Many optimal design problems are well-known to be ill-posed, leading to mesh-dependency and checkerboards, addressed heuristically through filtering techniques (Sigmund, 1997) where in the optimization algorithm, sensitivities are spatially averaged.
In (Bourdin, 2001), I proposed to apply the filter at the level of the equilibrium equation. The sensitivities for this problem are exactly that used in the original filtering algorithm (Sigmund, 1997), and one can prove existence of solutions, convergence of the finite element approximation and therefore mesh independence. As of today, this work has been cited over 350 times.
Optimal design of a cantilever beam, exhibiting mesh dependency (first row) and checkerboards (second row). Both of which can be addressed by using the filtering technique (third row).