A new approach based on spectral graph theory to avoiding enclosed holes in topology optimization

It is well-known that when maximizing the stiffness in structural design, several holes often appear distributed throughout the structure, leading to a distribution of the void phase that is not connected. Inspired by known results of spectral graph theory, this work proposes a new idea to prevent the appearance of those internal or encapsulated holes in topology optimized structures. The approach is density-based, where some sort of eigenvalue problem is solved in addition to the governing elasticity problem in each step of the iterative process. By means of the auxiliary eigenproblem, void phase connectivity may be imposed by constraining the second smallest eigenvalue, invariant known as algebraic connectivity in the context of graphs. Several examples in 2d and in 3d for minimum compliance are shown to corroborate our approach, thereby providing efficient performance together with improved manufacturability.