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Table of contents
Fluid-structure interaction in the cardiovascular system
1 Introduction
1.1 The cardiovascular system
1.2 Numerical simulation of uid-structure interaction problems arising in hemodynamics: state-of-the-art
1.3 Thesis outline and main contributions
1.3.1 A semi-implicit projection coupling for FSI problems: analysis and numerics (Part I)
1.3.2 Numerical simulation of FSI problems with cardiac valves (Part II)
1.3.3 Towards the uid-structure interaction in the heart (Part III) 15
2 Mathematical modeling and numerical discretization of the coupled uid-structure interaction problem
2.1 Introduction
2.2 Fluid and solid modeling
2.2.1 Fluid model
2.2.2 Solid models
2.3 The coupled FSI problem
2.4 Variational formulations of the coupled FSI problem
2.4.1 Elements of functional analysis
2.4.2 A rst variational formulation
2.4.3 A second formulation based on Lagrange multipliers
2.5 Numerical discretization of the coupled FSI problem
2.5.1 Semi-discretization in space
2.5.2 Semi-discretization in time
2.5.3 Partitioned schemes in FSI
2.6 Conclusion
I A projection semi-implicit coupling for uid-structure inter- action problems: analysis and numerics
3 Convergence analysis of a semi-implicit coupling scheme for uid- structure interaction problems
3.1 Introduction
3.2 Problem setting
3.2.1 Hypotheses and notations
3.2.2 Variational formulation
3.3 Semi-implicit projection scheme
3.3.1 Time semi-discrete scheme
3.3.2 Fully discrete scheme
3.3.3 Interface matching operators
3.4 Construction of the nite element approximations
3.5 Main result and error analysis
3.6 Numerical experiments
3.7 Conclusion
4 A Robin based semi-implicit coupling in uid-structure interaction
4.1 Introduction
4.2 Preliminaries
4.3 Robin based semi-implicit coupling
4.3.1 The coupling scheme
4.3.2 Pressure load computation
4.3.3 Variants
4.4 Stability analysis
4.4.1 A simplied model problem
4.4.2 Semi-implicit coupling with pressure-Darcy formulation .
4.4.3 Semi-implicit coupling with pressure-Poisson formulation .
4.5 Numerical experiments
4.5.1 Two-dimensional test cases
4.5.2 Three-dimensional test cases
4.6 Conclusion
II Numerical simulation of uid-structure interaction problems with cardiac valves
5 A partitioned scheme for FSI and multi-body contact
5.1 Introduction
5.2 Modeling and discretization
5.2.1 Fluid and solid models
5.2.2 Contact model
5.3 General algorithm
5.3.1 Fluid-structure interaction (loop 1)
5.3.2 Denition of a convex neighborhoods (loop 2)
5.3.3 Minimization with convex constraints (loop 3)
5.3.4 Remarks on implementation
5.4 Numerical experiments
5.5 Conclusion
6 Computational analysis of an aortic valve jet with Lagrangian co- herent structures
6.1 Introduction
6.2 Challenges in FSI simulations with cardiac valve
Contents ix
6.3 Computation of LCS
6.4 Numerical experiments
6.4.1 Two-dimensional simulation
6.4.2 Three-dimensional simulation
6.5 Discussion
III Towards the uid-structure interaction in the heart
7 Resistive immersed surfaces for heart valves modeling
7.1 Introduction
7.2 Lumped parameter models for heart valves: an overview
7.3 Resistive immersed surface model for heart valves
7.4 Numerical experiments
7.4.1 Pressure jump test.
7.4.2 Normal and stenotic congurations of the aortic valves.
7.4.3 Left ventricle with imposed analytical displacements.
7.4.4 Left ventricle with imposed realistic displacements.
7.5 Discussion
Conclusions and perspectives
A External tissue support for FSI simulations
Bibliography
