Standard Finite-difference Time-domain Research Articles

An Efficient FDTD Method Based on Subgridding Technique and One-Step Leapfrog ADI-FDTD

In this letter, a 3-D subgridding finite-difference time-domain (FDTD) approach is proposed. The calculation domain is divided into regions with dense meshes and regions with coarse meshes. By applying the proposed subgridding technique to dense grid regions, memory and computation resources can be significantly reduced. Furthermore, the Courant–Friedrich–Levy (CFL) limitation of dense mesh regions is broken by the one-step leapfrog alternately direction-implicit (ADI)-FDTD. Coarse and dense regions can adopt a uniform time step, which is limited only by the standard FDTD CFL condition for coarse meshes. In comparison with the standard FDTD approach and the subgridding FDTD approach, numerical results demonstrate the efficiency and accuracy of the proposed method.

Controlled-source electromagnetic modeling using a high-order finite-difference time-domain method on a nonuniform grid

Simulation of 3D low-frequency electromagnetic (EM) fields propagating in the earth is computationally expensive. We have developed a fictitious wave-domain high-order finite-difference time-domain (FDTD) modeling method on nonuniform grids to compute frequency-domain 3D controlled-source EM data. The method overcomes the inconsistency issue widely present in the conventional second-order staggered-grid finite-difference scheme over a nonuniform grid, achieving high accuracy with an arbitrarily high-order scheme. The finite-difference coefficients adaptive to the node spacings can be accurately computed by inverting a Vandermonde matrix system using an efficient algorithm. A generic stability condition applicable to nonuniform grids is established, revealing the dependence of the time step and these finite-difference coefficients. A recursion scheme using fixed-point iterations is designed to determine the stretching factor to generate the optimal nonuniform grid. The grid stretching in our method reduces the number of grid points required in the discretization, making it more efficient than the standard high-order FDTD with a densely sampled uniform grid. Instead of stretching in vertical and horizontal directions, better accuracy of our method is observed when the grid is stretched along the depth without horizontal stretching. The efficiency and accuracy of our method are demonstrated by numerical examples.

중간 주파수 해석을 위한 수정된 준정적 FDTD 기법과 이론 해의 비교

The Finite-Difference Time-Domain (FDTD) is widely used in electromagnetic exposure analysis in high-resolution models such as the human body. The required analysis time, however, extremely increases when a standard FDTD technique is used in a band below the intermediate frequency. In this paper, the difference between the modified quasi-static FDTD method and the theoretical solution was analyzed by calculating the induced electric field in the dielectric sphere from the incident electric field and magnetic field, respectively.

Simultaneous ThermoBrachytherapy: Electromagnetic Simulation Methods for Fast and Accurate Adaptive Treatment Planning.

The combination of interstitial hyperthermia treatment (IHT) with high dose rate brachytherapy (HDR-BT) can improve clinical outcomes since it highly enhances the efficiency of cell kill, especially when applied simultaneously. Therefore, we have developed the ThermoBrachy applicators. To effectively apply optimal targeted IHT, treatment planning is considered essential. However, treatment planning in IHT is rarely applied as it is regarded as difficult to accurately calculate the deposited energy in the tissue in a short enough time for clinical practice. In this study, we investigated various time-efficient methods for fast computation of the electromagnetic (EM) energy deposition resulting from the ThermoBrachy applicators. Initially, we investigated the use of an electro-quasistatic solver. Next, we extended our investigation to the application of geometric simplifications. Furthermore, we investigated the validity of the superpositioning principle, which can enable adaptive treatment plan optimization without the need for continuous recomputation of the EM field. Finally, we evaluated the accuracy of the methods by comparing them to the golden standard Finite-Difference Time-Domain calculation method using gamma-index analysis. The simplifications considerably reduced the computation time needed, improving from >12 h to a few seconds. All investigated methods showed excellent agreement with the golden standard by showing a >99% passing rate with 1%/0.5 mm Dose Difference and Distance-to-Agreement criteria. These results allow the proposed electromagnetic simulation method to be used for fast and accurate adaptive treatment planning.

Modified Finite-Difference Time-Domain Method for Hertzian Dipole Source under Low-Frequency Band

In this paper, a modified finite-difference time-domain (FDTD) method is proposed for the rapid analysis of a Hertzian dipole source in the low-frequency band. The FDTD technique is one of the most widely used methods for interpreting high-resolution problems such as those associated with the human body. However, this method has been difficult to use in the low-frequency band as the required number of iterations has increased significantly in such cases. To avoid this problem, FDTD techniques using quasi-static assumptions in low-frequency bands were used. However, this method was applied only to plane wave excitation, making it difficult to apply to near-field problems. Therefore, a modified approach is proposed, involving the application of the FDTD technique with a quasi-static approximation to an electric and magnetic dipole problem. The results when using the proposed method are in good agreement with those from a theoretical solution. An example of comparison with the standard FDTD method is shown for illustrating the proposed method’s performance.

A Fast and Fully Parallel Analog CMOS Solver for Nonlinear PDEs

A general-purpose analog computing method is proposed to compute the continuous-time solutions of nonlinear partial differential equations (PDEs). The discrete-time difference operator in the standard finite difference time domain (FDTD) method is replaced by continuous-time delay operators that can be realized using analog all-pass filters. The resulting spatially discrete time-continuous (SDTC) update equations are realized using analog circuits which compute continuous-time solutions of the PDE with prescribed initial and boundary conditions. The proposed concept is demonstrated in simulation via an integrated circuit (IC) design of a nonlinear acoustic wave equation solver in 180 nm CMOS technology. Analog arithmetic operations (multiply, scale, and add) are realized in parallel using fully differential op-amps and analog multipliers. The proposed IC computes the PDE solution in parallel at 33 discrete spatial points and has a simulated bandwidth and power consumption of approximately 2 MHz and 3 W, respectively. The performance of the IC is simulated using foundry-supplied device models and quantified using i) the mean squared difference between the circuit simulation results and FDTD simulations, and ii) the noise to signal energy ratio. Acceptable accuracy is obtained, with error metric values varying between -7 and -30 dB for various configurations of the problem. Comparison of the custom analog IC simulations with MATLAB- and C-based FDTD code running on a modern workstation shows an expected average speedup of 205× and 140×, respectively.

Quantifying Sub-gridding Errors in Standard and Hybrid Higher Order 2D FDTD Simulations

Sub-gridding errors for a 2D Finite-Difference Time-Domain (FDTD) simulation are compared for both the standard FDTD and Hybrid higher order FDTD cases. Subgridding contrast ratios of 1:3, 1:9, 1:15, and 1:27 are considered and analyzed. A correlation is seen between the increase of contrast ratio with the increase of sub-gridding errors for both standard and hybrid cases. However, a trend of errors reduction when using hybrid formulations over standard formulations is apparent for each contrast ratio.

Superluminal phase velocity approach for suppression of Numerical Cherenkov Instability in Maxwell solver

A modification of a standard finite-difference time-domain (FDTD) Maxwell solver is proposed based on the inclusion of the terms of the higher order in the finite differences approximation. By selecting the coefficients of this hybrid scheme the numeric dispersion of the electromagnetic waves can be fine-tuned. It is shown that a particular choice of the coefficients leads to the phase velocity of all the waves being slightly larger than speed of light. Results of the 3D Particles-In-Cells (PIC) simulations demonstrate that the proposed scheme of the Maxwell solver is efficient to mitigate the Numerical Cherenkov Instability (NCI) when simulating ultrarelativistic charged beams.

Faster and More Accurate Time Domain Electromagnetic Simulation Using Space Transformation

A novel finite difference time domain (FDTD) method based on space transformations is developed that overcomes the inherent obstacles of the conventional FDTD algorithm in a spatially complex domain. Our method leads to an adaptive mesh for the investigated structure based on its geometrical shape without adding additional numerical problems such as late-time instability. In this method, mesh boundaries can follow arbitrary geometrical shapes precisely meaning that discretization errors are minimized. Such errors can be considerable when dealing with large material differences or boundary conditions within the simulation domain. Different meshing and transformation techniques for a variety of different scenarios are presented. We show how boundaries discontinuity can be handled using our method without resulting artificial singularities or zeros. Unlike previous works no dispersive medium has been used so the simulation speed is similar to the standard FDTD method. The usability and superiority of this method in terms of simulation run time and accuracy are shown through a couple of scattering and plasmonic problems. Also, the result of any simulations are validated using finite element method.

Continuous-Time Algorithms for Solving Maxwell’s Equations Using Analog Circuits

Two analog computing methods are proposed to compute the continuous-time solutions of one-dimensional (1-D) Maxwell’s equations. In the first method, the spatial domain partial derivatives in the governing partial differential equation (PDE) are approximated using discrete finite differences while applying the Laplace transformation along the time dimension. The resulting spatially discrete time-continuous update equation is utilized to design an analog circuit that can compute the continuous-time solution. The second method replaces the discrete-time difference operators in the standard finite difference time domain (FDTD) cell (Yee cell) using continuous-time delay operators, which can be realized using analog all-pass filters. Both methods have been simulated using ideal analog circuits in Cadence Spectre for the Dirichlet, Neumann, and radiation boundary conditions. The performance of the proposed methods has been quantified using: i) mean squared differences between the results and fully discrete FDTD simulations and ii) the noise to signal energy ratio. Both methods have been extended to design the analog circuits that compute the continuous-time solution of the 1-D and 2-D wave equations. The 1-D wave equation solver is simulated with a dominant-pole model (which better approximates the non-ideal circuit behavior) along with a propagation delay compensation technique. The experimental results from a simplified board-level low-frequency implementation are also presented. The key challenges toward CMOS implementations of the proposed solvers are identified and briefly discussed with possible solutions.

Computation of High-Order Electromagnetic Field Sensitivities With FDTD and the Complex-Step Derivative Approximation

This paper introduces a new approach for the computation of electromagnetic field derivatives, up to any order, with respect to the material and geometric parameters of a given geometry, in a single Finite-Difference Time-Domain (FDTD) simulation. The proposed method is based on embedding the complex-step derivative (CSD) approximation into the standard FDTD update equations. Being finite-difference free, CSD provides accurate derivative approximations even for very small perturbations of the design parameters, unlike finite-difference approximations that are prone to subtractive cancellation errors. The availability of accurate approximations of field derivatives with respect to design parameters enables studies such as sensitivity analysis of multiple objective functions (as derivatives of those can be derived from field derivatives via the chain rule), uncertainty quantification, as well as multi-parametric modeling and optimization of electromagnetic structures. The theory, FDTD implementation and applications of this technique are presented.

Generalized tensor FDTD method for sloped dispersive interfaces and thin sheets.

We present a modified formulation of the Finite-Difference Time-Domain (FDTD) technique that facilitates the accurate modeling of curved plasmonic interfaces. These interfaces appear in structures of interest for the design of optical metamaterials, such as arrays of plasmonic nanorods. Our approach uses the standard rectangular FDTD mesh and tensor effective permittivities for the interface cells, implicitly enforcing field boundary conditions, and is readily applicable to thin curved dispersive layers. We demonstrate the accuracy and effectiveness of our approach with the periodic analysis of a silver nanorod array and the computation of scattering parameters from a thin dispersive ring in a waveguide.

Dispersive Contour-Path MNL-FDTD Scheme for Fast Analysis of Waveguide Metamaterials

A semi-implicit dispersive thin-wire finite-difference time-domain (FDTD) scheme is proposed for fast analysis of waveguide metamaterials. The proposed scheme is based on combining the Newmark-beta method with Makinen’s improved thin-wire FDTD model based on the contour-path integral of Maxwell’s equations around a wire corrected with scaling factors and Wait’s surface-impedance boundary condition for thin wires of finite conductivity. Its stability condition is determined by only the two meshes in the transverse plane to the wire axis. This feature allows using larger mesh steps than a wire radius without time step reducing. In order to verify the scheme, we demonstrate four device applications: plasmonic parallel-plate waveguide filled with only positive dielectrics, edge Fabry–Perot resonator, waveguide metatronic high-pass filter, and coaxial-to-waveguide matching with an epsilon-near-zero narrow channel. Its stability, accuracy, and computational efficiency are assessed in comparison with the standard FDTD explicit scheme, analytical solutions, and existing experimental data.

Comparison of the lattice-Boltzmann model with the finite-difference time-domain method for electrodynamics.

A three-dimensional lattice-Boltzmann model (LBM) for the simulation of the Maxwell equations is presented. The inclusion of media follows an extension of a special limit described in the literature which is applicable to this LBM and does not harm the stability of simulations. The focus of the present study lies on the properties of numerical accuracy and stability of the LBM in comparison to the standard finite-difference time-domain (FDTD) method based on Yee's method. Typical examples, often investigated in the context of numerical simulations, are considered. These include the propagation of electrodynamic (EM) fields in one- and three-dimensional systems. Results of this simulations are compared to the ones of their theoretical predictions. Further on, long-time simulations are done in systems with periodic boundary conditions to check if the total energy is conserved. To investigate the effect of the numeric impedance, the propagation of an EM pulse is monitored spatially and temporarily in a two-dimensional system. The simulation results indicate, in contrast to the one obtained from the FDTD method, that the presented LBM does fulfill the expected energy conservation and is not effected by the numerical impedance. This LBM therefore represents a valuable alternative for the simulation of EM problems like long-time simulations by avoiding intrinsic properties the FDTD method suffers from.

Double-helix array structure using phase controlled interference of 6+6 beams

We describe a novel single-step approach to form hexagonal array of DNA like intertwined double-helix structure. The lattice structure has been simulated by using interference of 6+6 mutually counter-propagating beams arranged in a double-cone geometry. This is a phase controlled interference technique where each of the twelve beams is assigned a suitable phase offset and the helicity of the double-helix structure depends on the phase offsets of the interfering beams. For the application of the structure as a broadband circular polarizer, a simulation study has been performed using standard FDTD (finite difference time domain) method. A comparative study between aluminium(Al) based single-helix and double-helix hexagonal lattice structures is also carried out, showing that the operating wavelength range of the double-helix structure is much larger as compared to the single-helix structure.

Non-standard finite-difference time-domain method for solving the Schrödinger equation

In this paper, an improvement of the finite-difference time-domain (FDTD) method using a non-standard finite-difference scheme for solving the Schrödinger equation is presented. The standard numerical scheme for a second derivative in the spatial domain is replaced by a non-standard numerical scheme. In order to apply the non-standard FDTD (NSFDTD), first, the estimates of eigenenergies of a system are needed and computed by the standard FDTD method. These first eigenenergies are then used by the NSFDTD method to obtain improved eigenenergies. The NSFDTD method can be employed iteratively using the resulting eigenenergies to obtain more accurate results. In this paper, the NSFDTD method is validated using infinite square well, harmonic oscillator and Morse potentials. Significant improvements are found when using the NSFDTD method.

An advanced numerical technique for a quasi-static electromagnetic field simulation based on the finite-difference time-domain method

In this paper, an advanced numerical technique is proposed for a low-frequency electromagnetic field simulation. The finite difference time domain (FDTD) method is one of the best methods for analyzing electromagnetic field problems, which require high levels of precision, such as a heterogeneous whole-body voxel human model. However, directly using the standard FDTD method in the quasi-static frequency band requires an extra-large number of iterations. We overcome this drawback using the hybrid of surface equivalence theorem and quasi-static FDTD (QS-FDTD) method. The surface equivalence theorem is used to consider arbitrary shape of source and the QS-FDTD is applied to obtain electromagnetic response quickly in the low-frequency region. The results of the proposed method are in good agreement with those from a commercial electromagnetic simulator. The proposed method is valid where the quasi-static approximation is satisfied.

Using of Generalized Goertzel Algorithm for FDTD Calculation of the Transmission and Reflection Spectra of Periodic Structures

Finite-difference time-domain (FDTD) method is frequently used for calculation of transmission and reflection characteristics of periodic structures. We propose a hybrid FDTD technique for evaluation of the transmittance and reflectance spectra of periodic structures that significantly reduces the usage of computer memory and time of the calculation. In the considered case, calculation time and memory occupation were reduced up to 30% and 20%, respectively, compared to the standard FDTD procedure. Calculated results fit well in the experimentally measured data.

Pseudospectral Maxwell solvers for an accurate modeling of Doppler harmonic generation on plasma mirrors with particle-in-cell codes.

With the advent of petawatt class lasers, the very large laser intensities attainable on target should enable the production of intense high-order Doppler harmonics from relativistic laser-plasma mirror interactions. At present, the modeling of these harmonics with particle-in-cell (PIC) codes is extremely challenging as it implies an accurate description of tens to hundreds of harmonic orders on a broad range of angles. In particular, we show here that due to the numerical dispersion of waves they induce in vacuum, standard finite difference time domain (FDTD) Maxwell solvers employed in most PIC codes can induce a spurious angular deviation of harmonic beams potentially degrading simulation results. This effect was extensively studied and a simple toy model based on the Snell-Descartes law was developed that allows us to finely predict the angular deviation of harmonics depending on the spatiotemporal resolution and the Maxwell solver used in the simulations. Our model demonstrates that the mitigation of this numerical artifact with FDTD solvers mandates very high spatiotemporal resolution preventing realistic three-dimensional (3D) simulations even on the largest computers available at the time of writing. We finally show that nondispersive pseudospectral analytical time domain solvers can considerably reduce the spatiotemporal resolution required to mitigate this spurious deviation and should enable in the near future 3D accurate modeling on supercomputers in a realistic time to solution.

Broadband terahertz half-wave plate based on anisotropic polarization conversion metamaterials

We propose a half wave plate in the terahertz region to realize broadband linear polarization conversion in the reflection mode through anisotropic metamaterials. The structure of this design is composed of two pairs of patches in one unit cell that can rotate the polarization direction of a linearly polarized incident wave for an angle of 2β in a broadband frequency range. This work concentrates mainly on numerical simulations by using the standard finite difference time domain method. The efficiency of the half wave plate is higher than 50%, and PCR is higher than 85% in a broadband frequency range for cross polarization conversion. In the operating frequency band, three polarization conversion peaks can be found with the conversion efficiency nearly 100%. A more outstanding performance can be achieved by using low-loss substrate. The in-depth physical mechanism is revealed by further analysis. Besides, the expressions referring to theoretical decomposed electric field are deduced.