The isotherm migration method in spherical coordinates with a moving heat source

An isotherm migration method is presented to track the isotherms at the surface of a semi-infinite workpiece that is being heated by a point source with a linear motion. The governing heat conduction equation is first transformed into a one-dimensional isotherm migration equation in spherical coordinates, which expresses the isotherm velocities as a function of the isotherm positions and their temperature derivatives. A finite difference formulation of this equation is then derived together with the corresponding boundary conditions. It is shown through a number of simulations that this system converges to the analytical solution when the temperature mesh is refined. The additional presence of a phase change results in a Stefan problem that can be solved approximately with some minor modifications to the method. The resulting system of equations can be used as a simple but accurate thermal model for some laser-material interaction processes such as laser heat treatments and laser cladding.