On the Optimization of Band Gaps in Periodic Waveguides

Abstract Purpose This work applies a computational framework for vibration attenuation in periodic structures by combining the established wave and finite element (WFE) method with nature-inspired optimization algorithms. The purpose of this work is to provide a systematic comparison of various nature-inspired algorithms across both low-cost analytical and high-fidelity simulation-driven design problems. Methodology Two numerical examples are investigated. The first considers a semi-infinite beam with periodic masses, where the dynamic stiffness matrix is obtained analytically. Two optimization scenarios are addressed within the first example. The second example involves a periodic beam with alternating thick and thin segments, modeled using WFE, where the objective is to minimize transmissibility over a lower frequency range. Five nature-inspired optimization algorithms: Genetic Algorithm (GA), Differential Evolution (DE), Grey Wolf Optimizer (GWO), Improved Grey Wolf Optimizer (IGWO), and Particle Swarm Optimization (PSO) are compared. In the first example, a fixed number of function evaluations ensures fair comparison; in the second, equal runtime account for the high computational cost of each function evaluation. Results In the first example, all algorithms achieved similar optimal solutions in the two optimization scenarios, with differences arising primarily in computational efficiency. For the first optimization scenario, distribution-free analysis showed that at intermediate function evaluation budgets, detectable differences emerge among algorithms, whereas in the second scenario, these differences diminish at higher evaluation budgets (with no significant pairwise contrasts), indicating convergence. GA incurred the highest computational overhead; DE was fast but tended to plateau while GWO, IGWO, and PSO exhibited strong accuracy-efficiency trade-offs within the tested budgets. In the second example, which operated under fixed runtime budgets, Kruskal–Wallis tests indicated significant differences at all time budgets: PSO consistently attained the lowest transmissibility, with GWO and IGWO close behind, GA showing modest improvement with longer runtime, and DE stagnating early. Across both cases, the effect-size analysis (Cliff’s $$\delta$$ ) confirms that observed gaps are generally small to medium at moderate budgets and tend to narrow at higher budgets as solutions converge. This highlights that the efficiency with which evaluations are translated into meaningful search progress is just as critical as raw accuracy in expensive, simulation-driven problems. The WFE-based framework therefore provides a general and effective tool for optimizing periodic structures for targeted vibration attenuation.