Classical Finite-difference Time-domain Research Articles

An Unstructured Mesh Coordinate Transformation‐Based FDTD Method

ABSTRACTWe propose a novel unstructured mesh finite‐difference time‐domain (FDTD) method for solving electromagnetics problems with complicated geometries. The method, which solves the TE‐mode reduced form of Maxwell's equations, can handle both material interfaces and anisotropic material. Using the transformation optics principle, which describes how fields and material tensors change under coordinate transformations, we locally transform each cell in the mesh to a reference unit‐square computational domain where the usual FDTD update is performed. This comes at a cost: employing unstructured grids and coordinate transformations requires more complicated data structures, a mesh orientation process, and potentially introduces an anisotropic material tensor at every mesh cell. Nonetheless, we find that the method maintains the same desirable properties of the classic FDTD method (explicit, divergence‐free B‐field, nondissapative, and second‐order accuracy) while also gaining conforming material interfaces and boundaries in complicated geometry. Even further, we prove that the method is stable under a Courant condition and a fairly nonrestrictive mesh condition, hence defeating the late‐time stability issue plaguing prior nonorthogonal FDTD methods. To verify the method, we conduct convergence studies on three electromagnetic cavity problems with known exact solutions. For these numerical studies, we find that the method maintains second‐order convergence and stability.

Analysis of Moving Bodies with a Direct Finite Difference Time Domain Method

This paper proposes an original and thorough analysis of the behavior of electromagnetic waves in the presence of moving bodies by using the finite difference time domain (FDTD) method. Movements are implemented by changing positions of the objects at each time step, through the classical FDTD time loop. This technique is suitable for non-relativistic speeds, thus for most encountered problems in antennas and propagation domain. The numerical aspects that need to be considered are studied. Then, different bodies in motion are examined: plane wave source with matching resistors, observation point, inclined partially reflecting surface (PRS), line source, and metallic cylinder illuminated by a plane wave. The results are compared with those of special relativity which are considered as the references. Some aspects of special relativity are present in the direct FDTD approach, such as the independence of the velocity of electromagnetic wave propagation with the speed of the source and Lorentz local time (with a different physical interpretation). It is shown that the amplitude of the electric field for a moving plane wave source does not increase with the speed of motion, if the impedance of the source is small. Moreover, for a moving scattering metallic wire, one can observe a phenomenon similar to shock waves.

Analysis of the oblique incidence of periodic structures in a sound field by the finite-difference time-domain method

A method to analyze the acoustic characteristics of periodic structures is investigated. When a continuous boundary condition is introduced, the field in which a plane wave is normally incident on the periodic structure can be easily analyzed by the finite-difference time-domain (FDTD) method. However, when a plane wave source is obliquely incident, field mapping with auxiliary variables is necessary to remove the phase shift between unit cells. Here, a technique originally proposed for an electromagnetic field is applied to a sound field. The effects of the geometric parameters of the periodic structures on the sound pressure distribution and transmission loss in the mapped sound field are shown. This technique is validated by comparing the results with those in a field solved with the classical FDTD scheme. The dispersion relation and the stability condition for the time interval of the FDTD method for each incident angle of the plane wave in the mapped field are studied.

Novel Time-Domain Electromagnetic Simulation Using Triangular Meshes by Applying Space Curvature

A novel stable, accurate and fast numerical method based on triangular meshing and space transformations for electrodynamics problems with arbitrary boundaries is developed. Using transformation optics it is possible to convert non uniform domains into uniform rectangular ones allowing Maxwell’s Equations to be solved quickly and simply using a modified finite difference time domain (FDTD) algorithm. A novel averaging method is presented to ensure the stability and accuracy of our algorithm when dealing with anisotropic and inhomogeneous materials in the transformed space. Finally we demonstrate the advantages of the present technique compared with conventional FDTD and transformation optics FDTD through a couple of scattering problems. These show that this method is more than one thousand times faster than classical FDTD in the presence of curved boundaries.

Unconditionally stable FDTD algorithm for 3-D electromagnetic simulation of nonlinear media.

For the first time, an unconditionally stable finite-difference time-domain (FDTD) method for 3-D simulation of dispersive nonlinear media is presented. By applying a new adopted alternating-direction implicit (ADI) time-splitting scheme and the auxiliary differential equation (ADE) technique, the time-step in the FDTD simulations can be increased much beyond the Courant-Friedrichs-Lewy (CFL) stability limit. Thus, in comparison to the classical nonlinear FDTD method, the computational time for the proposed approach is decreased significantly while maintaining a reasonable level of accuracy. Numerical examples are presented to demonstrate the validity, stability, accuracy and computational efficiency of the proposed method.

Von Neumann stability analysis of globally constraint-preserving DGTD and PNPM schemes for the Maxwell equations using multidimensional Riemann solvers

The time-dependent equations of computational electrodynamics (CED) are evolved consistent with the divergence constraints on the electric displacement and magnetic induction vector fields. Respecting these constraints has proved to be very useful in the classic finite-difference time-domain (FDTD) schemes. As a result, there has been a recent effort to design finite volume time domain (FVTD) and discontinuous Galerkin time domain (DGTD) schemes that satisfy the same constraints and, nevertheless, draw on recent advances in higher order Godunov methods. This paper catalogues the first step in the design of globally constraint-preserving DGTD schemes. The algorithms presented here are based on a novel DG-like method that is applied to a Yee-type staggering of the electromagnetic field variables in the faces of the mesh. The other two novel building blocks of the method include constraint-preserving reconstruction of the electromagnetic fields and multidimensional Riemann solvers; both of which have been developed in recent years by the first author.The resulting DGTD scheme is linear, at least when limiters are not applied to the DG scheme. As a result, it is possible to carry out a von Neumann stability analysis of the entire suite of DGTD schemes for CED at orders of accuracy ranging from second to fourth. The analysis requires some simplifications in order to make it analytically tractable, however, it proves to be extremely instructive. A von Neumann stability analysis is a necessary precursor to the design of a full DGTD scheme for CED. It gives us the maximal CFL numbers that can be sustained by the DGTD schemes presented here at all orders. It also enables us to understand the wave propagation characteristics of the schemes in various directions on a Cartesian mesh. We find that constraint-preserving DGTD schemes permit CFL numbers that are competitive with conventional DG schemes. However, like conventional DG schemes, the CFL of DGTD schemes decreases with increasing order. To counteract that, we also present constraint-preserving PNPM schemes for CED. We find that the third and fourth order constraint-preserving DGTD and P1PM schemes have some extremely attractive properties when it comes to low-dispersion, low-dissipation propagation of electromagnetic waves in multidimensions. Numerical accuracy tests are also provided to support the von Neumann stability analysis. We expect these methods to play a role in those problems of engineering CED where exceptional precision must be achieved at any cost.

A Hybrid Algorithm Based on FDTD and HIE-FDTD Methods for Simulating Shielding Enclosure

A novel hybrid technique based on the classical finite-difference time-domain (FDTD) algorithm and hybrid implicit explicit finite-difference time-domain (HIE-FDTD) algorithm is developed to simulate the complex objects, which have both fine and coarse structures. The classical FDTD method is utilized to update the field components in coarse main grids, and the HIE-FDTD is applied in the fine local grids. Through controlling the classical FDTD and the HIE-FDTD calculation regions stringently, this method employs the uniform marching time-step size in the entire computational space. This time step size satisfies the Courant–Friedrich–Levy condition of the classical FDTD in coarse grids. Therefore, without an excessive increase of the computational burden, the computation efficiency of the proposed method is increased. Practical applications of the hybrid algorithm in the analysis of the shielding effectiveness of the enclosure are presented, and the simulation results certify the high accuracy and efficiency of the novel scheme.

First-principles spectra of Au nanoparticles: from quantum to classical absorption

ABSTRACTAbsorption cross-section spectra for gold nanoparticles were calculated using fully quantum Stochastic Density Functional Theory and a classical Finite-Difference Time Domain Maxwell solver. Spectral shifts were monitored as a function of size (1.3–) and shape (octahedron, cubeoctahedron and truncated cube). Even though the classical approach is forced to fit the quantum time-dependent density functional theory at , at smaller sizes there is a significant deviation as the classical theory is unable to account for peak splitting and spectral blueshifts even after quantum spectral corrections. We attribute the failure of classical methods at predicting these features to quantum effects and low density of states in small nanoparticles. Classically, plasmon resonances are modelled as collective conduction electron excitations, but at small nanoparticle size these excitations transition to few or even individual conductive electron excitations, as indicated by our results.

Computational electrodynamics in material media with constraint-preservation, multidimensional Riemann solvers and sub-cell resolution – Part I, second-order FVTD schemes

While classic finite-difference time-domain (FDTD) solutions of Maxwell's equations have served the computational electrodynamics (CED) community very well, formulations based on Godunov methodology have begun to show advantages. We argue that the formulations presented so far are such that FDTD schemes and Godunov-based schemes each have their own unique advantages. However, there is currently not a single formulation that systematically integrates the strengths of both these major strains of development. While an early glimpse of such a formulation was offered in Balsara et al. [16], that paper focused on electrodynamics in plasma. Here, we present a synthesis that integrates the strengths of both FDTD and Godunov-based schemes into a robust single formulation for CED in material media.Three advances make this synthesis possible. First, from the FDTD method, we retain (but somewhat modify) a spatial staggering strategy for the primal variables. This provides a beneficial constraint preservation for the electric displacement and magnetic induction vector fields via reconstruction methods that were initially developed in some of the first author's papers for numerical magnetohydrodynamics (MHD). Second, from the Godunov method, we retain the idea of upwinding, except that this idea, too, has to be significantly modified to use the multi-dimensionally upwinded Riemann solvers developed by the first author. Third, we draw upon recent advances in arbitrary derivatives in space and time (ADER) time-stepping by the first author and his colleagues. We use the ADER predictor step to endow our method with sub-cell resolving capabilities so that the method can be stiffly stable and resolve significant sub-cell variation in the material properties within a zone.Overall, in this paper, we report a new scheme for numerically solving Maxwell's equations in material media, with special attention paid to a second-order-accurate formulation. Several numerical examples are presented to show that the proposed technique works. Because of its sub-cell resolving ability, the new method retains second-order accuracy even when material permeability and permittivity vary by an order-of-magnitude over just one or two zones. Furthermore, because the new method is also unconditionally stable in the presence of stiff source terms (i.e., in problems involving giant conductivity variations), it can handle several orders-of-magnitude variation in material conductivity over just one or two zones without any reduction of the time–step. Consequently, the CFL depends only on the propagation speed of light in the medium being studied.

A POD reduced-order finite difference time-domain extrapolating scheme for the 2D Maxwell equations in a lossy medium

This paper is concerned with establishing a reduced-order finite difference time-domain (FDTD) extrapolating scheme based on proper orthogonal decomposition (POD) method for the two-dimensional (2D) Maxwell equations in a lossy medium. For this purpose, we first review the classical FDTD scheme and associated results for the 2D Maxwell equations in a lossy medium. And then, we formulate three sets of POD bases by means of the solutions obtained from the classical FDTD scheme on the very short time interval, establish the POD reduced-order FDTD extrapolating scheme, and furnish the error estimates among the POD reduced-order FDTD solutions and the classical FDTD solutions as well as the accuracy solution for the 2D Maxwell equations in a lossy medium. Finally, we enumerate two numerical examples to confirm that the POD reduced-order FDTD extrapolating scheme is feasible and efficient for simulating the phenomena for the electric and magnetic fields in a lossy medium.

Efficient Approach to Accurately Analyze Wireless Propagation and Fading in Time Domain

To more accurately and efficiently analyze wireless propagation and fading in urban picocell, a new time-domain approach is developed by the conformal technique of time-domain finite-integral theory. In urban simulations, the practical buildings are always approximated as perfect electric conductors (PECs). For this reason, the cell matrix of classic finite-difference time-domain (FDTD) method is mostly filled with PEC full-fill cells. As electromagnetic fields are valueless in PEC full-fill cells, FDTD has been regarded as an extremely inefficient way in wireless predictions. To improve the efficiency and reduce the number of unknowns, these valueless elements are accurately recognized and eliminated by our new scheme. Meanwhile, a hybrid data container is presented to efficiently store the discontinuous valuable elements after eliminating. Moreover, a parallel technique is applied to further improve the simulation capability. Based on self-developed urban simulation tool, some important subjects in wireless communications have been discussed.

A Monte Carlo method for solving the one-dimensional telegraph equations with boundary conditions

A Monte Carlo algorithm is derived to solve the one-dimensional telegraph equations in a bounded domain subject to resistive and non-resistive boundary conditions. The proposed numerical scheme is more efficient than the classical Kac's theory because it does not require the discretization of time. The algorithm has been validated by comparing the results obtained with theory and the Finite-difference time domain (FDTD) method for a typical two-wire transmission line terminated at both ends with general boundary conditions. We have also tested transmission line heterogeneities to account for wave propagation in multiple media. The algorithm is inherently parallel, since it is based on Monte Carlo simulations, and does not suffer from the numerical dispersion and dissipation issues that arise in finite difference-based numerical schemes on a lossy medium. This allowed us to develop an efficient numerical method, capable of outperforming the classical FDTD method for large scale problems and high frequency signals.

Spectral Deferred Correction Method for FDTD

Finite difference time domain (FDTD) is widely used to compute the transient behavior of transmission lines. However, the time step of classic FDTD is limited by the Courant-Friedrich-Lewy (CFL) condition, which makes it not sufficiently efficient for relative low frequency (compared with computation cell size and wave length) signals. The FDTD with alternating direction implicit (ADI) technique is not restricted by the CFL condition, but suffers from numerical dispersions. This paper employs the spectral deferred correction method as the time marching scheme to solve the semidiscretized form of the telegraph equation. The method is unconditionally stable, which allows a larger time step, and can be essentially arbitrary high order accurate.

A new high order dispersive FDTD method for Drude material with complex interfaces

In this paper, motivated by the needs of tracking the transient change in the regularity of the electromagnetic fields across a Drude interface, we propose a new Maxwell–Drude formulation for transverse magnetic problems with inhomogeneous Drude dispersive materials. Based on the auxiliary differential equation approach, the proposed formulation couples the wave equation for the electric component with Maxwell’s equations for the magnetic components. A new finite-difference time-domain (FDTD) algorithm is introduced for solving the proposed Maxwell–Drude system, in which the time dependent jump conditions across the Drude interface are enforced through the matched interface and boundary (MIB) method. The proposed FDTD method achieves a second order of accuracy in solving Drude interfaces with fluctuating curvatures and non-smooth corners based on a simple Yee lattice, while it can be generalized to up to sixth order in dealing with a straight Drude interface. Therefore, the proposed FDTD method is more accurate and cost-efficient than the classical FDTD methods for Drude material with complex interfaces.

Numerical simulation of light propagation in silver nanowire films using time-harmonic inverse iterative method

The interaction between light and silver nanowires (Ag NWs) in a thin film is simulated by solving Maxwell's equations numerically. Time-harmonic inverse iterative method is implemented to overcome the problem of negative permittivity of silver, which makes the classical finite-difference time-domain iteration unstable. The method is validated by showing the correspondence between the plasmonic resonance of an Ag NW from a two dimensional simulation and the analytical solution. In agreement with previous experimental studies, the simulation results show that the transmissivity of the Ag NW films is higher than expected from the geometric aperture. The cause of this phenomenon is studied using TE/TM modes analysis for Ag NW films with different surface coverage of parallel-aligned Ag NWs. Furthermore, 3D simulation of Ag NW films with randomly arranged Ag NWs is performed by parallel computation on high performance computers. A binder layer is taken into account for a preliminary comparison between the simulation and experimental results. The agreements and disagreements between the simulated and measured spectra are discussed.

Field enhancement in subnanometer metallic gaps

Motivated by recent experiments [Ward et al., Nature Nanotech. 5, 732 (2010)], we present here a theoretical analysis of the optical response of sharp gold electrodes separated by a subnanometer gap. In particular, we have used classical finite difference time domain simulations to investigate the electric field distribution in these nanojunctions upon illumination. Our results show a strong confinement of the field within the gap region, resulting in a large enhancement compared to the incident field. Enhancement factors exceeding 1000 are found for interelectrode distances on the order of a few angstroms, which are fully compatible with the experimental findings. Such huge enhancements originate from the coupling of the incident light to the evanescent field of hybrid plasmons involving charge density oscillations in both electrodes.

Two-Dimensional Time-Domain Algorithm With Adaptive Spectral Properties

We develop a two-dimensional scheme, whose spectral properties exhibit a certain degree of controllability, as an alternative to the classic finite-difference time-domain method. The new approach adopts a nonstandard structure for the spatial operators, whose final form is determined through a consistent error-minimization procedure. It is shown that isotropic improvement of accuracy can be accomplished at a single frequency, while the case of wider bands is handled with the aid of a least-squares mechanism. Reliable performance is ensured by the proposed algorithm, without compromising the stability region or requiring excessive computational resources.

Multi-level dual-grid finite-difference time-domain approach for the analysis of body implanted antennas

This study presents an extension of the recently published dual-grid finite-difference time-domain (FDTD) method, which aims at simulating efficiently body area network problems. It consists in splitting the overall simulation into several classical FDTD simulations sequentially executed with an appropriate mesh. It is proposed as an accurate and fast way of modelling integrated antennas and especially analysing the transmission between a body implanted antenna and a receiving antenna located in the nearby environment.

SUB-NANOSECOND ELECTROMAGNETIC-MICROMAGNETIC DYNAMIC SIMULATIONS USING THE FINITE-DIFFERENCE TIME-DOMAIN METHOD

This paper presents an efficient and simple approach of implementing the Landau-Lifshitz-Gilbert (LLG) equation of mag- netisation motion within the Finite-difference Time-domain (FDTD) method. This combined electromagnetic-micromagnetic simulation technique is particularly important for modeling electromagnetic in- teraction with lossy magnetic material in the presence of current and magnetic sources, particularly at very high frequencies. The efficient implementation involves simple two-point spatial interpolations that are applicable to two and three-dimensional FDTD grids, and uses a stable iterative algorithm for the time integration of the LLG equa- tion. A ferromagnetic resonance numerical experiment on a rectangu- lar Permalloy prism excited through its cross-section by a non-uniform pulse field from a transmission line was carried out for the purpose of verifying the combined FDTD-LLG computations. The numerical results were in good agreement with linearised analytical solutions of the LLG equation for uniform and non-uniform precession modes. This paper also presents a brief investigation on the use of non-staggered FDTD grid schemes to model magnetic material using the LLG equa- tion, and indicates that the classical FDTD staggered scheme offers simplicity in implementation and more accuracy for modeling wave in- teraction with lossy magnetic material than the non-staggered schemes based on Maxwell's equations formulation.

A New Hybrid DGTD/FDTD Method in 2-D

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> In this letter, we present a method to efficiently hybridize the discontinuous Galerkin time domain method (DGTD) with the classical finite difference time domain method (FDTD). The main advantage of this hybrid method is that it maintains the generality of FDTD to treat the main part of a complex inhomogeneous problem, applying only DGTD for accurately taking into account the geometrical details of the objects which would not be properly treated with the usual staircased approximation of the classical FDTD. </para>