Dimensional reduction of field problems in a differential-forms framework

ISSN： 0332-1649

Source： COMPEL: Int J for Computation and Maths. in Electrical and Electronic Eng., Vol.28, Iss.4, 2009-07, pp. : 907-921

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Abstract

Purpose ‐ The purpose of this paper is to present a geometric approach to the problem of dimensional reduction. To derive (3?+?1) D formulations of 4D field problems in the relativistic theory of electromagnetism, as well as 2D formulations of 3D field problems with continuous symmetries. Design/methodology/approach ‐ The framework of differential-form calculus on manifolds is used. The formalism can thus be applied in arbitrary dimension, and with Minkowskian or Euclidean metrics alike. Findings ‐ The splitting of operators leads to dimensionally reduced versions of Maxwell's equations and constitutive laws. In the metric-incompatible case, the decomposition of the Hodge operator yields additional terms that can be treated like a magnetization and polarization of empty space. With this concept, the authors are able to solve Schiff's paradox without use of coordinates. Practical implications ‐ The present formalism can be used to generate concise formulations of complex field problems. The differential-form formulation can be readily translated into the language of discrete fields and operators, and is thus amenable to numerical field calculation. Originality/value ‐ The approach is an evolution of recent work, striving for a generalization of different approaches, and deliberately avoiding a mix of paradigms.