Maxwell's Equations Research Articles

The Effect of Object Thickness in Numerical Reflectance and Transmittance using Finite Difference Time Domain (FDTD)

The electromagnetic waves could be simulated using finite difference time domain. This simulation method based on Maxwell equations. Here, the fields were spatially discretized along time discretization. This was the basic concept for updating the fields in every time step. Since the reflection and transmission of electromagnetic waves were the basic phenomena for optical studies of the waves in materials, we studied the reflectance and transmittance. In FDTD, the reflection and transmission could be calculated using Fourier transformation. We found that the thickness of the object which was inserted to the system affected the reflection and transmission significantly. If the thickness of the object became thicker, then there also increased the appearance frequency of the reflectance with the similar values. It was also happened to the transmittance.

Teaching Maxwell Equations with LLM Assistance

Teaching Maxwell Equations with LLM Assistance

The method of calculating the vector antenna pattern based on the algebra of quaternions

The quaternion algebra contains the algebra of real and complex numbers, as well as three-dimensional vector algebra. The algebra of quaternions finds application in describing the rotations of coordinate systems, which is also of interest in describing problems of electrodynamics, in which it is often necessary to perform a transition from the basis of sources to the basis of observation. It is shown that using the algebra of quaternions, it is possible to express Maxwell's equations and their solutions in a coordinate system invariant form. A method for calculating a vector antenna based on the algebra of quaternions is proposed. It is shown that the algebra of quaternions is applicable not only for the calculation of electromagnetic fields, radiation patterns, but also for the calculation of polarization characteristics. All transformations define products of rotation functions, for which it is enough to find trigonometric functions sinx and cosx once, and all subsequent rotation operations are performed when implementing quaternion algebra procedures. The application of the quaternion algebra also avoids errors in determining the directions of turns. The proposed methodology can be used to design antennas placed on mobile objects when solving problems of analysis and synthesis.

The inverse problem for a class of implicit differential equations and the coisotropic embedding theorem

We carry on the approach used in [L. Schiavone, The inverse problem within free Electrodynamics and the coisotropic embedding theorem, Inter. J. Geom. Meth. Mod. Phys. 21(7) (2024) 2450131] to provide a solution for the inverse problem of the calculus of variations for Maxwell equations in vacuum and we provide an abstract theory including all implicit differential equations that can be formulated in terms of vector fields over pre-symplectic manifolds.

On the Regularity of Time-Harmonic Maxwell Equations with Impedance Boundary Conditions

On the Regularity of Time-Harmonic Maxwell Equations with Impedance Boundary Conditions

Simulation of Electromagnetic Waves Propagation Using Matlab

This research explores the simulation of electromagnetic wave propagation using MATLAB, focusing on both 2D and 3D environments. By deriving the wave equation from Maxwell's equations, we utilize the Finite Difference Time Domain (FDTD) method for numerical solutions. The MATLAB code implementation demonstrates the propagation of electric and magnetic fields and is validated through visualizations. Practical applications, including telecommunications, medical imaging, and microwave engineering, are examined through case studies on antenna design and waveguide analysis. Challenges in simulation, recent advances, and future research directions are discussed, providing a comprehensive overview of electromagnetic wave propagation and its technological implications.

Quantized tensor networks for solving the Vlasov–Maxwell equations

The Vlasov–Maxwell equations provide an ab initio description of collisionless plasmas, but solving them is often impractical because of the wide range of spatial and temporal scales that must be resolved and the high dimensionality of the problem. In this work, we present a quantum-inspired semi-implicit Vlasov–Maxwell solver that uses the quantized tensor network (QTN) framework. With this QTN solver, the cost of grid-based numerical simulation of size $N$ is reduced from $O(N)$ to $O(\text {poly}(D))$ , where $D$ is the ‘rank’ or ‘bond dimension’ of the QTN and is typically set to be much smaller than $N$ . We find that for the five-dimensional test problems considered here, a modest $D=64$ appears to be sufficient for capturing the expected physics despite the simulations using a total of $N=2^{36}$ grid points, which would require $D=2^{18}$ for full-rank calculations. Additionally, we observe that a QTN time evolution scheme based on the Dirac–Frenkel variational principle allows one to use somewhat larger time steps than prescribed by the Courant–Friedrichs–Lewy constraint. As such, this work demonstrates that the QTN format is a promising means of approximately solving the Vlasov–Maxwell equations with significantly reduced cost.

Numerical simulation of time-dependent viscous fluid flow with upward and downward fluctuation of spinning disk

The parametric approach towards time-dependent viscous fluid flow across a gyrating disk with upward and downward fluctuation. The major goal of this research is to assess fluid flow under the influence of magnetic fields and heat propagation processes. Because they provide a thorough description of electromagnetic interactions. Maxwell's equations are at the heart of all contemporary information and communication technologies. The governing equations comprising Navier Stokes equation, energy, concentration, and Maxwell equations have been represented appropriately for this purpose. The governing equations are turned down to the system of non-linear ODEs through a resemblance framework. The obtained system of differential equations has been resolved via numerical procedure Parametric Continuation Method (PCM). For the scale reliability purpose, the outcomes are compared to another numerical Matlab scheme boundary value solver. In the current analysis, the presence of convective boundary conditions correlated to mass and energy is of physical relevance. The numerical findings are provided in tabular and graphical forms. The consequences of suction and wall injection have been also highlighted. The upward motion of the spinning disc is thought to lead to comparable findings as in an injection scenario, whilst the downhill motion is thought to contribute to wall suction-like effects.

Coupling coefficient calculation method of circular coil with bilateral finite magnetic shields at horizontal misalignment in wireless power transfer systems

AbstractThe coupling coefficient determines the transfer efficiency of the wireless power transfer (WPT) system. The magnetic leakage can be reduced, and the coupling coefficient of the coil can be enhanced by adding a magnetic shield. Thus, the transfer efficiency of the WPT system can be improved. However, when the receiving coil is added with magnetic shield and occurs horizontally misaligned, the coupling coefficients between the coils with the magnetic shield can only be calculated by finite element simulation, and there is no calculation method for the coupling coefficient of two circular coils with bilateral finite magnetic shield at horizontal misalignment. Therefore, based on Maxwell's equations and Bessel's function, the formula of the coupling coefficient between circular coils at coaxial is derived by the subdomain hierarchical analysis (SHA) method. On this basis, the coupling coefficient calculation formula between circular coils at horizontal misalignment is further derived by using the boundary vector equivalence (BVE) method. The results show that the maximum experimental error is not more than 5.58%, and the longest average time of the algorithm is not more than 1.282 s, which verifies the effectiveness and rapidity of the proposed method.

FEM-Circuit co-simulation of superconducting synchronous wind generators connected to a DC network using the homogenized J–A formulation of the Maxwell equations

High-temperature superconductors (HTS) are greatly appealing for the development of high efficient, and high energy density power devices. They are particularly relevant for applications requiring light and compact machines such as wind power generation. In this context, to ensure the proper design of the superconducting machines and their reliable operation in power systems, it is then important to develop models that can accurately include their physics but also can describe properly their interaction with the system. To achieve such a goal, one approach is the co-simulation. This numerical technique can bring fine geometrical and physical details of the machines through a finite element model (FEM) meanwhile dealing with the operation of the whole system that incorporates the machine and a subset of the power grid represented by an external electrical circuit. The goal of the present work is to put to use this numerical technique when superconducting components are involved. Here, a case study is proposed involving a 15 MW hybrid superconducting synchronous generator (HTS rotor and conventional stator) coupled to a direct current network via a rectifier and its associated filter. The case study related to wind power application allows grasping the technical issues when employing co-simulation dealing with HTS machines. The FEM of the generator is done in the commercial software COMSOL Multiphysics, which interacts with the circuit simulator Simulink through the built-in Functional Mock-up Unit. For the present study, a new version of the latest J–A formulation combined with homogenization technique is introduced allowing an even faster computation time compared to the T–A formulation. Distributed variables and global variables such as current density, magnetic flux density, and local losses for the former and voltage, current, electromagnetic torque, and power quality for the latter are estimated and compared for both formulations. The idea is to find the best-suited combination FEM-circuit under criteria of computational speed, accuracy, and numerical stability. Thus, it is shown that all formulations generate an error of less than 5% on the machine parameters and that the J–A formulation with first order elements stands out with a significant 4-fold reduction in computational costs.

Propagation of elliptical Gaussian vortex beam based on angular spectrum representation

Vortex beams with hollow elliptical ring structure have attracted broad interest in applications such as micro-particle trapping and manipulation. The paper investigates the dependence of the hollow elliptic ring structure of vortex beams on the beam order and the topological charge, while propagating in the near- and far-field zones. The electric field of the beam is formulated by using the angular spectrum method, which satisfies rigorously Maxwell's equation. The electric field, linear momentum and orbital angular momentum are calculated. Numerical results show that the hollow elliptical ring structure approximately remains unchanged over a finite propagation distance, accompanied by a rotation whose angle depends on the topological charge. In the far-field zone, the ring structure is broken, leading to two bright spots and several dark islands. It is found that the beam profile in the near-field zone is mainly related to the beam order but in the far-field zone is primarily affected by the topological charge. The results presented in the paper may be useful in predicting and explaining the interaction between the elliptic hollow Gaussian vortex beam and the micro-particles, such as the movement of micro-particles along the orbit and the rolling rotation.

Splines finite element solver for one-dimensional time-dependent Maxwell's equations via Fourier Transform Discretization

In this article, we solve the time-dependent Maxwell coupled equations in their one-dimensional version relatively to space-variable. We effectuate a variable reduction via Fourier transform to make the time variable as a frequency parameter easy and quickly to manage. A Galerkin variational method based on higher-order spline interpolations is used to approximate the solution relatively to the spacial variable. So, the state of existence of the solution, its uniqueness, and its regularity are studied and proved, and the study is also provided by an error estimate and the order of convergence of the proposed method. Also, we use the critical Nyquist frequency to calculate numerically the solution of the Inverse Fourier Transform(IFT); and for all numerical computations, we consider several quadrature methods. Finally, we give some experiments to illustrate the success of such an approach.

Vortex Circular Dichroism: An experimental technique to assess the scalar/vectorial regime of diffraction.

In classical electrodynamics, light-matter interactions are modelled using Maxwell equations. The solution of Maxwell equations, which is typically given by means of the electric and magnetic field, is vectorial in nature. Yet it is well known that light-matter interactions can be approximately described in a scalar (polarization independent) way for many optical applications. While the accuracy of the scalar approximation can be theoretically computed, to the best of our knowledge, it has never been determined experimentally. Here, we introduce Vortex Circular Dichroism ( VCD), an optical measurement that has the required features to assess the vectoriality of diffraction. VCD is measured as the differential transmission (or absorption) of left and right circularly polarized vortex beams. We test the VCD measurement with two different systems: i) an experimental set of single circular nano-apertures drilled in a gold film with diameters ranging from 150 to 1950 nm; and ii) a theoretical set of golden spheres with the same diameters as the nano-apertures. We observe that in both systems, VCD > 0 for smaller diameters, VCD ≲ 0 for intermediate values and VCD ≈ 0 for larger values of the diameter. Furthermore, the simulations show that a diffraction process characterized by a VCD ≈ 0 ( VCD ≠ 0) is polarization-independent (polarization-dependent). As a result, we relate VCD ≠ 0 to a vectorial diffraction, and VCD ≈ 0 to a scalar one. Overall, our results show compelling evidence that it is possible to experimentally assess the scalar/vectorial regime of a diffraction process, and that the VCD technique possesses the required features to measure the vectoriality of diffraction processes involving plasmonic cylindrically symmetric structures.

Modified free energy generation using permanent Neodymium Magnet based on Bedini with Maxwell and Lorenz gauge conditions

An efficient, reliable, and real-time electricity generator is proposed by keeping in view the constraints, related to the electricity generation cost, resource reservoirs, greenhouse gas emission, energy storage, and power demand for a pre-defined residential area. Paramount importance is given to the rotor's design; coupled with movable neodymium magnets to increase the harvester's life span and system's net revenue. For the amplification in the stator output, a cost-effective approach is adopted that is based on the parallel winding of the bifilar coil. Furthermore, the coefficient of performance factor for the Bedini's smart school girl circuit based electricity harvester is modified by aiding the concepts of radiant energy incorporated with the electromagnetic phenomenon. Moreover, the mathematical model is formulated by the Maxwell equations using the Lorenz gauge condition to capture free vacuum energy from the environment. For the investigation, analysis, and corroboration of the proposed design, results are reported after performing extensive numerical computations and a comprehensive comparison between the original and modified design is stated in detail.

Inverse stable reconstruction of 3 coefficients for the heterogeneous Maxwell equations by finite number of partial interior observations

We consider an inverse problem of determining the isotropic inhomogeneous electromagnetic coefficients of the non-stationary Maxwell’s equations in a bounded domain of R3 by means of a finite number of interior data of as less as possible components of the solutions. Our main result is a Lipschitz stability estimate for the inverse problem and our proof relies on a Carleman estimate for the heterogeneous Maxwell’s equations.

Bootstrapping boundary QED. Part I

We use the numerical conformal bootstrap to study boundary quantum electrodynamics, the theory of a four dimensional photon in a half space coupled to charged conformal matter on the boundary. This system is believed to be a boundary conformal field theory with an exactly marginal coupling corresponding to the strength of the interaction between the photon and the matter degrees of freedom. In part one of this project, we present three results. We show how the Maxwell equations put severe constraints on boundary three-point functions involving two currents and a symmetric traceless tensor. We use semi-definite programming to show that any three dimensional conformal field theory with a global U(1) symmetry must have a spin two gap less than about 1.05. Finally, combining a numerical bound on an OPE coefficient and some Ward identities involving the current and the displacement operator, we bound the displacement operator two-point function above. This upper bound also constrains a boundary contribution to the anomaly in the trace of the stress tensor for these types of theories.

Quantum simulation of dissipation for Maxwell equations in dispersive media

The dissipative character of an electromagnetic medium breaks the unitary evolution structure that is present in lossless, dispersive optical media. In dispersive media, dissipation appears in the Schrödinger representation of classical Maxwell equations as a sparse diagonal operator occupying an r-dimensional subspace. A first order Suzuki-Trotter approximation for the evolution operator enables us to isolate the non-unitary operators (associated with dissipation) from the unitary operators (associated with lossless media). The unitary operators can be implemented through qubit lattice algorithm (QLA) on n qubits, based on the discretization and the dimensionality of the pertinent fields. However, the non-unitary-dissipative part poses a challenge both physically and computationally on how it should be implemented on a quantum computer. In this paper, two probabilistic dilation algorithms are considered for handling the dissipative operators. The first algorithm is based on treating the classical dissipation as a linear amplitude damping-type completely positive trace preserving (CPTP) quantum channel where an unspecified environment interacts with the system of interest and produces the non-unitary evolution. Therefore, the combined system-environment is now closed, and must undergo unitary evolution in the dilated space. The unspecified environment can be modeled by just one ancillary qubit, resulting in an implementation scaling of O(2n−1n2) elementary gates for the total system-environment unitary evolution operator. The second algorithm approximates the non-unitary operators by the Linear Combination of Unitaries (LCU). On exploiting the diagonal structure of the dissipation, we obtain an optimized representation of the non-unitary part, which requires O(2n) elementary gates. Applying the LCU method for a simple dielectric medium with homogeneous dissipation rate, the implementation scaling can be further reduced into O[poly(n)] basic gates. For the particular case of weak dissipation we show that our proposed post-selective dilation algorithms can efficiently delve into the transient evolution dynamics of dissipative systems by calculating the respective implementation circuit depth. A connection of our results with the non-linear-in-normalization-only (NINO) quantum channels is also presented.

Topological optimization in 3D-magnetostatics: development of adjoint methods using the equations of magnetic moments

PurposeThe purpose of this paper is mainly to develop the adjoint method within the method of magnetic moment (MMM) and thus, to provide an efficient new way to solve topology optimization problems in magnetostatic to design 3D-magnetic circuits.Design/methodology/approachFirst, the MMM is recalled and the optimization design problem is reformulated as a partial derivative equation-constrained optimization problem where the constraint is the Maxwell equation in magnetostatic. From the Karush–Khun–Tucker optimality conditions, a new problem is derived which depends on a Lagrangian parameter. This problem is called the adjoint problem and the Lagrangian parameter is called the adjoint parameter. Thus, solving the direct and the adjoint problems, the values of the objective function as well as its gradient can be efficiently obtained. To obtain a topology optimization code, a semi isotropic material with penalization (SIMP) relaxed-penalization approach associated with an optimization based on gradient descent steps has been developed and used.FindingsIn this paper, the authors provide theoretical results which make it possible to compute the gradient via the continuous adjoint of the MMMs. A code was developed and it was validated by comparing it with a finite difference method. Thus, a topology optimization code associating this adjoint based gradient computations and SIMP penalization technique was developed and its efficiency was shown by solving a 3D design problem in magnetostatic.Research limitations/implicationsThis research is limited to the design of systems in magnetostatic using the linearity of the materials. The simple examples, the authors provided, are just done to validate our theoretical results and some extensions of our topology optimization code have to be done to solve more interesting design cases.Originality/valueThe problem of design is a 3D magnetic circuit. The 2D optimization problems are well known and several methods of resolution have been introduced, but rare are the problems using the adjoint method in 3D. Moreover, the association with the MMMs has never been treated yet. The authors show in this paper that this association could provide gains in CPU time.

An unstructured body-of-revolution electromagnetic particle-in-cell algorithm with radial perfectly matched layers and dual polarizations

A novel electromagnetic particle-in-cell algorithm has been developed for fully kinetic plasma simulations on unstructured (irregular) meshes in complex body-of-revolution geometries. The algorithm, implemented in the BORPIC++ code, utilizes a set of field scalings and a coordinate mapping, reducing the Maxwell field problem in a cylindrical system to a Cartesian finite element Maxwell solver in the meridian plane. The latter obviates the cylindrical coordinate singularity in the symmetry axis. The choice of an unstructured finite element discretization enhances the geometrical flexibility of the BORPIC++ solver compared to the more traditional finite difference solvers. Symmetries in Maxwell's equations are explored to decompose the problem into two dual polarization states with isomorphic representations that enable code reuse. The particle-in-cell scatter and gather steps preserve charge-conservation at the discrete level. Our previous algorithm (BORPIC+) discretized the E and B field components of TEϕ and TMϕ polarizations on the finite element (primal) mesh [1,2]. Here, we employ a new field-update scheme. Using the same finite element (primal) mesh, this scheme advances two sets of field components independently: (1) E and B of TEϕ polarized fields, (Ez,Eρ,Bϕ) and (2) D and H of TMϕ polarized fields, (Dϕ,Hz,Hρ). Since these field updates are not explicitly coupled, the new field solver obviates the coordinate singularity, which otherwise arises at the cylindrical symmetric axis, ρ=0 when defining the discrete Hodge matrices (generalized finite element mass matrices). A cylindrical perfectly matched layer is implemented as a boundary condition in the radial direction to simulate open space problems, with periodic boundary conditions in the axial direction. We investigate effects of charged particles moving next to the cylindrical perfectly matched layer. We model azimuthal currents arising from rotational motion of charged rings, which produce TMϕ polarized fields. Several numerical examples are provided to illustrate the first application of the algorithm.

Entanglement between Micro-Magnetism, electromagnetism and the Tensor Magnetic Phase Theory (TMPT) – Symmetry, conservation and invariance laws analysis at low frequency

Ferromagnetic materials show magnetic structures with domains and walls. Thanks to decades of research regarding the origin and behaviour of magnetic domains, we now possess a general foundation which has been verified experimentally in single crystals and powders. The governing equations at the microscopic scale were built in the 1960 s when Brown published calculations of the magnetic moments distribution inside domain walls. This micromagnetic theory uses the so called LLG ‘Landau-Lifshitz-Gilbert’ equation and can include the damping effects. The LLG equation requires a coupling to the field derived from energy contributions: exchange, anisotropy, magnetostriction, stray-field … and the anti-eddy field. At the macroscopic scale, such behaviours are lumped in a homogenized magnetization law for the electromagnetic field equations inside larger polycrystals. Therefore, the inhomogeneous magnetic material nature is always ignored. The Tensor Magnetic Phase Theory (TMPT) describes the magnetic structure and the magnetization thanks to a tensor variable at a mesoscopic scale. The material structuring is then explained thanks to an energy balance which will be discussed.This paper presents the results of investigations on entangled relationships between the micromagnetic theory, the electromagnetic theory and what is called the Tensor Magnetic Phase Theory (TMPT), which statistically describes the magnetic structures of soft ferromagnetic materials. It examines a connection between the TMPT and the LLG by deriving the main energy terms. Then, the TMPT must stay compatible and coupled to the Maxwell equations at low frequencies with volume and surface connections. Additionally, this paper investigates the way to derive the domains structuring and magnetization laws through the Lagrange principle and the corresponding conservation laws with invariants linked to the Nœther theorem. Finally, the TMPT must be discussed while checking its coherence and formulation when changing the reference frame.