Mesh generation for Tokamaks and Automated Isogeometric Analysis

Computational Mathematics Seminar
Abstract : Due to the very large anisotropic character of strongly magnetized plasma, the use of flux aligned grid is generally believed to be highly useful (or even mandatory) to obtain accurate and reliable simulations for fusion applications. For real geometries, the magnetic topology can only be computed by the use of specialized equilibrium solvers solving the non-linear Grad-Shafranov equation.  The output of these solvers then have to be used as input to construct flux aligned meshes that respect the magnetic topology. This process usually requires some manual input and expertise from the final users to identify the relevant features of the magnetic topology (X points, magnetic axis).
In the first part of this talk, we will describe an original method for the automated construction of flux aligned grids. This method assumes that the magnetic flux is a Morse function and consequently that the results of Morse theory can be applied. Then we will describe a new method for constructing Adaptive and Anistotropic mappings by solving an Optimal Transport problem. This method leads to equidistributed meshes and ensures the one-to-one constraint.
In the second part of the talk, we will describe a new framework, à la FENICS/FreeFem++, for IsoGeometric Analysis.  Compared to existing solutions, our formal language captures the semantic of mathematical expressions and therefor can be presented as a static compiler. The framework is based on ideas from Type Theory and Category Theory, which allow us to do some verifications before discretization and keep track of different algebraic structures from the formal/continuous level to the discrete one.
Moreover, it also exposes DeRham complexes, which turns out to add additional levels of type checking.  Different examples will be shown both on the formal language, associated discretizations and parallel computing.