To proceed what we have introduced in “The first glance at homogenization“, this present note shall focus on the mathematical point of view of special domains in which the homogenization is conducted. Roughly speaking, we consider here the perforated domains.
Let be a bounded open set in with a smooth boundary . Let be the representative cell, i.e.
Let be an open subset of with a smooth boundary in such a way that . Setting and denoting by the set of all translated images of the form for and , we define the set, denoted by , of periodically distributed holes as follows:
Thus, the domain , called perforated, is defined by
Notice that its boundary is composed of two parts. The first part is denoted by describing the union of the boundaries of the holes strictly contained in . Mathematically, it is given by
Governed by this way, the second one is certainly its exterior boundary, denoted by , and given by
In order to reach our analysis in further notes, it suffices to make one of the following assumptions on the hole :
- is a smooth open set with a boundary;
- is locally on one side of , and is a union of a finite number of segments if or plane faces if .
In addition, the perforated domain is supposed to be connected. One important thing should be noticed here is that the second boundary is not necessarily smooth. In general, it may have angles, and the perforated one may not be locally on one side of this boundary.
Finally, we stop this brief note by showing some depicting figures closely related to practical applications. The first picture is usually seen in many building construction area, the perforated metal sheet while the second one describes the lower part of a wireless router, made of soft perforated plastic.