PSAFE: A Computationally Efficient Semi-Analytical Finite Element Scheme to Model Periodic Waveguides

Abstract

We present a Semi-Analytical Finite Element (SAFE) formulation capable of han- dling structural systems with spatial periodicity along the wave propagation direction. This overcomes a fundamental limitation of existing SAFE schemes, which require translational invariance along the propagation direction. The key premise of our formu- lation, denoted PSAFE, is the assumption of a Bloch-form solution for the unknown displacement field within the waveguide. This transforms the problem into Fourier space, replacing the spatial coordinates with their Fourier series expansion counterparts. As a result, we obtain an infinite set of coupled equations governing the waveguide’s dispersion relations, each corresponding to a specific Fourier coefficient. To solve this problem numerically, we truncate the infinite series to a finite number of terms and reformulate the system as a linear eigenvalue problem. We demonstrate that accurate results can be obtained even with low truncation orders and that increasing the system’s dimensionality improves accuracy, albeit at the cost of higher computational expense.