Skip to content

Methodology

Background

The Eagar–Tsai model (1983) provides an analytical solution for the steady-state temperature field produced by a Gaussian laser beam moving at constant velocity over a semi-infinite solid. It is widely used in additive manufacturing and laser welding research to estimate melt pool geometry without the cost of full finite-element simulations.

References:

  • T. W. Eagar and N.-S. Tsai, "Temperature Fields Produced by Traveling Distributed Heat Sources," Welding Journal (Research Supplement), December 1983, pp. 346-s–354-s.
  • C integrand reformulation: Sasha Rubenchik, LLNL, 2015.

Assumptions

  • Semi-infinite solid (no boundaries other than the top surface).
  • Constant material properties evaluated at the liquidus temperature.
  • Gaussian heat source with constant absorptivity.
  • Steady-state moving source (the temperature field moves with the beam).
  • No melt flow, vaporization, or latent heat effects.

Governing Equations

Thermal diffusivity

Material thermal diffusivity is computed from the three input properties:

alpha = k / (rho * cp)

where k is thermal conductivity (W/(m·K)), rho is density (kg/m³), and cp is specific heat (J/(kg·K)).

Non-dimensional parameter

The model uses a single non-dimensional parameter that captures the ratio of diffusive to advective transport:

p = alpha / (v * sigma)

where v is the scan velocity (m/s) and sigma = sqrt(2) * (d / 2) is the Gaussian width derived from beam diameter d.

Temperature prefactor

The overall temperature scale is set by:

Ts = (A * P) / (pi * (k / alpha) * sqrt(pi * alpha * v * sigma^3))

where A is absorptivity and P is laser power (W).

Temperature field

The temperature at any point (x, y, z) in the frame co-moving with the beam is:

T(x, y, z) = T0 + Ts * integral_0^inf  f(t, x, y, z, p)  dt

where T0 = 300 K is the ambient temperature and the integration variable t is a dimensionless time-like parameter.

Integrand

f(t, x, y, z, p) =  1 / ((4*p*t + 1) * sqrt(t))
                  * exp(-z^2 / (4*t)  -  (y^2 + (x - t)^2) / (4*p*t + 1))

The integrand is evaluated numerically using scipy.integrate.quad. For performance, the integrand is implemented as a C extension (_integrand_ext.c) and passed to QUADPACK as a LowLevelCallable, eliminating Python overhead on every function evaluation.

Melt Pool Extraction

The temperature field is evaluated on two planes:

  • the x–y plane (z = 0, top surface) to obtain melt pool length and half-width,
  • the x–z plane (y = 0, centerline) to obtain melt pool depth.

The melt pool boundary is the liquidus isotherm T = T_liquidus. The three dimensions are extracted as:

Dimension Definition
Length Extent of T >= T_liquidus along x
Width 2 × half-extent of T >= T_liquidus along y at the surface
Depth Extent of T >= T_liquidus along z at the centerline

If the melt pool reaches any domain boundary, the domain is automatically expanded and the computation is repeated (up to 20 iterations).