Electromagnetic applications of a new finite-difference calculus

Tsukerman, Igor

doi:10.1109/tmag.2005.847637

Cited by 54 publications

(96 citation statements)

References 90 publications

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“…We now use the effective parameters displayed in figure 7 to compute the transmission and reflection coefficients for a finite structure consisting of L = 10 layers of elementary cells and to compare T/R for the equivalent homogeneous slab with the exact results, which were obtained by the finite difference method previously referred to as FD-FLAME [24,25,27,28]. In p-polarization, the magnetic field has only one component.…”

Section: (B) Example 2: Dispersive Layered Metal-dielectric Mediummentioning

confidence: 99%

A non-asymptotic homogenization theory for periodic electromagnetic structures

Tsukerman

Markel

2014

Proc. R. Soc. A.

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Homogenization of electromagnetic periodic composites is treated as a two-scale problem and solved by approximating the fields on both scales with eigenmodes that satisfy Maxwell's equations and boundary conditions as accurately as possible. Built into this homogenization methodology is an error indicator whose value characterizes the accuracy of homogenization. The proposed theory allows one to define not only bulk, but also position-dependent material parameters (e.g. in proximity to a physical boundary) and to quantify the trade-off between the accuracy of homogenization and its range of applicability to various illumination conditions.

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Section: (B) Example 2: Dispersive Layered Metal-dielectric Mediummentioning

confidence: 99%

A non-asymptotic homogenization theory for periodic electromagnetic structures

Tsukerman

Markel

2014

Proc. R. Soc. A.

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“…For previous work on the UWVF for Maxwell, we refer to [8,17,19,34]; for different Trefftz-based approaches, we mention [20,47]. Taking cue from the UWVF and following [34], we study a class of Trefftz methods that rely on a DG formulation of the electric field-based Maxwell problem, where the divergence-free constraint is not imposed; the discrete solutions will be elementwise divergence-free, but not globally.…”

Section: Introductionmentioning

confidence: 99%

Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations

2012

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Abstract. In this paper, we extend to the time-harmonic Maxwell equations the p-version analysis technique developed in [R. Hiptmair, A. Moiola and I. Perugia, Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version, SIAM J. Numer. Anal., 49 (2011), 264-284] for Trefftz-discontinuous Galerkin approximations of the Helmholtz problem. While error estimates in a mesh-skeleton norm are derived parallel to the Helmholtz case, the derivation of estimates in a mesh-independent norm requires new twists in the duality argument. The particular case where the local Trefftz approximation spaces are built of vector-valued plane wave functions is considered, and convergence rates are derived.

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“…Classical FD methods are easy to understand and simple to implement on digital computers, but they have poor numerical dispersion characteristics [3,4]. Recent advancements in highly accurate 2D FDFD algorithms [5][6][7][8][9] have overcome dispersion problems of the classical methods. Along these lines in 2010 Chang and Mu published work on the method of connected local fields (CLF, [10,11]).…”

Section: Introductionmentioning

confidence: 99%

Semi-Analytical Solutions of the 3-D Homogeneous Helmholtz Equation by the Method of Connected Local Fields

Chang

Mu²

2013

PIER

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Abstract-We advance the theory of the two-dimensional method of connected local fields (CLF) to the three-dimensional cases. CLF is suitable for obtaining semi-analytical solutions of Helmholtz equation. The fundamental building block (cell) of the 3-D CLF is a cube consisting of a central point and twenty six points on the cube's surface. These surface points form three symmetry groups: six on the planar faces, twelve on the edges and eight on the vertices (corners). The local field within the unit cell is expanded in a truncated spherical FourierBessel series. From this representation we develop a closed-form, 3D local field expansion (LFE) coefficients that relate the central point to its immediate neighbors. We also compute the CLF-based FD-FD numerical solutions of the 3D Green's function in free space. Compared with the analytic solution, we found that even at a low three points per wavelength spatial sampling, the accumulated phase errors of the CLF 3D Green's function after propagating a distance of ten wavelengths are well under ten percent.

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Electromagnetic applications of a new finite-difference calculus

Cited by 54 publications

References 90 publications

A non-asymptotic homogenization theory for periodic electromagnetic structures

A non-asymptotic homogenization theory for periodic electromagnetic structures

Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations

Semi-Analytical Solutions of the 3-D Homogeneous Helmholtz Equation by the Method of Connected Local Fields

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