Temporal and Spatial Coupling Methods for the Efficient Modelling of Dynamic Solids

. 2025 Feb 28;18(5):1080. doi: 10.3390/ma18051080

Algorithm 1 Summary of Algorithm for Coupling in Time from N to

N + 1

1:
procedure a two-subdomain multi-time integration step
2:
while $t_{r a t i o}^{n + k + 1} \leq 1$ or ( $t_{r a t i o}^{n + k} \leq 1$ and $t_{r a t i o}^{n + k + 1} \leq 1 + TOL$ ) do
3:
Integrate subdomain $Ω_{s}$ with ${\ddot{u}}_{Γ}$ and compute force vectors $f_{s}^{int}, f_{s}^{ext}$ over $Δ t_{s}$
4:
Compute trial times $t_{T s}^{n + k}$ , $t_{T L}^{N + 1}$ and time step ratios $t_{r a t i o}^{n + k}$ , $t_{r a t i o}^{n + k + 1}$
5:
Compute time step reduction factors $α_{L}, α_{s}$
6:
if $α_{L} \geq α_{s}$ then
7:
$Δ t_{L} = α_{L} \cdot Δ t_{L}$
8:
else
9:
$Δ t_{s} = α_{s} \cdot Δ t_{s}$
10:
Integrate subdomain $Ω_{s}$ with ${\ddot{u}}_{Γ}$ and recompute force vectors $f_{s}^{int}, f_{s}^{ext}$ over $Δ t_{s}$
11:
Recompute trial times $t_{T s}^{n + k}$ , $t_{T L}^{N + 1}$ and time step ratios $t_{r a t i o}^{n + k}$ , $t_{r a t i o}^{n + k + 1}$
12:
Integrate subdomain $Ω_{L}$ with ${\ddot{u}}_{Γ}$ and compute force vectors $f_{L}^{int}, f_{L}^{ext}$ over $Δ t_{L}$
13:
Compute interface acceleration ${\ddot{u}}_{Γ}$ with Equations (15)–(17) for next time step
14:
Recompute trial times $t_{T s}^{n + k}$ , $t_{T L}^{N + 1}$ and time step ratios $t_{r a t i o}^{n + k}$ , $t_{r a t i o}^{n + k + 1}$