Homogenization via Harmonic Analysis

Homogenization, which reduces the cost of numerical simulation in materials with repetitive structure, is a promising approach to the design of photonic crystals and metamaterials. But its mathematical foundations may seem daunting. A link with standard harmonic analysis is indicated here ("homogenization as approximate Fourier analysis for lattice group actions"), which should help clarify the issue. New applications, to high-frequency problems are suggested

Homogenization is the technique by which a complex composite material with crystal-like symmetry can be replaced, as far as numerical simulations are concerned, by an. equivalent hOIDoge:neous tnaterial. Tbe procedure consists in solving on tbejundan1:enta.l donain or ce.l.l that ge:nerates the composite by spatial tral1Siatiolls; a “cell: probleln”; ‘\vith tl:ppropriate pseudo-periodic boundary conditions; thus obtaining constitutive laws far tile equivalent Inate:rial. After tllis standard, computational Inetbods a.pply to the “Tbole structure, ‘\vith a. coarser Inesb, and. subseque:nt eco:nO:1TIy : Holnogelnza:tio:n is esp;ecial1y proll1i.sing for tbe study of m;etam;a.te.rials,. sucb as photonjc crystals and “negative ind, ex” materiaJs a bot topic nowadays [1]; “vllere it is solnetilnes presented as lIn alternative to another popular metllod:; the “Bloch waves” analysis [2]