The fundamental advantages of applying Isogeometric Analysis (IGA) to shell analysis have been extensively demonstrated across a wide range of problems and formulations. However, a phenomenon called numerical locking is still a major challenge in IGA shell analysis, which can lead to dramatically deteriorated analysis accuracy. Additionally, for complex thin-walled structures, a simple and robust coupling technique is desired to sew together models composed of multiple patches. This dissertation focuses on addressing these challenges of IGA shell analysis. First, an isogeometric dual mortar method is developed for multi-patch coupling. This method is based on Be ?zier extraction and projection and can be employed during the creation and editing of geometry through properly modified extraction operators. It is applicable to any spline space which has a representation in Be ?zier form. The error in the method can be adaptively controlled, in some cases recovering optimal higher-order rates of convergence, by leveraging the exact refineability of the proposed dual spline basis without introducing any additional degrees-of-freedom into the linear system. This method can be used not only for shell elements but also for heat transfer and solid elements, etc. Next, a mixed formulation for IGA shell analysis is proposed that addresses both shear and membrane locking and improves the quality of computed stresses. The starting point of the formulation is the modified Hellinger-Reissner variational principle with independent displacement, membrane, and shear strains as the unknown fields. To overcome locking, the strain variables are interpolated with lower-order spline bases while the variations of the strain variables are interpolated with the proposed dual spline bases. As a result, the strain variables can be condensed out of the system with only a slight increase in the bandwidth of the resulting linear system and the condensed approach preserves the accuracy of the non-condensed mixed approach but with fewer degrees-of-freedom. Finally, as an alternative, new quadrature rules are developed to release membrane and shear locking. These quadrature rules asymptotically only require one point for Reissner-Mindlin (RM) shell elements and two points for Kirchhoff-Love (KL) shell elements in B-spline and NURBS-based isogeometric shell analysis, independent of the polynomial order p of the elements. The quadrature points are Greville abscissae and the quadrature weights are calculated by solving a linear moment fitting problem in each parametric direction. These quadrature rules are free of spurious zero-energy modes and any spurious finite-energy modes in membrane stiffness can be easily stabilized by using a higher-order Greville rule.

<p>In the finite element simulation of problems with contact there arises</p><p>the need to change the mesh and continue the simulation on a new mesh.</p><p>This is encountered when the mesh has to be changed because the original mesh experiences severe distortion or the mesh is adapted to minimize errors in the solution. In such instances a crucial component is the transfer of data from the old mesh to the new one. </p><p>This work proposes a strategy by which such remeshing can be accomplished in the presence of mortar-discretized contact, </p><p>focusing in particular on the remapping of contact variables which must occur to make the method robust and efficient. </p><p>By splitting the contact stress into normal and tangential components and transferring the normal component as a scalar and the tangential component by parallel transporting on the contact surface an accurate and consistent transfer scheme is obtained. Penalty and augmented Lagrangian formulations are considered. The approach is demonstrated by a number of two and three dimensional numerical examples.</p> / Dissertation

Dans cette thèse, nous développons une méthode locale de type "mortar" pour traiter le problème de contact avec maillages incompatibles de manière optimale dans un code de calcul par éléments finis de niveau industriel. Dans la première partie de la thèse, nous introduisons le cadre mathématique de la méthode intitulée "Local Average Contact" (LAC). Cette approche consiste à imposer la condition de non-interpénétration en moyenne sur chaque élément d'un macro-maillage défini de manière idoine. Nous commençons par développer une nouvelle technique de preuve pour démontrer l'optimalité des approches de type inéquation variationnelle discrétisée par éléments finis standards pour le problème de Signorini, sans hypothèse autre que la régularité Sobolev de la solution du problème continu. Puis nous définissons la méthode LAC et démontrons, à l'aide des nouveaux outils techniques, l'optimalité de cette approche locale modélisant le contact unilatéral dans le cas général des maillages incompatibles. Pour finir, nous introduisons la formulation mixte équivalente et démontrons son optimalité et sa stabilité. Dans la seconde partie de la thèse, nous nous intéressons à l'étude numérique de la méthode LAC. Nous confirmons sa capacité à gérer numériquement le contact unilatéral avec maillages incompatibles de manière optimale à l'instar des méthodes "mortar" classiques, tout en restant facilement implémentable dans un code de calcul industriel. On montre ainsi, entre autres, que la méthode passe avec succès le patch test de Taylor. Finalement, nous montrons son apport en terme de robustesse et au niveau de la qualité des pressions de contact sur une étude de type industrielle. / In this thesis, we develop a local "mortar" kind method to deal with the problem of contact with non-matching meshes in an optimal way into a finite element code of industrial level. In the first part of the thesis, we introduce the mathematical framework of the Local Average Contact method (LAC). This approach consists in satisfying the non-interpenetration condition in average on each element of a macro-mesh defined in a suitable way. We start by developing a new technique for proving the optimality of variational inequality approaches discretized by finite elements modeling Signorini problem without other hypothesis than the Sobolev regularity of the solution of the continuous problem. Then we define the LAC method and prove, using the new technical tools, the optimality of this local approach modeling the unilateral contact in the general case of non-matching meshes. Finally, we introduce the equivalent mixed formulation and prove its optimality and stability. In the second part of the thesis, we are interested in the numerical study of the LAC method. We confirm its ability to optimally treat the contact problem when considering non-matching meshes like standard "mortar" methods, while remaining easily implementable in an industrial finite element code. We show, for example, that the method successfully passes the Taylor patch test. Finally, we show its contribution in terms of robustness and at the quality of the contact pressures on an industrial study.