Approximate Analytical Solution for the Temperature Field in Welding

To determine the temperature fields associated with welding, significant efforts have been made to establish the relative merits of numerical approaches with variable material properties and the analytical approaches with constant material properties. Currently, analytical solutions are either based on the temperature field generated by a point source of heat or are developed for a finite domain derived approximately by using an infinite or semi-infinite heat kernel. Furthermore, the heat kernel applied in these solutions is derived from the Image method (for example, Nguyen’s book (Thermal Analysis of Welds, 2004)). The main problem with the heat kernels obtained from Image method is that they face the problem of singularity at and around the point where the heat source is located, and they do not satisfy the boundary condition accurately. That is why the Laplace transform method has been applied here instead of using the Image method to formulate a heat kernel that (1) converges rapidly, (2) avoids the problem of singularity, and (3) gives a good and robust approximation of the real analytic solution for the temperature field. The results obtained from the analytical solutions were compared with the results obtained from finite element method. The current work is believed to make a considerable contribution to the avoidance of previously mentioned problems by deriving a new approximate analytical solution for the temperature field on a three-dimensional finite body.