Finite-difference Time-domain Scheme Research Articles

Modelling of anisotropic beam for rotating composite wind turbine blade by using finite-difference time-domain (FDTD) method

Most modern large wind turbine blades are made of composite materials which are naturally anisotropic. Modern wind turbine blade design, such as BTC design tends to further enhance the anisotropy of a composite blade. As a result, the modelling of an anisotropic rotating wind turbine blade is an important topic in the wind energy industry. In this paper, the governing equations of an anisotropic rotating beam is derived using Newtonian theory. These governing equations are discretized and solved using a finite-difference time-domain (FDTD) method. This methodology is shown to be highly computationally efficient owing to the fact that the governing equations are solved element by element alternately and explicitly, so only a few operations are required per grid point. The anisotropic beam model developed in this paper is validated using four test cases: (1) modal analysis of an anisotropic box beam; (2) dynamic simulation of a spin-up maneuver; (3) simulation of the NREL 5 MW wind turbine blade; and, (4) simulation of the WindPACT wind turbine blade. The validation is conducted in terms of the predicted natural frequencies and tip displacements for both inertial and non-inertial frames. It is shown that the proposed model can be extended to deal with the case of large rotations. • A novel FDTD scheme is proposed for structural modelling of wind turbine blade. • The proposed FDTD scheme is simple to implement and computationally efficient. • Various aspects of FDTD scheme have been validated using three test cases. • Very good agreement of natural frequencies/mode shapes were obtained from validation.

Energy-preserving local mesh-refined splitting FDTD schemes for two dimensional Maxwell's equations

In this paper, we develop and analyze two types of new energy-preserving local mesh-refined splitting finite difference time-domain (EP-LMR-S-FDTD) schemes for two-dimensional Maxwell's equations. For the local mesh refinements, it is challenging to define the suitable local interface schemes which can preserve energy and guarantee high accuracy. The important feature of the work is that based on energy analysis, we propose the efficient local interface schemes on the interfaces of coarse and fine grids that ensure the energy conservation property, keep spatial high accuracy and avoid oscillations and meanwhile, we propose a fast implementation of the EP-LMR-S-FDTD schemes, which overcomes the difficulty in solving unknowns on the “trifuecate structure” of refinement by first solving the values of coarse mesh unknowns and the average values of fine mesh unknowns on a line-structure and then solving the values of fine mesh unknowns and the coarse mesh unknown on an inverted “U-form” structure for each loop. The EP-LMR-S-FDTD schemes can be solved in a series of tridiagonal linear systems of unknowns which can be efficiently implemented at each time step. We prove the EP-LMR-S-FDTD schemes to be energy preserving and unconditionally stable. We further prove the convergence of the schemes and obtain the error estimates. Numerical experiments are given to show the performance of the EP-LMR-S-FDTD schemes which confirm theoretical results.

Designing a wideband dielectric polygonal directional beam antenna using the ray inserting method.

This paper presents the design of a wideband polygonal directional beam antenna based on the ray inserting method. The wideband characteristic of the directional beam antenna is achieved thanks to the use of inhomogeneous dielectric material. Also, unlike most previous works, the present design can be implemented with the isotropic and above unity refractive index materials, consequently simplifying its fabrication process. The finite difference time domain scheme is used to evaluate the directional beam antenna.

An Efficient Numerical Formulation for Wave Propagation in Magnetized Plasma Using PITD Method

A modified precise-integration time-domain (PITD) formulation is presented to model the wave propagation in magnetized plasma based on the auxiliary differential equation (ADE). The most prominent advantage of this algorithm is using a time-step size which is larger than the maximum value of the Courant–Friedrich–Levy (CFL) condition to achieve the simulation with a satisfying accuracy. In this formulation, Maxwell’s equations in magnetized plasma are obtained by using the auxiliary variables and equations. Then, the spatial derivative is approximated by the second-order finite-difference method only, and the precise integration (PI) scheme is used to solve the resulting ordinary differential equations (ODEs). The numerical stability and dispersion error of this modified method are discussed in detail in magnetized plasma. The stability analysis validates that the simulated time-step size of this method can be chosen much larger than that of the CFL condition in the finite-difference time-domain (FDTD) simulations. According to the numerical dispersion analysis, the range of the relative error in this method is 10−6 to 5×10−4 when the electromagnetic wave frequency is from 1 GHz to 100 GHz. More particularly, it should be emphasized that the numerical dispersion error is almost invariant under different time-step sizes which is similar to the conventional PITD method in the free space. This means that with the increase of the time-step size, the presented method still has a lower computational error in the simulations. Numerical experiments verify that the presented method is reliable and efficient for the magnetized plasma problems. Compared with the formulations based on the FDTD method, e.g., the ADE-FDTD method and the JE convolution FDTD (JEC-FDTD) method, the modified algorithm in this paper can employ a larger time step and has simpler iterative formulas so as to reduce the execution time. Moreover, it is found that the presented method is more accurate than the methods based on the FDTD scheme, especially in the high frequency range, according to the results of the magnetized plasma slab. In conclusion, the presented method is efficient and accurate for simulating the wave propagation in magnetized plasma.

Solution of Maxwell’s Equations for Nonrectangular Electromagnetic Applications

The use of a fourth-order modified Runge–Kutta (MRK) scheme on a transformed coordinate system with Maxwell’s equations for nonrectangular domains and applications is presented. Maxwell’s equations are the governing equations for modeling electromagnetic wave propagation involving scattering, radiating structures, transmission lines, radar, biomedical applications, and nondestructive testing. Because of complex geometries in most problems where the material boundaries are not parallel to the grid axis, the application of finite differencing schemes in physical coordinates becomes nonviable. Therefore, by transforming the arbitrary-shaped structures to a uniform rectangular grid, numerical schemes and boundary conditions can be easily implemented. Numerical results for four cases are presented in this paper. Accuracy of the scheme has been established by comparing numerical results with the exact solution and error distribution plots. In the third case, scattering from the lossless and the lossy square cylindrical dielectric device where the plane wave source is injected using the total-field–scattered-field technique has been investigated. The results are compared with the results obtained from the benchmark finite difference time domain scheme. Lastly, the application of the numerical model to simulate scattering from curved boundaries has been presented in the fourth example.

Improved direct integration auxiliary differential equation FDTD scheme for modeling graphene drude dispersion

In this communication, an improved direct-integration auxiliary differential equation (IM-DI-ADE) scheme is introduced for incorporating graphene's Drude dispersion into the finite difference time domain (FDTD) simulations in the GHz and THz frequency ranges. Stability analysis is performed and it is shown that the time-step stability limit of the presented IM-DI-ADE scheme coincides with the conventional non-dispersive FDTD Courant–Friedrichs–Lewy (CFL) limit and removes the additional stability stringent of the classical DI-ADE counterpart. In addition, the presented scheme does not increase the memory storage overhead. Numerical dispersion analysis is also addressed and it is shown that the presented scheme provides high accuracy. The formulation is validated by numerical tests that investigate the tunneling of electromagnetic wave through a graphene sheet and the existence of the surface plasmon polaritons (SPPs) waves created at the interface between the graphene sheet and a dielectric material.

Analysis of the oblique incidence of periodic structures in a sound field by the finite-difference time-domain method

A method to analyze the acoustic characteristics of periodic structures is investigated. When a continuous boundary condition is introduced, the field in which a plane wave is normally incident on the periodic structure can be easily analyzed by the finite-difference time-domain (FDTD) method. However, when a plane wave source is obliquely incident, field mapping with auxiliary variables is necessary to remove the phase shift between unit cells. Here, a technique originally proposed for an electromagnetic field is applied to a sound field. The effects of the geometric parameters of the periodic structures on the sound pressure distribution and transmission loss in the mapped sound field are shown. This technique is validated by comparing the results with those in a field solved with the classical FDTD scheme. The dispersion relation and the stability condition for the time interval of the FDTD method for each incident angle of the plane wave in the mapped field are studied.

Fast Nonuniform Metasurface Analysis in FDTD Using Surface Susceptibility Model

Analysis of nonuniform metasurfaces requires enormous computational resources and even cannot be achieved in some cases, due to their multiscale property. To address this, we propose a fast analysis method in finite difference time domain (FDTD) scheme in this article. Two steps are required: first, nonuniform metasurfaces are modeled as a distribution of surface susceptibilities. Secondly, the surface susceptibility model (SSM) is inserted into FDTD equations in a form of surface polarization currents based on generalized sheet transition conditions (GSTCs). A detailed comparison is made between brute force simulation and our proposed method. Compared with the hours consumed for physical model simulation, surface susceptibility model simulation’s time consumption is in seconds while simultaneously maintaining similar field responses independent of the incident wave’s angle and polarization.

Quantum Electromagnetic Finite-Difference Time-Domain Solver

We employ another approach to quantize electromagnetic fields in the coordinate space, instead of the mode (or Fourier) space, such that local features of photons can be efficiently, physically, and more intuitively described. To do this, coordinate-ladder operators are defined from mode-ladder operators via the unitary transformation of systems involved in arbitrary inhomogeneous dielectric media. Then, one can expand electromagnetic field operators through the coordinate-ladder operators weighted by non-orthogonal and spatially-localized bases, which are propagators of initial quantum electromagnetic (complex-valued) field operators. Here, we call them QEM-CV-propagators. However, there are no general closed form solutions available for them. This inspires us to develop a quantum finite-difference time-domain (Q-FDTD) scheme to numerically time evolve QEM-CV-propagators. In order to check the validity of the proposed Q-FDTD scheme, we perform computer simulations to observe the Hong-Ou-Mandel effect resulting from the destructive interference of two photons in a 50/50 quantum beam splitter.

Dispersive FDTD Scheme and Surface Impedance Boundary Condition for Modeling Pulse Propagation in Very Lossy Medium

A dispersive surface impedance boundary condition (SIBC) is implemented in a dispersive finite-difference time-domain (FDTD) scheme to model pulse propagation and scattering from an aquifer, which has high permittivity and high conductivity. A quadratic complex rational function (QCRF) is adopted to model the dispersive media over a wide frequency band for implementing the dispersive FDTD scheme. The proposed method is demonstrated by simulating the operation of ground-penetrating radar (GPR) in detecting water table. Group delay and pulse-broadening parameter (PBP) are defined to analyze the deformation of pulses propagating in a dispersive lossy medium.

On the Analysis of the Dynamic Energy Losses in NGO Electrical Steels Under Non-Sinusoidal Polarization Waveforms

This article presents a discussion on the main approaches for the characterization and modeling of the dynamic energy losses for a non-grain-oriented (NGO) electrical steel excited with either sinusoidal or non-sinusoidal magnetic inductions. The experimental analysis is carried out via an Epstein frame, where an efficient feedback algorithm is implemented to control the magnetic-induction waveform. Different approaches are tested to reproduce the dynamic energy losses, including analytic equations and a dynamic hysteresis model, based on the solution of the diffusion equation via a finite-difference time-domain (FDTD) scheme, where the hysteretic constitutive law is modeled according to the Preisach model. Discussion on the accuracy of different approaches in the reproduction of the energy loss components is presented.

Fourth‐order Jameson–Schmidt–Turkel FDTD scheme for non‐magnetised cold plasma

A fourth-order finite-difference time-domain (FDTD) scheme is proposed for the solution of Maxwell's equations in cold plasma (Drude medium), based on the multistage method of Jameson, Schmidt and Turkel, which was originally introduced in the framework of fluid dynamics. First, the system of governing differential equations is formed as a general first-order, operator-based approach, and then a four-stage algorithm is established. The accuracy of the method is verified in benchmark problems compared with analytical solutions and with the conventional second-order FDTD algorithm.

FDTD propagation of VLF-LF waves in the presence of ions in the earth-ionosphere waveguide

The finite-difference time-domain (FDTD) method has been used for a long time to compute the propagation of very low frequency (VLF) and low frequency (LF) radio waves in the Earth-Ionosphere waveguide. In previously published FDTD schemes, only the electronic density of the ionosphere was accounted for, since in usual natural conditions the effect of the ion density can be neglected. In the present paper, the FDTD scheme is extended to the case where one or several ion species must be accounted for, which may occur in special natural conditions or in such artificial conditions as after high altitude nuclear bursts. The conditions that must hold for the effect of the ions not to be negligible are discussed, the FDTD scheme with ions is derived, and numerical experiments are provided to show that the effect of the ions may be significant when the ionosphere is disturbed by incident flows of γ or β rays.

A 3D Unstructured Mesh FDTD Scheme for EM Modelling

The Yee finite difference time domain (FDTD) algorithm is widely used in computational electromagnetics because of its simplicity, low computational costs and divergence free nature. The standard method uses a pair of staggered orthogonal cartesian meshes. However, accuracy losses result when it is used for modelling electromagnetic interactions with objects of arbitrary shape, because of the staircased representation of curved interfaces. For the solution of such problems, we generalise the approach and adopt an unstructured mesh FDTD method. This co-volume method is based upon the use of a Delaunay primal mesh and its high quality Voronoi dual. Computational efficiency is improved by employing a hybrid primal mesh, consisting of tetrahedral elements in the vicinity of curved interfaces and hexahedral elements elsewhere. Difficulties associated with ensuring the necessary quality of the generated meshes will be discussed. The power of the proposed solution approach is demonstrated by considering a range of scattering and/or transmission problems involving perfect electric conductors and isotropic lossy, anisotropic lossy and isotropic frequency dependent chiral materials.

Modeling full-wavefield time-varying sea-surface effects on seismic data: A mimetic finite-difference approach

Seismic-data processing flows often ignore spatial and temporal variations in the sea surface during marine seismic acquisition by assuming a flat free surface. However, weather patterns during data acquisition can generate rough sea conditions, which can significantly influence seismic full-wavefield source behavior, including ghost reflections and surface-related multiples, by introducing spatial and temporal distortions of the seismic wavelet. To investigate the effects of rough seas on seismic wave propagation, we have developed and solved a new acoustic wave equation using a mimetic finite-difference time-domain (MFDTD) scheme that uses a dynamic (i.e., moving) generalized coordinate system defined to be conformal to the assumed known time-varying free surface. This “sea-surface” coordinate system allows us to model the full dynamic effects associated with this complex boundary condition. Numerical examples demonstrate that the developed MFDTD method can accurately simulate seismic wavefield propagation on a moving mesh for significant wave heights of 5 m and beyond, and it is thus a reliable tool for applications involving modeling, processing, imaging, and inversion of seismic data acquired in rough seas.

Making a Synthesis of FDTD and DGTD Schemes for Computational Electromagnetics

A novel class of discontinuous Galerkin time-domain (DGTD) schemes, invented by the first author, are presented that are capable of globally preserving the constraints that are inherent in Maxwell's equations. The methods share the same Yee-type mesh structure as finite-difference time-domain (FDTD) schemes for computational electrodynamics. Since FDTD schemes also preserve global constraints, the novelty of this work consists of making a synthesis of FDTD and DGTD schemes. While previous DG methods were based on applying identities involving Gauss’ law in weak form to the volumetric elements of a mesh, the newer methods are based on applying identities involving Stokes’ law in weak form to the facial elements of the mesh. This fundamental paradigm shift is crucial for obtaining the globally constraint-preserving DGTD methods in this paper. The new DGTD methods meet their design accuracies. The more accurate schemes are indeed more accurate even at the lowest resolutions. Moreover, as the mesh is refined, the schemes reach their design accuracies much faster. These benefits are all attributable to the subcell resolving ability of the DGTD schemes presented here. The higher order methods offer the lowest time to solution, especially when very high accuracies are demanded. Excellent scalability is also demonstrated.

Efficient adjustment of finite graphene scattering properties via magnetic-bias control for advanced beam manipulation

In the present work, the scattering of an incident plane wave due to magnetically-biased graphene patches is thoroughly investigated at millimeter-wave and THz bands. Initially, the surface conductivity of graphene is evaluated at these spectral regions and a finite layer is placed perpendicular to the propagation of an incident plane wave. Then, the radar cross-section, at a plane normal to graphene, is numerically extracted and the anisotropic effects due to the magnetostatic bias Lorentz forces on electrons, reveal the influence of gyrotropy and magnetoplasmon excitation on the back-scattered wave. Specifically, the directivity of the latter is calculated as a function of the magnetostatic field considering a couple of electrostatic biases and frequencies. As expected, stronger fields are enabling graphene gyrotropic behaviour, while the propagating surface waves increase the edge effects of the finite sheet. Finally, the extracted results from the previous analyses are evaluated appropriately to design combinations of graphene patches, of different magnetic-bias fields in order to investigate the potential of advanced beam manipulation potential. The outcome of this part is promising since the variation of bias fields is able to adjust considerably the main-lobe direction of the back-scattered field. All numerical results are extracted via an accurate modification of the popular Finite-Difference Time-Domain scheme.

Modeling the Third-Order Electrodynamic Response of Graphene via an Efficient Finite-Difference Time-Domain Scheme

The third-order electrodynamic response of graphene and the subsequent non-linear effects are comprehensively investigated and numerically modeled in this article. Initially, the third-order conductivity of the medium is analyzed in order to transform it into a convenient expression for its efficient incorporation to explicit computational algorithms. Moreover, the simplified expression of the surface conductivity is imported into a suitably modified 3-D finite-difference time-domain scheme, where graphene is embodied as an equivalent surface current. Numerical examples, considering both an incident plane-wave toward graphene and surface-wave propagation, indicate the successful frequency upconversion, thus validating the proper third-order response modeling.

Coupling analysis of multiple linear antennas fed by coaxial lines on electronic system with a time domain hybrid method

AbstractDue to the shielding structures of electronic system and coaxial lines feeding the antennas are complex, full‐wave algorithms applied for the coupling analysis of multiple linear antennas on electronic system excited by ambient wave should occupy amount of memories by meshing the structures of coaxial lines and electronic system in fine grid, which may affect the computation efficiency. This paper presents a time domain hybrid method, which consisting of thin wire finite‐difference time domain (FDTD) method, transmission line (TL) equations, higher order FDTD scheme, and an auto mesh generator technique. In this method, the thin wire FDTD combined with the auto mesh generator technique is utilized to mesh the shielding structure of electronic system rapidly and obtain the current responses on the conductor columns of linear antennas, which are extracted equivalent current sources for the further coupling analysis of coaxial lines. And then the TL equations are adopted to build the coupling models of coaxial lines, and solved by the higher order FDTD scheme to compute the transient responses on the coaxial lines and terminal loads of the lines. This method has one significant feature is that it can avoid modeling the structures of linear antennas and coaxial lines directly. The accuracy of this method has been verified by comparing with the simulation software Computer Simulation Technology (CST) via the coupling simulations of one and three linear antennas on electronic system excited by ambient plane wave.

Superposition method for modelling boundaries between media in viscoelastic finite difference time domain simulations.

Finite-difference time domain (FDTD) techniques are widely used to model the propagation of viscoelastic waves through complex and heterogeneous structures. However, in the specific case of media mixing liquid and solid, attempts to model continuous media onto a Cartesian grid produces errors when the liquid-solid interface between different media do not align precisely with the Cartesian grid. The increase in spatial resolution required to eliminate this grid staircasing effect can be computationally prohibitive. Here, a modification to the Virieux staggered-grid FDTD scheme called the superposition method is presented. This method is intended to reduce this staircasing effect while keeping a manageable computational time. The method was validated by comparing low-spatial-resolution simulations against simulations with sufficiently high resolution to provide reasonably accurate results at any incident angle. The comparison of the root-mean-square of the stress amplitude maps showed that the amplitude of artifactual waves could be reduced by several orders of magnitude when compared to the Virieux staggered-grid FDTD method and that the superposition method helped to significantly reduce the staircasing effect in FDTD simulations.