**1. Introduction**

Over the years, increased computing power combined with proper software development, has allowed for the implementation of simulation tools to speed up product development [1]. Chief among the numerous advantages offered by simulation is the possibility of investigating complex processes. Especially for new processes, such as additive manufacturing (AM), process simulation looks to be the best tool for processing complex understandings [2]. Electron beam melting (EBM) is an AM powder-bed process for metal components used for the production of end-usable parts in several industrial sectors [2]. During the EBM process, a high energy electron beam melts metallic powders by moving across the powder bed [3]. After the powder distribution, the whole powder bed is preheated. The preheating step establishes a uniform temperature on the powder bed, which is approximately equal to 60% of the melting point of the processed material. Except for a low helium flow (0.002 mbar) [4], the whole process occurs under vacuum to avoid beam deflection and to maintain a hot working environment. Thanks to the high working temperature and vacuum environment, EBM guarantees the possibility of manufacturing parts with high-quality materials such as intermetallic titanium alloys that cannot be processed with other AM or conventional processes [5]. Although the EBM process has grea<sup>t</sup> potential, the difficulties in controlling and determining proper process conditions

limit the EBM application to niche industrial sectors. Process complexity is a result of numerous phenomena involved in the process [6], including the rapid material phase change [2]. The process conditions and implementation difficulties of process monitoring systems [2] make EBM attractive at the numerical scale. Figure 1 maps the relevant literature to an EBM simulation. Few models have adopted a mesoscopic approach in which particle to particle modeling is conducted [7–11]. The grea<sup>t</sup> level of detail included in powder modeling requires significant computational time and resources. Therefore, usually, only two-dimensional (2D) models were implemented. To reduce the computational time, most models in the literature are solved using the finite element (FE) method. In this case, the material is modeled as a continuum. FE models are mainly used to predict thermal distribution (pure thermal or uncoupled) [2,6,12–22]. These models simulated a single track or a single-layer and analyzed the process from a microscopic level (i.e., temperature distribution and melt pool size). Coupled models use a thermal solver coupled with at least one other solver. These models can be grouped into fully coupled thermomechanical [12,22–28], thermo-fluid flow [18,24,29], or weakly coupled [13,14]. A limited category of models analyzed the microscopic aspects of the process, such as the interaction between beam and powder particles [15,30,31], or even microstructural evolution [32]. Owing to the high number of elements that should be involved in the calculation, multilayer simulations have rarely been implemented [33] and instead have been primarily used for predicting residual stresses.

**Figure 1.** Classification of electron beam melting (EBM) process simulation models.

Except for a few studies [15,16,23], calibrating coefficients were often applied to the heat source model to improve the match between experimental and simulation results. None of the reviewed FE models have explored the possibility of predicting the macroscopic process quality issues, i.e., the lack of fusion or surface roughness profile. As far as surface roughness is concerned, all developed models are empirical and can only estimate the average roughness value of the surface according to its angle with respect to the build platform [34]. S. Shrestha and K. Chou [35] developed a finite volume model based on a free surface approach to predict only the top surface roughness. However, the top surface is usually less critical than the vertical one because it shows better texture and a lower Ra value [34]. Typical values for vertical surface roughness range between 24 and 30 μm [36], while the Ra value of the top surfaces is around 6 μm [34]. Vertical surface roughness is mainly affected by the heat transfer effect between the molten material and the surrounding powder [34,37].

The prediction of macroscopic quality issues such as the lack of fusion and surface roughness profile over a short time is an important topic for simplifying the optimization process. This work presents a new and fast three-dimensional (3D) uncoupled thermal model for the multilayer simulation of the EBM process. The heat source is modeled following the approach developed in previous work [15], in which no calibrating coefficient was used. The multilayer simulation emulates the creation of the part layer by layer and the electron beam control along the hatching path and layer rotation. A material state variable is introduced to represent the material, which is melted during the process. This variable allows the identification of both the surface roughness profile and the lack of fusion. A new procedure is developed to reduce the calculation time, which accounts for a minimum number of elements during the calculation. This procedure is based on the quiet element approach, in which certain elements are activated only when necessary. This approach has been investigated in the literature to activate elements within a predefined volume by the time [38], such as the heat-affected zone (HAZ). However, the HAZ can vary in size and severity depending on the process condition and can be determined only by running experiments. A rough estimation of this quantity involves an incorrect number of elements that participate in the calculation and, therefore, a heat transfer analysis can be inaccurate or slow. To overcome this issue, the proposed procedure automatically considers element activation as a function of a predefined temperature. The activation temperature is determined by observing the HAZ predicted by the microscale model developed in a previous work [15]. The accuracy of the heat transfer analysis carried out with the proposed model, in terms of melt pool size and temperature distribution, is verified against the microscale model [15]. The capability of the proposed model in predicting the lateral roughness is validated by an experimental comparison. Herein, we verify the model response to a critical process parameter condition that causes a lack of fusion.

#### **2. Materials and Methods**
