Finite-difference Time-domain Grid Research Articles

Explicit and unconditionally stable finite-difference time-domain subgridding algorithm based on hanging variables

Due to the limitation of the stability condition, the finite-difference time-domain (FDTD) method cannot efficiently deal with electromagnetic problems containing fine structures. The explicit and unconditionally stable (EUS) FDTD method can eliminate the constraint of the stability condition and improve the simulation efficiency of fine structures by filtering out the unstable modes for the system matrix. However, the EUS-FDTD method needs to solve the eigenvalues of the numerical system matrix, and the symmetry of the numerical system matrix needs to be ensured when the subgridding scheme is used to discretize targets containing fine structures. The existing EUS-FDTD subgridding method encounters some problems such as complex implementation and insufficient accuracy. In order to solve the above problems, in this work, the hanging variables subgridding (HVS) algorithm is applied to the EUS-FDTD algorithm. Starting from the symmetry of the system matrix, the stability of the hanging variables subgridding algorithm is proven, and a high-precision, stable, and easy-to-implement HVS-EUS-FDTD scheme is proposed. Numerical examples of the radiation of linear magnetic currents in free space, electromagnetic scattering of multiple dielectric objects, and a three-dimensional cavity containing a medium demonstrate the stability, high accuracy and efficiency of the proposed method. Numerical experiments show that the computational efficiency of the HVS-EUS-FDTD algorithm can be improved hundreds of times compared with that of the uniform fine grid FDTD algorithm, and the highest computational efficiency can be improved up to ratio (the size ratio of coarse grid to fine grid) times compared with that of the HVS-FDTD algorithm.

Optimized 3D Finite-Difference-Time-Domain Algorithm to Model the Plasmonic Properties of Metal Nanoparticles with Near-Unity Accuracy

The finite difference time domain (FDTD) method is a grid-based, robust, and straightforward method to model the optical properties of metal nanoparticles (MNPs). Modelling accuracy and optical properties can be enhanced by increasing FDTD grid resolution; however, the resolution of the grid size is limited by the memory and computational requirements. In this paper, a 3D optimized FDTD (OFDTD) was designed and developed, which introduced new FDTD approximation terms based on the physical events occurring during the plasmonic oscillations in MNP. The proposed method not only required ~52% less memory than conventional FDTD, but also reduced the calculation requirements by ~9%. The 3D OFDTD method was used to model and obtain the extinction spectrum, localized surface plasmon resonance (LSPR) frequency, and the electric field enhancement factor (EF) for spherical silver nanoparticles (Ag NPs). The model’s predicted results were compared with traditional FDTD as well as experimental results to validate the model. The OFDTD results were found to be in excellent agreement with the experimental results. The EF accuracy was improved by 74% with respect to FDTD simulation, which helped reaching a near-unity OFDTD accuracy of ~99%. The λLSPR discrepancy reduced from 20 nm to 3 nm. The EF peak position discrepancy improved from ±5.5 nm to only ±0.5 nm.

Perfectly Matched Layer Formulation of the INBC-FDTD Algorithm for Electromagnetic Analysis of Thin Film Materials

The impedance network boundary condition (INBC)-based finite-difference time-domain (FDTD) method has been widely used for electromagnetic analysis of highly conductive thin film materials. In the INBC-FDTD formulation, the electromagnetic field variations inside the thin film material are taken into account mathematically and thus extremely small FDTD grids are not necessary for the FDTD modeling of the material. Therefore, computational efficiency of the INBC-FDTD formulation is significantly better than other FDTD formulations. Albeit with this great advantage, the INBC-FDTD formulation cannot be fully employed for thin film materials because the corresponding perfectly matched layer (PML) formulation has not been reported in literature. In this work, we propose a PML formulation suitable for the INBC-FDTD algorithm. Numerical examples illustrate that the proposed PML-INBC-FDTD formulation can yield good absorption performance and also it can improve computational efficiency while maintaining numerical accuracy.

Lightning Currents on Fastening Assemblies of an Aircraft Fuel Tank—Part III: Validation of the Current Assessment Method With Measurement

This article proposes a new fastener model and validate the stochastic process for the lightning current assessment in the aircraft assemblies proposed in the previous parts. In particular, the stochastic method concerns the finite-difference time domain (FDTD) fastener modeling for which the variability of the contact resistance has to be taken into account. The principle is to merge an electromagnetic fastener model with an uncertainty model. The new fastener model proposed in this article is a wire with a lumped resistor on a low conductive plate. This model is an improvement of the model of Part I, where the fastener is only represented by a wire with a lumped resistor. The proposed model in this article is able to ensure a good shielding effectiveness without current distribution disruptions provided that the conductivity of the plate is weak in contrast to the lumped resistor. Moreover, it has the benefits to be able to represent any fastener without location constraint in the FDTD grid. A log-normal distribution has been previously established as uncertainty model that represents the variability of the resistive parameter. The fastener model is included in the FDTD modeling of a realistic fuel tank with each lumped resistor taking a probabilistic value according to the log-normal law. To take into account, the variability of resistors, several simulations with different sets of resistance values are performed. The aim is to surround current measurements with simulated currents at various locations of the fuel tank for a lightning injection performed directly on a fuel tank fastener. This article demonstrates that a specific attention must be given to the modeling of fasteners close to the lightning injection point for the current assessment. Moreover, the results highlight the relevance of the use of an uncertainty model and its reliability for the lightning current assessment.

Maloney and Smith Method for Modeling Debye-Media Thin Sheets in the FDTD Grid

The Maloney and Smith method used to account for sheets thinner than the cell size of the finite-difference time-domain (FDTD) method is extended to the case where both the sheet and the surrounding media are such dispersive Debye media as the Human Body tissues. The new method is then compared to the simpler Luebbers and Kuntz method that was previously extended to Debye media. The comparison relies on the reflection and transmission of plane waves and on 3-D experiments. It is concluded that the Maloney and Smith method permits a better accuracy to be achieved. It provides accurate reflection and transmission whatever may be the incidence angle of the wave that strikes the sheet.

Towards Reducing Computational Costs of Finite Difference Time Domain Algorithm in Plasmonic Optical Properties Modelling of Metal Nanoparticles

Finite difference time domain (FDTD) method is a grid-based, robust and straightforward method to model and study the optical properties of metal nanoparticles (MNPs). However, high computational costs of FDTD including simulation time and memory demand mitigate the interest in this algorithm. In this paper, FDTD algorithm is reviewed and reasons of high computational cost requirement in FDTD are investigated. Computational costs are directly characterised by the resolution and size of FDTD grid (known as Yee grid). High FDTD grid resolution is essentially required in MNPs plasmonic modelling to achieve promising accuracy. This results in increasing the dimension of FDTD operational parameters (stored in FDTD matrices) which increases memory and simulation time required and consumed by the algorithm.

Time Domain Hybrid Method for the Coupling Analysis of Oblique Transmission Line Network Excited by Ambient Wave

Efficient numerical methods which analyse the coupling within an oblique transmission line network (OTLN) excited by electromagnetic wave are still rare in the literature. A novel time domain hybrid method is proposed in this article, in which finite-difference time-domain (FDTD) method, transmission line (TL) equations, Norton's theorem, and interpolation techniques are organically combined. In this method, the terminal loads and connection nodes of OTLN are first shifted to the center lines of FDTD grids, where the loads and nodes are located. Then the OTLN is divided into several independent oblique transmission lines according to the connecting nodes. Finally, the FDTD method combined with TL equations and interpolation techniques is applied to compute the transient responses on each oblique transmission line, and the Norton's theorem is utilized to build the equivalent circuit models of connecting nodes to realize the transmissions of interference signals on these oblique transmission lines. Numerical simulations of OTLN on the perfect conductor (PEC) ground and in a shielded cavity excited by ambient wave are employed to exhibit the accuracy and efficiency of the proposed method. Because the structure of OTLN does not need to be meshed, the proposed method outperforms the conventional FDTD method in both memory usage and computation time.

Time domain hybrid method for coupling analysis of multi-conductor transmission lines on the lossy dielectric layer excited by ambient wave

At present, numerical methods applied to the coupling analysis of transmission lines on the lossy dielectric layer excited by ambient wave are still rare in the literature. As a temptation to fill this gap, a novel time domain hybrid method is proposed, in which the modified transmission line (TL) equations, finite-difference time-domain (FDTD) method, and some interpolation schemes are organically combined together. It can overcome the difficulty in building the coupling model of ambient wave to transmission lines on the lossy dielectric layer greatly. In this method, the modified transmission line (TL) equations suitable for the coupling analysis of multi-conductor transmission lines (MTLs) on the lossy dielectric layer are derived from the traditional TL equations firstly. Compared with the traditional TL equations, the electromagnetic fields in the lossy dielectric layer are introduced into the equivalent distribution sources of modified TL equations. Generally, the precision of TL equations is dependent on the accuracy of equivalent distribution sources, which are obtained from the incident electric fields parallel and perpendicular to the MTLs. Therefore, the FDTD method is utilized to model the structure of lossy dielectric layer to calculate the electromagnetic field distribution surrounding the MTLs and in the dielectric layer. Since the heights and distances of MTLs can be arbitrary values, the electric fields parallel and perpendicular to the MTLs cannot be obtained from the electric fields on the edges of FDTD grids directly, which should be computed via some interpolation schemes. Then the modified TL equations are established, which should be solved by the central difference scheme of FDTD method to obtain the voltages and currents on the MTLs and terminal loads. The significant feature of this proposed method is that it can realize the synchronous calculations of electromagnetic field radiation and transient responses on the MTLs. Finally, numerical simulations of single and multiconductor transmission lines on the lossy dielectric layer excited by ambient wave at different incident angles are employed to exhibit the accuracy and efficiency of the proposed method by comparing with the simulation software CST. Because the structures of MTLs do not need to be meshed, the proposed method outperforms the simulation software CST in both memory usage and computation time.

Origin of Parasitic Solutions With Holland and Simpson Thin Wires in the FDTD Grid

The Holland and Simpson thin wire formalism permits arbitrarily thin wires to be accounted for in the finite-difference time-domain (FDTD) method. However, it is known that a parasitic solution may be added to the physical solution when the wires are not parallel to the axes of coordinates. In this paper, we show that the spurious parasitic solutions originate in the distribution of the currents of the wires in the FDTD grid, which may produce nonphysical deposit of electric charges. A condition not to produce spurious solutions is derived, and an implementation of oblique wires that satisfies the condition is presented. Numerical experiments demonstrate the absence of parasitic solution.

FDTD Modeling of Scattered Ultra-Low Frequency Electromagnetic Waves From Objects Submerged in the Ocean

The skin depth of electromagnetic (EM) waves in ocean water is on the order of 160 m at 3 Hz and just 50 m at 30 Hz. The detection of objects in the ocean using EM waves is, thus, very challenging due to the high attenuation rate of the EM fields. We propose the usage of ultra-low frequencies (ULF: <3 Hz) to detect submerged objects due to the larger skin depths (1.5 km at 0.03 Hz) at those frequencies. Using the finite-difference time-domain (FDTD) method, ULF scatterings from objects submerged in the ocean are obtained. This is achieved by running the FDTD model in the scattered-field-only regime, in which EM field components in a 3-D FDTD grid are excited by a “hard” source, i.e., the electric field components are set to a sinusoid at ULF to represent the scatterings from the object. The propagation attenuation versus distance from the source (object) is recorded and compared to the propagation attenuation as predicted by the plane wave theory. A total round-trip propagation attenuation is estimated for ULF waves originating in the air region above the ocean. Frequencies ranging from 0.01 to 2 Hz are tested.

Wave localized finite-difference-time-domain modelling of scattering of elastic waves within a polycrystalline material.

Ultrasonic studies based on the first arrived signals are of utmost importance when dealing with heterogeneous material especially to seismology, biomedical imaging, as well as for nondestructive evaluation and structural health monitoring applications. Numerical modelling of elastic waves through polycrystalline features has been primarily held back by huge computational requirements. This article discusses the development of a robust and efficient numerical scheme based on finite-difference-time-domain (FDTD) by introducing wave-localized approach to simulate elastic waves in polycrystalline media. The numerical scheme adopts a rotated staggered grid in velocity-stress configuration. The numerical efficiency is improved by adopting parallel computing using efficient graphical processors and by introducing wave-localized computations. It is demonstrated that the proposed tool, especially with the introduction of wave-localized approach, is computationally faster and can handle large-scale grains in comparison with the commercial finite element software, especially when dealing with first arrived signals. This article reports an optimal ratio of FDTD grids per grain to minimize the staircasing effects at the polycrystalline boundaries and was found to be valid over a range of grain sizes. The article also addresses the orientation averaging requirements achieving statistically significant first arrived signal and suggests optimal averaging trials for various grain size models. The developed two-dimensional model shows good agreement with the prediction across the Rayleigh and Stochastic scattering regimes for the chosen model material (Inconel 600) having a cubic symmetry.

A high-precision electromagnetic technique for modeling and simulation in inhomogeneous media

ABSTRACTThe high-precision method for modeling inhomogeneous media and the precision limit of finite-difference time-domain (FDTD) are studied theoretically and numerically. Firstly, based on the transform-domain idea, a new layered modeling technique for inhomogeneous media is developed, with the control equation of the modeling precision given. Compared with other modeling methods, the paramount advantage of this method is that the modeling process is not restrained by the magnitude range of the data, and that the modeling precision can be controlled easily. Then, the shift operator finite-difference time-domain is adopted to determine the field values in inhomogeneous media, and a high-precision computational method is proposed. This method theoretically proves that the relationship between the FDTD grid size and the calculated field value satisfies a definite function, which is then verified by numerical examples. The significance of the results is that this relationship will provide an effective approach to predict field values during the modeling grid size tending to zero, which can be regarded in theory as an approximation to the true value of the electromagnetic (EM) fields. The advantages of this approach are that it not only inherits the advantages of the FDTD method, but also gives a way out of the difficulty that the EM algorithms available are difficult to provide a high-precision prediction of radio propagation; and it can be seen from the validation process in this paper that this approach could also be used in the algorithms based on the mesh modeling and in the inhomogeneous media for improving their calculation precision.

Performance Analysis of Perfectly Matched Layers Applied to Spherical FDTD Grids

A numerical analysis of the performance of the spherical perfectly matched layer (PML) applied to the finite-difference time-domain (FDTD) algorithm implemented in spherical staggered grids is presented in this communication. The spherical PML is derived through a complex transformation (analytical continuation) of the radial coordinate. We evaluate the performance of the spherical PML by simulating fields produced by electric dipoles within the spherical FDTD terminated by a spherical PML and computing the reflection error. The influence of several PML parameters on the reflection error is examined.

Anisotropic Reverse-Time Migration of Ground-Penetrating Radar Data Collected on the Sand Dunes in the Badain Jaran Desert

Ground-penetrating radar (GPR) profiling is the primary tool to provide detail information of the internal structure and characteristics inside a sand dune in the desert area. However, with the severe elevation change in a short horizontal distance and on the rugged and complex surface topography of the sand dunes, getting clear imaging of the internal structure of sand dunes from the GPR profiling is still a big challenge. The classic imaging technique such as the Kirchhoff migration has been applied to process GPR data on sand dunes, with limited success. The reverse-time migration (RTM) technique is the most advanced imaging technique that can handles GPR data acquired on rugged surface with severe topographic relief to generate subsurface structural images with high fidelity and has been tried to process GPR data on sand dunes. The results are encouraging, and the imaging quality is significantly improved. The finite-difference time-domain (FDTD) method is the major numerical tool for forward and backward continuations of the wave field during the RTM process. There are two aspects for using FDTD in RTM of GPR data on sand dunes still need improvement: the numerical scattering caused by the staircase approximation of the ground surface by using gridding in the Cartesian coordinate, and the negligence of the possible anisotropy of the electromagnetic material properties due to the calcareous cementation bedding inside a sand dune. In this paper, we develop the RTM algorithm based on the staggered grid FDTD that handles the rugged topographic surface by using the curvilinear coordinate, and the possible anisotropic radar wave velocity of the sand dune media. We first demonstrate the equivalency of the nonuniform, isotropic medium and the uniform, anisotropic medium for justifying using the uniform, anisotropic velocity in the RTM by synthetic modeling. Next, we validate our approach of using the synthetic data with the comparison of using the Cartesian coordinate and the curvilinear coordinate in an isotropic medium. The results indicate that the RTM algorithm using the curvilinear coordinate can efficiently eliminate the adverse effect of the staircase approximated boundary of the topography surface. Finally, we processed the real GPR data collected on a sand dune in the Badain Jaran desert by using the curvilinear coordinate and the uniform, anisotropic velocity in FDTD forward and backward wave field continuation. Comparison of the results indicates that the RTM imaging using the boundary-conforming curvilinear coordinate and anisotropic velocity gains more coherent and higher resolution images for the calcareous cementation layers and the water table.

Insulated Oblique Thin Wire Formalism in the FDTD Method

Recently, we have introduced a new approach based on Holland's formalism to deal with oblique thin wire in the finite-difference time-domain (FDTD) method. Here, we propose an original technique to model accurately dielectric coated thin wire, which can be freely inclined or shifted in the FDTD grid. In addition, the multiwire junction of oblique thin wire formalism is extended by means of a rigorous equivalent circuit diagram of the insulated wires. The approach facilitates complex modeling of cables embedded in lossy dielectric medium including buried insulated cables. It exhibits same accuracy as in integral equation method. However, the FDTD method shows better efficiency in time computational as soon as the application includes a soil or when increasing complexity and density of the modeling structure.

3-D Domain Decomposition Based Hybrid Finite-Difference Time-Domain/Finite-Element Time-Domain Method With Nonconformal Meshes

A new 3-D domain decomposition based hybrid finite-difference time-domain (FDTD)/finite-element time-domain (FETD) method is introduced to facilitate electromagnetic modeling by exploiting both the computational efficiency of FDTD and the meshing flexibility of FETD. The proposed hybrid method allows the FETD mesh and the FDTD grid to be nonconformal based on domain decomposition technique. It implements the hybridization with a buffer zone, which functions as a transition region between FDTD and FETD. The buffer zone helps the proposed hybrid method obviate the interpolation approach for field coupling of the nonconformal mesh and hence overcome the late-time instability issue. The discontinuous Galerkin method is utilized to couple different regions, thus improving the coupling accuracy compared with that using the Dirichlet boundary condition. Moreover, the hybrid method allows further division of the FETD region into multiple subdomains when the degrees of freedom in this region are large. For temporal discretization, a global leapfrog time integration scheme is implemented to sequentially update the fields in the FDTD, buffer, and FETD regions. The numerical results are shown to demonstrate the meshing flexibility and computational efficiency of the proposed hybrid method inherited from FETD and FDTD methods.

Capturing Plasmon–Molecule Dynamics in Dye Monolayers on Metal Nanoparticles Using Classical Electrodynamics with Quantum Embedding

A multiscale hybrid quantum/classical approach using classical electrodynamics and a collection of discrete three-level quantum systems is used to simulate the coupled dynamics and spectra of a malachite green monolayer adsorbed to the surface of a spherical gold nanoparticle (NP). This method utilizes finite difference time domain (FDTD) to describe the plasmonic response of the NP within the main FDTD framework and a three-level quantum description for the molecule via a Maxwell/Liouville framework. To avoid spurious self-excitation, each quantum molecule has its own auxiliary FDTD subregion embedded within the main FDTD grid. The molecular parameters are determined by fitting the experimental extinction spectrum to Lorentzians, yielding the energies, transition dipole moments, and the dephasing lifetimes. This approach can be potentially applied to modeling thousands of molecules on the surface of a plasmonic NP. In this paper, however, we first present results for two molecules with scaled oscillator ...

Analytical Expression for the Time-Domain Green’s Function of a Discrete Plane Wave Propagating in a 3-D FDTD Grid

In this paper, a closed-form expression for the time-domain dyadic Green’s function of a discrete plane wave (DPW) propagating in a 3-D finite-difference time-domain (FDTD) grid is derived. In order to verify our findings, the time-domain implementation of the DPW-injection technique is developed with the use of the derived expression for 3-D total-field/scattered-field (TFSF) FDTD simulations. This implementation requires computations of the time-domain Green’s function of DPW with the use of multiple-precision arithmetic. Then, excitations at the TFSF interface can be computed as a time-domain convolution of a source function with Green’s function of DPW. The developed time-domain implementation of the DPW-injection technique demonstrates the leakage error across the TFSF interface around the numerical noise level that verifies the correctness of the derivation.

Optimized (2, 4) Stencil Runge–Kutta ADE-ADI FDTD With Application to Plasma

This paper discusses the improvement of the numerical dispersion characteristics of alternating direction implicit (ADI) finite-difference time-domain (FDTD) aimed at acquiring more accurate electromagnetic information of plasma. Through adding the optimization method, which is based on the optimization of spatial derivative to the (2, 4) stencil ADI FDTD, the optimized (2, 4) stencil ADI FDTD is proposed, and its unconditional stability is proved theoretically. The phase velocity error of the optimized (2, 4) stencil ADI FDTD versus propagation angle and grid density is investigated. In addition, the Runge–Kutta auxiliary differential equation (RKADE) scheme for tackling the constitutive relation equation of plasma is deduced, which is without additional storage occupation and computational burden compared with ADE scheme. Its numerical conductivity error is analyzed under different incident frequencies and electron collision frequencies. Through incorporating the RKADE scheme into the optimized (2, 4) stencil ADI FDTD, the optimized (2, 4) stencil RKADE-ADI FDTD is presented. The accuracy and relatively wideband capability of the proposed method is validated by two numerical experiments.

Stochastic Reduced-Order Electromagnetic Macromodels in FDTD

To enable an efficient stochastic-based design optimization methodology for multiscale structures of electrical devices, circuits, and systems, we propose the infusion of stochastic modeling with the electromagnetic macromodel in the method of finite-difference time domain (FDTD). We enhance the computational efficiency of the stochastic macromodel by applying the model order reduction techniques to produce the stochastic reduced-order macromodel. In addition, we provide a methodology and algorithms for the efficient generation and application of the stochastic reduced-order macromodel in the FDTD grid. The proposed methodology quantifies the impact of uncertainty in the electromagnetic system response that is manifested as random material and structural variations, through modeling and simulation. We demonstrate the proposed methodology by applying it to the computation of the stochastic transient electromagnetic fields in a 3-D bandgap structure comprising a rectangular waveguide, which contains an array of dielectric posts that exhibit uncertainty.