A stable FDTD algorithm for non-diagonal, anisotropic dielectrics

Werner, Gregory R.; Cary, John R.

doi:10.1016/j.jcp.2007.05.008

Cited by 67 publications

(81 citation statements)

References 21 publications

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“…This extends our previous work for interfaces between isotropic media [3], combined with a recently proposed scheme for stable FDTD in anisotropic media [4], and replaces our previous heuristic proposal for anisotropic-material interfaces [5]. Ordinarily, the presence of discontinuous material interfaces degrades the accuracy of FDTD to first-order ͓O͑⌬x͔͒ from the usual second-order ͓O͑⌬x 2 ͔͒ accuracy [6], but our work demonstrates how an appropriate choice of subpixel smoothing can both restore second-order asymptotic accuracy and give the lowest errors compared with competing schemes even at modest resolutions.…”

supporting

confidence: 81%

“…1. One approach is to average the four adjacent D y values and use them in updating E x , along with ͑ xy ͒ −1 at the E x point [3,4]. This approach, however, is theoretically unstable and leads to divergences for a long simulation [4].…”

Section: ͑4͒mentioning

confidence: 99%

“…One approach is to average the four adjacent D y values and use them in updating E x , along with ͑ xy ͒ −1 at the E x point [3,4]. This approach, however, is theoretically unstable and leads to divergences for a long simulation [4]. Instead, a modified technique was recently shown to satisfy a necessary condition for stability with Hermitian [4]: as depicted in Fig.…”

Section: ͑4͒mentioning

confidence: 99%

“…Subpixel smoothing has an additional benefit: it allows the simulation to respond continuously to changes in the geometry, such as during optimization or parameter studies, rather than changing in discontinuous jumps as interfaces cross pixel boundaries. This technique additionally yields much smoother convergence of the error with resolution, which makes it easier to evaluate the accuracy and enables the possibility of extrapolation to gain another order of accuracy [4]. The ability to handle anisotropic materials is becoming increasingly important via the use of anisotropic materials to represent arbitrary coordinate transformations in Maxwell's equations [7], most prominently to design cloaking metamaterials [8].…”

mentioning

confidence: 99%

“…Advances in perturbation theory have enabled us to extend this scheme to interfaces between anisotropic materials, initially for a plane-wave method [2]. Here, we adapt the technique to FDTD, combined with a recent FDTD scheme with improved stability for anisotropic media [4]. Although this Letter focuses on the case of anisotropic electric permittivity , exactly the same smoothing and discretization schemes apply to magnetic permeabilities µ owing to the equivalence in Maxwell's equations under interchange of /µ and E / H.…”

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confidence: 99%

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Accurate finite-difference time-domain simulation of anisotropic media by subpixel smoothing

Oskooi¹,

Kottke²,

Johnson³

2009

Opt. Lett.

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Finite-difference time-domain methods suffer from reduced accuracy when discretizing discontinuous materials. We previously showed that accuracy can be significantly improved by using subpixel smoothing of the isotropic dielectric function, but only if the smoothing scheme is properly designed. Using recent developments in perturbation theory that were applied to spectral methods, we extend this idea to anisotropic media and demonstrate that the generalized smoothing consistently reduces the errors and even attains second-order convergence with resolution. , and replaces our previous heuristic proposal for anisotropic-material interfaces [5]. Ordinarily, the presence of discontinuous material interfaces degrades the accuracy of FDTD to first-order ͓O͑⌬x͔͒ from the usual second-order ͓O͑⌬x 2 ͔͒ accuracy [6], but our work demonstrates how an appropriate choice of subpixel smoothing can both restore second-order asymptotic accuracy and give the lowest errors compared with competing schemes even at modest resolutions. Subpixel smoothing has an additional benefit: it allows the simulation to respond continuously to changes in the geometry, such as during optimization or parameter studies, rather than changing in discontinuous jumps as interfaces cross pixel boundaries. This technique additionally yields much smoother convergence of the error with resolution, which makes it easier to evaluate the accuracy and enables the possibility of extrapolation to gain another order of accuracy [4]. The ability to handle anisotropic materials is becoming increasingly important via the use of anisotropic materials to represent arbitrary coordinate transformations in Maxwell's equations [7], most prominently to design cloaking metamaterials [8]. Our smoothing scheme requires preprocessing of the materials and does not otherwise modify the FDTD algorithm. It is therefore particularly simple to implement (free software is available [9]).Our basic approach, as described previously [2,3], is to smooth the structure to eliminate the discontinuity before discretizing, but because the smoothing itself changes the geometry we use first-order perturbation theory to select a smoothing with zero firstorder effect. For isotropic materials, this approach made rigorous a smoothing scheme that had previously been proposed heuristically [10][11][12] and explained its second-order accuracy [3]. Advances in perturbation theory have enabled us to extend this scheme to interfaces between anisotropic materials, initially for a plane-wave method [2]. Here, we adapt the technique to FDTD, combined with a recent FDTD scheme with improved stability for anisotropic media [4]. Although this Letter focuses on the case of anisotropic electric permittivity , exactly the same smoothing and discretization schemes apply to magnetic permeabilities µ owing to the equivalence in Maxwell's equations under interchange of /µ and E / H.We define an interface-relative coordinate frame as in Fig. 1, so that the first component "1" is the direction normal to the interface. ...

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confidence: 81%

Section: ͑4͒mentioning

confidence: 99%

Section: ͑4͒mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Accurate finite-difference time-domain simulation of anisotropic media by subpixel smoothing

Oskooi¹,

Kottke²,

Johnson³

2009

Opt. Lett.

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An overlapping Yee finite‐difference time‐domain method for material interfaces between anisotropic dielectrics and general dispersive or perfect electric conductor media

Liu

Brio

Moloney

2013

Int J Numerical Modelling

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SUMMARYA novel stable anisotropic finite‐difference time‐domain (FDTD) algorithm based on the overlapping cells is developed for solving Maxwell's equations of electrodynamics in anisotropic media with interfaces between different types of materials, such as the interface between anisotropic dielectrics and dispersive medium or perfect electric conductor (PEC). The previous proposed conventional anisotropic FDTD methods suffer from the late‐time instability due to the extrapolation of the field components near the material interface. The proposed anisotropic overlapping Yee FDTD method is stable, as it relies on the overlapping cells to provide the collocated field values without any interpolation or extrapolation. Our method has been applied to simulate electromagnetic invisibility cloaking devices with both anisotropic dielectrics and PEC included in the computational domain. Numerical results and eigenvalue analysis confirm that the conventional anisotropic FDTD method is weakly unstable, whereas our method is stable. Copyright © 2013 John Wiley & Sons, Ltd.

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Stable 3D FDTD method for arbitrary fully electric and magnetic anisotropic Maxwell's equations

Nehls

Dineen

Liu

et al. 2018

Int J Numerical Modelling

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We have developed a new fully anisotropic 3D FDTD Maxwell solver for arbitrary electrically and magnetically anisotropic media for piecewise constant electric and magnetic materials that are co-located over the primary computational cells. Two numerical methods were developed that are called non-averaged and averaged methods, respectively. The non-averaged method is first order accurate, while the averaged method is second order accurate for smoothly-varying materials and reduces to first order for discontinuous material distributions. For the standard FDTD field locations with the co-location of the electric and magnetic materials at the primary computational cells, the averaged method require development of the different inversion algorithms of the constitutive relations for the electric and magnetic fields. We provide a mathematically rigorous stability proof followed by extensive numerical testing that includes long-time integration, eigenvalue analysis, tests with extreme, randomly placed material parameters, and various boundary conditions. For accuracy evaluation we have constructed a test case with an explicit analytic solution. Using transformation optics, we have constructed complex, spatially inhomogeneous geometrical object with fully anisotropic materials and a large dynamic range of and µ, such that a plane wave incident on the object is perfectly reconstructed downstream. In our implementation, the considerable increase in accuracy of the averaged method only increases the computational run time by 20%.

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A stable FDTD algorithm for non-diagonal, anisotropic dielectrics

Cited by 67 publications

References 21 publications

Accurate finite-difference time-domain simulation of anisotropic media by subpixel smoothing

Accurate finite-difference time-domain simulation of anisotropic media by subpixel smoothing

An overlapping Yee finite‐difference time‐domain method for material interfaces between anisotropic dielectrics and general dispersive or perfect electric conductor media

Stable 3D FDTD method for arbitrary fully electric and magnetic anisotropic Maxwell's equations

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