Prescribed Temperature Heat Source Models
These models treat the weld heat source as a sub-domain in which the temperature or specific enthalpy in the weld pool is known as a function of (x, y,z, t). Since the temperature is known, it need not be solved. Instead, the boundary of this sub-domain can act as a Dirichlet BC for the complement of the weld heat source subdomain. Perhaps the simplest useful example of this class is parameterized by a double ellipsoid function that specifies the liquidus temperature on the liquid solid interface and a maximum temperature at the centroid of the weld pool and constrains the temperature to vary quadratically from the melting point to the maximum temperature.
Radaj [4 and 8] proposed a model based on the liquidus surface and liquidus temperature. For the continuous problem, i. e., before the problem is discretized for FEM or FDM, this interface partitions the domain into the weld heat source model and its complement. However, after discretization, the problem arises of how to map this surface into the discrete problem. If the weld pool is meshed so that the boundary of the weld pool lies on faces of elements and if the mesh moves with the weld pool, then a surface representation of a weld pool has distinct advantages. If the weld pool moves through the mesh in time, then the advantages of representing a weld pool by the liquid-solid surface diminish.
Sudnik  uses a FDM with a regular grid to solve for specific enthalpy in his weld heat source models. He chooses a rectangular Cartesian grid with edge lengths P*dx, Q*dy, and R*dz where P, Q and R are integers and dx, dy and dz are the lengths of edges of each cell. These models could be used directly in a FDM or FEM method for solving the transient temperature in a structure being welded. There is no need to restrict the weld heat source model to the weld pool. The farther the weld heat source model can be extended from the weld pool, the coarser the mesh required by the structure being welded and the lower the computing costs. In particular, ideas from A. Brandt's multi-grid method can be used which would treat Sudnik's model as a local fine grid solution. The thermal analysis of the structure could use a coarse grid with the fine grid solution in the weld pool region restricted to the coarse grid.