To simulate increasingly complex physical phenomena and systems, tightly integrated design-through-analysis (DTA) tools are essential. In this dissertation, the complementary strengths of isogeometric analysis and T-splines are coupled and enhanced to create a seamless DTA framework. In all cases, the technology de- veloped meets the demands of both design and analysis. In isogeometric analysis, the smooth geometric basis is used as the basis for analysis. It has been demonstrated that smoothness offers important computational advantages over standard finite elements. T-splines are a superior alternative to NURBS, the current geometry standard in computer-aided design systems. T-splines can be locally refined and can represent complicated designs as a single watertight geometry. These properties make T-splines an ideal discretization technology for isogeometric analysis and, on a higher level, a foundation upon which unified DTA technologies can be built. We characterize analysis-suitable T-splines and develop corresponding finite element technology, including the appropriate treatment of extraordinary points (i.e., unstructured meshing). Analysis-suitable T-splines form a practically useful subset of T-splines. They maintain the design flexibility of T-splines, including an efficient and highly localized refinement capability, while preserving the important analysis-suitable mathematical properties of the NURBS basis. We identify Bézier extraction as a unifying paradigm underlying all isogeometric element technology. Bézier extraction provides a finite element representation of NURBS or T-splines, and facilitates the incorporation of T-splines into existing finite element programs. Only the shape function subroutine needs to be modified. Additionally, Bézier extraction is automatic and can be applied to any T-spline regardless of topological complexity or polynomial degree. In particular, it represents an elegant treatment of T-junctions, referred to as "hanging nodes" in finite element analysis We then detail a highly localized analysis-suitable h-refinement algorithm. This algorithm introduces a minimal number of superfluous control points and preserves the properties of an analysis-suitable space. Importantly, our local refinement algorithm does not introduce a complex hierarchy of meshes. In other words, all local refinement is done on one control mesh on a single hierarchical “level” and all control points have similar influence on the shape of the surface. This feature is critical for its adoption and usefulness as a design tool. Finally, we explore the behavior of T-splines in finite element analysis. It is demonstrated that T-splines possess similar convergence properties to NURBS with far fewer degrees of freedom. We develop an adaptive isogeometric analysis framework which couples analysis-suitable T-splines, local refinement, and Bézier extraction and apply it to the modeling of damage and fracture processes. These examples demonstrate the feasibility of applying T-spline element technology to very large problems in two and three dimensions and parallel implementations. / text

In this thesis, a robust, reliable and efficient isogeometric analysis framework is presented that allows for an adaptive spatial discretization of non-linear and time-dependent multi-field problems. In detail, B\'ezier extraction of truncated hierarchical B-splines is proposed that allows for a strict element viewpoint, and in this way, for the application of standard finite element procedures. Furthermore, local mesh refinement and coarsening strategies are introduced to generate graded meshes that meet given minimum quality requirements. The different strategies are classified in two groups and compared in the adaptive isogeometric analysis of two- and three-dimensional, singular and non-singular problems of elasticity and the Poisson equation. Since a large class of boundary value problems is non-linear or time-dependent in nature and requires incremental solution schemes, projection and transfer operators are needed to transfer all state variables to the new locally refined or coarsened mesh. For field variables, two novel projection methods are proposed and compared to existing global and semi-local versions. For internal variables, two different transfer operators are discussed and compared in numerical examples. The developed analysis framework is than combined with the phase-field method. Numerous phase-field models are discussed including the simulation of structural evolution processes to verify the stability and efficiency of the whole adaptive framework and to compare the projection and transfer operators for the state variables. Furthermore, the phase-field method is used to develop an unified modelling approach for weak and strong discontinuities in solid mechanics as they arise in the numerical analysis of heterogeneous materials due to rapidly changing mechanical properties at material interfaces or due to propagation of cracks if a specific failure load is exceeded. To avoid the time consuming mesh generation, a diffuse representation of the material interface is proposed by introducing a static phase-field. The material in the resulting transition region is recomputed by a homogenization of the adjacent material parameters. The extension of this approach by a phase-field model for crack propagation that also accounts for interface failure allows for the computation of brittle fracture in heterogeneous materials using non-conforming meshes.

L’Analyse isogéométrique (AIG) est une méthode innovante de résolution numérique des équations différentielles, proposée à l’origine par Thomas Hughes, Austin Cottrell et Yuri Bazilevs en 2005. Cette technique de discrétisation est une généralisation de l’analyse par éléments finis classiques (AEF), conçue pour intégrer la conception assistée par ordinateur (CAO), afin de combler l’écart entre la description géométrique et l’analyse des problèmes d’ingénierie. Ceci est réalisé en utilisant des B-splines ou des B-splines rationnelles non uniformes (NURBS), pour la description des géométries ainsi que pour la représentation de champs de solutions inconnus.L’objet de cette thèse est d’étudier la méthode isogéométrique dans le contexte des problèmes hyperboliques en utilisant les fonctions B-splines comme fonctions de base. Nous proposons également une méthode combinant l’AIG avec la méthode de Galerkin discontinue (GD) pour résoudre les problèmes hyperboliques. Plus précisément, la méthodologie de GD est adoptée à travers les interfaces de patches, tandis que l’AIG traditionnelle est utilisée dans chaque patch. Notre méthode tire parti de la méthode de l’AIG et la méthode de GD.Les résultats numériques sont présentés jusqu’à l’ordre polynomial p= 4 à la fois pour une méthode deGalerkin continue et discontinue. Ces résultats numériques sont comparés pour un ensemble de problèmes de complexité croissante en 1D et 2D. / Isogeometric Analysis (IGA) is a modern strategy for numerical solution of partial differential equations, originally proposed by Thomas Hughes, Austin Cottrell and Yuri Bazilevs in 2005. This discretization technique is a generalization of classical finite element analysis (FEA), designed to integrate Computer Aided Design (CAD) and FEA, to close the gap between the geometrical description and the analysis of engineering problems. This is achieved by using B-splines or non-uniform rational B-splines (NURBS), for the description of geometries as well as for the representation of unknown solution fields.The purpose of this thesis is to study isogeometric methods in the context of hyperbolic problems usingB-splines as basis functions. We also propose a method that combines IGA with the discontinuous Galerkin(DG)method for solving hyperbolic problems. More precisely, DG methodology is adopted across the patchinterfaces, while the traditional IGA is employed within each patch. The proposed method takes advantageof both IGA and the DG method.Numerical results are presented up to polynomial order p= 4 both for a continuous and discontinuousGalerkin method. These numerical results are compared for a range of problems of increasing complexity,in 1D and 2D.