokes and electrokinetic equations Instituto de Matemáticas, CSIC Collaborators: C. Eck, G. Grün, F. Klingbeil (Erlangen Univertsität), O. Vantzos (Bonn)
More informationFrequency functions, monotonicity formulas, and the thin obstacle problem
Frequency functions, monotonicity formulas, and the thin obstacle problem IMA - University of Minnesota March 4, 2013 Thank you for the invitation! In this talk we will present an overview of the parabolic
More informationSolving PDEs with freefem++
Solving PDEs with freefem++ Tutorials at Basque Center BCA Olivier Pironneau 1 with Frederic Hecht, LJLL-University of Paris VI 1 March 13, 2011 Do not forget That everything about freefem++ is at www.freefem.org
More informationModelling crystal growth with the phase-field method Mathis Plapp
Modelling crystal growth with the phase-field method Mathis Plapp Laboratoire de Physique de la Matière Condensée CNRS/Ecole Polytechnique, 91128 Palaiseau, France Solidification microstructures Hexagonal
More informationWeak solutions for the Cahn-Hilliard equation with degenerate mobility
Archive for Rational Mechanics and Analysis manuscript No. (will be inserted by the editor) Shibin Dai Qiang Du Weak solutions for the Cahn-Hilliard equation with degenerate mobility Abstract In this paper,
More informationAPPROXIMATION OF CRYSTALLINE DENDRITE GROWTH IN TWO SPACE DIMENSIONS. Introduction
Acta Mat. Univ. Comenianae Vol. LXVII, 1(1998), pp. 57 68 57 APPROXIMATION OF CRYSTALLINE DENDRITE GROWTH IN TWO SPACE DIMENSIONS A. SCHMIDT Abstract. Te pase transition between solid and liquid in an
More informationSurface Evolution Equations a level set method. Yoshikazu Giga Department of Mathematics Hokkaido University Sapporo , Japan
Surface Evolution Equations a level set method Yoshikazu Giga Department of Mathematics Hokkaido University Sapporo 060-0810, Japan March, 2002 Preface This book is intended to be a self-contained introduction
More informationTHE CAHN-HILLIARD EQUATION WITH A LOGARITHMIC POTENTIAL AND DYNAMIC BOUNDARY CONDITIONS
THE CAHN-HILLIARD EQUATION WITH A LOGARITHMIC POTENTIAL AND DYNAMIC BOUNDARY CONDITIONS Alain Miranville Université de Poitiers, France Collaborators : L. Cherfils, G. Gilardi, G.R. Goldstein, G. Schimperna,
More informationLecture No 1 Introduction to Diffusion equations The heat equat
Lecture No 1 Introduction to Diffusion equations The heat equation Columbia University IAS summer program June, 2009 Outline of the lectures We will discuss some basic models of diffusion equations and
More informationA PRACTICALLY UNCONDITIONALLY GRADIENT STABLE SCHEME FOR THE N-COMPONENT CAHN HILLIARD SYSTEM
A PRACTICALLY UNCONDITIONALLY GRADIENT STABLE SCHEME FOR THE N-COMPONENT CAHN HILLIARD SYSTEM Hyun Geun LEE 1, Jeong-Whan CHOI 1 and Junseok KIM 1 1) Department of Mathematics, Korea University, Seoul
More informationQuantum Dots and Dislocations: Dynamics of Materials Defects
Quantum Dots and Dislocations: Dynamics of Materials Defects with N. Fusco, G. Leoni, M. Morini, A. Pratelli, B. Zwicknagl Irene Fonseca Department of Mathematical Sciences Center for Nonlinear Analysis
More informationApplied PDEs: Analysis and Computation
Applied PDEs: Analysis and Computation Hailiang Liu hliu@iastate.edu Iowa State University Tsinghua University May 07 June 16, 2012 1 / 15 Lecture #1: Introduction May 09, 2012 Model, Estimate and Algorithm=MEA
More informationModelling and analysis for contact angle hysteresis on rough surfaces
Modelling and analysis for contact angle hysteresis on rough surfaces Xianmin Xu Institute of Computational Mathematics, Chinese Academy of Sciences Collaborators: Xiaoping Wang(HKUST) Workshop on Modeling
More informationCoarsening Dynamics for the Cahn-Hilliard Equation. Jaudat Rashed
Coarsening Dynamics for the Cahn-Hilliard Equation Jaudat Rashed Coarsening Dynamics for the Cahn-Hilliard Equation Research Thesis Submitted in partial fulfillment of the requirements for the degree of
More informationMathematica Bohemica
Mathematica Bohemica Miroslav Kolář; Michal Beneš; Daniel Ševčovič Computational studies of conserved mean-curvature flow Mathematica Bohemica, Vol. 39 (4), No. 4, 677--684 Persistent URL: http://dml.cz/dmlcz/4444
More informationNumerical Simulations on Two Nonlinear Biharmonic Evolution Equations
Numerical Simulations on Two Nonlinear Biharmonic Evolution Equations Ming-Jun Lai, Chun Liu, and Paul Wenston Abstract We numerically simulate the following two nonlinear evolution equations with a fourth
More informationUniversität Regensburg Mathematik
Universität Regensburg Mathematik Optimal control of Allen-Cahn systems Luise Blank, M. Hassan Farshbaf-Shaker, Claudia Hecht Josef Michl and Christoph Rupprecht Preprint Nr. 21/2013 Optimal control of
More informationLevel-Set Minimization of Potential Controlled Hadwiger Valuations for Molecular Solvation
Level-Set Minimization of Potential Controlled Hadwiger Valuations for Molecular Solvation Bo Li Dept. of Math & NSF Center for Theoretical Biological Physics UC San Diego, USA Collaborators: Li-Tien Cheng
More informationControl of Interface Evolution in Multi-Phase Fluid Flows
Control of Interface Evolution in Multi-Phase Fluid Flows Markus Klein Department of Mathematics University of Tübingen Workshop on Numerical Methods for Optimal Control and Inverse Problems Garching,
More informationarxiv: v1 [math.oc] 9 Dec 2013
arxiv:1312.2356v1 [math.oc] 9 Dec 2013 Multi-material phase field approach to structural topology optimization Luise Blank, M. Hassan Farshbaf-Shaker, Harald Garcke, Christoph Rupprecht, Vanessa Styles
More informationThe main motivation of this paper comes from the following, rather surprising, result of Ecker and Huisken [13]: for any initial data u 0 W 1,
Quasilinear parabolic equations, unbounded solutions and geometrical equations II. Uniqueness without growth conditions and applications to the mean curvature flow in IR 2 Guy Barles, Samuel Biton and
More informationERROR ANALYSIS OF STABILIZED SEMI-IMPLICIT METHOD OF ALLEN-CAHN EQUATION. Xiaofeng Yang. (Communicated by Jie Shen)
DISCRETE AND CONTINUOUS doi:1.3934/dcdsb.29.11.157 DYNAMICAL SYSTEMS SERIES B Volume 11, Number 4, June 29 pp. 157 17 ERROR ANALYSIS OF STABILIZED SEMI-IMPLICIT METHOD OF ALLEN-CAHN EQUATION Xiaofeng Yang
More informationAdaptive evolution : a population approach Benoît Perthame
Adaptive evolution : a population approach Benoît Perthame Adaptive dynamic : selection principle d dt n(x, t) = n(x, t)r( x, ϱ(t) ), ϱ(t) = R d n(x, t)dx. given x, n(x) = ϱ δ(x x), R( x, ϱ) = 0, ϱ( x).
More informationSingular Diffusion Equations With Nonuniform Driving Force. Y. Giga University of Tokyo (Joint work with M.-H. Giga) July 2009
Singular Diffusion Equations With Nonuniform Driving Force Y. Giga University of Tokyo (Joint work with M.-H. Giga) July 2009 1 Contents 0. Introduction 1. Typical Problems 2. Variational Characterization
More informationSimulation of free surface fluids in incompressible dynamique
Simulation of free surface fluids in incompressible dynamique Dena Kazerani INRIA Paris Supervised by Pascal Frey at Laboratoire Jacques-Louis Lions-UPMC Workshop on Numerical Modeling of Liquid-Vapor
More informationRisk Averse Shape Optimization
Risk Averse Shape Optimization Sergio Conti 2 Martin Pach 1 Martin Rumpf 2 Rüdiger Schultz 1 1 Department of Mathematics University Duisburg-Essen 2 Rheinische Friedrich-Wilhelms-Universität Bonn Workshop
More informationNotes: Outline. Diffusive flux. Notes: Notes: Advection-diffusion
Outline This lecture Diffusion and advection-diffusion Riemann problem for advection Diagonalization of hyperbolic system, reduction to advection equations Characteristics and Riemann problem for acoustics
More informationExistence of viscosity solutions for a nonlocal equation modelling polymer
Existence of viscosity solutions for a nonlocal modelling Institut de Recherche Mathématique de Rennes CNRS UMR 6625 INSA de Rennes, France Joint work with P. Cardaliaguet (Univ. Paris-Dauphine) and A.
More informationPHASE-FIELD MODELLING OF NONEQUILIBRIUM PARTITIONING DURING RAPID SOLIDIFICATION IN A NON-DILUTE BINARY ALLOY. Denis Danilov, Britta Nestler
First published in: DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS Volume 15, Number 4, August 2006 pp. 1035 1047 PHASE-FIELD MODELLING OF NONEQUILIBRIUM PARTITIONING DURING
More informationApplied Math Qualifying Exam 11 October Instructions: Work 2 out of 3 problems in each of the 3 parts for a total of 6 problems.
Printed Name: Signature: Applied Math Qualifying Exam 11 October 2014 Instructions: Work 2 out of 3 problems in each of the 3 parts for a total of 6 problems. 2 Part 1 (1) Let Ω be an open subset of R
More informationINTRODUCTION TO PDEs
INTRODUCTION TO PDEs In this course we are interested in the numerical approximation of PDEs using finite difference methods (FDM). We will use some simple prototype boundary value problems (BVP) and initial
More informationAnisotropic multi-phase-field model: Interfaces and junctions
First published in: PHYSICAL REVIEW E VOLUME 57, UMBER 3 MARCH 1998 Anisotropic multi-phase-field model: Interfaces and junctions B. estler 1 and A. A. Wheeler 2 1 Foundry Institute, University of Aachen,
More informationInfo. No lecture on Thursday in a week (March 17) PSet back tonight
Lecture 0 8.086 Info No lecture on Thursday in a week (March 7) PSet back tonight Nonlinear transport & conservation laws What if transport becomes nonlinear? Remember: Nonlinear transport A first attempt
More informationThe maximum principle for degenerate parabolic PDEs with singularities
113 The maximum principle for degenerate parabolic PDEs with singularities Hitoshi Ishii 1. Introduction This is a preliminary version of [7]. Here we shall be concerned with the degenerate parabolic partial
More informationOn Two Nonlinear Biharmonic Evolution Equations: Existence, Uniqueness and Stability
On Two Nonlinear Biharmonic Evolution Equations: Existence, Uniqueness and Stability Ming-Jun Lai, Chun Liu, and Paul Wenston Abstract We study the following two nonlinear evolution equations with a fourth
More informationAllen Cahn Equation in Two Spatial Dimension
Allen Cahn Equation in Two Spatial Dimension Yoichiro Mori April 25, 216 Consider the Allen Cahn equation in two spatial dimension: ɛ u = ɛ2 u + fu) 1) where ɛ > is a small parameter and fu) is of cubic
More informationDeutsche Mathematiker-Vereinigung and Springer-Verlag Berlin Heidelberg 2013
Jahresber Dtsch Math-Ver (2013) 115:61 62 DOI 10.1365/s13291-013-0068-0 PREFACE Preface Issue 2-2013 Hans-Christoph Grunau Deutsche Mathematiker-Vereinigung and Springer-Verlag Berlin Heidelberg 2013 Curvature
More informationNonsmooth Schur Newton Methods and Applications
Nonsmooth Schur Newton Methods and Applications C. Gräser, R. Kornhuber, and U. Sack IMA Annual Program Year Workshop Numerical Solutions of Partial Differential Equations: Fast Solution Techniques November
More informationDerivation of continuum models for the moving contact line problem based on thermodynamic principles. Abstract
Derivation of continuum models for the moving contact line problem based on thermodynamic principles Weiqing Ren Courant Institute of Mathematical Sciences, New York University, New York, NY 002, USA Weinan
More informationProblem Set Number 01, MIT (Winter-Spring 2018)
Problem Set Number 01, 18.377 MIT (Winter-Spring 2018) Rodolfo R. Rosales (MIT, Math. Dept., room 2-337, Cambridge, MA 02139) February 28, 2018 Due Thursday, March 8, 2018. Turn it in (by 3PM) at the Math.
More informationDerivation and Analysis of a Phase Field Model for Alloy Solidification
Derivation and Analysis of a Phase Field Model for Alloy Solidification Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften Dr. rer. nat. der Naturwissenschaftlichen Fakultät I - Mathematik
More informationNumber of pages in the question paper : 05 Number of questions in the question paper : 48 Modeling Transport Phenomena of Micro-particles Note: Follow the notations used in the lectures. Symbols have their
More informationarxiv: v1 [cond-mat.mtrl-sci] 6 Feb 2012
Numerical computations of facetted pattern formation in snow crystal growth John W. Barrett a, Harald Garcke b,, Robert Nürnberg a a Department of Mathematics, Imperial College London, London SW7 2AZ,
More informationIntegro-differential equations: Regularity theory and Pohozaev identities
Integro-differential equations: Regularity theory and Pohozaev identities Xavier Ros Oton Departament Matemàtica Aplicada I, Universitat Politècnica de Catalunya PhD Thesis Advisor: Xavier Cabré Xavier
More informationAn Epiperimetric Inequality Approach to the Thin and Fractional Obstacle Problems
An Epiperimetric Inequality Approach to the Thin and Fractional Obstacle Problems Geometric Analysis Free Boundary Problems & Measure Theory MPI Leipzig, June 15 17, 2015 Arshak Petrosyan (joint with Nicola
More informationEntropy-dissipation methods I: Fokker-Planck equations
1 Entropy-dissipation methods I: Fokker-Planck equations Ansgar Jüngel Vienna University of Technology, Austria www.jungel.at.vu Introduction Boltzmann equation Fokker-Planck equations Degenerate parabolic
More informationBuilding Solutions to Nonlinear Elliptic and Parabolic Partial Differential Equations
Building Solutions to Nonlinear Elliptic and Parabolic Partial Differential Equations Adam Oberman University of Texas, Austin http://www.math.utexas.edu/~oberman Fields Institute Colloquium January 21,
More informationContinuum Methods 1. John Lowengrub Dept Math UC Irvine
Continuum Methods 1 John Lowengrub Dept Math UC Irvine Membranes, Cells, Tissues complex micro-structured soft matter Micro scale Cell: ~10 micron Sub cell (genes, large proteins): nanometer Macro scale
More informationEntropy and Relative Entropy
Entropy and Relative Entropy Joshua Ballew University of Maryland October 24, 2012 Outline Hyperbolic PDEs Entropy/Entropy Flux Pairs Relative Entropy Weak-Strong Uniqueness Weak-Strong Uniqueness for
More informationIntroduction to Partial Differential Equations
Introduction to Partial Differential Equations Philippe B. Laval KSU Current Semester Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 1 / 19 Introduction The derivation of the heat
More informationSurface phase separation and flow in a simple model of drops and vesicles
Surface phase separation and flow in a simple model of drops and vesicles Tutorial Lecture 4 John Lowengrub Department of Mathematics University of California at Irvine Joint with J.-J. Xu (UCI), S. Li
More informationCurvature driven interface evolution
Curvature driven interface evolution Harald Garcke Abstract Curvature driven surface evolution plays an important role in geometry, applied mathematics and in the natural sciences. In this paper geometric
More informationNonlinear elasticity and gels
Nonlinear elasticity and gels M. Carme Calderer School of Mathematics University of Minnesota New Mexico Analysis Seminar New Mexico State University April 4-6, 2008 1 / 23 Outline Balance laws for gels
More informationA Remark on IVP and TVP Non-Smooth Viscosity Solutions to Hamilton-Jacobi Equations
2005 American Control Conference June 8-10, 2005. Portland, OR, USA WeB10.3 A Remark on IVP and TVP Non-Smooth Viscosity Solutions to Hamilton-Jacobi Equations Arik Melikyan, Andrei Akhmetzhanov and Naira
More informationFoliations of hyperbolic space by constant mean curvature surfaces sharing ideal boundary
Foliations of hyperbolic space by constant mean curvature surfaces sharing ideal boundary David Chopp and John A. Velling December 1, 2003 Abstract Let γ be a Jordan curve in S 2, considered as the ideal
More informationTRANSPORT IN POROUS MEDIA
1 TRANSPORT IN POROUS MEDIA G. ALLAIRE CMAP, Ecole Polytechnique 1. Introduction 2. Main result in an unbounded domain 3. Asymptotic expansions with drift 4. Two-scale convergence with drift 5. The case
More informationIntrinsic finite element modeling of a linear membrane shell problem
arxiv:3.39v [math.na] 5 Mar Intrinsic finite element modeling of a linear membrane shell problem Peter Hansbo Mats G. Larson Abstract A Galerkin finite element method for the membrane elasticity problem
More informationPart 1 Introduction Degenerate Diffusion and Free-boundaries
Part 1 Introduction Degenerate Diffusion and Free-boundaries Columbia University De Giorgi Center - Pisa June 2012 Introduction We will discuss, in these lectures, certain geometric and analytical aspects
More informationCURVATURE DRIVEN FLOWS IN DEFORMABLE DOMAINS
CURVATURE DRIVEN FLOWS IN DEFORMABLE DOMAINS Department of Mathematics and Institute for Physical Science and Technology University of Maryland, USA BCAM EHU/UPV Basque Colloquium in Mathematics and its
More informationModelos de mudança de fase irreversíveis
Modelos de mudança de fase irreversíveis Gabriela Planas Departamento de Matemática Instituto de Matemática, Estatística e Computação Científica Universidade Estadual de Campinas, Brazil Em colaboração
More informationLecture No 2 Degenerate Diffusion Free boundary problems
Lecture No 2 Degenerate Diffusion Free boundary problems Columbia University IAS summer program June, 2009 Outline We will discuss non-linear parabolic equations of slow diffusion. Our model is the porous
More informationTHE CRYSTALLINE ALGORITHM FOR COMPUTING MOTION BY CURVATURE
THE CRYSTALLINE ALGORITHM FOR COMPUTING MOTION BY CURVATURE PEDRO MARTINS GIRÃO AND ROBERT V. KOHN Abstract. Motion by (weighted) mean curvature is a geometric evolution law for surfaces, representing
More informationA Nonlinear PDE in Mathematical Finance
A Nonlinear PDE in Mathematical Finance Sergio Polidoro Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40127 Bologna (Italy) polidoro@dm.unibo.it Summary. We study a non
More informationIntroduction to Partial Differential Equations
Introduction to Partial Differential Equations Partial differential equations arise in a number of physical problems, such as fluid flow, heat transfer, solid mechanics and biological processes. These
More informationPart IV: Numerical schemes for the phase-filed model
Part IV: Numerical schemes for the phase-filed model Jie Shen Department of Mathematics Purdue University IMS, Singapore July 29-3, 29 The complete set of governing equations Find u, p, (φ, ξ) such that
More informationMulti-dimensional Stochastic Singular Control Via Dynkin Game and Dirichlet Form
Multi-dimensional Stochastic Singular Control Via Dynkin Game and Dirichlet Form Yipeng Yang * Under the supervision of Dr. Michael Taksar Department of Mathematics University of Missouri-Columbia Oct
More informationMSc Dissertation topics:
.... MSc Dissertation topics: Omar Lakkis Mathematics University of Sussex Brighton, England November 6, 2013 Office Location: Pevensey 3 5C2 Office hours: Autumn: Tue & Fri 11:30 12:30; Spring: TBA. O
More informationMath background. Physics. Simulation. Related phenomena. Frontiers in graphics. Rigid fluids
Fluid dynamics Math background Physics Simulation Related phenomena Frontiers in graphics Rigid fluids Fields Domain Ω R2 Scalar field f :Ω R Vector field f : Ω R2 Types of derivatives Derivatives measure
More informationA Differential Variational Inequality Approach for the Simulation of Heterogeneous Materials
A Differential Variational Inequality Approach for the Simulation of Heterogeneous Materials Lei Wang 1, Jungho Lee 1, Mihai Anitescu 1, Anter El Azab 2, Lois Curfman McInnes 1, Todd Munson 1, Barry Smith
More informationComputers and Mathematics with Applications. A new application of He s variational iteration method for the solution of the one-phase Stefan problem
Computers and Mathematics with Applications 58 (29) 2489 2494 Contents lists available at ScienceDirect Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa A new
More informationLECTURE 3: DISCRETE GRADIENT FLOWS
LECTURE 3: DISCRETE GRADIENT FLOWS Department of Mathematics and Institute for Physical Science and Technology University of Maryland, USA Tutorial: Numerical Methods for FBPs Free Boundary Problems and
More informationFrom nonlocal to local Cahn-Hilliard equation. Stefano Melchionna Helene Ranetbauer Lara Trussardi. Uni Wien (Austria) September 18, 2018
From nonlocal to local Cahn-Hilliard equation Stefano Melchionna Helene Ranetbauer Lara Trussardi Uni Wien (Austria) September 18, 2018 SFB P D ME S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal
More informationSpace-time XFEM for two-phase mass transport
Space-time XFEM for two-phase mass transport Space-time XFEM for two-phase mass transport Christoph Lehrenfeld joint work with Arnold Reusken EFEF, Prague, June 5-6th 2015 Christoph Lehrenfeld EFEF, Prague,
More informationFinite Volume Method
Finite Volume Method An Introduction Praveen. C CTFD Division National Aerospace Laboratories Bangalore 560 037 email: praveen@cfdlab.net April 7, 2006 Praveen. C (CTFD, NAL) FVM CMMACS 1 / 65 Outline
More informationNavier-Stokes equations in thin domains with Navier friction boundary conditions
Navier-Stokes equations in thin domains with Navier friction boundary conditions Luan Thach Hoang Department of Mathematics and Statistics, Texas Tech University www.math.umn.edu/ lhoang/ luan.hoang@ttu.edu
More informationABSTRACT. ALANEZY, KHALID ALI. Optimal Control of Moving Interface and Phase-Field Separations. (Under the direction of Kazufumi Ito.
ABSTRACT ALANEZY, KHALID ALI. Optimal Control of Moving Interface and Phase-Field Separations. (Under the direction of Kazufumi Ito.) In this thesis we consider two moving interface problems of practical
More informationCandidates must show on each answer book the type of calculator used. Log Tables, Statistical Tables and Graph Paper are available on request.
UNIVERSITY OF EAST ANGLIA School of Mathematics Spring Semester Examination 2004 FLUID DYNAMICS Time allowed: 3 hours Attempt Question 1 and FOUR other questions. Candidates must show on each answer book
More informationA SHORT INTRODUCTION TO TWO-PHASE FLOWS Two-phase flows balance equations
A SHORT INTRODUCTION TO TWO-PHASE FLOWS Two-phase flows balance equations Hervé Lemonnier DM2S/STMF/LIEFT, CEA/Grenoble, 38054 Grenoble Cedex 9 Ph. +33(0)4 38 78 45 40 herve.lemonnier@cea.fr, herve.lemonnier.sci.free.fr/tpf/tpf.htm
More informationHigh Order Unfitted Finite Element Methods for Interface Problems and PDEs on Surfaces
N O V E M B E R 2 0 1 6 P R E P R I N T 4 5 9 High Order Unfitted Finite Element Methods for Interface Problems and PDEs on Surfaces Christoph Lehrenfeld and Arnold Reusken Institut für Geometrie und Praktische
More informationGame Physics. Game and Media Technology Master Program - Utrecht University. Dr. Nicolas Pronost
Game and Media Technology Master Program - Utrecht University Dr. Nicolas Pronost Soft body physics Soft bodies In reality, objects are not purely rigid for some it is a good approximation but if you hit
More informationDELFT UNIVERSITY OF TECHNOLOGY
DELFT UNIVERSITY OF TECHNOLOGY REPORT 03-16 LITERTURE STUDY: NUMERICAL METHODS FOR SOLVING STEFAN PROBLEMS E.JAVIERRE-PÉREZ1 ISSN 1389-6520 Reports of the Department of Applied Mathematical Analysis Delft
More information0.3.4 Burgers Equation and Nonlinear Wave
16 CONTENTS Solution to step (discontinuity) initial condition u(x, 0) = ul if X < 0 u r if X > 0, (80) u(x, t) = u L + (u L u R ) ( 1 1 π X 4νt e Y 2 dy ) (81) 0.3.4 Burgers Equation and Nonlinear Wave
More informationNUMERICAL ANALYSIS OF THE CAHN-HILLIARD EQUATION AND APPROXIMATION FOR THE HELE-SHAW PROBLEM, PART II: ERROR ANALYSIS AND CONVERGENCE OF THE INTERFACE
NUMERICAL ANALYSIS OF THE CAHN-HILLIARD EQUATION AND APPROXIMATION FOR THE HELE-SHAW PROBLEM, PART II: ERROR ANALYSIS AND CONVERGENCE OF THE INTERFACE XIAOBING FENG AND ANDREAS PROHL Abstract. In this
More informationHeat Transfer Equations The starting point is the conservation of mass, momentum and energy:
ICLASS 2012, 12 th Triennial International Conference on Liquid Atomization and Spray Systems, Heidelberg, Germany, September 2-6, 2012 On Computational Investigation of the Supercooled Stefan Problem
More informationFree boundary problems deal with solving
Free Boundary Problems in Science and Technology Avner Friedman Free boundary problems deal with solving partial differential equations (PDEs) in a domain, a part of whose boundary is unknown in advance;
More informationFDM for parabolic equations
FDM for parabolic equations Consider the heat equation where Well-posed problem Existence & Uniqueness Mass & Energy decreasing FDM for parabolic equations CNFD Crank-Nicolson + 2 nd order finite difference
More information