High-frequency homogenization for periodic media

Abstract

An asymptotic procedure based upon a two-scale approach is developed for wave propagation in a doubly periodic inhomogeneous medium with a characteristic length scale of microstructure far less than that of the macrostructure. In periodic media, there are frequencies for which standing waves, periodic with the period or double period of the cell, on the microscale emerge. These frequencies do not belong to the low-frequency range of validity covered by the classical homogenization theory, which motivates our use of the term 'high-frequency homogenization' when perturbing about these standing waves. The resulting long-wave equations are deduced only explicitly dependent upon the macroscale, with the microscale represented by integral quantities. These equations accurately reproduce the behaviour of the Bloch mode spectrum near the edges of the Brillouin zone, hence yielding an explicit way for homogenizing periodic media in the vicinity of 'cell resonances'. The similarity of such model equations to high-frequency long wavelength asymptotics, for homogeneous acoustic and elastic waveguides, valid in the vicinities of thickness resonances is emphasized. Several illustrative examples are considered and show the efficacy of the developed techniques.

Long-wave asymptotic theories: The connection between functionally graded waveguides and periodic media

“…2(a). This is the analogous case to the piecewise periodic string treated in [14] for which the assumption is that Floquet-Bloch conditions, u(1) = exp(i2κǫ)û(−1) and u ξ (1) = exp(i2κǫ)û ξ (−1) with 2ǫ being the relative width of the elementary cell, hold and the Kronig-Penney dispersion relation 2r[cos Ω cos rΩ − cos 2ǫκ] − (1 + r 2 ) sin Ω sin rΩ = 0 (6) follows relating the Bloch wavenumber, κ, to the frequency Ω. It is clear that the wave numbers in (5) and (6) are linked as k = ǫκ.…”

“…As in section 2.1, we adapt the ansatz (7) over the high-frequency domain, resulting in (14,15) and also …”

“…By the solvability condition for the first-order term, we now obtain two coupled ODEs for f (1,2) 0 from which the linear eigenvalue correction Ω 2 1 is derived, [14] ,…”

Elastic metamaterials and dynamic homogenization: a review

“…Asymptotic homogenization was also limited in application to describing only the behavior of the fundamental Bloch mode at low frequencies [72,73]. Recently, however, it has been extended to higher frequencies and higher Bloch modes [74][75][76][77] (see also [78]). Additionally, there have been other efforts to bridge the scales for the study of dispersive systems based upon variational formulations [79], micromechanical techniques [80], Fourier transform of the elastodynamic equations [81], and strain projection methods [82].…”

Evanescent wave boundary layers in metamaterials and sidestepping them through a variational approach

“…Currently, there are two main ways of doing this. The first is based upon asymptotic methods [8][9][10][11][12][13][14][15][16] and the second is based upon field averaging methods [17][18][19][20][21][22][23][24][25][26][27]. In addition to these, there are also scattering measurements based analytical and experimental techniques.…”