Electromagnetic Equations Research Articles

Numerical Analysis of Therapeutic Effects by Varying Slot Numbers and Slot‐to‐Slot Distance in Microwave Ablation Using Multislot Coaxial Antenna

Microwave ablation (MWA) is based on the energy absorbed by biological tissue through microwave emission, which raises the temperature of the tumor to perform the treatment. MWA has the advantage of causing minimal bleeding and treating deep tumors. In this study, MWA for various slot design situations in a microwave coaxial antenna (MCA) was simulated through numerical analysis. For the slot design, the number of slots and the distance between slots were determined by the analysis parameters. A tumor developed inside the liver tissue was implemented, and the temperature distribution of the tumor and surrounding liver tissue was calculated for all cases selected for numerical analysis. The Helmholtz electromagnetic equation was used to calculate the electromagnetic field inside the tissue, and the modified Pennes bioheat equation was used to calculate the temperature change in the tissue due to the emitted microwave. Treatment effects were quantitatively analyzed for each slot design condition through an apoptotic variable based on the calculated temperature distribution. Lastly, conditions that produce optimal treatment effects were derived depending on the number of slots. The analysis showed that as the distance between the slots increased from 0.5 to 3.5 mm, the optimal treatment effect was obtained when the number of slots was 4, 3, 3, and 2, respectively, and the microwave input power at that time was 3.0, 2.8, 2.6, and 3.0 W, respectively. This is expected to allow for more rigorous and more therapeutically effective MWA.

Electromagnetic duality and discrete symmetries for dyon in macroscopic media

This paper investigates discrete symmetries for dyon and the invariance of Maxwell’s field equations in the macroscopic media (material medium). Using advanced mathematical approaches, the paper derived electromagnetic duality and a unique form of Poynting theorem for dyon in macroscopic media. Here, we demonstrate that the combination of parity [Formula: see text], charge conjugation [Formula: see text], and time reversal [Formula: see text] symmetry: [Formula: see text], is an exact symmetry for dyon in macroscopic media. Furthermore, a comprehensive analysis of the electromagnetic wave equation for dyons in conducting media is also derived. These findings contribute significantly to the understanding of dyon behavior in electromagnetic fields.

An Educational Study of Electromagnetic Phenomena on Ferromagnetic Structure Using a Software Environment

AbstractThe solution to electromagnetic problems relies on complex numerical differential equations that are difficult to understand. Therefore, it is crucial to analyze them step by step and visualize them in a computer environment, even by developing a program. This approach enhances understanding of the subject and facilitates further research. In this study, an electromagnetic phenomenon is examined on a ferromagnetic structure using the Maxwell 2D static-elliptical partial differential equation within a developed software environment. The paper covers all aspects, ranging from the mathematical model and analysis solution to the fundamental operating parameters and characteristics, with a focus on educational purposes. The modeling and software development have been executed in a user-friendly manner, enabling students to apply various numerical techniques to practical problems while maintaining control over the entire modeling process. Through this study, it is anticipated that the transition from theory to practice can be facilitated more smoothly and directly. The theoretical analysis has been conducted using Mathematica software, and a comparison with finite element magnetic method simulation on a typical magnetic structure has been presented to illustrate the analogy. A comprehensive analysis of the Maxwell static-elliptical electromagnetic equation, based on a 2D numerical solution within a specific domain, has been performed in a detailed and unambiguous manner, utilizing different equation forms, boundary conditions, and ferromagnetic materials, all for educational purposes.

EUTERPE: A global gyrokinetic code for stellarator geometry

The current state of the EUTERPE code is described with emphasis on the implemented models and their numerical implementation. The code solves the multi-species electromagnetic gyrokinetic equations in the full volume of a three-dimensional domain. Noise reduction of the particle-in-cell method is achieved by using a δf-method and Fourier filters. The field equations are discretized with B-splines and the resulting system of equations is solved iteratively. For linear simulations a phase-factor transformation is applied in order to strongly reduce the necessary grid resolution. Apart from the full gyrokinetic model, other numerically less expensive hybrid models are also implemented. They are mainly tailored for comparison with fluid theory and for studying the interaction of the bulk plasma with fast particles. The code is parallelized for CPUs by particle and domain decomposition. Good scalability up to several thousand nodes is demonstrated.

Effect of the antenna slot numbers and position on the performance of microwave ablation

Microwave ablation (MWA) is a type of thermal ablation used for cancer treatment in interventional radiology. To induce localized tissue heating MWA employs electromagnetic waves within the microwave energy spectrum, which is done by the precisely designed antenna. This study substantially emphasizes the design and performance ameliorating of slot (both single and double) antennae and compares the results with conventional monopole antennae in terms of temperature distribution, specific absorption ratio (SAR), and thermal tissue damage rate. The simulation has been done in COMSOL by solving the Bioheat equation along with Maxwell electromagnetic equations using the finite element method. The simulation results reveal that the double-slot antenna has the most accurate and directional heat dissipation for liver tumors as well as the highest tissue damage rate and SAR. The highest SAR was found to be 3500 ​W/kg and 3350 ​W/kg at the implant depth of 61 ​mm and 63 ​mm for double and single-slot antennae, respectively. In addition, the fastest tissue damage occurred near the upper slot of the double-slot antenna. This study helps to understand the basic design parameters for enhancing single and double-slot antennae performance.

Physics‐Informed Machine Learning for Inverse Design of Optical Metamaterials

Optical metamaterials manipulate light through various confinement and scattering processes, offering unique advantages like high performance, small form factor and easy integration with semiconductor devices. However, designing metasurfaces with suitable optical responses for complex metamaterial systems remains challenging due to the exponentially growing computation cost and the ill‐posed nature of inverse problems. To expedite the computation for the inverse design of metasurfaces, a physics‐informed deep learning (DL) framework is used. A tandem DL architecture with physics‐based learning is used to select designs that are scientifically consistent, have low error in design prediction, and accurate reconstruction of optical responses. The authors focus on the inverse design of a representative plasmonic device and consider the prediction of design for the optical response of a single wavelength incident or a spectrum of wavelength in the visible light range. The physics‐based constraint is derived from solving the electromagnetic wave equations for a simplified homogenized model. The model converges with an accuracy up to 97% for inverse design prediction with the optical response for the visible light spectrum as input, and up to 96% for optical response of single wavelength of light as input, with optical response reconstruction accuracy of 99%.

ALLIANCE: Spectral solver for kinetic plasma turbulence

The ALLIANCE (ALLIANCE - spectrAL soLver for kInetic plAsma turbuleNCE) code is developed to solve a new set of four-dimensional electromagnetic drift-kinetic equations in slab geometry [1]. The nonlinear equations are useful for the study of magnetized plasma systems at scales comparable to, or larger than the ion gyroradius. In particular, it is suited for the study the kinetic turbulent cascade in astrophysical plasma, while preserving finite Larmor radius effects at the fluid-kinetic transition. The equations solved are in spectral Fourier-Laguerre-Hermite form, a pseudo-spectral approach is used for the nonlinear terms, and the code is parallelised over multiple directions. After a presentation of the code, validation runs are shown, and benchmarks for serial and parallel computations are presented.

Magnetohydrodynamic Analyses of a Miniature, Dual-Stage Electromagnetic Pump for Lead and Sodium In-Pile Test Loops

To support the development of Generation IV nuclear reactors, in-pile and out-of-pile test loops with miniature, submersible Direct-Current Electromagnetic Pumps (DC-EMPs) are used to investigate compatibility and corrosion issues of nuclear fuel and structure materials with flowing molten lead and alkali liquid metals. Owing to the absence of detailed experimental measurements and because of its simplicity and low computational cost, the Equivalent Circuit Model (ECM) is widely used to predict the pump characteristics. The simplifying assumptions in the ECM contribute to overpredicting the pump characteristics by >10%. To gain insight into the pump operation and assess the effect of various assumptions in ECM, not possible even experimentally, this work performed three-dimensional (3-D), Magneto-Hydro-Dynamic (MHD) analyses of a 66.8-mm-diameter, submersible, dual-stage DC-EMP, recently developed by the authors, for circulating molten Pb and liquid Na at up to 500°C. The solution of the coupled electromagnetism equations and the momentum and energy balance equations calculates the pump characteristics and provides 3-D images of the flow, electric current, and magnetic field strength distributions in the flow duct. The Grid Convergence Index (GCI) criterion confirmed the adequacy of the employed numerical mesh refinement and the results conversion. Results demonstrate strong dependence of the magnetic field strength distribution in the flow duct on the value and the distribution of the electric current but negligible effects of the fluid temperature on joule heating and pump characteristics. The Lorentz force highest densities occur at the entrance of the two pumping stages, and approximately 10.0% of the total force occurs in the fringe regions upstream and downstream of pumping stages. The MHD pump characteristics are in general agreement with, but consistently lower than, the ECM predictions. For molten lead and liquid sodium, the difference between the calculated characteristics increases with increased flow rate and input current, up to 12% and 14%, respectively.

On the mathematical structure of Einstein field equations and the existence of dark fields

In this work, we examine the possible existence of dark fields by showing that an energy‐momentum tensor associated with a dark field can be introduced into Einstein field equations of general relativity, in which the trace of the energy‐momentum of the dark field is identified with the cosmological constant. The introduction of dark fields into Einstein field equations is made possible by establishing field equations for the Ricci curvature tensor, which have similar mathematical structure to Maxwell field equations of electromagnetism. We also establish a system of field equations for the Riemann curvature tensor and investigate the possibility to represent physical systems consisting of dark fields and observable fields as spaces of constant scalar curvature, which are maximally symmetric spaces that admit the maximal number of Killing vectors. As an illustration, we show that if a dark field is considered as a dark fluid, then the pressure associated with the dark field can take negative values if the cosmological constant is assigned with positive values.

Vector wave solutions in electrodynamics: the Heaviside-Larmor symmetry and tiered potential invariance

Abstract The tiered symmetric structure of the electromagnetic equations, in particular the Heaviside Larmor (HL) symmetry, is used to obtain vector wave solutions of the wave equation obtained from Maxwell's equations. In this HL vector solutions procedure (HL-VS), the seed vector need not be constant but must satisfy an inhomogeneous Helmholtz equation. This seed can be used either to obtain the polarization basis in a given coordinate system or a particular polarization state. The spherical vector waves (SVW) basis is economically derived retaining the physical meaning throughout the procedure. Examples of low order modes are expounded. The cylindrical vector wave basis is also derived and particular polarization states are compared with previous results. Plane waves are used to elucidate the HL symmetry terms and their controlled manipulation in order to tailor specific structured light states. The angular momentum of plane waves with elliptical polarization is obtained. The controversy as to whether circularly polarized plane waves carry angular momentum is answered in the affirmative.

Simulation study of mobile phone radiation thermal effect in human heart tissue

The public's concern over potential health risks related to the absorption of electromagnetic radiation (EMR) has been developing due to the prolonged use of cell phones close to the human body, particularly the human heart. To address these issues, this study aims to present a numerical simulation of EMR in the spectral range (900 MHz, 1800 MHz, and 2400 MHz) on human heart tissue using a Matlab program and the Finite-Difference Time-Domain (FDTD) method in One Dimension (1D). A mathematical analysis of electromagnetic radiation heating equations in a one-dimensional, one-layer model has also been discussed by numerically calculating the transient bioheat transfer equation and Maxwell's equations by using the FDTD to predict the effects of thermal physics properties on the transient temperature of human heart tissue. The results revealed that FDTD is an efficient method to evaluate the thermal effect of EMR due to its perfect boundary condition. It was found that the heart tissue reacts more at 900 MHz compared to 1800 MHz and 2400 MHz, and the tissue absorption is higher at the lower frequency. The effects of various parameters on the temperature increase in the human heart tissue were considered, such as the electric field, magnetic field, thicknesses, and thermal conductivity of the heart tissue. It was found that the frequency most affecting the tissue was 900 MHz. Our study found that the temperature distribution decreases with an increase in tissue thickness. We note that the results are similar for the different frequencies. It turns out that as the thermal conductivity of the heart tissue increases, the temperature distribution decreases slightly, while no change occurs here in the relationship between thermal conductivity and the temperature distribution of tissue with the change in the frequency of a wave.

Formation and modulation of multi-pulse femtosecond laser-induced periodic surface structures by electromagnetic and partial differential equation simulations.

In order to demonstrate the formation of laser-induced periodic surface structures (LIPSS), simulations were performed to investigate the effect of multiple femtosecond laser pulses with different laser energy densities on a Ti6Al4V surface. In this work, a set of partial differential equations calculating the electron and lattice temperature variations, followed by coupling with an electric field, is used to analyze the evolution of the periodic surface structure induced by the interaction of the femtosecond laser with the material. As the number of pulses increases, the surface structure of the material changes from none to produce LIPSS structure and from low spatial frequency LIPSS (LSFL) structure to high spatial frequency LIPSS (HSFL) structure. In order to compare the results, single-point laser scanning ablation experiments were carried out at femtosecond laser energy. The experimental results are consistent with the simulation results.

Three Serious Mistakes in Einstein’s Original Paper of Special Relativity in 1905

It is revealed in this paper that there were three serious mistakes in the Einstein’s original paper in 1905. Einstein did not prove that the motion equation of classical electromagnetic field could satisfy the invariance of the Lorentz coordinate transformation. The Einstein’s derivations on the formulas of transverse and longitudinal masses, as well as the calculation on the mass-energy relation are wrong. 1. In order to prove that the classical Maxwell electromagnetic field equation satisfied the invariance of Lorentz transformation in free space without charged and current, Einstein introduced the transformations of electromagnetic fields themselves, called the Einstein’s transformations of electromagnetic fields. However, these transformations are completely different from the Lorentz transformations of electromagnetic fields themself, which leads to contradiction and does not hold. 2. For the electromagnetic field equations in non-free space with charge and current, the Einstein’s transformations can not make the electromagnetic fields unchanged under the Lorentz transformation. 3. The constitutive equations of electromagnetic theory in the medium do not satisfy the invariance of the Lorentz transformation too. Therefore, the classical electromagnetic field equations have no the invariance of the Lorentz transformation actually, and the most important theoretical and experimental basis of special relativity do not exist. 4. The Einstein's derivations on the formulas of transverse and longitudinal masses have a series of elementary mistakes in mathematics and physics. Einstein took the relative speed between two reference frames as the arbitrary moving velocity of a particle, and the obtained formulas were completely different from the existing mass-velocity of special relativity. 3. When Einstein derived the mass-energy relationship, he only calculated the work done by the force in the x-axis direction of particle’s motion, ignoring the work done by the force at the y- and z-axes directions. Meanwhile, the constant relative motion velocity between two reference frames was misused as the variable arbitrary velocity of a particle. Therefore, Einstein had not derived the mass-velocity formula and mass-energy relationship used in the present special relativity.

A Level Set-Based Solver for Two-Phase Incompressible Flows: Extension to Magnetic Fluids

Development of a two-phase incompressible solver for magnetic flows in the magnetostatic case is presented. The proposed numerical toolkit couples the Navier–Stokes equations of hydrodynamics with Maxwell's equations of electromagnetism to model the behaviour of magnetic flows in the presence of a magnetic field. To this end, a rigorous implementation of a second-order two-phase solver for incompressible nonmagnetic flows is introduced first. This solver is implemented in the finite-difference framework, where a fifth-order conservative level set method is employed to capture the evolution of the interface, along with an incompressible solver based on the projection scheme to model the fluids. The solver demonstrates excellent performance even with high density ratios across the interface (Atwood number ≈ 1 ), while effectively preserving the mass conservation property. Subsequently, the numerical discretisation of Maxwell's equations under the magnetostatic assumption is described in detail, utilising the vector potential formulation. The primary second-order solver for two-phase flows is extended to the case of magnetic flows, by incorporating the Lorentz force into the momentum equation, accounting for high magnetic permeability ratios across the interface. The implemented solver is then utilised for examining the deformation of ferrofluid droplets in both quiescent and shear flow regimes across various susceptibility values of the droplets. The results suggest that increasing the susceptibility value of the ferrofluid droplet can affect its deformation and rotation in low capillary regimes. In higher capillary flows, increasing the magnetic permeability jump across the interface can further lead to droplet breakup as well. The effect of this property is also investigated for the Rayleigh–Taylor instability growth in magnetic fluids.

Characterization of Excitation Effects and Data Interpretation of Combined Time-Domain Multiwaveform Transmission Currents

With the development of electronic technology, time-domain electromagnetic (TDEM) transmitters are now capable of emitting various waveforms such as trapezoidal, triangular, and semi-sinusoidal waveforms, as well as their combinations. However, the application of multi-waveform combination detection methods is still immature, and modeling methods are required to analyze the characteristics of different waveforms and complete data interpretation. Based on the Crank-Nicolson finite difference time-domain (CN-FDTD) method for discretizing the electromagnetic wave equations, the current waveform can be converted into current surface density for source-included calculations, enabling high-precision modeling of the responses to different current waveforms. The responses to multiple transmission current waveforms were compared and analyzed using various models, and the results showed that selecting targeted excitation parameters based on the characteristics of the target can improve detection sensitivity. We applied the multi-waveform combination detection method to the actual exploration in the field, obtained the pure secondary field response by subtracting the primary field of the actual current waveform from the measured data, converted the responses of different transmission current waveforms into step wave responses by deconvolution method, and performed resistivity imaging. The results of modeling and data interpretation demonstrate the effectiveness of the multi-waveform combination detection method in improving geological exploration accuracy, and promoted the development of fine-scale geophysical exploration.

Accelerated Ray Launching Method for Efficient Field Coverage Studies in Wide Urban Areas.

The implementation of a fast and efficient computer tool for field coverage studies in urban mobile radio systems is presented in this work. An accelerated and tailored ray launching method takes advantage of a ray tracing programmable framework optimized for massively parallel processing on GPUs. The PlotOptiX API is used to customize the code before applying the electromagnetic equations. The proposed code is described, and results are shown to demonstrate its correct operation. A high number of diffractions and reflections can be tracked in each ray from the transmitter to the receiver. In addition to the typical point-to-point simulation, measurement planes can also be set as receivers to provide fast predictions in wide urban areas.

Effects of interior angle and inclination angle of slot defect on electromagnetic-stress behaviours of superconducting swing system

Currently, copper-oxygen high-temperature superconducting materials have strongly anisotropic electromagnetic properties, which are difficult to describe in the equations. And in the rotating machine containing the permanent magnet (PM) rotor and bulk high-temperature superconductor (HTS) stator, when the HTSs have defects, their electromagnetic-stress behaviours may affect the mechanical stability of the equipment and even cause it to not work properly. In this paper, we proposed an anisotropic electromagnetic equation based on the H-formulation and established a three-dimensional coupled model with the magnetic, thermal and stress fields to study the electromagnetic-stress behaviours of a HTS with a slot defect during the swing of a PM, discussing the effects of the interior angle and inclination angle of the defect on the behaviours. The results show that the interior and inclination angles of the defect have a large influence on the electromagnetic-stress behaviours. For the electromagnetic characteristics, the main influence is on the rotational losses. Especially the losses at an inclination angle of 60° are 16.5 times those without damage. This is related to the ‘thin wall’ structure near the upper surface of the HTS. The stress concentration point appears on the defect boundary. The novelty of this paper is the proposal of the anisotropic electromagnetic equations based on the H-formulation and the study of the electromagnetic-stress behaviours of a superconducting swing system containing a PM and HTS with a slot defect from a three-dimensional perspective. The research results of this paper can be the references for the design and structural protection of superconducting rotating machines.

Simulation of Seismoelectric Waves Using Time-Domain Finite-Element Method in 2D PSVTM Mode

The study of the numerical simulation of seismoelectric effects is very helpful for understanding the theory and mechanism of seismoelectric activities. Quasi-static approximation is widely used in the numerical simulation of seismoelectric fields. However, numerical errors occur when the model domain is not within the near-field area of EM waves or the medium is of high salinity. To solve this problem, we propose a time-domain finite-element algorithm (FETD) based on the full-wave electromagnetic (EM) equation to simulate seismoelectric waves in 2D PSVTM mode. By decomposing the electrokinetic coupling equations into two independent ones, we can solve the seismoelectric waves separately. In our implementation, we focus our attention on the solution of EM waves based on vector–scalar potentials, while using the open-source code SPECFEM2D to explicitly solve Biot’s equations and obtain the relative fluid–solid displacement, which is taken as the source for the complete Maxwell’s equations. In the solution of EM wave fields, we use an unconditionally stable implicit method for time discretization. Computation efficiency can be improved by combining explicit and implicit recursions. After conducting the mathematical formulation, we first validate our method by comparing its results with the analytic solutions for a half-space and a two-layer model, as well as with a quasi-static approximation method. Moreover, we run numerical simulations and wavefield analyses on an elliptical hydrocarbon reservoir, and reveal that the interface responses are promising for the identification of underground interfaces and hydrocarbon reservoir exploration.

Curved spacetime as a dispersive multiferroic medium for an electromagnetic wave: Polarization and magnetization vectors in the Schwarzschild spacetime

We study one of the interesting properties of the electromagnetic wave propagation in the Schwarzschild background spacetime in the framework of general relativity (GR). The electromagnetic wave equation has been derived from vacuum general relativistic Maxwell’s equations. It is shown that the solutions for the electromagnetic field can be expanded in the spherical harmonic functions and all components of the electromagnetic fields can be expressed in terms of two radial profile functions. These radial profile functions can be expressed in terms of the confluent Heun function. The calculated behavior of the electric and magnetic susceptibilities near the event horizon appears to be similar to the susceptibilities of multiferroic materials near phase transition. The Curie temperature of this phase transition appears to coincide with the Hawking temperature.

Electromagnetic Waves in Cosmological Spacetime

We consider the propagation of electromagnetic waves in the Friedmann–Lemaître–Robertson–Walker metric. The exact solutions for plane and spherical wave are written down. The corresponding redshift, amplitude change, and dispersion are discussed. We also speculate about the connection of the electromagnetic wave equation to the Proca equation and its significance for the early Universe.