# A Numerical Investigation of Laser Beam Welding of Stainless Steel Sheets with a Gap
## 2 Simulation Model  
### Description of the Model  
The model is based on the open source C++ toolkit OpenFOAM [15], which provides a framework for using the finite-volume method to solve continuum mechanics problems described by partial differential equations. The solver is based on the OpenFOAM-native \"multiphaseInterFOAM\" solver, which itself is the n-phase extension of the well-known \"interFoam\" solver [22]. It uses the volume-of-fluid method to solve multiphase problems, involving an arbitrary number of different phases combined with the possibility of dynamic mesh refinement at run-time. This solver has been extensively modified and extended to make it suitable for the simulation of laser-material processing. An overview of the solver and incorporated models for other types of processes can be found in [17]. The models relevant for the simulation of laser beam welding are briefly explained in the following paragraph. The propagation of laser beams of arbitrary shape (e.g., rings, rectangular, or elliptical) and intensity distribution (e.g., Gaussian, tophat, or Bessel) is calculated by solving a differential radiative transfer equation [23]. A ray tracing algorithm takes into account multiple reflections, which is crucial in keyhole welding simulations. Incident angle-dependent transmission and reflection are modeled according to [PERSON]'s equations; absorption is calculated via [PERSON]'s law. The state and temperature-dependent complex refractive index (consisting of the refractive index \(n\) and the extinction coefficient \(\kappa\)) of the material therefore needs to be known to correctly model the absorption of laser power within the material. This index is not only dependent on the material but also on the wavelength of the laser [24] and reliable information in the literature is scarce, especially for high temperature applications. Therefore, in many cases, both estimation and calibration of this parameter using experimental results are needed. The amount of laser energy absorbed by the material, resulting from the propagation model described above, is included as a source term in the energy equation. The classical energy equation is split into one equation containing the conductive heat transfer and one transport equation for calculating the convective heat transport. Within each time step, the convective part of the energy equation is solved once the flow field is known. Then, the conductive part is solved, including the absorbed laser power as an additional source term. This approach is necessary because of the temperature-dependency of the material's heat conductivity and capacity, which are needed within the conductive part of the energy equation. Each material with its respective phases involved in the problem, i.e., solid, liquid and vapor, and ambient gas (e.g., surrounding air or shielding gas), is treated as a distinct phase in the multiphase solver. Phase changes are modeled as mass transfers between the respective phases via the Clapeyron equation and an accompanying energy transfer. The thus yielded phase change rates have an influence on the pressure, and therefore phase change and pressure have to be calculated together in an iterative manner. The movement restriction of the solid material is realized through a porous bed model via [PERSON]'s law [25], where a source term in the momentum equation increases with the fraction of solid material inside a cell, letting the velocity approach zero for a solid phase far from any interface. The solver is capable of dynamic mesh refinement to decrease computational costs. Initially a coarse hexahedral mesh is used, with the cells' size being in the range of half the gap size. Then, dynamically throughout the simulation, the mesh is refined iteratively depending on the presence of propagating or absorbed laser power, presence of a liquid phase and presence of a re-solidified phase. The initial (unrefined) computational grid was the same for all simulations presented within this work, except for the different gap widths.