Phononic band gap optimization in truss-like cellular structures using smooth P-norm approximations

The emergence of additive manufacturing and the advances in structural optimization have boosted the development of tailored cellular materials. These modern materials with complex architectures show higher structural efficiency when compared to traditional materials. In particular, truss-like cellular structures show great potential to be applied in lightweight applications due to their large strength/stiffness to mass ratio. Besides lightweight, these materials may exhibit incredible vibration isolation properties known as phononic band gaps. The present investigation addresses the topology optimization of two-dimensional (2D) truss-like cellular structures. The formulation aims to find the optimal geometrical and mechanical properties of each truss element to create a material exhibiting outstanding vibration (elastic wave) isolation at a certain frequency range (band gap). A new method to handle the non-differentiation of repeated eigenvalues, as well as mode switching, is proposed, where P-norms are used to create continuous approximation for extreme frequency values for all wave vectors of the band diagram. Results show that the proposed formulation is effective and avoids convergence problems associated to mode switching and to repeated eigenvalues.

Automated summary

The emergence of additive manufacturing and advances in structural optimization have led to the development of tailored cellular materials with complex architectures that show higher structural efficiency compared to traditional materials. Truss-like cellular structures, in particular, have great potential for lightweight applications due to their high strength/stiffness to mass ratio. These materials may also exhibit impressive vibration isolation properties known as phononic band gaps. The study focuses on the topology optimization of 2D truss-like cellular structures to create a material with outstanding vibration isolation at specific frequency ranges. The proposed formulation effectively addresses convergence problems associated with mode switching and repeated eigenvalues.