The study of myocardial motion is essential for understanding the normal heart function and developing new treatments for cardiovascular diseases. The goals of my PhD research is to develop new methods for cardiac deformation recovery from 3D magnetic resonance (MR) images. The main contribution of my work is that the proposed methods are guaranteed to generate exactly or nearly incompressible deformations. This is a desirable property since the myocardium has been shown to be close to incompressible. From the recovered deformation, one can directly compute a number of clinically useful parameters, including strains. The first method for 3D deformation recovery of the left ventricular wall (LV) from anatomical cine MRI is based on a deformable model that is incompressible. This method is not suitable for the deformation recovery of the biventricular wall. The second method is based on a 3D deformable model that is nearly incompressible. The model uses a matrix-valued radial basis function to represent divergence free displacement fields, which is a first order approximation of incompressibility. This representation allows for deformation modeling of arbitrary topologies with a relatively small number of parameters, which is suitable for representing the motion of the multi-chamber structure of the heart. The two methods have similar performance. A method to obtain a smooth and accurate surface of the myocardium wall is needed to illustrate the cardiac deformation recovery. I present a novel method for the generation of endocardial and epicardial surface meshes. The same algorithm is independently used to generate the surface meshes of the epicardium and endocardium of the four cardiac chambers. It provides smooth meshes despite the strong voxel anisotropy, which is not the case for the marching cubes algorithm. Phase velocity MRI is an acquisition technique that contains more information about the myocardial motion than cine MRI. I present a method to interpolate the velocity vector field in a phase velocity MRI sequence. The method uses an interpolation model that provides a continuous divergence free velocity vector field, which means that the corresponding deformation is incompressible.

L’étude des résonances géantes monopolaires isoscalaires (ISGMR) et des résonances géantes quadrupolaires isoscalaires (ISGQR) dans les noyaux stables, a permis d’obtenir ces dernières décennies des informations fondamentales sur la structure et la matière nucléaire. En particulier, le centroïde de la ISGMR peut être relié au module d’incompressibilité de la matière nucléaire infinie. Des données dans les noyaux exotiques nous aideraient à contraindre ce module d’incompressibilité. Dans les noyaux instables, une seule mesure a, à l’heure actuelle, été réalisée (56Ni). Afin d’étudier l’évolution de la ISGMR et de la ISGQR le long d’une chaîne isotopique, des mesures dans un noyau exotique riche en neutrons sont donc nécessaires.L’expérience étudiée dans cette thèse a été réalisée au Grand Accélérateur National d’Ions Lourds (GANIL) à Caen en septembre 2010. Un faisceau de 68Ni à 50 AMeV et d’une intensité de 10^4 pps a été produit et purifié sur la ligne LISE. Les réactions de diffusion inélastique de particules alpha (alpha, alpha’) et de deutons (d,d’) sur 68Ni en cinématique inverse ont été étudiées avec la cible active Maya. Il s’agit de la première mesure de la ISGMR et de la ISGQR dans un noyau instable riche en neutrons.Pour chaque expérience, le spectre en énergie d’excitation a été reconstruit et les distributions angulaires étudiées par deux méthodes indépendantes. Les sections efficaces expérimentales ont été comparées à celles issues de calculs DWBA utilisant des densités de transition RPA. L’analyse en (alpha, alpha’) a permis l’observation d’une ISGMR fragmentée avec un épaulement à 21.1+/-0.6 MeV, d’une ISGQR concentrée à 16.9+/-0.8 MeV qui épuise 61+/-17% de la règle de somme pondérée en énergie (EWSR). De plus, un mode « soft GMR », prédit mais jamais observé, a été identifié à 13.4+/-0.5 MeV. Tous ces résultats sont confirmés par l’analyse en (d,d’), à l’exception de l’observation de la ISGQR pour laquelle les conditions de fonctionnement n’étaient pas favorables. / The study of the Isoscalar Giant Monopole Resonance (ISGMR) and the Isoscalar Giant Quadrupole Resonance (ISGQR) in stable nuclei provided relevant information on both nuclear matter and nuclear structure in past decades. For instance the centroid of the ISGMR can be linked to the incompressibility modulus of the infinite nuclear matter. Values for exotic nuclei would help in constraining it. In unstable nuclei, only one measurement has been performed so far (56Ni) and in order to study the evolution of the ISGMR and the ISGQR along an isotopic chain, measurements in neutron-rich Ni are called for.To reach this goal, a dedicated experiment was performed at GANIL in Septembre 2010. A 68Ni beam at 50 AMeV and with an intensity of 10^4 pps has been produced on LISE beamline. The inelastic scattering of alpha (alpha, alpha’) and deuteron particles (d,d’) on 68Ni in inverse kinematics has been studied with the active target Maya. It is the first attempt to measure the ISGMR and ISGQR in an unstable neutron-rich nucleus. For each experiment, the excitation energy spectrum has been reconstructed and the angular distributions have been studied with two independant methods. Experimental differential cross sections have been compared to DWBA calculations using RPA transition density in order to deduce the properties of the states observed. Concerning the analysis in (alpha, alpha’), the ISGMR is fragmented with a shoulder at 21.1+/-0.6 MeV, the ISGQR is concentred at 16.9+/-0.8 MeV and exhausted 61+/-17% of the Energy Weighted Sum Rule (EWSR). Moreover, a « soft GMR », predicted but never observed, is identified at 13.4+/-0.5 MeV. All these results are confirmed with the experiment in (d,d’), with the exception of the ISGQR due to unadapted experimental conditions.

Ziel dieser Arbeit ist die Simulation der Deformation von Bauteilen, welche aus nahezu inkompressiblem Material bestehen. Dabei soll sich das Material sowohl linear als auch nichtlinear elastisch verhalten können. Zusätzlich soll die Belastung des Bauteils beliebig gewählt werden können, das heißt, es sollen kleine als auch große Deformationen möglich sein.:1. Einleitung 2. Grundlagen 3. Aufgabenstellung für linear elastisches Material unter kleinen Deformationen 4. Gemischte Methode der finiten Elemente 5. Herleitung der Fehlerschätzung 6. Aufgabenstellung für nichtlinear elastisches Material unter großen Deformationen 7. Lösungsstrategie A. Anhang

This thesis investigates the numerical simulation of three-dimensional, mechanical deformation problems in the context of large deformations. The main focus lies on the prediction of non-linearly elastic, incompressible material. Based on the equilibrium of forces, we present the weak formulation of the large deformation problem. The discrete version can be derived by using linearisation techniques and an adaptive mixed finite element method. This problem turns out to be a saddle point problem that can, among other methods, be solved via the Bramble-Pasciak conjugate gradient method or the minimal residual algorithm. With some modifications the resulting simulation can be improved but we also address remaining limitations. Some numerical examples show the capability of the final FEM software. In addition, we briefly discuss the special case of linear elasticity with small deformations. Here we directly derive a linear weak formulation with a saddle point structure and apply the adaptive mixed finite element method. It is shown that the presented findings can also be used to treat the nearly incompressible case.

Les simulations en mécanique du solide présentent des difficultés comme le traitement de l'incompressibilité ou les non-linéarités dues aux grandes déformations, aux lois de comportement et de contact. L'objectif principal de ce travail est de proposer des méthodes éléments finis capables de gérer l'incompressibilité en grandes déformations en utilisant des éléments de faible ordre. Parmi les approches de la littérature, les formulations mixtes offrent un cadre théorique intéressant. Dans ce travail, une formulation mixte à trois champs (déplacements, pression, gonflement) est introduite. Dans certains cas, cette formulation peut être condensée en formulation à deux champs. Cependant, il est connu que le problème discret obtenu par une approche éléments finis de type Galerkin n'hérite pas automatiquement de la condition de stabilité “inf-sup” du problème continu : les éléments finis utilisés, et notamment les ordres d'interpolation doivent être choisis de sorte à vérifier cette condition de stabilité. Cependant, il est possible de s'affranchir de cette contrainte en ajoutant des termes de stabilisation à la formulation EF Galerkin. Cette approche permet entre autres d'utiliser des ordres d'interpolation égaux. Dans ce travail, des éléments finis stables de type P2/P1 sont utilisés comme référence, et comparés à une formulation P1/P1, stabilisée soit avec une fonction bulle, soit avec une méthode VMS (Variational Multi-Scale) basée sur un espace sous-grille orthogonal à l'espace EF. Combinées à un modèle grandes déformations basé sur des déformations logarithmiques, ces approches sont d'abord validées sur des cas académiques puis sur des cas industriels. / Simulations in solid mechanics exhibit difficulties as dealing with incompressibility or nonlinearities due to finite strains, constitutive laws and contact. The basic motivation of our work is to propose efficient finite element methods capable of dealing with incompressibility in finite strain context, and using low order elements. Among the approaches in the literature, mixed formulations offer an interesting theoretical framework. In this work, a three-field mixed formulation (displacement, pressure, volumetric strain) is investigated. In some cases, this formulation can be condensed in a two-field formulation. However, it is well-known that the discrete problem given by the Galerkin finite element technique, does not inherit the “inf-sup” stability condition from the continuous problem: the finite elements used, and in particular the interpolation orders must be chosen so as to satisfy this stability condition. However, it is possible to circumvent it, by adding terms stabilizing the FE Galerkin formulation. The latter approach allows the use of equal order interpolation. In this work, stable finite elements of type P2/P1 are used as reference, and compared to a P1/P1 formulation, stabilized with a bubble function, or with a VMS method (Variational Multi-Scale) based on a sub-grid-space orthogonal to the FE space. Combined to a finite strain model based on logarithmic strain, these approaches are first validated on academic cases and then on industrial cases.

Material point method (MPM) is a numerical tool which was originally used for modelling large deformations of solid mechanics problems. Due to the particle based spatial discretiza- tion, MPM is naturally capable of handling large mass movements together with topological changes. Further, the Lagrangian particles in MPM allow an easy implementation of history dependent materials. So far, however, research on MPM has been mostly restricted to explicit dynamic formu- lations with linear approximation functions. This is because of the simplicity and the low computational cost of such explicit algorithms. Particularly in MPM analysis of geomechan- ics problems, a considerable attention is given to the standard explicit formulation to model dynamic large deformations of geomaterials. Nonetheless, several limitations exist. In the limit of incompressibility, a significantly small time step is required to ensure the stability of the explicit formulation. Time step size restriction is also present in low permeability cases in porous media analysis. Spurious pressure oscillations are another numerical instability present in nearly incompressible flow behaviours. This research considers an implicit treatment of the pressure in MPM algorithm to simu- late material incompressibility. The coupled velocity (v)-pressure (p) governing equations are solved by applying Chorin’s projection method which exhibits an inherent pressure stability. Hence, linear finite elements can be used in the MPM solver. The main purpose of this new MPM formulation is to mitigate artificial pressure oscillations and time step restrictions present in the explicit MPM approach. First, a single phase MPM solver is applied to free surface incompressible fluid flow problems. Numerical results show a better approximation of the pressure field compared to the results obtained from the explicit MPM. The proposed formulation is then extended to model fully saturated porous materials with incompress- ible constituents. A solid velocity(v S )-fluid velocity (v F )-pore pressure (p) formulation is presented within the framework of mixture theory. Comparing the numerical results for the one-dimensional consolidation problem shows that the proposed incompressible MPM algorithm provides a stable and accurate pore pressure field even without implementing damping in the solver. Finally, the coupled MPM is used to solve a two-dimensional wave propagation problem and a plain strain consolidation problem. One of the important features of the proposed hydro mechanical coupled MPM formulation is that the time step size is not dependent on the incompressibility and the permeability of the porous medium.

Ce travail de recherche traite du couplage entre un liquide incompressible, irrotationnel et son contenant : une structure élastique. Cette interaction fluide-structure est traitée dans le cadre des petites déformations autour d'un état d'équilibre.Dans un premier temps, on présente une méthode d'introduction des sources dissipatives visqueuses dans le liquide à partir des équations du système couplé conservatif en s'appuyant sur une approche de type fluide potentiel généralement utilisée pour traiter les problèmes de couplage fluide-structure linéarisés non amortis. Un modèle d'amortissement diagonal est alors choisi pour le liquide et les effets dissipatifs de celui-ci sont pris en compte en calculant les coefficients d'amortissement modaux. Seuls les effets dissipatifs liées à la viscosité du liquide sont alors pris en compte. Le système couplé dissipatif obtenu possède une matrice d'amortissement non symétrique. Une résolution de ce système à amortissement non classique est alors présentée et les expressions des réponses fréquentielle et temporelle linéarisées sont données pour différents types d'excitations.Dans un deuxième temps, le liquide est supposé non visqueux et les forces de tension surfacique sont prises en compte. Cette configuration concerne principalement les satellites où le système couplé est en situation de microgravité. Une formulation du problème conservatif permettant de prendre en compte l'incompressibilité du fluide, la condition de continuité à l'interface fluide structure, les effets de capillarité du fluide ainsi que les effets éventuels de précontraintes statiques est alors établie. On se propose pour cela d'utiliser une méthode énergétique via le Principe de Moindre Action. La démarche est alors décomposée en deux étapes : une étude statique afin de déterminer la position de référence, puis une étude dynamique linéarisée autour de cette position d'équilibre. Cette formulation forme notamment une base pour l'introduction des sources dissipatives liées aux effets de capillarité via la méthode précédemment introduite. / This study deals with the coupling between an incompressible, irrotational fluid and an elastic container in the context of small amplitude vibrations.Firstly, we present a method to introduce the viscous dissipative sources in the liquid directly from the equations of the conservative coupled problem using a fluid potential approach generally used to treat linear undamped problems. A diagonal damping model is chosen for the liquid and its dissipative effects are taken into account through modal damping coefficients. Only the viscous effects are considered here. The coupled system obtained has a non symmetric damping matrix. This system with non classical damping is solved and expressions of the frequency and linearized time responses are given for different load examples.Secondly, the liquid is supposed to be inviscid and surface tension forces are considered. This configuration is related to satellite applications where the coupled system is in microgravity conditions. A unified formulation of the conservative problem taking into account the fluid incompressibility, the contact condition at the fluid structure interface, capillarity and prestress effects is given. Thus, we propose to use an energy method via the Least Action Principle. The reasoning is then divided into two parts: a static study to determine the reference state and a linearized dynamic study around this equilibrium state. This formulation is a good framework to introduce the dissipative sources associated with the capillary effects by using the method previously introduced.

Indiana University-Purdue University Indianapolis (IUPUI) / The passive properties of the venous wall are important in the development of venous pathology. Increase in venous pressure due to retrograde flow (reflux) and obstruction of venous flow by intrinsic and extrinsic means are the two possible mechanisms for venous hypertension. Reflux is the prevailing theory in the etiology of venous insufficiency. The objective of this thesis is to quantify the passive biomechanical response and structural remodeling of veins subjected to chronic venous reflux and hypertension. To investigate the effects of venous reflux on venous mechanics, the tricuspid valve was injured chronically in canines by disrupting the chordae tendineae. The conventional inflation-extension protocol in conjunction with intravascular ultrasound (IVUS) was utilized to investigate the passive biomechanical response of both control common iliac veins (from 9 dogs) and common iliac veins subjected to chronic venous reflux and hypertension (from 9 dogs). The change in thickness and constituent composition as a result of chronic venous reflux and hypertension was quantified using multiphoton microscopy (MPM) and histological evaluation. Biomechanical results indicate that the veins stiffened and became less compliant when exposed to eight weeks of chronic venous reflux and hypertension. The mechanical stiffening was found to be a result of a significant increase in wall thickness (p < 0.05) and a significant increase in the collagen to elastin ratio (p < 0.05). After eight weeks of chronic reflux, the circumferential Cauchy stress significantly reduced (p < 0.05) due to wall thickening, but was not restored to control levels. This provided a useful model for development and further analysis of chronic venous insufficiency and assessment of possible intervention strategies.