Stationary Maxwell Equations Research Articles

A magnetic field coupling lattice Boltzmann model and its application on the merging process of multiple-ferrofluid-droplet system

The present study concerns the dynamics of ferrofluid droplets in a Newtonian fluid. The incompressible Navier-Stokes (NS) equations are applied for the dynamics of the immiscible magnetic and Newtonian fluids while the Cahn-Hilliard (C-H) equation is adopted to describe the behavior of their interface. A magnetic field coupling lattice Boltzmann (LB) model is developed to solve both the NS equations and the C-H equation. Specially, a mass-correcting term is introduced into the C-H equation, which strongly enforces the mass conservation. The magnetic field is evaluated by a Poisson equation solver with a self-correcting procedure for the static Maxwell equations. The magnetic dipole force is transformed into the magnetic surface force by a rigorous mathematical procedure, which can physically describe the magnetic effect on the interface. Moreover, the magnetic force after this treatment becomes easy-to-implement, which can be directly incorporated into the external force term of the LB model. The capability of the present combination method to simulate the magnetic multiphase flows is demonstrated by three typical numerical examples, i.e., Laplace law for a stationary droplet, a stationary cylinder under an external uniform magnetic field, the deformation of a single ferrofluid droplet. Further, the merging process of multiple-ferrofluid-droplet system in organic oil is investigated. It is found that the elongation of ferrofluid droplet in the direction of magnetic field shows positive impact on the merging process when the orientation of ferrofluid droplets is parallel to the external magnetic field, while negative impact on the merging process when their orientation is perpendicular to the external magnetic field.

Temporal convergence analysis of an energy preserving projection method for a coupled magnetohydrodynamics equations

In this paper, we study and analyze a semi-discrete projection scheme for solving a hybrid magnetohydrodynamics equations coupled by the nonstationary Navier–Stokes equations and the stationary Maxwell equations. The suggested projection scheme is energy preserving and satisfies the divergence free condition of the velocity field at the discrete level. As the time step size is sufficiently small, we establish optimal temporal error estimates under some regularity assumptions. Numerical results are provided to show the efficiency of the suggested scheme and verify the theoretical analysis.

Parallel block preconditioners for three-dimensional virtual element discretizations of saddle-point problems

Several physical phenomena are described by systems of partial differential equations (PDEs) that, after space discretization, yield the solution of saddle point algebraic linear systems. In realistic three-dimensional numerical simulations, these linear systems are large scale and ill-conditioned, thus they require the development of effective solvers. The aim of this work is the construction and numerical validation of parallel block preconditioners for a set of three-dimensional saddle point problems discretized by the low order virtual element method (VEM). VEM is a recent numerical technology for the approximation of PDEs on polygonal and polyhedral meshes. We focus on the following systems of PDEs: stationary Maxwell equations in the mixed Kikuchi formulation; elliptic equations in mixed form; Stokes system; linear elasticity in the mixed Hellinger–Reissner formulation. We provide two parallel block preconditioners: one based on the approximate Schur complement and the other on a regularization technique. Several numerical experiments are run in parallel on a Linux cluster. We analyze the performance of the iterative solvers in terms of GMRES iterations and computational time. We verify the robustness of the solvers with respect to different polyhedral meshes and the scalability of both the assembling and solution time by varying the number of processors. The performance of the two iterative solvers is also compared with state-of-the-art parallel direct linear solvers.

A Unified Error Analysis of Hybridizable Discontinuous Galerkin Methods for the Static Maxwell Equations

We propose a framework that allows us to analyze different variants of hybridizable discontinuous Galerkin (HDG) methods for the static Maxwell equations using one simple analysis. It reduces all t...

Numerical Solution to the 3D Static Maxwell Equations in Axisymmetric Singular Domains with Arbitrary Data

Abstract We propose a numerical method to solve the three-dimensional static Maxwell equations in a singular axisymmetric domain, generated by the rotation of a singular polygon around one of its sides. The mathematical tools and an in-depth study of the problem set in the meridian half-plane are exposed in [F. Assous, P. Ciarlet, Jr., S. Labrunie and J. Segré, Numerical solution to the time-dependent Maxwell equations in axisymmetric singular domains: the singular complement method, J. Comput. Phys. 191 2003, 1, 147–176] and [P. Ciarlet, Jr. and S. Labrunie, Numerical solution of Maxwell’s equations in axisymmetric domains with the Fourier singular complement method, Differ. Equ. Appl. 3 2011, 1, 113–155]. Here, we derive a variational formulation and the corresponding approximation method. Numerical experiments are proposed, and show that the approach is able to capture the singular part of the solution. This article can also be viewed as a generalization of the Singular Complement Method to three-dimensional axisymmetric problems.

Second-order semi-implicit Crank-Nicolson scheme for a coupled magnetohydrodynamics system

This paper presents a second-order semi-implicit Crank-Nicolson scheme for a hybrid magnetohydrodynamics system coupled by the nonstationary Navier-Stokes equations and the stationary Maxwell equations. Main features of proposed scheme are two-fold. One is that it is a decoupled scheme and the magnetic field and velocity field can be solved independently at the same time discrete level. Another is that two subproblems both are linear and are easy to be solved numerically. A rigorous temporal-spatial error analysis is done and we prove that the proposed scheme is of second-order convergence rate O((Δt)2) for approximations of the magnetic field, the velocity field and the pressure under the condition Δt=O(h1/2). Finally, a numerical result is displayed to illustrate the theoretical results.

Photonic Crystals Based on Media with Arbitrary Anisotropy of Permittivity and Permeability

A general approach to analyzing eigenmodes in anisotropic and gyrotropic 3D photonic crystals based on dielectric and magnetic media is proposed. This approach is based on the representation of stationary macroscopic Maxwell equations in the operator form corresponding to the quantum-mechanical equation for a photon with spin s = 1. In these equations, the strengths of electric and magnetic fields are put into correspondence with state vectors in the complex Hilbert space. The permittivity and permeability serve as operators acting on these vectors. It is shown that the problem of determining the eigenmodes of a photonic crystal is reduced to searching for eigenvectors and eigenvalues of the Hermitian operator characterizing the spin–orbit interaction of a photon in the periodic anisotropic medium under study. It is proposed to use photon states with a certain wave vector (certain momentum) and a certain linear or circular spin polarization as a basis for the representation operator equations. One-dimensional photonic crystals are considered as an example. The influence of anisotropy and gyrotropy on the dispersion of eigenmodes in these crystals is investigated. The group velocity of the eigenmodes, momentum transferred by them, and spin angular momentum are analyzed for the case of gyrotropic media.

A Riemann-Hilbert Problem for the Moisil-Teodorescu System

In a bounded domain with smooth boundary in ℝ3 we consider the stationary Maxwell equations for a function u with values in ℝ3 subject to a nonhomogeneous condition (u, v)x = u0 on the boundary, where v is a given vector field and u0 a function on the boundary. We specify this problem within the framework of the Riemann-Hilbert boundary value problems for the Moisil-Teodorescu system. This latter is proved to satisfy the Shapiro-Lopaniskij condition if an only if the vector v is at no point tangent to the boundary. The Riemann-Hilbert problem for the Moisil-Teodorescu system fails to possess an adjoint boundary value problem with respect to the Green formula, which satisfies the Shapiro-Lopatinskij condition. We develop the construction of Green formula to get a proper concept of adjoint boundary value problem.

Electron Motion in the Three-Dimensional Field of Undulator

In this paper we consider the motion of relativistic electrons in an ideal three-dimensional magnetic field of an undulator. The ideality of the magnetic field means that, on the undulator axis, the field is directed strictly vertically upward and has a strictly sinusoidal shape. In the overwhelming majority of cases, only this leading component of the field is taken into account in calculating the electron trajectory. In this paper, in the equations of motion of an electron in the magnetic field of an undulator, all three components of the field are taken into account, so that the undulator field under consideration satisfies the stationary Maxwell equations. In this case, the differential equations of motion of the electron are solved analytically with the help of perturbation theory, and not by the method of averaging over fast oscillations of the electron, as was done in a number of previous papers. These analytic expressions for trajectories describe the behavior of particles in the focusing magnetic field of an undulator much more completely. An analysis of these expressions shows that the behavior of electrons in such a three-dimensional field of the undulator is much more complicated than what follows from the equations obtained by the averaging method. In particular, there is a cross effect when changes in the initial vertical parameters of the electron trajectory cause changes in the horizontal component of its trajectory and vice versa. A comparison of the solutions obtained analytically with the results of numerical calculations of electron trajectories using the Runge–Kutta method demonstrates their high accuracy.

On the asymptotic behaviour of a variable exponent power law magnetostatic problem

We introduce a variational scheme to study the asymptotic behaviour of magnetostatic problems, with a -curl-type equation coming from the stationary Maxwell equations in magnetic media coupled with a nonstandard magnetic constitutive power law, which enables the explicit computation of the effective coefficients.

Increasing Stability for the Conductivity and Attenuation Coefficients

In this work we consider stability of recovery of the conductivity and attenuation coefficients of the stationary Maxwell and Schrodinger equations from a complete set of (Cauchy) boundary data. By using complex geometrical optics solutions we derive some bounds which can be viewed as an evidence of increasing stability in these inverse problems when the frequency is growing.

Interfacial deformation and jetting of a magnetic fluid

An attractive technique for forming and collecting aggregates of magnetic material at a liquid–air interface by an applied magnetic field gradient was recently proposed, and its underlying principle was studied theoretically and experimentally (Tsai et al., 2013): when the magnetic field is weak, the deflection of the liquid–air interface has a steady shape, while for sufficiently strong fields, the interface destabilizes and forms a jet that extracts magnetic material. Motivated by this work, we develop a numerical model for the closely related problem of solving two-phase Navier–Stokes equations coupled with the static Maxwell equations. We computationally model the forces generated by a magnetic field gradient produced by a permanent magnet and so determine the interfacial deflection of a magnetic fluid (a pure ferrofluid system) and the transition into a jet. We analyze the shape of the liquid–air interface during the deformation stage and the critical magnet distance for which the static interface transitions into a jet. We draw conclusions on the ability of our numerical model to predict the large interfacial deformation and the consequent jetting, free of fitting parameters.

Edge Element Methods for Maxwell's Equations with Strong Convergence for Gauss' Laws

In this paper we propose and investigate some edge element approximations for three Maxwell systems in three dimensions: the stationary Maxwell equations, the time-harmonic Maxwell equations and the time-dependent Maxwell equations. These approximations have three novel features. First, the resulting discrete edge element systems can be solved by some existing preconditioned solvers with optimal convergence rate independent of nite element meshes, including the stationary Maxwell equations. Second, they ensure the optimal strong convergence of the Gauss’ laws in some appropriate norm, in addition to the standard optimal convergence in energy-norm, under the general weak regularity assumptions that hold for both convex and non-convex polyhedral domains and for the discontinuous coecients that may have large jumps across the interfaces between dierent media. Finally, no saddle-point discrete systems are needed to solve for the stationary Maxwell equations, unlike most existing edge element schemes.

Another approach to the thin-film Γ-limit of the micromagnetic free energy in the regime of small samples

The asymptotic behavior of the micromagnetic free energy governing a ferromagnetic film is studied as its thickness gets smaller and smaller compared to its cross section. Here the static Maxwell equations are treated as a Murat’s constant-rank PDE constraint on the energy functional. In contrast to previous work, this approach allows us to keep track of the induced magnetic field without solving the magnetostatic equations. In particular, the mathematical results of Gioia and James [Proc. R. Soc. Lond. A 453 (1997), pp. 213–223] regarding convergence of minimizers are recovered by giving a characterization of the corresponding Γ \Gamma -limit.

A divergence-free finite element method for a type of 3D Maxwell equations

We seek a divergence-free finite element solution for the magnetic field governed by the static Maxwell equations. As usual, the solution is represented as a curl of a vector potential. Typically, this vector potential is uniquely defined in a divergence-free space. The novelty of our method is that we use some simple but non-divergence-free finite element spaces. In this way, the finite element vector potential does not approximate the divergence-free vector, but its curl is divergence-free and is exactly the same solution obtained by the divergence-free finite element potential. Computationally, the finite element solution for the magnetic field is obtained directly as a certain weighted L2-orthogonal projection within the divergence-free finite element subspace. Optimal order convergence is shown for the method. Numerical tests are provided.

Simulation of Magnetohydrodynamic and Thermal Coupling in the Linear Induction MHD Pump

This article is concerned with the study of a coupling between the stationary Maxwell equations, the transient state Navier Stokes and thermal equations. The model developed computes the magnetic field using the finite element method and calculates the velocity and the temperature using the finite volume method. The paper focuses on the analysis of the flux density, the electromagnetic thrust, the electric power density, the velocity, the pressure and the temperature in the channel of the MHD pump. Effect of the frequency is also presented.

Path integrals for light propagation in dielectric media

We develop a path integral approach for analyzing the stationary light propagation in general dielectric media. The Hermitian form of the stationary Maxwell equations is transformed into a quantum mechanical problem of a spin 1 particle with spin-orbit coupling and position dependent mass. After appropriate ordering several path integral representations of a solution are constructed. First we keep the propagation of the polarization degrees of freedom in an operator form integrated over paths in a coordinate space. The use of spin 1 coherent states allows representing this part as a path integral over such states. Finally a path integral in a transversal momentum space with explicit transversality enforced at every time slice is also given. As an example the geometrical optics limit is discussed and the ray equation is recovered together with the Rytov rotation of the polarization vector.

Two-sided a posteriori error bounds for electro-magnetostatic problems

This paper is concerned with the derivation of computable and guaranteed upper and lower bounds of the difference between exact and approximate solutions of a boundary value problem for static Maxwell equations. Our analysis is based upon purely functional argumentation and does not invoke specific properties of the approximation method. For this reason, the estimates derived in the paper at hand are applicable to any approximate solution that belongs to the corresponding energy space. Such estimates (also called error majorants of the functional type) have been derived earlier for elliptic problems. Bibliography: 24 titles.

Hodge decomposition to solve singular static Maxwell's equations in a non-convex polygon

We are concerned with the singular solution of the static Maxwell equation in a non-convex polygon. Thanks to a Hodge decomposition of the solution on a solenoidal and irrotational parts, one obtains an equivalent formulation to the static problem by solving two Laplace equations. Then a finite element formulation is derived, based on a Nitsche type method. This allows us to solve numerically the static Maxwell equation in domains with reentrant corners, where the solution can be singular. We formulate the method and report some numerical experiments. As a by product, this approach proves its ability to compute the dual singular functions of the Laplacian (see definition below).

Convergence analysis of an adaptive edge element method for Maxwell's equations

An efficient and reliable a posteriori error estimate is derived for solving three-dimensional static Maxwell's equations by using the edge elements of first family. Based on the a posteriori error estimates, an adaptive finite element method is constructed and its convergence is established. Compared with the existing results, an important advantage of the new theory lies in its feature that the usual marking of elements based on the oscillation is not needed in our adaptive algorithm, while the linear convergence of the algorithm can be still demonstrated in terms of the reduction of the energy-norm error and the oscillation. Numerical examples are provided which demonstrate the effectiveness and robustness of the adaptive methods.