The goal of this thesis is to develop a hybrid _nite element formulation to carry out stress analysis of arteries. To the best of our knowledge, a hybrid _nite element impel mentation of the Holzapfel-Ogden artery model has not been carried out before. Since arteries are thin `shell-type' structures, they are subjected to membrane, shear and volumetric locking in case when standard _nite elements are used. Since hybrid _nite elements are known to overcome these problems, we develop hybrid hexahedral element formulations (both lower and higher-order) for artery analysis. We demonstrate The better coarse mesh accuracy of hybrid elements, which are based on a two-_eld variational formulation, over conventional displacement based elements. Typically, wend that three or four extra levels of renement are required with conventional elements to achieve the ame accuracy as hybrid elements. The recently proposed Holzapfel-Ogden constitutive model for the artery and its implementation both within the conventional and hybrid _nite element frameworks is discussed. The numerical implementation is particularly challenging due to the presence of _bers which can only take tensile loads. The mathematically exact tangent stiness matrix that we have derived in this work is crucial in ensuring convergence of the numerical strategy.

The main focus of this work lies in the simulation of the deformation of mechanical components which consist of nonlinear elastic, incompressible material and that are subject to large deformations. Starting from a nonlinear formulation one can derive a discrete problem by using linearisation techniques and an adaptive mixed finite element method. This turns out to be a saddle point problem that can be solved via a Bramble-Pasciak conjugate gradient method. With some modifications the simulation can be improved.:1. Introduction 2. Basics 3. Mixed variational formulation 4. Solution method 5. Error estimation 6. LBB conditions 7. Improvement suggestions

L'objectif de ce travail est d'étudier la dé<br />formation d'un tissu posé sur un support bi- ou tri-dimensionnel et soumis à<br />son propre poids.<br />Dans la première partie, nous établissons les équations<br />d'équilibre de ce problème dans le cas général et<br />introduisons deux modèles mathématiques. Le premier est un<br />modèle membranaire non-linéaire, dont l'analyse mathématique<br />conduit au calcul de l'enveloppe quasi-convexe de la densité<br />d'énergie associée. Le deuxième modèle (modèle<br />membrane-flexion non-linéaire) est obtenu en ajoutant un terme<br />régularisant à une fonctionnelle énergie non coercive. Nous<br />prouvons l'existence d'au moins une solution de ce problème de<br />minimisation, en utilisant les techniques du Calcul des Variations. Enfin,<br />nous établissons l'existence de solutions pour le problème de<br />drapé tri-dimensionnel.<br />La seconde partie est consacrée à la résolution numérique des diffé%<br />rents modèles élaborés dans la première partie, au moyen d'une méthode ité%<br />rative de descente couplée avec une méthode multigrille, afin d'accélérer la<br />convergence de l'algorithme. Nous montrons que le problème discret admet au<br />moins une solution. Enfin, nous prouvons la convergence théorique d'une<br />sous-suite de solutions discrètes vers une solution du problème continu,<br />moyennant une hypothèse de densité.

Synthetic vascular grafts are useful to bypass diseased arteries. The long-term failure of synthetic grafts is primarily due to intimal hyperplasia at the anastomotic sites. The accelerated intimal hyperplasia may stem from a compliance mismatch between the host artery and the graft since commercially available synthetic conduits are much stiffer than an artery. The objective of this thesis is to design a method for fabricating a vascular graft that mechanically matches the patients native artery over the expected physiologic range of pressures. The creation of an optimized mechanical graft will hopefully lead to an improvement in patency rates. The mechanical equivalency between the graft and the host artery is defined locally by several criteria including the diameter upon inflation, the elasticity at mean pressure, and axial force. A single parameter mathematical for a thin-walled tube is used to describe of the final mechanical behavior of a synthetic graft. For the general problem, the objective would be to fabricate a mechanics-matching vascular graft for each host artery. Typically, fabrication parameters are set initially and the properties of the fabricated graft are measured. However, by modeling the entire fabrication process and final mechanical properties, it is possible to invert the situation and let the typical output mechanical values be used to define the fabrication parameters. The resultant fabricated graft will then be mechanically matching. As a proof-of-concept, several prototype synthetic grafts were manufactured and characterized by a single Invariant to match a canine artery. The resultant graft equaled the diameter upon inflation, the elasticity at mean pressure, and axial force of the native canine artery within 6%. An alternative to making an individual graft for each artery is also presented. A surgeon may choose the best graft from a set of pre-manufactured grafts, using a computer program algorithm for best fit using two parameters in a neighborhood. The design optimization problem was solved for both canine carotid and human coronary arteries. In conclusion, the overall process of design, fabrication and selection of a mechanics matching synthetic vascular graft is shown to be reliable and robust.