On the solution of the static Maxwell system in axially symmetric inhomogeneous media (original) (raw)

Analytic solutions of electromagnetic fields in inhomogeneous media

The International Journal of Electrical Engineering & Education, 2015

We present guidelines for teaching students how to analytically solve problems that involve inhomogeneous media in electrostatic fields, stationary current fields, and stationary magnetic fields. At the introductory level, the focus is on recognizing classes of problems that can be solved in closed form and applying simple rules, based on comparison with solutions in homogeneous media. At the intermediate level, the focus is on strict proofs based on vector calculus.

Solution of axisymmetric Maxwell equations

Mathematical Methods in the Applied Sciences, 2003

In this article, we study the static and time-dependent Maxwell equations in axisymmetric geometry. Using the mathematical tools introduced in (Math. Meth. Appl. Sci. 2002; 25:49), we investigate the decoupled problems induced in a meridian half-plane, and the splitting of the solution in a regular part and a singular part, the former being in the Sobolev space H 1 component-wise. It is proven that the singular parts are related to singularities of Laplace-like or wave-like operators. We infer from these characterizations: (i) the ÿnite dimension of the space of singular ÿelds; (ii) global space and spacetime regularity results for the electromagnetic ÿeld. This paper is the continuation of (Modà el. Math.

On the Uniqueness of a Solution to Anisotropic Maxwell’s Equations

Topics in Operator Theory, 2010

In the present paper we consider Maxwell's equations in an anisotropic media, when the dielectric permittivity ε and the magnetic permeability µ are 3 × 3 matrices. We formulate relevant boundary value problems, investigate a fundamental solution and find a Silver-Müller type radiation condition at infinity which ensures the uniqueness of solutions when permittivity and permeability matrices are real valued, symmetric, positive definite and proportional ε = κµ, κ > 0.

Symmetries and modes of wave fields in inhomogeneous media

Journal of Physics A: Mathematical and General, 2000

In this paper the 2D Helmholtz equation with space-dependent dielectric permittivity and/or magnetic permeability is considered. The structure of the determining equations for the symmetry generator is analysed. It allows one to find many nontrivial spatial profiles of dielectric functions for which one can still separate variables in both the Helmholtz equation and corresponding Hamilton-Jacobi equation. For the case of the Dirichlet problem of the half-plane in an inhomogeneous medium the diffractive modes of the field are characterized. Several specific examples are provided.

On the solution of Maxwell's equations in axisymmetric domains with edges

ZAMM, 2005

In this paper we present the basic mathematical tools for treating boundary value problems for the Maxwell equations in three-dimensional axisymmetric domains with reentrant edges using the Fourier-finite-element method. We consider both the classical and the regularized time-harmonic Maxwell equations subject to perfect conductor boundary conditions. The partial Fourier decomposition reduces the three-dimensional boundary value problem into an infinite sequence of twodimensional boundary value problems in the plane meridian domain of the axsiymmetric domain. Here, suitable weighted Sobolev spaces that characterize the solutions of the reduced problems are given, and their trace properties on the rotational axis are proved. In these spaces, it is proved that the reduced problems are well posed, and the asymptotic behavior of the solutions near reentrant corners of the meridian domain is explicitly described by suitable singularity functions. Finally, a finite number of the two-dimensional problems is considered and treated using H 1-conforming finite elements. An approximation of the solution of the three-dimensional problem is obtained by Fourier synthesis. For domains with reentrant edges, the singular field method is employed to compensate the singular behavior of the solutions of the reduced problems. Emphases are given to convergence analysis of the combined approximations in H 1 under different regularity assumptions on the solution.

Method for solution of Maxwell's equations in non-uniform media

It is shown that the L_2 norm of electric currents induced in a dissipative medium can never exceed the norm of the external currents. This allows the construction of a simple iteration method to solve the Maxwell's equations. The method produces a series converging to the solution for an arbitrary conductivity distribution and arbitrary frequency of field variations. The convergence is slow if the lateral contrast of the conductivity distribution is about 10^4 or higher. A modification significantly improving the convergence is described in this paper. As an example, electromagnetic fields induced in the model (including the western part of the Northern American continent and the adjacent part of the Pacific Ocean) are calculated.

Solution of Maxwell's equations

Computer Physics Communications, 1992

A numerical approach for the solution of Maxwell's equations is presented. Based on a finite difference Yee lattice the method transforms each of the four Maxwell equations into an equivalent matrix expression that can be subsequently treated by matrix mathematics and suitable numerical methods for solving matrix problems. The algorithm, although derived from integral equations, can be consideredto be a special case of finite difference formalisms. A large variety of two-and three-dimensional field problems can be solved by computer programs based on this approach: electrostatics and magnetostatics, low-frequency eddy currents in solid and laminated iron cores, high-frequency modes in resonators, waves on dielectric or metallic waveguides, transient fields of antennas and waveguide transitions, transient fields of free-moving bunches of charged particles etc.

Solving Numerically the Static Maxwell Equations in an Axisymmetric Singular Geometry

Mathematical Modelling and Analysis, 2015

We propose a new numerical method to compute the singular solution of the Maxwell equations in axisymmetric domains, as for example in non convex polygonal domains. As geometrical singularities are mainly related to the space dependent part of the model, we focus on the static field computation. We then introduce a new approach, that consists in decomposing the domain into two or more subdomains, and to derive an ad hoc variational formulation in each subdomain. The interface conditions are then imposed with a method deduced from a Nitsche method coupled with a specific “exchange” approach. An advantage of this domain decomposition method is that it does not require neither overlapping nor iteration process. Another advantage is that no particular mesh refinement is needed near the geometrical singularities. Numerical examples will be shown.