The interaction of a heat source (arc, electron beam or laser) with a weld pool is a complex physical phenomenon that still cannot be modeled rigorously. It is known that the distribution of pressure and shear from the arc source, droplets from the electrode the effects of surface tension, buoyancy forces and molten metal viscosity combine to cause weld puddle distortion and considerable stirring. Because of the arc ‘digging’ and stirring, it is clear that the heat input is effectively distributed throughout a volume in the workpiece.

The ‘disc’ model is more realistic than the point source because it distributes the heat input over a source area. In fact, for a preheat torch that causes no melting this may be a very accurate model indeed. However, in the absence of modeling the weld pool free boundary position, the applied tractions, and convective and radiative conditions between the weld pool and the arc, some form of idealization of the heat source is necessary to achieve as good an approximate solution as one can afford. The disc model does not account for the rapid transfer of heat throughout the fusion zone FZ. In particular, it is not possible to predict the deep penetration FZ of an electron beam EB or laser weld with the surface disc model. A comparison of calculated thermal history data (disc model) with measured values during author’s investigations [46] underscored the need for an ‘effective volume source’ such as the one suggested by Paley and Hibbert [44]. In addition, it was found necessary to generate a volume source with considerable flexibility, i. e., the double ellipsoid model. With less general shapes such as a hemisphere or a single ellipsoid significant discrepancies between the computed and measured temperature distributions could not be resolved.

The size and shape of the ‘double ellipsoid’ i. e., the semi-axes lengths, can be fixed by recognizing that the solid-liquid interface is the melting point isotherm. In reality the melting point is a function of curvature and the speed of the liquid-solid interface but the changes have been ignored in most models published to date. At the same time weld pool temperature measurements have shown that the peak temperature in the weld pool is often 300 to 500 °С above the melting point. The accuracy with which the heat source model predicts the size and shape of the FZ and the peak temperatures is probably the most stringent test of the performance of the model. In the author’s investigation [46] it was found that the most accuracy was obtained when the ellipsoid size and shape were equal to that of the weld pool. The non-dimensional system suggested by Christensen [48] can be used to estimate the ellipsoid parameters. Furthermore a Gaussian distribution is assumed centered at the origin of the heat source.