Local and Homogenized Equations

Homogenization theory is concerned with finding the appropriate homogenized (or averaged, or macroscopic) governing partial differential equations describing physical processes occurring in heterogeneous materials when the length scale of the heterogeneities tends to zero. In such instances it is desired that the effects of the microstructure reside wholly in the macroscopic or effective properties via certain weighted averages of the microstructure. In its simplest form, the method is based on the consideration of two length scales: the macroscopic scale L, characterizing the extent of the system, and the microscopic scale -, associated with the heterogeneities. Moreover, it is supposed that some external field is applied that varies on a characteristic length scale W. If is comparable in magnitude to W or L, then one must employ a microscopic description, i.e., one cannot homogenize the equations. The limit of interest for purposes of homogenization is L ≥ W " -.