Leapfrog Alternating Direction Implicit Research Articles

Unified Formulation of Leapfrog Complying Divergence Implicit FDTD Method for Lossy Media

The leapfrog alternating direction implicit finite-difference time-domain (ADI-FDTD) method for lossy media has been found to suffer from the non-physical field leakage problem in the non-penetrable structures, which gives rise to inaccuracies in modelling lossy media during scattering analyses. To address such problem, in this paper, we propose a unified formulation of leapfrog complying-divergence implicit (CDI)-FDTD method for lossy media without field leakage problem. Several different lossy schemes to incorporate the conductivity terms of lossy media, including exponential time differencing, averaging and forward-forward schemes are integrated into a unified formulation. The proposed leapfrog CDI-FDTD method not only resolves the problem of field leakage but also has complying divergence. Moreover, since the implicit update procedures in the proposed formulation are in the fundamental form with matrix-operator-free right-hand sides, the floating-point operations (flops) are fewer than that of the conventional leapfrog ADI-FDTD formulation for lossy media. Numerical results are given to demonstrate the advantages of the proposed method over the leapfrog ADI-FDTD method in accuracy, efficiency, complying divergence and resolving the field leakage problem for lossy media.

Higher Order CPML for Leapfrog Complying-Divergence Implicit FDTD Method and Its Numerical Properties

The leapfrog complying-divergence implicit finite-difference time-domain (CDI-FDTD) method is an implicit scheme that is unconditionally stable involving the simplest and most efficient leapfrog update procedures with minimum floating-point operations (flops). Moreover, the method possesses complying divergence satisfying Gauss’s law, while other methods, including the leapfrog alternating direction implicit (ADI)-FDTD counterpart, do not. These desirable features make the leapfrog CDI-FDTD useful for electromagnetic simulations when the unconditional stability is to be exploited. In this article, a higher order convolutional perfectly matched layer (CPML) for leapfrog CDI-FDTD is developed and implemented for simulating open space problems. To formulate the CPML for leapfrog CDI-FDTD, we have carefully considered the order of CPML update along with proper use of main and auxiliary fields to retain the unconditional stability and complying divergence. The proposed CPML for leapfrog CDI-FDTD remains stable beyond the Courant time step as ascertained through von Neumann analysis. By comparing the flops and computation time, the leapfrog CDI-FDTD is shown to be most efficient among common implicit FDTD methods with CPML. The formulation of numerical divergence for higher order CPML is also presented. Simulation results show that the proposed higher order CPML has many advantages of simplicity, complying divergence, and high absorption efficiency.

From Time-Collocated to Leapfrog Fundamental Schemes for ADI and CDI FDTD Methods

The leapfrog schemes have been developed for unconditionally stable alternating-direction implicit (ADI) finite-difference time-domain (FDTD) method, and recently the complying-divergence implicit (CDI) FDTD method. In this paper, the formulations from time-collocated to leapfrog fundamental schemes are presented for ADI and CDI FDTD methods. For the ADI FDTD method, the time-collocated fundamental schemes are implemented using implicit E-E and E-H update procedures, which comprise simple and concise right-hand sides (RHS) in their update equations. From the fundamental implicit E-H scheme, the leapfrog ADI FDTD method is formulated in conventional form, whose RHS are simplified into the leapfrog fundamental scheme with reduced operations and improved efficiency. For the CDI FDTD method, the time-collocated fundamental scheme is presented based on locally one-dimensional (LOD) FDTD method with complying divergence. The formulations from time-collocated to leapfrog schemes are provided, which result in the leapfrog fundamental scheme for CDI FDTD method. Based on their fundamental forms, further insights are given into the relations of leapfrog fundamental schemes for ADI and CDI FDTD methods. The time-collocated fundamental schemes require considerably fewer operations than all conventional ADI, LOD and leapfrog ADI FDTD methods, while the leapfrog fundamental schemes for ADI and CDI FDTD methods constitute the most efficient implicit FDTD schemes to date.

One‐step leapfrog alternating direction implicit procedure for left‐handed material in open region problems with enhanced absorption

AbstractBy incorporating higher order formulation and one‐step leapfrog alternating direction implicit (ADI) procedure, unconditionally stable perfectly matched layer implementation is proposed for terminating frequency‐dependent left‐handed material in finite‐difference time‐domain lattice. To be more specific, frequency‐dependent left‐handed material can be calculated by the piecewise linear recursive convolution method. The proposed scheme shows advantages of higher order formulation and one‐step leapfrog ADI procedure in terms of enhancing absorbing performance, improving computational efficiency and simulating open region problems. The effectiveness is demonstrated through the computational domain with dipole antenna model above the left‐handed material slab. From results, it can be concluded that the proposed formulation can obtain considerable absorption, outstanding efficiency, and improved entire performance. Meanwhile, the proposed scheme also keeps its stability when time step far exceeds the Courant‐Friedrich‐Levy condition.

A One-Step Leapfrog ADI Procedure with Improved Absorption for Fine Geometric Details

Based on the alternating direction implicit (ADI) procedure, leapfrog formulation, and a higher order PML scheme, we propose an unconditionally stable perfectly matched layer (PML) algorithm with improved absorption to treat open regions in a finite computational domain with improved overall performance. The proposed algorithm performed well compared to other algorithms during simulations. We further demonstrated the proposed scheme’s effectiveness using numerical examples. We found that the proposed scheme had enhanced effectiveness and improved the absorption during the whole simulation. Furthermore, it was able to break the stability condition, proving that it is an unconditionally stable scheme.

A Leapfrog Scheme for Complying-Divergence Implicit Finite-Difference Time-Domain Method

A leapfrog scheme for the unconditionally stable complying-divergence implicit (CDI) finite-difference time-domain (FDTD) method is presented. The leapfrog CDI-FDTD method is formulated to comprise distinctive implicit and explicit update procedures. The implicit update procedures are in the fundamental form with operator-free right-hand sides (RHS), whereas the explicit ones are compatible and correspond to the familiar explicit FDTD method. Using the Fourier (von Neumann) approach, the stability of the leapfrog CDI-FDTD method is analyzed. The eigenvalues of amplification matrix are obtained to prove the unconditional stability. Furthermore, the leapfrog alternating direction implicit (ADI) FDTD and CDI-FDTD methods are discussed and compared, including the RHS floating-point operations (flops) count, for-loops and memory. The leapfrog CDI-FDTD requires merely half the flops of leapfrog ADI-FDTD at RHS while retaining the same left-hand sides (LHS) of implicit update equations and the same number of for-loops without much extra memory. Discussions and numerical results are presented to demonstrate the advantages of leapfrog CDI-FDTD method including unconditional stability, complying divergence, and efficient leapfrog scheme with reduced RHS flops.

Conformal Subgridding and Application to One-Eighth Spherical Shell Dielectric Resonator Antenna Array

To model the object with concave and convex surfaces accurately, an economical conformal subgridding scheme is proposed in this article. Its double conforming procedure with coarse grid (CG) and fine grid (FG) guarantees the modeling accuracy. Fast linear interpolation effectively exchanges electromagnetic fields information on the interface between grids of two sizes. One-step leapfrog alternating direction implicit (ADI) finite-difference time-domain (FDTD) method overcomes the Courant stability condition in FG region; meanwhile, conventional FDTD ensures the efficiency of simulation in CG region. While we concern the unconditionally stable method, there is not any grid omitted in our irregularly profiled fine region and any extra time wasted in solving small-scale tridiagonal matrixes cell by cell. To verify our codes, a 2-D arc component has been simulated first. Then the performance of a probe-fed normal one-eighth spherical dielectric resonator antenna (DRA), a new designed conformal-microstrip-fed one-eighth spherical shell DRA, and a four-element array with the spherical shell DRAs connected by a T-shaped power divider has been calculated by our codes. Satisfactory performance of the antenna has been obtained in return loss and radiation patterns, which are verified by measured results. The shell antenna features smaller size, lighter weight, and more convenience in conforming to the corners of the instrument housing, even with different dimensions. Most particularly, the proposed conformal subgridding scheme and its hybrid FDTD implementation perform outstandingly in simulating the thin shell antenna in terms of computer memory, CPU time, as well as accuracy.

A Hybrid Method Based on Leapfrog ADI-FDTD and FDTD for Solving Multiscale Transmission Line Network

This article proposes a novel hybrid finite-difference time-domain (FDTD) method to calculate the transient responses of the transmission line network based on the leapfrog alternating direction implicit FDTD (leapfrog ADI-FDTD) algorithm and the traditional FDTD method. The proposed hybrid method can be implemented by dividing the transmission line network into two parts that are the interconnects part in a printed circuit board solved by the leapfrog ADI-FDTD and the cable part solved by the traditional FDTD with different mesh sizes and the same time step, respectively. In addition, some boundary conditions should be introduced based on the modified nodal approach and Kirchhoff's law at the terminals of the transmission line network. The numerical results show that the proposed hybrid method is in good agreement with HSPICE. Especially, it is more efficient and applicative for the multiscale transmission line network problems due to the use of the implicit FDTD solver with unconditional stability compared with the traditional FDTD 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.

Tunable hybrid metal–graphene frequency selective surfaces based on split‐ring resonators by leapfrog ADI‐FDTD method

A tunable hybrid metal–graphene frequency selective surface (FSS) based on split-ring resonators at mid-infrared frequencies is proposed and numerically investigated by an improved leapfrog alternating direction implicit (ADI)-finite-difference time-domain (FDTD) method. The graphene conductivity described by a new closed-form approximate expression is incorporated into the leapfrog ADI-FDTD through the vector-fitting technique and auxiliary differential equation approach. Numerical simulations show that the proposed method has good computational accuracy and higher efficiency compared to conventional FDTD method and analytical solutions. Besides, the transmission and reflection coefficients of the FSS can be effectively adjusted by changing the chemical potential of graphene, the layer number of graphene patch and operating temperature. This work provides a novel method for designing tunable devices for optical plasmonic applications.

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.

Efficient Three-Step LOD-FDTD Method in Lossy Saturated Ferrites With Arbitrary Magnetization

A three-step unconditionally stable locally one-dimensional finite-difference time-domain method is proposed to represent an arbitrarily magnetized lossy ferrite medium. The small loss in the saturated ferrite is introduced to the algorithm through a simple explicit relation. A step-by-step procedure is established to minimize the number of operations and variables. It is shown that the processing time and occupied memory variables in the proposed method are about 90% and 50% of those of the one-step leapfrog alternating direction implicit method, respectively. The accuracy and stability of the method are verified through the simulation of wave propagation in two ferrite-loaded devices, including a microstrip phase shifter, and a stripline circulator.

Unconditionally Stable Divergence-Free Vector-Based Meshless Method for Transient Electromagnetic Analysis

A vector-based meshless method or vector meshless method, which is proven to be divergence-free, was recently proposed for transient electromagnetic analysis. In this method, an explicit finite-difference scheme is used for approximating the time derivatives in Maxwell’s equations. Therefore, the stability criterion constraints the maximum time-step. This paper presents the alternating-direction implicit (ADI) formulation of the vector meshless method based on a different form of the ADI algorithm named the leapfrog ADI scheme. The computational cost of the leapfrog ADI scheme is less than the conventional ADI algorithm and algebraically equivalent. In the proposed vector meshless method, Courant–Friedrichs–Lewy limit does not constrain the time-step due to its implicit formulation. Moreover, the stability condition of the conventional vector meshless method and the proposed ADI vector meshless method are investigated through their amplification matrices in this paper. The results show that the proposed method has two features: unconditional stability and divergence-free property. The first one is obtained due to the incorporation of the leapfrog ADI algorithm, and the second one is achieved by the use of vector basis functions. The divergence-free property of the proposed method can lead to obtaining more accurate solutions than the conventional meshless method. In addition, some numerical examples are presented to verify the properties and the efficiency of the proposed method.

An Efficient Domain Decomposition Parallel Scheme for Leapfrog ADI-FDTD Method

A flexible and universal domain decomposition parallel scheme is proposed for the unconditionally stable finite-difference time-domain (FDTD) method. The leapfrog alternating direction implicit FDTD (ADI-FDTD) method is employed to eliminate the restriction of the Courant–Friedrichs–Lewy stability condition. The proposed domain decomposition parallel implementation of the leapfrog ADI-FDTD method is more flexible with process allocation and requires fewer data communications. A buffer region is introduced to decouple the interactions between neighboring subdomains at each time step. Electromagnetic simulations are presented to demonstrate the applicability, accuracy, and efficiency of the proposed method.

ADI Scheme of a Nonstandard FDTD Method and Its Numerical Properties

An alternating direction implicit (ADI) scheme for a nonstandard (NS) finite-difference time-domain (FDTD) method is proposed. The NS FDTD method is derived by analyzing the dispersion relation. Two different groups of update coefficients are used to update the electric field and magnetic field, respectively. The dispersion property of the NS method is similar with that of the standard higher order method, though it is only second order accurate in the space domain. The total number of nonzero update coefficients for this method is fewer than that of the latter. This small stencil property can be used to reduce computational burden in implicit formulation of this method. The one-step leapfrog ADI scheme is adopted to examine this idea. The detailed update equations of this ADI scheme are derived. The dispersion and stability properties are also analyzed and compared with the corresponding standard one. The results show that the computational efficiency of the proposed method is higher than that of the standard one.

Scattered Field Leapfrog ADI–FDTD Method for Drude Dispersive Media

This letter presents a new unconditionally stable scattered field (SF) finite-difference time-domain (FDTD) scheme for Drude dispersive medium based on leapfrog alternating direction implicit (ADI) technique. The proposed scheme introduces polarization current density and analytical incident field terms in the leapfrog scheme without adding extra computational complexities except causing minor changes in the coefficients for the central spatial locations. Unconditional stability and accuracy of the scheme are validated by numerical tests. When the Courant number is set to three or a higher value, the scheme can perform faster than the explicit FDTD scheme.

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.

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.

Compact 3-D Envelope ADI-FDTD Algorithm for Simulations of Coherent Radiation Sources

A modified complex-envelope (CE) alternating-direction-implicit (ADI) finite-difference time-domain solver for electromagnetic (EM) fields is presented. Like the conventional ADI scheme, the modified scheme is based on splitting the time step into two halves. In each substep, Maxwell's equations are solved implicitly in one direction and explicitly in the perpendicular direction. The new scheme is a combination of the conventional CE-ADI scheme and the leapfrog ADI scheme that was recently reported. The main features are a more compact form of the tridiagonal equations and greater flexibility in the application of boundary conditions. The boundary conditions on the six faces of the solution box are general and can be specified either as perfect conductor, symmetry boundary conditions, cavity modes, or arbitrary external specification. The new compact CE-ADI scheme is unconditionally stable; thus, the time step can be larger than the one imposed by the Courant-Friedrichs-Lewy (CFL) condition. Fast stable EM simulations of a planar slow-wave structure are demonstrated, with time steps exceeding the CFL limit by two orders of magnitude.