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1.2 The Finite Volume Method for Electromagnetics

by Jose Alves

Static and Quasi-static Quasi-static analyses focus on computing fields produced by slowly varying electric currents. In those situations, the temporal change of the sources (currents) is so slow that the displacement current in the Maxwell equation can be safely ignored. In particular, Finite Volume Methods (FVM) methods can be useful for quasi-static analyses where the magnetic permeability of the involved materials is uniform in space, or does not exhibit large discontinuities at interfaces between different regions. This requirement implies that FV methods cannot accurately model ferromagnetic materials, in which the magnetic permeability is not only non-linear but exhibiting large values, e.g. orders of magnitude respect to surrounding air/vacuum.

There are different applications honouring the above-mentioned constraints, typically in areas like power generation and transmission, switch gears and circuit breakers, cables and lines, plasma applications (torches, thermal spraying, arc welding) and induction heating.

In particular in multiphysics applications solved with a co-simulation approach, the main advantage of using FVM methods (when numerically allowed, see above) is that the electromagnetic computation can carried out on meshes designed for Computational Fluid Dynamics (CFD) purposes, which traditionally tend to use FV methods. Using the same mesh as in CFD, one can significantly reduce errors due to data mapping between a FVM-optimised mesh (usually a polyhedral mesh for CFD) and a FEM-optimised mesh (usually tetrahedral for computational electromagnetics). Moreover, co-simulation usability and run-time are kept under control.

An example of the above is Magnetohydrodynamics (MHD) simulations, where a conducting fluid responds to Ohmic heating and Lorentz forces generated by the electric currents flowing within the fluid. In this kind of simulation, FE methods may struggle to accurately include the motion-induced electromotive vxB term, typically found in MHD (where v is the fluid velocity field and B the magnetic induction field). This is due to the fact that the velocity field - typically solved by a FVM strategy - can be rather arbitrary and the resulting vxB term could lack the regularity that some FEM solvers require as some FEM solvers accept only divergence-free sources.

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