These lecture notes are dedicated to the mathematical modelling, analysis and computation of interfaces and free boundary problems appearing in geometry and in various applications, ranging from crystal growth, tumour growth, biological membranes to porous media, two-phase flows, fluid-structure interactions, and shape optimization.
We first give an introduction to classical methods from differential geometry and systematically derive the governing equations from physical principles. Then we will analyse parametric approaches to interface evolution problems and derive numerical methods which will be thoroughly analysed. In addition, implicit descriptions of interfaces such as phase field and level set methods will be analysed. Finally, we will discuss numerical methods for complex interface evolutions and will focus on two phase flow problems as an important example of such evolutions.
Author(s): Eberhard Bänsch, Klaus Deckelnick, Harald Garcke, Paola Pozzi
Series: Oberwolfach Seminars 51
Edition: 1
Publisher: Birkhäuser
Year: 2023
Language: English
Pages: 178
Tags: Interfaces, Mean Curvature Flow, Elastic Flow, Free Boundary Problems, Phase Field Methods, Level Set Method, Differential Geometry, Finite Element Methods, Geometric Evolution Equations, Two Phase Flow
Preface
Contents
About the Authors
1 Introduction
1.1 Grain Boundary Motion
1.2 Melting and Solidification
1.3 Flow Problems with Interfaces
1.4 Curvature Energies and Biomembranes
2 Some Notions from Differential Geometry
2.1 What Is a Surface?
2.2 Integration and Differentiation on a Surface
2.3 Weingarten Map
2.4 Signed Distance Function and Canonical Extension
2.5 Integration by Parts on Manifolds
2.6 Evolving Surfaces
2.7 Normal Velocity and Normal Time Derivative
2.8 Velocity Fields and Material Time Derivatives Induced by the Motion of Material Points
2.9 Jacobi's Formula for the Derivative of the Determinant
2.10 A Transport Theorem
2.11 Reynolds Transport Theorem
3 Modeling
3.1 Gradient Flows
3.1.1 Gradient Flows in Rn
3.1.2 Minimizing Movements for Gradient Flows
3.2 First Variation of Area
3.3 Mean Curvature Flow as a Gradient Flow of the Area Functional
3.4 Anisotropic Energies and Their Gradient Flows
3.5 The Gradient Flow of the Willmore Functional
3.6 The Stefan Problem
3.6.1 Governing Equations in the Bulk
3.6.2 Another Transport Theorem
3.6.3 Governing Equations on the Interface
3.7 Mathematical Modeling of Two-Phase Flows
3.7.1 Conservation of Mass for Individual Species
3.7.2 Conservation of Momentum
3.7.3 Jump Condition at the Interface
3.7.4 Surface Tension
Principle of Virtual Work
3.7.5 Conditions on the Free Surface
3.7.6 The Overall Two-Phase Flow System
3.7.7 Formal Energy Estimate
3.7.8 Contact Angle
3.8 Phase Field Models
3.8.1 The Ginzburg–Landau Energy
3.8.2 Phase Field Models as Gradient Flows
The Allen–Cahn Equation
The Cahn–Hilliard Equation
The Phase Field System
4 Parametric Approaches for Geometric Evolution Equationsand Interfaces
4.1 Curve Shortening Flow
4.1.1 Local and Global Existence
4.1.2 Spatial Discretization and Error Analysis
4.1.3 Fully Discrete Scheme and Stability
4.2 Fully Discrete Anisotropic Curve Shortening Flow
4.3 Mean Curvature Flow
4.3.1 Some Properties of Solutions
4.3.2 Existence of Solutions in the Graph Case
4.3.3 Existence in the General Parametric Case
4.3.4 Discretization
4.4 Elastic Flow for Curves
4.4.1 Long Time Existence
4.4.2 Stability for the Semi-discrete Problem
4.5 A General Strategy to Solve Interface Problems Involving Bulk Quantities in a Parametric Setting
5 Implicit Approaches for Interfaces
5.1 A Way to Handle Topological Changes: The Level Set Method
5.2 Viscosity Solutions for Mean Curvature Flow
5.3 An Existence Theorem for Viscosity Solutions of MeanCurvature Flow
5.4 A Level Set Approach for Numerically Solving Mean Curvature Flow
5.5 Relating Phase Field and Sharp Interface Energies
5.6 Solving Interface Evolution Problems in a BV-Setting
5.7 Phase Field Models for Two-Phase Flow: The Cahn–Hilliard–Navier–Stokes Model
5.8 Existence Theory for the Cahn–Hilliard Equation
5.9 The Mullins–Sekerka Problem as the Sharp Interface Limit of the Cahn–Hilliard Equation
5.9.1 The Governing Equations
5.9.2 Outer Expansions
5.9.3 Inner Expansions
New Coordinates in the Inner Region
Matching Conditions
The Equations to Leading Order
The Equation for the Chemical Potential at the Interface
Interfacial Flux Balance in the Sharp Interface Limit
5.10 How to Discretize the Cahn–Hilliard Equation?
5.10.1 The Time Discrete Setting
5.10.2 The Fully Discrete Setting
5.10.3 Existence of Solutions to the Fully Discrete System
5.10.4 An Energy Inequality in the Fully Discrete Setting
6 Numerical Methods for Complex Interface Evolutions
6.1 Introduction and General Remarks About the Different Methods
6.2 Interface Capturing
6.2.1 Level Set Methods
6.2.2 Phase Field Methods
6.3 Interface Tracking
6.3.1 Mesh Moving Approaches (Fitted Approaches)
6.3.2 Front Tracking Approaches (Unfitted Approaches)
6.4 Two Phase Flow
6.4.1 Mesh Moving
Arbitrary Lagrangian–Eulerian Coordinates (ALE)
Eulerian Coordinates
Lagrangian Coordinates
ALE Coordinates
Moving Mesh Method for the Two Phase Problem
6.4.2 Level Set Method for Two Phase Flow
Discontinuous Pressures
7 Exercises
References
