# Time Step Control Stability

The stability conditions of explicit scheme in SPH formulation can be written over cells or on nodes.

## Cell Time Step

In case of cell stability computation (when no nodal time step is used), the stable time step is computed as:(1)
$\text{Δ}t=\text{Δ}{t}_{sca}\cdot {\mathrm{min}}_{i}\left(\frac{{d}_{i}}{{c}_{i}\left({\alpha }_{i}+\sqrt{{\alpha }_{i}^{2}+1}\right)}\right),with\text{ }{\alpha }_{i}=\left({q}_{b}+\frac{{q}_{a}\cdot {\overline{\mu }}_{i}\cdot {d}_{i}}{{c}_{i}}\right),and\text{ }{\overline{\mu }}_{i}={\mathrm{max}}_{j}\left({\mu }_{ij}\right)$

$\text{Δ}{t}_{sca}$ is the user-defined coefficient (Radioss option /DT or /DT/SPHCEL). The value of $\text{Δ}{T}_{sca}$ =0.3 is recommended. 1

## Nodal Time Step

In case of nodal time step, stability time step is computed in a more robust way:(2)

$\text{Δ}{t}_{i}=\sqrt{\frac{2{m}_{i}}{{K}_{i}}}$ at particle $i$

Use the following notations, if kernel correction:(3)
${W}_{j}\left(i\right)=\stackrel{^}{W}\left({x}_{i}-{x}_{{j}^{\prime }}\frac{{d}_{i}+{d}_{j}}{2}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}and\text{\hspace{0.17em}}\nabla {W}_{j}\left(i\right)=grad|{}_{xi}\left[\stackrel{^}{W}\left(x-{x}_{{j}^{\prime }}\frac{{d}_{i}+{d}_{j}}{2}\right)\right]$
Or, if no kernel correction:(4)
${W}_{j}\left(i\right)=W\left({x}_{i}-{x}_{{j}^{\prime }}\frac{{d}_{i}+{d}_{j}}{2}\right)\text{\hspace{0.17em}}\text{ }and\text{\hspace{0.17em}}\nabla {W}_{j}\left(i\right)=grad|{}_{xi}\left[W\left(x-{x}_{{j}^{\prime }}\frac{{d}_{i}+{d}_{j}}{2}\right)\right]$
Recalling that apart from the artificial viscosity terms:(5)
${F}_{i}=\sum _{j}{F}_{ij},{F}_{ij}={V}_{i}{V}_{j}\left[{p}_{i}\nabla {W}_{j}\left(i\right)-{p}_{j}\nabla {W}_{j}\left(j\right)\right]$
write (6)
$|{K}_{ij}|=‖\frac{d{F}_{ij}}{d\left({u}_{i}-{u}_{j}\right)}‖\le \frac{d}{d\left({u}_{i}-{u}_{j}\right)}\left({V}_{i}{V}_{j}\left[{p}_{i}‖\nabla {W}_{j}\left(i\right)‖+{p}_{j}‖\nabla {W}_{i}\left(j\right)‖\right]\right)$
Where, ${u}_{i}-{u}_{j}$ is the relative displacement of particles $i$ and $j$ . Keeping the only first order terms leads to:(7)
$|{K}_{ij}|\le {V}_{i}{V}_{j}\left[\frac{d{p}_{i}}{d\left({u}_{i}-{u}_{j}\right)}‖\nabla {W}_{j}\left(i\right)‖+\frac{d{p}_{j}}{d\left({u}_{i}-{u}_{j}\right)}‖\nabla {W}_{i}\left(j\right)‖\right]$
Where, (8)
${V}_{i}{V}_{j}\frac{d{p}_{i}}{d\left({u}_{i}-{u}_{j}\right)}‖\nabla {W}_{j}\left(i\right)‖={V}_{i}{V}_{j}\frac{d{p}_{i}}{d{\rho }_{i}}\cdot \frac{d{\rho }_{i}}{d\left({u}_{i}-{u}_{j}\right)}‖\nabla {W}_{j}\left(i\right)‖={V}_{i}{V}_{j}{c}_{i}^{2}\frac{d{\rho }_{i}}{d\left({u}_{i}-{u}_{j}\right)}‖\nabla {W}_{j}\left(i\right)‖$
that is(9)
${V}_{i}{V}_{j}\frac{d{p}_{i}}{d\left({u}_{i}-{u}_{j}\right)}‖\nabla {W}_{j}\left(i\right)‖={m}_{i}{c}_{i}^{2}{\stackrel{˙}{V}}_{j}^{2}{‖\nabla {W}_{j}\left(i\right)‖}^{2}$
${V}_{i}{V}_{j}\frac{d{p}_{j}}{d\left({u}_{i}-{u}_{j}\right)}‖\nabla {W}_{i}\left(j\right)‖={m}_{j}{c}_{j}^{2}{\stackrel{˙}{V}}_{i}^{2}{‖\nabla {W}_{i}\left(j\right)‖}^{2}$
$|{K}_{ij}|\le {m}_{i}{c}_{i}^{2}{\stackrel{˙}{V}}_{j}^{2}{‖\nabla {W}_{j}\left(i\right)‖}^{2}+{m}_{j}{c}_{j}^{2}{\stackrel{˙}{V}}_{i}^{2}{‖\nabla {W}_{i}\left(j\right)‖}^{2}$
Stiffness around node $i$ is then estimated as:(12)
$|{K}_{i}|\le \sum _{j}|{K}_{ij}|$