Topology Optimization of Phononic Crystal with Prescribed Band Gaps

Phononic crystal, which can block the propagation of elastic waves in a certain frequency range, is widely used in acoustic devices, vibration reduction, and other fields. However, most of the previous studies on the design of phononic crystals only consider the problem of the maximum band gap of adjacent two orders and cannot give the design for a prescribed band gap, which is more applicability in practical engineering problems. To solve this issue, this study proposes a novel topology optimization method to realize the prescribed band gaps design of phononic crystal. Compared to the vibration structural design with prescribed band gaps, the difficulty of phononic crystals’ prescribed band gaps design lies in the correlation between stiffness matrix and wave vector, which is a typical eigenvalue problem with multi-class boundary conditions. Directly utilizing the frequency band constraints proposed by the authors in the previous papers may lead to numerical convergence difficulties. Thus an ipsilateral frequency constraint based on a modified Heaviside function is further proposed. Since the developed function is continuous and differentiable, the sensitivity of the constraint function can be derived for using the gradient-based optimization solver. Furthermore, in order to eliminate the checkerboard phenomena, grey elements and give the minimum size control in topology optimization, the “Robust formulation” is applied. Several numerical examples, including single-band gap and double-band gaps problems, are solved by the proposed method. The results validate the effectiveness of the developed method.