Comparative study of numerical explicit time integration algorithms

Abstract

The aim of the present research is to make a comparative study of numerical explicit time integration algorithms used in the domain of shock and impact. Numerical simulation of such problems, with explicit algorithms for time integration, involves minute time steps for reasons of stability. Consequently, due to spatial discretization, very high numerical frequencies are found in the final solution in displacement or stress. Usually, the high frequencies and mode shapes of the spatially discretized equations do not accurately represent the behavior of the original problem. It has been proven that algorithms such as these of Chung–Lee, Zhai, HHT, Tchamwa–Wielgosz and the central difference method, are useful in solving problems including high-speed phenomena. The consistency, the different regions of stability, but also the ability of each numerical scheme to smooth very high frequencies are compared. Finally, the integration schemes are implemented in the HEREZH++ finite element code developed at the LG2M laboratory [ http//www-lg2m.univ-ubs.fr]. For some simple problems, the solutions obtained from HEREZH++ with these different schemes are compared with continuous and discretized theoretical solutions. Conclusions are independent of the problems and can be applied to more complex geometry. For instance, following these simulations, it seems that the Tchamwa–Wielgosz and the Chung–Lee algorithms are particularly efficient in smoothing the highest frequencies.