We consider general shape optimization problems governed by Dirichlet boundary value problems. The proposed approach may be extended to other boundary conditions as well. It is based on a recent representation result for implicitly defined manifolds, due to the authors, and it is formulated as an optimal control problem. The discretized approximating problem is introduced and we give an explicit construction of the associated discrete gradient. Some numerical examples are also indicated.
"Topological optimization via cost penalization." Topol. Methods Nonlinear Anal. 54 (2B) 1023 - 1050, 2019. https://doi.org/10.12775/TMNA.2019.080