Dual magnetodynamic finite element formulations with natural definitions
of global quantities for electric circuit coupling *
Both general families of magnetodynamic
formulations, i.e. b- and h-conform formulations, are intended to be studied
in the frame of the finite element method, as well in 2D as in 3D. These
dual formulations are respectively weak forms of the Ampere and Faraday
equations, and respectively use unknowns directly associated with the magnetic
flux density b (e.g. magnetic vector potential a-formulations) and with
the magnetic field h (e.g. magnetic field - magnetic scalar potential h-formulations).
When such formulations are applied to systems coupled with electric circuits,
not only local quantities, characterizing the unknown field, are involved,
but also electric global quantities, i.e. currents and voltages. The physical
conducting regions with which these global quantities are associated can
be of massive or stranded types, each type necessitating a particular treatment
depending on the formulation. The mathematical and numerical tools for
naturally coupling local and global quantities will be studied for all
these variants. The results of this coupling are circuit relations characterizing
the conducting regions. The developed method uses edge finite elements
and benefit from their properties to define currents and voltages in strong
or weak senses, in accordance with the considered weak formulations, i.e.
with no additional approximation. For that, in some cases, they make use
of well defined source fields as mathematical tools. In particular, when
dealing with a stranded conductor, a h-formulation needs a source magnetic
field. The same kind of source field can also be used in a-formulations.
Other source fields, source electric scalar potentials, are proposed for
massive conductors in a-formulations. A global function of another type
is used in h-formulations for massive conductors. Advantages of such dual
formulations will be pointed out, in particular concerning the accuracy
obtained on local and global solutions and circuit parameters. The use
of both formulations can constitute a way to estimate the error on finite
element analyses with electric circuit coupling. The consistent approximations
proposed for circuit relations appear to be necessary conditions for correct
interpretations of the obtained results. A software environment allowing
various kinds of coupling has been developed. It will be shown to be particularly
well adapted for the multiple analysis undertaken as well for researchers
to develop various convenient methods as for users from industry to apply
them with a good level of generality.
* Research results of the Belgian programme on
Interuniversity Poles of Attraction initiated by the Belgian State, Prime
Minister's Office, Science
Policy Programming.
** Research Associate with the Belgian National
Fund for Scientific Research (F.N.R.S.).
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