# Simulation of the Marangoni Effect and Phase Change Using the Particle Finite Element Method
## 1 Introduction  
Phase change plays a key role in some state-of-the-art manufacturing processes, such as welding and additive manufacturing. Characteristic of these applications is that phase change occurs very locally at a specific length scale, which we call the mesoscale. The mesoscale is the length scale at which the melt pool and the concentrated heat source are well-resolved. Other phenomena relevant to engineers occur on all length scales while still strongly depending on phenomena at the mesoscale. Examples include residual stresses and material warping at the part scale and the crystallographic evolution upon freezing at the microscale, which are closely related to convective flows, heat transfer and melt pool evolution at the mesoscale. Few methods have been developed that can fully or even partially simulate these effects at the mesoscale. In the literature, there are some examples of simulation methods that neglect the fluid flow altogether and only concentrate on solid mechanics and heat transfer at the mesoscale. Some authors [1; 2; 3] use a Eulerian or Lagrangian finite element method (FEM) to obtain the temperature field around the laser heat source in selective laser melting (SLM). The latter includes elasto-perfectly plastic behavior of the liquid melt pool; however, no convective flows are modeled that would greatly influence the heat transfer. The focus lies more on the accurate representation of the solid behavior under strong temperature gradients and fast cooling. Conversely, much work can be found investigating the fluid flow in the melt pool and the evolution of the melt pool geometry. Several authors [4; 5] use an Arbitrary Lagrangian-Eulerian (ALE) description of the liquid melt pool, with the former example assuming the powder to be a continuum, and the latter using the discrete element method (DEM) to create a powder bed where individual particles are resolved. The aim of such studies is to explore the physics of the melt pool, such as the recoil pressure from evaporation and the Marangoni effect, as well as other phenomena such as denudation of the powder bed, spattering, pore formation and keyhole evolution. The most common approach for simulating melt pool fluid dynamics is the use of the finite volume method (FVM) in combination with an interface tracking method such as Level set (LS) or Volume of Fluid (VoF). Often, the powder bed is created using the DEM. This is done by many groups, and a good summary can be found in [PERSON] and [PERSON]'s review paper [6]. The modelling of the solid mechanics is omitted in all of these works, which are thus geared towards predicting melt track topology and provide impressive results thereof. There are limited examples of other methods used for welding and additive manufacturing, such as smoothed particle hydrodynamics (SPH) and the lattice Boltzmann method (LBM). Both share the drawback that some boundary conditions and the Marangoni effect are difficult to implement, making these a rare choice in the context of additive manufacturing. See also [PERSON] and [PERSON] [6] for details. One particularly noteworthy work devoted to the modeling of the additive manufacturing process in which the problem is split into a solid and fluid parts is that of [PERSON] [7]. The powder is modeled as a flowable continuum, and hence included in the fluid part. For the fluid phase, a Eulerian FEM code for Newtonian fluids with a level set method for interface tracking together with remeshing at regular intervals is used. For the solid, a separate Eulerian FEM code with an elasto-plastic solid model is used. Both are coupled and solved in sequence for each time increment. While the fluid and powder are solved using the Navier-Stokes equations, the solid region in the domain is also included, although with an extremely high viscosity. Based on the fluid flow and the heat transfer, the melt pool can evolve in this step. Then, only the new outlines of the melt pool and the temperature profile are transferred to the solid solver.