Review and analysis of heat source models for additive manufacturing

As additive manufacturing (AM) becomes a viable manufacturing solution, demand for an accurate thermo-structural model of the process increases. Iteratively correcting discrepancies between the CAD model and additively manufactured product through trial and error can be an expensive and time-consuming process, taking up to several hours to build and costing up to tens of thousands of dollars Lindgren et al. (Addit Manuf 12:144–158, 2016). A numerical model reduces manufacturing cost and time considerably by predicting discrepancies that will arise due to the complex thermal history induced by the AM process, thus reducing the need for iterative manufacturing. An important part of any additive manufacturing model is the heat source model. The heat source model is a mathematical function which represents how much of a heat source’s power actually goes into heating the powdered metal and how this heat is distributed across the heat-affected zone (HAZ). This paper provides a review and analysis of heat source models in the AM literature to date in order to alleviate some of the confusion and provide emerging researchers in the field with perspective on the issue. Both two-dimensional surface models and three dimensional volumetric models are explored. Next, an analysis of the models was performed and presented in an effort to validate their physical accuracy and mathematical usability. This analysis consisted of checking for sensible boundary conditions and ensuring that energy conservation is upheld. In surface models, the TEM₀₀ model is a classic representation of the Gaussian power distribution of most heat sources used in AM. Researchers interested in simply modeling the heat distribution, without accounting for any other phenomena that intervene in the heat transfer process (such as molten pool dynamics) will find the TEM₀₀ model suitable. The literature also shows cases where the TEM₀₀ model has been modified to have a sharper radial gradient, and these modifications can be suitable for high-powered heat sources. For volumetric models, Goldak’s ellipsoidal model (Metall Trans B 15(2)299–305, 1984) remains a straightforward and accurate model that is physically sound and applicable to a variety of cases. The Gaussian cone model presented by Rogeon et al. [48] also performs well, meeting all the required physical and mathematical restrictions. This model’s linearly decaying penetration is better suited for high-energy applications. The non-Gaussian cone proposed by Tsirkas et al. (J Mater Process Technol 134(1):59–69, 2003) imposes inaccurate boundary conditions and violates the first law of thermodynamics, and is thus deemed an inadequate model. Other novel models have been introduced in recent years, most notably the line model and the elongated ellipsoidal model presented by Irwin and Michaleris (J Manuf Sci Eng 138(11):111004, 2016). Both of these models are based on Goldak’s ellipsoidal model and attempt to maintain the accuracy of that model while allowing for fewer time steps and requiring less computational resources. These models appear to function well and can be used effectively in some applications, but could benefit from further study and validation. Care must be taken to ensure that the parameters used with these models do not result in averaging errors or a discontinuous thermal field. These tools must be used carefully with a thorough understanding of the underlying mathematics.