Level Set Methods for Fluid-Structure Interaction

This monograph is devoted to the study of Eulerian models for fluid-structure interaction from the original point of view of level set methods. In the last 15 years, Eulerian models have become popular tools for studying fluid-structure interaction problems. One major advantage compared to more conventional methods such as ALE methods is that they allow the use of a single grid and a single discretization method for the different media. Level set methods in addition provide a general framework to follow the fluid-solid interfaces, to represent the elastic stresses of solids, and to model the contact forces between solids. This book offers a combination of mathematical modeling, aspects of numerical analysis, elementary codes and numerical illustrations, providing the reader with insights into the applications and performance of these models. Assuming background at the level of a Master's degree, Level Set Methods for Fluid-Structure Interaction provides researchers in the fields of numerical analysis of PDEs, theoretical and computational mechanics with a basic reference on the topic. Its pedagogical style and organization make it particularly suitable for graduate students and young researchers--back cover Foreword 6 Contents 8 1 Level Set Methods and Lagrangian Interfaces 11 1.1 Interface Tracking or Interface Capturing 11 1.2 Level Set Methods and Geometry of Surfaces 14 1.3 Level Set Methods and Geometry of Curves in R3 18 1.4 Expression of Surface Forces Using the Level Set Function 20 1.4.1 Example 1: Image Processing 25 1.4.2 Exemple 2: Surface Tension 27 1.5 Numerical Aspects I: Consistency and Accuracy 29 1.5.1 Redistancing of φ 31 1.5.2 Renormalization of φ 33 1.5.3 Comparison of the Two Approaches 34 1.5.4 Towers Method to Approximate Surface Integrals 37 1.6 Numerical Aspects II: Stability 38 1.6.1 Explicit Scheme 41 1.6.2 Implicit Scheme 44 1.6.3 Semi-Implicit Scheme 45 2 Mathematical Tools for Continuum Mechanics 48 2.1 Characteristics and Flows Associated with a Velocity Field 48 2.2 Change of Variables 52 2.3 Reynolds Formulas 55 2.4 Conservation of Mass 59 2.4.1 Mass Conservation in Eulerian Formulation 59 2.4.2 Mass Conservation in Lagrangian Formulation 59 2.5 Conservation of Momentum 60 2.5.1 Momentum Conservation in Eulerian Formulation 60 2.5.2 Momentum Conservation in Lagrangian Formulation 60 3 Interaction of an Incompressible Fluid with an Elastic Membrane 62 3.1 From the Immersed Boundary Method to Level Set Methods 63 3.2 Immersed Membrane: Case Without Shear 66 3.2.1 Level Set Formulation of the Elastic Deformation of a Hypersurface Immersed in a Incompressible Fluid 66 3.2.2 Level Set Formulation of Elastic Energy and Fluid-Structure Coupling in the Incompressible Case 70 3.2.3 Generalization to Compressible Flows 73 3.2.4 Taking into Account Curvature Forces 75 3.2.5 Korteweg Models and Existence of Solutions 77 3.3 Immersed Membrane: The Case with Surface Shear 79 3.3.1 Level Set Approach for Surfaces 79 3.3.2 An Eulerian Tensor to Measure Surface Deformation 81 3.3.3 Invariants and Associated Elastic Force 82 3.3.4 Energy and Coupling Model 86 3.4 Curves Immersed in R3 88 3.4.1 An Eulerian Tensor to Measure Strains Along Curves 88 3.4.2 Invariants and Associated Elastic Force 89 3.5 Explicit and Semi-implicit Time Discretizations 91 3.5.1 Explicit Schemes 91 3.5.2 Semi-implicit Scheme 92 3.5.3 Numerical Validation 93 3.6 Numerical Illustrations and Sample Code 96 3.6.1 Shear-Free Membrane 96 3.6.1.1 2D Oscillating Elastic Membrane: FreeFEM++ and Matlab Codes 96 3.6.1.2 Membrane with Bending Energy 100 3.6.2 Membrane with Shear 101 4 Immersed Bodies in a Fluid: The Case of Elastic Bodies 107 4.1 Hyperelastic Materials in Lagrangian Formulation 108 4.1.1 Principle of Material Indifference 109 4.1.2 Isotropic Materials 110 4.1.3 Computation of the Stress Tensor in a Lagrangian Framework 111 4.2 Hyperelastic Materials in Eulerian Formulation 112 4.2.1 Computation of the Stress Tensor in an Eulerian Framework 112 4.2.2 Elastic Constitutive Laws for Elastic Media 114 4.2.3 Eulerian Elasticity in the Incompressible Case 114 4.2.4 Eulerian Elasticity in the Compressible Case 115 4.3 Fluid-Structure Coupling Model in the Incompressible Case 116 4.3.1 Model and Constitutive Law in the Incompressible Case 117 4.3.2 Numerical Illustrations 118 4.3.2.1 Elastic Ball in a Driven Cavity 118 4.3.2.2 Flapping of an Elastic Rod 119 4.3.2.3 Wave Damping by Elastic Structures 121 4.3.2.4 Fluid-Structure Interaction in the Contraction of a Cardiac Muscle Cell 122 4.4 Fluid-Structure Coupling in the Compressible Case 124 4.4.1 Model and Constitutive Law in the Compressible Case 127 4.4.2 Numerical Scheme 128 4.4.3 Numerical Illustration 130 5 Immersed Bodies in Incompressible Fluids: The Case of Rigid Bodies 133 5.1 The Penalization Method for Flow Around Bodies with Given Velocity 134 5.2 The Case of the Two-Ways Fluid-Solid Interaction 135 5.3 Remarks on the Numerical Implementation 138 5.4 Extensions of the Penalization Method 140 5.5 Numerical Illustrations 141 5.5.1 Kissing and Tumbling of Two Spheres 142 5.5.2 Flows Around Oscillating Obstacles 143 5.5.3 Anguilliform Swimmers 146 6 Computing Interactions Between Solids by Level Set Methods 150 6.1 Level Set Method to Model Interaction Forces 151 6.1.1 Point Repulsion Model 151 6.1.2 Surface Repulsion Model by Level Set Method 152 6.1.3 Taking into Account Cohesion and Damping Forces 153 6.1.4 Numerical Illustrations 154 6.2 An Efficient Method for Dealing with Contacts Between Multiple Objects 156 6.2.1 Motivation 156 6.2.2 The Algorithm 157 6.2.2.1 Label Functions 158 6.2.2.2 Distance Functions 159 6.2.2.3 Dealing with Contact Forces 159 6.2.2.4 Penalization and Complete Model 161 6.2.3 Computational Efficiency of the Method 162 6.2.4 Numerical Illustrations 163 7 Annex 168 7.1 Examples of Curvature Calculations Using a Level Set Function 168 7.1.1 The Case of the Ellipsoid 168 7.1.2 The Case of the Torus 169 7.2 Justification of the Results Used for Membranes with Shear 170 7.2.1 Proof of the Results Concerning the Z1 Invariant 170 7.2.2 Analytical Illustrations for Z2 175 7.3 Justification of the Results Used for the Curves Parameterized in R3 180 7.3.1 Proof of the Results Concerning the Invariant Z3 181 7.3.2 Area and Co-area Formulas 184 7.3.3 Volume Approximation of Line Integrals and Calculation of the Elastic Force 185 7.4 WENO Schemes for the Transport Equation 189 7.5 Some Ideas to Go Further 193 Credits of Figures Reproduced with Permission 195 References 197 This monograph is devoted to Eulerian models for fluid-structure interaction by applying the original point of view of level set methods. In the last 15 years, Eulerian models have become popular tools for studying fluid-structure interaction problems. One major advantage compared to more conventional methods such as ALE methods is that they allow the use of a single grid and a single discretization method for the different media. Level set methods in addition provide a general framework to follow the fluid-solid interfaces, to represent the elastic stresses of solids, and to model the contact forces between solids. This book offers a combination of mathematical modeling, aspects of numerical analysis, elementary codes and numerical illustrations, providing the reader with insights into ​​the applications and performance of these models. Assuming background at the level of a Master's degree, Level Set Methods for Fluid-Structure Interaction provides researchers in the fields of numerical analysis of PDEs, theoretical and computational mechanics with a basic reference on the topic. Its pedagogical style and organization make it particularly suitable for graduate students and young researchers.