Analytical and numerical modelling of laser powder bed fusion (L-PBF) of polymers – a review of the key areas of focus of the process

. The uptake of laser powder bed fusion for polymers has remained limited mainly because the interaction between material properties and process parameters is not well understood. The constraints of experimentally determining the optimal process parameters for new polymers in laser powder bed fusion include high expense, time-consumption, errors, and considerable effort. Hence, the need for using analytical and numerical models as alternatives. This paper starts with a summary on laser powder bed fusion of polymers, reviews the aspects of the process requiring the use of analytical and numerical tools, limitations, and possible improvements of the existing studies on the analytical models, and finally briefly explores approaches for numerical modelling of laser powder bed fusion of polymers. Some of the key aspects of the process that have been identified as being amenable to modelling include powder spreading and deposition of the layers, interaction between the laser beam and powder particles, melting and fusion of the particles, powder bed surface temperature, heat transfer through the powder, cooling phase, and the properties of printed parts. It is suggested in the study that the existing analytical and/or numerical models can be improved by increasing relevant variables (process parameters and material characteristics) used in them.


Introduction
Laser Powder Bed Fusion (L-PBF) is a type of Additive Manufacturing (AM) technology that utilises a laser beam energy source to selectively coalesce powder particles to form threedimensional parts through the addition of material layer-upon-layer [1]. Laser powder bed fusion is among the leading-edge technologies for AM of polymers. It is commonly referred to as Selective Laser Sintering (SLS) or Laser Sintering (LS). Selective Laser Sintering was invented by Carl Deckard and Joseph Beaman at the University of Texas, Austin, in the 1980s [2]. The technology was commercialised by DTM Corporation60 and EOS GmbH Electro Optical Systems, and the first commercial machines were developed and shipped by the two companies in 1992 and 1994, respectively [2]. The technique has undergone significant improvements in the last three decades and has matured to developing functional products for commercial and domestic use. The term Selective Laser Sintering is patented, and its use restricted. Hence, this study adopts the use of Polymer Laser Sintering (PLS) to represent selective laser sintering/laser sintering/ laser powder bed fusion of polymers.
The process begins with development of a CAD model, which is then converted to STL format. The sliced data is then transferred to a machine ready to print. The printing process begins by spreading and levelling a few layers (about 6mm, specified in the P380, P385, P395 machines) of polymeric powder over the powder bed using a roller or recoater blade. The number of layers spread and levelled at this stage is at the discretion of the operator. The powder bed is then pre-heated to a temperature close to the melting point of the respective thermoplastic polymer. The pre-heating time varies for different machines, and for P380 and P385 machines from EOS can be 0.5, 1, 2, 3, or 4 hours. A pre-heating duration of 2 and 3-4 hours is recommended for PA 12 and PP materials, respectively. The pre-heating phase is meant to homogenise and reduce the temperature gradient in the build chamber [10]. Once the pre-heating period is completed, building of parts begins. During the exposure phase, the material in the powder bed is first pre-heated to temperatures close to the melting point. After this, CO2 laser beam is then applied to tip the material to melt, thus allowing particles to fuse and leading to densification of the material. The building platform then recedes downwards by a pre-determined depth of one layer thickness, and another layer of material from the supply bin is spread on the build platform, and the building process is repeated. This process continues until the part to be built is completed. Afterwards, the process chamber is cooled in a regulated manner to room temperature and the printed parts are separated from the cake powder and removed from the build chamber. The parts are then cleaned using bead blasting or brushing, and the un-sintered powder material kept aside for recycling. Figure 1 shows a schematic of a PLS machine and some key parameters of the process. Schematic of a PLS machine and some parameters of the process [14,15] In Figure 1, the symbols given are defined as: I = distribution of the intensity of the beam in a material x' = path of the beam = scanning velocity of the laser beam r = radius of the laser beam spot 370, 06001 (2022) https://doi.org/10.1051/matecconf/202237006001 MATEC Web of Conferences 2022 RAPDASA-RobMech-PRASA-CoSAAMI Conference Db = diameter of the laser beam t = instantaneous scanning time s = hatch distance T = temperature of the build chamber Tb= pre-heating temperature qconv = heat flux due to convention qrad = heat flux due to radiation qcond = heat flux due to conduction qb = laser heat flux density

Aspects of PLS requiring analytical and numerical modelling
Polymer Laser Sintering can be summarised into three main steps: powder-spreading, sintering, and cooling [16]. It is a multifactorial phenomenon subject to process parameters and material properties. Some aspects of the process are indispensable when developing analytical and numerical models to describe and optimise this technique. The key features that should be modelled include the spreading of powder and deposition of layers of powder, interaction between the laser beam and powder, melting of powder, fusion of the melted particles of powder, cooling (solidification and crystallisation) of the melt, volumeshrinkage, and the physical and mechanical attributes of the finished product [10,16]. The ensuing discussion outlines the reasons for using analytical and numerical models to describe and expound on the mentioned aspects of PLS.

Spreading of powder and deposition of layers
The quality of spreading of powder on the powder bed is crucial because loose powder causes cavities and defects in printed parts [17]. Powder bed irregularities arising from spreading affect the distribution of energy into the powder during sintering, leading to poor quality of parts [18,19]. Le et al. [20] noted that uniformity of the layer thickness and packing density of powder determine the quality and performance of the final components. Therefore, it is imperative to investigate the influence of material properties and process parameters on the quality of the process of spreading powder, as this affects the physical and mechanical properties of the printed parts. The spreading of powder depends on different properties of materials and process parameters, which include [20,21,22]: Particle size, shape, and distribution ii.
Cohesiveness of the particles iii. Coefficient of restitution iv. Rolling friction coefficient v.
Geometry of the spreader (roller or blade) vi. Recoating speed vii. Layer thickness Some of these parameters (cohesiveness of particles, coefficient of restitution, rolling friction coefficient) are difficult to measure, necessitating the use of analytical and numerical tools to determine how these, and the other factors, affect the quality of spreading of powder.
Moreover, current models assume that layer thickness is equal to the depth that the built platform moves downwards per print cycle. However, the layer thickness for the first few layers subsequent to the first layer are not precisely equal to the depth that the laser bed platform recedes downwards because after sintering the first layer shrinks. Wischeropp et al.  [29] developed models describing powder layering, as illustrated in Figure 2, that illustrate the layer thickness for the first, second, third and i-layers, where a steady state is attained.

Figure 2
Powder layer heights for different layers on a build platform [29] Based on this figure, the authors observed that after sintering, the height of the first layer is smaller than the initial height of unsintered powder (h1), the levelling height of the build platform. This difference was represented as Δh1 in Figure 2. The correct thickness of the second layer before sintering was thus noted to be equal to (h1 + Δh1). The authors observed that after sintering, the second layer would also shrink as well, and so on. They concluded that if the process was repeated for i-layers, a steady state was attained, where the height of sintered powder was equal to hpow-theo (theoretical powder layer height after a steady state has been attained). They stated that the actual layer thickness before attainment of a steady state is subject to the number of layers (i), which was summarised by Equation 1 as follows: if i = 1, then layer thickness = h1 if i > 1, then layer thickness = h1 + Δhi, until a steady state is attained (1) Wischeropp et al. [29] stated that the theoretical powder layer height, after a steady state has been attained, can be calculated using Equation 2, where a is the packing density in a single layer, and h1 is the layer thickness.
The amount of material added during powder deposition influences the amount of energy absorbed, which subsequently affects fusion of the particles of powder and the resulting density, as well as physical and mechanical properties of printed parts. Hence, more research in analytical and numerical models should be directed to elaborate, suitable, laser energy power consistent with different layer thicknesses at different stages of the sintering process.

Interaction between the laser beam and particles of powder
Polymer Laser Sintering involves the use of a laser beam to selectively melt particles on a powder bed. Figure 3 illustrates the optical behaviour of powder material upon impact by a laser beam. The energy of a laser beam is absorbed, reflected, and transmitted upon interacting with particles of powder ( Figure 3a). The amount of laser energy absorbed by a polymeric material during sintering should be sufficient to ensure complete fusion of the particles of powder, and mechanical integrity of the printed parts [23]. Additionally, the heat energy should be transmitted through the top layer into the previously sintered layer to ensure satisfactory coalescence of two adjacent layers ( Figure 3b) [23].

Figure 3
Interaction between a laser beam and polymeric powder materials during sintering [26] The interaction of the beam and powder particles also results in dynamic thermal effects (melting and evaporation of the polymer particles), which is a determinant of the properties of the parts developed [24]. Lupone et al. [27] indicate that high temperatures might cause evaporation of polymeric material used, resulting in formation of bubbles that introduce part porosity, undermining the mechanical strength of printed components. Different models have been developed over the years to describe the distribution and intensity of the laser beam over the powder bed. The absorption of the laser beam is typically modelled using a ray tracer [24]. In this approach, the model traces the path of photons from a light source with a single wavelength. The intensity of radiation in a material is described using the Beer-Lambert Law, given in Equations 3 and 4 [25].
where, = intensity of the transmitted radiation at the maximum depth of penetration from the material surface (Z) = absorptivity I0 = intensity of the incident radiation z = maximum depth of penetration of a laser beam from the surface of a material l = absorption length (depth into a specific material at which the intensity of the beam drops to 1/e) λ = wavelength of the radiation = refractive index of the material as a function of the wavelength of the material.
The Beer-Lambert law applies when moving in a straight line in a homogenous medium. It is evident that the interaction of the laser beam and the polymeric material during sintering is crucial because it determines the degree of fusion of particles and influences the mechanical and physical integrity of printed parts. Analytical and numerical models provide more information into this aspect. Equation 3 can be improved further by linking process 370, 06001 (2022) https://doi.org/10.1051/matecconf/202237006001 MATEC Web of Conferences 2022 RAPDASA-RobMech-PRASA-CoSAAMI Conference parameters (laser power, layer thickness, scanning speed, diameter of the laser beam, hatch distance) and the material properties (absorption, reflection, and transmission coefficients).

Fusion of melted particles of powder
The fusion of particles in PLS is dependent on the amount of laser anergy absorbed and on their material properties, such as viscosity and surface tension. Coalescence of molten polymeric materials is a critical aspect of PLS because it determines the mechanical integrity of printed components. Lupone et al. [27] predicted the coalescence behaviour of viscous materials using the Frenkel's model. The results of their work indicated that the level of fusion of two spherical polymer particles is directly proportional to the surface tension and inversely proportional to the viscosity of the material, as outlined in Equation 5 [14,27]. Figure 3 illustrates Frenkel's model for the fusion of two spherical particles. According to the model, fusion begins when the particles are heated above their glass transition or melting temperatures. Two adjacent particles (Figure 4(1)) start to join through formation of a neck ( Figure 4(2)). Ideally, full fusion is attained when the two adjacent particles join to form a single sphere (Figure 4(3)). Perfect fusion is achieved when the ratio ( ⁄ ) is equal to 1.26, where di is the diameter of the particles at room temperature and l is the neck radius [27]. where, l = length of the growing neck between two particles undergoing fusion 0 = initial radius of the particles a = powder particle radius after fusion γ = surface tension η = viscosity It is noted that Frenkel's model applies to two particles of the same size and coalescence where the particles maintain a spherical shape. In addition, the model assumes a zero thermal gradient within the particles [28]. This is very limiting and therefore, the need for a modified model that describes the coalescence of two non-spherical particles through the entire process of fusion.

Powder bed surface temperature
The lack of reproducibility of geometrical and mechanical qualities of the printed parts is one of the significant challenges of PLS, especially for new materials [30]. Greiner et al. [30] noted that regulation of the surface temperature affects the geometrical accuracy of the final product. According to the authors, a homogenous temperature field is recommended because if the chamber temperatures are below the crystallisation point for semi-crystalline polymers, curling of the layers will occur, thus disrupting the printing process. Hence, the temperature of the build chamber should be maintained within a processing range, referred to as a sintering window, which is the difference between the onset of melting and crystallisation temperatures for a given polymer, as described in Equation 6.
where, ∆ = sintering window = onset of the melting point = onset of the crystallisation point Figure 5 shows the processing range (sintering window) for polypropylene powder (Laser PP CP 75) from Diamond Plastics. Greiner et al. [30] further indicated that the chamber temperature is influenced by the different sources of heat, which include powder pre-heating, surface temperature heaters, extraction chamber heaters, and the laser beam, as illustrated in Figure 6, which also shows 370, 06001 (2022) https://doi.org/10.1051/matecconf/202237006001 MATEC Web of Conferences 2022 RAPDASA-RobMech-PRASA-CoSAAMI Conference various components of the EOSINT P380 machine from EOS, GmbH. Moreover, the aspects of cooling through conduction, convection, and radiation also affect the temperatures in the build chamber. Measuring the surface temperature of the build chamber is difficult. Currently, optical temperature sensors are used, but they are prone to fogging when new material is introduced into the chamber from the supply bins, which affects their operation. Therefore, numerical and analytical models provide an alternative avenue to model powder surface temperatures during sintering and create opportunity to regulate the temperatures of powder surfaces.

Heat transfer through the powder
The mechanical integrity of parts produced using PLS is subject to the extent of fusion of particles, which is dependent on the amount of laser beam energy absorbed. Considerations of heat transfer through the powder can be used to describe the amount of energy absorbed by the powder and can subsequently be used to determine the degree of fusion for a particular polymeric material. Most of the current models that focus on heat transfer through a polymeric powder are based on Fourier's law in Equation 7, adopted from the work of Brighenti et al. [14]. The existing models are all based on heat from the laser source and can be improved by including heat from pre-heating, surface temperature heaters, and extraction chamber heaters.

Cooling (solidification and crystallisation)
Schmidt et al. [32] stated that the cooling rate after sintering impacts the material microstructures, which influences the final properties of the product such as density, tensile strength, and porosity of the printed parts. These properties are all subject to the degree of crystallisation of the printed parts, which is also a function of the cooling rate [33,34]. The Nakamura model is the most used tool for describing crystallisation kinetics [35]. It describes the evolution of the degree of crystallisation as a function of time and temperature, according to Equation 8.  (8) where, = degree of crystallization t = cooling time T = temperature n = constant value (a typical value for PA 12 is 6.8 [36] ( ) = is the non-isothermal crystallisation rate (Nakamura rate constant) and can be determined using Equation 9.
where, ( ) = Avrami rate constant 1 2 = half crystallisation time that is calculated using Equation 10.  The Nakamura model is detailed and captures different aspects, such as material properties (melting point, glass transition temperature, nucleation properties, and activation energy of crystallisation) and process parameters such as cooling time and temperature. However, it fails to link the cooling rate to the final characteristics of the product (density, tensile strength, and porosity). Hence, there is a need for models that relate the cooling rate and properties of final printed parts.

Properties of the printed parts (curling)
Crucial properties of finished polymer products, such as surface roughness, dimensional accuracy, and mechanical strength, are influenced by different process parameters [37]. Pilipović et al. [37] noted that some of the essential process parameters include scan pattern, laser power, laser scanning speed, laser spot diameter, layer thickness, laser beam offset, hatch distance, scaling factor, and temperature of the working chamber. The authors observed that the tensile properties of parts printed using PA 12 increase with increasing laser energy density. However, they cautioned that optimal laser energy density should be utilised because too high values promoted warping, which affected the dimensions of the final product. Beitz et al. [38] found that the density of a printed part using PA 12 increased with increasing laser power and decreased with increasing scanning speed and hatch distance. It is known that layers do not fuse fully when laser energy density is utilised that is too low to penetrate through to the previous layer adequately, thus compromising the mechanical integrity of printed parts. Moreover, thicker layers promote faster printing but reduce the mechanical properties of produced parts because of limited melt depth relative to the thickness of the layers and improper bonding of adjacent layers [39]. It is also known that high powder bed temperature causes poor dimensional accuracy of built parts and results in a baked cake, thus reducing recyclability of the powder [40]. Too low powder bed temperature causes insufficient melting, resulting in high porosity of built parts and rapid cooling of the molten component leading to warping [40]. Hence, the powder bed temperature should be kept between crystallisation and melting points of the materials to be printed [41]. Moreover, the amount of laser energy beam required is also subject to the powder bed temperature, according to the observation of Leu et al. [42] that, lower incident laser beam energies are required for higher powder bed temperatures and the vice versa.
Warping/curling is one of the major hindrances to PLS because it adversely affects the dimensional and geometrical accuracy of fabricated parts. Printed parts must be dimensionally and geometrically accurate to ensure overall acceptance in the market. Armillotta et al. [43] investigated the effects of warpage for details printed using Fused Deposition Modelling (FDM), which subjects materials to thermal cycles like the PLS technology. The authors inferred that factors that influence warpage/curling of parts printed 370, 06001 (2022) https://doi.org/10.1051/matecconf/202237006001 MATEC Web of Conferences 2022 RAPDASA-RobMech-PRASA-CoSAAMI Conference using FDM can be categorised into part geometry, material, and process parameters, as summarised in Table 1. Most of the parameters outlined in Table 1 apply to the PLS. It is clear from Table 1 that curling/warpage is affected by numerous parameters. Hence, analytical and numerical models relating to curling/warpage and the process parameters can be of use in optimising PLS. The models can also analyse the effectiveness of various remedial actions for curling/warpage, such as using fillers (talc) or adding ribs in parts printed using PLS.
According to Li et al. [44], polymeric materials undergo fusion, solidification, and crystallisation during the heating and cooling phases of the PLS, leading to changes in volume that encourage shrinkage. Figure 7 illustrates changes in volume that occur during the heating and cooling stages.

Figure 7
Changes in the volume of polymers during the heating and cooling phases of polymer laser sintering [44] Shrinkage of the material can be attributed to the fusion of powder, thermal contraction during cooling, and volumetric changes due to phase transition from amorphous to crystalline phases [44,45]. Schmutzler et al. [45] stated that residual stresses lead to upward bending, referred to as the curling effect. The upward bending of printed parts can be attributed to uneven cooling of the upper and lower sections of the parts. The upper portions of the parts experience faster cooling than the lower sections, which are embedded in powder, resulting into a greater shrinkage in the upper parts, leading to curling of printed components. Li et al. [44] used Equation 11 to represent the total shrinkage of a part printed using PLS technology.
where, = total shrinkage = shrinkage caused by recrystallisation of a polymer = shrinkage attributed to the phase transition of the powder

Limitations and possible improvements to the existing studies on analytical models describing laser sintering of polymers
Numerous studies have modelled the distribution and intensity of a laser beam. Mokrane et al. [10], Brighenti et al. [14], Peyre et al. [46], and Yaagoubi et al. [47] developed models to predict the distribution and intensity of a laser beam over a powder bed. Though the models differ, they all consider optical material properties (surface reflectivity and refractive index of the material) and laser beam characteristics (beam intensity, diameter of the laser beam, path of the beam, and wavelength of the radiation). The model by Mokrane et al. [10] is superior to the others because it considered variables that are more significant in respect to modelling the distribution and intensity of a laser beam. However, the model can still be improved by integrating scanning speed into it.
Different studies have modelled heat transfer of PLS. The models consider the input heat flux as a result of laser-material interaction, heat transfer in the powder bed, and the change in the phase of the material from solid to liquid phases. The combination of the heat from the laser beam and repeated temperature cycles arising from the layerupon -layer deposition, both result in a non-linear thermal field within a printed part [48]. Different defects such as residual stresses, part distortion, and incomplete melting have been observed due to the existence of non-linear thermal environments [48]. The transfer of heat can be represented as a combination of gradients of the density of a material and internal energy as outlined by Fourier's law of heat transfer [49]. More work on the topic was performed by Mokrane et al. [10], Brighenti et al. [14], Peyre et al. [46], Yaagoubi et al. [47], and Liu et al. [48]. Of these, the models by Mokrane et al. [10], Brighenti et al. [14], and Liu et al. [48] are more comprehensive. This is because they incorporate more variables than the models proposed by Peyre et al. [46] and Yaagoubi et al. [47]. Mokrane et al. [10], Li et al. [36], Peyre et al. [46], and Liu et al. [48] developed models for initial boundary conditions which concur that heat loss on the top surface of a printed part is due to radiation and convention, while there is no heat loss at the bottom of the powder bed. This last assumption is incorrect, as the extraction chamber temperature is always set lower than that of the build chamber. All these models can, therefore, be improved by considering heat loss at the bottom of the powder bed through conduction.
Various models have been developed to describe the physical and mechanical properties of printed parts. Li et al. [44] and Schmutzler et al. [45] developed models to elucidate the shrinkage of printed parts. Jamal [50] conducted a study to model the curling of polycarbonate when subjected to PLS. More work is still required to provide further insights into the impact of process parameters and material properties on the physical and mechanical characteristics of polymeric parts printed using PLS.

Approaches for numerical modelling of the selective laser sintering of polymers
Numerical models can be used to optimise PLS process parameters and predict the properties of printed parts. Thermal-recrystallisation-mechanical models have been employed extensively to simulate the PLS process of polymers [36]. Li et al. [36] proposed a theoretical framework of the thermal-recrystallisation-mechanical model, as illustrated in Figure 8.  [36] The framework in Figure 8 illustrates that the distribution of temperature, which affects the macroscopic structure and mesoscopic melt pool, is subject to the prevailing thermal conditions (powder bed temperature), material properties (melting point, sintering window, viscosity, surface tension), and heat sources. The cooling rate is shown in the figure to affect the crystallinity and crystallite size of printed parts, both of which influence the mechanical properties of printed components. According to Li et al. [44], a recrystallisation model can describe the impact of cooling rate on parts printed using the PLS technology. Lastly, Li et al. [44] proposed a mechanical model that links thermal strain, elastic strain, plastic strain, and recrystallisation induced strain. The model can estimate displacement and residual stress in printed components.
Brighenti et al. [14] proposed a model to predict the properties of parts printed by laser sintering. The authors considered optical, thermal, sintering, and kinetic models in this work, as illustrated in Figure 9. Model for predicting the properties of parts printed by laser sintering [14] The optical model in this model elucidates the interaction between the laser beam and polymeric materials. They described the quantities of the beam energy that are absorbed, reflected, and transmitted. Enough laser beam energy should be supplied to ensure sufficient fusion of polymer particles, in order to ensure suitable mechanical properties of the printed parts [26]. The thermal model in this model describes the distribution of heat during preheating, sintering, and cooling phases of PLS. The authors noted that the distribution of heat affects the physical properties of the finished product, for example, a high cooling rate induces warpage or curling due to rapid crystallisation [46]. In addition, low bed temperature was observed to promote shrinkage that affects the geometric properties of printed parts and can lead to stoppage of the process [46]. The sintering model in this model describes different changes that occur during PLS, through which powdered polymeric material is transformed into a porous solid [14]. The kinetic model in this model describes the impact of cooling during the last phase of PLS. The approach also mentioned the need to model the mechanical properties of the printed parts to ascertain their performance.
The two approaches (outlined in Figures 8 and 9) are comprehensive and cover the entire PLS process and properties of the finished parts. However, they do not address the preexposer steps like powder-spreading and layering phases. The frameworks also do not mention physical models that describe the properties of finished products like curling, which is a major hindrance to PLS. These missing stages can be modelled and physical models for the printed parts added to improve this model for PLS. Figure 10 is a summary of models recommended for PLS. The framework shown in this figure is a modification of previous work by Brighenti et al. [14] and Li et al. [36]. A summary of models recommended for describing PLS Different multi-scale and multi-physical models have been developed over the years to predict the properties of AM parts. The models can be grouped into micro-, meso-, and macro-scale types based on their respective scales [13,36]. The microscale models are used to evaluate laser beam penetration, calculate grain growth, and predict solidification of microstructure. In contrast, the mesoscale models predict the flow of the melt pool, the fusion of powder particles, and solidification. The macroscale models are used to determine residual stresses and deformation of printed components. According to Sjöström [51], microscale level models consider detailed parameters, such as a moving heat source to describe thermal history and microstructural properties of printed parts. The models are more sophisticated compared to the meso-scale models that are simplified and consider a stationary heat source. Meso-scale models require less computational effort and time compared to the microscale models. Moreover, some physics are neglected in meso-and macro-scale modelling, thus reducing their accuracy [51]. However, the three modelling scales should be employed together in PLS since they yield different but essential information [52]. Multi-scale, multiphysics modelling and experimental validation are crucial to understanding PLS.

Conclusions
Several conclusions arise from the foregoing material including: -Analytical and numerical tools are gaining popularity for use in describing PLS to better understand this technique, which involves interactions of numerous materials and process parameters. -Currently, process optimisation for new materials is primarily done experimentally, which is complicated, tedious, costly, and time-consuming. -Powder spreading and deposition of layers, interaction between the laser beam and powder particles, fusion of the melted powder particles, powder bed surface temperatures, heat transfer through powder, cooling, and properties of the final components (surface roughness, dimensional accuracy, curling, and mechanical strength) are all aspects of PLS that are open for study through analytical and numerical modelling. -A good number of models can be improved quickly by increasing the relevant variables (process parameters and material characteristics) used in them. -The current approaches for numerical modelling of PLS do not address the pre-exposer steps (powder spreading and layering phases), and do not incorporate physical models describing phenomenon such as curling. -Future numerical models should include these missing stages and should also describe the physical properties of the printed parts. -The numerical models for PLS can be grouped into micro-, meso-, and macro-scale types based on the simulation scale. All the models should be linked to provide a comprehensive description of PLS.