A phase-field based robust topology optimization method for phononic crystals design considering uncertain diffuse regions

Uncertain manufacturing errors in fabricating phononic crystals (PnCs) may notably affect their band gap properties. One thus needs to consider uncertainties in the design and optimization of microstructures formed from the unit cell of a PnC. This study focuses on proposing a robust topology optimization method of PnC microstructures against random diffuse regions between material phases. The material distribution of the unit cell is performed using the phase-field method, which is able to simulate the motion and the uncertain width of the diffuse regions. A stochastic band-gap analysis is performed using the expansion optimal linear estimation method and the polynomial chaos expansion (PCE)-based framework. The objective function of the robust optimization formulation is taken as the weighted sum of the mean value and standard deviation of the specified band gap. With the sensitivities of the stochastic band gap obtained by the nonintrusive PCE method, the phase-field function is updated with the Allen–Cahn equation. Numerical examples show that the proposed method provides meaningful optimal designs of microstructures of the unit cell in achieving a robust band gap.