A consistent approximation of the total perimeter functional for topology optimization algorithms

This article revolves around the total perimeter functional, one particular version of the perimeter of a shape [[EQUATION]] contained in a fixed computational domain D measuring the total area of its boundary [[EQUATION]], as opposed to its relative perimeter, which only takes into account the regions of [[EQUATION]] strictly inside [[EQUATION]]. We analyze approximate versions of the total perimeter which make sense for general ``density functions'' [[EQUATION]]. Their use in the context of density-based topology optimization is particularly convenient as they do not involve the gradient of the optimized function [[EQUATION]]. Two different constructions are proposed: while the first one involves the convolution of the function [[EQUATION]] with a smooth mollifier, the second one relies on the resolution of an elliptic boundary-value problem featuring Robin boundary conditions. The ``consistency'' of these approximations is appraised from various points of view. At first, we prove the pointwise convergence of our approximate functionals, then the convergence of their derivatives, as the level of smoothing tends to [[EQUATION]], when the considered density function [[EQUATION]] is the characteristic function of a shape [[EQUATION]]. Then, we focus on the [[EQUATION]]-convergence of the second type of approximate total perimeter functional. Several numerical examples are eventually presented.