Finite-difference Time-domain Scheme Research Articles

A new variable-order fractional momentum operator for wave absorption when solving Schrödinger equations

Absorbing boundary conditions are often needed for solving Schrödinger equations when the analytical solution is unknown. This article presents a new wave-absorption method by unifying the Schrödinger equations with associated one-way wave equations via a fractional-order momentum operator pˆα+1 where 0≤α≤1. By gradually varying the fractional order α, the computational boundary is made to be transparent. This method allows for incident waves to propagate freely and forces reflected waves to decay exponentially with time. The unified equation is then solved using the Finite-Difference Time-Domain (FDTD) method so that the scheme is iteratively explicit and can be parallelized easily. Finally, the obtained fractional FDTD scheme is tested to simulate soliton and particle propagations and is compared with the existing Perfectly Matched Layer (PML) method. Results show that the new method is promising.

On the Application of Fractional Derivative Operator Theory to the Electromagnetic Modeling of Frequency Dispersive Media

Fractional derivative operators are finding applications in a wide variety of fields with their ability to better model certain phenomena exhibiting spatial and temporal nonlocality. One area in which these operators are applicable is in the field of electromagnetism, thereby modelling transient wave propagation in complex media. To apply fractional derivative operators to electromagnetic problems, the operator must adhere to certain principles, like the trigonometric functions invariance property. The Grünwald–Letnikov and Marchaud fractional derivative operators comply with these principles and therefore could be applied. The fractional derivative arises when modelling frequency-dispersive dielectric media. The time-domain convolution integral in the relation between the electric displacement and the polarisation density, containing an empirical extension of the Debye model, is approximated directly. A common approach is to recursively update the convolution integral by approximating the time series by a truncated sum of decaying exponentials, with the coefficients found through means of optimisation or fitting. The finite-difference time-domain schemes using this approach have shown to be more computationally efficient compared to other approaches using auxiliary differential equation methods.

Wide-viewing-angle panoramic inhomogeneous free-shape lens

This paper presents a design of an inhomogeneous lens with an arbitrary curvature supporting a high field of view without distortion and aberration. The lens profile is designed based on the ray-inserting method (RIM). The advantage of this proposed structure is matching the input and output of the rays. No approximation or optimization has been considered in the innovation of the lens. Unlike most previous works, the present design can be implemented with isotropic and above unity refractive index materials. The Finite Difference Time Domain (FDTD) scheme and multi-physics simulations are employed to validate the design lens.

The low‐cost inhomogeneous multi‐section flat lens antenna with minimum F over D design in Ku‐band

AbstractThis article uses the ray inserting method to design a low‐cost inhomogeneous lens in the Ku‐band. The focal length to the diametre ratio of the designed lens is 0.45, which is lower than most designed lenses in this area. Also, no approximation is considered to yield the refractive index profile. The refractive index changes from the centre to the sides have been made in two sections to limit the maximum calculated refractive index of the structure to the Plexiglass refractive index. The designed lens proposed a good match to the source and surroundings. The theoretical design is validated by the finite difference time domain scheme and full‐wave simulation. The lens structure is realised and fabricated on a multilayer Plexiglass sheet. The results of simulations and experiments indicate good performances of the designed lens in the wide frequency bandwidth.

Numerical model of nonlinear elastic bulk wave propagation in solids for non-destructive evaluation

Nonlinear ultrasonic techniques can be difficult and non-intuitive to understand due to the range of wave mixing combinations available between similar or different wave modes. To overcome this, a numerical model that uses a Finite Difference Time Domain (FDTD) scheme, without a staggered grid system, to solve the nonlinear elastic bulk wave equations in two dimensions is proposed in this paper, with the purpose of better understanding nonlinear ultrasonic techniques. Both material and geometrical nonlinearities are considered and a stress-type boundary condition is used to model the excitation. The energy transfer between frequency bands is shown and the amplitude trend of the harmonics are validated against available theories. It is then used to simulate the nonlinear ultrasonic field generated by a finite-size element of an array in a solid to better understand its behaviour. An array-element directivity at the second harmonic frequency is shown. Since the FDTD model does not account for attenuation, a correction using a ray-based approach is applied to the simulated result for direct comparison against experimental measurements. The field obtained from a typical array used in non-destructive testing is then simulated to characterise its behaviour. Experimental comparison of the nonlinear displacement fields for different focal positions showed good agreement, further validating the FDTD model. This FDTD model opens opportunities to virtually experiment and design appropriate nonlinear ultrasonic array inspections whilst isolating and understanding contribution from material nonlinearity.

Modeling impedance boundary conditions and acoustic barriers using the immersed boundary method: The three-dimensional case.

One of the main challenges in time domain wave-based acoustics is the accurate simulation of both boundary conditions and barriers capable of reflecting and transmitting energy. Such scattering structures are generally of irregular geometry and characterised in terms of frequency-dependent reflectances and transittances. Conditions for numerical stability can be difficult to obtain in either case. Immersed boundary methods, which are heavily used in computational fluid dynamics applications, replace boundaries by discrete driving terms, avoiding volumetric meshing and staircasing approaches altogether. The main contribution of this article is a unified numerical treatment of both impedance boundary conditions and barriers capable of transmitting energy and suitable for use in the setting of wave-based acoustics. It is framed in terms of the immersed boundary method within a finite difference time domain scheme, using a dual set of matched discrete driving terms in both the conservation of mass and momentum equations that can be tuned against a desired reflectance or transmittance. Numerical results in three dimensions are presented, illustrating non-porous barriers and impedance boundary conditions, and highlight important features such as spurious leakage through an immersed boundary. A brief demonstration of conditions for numerical stability of the immersed boundary method in this context is provided in an appendix.

Newton-ADE FDTD Method for Time-Varying Plasma

A novel finite-difference time-domain (FDTD) method is proposed in this article for electromagnetic (EM) wave propagation in time-varying plasma. It is formulated based on the fundamental Newton’s equation of motion, which governs the relationships among the time-varying electron density, current density and electric field. Utilizing the auxiliary differential equation (ADE) FDTD scheme, the method is aptly referred to as the Newton-ADE FDTD method for time-varying plasma. Traditionally, the previous ADE FDTD methods directly apply the time-varying electron density for the plasma frequency in the update equations of current density and electric field. This is inadequate and incorrect in general time-varying plasma conditions. The formulation of the Newton-ADE FDTD method is provided and compared to that of the traditional ADE FDTD method. It is found that the key difference lies in the new compact term being the time-derivative of logarithm of electron density. Two discretization schemes for this term are provided. The Newton-ADE FDTD method is validated based on the matrix exponential method. Novel stability and convergence analyses are provided for the proposed method in time-varying dispersive media. These analyses show that our proposed method is stable and achieves second-order temporal accuracy. Numerical results are presented for various time-varying plasma conditions. It is demonstrated that when the absolute value for time-derivative of logarithm of electron density is large, e.g., greater than the collision frequency, the Newton-ADE FDTD method can provide correct results, while the traditional ADE FDTD method yields significant differences.

Rupture-induced dynamic pore pressure effect on rupture

The ordinary elasticity model is often used in dynamic rupture earthquake simulation without incorporating the poroelasticity effect in rocks. In consideration that rocks are porous and saturated, we carry out a two-dimensional simulation of spontaneous rupture in a poroelastic medium with a finite difference time domain scheme. In particular, the influence of the rupture-induced pore pressure on the rupture process is investigated. The fault plane is idealized as an impermeable layer of negligible thickness, and the rupture is assumed to be governed by the slip-weakening friction law. Results show that an impermeable fault plane leads to a rupture-induced pore pressure change, which promotes or hinders the rupture process by altering effective normal compressive stress (normal compressive stress minus pore pressure). It is remarkable that super-shear rupturing will occur when the porosity of the rock around the fault increases to some extent. The comparison with the permeable fault model suggests that the permeability of the fault and the porosity of rocks around the fault significantly influence the dynamic rupture process.

Investigating the stability of an explicit ADE-FDTD scheme for modeling graphene: Avoiding erroneous conclusions

In recent years, an explicit auxiliary differential equation finite difference time domain (ADE-FDTD) scheme has been frequently used in graphene simulations. In this respect, a differencing scheme in which the magnetic field and the current density are collocated in time was developed and it has been recently stated that this explicit ADE-FDTD implementation is “conditionally stable with the maximum time-step size bounded by the common Courant–Friedrichs–Lewy (CFL) limit”. In this communication, the stability of this explicit ADE-FDTD scheme is revisited and it is shown that the conventional CFL constraint is not retained, and a time-step less than the CFL limit must be used in all domain grids to insure the stability in the whole system. Although this time-step stringent can be of little effect for some graphene applications, and in order to avoid drawing erroneous conclusions, it is shown that when the given ADE-FDTD scheme is used for simulating graphene structures that include noble and other plasmonic materials the derived time-step stringent can be of significant effect. To retain the CFL stability constraint, alternative formulation based on the Runge–Kutta (RK) time-differencing scheme is also presented in this communication. Contrary to the other existing CFL-stable ADE-FDTD approaches, the presented RK-FDTD method is equally applicable to both magnetized and unmagnetized graphene structures without incurring additional computational cost. Furthermore, the RK-FDTD scheme is extended for modeling the complex-frequency-shifted perfectly matched layer (CFS-PML) mesh truncating technique. Finally, the stability and accuracy of the explicit ADE/RK-FDTD schemes are demonstrated through numerical examples.

Consistent FDTD Modeling of Dispersive Dielectric Media via Multiple Debye Terms Derived Rigorously by Padé Approximants

An finite-difference time-domain (FDTD) scheme for simulating wave propagation inside Cole-Cole (CC), Havriliak-Negami (HN), or Raicu (R) dispersive media is proposed. The main difficulty in FDTD implementations for such media is the fact that the time-domain polarization relations are fractional integro-differential equations. To circumvent this difficulty, the dispersive susceptibility of the medium is modeled by Padé approximants. It is proven that under certain conditions the Padé approximant is equal to a weighted sum of Debye terms, while the physical restrictions for positive weights and relaxation times are fulfilled with certainty. Hence, a set of first-order auxiliary differential equations (ADEs), which approximate the original fractional polarization relation, is derived and directly embedded in the FDTD scheme. In contrast to previous works, the derivation of the Debye expansion of the original dispersive permittivity is carried out in a straightforward manner without involving nonlinear optimization techniques. Comparisons between analytical and FDTD results of the reflection and transmission coefficients as well as of the transfer functions inside the dispersive media illustrate the efficiency of the proposed scheme, over wideband frequency domains.

Pre-stack elastic reverse time migration in tunnels based on cylindrical coordinates

Seismic forward-prospecting in tunnels is an important step to ensure excavation safety. Nowadays, most advanced imaging techniques in seismic exploration involve calculating the solution of elastic wave equation in a certain coordinate system. However, considering the cylindrical geometry of common tunnel body, Cartesian coordinate system seemingly has limited applicability in tunnel seismic forward-prospecting. To accurately simulate the seismic signal received in tunnels, previous imaging method using decoupled non-conversion elastic wave equation is extended from Cartesian coordinates to cylindrical coordinates. The proposed method preserves the general finite-difference time-domain (FDTD) scheme in Cartesian coordinates, except for a novel wavefield calculation strategy addressing the singularity issue inherited at the cylindrical axis. Moreover, the procedure of cylindrical elastic reverse time migration (CERTM) in tunnels is introduced based on the decoupled non-conversion elastic wavefield. Its imaging effect is further validated via numerical experiments on typical tunnel models. As indicated in the synthetic examples, both the PP- and SS-images could clearly show the geological structure in front of the tunnel face without obvious crosstalk artifacts. Migration imaging using PP-waves can present satisfactory results with higher resolution information supplemented by the SS-images. The potential of applying the proposed method in real-world cases is demonstrated in a water diversion tunnel. In the end, we share our insights regarding the singularity treatment and further improvement of the proposed method.

Efficient high-resolution electric and magnetic field simulations inside the human body in the vicinity of wireless power transfer systems with varying models

PurposeMagneto-quasi-static fields emanated by inductive charging systems can be potentially harmful to the human body. Recent projects, such as TALAKO and MILAS, use the technique of wireless power transfer (WPT) to charge batteries of electrically powered vehicles. To ensure the safety of passengers, the exposing magnetic flux density needs to be measured in situ and compared to reference limit values. However, in the design phase of these systems, numerical simulations of the emanated magnetic flux density are inevitable. This study aims to present a tool along with a workflow, based on the Scaled-Frequency Finite Difference Time-Domain and Co-Simulation Scalar Potential Finite Difference schemes, to determine body-internal magnetic flux densities, electric field strengths and induced voltages into cardiac pacemakers. The simulations should be time efficient, with lower computational costs and minimal human workload.Design/methodology/approachThe numerical assessment of the human exposure to magneto-quasi-static fields is computationally expensive, especially when considering high-resolution discretization models of vehicles and WPT systems. Incorporating human body models into the simulation further enhances the number of mesh cells by multiple millions. Hence, the number of simulations including all components and human models needs to be limited while efficient numerical schemes need to be applied.FindingsThis work presents and compares four exposure scenarios using the presented numerical methods. By efficiently combining numerical methods, the simulation time can be reduced by a factor of 3.5 and the required storage space by almost a factor of 4.Originality/valueThis work presents and discusses an efficient way to determine the exposure of human beings in the vicinity of wireless power transfer systems that saves computer simulation resources and human workload.

A novel 1D-FDTD scheme to solve the nonlinear second-order thermoviscous hydrodynamic model

In this paper, we present a novel and simple Yee Finite-Difference Time-Domain (FDTD) scheme to solve numerically the nonlinear second-order thermoviscous Navier–Stokes and the Continuity equations. In their original form, these equations cannot be discretized by using the Yee’s mesh, at least, easily. As it is known, the use of the Yee’s mesh is recommended because it is optimized in order to obtain higher computational performance and remains at the core of many current acoustic FDTD softwares. In order to use the Yee’s mesh, we propose to rewrite the aforementioned equations in a novel form. To achieve this, we will use the substitution corollary. This procedure is novel in the literature. Although the scheme can be extended to more than one dimension, in this paper, we will focus only on the one-dimensional solution because it can be validated with two analytical solutions to the Burgers equation: the Mendousse mono-frequency solution and the Lardner bi-frequency solution. Numerical solutions are excellently consistent with the analytical solution, which demonstrates the effectiveness of our formulation.

Heterogeneous CPU-GPU Accelerated Subgridding in the FDTD Modelling of Microwave Breakdown

Microwave breakdown is crucial to the transmission of high-power microwave (HPM) devices, where a growing number of studies have analyzed the complex interactions between electromagnetic waves and the evolving plasma from theoretical and analytical perspectives. In this paper, we propose a finite-difference time-domain (FDTD) scheme to numerically solve Maxwell’s equation, coupled with a fluid plasma equation for simulating the plasma formation during HPM air breakdown. A subgridding method is adopted to obtain accurate results with lower computational resources. Moreover, the three-dimensional subgridding Maxwell–plasma algorithm is efficiently accelerated by utilizing heterogeneous computing technique based on graphics processing units (GPUs) and multiple central processing units (CPUs), which can be applied as an efficient method for the investigation of the HPM air breakdown phenomena.

Exploiting temporal data reuse and asynchrony in the reverse time migration

Reverse Time Migration (RTM) is a state-of-the-art algorithm used in seismic depth imaging in complex geological environments for the oil and gas exploration industry. It calculates high-resolution images by solving the three-dimensional acoustic wave equation using seismic datasets recorded at various receiver locations. Reverse Time Migration’s computational phases are predominantly composed of stencil computational kernels for the finite-difference time-domain scheme, applying the absorbing boundary conditions, and I/O operations needed for the imaging condition. In this paper, we integrate the asynchronous Multicore Wavefront Diamond (MWD) tiling approach into the full RTM workflow. Multicore Wavefront Diamond permits to further increase data reuse by leveraging spatial with Temporal Blocking (TB) during the stencil computations. This integration engenders new challenges with a snowball effect on the legacy synchronous RTM workflow as it requires rethinking of how the absorbing boundary conditions, the I/O operations, and the imaging condition operate. These disruptive changes are necessary to maintain the performance superiority of asynchronous stencil execution throughout the time integration, while ensuring the quality of the subsurface image does not deteriorate. We assess the overall performance of the new MWD-based RTM and compare against traditional Spatial Blocking (SB)-based RTM on various shared-memory systems using the SEG Salt3D model. The MWD-based RTM achieves up to 70% performance speedup compared to SB-based RTM. To our knowledge, this paper highlights for the first time the applicability of asynchronous executions with temporal blocking throughout the whole RTM. This may eventually create new research opportunities in improving hydrocarbon extraction for the petroleum industry.

Extended FDTD Method for Coanalysis of Superconducting Passive Transmission Lines and Josephson Junction Drivers and Receivers

In rapid single-flux quantum circuits, the signal integrity problem of superconducting passive transmission lines (PTLs) becomes critical. In this article, an extended 3-D finite-difference time-domain (FDTD) method is proposed for coanalysis of superconducting PTLs and Josephson junction (JJ) drivers and receivers. The central difference FDTD scheme is used for PTLs, and the backward difference method is adopted for JJs, which are introduced to FDTD cells as lumped elements. Due to the nonlinearity of JJs, the Newton–Raphson method is introduced to solve the boundary condition of the FDTD method. Compared with the 1-D FDTD method of the transmission line model of PTLs, the proposed method performs full-wave analysis of the physical structure of PTLs. Therefore, the proposed 3-D FDTD method is theoretically more accurate than the conventional method by avoiding parameter extraction and circuit analysis.

Beam steered graphene-based Yagi-Uda array antenna with a transverse magnetic to hybrid mode conversion approach.

This work presents the existence of the mode conversion technique in a graphene-based Yagi-Uda array antenna. It comprises four arrays whose strands are placed on a silicon dioxide substrate, and are eventually connected with the graphene ring. All four driven elements of the Yagi-Uda array antenna are excited through the 50Ω silver nanostrip feedline. The proposed antenna offers mode conversion due to variation of the chemical potential of graphene. It controls conversion of the TM32δ to HEM21δ mode of the antenna. This is attributed to the change in the biasing voltage of graphene. This in turn shifts the radiation pattern from the end-fire to the broadside direction, which effectively confirms the beam reconfigurability. This antenna provides a high directivity of 12.21 dBi at 4.55 THz center frequency. The proposed antenna is designed and analyzed by using CST Microwave Studio, which is based on the finite difference time domain scheme. The beam steered graphene-based antenna has been utilized for several terahertz communication systems.

A Novel FDTD–PML Scheme for Noise Propagation Generated by Biomimetic Flapping Thrusters in the Ocean Environment

Biomimetic flapping-foil thrusters can operate efficiently while offering desirable levels of thrust required for the propulsion of a small vessel or an Autonomous Underwater Vehicle (AUV). These systems have been studied both as main propulsion devices and for augmenting ship propulsion in waves. In this work, the unsteady hydrofoil loads are used to calculate the source terms of the Ffowcs Williams–Hawkings (FW-H) equation which is applied to model noise propagation in the underwater ocean acoustic environment. The solution provided by a simplified version of the Farassat formulation in free space is extended to account for a bounded domain and an inhomogeneous medium, characterizing the sea acoustic waveguide. Assuming the simplicity azimuthal symmetry of the environmental parameters, a numerical model is developed based on a Finite Difference Time Domain (FDTD) scheme, incorporating free-surface and seabed effects, in the presence of a variable sound speed profile. For the treatment of the outgoing radiating field, a Perfectly Matched Layer (PML) technique is implemented. Numerical results are presented illustrating the applicability of the method.

Implicit finite-difference time-domain schemes for TDEM modeling in three dimensions

Geophysical exploration methods that use controlled electromagnetic sources in time domain are becoming ubiquitous because of their ease of deployment and data coverage capabilities. Although these prospecting methodologies have evolved significantly, most computational numerical modeling techniques have been developed in the frequency domain. Given this evolution, it is pertinent to ask whether some known advantages of implicit finite-difference (FD) modeling techniques, which are common among other disciplines, apply to the time-domain electromagnetic (TDEM) problem in geophysics. To explore the potential advantages of 3D time-domain modeling for TDEM, we have analyzed the differences between two implicit FD formulations: a single-order backward Euler scheme and a second-order Crank-Nicolson scheme. To validate our algorithms, we tested them with existing analytical solutions for simple geometries and performed detectability tests for various electromagnetic sources. The results find an acceptable match of both numerical schemes to the analytical solutions and higher accuracy of the second-order discrete operator without a significant difference in computing time.

A Unified View of DI- and ETD-FDTD Methods for Drude Media

A unified view of direct-integration (DI) and exponential-time-differencing (ETD) methods to incorporate Drude media, such as isotropic plasma and microwave graphene, into finite-difference time-domain (FDTD) simulators is provided. To this end, the Drude constitutive relation is expressed in integral form and the DI integrators are obtained by applying quadrature rules. Analogously, the ETD integrators are obtained by starting from the variation of constants formula and applying the same quadrature rules as in the DI case. This approach allows one to directly compare the two families of methods. In addition, the accuracy of each integrator is discussed and the stability condition of the resulting FDTD schemes is derived in exact closed form by applying the von Neumann method.