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Absorbing Boundary Conditions for Finite Difference Approximation of the Time Domain Electromagnetic Field Equations
Abstract
When time-domain electromagnetic-field equations are solved using finite-difference techniques in unbounded space, there must be a method limiting the domain in which the field is computed. This is achieved by truncating the mesh and using absorbing boundary conditions at its artificial boundaries to simulate the unbounded surroundings. This paper presents highly absorbing boundary conditions for electromagnetic-field equations that can be used for both two-and three-dimensional configurations. Numerical results are given that clearly exhibit the accuracy and limits of applicability of highly absorbing boundary conditions. A simplified, but equally accurate, absorbing condition is derived for two- dimensional time-domain electromagnetic-field problems.
... Journal of science, Vol. 16, 2023 and MUR's absorbing boundary condition which was introduced by G. Mur in 1981 [8]. The boundary condition is used to limit a computational domain, which have the advantage to save a computational time. ...
... In this study, we calculated a number of examples to make a comparison between the simulations and in all the calculations the grid terminated by the second order MUR's absorbing boundary condition (ABCs). In two-dimensional system, the second order MUR's Boundary condition can be expressed by four differential equations (6) as the following [8]: ...
... At x=0 The boundary condition must be applied in the four edges as at x=0, x=xmax, y=0 and y=ymax when a uniform mesh is used (∆ = ∆ = ) [8]. The equation (7) will be implemented in a computer program to calculate the new values at the four edges and the equation (7) is called updating equation in each time step which can be written as [8]: ...
In this paper, Finite difference time domain technique was applied to find the solutions of Maxwell’s curl equations numerically. We calculated the Transverse magnetic (TMz) wave propagation in twodimensional (2-D) system in order to describe the propagations of the electric and magnetic waves for the example in Y-branch shape and also structure consists of two elements, each element is constructed as two parallel strips. The results of simulations can describe that the waves propagated and also controlled in a computational domain in 2-D. It was found that the distributions of the TMz wave can be changed when excited in different phases. The sources of excitations set in the phase and out of phase in two elements to make a comparison between the simulations. Instead of using a metal material such as a copper, we used the perfect electric conductors (PECs) to construct the strips. Therefore, the simulations results indicated that very good distributions were obtained and the waves of propagation controlled between the PECs strips as the electric field component must be set to equal zeros at boundaries in the PECs regions during the calculations. Moreover, this numerical study has demonstrated that the signals appeared identical, equally divided between the elements and propagated in the same phase and amplitude into the upper and lower elements in the Y-branch. The results have proved that the intensities of the TMz field components can be changed when varied the phases in the calculations.
... Mur's firstorder absorbing boundary condition (ABC) can utilize on the boundaries. For demonstrating this approach, it requires to consider the ABC at x=one [11]: ...
... The results illustrated in this research programmed by utilizing MATLAB. The equation (6) and equation (11) wrote in the programs to simulate electromagnetic in two and one dimensions, respectively as described in the method section and explained how to the FDTD scheme utilizes to simulate the propagation of electromagnetic wave in the 1D and 2D. ...
... Numerical Solutions of one dimension system (1D): In a one-dimensional problem space, the domain contains of 300 points and a mesh filled with a free space as well as a relative permittivity. This acquired by the calculation of the field coefficient in equation (11). The kinds of sources utilize to radiate classified as a hard and soft sources [13 14]. ...
This research focuses on electromagnetic waves propagation when the incident waves interact with different materials. The finite difference time domain (FDTD) technique is utilized for studying the propagation of the electromagnetic waves when the wave interacted with obstacles such as a dielectric and perfect electric conductor (PEC) inserted in the domain. The electromagnetic problems could be solved by discretizing the differential form of Maxwell’s curl equations. The propagations of electromagnetic waves in a dielectric material in a one dimension (1D) demonstrated that the symmetry of distributions appeared when the results compared as the sources assigned to different electric field node. Moreover, we simulated the problems in two dimensions (2D) for calculating the electric and magnetic field components as pixel by pixel in the x-y plane. In this case, two different types of obstacles utilized to compare the results. The results illustrated that the waves totally reflected back into the space and the waves propagated in the opposite direction compared with the incident signal once the PEC inserted. There is no field appeared inside the PEC obstacle whereas utilizinga dielectric obstacle, the fields generated inside a dielectric and everywhere nearby the obstacle. The results compared between the calculations when the dielectric slab and PEC slab placed in the same area. The first simulation showed that part of signal transmitted into next area whereas the second simulation result illustratedthat the field reflected, there is no signal appeared and produced in the next area after the slab. The results have explained that the electric and magnetic fields reflected back and updated after the obstacles in the space when dielectric slab in the same area of the PEC slab.
... In this regard, we consider here natural first-order ABCs involving first spatial and temporal partial derivatives only (see, e.g. [19,21]): ...
... with initial conditions u(0) = U 0 ,u(0) = W 0 (see [27] for details). In system (19), M and A are the mass and stiffness IGA Galerkin matrices associated to a h (·, ·) and (·, ·) h respectively, whereas C accounts for the boundary term √ c 0 < ·, · > h,∂Ω . We recall that M, C and A are symmetric, M is positive definite, whereas A is positive semi-definite, C is positive semi-definite with most elements equal to zero. ...
... We enforce the average of normal derivatives at any corner point. If we consider now the IGA Galerkin approximation (19), the application of the Newmark scheme gives the recurrence relations: ...
... This configuration allows for accurate time-domain solving of Maxwell's equations in three dimensions. To achieve an accurate distribution of the electric and magnetic fields for the studied structure, the Yee cell size is considered to be less than 10% of the shortest wavelength [38]. ...
... Fig. 1The electrical element in three dimensions[38] Content courtesy of Springer Nature, terms of use apply. Rights reserved. ...
Surface enhanced Raman scattering (SERS) is a highly sensitive and precise technique that enables the acquisition of high-quality spectra from materials in tiny quantities, down to the single-molecule level. Today, this technique, with its high signal enhancement factor, is considered an efficient tool for molecular detection in biosensors. This article will present and investigate the impact of uses of a novel hybrid silver-aluminum structure, a sensor, and also study the enhancement of SERS signal due to the plasmonic coupling effects. The results indicate that the hybrid multi-shape structure enhances the signal and enables a noticeable frequency shift in the Raman spectrum. Furthermore, the role of localized surface plasmon resonance (LSPR) and surface plasmon resonance (SPR) coupling in improving SERS performance is being investigated. It is found that the presence of multi-shape nanoparticles and the proposed arrangements optimize the plasmonic coupling between the localized surface plasmon and the surface plasmons of silver (Ag) and aluminum (Al), leading to an increased concentration of the electromagnetic field in the hotspot regions. We also present the results, the influence of various parameters on the spectral characteristics, and signal enhancement of SERS in the hybrid multi-shape structure. In this regard, the effect of the change in the inner radius of the nanostar from 10 to 100 nm and the use of different sequential arrangements of plasmonic materials, including gold (Au), silver (Ag), copper (Cu), aluminum (Al), and platinum (Pt), on electromagnetic field enhancement and SERS signal are being investigated. We show that in all configurations, the maximum SERS signal intensity occurs at an inner radius of 17.2 nm, reaching a value of 3.884 × 10⁷ /m. Additionally, the results for different sequential plasmonic material arrangements indicate that the Ag _ Al _ Ag configuration achieves the highest SERS signal intensity, with a value of 3.67328 × 10⁸ V/m. The Ag_Al_Ag structure achieves a sensitivity of 20.4 nm/RIU and a quality factor of 23.93. The corresponding enhancement factor (EF) for this structure has been calculated to be approximately 1.1 × 10⁷, indicating a significant enhancement in plasmonic sensor performance compared to previous designs.
... Usually, simulating electromagnetic fields generated by lightning typically involves transforming the infinite real domain into a computationally finite domain. In this paper, we utilize first-order Mur absorbing boundary conditions (ABCs) [40] at the artificial boundaries of the chosen geometry. ...
... Progress In Electromagnetics Research B, Vol. 112,[29][30][31][32][33][34][35][36][37][38][39][40][41] 2025 ...
... The authors of reference [15] simulated the propagation of electromagnetic pulses in the ionosphere using the FDTD method. The authors of reference [16] applied Z-transform to obtain the iterative relationship between the electric displacement vector D and the electric-field strength E in FDTD derivation, and analyzed the propagation characteristics of high-frequency radar signals in the ionosphere. When simulating the propagation characteristics of electromagnetic waves in the ionosphere, it is usually necessary to divide the ionosphere into many thin layers. ...
... When simulating the propagation characteristics of electromagnetic waves in the ionosphere, it is usually necessary to divide the ionosphere into many thin layers. With the continuous increase in ionospheric calculation altitude, the time step of FDTD model is limited by the minimum grid size in space, which leads to high computational complexity and memory-space occupation, resulting in error accumulation and inaccurate calculation results [16][17][18]. Therefore, how to quickly and accurately analyze the propagation characteristics of very-low-frequency electromagnetic waves in the ionosphere has become a difficult problem in the fields of computational electromagnetics and radio-wave propagation in recent years [19]. ...
Very-low-frequency electromagnetic waves have low propagation loss, slow attenuation, a stable phase and amplitude in the Earth ionosphere waveguide cavity, and are widely used in VLF communication and navigation, ionospheric heating, global lightning distribution inversion, and other fields. Studying the transmission characteristics of very-low-frequency (VLF) signals in the ionosphere is of great significance in spaceborne VLF communication technology. The existing research on ionospheric transmission characteristics using the finite-difference time domain (FDTD) algorithm is mostly based on high-frequency pulse signals, and the propagation model is relatively rough, resulting in certain calculation errors. To this end, a time-domain finite-difference algorithm model based on a uniaxial anisotropic perfectly matched layer (UPML) boundary in a spherical coordinate system was established, effectively solving the reflection problem existing in PEC boundary. The algorithm was used to numerically calculate the field-strength attenuation of VLF waves in the ionosphere. The simulation results showed that in the VLF frequency band, reducing the frequency is beneficial for electromagnetic waves to penetrate the ionosphere. Although the attenuation trend in the VLF waves is roughly the same during the day and night, the attenuation during the day is significantly greater than that at night, and this was compared and analyzed with traditional algorithms to verify the accuracy of the algorithm.
... In the numerical experiment, errors of the conservation laws were strictly maintained at the round-off level when the tolerance of the Crank-Nicolson method was small enough. However, the spectral method cannot employ non-periodic boundary conditions [13], so that these algorithms are applicable only to restricted situations. To perform kinetic simulations with non-periodic boundaries, a conservative Vlasov-Maxwell scheme based on the finite-difference manner is required. ...
... To obtain the momentum of the electromagnetic field in discrete form, the temporal product rule Eq. (26), and the spatial product rule Eq. (23) are applied to Eqs. (12), (13), (15), and (16): ...
For more than half a century, most of the plasma scientists have encountered a violation of the conservation laws of charge, momentum, and energy whenever they have numerically solve the first-principle equations of kinetic plasmas, such as the relativistic Vlasov--Maxwell system. This fatal problem is brought by the fact that both the Vlasov and Maxwell equations are indirectly associated with the conservation laws by means of some mathematical manipulations. Here we propose a quadratic conservative scheme, which can strictly maintain the conservation laws by discretizing the relativistic Vlasov--Maxwell system. A discrete product rule and summation-by-parts are the key players in the construction of the quadratic conservative scheme. Numerical experiments of the relativistic two-stream instability and relativistic Weibel instability prove the validity of our computational theory, and the proposed strategy will open the doors to the first-principle studies of mesoscopic and macroscopic plasma physics.
... If the integrand is a scalar then it can be written as (e 1 ∧ · · · ∧ e n ) · A n , (17) and the integral ...
... Consequently, one would prefer to relate to d and b as they relate to the stationary observer. For example, extension of Mur's ABCs to a 3D Cartesian grid rotating with arbitrary high rotation rate is straightforward; namely, one needs to replace e by d in [17,. ...
We employ Maxwell's equations formulated in Space-Time Algebra to perform discretization of moving geometries directly in space-time. All the derivations are carried out without any non-relativistic assumptions, thus the application area of the scheme is not restricted to low velocities. The 4D mesh construction is based on a 3D mesh stemming from a conventional 3D mesh generator. The movement of the system is encoded in the 4D mesh geometry, enabling an easy extension of well-known 3D approaches to the space-time setting. As a research example, we study a manifestation of Sagnac's effect in a rotating ring resonator. In case of constant rotation, the space-time approach enhances the efficiency of the scheme, as the material matrices are constant for every time step, without abandoning the relativistic framework.
... Therefore, the previous numerical method must be implemented in a spatial interval large enough to avoid the radiation to corrupt the results. In order to gain more control over the radiation evolution, the previous scheme will be complemented with the use of the energy conservative second-order finite difference Strauss-Vazquez algorithm [63] implemented with Mur boundary conditions [64], which absorb the linear plane waves at the boundaries. The efficiency of this numerical scheme has been proved in [65], and its global stability and convergence were established in [66]. ...
... If the spatial interval is large enough to the kink scattering occurs far away from the boundaries the dynamics in these peripheral regions is described by the linear partial differential equations ∂ 2 φ ∂t 2 − ∂ 2 φ ∂x 2 = 0 , ∂ 2 ψ ∂t 2 − ∂ 2 ψ ∂x 2 = 0 . This fact suggests the use of second order absorbing Mur contour conditions for our problem [64], which are given by the relations ...
In this paper we examine the scattering processes among the members of a rich family of kinks which arise in a (1+1)-dimensional relativistic two scalar field theory. These kinks carry two different topological charges that determine the mutual interactions between the basic energy lumps (extended particles) described by these topological defects. Processes like topological charge exchange, kink-antikink bound state formation or kink repulsion emerge depending on the charges of the scattered particles. Two-bounce resonant windows have been found in the antikink-kink scattering processes, but not in the kink-antikink interactions.
... We note that using larger boxes did not yield substantial differences in our simulation. To prevent radiation reflecting from the boundary at r ¼ L, Mur absorbing boundary conditions have been introduced [48]. Additionally, we introduced damping terms −ϵðrÞ dF n dt and −ϵðrÞ dB n dt , respectively, in (50) and (51), where ...
... However, due to the ratio between their oscillation periods and the total simulation time, they appear as a continuous blue area. On the other hand, the analytical response of the shape mode amplitudes (48) are represented by the dashed red curves for the same cases in Fig. 8. This figure shows a close match between the envelope of the numerical oscillations of the shape mode and the analytical amplitude. ...
The evolution of 1 vortices when their massive bound mode is excited is investigated in detail (both analytically and numerically) in the Abelian-Higgs model for different ranges of the self-coupling constant. The dependence of the spectrum of the 1 vortex fluctuation operator on the model parameter is discussed initially. A perturbative approach is employed to study the radiation emission in both the scalar and the vector channels. Our findings reveal that the oscillating initial configuration of the 1 vortex radiates at a frequency twice that of the internal mode. Through energy conservation considerations, we derive the decay law of the massive mode. Finally, these analytical results are compared with numerical simulations in field theory.
Published by the American Physical Society 2024
... The FDTD approach was used throughout the late 1970s to model a variety of transient and frequency-domain problems [72], [73], [74]. The development of the Mur ABC facilitated the treatment of open problems [75]. ...
... -Articles by Lindman [63] and Enquist and Majda [64] on ABCs led to their use with FDTD [75]. ...
Computational electromagnetics (CEM) is heavily intertwined with the IEEE Antennas and Propagation Society (AP-S). Effective designs for antennas and electromagnetic systems have motivated accurate simulation tools that have continuously exhausted available computer resources. This two-part article traces the development of computational tools and techniques and ties them to milestones in computer hardware, the growth and development of the AP-S, and some historical events throughout the world.
... Using the 3-D FDTD method, a two-step approach [41] is employed to calculate the LIVs on an overhead distribution line: first, the lightning EM fields are calculated over mountainous terrain, and then EM fields along the wire are used to compute the LIVs by using the Agrawal coupling model. As shown in Figure 1, the simulation space is 2900 × 2500 × 1500 m 3 , which is divided into ∆x × ∆y × ∆z = 5 × 5 × 5 m 3 cells, and surrounded by the second-order Mur absorbing boundary condition [42], the time step is set to 8.33 ns. Three different lightning strike locations (SL) are considered in the analysis: SL1 represents the side point at a distance d from the mountain center, SL2 is located at the mountain top, and SL3 represents the location of the lighting strike behind the mountain at a distance d from the mountain center. ...
Lightning-induced voltages (LIVs) computation is crucial for lightning protection of power systems and equipment, yet the effect of complex terrain on LIVs remains not fully evaluated. This study establishes a three-dimensional finite-difference time-domain model to investigate the LIVs over Gaussian-shaped mountainous terrain, considering different lightning strike locations. Simulation results show that the influence of Gaussian-shaped mountains on LIVs is directly related to the lightning strike location. Compared with the flat ground scenario, the LIVs’ amplitude can increase by approximately 56% when lightning strikes the mountain top. However, for lightning strikes to the ground adjacent to the mountain, the LIVs’ amplitude is attenuated to varying degrees due to the shielding effect of the mountain. Additionally, the influences of line configuration, as well as mountain height and width on the LIVs, are evaluated.
... The 3D domain had computational grids whose interval of λ/15, where λ was the 28-GHz beam wavelength. For the outer boundaries, a 2nd-order Mur absorbing boundary was used to reproduce the open boundary condition 51 . The PTFE lens with a curvature radius of 32.25 mm and a thickness of 16.9 mm was set at (x, y, z) = (50, 50, 5) mm, reproducing the experiment for the F = 75 mm lens. ...
Tractor millimeter-wave beam propulsion (TMiP), which obtains a propulsive force by receiving a high-power tractor beam from the front side of the vehicle, was proposed for space missions such as rocket launches from Earth and planetary takeoff missions. A beam irradiation experiment with a gyrotron device was conducted for a model rocket with a polytetrafluoroethylene (PTFE) lens to collect the tractor beam power, facilitating gas breakdown and plasma generation at the focal point. A high-pressure gas was formed via plasma heating, interacting with the PTFE lens mounted on the front of the vehicle and generating a propulsive force for rocket launching. A parametric study was conducted by changing the pulse width, which showed that the maximum thrust performance was achieved when the plasma front propagating toward the beam source did not protrude from the air-breathing intake at the front of the vehicle. Additionally, computational simulations for electromagnetic wave propagation and compressible fluid dynamics indicated that the thrust performance could be improved by decreasing the rocket diameter due to shock wave concentration.
... The boundary condition at the dielectric is formulated in the form of the Mur condition for the outgoing TEM wave, [43]. Then, recalling that H − θ = H θ − H + θ with H θ the total magnetic field intensity and H + θ the incoming wave at η = 0 described by H + θ = A(t) cos(ωt) with A(t) slowly varying amplitude, one can write the boundary condition for H θ at the port as [12,44,45] ( ...
A low pressure discharge sustained in molecular hydrogen with help of the electron cyclotron resonance heating at a frequency of 2.45 GHz is simulated using a fully electromagnetic implicit charge- and energy-conserving particle-in-cell/Monte Carlo code. The simulations show a number of kinetic effects, and the results are in good agreement with various experimentally measured data such as electron density, electron temperature and degree of dissociation. The electron energy distribution shows a tri-Maxwellian form due to a number of different electron heating mechanisms, agreeing with the experimental data in the measured electron energy interval. The simulation results are also compared with output data of a drift-diffusion model and proximity is observed between the computational results for the plasma density at the location of experimental measurement. However, the fluid approximation fails to accurately predict radical density and electron temperature because of the assumption of a single electron temperature. Special attention is paid to the characteristics of hydrogen radicals, whose production is strongly underestimated by the fluid model, whereas it is well predicted by the model considered here. The energy distribution of such radicals demonstrates the presence of a relatively large number of energetic hydrogen atoms produced by the dissociation of molecular hydrogen. The new insights are of significance for practical applications of hydrogen plasmas.
... The particular absorbing boundary employed is Mur's second-order approximation. 44 The bubble concentration (N) of 10 12 MBs per cubic meter is selected, equivalent to 10 6 MBs per milliliter. The numerical model is implemented in Cþþ programing language, and the simulated data are analyzed using the MATLAB software (MathWorks Inc., R2023b, Massachusetts, United States). ...
Ultrasound-mediated techniques are very promising tools, and efforts are needed to investigate ultrasound-responsive microbubbles (MBs) for medical applications. Serving a crucial role in optimizing the therapeutic effect, the dynamics of MBs have been a particular focus in present studies. Because MBs often exist in the form of bubble clusters in the ultrasonic field, a precise understanding of the interaction between ultrasound propagation and oscillations of MBs should be paid attention to. In this paper, a model coupling multi-bubble dynamics with nonlinear acoustic wave equations is proposed, and the oscillation of coated MBs is analyzed under different circumstances. In general agreement with experimental results, numerical studies indicate that the MB oscillations vary due to the bubble's initial radius, bubble property, and MB concentration. The promotion or suppression of the concentration on the MB oscillations varies alternately on larger MBs due to changes in the resonance frequency. The stable cavitation dose (SCD) increases with increasing peak negative pressure and pulse length. Moreover, the SCD initially increased with increasing MB concentration and then decreased rapidly as the concentration further increased. This study presents an opportunity for the interplay between MB cavitation, ultrasound parameters and the biological effects for future research from laboratory bench to patient bedside.
... The initial wave form in vacuum is given as a spatial distribution of the vector potential. At the boundary of the simulation box for the Maxwell equation, which is extended longer than Ω along the optical axis, Mur's absorbing boundary conditions are imposed to suppress unphysical reflection [48]. Figure 1 shows a schematic diagram in x-y plane. ...
The description of electron-electron scattering presents challenges in the microscopic modeling of the interaction of ultrashort intense laser pulses with solids. We extend the semiclassical approach based on the Vlasov equation [Phys. Rev. B 104, 075157(2021)] to account for dynamic electron-electron scattering by introducing the Vlasov-Uehling-Uhlenbeck (VUU) equation. We further couple the VUU equation with Maxwell's equations to describe the laser pulse propagation. We apply the present approach to simulate laser-electron interactions in bulk and thin-film aluminum, focusing on energy absorption and transport. Our calculation results reveal that electron-electron scattering affects energy absorption more significantly under p-polarization than under s-polarization, highlighting the role of the non-uniform surface potential. Our simulations also show that the energy transport extends beyond the optical penetration depth, which is consistent with observations in previous laser ablation experiments. The developed Maxwell-VUU approach is expected to advance the understanding of intense laser-material interactions not only as a cost-effective alternative to the time-dependent density functional theory (TDDFT), but also by incorporating fermionic two-body collisions whose description is limited in TDDFT.
... In order to simulate an infinite open space, truncation must be made at the edge of the space, and absorption boundary conditions must be set. Here, the first-order Mur absorption boundary conditions are adopted [26]. The grid space of the lightning electric field calculated by the FDTD method is shown in Fig. 1(a). ...
Spectrographic analysis was performed to determine the plasma composition of artificially triggered lightning in Guangzhou, China, by about N I, N II, O I, O II, and Fe II. Different engineering models were adopted to simulate the channel current ( i ) of triggered lightning, the changing rate of current over time ( ), as well as the changing rate of current over time and height [ ]. It is indicated that the modified transmission line model with linear current decay with height (MTLL) can simulate a moderate channel current. The experimental current is found to be stronger than the simulation at the bottom of the channel, and it is predicted that the addition of iron ions affects the channel current. Furthermore, based on a finite-difference time-domain simulation, the radiated electric field of the return stroke of triggered lightning is found more varied than that of natural lightning. This forms a phenomenon of an overshoot amplitude electric field in triggered lightning. It is predicted that the addition of iron ions in the channel current affects the distribution of the radiated electric field.
... Among these, spherical particles align closely with many natural models, so many researchers have carried out a lot of research on the shaped beam scattering of spherical particles [1,2]. The common methods to solve this problem include discrete dipole approximation [3], the finite-difference time-domain method [4], and the T-matrix method [5]. The GLMT [6,7] is an accurate solution of Maxwell's equation; there is no limit to the size of the particle studied, and it is recognized as an important theoretical method to solve the interaction between shaped beams and particles. ...
Photonic jets (PJs) are generated by the interaction between beams and dielectric particles, resulting in narrow, high-intensity beams produced behind the medium. Silicon nitride exhibits excellent optical properties; we aim to explore the potential of silicon nitride in PJ applications. In this paper, the beam is expanded by spherical vector wave functions with the framework of the generalized Lorenz–Mie theory (GLMT). Numerical simulations are performed to investigate the PJs generated by silicon nitride spherical particles illuminated by the plane wave, Gaussian beam, and zero-order Bessel beam under water; the effects of various parameters on the characteristics of PJs are further examined.
... The boundary condition at the dielectric is formulated in the form of the Mur condition for the outgoing TEM wave, (J −1 z ∂/∂η − ( √ ϵµ/c)∂/∂t)H − θ = 0 [43]. Then, recalling that H − θ = H θ − H + θ with H θ the total magnetic field intensity and H + θ the incoming wave, one can write the boundary condition for H θ at the port as [12,44,45] ...
A low pressure discharge sustained in molecular hydrogen with help of the electron cyclotron resonance heating at a frequency of 2.45 GHz is simulated using a fully electromagnetic implicit charge- and energy-conserving particle-in-cell/Monte Carlo code. The simulations show a number of kinetic effects, and the results are in good agreement with various experimentally measured data such as electron density, electron temperature and degree of dissociation. The electron energy distribution shows a tri-Maxwellian form due to a number of different electron heating mechanisms, agreeing with the experimental data in the measured electron energy interval. The simulation results are also verified against a drift-diffusion model and proximity is observed between the computational results for the plasma density at the location of experimental measurement. However, the fluid approximation fails to accurately predict radical density and electron temperature because of the assumption of a single electron temperature. Special attention is paid to the characteristics of hydrogen radicals, whose production is strongly underestimated by the fluid model, whereas it is well predicted by the model considered here. The energy distribution of such radicals demonstrates the presence of a relatively large number of energetic hydrogen atoms produced by the dissociation of molecular hydrogen. The new insights are of significance for practical applications of hydrogen plasmas.
... However, analytical solutions are limited to simple geometries with well-defined boundaries. The finite-difference time-domain (FDTD) method simplifies solving Maxwell's equations by discretization, enabling numerical solutions [36][37][38]. Ray tracing (RT) rooted in geometric optics, models electromagnetic wave propagation in real-world environment by computing multipath effects such as direct paths, reflections, and scattering [39][40][41][42]. Electromagnetic information theory (EMIT) extends computational electromagnetics by integrating circuit and antenna effects to enhance communication degrees of freedom [43][44][45][46][47]. Recently, by integrating prior environmental information, RT improves accuracy and reduces complexity, establishing itself as a widely used deterministic channel modeling method [52][53][54]. ...
The channel is one of the five critical components of a communication system, and its ergodic capacity is based on all realizations of statistic channel model. This statistical paradigm has successfully guided the design of mobile communication systems from 1G to 5G. However, this approach relies on offline channel measurements in specific environments, and the system passively adapts to new environments, resulting in deviation from the optimal performance. With the pursuit of higher capacity and data rate of 6G, especially facing the ubiquitous environments, there is an urgent need for a new paradigm to combat the randomness of channel, i.e., more proactive and online manner. Motivated by this, we propose an environment intelligence communication (EIC) based on wireless environmental information theory (WEIT) for 6G. The proposed EIC architecture is composed of three steps: Firstly, wireless environmental information (WEI) is acquired using sensing techniques. Then, leveraging WEI and channel data, AI techniques are employed to predict channel fading, thereby mitigating channel uncertainty. Thirdly, the communication system autonomously determines the optimal air-interface transmission strategy based on real-time channel predictions, enabling intelligent interaction with the physical environment. To make this attractive paradigm shift from theory to practice, we answer three key problems to establish WEIT for the first time. How should WEI be defined? Can it be quantified? Does it hold the same properties as statistical communication information? Furthermore, EIC aided by WEI (EIC-WEI) is validated across multiple air-interface tasks, including CSI prediction, beam prediction, and radio resource management. Simulation results demonstrate that the proposed EIC-WEI significantly outperforms the statistical paradigm in decreasing overhead and performance optimization.
... For the Maxwell equations, many numerical methods, such as the finite-difference time-domain method has been developed [50][51][52]. For the Schrödinger equation, unitary algorithm has been proposed [49,[53][54][55]. ...
An infinite dimensional canonical symplectic structure and structure-preserving geometric algorithms are developed for the photon-matter interactions described by the Schr\"odinger-Maxwell equations. The algorithms preserve the symplectic structure of the system and the unitary nature of the wavefunctions, and bound the energy error of the simulation for all time-steps. This new numerical capability enables us to carry out first-principle based simulation study of important photon-matter interactions, such as the high harmonic generation and stabilization of ionization, with long-term accuracy and fidelity.
... The numerical scheme that has been employed in this paper follows the algorithm introduced in [62] by Kassam and Trefethen, which is spectral in space and fourth order in time. As a complement to the previous scheme, an energy conservative second-order finite difference algorithm [63,60] implemented with Mur boundary conditions [64] has also been used. This algorithm lets to control the effect of radiation in the simulation because it absorbs the linear plane waves at the boundaries. ...
In this paper kink scattering processes are investigated in the Montonen-Sarker-Trullinger-Bishop model. The MSTB model is in fact a one-parametric family of relativistic scalar field theories living in a one-time one-space Minkowski space-time which encompasses two coupled scalar fields. Between the static solutions of the model two kinds of topological kinks are distinguished in a precise range of the family parameter. In that regime there exists one unstable kink exhibiting only one non-null component of the scalar field. Another type of topological kink solutions, stable in this case, includes two different kinks for which the two-components of the scalar field are non-null. Both one-component and two-component topological kinks are accompanied by their antikink partner. The decay of disintegration of the unstable kink to one of the stable pair plus radiation is numerically computed. The pair of stable two-component kinks living respectively on upper and lower half-ellipses in field space belong to identical topological sectors in configuration space and provides an ideal playground to address several scattering events involving one kink and either its own antikinks or either the antikink of the other stable kink of the pair. By means of a numerical computation procedure we shall find and describe interesting physical phenomena. Bion (kink-antikink oscillations) formation, kink reflection, kink-antikink annihilation, kink transmutation and resonances are examples of these type of events. The appearance of these special phenomena emerging in kink-antikink scattering configurations depends critically on the initial collision velocity and the chosen value of the coupling constant parametrizing the family of MSTB models.
... Symmetries in the structure can be enforced by setting E = 0 (V = 0) on selected boundaries, see Section 4. The last equation imposes the radiation conditions required to truncate the infinite space. In particular, we employ Silver-Müller conditions, which are first order absorbing boundary conditions [32,53], ...
The interaction of electromagnetic waves with metallic nanostructures generates resonant oscillations of the conduction-band electrons at the metal surface. These resonances can lead to large enhancements of the incident field and to the confinement of light to small regions, typically several orders of magnitude smaller than the incident wavelength. The accurate prediction of these resonances entails several challenges. Small geometric variations in the plasmonic structure may lead to large variations in the electromagnetic field responses. Furthermore, the material parameters that characterize the optical behavior of metals at the nanoscale need to be determined experimentally and are consequently subject to measurement errors. It then becomes essential that any predictive tool for the simulation and design of plasmonic structures accounts for fabrication tolerances and measurement uncertainties. In this paper, we develop a reduced order modeling framework that is capable of real-time accurate electromagnetic responses of plasmonic nanogap structures for a wide range of geometry and material parameters. The main ingredients of the proposed method are: (i) the hybridizable discontinuous Galerkin method to numerically solve the equations governing electromagnetic wave propagation in dielectric and metallic media, (ii) a reference domain formulation of the time-harmonic Maxwell's equations to account for geometry variations; and (iii) proper orthogonal decomposition and empirical interpolation techniques to construct an efficient reduced model. To demonstrate effectiveness of the models developed, we analyze geometry sensitivities and explore optimal designs of a 3D periodic annular nanogap structure.
... Conventional BCs are the usual boundary conditions at an interface between two media, including the Perfect Electric Conductor (PEC) BC, the Perfect Magnetic Conductor (PMC) BC, the Perfect Electromagnetic Conductor (PEMC) BC [48]- [50], the Absorbing Boundary Condition (ABC) [51], [52] and the Perfectly Matched Layers (PMLs) BC [53]. These BCs address the problem of a discontinuity formed by the juxtaposition of two different media. ...
Metasurfaces represent one of the most vibrant fields of modern science and technology. A metasurface is a complex electromagnetic structure, that is typically deeply subwavelength in thickness, electrically large in transverse size and composed of subwavelength scattering particles with extremely small features; it may generally be bianisotropic, spacevarying and time-varying, nonlinear, curved and multiphysics. With such complexity, the design of a metasurface requires a holistic approach, involving synergistic synthesis and analysis operations, based on a solid model. The Generalized Sheet Transition Conditions (GSTCs), combined with bianisotropic surface susceptibility functions, provide such a model, and allow now for the design of sophisticated metasurfaces, which still represented a major challenge a couple of years ago. This paper presents this problematic, focusing on the computational analysis of metasurfaces via the GSTC-susceptibility approach. It shows that this analysis plays a crucial role in the holistic design of metasurfaces, and overviews recently reported related frequency-domain (FDFD, SD-IE, FEM) and time-domain (FDTD) computational techniques.
... Absorbing boundary conditions (ABCs) are extremely important numerical tools [29,6,27] to efficiently simulate wave propagation phenomena in large or infinite domains. In these methods, the computational cost is greatly reduced by truncating the entire domain into a much smaller region of interest. ...
A new type of absorbing boundary conditions for molecular dynamics simulations are presented. The exact boundary conditions for crystalline solids with harmonic approximation are expressed as a dynamic Dirichlet- to-Neumann (DtN) map. It connects the displacement of the atoms at the boundary to the traction on these atoms. The DtN map is valid for a domain with general geometry. To avoid evaluating the time convo- lution of the dynamic DtN map, we approximate the associated kernel function by rational functions in the Laplace domain. The parameters in the approximations are determined by interpolations. The explicit forms of the zeroth, first, and second order approximations will be presented. The stability of the molecular dynamics model, supplemented with these absorbing boundary conditions is established. Two numerical simulations are performed to demonstrate the effectiveness of the methods.
... The HDG discretization of the time-harmonic Maxwell's equations is described in detail in [46]. The boundaries of the surrounding dielectric medium Ω represent the farfield truncation of the infinite space, where radiation is imposed with the Silver-Müller conditions, which are first order absorbing boundary conditions [41,61], namely ...
The interaction of light with metallic nanostructures produces a collective excitation of electrons at the metal surface, also known as surface plasmons. These collective excitations lead to resonances that enable the confinement of light in deep-subwavelength regions, thereby leading to large near-field enhancements. The simulation of plasmon resonances presents notable challenges. From the modeling perspective, the realistic behavior of conduction-band electrons in metallic nanostructures is not captured by Maxwell's equations, thus requiring additional modeling. From the simulation perspective, the disparity in length scales stemming from the extreme field localization demands efficient and accurate numerical methods. In this paper, we develop the hybridizable discontinuous Galerkin (HDG) method to solve Maxwell's equations augmented with the hydrodynamic model for the conduction-band electrons in noble metals. This method enables the efficient simulation of plasmonic nanostructures while accounting for the nonlocal interactions between electrons and the incident light. We introduce a novel postprocessing scheme to recover superconvergent solutions and demonstrate the convergence of the proposed HDG method for the simulation of a 2D gold nanowire and a 3D periodic annular nanogap structure. The results of the hydrodynamic model are compared to those of a simplified local response model, showing that differences between them can be significant at the nanoscale.
... The dark gray area is the input grid with Mur's absorbing boundary condition [32]. In the CE-FDTD simulation, sound pressure propagates omnidirectionally from the input grid. ...
The finite difference time domain (FDTD) method has been proposed and used for sound field simulation. To reproduce actual sound wave propagation in sound field simulations, it is necessary to apply the radiation characteristics. With the FDTD method, radiation characteristics can be applied by setting sound pressure in a dense grid arrangement. However, conventional techniques for capturing radiation characteristics use a sparse array of microphones and are considered insufficient for the FDTD simulation. Furthermore, the technique required to apply captured acoustic signals in a dense grid arrangement with the FDTD method has not been considered. In this paper, we propose a novel hardware and software system that captures the radiation characteristics for a dense grid arrangement and applies them to the FDTD method, while controlling the sound wave propagation with the non-propagation region. The proposed system produces the average differences from measured values of sound pressure, propagation time, center frequency, and log-spectral distortion of 1.8 dB, 0.04 ms, 700 Hz, and 3.5 dB, respectively, which is more accurate than the conventional techniques. The result shows that this system is useful for improving the accuracy of sound wave propagation reproduction with the sound field simulation.
In this chapter, we will first introduce the time-dependent Maxwell’s equations and solution representations using scalar and vector potentials. Then, physical boundary conditions involving interfaces between dielectrics and conductors will be discussed. For computing scattering fields in infinite domains, several types of local artificial boundary conditions for computational domains will be presented, including the Engquist–Majda one-way boundary conditions, the high-order Bayliss–Turkel boundary conditions in auxiliary variable forms, and the uniaxial perfectly matched layered (UPML) boundary conditions.
In this chapter, use of the PML ABC in two typical applications of the FDTD method is described with details. The numerical reflection observed from the PML is interpreted and the PML parameters are optimized so as to reduce the computational cost of the PML while preserving a satisfactory simulation of free space.
Answering two questions is the principal objective of this introductory chapter. The first question is: why is the simulation of free space needed in numerical electromagnetics? The second one is: which requirements have to be satisfied by the methods that simulate free space? In addition, the methods developed for simulating free space before the introduction of the Perfectly Matched Layer concept are briefly reviewed.
The finite‐difference time‐domain (FDTD) method is one of key numerical techniques to solve Maxwell's equations, which has the broad applications for a variety of electromagnetic problems such as electromagnetic propagation, electromagnetic compatibility (EMC), and electromagnetic interference (EMI). This article first reviews the basic FDTD techniques such as update equations, numerical dispersion, stability properties, boundary conditions and conformal techniques, and then focuses on some enhanced FDTD techniques such as parallelization techniques and hardware acceleration techniques that can be greatly used to extend applications of the regular FDTD methods.
Electromagnetic modeling is a critical analysis and design methodology in various fields such as radio‐frequency (RF) and microwave engineering, antennas, radar, remote sensing, medical diagnostics, bio‐electromagnetics, physics and radio astronomy, and electromagnetic compatibility (EMC) and interference. It formulates and solves electromagnetic field problems using mathematical, physical, or computational models that relate the electromagnetic functionality of devices to their geometrical and material parameters or predict how electromagnetic fields interact with physical objects and materials. This chapter presents the most common methods for modeling electromagnetic scenarios in the frequency, the spectral, and the time domain, as well as their hybrid combinations. They are introduced as projective approximations of field solutions using the method of weighted residuals (MWR). This unified perspective best illustrates their properties, differences, and their suitability for solving a particular electromagnetic problem, thus helping the reader to appreciate the connections, similarities, and differences among the various methods. It also reflects historical and methodological developments and aims to provide a foundational understanding of current electromagnetic modeling methods.
Представлена математическая модель расчeта эффективной площади рассеяния летательного аппарата. Модель базируется на использовании двухэтапного подхода. 1 этап - получение параметров отражeнной электромагнитной волны в окрестности летательного аппарата (используется классический конечно-разностный метод интегрирования уравнений Максвелла в ограниченной области пространства, включающей летательный аппарат). 2 этап - расчeт ослабления отражeнной электромагнитной волны при еe распространении от летательного аппарата до приeмной антенны радиолокатора (используется метод функций Грина, источниками в которых являются параметры электромагнитных полей на границе расчeтной области первого этапа расчeта). Данный подход позволяет осуществить расчeт эффективной площади рассеяния летательных аппаратов, окружeнных формирующимися за счeт различных физических эффектов ионизованными образованиями. Расчeтная модель протестирована на известных аналитических решениях расчeта эффективной площади рассеяния для простых геометрических тел.
Electromagnetic modeling is a critical analysis and design methodology in various fields such as radio‐frequency (RF) and microwave engineering, antennas, radar, remote sensing, medical diagnostics, bio‐electromagnetics, physics and radio astronomy, and electromagnetic compatibility (EMC) and interference. It formulates and solves electromagnetic field problems using mathematical, physical, or computational models that relate the electromagnetic functionality of devices to their geometrical and material parameters or predict how electromagnetic fields interact with physical objects and materials. This chapter presents the most common methods for modeling electromagnetic scenarios in the frequency, the spectral, and the time domain, as well as their hybrid combinations. They are introduced as projective approximations of field solutions using the method of weighted residuals (MWR). This unified perspective best illustrates their properties, differences, and their suitability for solving a particular electromagnetic problem, thus helping the reader to appreciate the connections, similarities, and differences among the various methods. It also reflects historical and methodological developments and aims to provide a foundational understanding of current electromagnetic modeling methods.
The finite‐difference time‐domain (FDTD) method is one of key numerical techniques to solve Maxwell's equations, which has the broad applications for a variety of electromagnetic problems such as electromagnetic propagation, electromagnetic compatibility (EMC), and electromagnetic interference (EMI). This article first reviews the basic FDTD techniques such as update equations, numerical dispersion, stability properties, boundary conditions and conformal techniques, and then focuses on some enhanced FDTD techniques such as parallelization techniques and hardware acceleration techniques that can be greatly used to extend applications of the regular FDTD methods.
In this article, various time‐domain electromagnetic simulation methods, including the finite‐difference time‐domain (FDTD), transmission‐line matrix (TLM), finite‐element time‐domain (FETD), and multiresolution time‐domain (MRTD) techniques are discussed in detail along with grid optimization approaches. In addition, conformal meshing techniques, specifically finite‐volume time‐domain (FVTD) and generalized Yee FDTD methods are briefly descirbed. Also included is a discussion of code parallelization, grid excitation, absorbing boundary conditions, antenna modeling, and methods that can be used to derive S parameters from time‐domain output data.
The rapid growth of technology in computing architecture and networking has resulted in fundamental changes in the way we perform computational analysis in science and engineering. Some problems that are intractable because they are computation‐intensive in memory and/or CPU time can now be solved in an effective manner with the help of parallel computing. The aim of this article is to introduce to readers some examples of parallel computing algorithms that were developed for the computational electromagnetics community but are also applicable to other fields in science and engineering. In this article, the construction components of parallel computing platforms are introduced. A Beowulf cluster, which is a system with distributed memory multiprocessor architecture, will be briefly discussed. Some of the most popular simulation techniques in computational electromagnetics such as the finite‐difference time‐domain (FDTD) method and the method of moments (MoM) are discussed. Several methods of solving a full matrix either directly or iteratively will also be discussed. Illustrative examples are given to demonstrate the use of these algorithms and their implementations on the parallel computing platform for solving large‐scale electromagnetic problems.
The FDTD method was used to simulate focused Gaussian beam propagation through multiple inhomogeneous biological cells. To our knowledge this is the first three dimensional computational investigation of a focused beam interacting with multiple biological cells using FDTD. A parametric study was performed whereby three simulated cells were varied by organelle density, nuclear type and arrangement of internal cellular structure and the beam focus depth was varied within the cluster of cells. Of the organelle types investigated, it appears that the cell nuclei are responsible for the greatest scattering of the focused beam in the configurations studied. Additional simulations to determine the optical scattering from 27 cells were also run and compared to the three cell case. No significant degradation of two-photon lateral imaging resolution was predicted to occur within the first 40 µm of imaging depth.
Antenna miniaturization is currently facing increased performance demands while simultaneously lacking a computational framework to drive robust designs. Future platforms must radiate farther, at lower frequency, and be increasingly compact. While mechanical resonance based piezoelectric antenna arrays are a viable candidate, detrimental mutual depolarization effects arise that must be characterized by multi-scale simulations, coupling the elastodynamic and EM wave physics. This work presents an algorithm capable of performing such full-wave simulations to provide design guidance to engineers wishing to mitigate mutual depolarization. The relevant dynamic systems of equations are discretized and put into a Finite Difference Time Domain (FDTD) scheme. This scheme exhibits electrodynamic unconditional stability and features heavily graded meshes to directly tackle the time and length scale disparity between the mechanical and EM waves. The algorithm was validated by comparison with experimental data and analytical solutions. Additionally, the algorithm compared well with predicted values for depolarization. Simulations demonstrated that spacing within piezoelectric antenna arrays should not be made too small, as to induce undue mutual depolarization, or too large, as to not allow sufficient elements to contribute to array dipole moment. Computational guidance is also provided based on the authors' own experiences
In this paper, a flexible coplanar waveguide (CPW)-fed circular ring antenna is demonstrated as a key component in a smart packaging solution. The CPW-fed circular ring antenna is fabricated on a Polyethylene terephthalate (PET) substrate. The proposed sensing antenna is developed for operation in the C-band frequency ( 3.9 GHz). The antenna exhibits a frequency shift and changes in return loss (S11) with bending and deformation. It is observed that the change in S11 is more reliable for quantitative measurements and the change in frequency can be utilized for qualitative analysis. A sensitivity of 8.25 and 7.37 is recorded for lower and higher longitudinal bending conditions in the experimental set-up. The sensing antenna demonstrated satisfactory performance with smart chemical packaging to detect and distinguish the bending deformations in the lateral and longitudinal directions. A maximum of 1.5 cm lateral and longitudinal bending with a resolution of 25 mm is detected using the sensing antenna. The measured sensitivities for packaging experiments are 16.8 and 15.22 for transverse direction and longitudinal direction. The flexible nature of the antenna enables seamless integration into various packaging configurations, ensuring adaptability to diverse environments and applications.
Solving partial differential equations (PDEs) is omnipresent in scientific research and engineering and requires expensive numerical iteration for memory and computation. The primary concerns for solving PDEs are convergence speed, data movement, and power consumption. This work proposed the first fast-convergence PDE solver with an automatic adjustment multiple-stride iteration method, significantly increasing the PDE convergence speed. A dynamic-precision near-memory-computing architecture with booth encoding is proposed to reduce iterated intermediate data movement. A customized 32T compressor and a 14T full adder are designed to reduce the power and hardware cost of the solver. The processor is fabricated using 65-nm CMOS technology and occupies a 6.25 mm
die area. It can achieve a convergence speedup by 4
compared with the existing work.
Suppose a PV array installed on flat ground, twenty PV individual panels are connected into a group, and eight groups are wired together and then linked to the inverter, as shown in Fig. 3.1a. Considering the lightning channel is vertical and perpendicular to the ground, the horizontal distance R, from PV array to the lightning channel, is 5 m. The detailed dimension of single PV panel is depicted in Fig. 3.1b.
This paper presents the finite-difference time-domain (FDTD) method for gauge-invariant field-impulses, which constitute the fundamental physical quantities of electromagnetics (EM) in lieu of fields and potentials. The field-impulses not only provide satisfactory resolution to various shortcomings of fields/potentials, but also facilitate simple and versatile numerical computations. They overcome the inadequacy of fields and the issues of gauge-dependent potentials in classical and quantum EM. The significance and advantages of field-impulses are discussed and compared against those of fields and potentials, in physical, mathematical and computational aspects. The single electric field-impulse is sufficient to aptly-describe all EM, including electrostatics, magnetostatics and electrodynamics. For the excitation of field-impulses, we resort to a universal current-impulse source in lieu of traditional charges and currents. Such current-impulse can encapsulate simultaneously all static charges, direct currents and time-varying dynamic sources. Moreover, it is inherently complying with charge conservation and current continuity condition. There is no need to explicitly impose such condition as required for traditional charges and currents. The FDTD update equations for field-impulses with current-impulse source are much simpler with significantly fewer terms and arithmetic operations compared to those for potentials. Numerical experiments demonstrate the full versatility of field-impulse FDTD method for computing all electrostatics, magnetostatics and electrodynamics, which would be useful for classical and quantum EM.
In practical calculations, it is often essential to introduce artificial boundaries to limit the area of computation. Here we develop a systematic method for obtaining a hierarchy of local boundary conditions at these artifical boundaries. These boundary conditions not only guarantee stable difference approximations, but also minimize the (unphysical) artificial reflections that occur at the boundaries.
In practical calculations, it is often essential to introduce artificial boundaries to limit the area of computation. Here we develop a systematic method for obtaining a hierarchy of local boundary conditions at these artificial boundaries. These boundary conditions not only guarantee stable difference approximations but also minimize the (unphysical) artificial reflections which occur at the boundaries.
The interaction of nuclear EMP with a large aeronautical or
communication system involves the penetration of structures to generate
electric currents and voltages in electronic subsystems. Attention is
presently given to experimentally and analytically derived quantitative
data on the EMP coupling, penetration, and propagation interactions
which can serve as bases for system hardness design and implementation.
The general process of EMP penetration into a large system is
generically viewed as one of EMP transfer across successive strata of a
layered topological model, where the crossing of each surface can be
quantified by a transfer function which is largely dependent on the
local surface geometry and which can be determined by solving an
appropriate boundary-value problem in EM theory. Analytical formulas and
illustrative samples are also presented.
The characteristics of the waves guided along a plane [I] P. S. Epstein, “On the possibility of electromagnetic surface waves, ” Proc. Nat’l dcad. Sciences, vol. 40, pp. 1158-1165, Deinterface which separates a semi-infinite region of free cember 1954. space from that of a magnetoionic medium are investi- [2] T. Tamir and A. A. Oliner, “The spectrum of electromagnetic waves guided by a plasma layer, ” Proc. IEEE, vol. 51, pp. 317gated for the case in which the static magnetic field is 332, February 1963. oriented perpendicular to the plane interface. It is [3] &I. A. Gintsburg, “Surface waves on the boundary of a plasma in a magnetic field, ” Rasprost. Radwvoln i Ionosf., Trudy found that surface waves exist only when w,<wp and NIZMIRAN L’SSR, no. 17(27), pp. 208-215, 1960. that also only for angular frequencies which lie bet\\-een [4] S. R. Seshadri and A. Hessel, “Radiation from a source near a plane interface between an isotropic and a gyrotropic dielectric,” we and 1/42 times the upper hybrid resonant frequency. Canad. J. Phys., vol. 42, pp. 2153-2172, November 1964. The surface waves propagate with a phase velocity [5] G. H. Owpang and S. R. Seshadri, “Guided waves propagating along the magnetostatic field at a plane boundary of a semiwhich is always less than the velocity of electromagnetic infinite magnetoionic medium, ” IEEE Trans. on Miomave waves in free space. The attenuation rates normal to the Tbory and Techniques, vol. MTT-14, pp. 136144, March 1966. [6] S. R. Seshadri and T. T. \Vu, “Radiation condition for a maginterface of the surface wave fields in both the media are netoionic medium. ” to be Dublished. examined. Kumerical results of the surface wave characteristics are given for one typical case.
Preface. Introduction. 1. One-dimensional motion of an elastic continuum. 2. The linearized theory of elasticity. 3. Elastodynamic theory. 4. Elastic waves in an unbound medium. 5. Plane harmonic waves in elastic half-spaces. 6. Harmonic waves in waveguides. 7. Forced motions of a half-space. 8. Transient waves in layers and rods. 9. Diffraction of waves by a slit. 10. Thermal and viscoelastic effects, and effects of anisotrophy and non-linearity. Author Index. Subject Index.
The present state of calculational techniques for IEMP and SGEMP phenomena is reviewed, with emphasis on the physical, mathematical, and geometrical approximations made in various approaches. Some speculations are made on the ultimate capabilities of computer calculations, and the need for analytical calculations, especially simple methods for making estimates, is stressed. Finally, the possibly fruitful competition and complementation between digital (computer) and analogue (electrical simulation) techniques are discussed.
A numerical method is described for the solution of the electromagnetic fields within an arbitrary dielectric scatterer of the order of one wavelength in diameter. The method treats the irradiation of the scatterer as an initial value problem. At t = 0, a plane-wave source of frequency f is assumed to be turned on. The diffraction of waves from this source is modeled by repeatedly solving a finite-difference analog of the time-dependent Maxwell's equations. Time stepping is continued until sinusoidual steady-state field values are observed at all points within the scatterer. The envelope of the standing wave is taken as the steady-state scattered field. As an example of this method, the computed results for a dielectric cylinder scatterer are presented. An error of less than ±10 percent in locating and evaluating the standing-wave peaks within the cylinder is achieved for a program execution time of 1 min. The extension of this method to the solution of the fields within three-dimensional dielectric scatterers is outlined.
Low-frequency electromagnetic penetration of a closed shielded region through an aperture in the shield is considered by investigating the canonical problems in which the shield is a perfectly conducting spherical shell, the aperture is circular, and the applied field is uniform. Each of these problems reduces to that of solving a set of dual series equations. The solutions of previously solved problems are presented as well as those of heretofore unsolved problems. The penetration of the shielded region is measured by the ratio of the field at the center of the shell to the external applied uniform field. It has been previously shown that these ratios are the same for an applied magnetic field parallel to the symmetry axis and an applied electric field perpendicular to this axis. In this paper it is shown that the ratios are the same for an applied electric field parallel to the axis when the shell is uncharged and for an applied magnetic field perpendicular to the axis. In addition, a new approach to the solution of a certain class of dual series equations is found and exploited in the solution of one of the canonical problems.
This article describes the 3-D EMP finite-difference time-domain computer code THREDE as generalized to calculate coupling to, and scattering from, lossy dielectric objects. The code primarily treats the scattered component of the electromagnetic fields (thus presuming linearity) and employs a radiating outer boundary. As sample scatterers, we use dielectric spheres of ¿ = 2¿0 and 9¿0 illuninated by an EMP plane wave of double exponential profile. Comparitive calculations were made using the inverse-Fourier transform of the Rayleigh-Mie spherical-hannonic expansion solution-agreement of the two solutions is very good.
A numerical method for predicting the sinusoidal steady-state electromagnetic fields penetrating an arbitrary dielectric or conducting body is described here. The method employs the finite-difference time-domain (FD-TD) solution of Maxwell's curl equations implemented on a cubic-unit-cell space lattice. Small air-dielectric loss factors are introduced to improve the lattice truncation conditions and to accelerate convergence of cavity interior fields to the sinusoidal steady state. This method is evaluated with comparison to classical theory, method-of-moment frequency-domain numerical theory, and experimental results via application to a dielectric sphere and acylindrical metal cavity with an aperture. Results are also given for a missile-like cavity with two different types of apertures illuminated by an axial-incidence plane wave.
Experimental charge and current measurements have recently been performed on an aircraft when it was exposed to the transient electromagnetic fi'eld of an electromagnetic pulse (EMP) simulator. These new data allow a test of the predictive capabilities of the three-dimensional finite-difference method for realistic aircraft simulator test problems. In the paper, the workings of the threedimensional finite-difference method and its required inputs and sensitivity to variations in the inputs are discussed in sufficient detail to enable others to employ the method. A companion paper compares the experimental measurements to predictions for a large variety of measurement locations. Agreement is shown to be good for all major response measurements and satisfactory for a number of other measurements.
A numerical technique is presented that may be used to predict the current induced on a thin metallic body of revolution excited by an electromagnetic pulse. Examples are given. Introduced here is the use of the radiation condition in a finite difference solution. This development alleviates the requirement that finite difference techniques be applied to a bounded region of space.
In this paper, a more general form of Babinet's principle for electromagnetic waves and perfectly conducting screens with apertures covered by resistive sheets is presented. It shows that the field components parallel to the screens are now further related to one another by the resistance of the resistive sheets whereas the field components normal to the screens retain the same relationship as in the absence of the resistive sheets.
A finite difference solution technique is used to solve Maxwell's equations directly in the treatment of electromagnetic pulse scattering in a time-varying inhomogeneous medium. In particular the scattering from a cylindrical rod inside a cylindrical waveguide is considered where axial symmetry is obtained.
Small holes in cable shields
- R W Latham
R. W. Latham, "Small holes in cable shields," Interaction Notes, Note 1 18, Sept. 1972.
Absorbing boundary conditions for the numerical simulation of wavesFinite-difference analysis of EMP coupling to lossy dielectric structures
- B Engquist
- A Majda
- R Holland
- L Simpson
- K S Kunz
B. Engquist and A. Majda, "Absorbing boundary conditions for the numerical simulation of waves," Math. Comp., vol. 31, pp. 629-651, July 1977. [91 R. Holland, L. Simpson, and K. S. Kunz, "Finite-difference analysis of EMP coupling to lossy dielectric structures," IEEE Trans. Electromagn. Compat., vol. EMC-22, pp. 203-209, 1980.
Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic mediaState of the art in IEMP and SGEMP cal-culations
- K S Yee
- C L Longmire
K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propagat., vol. AP-14, pp. 302-307, May 1966. [21 C. L. Longmire, "State of the art in IEMP and SGEMP cal-culations," IEEE Trans. Nucl. Sci., vol. NS-22, pp. 2340-2344, Dec. 1975.