Error-based derivation of complementary formulations for the eddy-current problem

Some key developments in computational electromagnetics and their attribution

“…This gives rise to dual energy formulations. There have been several contributions in this area and one should mention in particular works by Hammond and Penman [46], Rikabi [47], and Tonti [48]. Variational methods may be applied, based on the equilibrium energy, providing either a minimum or-through a dual-complementary principle-a maximum for the system energy.…”

An Energy Based Approach of Electromagnetism Applied to Adaptive Meshing and Error Criteria

“…• [6] is an extension of [5] to eddy-current problems, still using complementary formulations; • [7] proposes two error estimators, one based on the principle of the Ligurian and the other based on the numerical discontinuity of the field; • [8] uses a local solution in the conducting region; • [9] uses an error estimator based on the precision on the Mawxell-Ampère equation. However, adaptive meshing is not widely spread for timevarying cases.…”

“…The new error criterion proposed evaluates the local conservation of the power in each element: (6) After some calculation, using the Green theorem, the error reads:…”

“…The criterion allows to build an error map that can drive the bubble mesh generator. (6) in the conducting region for the first mesh (top) and for the refined one (bottom). The local error is divided by 10.…”

Upper and lower bounds in eddy-current calculations

“…In an early paper written with Dr. J. Penman [5], I put forward the view that bounded values for the equivalent resistance and reactance could be obtained, and this view was restated and amplified in a book [6]. More recently, Ferrari [7] has given support to this view, in general terms, by stating that bounded values should be obtainable, but Rikabi* [8] has expressed doubts about the existence of bounded solutions at least as far as his own variational procedure is concerned. Penman and Fraser give dual bounds for static but not for eddy-current problems [9].…”