From Whitney Forms to Metamaterials: a Rigorous Homogenization Theory

TL;DR

This paper develops a rigorous homogenization theory for metamaterials using Whitney forms, enabling precise effective parameter derivation and error analysis, with broad applicability beyond electromagnetics.

Contribution

It introduces a mathematically rigorous framework combining Whitney forms with physical insights, improving accuracy and clarity in homogenization of metamaterials.

Findings

The theory aligns with classical results like Maxwell-Garnett formula.

It clearly identifies sources of approximation errors.

The approach is applicable to acoustics and elasticity.

Abstract

A rigorous homogenization theory of metamaterials -- artificial periodic structures judiciously designed to control the propagation of electromagnetic waves -- is developed. All coarse-grained fields are unambiguously defined and effective parameters are then derived without any heuristic assumptions. The theory is an amalgamation of two concepts: Smith & Pendry's physical insight into field averaging and the mathematical framework of Whitney-Nedelec-Bossavit-Kotiuga interpolation. All coarse-grained fields are defined via Whitney forms and satisfy Maxwell's equations exactly. The new approach is illustrated with several analytical and numerical examples and agrees well with the established results (e.g. the Maxwell-Garnett formula and the zero cell-size limit) within the range of applicability of the latter. The sources of approximation error and the respective suitable error indicators are clearly identified, along with systematic routes for improving the accuracy further. The proposed approach should be applicable in areas beyond metamaterials and electromagnetic waves -- e.g. in acoustics and elasticity.