Maxwell's Curl Equations Research Articles

Accelerating Electromagnetic Field Simulations Based on Memory-Optimized CPML-FDTD with OpenACC

Although GPUs can offer higher computing power at low power consumption, their low-level programming can be relatively complex and consume programming time. For this reason, directive-based alternatives such as OpenACC could be used to specify high-level parallelism without original code modification, giving very accurate results. Nevertheless, in the FDTD method, absorbing boundary conditions are commonly used. The key to successful performance is correctly implementing the boundary conditions that play an essential role in memory use. This work accelerates the simulations of electromagnetic wave propagation that solve the Maxwell curl equations by FDTD using CMPL boundary in TE mode using OpenACC directives. A gain of acceleration optimizing the use of memory is shows, checking the loops intensities, and the use of single precision to improve the performance is also analyzed, producing an acceleration of around 5X for double precision and 11X for single precision respectively, comparing with the serial vectorized version, without introducing errors in long-term simulations. The scenarios of simulation established are common of interest and are solved at different frequencies supported by a Mid-range cards GeForce RTX 3060 and Titan RTX.

Hybrid Simulation Method for EM Wave Generation and Propagation of Streamer Discharges

The finite-difference time-domain (FDTD) method has been widely used for analyzing the EM wave propagation property of partial discharge (PD). However, many studies use Gaussian pulse as the excitation source, which shows few theoretical relations with the EM wave generation mechanism of PD. To simulate the EM wave generated by PD, the electric field distortion caused by the streamer discharges is analyzed. The finite element method (FEM) is used to calculate the fluid model to get the electric distribution alongside the charged air gap. This distorted electric field obtained from the fluid model is applicable as an excitation source of EM wave, which also explains its generation mechanism caused by the time-varying electric field. Because of the limitations of FEM, such as one frequency response and the huge computation complexity for large electric size. FDTD method is a useful tool to approximate Maxwell's curl equations and simulate the EM wave generated by the distorted electric field. In contrast to typical FDTD applications with one excitation, several excitation points should be chosen to simulate the distorted electric field, and they are evenly distributed alongside the streamer path since the EM wave is emitted by a charged gap rather than a point. Thus, this paper proposes a hybrid method that combines FEM and FDTD for the whole process simulation of the EM wave caused by PD. A positive streamer under two types of electrodes is simulated and the frequency spectrums of generated EM time-domain waveforms are also obtained. The proposed method simulates the EM radiation of PDs digitally, which provides a reference for PD monitoring.

A Simple Solver for the Two-Fluid Plasma Model Based on PseudoSpectral Time-Domain Algorithm

We present a solver of 3D two-fluid plasma model for the simulation of short-pulse laser interactions with plasma. This solver resolves the equations of the two-fluid plasma model with ideal gas closure. We also include the Bhatnagar-Gross-Krook collision model. Our solver is based on PseudoSpectral Time-Domain (PSTD) method to solve Maxwell's curl equations. We use a Strang splitting to integrate Euler equations with source term: while Euler equations are solved with a composite scheme mixing Lax-Friedrichs and Lax-Wendroff schemes, the source term is integrated with a fourth-order Runge-Kutta scheme. This two-fluid plasma model solver is simple to implement because it only relies on finite difference schemes and Fast Fourier Transforms. It does not require spatially staggered grids. The solver was tested against several well-known problems of plasma physics. Numerical simulations gave results in excellent agreement with analytical solutions or with previous results from the literature.

A Numerical Technique for Estimating High-Frequency Radiated Emissions From Railway System

This article presents a finite-difference time-domain technique to analyze the characteristics of the railway system emission to the outside environment. The 2-D simulation implements algorithms accommodating the interactions of Poisson's equation for the cross-component electrostatic coupling between the train and infrastructure with Maxwell's curl equations accounting for the time-varying wave propagation across the discontinuous boundaries excited by the on-board 400-MHz terrestrial trunked radio system. The 2-D near-field to far-field conversion using the generated surface currents concludes the investigation of system emission characteristics in terms of the radiation pattern and emission level in accordance with the EN 50121-2 criteria. The numerical results of the emission spectrum from 300 MHz to 1 GHz influenced by the integrated railway system components and on-site measurement records during the 750-Vdc metro system integration tests provide a validation of technique for estimating the railway system radiation prior to the start of the system installation phase.

An Unconditionally Stable Single-Field Finite-Difference Time-Domain Method for the Solution of Maxwell Equations in Three Dimensions

The recently developed two-directional unconditionally stable single-field (US-SF) finite-difference time-domain (FDTD) method is generalized to a 3-D. The method is based on the application of the Crank–Nicolson scheme to only one of Maxwell curl equations which leads to an unconditionally stable finite-difference solution of the 3-D vector wave equation in the time domain. The method is designated as a single field because only the electric (or magnetic) field is updated in each time step. The implicit equations for each time step can be solved using a tridiagonal matrix algorithm without a need for matrix inversion. To achieve this, we introduced a new modified time-splitting scheme for the multilevel difference equations. Unlike the existing time-splitting method used to solve such equations which results in instability when dealing with inhomogeneous media, the presented method remains stable. As an important feature of the proposed US-SF-FDTD method, the updating of the three field components can be executed simultaneously (in parallel) by applying multithreading, thereby significantly reducing runtime. The unconditional stability of the proposed method is proved analytically. The accuracy and computational efficiency of the proposed method are demonstrated by providing numerical examples and by comparison to other FDTD methods and to the analytic solution when available.

K-domain method for the fast calculation of electromagnetic fields propagating in graded-index media.

A conceptually straightforward method for the fast calculation of electromagnetic fields propagating in graded-index media is presented. More specifically, in this method, we convert Maxwell's curl equations into the spatial-frequency domain to obtain an ordinary differential equation (ODE), and subsequently solve the ODE with the 4th-order Runge-Kutta method. Compared to the traditional beam propagation methods, this method deals with vectorial fields accurately, without physical approximations, like the scalar field approximation or the paraxial approximation; numerically, this method takes advantage of the fast Fourier transform and the convolution theorem to achieve an efficient calculation.

An Improved Fast Finite Element Time-Domain Method Based on Compressive Sensing for Cavity Problems

To avoid directly solving linear equations for major time marching in the finite element time-domain (FETD) method, the compressive sensing (CS) theory is introduced to construct an explicit iterative scheme instead of an implicit form. In this method, Whitney elements and a leapfrog model are applied to time-dependent Maxwell curl equations for space and time discretization. After a small number of iterations, steps for linear equations in the initial time, the electric field intensity and magnetic flux density are updated just by the linear combination of results of previous time steps. Furthermore, a restart mechanism is built to control the computational complexity and accuracy over a long simulation time. The effectiveness of the new algorithm is verified by analyzing 3-D resonant cavities.

Impact of Transient Population Grating on the Superradiant Phase Transition in Semiconductor Laser Structures

We study theoretically and experimentally superradiant dynamics in multiple-section GaAs/AlGaAs laser structures. We develop a model based on the one-dimensional Maxwell curl equations without making use of the slowly-varying envelope approximation and demonstrate that strong transient periodic modulation of both e – h density and refractive index drastically affect the dynamics of emission generation in semiconductor media. We show that the coherent λ/2 transient population and the relative permittivity gratings facilitate e – h self-ordering and the onset of the superradiant phase transition in the e – h system.

A Memory-Efficient Formulation of Precise-Integration Time-Domain Method With Riccati Matrix Differential Equations

A novel numerical method, referred to as the Riccati precise-integration time-domain (Riccati-PITD) method, is proposed to reduce the memory requirement of the conventional PITD method. In the Riccati-PITD method, the electromagnetic field components of the transverse electric/transverse magnetic (TE/TM) wave are arranged in a matrix for constructing the Riccati matrix differential equations (RDEs) about Maxwell's curl equations. Then the precise integration technique is adopted for the numerical integration of the RDEs. Theoretical analyses about the memory requirements of the conventional and the Riccati-PITD methods are presented. It is found that the memory requirement can be more greatly reduced by solving the electromagnetic-field components in the form of the matrices in the Riccati-PITD method rather than in the form of the vectors in the conventional PITD method. Numerical examples are also given to confirm that the proposed method has significant advantages over the conventional PITD method with respect to the execution time and the memory requirement.

Unmagnetized plasma media parameters reconstruction using FDTD and gradient‐based minimization method

AbstractA time‐domain inverse scattering algorithm for determining unknown plasma resonance frequency and electron collision frequency of unmagnetized plasma media object has been proposed. Finite‐difference time‐domain (FDTD) method in conjunction with Lagrange multipliers and gradient‐based minimization method are used in the proposed algorithm. The reconstruction of the parameters is based on the minimization of a cost function that describes the discrepancy between simulation and measured data of the electric field at the observation points. In order to obtain the gradient value of the unknown plasma parameters, an augmented cost function has been given by means of Lagrange multipliers with the Maxwell curl equations and constitutive relation of the unmagnetized plasma media. Then, the gradient value could be obtained after a series of transformations. At last, the Ploak‐Ribiere conjugate gradient algorithm has been applied to update the plasma parameters iteratively. The effectiveness of the proposed algorithm has been validated through the simulation test.

A Novel ADE-CN-FDTD With Improved Computational Efficiency for Dispersive Media

A novel auxiliary difference equation (ADE) finite-difference time-domain (FDTD) method with Crank–Nicolson (CN) scheme is proposed for electromagnetic simulation of dispersive materials in this letter. The proposed method introduces an ADE technique that establishes the relationship between the electric displacement vector and electric field intensity in dispersive media with a difference equation. Hence, the CN scheme is only applied to one of the Maxwell curl equations, resulting in reduced memory usage and computing time. To validate the advantages of the proposed approach, an example with plane wave propagation in a dispersive media is presented. Compared with the conventional ADE-CN-FDTD method, the proposed method consumes much less computing time while preserves excellent simulation accuracy.

Nonlinear optical effects during femtosecond superradiant emission generation in semiconductor laser structures.

This paper presents theoretical and experimental studies of ultrabright internal second harmonic during femtosecond superradiant emission generation in multiple sections GaAs/AlGaAs laser structures at room temperature. Experimentally measured conversion efficiencies are by 1-2 orders of magnitude greater than expected. To explain this fact, a model based on one-dimensional nonlinear Maxwell curl equations without taking into consideration the slowly-varying envelope approximation has been developed. It has been demonstrated that strong transient periodic modulation of e-h density and refraction index dramatically affects the process of superradiance in semiconductor media and can explain the ultrastrong internal second harmonic generation.

A New Efficient Unconditionally Stable Finite-Difference Time-Domain Solution of the Wave Equation

This paper presents a new 2-D unconditionally stable finite-difference time-domain (FDTD) method based on the Crank–Nicolson (CN) scheme. Unlike other CN-based unconditionally stable FDTD algorithms, the CN scheme in the proposed method applies only to one of Maxwell curl equations. In fact, the proposed scheme is essentially a finite-difference solution of the vector wave equation in time domain. Therefore, only one of the electric or magnetic fields needs to be updated during the iterations of the algorithm. Most importantly, the implicit updating equations of the proposed methods can be solved simultaneously thereby significantly reducing run time. The stability of the proposed method is proved analytically. For the validation and demonstration of the accuracy of the proposed method and its efficiency, numerical examples are presented and the results are compared with those obtained by analytic solutions and other FDTD methods.

Light scattering computation model for nonspherical aerosol particles based on multi-resolution time-domain scheme: model development and validation

Due to the inadequate understanding of the scattering properties of nonspherical aerosols, considerable uncertainties still exist in the radiative transfer numerical simulation. To this end, a new scattering model for nonspherical aerosols is established based on Multi-Resolution Time-Domain (MRTD) scheme. The model is comprised of three modules: near field calculation module, near-to-far transformation module and scattering parameters computation module, in which, the near electromagnetic field is calculated by MRTD technique, the near-to-far transformation scheme is performed by volume integral method, and the calculation models for extinction and absorption cross section are directly derived from Maxwell's curl equations in the frequency domain. To achieve higher computational efficiency, the model is further parallelized by MPI non-blocking repeated communication technique. The accuracy of the scattering model is validated against Lorenz-Mie, Aden-Kerker and T-matrix theories for spherical particles, particles with inclusions and nonspherical particles. At last, the parallel computational efficiency of the MRTD scattering model is quantitatively discussed as well. The results obtained by parallel MRTD scattering model show an excellent agreement with those of the well-tested scattering theories, where the relative simulation errors of the phase function are less than 5% for most scattering angles. In backward directions, the simulation errors are much larger than that in forward scattering directions due to the stair approximation in particle construction. The computational accuracy of the integral scattering parameters like extinction and absorption efficiencies is higher than phase matrix, where the simulation errors of extinction and absorption efficiencies for the particle with a size parameter of 10 achieve -0.4891% and -1.6933%, respectively.

Simulating scattering properties of nonspherical aerosol particles using multiresolution timedomain method

Scattering process of aerosol particles plays an important role in atmospheric radiative transfer since it can modify the transmission, reflection and absorption ability of atmospheric system. Owning to the uncertainty of aerosol particles' scattering properties, which results from their complicated geometries and inhomogeneous compositions, there still exists a considerable uncertainty in the radiative transfer numerical simulation, and simulating the scattering properties of aerosol with irregular shapes has become a hotspot in meteorological study. To this end, a new aerosol scattering model is developed based on multi-resolution time-domain (MRTD), by which the scattering processes of nonspherical and inhomogeneous particles can be simulated. In this model, the near electromagnetic field is calculated by MRTD technique. Considering the particularity of aerosol medium, a transformation technique from near field to far field is derived based on volume integration method, and then the scattering amplitude matrix and Meller matrix can be calculated by the obtained far electric field as well. The models for particle extinction and absorption cross section are derived from Maxwell's curl equations in the frequency domain, by which the integration scattering properties can be simulated accurately. The MRTD scattering model is validated by comparing with Mie theory and T matrix method for spherical particle, ellipsoidal particle and cylindrical particle, and the influence of grid size on the simulation accuracy is analyzed subsequently. In the last part, the efficiency of the MRTD scattering model is quantitatively discussed. The simulation results show that the relative errors of scattering phase function simulated by our model are less than 8%, and the errors in forward scattering direction are much smaller, which are less than 4%. The precisions for extinction and absorption efficiency are much higher than the results from the scattering phase function, and the relative errors can reduce to 0.1% for particles with their radii comparable to the wavelength of incident light. The gird size has a significant influence on model precision; to achieve the same accuracy, the grid size first increases with increasing particle radius, and then decreases as a function of particle size for particles with size parameter less than 20. In the next step, we will try to establish the scattering property database of nonspherical particles based on the MRTD scattering model developed here.

Simulation of fluorescence enhancement by an AFM tip on a gold particle quenched emitter.

The local field enhancement in proximity of metallic nanostructures can strongly modify the excitation and emission behaviors for the nearby fluorophore. In this paper, Maxwell's time-dependent curl equations are solved by the finite-difference time-domain method to investigate the electric field enhancement around an atomic-force microscopy (AFM) tip and a Au nanosphere (NS). To lower the background fluorescence signal, we proposed to induce the fluorescence quenching by placing the emitter at an optimized position that is 2 nm away from the Au NS. The AFM tip is thereby moved to the vicinity of the emitter quenched by the Au NS. The fluorescence enhancement factor (FEF) increases rapidly when the tip approaches the Au NS. A maximum FEF of 1500-fold is obtained when their separation is 4 nm. By laterally scanning the tip over the Au NS at a constant height, the full width at half-maximum of fluorescence's signal peak with respect to tip position is around 20 nm. This high sensitivity of the FEF on the relative position of the tip and Au NS provides valuable information to guide future experiments on high-resolution optical imaging and fluorescence enhancement for high quantum yield emitters.

Finite difference time domain formulation for epsilon-negative medium using wave equation

A novel approach utilizing one-dimensional (1D) wave equation based finite difference time domain (FDTD) formulations for Lorentz-type Epsilon-Negative (ENG) medium is developed. To demonstrate the validity and accuracy of the formulation, reflection of a plane wave normally incident at free space—ENG interface is compared with the analytical solution. Moreover, propagation characteristics of Lorentz-type ENG medium is investigated using the FDTD formulations both for wave equation and Maxwell's curl equations in 1D case. Comparisons of memory and computational time requirements of both formulations are given. © 2016 Wiley Periodicals, Inc. Int J RF and Microwave CAE 26:304–310, 2016.

A Nodal Continuous-Discontinuous Galerkin Time-Domain Method for Maxwell's Equations

A new nodal hybrid continuous-discontinuous Galerkin time-domain (CDGTD) method for the solution of Maxwell's curl equations is proposed and analyzed. This hybridization is made by clustering small collections of elements with a continuous Galerkin (CG) formalism. These clusters exchange information with their exterior through a discontinuous Galerkin (DG) numerical flux. This scheme shows reduced numerical dispersion error with respect to classical DG formulations for certain orders and numbers of clustered elements. The spectral radius of the clustered semi-discretized operator is smaller than its DG counterpart allowing for larger time steps in explicit time integrators. Additionally, the continuity across the element boundaries allows us a reduction of the number of degrees of freedom of up to about 80% for a low-order three-dimensional implementation.

Three-dimensional efficient dispersive alternating-direction-implicit finite-difference time-domain algorithm using a quadratic complex rational function.

Efficient unconditionally stable FDTD method is developed for the electromagnetic analysis of dispersive media. Toward this purpose, a quadratic complex rational function (QCRF) dispersion model is applied to the alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method. The 3-D update equations of QCRF-ADI-FDTD are derived using Maxwell's curl equations and the constitutive relation. The periodic boundary condition of QCRF-ADI-FDTD is discussed in detail. A 3-D numerical example shows that the time-step size can be increased by the proposed QCRF-ADI-FDTD beyond the Courant-Friedrich-Levy (CFL) number, without numerical instability. It is observed that, for refined computational cells, the computational time of QCRF-ADI-FDTD is reduced to 28.08 % of QCRF-FDTD, while the L2 relative error norm of a field distribution is 6.92 %.

THE TIME-HARMONIC DISCONTINUOUS GALERKIN METHOD AS A ROBUST FORWARD SOLVER FOR MICROWAVE IMAGING APPLICATIONS

Novel microwave imaging systems require flexible forward solvers capable of incorporating arbitrary boundary conditions and inhomogeneous background constitutive parameters. In this work we focus on the implementation of a time-harmonic Discontinuous Galerkin Method (DGM) forward solver with a number of features that aim to benefit tomographic microwave imaging algorithms: locally varying high-order polynomial field expansions, locally varying high-order representations of the complex constitutive parameters, and exact radiating boundary conditions. The DGM formulated directly from Maxwell's curl equations facilitates including both electric and magnetic contrast functions, the latter being important when considering quantitative imaging with magnetic contrast agents. To improve forward solver performance we formulate the DGM for time-harmonic electric and magnetic vector wave equations driven by both electric and magnetic sources. Sufficient implementation details are provided to permit existing DGM codes based on nodal expansions of Maxwell's curl equations to be converted to the wave equation formulations. Results are shown to validate the DGM forward solver framework for transverse magnetic problems that might typically be found in tomographic imaging systems, illustrating how high-order expansions of the constitutive parameters can be used to improve forward solver performance.