Boundary perturbation methods for high-frequency acoustic scattering: shallow periodic gratings

Despite significant recent advances in numerical methodologies for simulating rough-surface acoustic scattering, their applicability has been constrained by the limitations of state-of-the-art computational resources. This has been particularly true in high-frequency applications where the sheer size of the full-wave simulations render them impractical, and engineering processes must therefore rely on asymptotic models [e.g., Kirchhoff approximation (KA)]. However, the demands for high precision can make the latter inappropriate, thus efficient, error-controllable methodologies must be devised. This paper presents a computational strategy that combines the virtues of rigorous solvers (error control) with those of high-frequency asymptotic models (frequency-independent computational costs). These methods are based on high-order "boundary perturbations," which display high precision and unparalleled efficiency. This is accomplished by incorporating asymptotic phase information to effect a significant decrease in computational effort, simultaneously retaining the full-wave nature of the approach. The developments of this contribution are constrained to configurations that preclude multiple scattering; it is further explained how the schemes can be made applicable to general scattering scenarios, though implementation details are left for future work. Even for single-scattering configurations, the approach presented here gives significant gains in accuracy when compared to asymptotic theories (e.g., KA) with modest additional computational cost.