ADI-FDTD Method Research Articles

Energy analysis of the ADI-FDTD method with fourth-order accuracy in time for Maxwell's equations

<abstract><p>In this work, the ADI-FDTD method with fourth-order accuracy in time for the 2-D Maxwell's equations without sources and charges is proposed. We mainly focus on energy analysis of the proposed ADI-FDTD method. By using the energy method, we derive the numerical energy identity of the ADI-FDTD method and show that the ADI-FDTD method is approximately energy-preserving. In comparison with the energy in theory, the numerical one has two perturbation terms and can be used in computation in order to keep it approximately energy-preserving. Numerical experiments are given to show the performance of the proposed ADI-FDTD method which confirm the theoretical results.</p></abstract>

Partially Implicit FDTD (PI-FDTD) Method for Lower Dispersion and Anisotropic Errors

A new partially implicit 3D FDTD (PI-FDTD) method is introduced. The method is not unconditionally stable, but its stability limit is 6 times higher than that of the classic Yee&#x2019;s FDTD method. The method has only one implicit update equation and uses four split time steps. A study has been carried out to compare the performances of the new method and the ADI-FDTD, LOD-FDTD, and HIE-FDTD methods. ADI-FDTD and LOD-FDTD methods are unconditionally stable and the HIE-FDTD method is conditionally stable with a stability limit of 3.46 times above the Yee&#x2019;s FDTD method. The dispersion and anisotropic errors of the PI-FDTD method are considerably lower than that of these methods and for small cell sizes, the new method is almost isotropic. For example at the stability limit of the PI-FDTD method with a uniform cell size of <inline-formula> <tex-math notation="LaTeX">$\lambda \mathord {\left /{ {{ {50}}} }\right. }$ </tex-math></inline-formula> the maximum dispersion errors and anisotropic errors are 0.15&#x0025; and 0.031&#x0025; for the new method and 1.66&#x0025; and 0.64&#x0025; for the ADI-FDTD and LOD-FDTD methods. At the stability limit of the HIE-FDTD method, the maximum dispersion errors and anisotropic errors of the PI-FDTD method are also less than HIE-FDTD, ADI-FDTD, and LOD-FDTD methods. The differences in errors are more pronounced for larger cell sizes. Although the new method uses a four time split step scheme and the ADI-FDTD and LOD-FDTD methods use a two time split step scheme, the computational time of the PI-FDTD method is less, as it uses an implicit scheme for only one of the field components when updating. The computational time of the PI-FDTD method is the same as that of the HIE-FDTD method. The study has revealed that the new method can be used over much wider frequency bandwidths.

Modeling of dispersive media in ADI-FDTD method with complex–conjugate pole residue pairs

This work presents an alternating-direction implicit (ADI) finite-difference time-domain (FDTD) scheme for the study of structures that involve materials with arbitrary frequency dispersion. The material dispersion is fitted to the complex–conjugate pole-residue (CCPR) terms model, and a novel, to the best of our knowledge, numerical formulation is presented based on auxiliary differential equations and two-step ADI methodology. Additionally, the proposed technique is combined with the concept of the perfectly matched layer, and a new implicit scheme is introduced for the termination of media with CCPR dispersion in the ADI-FDTD framework. The ADI-FDTD formulation is compared with the explicit FDTD scheme for several benchmark two-dimensional problems in terms of accuracy and efficiency. The suggested algorithm is proven to be robust and capable of simulating applications in different frequency regions, spanning from microwaves to optical frequencies. It can provide a powerful tool for the analysis of nanostructures involving both strongly dispersive and nanosized materials, such as plasmonic metasurfaces, antennas, core–shell nanoparticle systems, light-trapping plasmonic solar cells, surface-enhanced Raman spectroscopy substrates, or nanodevices based on epsilon-near-zero materials.

High-Accurate Non-Uniform Grids for System-Combined ADI-FDTD Method in Near-Field Scattering With Proper CFL Factor

In this paper, a high-accurate technique with non-uniform grids is introduced into a system-combined alternative-direction-implicit finite-difference time-domain (SC-ADI-FDTD) algorithm, and then successfully used to analyze electromagnetic propagations. To our knowledge, the conventional FDTD with non-uniform grids can be effectively deal with some edges of the three-dimensional cubes and complicated structures of the tiny objects by modulating the local grid scales, which to the extent improves its reliability and accuracy. However, due to existing the finer grids in the local computational region and the inevitable Courant-Friedrichs-Lewy (CFL) limit in the conventional FDTD, the temporal interval must be determined by the minimum fine spatial grid, resulting in much larger temporal sampling density required during the whole computation process. As the advantage of circumventing the repeated variables, the non-uniform SC-ADI-FDTD (NUSC-ADI-FDTD) cannot only break through the CFL limit to implement the high-efficient computation, but also further save more CPU time in the local microstructure cases. Furthermore, the empirical formula between the spatial sampling density and the CFL factors can be obtained from the numerical fitting method after errors analysis. The numerical simulations of the electromagnetic scattering have been executed to illustrate feasibility and validity of our proposed method.

Multiple 1-D Fundamental ADI-FDTD Method for Coupled Transmission Lines on Mobile Devices

This article presents the multiple one-dimensional (M1-D) fundamental alternating direction implicit (FADI) finite-difference time-domain (FDTD) method for coupled transmission lines on mobile devices. The method is aptly called the M1-D FADI coupled line (CL)-FDTD method. It is based on the fundamental implicit scheme, which features matrix-operator-free right-hand sides (RHS). The formulations of the M1-D FADI CL-FDTD method and update equations are provided. Various sets of split matrices, including those with self-symmetry, mutual symmetry, and self-mutual separation, are proposed and discussed. Their stability analyses are performed using the Fourier amplification matrix formulated in fundamental form, which is simpler with only one inverse term each without RHS operator. It is found that the sets of split matrices with self-symmetry are unconditionally stable, while those with mutual symmetry and self-mutual separation are not. Using two auxiliary quadratic polynomials along with single necessary and sufficient condition, the analytical proof of unconditional stability is made more convenient and concise. The efficiency of both stable sets of split matrices with self-symmetry is discussed based on the number of RHS terms and floating-point operations count. Numerical results are presented to validate the accuracy of the proposed method at time step larger than the stability limit. Several electromagnetic (EM) simulations of coupled line structures are demonstrated on mobile devices. The CPU time incurred on various platforms is provided for the M1-D FADI CL-FDTD method using both sets of split matrices with self-symmetry. Using mobile devices, the EM simulations can be performed efficiently and ubiquitously anytime, anywhere.

Analysis of FSS Using NWCS-FDTD Method

In this paper, the frequency selective surfaces (FSS) are analyzed by using a novel weakly conditional Stable(NWCS) finite-difference time-domain (FDTD) method, which is extremely useful for periodic problems with thin slots in two directions. Compared with the ADI-FDTD method, the presented method has higher accuracy and efficiency. Two examples are provided

System-Combined ADI-FDTD Method and Its Electromagnetic Applications in Microwave Circuits and Antennas

A system-combined alternating-direction-implicit finite-difference time-domain (SC-ADI-FDTD) is fully presented and well corroborated for 3-D electromagnetic applications. Spatial differential operators of Maxwell’s equations can be expressed as two matrices with the different time-steps in original ADI-FDTD method. From this matrix form, the system chart can be drawn by combining with digital signal processing (DSP) techniques. Based on the intuitive process of the original ADI-FDTD method, we can avoid the repeated computation. Due to the limited computer memory, we employ the unsplited-field convolution perfectly matched layer (CPML) to terminate the FDTD computational domain. Compared with HFSS commercial software, conventional FDTD and original ADI-FDTD methods, the proposed SC-ADI-FDTD method can be well demonstrated and further validated by adopting electromagnetic applications, such as microwave circuits and antennas.

Wide-Band Modeling On-Chip Spiral Inductors Using Frequency-Dependent Conformal ADI-FDTD Method

To analyze on-chip spiral inductors efficiently, an alternating direction implicit (ADI) finite difference time-domain (FDTD) method is proposed for general dispersive media. The time-domain recursive-convolution iteration equation can be used to calculate the Debye, cold Plasma, and Lorentz media in the same formula. A locally conformal technique, named medium parameters weighting method is proposed to modeling complex structure accurately. The spiral inductor integrated on the silicon substrate has been analyzed by using conformal ADI-FDTD method. And the simulation results are in agreement with the measured results. These data have been obtained over a wide frequency range from 0.1–20 GHz. The proposed method can easily be used as an accurate computer-aided design tool for radio-frequency integrated circuits.

Unconditionally Stable Multiple One-Dimensional ADI-FDTD Method for Coupled Transmission Lines

The unconditionally stable multiple one-dimensional alternating direction implicit finite-difference time-domain (M1-D ADI-FDTD) method for coupled transmission lines is presented. The differential equations for coupled transmission lines are formulated in compact matrix form. Proper $4 \times 4$ split matrices are introduced, which feature unconditional stability for the ADI procedures. These matrices also lead to update equations with the left-hand sides involving tridiagonal matrices that can be solved efficiently. The update equations of the M1-D ADI-FDTD method are given, which comprise one implicit and three explicit updates in each procedure. The analytical proof of unconditional stability is also provided based on the von Neumann method. The M1-D ADI-FDTD method allows quick simulation and visualization of electromagnetic fields everywhere along the coupled transmission lines. To demonstrate the usefulness of the M1-D ADI-FDTD method, various coupled line structures for filter and filtering antenna are analyzed using the method.

Leapfrog ADI FDTD Method for Lumped Elements

Leapfrog alternating direction implicit finite-difference time-domain (Leapfrog ADI-FDTD), a method with high computational efficiency, as well as it was proved to overcome CFL stability condition, the unconditional stability was given numerically, thus having a widely application in simulation. Lumped elements are introduced into Leapfrog ADI-FDTD method in this paper to further improvement of computational efficiency, single-grid updating equations of typical lumped elements are presented such as resistance, capacitor and inductor, what’s more, multi-grid model are provided to simulate special lumped elements which owned more than one grid, and situation of application is analyzed through iterated equations. Its high efficiency and stability were verified by numerical experiments, thus providing a foundation for Leapfrog ADI-FDTD applied to integrated circuits.

Stability analysis and improvement of conformal leapfrog alternating direction implicit finite-difference time-domain method

A conformal leapfrog alternating direction implicit finite-difference time-domain (CLeapfrog ADI-FDTD) method based on conformal technique was proposed in the article. Compared with the conventional FDTD method, the proposed method decreased the step approximation error, it was used to simulate the irregular object whose boundary couldnt match the orthogonal grid; at the same time, this method could have a high efficiency because leapfrog alternating direction implicit finite-difference time-domain (Leapfrog ADI-FDTD) is a method with unconditional stability. However, CLeapfrog ADI-FDTD method may lose the stability expected with the Leapfrog ADI-FDTD schemes, and instability factor in CLeapfrog ADI-FDTD was analyzed through eigenvalue of the growth matrix, then a new method named improved conformal leapfrog alternating direction implicit finite-difference time-domain (ICLeapfrog ADI-FDTD) with a modified conformal technique was proposed, which could improve the stability without losing the calculation accuracy. The accuracy and efficiency of the proposed ICLeapfrog ADI-FDTD method were verified by numerical results.

Numerical analysis of a leapfrog ADI–FDTD method for Maxwell’s equations in lossy media

Recently, a so-called one-step leapfrog ADI–FDTD method has been developed in engineering community for solving the 3D time-dependent Maxwell’s equations. This method becomes quite popular in simulation wave propagation in graphene-based devices due to its efficiency. We investigate this method from a theoretical point of view by proving the energy conservation property, the unconditional stability of this ADI–FDTD method, and establishing the optimal second-order convergence rate in both time and space on non-uniform cubic grids. Numerical results are presented justifying our analysis.

Efficient Implementation for the AH FDTD Method With Iterative Procedure and CFS-PML

An efficient implementation for the associated Hermite finite-difference time-domain (AH FDTD) method with iterative procedure is proposed, which is an extension of previous works: the original and the paralleling-in-order-based AH FDTD methods. By introducing the alternating direction implicit method, the direct calculation of five-point matrix equation is replaced with iterative solving of tridiagonal matrices equations. For the 2-D case, only two tridiagonal matrixes need to be calculated in a full iterative cycle. In addition, the complex frequency shifted (CFS) perfectly matched layer (PML) is extended to the AH FDTD method and implemented together with this efficient solution scheme. Numerical examples show the superior computational performance of the proposed iterative procedure by comparison with the conventional FDTD method, the ADI FDTD method, and previous AH FDTD methods. The effectiveness of CFS-PML in AH FDTD is also verified.

Modified Splitting FDTD Methods for Two-Dimensional Maxwell’s Equations

In this paper, we develop a new method to reduce the error in the splitting finite-difference method of Maxwell’s equations. By this method two modified splitting FDTD methods (MS-FDTDI, MS-FDTDII) for the two-dimensional Maxwell equations are proposed. It is shown that the two methods are second-order accurate in time and space and unconditionally stable by Fourier methods. By energy method, it is proved that MS-FDTDI is second-order convergent. By deriving the numerical dispersion (ND) relations, we prove rigorously that MS-FDTDI has less ND errors than the ADI-FDTD method and the ND errors of ADI-FDTD are less than those of MS-FDTDII. Numerical experiments for computing ND errors and simulating a wave guide problem and a scattering problem are carried out and the efficiency of the MS-FDTDI and MS-FDTDII methods is confirmed.

An Efficient 3-D HIE-FDTD Method With Weaker Stability Condition

For analyzing the electromagnetic objects with the fine structures in one direction efficiently, a three-dimensional (3-D) hybrid implicit–explicit finite-difference time-domain (HIE-FDTD) method with weaker stability condition is proposed in this paper. First, by adopting the alternating-direction implicit (ADI) scheme, the 3-D Maxwell’s equations are subdivided into two substep procedures, and then by adding some disturbance terms into two substeps, respectively, the proposed method is derived. Compared with the existing HIE-FDTD methods, the Courant–Friedrichs–Lewy (CFL) stability condition of the proposed HIE-FDTD method is weaker, which means the proposed method is more efficient due to the possibility of choosing the larger time step size. In addition, the relationships between the CFL stability conditions and the numerical dispersion errors of the traditional FDTD method, the three HIE-FDTD methods, and the ADI-FDTD method are summarized in order to provide a basis for the choice among them.

Energy Identities of ADI-FDTD Method with Periodic Structure

In this paper, a new kind of energy identities for the Maxwell equations with periodic boundary conditions is proposed and then proved rigorously by the energy methods. By these identities, several modified energy identities of the ADI-FDTD scheme for the two dimensional (2D) Maxwell equations with the periodic boundary conditions are derived. Also by these identities it is proved that 2D-ADI-FDTD is approximately energy conserved and unconditionally stable in the discrete L2 and H1 norms. Experiments are provided and the numerical results confirm the theoretical analysis on stability and energy conservation.

The One-Step ADI-FDTD for Periodic Structures and its Stability

In this paper, the stability analysis of the unconditionally stable one-step leapfrog alternating direction implicit (ADI) finite-difference time-domain (FDTD) method for periodic structures is presented. The amplification matrix of the proposed leap-frog ADI-FDTD method is obtained through the spatial domain with Fourier method and eigenvalues of the Fourier amplification matrix are obtained analytically to prove the unconditional stability of the periodic leapfrog ADI-FDTD method. Numerical verification is proposed to confirm the theoretical result.

Efficient Parallel LOD-FDTD Method for Debye-Dispersive Media

The locally one-dimensional finite-difference time-domain (LOD-FDTD) method is a promising implicit technique for solving Maxwell's equations in numerical electromagnetics. This paper describes an efficient message passing interface (MPI)-parallel implementation of the LOD-FDTD method for Debye-dispersive media. Its computational efficiency is demonstrated to be superior to that of the parallel ADI-FDTD method. We demonstrate the effectiveness of the proposed parallel algorithm in the simulation of a bio-electromagnetic problem: the deep brain stimulation (DBS) in the human body.

Stable Formulation of FADI-FDTD Method for Multiterm, Doubly, Second-Order Dispersive Media

This paper presents stable formulation of fundamental alternating-direction-implicit finite-difference time-domain (FADI-FDTD) method for multiterm, doubly, second-order dispersive media. The formulation is achieved by decomposing the electric and magnetic susceptibility functions into first-order poles and it is derived based on currents constitutive relations. It is conveniently applicable for most commonly used second-order dispersive models, such as Lorentz and Drude models, and equally applicable to first-order Debye model. The extension for full 3-D doubly dispersive media using FADI-FDTD method makes the resultant update equations much more concise and simpler than using conventional ADI-FDTD method. The number of field variables and update coefficients are greatly reduced at the right-hand sides, with only first-order spatial differencing and no dispersive terms and magnetic update coefficients in the implicit E update equations. All these contribute to conciseness and programming simplicity, as well as leading to higher efficiency due to much less memory indexing overhead and fewer arithmetic operations. Most importantly, our present formulation is stable, while the contemporary second-order differential equation formulation has potential instability as to be demonstrated. The stability analysis is performed using Fourier method by examining the eigenvalues of the Fourier amplification matrix numerically.

Application of the Leapfrog ADI FDTD Method to Periodic Structures

In this letter, the one-step leapfrog alternating direction implicit (ADI) finite-difference time-domain (FDTD) method has been introduced to solve periodic structures, resulting in a one-step leapfrog periodic ADI-FDTD method. In comparison to the original ADI-FDTD method, the one-step leapfrog ADI-FDTD method retains almost the same numerical modeling accuracy, but with higher computational efficiency. To simplify the issue, a reformation of the periodic one-step leapfrog ADI-FDTD method is also presented. Numerical results are given to demonstrate the proposed formulation. It is found that the periodic one-step leapfrog ADI-FDTD method requires less memory and CPU time than the conventional periodic ADI-FDTD method. To reduce the numerical dispersion error, an optimization procedure is applied.