Recommended Space and Time Steps

Recommended Values

\(\Delta x = 2\) μm	\(\Delta t = 15\) ns
\(\Delta x = 3\) μm	\(\Delta t = 30\) ns
\(\Delta x = 4\) μm	\(\Delta t = 50\) ns
\(\Delta x = 5\) μm	\(\Delta t = 80\) ns
\(\Delta x = 6\) μm	\(\Delta t = 120\) ns
\(\Delta x = 7\) μm	\(\Delta t = 160\) ns

For laser beam melting the recommended steps are \(\Delta r=3\) μm (dr=3e-6) and \(\Delta t=30\) ns (dt=30e-9).

For electron beam melting recommended steps are \(\Delta r=5\) μm (dr=5e-6) and \(\Delta t=80\) ns (dt=80e-9).

Explanation

LBM generally has no strictly defined stability criteria. Standard BGK LBM is usually stable if the Reynolds number \(Re < 1000\) and Mach number \(Ma \lesssim 0.3\). Here the value of the Mach number is \(Ma=\cfrac{u}{c_s}=\cfrac{u\sqrt{3}\Delta t}{\Delta x}\). LBM has the second order accuracy over the Mach number, so it must be as low as possible.

So, in total, we have two restrictions for the space and time steps, \(\Delta x\) and \(\Delta t\) respectively:

\[1000 > Re = \cfrac{uL}{\nu}\]

After conversion to dimensionless units:

\[\cfrac{c_s Ma \cdot 10 \Delta x}{(\tau-1/2)c_s^2} = \cfrac{Ma\cdot 10\Delta t\sqrt{3}}{\tau-1/2} \lesssim \cfrac{10}{\tau-1/2}< 1000;\]

\[Ma = \cfrac{u\Delta t}{\Delta x} \ll 1\]

The first condition requires the relaxation parameter \(\tau\) should be more than 0.51, i.e. \(3\hat \nu+0.5 = \cfrac{ 3\nu \Delta t}{\Delta x^2}+0.5>0.51\), where \(\hat \nu\) is kinematic viscosity in numerical LBM units and \(\nu\) is the kinematic viscosity in physical units (SI). Therefore, we get

\[\Delta t > 0.003\frac{\Delta x^2}{\nu}.\]

From the second condition, \(Ma \ll 1\), assuming that typical velocities in the melt pool are about 1 m/s we obtain one more condition:

\[\Delta t \ll \cfrac{\Delta x}{1 m/s}.\]

 The CFL condition for heat solver is:

\[\Delta t < \cfrac{1}{2} \cfrac{1}{\kappa \left(\frac{1}{\Delta x^2}+\frac{1}{\Delta y^2}+\frac{1}{\Delta z^2}\right)}\]

If \(\Delta x=\Delta y =\Delta z\),

\[\Delta t < \cfrac{1}{6} \cfrac{\Delta x^2}{\kappa},\]

where \(\kappa\) is the thermal diffusivity of material in solid and low-temperature liquid phases.  Combining this, the time step \(\Delta t\) should be chosen after the space step \(\Delta x\) as

\[\Delta t = \min \left( \cfrac{0.003\Delta x^2}{\nu},\quad \cfrac{\Delta x^2}{6\kappa_{max}} \right)\]

where \(\kappa_{max}\) is the maximum value of thermal diffusivity for temperature ranges \(T<1.1T_l\) (\(T_l\) is the liquidus temperature). The recommended space/time steps parameters set for the most common materials are following: