A simple approach to deal with zero densities in topology optimisation

Topology optimisation techniques typically use a very small positive density $$\varepsilon $$to model voids. Despite its simplicity and generally acceptable results, this approach can impose a number of difficulties. The weak material should be weak enough to validate the approximation of void areas, but on the other hand, using a very weak material can result in ill-conditioning of the stiffness matrix. Further and more serious complications can arise, for example in non-linear problems where weak elements cause numerical instabilities in the solution procedure. By studying the mechanical responses of structures when $$\varepsilon \rightarrow 0$$, this paper presents a simple approach to use arbitrarily weak material properties in void areas. This approach would effectively allow us to actually remove the void areas from the mesh in a range of problems and avoid the above-mentioned complexities.