Thermal Model

Governing problem

Condor models linear heat conduction in a homogeneous material,

\[\rho c_p\frac{\partial T}{\partial t}=k\nabla^2T+q(\mathbf{x},t), \qquad \alpha=\frac{k}{\rho c_p},\]

where $T$ is temperature [K], $k$ is conductivity [W m$^{-1}$ K$^{-1}$], $\rho$ is density [kg m$^{-3}$], $c_p$ is specific heat [J kg$^{-1}$ K$^{-1}$], and $\alpha$ is diffusivity [m$^2$ s$^{-1}$]. All four material quantities are spatially and temporally constant.

For absorbed power $\eta Q$, the ellipsoidal Gaussian Green-function convolution used by Condor is

\[T(\mathbf{x},t)-T_0 = \frac{2\eta Q}{\rho c_p(\pi/3)^{3/2}} \int_0^t \frac{m(t')}{\sqrt{\Phi_x\Phi_y\Phi_z}} \exp\!\left[-3\sum_{i\in\{x,y,z\}} \frac{(x_i-x_{b,i}(t'))^2}{\Phi_i}\right]dt',\]

with

\[\Phi_i(t,t')=\sigma_i^2+12\alpha(t-t').\]

$T_0$ is the initial temperature, $m(t’)$ is the path power modifier, $\mathbf{x}_b(t’)$ is the beam location, and $\sigma_i$ is the input source width along axis $i$ [m]. The width_x, width_y, and width_z values are this model’s Gaussian width parameters; they are not beam diameters. At zero diffusion time, comparison with the conventional form $\exp[-x_i^2/(2s_i^2)]$ gives $\sigma_i=\sqrt{6}\,s_i$.

Source paths and superposition

Temperature is linear in source history. Contributions from every enabled segment and every beam/path pair are summed. A line segment interpolates $\mathbf{x}_b(t’)$ from its previous endpoint to its endpoint at the specified speed. A spot segment holds its endpoint for its dwell time. Pmod multiplies absorbed power over that segment.

Configured boundary planes add reflected copies of the quadrature nodes. These are same-sign image sources and therefore approximate insulating/zero-normal-flux planes.

Adaptive history integration

Condor uses the adaptive strategy developed by Stump and evaluates the time convolution with Gauss–Legendre rules of order 2, 4, 8, or 16. It uses the nondimensional lag

\[s=\frac{t-t'}{t_c},\qquad t_c=\frac{\sigma_x^2}{\alpha}.\]

Near-source intervals receive higher order. Stationary-spot history intervals grow by powers of two in $s$. For a moving line, the implemented physical step scale is

\[\Delta t' \sim \frac{\sqrt{\ln\sqrt{2}}\,\sigma_x}{v}\sqrt{12s+1},\]

clipped to segment boundaries; older history is smoother and can use wider, lower-order intervals.

The settings quadrature.cutoff_peak and cutoff_t0tl are parsed and used in spatial search-radius estimates, but the current t0calc implementation resets the history start to zero. They do not currently truncate the temporal convolution.

Scope and assumptions

The solution assumes linear conduction, uniform constant properties, and the Gaussian effective source. It omits fluid advection in the liquid, latent heat, radiative and convective surface losses, evaporation, recoil pressure, powder resolution, and deformation. The source is consequently an effective model: calibration can absorb some omitted physics, but does not add that physics.

See Papers for the derivation lineage.