The emergence of freeform architecture provides interesting geometric challenges with regards to the design and manufacturing of large-scale structures. To design these architectural structures, we have to consider two types of constraints. First, aesthetic constraints are important because the buildings have to be visually impressive. Sec- ond, functional constraints are important for the performance of a building and its e cient construction. This thesis contributes to the area of architectural geometry. Specifically, we are interested in the geometric rationalization of freeform architec- ture with the goal of combining aesthetic and functional constraints and construction requirements. Aesthetic requirements typically come from designers and architects. To obtain visually pleasing structures, they favor smoothness of the building shape, but also smoothness of the visible patterns on the surface. Functional requirements typically come from the engineers involved in the construction process. For exam- ple, covering freeform structures using planar panels is much cheaper than using non-planar ones. Further, constructed buildings have to be stable and should not collapse. In this thesis, we explore the geometric rationalization of freeform archi- tecture using four specific example problems inspired by real life applications. We achieve our results by developing optimization algorithms and a theoretical study of the underlying geometrical structure of the problems. The four example problems are the following: (1) The design of shading and lighting systems which are torsion-free structures with planar beams based on quad meshes. They satisfy the functionality requirements of preventing light from going inside a building as shad- ing systems or reflecting light into a building as lighting systems. (2) The Design of freeform honeycomb structures that are constructed based on hex-dominant meshes with a planar beam mounted along each edge. The beams intersect without torsion at each node and create identical angles between any two neighbors. (3) The design of polyhedral patterns on freeform surfaces, which are aesthetic designs created by planar panels. (4) The design of space frame structures that are statically-sound and material-e cient structures constructed by connected beams. Rationalization of cross sections of beams aims at minimizing production cost and ensuring force equilibrium as a functional constraint.

The field of algebraic tropical geometry establishes a deep connection between algebraic geometry and combinatorics by associating to certain classical algebraic varieties so called tropical varieties, which are polyhedral complexes in some real vectorspaces. Tropical varieties are closely related to the Groebner complexes of the ideal defining the classical variety. In this thesis the tropical variety of an ideal is studied under a generic change of coodinates. Analogously to the existence of generic initial ideals the existence of generic Groebner complexes and generic tropical varieties is proved. Moreover, it is shown that in the constant coefficient case information on the invariants dimension, Hilbert-Samuel multiplicity and depth of the corresponding coordinate rings can be obtained from generic tropical varieties.

Les processeurs multi-coeurs sont maintenant largement utilisés presque partout en informatique: ordinateurs de bureau, ordinateurs portables et accélérateurs tels que les GPGPU (General Purpose Graphics Processing Units). La difficulté de la programmation des systèmes parallèles est considérée comme un problème majeur qui va empêcher l'exploitation de leurs capacités dans le futur. Pour exploiter la puissance des processeurs multi-coeurs et les hiérarchies complexes de mémoire, il y a une grande nécessité pour utiliser des outils de parallélisation et d'optimisation automatique de code. L'optimisation polyédrique est un axe de recherche qui a comme but de résoudre ces problèmes. C'est est une représentation algébrique du programme et un ensemble d'analyses, de transformations et d'algorithmes de génération de code qui permettent à un compilateur de raisonner sur des transformations avancées de nids de boucle. Dans cette thèse, nous abordons certaines des limites du modèle polyédrique. Nous nous intéréssons particulièrement à trois problèmes et nous proposons des solutions pratiques à ces trois problèmes. Le premier problème est lié à la capacité d'appliquer l'optimisation de tuilage sur un code qui contient des fausses dépendances. Nous proposons une téchnique qui permet d'ignorer certaines fausses dépendences et donc qui permet d'appliquer l'optimisation de tuilage qui n'est pas possible sinon. Le second problème est lié au temps de compilation qui peut être trés long pour certains programmes. Nous proposons une téchnique qui transforme la représentation originale du programme à une nouvelle representation dans laquelle il y a moins d'instructions. L'optimisation de cette nouvelle représentation du programme est moins couteuse en terme de temps de compilation en comparaison avec l'optimisation de la représentation originale du programme. Le troisième problème est lié à deux limites: la première limite concerne la possibilité d'utiliser la compilation polyédrique sur des programmes qui ne resepectent pas les restrictions classiques du modèle polyédrique (un programme peut être représenté de façon précise dans le modèle polyédrique s'il ne contient pas des conditionnelles non-affines, des bornes de boucles non-affines et des accés non-affines). La seconde limite est liée à l'aptitude des outils à générer un code performant dans les performances se rapprochent des performances du code écrit à la main. Pour éviter ces deux limites, nous proposons un language de programmation que l'on appelle PENCIL, c'est un sous-ensemble de GNU C99 avec des règles de programmation spécifiques et quelques extensions. L'utilisation de ce sous-ensemble et l'utilisation de ces extensions permettent aux compilateurs de mieux exploiter le parallélisme et de mieux optimiser le code. / Multi-core processors are now in widespread use in almost all areas of computing: desktops, laptops and accelerators such as GPGPUs (General Purpose Graphics Processing Units). To harness the power of multi-core processors and complex memory hierarchies, the need for powerful compiler optimizations and especially loop nest transformations is now in high demand. The polyhedral optimization framework is showing promising results in addressing such a problem. It's an algebraic program representation and a set of analyses, transformations and code generation algorithms that enable a compiler to reason about advanced loop nest transformations addressing most of the parallelism and locality-enhancing challenges.In this thesis we address some of the limitations of the polyhedral framework. We address three problems and propose practical solutions to these three problems.The first problem is related to the ability to apply tiling on code that has false dependences (loop nest tiling is an optimization that changes the order of execution of statements in a loop nest in order to enhance data locality; false dependences are induced by the reuse of a single memory location to store multiple values during the life of the program). To preserve the validity of loop nest transformations and parallelization, data-dependences need to be analyzed. Memory dependences come in two varieties: true dependences (a.k.a. flow dependences) and false dependences (a.k.a. output and anti dependences). While true dependences must be satisfied in order to preserve the correct order of computations. False dependences reduce the degrees of freedom for loop transformations. In particular, loop tiling is severely limited in the presence of these dependences. While array expansion, a transformation that transforms scalars into arrays and arrays into higher dimensional arrays, removes all false dependences, the overhead of this transformation on memory and the detrimental impact on register-level reuse can be catastrophic. We propose and evaluate a compilation technique to safely ignore a large number of false dependences in order to enable loop nest tiling in the polyhedral model. It is based on the precise characterization of interferences between live range intervals, and it does not incur any scalar or array expansion.The second problem is related to the long compilation time that one may experience when using polyhedral tools to optimize a program. Particularly, the long execution time of the Pluto affine scheduling algorithm. The Pluto affine scheduling algorithm is the algorithm that is responsible for changing the schedule (order of execution) of statements in order to optimize the code (maximize parallelism and data locality). Reducing the execution time of this affine scheduling algorithm enhances the overall compilation time. We introduce and evaluate a technique called offline statement clustering. It is a practical technique designed to reduce the execution time of the Pluto affine scheduling algorithm without much loss in optimization opportunities. Using this technique, the statements of the program are clustered into macro-statements, the Pluto affine scheduling algorithm is then used to schedule the macro-statements instead of scheduling the original statements of the program. Since the number of macro-statements is less than the number of statements in the original program, scheduling the macro-statements is in general faster than scheduling the original statements of the program. We present the statement clustering algorithm, we show how offline statement clustering integrates transparently with the work-flow of a state-of-the-art polyhedral compiler and present two heuristics for choosing how statements should be clustered together. We show experimentally that statement clustering can reduce the scheduling time by a factor of 8x (in median) without a significant loss in optimization opportunities...

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This paper presents a robust and artist driven method for fracturing a surface polyhedral mesh via fracture maps. A fracture map is an undirected simple graph with nodes representing positions in UV-space and fracture lines along the surface of a mesh. Fracture maps allow artists to concisely and rapidly define, edit, and apply fracture patterns onto the surface of their mesh. The method projects a fracture map onto a polyhedral surface and splits its triangles accordingly. The polyhedral mesh is then segmented based on fracture lines to produce a set of independent surfaces called fracture components, containing the visible surface of each fractured mesh fragment. Subsequently, we utilize a Voronoi-based approximation of the input polyhedral mesh’s medial axis to derive a hidden surface for each fragment. The result is a new watertight polyhedral mesh representing the full fracture component. Results are aquired after a delay sufficiently brief for interactive design. As the size of the input mesh increases, the computation time has shown to grow linearly. A large mesh of 41,000 triangles requires approximately 3.4 seconds to perform a complete fracture of a complex pattern. For a wide variety of practices, the resulting fractures allows users to provide realistic feedback upon the application of extraneous forces.

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