A Dissipative Systems Theory for FDTD With Application to Stability Analysis and Subgridding

Bekmambetova, Fadime; Zhang, Xinyue; Triverio, Piero

doi:10.1109/tap.2016.2637867

Cited by 19 publications

(32 citation statements)

References 28 publications

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“…The definitions given in this section allow us to write the FDTD equations for a 3-D region in the compact matrix form (16a)-(16b). Remarkably, equations (16a)-(16b) have the same structure as in the 2-D case [13]. This fact allows us to generalize previous results valid in two dimensions to the most general 3-D case.…”

Section: Dynamical System Formulationsupporting

confidence: 64%

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A Dissipation Theory for Three-Dimensional FDTD With Application to Stability Analysis and Subgridding

Bekmambetova

Zhang

Triverio

2018

IEEE Trans. Antennas Propagat.

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The finite-difference time-domain (FDTD) algorithm is a popular numerical method for solving electromagnetic problems. FDTD simulations can suffer from instability due to the explicit nature of the method. Stability enforcement can be particularly challenging in scenarios where a setup is composed of multiple components, such as grids of different resolution, advanced boundary conditions, reduced-order models, and lumped elements. We propose a dissipation theory for 3-D FDTD inspired by the principle of energy conservation. We view the FDTD update equations for a 3-D region as a dynamical system, and show under which conditions the system is dissipative. By requiring each component of an FDTD-like scheme to be dissipative, the stability of the overall coupled scheme follows by construction. The proposed framework enables the creation of provably stable schemes in an easy and modular fashion, since conditions are imposed on the individual components, rather than on the overall coupled scheme as in existing approaches. With the proposed framework, we derive a new subgridding scheme with guaranteed stability, low reflections, support for material traverse and arbitrary (odd) grid refinement ratio.

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Section: Dynamical System Formulationsupporting

confidence: 64%

“…In this paper, we present a dissipation theory for modular stability enforcement of complex 3-D FDTD systems, generalizing previous work in two dimensions [13]. The method is based on the theory of dissipative dynamical systems [14].…”

Section: Introductionmentioning

confidence: 99%

A Dissipation Theory for Three-Dimensional FDTD With Application to Stability Analysis and Subgridding

Bekmambetova

Zhang

Triverio

2018

IEEE Trans. Antennas Propagat.

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“…The left domain is discretized with the uniform meshes with the mesh size ∆=4 × 10 −2 m, and the uniform meshes with the mesh size ∆=2 × 10 −2 m are used in the right domain. The time steps used in the simulations are 0.99 times of the CFL condition in the fine mesh domain defined in (25).…”

Section: A a Two-dimensional Cavity With Pec Boundariesmentioning

confidence: 99%

“…σ and ρ denote the specific conductance and density of the corresponding tissue respectively. According to [25], the surface integral is calculated as a measure of the accuracy…”

Section: The Sar Calculationmentioning

confidence: 99%

A Stable FDTD Subgridding Scheme with SBP-SAT for Transient Electromagnetic Analysis

Cheng,

Wang,

Liu

et al. 2021

Preprint

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We proposed a provably stable FDTD subgridding method for accurate and efficient transient electromagnetic analysis. In the proposed method, several field components are properly added to the boundaries of Yee's grid to make sure that the discrete operators meet the summation-by-parts (SBP) property. Then, by incorporating the simultaneous approximation terms (SATs) into the finite-difference time-domain (FDTD) method, the proposed FDTD subgridding method mimics the energy estimate of the continuous Maxwell's equations at the semi-discrete level to guarantee its stability. Further, to couple multiple mesh blocks with different mesh sizes, the interpolation matrices are also derived. The proposed FDTD subgridding method is accurate, efficient, easy to implement and be integrated into the existing FDTD codes with only simple modifications. At last, three numerical examples with fine structures are carried out to validate the effectiveness of the proposed method.

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“…E zf1 and E zf2 , it is difficult to recover the absent H w via simple interpolation/extrapolation approaches. We adopt the strategy in Bekmambetova et al (2017), reading:…”

Section: Field Projection Between the Subgrid And The Main Gridmentioning

confidence: 99%

A hybrid Krylov-subspace-exponential and finite-difference time integration approach for multiscale electromagnetic simulations

Wang

Chen

Zhang

et al. 2017

COMPEL

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Purpose This paper aims to present a novel hybrid time integration approach for efficient numerical simulations of multiscale problems involving interactions of electromagnetic fields with fine structures. Design/methodology/approach The entire computational domain is discretized with a coarse grid and a locally refined subgrid containing the tiny objects. On the coarse grid, the time integration of Maxwell’s equations is realized by the conventional finite-difference technique, while on the subgrid, the unconditionally stable Krylov-subspace-exponential method is adopted to breakthrough the Courant–Friedrichs–Lewy stability condition. Findings It is shown that in contrast with the conventional finite-difference time-domain method, the proposed approach significantly reduces the memory costs and computation time while providing comparative results. Originality/value An efficient hybrid time integration approach for numerical simulations of multiscale electromagnetic problems is presented.

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A Dissipative Systems Theory for FDTD With Application to Stability Analysis and Subgridding

Cited by 19 publications

References 28 publications

A Dissipation Theory for Three-Dimensional FDTD With Application to Stability Analysis and Subgridding

A Dissipation Theory for Three-Dimensional FDTD With Application to Stability Analysis and Subgridding

A Stable FDTD Subgridding Scheme with SBP-SAT for Transient Electromagnetic Analysis

A hybrid Krylov-subspace-exponential and finite-difference time integration approach for multiscale electromagnetic simulations

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