| |
In many engineering applications the interaction between the electromagnetic field and moving bodies is of great interest. It is natural to use a Lagrangian description, where the unknowns are defined on a mesh which moves and deforms together with the considered objects. What is the correct form of Maxwell’s equations and the material laws under such circumstances? The aim of the present paper is to tackle this question by using the language of differential forms. We first provide a review of the formulations of electrodynamics in terms of vector fields, as well as differential forms in the (1+3)- and four-dimensional setting. In order to keep both Maxwell’s and the constitutive equations as simple as possible, we set up two reference frames. In the natural material frame, the (1+3)-Maxwell’s equations have their simple form, whereas in the co-moving inertial frame, the material laws are canonical. In contrast to existing literature these frames are both retained to benefit from their individual advantages. It remains to construct transformation laws connecting the considered frames. To achieve this, we use a (1+3)-decomposition in terms of general projection operators which do primarily not depend on an underlying metric or on the choice of a spatial coordinate system [1]. The desired transformation laws are established by comparing the different decompositions of an arbitrary p-form with respect to the considered frames. We provide an interpretation in terms of vector fields, and consider the low frequency limit, which is the most relevant case for an implementation into numerical codes. For the description of low frequency electromagnetism, all rigid frames are equivalent. This goes beyond the standard principle of Galilean relativity, where only inertial frames are regarded as equivalent. The proper treatment in the general case is demonstrated by means of an example in rotating coordinates, where the classical paradox by Schiff [2] is resolved.
[1] F. Hehl and Y. Obukhov, Foundations of Classical Electrodynamics. Boston: Birkhäuser, 2003.
[2] L. Schiff, "A question in general relativity," Proc. Nat. Acad. Sci. USA, vol. 25, 1939.
|
|
| |
The simulation of pulsed superconducting magnets has gained importance at the verge of fast-ramping cyclotron projects. The ROXIE program has been devised for the design and optimization of superconducting magnets. The 2-D electromagnetic model of a fast-ramping magnet in ROXIE consists of- a representation of strands by line currents,
- a coupling of the finite element method and the boundary element method to take into account the field contribution of the magnet yoke, as well as eddy-current effects in conductive bulk material,
- a model for persistent currents,
- a model for inter-filament coupling currents, and
- a model for inter-strand coupling currents in Rutherford-type cables.
We will present the coupling of all these effects in the mathematical framework of the theory of discrete electromagnetism. We will then proceed to demonstrate how the coupled approach helps to understand a pulsed magnet's behavior. Each of the above effects leaves an identifiable signature in the measured field quality and contributes to the losses. With ROXIE, we can trace measurements to their origin and make predictions based on experience and simulation.
|
|
