Electromagnetic Scattering by Periodic Structures

This paper is devoted to the scattering of electromagnetic waves by quite general biperiodic structures which may consist of anisotropic optical materials and separate two regions with constant dielectric coefficients. The time-harmonic Maxwell equations are transformed to an equivalent H ¹-variational problem for the magnetic field in a bounded biperiodic cell with nonlocal boundary conditions. The existence of solutions is shown for all physically relevant material parameters. The uniqueness is proved for all frequencies excluding possibly a discrete set. The results of the general problem are compared with known results for a special case, the conical diffraction.     

Analysis of the diffraction from periodic chiral structures

Consider a time-harmonic electromagnetic plane wave incident on a biperiodic structure in R³. The periodic structure separates two homogeneous regions. The medium inside the structure is chiral and heterogeneous. In general, wave propagation in the chiral medium is governed by Maxwell's equations together with the Drude-Born-Fedorov (constitutive) equations. In this Note, it is shown that for all but possibly a discrete set of parameters, there is a unique quasiperiodic weak solution to the diffraction problem. Our proof is based on a Hodge decomposition, a compact imbedding result, as well as the Lax-Milgram Lemma.     

Maxwell's Equations in a Perturbed Periodic Structure

The paper is concerned with electromagnetic wave propagation in a perturbed periodic structure. Consider a time-harmonic electromagnetic plane wave incident on a biperiodic (grating) structure. A single inhomogeneous object is placed inside the biperiodic structure. The scattering problem is to study the electromagnetic field distributions. The problem arises in the study of near-field optics and has recently found many applications in physics and biology. In this paper, an integral representation approach is presented to solve the problem. Using the approach, well-posedness of the model problem is established. In addition, it is shown that the perturbation due to the object is exponentially decaying along the periodic directions of the structure, provided that no surface waves occur.     

Existence and uniqueness of optimal solutions for multirate partial differential algebraic equations

The numerical simulation of electric circuits including multirate signals can be done by a model based on partial differential algebraic equations. In the case of frequency modulated signals, a local frequency function appears as a degree of freedom in the model. Thus the determination of a solution with a minimum amount of variation is feasible, which allows for resolving on relatively coarse grids. We prove the existence and uniqueness of the optimal solutions in the case of initial-boundary value problems as well as biperiodic boundary value problems. The minimisation problems are also investigated and interpreted in the context of optimal control. Furthermore, we construct a method of characteristics for the computation of optimal solutions in biperiodic problems. Numerical simulations of test examples are presented.     

Complex variable equations and the numerical solution of harmonic problems for piecewise-homogeneous media

Complex variable boundary integral equations are derived using of holomorphicity theorems for plane harmonic problems concerning unit structures with inclusions, pores and lines of discontinuity of the potential and/or the flow. Unlike the method of analytical elements, the equations cover problems in which discontinuities in the potential, flow and conductance can simultaneously be encountered at the contact points. Versions of the equations are given for connected half planes and for periodic and biperiodic problems. Formulae are obtained which determine the effective impedance tensor of the equivalent homogeneous medium in cases when the unit structure is biperiodic or when the representative volume of a structured medium is identified with the basic cell of a biperiodic system. Recurrence quadrature formulae are proposed which enable one to solve the resulting equations effectively using the complex variable boundary element method. They indicate the computational advantages of using the complex variable method compared with the real variable method: the three integrals appearing in the resulting equations are evaluated analytically in the case of linear elements (regular and singular) with the densities approximated using algebraic polynomials of arbitrary degree. In the case of elements (regular and singular) in the form of an arc of a circle, only one integral requires numerical integration when the densities are approximated using complex trigonometrical polynomials of arbitrary degree. It is emphasized that the combination of the linear and curved boundary elements which have been developed enables the smooth part of a contour to be approximated while retaining the continuity of the tangent and avoiding the complications which arise when the smoothness of the approximation of a contour is ensured using conformal mapping. Examples are presented which illustrate the computational merits of the method developed. They show a sharp increase in accuracy (by orders of magnitude) when curved elements are used for the curvilinear parts of a contour and when terminal elements are used to calculate the flow intensity coefficient at singular points (the crack tips the vertices of angular notches and the common vertices of the units of the medium).     

Generalized Electrodynamics With Ternary Internal Structure

In Refs. [2]–[7] we suggested generalized dynamic equations of motion of relativistic charged particles inside electromagnetic fields. The dynamic equations had been formulated in terms of external as well as internal momenta. Evolution equations for external momenta, the Lorentz-force equations, had been derived from evolution equations for internal momenta. In this paper, along with relativistic dynamics we generalize electromagnetic fields within the scope of ternary algebras. The full theory is constructed in 4D euclidean space. This space possesses an advantage to build ternary mappings from three vectors onto one. The dynamics is given by non-linear evolution equations with cubic characteristic polynomial. In polar representation the internal momenta obey the Jacobi equations whereas external momenta obey the Weierstrass equations for elliptic functions. The generalized electromagnetic fields are defined by the triple fields where the first one has properties of the electric field and the other two have properties of the magnetic field. The field equations for the triple fields analogous to the Maxwell equations are suggested.     

Asymptotics of the spectrum and eigenfunctions of the magnetic induction operator on a compact two-dimensional surface of revolution

Magnetic fields in conducting liquids (in particular, magnetic fields of galaxies, stars, and planets) are described by the magnetic induction operator. In this paper, we study the spectrum and eigenfunctions of this operator on a compact two-dimensional surface of revolution. For large magnetic Reynolds numbers, the asymptotics of the spectrum is studied; equations defining the eigenvalues (quantization conditions) are obtained; and examples of spectral graphs near which these points are located are given. The spatial structure of the eigenfunctions is studied.     

Global strong solutions to the Shliomis system for ferrofluids in a bounded domain

We consider the partial differential equations proposed by Shliomis to model the dynamics of an incompressible viscous ferrofluid submitted to an external magnetic field. The Shliomis system consists of the incompressible Navier‐Stokes equations, the magnetization equations, and the magnetostatic equations. The magnetization equations is of Bloch type, and no regularizing term is added. We prove the global existence of unique strong solution to the initial boundary value problem for the system in a bounded domain, with the small initial data and external magnetic field but without any restrictions on the physical parameters. The novelty of the analysis is to introduce a linear combination of magnetic fields.     

A lattice Boltzmann model for Maxwell’s equations

In this paper, a lattice Boltzmann model for the Maxwell’s equations is proposed by taking separate sets of distribution functions for the electric and magnetic fields, and a lattice Boltzmann model for the Maxwell vorticity equations with third order truncation error is proposed by using the higher-order moment method. At the same time, the expressions of the equilibrium distribution function and the stability conditions for this model are given. As numerical examples, some classical electromagnetic phenomena, such as the electric and magnetic fields around a line current source, the electric field and equipotential lines around an electrostatic dipole, the electric and magnetic fields around oscillating dipoles are given. These numerical results agree well with classical ones.     

Effective Maxwell equations from time-dependent density functional theory

The behavior of interacting electrons in a perfect crystal under macroscopic external electric and magnetic fields is studied. Effective Maxwell equations for the macroscopic electric and magnetic fields are derived starting from time-dependent density functional theory. Effective permittivity and permeability coefficients are obtained.     

Covariance and the hierarchy of frame bundles

This is an essay on the general concept of covariance, and its connection with the structure of the nested set of higher frame bundles over a differentiable manifold. Examples of covariant geometric objects include not only linear tensor fields, densities and forms, but affinity fields, sectors and sector forms, higher order frame fields, etc., often having nonlinear transformation rules and Lie derivatives. The intrinsic, or invariant, sets of forms that arise on frame bundles satisfy the graded Cartan-Maurer structure equations of an infinite lie algebra. Reduction of these gives invariant structure equations for Lie pseudogroups, and forG-structures of various orders. Some new results are introduced for prolongation of structure equations, and for treatment of Riemannian geometry with higher-order moving frames. The use of invariant form equations for nonlinear field physics is implicitly advocated.Research sponsored by the U.S. Army Research Office through an agreement with the National Aeronautics and Space Administration.     

Modeling of electrodynamic-mechanical coupling phenomena in smart magnetoelectroelastic materials

The coupling of electric, magnetic and mechanical phenomena may have various reasons. The famous Maxwell equations of electrodynamics describe the interaction of transient magnetic and electric fields. On the constitutive level of dielectric materials, coupling mechanisms are manyfold comprising piezoelectric, magnetostrictive or magnetoelectric effects. Electromagnetically induced specific forces acting at the boundary and within the domain of a dielectric body are, within a continuum mechanics framework, commonly denoted as Maxwell stresses. In transient electromagnetic fields, the Poynting vector gives another contribution to mechanical stresses. First, a system of transient partial differential equations is presented. Introducing scalar and vector potentials for the electromagnetic fields and representing the mechanical strain by displacement fields, seven coupled differential equations govern the boundary value problem, accounting for linear constitutive equations of magnetoelectroelasticity. To reduce the effort of numerical solution, the system of equations is partly decoupled applying generalized forms of Coulomb and Lorenz gauge transformations [1,2]. A weak formulation is given to establish a basis for a finite element solution. The influence of constitutive magnetoelectric coupling on electromagnetic wave propagation is finally demonstrated with a simple one-dimensional example. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Wavelet-based Adaptive Grids for the Simulation of Multirate Partial Differential-Algebraic Equations

We present an adaptive method to solve biperiodic problems in chip design with widely separated time scales and steep gradients due to digital-like signal structures. Motivated from the example of the switched capacitor filter, we explain the multidimensional signal model and the transformation of network equations to the hyperbolic multirate differential-algebraic system. Then adapted hat-wavelets are introduced to detect the steep gradients. In this way, an adaptive grid is defined for the efficient application of a finite difference method to solve the limit cycle problem. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some problems in nonlinear magnetoelasticity

In this paper we examine the influence of magnetic fields on the static response of magnetoelastic materials, such as magneto-sensitive elastomers, that are capable of large deformations. The analysis is based on a simple formulation of the mechanical equilibrium equations and constitutive law for such materials developed recently by the authors, coupled with the governing magnetic field equations. The equations are applied in the solution of some simple representative and illustrative problems, with the focus on incompressible materials. First, we consider the pure homogeneous deformation of a slab of material in the presence of a magnetic field normal to its faces. This is followed by a review of the problem of simple shear of the slab in the presence of the same magnetic field. Next we examine a problem involving non-homogeneous deformations, namely the extension and inflation of a circular cylindrical tube. In this problem the magnetic field is taken to be either axial (a uniform field) or circumferential. For each problem we give a general formulation for the case of an isotropic magnetoelastic constitutive law, and then, for illustration, specific results are derived for a prototype constitutive law. We emphasize that in general there are significant differences in the results for formulations in which the magnetic field or the magnetic induction is taken as the independent magnetic variable. This is demonstrated for one particular problem, in which restrictions are placed on the admissible class of constitutive laws if the magnetic induction is the independent variable but no restrictions if the magnetic field is the independent variable.Received: May 17, 2004     

Some problems in nonlinear magnetoelasticity

In this paper we examine the influence of magnetic fields on the static response of magnetoelastic materials, such as magneto-sensitive elastomers, that are capable of large deformations. The analysis is based on a simple formulation of the mechanical equilibrium equations and constitutive law for such materials developed recently by the authors, coupled with the governing magnetic field equations. The equations are applied in the solution of some simple representative and illustrative problems, with the focus on incompressible materials. First, we consider the pure homogeneous deformation of a slab of material in the presence of a magnetic field normal to its faces. This is followed by a review of the problem of simple shear of the slab in the presence of the same magnetic field. Next we examine a problem involving non-homogeneous deformations, namely the extension and inflation of a circular cylindrical tube. In this problem the magnetic field is taken to be either axial (a uniform field) or circumferential. For each problem we give a general formulation for the case of an isotropic magnetoelastic constitutive law, and then, for illustration, specific results are derived for a prototype constitutive law. We emphasize that in general there are significant differences in the results for formulations in which the magnetic field or the magnetic induction is taken as the independent magnetic variable. This is demonstrated for one particular problem, in which restrictions are placed on the admissible class of constitutive laws if the magnetic induction is the independent variable but no restrictions if the magnetic field is the independent variable.     

Sur les équations de la magnétothermoélasticité

Summary Writing the MTE equations in non-dimensional variables, the author points out the essential parameters and the structure of the equations. The equation of the MTE system characteristics is determined, as well as the displacement velocities of the waves of weak discontinuity. Furthermore, the equation which is satisfied by the current density vectorJ is determined.The paper considers the Cauchy problem for the infinite perfectly conductive plane, in the presence of a temperature and of a magnetic field. The general solution for the current density and the magnetic field are determined from Maxwell's equation. The solution obtained permits one to determine the displacement and the temperature fields by means of the classical thermoelasticity.The method of solving the Cauchy problem is based upon the representation of the general solution by a continous superposition of plane waves.     

The effect of magnetic field on dynamics of gas bubbles in visco-elastic fluids

The effect of magnetic field on nonlinear oscillations of a spherical, acoustically forced gas bubble in nonlinear visco-elastic media is studied. The constitutive equation UCM used for modeling the rheological behaviors of the fluid. By starting from the momentum equations for bubbles considering the magnetic force and considering some simplifying assumptions, the modified bubble dynamics equation (the modified Rayleigh–Plesset equation) has been achieved. Assumptions concerning the trace of the stress tensor are addressed in light of the incorporation of visco-elastic constitutive equations into modified bubble dynamics equations. The governing equations are non-dimesionalized and numerically solved by using 4th order Runge–Kutta method. The accuracy of the calculations and the formulation is compared with the previous works done for models without the presence of magnetic field. Furthermore, the bubble size variations due to acoustic motivations and stress tensor components variations in presence of different magnitudes of magnetic fields are studied. Also, the bubble size dependence on fluid conductivity variations is declared. The relevance and importance of this approach to biomedical ultrasound applications are highlighted. Preliminary results indicate that magnetic field may be an important consideration for the risk assessment of potential cavitations and also it could be possible to damp the bubble oscillations by using magnetic fields or in opposite case amplify the oscillations which could result in higher level light emissions in sonoluminescence approach.     

Transport equations with disparate advection fields. Application to the gyrokinetic models in plasma physics

The subject matter of this paper concerns the asymptotic regimes for transport equations with advection fields having components of very disparate orders of magnitude. The main purpose is to derive the limit models: we justify rigorously the convergence towards these limit models and we investigate the well-posedness of them. Such asymptotic analysis arises in the magnetic confinement context, where charged particles move under the action of strong magnetic fields. In these situations we distinguish between a slow motion driven by the electric field and a fast motion around the magnetic lines.     

The vector integral-equation method for computing three-dimensional magnetic fields

The vector integral-equation method for computing three-dimensional, quasistatic magnetic fields is developed with a view to its application to configurations of the type that occur in magnetic recording. Starting from appropriate Green-type vector integral relations for the magnetic-field quantities, the relevant integral equations are derived. Their numerical handling is briefly discussed.     

On the one- and one-half-dimensional relativistic Vlasov-Fokker-Planck-Maxwell system

In this paper, we study the relativistic Vlasov-Fokker-Planck-Maxwell system in one space variable and two momentum variables. This non-linear system of equations consists of a transport equation for the phase space distribution function combined with Maxwell's equations for the electric and magnetic fields. It is important in modelling distribution of charged particles in the kinetic theory of plasma. We prove the existence of a classical solution when the initial density decays fast enough with respect to the momentum variables. The solution which shares this same decay condition along with its first derivatives in the momentum variables is unique.