A wave finite element-based approach for the modeling of periodic structures with local perturbations

• The wave finite element method is used for modeling periodic structures with local perturbations.
• A wave-based model reduction technique is proposed for fast forced response computation.
• The sensitivity of periodic structures to the occurrence of arbitrarily located perturbations is assessed.
• Robust design of periodic structures is analyzed.

The wave finite element (WFE) method is investigated to describe the dynamic behavior of finite-length periodic structures with local perturbations. The structures under concern are made up of identical substructures along a certain straight direction, but also contain several perturbed substructures whose material and geometric characteristics undergo arbitrary slight variations. Those substructures are described through finite element (FE) models in time-harmonic elasticity. Emphasis is on the development of a numerical tool which is fast and accurate for computing the related forced responses. To achieve this task, a model reduction technique is proposed which involves partitioning a whole periodic structure into one central structure surrounded by two unperturbed substructures, and considering perturbed parts which are composed of perturbed substructures surrounded by two unperturbed ones. In doing so, a few wave modes are only required for modeling the central periodic structure, outside the perturbed parts. For forced response computation purpose, a reduced wave-based matrix formulation is established which follows from the consideration of transfer matrices between the right and left sides of the perturbed parts. Numerical experiments are carried out on a periodic 2D structure with one or two perturbed substructures to validate the proposed approach in comparison with the FE method. Also, Monte Carlo (MC) simulations are performed with a view to assessing the sensitivity of a purely periodic structure to the occurrence of arbitrarily located perturbations. A strategy is finally proposed for improving the robustness of periodic structures. It involves artificially adding several “controlled” perturbations for lowering the sensitivity of the dynamic response to the occurrence of other uncontrolled perturbations.