Three dimensional (3D) curved objects are ubiquitous in computational science and engineering problems such as geometry acquisition, fabrication of shapes, virtual/augmented reality and computer aided design, to name a few. Often, only the surface, namely the surrounding shell of the object, is required in practice. To represent these curved surfaces in a computer, many algorithms approximate the object using a triangle mesh. That is, the given surface is tiled using small triangles glued over their edges. While triangle meshes are useful in many scenarios, in some cases such as computer modeling and numerical simulation, quad meshes, i.e. meshes with quadrangular faces are preferred. To this end, several conversion techniques were devised, allowing to transform an input triangle mesh to a quad mesh. To date, existing conversion methods work well on certain examples, but in other cases, the results could be greatly improved, motivating the following research. The main goal of this action is to explore new methods for generating a quadrangular surface that approximates a given triangle mesh.
Our investigation shows that the Ginzubrg—Landau potential is somewhat too sensitive to work with in practice. The main reason for the increased sensitivity is non-convexity of the potential which leads to less robust numerical schemes. Instead, we discovered that functionals based on the elastic and membrane energies are much more robust and easier to work with. While both energy functionals can be used for similar applications, the elastic energy we developed is especially useful to handle non-smooth surfaces which contain sharp corners or edges. In addition, we demonstrated that our core method can be utilized on data that originate from complex dynamical systems such as fluid flows and weather systems. Thus, we developed a simple method for analyzing the behavior of such complex systems in terms of their dominating main modes and their decay and growth rates
