Parameterizations for mesh generation : discrete atlases, crossfields and 3D frames

Numerical simulation is a jewel for industrial companies and research programs. During the last decades, most of design processes have radically changed thanks to numerical simulation. The Finite Element Method is a versatile and powerful framework for numerical simulation, which is broadly used by industrial companies: one million analyses a day. The main and seemingly only drawback is that the Finite Element Method comes with the tremendous price of mesh generation: it is 80% of the human time devoted to numerical analysis. Mesh generation consists of dividing a given object into simple shapes such as triangles and cubes. This laborious task uses auxiliary tools. This thesis considers such three tools: Discrete Atlases, Crossfields and 3D Frames. Usually, mesh generation is triggered from a computer aided design of the given object. Nowadays, more and more discrete models of objects are available, which lack such a design. Discrete atlases recover such a design. This thesis proposes to compute those discrete atlases with the mean value coordinates, leveraging a robust and automatic pipeline of mesh generation for discrete models. Some shapes have better numerical features than others. It is the case of squares and cubes. However, paving an object with only such shapes is challenging. A way to proceed is to use a crossfield and 3D frames respectively. They guide the corresponding pavement with squares/cubes. This thesis computes crossfields with the Ginzburg-Landau model giving an explicit parameterization of pavement irregularities. Finally, a quaternionic parameterization of 3D frames is derived with the special unitary group, yielding the smallest number of unique parameters.(FSA - Sciences de l'ingénieur) -- UCL, 201