Algorithm 2 Summary of Non-Matching Mesh Algorithm with Multi-Time Stepping

1:procedureintegrate a two-domain non-matching mesh with MTS   2:      while $t_{ratio}^{n+k+1}\le 1$ or ($t_{ratio}^{n+k}\le 1$ and $t_{ratio}^{n+k+1}\le 1+TOL$) do   3:         Compute ${\ddot{\mathbf{u}}}_{{\Gamma }_s}$ with operator ${\mathbf{N}}_s$ in Equation (34) on ${\Gamma }_s$   4:         Integrate small domain ${\Omega }_s$ and compute vectors ${\mathbf{f}}_s^{\mathrm{int}},{\mathbf{f}}_s^{\mathrm{ext}}$   5:         Compute trial times $t_{Ts}^{n+k}$, $t_{TL}^{N+1}$ and time step ratios $t_{ratio}^{n+k}$, $t_{ratio}^{n+k+1}$   6:      Compute time step reduction factors ${\alpha }_L,{\alpha }_s$   7:      if ${\alpha }_L\ge {\alpha }_s$ then   8:         $\Delta t_L={\alpha }_L·\Delta t_L$   9:      else 10:         $\Delta t_s={\alpha }_s·\Delta t_s$ 11:         Compute ${\ddot{\mathbf{u}}}_{{\Gamma }_s}$ with operator ${\mathbf{N}}_s$ in Equation (34) on ${\Gamma }_s$ 12:         Integrate small domain ${\Omega }_s$ and compute vectors ${\mathbf{f}}_s^{\mathrm{int}},{\mathbf{f}}_s^{\mathrm{ext}}$ 13:         Recompute trial times $t_{Ts}^{n+k}$, $t_{TL}^{N+1}$ and time step ratios $t_{ratio}^{n+k}$, $t_{ratio}^{n+k+1}$ 14:      Compute ${\ddot{\mathbf{u}}}_{{\Gamma }_L}$ with operator ${\mathbf{N}}_L$ in Equation (34) on ${\Gamma }_L$ 15:      Integrate large domain ${\Omega }_L$ and compute vectors ${\mathbf{f}}_L^{\mathrm{int}},{\mathbf{f}}_L^{\mathrm{ext}}$ 16:      Summate kinetics with ${\mathbf{R}}_L,{\mathbf{R}}_s,{\mathbf{C}}_L,{\mathbf{C}}_s$ to find ${\mathbf{M}}_{\Gamma },{\mathbf{f}}_{\Gamma }^{\mathrm{int}}$ and ${\mathbf{f}}_{\Gamma }^{\mathrm{ext}}$ with Equation (31)–(33) 17:      Compute trial times $t_{Ts}^{n+k}$, $t_{TL}^{N+1}$ and time step ratios $t_{ratio}^{n+k}$, $t_{ratio}^{n+k+1}$