Energy Stable Nodal DG Methods for Maxwell’s Equations of Mixed-Order Form in Nonlinear Optical Media

In this work,we develop energy stable numerical methods to simulate electromagnetic waves propagating in optical media where the media responses include the linear Lorentz dispersion,the instantaneous nonlinear cubic Kerr response,and the nonlinear delayed Raman molecular vibrational response.Unlike the first-order PDE-ODE governing equations considered previously in Bokil et al.(J Comput Phys 350:420–452,2017)and Lyu et al.(J Sci Comput 89:1–42,2021),a model of mixed-order form is adopted here that consists of the first-order PDE part for Maxwell’s equations coupled with the second-order ODE part(i.e.,the auxiliary differential equations)modeling the linear and nonlinear dispersion in the material.The main contribution is a new numerical strategy to treat the Kerr and Raman nonlinearities to achieve provable energy stability property within a second-order temporal discretization.A nodal discontinuous Galerkin(DG)method is further applied in space for efficiently handling nonlinear terms at the algebraic level,while preserving the energy stability and achieving high-order accuracy.Indeed with d_(E)as the number of the components of the electric field,only a d_(E)×d_(E)nonlinear algebraic system needs to be solved at each interpolation node,and more importantly,all these small nonlinear systems are completely decoupled over one time step,rendering very high parallel efficiency.We evaluate the proposed schemes by comparing them with the methods in Bokil et al.(2017)and Lyu et al.(2021)(implemented in nodal form)regarding the accuracy,computational efficiency,and energy stability,by a parallel scalability study,and also through the simulations of the soliton-like wave propagation in one dimension,as well as the spatial-soliton propagation and two-beam interactions modeled by the two-dimensional transverse electric(TE)mode of the equations.

High-Order Spatial FDTD Solver of Maxwell’s Equations for Terahertz Radiation Production

We applied a spatial high-order finite-difference-time-domain (HO-FDTD) scheme to solve 2D Maxwell’s equations in order to develop a fluid model employed to study the production of terahertz radiation by the filamentation of two femtosecond lasers in air plasma. We examined the performance of the applied scheme, in this context, we implemented the developed model to study selected phenomena in terahertz radiation production, such as the excitation energy and conversion efficiency of the produced THz radiation, in addition to the influence of the pulse chirping on properties of the produced radiation. The obtained numerical results have clarified that the applied HO-FDTD scheme is precisely accurate to solve Maxwell’s equations and sufficiently valid to study the production of terahertz radiation by the filamentation of two femtosecond lasers in air plasma.

ANALYSIS OF FDTD TO UPML FOR MAXWELL EQUATIONS IN POLAR COORDINATES

An FDTD system associated with uniaxial perfectly matched layer（UPML） for an electromagnetic scattering problem in two-dimensional space in polar coordinates is considered.Particularly the FDTD system of an initial-boundary value problems of the transverse magnetic（TM） mode to Maxwell＇s equations is obtained by Yee＇s algorithm,and the open domain of the scattering problem is truncated by a circle with a UPML.Besides,an artificial boundary condition is imposed on the outer boundary of the UPML.Afterwards,stability of the FDTD system on the truncated domain is established through energy estimates by the Gronwall inequality.Numerical experiments are designed to approve the theoretical analysis.

Adaptive Maxwell’s Equations Derived Optimization and Its Application in Antenna Array Synthesis

In this paper,a self-adaptive method for the Maxwell’s Equations Derived Optimization(MEDO)is proposed.It is implemented by applying the Sequential Model-Based Optimization(SMBO)algorithm to the iterations of the MEDO,and achieves the automatic adjustment of the parameters.The proposed method is named as adaptive Maxwell’s equations derived optimization(AMEDO).In order to evaluate the performance of AMEDO,eight benchmarks are used and the results are compared with the original MEDO method.The results show that AMEDO can greatly reduce the workload of manual adjustment of parameters,and at the same time can keep the accuracy and stability.Moreover,the convergence of the optimization can be accelerated due to the dynamical adjustment of the parameters.In the end,the proposed AMEDO is applied to the side lobe level suppression and array failure correction of a linear antenna array,and shows great potential in antenna array synthesis.

New Reflection Principles for Maxwell's Equations and Their Applications

Some new reflection principles for Maxwell's equations are first established, which are then applied to derive two novel identifiability results in inverse electromagnetic obstacle scattering problems with polyhedral scatterers.

High-Order Leap-Frog Based Discontinuous Galerkin Method for the Time-Domain Maxwell Equations on Non-Conforming Simplicial Meshes

A high-order leap-frog based non-dissipative discontinuous Galerkin time- domain method for solving Maxwell＇s equations is introduced and analyzed. The pro- posed method combines a centered approximation for the evaluation of fluxes at the in- terface between neighboring elements, with a Nth-order leap-frog time scheme. More- over, the interpolation degree is defined at the element level and the mesh is refined locally in a non-conforming way resulting in arbitrary level hanging nodes. The method is proved to be stable under some CFL-like condition on the time step. The convergence of the semi-discrete approximation to Maxwelrs equations is established rigorously and bounds on the global divergence error are provided. Numerical experiments with high- order elements show the potential of the method.

Complex Maxwell's equations

A unified complex model of Maxwell＇s equations is presented.The wave nature of the electromagnetic field vector is related to the temporal and spatial distributions and the circulation of charge and current densities.A new vacuum solution is obtained,and a new transformation under which Maxwell＇s equations are invariant is proposed.This transformation extends ordinary gauge transformation to include charge-current as well as scalar-vector potential.An electric dipole moment is found to be related to the magnetic charges,and Dirac＇s quantization is found to determine an uncertainty relation expressing the indeterminacy of electric and magnetic charges.We generalize Maxwell＇s equations to include longitudinal waves.A formal analogy between this formulation and Dirac＇s equation is also discussed.

Theoretical Maxwell's Equations, Gauge Field and Their Universality Based on One Conservation Law

The notion of the inner product of vectors is extended to tensors of different orders, which may replace the vector product usually. The essences of the differential and the codiffcrential forms are pointed out： they represent the tangent surface and the normal surface fluxes of a tensor, reslpetivcly. The definitions of the divergence and the curl of a 2D surface flux of a tensor arc obtained. Maxwell＇s equations, namely, the constraction law of field, which were usually established based on two conservation laws of electric charge and imaginary magnetic charge, are derived by the author only by using one conservation law （ mass or fluid flux quantity and so on） and the feature of central field （or its composition）. By the feature of central field （or its composition）, the curl of 2D flux is zero. Both universality of gauge field and the difficulty of magnetic monopole theory （ a magnetic monopole has no effect on electric current just like a couple hasing no effect on the sum of forces） axe presented： magnetic monopole has no the feature of magnet. Finally it is pointed out that the base of relation of mass and energy is already involved in Maxwell＇s equations.

Determination of Photon and Elementary Particles Rest Masses Using Maxwell’s Equations and Generalized Potential Dependent Special Relativity

The nature and origin of the photon and elementary rest masses are some of the challeng-ing problems that physics face. The approaches used to solve these problems are complex and time-consuming. Specifically, the photon rest mass pays attention to theoretical physi-cists. Many experimental works show that the photon rest mass is non zero. This problem can be solved using generalized potential dependent special relativity, which has been de-rived using simple arguments, and Maxwell’s equations, besides the conventional Einstein energy-momentum relation. The results obtained show that the rest mass of photons and elementary particles are strongly dependent on the vacuum energy and a universal con-stant. This result conforms with the models that predict time decaying vacuum energy as-sociated with production of smaller rest mass particles followed by larger masses. The two potential dependent mass expressions conform with the cosmological models that suggest the photon is generated first by assuming the universe consisting of total constant vacuum with decaying cosmological part and mass generating part. Using Maxwell’s equations, beside plank and De Broglie hypothesis together with special relativity energy-momentum relation the photon rest mass is estimated. It was shown that the photon rest mass is ex-tremely small compared to the electron mass.

Complex Maxwell’s Equations

Maxwell’s equations in electromagnetism can be categorized into three dis-tinct groups based on the electromagnetic source when employing quaterni-ons. Each group represents a self-contained system in which Maxwell’s equations are applied and validated concurrently, in contrast to the previous approach that did not account for this. It has been noted that the formulation of these Maxwell equations ultimately results in the formulation of Max-well’s equations utilizing the scalar function.

Exact Inverse Operator on Field Equations

Differential equations of electromagnetic and similar physical fields are generally solved via antiderivative Green’s functions involving integration over a region and its boundary. Research on the Kasner metric reveals a variable boundary deemed inappropriate for standard anti-derivatives, suggesting the need for an alternative solution technique. In this work I derive such a solution and prove its existence, based on circulation equations in which the curl of the field is induced by source current density and possibly changes in associated fields. We present an anti-curl operator that is believed novel and we prove that it solves for the field without integration required.

Analysis of the Electromagnetic Characteristics and the Mechanism Underlying Bio-Medical Function of Longitudinal Electromagnetic (LEM) Waves

Based on theoretical system of current Maxwell’s equations, the Maxwell’s equations for LEM waves concealed in full current law and Faraday’s law of electromagnetic induction (Faraday’s law) are proposed. Then, taking them as the fundamental equations, the wave equation and energy equation of LEM waves are established, and a new electromagnetic wave propagation mode based on the mutual induction of scalar electromagnetic fields/vortex magneto-electric fields, which was overlooked in current Maxwell’s equations, are put forward. Moreover, through theoretical derivation based on vacuum LEM waves, the Maxwell’s equations of the gravitational field generated by vacuum LEM waves, the wave equations of the electromagnetic scalar potential/magnetic vector potential and the constraint equation governing the wave phase-velocities between LEM/TEM waves are discovered. Finally, on the basis of these theoretical research results, the electromagnetic properties of vacuum LEM waves are analyzed in detail, encompassing the speed of light, harmless penetrability to the human body, absorption and stable storage by water, the possibility of generating artificial gravitational fields, and the capability of extracting free energy. This reveals the medical functional mechanism of LEM waves and establishes a solid theoretical basis for the application of LEM waves in the fields of medicine and energy.

Elements of Electromagnetic Matter

Originally, Maxwell attempted to express his electromagnetic theory using four-dimensional mathematics of quaternions. Maxwell’s equations were later re-written in a three-dimensional real vector form, which is how the theory is presented today. Thus, an interesting question remains whether we can derive electromagnetic equations analytically from the basic mathematical principles of quaternion algebra and calculus, resulting in general and analytic matter equations. This question seems highly intriguing. Previously, we developed a mathematical theory of time using a normed division algebra of real quaternions [1]. In this study, we extend the theory of time by presenting a new analytical derivation of electromagnetic matter equations using the calculus of real quaternions, as originally intended by Maxwell. Therefore, we propose a novel mathematical definition of the quaternion path derivative using the properties of quaternion division. We then apply the quaternion derivative to an external electromagnetic potential and assume that the first quaternion derivative represents the quaternion electromagnetic force. Next, we assume that the second derivative, or quaternion Laplacian operator, applied to an external electromagnetic potential leads to the quaternion electromagnetic current density. The new analytical expressions are similar to the original empirical Maxwell equations, except for an additional scalar electric field, which allows for a novel formulation of Ohm’s conductivity law. We demonstrate that the resulting analytical equations can be written equivalently using either electromagnetic potentials or fields. Finally, we summarize the key postulates and equations of the new electromagnetic matter theory, which were based on normed division algebra and the calculus of quaternions. The resulting theory appears to be a useful analytical enhancement of the original Maxwell equations, and therefore, seems highly comprehensive, logical, and compelling.

Efficient finite-volume simulation of the LWD orthogonal azimuth electromagnetic response in a three-dimensional anisotropic formation using potentials on cylindrical meshes

In this study,the cylindrical finite-volume method(FVM)is advanced for the efficient and high-precision simulation of the logging while drilling(LWD)orthogonal azimuth electromagnetic tool(OAEMT)response in a three-dimensional(3 D)anisotropic formation.To overcome the ill-condition and convergence problems arising from the low induction number,Maxwell’s equations are reformulated into a mixed Helmholtz equation for the coupled potentials in a cylindrical coordinate system.The electrical fi eld continuation method is applied to approximate the perfectly electrical conducting(PEC)boundary condition,to improve the discretization accuracy of the Helmholtz equation on the surface of metal mandrels.On the base,the 3 D FVM on Lebedev’s staggered grids in the cylindrical coordinates is employed to discretize the mixed equations to ensure good conformity with typical well-logging tool geometries.The equivalent conductivity in a non-uniform element is determined by a standardization technique.The direct solver,PARDISO,is applied to efficiently solve the sparse linear equation systems for the multi-transmitter problem.To reduce the number of calls to PARDISO,the whole computational domain is divided into small windows that contain multiple measuring points.The electromagnetic(EM)solutions produced by all the transmitters per window are simultaneously solved because the discrete matrix,relevant to all the transmitters in the same window,is changed.Finally,the 3 D FVM is validated against the numerical mode matching method(NMM),and the characteristics of both the coaxial and coplanar responses of the EM field tool are investigated using the numerical results.

A DIAGONAL SPLIT-CELL MODEL FOR THE OVERLAPPING YEE FDTD METHOD

In this paper, we present a nonorthogonal overlapping Yee method for solv- ing Maxwell＇s equations using the diagonal split-cell model. When material interface is presented, the diagonal split-cell model does not require permittivity averaging so that better accuracy can be achieved. Our numerical results on optical force computation show that the standard FDTD method converges linearly, while the proposed method achieves quadratic convergence and better accuracy.

ANALYSIS OF TWO-DIMENSIONAL ELECTROMAGNETIC SCATTERING BY A PERFECTLY CONDUCTING OBSTACLE IN A HOMOGENEOUS CHIRAL ENVIRONMENT

The time-harmonic electromagnetic plane waves incident on a perfectly conducting obstacle in a homogeneous chiral environment are considered. A two-dimensional direct scattering model is established and the existence and uniqueness of solutions to the problem are discussed by an integral equation approach. The inverse scattering problem to find the shape of scatterer with the given far-field data is formulated. Result on the uniqueness of the inverse problem is proved.

Retrieval of Electron Return Time from High-order Harmonics Generated in a Mixture of He and Ne Gases

In terms of single-atom induced dipole moment by Lewenstein model, we present the macroscopic high-order harmonic generation from mixed He and Ne gases with different mixture ratios by solving three-dimensional Maxwell＇s equation of harmonic field. And then we show the validity of mixture formulation by Wagner et al. [Phys. Rev. A 76 （2007） 061403（R）] in macroscopic response level. Finally, using/east squares fitting we retrieve the electron return time of short trajectory by formulation in Kanai et al. [Phys. Rev. Lett. 98 （2007） 153904] when the gas jet is put after the laser focus.

Analysis of the Linear Sampling Method for Boundary Reconstruction of Chiral Obstacle in Electromagnetic Scattering Problems

In this paper, we consider the inverse scattering by chiral obstacle in electromagnetic fields, and prove that the linear sampling method is also effective to determine the support of a chiral obstacle from the noisy far field data.

Unification of Gravitational and Electromagnetic Fields in Curved Space-Time Using Gauge Symmetry of Bianchi Identities

This paper deals with the generalization of the linear theory of the unification of gravitational and electromagnetic fields using 4-dimensional gauge symmetry in order to solve the contradictions from the Kaluza-Klein theory’s unification of the gravitational and electromagnetic fields. The unification of gravitational and electromagnetic fields in curved space-time starts from the Bianchi identity, which is well known as a mathematical generalization of the gravitational equation, and by using the existing gauge symmetry condition, equations for the gravitational and electromagnetic fields can be obtained. In particular, the homogeneous Maxwell’s equation can be obtained from the first Bianchi identity, and the inhomogeneous Maxwell’s equation can be obtained from the second Bianchi identity by using Killing’s equation condition of the curved space-time. This paper demonstrates that gravitational and electromagnetic fields can be derived from one equation without contradiction even in curved space-time, thus proving that the 4-dimensional metric tensor using the gauge used for this unification is more complete. In addition, geodesic equations can also be derived in the form of coordinate transformation, showing that they are consistent with the existing equations, and as a result, they are consistent with the existing physical equations.

Analysis and FDTD Modeling of the Influences of Microwave Electromagnetic Waves on Human Biological Systems

The interactions of electromagnetic waves with the human body are complex and depend on several factors related to the characteristics of the incident wave, including its frequency, its intensity, the polarization of the tissue encountered, the geometry of the tissue and its electromagnetic properties. That’s to say, the dielectric permittivity, the conductivity and the type of coupling between the field and the exposed body. A biological system irradiated by an electromagnetic wave is traversed by induced currents of non-negligible density;the water molecules present in the biological tissues exposed to the electromagnetic field will begin to oscillate at the frequency of the incident wave, thus creating internal friction responsible for the heating of the irradiated tissues. This heating will be all the more important as the tissues are rich in water. This article presents the establishment from a mathematical and numerical analysis explaining the phenomena of interaction and consequences between electromagnetic waves and health. Since the total electric field in the biological system is unknown, that is why it can be determined by the Finite Difference Time Domain FDTD method to assess the electromagnetic power distribution in the biological system under study. For this purpose, the detailed on the mechanisms of interaction of microwave electromagnetic waves with the human body have been presented. Mathematical analysis using Maxwell’s equations as well as bio-heat equations is the basis of this study for a consistent result. Therefore, a thermal model of biological tissues based on an electrical analogy has been developed. By the principle of duality, an electrical model in the dielectric form of a multilayered human tissue was used in order to obtain a corresponding thermal model. This thermal model made it possible to evaluate the temperature profile of biological tissues during exposure to electromagnetic waves. The simulation results obtained from computer tools show that the temperature in the biological tissue is a linear function of the duration of exposure to microwave electromagnetic waves.