A polynomial-based method for topology optimization of phononic crystals with unknown-but-bounded parameters

The design and analysis of phononic crystals (PnCs) are generally based on the deterministic models without considering the effects of uncertainties. However, uncertainties that existed in PnCs may have a nontrivial impact on their band structure characteristics. In this paper, a sparse point sampling-based Chebyshev polynomial expansion (SPSCPE) method is proposed to estimate the extreme bounds of the band structures of PnCs. In the SPSCPE, the interval model is introduced to handle the unknown-but-bounded parameters. Then, the sparse point sampling scheme and the finite element method are used to calculate the coefficients of the Chebyshev polynomial expansion. After that, the SPSCPE method is applied for the band structure analysis of PnCs. Meanwhile, the checkerboard and hinge phenomena are eliminated by the hybrid discretization model. In the end, the genetic algorithm is introduced for the topology optimization of PnCs with unknown-but-bounded parameters. The specific frequency constraint is considered. Two numerical examples are investigated to demonstrate the effectiveness of the proposed method.