Alternating Direction Implicit Finite-difference Time-domain Research Articles

Kinetic simulations of low-pressure inductively coupled plasma: an implicit electromagnetic PIC/MCC model with the ADI-FDTD method

Low-pressure inductively coupled plasma (ICP) is promising for space electric propulsion. For the first time, an implicit electromagnetic particle-in-cell Monte Carlo collision model based on the alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method is developed to investigate low-pressure xenon plasma characteristics of a miniature ICP source. The induced simulated electric field is well consistent with that calculated by the finite element method, indicating that this method can provide an accurate estimation of the electromagnetic field. The simulation time step used in the ADI-FDTD method is no longer restricted by the Courant–Friedrichs–Lewy constraints. Compared with the FDTD method, the ADI-FDTD method increases the size of the time step and significantly improves computational efficiency. The method is validated by comparing the simulated and measured electron density and plasma potential profile and reasonable agreement is reached. Therefore, the model is used to investigate the temporal and spatial distribution of plasma properties and the influence of the current amplitude of radio frequency (RF) coil, applied frequency of RF coil and neutral gas pressure on the plasma dynamics in the ionization chamber of a miniature gridded RF ion thruster. To explain the influence of the operating parameters, a concept called ‘the energy relaxation characteristics of electrons in response to the change of electric field’ is proposed and verified. The simulations also find that the oscillation frequency of plasma properties is twice the applied frequency of RF coil. The oscillation characteristics reveal the dynamic energy balance in the ICP. The experiment on the gridded RF ion thruster BHRIT-4 confirms the oscillation by measuring the plasma sheath potential.

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.

Current Source Implementations for the HIE-FDTD Methods

In this letter, the current source implementations are presented for the hybrid implicit–explicit finite-difference time-domain (HIE-FDTD), the improved HIE-FDTD, and the leapfrog HIE-FDTD methods. The asymmetry and the field errors of these HIE-FDTD methods are investigated for all possible values of the time index parameters. The proposed implementations give very low asymmetry and field errors for all the three HIE–FDTD methods. The field errors given by these HIE-FDTD methods are lower than those given by the alternating direction implicit finite-difference time-domain (ADI-FDTD) method.

Modeling GPR Wave Propagation in Complex Underground Structures Using Conformal ADI-FDTD Algorithm

Ground Penetrating Radar (GPR) is a shallow geophysical method for detecting and locating subsurface targets. The GPR image echo characteristics of complex underground structures can be obtained by carrying out GPR forward modeling research. The traditional finite-difference time-domain (FDTD) method has low efficiency and accuracy. The alternating direction implicit FDTD (ADI-FDTD) algorithm surmounts the stability limitations of the traditional FDTD method, making it possible to select a larger time step for higher computational efficiency. For circular underground structures, a pseudowave produced by the ladder approximation method can be corrected using the surface conformal technique. This paper proposes a high-efficiency and high-accuracy GPR forward modeling method that combines the ADI-FDTD algorithm and surface conformal technology. The performance of the conformal ADI-FDTD algorithm is verified by a simple two-layer model. Based on the proposed algorithm, the GPR image features of three complex underground structure models are obtained. Finally, a field experiment is used to support the accuracy and usefulness of the conformal ADI-FDTD algorithm. The numerical simulation results and experimental results show that the conformal ADI-FDTD algorithm reduces the pseudodiffraction wave caused by the ladder approximation method and can significantly improve the computing efficiency for complex underground structure models.

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.

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.

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.

Three-Dimensional Higher Order PML Based on Alternating Direction Implicit Algorithm

Based on the alternating direction implicit finite-difference time-domain (ADI-FDTD) algorithm, a higher order perfectly matched layer (HO-PML) is proposed to solve the problems in a three-dimensional computational domain. The proposed scheme can not only take advantage of the HO-PML for improving the absorbing performance but also has the advantage of the ADI-FDTD algorithm in terms of the unconditional stability. A dielectric resonator antenna example is carried out to validate its effectiveness. The results show that the Courant–Friedrich–Levy limit can be eliminated and the absorbing performance can be further enhanced.

Extended ADI‐FDTD for anisotropic magnetised graphene sheets

The unconditionally stable alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method is extended for modelling anisotropic magnetised graphene sheets. In contrast to the conventional method for isotropic media, two additional sub-stepping calculations are applied to incorporate anisotropic medium properties. The unconditional stability of the proposed extended method is examined numerically. The method is applied to study the transmission properties of graphene sheets. Numerical results show that the proposed method is more accurate than the recently developed locally one-dimensional 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.

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.

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.

A Multiscale Unconditionally Stable Time-Domain (MUST) Solver Unifying Electrodynamics and Micromagnetics

A rigorous yet computationally efficient 3-D numerical method has been proposed based on modified alternatingdirection-implicit finite-difference time-domain (ADI FDTD) methods. It has the capability of modeling the anisotropic and dispersive properties of magnetic material. The proposed algorithm solves Maxwell's equations and Landau-Lifshitz-Gilbert equation jointly and simultaneously, requiring only tridiagonal matrix inversion as in ADI FDTD. The accuracy of the modeling has been validated by: 1) the nonreciprocity of an X-band ferrite resonance isolator; 2) the attenuation constant of a magnetically tunable waveguide filter; and 3) the dispersive permeability of a 2-μm-thick magnetic thin film. Time steps that are up to 5000 times larger than the Courant-Friedrichs-Lewy limit have been used in these simulations without encountering stability issues. The simulation results agree with the predictions made from the theory, the commercial software or the experiments. Moreover, the algorithm has been applied to predict the effect of high permeability thin films in platform effect reduction. An electric current sheet close to a perfect electrically conducting plane coated with a 2-μm-thick magnetic thin film is simulated, which exhibits an enhanced surface resistance by three orders of magnitude higher than that without the magnetic thin film.

Numerical modeling of electromagnetic wave logging while drilling in deviated well

Electromagnetic wave logging-while-drilling (LWD) tools are used in geosteering for hydrocarbon exploration. Comparing with vertical wells, the response of logging has widely changed because the electrical parameters surrounding the borehole are nonaxisymmetrical distribution in deviated well. In this paper, with the method of alternating-direction-implicit finite-difference time-domain (ADI-FDTD), we discuss the logging response of electromagnetic wave LWD in deviated well, three-dimensional Yee’s non-uniform staggered grid is used in cylindrical coordinates, transfer coordinate system is built between strata space and instrument space, which the conductivity tensor of the anisotropic and dipping formation can be expressed in coordinates of instrument, the method of area weighted average is used to compute the effective conductivity of partially-filled grid cells at interfaces, and uniaxial perfectly matched layer (UPML) absorbing boundary conditions is used to truncate the computational domain. Result shows that horn of logging response curve is appeared on both upper and lower boundaries, when electromagnetic wave LWD tool penetrating through the boundary with large dipping angle, and this horn is used to indicate the presence of formation boundaries. What’s more, with the eccentric distance increases, horns effect of the boundary is more obvious.

Reduced shift operator alternating direction implicit FDTD applied to plasma

ABSTRACTIn order to analyze the electromagnetic characteristics of plasma more efficiently, a novel FDTD called R-SO-ADI FDTD is proposed. The proposed method which is based on the shift operator technique has clear conceptions and simple formulae, and it saves CPU time and reduces the storage of the field components through combining the reduced formulation technique and the ADI scheme. Calculation results of the RCS of a perfectly conducting cylinder coated by plasma confirm the validity of the proposed method.

Memory-optimized shift operator alternating direction implicit finite difference time domain method for plasma

Through introducing the alternating direction implicit (ADI) technique and the memory-optimized algorithm to the shift operator (SO) finite difference time domain (FDTD) method, the memory-optimized SO-ADI FDTD for nonmagnetized collisional plasma is proposed and the corresponding formulae of the proposed method for programming are deduced. In order to further the computational efficiency, the iteration method rather than Gauss elimination method is employed to solve the equation set in the derivation of the formulae. Complicated transformations and convolutions are avoided in the proposed method compared with the Z transforms (ZT) ADI FDTD method and the piecewise linear JE recursive convolution (PLJERC) ADI FDTD method. The numerical dispersion of the SO-ADI FDTD method with different plasma frequencies and electron collision frequencies is analyzed and the appropriate ratio of grid size to the minimum wavelength is given. The accuracy of the proposed method is validated by the reflection coefficient test on a nonmagnetized collisional plasma sheet. The testing results show that the proposed method is advantageous for improving computational efficiency and saving computer memory. The reflection coefficient of a perfect electric conductor (PEC) sheet covered by multilayer plasma and the RCS of the objects coated by plasma are calculated by the proposed method and the simulation results are analyzed.

Using the ADI-FDTD Method to Simulate Graphene-Based FSS at Terahertz Frequency

The alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method is modified to simulate a graphene-based frequency selective surface. By using the auxiliary differential equation and Pade fitting method, both the interband and intraband conductivity of the graphene are incorporated into the ADI-FDTD method. The ADI-FDTD method has excellent computational accuracy and its computational efficiency is considerably improved from that of the conventional FDTD method. By using the proposed ADI-FDTD method, a graphene-based frequency selective surface is simulated and analyzed at terahertz frequency. The numerical results show that the graphene can achieve a tunable frequency selective surface through controlling its chemical potential, and the interband conductivity of the graphene has important effect on the performance of the frequency selective surface.

Optimized (2, 4) Stencil Runge–Kutta ADE-ADI FDTD With Application to Plasma

This paper discusses the improvement of the numerical dispersion characteristics of alternating direction implicit (ADI) finite-difference time-domain (FDTD) aimed at acquiring more accurate electromagnetic information of plasma. Through adding the optimization method, which is based on the optimization of spatial derivative to the (2, 4) stencil ADI FDTD, the optimized (2, 4) stencil ADI FDTD is proposed, and its unconditional stability is proved theoretically. The phase velocity error of the optimized (2, 4) stencil ADI FDTD versus propagation angle and grid density is investigated. In addition, the Runge–Kutta auxiliary differential equation (RKADE) scheme for tackling the constitutive relation equation of plasma is deduced, which is without additional storage occupation and computational burden compared with ADE scheme. Its numerical conductivity error is analyzed under different incident frequencies and electron collision frequencies. Through incorporating the RKADE scheme into the optimized (2, 4) stencil ADI FDTD, the optimized (2, 4) stencil RKADE-ADI FDTD is presented. The accuracy and relatively wideband capability of the proposed method is validated by two numerical experiments.

Improved Hybrid Leapfrog ADI-FDTD Method for Simulating Near-Field Coupling Effects Among Multiple Thin Wire Monopole Antennas on a Complex Platform

A hybrid one-step leapfrog alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method integrated with a thin wire algorithm is developed for simulating the near-field mutual coupling effects among multiple wire antennas on an electrically large warship platform, where induced surface current and electric field distributions are also captured. Its stability analysis, based on von Neumann method, is performed in a semi-analytical way and verified numerically. In order to demonstrate its accuracy and efficiency, the near-field mutual coupling effect between two thin wire antennas is at first investigated. Then, it is applied to accurately characterize complex electromagnetic environment effects on a warship with multiple thin wire antennas, and the results are compared with those obtained by the other commercial simulators.