A partitioned solution approach for fluid-structure interaction problems in the arterial system

Produktbeschreibung

The present work is concerned with the partitioned solution of the multifeld problem arising from a hierarchical modeling approach to cardiovascular fluid-structure interaction. Different strategies to couple the participating feld solvers are investigated in detail. This includes staggered and parallel coupling algorithms as well as different methods for convergence acceleration, spatial interpolation and temporal extrapolation of coupling quantities. In the developed modeling and simulation approach, a fully resolved model of a segment of the arterial network is coupled to reduced order models in order to account for the influence of the surrounding.

There is experimental evidence that hemodynamic quantities such as the wall shear stress promote the progression cardiovascular disease. Cardiovascular FSI simulations, that can predict these quantities, are therefore of great interest and can aid in surgical planning and optimization of anastomoses shapes and graft materials.

Contents
List of medical terms IX
Abstract X
1 Introduction 1
2 Fluid-structure interaction in the arterial system 6
2.1 The cardiovascular system . . . . . . . . . . . . . . . . . . . 7
2.1.1 Anatomy of the larger arteries . . . . . . . . . . . . . 8
2.1.2 Physical characteristics of arterial blood flow . . . . . 10
2.1.3 Cardiovascular diseases . . . . . . . . . . . . . . . . . 12
2.1.4 Vascular bypass grafts . . . . . . . . . . . . . . . . . 14
2.2 Computational modeling . . . . . . . . . . . . . . . . . . . . 16
2.2.1 Fluid-structure interaction . . . . . . . . . . . . . . . 17
2.2.2 Arterial hemodynamics . . . . . . . . . . . . . . . . . 18
3 Mechanical modeling of the arterial system 23
3.1 Coupled problems . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1.1 Solution approaches . . . . . . . . . . . . . . . . . . . 25
3.2 Continuum mechanics . . . . . . . . . . . . . . . . . . . . . 28
3.2.1 Conservation laws on moving domains . . . . . . . . 29
3.2.2 Structural mechanics . . . . . . . . . . . . . . . . . . 35
3.2.3 Fluid mechanics . . . . . . . . . . . . . . . . . . . . . 45
3.2.4 Interface constraints and domain motion . . . . . . . 47
3.3 Mechanical models for the cardiovascular system . . . . . . . 50
3.3.1 Constitutive equations for soft tissue . . . . . . . . . 50
3.3.2 Constitutive equations for blood . . . . . . . . . . . . 56
3.3.3 One-dimensional models . . . . . . . . . . . . . . . . 58
3.3.4 Windkessel models . . . . . . . . . . . . . . . . . . . 61
3.3.5 Models for the surrounding tissue . . . . . . . . . . . 62
3.3.6 Velocity profiles . . . . . . . . . . . . . . . . . . . . . 64
3.3.7 Hemodynamic quantities . . . . . . . . . . . . . . . . 66
4 Numerical methods 69
4.1 Space and time discretization . . . . . . . . . . . . . . . . . 70
4.1.1 High-order finite elements for structural mechanics . 70
4.1.2 Finite volumes for fluid mechanics in moving domains 83
4.1.3 Taylor-Galerkin method for one-dimensional blood flow 88
4.1.4 Solvers for ordinary differential equations . . . . . . . 90
4.2 Geometry and mesh generation . . . . . . . . . . . . . . . . 93
4.2.1 G1-continuous surface construction . . . . . . . . . . 95
4.2.2 Polynomial G1 PN quads . . . . . . . . . . . . . . . . 99
4.2.3 General polynomial G1 quads . . . . . . . . . . . . . 104
4.3 Partitioned solution approach . . . . . . . . . . . . . . . . . 112
4.3.1 Coupling algorithms . . . . . . . . . . . . . . . . . . 113
4.3.2 Convergence acceleration . . . . . . . . . . . . . . . . 117
4.3.3 Predictors . . . . . . . . . . . . . . . . . . . . . . . . 124
4.3.4 Convergence criteria . . . . . . . . . . . . . . . . . . 127
4.3.5 Interpolation . . . . . . . . . . . . . . . . . . . . . . 128
4.4 Coupling software . . . . . . . . . . . . . . . . . . . . . . . . 141
4.4.1 Software design . . . . . . . . . . . . . . . . . . . . . 143
4.4.2 Inter process communication . . . . . . . . . . . . . . 145
4.4.3 Implementation of coupling algorithms . . . . . . . . 147
4.4.4 Field solver manipulation . . . . . . . . . . . . . . . 147
5 Numerical investigations 153
5.1 Preliminary analyses . . . . . . . . . . . . . . . . . . . . . . 153
5.1.1 Structural mechanics . . . . . . . . . . . . . . . . . . 153
5.1.2 Fluid dynamics . . . . . . . . . . . . . . . . . . . . . 168
5.1.3 Reduced models . . . . . . . . . . . . . . . . . . . . . 171
5.1.4 Interpolation . . . . . . . . . . . . . . . . . . . . . . 174
5.1.5 Load integration . . . . . . . . . . . . . . . . . . . . 179
5.2 Coupled benchmark problems . . . . . . . . . . . . . . . . . 183
5.2.1 Multi-body system . . . . . . . . . . . . . . . . . . . 184
5.2.2 Lid-driven cavity flow . . . . . . . . . . . . . . . . . . 199
5.2.3 Two-dimensional flag in channel flow . . . . . . . . . 204
5.2.4 Pulse wave in an elastic tube . . . . . . . . . . . . . 208
5.3 Arterial fluid-structure interaction . . . . . . . . . . . . . . . 212
5.3.1 Initial boundary value problem . . . . . . . . . . . . 213
5.3.2 Coupling algorithm . . . . . . . . . . . . . . . . . . . 214
5.3.3 Test case . . . . . . . . . . . . . . . . . . . . . . . . . 215
5.3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 217
6 Applications 220
6.1 Hemodynamics in the scope of vessel geometry and material 220
6.1.1 Decoupled simulations . . . . . . . . . . . . . . . . . 222
6.1.2 Coupled simulations . . . . . . . . . . . . . . . . . . 225
6.2 Hemodynamics in idealized end-to-side anastomoses . . . . . 227
6.2.1 Simulation setup . . . . . . . . . . . . . . . . . . . . 228
6.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 230
6.3 Hemodynamics in a patient specific anastomosis . . . . . . . 235
6.3.1 Study case . . . . . . . . . . . . . . . . . . . . . . . . 235
6.3.2 Modeling and simulation approach . . . . . . . . . . 237
6.3.3 Results { one-dimensional analysis . . . . . . . . . . 242
6.3.4 Results { three-dimensional analysis . . . . . . . . . . 247
6.3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . 251
7 Summary and Outlook 253
Appendix 257
A.1 Tensor algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 257
A.1.1 Contractions and scalar products . . . . . . . . . . . 257
A.1.2 Dyadic products of second order tensors . . . . . . . 257
A.1.3 Special fourth order tensors . . . . . . . . . . . . . . 258
A.2 Continuum mechanics . . . . . . . . . . . . . . . . . . . . . 258
A.2.1 Neo-Hookean elasticity tensor . . . . . . . . . . . . . 258
A.3 Finite elements . . . . . . . . . . . . . . . . . . . . . . . . . 259
A.3.1 Weak form . . . . . . . . . . . . . . . . . . . . . . . . 259
A.3.2 Special matrices . . . . . . . . . . . . . . . . . . . . . 259
A.3.3 Assembly . . . . . . . . . . . . . . . . . . . . . . . . 260
A.3.4 Voigt notation . . . . . . . . . . . . . . . . . . . . . . 261
A.3.5 Nodal shape function indices . . . . . . . . . . . . . . 261
A.3.6 Face and edge coordinates . . . . . . . . . . . . . . . 261
A.4 Taylor-Galerkin method . . . . . . . . . . . . . . . . . . . . 262
A.4.1 Left-hand side . . . . . . . . . . . . . . . . . . . . . . 262
A.4.2 Right-hand side . . . . . . . . . . . . . . . . . . . . . 263
A.5 Radial basis functions . . . . . . . . . . . . . . . . . . . . . 264

A.6 Multi body system . . . . . . . . . . . . . . . . . . . . . . . 264
A.7 Coupling software . . . . . . . . . . . . . . . . . . . . . . . . 266
A.8 Preliminary investigations . . . . . . . . . . . . . . . . . . . 266
A.9 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

Bibliography 269

Keywords: fluid-structure interaction, blood flow, partitioned coupling, high-order fnite elements,