Hybrid cubic/tetrahedral DGTD method

A discontinuous Galerkin formulation can be seen as a classical finite element method for which the global continuity of the approximation has been lifted. This implies that the support of each basis function is restrained to a discretization cell, which leads to local formulations implying no large mass matrix inversion if an explicit time-marching scheme is adopted. Then, connexion between neighboring cells is restored by the use of a numerical flux as in a finite volume method. The form of the numerical flux impacts the mathematical properties of the resulting DGTD scheme. The discontinuity of the approximation allows for several methodological improvements among which the local adaptation of the approximation order, and the use of non-conforming meshes.

This study is concerned with the development of a non-conforming multi-element DGTD method for the solution of the 3D time-domain Maxwell equations coupled to a Drude dispersion model for the simulation of the scattering of an electromagnetic wave by metallic nanoparticles. Such nanoparticles most often have curvilinear shapes, therefore we propose a numerical modeling strategy which combines the use of an unstructured tetrahedral mesh for the discretization of the scattering structures with a structured (uniform cartesian) mesh for treating the rest of the domain.

Partial views of the tetrahedral and hybrid hexahedral-tetrahedral meshes used for the simulation of the scattering of a plane wave by a single nanosphere

Simulation of the scattering of a plane wave by a single nanosphere. Module of the electric field in the Fourier domain: (left) Mie analytical solution / (middle) DGTD-P₂Q₂ result / (right) DGTD-P₂ result (the field distribution in the Au-sphere is hidden)

The L-shaped waveguide is formed of seven 50 nm diameter Au spheres in vacuum, with a 75 nm center-to-center spacing while the whole computational domain consists of a 550 nm × 750 nm × 400 nm parallelepipedic domain.

Fully tetrahedral mesh	Hybrid cubic/tetrahedral mesh
Fully tetrahedral mesh	Hybrid cubic/tetrahedral mesh method
DGTD-P2 method	DGTD-P2Q2 method

Characteristics of the tetrahedral and hybrid hexahedral-tetrahedral meshes used for simulation of the L-shaped waveguide. Elapsed time is for a physical time of 1 fs (total physical time is about 30 fs) on 8 CPU cores of an Intel Xeon 2.66 GHz node

Related publications

C. Durochat, S. Lanteri and C. Scheid
High order non-conforming multi-element Discontinuous Galerkin method for time domain electromagnetics
Appl. Math. Comput., Vol. 224, pp. 681–704 (2013)
Available as INRIA RR-8257 on Hyper Article Online

R. Léger, J. Viquerat, C. Durochat, C. Scheid and S. Lanteri
A parallel non-conforming multi-element DGTD method for the simulation of electromagnetic wave interaction with metallic nanoparticles
J. Comput. Appl. Math., Vol. 270, pp. 330–342 (2014)

C. Durochat
High order non-conforming multi-element discontinuous Galerkin method for time-domain electromagnetics
Doctoral thesis, University of Nice - Sophia Antipolis (2013)