Optimization of Explicit Runge–Kutta–Nyström (RKN) Schemes to Minimize Spurious Reflections From Shock Wave Propagation on Nonuniform Grids

ABSTRACT The propagation of shock waves on nonuniform grids appears, for example, in some impact problems of the fast‐dynamics type, where a finer mesh can typically be employed in regions of interest, leading to mesh ratios that can range from 10 to 20 or even more. Unfortunately, unwanted spurious reflections occur for this type of problem using (standard) explicit finite element (FE) software. An optimized explicit time‐integration scheme is therefore built to minimize spurious wave reflections for shock wave propagation on nonuniform grids. Specifically, we consider RKN (Runge–Kutta–Nyström) time‐integration methods and optimize a parameter appearing in the and schemes by minimizing the strain error that occurs when a shock wave passes through the mesh break. The effectiveness of the proposed approach is highlighted using some discrete indicators in a one‐dimensional (1D) academic test case. Moreover, our method is compared to Hulbert and Chung's and Noh and Bathe's time‐integration schemes using a two‐dimensional (2D) test case from the literature. All schemes exhibited merits in mitigating spurious reflections of shock waves propagating on nonuniform grids. However, the proposed optimized was the most efficient one.