Finite-difference Time-domain Scheme Research Articles

Bayesian tomography with prior-knowledge-based parametrization and surrogate modelling

SUMMARYWe present a Bayesian tomography framework operating with prior-knowledge-based parametrization that is accelerated by surrogate models. Standard high-fidelity forward solvers (e.g. finite-difference time-domain schemes) solve wave equations with natural spatial parametrizations based on fine discretization. Similar parametrizations, typically involving tens of thousand of variables, are usually employed to parametrize the subsurface in tomography applications. When the data do not allow to resolve details at such finely parametrized scales, it is often beneficial to instead rely on a prior-knowledge-based parametrization defined on a lower dimension domain (or manifold). Due to the increased identifiability in the reduced domain, the concomitant inversion is better constrained and generally faster. We illustrate the potential of a prior-knowledge-based approach by considering ground penetrating radar (GPR) traveltime tomography in a crosshole configuration with synthetic data. An effective parametrization of the input (i.e. the permittivity distributions determining the slowness field) and compression in the output (i.e. the traveltime gathers) spaces are achieved via data-driven principal component decomposition based on random realizations of the prior Gaussian-process model with a truncation determined by the performances of the standard solver on the full and reduced model domains. To accelerate the inversion process, we employ a high-fidelity polynomial chaos expansion (PCE) surrogate model. We investigate the impact of the size of the training set on the performance of the PCE and show that a few hundreds design data sets is sufficient to provide reliable Markov chain Monte Carlo inversion at a fraction of the cost associated with a standard approach involving a fine discretization and physics-based forward solvers. Appropriate uncertainty quantification is achieved by reintroducing the truncated higher order principle components in the original model space after inversion on the manifold and by adapting a likelihood function that accounts for the fact that the truncated higher order components are not completely located in the null space.

Modeling Graphene-Based THz Devices With an Improved Surface Boundary Condition in Leapfrog FDTD Scheme

Finite-difference time-domain (FDTD) method is an efficient full-wave numerical method for simulating graphene-based optoelectronic devices. The previous investigation on modeling graphene in leapfrog FDTD methods shows that dispersive errors increase with tuning up the chemical potential of graphene. In this article, a new implementation of graphene’s surface boundary condition (SBC) is proposed to reduce the dispersive errors in the leapfrog FDTD update scheme. To achieve lower dispersive errors, the graphene layer is positioned exactly on the grid lattice of transverse magnetic fields, and each of the transverse magnetic fields is split into two fields on the two sides of graphene. The split magnetic fields are updated using forward and backward differences in the electric fields on the normal direction of graphene plane. Numerical analysis shows that the maximum dispersive error of the proposed graphene model is reduced to only 37.5% of the other publicly reported graphene models in the leapfrog FDTD update scheme. However, the CPU time cost keeps almost the same as the other graphene models. The proposed implementation of the graphene SBC model also shows less CPU time consumption compared with the graphene SBC model which requires to solve a linear system and breaks the leapfrog FDTD update scheme, while the maximum numerical error is the same. The proposed graphene model is applied to simulate graphene terahertz (THz) surface plasmon resonances and modulators to demonstrate the applications in accurately modeling and analyzing graphene-based THz devices.

A Dissipation Theory for Potentials-Based FDTD for Lossless Inhomogeneous Media

A dissipation theory is proposed for the potentials-based finite-difference time-domain (FDTD) algorithm for the case of inhomogeneous lossless media. We show that under the Courant–Friedrichs–Lewy limit, the equations describing the time evolution of scalar and vector potentials can be seen as a lossless system. The developed theory provides insights into how electromagnetic energy and power flow are approximated in FDTD schemes. It can also be used to create new algorithms with guaranteed stability.

High-order accurate schemes for Maxwell's equations with nonlinear active media and material interfaces

We describe a fourth-order accurate finite-difference time-domain scheme for solving dispersive Maxwell's equations with nonlinear multi-level carrier kinetics models. The scheme is based on an efficient single-step three time-level modified equation approach for Maxwell's equations in second-order form for the electric field coupled to ODEs for the polarization vectors and population densities of the atomic levels. The resulting scheme has a large CFL-one time-step. Curved interfaces between different materials are accurately treated with curvilinear grids and compatibility conditions. A novel hierarchical modified equation approach leads to an explicit scheme that does not require any nonlinear iterations. The hierarchical approach at interfaces leads to local updates at the interface with no coupling in the tangential directions. Complex geometry is treated with overset grids. Numerical stability is maintained using high-order upwind dissipation designed for Maxwell's equations in second-order form. The scheme is carefully verified for a number of two and three-dimensional problems. The resulting numerical model with generalized dispersion and arbitrary nonlinear multi-level system can be used for many plasmonic applications such as for ab initio time domain modeling of nonlinear engineered materials for nanolasing applications, where nano-patterned plasmonic dispersive arrays are used to enhance otherwise weak nonlinearity in the active media.

Acoustic Wave Propagation in a Borehole with a Gas Hydrate-Bearing Sediment

A knowledge of wave propagation in boreholes with gas hydrate-bearing sediments, a typical three-phase porous medium, is of great significance for better applications of acoustic logging information on the exploitation of gas hydrate. To study the wave propagation in such waveguides based on the Carcione–Leclaire three-phase theory, according to the equations of motion and constitutive relations, a staggered-grid finite-difference time-domain (FDTD) scheme and a real axis integration (RAI) algorithm in a two-dimensional (2D) cylindrical coordinate system are proposed. In the FDTD scheme, the partition method is used to solve the stiff problem, and the nonsplitting perfect matched layer (NPML) scheme is extended to solve the problem of the false reflection waves from the artificial boundaries of the computational region. In the RAI algorithm, combined with six boundary conditions, the displacement potentials of waves are studied to calculate the borehole acoustic wavefields. The effectiveness is verified by comparing the results of the two algorithms. On this basis, the acoustic logs within a gas hydrate-bearing sediment are investigated. In particular, the wave field in a borehole is analyzed and the amplitude of a Stoneley wave under different hydrate saturations is studied. The results indicate that the attenuation coefficient of the Stoneley wave increases with the increase of gas hydrate saturation. The acoustic responses in a borehole embedded in a horizontally stratified hydrate formation are also simulated by using the proposed FDTD scheme. The result shows that the amplitude of the Stoneley wave from the upper interface is smaller than that from the bottom interface.

A Novel High-Order Symplectic Compact FDTD Schemes for Optical Waveguide Simulation

As a 2-D full-wave numerical algorithm in the time domain, the compact Finite Difference Time Domain (FDTD) is an efficient algorithm for eigenvalue analysis of optical waveguide system. However, the numerical dispersion accuracy and stability of fast algorithm need to be improved while simulating at high frequency. A novel high-order symplectic compact FDTD scheme is developed and validated for optical waveguide modal analysis. The stability condition and the numerical dispersion of schemes with fourth-order accuracy in temporal and spatial using the symplectic integrator and compact scheme are analyzed. By comparisons with other time-domain schemes, their stable and accurate performance is qualitatively verified. The proposed high-order SC-FDTD method can be used for efficiently simulating electrically large and longitudinally invariant optical devices since the reduction of simulation dimensionality and the novel high-order symplectic algorithm can greatly reduce the memory cost and the numerical dispersive errors.

Finite-difference wave-propagation models for dispersive media: impact of space-time discretization

PurposeThe computational accuracy and performance of finite-difference time-domain (FDTD) methods are affected by the implementation of approximating derivative formulae in diverse ways. This study aims to focus on FDTD models featuring material dispersion with negligible losses and investigates two specific aspects that, until today, are usually examined in the context of non-dispersive media only. These aspects pertain to certain abnormal characteristics of coarsely resolved electromagnetic waves and the selection of the proper time-step size, in the case of a high-order discretization scheme.Design/methodology/approachConsidering a Lorentz medium with negligible losses, the propagation characteristics of coarsely resolved waves is examined first, by investigating thoroughly the numerical dispersion relation of a typical discretization scheme. The second part of the study is related to the unbalanced space-time errors in FDTD schemes with dissimilar space-time approximation orders. The authors propose a remedy via the suitable choice of the time-step size, based on the single-frequency minimization of an error expression extracted, again, from the scheme’s numerical dispersion formula.FindingsUnlike wave propagation in free space, there exist two parts of the frequency spectrum where waves in a Lorentz medium experience non-physical attenuation and display non-changing propagation constants, due to coarse discretization. The authors also show that an optimum time-step size can be determined, in the case of the (2,4) FDTD scheme, which minimizes the selected error formula at a specific frequency point, promoting more efficient implementations.Originality/valueUnique characteristics displayed by discretized waves, which have been known for non-dispersive media, are examined and verified for the first time in the case of dispersive materials, thus completing the comprehension of the space-time discretization impact on simulated quantities. In addition, the closed-form formula of the optimum time-step enables the efficient implementation of the (2,4) FDTD method, minimizing the detrimental influence of the low-order temporal integration.

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.

Three-Dimensional Elastic Full-Waveform Inversion Using Temporal Fourth-Order Finite-Difference Approximation

Full-waveform inversion (FWI) serves as a useful tool to quantitatively investigate the properties of the subsurface. Presently, 3-D elastic FWI uses a finite-difference time-domain (FDTD) approach in numerical simulation. However, such an FDTD scheme often includes only second-order temporal approximations, causing errors in temporal dispersion in the case of a large time-stepping size. Such temporal dispersion will affect the inversion results and reduce the inversion quality. We introduce a unique 3-D elastic FWI using a temporal fourth-order finite-difference (FD) approximation. A new quasi-stress–velocity elastic equation is solved by the temporal fourth-order and spatial arbitrary even-order FDTD method, and a novel inversion procedure for the convolutional objective function based on this equation is derived. The multiscale strategy is used to enhance the robustness of our algorithm. The forward modeling and FWI examples presented here demonstrate that our method can achieve modeling and inversion with a high degree of accuracy.

Developing a leap-frog meshless methods with radial basis functions for modeling of electromagnetic concentrator

<abstract><p>The main goal of this paper is to develop a fast and effective meshless method by using radial basis function (RBF) for the time domain model equations of electromagnetic wave concentration device. This is mainly because the complex model equations involve different partial differential equations in different subdomains, which makes the meshless method very attractive and also very challenging. In order to simulate the propagation of electromagnetic waves in the electromagnetic concentrator, perfect matching layer technology was used to reduce an unbounded domain problem into a bounded domain problem. Borrowing the idea of the leap-frog finite-difference time-domain scheme, I develop the leap-frog RBF meshless method to solve the coupled complex modeling equations. The numerical results obtained by using a multiquadric RBF and Gaussian RBF demonstrate that our RBF method is very effective.</p></abstract>

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 Numerical Method for Pulse Propagation in Nonlinear Dispersive Optical Media

In this work, the pulse propagation in a nonlinear dispersive optical medium is numerically investigated. The finite difference time-domain scheme of third order and periodic boundary conditions are used to solve generalized nonlinear Schr¨odinger equation governing the propagation of the pulse. As a result a discrete system of ordinary differerential equations is obtained and solved numerically by fourth order Runge-Kutta algorithm. Varied input ultrashort laser pulses are used. Accurate results of the solutions are obtained and the comparison with other results is excellent.

Time Domain Response Analyses of Multiconductor Transmission Lines Using L-Stable Single Step Multistage Block Method

In order to solve the non-physical oscillation problem of the traditional finite difference time domain(FDTD) scheme at the discontinuous solution, the L-stable single step multistage block method is proposed to analyze the time domain transient response of multiconductor transmission lines. In order to avoid the repeated inversion of the coefficient matrix of the system state equation when the boundary conditions are imposed, the more general Sherman-Morrison formula is used to solve the inverse matrix of the coefficient matrix after the boundary conditions are applied. This method is used to simulate uniform lossless multiconductor transmission lines (MTLs) and non-uniform lossy MTLs, and its simulation result is compared with the results given by FDTD algorithm. The results showed that the single step multistage block method can effectively suppress the numerical oscillation and is not limited by step size of simulation in time domain.

A three-dimensional multi-grid staggered finite-difference time-domain method for underwater target acoustic scattering

Solving the linearized equations of continuum mechanics to model the acoustic scattering of targets close to the seafloor is a challenging task. Among the traditional algorithms, staggered second-order finite-difference time-domain (FDTD) schemes combine simplicity and ease of implementation and parallelization, especially on modern GPUs. They have been successfully applied in two-dimensional configurations. Their major drawback is their inherent low resolution, due to their formal low order and the stair-step representation of the interfaces between media (e.g., target/water). To avoid excessive computational costs in three-dimensional configurations, while maintaining high numerical accuracy, we propose a FDTD multi-grid method. The mesh is refined locally near the interfaces, while a coarse grid is employed in homogeneous regions. In the fine grids, the aforesaid FDTD is employed, whereas, in the coarse grid, higher-order staggered schemes are used to compensate for the decrease in resolution due to the larger spacing. High-order interpolation allows matching the solutions at the grid interfaces. In this presentation, the performances of the new method on classical benchmark tests are analyzed and discussed. The numerical and computational capabilities are then illustrated by applying the proposed technique to the scattering of buried objects.

Protection From Indirect Lightning Effects for Power Converters in Avionic Environment: Modeling and Experimental Validation

This article focuses on a protection system from lightning indirect effects applicable to power converters for avionic applications. The case study considered in this article is a dc-dc pulsewidth modulation buck-boost converter protected by a metal oxide varistor and a series inductive blocking element. The converter is investigated when operated in normal operating conditions, under lightning indirect strokes without protection, and under lightning indirect strokes with protection. A numerical model of the system based on a finite-difference time-domain scheme is proposed. Validation of the system is performed through a comparison with experimental measurements and alternative numerical modeling techniques, showing the effectiveness of the blocking system and the accuracy of the proposed model.

The Focusing Lens Design in Ku-Band Using Ray Inserting Method (RIM)

In this article, types of broadband focusing lens are presented based on the ray inserting method (RIM) in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Ku$ </tex-math></inline-formula> -band. The realizable inhomogeneous isotropic dielectric media is obtained to converge wavefront produced by a source on a focal point. The proposed three lenses are broadband because of good matching to the source and the surroundings. The finite-difference time-domain (FDTD) scheme and the full-wave simulation are used to confirm the results in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Ku$ </tex-math></inline-formula> -band. To validate designed lenses’ performance, the circular focusing lens is realized and fabricated. The results of simulations and experiments indicate good performances of the designed lens in the wide frequency bandwidth.

Using the EC-S-FDTD Scheme to Approximate Eddy Currents Induced by Z-Gradient Coils

Eddy currents are induced by rapidly switching currents of the gradient coils in an magnetic resonance imaging (MRI) scanner, which is typically cylindrical in shape; hence, the use of cylindrical coordinates is more numerically efficient. We develop the energy conserved splitting finite-difference time-domain (EC-S-FDTD) scheme in the cylindrical coordinates to estimate the eddy currents. The important feature is that this scheme conserves energy and is unconditionally stable. The transformation results in numerical singularities are formed at the internal boundary, and we formulate methods to handle these singularities and efficiently solve the equations. The accuracy of the scheme and energy conversation property are tested, and the performance of the proposed method will be evaluated using two numerical simulations, cylindrical transmission lines, and eddy currents induced by Z-gradient coils. Our numerical results indicate that current density distribution reflects the differences in the tissues by their dielectric properties and has second-order convergence in time and space while preserving energy conservation.

Radiation Efficiency Enhancement of Graphene Plasmonic Devices Using Matching Circuits

In the present work, the radiation properties of a graphene plasmonic patch antenna are investigated and enhanced in terms of efficiency, utilizing circuit-matching techniques. Initially, the reflection coefficient of graphene surface waves due to discontinuities is studied, while the power flow towards free-space is numerically extracted. This analysis indicates that the radiated power is increased for higher values of the chemical potential, although the surface wave is weakly confined and prone to degradation due to interference. For this reason, a graphene sheet that supports strongly confined surface waves is terminated via a matching layer, in order to enhance the radiating power. In particular, the matching layer consists of an appropriately selected larger chemical potential value, in order to minimize the reflection coefficient and boost the radiation performance. The numerical investigation of this setup validates the upgraded performance, since the radiating power is significantly increased. Then, a realistic setup that includes a graphene patch antenna is examined numerically, proving the augmentation of the radiation efficiency when the matching layer is utilized. Finally, the latter is designed with a graded increment in the chemical potential, and the computational analysis highlights the significant enhancement of the graphene plasmonic antenna gain towards the desired direction. Consequently, a more reliable framework for wireless communications between plasmonic devices at THz frequencies is established, which corresponds to the practical significance of the proposed methodology for improved radiation efficiency. All numerical results are extracted by means of an efficient modification of the Finite-Difference Time-Domain (FDTD) scheme, which models graphene accurately.

Seabed seismic wave simulation by finite difference time domain scheme in marine environment with complex seafloor topography

The studying of the excitation and propagation characteristics of seabed seismic waves in a complex marine environment is of great significance in investigating seafloor physical and mechanical properties and exploring resources. At present, the research of time-domain seabed seismic waves is mostly restricted in a marine environment with horizontal stratification, but the actual geological conditions of seafloor are relatively complex, and the numerical solutions obtained under ideal assumption are quite different from those in an actual complex environment. To master the propagation characteristics of seabed seismic wave in the environment that is closer to the actual one, a complex and long range model including layers of water, soft mud and bedrocks is designed in the paper, where non-horizontal seafloor topography with a dipping and uplifting structure is considered. The staggered-grid finite difference method with 2nd-order accuracy in time and 10th-order accuracy in space is used to simulate the seabed seismic waves under such a complex marine environment. Meanwhile, multi axial perfectly matched layer is used as an artificial boundary condition to ensure the numerical long-term stability in a liquid-solid medium. Considering the dipping structure, the acoustic signals excited by sources at different positions of the model are compared to determine the favorable style of source excitation for Scholte interface wave receiving. Through the time-domain waveform of the calculated acoustic field, the propagation characteristics of the seabed seismic wave in the complex marine environment are analyzed. The results show that the staggered-grid finite difference method with high-order spatial accuracy can improve the dispersion problem in numerical calculation. The multi-axial perfectly matched layer used to replace the traditional perfectly matched layer can solve the instability problem in the numerical simulation of acoustic field in liquid-solid media for a long range. Through the comparison among the acoustic signal amplitudes excited by sources at different positions, a better performance can be achieved when the source-receiver is placed along the updip direction. In such a case, the acoustic signal is stronger, which is more advantageous to receive and analyze the Scholte interface wave. In the complex seabed model with a dipping and uplifting structure, the uplift of seafloor bedrock changes the propagation direction of Scholte wave, which makes it possible to receive Scholte wave at shallower depth.

Numerical analysis of high frequency GaAs FET transistor using alternating direction implicit (ADI) finite-difference time-domain method; Distributed modeling approach

The unconditionally stable ADI Finite Difference Time Domain (FDTD) method is used to govern the distributed model equations of high frequency GaAs FET transistor. This implicit scheme with no constraint by a stability criterion is exploited to increase the time performance of the CPU time considerably compared with the conditionally stable methods such as explicit Leap-Frog (LF) FDTD scheme. Moreover, the ADI method can effectively decrease the complexity of the sparse matrix structure emerging in Crank–Nicolson (CN) FDTD scheme by splitting it into two time increments. The accuracy of the ADI is appreciated to the LF, CN and measured results.