  Simulating physical problems involving multi-time scale coupling is
challenging due to the need of solving these multi-time scale processes
simultaneously. In response to this challenge, this paper proposed an explicit
multi-time step algorithm coupled with a solid dynamic relaxation scheme. The
explicit scheme simplifies the equation system in contrast to the implicit
scheme, while the multi-time step algorithm allows the equations of different
physical processes to be solved under different time step sizes. Furthermore,
an implicit viscous damping relaxation technique is applied to significantly
reduce computational iterations required to achieve equilibrium in the
comparatively fast solid response process. To validate the accuracy and
efficiency of the proposed algorithm, two distinct scenarios, i.e., a nonlinear
hardening bar stretching and a fluid diffusion coupled with Nafion membrane
flexure, are simulated. The results show good agreement with experimental data
and results from other numerical methods, and the simulation time is reduced
firstly by independently addressing different processes with the multi-time
step algorithm and secondly decreasing solid dynamic relaxation time through
the incorporation of damping techniques.

Computational Engineering, Finance, and Science