Crank-Nicolson Finite-difference Time-domain Research Articles

Characterization of Excitation Effects and Data Interpretation of Combined Time-Domain Multiwaveform Transmission Currents

With the development of electronic technology, time-domain electromagnetic (TDEM) transmitters are now capable of emitting various waveforms such as trapezoidal, triangular, and semi-sinusoidal waveforms, as well as their combinations. However, the application of multi-waveform combination detection methods is still immature, and modeling methods are required to analyze the characteristics of different waveforms and complete data interpretation. Based on the Crank-Nicolson finite difference time-domain (CN-FDTD) method for discretizing the electromagnetic wave equations, the current waveform can be converted into current surface density for source-included calculations, enabling high-precision modeling of the responses to different current waveforms. The responses to multiple transmission current waveforms were compared and analyzed using various models, and the results showed that selecting targeted excitation parameters based on the characteristics of the target can improve detection sensitivity. We applied the multi-waveform combination detection method to the actual exploration in the field, obtained the pure secondary field response by subtracting the primary field of the actual current waveform from the measured data, converted the responses of different transmission current waveforms into step wave responses by deconvolution method, and performed resistivity imaging. The results of modeling and data interpretation demonstrate the effectiveness of the multi-waveform combination detection method in improving geological exploration accuracy, and promoted the development of fine-scale geophysical exploration.

Compact System-Combined-Based FDTD Implementations With Approximate Crank–Nicolson Scheme and Electromagnetic Scattering

In this article, we develop the efficacious system-combined-based approximate Crank–Nicolson finite-difference time-domain (SC-ACN-FDTD) method which can be applied to electromagnetic scattering. After being processed by the matrix form with the discrete time, the electric and magnetic fields from Maxwell’s equations are both unified as the unique vector at the same time step. By means of the vector transformation, iterative procedures in the time domain of both electric and magnetic fields can be obtained in the Crank–Nicolson format. To model an open-domain problem, we apply convolutional perfectly matched layer (CPML) to truncate the finite computational region. To guarantee the numerical accuracy of the SC-ACN-FDTD method, we employ the empirical formula to forecast the maximum Courant–Friedrichs–Lewy (CFL) factor based on the spatial sampling density. Compared with the conventional finite-difference time-domain (FDTD) methods, the proposed SC-ACN-FDTD method can be well demonstrated and further validated by adopting electromagnetic scattering in the near and far fields. To further validate our results, the commercial software COMSOL is also adopted for the far field’s pattern in the single frequency.

Explicit FDTD method based on iterated Crank–Nicolson scheme

An implicit Crank–Nicolson procedure can be replaced with an explicit iteration process. An explicit finite-difference time-domain method based on the iterated Crank–Nicolson scheme that has been widely used for solving Einstein's equations is newly developed. The formulation is presented with two iterations and its stability condition is also derived. Numerical results are found to agree well with those obtained from the traditional explicit finite-difference time-domain method, showing the validity of the present iterated Crank–Nicolson–finite-difference time-domain method.

The impact of different corona models on FD algorithms for the solution of multiconductor transmission lines equations

In this study, the implementation of different corona discharge models in a transient program for overhead multiconductor lines based on the implicit Crank-Nicolson Finite Difference Time Domain method is presented. Among different corona models available in the literature, two empirical models (Gary's and Suliciu's models) and a physics-based model (Malik's model) are discussed, which are broadly used in EMT programs, and their predictions under fast-front lightning surges and slow-front switching impulses are assessed. The authors critically review the dynamic capacitance approach in terms of numerical accuracy, when dealing with slowly varying voltages, and applicability, regarding dispersive models assuming a non-instantaneous charge-voltage relation. The alternative voltage-controlled generator approach is proposed, which is successfully included in the implicit Crank-Nicolson scheme. One of the objectives of this study is to show how different corona models may predict overvoltages with huge discrepancies, depending on the transient waveform and the propagation distance, despite the similarity of the q-v hysteresis curves. More evident discrepancies between models under fast-front voltage waves are reported, rather than for switching impulses. The consistency of these results in terms of the models' features and q-v curves is analysed, highlighting the necessity for new engineering tools for corona simulation of general predictive capability.

Reply to Comments on “Direct-Splitting-Based CN-FDTD for Modeling 2D Material Nanostructure Problems”

In this response letter, we focus on addressing Yu Hang Yang's comments so that the correctness and validity of the proposed auxiliary-differential-equation-based (ADE) Crank-Nicolson finite-difference time-domain method with direct-splitting technique (CNDS-FDTD) are demonstrated. Moreover, related applications in modeling 2D material nanostructure problems are shown.

Analysis of Metal Oxide Varistor Arresters for Protection of Multiconductor Transmission Lines Using Unconditionally-Stable Crank–Nicolson FDTD

Surge arresters may represent an efficient choice for limiting lightning surge effects, significantly reducing the outage rate of power lines. The present work firstly presents an efficient numerical approach suitable for insulation coordination studies based on an implicit Crank–Nicolson finite difference time domain method; then, the IEEE recommended surge arrester model is reviewed and implemented by means of a local implicit scheme, based on a set of non-linear equations, that are recast in a suitable form for efficient solution. The model is proven to ensure robustness and second-order accuracy. The implementation of the arrester model in the implicit Crank–Nicolson scheme represents the added value brought by the present study. Indeed, its preserved stability for larger time steps allows reducing running time by more than 60 % compared to the well-known finite difference time domain method based on the explicit leap-frog scheme. The reduced computation time allows faster repeated solutions, which need to be looked for on assessing the lightning performance (randomly changing, parameters such as peak current, rise time, tail time, location of the vertical leader channel, phase conductor voltages, footing resistance, insulator strength, etc. would need to be changed thousands of times).

Direct-Splitting-Based CN-FDTD for Modeling 2D Material Nanostructure Problems

Incorporating a truncation of the complex-frequency-shifted perfectly matched layer (CFS-PML), the direct-splitting- based Crank-Nicolson finite-difference time-domain (CNDS-FDTD) is developed and applied to the infrared two-dimensional layered material (2DLM) black phosphorous (BP) metasurface implementations on the all-dielectric nanostructure. To improve extremely low efficiencies in solving infrared terahertz (THz) problems with the few-atomic-layer thickness of 2DLMs, the CFS-CNDS-FDTD is proposed in demand due to the fact that it possesses capabilities of implicit FDTD method and unsplit-field CFS-PML truncation, respectively, in completely conquering the Courant-Friedrich-Levy condition (CFL) limit and holding good performance. The temporal incremental in the CFS-CNDS-FDTD can reach 1000 times larger than that in the regular FDTD for infrared nanoscale problems centered at the 2.5 THz and then keep accurate. Three-dimensional (3D) numerical cases have been carried out to corroborate the proposed method. The CFS-CNDS- FDTD can not only achieve high accuracies and then saves several dozen times of CPU time as compared to the regular FDTD, but also pave the way for designing all-dielectric nanostructures with other 2DLM metasurfaces.

Unconditionally stable higher order perfectly matched layer applied to terminate anisotropic magnetized plasma

An effective and efficient absorbing boundary condition applied to terminate anisotropic magnetized plasma is proposed in this article. As its mathematical derivations are based on the high order perfectly matched layer (HO-PML) and Crank-Nicolson finite-difference time-domain method, it has the advantages of reducing late-time reflections, attenuating evanescent waves, absorbing low-frequency propagation waves and overcoming the Courant-Friedrichs-Levy condition. To validate the efficiency and effectiveness of the proposed HO-PML, a ridge waveguide model composed of vacuum and the anisotropic magnetized plasma, a dielectric waveguide model composed of the dielectric and the anisotropic magnetized plasma are employed. The numerical results show that the proposal can not only maintain the stability of the algorithm with the increment of the time step but also further enhance the absorbing performance.

Time-domain analysis of a CRLH coupled-line coupler using the CN-FDTD method

In this study, the implicit Crank–Nicolson finite-difference time-domain (CN-FDTD) method is applied to discretize the governing telegrapher's equations of a composite right-/left-handed (CRLH) coupled-line coupler. The unconditionally stable CN-FDTD is compared with the conventional leap-frog (LF) FDTD method. The results obtained from the CN-FDTD scheme show up to 10 times increase in the temporal step size, reflecting in a dramatic decrease in processing time; in addition to having a good agreement with the LF method and the measurements.

Subgridding Boundary Conditions to Model Arbitrarily Dispersive Thin Planar Materials

In a previous work, we presented a hybrid implicit–explicit Crank–Nicolson finite-difference time-domain method for treating multilayered lossy thin slabs. The main advantage of this method was its capability to overcome certain late-time stability issues of the conventional surface impedance boundary condition approaches. In this communication, we extend this method to deal with thin slabs having arbitrarily dispersive profiles. This approach is validated with the analysis of a spherical shell made of a metallic wire mesh whose macroscopic equivalent constitutive parameters are derived from its microscopic structure. The results for the electric field inside the sphere are compared against the analytical data and show good agreement with them.

Corrections to “Linear Analysis of High-Frequency Field-Effect Transistors Using the CN-FDTD Method” [Jun 17 1946-1954

In the above-cited paper [1] , the Crank–Nicolson finite-difference time-domain method is used for the analysis of a high-frequency field-effect transistor (FET). However, two references are missed. In addition, there are some errors in (6b), Fig. 2 , and Table I . The reference of Figs. 1 and 2 is [2] . The reference of Table I is [3] . In (6b), $L_{\mathrm {dd}}$ , $L_{\mathrm {gg}}$ , $L_{\mathrm {ss}}$ , $M_{\mathrm {gd}}$ , and $G_{m}$ should be, respectively, $L_{\mathrm {d}}$ , $L_{\mathrm {g}}$ , $L_{\mathrm {s}}$ , $M_{\mathrm {dg}}$ , and $g_{m}$ . The corrected version of Fig. 2 is depicted here, wherein the designations are consistent with the present correction. The corrected table is also reported here, subsequently, wherein the values are valid when the transistor is biased at $V_{\mathrm {ds}} = 3$ V and $I_{\mathrm {ds}} = 10$ mA.

Analysis of dual‐gate high electron mobility transistor using an unconditionally stable time domain method

A dual-gate aluminium gallium arsenide (GaAs)/GaAs pseudomorphic high electron mobility transistor is analysed based on the distributed modelling approach, wherein the transistor is considered as an active multi-conductor transmission line. Discretisation of the governing matrix Telegrapher's equation is carried out using the implicit Crank–Nicolson-finite-difference time-domain (CN-FDTD) method. The results obtained from the proposed method are compared with the results of the conventional leap-frog (LF) FDTD scheme. It is observed that the unconditionally stable CN method is in good agreement with the LF method even by increasing the time step size by the factor of 5000, leading to a dramatic decrease in the central processing unit time.

Domain Decomposition CN-FDTD Method for Analyzing Dispersive Metallic Gratings

In this paper, an efficient domain decomposition (DD) technique is originally introduced into the unconditionally stable Crank–Nicolson finite-difference time-domain (CN-FDTD) method for analyzing the extraordinary optical transmission (EOT) phenomenon of periodic metallic gratings. Being caused by the evanescent waves propagating along the metal-dielectric interface in the visible and near infrared regions, the dispersion of the considered metal in this case is expressed by the Drude model and solved with a generalized auxiliary differential equation (ADE) technique. The periodic boundary condition is applied to the two-dimensional periodic metallic grating structures due to their periodicity. Then, the standard unsplit-field perfectly matched layer is derived as the absorbing boundary condition for CN-FDTD based on the ADE concept. In DD structures, the whole computational domain is divided into several subdomains to reduce the matrix size and save calculating time. Furthermore, the reverse Cuthill–Mckee scheme for the lower–upper decomposition is applied to the subdomain matrices individually to improve the efficiency of DD-CN-FDTD. Finally, two numerical examples are calculated and the physical mechanism of the EOT phenomenon is investigated. The results from the proposed method demonstrate its accuracy and efficiency for the nanostructures.

Linear Analysis of High-Frequency Field-Effect Transistors Using the CN-FDTD Method

The Crank–Nicolson finite-difference time-domain (CN-FDTD) method is used for the linear analysis of a field-effect transistor (FET) based on the distributed model, wherein the FET is considered as an active multiconductor transmission line (AMCTL). The telegrapher’s equations governing the AMCTL are derived and discretized by the CN-FDTD method. The results are compared with measurements and the leapfrog (LF)-FDTD method. Unconditional stability of the CN scheme is investigated. Agreement with measured results is preserved in an acceptable level even by increasing the time step-size by a factor of 1500 to that of the LF-FDTD method.

Application of the Crank–Nicolson FDTD method for analysis of a wideband multi-section hybrid coupler

In this paper the Crank---Nicolson (CN) finite-difference time-domain (FDTD) method is applied for the analysis of a planar wideband hybrid coupler. The proposed method is applied for analysis of transmission lines. Stability issue is investigated for different cases, including lossless and lossy transmission lines. Sufficient conditions for unconditional stability are derived. As a practical problem, a wideband compact hybrid coupler working on 1---6 GHz frequency band is analyzed using the CN-FDTD method. Measurement results show that the proposed coupler provides coupling of 3 ± 0.8 dB with 90° ± 0.85° phase shift and demonstrate the return loss and isolation better than 16 dB over 1---6 GHz. To validate the accuracy of the method, the results of this scheme are compared with measurements and the conditional stable leap-frog (LF) FDTD method. It is observed that using the CN-FDTD, the temporal step-size can be increased up to 1500 time compared with the LF-FDTD method with still good agreement with the measured results.

A research for plasma electromagnetic character using JEC-CN-FDTD algorithm based on ICCG method

For the first time, the current density convolution finite-difference time-domain (JEC-FDTD) method is extended to dispersive media using Crank–Nicolson difference scheme, and derive the one-dimensional JEC-CN-FDTD iterative equation of plasma. The incomplete Cholesky conjugate gradient (ICCG) is proposed to solve the large sparse matrix equation in the Crank–Nicolson finite-difference time-domain (CN-FDTD) method as the ICCG method improves speed of convergence, enhances stability and reduces memory consumption. The high accuracy and efficiency of the JEC-CN-FDTD method are confirmed by computing the characteristic parameters of electromagnetic wave through a collision plasma slab in one dimension, such as the reflection electric field, transmission electric field, magnitudes and phases of reflection coefficient and transmission coefficient. The results prove that the method performs stably, is of accuracy, and has certain advantages.

Two Approximate Crank–Nicolson Finite -Difference Time-Domain Method for $TE_z$ Waves

Two approximate Crank-Nicolson finite-difference time-domain (CN-FDTD) methods are presented for a two-dimensional TEz wave. The stability conditions of the proposed methods are presented analytically and the numerical performance of these methods is demonstrated by comparing with those of the alternating-direction implicit (ADI) FDTD and conventional FDTD methods.

An improved formalism for the Crank‐Nicolson finite‐difference time‐domain by conformal mapping method

AbstractIn this article, a three‐dimensional (3‐D) Crank‐Nicolson finite‐difference time‐domain (CN‐FDTD) method combined with conformal mapping method is used to analyze the cavities with curved surface. The improved formalism in the CN‐FDTD is described and both time‐domain waveforms and resonant frequencies are presented. The numerical simulations of the typical examples show that the time step with this CN‐FDTD method is 10 times larger than the Courant‐Friedrich‐Levy condition without the large loss of accuracy. © 2009 Wiley Periodicals, Inc. Microwave Opt Technol Lett 51: 2378–2382, 2009; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.24613

Extend two-step preconditioning technique for the Crank-Nicolson finite-difference time-domain method to analyze the 3D microwave circuits

The increase of the time-step size significantly deteriorates the property of the coefficient matrix generated from the Crank–Nicolson finite-difference time-domain method. As a result, the convergence of classical iterative methods, such as generalized minimal residual method (GMRES) would be substantially slowed down. The main purpose of this article is to show that the two-step preconditioner can greatly accelerate the convergence of the iterative method, such as GMRES. Numerical simulations show that this two-step preconditioned GMRES iterative method has a faster convergence speed than those methods when the preconditioner is used alone. © 2009 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2009.

3D Crank-Nicolson finite difference time domain method for dispersive media

The unconditionally stable Crank-Nicolson finite difference time domain (CN-FDTD) method is extended to incorporate frequency-dependent media in three dimensions. A Gaussian-elimination-based direct sparse solver is used to deal with the large sparse matrix system arising from the formulation. Numerical results validate and confirm that the scheme is unconditionally stable for time steps over the Courant-Friedrich-Lewy limit of classical FDTD.