Asymptotic homogenization with a macroscale variation in the microscale

Abstract

Asymptotic homogenization is a useful mathematical tool that can be used to reduce the complexity of a problem with a periodic geometry. Generally, for asymptotic homogenization to be applicable, the full problem must have: (i) a periodic microstructure and (ii) a small ratio between the typical lengths of the periodic cell and the macroscale variation. In this chapter we consider a model for drug delivery. Namely, we discuss an asymptotic homogenization for the concentration field of a drug diffusing within a domain that contains a near-periodic array of circular obstructions whose boundaries can absorb the drug. In particular, the radii of these circular obstructions can slowly vary in space, and thus the microscale geometry varies in the macroscale. Constraining the shape of the obstacles to a one-parameter family, where the only variation is circle radius, allows us to homogenize this problem in a computationally efficient manner. Moreover, the method we present allows us to determine the homogenized equation for any arrangement of the microstructure within the one-parameter constraint.