Hybrid cubic/tetrahedral DGTD method

High order non-conforming multi-element discontinuous Galerkin method for time-domain electromagnetics\\

Doctoral thesis, University of Nice - Sophia Antipolis (2013)

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

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.

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)

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.

(:title Hybrid cubic/tetrahedral DGTD method:)

PhD disseration, University of Nice - Sophia Antipolis (2013)

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Available as INRIA RR-8257 on Hyper Article Online

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 PhD disseration

Related publications

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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)\\\

Related publications

C. Durochat, S. Lanteri and C. Scheid\\

Available as INRIA RR-8257 on Hyper Article Online

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

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Related publications

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 a Intel Xeon 2.66 GHz node

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(:cell align='center':) Elapsed time

(:cell align='center':) 11420 s

(:cell align='center':) 5680 s

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

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(:cellnr align='left':) DGTD-P₂ method

(:cellnr align='left':) DGTD-P₂Q₂ method

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(:cell align='center':) # vertices (:cell align='center':) # tetrahedra (:cell align='center':) # hexahedra (:cell align='center':) # DOF (:cellnr align='center':) DGTD-P₂ method (:cell align='center':) 222,175 (:cell align='center':) 1,306,356 (:cell align='center':) 0 (:cell align='center':) 13,063,560 (:cellnr align='center':) DGTD-P₂Q₂ method (:cell align='center':) 211,214 (:cell align='center':) 706,012 (:cell align='center':) 81,280 (:cell align='center':) 9,264,660

(:cellnr align='center':) (:cell align='center':)# vertices (:cell align='center':)# tetrahedra (:cell align='center':)# hexahedra (:cell align='center':)# DOF (:cellnr align='center':)DGTD-P₂ method (:cell align='center':)222,175 (:cell align='center':)1,306,356 (:cell align='center':)0 (:cell align='center':)13,063,560 (:cellnr align='center':)DGTD-P₂Q₂ method (:cell align='center':)211,214 (:cell align='center':)706,012 (:cell align='center':)81,280 (:cell align='center':)9,264,660

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(:cell:)# vertices

(:cell:) # vertices

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(:cell:)# vertices (:cell:)# tetrahedra (:cell:)# hexahedra (:cell:)# DOF (:cellnr:)DGTD-P₂ method (:cell:)222,175 (:cell:)1,306,356 (:cell:)0 (:cell:)13,063,560 (:cellnr:)DGTD-P₂Q₂ method (:cell:)211,214 (:cell:)706,012 (:cell:)81,280 (:cell:)9,264,660

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Characteristics of the tetrahedral and hybrid hexahedral-tetrahedral meshes used for simulation of the L-shaped waveguide

(:table border='1' align='center' width='80%':)

(:cell:) # vertices (:cell:) # tetrahedra (:cell:) # hexahedra (:cell:) # DOF (:cellnr:) DGTD-P₂ method (:cell:) 222,175 (:cell:) 1,306,356 (:cell:) 0 (:cell:) 13,063,560 (:cellnr:) DGTD-P₂Q₂ method (:cell:) 211,214 (:cell:) 706,012 (:cell:) 81,280 (:cell:) 9,264,660

(:table border='1' align='center' width='80%':)

(:cell:)# vertices (:cell:)# tetrahedra (:cell:)# hexahedra (:cell:)# DOF (:cellnr:)DGTD-P₂ method (:cell:)222,175 (:cell:)1,306,356 (:cell:)0 (:cell:)13,063,560

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(:table now border=1 align='center' width=80%:) (:cellnr:) (:cell:) # vertices (:cell:) # tetrahedra (:cell:) # hexahedra (:cell:) # DOF (:cellnr:) DGTD-P₂ method (:cell:) 222,175 (:cell:) 1,306,356 (:cell:) 0 (:cell:) 13,063,560 (:tableend:)

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(:table now border=1 width=80%:) (:cellnr:) (:cell:) (:cell:) (:cellnr:) (:cell:) (:cell:) (:tableend:)

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

(:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_L-guide/Lguide_X_g1_R.png (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_L-guide/Lguide_X_g1_L.png

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 Au-sphere is hidden)

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(:table border='0' width='100%' align='center' cellspacing='0px':) (:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_L-guide/mie_final.png (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_L-guide/hyb_final.png (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_L-guide/tet_final.png (:tableend:)

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

(:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_L-guide/nanosphere_tet.png (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_L-guide/nanosphere_hyb.png (:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_L-guide/nanosphere_tet-zoom.png (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_L-guide/nanosphere_hyb-zoom.png

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(:table border='0' width='100%' align='center' cellspacing='0px':) (:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_L-guide/nanosphere_tet.png (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_L-guide/nanosphere_hyb.png (:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_L-guide/nanosphere_tet-zoom.pn (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_L-guide/nanosphere_hyb-zoom.pn (:tableend:)

Left: type of non-conformity considered in 3D, between a hexahedron (q2) and two tetrahedra (t1 and t2). - Middle: 2D view of the non-conforming hybrid face between q2 and, t1 and t2. - Right: refined example (2D view only) of non-conformity between one hexahedron and eight tetrahedra.

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(:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_L-guide/non_conf_2D.png (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_L-guide/non_conf_dec_3D.png

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(:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_L-guide/non_conf_3D.png (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_L-guide/non_conf_2D.png (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_L-guide/non_conf_dec_3D.png

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. The emphasis of this work is on increasing the flexibility in the meshing process of nanophotonic configurations while decreasing the needs in computational resources for the

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(:table border='0' width='100%' align='center' cellspacing='0px':) (:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_L-guide/non_conf_3D.png (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_L-guide/non_conf_2D.png (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_L-guide/non_conf_dec_3D.png (:tableend:)

(:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_L-guide/Lguide_X_g1_R.png (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_L-guide/Lguide_X_g1_L.png

(:cellnr align='center':) Fully tetrahedral mesh (:cell align='center':) Hybrid cubic/tetrahedral mesh method (:cellnr align='center':) DGTD-P₂ method (:cell align='center':) DGTD-P₂Q₂ method

(:cell align='center':) Hybrid cubic/tetrahedral mesh

(:cell align='center':) Hybrid cubic/tetrahedral mesh - DGTD-P₂Q₂ method

(:cellnr align='center':) Fully tetrahedral mesh (:cell align='center':) Hybrid cubic/tetrahedral meshmethod

(:cellnr align='center':) Fully tetrahedral mesh - DGTD-P₂ method (:cell align='center':) Hybrid cubic/tetrahedral mesh - DGTD-P₂Q₂

(:cellnr align='center':) Fully tetrahedral mesh - DGTD-P₂ method (:cell align='center':) Hybrid cubic/tetrahedral mesh - DGTD-P₂Q₂ method

(:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_L-guide/Lguide_X_g1_R.png (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_L-guide/Lguide_X_g1_L.png

(:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_L-guide/Lguide_X_g1_R.png

(:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_L-guide/Lguide_X_g1_R.jpg

(:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_L-guide/Lguide_X_1_short.png (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_L-guide/Lguide_X_g1_L.jpg

(:cellnr align='center':) Fully tetrahedral mesh (:cell align='center':) Hybrid cubic/tetrahedral mesh (:tableend:)

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(:table border='0' width='100%' align='center' cellspacing='0px':) (:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_L-guide/Lguide_X_1_short.png (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_L-guide/Lguide_X_g1_L.jpg

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(:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_L-guide/Lguide_tet_mesh.png (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_L-guide/Lguide_hyb_mesh.png

(:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_L-guide/Lguide_tet_mesh.png (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_L-guide/Lguide_hyb_mesh.png

(:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_L-guide/Lguide_tet_mesh.png (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_L-guide/Lguide_hyb_mesh.png

(:cell align='center':) Hybrid cubic/tetrahedral mesh

This study is concerned with the development of a parallel non-conforming multi-element DGTD method for the solution of the 3D time-domain Maxwell equations coupled to a Drude dispersion model for metals at frequencies relevant to nanophotonic applications, and in particular 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. The emphasis of this work is on increasing the flexibility in the meshing process of nanophotonic configurations while decreasing the needs in computational resources for the target applications.

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(:table border='0' width='100%' align='center' cellspacing='1px':) (:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_L-guide/Lguide_tet_mesh.png (:cellnr align='center':) Fully tetrahedral mesh (:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_L-guide/Lguide_hyb_mesh.png (:cellnr align='center':) Hybrid cubic/tetrahedral mesh (:tableend:)

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This study is concerned with the development of a parallel non-conforming multi-element DGTD method for the solution of the 3D time-domain Maxwell equations coupled to a Drude dispersion model for metals at frequencies relevant to nanophotonic applications, and in particular 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. The emphasis of this work is on increasing the flexibility in the meshing process of nanophotonic configurations while decreasing the needs in computational resources for the target applications.

This study development of a parallel non-conforming multi-element DGTD method for the solution of the 3D time-domain Maxwell equations coupled to a Drude dispersion model for metals at frequencies relevant to nanophotonic applications, and in particular 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. The emphasis of this work is on increasing the flexibility in the meshing process of nanophotonic configurations while decreasing the needs in computational resources for the target applications.