A Robin based semi-implicit coupling in uid-structure interaction 

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A semi-implicit projection coupling for FSI problems: anal- ysis and numerics (Part I)

Among the dierent partitioned schemes, the semi-implicit coupling schemes of- fer an excellent compromise between stability and eciency. As a matter of fact these are faster than the common implicit procedures, but more stable than stan- dard1 explicit coupling schemes, which are unstable when the added-mass eect becomes important [CGN05]. Examples of semi-implicit coupling schemes are given in the following works [FGG06, FGG07, QQ07, BQQ08, SM08]. Consid- ering the rst semi-implicit scheme [FGG06, FGG07], its eciency relies upon a convenient implicit-explicit splitting performed with the Chorin-Temam projection scheme [Cho68, Tem68, Cho69] in the uid: at each time step the projection sub- step (carried out in a known uid domain) is implicitly coupled with the structure, so accounting for the added-mass eect in an implicit way, while the expensive ALE-advection-viscous sub-step is explicit.
For a linearized version of problem (1.1)-(1.3), the authors proved that the scheme is stable under the condition (see [FGG07, Theorem 1]).

Numerical simulation of FSI problems with cardiac valves (Part II)

In Part II of this thesis we consider the numerical simulation of the interaction between blood and cardiac valves. This problem oers extraordinary challenges from the modeling, the mathematical and the numerical viewpoints. Examples are the highly non-linear constitutive laws, the intense unsteadiness and strong pressure gradients in the blood ow, but also the contact among the leaets and the modeling of the chordae tendineae (for the mitral valves).
In this eld, many works consider two-dimensional uid-structure problems, only a few three-dimensional problems. The ones considering three dimensional uid- structure interaction problems frequently assume simplications in the model in order to reduce its complexity. A typical simplication is the use of planes of sym- metry for the leaets. As a result, only one leaet is eectively simulated, the behaviors of the others, as well as the ow distribution, are retrieved by symmetry.
This approach could be used for example in the simulation prosthetic valves, the leaets being symmetric. Nonetheless, native valves are naturally non-symmetric and the ow behavior is inherently three dimensional. In addition, in view of the simulation of particular diseases, such as stenosis for example, the whole three- dimensional complexity of the problem has to be taken into account. In this work we consider fully 3D valves simulations, focusing in particular on the aortic valve. The other valves can be clearly handled within the same general approach.

Towards the uid-structure interaction in the heart (Part III)

The uid-structure interaction in the heart is a very fascinating problem which contains itself all the diculties related to the interaction of the blood with the wall and with the cardiac valves. From the modeling point of view, an accurate  description of the heart and valves mechanics is required. Advances in this direction are for example given in [HPS03, CFG+09] for the heart and in [WKM05, PSH07] for valves. From the numerical point of view, techniques such as the ones introduced in Chapters 4 and 5 have to be included in a single framework in order to consider all the possible interactions: blood – heart wall and blood – heart valves. This is feasible (see for example [dS07, Chapter 6] for preliminary results in two-dimensions) but in three-dimensions it could become so computationally intensive that it may not be the best option to address some clinical problems for which a precise mechanical description of all the elements is not required. If the mechanics of the heart itself is the principal point of interest, one could be motivated to replace the complex three-dimensional simulations of the valves with reduced valve models, which take into account the opening and closing behavior of the heart valves. Nonetheless the use of standard lumped parameter models has inherent limitations due to the introduction of articial boundaries in regions where high variability in the uid dynamics quantities is experienced. In the last part of the thesis, we propose a new reduced model for cardiac valves, which improves the accuracy of standard lumped models and the robustness and eciency of 3D FSI models.

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Variational formulations of the coupled FSI problem

We give here some elements of functional analysis necessary to introduce the vari- ational setting for the coupled problem (2.33)-(2.35). We refer to any standard functional analysis text (e.g. [Bre83]) for a more comprehensive presentation.
Let ⊂ Rd, d = 2, 3, be a bounded domain, Lipschitz at least. We denote with C0( ) the space of functions that are continuous in and with Ck( ) the space of functions that are k-times continuously Fréchet-dierentiable on . The space D( ) represents the set of C∞ functions whose support is compact in . D( ) is dense in Lp( ), 1 ≤ p < +∞, which denotes the space of functions whose p-th power is absolutely integrable with respect to the Lebesgue measure on . Let M( ) be the space of scalar-valued functions on that are Lebesgue-measurable, the spaces Lp( ), 1 ≤ p ≤ ∞, are dened as Lp( ) def = {f ∈ M( ) : kfkLp( ) < +∞}.

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 

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