Fundamental Trends in Fluid-structure Interaction (Contemporary Challenges in Mathematical Fluid Dynamics and Its Applications)

CONTENTS......Page 8
PREFACE......Page 6
1. Introduction......Page 10
2. Notation......Page 16
3.1. Lagrangian framework......Page 17
3.2. Eulerian framework......Page 18
3.3. Arbitrary Lagrangian framework......Page 19
3.4. Conservation equations and boundary conditions......Page 21
4.1.1. Fluid model in Eulerian formulation......Page 23
4.1.2. Fluid model in ALE formulation......Page 24
4.2. Structure model......Page 25
4.2.2. Incompressible Neo–Hookean (INH) material......Page 28
4.2.3. Structure model in Eulerian framework......Page 29
5.1. The ALE formulation of the FSI problem......Page 32
5.2. The fully Eulerian formulation of the FSI problem......Page 34
5.2.1. The IP set approach......Page 35
5.2.2. Eulerian formulation of the FSI problem......Page 37
6.1. Mesh notation and finite element spaces......Page 40
6.2. Galerkin formulation......Page 43
6.3. Time discretization......Page 46
6.4. Solution of the algebraic systems......Page 47
6.5. Evaluation of directional derivatives......Page 48
7. Mesh Adaptation......Page 50
7.1. The DWR method......Page 51
7.2. Mesh adaptation algorithm......Page 56
7.3. A stationary special case......Page 57
7.4. Numerical integration along the interface......Page 60
8. Numerical Test 1: Elastic Flow Cavity......Page 62
8.1. Computations on globally refined meshes......Page 63
8.2. Computations on locally adapted meshes......Page 64
9. Numerical Test 2: FSI Benchmark FLUSTRUK-A......Page 65
9.2. CSM test......Page 69
9.3. FSI tests......Page 72
9.4. FSI test with large deformations......Page 75
10. Summary and Future Development......Page 78
References......Page 80
1. Introduction......Page 86
1.1. Applications......Page 88
1.2. Flight at low Reynolds numbers......Page 89
2. Large Time Asymptotics......Page 91
2.1. Introduction......Page 92
2.2. Other function spaces: a counter example......Page 96
2.3. Power counting, asymptotic expansions......Page 97
2.3.1. The case of compact support......Page 98
2.3.2. Weighted Lp spaces......Page 99
2.3.3. Power counting......Page 100
2.4. Function spaces......Page 101
2.5. The renormalization group......Page 102
2.6. Technical lemmas......Page 104
2.7. Nonlinear problems......Page 105
2.7.1. The case of irrelevant perturbations, I......Page 106
2.7.2. The case of irrelevant perturbations, II......Page 110
2.7.3. The case of marginal perturbations......Page 114
2.7.5. The case of irrelevant perturbations, III......Page 115
2.8. Appendix I: Construction of a counter example......Page 119
2.9.1. Proof of Lemma 10......Page 121
2.9.2. Proof of Lemma 11......Page 122
2.9.3. Proof of Proposition 12......Page 123
2.10. Bibliographic notes......Page 125
3. Down-stream Asymptotics of Stationary Navier–Stokes Flows......Page 127
3.1. Leading order term in two dimensions......Page 131
3.2. Leading order term in three dimensions......Page 134
3.3. Connection with existing results......Page 136
3.4. Higher order terms in two dimensions......Page 137
3.5.1. The time periodic case......Page 142
3.5.3. Motion in the presence of a nearby wall......Page 143
4.1. The mathematical problem......Page 144
4.2. Consequences of incompressibility......Page 146
4.3. Connection between global and downstream asymptotics......Page 147
4.4. Drag, lift, and torque......Page 149
5. Artificial Boundary Conditions......Page 151
5.1. Stationary flows in two dimensions......Page 153
5.2. Stationary flows in three dimensions......Page 154
5.3. Bibliographic notes......Page 155
6.1. Galerkin finite element discretization......Page 157
6.2. The solver......Page 159
6.3. Numerical results......Page 160
7.2. Boundary layer theory, wakes......Page 161
7.5. Numerical studies......Page 162
7.7. Other references on exterior flows......Page 163
References......Page 164
3. Numerical Simulation and Benchmarking of Fluid-Structure Interaction with Application to Hemodynamics M. Razzaq, S. Turek, J. Hron, J. F. Acker, F. Weichert, I. Q. Grunwald, C. Roth, M. Wagner and B. F. Romeike......Page 180
1. Introduction......Page 181
2. Fluid-Structure Interaction Problem Formulation......Page 182
2.1. Fluid mechanics......Page 183
2.2. Structural mechanics......Page 184
2.3. Interaction conditions......Page 186
3.1. The conforming Stokes element Q2P1......Page 187
3.2. Time discretization......Page 189
3.3. Solution algorithms......Page 190
4. FSI Benchmarking......Page 192
5. Applications to Hemodynamics......Page 195
5.1. Geometry of the problem......Page 198
5.2. Boundary and initial conditions......Page 199
5.3. Numerical results......Page 200
5.4. Steady configurations......Page 201
6. Summary and Future Developments......Page 203
References......Page 206
1.1. Introduction......Page 210
1.2. The case of an inviscid fluid......Page 214
2.1. Governing equations and main results......Page 219
2.2. A change of variables......Page 223
2.3. Estimates on the coefficients......Page 228
2.4. Analysis of the linearized problem......Page 234
2.5. Proof of the main result......Page 240
2.6. Remarks and bibliographical notes on Sec. 2......Page 242
3.1. Notation and preliminaries......Page 243
3.2. Weak formulation of the governing equations......Page 247
3.3. Main steps of the proof Theorem 3.2.3......Page 251
3.4.1. Some background on the transport equation......Page 253
3.4.2. Passage to the limit in the transport equations......Page 254
3.5. Some technical results......Page 258
3.6. Proof of Theorem 3.3.3......Page 261
3.7.1. Proof of Theorem 3.2.3.......Page 265
3.8. Remarks and bibliographical notes on Sec. 3......Page 266
References......Page 267
1. Introduction......Page 270
2. Kinematics and Dynamics of Continuous Media......Page 271
3. Lagrangian, Eulerian and ALE Formulations......Page 274
4. Mass and Momentum Conservation Principles......Page 279
5. Navier–Stokes Equations for Blood Flow in the ALE Frame......Page 283
6. Elastodynamics Equations for the VesselWall Deformation......Page 284
7. Reduced Structural Models......Page 289
8. The Coupled Fluid-Structure Problem......Page 290
9. Algorithms of FSI......Page 294
References......Page 297
Index......Page 300