Level Set Methods for Fluid-Structure Interaction

Foreword
Contents
1 Level Set Methods and Lagrangian Interfaces
1.1 Interface Tracking or Interface Capturing
1.2 Level Set Methods and Geometry of Surfaces
1.3 Level Set Methods and Geometry of Curves in R3
1.4 Expression of Surface Forces Using the Level Set Function
1.4.1 Example 1: Image Processing
1.4.2 Exemple 2: Surface Tension
1.5 Numerical Aspects I: Consistency and Accuracy
1.5.1 Redistancing of φ
1.5.2 Renormalization of φ
1.5.3 Comparison of the Two Approaches
1.5.4 Towers Method to Approximate Surface Integrals
1.6 Numerical Aspects II: Stability
1.6.1 Explicit Scheme
1.6.2 Implicit Scheme
1.6.3 Semi-Implicit Scheme
2 Mathematical Tools for Continuum Mechanics
2.1 Characteristics and Flows Associated with a Velocity Field
2.2 Change of Variables
2.3 Reynolds Formulas
2.4 Conservation of Mass
2.4.1 Mass Conservation in Eulerian Formulation
2.4.2 Mass Conservation in Lagrangian Formulation
2.5 Conservation of Momentum
2.5.1 Momentum Conservation in Eulerian Formulation
2.5.2 Momentum Conservation in Lagrangian Formulation
3 Interaction of an Incompressible Fluid with an Elastic Membrane
3.1 From the Immersed Boundary Method to Level Set Methods
3.2 Immersed Membrane: Case Without Shear
3.2.1 Level Set Formulation of the Elastic Deformation of a Hypersurface Immersed in a Incompressible Fluid
3.2.2 Level Set Formulation of Elastic Energy and Fluid-Structure Coupling in the Incompressible Case
3.2.3 Generalization to Compressible Flows
3.2.4 Taking into Account Curvature Forces
3.2.5 Korteweg Models and Existence of Solutions
3.3 Immersed Membrane: The Case with Surface Shear
3.3.1 Level Set Approach for Surfaces
3.3.2 An Eulerian Tensor to Measure Surface Deformation
3.3.3 Invariants and Associated Elastic Force
3.3.4 Energy and Coupling Model
3.4 Curves Immersed in R3
3.4.1 An Eulerian Tensor to Measure Strains Along Curves
3.4.2 Invariants and Associated Elastic Force
3.5 Explicit and Semi-implicit Time Discretizations
3.5.1 Explicit Schemes
3.5.2 Semi-implicit Scheme
3.5.3 Numerical Validation
3.6 Numerical Illustrations and Sample Code
3.6.1 Shear-Free Membrane
3.6.1.1 2D Oscillating Elastic Membrane: FreeFEM++ and Matlab Codes
3.6.1.2 Membrane with Bending Energy
3.6.2 Membrane with Shear
4 Immersed Bodies in a Fluid: The Case of Elastic Bodies
4.1 Hyperelastic Materials in Lagrangian Formulation
4.1.1 Principle of Material Indifference
4.1.2 Isotropic Materials
4.1.3 Computation of the Stress Tensor in a Lagrangian Framework
4.2 Hyperelastic Materials in Eulerian Formulation
4.2.1 Computation of the Stress Tensor in an Eulerian Framework
4.2.2 Elastic Constitutive Laws for Elastic Media
4.2.3 Eulerian Elasticity in the Incompressible Case
4.2.4 Eulerian Elasticity in the Compressible Case
4.3 Fluid-Structure Coupling Model in the Incompressible Case
4.3.1 Model and Constitutive Law in the Incompressible Case
4.3.2 Numerical Illustrations
4.3.2.1 Elastic Ball in a Driven Cavity
4.3.2.2 Flapping of an Elastic Rod
4.3.2.3 Wave Damping by Elastic Structures
4.3.2.4 Fluid-Structure Interaction in the Contraction of a Cardiac Muscle Cell
4.4 Fluid-Structure Coupling in the Compressible Case
4.4.1 Model and Constitutive Law in the Compressible Case
4.4.2 Numerical Scheme
4.4.3 Numerical Illustration
5 Immersed Bodies in Incompressible Fluids: The Case of Rigid Bodies
5.1 The Penalization Method for Flow Around Bodies with Given Velocity
5.2 The Case of the Two-Ways Fluid-Solid Interaction
5.3 Remarks on the Numerical Implementation
5.4 Extensions of the Penalization Method
5.5 Numerical Illustrations
5.5.1 Kissing and Tumbling of Two Spheres
5.5.2 Flows Around Oscillating Obstacles
5.5.3 Anguilliform Swimmers
6 Computing Interactions Between Solids by Level Set Methods
6.1 Level Set Method to Model Interaction Forces
6.1.1 Point Repulsion Model
6.1.2 Surface Repulsion Model by Level Set Method
6.1.3 Taking into Account Cohesion and Damping Forces
6.1.4 Numerical Illustrations
6.2 An Efficient Method for Dealing with Contacts Between Multiple Objects
6.2.1 Motivation
6.2.2 The Algorithm
6.2.2.1 Label Functions
6.2.2.2 Distance Functions
6.2.2.3 Dealing with Contact Forces
6.2.2.4 Penalization and Complete Model
6.2.3 Computational Efficiency of the Method
6.2.4 Numerical Illustrations
7 Annex
7.1 Examples of Curvature Calculations Using a Level Set Function
7.1.1 The Case of the Ellipsoid
7.1.2 The Case of the Torus
7.2 Justification of the Results Used for Membranes with Shear
7.2.1 Proof of the Results Concerning the Z1 Invariant
7.2.2 Analytical Illustrations for Z2
7.3 Justification of the Results Used for the Curves Parameterized in R3
7.3.1 Proof of the Results Concerning the Invariant Z3
7.3.2 Area and Co-area Formulas
7.3.3 Volume Approximation of Line Integrals and Calculation of the Elastic Force
7.4 WENO Schemes for the Transport Equation
7.5 Some Ideas to Go Further
Credits of Figures Reproduced with Permission
References