Conformal and multi-scale time-domain methods: From unstructured meshes to meshless discretisations

This chapter reviews recent advances in numerical time-domain techniques for solving Maxwell’s equations in non-Cartesian discretisations. This class of techniques, which can be denoted as conformal time-domain methods, provides an approach particularly advantageous for geometries comprising curved surfaces and multi-scale features. The first part of the chapter reviews the developments of a particular time-domain method applied in tetrahedral meshes, namely the Finite- Volume Time-Domain (FVTD) method. Different associated techniques aiming at enhancing the capability of the method are described, and the potential of the FVTD method for solving multi-scale problems is illustrated with the example of a 31-antenna breast cancer imaging system. The successful solution of this particular example demonstrates the benefits of the approach for problems which might challenge time-domain methods applied in Cartesian grids, even when coupled to sub-cell models and sub-gridding schemes. The second part of the chapter points out to a novel class of methods which are amenable to conformal time-domain implementation on clouds of points. These so-called ‘‘meshless methods’’ do not require an explicit mesh definition, and open new perspectives towards future applications involving multi-scale multi-physics problems.Christophe Fumeaux, Thomas Kaufmann, Zahra Shaterian, Dirk Baumann and Maciej Klem