Showing papers on "Maxwell's equations published in 2016"

Breaking spaces and forms for the DPG method and applications including Maxwell equations

Abstract: Discontinuous Petrov-Galerkin (DPG) methods are made easily implementable using "broken" test spaces, i.e., spaces of functions with no continuity constraints across mesh element interfaces. Broken spaces derivable from a standard exact sequence of first order (unbroken) Sobolev spaces are of particular interest. A characterization of interface spaces that connect the broken spaces to their unbroken counterparts is provided. Stability of certain formulations using the broken spaces can be derived from the stability of analogues that use unbroken spaces. This technique is used to provide a complete error analysis of DPG methods for Maxwell equations with perfect electric boundary conditions. The technique also permits considerable simplifications of previous analyses of DPG methods for other equations. Reliability and efficiency estimates for an error indicator also follow. Finally, the equivalence of stability for various formulations of the same Maxwell problem is proved, including the strong form, the ultraweak form, and various forms in between.

Canonical symplectic particle-in-cell method for long-term large-scale simulations of the Vlasov-Maxwell equations

Abstract: Particle-in-cell (PIC) simulation is the most important numerical tool in plasma physics. However, its long-term accuracy has not been established. To overcome this difficulty, we developed a canonical symplectic PIC method for the Vlasov-Maxwell system by discretising its canonical Poisson bracket. A fast local algorithm to solve the symplectic implicit time advance is discovered without root searching or global matrix inversion, enabling applications of the proposed method to very large-scale plasma simulations with many, e.g. 10(9), degrees of freedom. The long-term accuracy and fidelity of the algorithm enables us to numerically confirm Mouhot and Villani's theory and conjecture on nonlinear Landau damping over several orders of magnitude using the PIC method, and to calculate the nonlinear evolution of the reflectivity during the mode conversion process from extraordinary waves to Bernstein waves.

The Mathematical Theory of Time-Harmonic Maxwell's Equations : Expansion-, Integral-, and Variational Methods

Abstract: Introduction.- Expansion into Wave Functions.- Scattering From a Perfect Conductor.- The Variational Approach to the Cavity Problem.- Boundary Integral Equation Methods for Lipschitz Domains.- Appendix.- References.- Index.

Initial value problem for magnetic fields in heavy ion collisions

Abstract: When the quark-gluon plasma emerges in the wake of a heavy-ion collision, a magnetic field created by the valence charges has already permeated the entire interaction region. Evolution of this ``initial'' field in the plasma is governed by the Maxwell equations in an electrically conducting medium. As the plasma expands, external valence charges induce a magnetic field that also contributes to the total magnetic field in the plasma. I solve the initial value problem describing these processes and argue that the initial magnetic field often dominates over the one induced by the valence charges. In particular, it grows approximately proportional to the collision energy, unlike the induced component, which is energy independent. As a result, the magnetic field has a significant phenomenological influence on the quark-gluon plasma at CERN Large Hadron Collider energies over its entire lifetime.

Advances in imaging and electron physics

Abstract: The purpose of this chapter is to present a unified theory for the numerical implementation of modal methods for the analysis of electromagnetic phenomena with specific boundary conditions. All the fundamental concepts that form the basis of our study will be detailed. In plasmonics and photonics in general, solving Maxwell equations involving irregular functions is common. For example, when the relative permittivity is a piecewise constant function describing a dielectric–metal interface, the eigenmodes of the propagation equation are solutions of Maxwell's equations subject to specific boundary conditions at the interfaces between homogenous media. Prior knowledge about the eigenmodes allows one to define more appropriate expansion functions, and the rate of convergence of the numerical scheme will depend on the choice of these functions. In this chapter, we present and explain, a unified numerical formalism that allows one to build, from a set of subsectional functions defined on a set of subintervals, expansion functions defined on a global domain by enforcing certain stresses deduced from electromagnetic field properties. Then numerical modal analysis of a plasmonic device, such as a ring resonator, is presented as an example of an application.

Anomalous Maxwell equations for inhomogeneous chiral plasma

Abstract: Using the chiral kinetic theory we derive the electric and chiral current densities in inhomogeneous relativistic plasma. We also derive equations for the electric and chiral charge chemical potentials that close the Maxwell equations in such a plasma. The analysis is done in the regimes with and without a drift of the plasma as a whole. In addition to the currents present in the homogeneous plasma (Hall current, chiral magnetic, chiral separation, and chiral electric separation effects, as well as Ohm's current) we derive several new terms associated with inhomogeneities of the plasma. Apart from various diffusion-like terms, we find also new dissipation-less terms that are independent of relaxation time. Their origin can be traced to the Berry curvature modifications of the kinetic theory.

Quantum walks and discrete gauge theories

Abstract: A particular example is produced to prove that quantum walks can be used to simulate full-fledged discrete gauge theories. A family of two-dimensional walks is introduced and its continuous limit is shown to coincide with the dynamics of a Dirac fermion coupled to arbitrary electromagnetic fields. The electromagnetic interpretation is extended beyond the continuous limit by proving that these discrete-time quantum walks (DTQWs) exhibit an exact discrete local U(1) gauge invariance and possess a discrete gauge-invariant conserved current. A discrete gauge-invariant electromagnetic field is also constructed and that field is coupled to the conserved current by a discrete generalization of Maxwell equations. The dynamics of the DTQWs under crossed electric and magnetic fields is finally explored outside the continuous limit by numerical simulations. Bloch oscillations and the so-called $\mathbf{E}\ifmmode\times\else\texttimes\fi{}\mathbf{B}$ drift are recovered in the weak-field limit. Localization is observed for some values of the gauge fields.

Symmetries and couplings of non-relativistic electrodynamics

Abstract: We examine three versions of non-relativistic electrodynamics, known as the electric and magnetic limit theories of Maxwell’s equations and Galilean electrodynamics (GED) which is the off-shell non-relativistic limit of Maxwell plus a free scalar field. For each of these three cases we study the couplings to non-relativistic dynamical charged matter (point particles and charged complex scalars). The GED theory contains besides the electric and magnetic potentials a so-called mass potential making the mass parameter a local function. The electric and magnetic limit theories can be coupled to twistless torsional Newton-Cartan geometry while GED can be coupled to an arbitrary torsional Newton-Cartan background. The global symmetries of the electric and magnetic limit theories on flat space consist in any dimension of the infinite dimensional Galilean conformal algebra and a U(1) current algebra. For the on-shell GED theory this symmetry is reduced but still infinite dimensional, while off-shell only the Galilei algebra plus two dilatations remain. Hence one can scale time and space independently, allowing Lifshitz scale symmetries for any value of the critical exponent z.

Hamiltonian particle-in-cell methods for Vlasov-Maxwell equations

Abstract: In this paper, we study the Vlasov-Maxwell equations based on the Morrison-Marsden-Weinstein bracket. We develop Hamiltonian particle-in-cell methods for this system by employing finite element methods in space and splitting methods in time. In order to derive the semi-discrete system that possesses a discrete non-canonical Poisson structure, we present a criterion for choosing the appropriate finite element spaces. It is confirmed that some conforming elements, e.g., Nedelec's mixed elements, satisfy this requirement. When the Hamiltonian splitting method is used to discretize this semi-discrete system in time, the resulting algorithm is explicit and preserves the discrete Poisson structure. The structure-preserving nature of the algorithm ensures accuracy and fidelity of the numerical simulations over long time.

Dielectric Haloscopes to Search for Axion Dark Matter: Theoretical Foundations

Abstract: We study the underlying theory of dielectric haloscopes, a new way to detect dark matter axions. When an interface between different dielectric media is inside a magnetic field, the oscillating axion field acts as a source of electromagnetic waves, which emerge in both directions perpendicular to the surface. The emission rate can be boosted by multiple layers judiciously placed to achieve constructive interference and by a large transverse area. Starting from the axion-modified Maxwell equations, we calculate the efficiency of this new dielectric haloscope approach. This technique could potentially search the unexplored high-frequency range of 10--100 GHz (axion mass 40--400 $\mu$eV), where traditional cavity resonators have difficulties reaching the required volume.

General-relativistic force-free pulsar magnetospheres

Abstract: Pulsar magnetospheres are shaped by ultra-relativistic electron/positron plasmas flowing in a strong magnetic field and subject to strong gravitational fields. The former induces magnetospheric currents and space charges responsible for the distortion of the electromagnetic field based on pure electrodynamics. The latter induces other perturbations in these fields based on space-time curvature. The force-free approximation describes the response of this magnetosphere to the presence of currents and charges and has been investigated by many authors. In this context, general relativity has been less discussed to quantify its influence on the neutron star electrodynamics. It is the purpose of this paper to compute general-relativistic force-free pulsar magnetospheres for realistic magnetic field configurations such as the inclined dipole. We performed time-dependent simulations of Maxwell equations in the 3+1 formalism of a stationary background metric in the slow-rotation approximation. We computed the resulting Poynting flux depending on the ratio R/rL and on frame-dragging through the spin parameter as, R is the neutron star radius and rL the light-cylinder radius. Both effects act together to increase the total Poynting flux seen by a distant observer by a factor up to 2 depending on the rotation rate. Moreover we retrieve the sin 2 χ dependence of this luminosity, χ being the obliquity of the pulsar, as well as a braking index close to n = 3. We also show that the angular dependence of the Poynting flux scales like sin 2 ϑ for the aligned rotator but like sin 4 ϑ for the orthogonal rotator, ϑ being the colatitude.

Magnetic materials and 3D finite element modeling

Abstract: Statics and Quasi-Statics Electromagnetics - Brief Presentation Introduction The Maxwell Equations The Maxwell Equations: Local Form The Maxwell Equations: Integral Form The Maxwell Equations in Low Frequency The Electrostatics Magnetostatic Fields Magnetic Materials Inductance and Mutual Inductance Magnetodynamic Fields Fields Defined by Potentials Final Considerations References Ferromagnetic Materials and Iron Losses Introduction Basic Concepts Losses Components Iron Losses under Alternating, Rotating and DC Biased Inductions Final Considerations References Scalar Hysteresis Modeling Introduction The Preisach's Scalar Model The Jiles-Atherton Scalar Model Final Considerations References Vector Hysteresis Modeling Introduction Vector Model Obtained with the Superposition of Scalar Models Vector Generalizations of the Jiles-Atherton Scalar Models Some Remarks Concerning the Vector Behavior of Hysteresis Final Considerations References Brief Presentation of the Finite Element Method Introduction The Galerkin Method: Basic Concepts using Real Coordinates Generalization of the FEM: Using Reference Coordinates Numerical Integration Some Finite Elements Using Edge Elements References Using Nodal Elements with Magnetic Vector Potential Introduction Main Equations Applying Galerkin Method Uniqueness of the Solution the Coulomb's Gauge Implementation Example and Comparisons Final Considerations References The Source-Field Method for 3D Magnetostatic Fields Introduction The Magnetostatic Case - Scalar Potential The Magnetostatic Case - Vector Potential Implementation Aspects and Conventions Computational Implementation Example and Results References The Source-Field Method for 3D Magnetodynamic Fields Introduction Formulation Considering Eddy Currents - Time Stepping Formulation Considering Eddy Currents - Complex Formulation Field-Circuit Coupling Computational Implementation The Differential Permeability Method Example and Results References A Matrix-Free Iterative Solution Procedure for Finite Element Problems Introduction The Classical FEM: T-Scheme The Proposed Technique: N-Scheme Implementation Convergence Implementation of N-Scheme with SOR Applying Non-Stationary Iterative Solver to the N-Scheme CG Algorithm Implementation Examples and Results Results and Discussion References

Numerical solutions of the macroscopic Maxwell equations for scattering by non-spherical particles: A tutorial review

Abstract: Numerical solution methods for electromagnetic scattering by non-spherical particles comprise a variety of different techniques, which can be traced back to different assumptions and solution strategies applied to the macroscopic Maxwell equations. One can distinguish between time- and frequency-domain methods; further, one can divide numerical techniques into finite-difference methods (which are based on approximating the differential operators), separation-of-variables methods (which are based on expanding the solution in a complete set of functions, thus approximating the fields), and volume integral-equation methods (which are usually solved by discretisation of the target volume and invoking the long-wave approximation in each volume cell). While existing reviews of the topic often tend to have a target audience of program developers and expert users, this tutorial review is intended to accommodate the needs of practitioners as well as novices to the field. The required conciseness is achieved by limiting the presentation to a selection of illustrative methods, and by omitting many technical details that are not essential at a first exposure to the subject. On the other hand, the theoretical basis of numerical methods is explained with little compromises in mathematical rigour; the rationale is that a good grasp of numerical light scattering methods is best achieved by understanding their foundation in Maxwell's theory.

Thermodynamically admissible boundary conditions for the regularized 13 moment equations

Abstract: A phenomenological approach to the boundary conditions for linearized R13 equations is derived using the second law of thermodynamics. The phenomenological coefficients appearing in the boundary conditions are calculated by comparing the slip, jump, and thermal creep coefficients with linearized Boltzmann solutions for Maxwell’s accommodation model for different values of the accommodation coefficient. For this, the linearized R13 equations are solved for viscous slip, thermal creep, and temperature jump problems and the results are compared to the solutions of the linearized Boltzmann equation. The influence of different collision models (hard-sphere, Bhatnagar–Gross–Krook, and Maxwell molecules) and accommodation coefficients on the phenomenological coefficients is studied.

Numerical study of chiral plasma instability within the classical statistical field theory approach

Abstract: We report on a numerical study of real-time dynamics of electromagnetically interacting chirally imbalanced lattice Dirac fermions within the classical statistical field theory approach. Namely, we perform exact simulations of the real-time quantum evolution of fermionic fields coupled to classical electromagnetic fields, which are in turn coupled to the vacuum expectation value of the fermionic electric current. We use Wilson-Dirac Hamiltonian for fermions, and noncompact action for the gauge field. In general, we observe that the backreaction of fermions on the electromagnetic field prevents the system from acquiring chirality imbalance. In the case of chirality pumping in parallel electric and magnetic fields, the electric field is screened by the produced on-shell fermions and the accumulation of chirality is hence stopped. In the case of evolution with initially present chirality imbalance, axial charge tends to transform to helicity of the electromagnetic field. By performing simulations on large lattices we show that in most cases this decay process is accompanied by the inverse cascade phenomenon, which transfers energy from short-wavelength to long-wavelength electromagnetic fields. In some simulations, however, we observe a very clear signature of inverse cascade for the helical magnetic fields that is not accompanied by the axial charge decay. This suggests that the relation between the inverse cascade and axial charge decay is not as straightforward as predicted by the simplest form of anomalous Maxwell equations.

Mosco convergence for H(curl) spaces, higher integrability for Maxwell's equations, and stability in direct and inverse EM scattering problems

Abstract: This paper is concerned with the scattering problem for time-harmonic electromagnetic waves, due to the presence of scatterers and of inhomogeneities in the medium. We prove a sharp stability result for the solutions to the direct electromagnetic scattering problem, with respect to variations of the scatterer and of the inhomogeneity, under minimal regularity assumptions for both of them. The stability result leads to uniform bounds on solutions to the scattering problems for an extremely general class of admissible scatterers and inhomogeneities. The uniform bounds are a key step to tackle the challenging stability issue for the corresponding inverse electromagnetic scattering problem. In this paper we establish two optimal stability results of logarithmic type for the determination of polyhedral scatterers by a minimal number of electromagnetic scattering measurements. In order to prove the stability result for the direct electromagnetic scattering problem, we study two fundamental issues in the theory of Maxwell equations: Mosco convergence for H(curl) spaces and higher integrability properties of solutions to Maxwell equations in nonsmooth domains.

Continuum Kinetic and Multi-Fluid Simulations of Classical Sheaths

Abstract: The kinetic study of plasma sheaths is critical, among other things, to understand the deposition of heat on walls, the effect of sputtering, and contamination of the plasma with detrimental impurities. The plasma sheath also provides a boundary condition and can often have a significant global impact on the bulk plasma. In this paper, kinetic studies of classical sheaths are performed with the continuum code, Gkeyll, that directly solves the Vlasov-Poisson/Maxwell equations. The code uses a novel version of the finite-element discontinuous Galerkin (DG) scheme that conserves energy in the continuous-time limit. The electrostatic field is computed using the Poisson equation. Ionization and scattering collisions are included, however, surface effects are neglected. The aim of this work is to introduce the continuum-kinetic method and compare its results to those obtained from an already established finite-volume multi-fluid model also implemented in Gkeyll. Novel boundary conditions on the fluids allow the sheath to form without specifying wall fluxes, so the fluids and fields adjust self-consistently at the wall. The work presented here demonstrates that the kinetic and fluid results are in agreement for the momentum flux, showing that in certain regimes, a multi-fluid model can be a useful approximation for simulating the plasma boundary. There are differences in the electrostatic potential between the fluid and kinetic results. Further, the direct solutions of the distribution function presented here highlight the non-Maxwellian distribution of electrons in the sheath, emphasizing the need for a kinetic model.

Block Preconditioners for Stable Mixed Nodal and Edge finite element Representations of Incompressible Resistive MHD

Abstract: The scalable iterative solution of strongly coupled three-dimensional incompressible resistive magnetohydrodynamics (MHD) equations is very challenging because disparate time scales arise from the electromagnetics, the hydrodynamics, as well as the coupling between these systems. This study considers a mixed finite element discretization of a dual saddle point formulation of the incompressible resistive MHD equations using a stable nodal (Q2/Q1) discretization for the hydrodynamics and a stable edge-node discretization of a reduced form of the Maxwell equations. This paper presents new approximate block factorization preconditioners for this system which reduce the system to approximate Schur complement systems that can be solved using algebraic multilevel methods. These preconditioners include a new augmentation-based approximation for the magnetic induction saddle point system as well as efficient approximations of the Schur complements that arise from the complex coupling between the Navier--Stokes eq...

Novel propagation regimes for TE waves guided by a waveguide filled with Kerr medium

Abstract: A fully analytical approach to the problem of TE wave propagation in an open lossless waveguide filled with Kerr medium is suggested. It is shown that the waveguide supports two physically interesting guided regimes in the focusing case. In each of the regimes, there exists an infinite number of guided waves that do not have linear counterparts. It is also shown that in the defocusing case only one regime arises with a finite number of guided waves; all these solutions have linear counterparts. Numerical illustrations and discussion of the found results are presented.

Variational formulations of guiding-center Vlasov-Maxwell theory

Abstract: The variational formulations of guiding-center Vlasov-Maxwell theory based on Lagrange, Euler, and Euler-Poincare variational principles are presented. Each variational principle yields a different approach to deriving guiding-center polarization and magnetization effects into the guiding-center Maxwell equations. The conservation laws of energy, momentum, and angular momentum are also derived by Noether method, where the guiding-center stress tensor is now shown to be explicitly symmetric.

On Kottler's path: origin and evolution of the premetric program in gravity and in electrodynamics ∗

Abstract: In 1922, Kottler put forward the program to remove the gravitational potential, the metric of spacetime, from the fundamental equations in physics as far as possible. He successfully applied this idea to Newton’s gravitostatics and to Maxwell’s electrodynamics, where Kottler recast the field equations in premetric form and specified a metric-dependent constitutive law. We will discuss the basics of the premetric approach and some of its beautiful consequences, like the division of universal constants into two classes. We show that classical electrodynamics can be developed without a metric quite straightforwardly: the Maxwell equations, together with a local and linear response law for electromagnetic media, admit a consistent premetric formulation. Kottler’s program succeeds here without provisos. In Kottler’s approach to gravity, making the theory relativistic, two premetric quasi-Maxwellian field equations arise, but their field variables, if interpreted in terms of general relativity, do depend on the...

Modelling and compensation of hysteresis in piezoelectric actuators based on Maxwell approach

Abstract: A Maxwell model is presented to describe the hysteresis in a piezoelectric actuator. The effect of the number of elements is studied and both of the forward and inverse algorithms are presented. Using the inverse Maxwell model, hysteresis is compensated for and an almost linear performance is obtained. Experimental results validate the effectiveness of the proposed algorithm and show that hysteresis nonlinearity reduces from 13.8 to 0.4%.

A priori error analysis of space–time Trefftz discontinuous Galerkin methods for wave problems

Abstract: We present and analyse a space–time discontinuous Galerkin method for wave propagation problems. The special feature of the scheme is that it is a Trefftz method, namely that trial and test functions are solution of the partial differential equation to be discretised in each element of the (space–time) mesh. The method considered is a modification of the discontinuous Galerkin schemes of Kretzschmar et al. (2014) and of Monk & Richter (2005). For Maxwell’s equations in one space dimension, we prove stability of the method, quasi-optimality, best approximation estimates for polynomial Trefftz spaces and (fully explicit) error bounds with high order in the meshwidth and in the polynomial degree. The analysis framework also applies to scalar wave problems and Maxwell’s equations in higher space dimensions. Some numerical experiments demonstrate the theoretical results proved and the faster convergence compared to the non-Trefftz version of the scheme.