Solver and Time Controls

This section controls four parts of the solution procedure: nonlinear temperature–phase correction, temporal discretization, optional PIMPLE fluid-flow coupling, and adaptive time-step selection. The controls are divided between the PIMPLE and linear-solver dictionaries in system/fvSolution, the ddtSchemes dictionary in system/fvSchemes, and the time controls in system/controlDict. A thermal-only case uses nOuterCorrectors 0; a coupled melt-pool-flow case uses a positive outer-corrector count.

Thermodynamic coupling

These controls determine how many temperature–solid-fraction corrections are performed within each CFD time step. Keep nThermoCorrectors large enough for the reported maximum solid-fraction update to fall below thermoTolerance; the solver log reports that update for every correction. Tmax limits the temperature accepted by the implicit solve, while explicitSolve permits forward Euler only when the instantaneous diffusion criterion is below one.

AdditiveFOAM reads these entries from the PIMPLE dictionary in system/fvSolution:

Entry	Default	Meaning
nThermoCorrectors	20	Maximum temperature–solid-fraction iterations per time step
thermoTolerance	1e-6	Maximum solid-fraction update for convergence
Tmax	Unlimited	Optional implicit upper-temperature limiter
explicitSolve	false	Request explicit Euler when the diffusion criterion permits it

PIMPLE
{
    momentumPredictor        no;
    nOuterCorrectors         0;
    nCorrectors              1;
    nNonOrthogonalCorrectors 0;
    pRefCell                 0;
    pRefValue                0;

    nThermoCorrectors        20;
    thermoTolerance          1e-8;
    Tmax                     4000;
    explicitSolve            false;
}

The thermodynamic loop performs at least two iterations and accepts convergence when

\[\max_c\left|f_{s,c}^{(m+1)}-f_{s,c}^{(m)}\right|<\varepsilon_f,\]

where \(c\) indexes cells, \(m\) is the thermodynamic-correction index, \(f_{s,c}^{(m)}\) is the iterated solid fraction, and \(\varepsilon_f\) is thermoTolerance; nThermoCorrectors bounds the iteration count.

Time discretization

Select the temperature time scheme under ddtSchemes in system/fvSchemes. Use Euler for first-order implicit stepping, backward for second-order backward differencing, or CrankNicolson for an off-centred second-order scheme. Other schemes cause a fatal error. When the explicit path is active, AdditiveFOAM uses forward Euler regardless of the configured scheme. See temperature and phase coupling for the coefficients used by each scheme.

Flow coupling

AdditiveFOAM solves velocity and pressure with OpenFOAM’s PIMPLE algorithm when nOuterCorrectors is positive. Each outer correction solves the momentum equation for U, applies the configured pressure corrections to p_rgh, and updates the velocity–pressure coupling.

Set nOuterCorrectors 0 for a thermal-only simulation. Set nOuterCorrectors to 1 or greater to enable fluid flow through the PIMPLE momentum–pressure loop. If there is no liquid in the domain, AdditiveFOAM skips that loop because the velocity is constrained to zero in fully solid material.

A thermal-only configuration is

PIMPLE
{
    momentumPredictor        no;
    nOuterCorrectors         0;
    nCorrectors              1;
    nNonOrthogonalCorrectors 0;
}

A coupled thermal-fluid configuration is, for example,

PIMPLE
{
    momentumPredictor        yes;
    nOuterCorrectors         1;
    nCorrectors              2;
    nNonOrthogonalCorrectors 0;
}

Entry	Effect
momentumPredictor	Solves a momentum predictor before the pressure-correction loop when enabled
nOuterCorrectors	Number of complete momentum–pressure coupling loops; zero disables flow
nCorrectors	Pressure corrections performed within each outer loop
nNonOrthogonalCorrectors	Additional pressure corrections for mesh non-orthogonality

Linear solvers

U and p_rgh use the configured OpenFOAM linear solvers. T can be matched with a regular-expression solver entry:

solvers
{
    "T.*"
    {
        solver         PBiCGStab;
        preconditioner DILU;
        tolerance      1e-12;
        relTol         0;
    }
}

In this entry, PBiCGStab is the temperature-equation linear solver, DILU is its preconditioner, tolerance is the absolute residual stopping criterion, and relTol 0 disables termination based on a relative residual reduction. Separate U and p_rgh entries are required when fluid flow is enabled.

Adaptive time stepping

Set adjustTimeStep yes in system/controlDict to let AdditiveFOAM select the next time step from fluid advection, phase change, thermal diffusion, and scan-path event times. The solver prints the current Courant, thermo, and diffusion numbers and the accepted deltaT; compare those values with maxCo, maxAlphaCo, maxDi, and maxDeltaT to identify which limit controls the run.

adjustTimeStep yes;
maxCo          0.5;
maxDi          1;
maxAlphaCo     1;
maxDeltaT      1e-5;

For the current time step \(\Delta t^n\), define the maximum phase-change number

\[A^n=\max_c\left( \frac{|(\partial T/\partial t)_c|\Delta t^n} {T_{\mathrm{liquidus}}-T_{\mathrm{solidus}}} \right),\]

where \(n\) is the current time-level index, \(T\) is temperature, and \(T_{\mathrm{solidus}}\) and \(T_{\mathrm{liquidus}}\) bound the phase-change interval. The maximum finite-volume diffusion number is

\[D^n=\max_c\left[ \frac{\Delta t^n}{V_c\rho c_{p,c}} \sum_{f\in\partial c}|\mathbf{S}_f|k_f\delta_f \right].\]

In these definitions, \(A^n\) is the maximum phase-change number, \(D^n\) is the maximum diffusion number, \(c\) indexes cells, and \(f\in\partial c\) indexes the faces bounding cell \(c\). The quantities \(V_c\), \(c_{p,c}\), and \(\rho\) are cell volume, cell heat capacity, and density; \(\mathbf S_f\) is the outward face-area vector, \(k_f\) is face thermal conductivity, and \(\delta_f\) is the inverse cell-centre spacing normal to the face. With fluid Courant number \(C^n\), the unrestricted factor is

\[r=\min\left( \frac{C_{\max}}{C^n+\epsilon}, \frac{A_{\max}}{A^n+\epsilon}, \frac{D_{\max}}{D^n+\epsilon} \right).\]

Here \(C_{\max}\), \(A_{\max}\), and \(D_{\max}\) are maxCo, maxAlphaCo, and maxDi; \(\epsilon\) is a small positive denominator safeguard and \(r\) is the unrestricted time-step ratio.

The applied growth factor and provisional step are

\[g=\min[r,1+0.1r,1.2], \qquad \Delta t^{n+1}=\min(g\Delta t^n,\Delta t_{\max}).\]

Here \(g\) is the applied growth factor and \(\Delta t_{\max}\) is maxDeltaT.

When hitPathIntervals true, the source model reduces this provisional value so that a step endpoint coincides with the next scan-path event. With explicitSolve true, forward Euler is selected only when the instantaneous \(D^n<1\); otherwise the configured implicit scheme is used.