Transformation conforme

Résumé

En mathématiques, et plus précisément en géométrie et en analyse complexe, une transformation conforme est une bijection qui conserve localement les angles, c'est-à-dire qui se comporte au voisinage de chaque point où elle est définie presque comme une similitude. Dans le plan, les transformations conformes qui conservent les angles orientés ont une telle utilité qu'il est fréquent qu'elles soient les seules baptisées du terme de conformes. Elles se confondent alors avec les bijections holomorphes. Les transformations conformes indirectes sont, dans ce cas, appelées transformations anticonformes. On rencontre les transformations conformes en géométrie différentielle, dans des problèmes d'électrostatique ou dans la résolution de l'équation de Poisson, en mécanique des fluides pour modéliser des écoulements, et en cartographie. La notion de transformation conforme se généralise à des espaces de dimension supérieure à 2, mais elle y perd un peu de sa diversité. Cas du plan

Source officielle

https://fr.wikipedia.org/wiki/Transformation_conforme

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An intrinsic approach to Finsler geometry is proposed. A concept of Finsler- Ehresmann manifold, denoted by (M,F,H), is introduced and a generalized Chern connection is built for this manifold. Conformal deformations on this manifold are considered. First, we have an analogous of Chern's theorem: we prove the existence and uniqueness of a generalized Chern connection for the manifold (M,F,H). Similarly, within an essentially koszulian formalism, we present two curvatures associated to this generalized connection, namely a R curvature and a P one. The second result is the deduction of conformal transformations laws for the generalized Chern connection and associated curvatures. The transformation of R seems to have very similar properties as that of the Riemannian curvature while that of P reveals other objects of pure Finslerian nature. Third, we construct the finsler Weyl and Schouten tensors W and S respectively and we study their conformal transformations. Furthermore, we show that for the dimension 3, the horizontal component of W for generalized Berwald manifolds is identically zero. The next result is a theorem of Weyl-Schouten type giving necessary and sufficient conditions for a Finsler-Ehresmann manifold to be conformaly R-flat. We complete this result by exploring the case of dimension 3 for Berwald spaces which gives a result very similar to the Riemannian case. In addition, we announce some necessary conditions to characterize conformal flatness of Finsler-Ehresmann manifolds.

EPFL2011

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Computational Design of Auxetic Shells

Mina Konakovic Lukovic

Recent advances in material science and digital fabrication provide promising opportunities for product design, mechanical and biomedical engineering, robotics, architecture, art, and science. Engineered materials and personalized fabrication are revolutionizing manufacturing culture and having a significant impact on various scientific and industrial works. As new fabrication technologies emerge, effective computational tools are needed to fully exploit the potential of digital fabrication. This thesis introduces a novel computational method for design and fabrication with auxetic materials. The term auxetic refers to solid materials with negative Poisson ratio â when the material is stretched in one direction, it also expands in all other directions. In particular, we study 2D auxetic materials in the form of a triangular linkage which exhibits auxetic behavior at the macro scale. This stretching, in turn, allows the flat material to approximate doubly-curved surfaces, making it attractive for fabrication. We physically realize auxetic materials by introducing a specific pattern of cuts into approximately inextensible material such as sheet metal, plastic, or leather. On a larger scale, we use individual rigid triangular elements and connect them with joints. First, this thesis focuses on a regular triangular linkage. When deformed into a curved shape, the linkage yields spatially-varying hexagonal openings. However, the global coupling of the linkage elements makes manual, incremental approach unlikely to succeed when trying to approximate a given curved surface. Thus, we leverage conformal geometry to enable complex surface design. In particular, we compute a global conformal map with bounded scale factor to initialize an otherwise intractable non-linear optimization. Constraint-based optimization is used to find the final linkage configuration that closely approximates a target 3D surface. Furthermore, we develop a computational method for designing novel deployable structures via programmable auxetics, i.e., spatially varying triangular linkage optimized to directly and uniquely encode the target 3D surface in the 2D pattern. The target surface is rapidly deployed from a flat initial state via inflation or gravitational loading. The thesis presents both inverse and forward design tools for interactive surface design with programmable auxetics. This allows the user to efficiently approximate a given shape and directly edit and adapt the auxetic linkage structure to explore the design alternatives. In addition, our solution enables simulation-based form-finding that uses deployment forces for interactive exploration of feasible shapes. The resulting designs can be easily manufactured via digital fabrication technologies such as laser cutting, CNC milling, or 3D printing. Auxetic materials and deployable structures enable scientific, industrial, and consumer applications across a wide variety of scales and usages. We validate our computational methods through a series of physical prototypes and application case studies, ranging from surgical implants, through art pieces, to large-scale architectural structures.

EPFL2019

Chargement

Beyond Developable: Computational Design and Fabrication with Auxetic Materials

Sofien Bouaziz, Bailin Deng, Mina Konakovic Lukovic, Mark Pauly

We present a computational method for interactive 3D design and rationalization of surfaces via auxetic materials, i.e., flat flexible material that can stretch uniformly up to a certain extent. A key motivation for studying such material is that one can approximate doubly-curved surfaces (such as the sphere) using only flat pieces, making it attractive for fabrication. We physically realize surfaces by introducing cuts into approximately inextensible material such as sheet metal, plastic, or leather. The cutting pattern is modeled as a regular triangular linkage that yields hexagonal openings of spatially-varying radius when stretched. In the same way that isometry is fundamental to modeling developable surfaces, we leverage conformal geometry to understand auxetic design. In particular, we compute a global conformal map with bounded scale factor to initialize an otherwise intractable non-linear optimization. We demonstrate that this global approach can handle non-trivial topology and non-local dependencies inherent in auxetic material. Design studies and physical prototypes are used to illustrate a wide range of possible applications.

Assoc Computing Machinery2016

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Computational Design of Auxetic Shells

Mina Konakovic Lukovic

EPFL2019