  In density-based topology optimization, design variables associated to the
boundaries of the design domain require unique treatment to negate boundary
effects arising from the filtering technique. An effective approach to deal
with filtering boundary conditions is to extend the design domain beyond the
borders using void densities. Although the approach is easy to implement, it
introduces extra computational cost required to store the additional Finite
Elements (FEs). This work proposes a numerical technique for the density filter
that emulates an extension of the design domain, thus, it avoids boundary
effects without demanding additional computational cost, besides it is very
simple to implement on both regular and irregular meshes. The numerical scheme
is demonstrated using the compliance minimization problem on two-dimensional
design domains. In addition, this article presents a discussion regarding the
use of a real extension of the design domain, where the Finite Element Analysis
(FEA) and volume restriction are influenced. Through a quantitative study, it
is shown that affecting FEA or volume restriction with an extension of the FE
mesh promotes the disconnection of material from the boundaries of the design
domain, which is interpreted as a numerical instability promoting convergence
towards local optimums.
