Third Generation Weld Heat Source Models
The next advance in weld heat source models, Third Generation models, was initiated by Ohji et al. . They predicted the liquid weld pool shape. The distinguishing feature of Third Generation models is that they must solve the Stefan problem for the weld pool liquid-solid free boundary. Recall that First and Second Generation models need not and usually do not solve the weld pool liquid-solid free boundary problem. In these Third Generation models the specific enthalpy is discontinuous across the liquid-solid interface. The melting temperature is usually defined for a flat static interface. Increasing curvature decreases the melting point. Interface velocity increases the melting point in the front facing part of the weld pool and decreases it in the backward facing part of the weld pool.
In addition to the energy equation with liquid-solid free boundary, these models include the hydrostatic stress in the liquid pool, a pressure distribution on the weld pool surface from the arc, surface tension forces, and a volume constraint to enforce conservation of mass, i. e., the mass entering the weld pool equals the mass leaving the weld pool. This force balance is the conservation of linear momentum with velocity set to zero or the momentum equation for a hydrostatic fluid. Thus this model couples the heat equation with the momentum equation for a hydrostatic fluid. Given the input data, these models can predict the weld pool shape and the shape of the weld bead created by the weld pool. Sudnik has developed these weld heat source models to the current state of the art. Weiss  extended Sudnik's ideas to include some effects of the arc interaction with the weld pool shape. Weiss was able to predict effects of vertical, horizontal welding on weld pool shape in addition to flat welding.
The Third Generation Models ignore the Lorenz force, the Marangoni force and the force due to the momentum flux from any droplets added to the weld pool from a consumable electrode. Except for the model by Weiss, they usually assume that the flux and pressure distribution from the arc are constant and independent of the weld pool free surface shape. Numerically, these models are robust and computational costs are only slightly higher than First or Second Generation Models. While the data required by the First and Second Generation Models was the distribution of power density, flux or temperature, the data required by these Third Generation Models usually include a pressure distribution from the arc, a mass flow rate into the weld pool and surface tension on the liquid surface of the weld pool. Now the geometry of the weld pool is not input data but output data.