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.

翻译：模拟涉及多时间尺度耦合的物理问题具有挑战性，因为需要同时求解这些多时间尺度过程。针对这一挑战，本文提出了一种显式多时间步长算法，并与固体动态松弛方案相结合。与隐式方案相比，显式方案简化了方程组，而多时间步长算法允许不同物理过程的方程在不同时间步长下求解。此外，应用了隐式黏性阻尼松弛技术，以显著减少在相对较快的固体响应过程中达到平衡所需的计算迭代次数。为了验证所提算法的准确性和效率，模拟了两个不同的场景，即非线性硬化杆的拉伸以及流体扩散与Nafion膜弯曲的耦合。结果显示，与实验数据及其他数值方法的结果吻合良好，并且模拟时间首先通过使用多时间步长算法独立处理不同过程而减少，其次通过引入阻尼技术缩短了固体动态松弛时间。