, pour obtenir des paramètres d'encastrement tels que ? R ? ?

, exécuter l'algorithme de recherche à partir de la valeur initiale donnée par la grille

, L'algorithme de Newton-Raphson est ensuite exécuté, avec f comme variable. L'absence de bornes dans le domaine de définition des paramètres de chargement est ici un atout, puiqu'on peut s'affranchir d'une minimisation sous contrainte. Les itérations sont présentées dans les algorithmes 1 et 2. La matrice jacobienne J est calculée numériquement à chaque incrément de la variable f s avec un schéma de différence finie centré d'ordre 2, En définitive, on utilise une grille deux fois plus petite, sans modifier les frontières ni la résolution. 2.3 Inversion Après avoir obtenu des valeurs initiales pour les paramètres physiques, les valeurs correspondantes de paramètres de chargement

, ? ? = ?q 2 la composée d'une des fonctions erreur de la sous-section 2.1

, Ensuite, les trajectoires particulières, dont certaines sont proches de structures d'acides nucléiques, sont localisées dans ce demi-espace : tiges rectilignes, elastica 2D, cercles, tiges fermées, hélices, homoclines. Leurs caractéristiques (rayon, pas, module, etc.) sont exprimées analytiquement avec des fonctions simples. De plus, les paramètres de Landau d'une tige en hélice sont obtenus à partir de la donnée du rayon, du pas et de l'orientation des sections en deux points de la trajectoire. Cela offre un accès immédiat à l'état de contrainte et d'énergie de ces hélices. Enfin, des propriétés géométriques générales des configurations d'équilibre sont mises en évidence. Les trajectoires infinies sont enveloppées à l'intérieur d'un tube hélicoïdal et s'enroulent autour de l'axe de ce tube, appelé hélice centrale. Les caractéristiques du tube enveloppe (deux rayons ? H , ? G et un pas p H ) sont exprimées analytiquement ainsi que le pas p G d'enroulement des tiges autour de l'hélice centrale. Trois propriétés de chiralité, liées au sens des enroulements, sont définies et exprimées analytiquement. Les configurations d'équilibre sont alors classées, d'abord en fonction des chiralités, Chapitre 4. Résolution du problème aux limites en position et orientation ces paramètres est déterminé pour la première fois. Nous avons montré qu'ils évoluent dans un demi-espace dont la borne inférieure est une surface a M in (?, t P ) relativement complexe, exprimée analytiquement

, Grâce à une paramétrisation de l'elastica idéale par (? H , ? G , a), celles-ci sont obtenues à partir de l'inversion numérique 1D de la fonction p H (a)

S. , S. , and S. , Tous les sens d'enroulement de l'hélice centrale autour de l'axe de la force, de la trajectoire autour de l'hélice centrale et des sections autour de la trajectoire sont possibles. Ces tiges ont donc une richesse structurelle suffisante pour décrire les biopolymères, y compris les protéines. Dans le cadre d'une modélisation quasi-statique, où les effets d'inertie sont négligés, les configurations d'équilibre peuvent être utilisées pour décrire la molécule, En définitive, les tiges élastiques sont des modèles prometteurs pour les acides nucléiques. La structure géométrique de leurs configurations d'équilibre

. C'est, ce qui a motivé les classifications proposées dans cette thèse. D'autre part, la résolution du problème aux limites avec des temps de calcul qui autorisent l'interactivité fait de ces tiges des éléments efficaces de construction et de déformation des acides nucléiques

, Comme la théorie de l'élasticité donne accès aux forces et moments appliqués sur la tige, un des projets en cours consiste à mettre en place un retour d'effort sur les simulations. Par l'intermédiaire d'un bras robotique, l'utilisateur spécifie les conditions d'encastrement ciblées et ressent l'action de la tige qui résiste à ses sollicitations. L'ajout d'une scène virtuelle stéréoscopique 3D et l'utilisation de dispositifs haptiques plus élaborés (interfaces à contacts intermittents, interfaces à câbles, etc.) peuvent encore améliorer l'immersion dans la vie de la molécule, et constituer des outils de modélisation intuitifs et innovants. Pour décrire plus exactement les contraintes imposées par l'environnement physico-chimique et les effets de la température, le modèle devra être enrichi, La recherche de plusieurs configurations sous des conditions d'encastrement fixées est encore à améliorer, en termes de rapidité notamment

, On souhaite notamment utiliser les résultats de cette thèse pour les protéines, mais aussi dans divers sujets de recherche attachés à la biologie, la mécanique et la robotique : insertion de cathéters pour l'endoscopie ou la chirurgie, création de robots compliants, conception de ressorts non linéaires, etc. La possibilité d'imposer des contraintes géométriques aux tiges (caractéristiques du tube enveloppe, conditions d'encastrement) constitue en effet un atout significatif dans la résolution de nombreux problèmes, Nous espérons ainsi avoir apporté des éléments favorables à l'accomplissement des projets qui sont et seront formés autour des tiges élastiques

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