DGTD method on curvilinear tetrahedral meshes

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http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_sphere-mesh.jpg |

http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_sphere-mesh.jpg |

Scattering cross-section of a gold nanosphere obtained with P₂ and P₃ interpolation of the EM field components, using affine (linear) and curvilinear meshes with various refinement levels

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J. Viquerat and C. Scheid
A 3D curvilinear discontinuous Galerkin time-domain solver for nanoscale light–matter interactions
J. Comput. Appl. Math., Vol. 289, pp. 35-70 (2015)\\

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Related publications

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(:title DGTD method on curvilinear tetrahedral meshes:)

(:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_dimer-cross.png

Classical finite element methods rely on tessellations composed of straight-edged elements mapped linearly from a reference element on domains which physical boundaries and interfaces are indifferently straight or curved. This geometrical approximation represents a serious hindrance for high order methods, since it limits the accuracy of the spatial discretization to second order. Thus, exploiting an enhanced representation of the geometrical features of a given electromagnetic wave propagation problem is an important issue in the design of high order methods, such as the Discontinuous Galerkin Time-Domain (DGTD) method. In this study, we exploit a high order mapping for tetrahedra, and focus on specific nanophotonics setups to assess numerically the gains in terms of accuracy and performance.

Classical finite element methods rely on tessellations composed of straight-edged elements mapped linearly from a reference element on domains which physical boundaries and interfaces are indifferently straight or curved. This geometrical approximation represents a serious hindrance for high order methods, since it limits the accuracy of the spatial discretization to second order. Thus, exploiting an enhanced representation of the geometrical features of a given electromagnetic wave propagation problem is an important issue in the design of high order methods, such as the Discontinuous Galerkin Time-Domain (DGTD) method. In this study, we exploit a high order mapping for tetrahedra, and focus on specific nanophotonics setups to assess numerically the impact of this modeling improvement in terms of accuracy and performance.

(:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_dimer-cross.png

(:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_sphere-cross-lin.png

(:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_sphere-cross-cur.png

(:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_dimer-cross.png (:cellnr align='center':) Absorption cross-section of a gold nanosphere dimer obtained with P₄ approximation (:cellnr align='center':) using affine and curvilinear meshes

(:table border='0' width='100%' align='center' cellspacing='1px':) (:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_dimer-cross.png (:cellnr align='center':) Absorption cross-section of a gold nanosphere dimer obtained with P₄ approximation using affine and curvilinear meshes (:tableend:)

(:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_sphere-cross-cur.png (:cellnr align='center':) DGTD method with curvilinear elements

http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_dimer-cross.png | Absorption cross-section of a gold nanosphere dimer obtained with P₄ approximation using affine and curvilinear meshes.

(:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_sphere-cross-lin.png (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_sphere-cross-cur.png

http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_dimer-cross.png | Absorption cross-section

http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_dimer-cross.png Absorption cross-section of a gold nanosphere dimer obtained with P₄ approximation using affine and curvilinear meshes.

(:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_sphere-cross-lin.png (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_sphere-cross-cur.png

http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_dimer-cross.png | Absorption cross-section \\ of a gold nanosphere dimer
obtained with P₄ approximation using affine and curvilinear meshes.

http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_dimer-cross.png | Absorption cross-section of a gold nanosphere dimer obtained with P₄ approximation using affine and curvilinear meshes.

http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_dimer-cross.jpg | Absorption cross-section of a gold nanosphere dimer obtained with P₄ approximation using affine and curvilinear meshes.

http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_dimer-cross.jpg | Absorption cross-section of a gold nanosphere dimer obtained with P₄ approximation using affine and curvilinear meshes.

http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_sphere-mesh.jpg |

http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_dimer-cross.jpg | Absorption cross-section of a gold nanosphere dimer obtained with P₄ approximation using affine and curvilinear meshes.

Near-field visualization of the electric field Fourier transform for a gold nanosphere dimer

http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_dimer-cross.jpg

Plasmonic coupling between nanoparticles is at the heart of many applications in nano-optics. We illustrate the coupling of two identical gold nanospheres which are aligned along the polarization direction of the incident field, and the surface-to-surface distance is set to 4 nm. In this configuration, the coupled plasmon resonance induces very intense fields in the gap between the particles. Then, a proper near-field resolution is essential to a good understanding of the properties of such coupled structures. Here, we use P₄ polynomial approximation with upwind fluxes and a fourth order low storage Runge Kutta time scheme (LSRK4).

Tetrahedral mesh for plasmonic resonance of a gold nanosphere with radius 50 nm. The scatterer (in red) is enclosed by the total field (TF) region (in blue), delimited by the TF/SF interface on which the incident field is imposed. Then we find the scattered field (SF) region (in purple), surrounded by UPMLs (in gray).

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Plasmonic coupling between nanoparticles is at the heart of many applications in nano-optics. We consider the coupling of two identical gold nanospheres which are aligned along the polarization direction of the incident field, and the surface-to-surface distance is set to 4 nm. In this configuration, the coupled plasmon resonance induces very intense fields in the gap between the particles. Then, a proper near-field resolution is essential to a good understanding of the properties of such coupled structures. To properly account for the high intensity of the fields, we use P4 polynomial approximation with upwind fluxes and a low storage Runge Kutta time-scheme of order 4 (LSRK4).

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Near-field visualization of the electric field Fourier transform for a gold nanosphere dimer

(:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_dimer-curP4.png

(:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_dimer-linP4.png (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_dimer-curP4.png

(:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_dimer-linP4.png (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_dimer-curP4.png

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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_sphere/nano_sphere-cross-lin.jpg (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_sphere-cross-cur.jp (:cellnr align='center':) DGTD method with affine elements (:cell align='center':) DGTD method with curvilinear elements (:tableend:)

Scattering cross-section of a gold nanosphere obtained with P₂ and P₃ interpolation of the EM filed compenents, using affine (linear) and curvilinear meshes with various refinement levels

Scattering cross-section of a gold nanosphere obtained with P₂ and P₃ interpolation of the EM filed compenents, using affine (linear) and curvilinear meshes with various refinement levels.

Scattering cross-section of a gold nanosphere

(:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_sphere-cross-lin.jpg (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_sphere-cross-cur.jpg

(:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_sphere-cross-lin.jpg (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_sphere-cross-cur.jpg

http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_sphere-mesh.jpg Tetrahedral mesh for plasmonic resonance of a gold nanosphere. The scatterer (in red) is enclosed by the total field (TF) region (in blue), delimited by the TF/SF interface on which the incident field is imposed. Then we find the scattered field (SF) region (in purple), surrounded by UPMLs (in gray).

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(:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_sphere-cross-lin.jpg

(:cellnr align='center':) DGTD method with affine elements (:cell align='center':) DGTD method with curvilinear elements

(:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_sphere-cross-lin.jpg (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_sphere-cross-cur.jpg

(:cell align='center':)Non-local model, modulus of Ex component

(:cellnr align='center':)% http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_sphere-cross-lin.jpg

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(:table border='0' width='100%' align='center' cellspacing='1px':)

(:table border='0' width='50%' align='center' cellspacing='1px':) (:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_sphere-cross-lin.jpg (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_sphere-cross-cur.jpg (:cellnr align='center':)Local model modulus of Ex component (:cell align='center':)Non-local model, modulus of Ex component

(:table border='0' width='500px%' align='center' cellspacing='1px':) (:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_sphere-cross-lin.jpg (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_sphere-cross-cur.jpg (:cellnr align='center':) Local model modulus of Ex component (:cell align='center':) Non-local model, modulus of Ex component (:tableend:)

(:table border='0' width='500px%' align='center' cellspacing='1px':)

(:table border='0' width='250px%' align='center' cellspacing='1px':)

(:table border='0' width='250%' align='center' cellspacing='1px':)

Tetrahedral mesh for plasmonic resonance of a gold nanosphere. The scatterer (in red) is enclosed by the total field (TF) region (in blue), delimited by the TF/SF interface on which the incident field is imposed. Then we find the scattered field (SF) region (in purple), surrounded by UPMLs (in gray).

Tetrahedral mesh for plasmonic resonance of a gold nanosphere. The scatterer (in red) is enclosed by the total field (TF) region (in blue), delimited by the TF/SF interface on which the incident field is imposed. Then we find the scattered field (SF) region (in purple), surrounded by UPMLs (in gray). \\

(:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_sphere-cross-lin.jpg (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_sphere-cross-cur.jpg (:cellnr align='center':) Local model modulus of Ex component

Classical finite element methods rely on tessellations composed of straight-edged elements mapped linearly from a reference element on domains which physical boundaries and interfaces are indifferently straight or curved. This geometrical approximation represents a serious hindrance for high order methods, since it limits the accuracy of the spatial discretization to second order. Thus, exploiting an enhanced representation of the geometrical features of a given electromagnetic wave propagation problem is an important issue in the design of high order methods, such as the Discontinuous Galerkin Time-Domain (DGTD) method. In this study, we exploit a high order mapping for tetrahedra, and focus on specific nanophotonics setups to assess the gains of in terms of accuracy and performance.

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(:table border='0' width='50%' align='center' cellspacing='1px':) (:cellnr align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_sphere-lin.jpg (:cell align='center':) http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_sphere-cur.jpg (:cellnr align='center':) Local model modulus of Ex component (:cell align='center':) Non-local model, modulus of Ex component (:tableend:)

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Classical finite element methods rely on tessellations composed of straight-edged elements mapped linearly from a reference element, on domains which physical boundaries and interfaces are indifferently straight or curved. This approximation represents serious hindrance for high-order methods, since it limits the accuracy of the spatial discretization to second order. Thus, exploiting an enhanced representation of the geometrical features of a given electromagnetic wave propagation problem is an important issue in the design of high-order methods, such as the Discontinuous Galerkin Time-Domain (DGTD) method. In the latter framework, we propose and validate an implementation of a high-order mapping for tetrahedra, and then focus on specific nanophotonics setups to assess the gains of the method in terms of memory and performances.

Classical finite element methods rely on tessellations composed of straight-edged elements mapped linearly from a reference element, on domains which physical boundaries and interfaces are indifferently straight or curved. This approximation represents serious hindrance for high-order methods, since they limit the precision of the spatial discretization to second order. Thus, exploiting an enhanced representation of the physical geometry of a considered problem is in agreement with the natural procedure of high-order methods, such as the discontinuous Galerkin method. In the latter framework, we propose and validate an implementation of a high-order mapping for tetrahedra, and then focus on specific nanophotonics setups to assess the gains of the method in terms of memory and performances.

http://www-sop.inria.fr/nachos/pics/results/nano_sphere/nano_sphere-mesh.jpg Tetrahedral mesh for plasmonic resonance of a gold nanosphere. The scatterer (in red) is enclosed by the total field (TF) region (in blue), delimited by the TF/SF interface on which the incident field is imposed. Then we find the scattered field (SF) region (in purple), surrounded by UPMLs (in gray).