Electromagnetics
In this chapter we briefly study finite elements for electromagnetic applications. We start off by recapitulating Maxwell’s equations and look at some special cases of these, including the time harmonic and steady state case. Without further ado we then d
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Electromagnetics
Abstract In this chapter we briefly study finite elements for electromagnetic applications. We start off by recapitulating Maxwell’s equations and look at some special cases of these, including the time harmonic and steady state case. Without further ado we then discretize the time harmonic electric wave equation using Nédélec edge elements. The computer implementation of the resulting finite element method is discussed in detail, and a simple application involving scattering from a perfectly conducting cylinder is studied numerically. Next, we introduce the magnetostatic potential equation as model problem. Using this equation we study the basic properties of the curl-curl operator r r, which frequently occurs in electromagnetic problems. In connection to this we also discuss the Helmholtz decomposition and its importance for characterizing the Hilbert space H.curlI ˝/. The concept of a gauge is also discussed. For the mathematical analysis we reuse the theory of saddle-point problems and prove existence and uniqueness of the solution as well as derive both a priori and a posteriori error estimates.
13.1 Governing Equations Electromagnetism is the area of science describing how stationary and moving charges affect each other. Loosely speaking stationary charges give rise to electric vector fields, whereas moving charges (i.e., current) give rise to magnetic vector fields. The electric vector field is commonly denoted by E, and the magnetic vector field by B. These fields extend infinitely in space and time and models the force felt by a small particle of charge q traveling with velocity v. This so-called Lorentz force is given by F D q.E C v B/
(13.1)
and is used to define the E and B vectors at any given point and time.
M.G. Larson and F. Bengzon, The Finite Element Method: Theory, Implementation, and Applications, Texts in Computational Science and Engineering 10, DOI 10.1007/978-3-642-33287-6__13, © Springer-Verlag Berlin Heidelberg 2013
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13 Electromagnetics
13.1.1 Constitutive Equations Certain media called dielectric materials have charges trapped inside their atomic structure. Under the influence of an externally applied electric field E these charges are slightly dislocated and form dipoles. The resulting so-called dipole moment polarizes the material. This polarization can be thought of as a frozen-in electric field, and is described by a polarization field P . Because it may be hard to know anything about P beforehand, it is desirable to try to separate the effects of free and bound charges within the material. This can sometimes be done by working with the electric displacement field D, defined by D D 0 E C P with 0 the electric permittivity of free space (i.e., vacuum). In linear, isotropic, homogeneous materials P D 0 E, where is the electric susceptibility indicating the degree of polarization of the dielectric material. This yields the constitutive relation D D E
(13.2)
where D 0 .1 C /. In free space is a positive scalar, but for a general linear
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