Implicit Finite-difference Time-domain Method Research Articles

Implementation of the Hybrid ADI-FDTD Scheme to Maxwell Equation for Mathematical Modeling of Breast Tumor

Breast cancer is the most common cancer in women, and non-destructive detection of the tumor is vital. The interaction of electromagnetic waves with breast tissue and the behavior of waves after interaction are used to model tumor detection mathematically. The behavior of electromagnetic waves in a medium is described using Maxwell's equations. Electromagnetic waves propagate according to the electrical properties of a medium. Since the electrical properties of tumor tissue are different from those of normal breast tissue, it is assumed that the tumor is a lossy dielectric sphere, and the breast is a lossy dielectric medium. Under this assumption, Maxwell's equations are used to calculate the scattered field from the tumor. The field scattered by the tumor is different from other tissues because their dielectric properties are different. The location and size of the tumor can be determined by utilizing the difference in scattering from the tissues. While the scattering field from the tumor in spherical geometric form is analytically calculated, it is not analytically possible to calculate the scattering field from the tumor in different geometric shapes. In addition to non-destructive detection of the tumor, an efficient numerical method, the finite difference time domain method (FDTD), is used to simulate the field distribution. After the location of the tumor is determined, the Alternating Direction Implicit (ADI) FDTD method, which gives simulation results by dividing the computation domain into smaller sub-intervals, can be used. Scattered fields are calculated analytically in the geometry where the tumor is in the form of a smooth sphere, and in more complex geometry, the field distributions are successfully obtained with the help of MATLAB using FDTD and ADI-FDTD algorithms.

Optimized Traditional and Leapfrog CDI-FDTD Schemes With Reduced Dispersion and Numerical Properties Analysis

In order to improve the numerical dispersion property of the finite-difference time-domain (FDTD) method further, two optimized low-dispersion schemes are herein proposed, which are based on the complying-divergence implicit (CDI) FDTD method and its one-step leapfrog method. These proposed approaches effectively reduce the numerical dispersion errors by intervening with anisotropic parameters while maintaining unconditional stability. The numerical stability and dispersion properties are analyzed in detail to illustrate their effectiveness. The factors, which impact the numerical dispersion including propagation angles, Courant–Friedrichs–Lewy (CFL) number, nonuniform multiscale, mesh resolution, and frequency are carefully examined. In addition, by using the dual-/triple-band multiple-input multiple-output (MIMO) antennas as numerical examples, the practicality of the optimization schemes is comprehensively validated by comparing the calculation time, memory resource, and relative error with those from the traditional and leapfrog CDI-FDTD methods. It is worth celebrating the superiority of the proposed schemes not only in terms of numerical dispersion, but also in calculation efficiency, solution accuracy, and floating-point operands.

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.

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.

EMI Analysis of Multiscale Transmission Line Network Using a Hybrid FDTD Method

A hybrid time-domain numerical method based on the full wave finite-difference time-domain (FDTD) method combined with a modified 1-D FDTD method has been proposed to analyze the electromagnetic interference (EMI) of multiscale transmission line networks in a metallic cabin in this article. This method is implemented through two numerical solvers for full wave and transmission line simulations, respectively, by adopting a set of reasonable physical boundary conditions between external fields and transmission line network. By using Ampere's law, the external fields can be transferred to a voltage source that can be further used as the forcing terms in the telegraph equation of the transmission line network. To overcome the challenge introduced by the multiscale size of the transmission line network, we combine 1-D explicit and implicit FDTD methods together to solve the coaxial cables and interconnects in PCB, respectively. Thus, among different subdomains, the hybrid method can be carried with the same time step without the restriction of the stability condition. Numerical examples show the accuracy as well as efficiency of the proposed method compared with the HSPICE and traditional FDTD method. In addition, the method is used successfully in analyzing the EMI effects of a complex transmission line network inside a cabin illuminated by external electromagnetic wave.

Hybrid Newmark-conformal FDTD modeling of thin spoof plasmonic metamaterials

The aim of this paper is to present a new efficient and accurate semi-implicit finite-difference time-domain (FDTD) method for modeling thin plasmonic metamaterials supporting spoof surface plasmons. The presented method is formulated by combining Yu–Mittra's conformal FDTD technique for dielectrics, the simplified conformal finite integration technique for perfectly electric conductors (PECs) and the Newmark-Beta method. The resultant scheme is magnetically implicit, has a more relaxed stability condition than that of Yee's FDTD scheme, and allows for accurate modeling of both PEC and dielectric curved surfaces. Energy conservation and stability of the presented scheme is theoretically proved with the energy inequality method. Its formulation can be generalized to a class of magnetically implicit FDTD schemes with weakly conditional and unconditional stability in a unified manner. The presented scheme is used to simulate thin structures supporting spoof surface plasmon polaritons and spoof localized surface plasmons, and also applied to recent plasmonic devices such as planar frequency splitter, ultrathin rainbow trapping device and compact planar metadisk with split-ring resonators. The accuracy and efficiency of the presented scheme are assessed in comparison with the conventional explicit and implicit FDTD methods, and their conformal variants.

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.

On the Behavior of the LOD-FDTD Method at Dielectric Interfaces

The electric field computed by the locally 1-D finite-difference time-domain (FDTD) method at dielectric interfaces is investigated. To this end, two comprehensive problems are considered, namely a dielectric step in a rectangular waveguide and a dielectric-loaded metallic cavity. We found that, in both problems, the electric field patterns exhibit an unexpected phase shift at the dielectric interface. By contrast, the alternating-direction implicit FDTD method does not present this unphysical behavior.

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.

Microwave tomography for early breast cancer detection based on the alternating direction implicit finite-difference time-domain method

In microwave tomography (MWT), electric-parameter distributions of the breast can be reconstructed to detect the early-breast-cancer, which has a specific physical explanation and medical diagnostic value. In time-domain, the finite-difference time-domain (FDTD) method is usually applied to these problems. However, due to the constraint of Courant-Friedrich-Levy (CFL) stability condition, the time step should be small enough to well match the small fine cells, which begets an increasing computational cost, such as the central processing unit (CPU) time. For real-time clinical, it is very important and essential to look for efficient methods to improve the computational efficiency. The alternating-direction implicit finite-difference time-domain (ADI-FDTD) method, on the other hand, provides a larger time step than that the CFL stability condition limitation could set. In order to shorten the time of imaging and improve the detection efficiency, the ADI-FDTD method is first used for the early-breast-cancer detection in this paper. MWT for breast cancer detection requires solving nonlinear inverse scattering problems. Most nonlinear inversion algorithms require solving a number of forward scattering problems followed by an optimization procedure. Therefore, we turn the inverse scattering problem into an optimization question according to the least squares criterion. The optimization procedure aims at minimizing the error between measured scattered fields and estimated scattered fields by the forward solver. Nonlinear reconstruction algorithm is used to solve an update for the scattering object properties used in our breast model. This iteration process is repeated until the convergence between the measured and estimated data is obtained. The specific process of the iteration method is divided into two steps: the forward step, which is to solve a forward problem for a scattering object with estimated electrical properties, and the backward step, which is to solve adjoint fields by introducing the Lagrange multiplier penalty function. Both the forward and backward calculations are conducted by using the ADI-FDTD method. The algorithm is evaluated for a two-dimensional (2D) semicircle breast model with tumors. We compare the imaging results obtained by the ADI-FDTD method for various time steps with the results obtained by the conventional FDTD method and the real distribution. The results agree well, the simulation results prove that the imaging time by using this ADI-FDTD method can be reduced to 23% that by the conventional FDTD method. In addition, the simulation results suggest that the ADI-FDTD method can be more efficient if higher resolution is required, thus further enhancing the clinical applicability of MWT.

Stability property of numerical Cherenkov radiation and its application to relativistic shock simulations

Abstract We studied the stability property of numerical Cherenkov radiation in relativistic plasma flows employing particle-in-cell simulations. Using the implicit finite-difference time-domain method to solve the Maxwell equations, we found that nonphysical instability was greatly inhibited with a Courant–Friedrichs–Lewy (CFL) number of 1.0. The present result contrasts with recently reported results (Vay et al. 2011, J. Comp. Phys., 230, 5908; Godfrey & Vay 2013, J. Comp. Phys., 248, 33; Xu et al. 2013, Comput. Phys. Commun., 184, 2503) in which magical CFL numbers in the range 0.5–0.7 were obtained with explicit field solvers. In addition, we found employing higher-order shape functions and an optimal implicitness factor further suppressed long-wavelength modes of the instability. The findings allowed the examination of the long-term evolution of a relativistic collisionless shock without the generation of nonphysical wave excitations in the upstream. This achievement will allow us to investigate particle accelerations in relativistic shocks associated with, for example, gamma-ray bursts.

A Partially Implicit FDTD Method for the Wideband Analysis of Spoof Localized Surface Plasmons

A partially implicit finite-difference time- domain (FDTD) method with relaxed stability condition is formulated on the basis of combining the implicit Newmark-Beta and the explicit leapfrog time integrations. Its stability is investigated by the Fourier method. The 2-D open and closed structures supporting spoof localized surface plasmons are analyzed with the presented method. The computational time is reduced in comparison with the traditional explicit FDTD while maintaining the same level of accuracy.

Structure-Preserving Reduction of Finite-Difference Time-Domain Equations With Controllable Stability Beyond the CFL Limit

The timestep of the finite-difference time-domain method (FDTD) is constrained by the stability limit known as the Courant-Friedrichs-Lewy (CFL) condition. This limit can make FDTD simulations quite time consuming for structures containing small geometrical details. Several methods have been proposed in the literature to extend the CFL limit, including implicit FDTD methods and filtering techniques. In this paper, we propose a novel approach, which combines model-order reduction and a perturbation algorithm to accelerate FDTD simulations beyond the CFL barrier. We compare the proposed algorithm against existing implicit and explicit CFL extension techniques, demonstrating increased accuracy and performance on a large number of test cases, including resonant cavities, a waveguide structure, a focusing metascreen, and a microstrip filter.

A New Efficient Algorithm for 3-D Laguerre-Based Finite-Difference Time-Domain Method

We previously introduced a new efficient algorithm for implementing the 2-D Laguerre-based finite-difference time-domain (FDTD) method. The new 2-D efficient algorithm is based on the use of an iterative procedure to reduce the splitting error associated with the perturbation term, and it does not involve any nonphysical intermediate variables. Numerical results indicated that the new efficient algorithm shows better performance for modeling some regions with larger spatial derivatives of the field. In this paper, we extend this approach to a full 3-D wave. Numerical formulations of the new 3-D Laguerre-based FDTD method are devised and simulation results are compared to those using the conventional 3-D FDTD method and the alternating-direction implicit (ADI) FDTD method. We numerically verify that, at the comparable accuracy, the efficiency of the proposed method with an iterative procedure is superior to the FDTD method and the ADI -FDTD method. Also, in order to verify the stability of the iterative procedure, we present a convergence analysis and a long-time simulation to it in the paper.

ANALYSIS OF PLANAR CIRCUITS USING AN EFFICIENT LAGUERRE-BASED FDTD METHOD

In this paper, an efficient three-dimensional Laguerre-based finite-difference time-domain (FDTD) method is used to analyze planar circuits. An iterative procedure is introduced to improve the accuracy. Both the time-domain waveforms and the S-parameters are presented. The numerical results show that at the comparable accuracy, the efficiency of the Laguerre-based FDTD method with an iterative procedure is superior to the FDTD method and alternating-direction implicit (ADI) FDTD method.

A Novel Weakly Conditionally Stable FDTD Method for Periodic Structures

In this letter, a novel weakly conditionally stable finite-difference time-domain (FDTD) method for solving periodic structures is presented, which is extremely useful for periodic problem with thin slots in one or two directions. To verify the validity of the proposed formulations, a numerical example of electromagnetic band-gap (EBG) structure with thin slots is given. Compared to the alternating-direction implicit FDTD (ADI-FDTD) method, the presented method has higher accuracy and efficiency. Results show the running time can be reduced to about 2/3 of the periodic ADI-FDTD method in the same simulation. To reduce the numerical dispersion error, the optimized procedure is applied.

Reformulation of the ADI-BPM using a fundamental scheme

A fundamental scheme, developed for implicit finite-difference time-domain methods, is utilized to reformulate the beam-propagation method based on the alternating-direction implicit scheme (ADI-BPM). A derivative-free formulation is performed in the right-hand sides of equations to be solved, leading to quite efficient implementation of the algorithm. Numerical results reveal that the computation time and memory requirements are reduced to ≅ 80%, providing the results equivalent to those of the conventional ADI-BPM.

High-Order Split-Step Unconditionally-Stable FDTD Methods and Numerical Analysis

High-order split-step unconditionally-stable finite-difference time-domain (FDTD) methods in three-dimensional (3-D) domains are presented. Symmetric operator and uniform splitting are adopted simultaneously to split the matrix derived from the classical Maxwell's equations into four sub-matrices. Accordingly, the time step is divided into four sub-steps. In addition, high-order central finite-difference operators based on the Taylor central finite-difference method are used to approximate the spatial differential operators first, and then the uniform formulation of the proposed high-order schemes is generalized. Subsequently, the analysis shows that all the proposed high-order methods are unconditionally stable. The generalized form of the dispersion relations of the proposed high-order methods is carried out. Moreover, the effects of the mesh size, the time step and the order of schemes on the dispersion are illustrated through numerical results. Specifically, the normalized numerical phase velocity error (NNPVE) and the maximum NNPVE of the proposed second-order scheme are lower than that of the alternating direction implicit (ADI) FDTD method. Furthermore, the analysis of the accuracy of the proposed methods is presented. In order to demonstrate the efficiency of the proposed methods, numerical experiments are presented.

The ADI-FDTD Method for Transverse-Magnetic Waves in Conductive Materials

A numerical dispersion analysis of the alternating-direction implicit finite-difference time-domain method is introduced for transverse-magnetic waves in lossy materials. To provide a general study, the conduction terms are discretized by using a weighted average in time. It is found that the numerical dispersion relation is different to that of the transverse-electric (TEz) case. Moreover, contrary to what happens in the TEz case, no set of weighted-average parameters makes the accuracy of the formulation independent of how well the time step resolves the material relaxation-time constant. To overcome this problem, a split-field version of Maxwell's equations is considered as the governing equations. Then, by synchronizing in time the lossy terms with the spatial derivatives, the resulting formulation has the same numerical dispersion relation as in the TEz case. Moreover, its accuracy becomes independent of the quality of the resolution of the relaxation-time constant by the time step.

An implicit discontinuous Galerkin time‐domain method for two‐dimensional electromagnetic wave propagation

PurposeThe purpose of this paper is to develop a time implicit discontinuous Galerkin method for the simulation of two‐dimensional time‐domain electromagnetic wave propagation on non‐uniform triangular meshes.Design/methodology/approachThe proposed method combines an arbitrary high‐order discontinuous Galerkin method for the discretization in space designed on triangular meshes, with a second‐order Cranck‐Nicolson scheme for time integration. At each time step, a multifrontal sparse LU method is used for solving the linear system resulting from the discretization of the TE Maxwell equations.FindingsDespite the computational overhead of the solution of a linear system at each time step, the resulting implicit discontinuous Galerkin time‐domain method allows for a noticeable reduction of the computing time as compared to its explicit counterpart based on a leap‐frog time integration scheme.Research limitations/implicationsThe proposed method is useful if the underlying mesh is non‐uniform or locally refined such as when dealing with complex geometric features or with heterogeneous propagation media.Practical implicationsThe paper is a first step towards the development of an efficient discontinuous Galerkin method for the simulation of three‐dimensional time‐domain electromagnetic wave propagation on non‐uniform tetrahedral meshes. It yields first insights of the capabilities of implicit time stepping through a detailed numerical assessment of accuracy properties and computational performances.Originality/valueIn the field of high‐frequency computational electromagnetism, the use of implicit time stepping has so far been limited to Cartesian meshes in conjunction with the finite difference time‐domain (FDTD) method (e.g. the alternating direction implicit FDTD method). The paper is the first attempt to combine implicit time stepping with a discontinuous Galerkin discretization method designed on simplex meshes.