Solidification Quantities

Interface mode detects a cooling crossing between one timestep’s liquid temperature and the next timestep’s solid temperature. It refines that bracket with a safeguarded secant iteration: a secant proposal is used when it remains inside the bracket, otherwise the midpoint is used. Refinement stops when

\[\left|1-\frac{T}{T_L}\right| < \epsilon_T\]

or max_iter is reached. dT_err is $\epsilon_T$. The reported tSol is a refined estimate, not an unconditional exact root.

Gradient, cooling rate, and interface speed

For one quadrature contribution

\[f=A\exp\!\left[-3\sum_i \Delta x_i^2\phi_i\right], \qquad \phi_i=\Phi_i^{-1},\]

the mathematically consistent derivatives are

\[\frac{\partial f}{\partial x_i}=f(-6\Delta x_i\phi_i),\] \[\frac{\partial^2 f}{\partial x_i^2} =f\left[36\Delta x_i^2\phi_i^2-6\phi_i\right].\]

After summing contributions, the intended solidification outputs are

\[G=\lVert\nabla T\rVert \quad [\mathrm{K/m}], \qquad \widehat{\mathbf{G}}=\frac{\nabla T}{G},\] \[\dot T=\alpha\nabla^2T+\dot T_{\mathrm{source}}, \qquad \text{cooling rate}=|\dot T| \quad [\mathrm{K/s}],\] \[V=\frac{|\dot T|}{G}\quad [\mathrm{m/s}].\]

This $V$ follows from differentiating the isotherm condition $T(\mathbf{x}(t),t)=T_L$ in the local normal direction.

Melting, remelting, and stored fields

Standalone Interface output treats the solidification columns as the current state of each tracked point, rather than as an archive of every crossing. Condor buffers at most one pending update per point between transfers to the host:

• A solid-to-liquid transition increments numMelt and stores a molten marker. When that marker reaches the host, tSol, G, V, dTdt, and the requested gradient components are reset to zero without running the solidification refinement.
• A subsequent liquid-to-solid transition replaces the pending molten marker with its temperature bracket. Condor then refines the crossing and stores the new solidification quantities.
• If several transitions occur before the next transfer, the sparse entry holds the point’s latest state while numMelt retains the total number of melting transitions.

Consequently, a liquid point has zero-valued solidification fields in ordinary CSV output, and a resolidified point contains values from its most recent crossing. numMelt counts solid-to-liquid transitions; it is not a count of completed melt/solidification pairs.

Coupled RDF output remains event-oriented. Melting updates the point’s melt time and count, but RDF publishes the resulting completed solidification event rather than a separate molten-marker event.

Columnar-to-equiaxed estimate

The optional CET hook evaluates

\[\phi_{eq}=1-\exp\!\left[ -\frac{4\pi N_0}{3} \frac{(aV)^{3/n}}{[G(n+1)]^3} \right].\]

$N_0$ is the nucleation-site density [m$^{-3}$], $n$ is dimensionless, and $a$ belongs to the convention $\Delta T_c=(aV)^{1/n}$, or $V=(\Delta T_c)^n/a$. Its units are K$^n$ s m$^{-1}$. This is the convention used by Condor’s pow(a*V, 3/n) implementation and the cited scan-optimization paper. The expression is an empirical, local CET estimate and inherits the accuracy and assumptions of $G$ and $V$. Requesting eqFrac automatically requests those two solidification fields internally. eqFrac is meaningful only where a valid solidification crossing exists; values at liquid points should not be interpreted as solidification conditions.