Upper and lower bounds in eddy-current calculations

Abstract

After reviewing the arguments for and against the existence of bounded solutions, the paper introduces a constrained variation which produces bounds for the electric and magnetic energies. It is shown that this principle has a geometrical foundation which can be used irrespective of the shape of the body in which the eddycurrents flow. The basis of the method encourages the expectation of economy in computation.

Calculation of Machine-Slot Parameters by Complementary Variational Technique

“…For conciseness, comparisons are made in terms of resistance and inductance estimates. Once the finite element field distributions are determined, the H-system parameter and [10]. However, on closer examination, it will be seen that the complementary estimates cross-over at low frequency: at 5 Hz for resistance, and 5-10 Hz for inductance.…”

“…The difference between 'neck' lengths (dimension c in Table 1) for the two conductors probably accounts for the apparent difference in behaviour of their respective curves [2,7,10]. The depth of penetration, or skin depth, is 31.8 rom at 5 Hz, so that it is well within the neck of the first conductor over most of the frequency range considered; the problem is thus essentially one-dimensional.…”

“…Thus, Ligurian minimization through BA=O (10) enforces the constitutive relationships on the trial variables for the fields. If the latter are preconstrained to satisfy the remaining governing equations (3)-(6), each within its own system, then eqn.…”

“…The necessary functionals can be derived in a number of ways [7][8][9][10][11][12]. Here we adopt an approach based on an error form, the Ligurian [11], that quantifies mismatch in constitutive relationships.…”

“…For the eddy-current problem, on the other hand, claims of boundedness [2,[7][8][9][10] have not been adequately supported in theory or application. A general error-based derivation of complementary formulation [11] establishes the absence of bounds on theoretical grounds.…”